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https://stackoverflow.com/questions/1102692/how-to-alpha-blend-rgba-unsigned-byte-color-fast/61943155#61943155	[ "# How to alpha blend RGBA unsigned byte color fast?\n\nI am using c++ , I want to do alpha blend using the following code.\n\n``````#define CLAMPTOBYTE(color) \\\nif ((color) & (~255)) { \\\ncolor = (BYTE)((-(color)) >> 31); \\\n} else { \\\ncolor = (BYTE)(color); \\\n}\n#define GET_BYTE(accessPixel, x, y, scanline, bpp) \\\n((BYTE)((accessPixel) + (y) (scanline) + (x) * (bpp)))\n\nfor (int y = top ; y < bottom; ++y)\n{\nBYTE* resultByte = GET_BYTE(resultBits, left, y, stride, bytepp);\nBYTE* srcByte = GET_BYTE(srcBits, left, y, stride, bytepp);\nBYTE* srcByteTop = GET_BYTE(srcBitsTop, left, y, stride, bytepp);\nint alpha = 0;\nint red = 0;\nint green = 0;\nint blue = 0;\nfor (int x = left; x < right; ++x)\n{\nred = (srcByteTop[R] * alpha + srcByte[R] * (255 - alpha)) / 255;\ngreen = (srcByteTop[G] * alpha + srcByte[G] * (255 - alpha)) / 255;\nblue = (srcByteTop[B] * alpha + srcByte[B] * (255 - alpha)) / 255;\nCLAMPTOBYTE(red);\nCLAMPTOBYTE(green);\nCLAMPTOBYTE(blue);\nresultByte[R] = red;\nresultByte[G] = green;\nresultByte[B] = blue;\nsrcByte += bytepp;\nsrcByteTop += bytepp;\nresultByte += bytepp;\n}\n}\n``````\n\nhowever I find it is still slow, it takes about 40 - 60 ms when compose two 600 * 600 image. Is there any method to improve the speed to less then 16ms?\n\nCan any body help me to speed this code? Many thanks!\n\n• What compiler are you using? What platform are you developing this software for? Are you willing to use off the shelf tools? Jul 9, 2009 at 23:36\n• I am using VS2005, the software is designed for windows platform. I am willing to use any method to accelerate this code. I think maybe it can be accelerated alot Jul 10, 2009 at 2:13\n• Let me know if you have trouble coding up the rest of the SIMD instructions in my solution Jul 11, 2009 at 21:40\n\nUse SSE - start around page 131.\n\nThe basic workflow\n\n1. Load 4 pixels from src (16 1 byte numbers) RGBA RGBA RGBA RGBA (streaming load)\n\n2. Load 4 more which you want to blend with srcbytetop RGBx RGBx RGBx RGBx\n\n3. Do some swizzling so that the A term in 1 fills every slot I.e\n\nxxxA xxxB xxxC xxxD -> AAAA BBBB CCCC DDDD\n\nIn my solution below I opted instead to re-use your existing \"maskcurrent\" array but having alpha integrated into the \"A\" field of 1 will require less loads from memory and thus be faster. Swizzling in this case would probably be: And with mask to select A, B, C, D. Shift right 8, Or with origional, shift right 16, or again.\n\n4. Add the above to a vector that is all -255 in every slot\n\n5. Multiply 1 * 4 (source with 255-alpha) and 2 * 3 (result with alpha).\n\nYou should be able to use the \"multiply and discard bottom 8 bits\" SSE2 instruction for this.\n\n6. add those two (4 and 5) together\n\n7. Store those somewhere else (if possible) or on top of your destination (if you must)\n\nHere is a starting point for you:\n\n`````` //Define your image with __declspec(align(16)) i.e char __declspec(align(16)) image[640480]\n// so the first byte is aligned correctly for SIMD.\n// Stride must be a multiple of 16.\n\nfor (int y = top ; y < bottom; ++y)\n{\nBYTE resultByte = GET_BYTE(resultBits, left, y, stride, bytepp);\nBYTE* srcByte = GET_BYTE(srcBits, left, y, stride, bytepp);\nBYTE* srcByteTop = GET_BYTE(srcBitsTop, left, y, stride, bytepp);\nfor (int x = left; x < right; x += 4)\n{\n//If you can't align, use _mm_loadu_si128()\n// Step 1\n// Step 2\n\n// Step 3\n// Fill the 4 positions for the first pixel with maskCurrent, etc\n// Could do better with shifts and so on, but this is clear\n)\n\n// step 4\n\n//Todo : Multiply, with saturate - find correct instructions for 4..6\n\nred = (srcByteTop[R] * alpha + srcByte[R] * (255 - alpha)) / 255;\ngreen = (srcByteTop[G] * alpha + srcByte[G] * (255 - alpha)) / 255;\nblue = (srcByteTop[B] * alpha + srcByte[B] * (255 - alpha)) / 255;\nCLAMPTOBYTE(red);\nCLAMPTOBYTE(green);\nCLAMPTOBYTE(blue);\nresultByte[R] = red;\nresultByte[G] = green;\nresultByte[B] = blue;\n//----\n\n// Step 7 - store result.\n//Store aligned if output is aligned on 16 byte boundrary\n_mm_store_si128(reinterpret_cast<__mm128i>(resultByte), result)\n//Slow version if you can't guarantee alignment\n//_mm_storeu_si128(reinterpret_cast<__mm128i>(resultByte), result)\n\n//Move pointers forward 4 places\nsrcByte += bytepp * 4;\nsrcByteTop += bytepp * 4;\nresultByte += bytepp * 4;\n}\n}\n``````\n\nTo find out which AMD processors will run this code (currently it is using SSE2 instructions) see Wikipedia's List of AMD Turion microprocessors. You could also look at other lists of processors on Wikipedia but my research shows that AMD cpus from around 4 years ago all support at least SSE2.\n\nYou should expect a good SSE2 implimentation to run around 8-16 times faster than your current code. That is because we eliminate branches in the loop, process 4 pixels (or 12 channels) at once and improve cache performance by using streaming instructions. As an alternative to SSE, you could probably make your existing code run much faster by eliminating the if checks you are using for saturation. Beyond that I would need to run a profiler on your workload.\n\nOf course, the best solution is to use hardware support (i.e code your problem up in DirectX) and have it done on the video card.\n\n• See edits to my origional post to address your question. Short answer - yes if not an ancient CPU. Jul 9, 2009 at 23:23\n• Will it work only on Windows or on other platforms as well ? (If I will define BYTe ofc) Main question is: are SIMD instructions crossplatform ? Aug 18, 2017 at 1:28\n• Yes, SIMD instructions require the CPU support them but they don't care about the OS (Windows, etc). The compiler also needs to translate the intrinsics (such as `_mm_set_epi8`) but I believe that GCC can do this. Aug 21, 2017 at 8:03\n\nYou can always calculate the alpha of red and blue at the same time. You can also use this trick with the SIMD implementation mentioned before.\n\n``````unsigned int blendPreMulAlpha(unsigned int colora, unsigned int colorb, unsigned int alpha)\n{\nunsigned int rb = (colora & 0xFF00FF) + ( (alpha * (colorb & 0xFF00FF)) >> 8 );\nunsigned int g = (colora & 0x00FF00) + ( (alpha * (colorb & 0x00FF00)) >> 8 );\nreturn (rb & 0xFF00FF) + (g & 0x00FF00);\n}\n\nunsigned int blendAlpha(unsigned int colora, unsigned int colorb, unsigned int alpha)\n{\nunsigned int rb1 = ((0x100 - alpha) * (colora & 0xFF00FF)) >> 8;\nunsigned int rb2 = (alpha * (colorb & 0xFF00FF)) >> 8;\nunsigned int g1 = ((0x100 - alpha) * (colora & 0x00FF00)) >> 8;\nunsigned int g2 = (alpha * (colorb & 0x00FF00)) >> 8;\nreturn ((rb1 \| rb2) & 0xFF00FF) + ((g1 \| g2) & 0x00FF00);\n}\n``````\n\n0 <= alpha <= 0x100\n\n• Nice trick. You should add handling of saturation in there too (right now it overflows) Jul 9, 2009 at 23:29\n• The overflow is intentional, it's handled in the return statement. Jul 10, 2009 at 0:41\n• It's got a rather rude handling of overflow: wraparound instead of saturation. Jul 10, 2009 at 8:55\n• @MSalters, could be because of the hangover, but I don't see the overflow; or well, I see an intentional overflow in rb and g, but they're masked out in the return statement. (As long as int is 32 bits). Jul 10, 2009 at 11:57\n• @JasperBekkers, did you actually try your example? With alpha=0xff (opaque), the result is 0xff80 (the red completely disappears, other colours wrong as well). Aug 17, 2012 at 20:10\n\nFor people that want to divide by 255, i found a perfect formula:\n\n``````pt->r = (r+1 + (r >> 8)) >> 8; // fast way to divide by 255\n``````\n• This can be extended to two 16bits words: ((r+0x10001+((r>>8)&0xFF00FF))>>8) & 0xFF00FF and this allow multiplexing xRxB and AxGx ops in ARGB, similar in RGBA and other variants Dec 23, 2013 at 17:14\n• `(x+1+((x+1)>>8))>>8 // integer div 255 for [0..65790)` -- slightly better Apr 1, 2015 at 16:13\n• `((x+1)257)>>16 // integer div 255 for [0..65790)` -- alternative formulation which might be faster on some platforms -- interesting notes: Division via Multiplication Apr 1, 2015 at 16:42\n• @nobar: The standard compiler trick of doing division with a multiplicative inverse is also worth considering: `n/255` compiles to = asm that does `(n0x8081) >> 23`. That also works for all 16-bit `n`. (I just noticed your upper-bound was higher than 65536). With x86 SSE2, that's one `_mm_mulhi_epu16` and one `_mm_srli_epu16(mul, 23-16)`. `x+1 * 257` is one paddw and one pmulhuw, so that's actually better (since mul and shift may compete for the same port). Apr 27, 2017 at 4:38\n\nHere's some pointers.\n\nConsider using pre-multiplied foreground images as described by Porter and Duff. As well as potentially being faster, you avoid a lot of potential colour-fringing effects.\n\nThe compositing equation changes from\n\n``````r = kA + (1-k)B\n``````\n\n... to ...\n\n``````r = A + (1-k)B\n``````\n\nAlternatively, you can rework the standard equation to remove one multiply.\n\n``````r = kA + (1-k)B\n== kA + B - kB\n== k(A-B) + B\n``````\n\nI may be wrong, but I think you shouldn't need the clamping either...\n\nI can't comment because I don't have enough reputation, but I want to say that Jasper's version will not overflow for valid input. Masking the multiplication result is necessary because otherwise the red+blue multiplication would leave bits in the green channel (this would also be true if you multiplied red and blue separately, you'd still need to mask out bits in the blue channel) and the green multiplication would leave bits in the blue channel. These are bits that are lost to right shift if you separate the components out, as is often the case with alpha blending. So they're not overflow, or underflow. They're just useless bits that need to be masked out to achieve expected results.\n\nThat said, Jasper's version is incorrect. It should be 0xFF-alpha (255-alpha), not 0x100-alpha (256-alpha). This would probably not produce a visible error.\n\nI've found an adaptation of Jasper's code to be be faster than my old alpha blending code, which was already decent, and am currently using it in my software renderer project. I work with 32-bit ARGB pixels:\n\n``````Pixel AlphaBlendPixels(Pixel p1, Pixel p2)\n{\nstatic const int AMASK = 0xFF000000;\nstatic const int RBMASK = 0x00FF00FF;\nstatic const int GMASK = 0x0000FF00;\nstatic const int ONEALPHA = 0x01000000;\nunsigned int a = (p2 & AMASK) >> 24;\nunsigned int na = 255 - a;\nunsigned int rb = ((na * (p1 & RBMASK)) + (a * (p2 & RBMASK))) >> 8;\nunsigned int ag = (na * ((p1 & AGMASK) >> 8)) + (a * (ONEALPHA \| ((p2 & GMASK) >> 8)));\n}\n``````\n• This is exactly what I was looking for. Are you certain about the precision with `na = 255 - a` rather than 256 or is it something that can't be helped in this case?\n– Nolo\nDec 10, 2016 at 11:56\n• Sorry for late response, haven't had anything to contribute on SO in years until tonight and have been going through old notifications. There's inaccuracies either way due to c/256 not always equaling c/255 but 256 - a is more inaccurate than 255-a. Rounding up by adding to the highest of the bits you lose could reduce the inaccuracy but also isn't perfectly accurate. I'm pretty sure the only way to get perfect accuracy is to divide each color channel individually by 255 which is costly. Jasper's code saturates quickly, while mine tends towards black. Dec 3, 2021 at 8:42\n\nNo exactly answering the question but...\n\nOne thing is to do it fast, the other thing is to do it right. Alpha compositing is a dangerous beast, it looks straight forward and intuitive but common errors have been widespread for decades without anybody noticing it (almost)!\n\nThe most famous and common mistake is about NOT using premultiplied alpha. I highly recommend this: Alpha Blending for Leaves\n\n• It's not necessary to use premultiplied alpha, only to make sure the background color is removed from partially transparent pixels. Removing the background color may be part of the process of converting to premultiplied alpha, but it can be done independently as well. Nov 9, 2011 at 19:29\n\nYou can use 4 bytes per pixel in both images (for memory alignment), and then use SSE instructions to process all channels together. Search \"visual studio sse intrinsics\".\n\nFirst of all lets use the proper formula for each color component\n\n`````` v = ( 1-t ) * v0 + t * v1\n``````\n\nwhere t=interpolation parameter [0..1] v0=source color value v1=transfer color value v=output value\n\nReshuffling the terms, we can reduce the number of operations:\n\n`````` v = v0 + t * (v1 - v0)\n``````\n\nYou would need to perform this calculation once per color channel (3 times for RGB).\n\nFor 8-bit unsigned color components, you need to use correct fixed point math:\n\n`````` i = i0 + t * ( ( i1 - i0 ) + 127 ) / 255\n``````\n\nwhere t = interpolation parameter [0..255] i0= source color value [0..255] i1= transfer color value [0..255] i = output color\n\nIf you leave out the +127 then your colors will be biased towards the darker end. Very often, people use /256 or >> 8 for speed. This is not correct! If you divide by 256, you will never be able to reach pure white (255,255,255) because 255/256 is slightly less than one.\n\nI hope this helps.\n\n• Interesting ideas there, but you do pay a steep price for your / 255. You have to calculate an intermediate 16 bit result using t * ( ( v1 - v0 ) + 127 ) that you then divide. Are you sure that your formula is really simpler than ( 1-t ) * v0 + t * v1 ? Remember that 1-t is pre-calculated and that / is often more expensive than * Aug 5, 2009 at 23:34\n• The formula is a reference for what the numerically correct formula looks like. It is certainly slower, however the results are accurate. It is useful to know what the right answer looks like in order to determine if the error in the optimized result is acceptible or not. Aug 7, 2009 at 9:39\n• Yes, i = i0 + t * ( ( i1 - i0 ) + 127 ) / 255 is more efficient than your formula, which for integers would be (I think) : i = ( ( 255 - t ) * i0 + ( t * i1 ) ) / 255 Aug 7, 2009 at 9:42\n• Most images on the PC have Gamma burnt in. So if it's pixel value is say 127, that's NOT exactly half way between white and black. It's actual brighness is.. powf( (c) / 255.f, gamma) .. or about 0.19 So all your calculations that assumes pixels brightness is linear are wrong. Mar 31, 2010 at 23:15\n• With Guilerme answer this makes for a correct and fast blending. Thanks! Dec 11, 2011 at 19:25\n\nI think hardware support will help you. try to move the logic from software to hardware if feasible\n\nI've done similar code in unsafe C#. Is there any reason you aren't looping through each pixel directly? Why use all the BYTE* and GET_BYTE() calls? That is probably part of the speed issue.\n\nWhat does GET_GRAY look like?\n\nMore importantly, are you sure your platform doesn't expose alpha blending capabilities? What platform are you targeting? Wiki informs me that the following support it out of the box:\n\n• Mac OS X\n• Windows 2000, XP, Server 2003, Windows CE, Vista and Windows 7\n• The XRender extension to the X Window System (this includes modern Linux systems)\n• QNX Neutrino\n• Plan 9\n• Inferno\n• AmigaOS 4.1\n• BeOS, Zeta and Haiku\n• Syllable\n• MorphOS\n• This alpha blend is used for a certain image enhancement algorithm, not for displaying. So I can not use platform capabilities. Thanks! remove most GET_BYTE() seems useless, maybe the multiply operation and divid 255 operation is the problem. Jul 9, 2009 at 13:27\n• Even if you aren't displaying the image you can still definitely use platform capabilities. For example, on Windows you can use GDI+ or the .NET wrappers to do alpha blending without ever displaying it. I'd assume other platforms are similar. Jul 9, 2009 at 21:26\n\nThe main problem will be the poor loop construct, possibly made worse by a compiler failing to eliminate CSE's. Move the real common bits outside the loops. `int red` isn't common, thouigh - that should be inside the inner loop.\n\nFurthermore, red, green and blue are independent. If you calculate them in turn, you don't need to keep interim red results in registers when you are calculating green results. This is especially important on CPUs with limited registers like x86.\n\nThere will be only a limited number of values allowed for bytepp. Make it a template parameter, and then call the right instantiation from a switch. This will produce multiple copies of your function, but each can be optimized a lot better.\n\nAs noted, clamping is not needed. In alphablending, you're creating a linear combination of two images a[x][y] and b[x][y]. Since 0<=alpha<=255, you know that each output is bound by max(255a[x][y], 255b[x][y]). And since your output range is the same as both input ranges (0-255), this is OK.\n\nWith a small loss of precision, you could calculate `(a[x][y]alpha b[x][y](256-alpha))>>8`. Bitshifts are often faster than division.\n\n• Modern CPUs prefer interleaved instructions as much as possible. This is because independent work (i.e calculating R while G is processing) suits the pipelined nature of modern CPUs well. See Intel optimisation manual : intel.com/Assets/PDF/manual/248966.pdf. - The registers might seem limited to you, but the CPU has many more actual registers than you think using \"register renaming\" Jul 9, 2009 at 23:28\n\nDepending on the target architecture, you could try either vectorize or parallellize the function.\n\nOther than that, try to linearize the whole method (i.e. no loop-in-loop) and work with a quadruple of bytes at once, that would lose the overhead of working with single bytes plus make it easier for the compiler to optimize the code.\n\nMove it to the GPU.\n\nI am assuming that you want to do this in a completely portable way, without the help of a GPU, the use of a proprietry intel SIMD library (which may not work as efficiently on AMD processors).\n\nPut the following inplace of your calculation for RGB\n\n``````R = TopR + (SourceR alpha) >> 8;\nG = TopG + (SourceG * alpha) >> 8;\nB = TopB + (SourceB * alpha) >> 8;\n``````\n\nIt is a more efficient calculation.\n\nAlso use shift left instruction on your get pixel macro instead of multiplying by the BPP.\n\n• SSE is pretty well adopted by both programmers and chip manufacturers. Jul 9, 2009 at 23:25\n\nThis one works when the first color, (colora, the destination) has also alpha channel (blending two transparent ARGB colors) The alpha is in the second color's alpha (colorb, the source)\n\nThis adds the two alphas (0 = transparent, 255 = fully opaque) It is a modified version of Jasper Bekkers' answer.\n\nI use it to blend transparent pixel art on to a transparent screen.\n\n``````Uint32 alphaBlend(unsigned int colora, unsigned int colorb) {\nunsigned int a2 = (colorb & 0xFF000000) >> 24;\nunsigned int alpha = a2;\nif (alpha == 0) return colora;\nif (alpha == 255) return colorb;\nunsigned int a1 = (colora & 0xFF000000) >> 24;\nunsigned int nalpha = 0x100 - alpha;\nunsigned int rb1 = (nalpha * (colora & 0xFF00FF)) >> 8;\nunsigned int rb2 = (alpha * (colorb & 0xFF00FF)) >> 8;\nunsigned int g1 = (nalpha * (colora & 0x00FF00)) >> 8;\nunsigned int g2 = (alpha * (colorb & 0x00FF00)) >> 8;\nunsigned int anew = a1 + a2;\nif (anew > 255) {anew = 255;}\nreturn ((rb1 + rb2) & 0xFF00FF) + ((g1 + g2) & 0x00FF00) + (anew << 24);\n}\n``````\n\nHere's my adaption of a software alpha blend that works well for 2 unsigned integers.\n\nMy code differs a bit as the code above is basically always assuming the destination alpha is 255.\n\nWith a decent optimizing compiler most calculations should be in registers as the scope of most variables is very short. I also opted to progressively shift the result << 8 incrementally to avoid << 24, << 16 when putting the ARGB back together. I know it's a long time ago... but I remember on the 286 cycles for a shift was (1 + 1each bit shifted) so assume there is still some sort of penalty for larger shifts.\n\nAlso... instead of \"/ 255\" I opted for \">> 8\" which can be changed as desired.\n\n``````/\nalpha blend source and destination, either may have an alpha!!!!\n\nSrc AAAAAAAA RRRRRRRR GGGGGGGG BBBBBBBB\nDest AAAAAAAA RRRRRRRR GGGGGGGG BBBBBBBB\n\nres AAAAAAAA RRRRRRRR GGGGGGGG BBBBBBBB\n\nNOTE - α = αsrc + αdest(1.0-αsrc) where α = 0.0 - 1.0\n\nALSO - DWORD is unsigned int so (F8000000 >> 24) = F8 not FFFFFFF8 as it would with int (signed)\n/\n\ninline DWORD raw_blend(const DWORD src, const DWORD dest)\n{\n// setup and calculate α\n\nDWORD src_a = src >> 24;\nDWORD src_a_neg = 255 - src_a;\nDWORD dest_a = dest >> 24;\n\nDWORD res = src_a + ((dest_a src_a_neg) >> 8);\n\n// setup and calculate R\n\nDWORD src_r = (src >> 16) & 255;\nDWORD dest_r = (dest >> 16) & 255;\n\nres = (res << 8) \| (((src_r * src_a) + (dest_r * src_a_neg)) >> 8);\n\n// setup and calculate G\n\nDWORD src_g = (src >> 8) & 255;\nDWORD dest_g = (dest >> 8) & 255;\n\nres = (res << 8) \| (((src_g * src_a) + (dest_g * src_a_neg)) >> 8);\n\n// setup and calculate B\n\nDWORD src_b = src & 255;\nDWORD dest_b = dest & 255;\n\nreturn (res << 8) \| (((src_b * src_a) + (dest_b * src_a_neg)) >> 8);\n}\n``````\n``````; In\\ EAX = background color (ZRBG) 32bit (Z mean zero, always is zero)\n; In\\ EDX = foreground color (RBGA) 32bit\n; Out\\ EAX = new color\n; free registers (R10, RDI, RSI, RSP, RBP)\nabg2:\nmov r15b, dl ; av\nmovzx ecx, dl\nnot ecx ; faster than 255 - dl\nmov r14b, cl ; rem\n\nshr edx, 8\nand edx, 0x00FFFFFF\nmov r12d, edx\nmov r13d, eax ; RBGA ---> ZRGB\n\n; s: eax\n; d: edx\n\n;=============================red = ((s >> 16) * rem + (d >> 16) * av) >> 8;\nmov edx, r12d\nshr edx, 0x10\nmovzx eax, r14b\nimul edx, eax\nmov ecx, r13d\nshr ecx, 0x10\nmovzx eax, r15b\nimul eax, ecx\nlea eax, [eax + edx] ; faster than add eax, edx\nshr eax, 0x8\nmov r9b, al\nshl r9d, 8\n\n;=============================green = (((s >> 8) & 0x0000ff) * rem + ((d >> 8) & 0x0000ff) * av) >> 8;\nmov eax, r12d\nshr eax, 0x8\nmovzx edx, al\nmovzx eax, r14b\nimul edx, eax\nmov eax, r13d\nshr eax, 0x8\nmovzx ecx, al\nmovzx eax, r15b\nimul eax, ecx\nlea eax, [eax, + edx] ; faster than add eax, edx\nshr eax, 0x8\nmov r9b, al\nshl r9d, 8\n\n;=============================blue = ((s & 0x0000ff) * rem + (d & 0x0000ff) * av) >> 8;\nmovzx edx, r12b\nmovzx eax, r14b\nimul edx, eax\nmovzx ecx, r13b\nmovzx eax, r15b\nimul eax, ecx\nlea eax, [eax + edx] ; faster than add eax, edx\nshr eax, 0x8\nmov r9b, al\n\nmov eax, r9d\nret\n``````" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.69429976,"math_prob":0.9683434,"size":3974,"snap":"2023-40-2023-50","text_gpt3_token_len":1130,"char_repetition_ratio":0.1256927,"word_repetition_ratio":0.014423077,"special_character_ratio":0.30120784,"punctuation_ratio":0.12880887,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97774804,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-30T14:29:24Z\",\"WARC-Record-ID\":\"<urn:uuid:3c2a4d29-d06d-4166-867c-4167aacba5f1>\",\"Content-Length\":\"378761\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0cc6891e-5de5-4b72-bbfb-e6eedf0d02b7>\",\"WARC-Concurrent-To\":\"<urn:uuid:c72cf01a-7a49-43e1-9745-2f6fe9bf54d1>\",\"WARC-IP-Address\":\"104.18.23.201\",\"WARC-Target-URI\":\"https://stackoverflow.com/questions/1102692/how-to-alpha-blend-rgba-unsigned-byte-color-fast/61943155#61943155\",\"WARC-Payload-Digest\":\"sha1:LFJVJES7BX236HDCYQD665XYDNQBR7OQ\",\"WARC-Block-Digest\":\"sha1:ZLIN37ROVZCOZT5FRQEYERLWAYC5OWPS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510676.40_warc_CC-MAIN-20230930113949-20230930143949-00130.warc.gz\"}"}
https://numberworld.info/11001010	[ "# Number 11001010\n\n### Properties of number 11001010\n\nCross Sum:\nFactorization:\nDivisors:\nCount of divisors:\nSum of divisors:\nPrime number?\nNo\nFibonacci number?\nNo\nBell Number?\nNo\nCatalan Number?\nNo\nBase 2 (Binary):\nBase 3 (Ternary):\nBase 4 (Quaternary):\nBase 5 (Quintal):\nBase 8 (Octal):\na7dcb2\nBase 32:\nafn5i\nsin(11001010)\n0.68684924308834\ncos(11001010)\n0.72679991556753\ntan(11001010)\n0.94503208981801\nln(11001010)\n16.213497644729\nlg(11001010)\n7.0414325594574\nsqrt(11001010)\n3316.7770500894\nSquare(11001010)\n\n### Number Look Up\n\nLook Up\n\n11001010 which is pronounced (eleven million one thousand ten) is a very impressive figure. The cross sum of 11001010 is 4. If you factorisate 11001010 you will get these result 2 * 5 * 1100101. The figure 11001010 has 8 divisors ( 1, 2, 5, 10, 1100101, 2200202, 5500505, 11001010 ) whith a sum of 19801836. The figure 11001010 is not a prime number. The figure 11001010 is not a fibonacci number. The number 11001010 is not a Bell Number. 11001010 is not a Catalan Number. The convertion of 11001010 to base 2 (Binary) is 101001111101110010110010. The convertion of 11001010 to base 3 (Ternary) is 202200220112211. The convertion of 11001010 to base 4 (Quaternary) is 221331302302. The convertion of 11001010 to base 5 (Quintal) is 10304013020. The convertion of 11001010 to base 8 (Octal) is 51756262. The convertion of 11001010 to base 16 (Hexadecimal) is a7dcb2. The convertion of 11001010 to base 32 is afn5i. The sine of the number 11001010 is 0.68684924308834. The cosine of 11001010 is 0.72679991556753. The tangent of 11001010 is 0.94503208981801. The root of 11001010 is 3316.7770500894.\nIf you square 11001010 you will get the following result 121022221020100. The natural logarithm of 11001010 is 16.213497644729 and the decimal logarithm is 7.0414325594574. You should now know that 11001010 is very amazing number!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7442872,"math_prob":0.88571316,"size":2110,"snap":"2019-51-2020-05","text_gpt3_token_len":731,"char_repetition_ratio":0.21699905,"word_repetition_ratio":0.24683544,"special_character_ratio":0.5099526,"punctuation_ratio":0.15189873,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99845153,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-20T15:18:02Z\",\"WARC-Record-ID\":\"<urn:uuid:74191947-60cd-4be8-8e4f-b1fee9b62d01>\",\"Content-Length\":\"13671\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b5531bf4-f621-4325-b98d-6e538cd25c5b>\",\"WARC-Concurrent-To\":\"<urn:uuid:70e1661d-ade1-4248-8902-ae385265432a>\",\"WARC-IP-Address\":\"176.9.140.13\",\"WARC-Target-URI\":\"https://numberworld.info/11001010\",\"WARC-Payload-Digest\":\"sha1:57TL72OBXZMKRFVIEGWPRAPNAFD4WMLF\",\"WARC-Block-Digest\":\"sha1:NR7NBIO6OSHUP7IISD7ZIEEHARLAKDL4\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250598800.30_warc_CC-MAIN-20200120135447-20200120164447-00090.warc.gz\"}"}
https://elimdigital.com/topic-9-ratio-profit-and-loss/	[ "Home MATHEMATICS TOPIC 9: RATIO, PROFIT AND LOSS~ MATHEMATICS FORM 1\n\n# TOPIC 9: RATIO, PROFIT AND LOSS~ MATHEMATICS FORM 1\n\n133\n0\nSHARE", null, "Ratio\n\nA ratio – is a way of comparing quantities measured in the same units\nExamples of ratios\n1. A class has 45 girls and 40 boys. The ratio of number of boys to the number of girls = 40: 45\n2. A football ground 100 𝑚 long and 50 𝑚 wide. The ratio of length to the width = 100: 50\nNOTE: Ratios can be simplified like fractions\n1. 40: 45 = 8: 9\n2. 100: 50 = 2: 1\nA Ratio in its Simplest Form\nExpress a ratio in its simplest form\nExample 1\nSimplify the following ratios, giving answers as whole numbers", null, "Solution", null, "A Given Quantity into Proportional Parts\nDivide a given quantity into proportional parts\nExample 2\nExpress the following ratios in the form of", null, "Solution", null, "To increase or decrease a certain quantity in a given ratio, multiply the quantity with that ratio\nExample 3\n1. Increase 6 𝑚 in the ratio 4 ∶ 3\n2. Decrease 800 /− in the ratio 4 ∶ 5\nSolution", null, "Profit or Loss\nFind profit or loss\nIf you buy something and then sell it at a higher price, then you have a profit which is given by: Profit = selling price − buying price\nIf you buy something and then sell it at a lower price, then you have a loss which is given by: Loss = buying price − selling price\nThe profit or loss can also be expressed as a percentage of buying price as follows:", null, "Percentage Profit and Percentage Loss\nCalculate percentage profit and percentage profit and percentage loss\nExample 4\nMr. Richard bought a car for 3, 000, 000/− and sold for 3, 500, 000/−. What is the profit and percentage profit obtained?\nSolution\nProfit= selling price − buying price = 3,500,000-3,000,000=500,000\nTherefore the profit obtained is 500,000/-", null, "Example 5\nSolution", null, "But buying price = 780, 000/− and loss = buying price − selling price = 780, 000 − 720, 000 = 60, 000/−", null, "Simple Interest\nCalculate simple interest\nThe amount of money charged when a person borrows money e. g from a bank is called interest (I)\nThe amount of money borrowed is called principle (P)\nTo calculate interest, we use interest rate (R) given as a percentage and is usually taken per year or per annum (p.a)", null, "Example 6\nCalculate the simple interest charged on the following\n1. 850, 000/− at 15% per annum for 9 months\n2. 200, 000/− at 8% per annum for 2 years\nSolution", null, "Real Life Problems Related to Simple Interest\nSolve real life problems related to simple interest\nExample 7\nMrs. Mihambo deposited money in CRDB bank for 3 years and 4 months. A t the end of this time she earned a simple interest of 87, 750/− at 4.5% per annum. How much had she deposited in the bank?\nSolution\nGiven I = 87, 750/− R = 4.5% % T = 3 years and 4 months\nChange months to years", null, "SHARE\nPrevious articleTOPIC 8: NUMBERS (II)~ MATHEMATICS FORM 1" ]	[ null, "https://elimdigital.com/wp-content/uploads/2021/06/RATIO-PROFIT-AND-LOS-640x346.jpg", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_15.38.43_1465389584390.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_15.40.47_1465389677439.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_15.46.27_1465390024092.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_15.49.44_1465390210793.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_15.56.23_1465390605588.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_15.59.51_1465390825484.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_16.06.24_1465391264034.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_16.09.47_1465391412914.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_16.12.15_1465391562368.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_16.15.55_1465391834547.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_16.19.14_1465391982423.png", null, "https://sdimg.blob.core.windows.net/images/ShuleDirect/22472/Original/Screen_Shot_2016-06-08_at_16.30.40_1465392677105.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8926568,"math_prob":0.9859122,"size":2603,"snap":"2023-14-2023-23","text_gpt3_token_len":721,"char_repetition_ratio":0.13813005,"word_repetition_ratio":0.04453441,"special_character_ratio":0.28851324,"punctuation_ratio":0.08795411,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9992948,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26],"im_url_duplicate_count":[null,2,null,null,null,null,null,null,null,null,null,null,null,9,null,9,null,9,null,9,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-03-26T09:26:50Z\",\"WARC-Record-ID\":\"<urn:uuid:22da9295-70b9-496d-ac59-8d5b41c480d1>\",\"Content-Length\":\"148715\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e237d9aa-a532-4c14-ae5e-ae21044160fa>\",\"WARC-Concurrent-To\":\"<urn:uuid:5be7f605-a3c0-4848-8eb5-724f3ed7f429>\",\"WARC-IP-Address\":\"67.223.118.26\",\"WARC-Target-URI\":\"https://elimdigital.com/topic-9-ratio-profit-and-loss/\",\"WARC-Payload-Digest\":\"sha1:N6KWR7F75DTSR55LBXV6UE3GG4NBDTJ6\",\"WARC-Block-Digest\":\"sha1:C2GPQMKLQRGDDAEKIMTVM6VLSWDCIB5I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-14/CC-MAIN-2023-14_segments_1679296945440.67_warc_CC-MAIN-20230326075911-20230326105911-00384.warc.gz\"}"}
https://studysoup.com/tsg/21984/an-introduction-to-thermal-physics-1st-edition-chapter-2-problem-29p	[ "# Consider a system of two Einstein solids, with NA = 300,", null, "## Problem 29P Chapter 2\n\nAn Introduction to Thermal Physics \| 1st Edition\n\n• 2901 Step-by-step solutions solved by professors and subject experts\n• Get 24/7 help from StudySoup virtual teaching assistants", null, "An Introduction to Thermal Physics \| 1st Edition\n\n4 5 0 397 Reviews\n14\n4\nProblem 29P\n\nConsider a system of two Einstein solids, with NA = 300, NB = 200, and (qtotal = 100 (as discussed in Section 2.3). Compute the entropy of the most likely macrostate and of the least likely macrostate. Also compute the entropy over long time scales, assuming that all microstates are accessible. (Neglect the factor of Boltzmann’s constant in the definition of entropy; for systems this small it is best to think of entropy as a pure number.)\n\nStep-by-Step Solution:\nStep 1<p>In this system we have", null, "and", null, "and", null, ".\n\nThe maximum probable microstate is when", null, ". And minimum probable microstate is when", null, ".\n\nNow we have to calculate the entropy as pure number (ignoring the boltzmann's constant) for the both states.\n\nStep 2<p>The total number of microstates in the most probable state is", null, "(given in the book)\n\nHence the entropy is", null, "Hence the entropy of the most probable state is 264.\n\nStep 3 of 3\n\n##### ISBN: 9780201380279\n\nThis full solution covers the following key subjects: entropy, likely, macrostate, compute, accessible. This expansive textbook survival guide covers 10 chapters, and 454 solutions. Since the solution to 29P from 2 chapter was answered, more than 233 students have viewed the full step-by-step answer. The full step-by-step solution to problem: 29P from chapter: 2 was answered by Sieva Kozinsky, our top Physics solution expert on 07/05/17, 04:29AM. This textbook survival guide was created for the textbook: An Introduction to Thermal Physics , edition: 1st. The answer to “Consider a system of two Einstein solids, with NA = 300, NB = 200, and (qtotal = 100 (as discussed in Section 2.3). Compute the entropy of the most likely macrostate and of the least likely macrostate. Also compute the entropy over long time scales, assuming that all microstates are accessible. (Neglect the factor of Boltzmann’s constant in the definition of entropy; for systems this small it is best to think of entropy as a pure number.)” is broken down into a number of easy to follow steps, and 77 words. An Introduction to Thermal Physics was written by Sieva Kozinsky and is associated to the ISBN: 9780201380279.\n\n#### Related chapters\n\nUnlock Textbook Solution\n\nConsider a system of two Einstein solids, with NA = 300,\n\n×\nGet Full Access to An Introduction To Thermal Physics - 1st Edition - Chapter 2 - Problem 29p\n\nGet Full Access to An Introduction To Thermal Physics - 1st Edition - Chapter 2 - Problem 29p\n\nI don't want to reset my password\n\nNeed help? Contact support\n\nNeed an Account? Is not associated with an account\nWe're here to help" ]	[ null, "https://studysoup.com/cdn/56cover_2610073", null, "https://studysoup.com/cdn/56cover_2610073", null, "https://lh3.googleusercontent.com/YrSwg4YpXLGB-1RbG3i1ZRtAbcFnVexT8DO5l3seaq1PIp2MLU6fgTW-vk2GgZ9RhHoRa1DEijaLTH8_dSJM4HCLR4IAMH6UmGfwhj05J-NE8Kkq75MQyL25CMiNjUegMve6kTtw", null, "https://lh5.googleusercontent.com/ROyMFdKbv_hoOfK4EKKIhZHJCAVa6sMx8wRELGA-E5cASetUZ-NkEBWgRC8K80BJlf44rI9h8Sh3g_zAHNYMGWXL7TxQU4u8ZfHh4S5TSek8FwxclNuQya7x6EC2mnKwPfu_R836", null, "https://lh3.googleusercontent.com/ApXyNNAvwYrQTGnuDwt7JGIhwTIn9qL_OEr5CzdYjjuXahV44UDWKhw2XiPE7IJbJPi6ka0cNCwxXsYO1WwcUKDwOHpnXgTuugI35G9AySDMf-OYq6N0sy6Ys0ne5vSGZ_8ddQdb", null, "https://lh6.googleusercontent.com/9uSh3h8xyOTm0iexFANdh39BljWK7gxoK9r1mf9_4WHbYJdSmw1QhNal1LHpFVEOg50dabAjqb4XuIeYZjOiTidnVnMrCTiMQ1t_wf3vzU-Sv13Md-4Fxsd4afLG2tjgwmjzANLs", null, "https://lh4.googleusercontent.com/mAjAAcX_UBQ70rbsFUTeppL4YNXHvoH8jEM44GKexK9h8ENh60MOCvNAMOhPrxhm2ZWD4eG2yqzGt6UHFCSV-iraZpxCZ-4WJD0Yrc_8F40vNlcK6FQPdKB4pARKiUIqzCRggsAU", null, "https://lh6.googleusercontent.com/wpktHHpZsLvy_3VSyusYGh7DzLcduSLGJlPxKsl0qDGa8nCTVN3xyKTtybi4bZzIlbRgjKn3DrnHHDzkSyWvvDu21XPCdnBRycmn33f5CNaaurEdYF2FQEzJ4uDIjcvwQp_EPYO_", null, "https://lh5.googleusercontent.com/TeFOEgYVUwxzxt8oNRaHgK9qVcfvP06gh_vdJBgdMPYH3DY4wfewFJeeRw05viVJ5di0E4l7qPR2FCZOsII-iusmAC9WpMxlRTrLfc-ZxTofFMWsvIz3CeC5wMXrryGHBe5WkLEc", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.89109564,"math_prob":0.69278866,"size":809,"snap":"2019-43-2019-47","text_gpt3_token_len":188,"char_repetition_ratio":0.16397515,"word_repetition_ratio":0.02919708,"special_character_ratio":0.22991347,"punctuation_ratio":0.09677419,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99605095,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18],"im_url_duplicate_count":[null,null,null,null,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T10:53:46Z\",\"WARC-Record-ID\":\"<urn:uuid:4e4cfb5e-c204-4a5c-a2f7-a56d1a69a58b>\",\"Content-Length\":\"84056\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:45c9de49-c37f-4c31-a95e-2e7ad2a1c2ba>\",\"WARC-Concurrent-To\":\"<urn:uuid:aa87d7ba-5d63-45f7-a729-32052da5d080>\",\"WARC-IP-Address\":\"54.189.254.180\",\"WARC-Target-URI\":\"https://studysoup.com/tsg/21984/an-introduction-to-thermal-physics-1st-edition-chapter-2-problem-29p\",\"WARC-Payload-Digest\":\"sha1:X7J24CRUCMTXKHLYDVFDBJMTG5DHNV46\",\"WARC-Block-Digest\":\"sha1:HF2CBOZOP2HQN4Y4DGJFXCIRYFYQHVB2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670770.21_warc_CC-MAIN-20191121101711-20191121125711-00487.warc.gz\"}"}
http://www.kejimatou.com/xinwen/41.html	[ "# sumif函数多条件求和案例汇总\n\nSUMIF函数是Excel使用频率很高的函数，使用SUMIF函数可以对报表范围中符合指定条件的值求和。Excel中SUMIF函数的用法是根据指定条件对若干单元格、区域或引用求和。\n\n=SUMIF(C2:C9,\"小米\",D2:D9)", null, "=SUMIF(C2:C9,\"<>小米\",D2:D9)", null, "=SUMIF(D2:D9,\">300\")", null, "=SUMIF(D2:D9,\"<\"&AVERAGE(D2:D9))", null, "=SUMIF(C2:C9,F1,D2:D9)", null, "=SUMIF(C2:C9,\"\",D2:D9)", null, "=SUMIF(C2:C9,\"???\",D2:D9)", null, "=SUMIF(C2:C9,\"米\",D2:D9)", null, "=SUMIF(C2:C9,\"大\",D2:D9)", null, "=SUMIF(B2:B9,TODAY(),D2:D9)", null, "=SUMIF(D2:D9,\"<9e307\")", null, "=SUM(SUMIF(C2:C9,{\"小米\",\"大米\"},D2:D9))", null, "=SUMIF(B3:F10,\"\",B2:F9)/5", null, "=SUMIF(C2:F9,\"小米\",D2:G9)", null, "" ]	[ null, "http://p3.pstatp.com/large/pgc-image/dc385b105773439991658511f641b2d7", null, "http://p9.pstatp.com/large/pgc-image/94c89a32854f41058a479667b993abab", null, "http://p3.pstatp.com/large/pgc-image/6d899bc0bc614abf84fb40865c0358b1", null, "http://p99.pstatp.com/large/pgc-image/f5f017d6794c496199a0436c03ba5615", null, "http://p99.pstatp.com/large/pgc-image/1a83815107b24f808e5bfc0e6210f229", null, "http://p99.pstatp.com/large/pgc-image/8e1ecfa8bfb74c028cca29e6c79978e0", null, "http://p1.pstatp.com/large/pgc-image/64ec41beaae944cbbcf72fe9087c972d", null, "http://p99.pstatp.com/large/pgc-image/83a1f569d0274a41bf0fb61fcec0ce67", null, "http://p3.pstatp.com/large/pgc-image/d19c7120078040bd9242bb9e32edaf85", null, "http://p99.pstatp.com/large/pgc-image/c8ed904df7d6488a99d3b57b07bfefe9", null, "http://p99.pstatp.com/large/pgc-image/18f5e7c0eb0a42ad9054864cabc795b2", null, "http://p1.pstatp.com/large/pgc-image/9a65995983b84ef88bb1ace4b72cc64e", null, "http://p99.pstatp.com/large/pgc-image/e571bcb1311d408aafee41f7a7287b17", null, "http://p1.pstatp.com/large/pgc-image/7606a19f5656445094efccdc5237ade3", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.64635915,"math_prob":0.9980608,"size":965,"snap":"2019-26-2019-30","text_gpt3_token_len":789,"char_repetition_ratio":0.22372529,"word_repetition_ratio":0.0,"special_character_ratio":0.34404144,"punctuation_ratio":0.2961165,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9942409,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28],"im_url_duplicate_count":[null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-07-17T06:24:18Z\",\"WARC-Record-ID\":\"<urn:uuid:156fd5a3-c015-4549-8a24-3673a45abc41>\",\"Content-Length\":\"21092\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:79bffbed-e27f-4570-b52a-4791115b2431>\",\"WARC-Concurrent-To\":\"<urn:uuid:911aef63-6146-4554-81f2-bfa8a0313ca3>\",\"WARC-IP-Address\":\"154.211.144.187\",\"WARC-Target-URI\":\"http://www.kejimatou.com/xinwen/41.html\",\"WARC-Payload-Digest\":\"sha1:QEQWJPPLY3LA2AQXDXLXEZBZMOPSFA2L\",\"WARC-Block-Digest\":\"sha1:IJY5IJWS2OVDUKN6FN6O2VUWQB4CEZSS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-30/CC-MAIN-2019-30_segments_1563195525094.53_warc_CC-MAIN-20190717061451-20190717083451-00007.warc.gz\"}"}
https://foreach.id/EN/fluids/viscositydynamic/decipoise-to-pound_force_second%7Csq_inch.html	[ "# Convert decipoise to pound-force second/inch² (dP to lbf s/in²)\n\nBatch Convert\n• decipoise [dP]\n• pound-force second/inch² [lbf s/in²]\nCopy\n_\nCopy\n• decipoise [dP]\n• pound-force second/inch² [lbf s/in²]\n\n## Decipoise to Pound-force second/inch² (dP to lbf s/in²)\n\n### Decipoise (Symbol or Abbreviation: dP)\n\nDecipoise is one of dynamic viscosity units. Decipoise abbreviated or symbolized by dP. The value of 1 decipoise is equal to 0.01 pascal second. In its relation with pound-force second/inch², 1 decipoise is equal to 0.0000014504 pound-force second/inch².\n\n#### Relation with other units\n\n1 decipoise equals to 0.01 pascal second\n\n1 decipoise equals to 0.0010197 kilogram-force second/meter²\n\n1 decipoise equals to 0.01 newton second/meter²\n\n1 decipoise equals to 10 millinewton second/meter²\n\n1 decipoise equals to 0.1 dyne second/centimeter²\n\n1 decipoise equals to 0.1 poise\n\n1 decipoise equals to 1e-19 exapoise\n\n1 decipoise equals to 1e-16 petapoise\n\n1 decipoise equals to 1e-13 terapoise\n\n1 decipoise equals to 1e-10 gigapoise\n\n1 decipoise equals to 1e-7 megapoise\n\n1 decipoise equals to 0.0001 kilopoise\n\n1 decipoise equals to 0.001 hectopoise\n\n1 decipoise equals to 0.01 dekapoise\n\n1 decipoise equals to 10 centipoise\n\n1 decipoise equals to 100 millipoise\n\n1 decipoise equals to 100,000 micropoise\n\n1 decipoise equals to 100,000,000 nanopoise\n\n1 decipoise equals to 100,000,000,000 picopoise\n\n1 decipoise equals to 100,000,000,000,000 femtopoise\n\n1 decipoise equals to 100,000,000,000,000,000 attopoise\n\n1 decipoise equals to 0.0000014504 pound-force second/inch²\n\n1 decipoise equals to 0.00020885 pound-force second/foot²\n\n1 decipoise equals to 0.0067197 poundal second/foot²\n\n1 decipoise equals to 0.1 gram/(centimetersecond)\n\n1 decipoise equals to 0.00020885 slug/(footsecond)\n\n1 decipoise equals to 0.0067197 pound/(footsecond)\n\n1 decipoise equals to 24.191 pound/(foothour)\n\n1 decipoise equals to 0.0000014514 reyn\n\n### Pound-force second/inch² (Symbol or Abbreviation: lbf s/in²)\n\nPound-force second/inch² is one of dynamic viscosity units. Pound-force second/inch² abbreviated or symbolized by lbf s/in². The value of 1 pound-force second/inch² is equal to 6894.8 pascal second. In its relation with decipoise, 1 pound-force second/inch² is equal to 689480 decipoise.\n\n#### Relation with other units\n\n1 pound-force second/inch² equals to 6,894.8 pascal second\n\n1 pound-force second/inch² equals to 703.07 kilogram-force second/meter²\n\n1 pound-force second/inch² equals to 6,894.8 newton second/meter²\n\n1 pound-force second/inch² equals to 6,894,800 millinewton second/meter²\n\n1 pound-force second/inch² equals to 68,948 dyne second/centimeter²\n\n1 pound-force second/inch² equals to 68,948 poise\n\n1 pound-force second/inch² equals to 6.8948e-14 exapoise\n\n1 pound-force second/inch² equals to 6.8948e-11 petapoise\n\n1 pound-force second/inch² equals to 6.8948e-8 terapoise\n\n1 pound-force second/inch² equals to 0.000068948 gigapoise\n\n1 pound-force second/inch² equals to 0.068948 megapoise\n\n1 pound-force second/inch² equals to 68.948 kilopoise\n\n1 pound-force second/inch² equals to 689.48 hectopoise\n\n1 pound-force second/inch² equals to 6,894.8 dekapoise\n\n1 pound-force second/inch² equals to 689,480 decipoise\n\n1 pound-force second/inch² equals to 6,894,800 centipoise\n\n1 pound-force second/inch² equals to 68,948,000 millipoise\n\n1 pound-force second/inch² equals to 68,948,000,000 micropoise\n\n1 pound-force second/inch² equals to 68,948,000,000,000 nanopoise\n\n1 pound-force second/inch² equals to 68,948,000,000,000,000 picopoise\n\n1 pound-force second/inch² equals to 68,948,000,000,000,000,000 femtopoise\n\n1 pound-force second/inch² equals to 6.8948e+22 attopoise\n\n1 pound-force second/inch² equals to 144 pound-force second/foot²\n\n1 pound-force second/inch² equals to 4,633.1 poundal second/foot²\n\n1 pound-force second/inch² equals to 68,948 gram/(centimetersecond)\n\n1 pound-force second/inch² equals to 144 slug/(footsecond)\n\n1 pound-force second/inch² equals to 4,633.1 pound/(footsecond)\n\n1 pound-force second/inch² equals to 16,679,000 pound/(foothour)\n\n1 pound-force second/inch² equals to 1.0007 reyn\n\n### How to convert Decipoise to Pound-force second/inch² (dP to lbf s/in²):\n\n#### Conversion Table for Decipoise to Pound-force second/inch² (dP to lbf s/in²)\n\ndecipoise (dP) pound-force second/inch² (lbf s/in²)\n0.01 dP 1.4504e-8 lbf s/in²\n0.1 dP 1.4504e-7 lbf s/in²\n1 dP 0.0000014504 lbf s/in²\n2 dP 0.0000029008 lbf s/in²\n3 dP 0.0000043511 lbf s/in²\n4 dP 0.0000058015 lbf s/in²\n5 dP 0.0000072519 lbf s/in²\n6 dP 0.0000087023 lbf s/in²\n7 dP 0.000010153 lbf s/in²\n8 dP 0.000011603 lbf s/in²\n9 dP 0.000013053 lbf s/in²\n10 dP 0.000014504 lbf s/in²\n20 dP 0.000029008 lbf s/in²\n25 dP 0.000036259 lbf s/in²\n50 dP 0.000072519 lbf s/in²\n75 dP 0.00010878 lbf s/in²\n100 dP 0.00014504 lbf s/in²\n250 dP 0.00036259 lbf s/in²\n500 dP 0.00072519 lbf s/in²\n750 dP 0.0010878 lbf s/in²\n1,000 dP 0.0014504 lbf s/in²\n100,000 dP 0.14504 lbf s/in²\n1,000,000,000 dP 1,450.4 lbf s/in²\n1,000,000,000,000 dP 1,450,400 lbf s/in²\n\n#### Conversion Table for Pound-force second/inch² to Decipoise (lbf s/in² to dP)\n\npound-force second/inch² (lbf s/in²) decipoise (dP)\n0.01 lbf s/in² 6,894.8 dP\n0.1 lbf s/in² 68,948 dP\n1 lbf s/in² 689,480 dP\n2 lbf s/in² 1,379,000 dP\n3 lbf s/in² 2,068,400 dP\n4 lbf s/in² 2,757,900 dP\n5 lbf s/in² 3,447,400 dP\n6 lbf s/in² 4,136,900 dP\n7 lbf s/in² 4,826,300 dP\n8 lbf s/in² 5,515,800 dP\n9 lbf s/in² 6,205,300 dP\n10 lbf s/in² 6,894,800 dP\n20 lbf s/in² 13,790,000 dP\n25 lbf s/in² 17,237,000 dP\n50 lbf s/in² 34,474,000 dP\n75 lbf s/in² 51,711,000 dP\n100 lbf s/in² 68,948,000 dP\n250 lbf s/in² 172,370,000 dP\n500 lbf s/in² 344,740,000 dP\n750 lbf s/in² 517,110,000 dP\n1,000 lbf s/in² 689,480,000 dP\n100,000 lbf s/in² 68,948,000,000 dP\n1,000,000,000 lbf s/in² 689,480,000,000,000 dP\n1,000,000,000,000 lbf s/in² 689,480,000,000,000,000 dP\n\n#### Steps to Convert Decipoise to Pound-force second/inch² (dP to lbf s/in²)\n\n1. Example: Convert 442 decipoise to pound-force second/inch² (442 dP to lbf s/in²).\n2. 1 decipoise is equivalent to 0.0000014504 pound-force second/inch² (1 dP is equivalent to 0.0000014504 lbf s/in²).\n3. 442 decipoise (dP) is equivalent to 442 times 0.0000014504 pound-force second/inch² (lbf s/in²).\n4. Retrieved 442 decipoise is equivalent to 0.00064107 pound-force second/inch² (442 dP is equivalent to 0.00064107 lbf s/in²).\n\n▸▸" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.50206625,"math_prob":0.9990213,"size":6158,"snap":"2021-21-2021-25","text_gpt3_token_len":2338,"char_repetition_ratio":0.33685407,"word_repetition_ratio":0.110199295,"special_character_ratio":0.4272491,"punctuation_ratio":0.14285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9838038,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-19T04:17:54Z\",\"WARC-Record-ID\":\"<urn:uuid:a0d5d940-5503-4937-80c5-36cc9054ddf1>\",\"Content-Length\":\"60610\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:14b04439-25e6-4f76-a1c3-b0f771e6b72c>\",\"WARC-Concurrent-To\":\"<urn:uuid:18c51859-7476-4189-81bd-2d73f5945e00>\",\"WARC-IP-Address\":\"104.21.3.52\",\"WARC-Target-URI\":\"https://foreach.id/EN/fluids/viscositydynamic/decipoise-to-pound_force_second%7Csq_inch.html\",\"WARC-Payload-Digest\":\"sha1:4COOFWV5FIMXI5JMJXG4ZI6GKOENRB3I\",\"WARC-Block-Digest\":\"sha1:LJKQXLQVIL3EWWOUZNIBTUGHJNMB4QRU\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623487643380.40_warc_CC-MAIN-20210619020602-20210619050602-00033.warc.gz\"}"}
https://socratic.org/questions/the-ratio-of-the-measures-of-two-supplementary-angles-is-2-7-how-do-you-find-the	[ "# The ratio of the measures of two supplementary angles is 2:7. How do you find the measures of the angles?\n\nMar 29, 2017\n\n${40}^{\\circ} \\text{ and } {140}^{\\circ}$\n\n#### Explanation:\n\ncolor(orange)\"Reminder \" color(red)(bar(ul(\|color(white)(2/2)color(black)(\" the sum of 2 supplementary angles\" = 180^@)color(white)(2/2)\|)))\n\n$\\text{sum the parts of the ratio}$\n\n$\\Rightarrow 2 + 7 = 9 \\text{ parts in total}$\n\nFind the value of 1 part by dividing ${180}^{\\circ} \\text{ by } 9$\n\n$\\Rightarrow {180}^{\\circ} / 9 = {20}^{\\circ} \\leftarrow \\textcolor{red}{\\text{ value of 1 part}}$\n\n$\\Rightarrow \\text{2 parts } = 2 \\times {20}^{\\circ} = {40}^{\\circ}$\n\n$\\Rightarrow \\text{7 parts } = 7 \\times {20}^{\\circ} = {140}^{\\circ}$\n\n$\\text{Thus the supplementary angles are \" 40^@\" and } {140}^{\\circ}$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7614711,"math_prob":0.9997359,"size":703,"snap":"2019-43-2019-47","text_gpt3_token_len":235,"char_repetition_ratio":0.13161659,"word_repetition_ratio":0.0,"special_character_ratio":0.36273116,"punctuation_ratio":0.046875,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998309,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-22T04:29:43Z\",\"WARC-Record-ID\":\"<urn:uuid:8eeb20bd-2838-44b9-8a68-fe66ec168820>\",\"Content-Length\":\"33885\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:dcea5486-b0e8-4452-bddc-3ad912a355be>\",\"WARC-Concurrent-To\":\"<urn:uuid:c19e3d94-02bd-402e-9230-3a17b3f08afa>\",\"WARC-IP-Address\":\"54.221.217.175\",\"WARC-Target-URI\":\"https://socratic.org/questions/the-ratio-of-the-measures-of-two-supplementary-angles-is-2-7-how-do-you-find-the\",\"WARC-Payload-Digest\":\"sha1:H5U7LQCNMXVE4E3RKXRW77DW7BTPTELC\",\"WARC-Block-Digest\":\"sha1:GLHPX3NLRWSANAGSEDID75YN5L73GQNW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496671239.99_warc_CC-MAIN-20191122042047-20191122070047-00363.warc.gz\"}"}
https://targetmol.com/compound/CX-157	[ "# CX-157\n\nCatalog No. T15023 CAS 205187-53-7\n\nCX-157 is a reversible Monoamine Oxidase-A (MAO-A) inhibitor (EC50:19.3 ng/mL).\n\nAll products from TargetMol are for Research Use Only. Not for Human or Veterinary or Therapeutic Use.", null, "CX-157, CAS 205187-53-7\nProduct consultation\nGet quote\nPurity: 98%\nBiological Description\nChemical Properties\nStorage & Solubility Information\n Description CX-157 is a reversible monoamine oxidase-A (MAO-A) inhibitor (EC50:19.3 ng/mL). Targets&IC50 MAO-A:（EC50）19.3 ng/ml In vivo CX-157 is an investigational reagent currently in development for the treatment of major depressive disorder (MDD). Mechanistic studies in animals have shown that CX-157 acts to inhibit MAO-A activity in a reversible and competitive manner which improving brain levels of monoamine neurotransmitters.\n Molecular Weight 348.27 Formula C14H8F4O4S CAS No. 205187-53-7\n\n#### Storage\n\nPowder: -20°C for 3 years\n\nIn solvent: -80°C for 2 years\n\n#### Solubility Information\n\n( < 1 mg/ml refers to the product slightly soluble or insoluble )\n\n##", null, "Dose Conversion\n\nYou can also refer to dose conversion for different animals. More\n\n##", null, "In vivo Formulation Calculator (Clear solution)\n\nStep One: Enter information below\nDosage\nmg/kg\nAverage weight of animals\ng\nDosing volume per animal\nul\nNumber of animals\nStep Two: Enter the in vivo formulation\n% DMSO\n%\n% Tween 80\n% ddH2O\n\n##", null, "Calculator\n\nMolarity Calculator\nDilution Calculator\nReconstitution Calculation\nMolecular Weight Calculator\n=\nX\nX\n\n### Molarity Calculator allows you to calculate the\n\n• Mass of a compound required to prepare a solution of known volume and concentration\n• Volume of solution required to dissolve a compound of known mass to a desired concentration\n• Concentration of a solution resulting from a known mass of compound in a specific volume\nSee Example\n\nAn example of a molarity calculation using the molarity calculator\nWhat is the mass of compound required to make a 10 mM stock solution in 10 ml of water given that the molecular weight of the compound is 197.13 g/mol?\nEnter 197.13 into the Molecular Weight (MW) box\nEnter 10 into the Concentration box and select the correct unit (millimolar)\nEnter 10 into the Volume box and select the correct unit (milliliter)\nPress calculate\nThe answer of 19.713 mg appears in the Mass box\n\nX\n=\nX\n\n### Calculator the dilution required to prepare a stock solution\n\nCalculate the dilution required to prepare a stock solution\nThe dilution calculator is a useful tool which allows you to calculate how to dilute a stock solution of known concentration. Enter C1, C2 & V2 to calculate V1.\n\nSee Example\n\nAn example of a dilution calculation using the Tocris dilution calculator\nWhat volume of a given 10 mM stock solution is required to make 20ml of a 50 μM solution?\nUsing the equation C1V1 = C2V2, where C1=10 mM, C2=50 μM, V2=20 ml and V1 is the unknown:\nEnter 10 into the Concentration (start) box and select the correct unit (millimolar)\nEnter 50 into the Concentration (final) box and select the correct unit (micromolar)\nEnter 20 into the Volume (final) box and select the correct unit (milliliter)\nPress calculate\nThe answer of 100 microliter (0.1 ml) appears in the Volume (start) box\n\n=\n/\n\n### Calculate the volume of solvent required to reconstitute your vial.\n\nThe reconstitution calculator allows you to quickly calculate the volume of a reagent to reconstitute your vial.\nSimply enter the mass of reagent and the target concentration and the calculator will determine the rest.\n\ng/mol\n\n### Enter the chemical formula of a compound to calculate its molar mass and elemental composition\n\nTip: Chemical formula is case sensitive: C10H16N2O2 c10h16n2o2\n\nInstructions to calculate molar mass (molecular weight) of a chemical compound:\nTo calculate molar mass of a chemical compound, please enter its chemical formula and click 'Calculate'.\nDefinitions of molecular mass, molecular weight, molar mass and molar weight:\nMolecular mass (molecular weight) is the mass of one molecule of a substance and is expressed n the unified atomic mass units (u). (1 u is equal to 1/12 the mass of one atom of carbon-12)\nMolar mass (molar weight) is the mass of one mole of a substance and is expressed in g/mol.\n\nbottom\n\n## Tech Support\n\nPlease see Inhibitor Handling Instructions for more frequently ask questions. Topics include: how to prepare stock solutions, how to store products, and cautions on cell-based assays & animal experiments, etc." ]	[ null, "https://www.targetmol.com/file/group1/M00/03/7D/CgoaDmTA9kmENx1IAAAAAPdWbvw363.png", null, "https://targetmol.com/images/icons2/calculator.svg", null, "https://targetmol.com/images/icons2/calculator.svg", null, "https://targetmol.com/images/icons2/calculator.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8083452,"math_prob":0.9483172,"size":1140,"snap":"2023-40-2023-50","text_gpt3_token_len":318,"char_repetition_ratio":0.10915493,"word_repetition_ratio":0.0875,"special_character_ratio":0.28245613,"punctuation_ratio":0.106481485,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97000796,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,1,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-29T17:15:31Z\",\"WARC-Record-ID\":\"<urn:uuid:0035228e-5558-43a4-8098-ead672669cb3>\",\"Content-Length\":\"111296\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4bcece40-b248-4779-ada2-9b9b3dd8c310>\",\"WARC-Concurrent-To\":\"<urn:uuid:66ef3d63-361f-4345-91af-93b93caae1a5>\",\"WARC-IP-Address\":\"49.51.37.151\",\"WARC-Target-URI\":\"https://targetmol.com/compound/CX-157\",\"WARC-Payload-Digest\":\"sha1:77PDV2OT4F53IQHL62X4YXAXQAUHT7OT\",\"WARC-Block-Digest\":\"sha1:ZUUIVWEV5BZWZPIUAKF5OY5R53SF7FUI\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510520.98_warc_CC-MAIN-20230929154432-20230929184432-00408.warc.gz\"}"}
https://mathemania.com/lesson/exponential-function/	[ "# Exponential function\n\nIf $a$ is the given base, $a>0$, $a \\neq 1$, and if $x$ is any real number, then the function\n\n$$f(x) = a^x$$\n\nis called the exponential function.\n\nThe base $a = 10^x$ has the special role in the calculating of powers. Let’s observe powers of the shape $10^x$.\n\nFor this purpose, in the coordinate plane we will draw the graph of the function $f(x) = 10^x$.", null, "Therefore, the graph of the function $f(x) = 10^x$ is:", null, "As we can see, for the positive real numbers $x$, this exponential function grows very fast, and for the negative real numbers $x$ the function falls to the zero and it’s very close to the negative part of the $x$ – axis.\n\nSimilarly, we can draw the graph of the function $f(x) = a ^x$, for any base $a$, $a>0$, $a \\neq 1$. For instance, we will draw the graph of the function $f(x) = 3^x$:", null, "Now we will observe the exponential function with the base $0 < a < 1$. For instance, let be $a = \\displaystyle{\\frac{1}{3}}$. In the same coordinate plane we will draw functions $f(x) = 3^x$ and $g(x) = \\left( \\displaystyle{\\frac{1}{3}} \\right)^x$:", null, "We can notice that functions $f$ and $g$ have the same values for numbers of opposite signs, because $f(-x) = 3^{-x} = g(x)$. Therefore, graphs of these functions are symmetrical to the respect of the $y$- axis.\n\nProperties of exponential function\n\nThe exponential function $f(x) = a^x$, $a>0$, $a \\neq 1$, has the following properties:\n\n1. The function is defined for every real number $x$.\n2. All values of the function are positive real numbers.\n3. $a^x \\cdot a^y = a^{x+y}$\n4. $(a^x)^y = a^{xy}$\n5. $(a \\cdot b)^x = a^x \\cdot b^x$\n6. $a^0 =1$\n7. If $a>1$, then for $x_1 < x_2$ is valid $a^{x_1} < a^{x_2}$\n8. If $0<a<1$, then for $x_1 > x_2$ is valid $a^{x_1}> a^{x_2}$" ]	[ null, "https://mathemania.com/wp-content/uploads/2017/07/tablica.png", null, "https://mathemania.com/wp-content/uploads/2017/07/eksponencijalna.png", null, "https://mathemania.com/wp-content/uploads/2017/07/eksp1.png", null, "https://mathemania.com/wp-content/uploads/2017/07/eksp2.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.66328144,"math_prob":1.000009,"size":1711,"snap":"2022-40-2023-06","text_gpt3_token_len":567,"char_repetition_ratio":0.16285881,"word_repetition_ratio":0.06774194,"special_character_ratio":0.3547633,"punctuation_ratio":0.10215054,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000099,"pos_list":[0,1,2,3,4,5,6,7,8],"im_url_duplicate_count":[null,2,null,3,null,2,null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-01T19:44:21Z\",\"WARC-Record-ID\":\"<urn:uuid:eb675da2-736e-402f-871f-e3b3e6a4c243>\",\"Content-Length\":\"92995\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8689bedc-eb5b-4961-8501-3660e175cb6c>\",\"WARC-Concurrent-To\":\"<urn:uuid:97d2f984-6e15-4498-ba2b-03655685a190>\",\"WARC-IP-Address\":\"137.184.39.193\",\"WARC-Target-URI\":\"https://mathemania.com/lesson/exponential-function/\",\"WARC-Payload-Digest\":\"sha1:3CLXTPZONNJLIC5677JKMQSLSSRETGFZ\",\"WARC-Block-Digest\":\"sha1:SMFIOJ4MT2OZOCHMQBLAMR2DGBVND5QN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764499949.24_warc_CC-MAIN-20230201180036-20230201210036-00591.warc.gz\"}"}
https://www.sourcetrail.com/python/programm-the-fibonacci-sequence/	[ "# Solved: programm the fibonacci sequence\n\nThe main problem with programming the Fibonacci sequence is that it is not a precise sequence. The first two numbers in the sequence are always the same, but the next two numbers are not always equal. This can cause problems when trying to create a program to calculate the next number in the sequence.\n\n```\ndef Fibonacci(n):\nif n<0:\nprint(\"Incorrect input\")\n\nelif n==1:\nreturn 0\n\nelif n==2:\nreturn 1\nelse:\nreturn Fibonacci(n-1)+Fibonacci(n-2)```\n\nThis is a recursive function for generating Fibonacci numbers. The function takes an integer input, n, and returns the nth Fibonacci number. If the input is less than 0, it prints an error message. If the input is 1 or 2, it returns the first or second Fibonacci number, respectively. Otherwise, it returns the sum of the previous two Fibonacci numbers.\n\nContents\n\n## Fibonacci\n\nIn mathematics, Fibonacci is a sequence of numbers which starts with 0 and 1, and goes on to each successive number by adding the previous two numbers together. The sequence is named after Leonardo Fibonacci, who introduced it in 1202.\n\n## Sequences\n\nSequences are a powerful data structure in Python. They allow you to store multiple values in a single location, and access them sequentially.\n\nFor example, you can create a sequence of numbers using the range() function:\n\n1, 2, 3, 4, 5\n\nYou can also create a sequence of strings using the string() function:\n\n“one”, “two”, “three”, “four”, “five”\n\nRelated posts:" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.81675607,"math_prob":0.97630256,"size":1399,"snap":"2022-27-2022-33","text_gpt3_token_len":322,"char_repetition_ratio":0.1734767,"word_repetition_ratio":0.0,"special_character_ratio":0.231594,"punctuation_ratio":0.13829787,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9992923,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-04T21:06:12Z\",\"WARC-Record-ID\":\"<urn:uuid:ea01398b-fb74-4e2a-838c-ace9dd9a2c28>\",\"Content-Length\":\"98162\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f68cb3b1-1679-47cf-b9b6-c373acda25ac>\",\"WARC-Concurrent-To\":\"<urn:uuid:2392a1fd-f9d2-4b26-85f5-d9131dc57b61>\",\"WARC-IP-Address\":\"178.255.231.122\",\"WARC-Target-URI\":\"https://www.sourcetrail.com/python/programm-the-fibonacci-sequence/\",\"WARC-Payload-Digest\":\"sha1:AAK2IUILXCSRAPGT6Z6Y5TLWLORNXDQR\",\"WARC-Block-Digest\":\"sha1:FY7BWTFSZDUFA2W6V3XI4Q6WY3VXMBZN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104496688.78_warc_CC-MAIN-20220704202455-20220704232455-00706.warc.gz\"}"}
https://etoobusy.polettix.it/2022/02/03/pwc150-square-free-integer/	[ "TL;DR\n\nOn with TASK #2 from The Weekly Challenge #150. Enjoy!\n\n# The challenge\n\nWrite a script to generate all square-free integers <= 500.\n\nIn mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1. That is, its prime factorization has exactly one factor for each prime that appears in it. For example, 10 = 2 ⋅ 5 is square-free, but 18 = 2 ⋅ 3 ⋅ 3 is not, because 18 is divisible by 9 = 3*2.\n\nExample\n\nThe smallest positive square-free integers are\n1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 26, 29, 30, ...\n\n\n# The questions\n\nAs it often happens, I’m nitpicking on the details about the domain of our investigation: should we consider negative values? I guess not by the examples…\n\n# The solution\n\nsub is_square_free ($N) { return unless$N % 4;\nmy $divisor = 3; while ($N > $divisor) { if ($N % $divisor == 0) {$N /= $divisor; return unless$N % $divisor; }$divisor += 2; # go through odd candidates only\n}\nreturn 1;\n}\n\n\nThe goal is not to find all divisors, so… we don’t find them and we take every possible chance to bail out with a false value. It can happen in two cases:\n\n• if the number is a multiple of 4 because… 4 is a square, you know;\n• otherwise, if the number happens to have the same divisor twice.\n\nWhy the explicit check on 4? Well, in this way we can get the prime number 2 out of the way, and iterate only through odd divisors, starting at 3. Actually, we might start at 7 because the first positive integer that is neither a multiple of 4 nor square-free is 9. Whatever.\n\nI like the Raku translation better because it allows us to use the is-divisible-by operator %%, instead of its “contrary” (sort of) remainder-in-the-division-by %:\n\nsub is-square-free ($N is copy) { return False if$N %% 4;\nmy $divisor = 3; while$N > $divisor { if$N %% $divisor {$N = ($N /$divisor).Int;\nreturn False if $N %%$divisor;\n}\n$divisor += 2; # go through odd candidates only } return True; } This makes the whole thing more readable, but at the end of the day it was pretty readable also to begin with. I have a little itch in the fact that the division between the two integers gives out a rational even when the result is an integer… but whatever. I also like the availability of proper boolean constants, again I think it adds to the readability. The Raku version also allowed me to play a bit with multi subroutines, in the MAIN: multi sub MAIN (Int$limit = 500) {\nmy @list = (1 .. $limit).grep({is-square-free($_)});\nwhile @list {\n@list.splice(0, 20).join(', ').print;\nput @list ?? ',' !! '';\n}\n}\n\nmulti sub MAIN (@args) {\nput $_, ' ', (is-square-free($_) ?? 'is' !! 'is not'), ' square free'\nfor @args;\n}\n\n\nI’m providing three different ways to call the program:\n\n• with no parameter, the limit is set to 500 like the challenge asks;\n• with one single parameter, the limit is set by the parameter itself;\n• with multiple parameters, each is checked for being square-free or not.\n\nThe multi helps distinguishing the first two cases from the last, which is functionally nifty.\n\nOK, I didn’t include the full programs for both languages… but you know where to find them should you be curious.\n\nStay safe and have fun!\n\nComments? Octodon, , GitHub, Reddit, or drop me a line!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8645165,"math_prob":0.967691,"size":3298,"snap":"2023-40-2023-50","text_gpt3_token_len":916,"char_repetition_ratio":0.11596843,"word_repetition_ratio":0.042414356,"special_character_ratio":0.3020012,"punctuation_ratio":0.15767045,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98852044,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-28T23:54:23Z\",\"WARC-Record-ID\":\"<urn:uuid:b4de5514-45ea-4fa7-9247-3489c3d48e54>\",\"Content-Length\":\"9801\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:292d8052-2fd0-44fd-b4c7-1d07c0088edc>\",\"WARC-Concurrent-To\":\"<urn:uuid:32cca44f-8448-4b0d-9531-f0c7d1a5bb78>\",\"WARC-IP-Address\":\"217.197.91.145\",\"WARC-Target-URI\":\"https://etoobusy.polettix.it/2022/02/03/pwc150-square-free-integer/\",\"WARC-Payload-Digest\":\"sha1:DWL6XF7M2GAST7SXCRN46Q25Y2ASVASC\",\"WARC-Block-Digest\":\"sha1:5C5DYQ3DMCAORSHAK2RBI4AG56K34ALX\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510462.75_warc_CC-MAIN-20230928230810-20230929020810-00018.warc.gz\"}"}
https://metanumbers.com/1076153	[ "# 1076153 (number)\n\n1,076,153 (one million seventy-six thousand one hundred fifty-three) is an odd seven-digits composite number following 1076152 and preceding 1076154. In scientific notation, it is written as 1.076153 × 106. The sum of its digits is 23. It has a total of 2 prime factors and 4 positive divisors. There are 993,360 positive integers (up to 1076153) that are relatively prime to 1076153.\n\n## Basic properties\n\n• Is Prime? No\n• Number parity Odd\n• Number length 7\n• Sum of Digits 23\n• Digital Root 5\n\n## Name\n\nShort name 1 million 76 thousand 153 one million seventy-six thousand one hundred fifty-three\n\n## Notation\n\nScientific notation 1.076153 × 106 1.076153 × 106\n\n## Prime Factorization of 1076153\n\nPrime Factorization 13 × 82781\n\nComposite number\nDistinct Factors Total Factors Radical ω(n) 2 Total number of distinct prime factors Ω(n) 2 Total number of prime factors rad(n) 1076153 Product of the distinct prime numbers λ(n) 1 Returns the parity of Ω(n), such that λ(n) = (-1)Ω(n) μ(n) 1 Returns: 1, if n has an even number of prime factors (and is square free) −1, if n has an odd number of prime factors (and is square free) 0, if n has a squared prime factor Λ(n) 0 Returns log(p) if n is a power pk of any prime p (for any k >= 1), else returns 0\n\nThe prime factorization of 1,076,153 is 13 × 82781. Since it has a total of 2 prime factors, 1,076,153 is a composite number.\n\n## Divisors of 1076153\n\n4 divisors\n\n Even divisors 0 4 4 0\nTotal Divisors Sum of Divisors Aliquot Sum τ(n) 4 Total number of the positive divisors of n σ(n) 1.15895e+06 Sum of all the positive divisors of n s(n) 82795 Sum of the proper positive divisors of n A(n) 289737 Returns the sum of divisors (σ(n)) divided by the total number of divisors (τ(n)) G(n) 1037.38 Returns the nth root of the product of n divisors H(n) 3.71424 Returns the total number of divisors (τ(n)) divided by the sum of the reciprocal of each divisors\n\nThe number 1,076,153 can be divided by 4 positive divisors (out of which 0 are even, and 4 are odd). The sum of these divisors (counting 1,076,153) is 1,158,948, the average is 289,737.\n\n## Other Arithmetic Functions (n = 1076153)\n\n1 φ(n) n\nEuler Totient Carmichael Lambda Prime Pi φ(n) 993360 Total number of positive integers not greater than n that are coprime to n λ(n) 248340 Smallest positive number such that aλ(n) ≡ 1 (mod n) for all a coprime to n π(n) ≈ 83828 Total number of primes less than or equal to n r2(n) 16 The number of ways n can be represented as the sum of 2 squares\n\nThere are 993,360 positive integers (less than 1,076,153) that are coprime with 1,076,153. And there are approximately 83,828 prime numbers less than or equal to 1,076,153.\n\n## Divisibility of 1076153\n\n m n mod m 2 3 4 5 6 7 8 9 1 2 1 3 5 1 1 5\n\n1,076,153 is not divisible by any number less than or equal to 9.\n\n## Classification of 1076153\n\n• Arithmetic\n• Semiprime\n• Deficient\n\n• Polite\n\n• Square Free\n\n### Other numbers\n\n• LucasCarmichael\n\n## Base conversion (1076153)\n\nBase System Value\n2 Binary 100000110101110111001\n3 Ternary 2000200012112\n4 Quaternary 10012232321\n5 Quinary 233414103\n6 Senary 35022105\n8 Octal 4065671\n10 Decimal 1076153\n12 Duodecimal 43a935\n20 Vigesimal 6ea7d\n36 Base36 n2d5\n\n## Basic calculations (n = 1076153)\n\n### Multiplication\n\nn×y\n n×2 2152306 3228459 4304612 5380765\n\n### Division\n\nn÷y\n n÷2 538076 358718 269038 215231\n\n### Exponentiation\n\nny\n n2 1158105279409 1246298470751833577 1341207838194997959389281 1443344838697061638990652915993\n\n### Nth Root\n\ny√n\n 2√n 1037.38 102.477 32.2084 16.0833\n\n## 1076153 as geometric shapes\n\n### Circle\n\n Diameter 2.15231e+06 6.76167e+06 3.6383e+12\n\n### Sphere\n\n Volume 5.22048e+18 1.45532e+13 6.76167e+06\n\n### Square\n\nLength = n\n Perimeter 4.30461e+06 1.15811e+12 1.52191e+06\n\n### Cube\n\nLength = n\n Surface area 6.94863e+12 1.2463e+18 1.86395e+06\n\n### Equilateral Triangle\n\nLength = n\n Perimeter 3.22846e+06 5.01474e+11 931976\n\n### Triangular Pyramid\n\nLength = n\n Surface area 2.0059e+12 1.46878e+17 878675\n\n## Cryptographic Hash Functions\n\nmd5 f6c66c397bb75510a95a78378d97039c bde2601d043239c32023e931fdcbfba9faf017bf 776516786a70c3c1fd09bbbd36fbf70ab789fd96546e67351067172c2b6270f9 6179aaa6ce483b4c56e239eba58913ff3ebd795e713a4913dd460c365582c84b850c2e0dfd921be1d59efab721485df03d695c04da95071089abb0dcee425a61 bc1585eb4d0af71ada35e4dd843e35482000df00" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.62649095,"math_prob":0.9814391,"size":4768,"snap":"2021-43-2021-49","text_gpt3_token_len":1681,"char_repetition_ratio":0.12153652,"word_repetition_ratio":0.03211679,"special_character_ratio":0.46371645,"punctuation_ratio":0.08405439,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99567026,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-07T06:41:59Z\",\"WARC-Record-ID\":\"<urn:uuid:1a8d46d6-8504-4e94-99f2-876a84ca8575>\",\"Content-Length\":\"40121\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:51521b20-c73a-451d-88dd-eb5b74cef83a>\",\"WARC-Concurrent-To\":\"<urn:uuid:a12242f4-fcbe-4b17-8b6b-e63a8164e3d6>\",\"WARC-IP-Address\":\"46.105.53.190\",\"WARC-Target-URI\":\"https://metanumbers.com/1076153\",\"WARC-Payload-Digest\":\"sha1:3TT6MJA3IZMJZKE37YIA447TG3577MU5\",\"WARC-Block-Digest\":\"sha1:LFNB5LUUTSATILX6JJRH2STWY6S2XBCO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363336.93_warc_CC-MAIN-20211207045002-20211207075002-00106.warc.gz\"}"}
https://www.softmath.com/math-book-answers/sum-of-cubes/how-to-use-algebrator.html	[ "", null, "## What our customers say...\n\nThousands of users are using our software to conquer their algebra homework. Here are some of their experiences:\n\nAll in all, this is a very useful, well-designed algebra help tool for school classes and homework.\nReese Pontoon, MO\n\nOK here is what I like: much friendlier interface, coverage of functions, trig. better graphing, wizards. However, still no word problems, pre-calc, calc. (Please tell me that you are working on it - who is going to do my homework when I am past College Algebra?!?\nNathan Lane, AZ\n\nI have tried many other programs that did not deliver on what they promised. I decided to take a chance with the Algebrator. All I can say is WOW! Thank you!\nAlan Cox, TX\n\n## Search phrases used on 2013-07-10:\n\nStudents struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. Can you find yours among them?\n\n• solving for 2nd order equation\n• second order linear nonhomogeneous differential equations\n• abstract algebra tests quizzes\n• simplify square roots calculator\n• solve and graph\n• mixed numbers to decimals\n• algebra 1 homework\n• what is the method to solve mixed numbers\n• online ratio simplifier\n• standard form to vertex form\n• chemistry points all free for teachers\n• a two digit number that is a perfect square\n• free example beginner algebra\n• free samples of iq questions for written bank exam\n• how do i write a standard form equation into vertex form\n• accounting notes calc ti programs\n• introduction to slope graphing calculators\n• algebra for beginners PDF\n• Who is the person who discovered greatest common factors\n• free college math tutorials - video\n• answer key to glencoe algebra 2\n• finding the roots of an expression calculator\n• least common denominator worksheets\n• Abstract algebra help\n• free worksheet for application of proportion - 6 grade\n• free algebra expressions and equations worksheets\n• applications of common logarithms worksheet\n• asset examination maths practice papers for class V\n• online t83 calculator\n• worksheets on adding and subtracting integers\n• middle school math with pizzazz order of operations\n• 5th grade algebraic expressions worksheets\n• aptitude questions pdf\n• 8th grade math solved papers\n• least common denominator calculator online\n• factor cubed polynomials\n• continuous discrete data activity hands-on 5th grade\n• worksheet algebra\n• merrill chemistry worksheet answer keys\n• solve my fraction\n• intermediate algebra proofs\n• java aptitude test\n• exponential simplify square root\n• free online college algebra solver\n• exponential equation excel\n• multiplying and dividing in scientific notation\n• finding the vertex of a absolute value\n• identify skew segments, pre algebra\n• www.algebrahelp.net\n• saxon math algebra 1 answers\n• McDougal Littell online Answer Key\n• scientific notation ( radical expressions)\n• what is a non factorable polynomial called?\n• algebra 2 statistic worksheets\n• free printable pictograph worksheets\n• factoring program for ti 84\n• factorize complex number\n• how to convert repeated decimals into fraction\n• subtract worksheets\n• logarithms ti-83 plus\n• lineal calculator\n• Order Operations Math Worksheet\n• multiplying binomials and monomials calculator online" ]	[ null, "https://www.softmath.com/r-solver/images/tutor.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8442152,"math_prob":0.8939636,"size":3985,"snap":"2021-43-2021-49","text_gpt3_token_len":890,"char_repetition_ratio":0.11931676,"word_repetition_ratio":0.0,"special_character_ratio":0.20828106,"punctuation_ratio":0.0576,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99787337,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-11-27T02:59:39Z\",\"WARC-Record-ID\":\"<urn:uuid:5c016bed-13c1-4ca8-b7dd-20b06f23c3d7>\",\"Content-Length\":\"35078\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ae71a6bc-240c-4be3-bfb7-3eaf853b1893>\",\"WARC-Concurrent-To\":\"<urn:uuid:27f891cd-0741-4866-9789-4cf9124ce1e2>\",\"WARC-IP-Address\":\"52.43.142.96\",\"WARC-Target-URI\":\"https://www.softmath.com/math-book-answers/sum-of-cubes/how-to-use-algebrator.html\",\"WARC-Payload-Digest\":\"sha1:62VWQY4PHWTMJ6E5J6BZBQY42D6SW2KQ\",\"WARC-Block-Digest\":\"sha1:DDMFOSEPRIQPQ22LYGMZB4TQ2XRB47B2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964358078.2_warc_CC-MAIN-20211127013935-20211127043935-00504.warc.gz\"}"}
https://wiki.bash-hackers.org/syntax/arith_expr?rev=1351878670&do=diff	[ "# Differences\n\nThis shows you the differences between two versions of the page.\n\n syntax:arith_expr [2012/11/02 17:51]techlivezheng [Arithmetic expressions] syntax:arith_expr [2017/02/11 14:22] (current)fgrose [Table] meaning of ternary operator 2017/02/11 14:22 fgrose [Table] meaning of ternary operator2013/04/19 18:49 thebonsai [Arithmetic expressions] change link to Greg's wiki, thanks Joan2012/11/02 18:00 techlivezheng [Arithmetic expressions and return codes] Fix a mistake2012/11/02 17:51 techlivezheng [Arithmetic expressions] 2011/11/17 23:13 ormaaj declare -i foo=[arith]2011/03/22 06:12 fgrose [Arithmetic expressions and return codes] 2011/03/21 05:29 fgrose [Arithmetic expressions and return codes] 2011/03/21 05:27 fgrose [Arithmetic expressions and return codes] 2011/03/21 05:16 fgrose [Arithmetic expressions and return codes] 2010/11/23 21:40 external edit 2017/02/11 14:22 fgrose [Table] meaning of ternary operator2013/04/19 18:49 thebonsai [Arithmetic expressions] change link to Greg's wiki, thanks Joan2012/11/02 18:00 techlivezheng [Arithmetic expressions and return codes] Fix a mistake2012/11/02 17:51 techlivezheng [Arithmetic expressions] 2011/11/17 23:13 ormaaj declare -i foo=[arith]2011/03/22 06:12 fgrose [Arithmetic expressions and return codes] 2011/03/21 05:29 fgrose [Arithmetic expressions and return codes] 2011/03/21 05:27 fgrose [Arithmetic expressions and return codes] 2011/03/21 05:16 fgrose [Arithmetic expressions and return codes] 2010/11/23 21:40 external edit Line 15: Line 15: These expressions are evaluated following some rules described below. The operators and rules of arithmetic expressions are mainly derived from the C programming language. These expressions are evaluated following some rules described below. The operators and rules of arithmetic expressions are mainly derived from the C programming language. - This article describes the theory of the used syntax and the behaviour. To get practical examples without big explanations, see [[http://wooledge.org/mywiki/ArithmeticExpression \| the article on Greg's wiki]]. + This article describes the theory of the used syntax and the behaviour. To get practical examples without big explanations, see [[http://mywiki.wooledge.org/BashGuide/CompoundCommands#Arithmetic_Evaluation \| this page on Greg's wiki]]. ===== Constants ===== ===== Constants ===== Line 164: Line 164: ==== Misc ==== ==== Misc ==== - ^Operator^Description^ + ^ Operator ^ Description ^ - \|''id++''\|post-increment of the variable ''id'' (not required by POSIX(r))\| + \| ''id++'' \| post-increment of the variable ''id'' (not required by POSIX(r)) \| - \|''id<nowiki>--''\|post-decrement of the variable ''id'' (not required by POSIX(r))\| + \| ''id%%--%%'' \| post-decrement of the variable ''id'' (not required by POSIX(r)) \| - \|''++id''\|pre-increment of the variable ''id'' (not required by POSIX(r))\| + \| ''++id'' \| pre-increment of the variable ''id'' (not required by POSIX(r)) \| - \|''<nowiki>--id''\|pre-decrement of the variable ''id'' (not required by POSIX(r))\| + \| ''%%--%%id'' \| pre-decrement of the variable ''id'' (not required by POSIX(r)) \| - \|''+''\|unary plus\| + \| ''+'' \| unary plus \| - \|''-''\|unary minus\| + \| ''-'' \| unary minus \| - \|''<EXPR> ? <EXPR> : <EXPR>''\|conditional (ternary) operator\| + \| ''<EXPR> ? <EXPR> : <EXPR>'' \| conditional (ternary) operator \\\\ <condition> ? <result-if-true> : <result-if-false> \| - \|''<EXPR> , <EXPR>''\|expression list\| + \| ''<EXPR> , <EXPR>'' \| expression list \| - \|''( <EXPR> )''\|subexpression (to force precedence)\| + \| ''( <EXPR> )'' \| subexpression (to force precedence) \| Line 205: Line 205: Bash's overall language construct is based on exit codes or return codes of commands or functions to be executed. ''if'' statements, ''while'' loops, etc., they all take the return codes of commands as conditions. Bash's overall language construct is based on exit codes or return codes of commands or functions to be executed. ''if'' statements, ''while'' loops, etc., they all take the return codes of commands as conditions. - Now the problem is: The return codes (0 means \"TRUE\" or \"SUCCESS\", not 0 means \"FALSE\" or \"FAILURE\") don't correspond to the meaning of the result of an arithmetic expression (0 means \"TRUE\", not 0 means \"FALSE\"). + Now the problem is: The return codes (0 means \"TRUE\" or \"SUCCESS\", not 0 means \"FALSE\" or \"FAILURE\") don't correspond to the meaning of the result of an arithmetic expression (0 means \"FALSE\", not 0 means \"TRUE\"). That's why all commands and keywords that do arithmetic operations attempt to translate the arithmetical meaning into an equivalent return code. This simply means: That's why all commands and keywords that do arithmetic operations attempt to translate the arithmetical meaning into an equivalent return code. This simply means:\n• syntax/arith_expr.1351878670.txt" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.755238,"math_prob":0.82278764,"size":3753,"snap":"2019-51-2020-05","text_gpt3_token_len":984,"char_repetition_ratio":0.1157642,"word_repetition_ratio":0.51492536,"special_character_ratio":0.31841195,"punctuation_ratio":0.09715243,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98035944,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-06T01:42:34Z\",\"WARC-Record-ID\":\"<urn:uuid:5448fdf0-cf47-4bfe-be68-9c8f82e72537>\",\"Content-Length\":\"33534\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9d74d6a9-5d3b-4ab7-bd97-af0fa79704c5>\",\"WARC-Concurrent-To\":\"<urn:uuid:dde135e2-1935-4158-a993-f5cb902849ae>\",\"WARC-IP-Address\":\"83.243.40.67\",\"WARC-Target-URI\":\"https://wiki.bash-hackers.org/syntax/arith_expr?rev=1351878670&do=diff\",\"WARC-Payload-Digest\":\"sha1:AQCOT7B4TGHVG3CW6YMIUVRGCLOSHTQ7\",\"WARC-Block-Digest\":\"sha1:4OUCHVBNB5PECE4YBOWY3VHH4K2EJGGS\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540482954.0_warc_CC-MAIN-20191206000309-20191206024309-00073.warc.gz\"}"}
https://www.calendarbede.com/book/calculation-orthodox-easter-sunday	[ "# Calculation of Orthodox Easter Sunday in the Julian calendar\n\nNot only does the Orthodox Church use the old Julian calendar, but other chronological cycles of the year to calculate the calendar date for Easter Sunday (and Passover) as well. Of course, the final date of Easter Sunday coincides with the western calculation of Easter Sunday in the Julian calendar (see the page Calculating Easter Sunday in the Julian calendar). The principle of the calculation is the same: first, we calculate the date of the first spring ecclesiastical full moon, and then the following Sunday is Easter Sunday. In the following calculations, only integers (whole numbers) are counted, with the character '%' (modulo) indicating that only the remainder after division is sought (e.g. 23/5 = 4 and 23 % 5 = 3).\n\nTo start with, we need to convert the given year to the year in the Byzantine calendar (in English, this means “from the creation of the world”, while in Russian it’s лето от сотворения мира nebo i лето от Адама). According to the Orthodox Church, the world was created on 1 September, year 1 of the Byzantine era. Sometimes, two values for the Orthodox calendar year’s foundations are given, with some calendars listing home many days until 1 September (Julian calendar) and how many days since 1 September (in brackets). In the case of Easter, we’re only interested in the first value. The leap year system is the same as it is for the Julian calendar. The conversion to the Byzantine date is then made by simply adding the constant 5508 to our year:\n\ncreation_of_world = year + 5508\n\nIn year 1 of the Byzantine era, three Orthodox calendar cycles (Russian: круг Луны, круг Солнцу, индикт) had a value of 1. By simply dividing by the length of the selected cycle and finding the remainder after this division, we obtain the value of the cycle. If the resulting value is 0, then this 0 is adjusted to a value equal to the cycle length. The first cycle is the lunar cycle (Russian: Kруг Луны), which is the equivalent of our Golden Number with the same 19-year long duration. It only one needs to be shifted by three years, which makes for an easy calculation:\n\nlunar_cycle = (creation_of_world - 1) % 19 + 1\n\nFor the sake of comparison, below is a table with the values of the Golden Number and the Kруг Луны. Unlike the Golden Number cycle, where the lunar leap (Latin: saltus lunae) is performed at the end of the cycle, the same leap (Russian: скачок Луны) is performed at the end of the 16th year of their lunar cycle. The table shows that although the monthly leap is performed at different values in both cycles, they are in fact, timed out the same.\n\n Golden Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Lunar cycle 17 18 19 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16\n\nFrom the value of the lunar cycle, we then calculate the foundation (Russian: основание), which indicates the age of the moon on 1 March. If the lunar cycle is larger than 16, a minor correction is required due to the monthly leap:\n\nfoundation = (11 × lunar_cycle + 3) % 30\nif lunar_cycle is greater than 16, foundation = foundation + 1\n\nIf we already know the lunar phase on 1 March, by subtracting 30 from it, we get the March date of the New Moon (Russian: мартовское новолуние), and by adding 14 to that we’ll get the date of the half moon (Russian: мартовское полнолуние). From here, it’s necessary to apply a special rule: in order to compare the calculated phases of the Moon with the actual phases of the moon during the time of the Council of Nicaea, three more days need to be added. If what we get for the final date of the full moon (Russian: пасхальное полнолуние) is before 21 March (i.e. before the date declared by the church as the beginning of spring) we choose the next full moon, which can be obtained by simply adding the entire length of a lunation (30 days). Everything can be summarized in a fairly simple calculation:\n\necclesiastical_full_moon = 47 - foundation\nif the ecclesiastical_full_moon is less than 21, then ecclesiastical_full_moon = ecclesiastical_full_moon + 30\n\nThe result is the March date of the first spring ecclesiastical full moon. If the number exceeds 31, for example the number 33, then subtract 31 and set it as an April date (i.e. 2 April in this example). Another Orthodox foundation of the year is the epact (Russian: епакта), the value of which shows the March date that falls on the twentieth day of the lunar month (this coincides with the end of the Passover celebration). For example, if the lunar cycle (Orthodox counterpart to the Golden Number) is equal to 1, then the half moon is equal to 14, which will put the ecclesiastical new moon on 1 March, thus making the lunar cycle 14 days old. In 6 more days (7 March) the Moon will be 20 days old, which means that the epact has a value of 7. This Orthodox epact (do not confuse it with the Gregorian epact) is obtained from the half moon by simply taking 21 and subtracting the half moon date. If we get a number less than 1, we add 30. However, this epact is not needed for this calculation of Easter Sunday. The term “correct date” (Russian: исправная дата) is defined as the date in a given year before which Passover is not possible, can be found throughout the sources. In essence, it is the date of the ecclesiastical full moon plus one day.\n\nNow we can make a simple table, for each value of the lunar cycle foundational elements in the Orthodox calculation. The half moon increases regularly by 11, and if the value is greater than 30 we subtract 30. After 16 years of the lunar cycle, a monthly leap is made (this is highlighted in the table) and the value of the half moon increases by 12. Similarly, the epact decreases again by 11 (for the monthly leap of 12), and if the result is less than 1, add 30.\n\nLunar cycle\n(круг Луны)\nfoundation\n(основание)\nepact\n(епакта)\necclesiastical new moon\n(пасхальное\nноволуние)\n114733\n2252622\n361541\n417430\n5282349\n691238\n720127\n812046\n912935\n10232824\n1141743\n1215632\n13262521\n1471440\n1518329\n16292248\n17111036\n18222925\n1931844\n\nWe already know the date for the first full moon of spring, but now it is necessary to find out when the next Sunday occurs. The equivalent of our solar cycle in the Orthodox calendar is the круг Солнцу, which also takes values from 1 to 28. Its calculation is rather simple:\n\nsolar_cycle = (creation_of_world - 1) % 28 + 1\n\nAnother Orthodox foundation for calculating its year is its 'vruceleto' (Russian: вруцелето), which determines which day of the week (день недели) falls on 1 September (the beginning of the Byzantine year). Its value is easily obtained using the following table:\n\nvruceleto\n(вруцелето)\nSolar cycle\n(круг Солнцу)\n1171218\n22131924\n3381425\n49152026\n54102127\n65111622\n76172328\n\nThe value for vruceleto can also be obtained by a small calculation:\n\nvruceleto = (solar_cycle + solar_cycle / 4 - 1) % 7 + 1\n\nInstead of a number, the first letters (Russian: буква) of the Cyrillic alphabet (азбука) were most frequently used. Often a word was added to the letter, all in accordance with the following table:\n\nvruceleto\n(вруцелето)\nletter\n(буква)\n1st of September\n1А (аз)Sunday\n2В (веди)Monday\n3Г (глаголь)Tuesday\n4Д (добро)Wednesday\n5Е (есть)Thursday\n6S (зело)Friday\n7З (земля)Saturday\n\nBy following along with this calculation, we’ll find the day in March, when the first Sunday of the month, or the 'first resurrection' (Russian: первое воскресение), occurs:\n\nif vruceleto is less than 4 first_resurrection = 4 - vruceleto\notherwise, first_resurrection = 11 - vruceleto\n\nThe result is shown in this small table:\n\nvruceleto\n(вруцелето)\nfirst resurrection\n(первое воскресенье)\n13rd of March\n22nd of March\n31st of March\n47th of March\n56th of March\n65th of March\n74th of March\n\nFinally, the last calculation finds the next Sunday after the first spring ecclesiastical full moon. This is a March date, so if the result is greater than 31, it is April, and in order to obtain the April date, it’s necessary to subtract from the result 31. This Sunday is what we’ve been looking for all along, and gives us Orthodox Easter Sunday (Russian: Христианская Пасха):\n\npassover = ecclesiastical_full_moon + 7 - (ecclesiastical_full_moon - first_resurrection) % 7\nif we get a number greater than 31, it’s an April date and it’is necessary to subtract 31; otherwise it’s a March date\n\nIn printed calendars, instead of the calendar date for Orthodox Easter Sunday, the 'boundary key' (Russian: ключ границ) was mentioned. It shows how many days from 21 March that Easter Sunday is. This number can take values from 1 to 35. The earliest date is Easter Sunday on 22 March, while at the value of 35, we have the latest possible date for Easter as 25 April. Each number is assigned a letter according to the table:\n\n 1 (А) March 22 2 (Б) March 23 3 (В) March 24 4 (Г) March 25 5 (Д) March 26 6 (Е) March 27 7 (Ж) March 28 8 (Ѕ) March 29 9 (З) March 30 10 (И) March 31 11 (І) Apr 1 12 (К) Apr 2 13 (Л) Apr 3 14 (М) Apr 4 15 (Н) Apr 5 16 (О) Apr 6 17 (П) Apr 7 18 (Р) Apr 8 19 (С) Apr 9 20 (Т) Apr 10 21 (У) Apr 11 22 (Ф) Apr 12 23 (Х) Apr 13 24 (Ѿ) Apr 14 25 (Ц) Apr 15 26 (Ч) Apr 16 27 (Ш) Apr 17 28 (Щ) Apr 18 22 (Ф) Apr 12 29 (Ъ) Apr 19 30 (Ы) Apr 20 31 (Ь) Apr 21 32 (Ѣ) Apr 22 33 (Ю) Apr 23 34 (Ѫ) Apr 24 35 (Ѧ) Apr 25\n\nNote: in some calendars, the “indiction” (Russian: индикт) was sometimes displayed, the value of which is always the same as the Western indiction. However, Easter Sunday has nothing to do with the calculation. Do not confuse this Western indiction with the term “large indiction” (Russian: великий индиктион or миротворный круг), or even more rarely as круг великой альфы), which is a cycle of 532 years. After this length of time, the calendar dates for Passover are repeated in the Julian calendar, and in the same order. In the Julian calendar, the calendar dates are repeated in the same order after 28 years (7 × 4, 7 days a week, and on every fourth year leap), which is a non-consecutive number with 19 values of the lunar cycle. It is from this that we calculate the large indiction as 28 × 19 = 532. We are currently in the fifteenth cycle since year 1 of the Byzantine era. This cycle began in 1941 AD and ends in 2472 AD.\n\nThe above calculations and tables apply only to the old Julian calendar!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.86355066,"math_prob":0.9752707,"size":10209,"snap":"2022-40-2023-06","text_gpt3_token_len":3192,"char_repetition_ratio":0.14483097,"word_repetition_ratio":0.027983105,"special_character_ratio":0.30424136,"punctuation_ratio":0.08593375,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9518115,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-05T10:38:56Z\",\"WARC-Record-ID\":\"<urn:uuid:3f6e2176-fb02-47aa-9ee1-9abc02e9c298>\",\"Content-Length\":\"41845\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:53ca39f2-2f3b-45fd-8aeb-964ce53c24c7>\",\"WARC-Concurrent-To\":\"<urn:uuid:07f9df98-9293-4b18-a62f-6c170bac472a>\",\"WARC-IP-Address\":\"80.211.207.141\",\"WARC-Target-URI\":\"https://www.calendarbede.com/book/calculation-orthodox-easter-sunday\",\"WARC-Payload-Digest\":\"sha1:5UWBSYXCVZ5PWYFJER37L3LBOYGEXQ6Z\",\"WARC-Block-Digest\":\"sha1:CVBDOVE2YDFTFA4KBCLE6HP4RNCAQR5R\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500251.38_warc_CC-MAIN-20230205094841-20230205124841-00226.warc.gz\"}"}
https://stackoverflow.com/questions/26355942/why-is-the-f-measure-a-harmonic-mean-and-not-an-arithmetic-mean-of-the-precision/26360501	[ "# Why is the F-Measure a harmonic mean and not an arithmetic mean of the Precision and Recall measures?\n\nWhen we calculate the F-Measure considering both Precision and Recall, we take the harmonic mean of the two measures instead of a simple arithmetic mean.\n\nWhat is the intuitive reason behind taking the harmonic mean and not a simple average?\n\nAccording to the theory of measurement the composite measure should satisfy the following 6 definitions:\n\n1. Connectedness(two pairs can be ordered) and transitivity(if e1 >= e2 and e2 >= e3 then e1 >= e3)\n2. Independence: two components contribute their effects independently to the effectiveness.\n3. Thomsen condition: Given that at a constant recall (precision) we find a difference in effectiveness for two values of precision (recall) then this difference cannot be removed or reversed by changing the constant value.\n4. Restricted solvability.\n5. Each component is essential: Variation in one while leaving the other constant gives a variation in effectiveness.\n6. Archimedean property for each component. It merely ensures that the intervals on a component are comparable.\n\nWe can then derive and get the function of the effectiveness:", null, "And normally we don't use the effectiveness but the much simper F score because:", null, "Now that we have the general formula of F measure:", null, "where we can place more emphesis on recall or precision by setting beta, because beta is defined as follows:", null, "If we weight recall more important than precision(all relevant are selected) we can set beta as 2 and we get the F2 measure. And if we do the reverse and weight precision higher than recall(as much selected elements are relevant as possible, for instance in some grammar error correction scenarios like CoNLL) we just set beta as 0.5 and get the F0.5 measure. And obviously we can set beta as 1 to get the mostly used F1 measure(harmonic mean of precision and recall).\n\nI think to some extent I have already answered why we do not use the arithmetic mean.\n\nTo explain, consider for example, what the average of 30mph and 40mph is? if you drive for 1 hour at each speed, the average speed over the 2 hours is indeed the arithmetic average, 35mph.\n\nHowever if you drive for the same distance at each speed -- say 10 miles -- then the average speed over 20 miles is the harmonic mean of 30 and 40, about 34.3mph.\n\nThe reason is that for the average to be valid, you really need the values to be in the same scaled units. Miles per hour need to be compared over the same number of hours; to compare over the same number of miles you need to average hours per mile instead, which is exactly what the harmonic mean does.\n\nPrecision and recall both have true positives in the numerator, and different denominators. To average them it really only makes sense to average their reciprocals, thus the harmonic mean.\n\n• Thanks, that is a good argument on why this is supported from theory; my answer was more on the pragmatic side. – Anony-Mousse Oct 14 '14 at 20:51\n\nBecause it punishes extreme values more.\n\nConsider a trivial method (e.g. always returning class A). There are infinite data elements of class B, and a single element of class A:\n\n``````Precision: 0.0\nRecall: 1.0\n``````\n\nWhen taking the arithmetic mean, it would have 50% correct. Despite being the worst possible outcome! With the harmonic mean, the F1-measure is 0.\n\n``````Arithmetic mean: 0.5\nHarmonic mean: 0.0\n``````\n\nIn other words, to have a high F1, you need to both have a high precision and recall.\n\n• When the recall is 0.0 the precision has to be greater than 0.0 right? But I get the point in your example. Nicely explained - Thanks. – London guy Oct 14 '14 at 12:29\n• In your example, precision for class A is 0.5 instead of 0 and recall of class A is 1; precision for class B is 0 and recall of class B is 0 as we'll. I assume your balanced class means the true labels are A and B; each applies to 50% of data. – greeness Oct 15 '14 at 8:52\n• Let's make infinite elements of class B, and a single element of class A. It doesn't change the math behind F1. – Anony-Mousse Oct 15 '14 at 13:20\n• It is not just a heuristic to select more balance. Harmonic mean is there only way that makes sense given the units of these ratios. Mean wouldn't have a meaning in comparison – Sean Owen Mar 2 '16 at 9:02\n• Where does it say \"heuristic\", and where does your comment differ from my answer? But: F-measure is a heuristic in that it assumes precision and recall are equally important. That is why the beta term needs to be chosen - heuristically, one usually uses beta=1. – Anony-Mousse Mar 2 '16 at 9:06\n\nThe harmonic mean is the equivalent of the arithmetic mean for reciprocals of quantities that should be averaged by the arithmetic mean. More precisely, with the harmonic mean, you transform all your numbers to the \"averageable\" form (by taking the reciprocal), you take their arithmetic mean and then transform the result back to the original representation (by taking the reciprocal again).\n\nPrecision and the recall are \"naturally\" reciprocals because their numerator is the same and their denominators are different. Fractions are more sensible to average by arithmetic mean when they have the same denominator.\n\nFor more intuition, suppose that we keep the number of true positive items constant. Then by taking the harmonic mean of the precision and the recall, you implicitly take the arithmetic mean of the false positives and the false negatives. It basically means that false positives and false negatives are equally important to you when the true positives stay the same. If an algorithm has N more false positive items but N less false negatives (while having the same true positives), the F-measure stays the same.\n\nIn other words, the F-measure is suitable when:\n\n1. mistakes are equally bad, whether they are false positives or false negatives\n2. the number of mistakes is measured relative to the number of true positives\n3. true negatives are uninteresting\n\nPoint 1 may or may not be true, there are weighted variants of the F-measure that can be used if this assumption isn't true. Point 2 is quite natural since we can expect the results to scale if we just classify more and more points. The relative numbers should stay the same.\n\nPoint 3 is quite interesting. In many applications negatives are the natural default and it may even be hard or arbitrary to specify what really counts as a true negative. For example a fire alarm is having a true negative event every second, every nanosecond, every time a Planck time has passed etc. Even a piece of rock has these true negative fire-detection events all the time.\n\nOr in a face detection case, most of the time you \"correctly don't return\" billions of possible areas in the image but this is not interesting. The interesting cases are when you do return a proposed detection or when you should return it.\n\nBy contrast the classification accuracy cares equally about true positives and true negatives and is more suitable if the total number of samples (classification events) is well-defined and rather small.\n\n• Very well explained! – Oren Yosifon Jul 30 '16 at 9:30\n\nThe above answers are well explained. This is just for a quick reference to understand the nature of the arithmetic mean and the harmonic mean with plots. As you can see from the plot, consider the X axis and Y axis as precision and recall, and the Z axis as the F1 Score. So, from the plot of the harmonic mean, both the precision and recall should contribute evenly for the F1 score to rise up unlike the Arithmetic mean.\n\nThis is for the arithmetic mean.", null, "This is for the Harmonic mean.", null, "• Please use formatting tools to properly edit and format your answer. Image should be displayed here , its not a hyperlink. – Prateek Mar 28 '18 at 13:30" ]	[ null, "https://i.stack.imgur.com/VX8co.png", null, "https://i.stack.imgur.com/ACesD.png", null, "https://i.stack.imgur.com/VGDMI.png", null, "https://i.stack.imgur.com/2hCM1.png", null, "https://i.stack.imgur.com/SYA7h.jpg", null, "https://i.stack.imgur.com/slxFc.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.92572194,"math_prob":0.9776598,"size":5813,"snap":"2019-43-2019-47","text_gpt3_token_len":1229,"char_repetition_ratio":0.15028404,"word_repetition_ratio":0.07931727,"special_character_ratio":0.20677792,"punctuation_ratio":0.09460654,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9949066,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,4,null,2,null,3,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-19T17:36:18Z\",\"WARC-Record-ID\":\"<urn:uuid:aae38ff8-3fe6-4da0-8dc2-877ec4eed303>\",\"Content-Length\":\"182841\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c73d5659-dfbf-419b-b424-13278985fb86>\",\"WARC-Concurrent-To\":\"<urn:uuid:f76b21da-1b6f-4464-89cf-ac79884c8c37>\",\"WARC-IP-Address\":\"151.101.193.69\",\"WARC-Target-URI\":\"https://stackoverflow.com/questions/26355942/why-is-the-f-measure-a-harmonic-mean-and-not-an-arithmetic-mean-of-the-precision/26360501\",\"WARC-Payload-Digest\":\"sha1:5L4JWZIWUEHMX22GJT7VPMYGNDBUV762\",\"WARC-Block-Digest\":\"sha1:W7QECH3JATMKT2GGHX4PIB6OL4W5VB2Q\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670162.76_warc_CC-MAIN-20191119172137-20191119200137-00492.warc.gz\"}"}
https://socratic.org/questions/how-do-you-express-x-2-x-3-in-partial-fractions	[ "# How do you express (x²+2) / (x+3) in partial fractions?\n\nFeb 22, 2016\n\n$\\frac{x}{1} + \\frac{- 3 x + 2}{x + 3}$\n\n#### Explanation:\n\nbecause the top quadratic and the bottom is linear you're looking for something or the form\n\n$\\frac{A}{1} + \\frac{B}{x + 3}$, were $A$ and $B$ will both be linear functions of $x$ (like 2x+4 or similar).\n\nWe know one bottom must be one because x+3 is linear.\n\nWe're starting with\n$\\frac{A}{1} + \\frac{B}{x + 3}$.\nWe then apply standard fraction addition rules. We need to get then to a common base.\n\nThis is just like numerical fractions $\\frac{1}{3} + \\frac{1}{4} = \\frac{3}{12} + \\frac{4}{12} = \\frac{7}{12.}$\n\n$\\frac{A}{1} + \\frac{B}{x + 3} \\implies \\frac{A \\cdot \\left(x + 3\\right)}{1 \\cdot \\left(x + 3\\right)} + \\frac{B}{x + 3} = \\frac{A \\cdot \\left(x + 3\\right) + B}{x + 3}$.\nSo we get the bottom automatically.\n\nNow we set $A \\cdot \\left(x + 3\\right) + B = {x}^{2} + 2$\n$A x + 3 A + B = {x}^{2} + 2$\n$A$ and $B$ are linear terms so the ${x}^{2}$ must come from $A x$.\nlet $A x = {x}^{2}$ $\\implies$ $A = x$\nThen\n$3 A + B = 2$\nsubstituting $A = x$, gives\n$3 x + B = 2$\nor\n$B = 2 - 3 x$\nin standard from this is $B = - 3 x + 2$.\nPutting it all together we have\n$\\frac{x}{1} + \\frac{- 3 x + 2}{x + 3}$" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.76485986,"math_prob":1.0000099,"size":547,"snap":"2020-34-2020-40","text_gpt3_token_len":129,"char_repetition_ratio":0.09944751,"word_repetition_ratio":0.0,"special_character_ratio":0.21572211,"punctuation_ratio":0.057692308,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.000009,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-10-01T13:09:00Z\",\"WARC-Record-ID\":\"<urn:uuid:2816bdfc-5ca5-4723-aff4-daa9a640071d>\",\"Content-Length\":\"35375\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:508b2d5b-dbf1-4f48-b998-8311c16b58d9>\",\"WARC-Concurrent-To\":\"<urn:uuid:ec6bd552-b4d3-4fe6-ae7d-ef6138f974aa>\",\"WARC-IP-Address\":\"216.239.34.21\",\"WARC-Target-URI\":\"https://socratic.org/questions/how-do-you-express-x-2-x-3-in-partial-fractions\",\"WARC-Payload-Digest\":\"sha1:7S3FLCYTOOHKVCYX7BGBHTUJ2QKC5CU3\",\"WARC-Block-Digest\":\"sha1:MYIFAGY4PRIDDQPFJZNQ3WEXWUY6P735\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-40/CC-MAIN-2020-40_segments_1600402131412.93_warc_CC-MAIN-20201001112433-20201001142433-00345.warc.gz\"}"}
https://onlineforexcharts.org/general/all-about-fibonacci-retracements-and-fibonacci-ratios/	[ "# All about Fibonacci Retracements and Fibonacci Ratios\n\nFibonacci retracements are a common tool among traders and are based on the Fibonacci series identified in the 13th century by the famous Leonardo Fibonacci. The Fibonacci sequence is a set of numbers that is also related to nature, and also called the golden ratio.\n\nA Fibonacci retracement tool is used by technical analysts as a guide to the behavior of the market. Traders use the tool for identification of resistance levels, drawing support lines setting target prices, and placing stop-loss orders.\n\nDuring a technical analysis, Fibonacci retracement is done by picking a different extreme point on the stock chart, mostly peak and trough. The vertical distance is then divided by the Fibonacci ratios, i.e., 23.6%, 38.2%, 50%, 61,8% and 100%. After identifying the levels, a horizontal line is drawn and identifies the hypothetical resistance and support levels.\n\n## How the Fibonacci Ratio Works\n\nThe Fibonacci series is: 0,1,1,2,3,5,8,13,21,34,55, etc. Each term in the series is a sum of the preceding terms, and the sequence goes on to infinity. One characteristic of the Fibonacci sequence is that every number in the series is approximately 1.618 times greater than the previous number. This is the very foundation of the ratio’s traders use to work out the retracement level.\n\nThe key ratio of 61.8% is derived by the division of one number in the sequence, by the number following it. For instance, 21 divided by 34= 0.6176, 55 divided by 89= 0.61798. The 38.2% is arrived at by dividing one number in the sequence by the number falling two places to its right.\n\nFor example, 55 divided by 144= around 0.38194. The 23,6% ratio is arrived at by dividing one number in the sequence, by the number three spots to its right. The 23.6% ratio is arrived at by dividing one number in the sequence by a number three spots to its right. For instance, 8/32= approximately 0.23529.\n\n## Fibonacci Retracing and How it Predicts Stock Prices\n\nFor some reason, the Fibonacci ratios work in the stock market as they work in nature. Traders try to use them for the determination of crucial points where the price momentum of an asset is likely to go backward.\n\nFibonacci retracements are the most common of the Fibonacci trading tools. This is because of their simplicity in part, and because they apply to almost all trading tools. They are used as primary mechanisms in countertrend trading strategies.\n\nFibonacci retracement levels are horizontal lines, which show the hypothetical support and resistance levels. Every level is related to one of the percentages or ratios. It indicates how much or the previous move the price retraces, with the previous trend’s direction likely to go on. The asset price normally retraces to one of the ratios before this happens.\n\n## Pros and Cons of the Fibonacci Retracement\n\nAs popular as the Fibonacci retracements are, these tools have some disadvantages.\n\n• Using the Fibonacci retracement tool is subjective, and can be used in different ways. Some people who make money using it swear on its effectiveness, while those who make losses swear on its unreliability.\n• Some traders consider the Fibonacci retracement as an illusion. Since the tool is repeatedly used by most traders, they often get similar results every time. This means that orders are stuck at similar price levels, which pushes the price in the direction they want.\n• To use the Fibonacci tool, you have to have an in-depth understanding of the tool. Lines drawn on price charts at the percentages will not give you results unless you know what to look for. Beginner traders should be wary of using these tools and be sure that an asset price’s dip is temporary, not a permanent reversal.\n\n## Conclusion\n\nA Fibonacci retracement is a powerful tool when used alongside other technical or indicator signals. The support and resistance levels are an indicator of potential rises or dips in market trends. This could show traders when it is lucrative to open or close positions. The Fibonacci retracement can be very useful and yield rewards for any trader who knows how to use the tool properly. Beginners are advised to approach the tool with caution or lose money while using the complex tool." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.94367844,"math_prob":0.91512454,"size":4176,"snap":"2021-43-2021-49","text_gpt3_token_len":895,"char_repetition_ratio":0.15604027,"word_repetition_ratio":0.047482014,"special_character_ratio":0.21599618,"punctuation_ratio":0.116930574,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98745584,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-22T04:15:32Z\",\"WARC-Record-ID\":\"<urn:uuid:5ead9b50-ec5f-48c7-a88f-46c9cd5a5a12>\",\"Content-Length\":\"37757\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ee41477f-7acb-4ace-a6ad-674578bc850c>\",\"WARC-Concurrent-To\":\"<urn:uuid:84ed569b-610e-407c-bb03-2201b791cbf4>\",\"WARC-IP-Address\":\"188.165.139.11\",\"WARC-Target-URI\":\"https://onlineforexcharts.org/general/all-about-fibonacci-retracements-and-fibonacci-ratios/\",\"WARC-Payload-Digest\":\"sha1:AY3S77RAFGDUFJJWA3CZ3UZBRE4TFX5F\",\"WARC-Block-Digest\":\"sha1:JTPV5OHPWFX2A4GYBASYGKUCGTEULSH2\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585450.39_warc_CC-MAIN-20211022021705-20211022051705-00053.warc.gz\"}"}
https://api.dart.cn/stable/2.18.5/dart-collection/ListMixin/reduce.html	[ "# reduce method Null safety\n\nE reduce(\n1. E combine(\n1. E previousValue,\n2. E element\n)\n)\noverride\n\nReduces a collection to a single value by iteratively combining elements of the collection using the provided function.\n\nThe iterable must have at least one element. If it has only one element, that element is returned.\n\nOtherwise this method starts with the first element from the iterator, and then combines it with the remaining elements in iteration order, as if by:\n\n``````E value = iterable.first;\niterable.skip(1).forEach((element) {\nvalue = combine(value, element);\n});\nreturn value;\n``````\n\nExample of calculating the sum of an iterable:\n\n``````final numbers = <double>[10, 2, 5, 0.5];\nfinal result = numbers.reduce((value, element) => value + element);\nprint(result); // 17.5\n``````\n\n## Implementation\n\n``````E reduce(E combine(E previousValue, E element)) {\nint length = this.length;\nif (length == 0) throw IterableElementError.noElement();\nE value = this;\nfor (int i = 1; i < length; i++) {\nvalue = combine(value, this[i]);\nif (length != this.length) {\nthrow ConcurrentModificationError(this);\n}\n}\nreturn value;\n}``````" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6023763,"math_prob":0.9971267,"size":1065,"snap":"2023-40-2023-50","text_gpt3_token_len":254,"char_repetition_ratio":0.16965127,"word_repetition_ratio":0.0,"special_character_ratio":0.26760563,"punctuation_ratio":0.205,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9879374,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-06T05:26:03Z\",\"WARC-Record-ID\":\"<urn:uuid:3b6d8355-46c9-46f2-93d4-38f7901d21de>\",\"Content-Length\":\"12241\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:19481e80-9156-4364-83c2-cbc7afa314de>\",\"WARC-Concurrent-To\":\"<urn:uuid:36ad5cb7-617c-4390-a202-eee3cda59951>\",\"WARC-IP-Address\":\"8.45.176.211\",\"WARC-Target-URI\":\"https://api.dart.cn/stable/2.18.5/dart-collection/ListMixin/reduce.html\",\"WARC-Payload-Digest\":\"sha1:B2ZQUZWYSWFSTR4HWQ35JHNAD5U54LQG\",\"WARC-Block-Digest\":\"sha1:DZOVX7WA5DKR4EQQSDKNACXEN6MFILEV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100583.13_warc_CC-MAIN-20231206031946-20231206061946-00306.warc.gz\"}"}
https://scipost.org/SciPostPhys.11.3.058	[ "## First law and quantum correction for holographic entanglement contour\n\nMuxin Han, Qiang Wen\n\nSciPost Phys. 11, 058 (2021) · published 14 September 2021\n\n### Abstract\n\nEntanglement entropy satisfies a first law-like relation, which equates the first order perturbation of the entanglement entropy for the region $A$ to the first order perturbation of the expectation value of the modular Hamiltonian, $\\delta S_{A}=\\delta \\langle K_A \\rangle$. We propose that this relation has a finer version which states that, the first order perturbation of the entanglement contour equals to the first order perturbation of the contour of the modular Hamiltonian, i.e. $\\delta s_{A}(\\textbf{x})=\\delta \\langle k_{A}(\\textbf{x})\\rangle$. Here the contour functions $s_{A}(\\textbf{x})$ and $k_{A}(\\textbf{x})$ capture the contribution from the degrees of freedom at $\\textbf{x}$ to $S_{A}$ and $K_A$ respectively. In some simple cases $k_{A}(\\textbf{x})$ is determined by the stress tensor. We also evaluate the quantum correction to the entanglement contour using the fine structure of the entanglement wedge and the additive linear combination (ALC) proposal for partial entanglement entropy (PEE) respectively. The fine structure picture shows that, the quantum correction to the boundary PEE can be identified as a bulk PEE of certain bulk region. While the \\textit{ALC proposal} shows that the quantum correction to the boundary PEE comes from the linear combination of bulk entanglement entropy. We focus on holographic theories with local modular Hamiltonian and configurations of quantum field theories where the \\textit{ALC proposal} applies.\n\n### Cited by 7", null, "### Authors / Affiliations: mappings to Contributors and Organizations\n\nSee all Organizations.\nFunder for the research work leading to this publication" ]	[ null, "https://scipost.org/static/scipost/images/citedby.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7457514,"math_prob":0.96513194,"size":1960,"snap":"2022-40-2023-06","text_gpt3_token_len":486,"char_repetition_ratio":0.13445808,"word_repetition_ratio":0.07089552,"special_character_ratio":0.22551021,"punctuation_ratio":0.07763975,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9928776,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-05T21:07:01Z\",\"WARC-Record-ID\":\"<urn:uuid:91474bea-2bbf-4204-8fdc-2375f8364fe3>\",\"Content-Length\":\"31029\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:02ad15e2-5a12-4d39-ae5d-7c3875a2e0b1>\",\"WARC-Concurrent-To\":\"<urn:uuid:95f00396-7d53-4abb-b3a0-f843d1637623>\",\"WARC-IP-Address\":\"142.93.224.252\",\"WARC-Target-URI\":\"https://scipost.org/SciPostPhys.11.3.058\",\"WARC-Payload-Digest\":\"sha1:D7ND7RBA36AFYWKUS4LWQVJHOMKHSBEL\",\"WARC-Block-Digest\":\"sha1:AF7FNEWSIBO6PWLNCPDMUKGJ77J35ELT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030337668.62_warc_CC-MAIN-20221005203530-20221005233530-00445.warc.gz\"}"}
https://www.apsnet.org/edcenter/disimpactmngmnt/topc/EcologyAndEpidemiologyInR/DiseaseProgress/Pages/LinearRegression.aspx	[ "# Linear regression in R\n\nThe AUDPC is not the only method for summarizing disease progress; regression analyses are also often applied. Regression analysis is a statistical tool for describing the relationship between two or more quantitative variables such that one variable (the dependent or response variable) may be predicted from other variable(s) (the independent or predictor variable(s)). For example, if the relationship between the severity of disease and time is known, disease severity can be predicted at a specified time. If we have only one predictor variable and the response and the predictor variable have a linear relationship, the data can be analyzed with a simple linear model. When there is more than one predictor variable, multiple regression may be used. When linear models are not sufficient to describe the data, a nonlinear regression model may be useful. In this section, some examples of simple linear regression are presented using R.\n\n## Linear regression\n\nLinear regression compares two variables x and y to answer the question, 'how does y change with x?' ' For example, what is disease severity (y) at two weeks (x)? Or, what will be the expected disease severity at the end of the growing season if no treatment is applied?\n\nA simple linear regression model is of the form: yi = β0ixii, where εi is iid Normal (0, σ2); that is, the error terms are independent and identically distributed following a normal distribution with mean 0 and variance σ2. The word \"linear\" in the model refers to the linear influence of the parameters β0 and β1, which are the regression coefficients. Specifically, β1 is the slope of the regression line, that is, the change in y corresponding to a unit change in x. β0 is the intercept, or the y value when x=0. The intercept has no practical meaning if the condition x=0 cannot occur, but is necessary to specify the model.\n\nIn simple linear regression the influence of random error in the model is allowed only in the error term ε. The predictor variable, x, is considered to be a deterministic quantity. The validity of the result for a typical linear regression model requires the fulfillment of the following assumptions:\n\n• the linear model is appropriate,\n• the error terms are independent,\n• the error terms are approximately normally distributed,\n• and the error terms have a common variance.\n\nFor more discussion on simple linear regression the reader should refer to a text for regression analysis (e.g. Harrell 2001). If model checking for a simple linear regression model indicates that a linear model is not appropriate, one could consider transformation, nonlinear regression, or other nonparametric options.\n\n## How to perform linear regression in R\n\nR makes the function `lm(formula, data, subset)` available.\nFor a complete description of `lm`, type: `help(lm)`.\nHere is a simple example of the relationship between disease severity and temperature:\n\n`## Disease severity as a function of temperature`\n`# Response variable, disease severitydiseasesev<-c(1.9,3.1,3.3,4.8,5.3,6.1,6.4,7.6,9.8,12.4)`\n`# Predictor variable, (Centigrade)temperature<-c(2,1,5,5,20,20,23,10,30,25)`\n`# Take a look at the dataplot(temperature,diseasesev)`\n\n## Output\n\n`## For convenience, the data may be formatted into a dataframeseverity <- as.data.frame(cbind(diseasesev,temperature))`\n`## Fit a linear model for the data and summarize## the output from function lm()severity.lm <- lm(diseasesev~temperature,data=severity)`\n`## Generate a summary of the linear modelsummary(severity.lm)`\n\n## Output\n\n`Residuals:Min 1Q Median 3Q Max -2.1959 -1.3584 -0.3417 0.7460 3.6957 `\n`Coefficients: Estimate Std. Error t value Pr(>\|t\|) (Intercept) 2.66233 1.10082 2.418 0.04195 temperature 0.24168 0.06346 3.808 0.00518 ---Signif. codes: 0 ‘’ 0.001 ‘’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 `\n`Residual standard error: 2.028 on 8 degrees of freedomMultiple R-Squared: 0.6445, Adjusted R-squared: 0.6001 F-statistic: 14.5 on 1 and 8 DF, p-value: 0.005175`\n\n## Graphical tools for testing assumptions\n\nCommon graphical tools for testing assumptions include plots evaluating the characteristics of residuals, the part of the observations that is not explained by the model:\n\n1. scatter plots of the residuals vs. x or the fitted value\n2. normal probability plots of the residuals.\n\nLook for a pattern in the graph of residual vs. fitted values that might suggest a non-constant variance. Equal variances should be dispersed evenly around zero. Other patterns can indicate that a linear regression model may not be appropriate for the data. If the pattern indicates unequal variances, a more complicated model that specifies unequal variances may be appropriate, or transformation of y might be useful.\n\n`## which=1 produces a graph of residuals vs fitted valuesplot(severity.lm, which=1)`\n\nThe graph shown below indicates some pattern in residuals which could be explored further, though this level of pattern relative to the magnitude of changes in the observed data may be unimportant.\n\n## Output\n\n`## which=2 produces a graph of a quantile-quantile (QQ) plotplot(severity.lm,which=2)`\n\n## Output\n\nClick on the image for larger version.\n\nPoints should lie close to a line on the QQ plot if residuals are from a normal distribution. The QQ plot above shows a very slight indication that the residual might not come from a normal distribution, since the last two observations deviate farther from the fitted line.\n\nAfter fitting the linear model, the function `predict()` can be used to generate fitted values. The function `data.frame()` is used below to create a table of original data and fitted values.\n\n`options(digits=4)fit.with.se<-predict( severity.lm, se.fit=TRUE)data.frame( severity, fitted.value=predict(severity.lm), residual=resid(severity.lm), fit.with.se)`\n\n## Output\n\n`disease temperature fitted.value residual fit se.fit df residual.scale1 1.9 2 3.146 -1.2457 3.146 1.0004 8 2.0282 3.1 1 2.904 0.1960 2.904 1.0499 8 2.0283 3.3 5 3.871 -0.5707 3.871 0.8629 8 2.0284 4.8 5 3.871 0.9293 3.871 0.8629 8 2.0285 5.3 20 7.496 -2.1959 7.496 0.7425 8 2.0286 6.1 20 7.496 -1.3959 7.496 0.7425 8 2.0287 6.4 23 8.221 -1.8209 8.221 0.8545 8 2.0288 7.6 10 5.079 2.5209 5.079 0.6920 8 2.0289 9.8 30 9.913 -0.1127 9.913 1.1955 8 2.02810 12.4 25 8.704 3.6957 8.704 0.9432 8 2.028`\n\nWe can also plot the data set together with the fitted line.\n\n`plot( diseasesev~temperature, data=severity, xlab=\"Temperature\", ylab=\"% Disease Severity\", pch=16)abline(severity.lm,lty=1)title(main=\"Graph of % Disease Severity vs Temperature\")`\n\n## Output\n\nClick on the image for larger version.\n\nIf the process of checking model assumptions indicates that the linear model is adequate, the linear model can be used to predict the response variable for a given predictor variable.\n\n`## Predict disease severity for three new temperaturesnew <- data.frame(temperature=c(15,16,17))predict( severity.lm, newdata=new, interval=\"confidence\")`\n\n## Output\n\n`fit lwr upr1 6.288 4.803 7.7722 6.529 5.025 8.0343 6.771 5.233 8.309`\n\nLinear regression with intercept and slope parameters can be used to describe straight-line relationships between variables. For disease progress measured over a limited time period, a straight line may provide a perfectly adequate description. Linear regression that includes additional parameters can be used to describe curved relationships, for example by including a squared term (quadratic term) in the set of predictors, such as time squared. More complicated relationships can readily be described using nonlinear regression.\n\nNext, nonlinear regression in R" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.775284,"math_prob":0.96467555,"size":7595,"snap":"2019-51-2020-05","text_gpt3_token_len":1986,"char_repetition_ratio":0.14688447,"word_repetition_ratio":0.020495303,"special_character_ratio":0.280711,"punctuation_ratio":0.16687346,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99690074,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-10T15:47:29Z\",\"WARC-Record-ID\":\"<urn:uuid:6e023c6a-4116-493d-8e5c-24be8695067f>\",\"Content-Length\":\"111147\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fc582084-1abd-459d-af28-e86cbae02381>\",\"WARC-Concurrent-To\":\"<urn:uuid:76bc14c4-ca42-4258-b321-8c0034acb854>\",\"WARC-IP-Address\":\"199.86.26.56\",\"WARC-Target-URI\":\"https://www.apsnet.org/edcenter/disimpactmngmnt/topc/EcologyAndEpidemiologyInR/DiseaseProgress/Pages/LinearRegression.aspx\",\"WARC-Payload-Digest\":\"sha1:PMFYQ6LKVUJWFEFF62A36STCDZ5FQDDU\",\"WARC-Block-Digest\":\"sha1:6VSVEBYFN3ATRR63M3UNCOSCYR3UOE55\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540528457.66_warc_CC-MAIN-20191210152154-20191210180154-00228.warc.gz\"}"}
https://browse.dgit.debian.org/yosys.git/commit/?id=3377a04bf26b9310017a71b2df587bad661e0da2	[ "summaryrefslogtreecommitdiff log msg author committer range\ndiff options\n context: 12345678910152025303540 space: includeignore mode: unifiedssdiffstat only\nauthor committer Clifford Wolf 2013-03-14 16:15:24 +0100 Clifford Wolf 2013-03-14 16:15:24 +0100 3377a04bf26b9310017a71b2df587bad661e0da2 (patch) f652d63a7614bf7d5e13ab3067beca2a5d5cd6ba 697cf1eb807ce48b69946a13769c647c82869efb (diff)\nChanged prefix for selection operators from # to %\n-rw-r--r--kernel/select.cc56\n1 files changed, 28 insertions, 28 deletions\n diff --git a/kernel/select.cc b/kernel/select.ccindex 650774c0..3c6fd669 100644--- a/kernel/select.cc+++ b/kernel/select.cc@@ -421,53 +421,53 @@ static void select_stmt(RTLIL::Design design, std::string arg) if (arg.size() == 0) return; - if (arg == '#') {- if (arg == \"#\") {+ if (arg == '%') {+ if (arg == \"%\") { if (design->selection_stack.size() > 0) work_stack.push_back(design->selection_stack.back()); } else- if (arg == \"##\") {+ if (arg == \"%%\") { while (work_stack.size() > 1) { select_op_union(design, work_stack.front(), work_stack.back()); work_stack.pop_back(); } } else- if (arg == \"#n\") {+ if (arg == \"%n\") { if (work_stack.size() < 1)- log_cmd_error(\"Must have at least one element on the stack for operator #n.\\n\");+ log_cmd_error(\"Must have at least one element on the stack for operator %%n.\\n\"); select_op_neg(design, work_stack[work_stack.size()-1]); } else- if (arg == \"#u\") {+ if (arg == \"%u\") { if (work_stack.size() < 2)- log_cmd_error(\"Must have at least two elements on the stack for operator #u.\\n\");+ log_cmd_error(\"Must have at least two elements on the stack for operator %%u.\\n\"); select_op_union(design, work_stack[work_stack.size()-2], work_stack[work_stack.size()-1]); work_stack.pop_back(); } else- if (arg == \"#d\") {+ if (arg == \"%d\") { if (work_stack.size() < 2)- log_cmd_error(\"Must have at least two elements on the stack for operator #d.\\n\");+ log_cmd_error(\"Must have at least two elements on the stack for operator %%d.\\n\"); select_op_diff(design, work_stack[work_stack.size()-2], work_stack[work_stack.size()-1]); work_stack.pop_back(); } else- if (arg == \"#i\") {+ if (arg == \"%i\") { if (work_stack.size() < 2)- log_cmd_error(\"Must have at least two elements on the stack for operator #i.\\n\");+ log_cmd_error(\"Must have at least two elements on the stack for operator %%i.\\n\"); select_op_intersect(design, work_stack[work_stack.size()-2], work_stack[work_stack.size()-1]); work_stack.pop_back(); } else- if (arg == \"#x\" \|\| (arg.size() > 2 && arg.substr(0, 2) == \"#x\" && (arg == ':' \|\| arg == '' \|\| arg == '.' \|\| ('0' <= arg && arg <= '9')))) {+ if (arg == \"%x\" \|\| (arg.size() > 2 && arg.substr(0, 2) == \"%x\" && (arg == ':' \|\| arg == '' \|\| arg == '.' \|\| ('0' <= arg && arg <= '9')))) { if (work_stack.size() < 1)- log_cmd_error(\"Must have at least one element on the stack for operator #x.\\n\");+ log_cmd_error(\"Must have at least one element on the stack for operator %%x.\\n\"); select_op_expand(design, arg, 'x'); } else- if (arg == \"#ci\" \|\| (arg.size() > 3 && arg.substr(0, 3) == \"#ci\" && (arg == ':' \|\| arg == '' \|\| arg == '.' \|\| ('0' <= arg && arg <= '9')))) {+ if (arg == \"%ci\" \|\| (arg.size() > 3 && arg.substr(0, 3) == \"%ci\" && (arg == ':' \|\| arg == '' \|\| arg == '.' \|\| ('0' <= arg && arg <= '9')))) { if (work_stack.size() < 1)- log_cmd_error(\"Must have at least one element on the stack for operator #ci.\\n\");+ log_cmd_error(\"Must have at least one element on the stack for operator %%ci.\\n\"); select_op_expand(design, arg, 'i'); } else- if (arg == \"#co\" \|\| (arg.size() > 3 && arg.substr(0, 3) == \"#co\" && (arg == ':' \|\| arg == '' \|\| arg == '.' \|\| ('0' <= arg && arg <= '9')))) {+ if (arg == \"%co\" \|\| (arg.size() > 3 && arg.substr(0, 3) == \"%co\" && (arg == ':' \|\| arg == '' \|\| arg == '.' \|\| ('0' <= arg && arg <= '9')))) { if (work_stack.size() < 1)- log_cmd_error(\"Must have at least one element on the stack for operator #co.\\n\");+ log_cmd_error(\"Must have at least one element on the stack for operator %%co.\\n\"); select_op_expand(design, arg, 'o'); } else log_cmd_error(\"Unknown selection operator '%s'.\\n\", arg.c_str());@@ -705,25 +705,25 @@ struct SelectPass : public Pass { log(\"\\n\"); log(\"The following actions can be performed on the top sets on the stack:\\n\"); log(\"\\n\");- log(\" #\\n\");+ log(\" %%\\n\"); log(\" push a copy of the current selection to the stack\\n\"); log(\"\\n\");- log(\" ##\\n\");+ log(\" %%%%\\n\"); log(\" replace the stack with a union of all elements on it\\n\"); log(\"\\n\");- log(\" #n\\n\");+ log(\" %%n\\n\"); log(\" replace top set with its invert\\n\"); log(\"\\n\");- log(\" #u\\n\");+ log(\" %%u\\n\"); log(\" replace the two top sets on the stack with their union\\n\"); log(\"\\n\");- log(\" #i\\n\");+ log(\" %%i\\n\"); log(\" replace the two top sets on the stack with their intersection\\n\"); log(\"\\n\");- log(\" #d\\n\");+ log(\" %%d\\n\"); log(\" pop the top set from the stack and subtract it from the new top\\n\"); log(\"\\n\");- log(\" #x[\|][.][:[:..]]\\n\");+ log(\" %%x[\|][.][:[:..]]\\n\"); log(\" expand top set num times accorind to the specified rules.\\n\"); log(\" (i.e. select all cells connected to selected wires and select all\\n\"); log(\" wires connected to selected cells) The rules specify which cell\\n\");@@ -736,14 +736,14 @@ struct SelectPass : public Pass { log(\" limit is reached. When '' is used instead of then the process\\n\"); log(\" is repeated until no further object are selected.\\n\"); log(\"\\n\");- log(\" #ci[\|][.][:[:..]]\\n\");- log(\" #co[\|][.][:[:..]]\\n\");- log(\" simmilar to #x, but only select input (#ci) or output cones (#co)\\n\");+ log(\" %%ci[\|][.][:[:..]]\\n\");+ log(\" %%co[\|][.][:[:..]]\\n\");+ log(\" simmilar to %%x, but only select input (%%ci) or output cones (%%co)\\n\"); log(\"\\n\"); log(\"Example: the following command selects all wires that are connected to a\\n\"); log(\"'GATE' input of a 'SWITCH' cell:\\n\"); log(\"\\n\");- log(\" select /t:SWITCH #x:+[GATE] /t:SWITCH #d\\n\");+ log(\" select /t:SWITCH %%x:+[GATE] /t:SWITCH %%d\\n\"); log(\"\\n\"); } virtual void execute(std::vector args, RTLIL::Design *design)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.59042686,"math_prob":0.9414322,"size":6256,"snap":"2023-40-2023-50","text_gpt3_token_len":2116,"char_repetition_ratio":0.1877799,"word_repetition_ratio":0.37594798,"special_character_ratio":0.45620206,"punctuation_ratio":0.19983687,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9894163,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-21T16:45:05Z\",\"WARC-Record-ID\":\"<urn:uuid:f727ae7c-433e-4f23-ae6e-5394610ef9e9>\",\"Content-Length\":\"15756\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b70de2a1-cf37-4804-bf3e-d1796078a64f>\",\"WARC-Concurrent-To\":\"<urn:uuid:305d842d-3744-4a25-8b25-90fd44e5f958>\",\"WARC-IP-Address\":\"194.177.211.202\",\"WARC-Target-URI\":\"https://browse.dgit.debian.org/yosys.git/commit/?id=3377a04bf26b9310017a71b2df587bad661e0da2\",\"WARC-Payload-Digest\":\"sha1:SDCLAJJPL5YMXUG6OEF4U7MCQRBITTDQ\",\"WARC-Block-Digest\":\"sha1:LV6MCWXYV4LHPBF5XUL6DW7EHQMKN72P\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506028.36_warc_CC-MAIN-20230921141907-20230921171907-00790.warc.gz\"}"}
https://tipcalc.net/how-much-is-a-5-percent-tip-on-400.26	[ "# Tip Calculator\n\nHow much is a 5 percent tip on \\$400.26?\n\nTIP:\n\\$ 0\nTOTAL:\n\\$ 0\nTIP PER PERSON:\n\\$ 0\nTOTAL PER PERSON:\n\\$ 0\n\n## How much is a 5 percent tip on \\$400.26? How to calculate this tip?\n\nAre you looking for the answer to this question: How much is a 5 percent tip on \\$400.26? Here is the answer.\n\nLet's see how to calculate a 5 percent tip when the amount to be paid is 400.26. Tip is a percentage, and a percentage is a number or ratio expressed as a fraction of 100. This means that a 5 percent tip can also be expressed as follows: 5/100 = 0.05 . To get the tip value for a \\$400.26 bill, the amount of the bill must be multiplied by 0.05, so the calculation is as follows:\n\n1. TIP = 400.265% = 400.260.05 = 20.013\n\n2. TOTAL = 400.26+20.013 = 420.273\n\n3. Rounded to the nearest whole number: 420\n\nIf you want to know how to calculate the tip in your head in a few seconds, visit the Tip Calculator Home.\n\n## So what is a 5 percent tip on a \\$400.26? The answer is 20.01!\n\nOf course, it may happen that you do not pay the bill or the tip alone. A typical case is when you order a pizza with your friends and you want to split the amount of the order. For example, if you are three, you simply need to split the tip and the amount into three. In this example it means:\n\n1. Total amount rounded to the nearest whole number: 420\n\n2. Split into 3: 140\n\nSo in the example above, if the pizza order is to be split into 3, you’ll have to pay \\$140 . Of course, you can do these settings in Tip Calculator. You can split the tip and the total amount payable among the members of the company as you wish. So the TipCalc.net page basically serves as a Pizza Tip Calculator, as well.\n\n## Tip Calculator Examples (BILL: \\$400.26)\n\nHow much is a 5% tip on \\$400.26?\nHow much is a 10% tip on \\$400.26?\nHow much is a 15% tip on \\$400.26?\nHow much is a 20% tip on \\$400.26?\nHow much is a 25% tip on \\$400.26?\nHow much is a 30% tip on \\$400.26?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9622917,"math_prob":0.9946978,"size":2605,"snap":"2023-40-2023-50","text_gpt3_token_len":854,"char_repetition_ratio":0.31987697,"word_repetition_ratio":0.24064171,"special_character_ratio":0.40460652,"punctuation_ratio":0.16235632,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99777305,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-11T02:41:47Z\",\"WARC-Record-ID\":\"<urn:uuid:a420fc40-2454-40b3-9c33-be432aaaa0d8>\",\"Content-Length\":\"13253\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c87edd0b-dddc-49a7-807a-72754918ce7c>\",\"WARC-Concurrent-To\":\"<urn:uuid:4a9aed67-e249-44a4-b0c9-4031b6620475>\",\"WARC-IP-Address\":\"161.35.97.186\",\"WARC-Target-URI\":\"https://tipcalc.net/how-much-is-a-5-percent-tip-on-400.26\",\"WARC-Payload-Digest\":\"sha1:JUTCQX462WNJWZNNXTU4H7SFHUQY4KNN\",\"WARC-Block-Digest\":\"sha1:MDJU6CZQ4HXCRZPEG5M5SYLD7K24B55E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679103464.86_warc_CC-MAIN-20231211013452-20231211043452-00198.warc.gz\"}"}
http://fstate.org/knowndecays/K*(892)~0	[ "# Known decays of K*(892)~0\n\n## (Only the first generation of decays, no cascades)\n\n Final state Branching $\\bar{K}'^{0}_{}(892)$ $\\to$ $K^{-}_{}$ $\\pi^{+}_{}$ 1.0e-0" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5091949,"math_prob":0.99996364,"size":285,"snap":"2020-24-2020-29","text_gpt3_token_len":97,"char_repetition_ratio":0.085409254,"word_repetition_ratio":0.0,"special_character_ratio":0.34385964,"punctuation_ratio":0.074074075,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9655584,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-05-27T22:41:43Z\",\"WARC-Record-ID\":\"<urn:uuid:78c6eb43-2019-4982-9304-b801e97eace9>\",\"Content-Length\":\"10428\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e982f12a-5340-4bf1-89db-d00a8d42d2ca>\",\"WARC-Concurrent-To\":\"<urn:uuid:77cdcd23-d482-401c-814a-d1cbbe5a9368>\",\"WARC-IP-Address\":\"45.79.85.160\",\"WARC-Target-URI\":\"http://fstate.org/knowndecays/K*(892)~0\",\"WARC-Payload-Digest\":\"sha1:UOP3GRQCZ6Y22WE327AY25E52WXIVPXI\",\"WARC-Block-Digest\":\"sha1:YHGV3H5DQWNFHBYIBPKFN7V2LWXIC6LF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347396163.18_warc_CC-MAIN-20200527204212-20200527234212-00459.warc.gz\"}"}
https://pinoybix.org/2014/10/mcqs-in-bjt-and-fet-frequency-response.html	[ "", null, "# MCQs in BJT and FET Frequency Response\n\n(Last Updated On: December 8, 2017)\n\nThis is the Multiple Choice Questions in BJT and FET Frequency Respons from the book Electronic Devices and Circuit Theory 10th Edition by Robert L. Boylestad. If you are looking for a reviewer in Electronics Engineering this will definitely help. I can assure you that this will be a great help in reviewing the book in preparation for your Board Exam. Make sure to familiarize each and every questions to increase the chance of passing the ECE Board Exam.\n\n### Online Questions and Answers Topic Outline\n\n• MCQs in Logarithms\n• MCQs in Decibels\n• MCQs in General Frequency Considerations\n• MCQs in Low-frequency analysis – Bode Plot\n• MCQs in Low-frequency response – BJT Amplifier\n• MCQs in Low-frequency response – FET Amplifier\n• MCQs in Miller Effect Capacitance\n• MCQs in High-frequency response – BJT Amplifier\n• MCQs in High-frequency response – FET Amplifier\n• MCQs in Multistage Frequency Effects\n• MCQs in Square wave testing\n• MCQs in Computer Analysis\n\n### Practice Exam Test Questions\n\nChoose the letter of the best answer in each questions.\n\n1. What is the ratio of the common logarithm of a number to its natural logarithm?\n\n• a. 0.435\n• b. 2\n• c. 2.3\n• d. 3.2\n\n2. logea = _____ log10a\n\n• a. 2.3\n• b. 2.718\n• c. e\n• d. 1.414\n\n3. By what factor does an audio level change if the power level changes from 4 W to 4096 W?\n\n• a. 2\n• b. 4\n• c. 6\n• d. 8\n\n4. The input power to a device is 10,000 W at 1000 V. The output power is 500 W, and the output impedance is 100 Ω. Find the voltage gain in decibels.\n\n• a. –30.01 dB\n• b. –20.0 dB\n• c. –13.01 dB\n• d. –3.01 dB\n\n5. What magnitude voltage gain corresponds to a decibel gain of 50?\n\n• a. 31.6238\n• b. 316.228\n• c. 3162.38\n• d. 31623.8\n\n6. An amplifier rated at 30-W output is connected to a 5-Ω speaker. Calculate the input power required for full power output if the power gain is 20 dB.\n\n• a. 3 mW\n• b. 30 mW\n• c. 300 mW\n• d. 3 W\n\n7. An amplifier rated at 30-W output is connected to a 5-Ω speaker. Calculate the input voltage for the rated output if the amplifier voltage gain is 20 dB.\n\n• a. 1.225 mV\n• b. 12.25 mV\n• c. 122.5 mV\n• d. 1.225 V\n\n8. For audio systems, the reference level is generally accepted as\n\n• a. 1 mW\n• b. 1 W\n• c. 10 mW\n• d. 100 mW\n\n9. For which of the following frequency region(s) can the coupling and bypass capacitors no longer be replaced by the short-circuit approximation?\n\n• a. Low-frequency\n• b. Mid-frequency\n• c. High-frequency\n• d. All of the above\n\n10. By what other name(s) are the cut-off frequencies in a frequency response plot called?\n\n• a. Corner frequency\n• b. Break frequency\n• c. Half-power frequency\n• d. All of the above\n\n11. What is the ratio of the output power to the input power at the cut-off frequencies in a normalized frequency response plot?\n\n• a. 0.25\n• b. 0.50\n• c. 0.707\n• d. 1\n\n12. What is the ratio of the output voltage to the input voltage at the cut-off frequencies in a normalized frequency response plot?\n\n• a. 0.25\n• b. 0.50\n• c. 0.707\n• d. 1\n\n13. What is the normalized gain expressed in dB for the cut-off frequencies?\n\n• a. –3 dB\n• b. +3 dB\n• c. –6 dB\n• d. –20 dB\n\n14. The ________-frequency response of a transformer-coupled system is calculated primarily by the stray capacitance between the turns of the primary and secondary windings.\n\n• a. Low\n• b. Mid\n• c. High\n\n15. The larger capacitive elements of the design will determine the ________ cut-off frequency.\n\n• a. Low\n• b. Mid\n• c. High\n\n16. The smaller capacitive elements of the design will determine the ________ cut-off frequencies.\n\n• a. Low\n• b. Mid\n• c. High\n\n17. What is the ratio of the capacitive reactance XCS to the input resistance RI of the input RC circuit of a single-stage BJT amplifier at the low-frequency cut-off?\n\n• a. 0.25\n• b. 0.50\n• c. 0.75\n• d. 1.0\n\n18. In the input RC circuit of a single-stage BJT, by how much does the base voltage lead the input voltage for frequencies much larger than the cut-off frequency in the low-frequency region?\n\n• a. About 0º\n• b. 45º\n• c. About 90º\n• d. None of the above\n\n19. In the input RC circuit of a single-stage BJT, by how much does the base voltage lead the input voltage at the cut-off frequency in the low-frequency region?\n\n• a. About 0º\n• b. 45º\n• c. About 90º\n• d. None of the above\n\n20. Determine the break frequency for this circuit.", null, "• a. 15.915 Hz\n• b. 159.15 Hz\n• c. 31.85 Hz\n• d. 318.5 Hz\n\n21. Refer to Figure 9.19. Calculate θ at 0.5f1.\n\n• a. 63.43º\n• b. 26.56º\n• c. 45º\n• d. Undefined\n\n22. A change in frequency by a factor of ________ is equivalent to 1 octave.\n\n• a. 2\n• b. 10\n• c. 5\n• d. 20\n\n23. A change in frequency by a factor of ________ is equivalent to 1 decade.\n\n• a. 2\n• b. 10\n• c. 5\n• d. 20\n\n24. For the low-frequency response of a BJT amplifier, the maximum gain is where ________ .\n\n• a. RB = 0 Ω\n• b. RC = 0 Ω\n• c. RE = 0 Ω\n\n25. Which of the low-frequency cutoffs determined by CS, CC, or CE will be the predominant factor in determining the low-frequency response for the complete system?\n\n• a. Lowest\n• b. Middle\n• c. Highest\n• d. None of the above\n\n26. Determine the lower cut-off frequency of this network.\n\n• a. 15.8 Hz\n• b. 46.13 Hz\n• c. 238.73 Hz\n• d. 1575.8 Hz\n\n27. Which of the following elements is (are) important in determining the gain of the system in the high-frequency region?\n\n• a. Interelectrode capacitances\n• b. Wiring capacitances\n• c. Miller effect capacitance\n• d. All of the above\n\n28. In the ________-frequency region, the capacitive elements of importance are the interelectrode (between terminals) capacitances internal to the active device and the wiring capacitance between the leads of the network.\n\n• a. Low\n• b. Mid\n• c. High\n\n29. Which of the following capacitors is (are) included in Ci for the high-frequency region of a BJT or FET amplifier?\n\n30. In the hybrid", null, "or Giacoletto model, which one of the following does rb include?\n\n• a. Base spreading resistance\n• b. Base contact\n• c. Base bulk\n• d. All of the above\n\n31. Which of the following configurations does (do) not involve the Miller effect capacitance?\n\n• a. Common-emitter\n• b. Common-base\n• c. Common-collector\n• d. All of the above\n\n32. A 3-dB drop in hfe will occur at a frequency defined by ________.\n\n33. What is the range of the capacitors Cgs and Cgd?\n\n• a. 1 to 10 pF\n• b. 1 to 10 nF\n• c. 1 to 10 F\n• d. 1 to 10 F\n\n34. What is the range of the capacitor Cds?\n\n• a. 0.01 to 0.1 pF\n• b. 0.1 to 1 pF\n• c. 0.1 to 1 nF\n• d. 0.1 to 1 F\n\n35. Which of the following statements is true for a square-wave signal?\n\n• a. It is composed of both even and odd harmonics.\n• b. It is composed only of odd harmonics.\n• c. It is composed only of even harmonics.\n• d. The harmonics waveforms are also square waves.\n\n### FILL-IN-THE-BLANKS\n\n1. Logarithms taken to the base _____ are referred to as common logarithms, while logarithms taken to the base _____ are referred to as natural logarithms.\n\n• A. 10, e\n• B e, 10\n• C. 5, e\n• D. 10, 5\n\n2. The logarithm of a number _____ than 1 is always _____.\n\n• A. greater, negative\n• B. less, positive\n• C. less, negative\n• D. None of the above\n\n3. The decibel (dB) is defined such that _____ decibel(s) = _____ bel(s).\n\n• A. 1, 10\n• B. 10, 1\n• C. 1, 1\n• D. 10, 10\n\n4. The resistance associated with the 1-mW power level is _____ , chosen because it is the characteristic impedance of audio transmission lines.\n\n• A. 100\n• B. 250\n• C. 400\n• D. 600\n\n5. The decibel gain of a cascaded system is the _____ of the decibel gains of each stage.\n\n• A. sum\n• B. difference\n• C. product\n• D. quotient\n\n6. Voltage gains of _____ dB or higher should immediately be recognized as being quite high.\n\n• A. 3\n• B. 6\n• C. 20\n• D. 50\n\n7. For the RC-coupled amplifier, the drop in gain at low frequencies is due to the increasing reactance of _____.\n\n• A. CC\n• B. Cs\n• C. CE\n• D. All of the above\n\n8. To fix the frequency boundaries of relatively high gain, _____ was chosen to be the gain at the cut-off levels.\n\n• A. 0.5Av mid\n• B. 0.707Av mid\n• C. Av low\n• D. 0.5Av high\n\n9. In the input RC circuit of a single-stage BJT or FET amplifier, as the frequency _____, the capacitive reactance _____ and _____ of the input voltage appears across the output terminals.\n\n• A. increases, decreases, more\n• B. increases, decreases, less\n• C. increases, increases, more\n• D. decreases, decreases, less\n\n10. A change in frequency by a factor of 2 results in a _____ change in the ratio of the normalized gain.\n\n• A. 3-dB\n• B. 6-dB\n• C. 10-dB\n• D. 20-dB\n\n11. A change in frequency by a factor of 10 results in a _____ change in the ratio of the normalized gain.\n\n• A. 3-dB\n• B. 6-dB\n• C. 10-dB\n• D. 20-dB\n\n12. In the low-frequency region, the _____ low-frequency cut-off determined by CS, CC, or CE will have the greatest impact on the network.\n\n• A. highest\n• B. average\n• C. lowest\n• D. None of the above\n\n13. The _____ region produces the maximum voltage gain in a single-stage BJT or FET amplifier.\n\n• A. low-frequency\n• B. mid-frequency\n• C. high-frequency\n• D. None of the above\n\n14. For any inverting amplifier, the impedance capacitance will be _____ by a Miller effect capacitance sensitive to the gain of the amplifier and the interelectrode capacitance.\n\n• A. unaffected\n• B. increased\n• C. decreased\n• D. None of the above\n\n15. The Miller effect is meaningful in the _____ amplifier.\n\n• A. inverting\n• B. noninverting\n• C. inverting/noninverting\n• D. None of the above\n\n16. With a BJT amplifier in the high-frequency region, the capacitance Cbe is the _____ of the parasitic capacitances while Cce is the _____.\n\n• A. smallest, largest\n• B. largest, smallest\n• C. smallest, medium\n• D. None of the above\n\n17. At very high frequencies, the effect of Ci is to _____ the total impedance of the parallel combination of R1, R2, R3, and Ci.\n\n• A. increase\n• B. maintain\n• C. decrease\n• D. None of the above\n\n18. If the parasitic capacitors were the only elements to determine the high cut-off frequency, the _____ frequency would be the determining factor.\n\n• A. lowest\n• B. highest\n• C. lowest or highest\n• D. None of the above\n\n19. The _____ configuration displays improved high-frequency characteristics over the _____ configuration.\n\n• A. common-collector, common-emitter\n• B. common-emitter, common-base\n• C. common-emitter, common-collector\n• D. common-base, common-emitter\n\n20. The _____ of the upper cut-off frequencies defines a _____ possible bandwidth for a system.\n\n• A. highest, maximum\n• B. lowest, maximum\n• C. lowest, minimum\n• D. None of the above\n• A. significantly smaller\n• B. smaller\n• C. significantly greater\n• D. None of the above\n\n22. For two identical stages in cascade, the drop-off rate in the high- and low-frequency regions has increased to _____ per decade.\n\n• A. –3 dB\n• B. –6 dB\n• C. –20 dB\n• D. –40 dB\n\n23. The bandwidth _____ in a multistage amplifier compared to an identical single-stage amplifier.\n\n• A. increases\n• B. decreases\n• C. remains the same\n• D. None of the above\n\n24. The _____ in the Fourier series has the same frequency as the square wave itself.\n\n• A. fundamental\n• B. third harmonic\n• C. fifth harmonic\n• D. seventh harmonic\n\n25. The magnitude of the third harmonic is _____ of the magnitude of the fundamental.\n\n• A. 1\n• B. 0.5\n• C. 0.33\n• D. 0.25\n\nRate this:" ]	[ null, "https://www.facebook.com/tr", null, "https://lh4.ggpht.com/-1NYiPg1xQOs/VFHMgdZdUgI/AAAAAAAABm4/UnAGUUS5Yfc/clip_image002_thumb4%25255B6%25255D.png", null, "https://lh6.ggpht.com/-r00jB2kgyuU/VFHMlgES-EI/AAAAAAAABng/90BojUoDJKQ/clip_image008_thumb%25255B2%25255D.gif", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8308556,"math_prob":0.95234483,"size":11215,"snap":"2019-13-2019-22","text_gpt3_token_len":3445,"char_repetition_ratio":0.17955579,"word_repetition_ratio":0.17296313,"special_character_ratio":0.33294696,"punctuation_ratio":0.18409714,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.9950952,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,3,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-25T03:15:49Z\",\"WARC-Record-ID\":\"<urn:uuid:f2879175-9b5c-4e4e-bd53-9df8f75d4542>\",\"Content-Length\":\"85258\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9b424184-e038-4ba4-8a8b-5bd3f387de84>\",\"WARC-Concurrent-To\":\"<urn:uuid:1c630e15-7154-48cb-a8cd-c84246a60fb2>\",\"WARC-IP-Address\":\"104.24.99.66\",\"WARC-Target-URI\":\"https://pinoybix.org/2014/10/mcqs-in-bjt-and-fet-frequency-response.html\",\"WARC-Payload-Digest\":\"sha1:T7TYMN7T5YP5CDH5FUYTMY4QKNEWT743\",\"WARC-Block-Digest\":\"sha1:RY3A2L4VQL5LTZP6BMQIV57M5KVOL47I\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912203548.81_warc_CC-MAIN-20190325031213-20190325053213-00140.warc.gz\"}"}
http://jasss.soc.surrey.ac.uk/19/4/7.html	[ "Home > 19 (4), 7\n\n# Robust Clustering in Generalized Bounded Confidence Models", null, ", and\n\naJacobs University Bremen, Germany; bUniversity of Groningen, Netherlands\n\nJournal of Artificial Societies and Social Simulation 19 (4) 7", null, "<http://jasss.soc.surrey.ac.uk/19/4/7.html>\nDOI: 10.18564/jasss.3220", null, "Received: 10-Jul-2016 Accepted: 03-Sep-2016 Published: 31-Oct-2016\n\n### Abstract\n\nBounded confidence models add a critical theoretical ingredient to the explanation of opinion clustering, opinion polarisation, and the persistence of opinion diversity, assuming that individuals are only influenced by others who are sufficiently similar and neglect actors with too different views. However, despite its enormous recognition in the literature, the bounded confidence assumption has been criticized for being able to explain diversity only when implemented in a very strict and unrealistic way. The model is unable to explain patterns of opinion diversity when actors are sometimes influenced also by others who hold distant views, even when these deviations from the bounded-confidence assumption are rare and random. Here, we echo this criticism but we also show that the model's ability to explain opinion diversity can be regained when another assumption is relaxed. Building on modeling work from statistical mechanics, we include that actors' opinion changes do not only result from social influence. When other influences are modelled as random, uniformly distributed draws, then robust patterns of opinion clustering emerge also with the relaxed implementations of bounded confidence. The results holds under both communication regimes: the updating to the average of all acceptable opinions as in the model of Hegselmann and Krause (2002) and random pair-wise communication as in the model of Deffuant et al. (2000). We discuss implications for future modelling work and point to gaps in empirical research on influence.\nKeywords: Opinion Dynamics, Continuous Opinions, Noise, Diversity Puzzle, Facilitation, Probability of Acceptance\n\n### Introduction\n\nThe bounded-confidence model (Hegselmann & Krause 2002; Deffuant et al. 2000) is one of the success stories of formal modeling work in the social sciences. It is build on very few, simple assumptions about individual behavior but generates complex and surprising dynamics and clustering patterns. This is certainly the main reason why it has attracted impressive scholarly attention in fields as diverse as sociology, physics, computer science, philosophy, economics, communication science, and political science.\n\nThe bounded-confidence model proposes a prominent solution to one of the most persistent research puzzles of the social sciences, Abelson’s diversity puzzle (Abelson 1964). Building on Harary (1959), Abelson studied social-influence dynamics in networks where connected nodes exert influence on each others’ opinions in that they grow more similar through repeated averaging. He analytically proved that social influence will always generate perfect consensus unless the network consists of multiple unconnected subsets with zero influence between them. This result was later extended to a theory of rational consensus in science and society (DeGroot 1974; Lehrer & Wagner 1981). Stunned by his finding, Abelson formulated an intriguing research puzzle: “Since universal ultimate agreement is an ubiquitous outcome of a very broad class of mathematical models, we are naturally led to inquire what on earth one must assume in order to generate the bimodal outcome of community cleavage studies.” Abelson’s puzzle is challenging, because consensus is a very robust pattern. For instance, Abelson (1964) showed that consensus is inevitable even when one also includes the assumption that social influence between actors is weaker when they hold very discrepant opinions prior to the interaction, an assumption that was supported by laboratory studies (Fisher & Lubin 1958; Aronson et al. 1963).\n\nPanel (a) of Figure 1 illustrates the inevitable trend towards perfect uniformity that social influence generates in a fixed network with 100 agents. For this ideal-typical simulation run, we assumed that agents’ initial opinions are randomly drawn from a uniform distribution ranging from zero to one. Agents with an initial opinion distance of less 0.15 were connected by a fixed network link. In each time step, agents update their opinions, adopting the arithmetic mean of their own opinion and the opinions of their network contacts. The figure shows that opinions converge to consensus despite the high initial opinion diversity and the relatively sparse influence network. Clustering of opinions can only be observed transiently.", null, "Figure 1. Panel (a) shows a run of opinion dynamics with a fixed network of 100 agents as in (Abelson 1964). Agents influence each other when their initial opinions differ by not more than 0.15. Nevertheless, this model generates consensus. Panel (b) shows a run with the model of Hegselmann & Krause (2002) with a bounded-confidence threshold of $$\\epsilon$$ = 0.15. The influence network is not fixed. This model generates multiple clusters of opinions. Panel (c) also assumes $$\\epsilon$$ = 0.15 but adds the assumption that influence weights turn 1 with a probability of 0.001. This model generates opinion clusters but clusters ultimately converge as a result of the random deviations from the bounded-confidence assumption.\n\nThe solution to Abelson’s puzzle proposed by the bounded-confidence model is strikingly simple, because it sharpens just one of Abelson’s own modelling assumptions: The influence of the opinions of others on the individual is not only declining with opinion discrepancy but individuals ignore influence by actors when opinion discrepancy exceeds a threshold – the bound of confidence. Panel (b) of Figure 1 shows opinion dynamics of the bounded confidence model of Hegselmann & Krause (2002) with initial conditions identical to the one from Panel (a). The crucial difference between the two simulation runs is that the influence network in Panel (b) is not fixed. New ties are created when two agents’ opinion distance has decreased to a value below the bounded-confidence threshold of $$\\epsilon$$ = 0.15, and ties are dissolved whenever opinion differences exceed 0.15. Panel (b) illustrates that the bounded-confidence models predicts the emergence of increasingly homogenous clusters, which at some point in time adopt opinions that differ too much from each other. As a consequence, opinions cannot converge further and clusters remain stable.\n\nOther approaches to Abelson’s puzzle included assumptions about negative social influence (Macy et al. 2003; Mark 2003; Salzarulo 2006; Flache & Mäs 2008), the communication of persuasive arguments (Mäs et al. 2013; Mäs & Flache 2013), biased assimilation (Dandekar et al. 2013), information accumulation (Shin & Lorenz 2010), striving for uniqueness (Mäs et al. 2010), attraction by initial views (Friedkin 2015), influences by external forces such as media and opinion leaders (Watts & Dodds 2007) and the assumption that opinions are measured on nominal scales (Axelrod 1997; Liggett 2013). However, none of these approaches has attracted as much scholarly attention as the bounded confidence model.\n\nThe bounded-confidence assumption has also been subject to important criticism (Mäs et al. 2013; Mäs & Flache 2013). To be sure, the assumption that individuals tend to be influenced by similar others is certainly very plausible in many social settings and has been supported by sociological research on homophily (McPherson et al. 2001; Lazarsfeld & Merton 1954) and social-psychological studies in the Similarity-Attraction-Paradigm (Byrne 1971). However, bounded-confidence models assume that individuals always ignore opinions that differ from their own views, which is an extreme and unrealistic interpretation. In other words, it is certainly plausible that individuals’ confidence in others is bounded but it is certainly not plausible that individuals are never influenced by actors with very different opinions.\n\nIn fact, even a tiny relaxation of the bounded-confidence assumption fundamentally changes model predictions, as Panel (c) of Figure 1 illustrates. These dynamics were generated with a model identical to the model of Panel (b) but with a slightly relaxed bounded-confidence assumption. We included that with a tiny probability of 0.001 also agents with otherwise too big opinion differences (difference exceeds 0.15) take each others’ opinions into account when they update their views. Panel (c) shows that this seemingly innocent relaxation of the bounded-confidence assumption results in the emergence of opinion consensus and, thus, suggests that the bounded-confidence assumption might not be a satisfactory solution to Abelson’s puzzle.\n\nEven though we emphasize this criticism of bounded-confidence models, we show here that the model’s ability to solve Abelson’s puzzle can be regained if another model assumption is relaxed. In particular, we relax the common assumption of social-influence models that actors’ opinions are only affected by social influence. Inspired by modelling work published in the field of statistical mechanics (Pineda et al. 2009, 2013; Carro et al. 2013), we include the assumption that an agent’s opinion might sometimes happen to switch to another, new opinion not as a result of social influence but at random. These random opinion changes might reflect, for instance, influences on agents’ opinions from external sources, or turnover dynamics in an organization where employees leave the organization and are replaced by new workers who hold opinions that are unrelated to the opinion of the leaving person. We study the conditions under which this mechanism leads to the emergence and persistence of opinion diversity and opinion clustering even when the bounded-confidence assumption is implemented in a weaker fashion as in its original strict way but accompanyied by random opinion replacement.\n\n### The model\n\nWe adopt an agent-based modeling framework and assume that each agent i of N agents is described by a continuous opinion xi (t) ranging from 0 to 1. At every time step t agents’ opinions are updated in two ways. First, agents are socially influenced by the opinions of others, as assumed by classical social-influence models and the bounded-confidence models. Second, we included that agents’ opinions may also change due to other forces. To this end, we added that agents can adopt a random opinion as studied in models inspired by statistical mechanics (Pineda et al. 2009, 2013; Carro et al. 2013) .\n\nSocial influence is implemented as follows. First, the computer program identifies those agents who will exert influence on i’s opinion. To this end, the program first calculates what Fishbein & Ajzen (1975) called the probability of acceptance, i.e. the probability that an agent j is influential, using Equation 1. Next, agent i updates her opinion, adapting to the average of the opinions of the influential agents and i’s own opinion.\n\nFollowing psychological literature on opinion and attitude change (Abelson 1964; Fishbein & Ajzen 1975; Hunter et al. 1984, e.g.), we define opinion discrepancy between agents i and j as the distance in opinion D(i, j) = \|xi(t) - xj(t)\| and assume that the probability that i is influenced by j declines with opinion discrepancy. To that end, we define the probability of acceptance for i and j as\n\n $$P(i,j) = \\begin{cases} \\max\\{\\beta, \\displaystyle{\\left(1 - \\frac{D(i,j)}{\\varepsilon}\\right)^{1/f}}\\} & \\text{if } D(i,j) \\leq \\varepsilon,\\\\ \\\\ \\beta & \\text{otherwise,} \\end{cases}$$ (1)\nwhere $$\\varepsilon \\in[0, 1]$$ is the bound of confidence, f reflects the facilitation, and $$\\beta$$ the default influence probability. We adopt the concept of “facilitation” from Fishbein & Ajzen (1975)’s seminal attitude theory. Facilitation determines the strength of the decay of the probability of acceptance with increasing discrepancy. Lower facilitation f gives an initially steeper downward slope. When facilitation is maximal 1 / f = 0, or f = ∞, however, P (i, j) is constant at one for discrepancies smaller than or equal to $$\\varepsilon$$. The default influence probability is the minimal probability of acceptance. Thus, for $$\\beta$$ > 0 there is always a certain chance that i is influenced by j. For the simulation run shown in Panel (c) of Figure 1, for instance, we assumed that $$\\beta$$ = 0.001. Thus, we have extended the bounded-confidence assumption to a functional form of a probability of acceptance with three parameters $$\\varepsilon$$, $$f$$ and $$\\beta$$. For $$\\beta$$= 0 and maximal facilitation Equation 1 turns into a step function identical to the model of (Hegselmann & Krause 2002).\n\nThe update of the opinion of agent i is modelled as follows\n\n $$x_{i}(t+1) = \\frac{1}{n}\\sum_{j'}x_{j'}(t)$$ (2)\nwhere n is the number of agents $$j'$$who are accepted by i through decisions based on the probability of acceptance. This implements the notion that an agent checks all other agents for inclusion in its confidence set as assumed by Hegselmann & Krause (2002). Therefore, we call this model the generalized Hegselmann-Krause model.\n\nFor $$\\varepsilon$$ = 1 and $$\\beta$$ = 0, the definition of P is identical to the definition in the seminal model in social psychology of Fishbein & Ajzen (1975) from which also the concept “probability of acceptance” is adopted.\n\nThe second model ingredient are random opinion replacements. We implemented that agents adopt a randomly picked opinion with probability m. The new opinion is a random value in the range of [0,1] and is drawn from a uniform distribution (Pineda et al. 2009, 2013; Carro et al. 2013). The rate of random opinion influx is given by $$m\\cdot N$$ per N time steps. This means that $$m\\cdot N$$ independent opinion renewals would occur on average during N interdependent opinion renewals, hence m gives the ratio, or relative importance, of independent opinion replacement.\n\nIn a nutshell, the simulation program proceeds as follows. In the iteration from time step t to t + 1, the program decides for each individual agent i whether each of the other agents exerts influence on i by random events with respect to the probability of acceptance. Next, the program computes the updated opinion of agent i at time step t + 1 as the average of all those agents that have been selected to exert influence. Next, the program decides through a random event with respect to the probability of random opinion replacements whether this updated opinion is replaced by a completely new, randomly drawn opinion. The order in which agents are updated does not matter in this process, because we always use the opinions from time step t to compute the opinion for time step t + 1 (synchronous updating).\n\nIn the following we assumed one thousand randomly distributed agents along the entire opinion axis for the initial state (N = 1000). Effects of di erent initial conditions (i.e., hysteresis problems) are not discussed here.\n\n### Results\n\nFigure 2 shows three ideal-typical opinion dynamics to illustrate the effects of random opinion changes on opinion dynamics. In the figures, the agent distribution is shown as the probability density, which is estimated by the kernel density estimation with the bandwidth of 0.005. For all three simulation runs, we assigned positive values to parameters $$\\beta$$ and f, implementing that there is always a small but positive probability that an agent j exerts influence on i’s opinion. When agents’ opinions only depend on social influence and there are no random opinion changes (m = 0), then this parameter setting implies that dynamics will always generate opinion consensus. Panel (a) of Figure 2 shows that social influence leads to the formation of clusters. However, as there is always a positive chance that members of distinct clusters exert influence on each other, clusters merge and the population reaches a perfect opinion consensus.\n\nPanel (b) of Figure 2, in contrast, shows that including random opinion replacements (m = 0.1) leads to a very different outcome. Like in the dynamics shown in Panel (a), social influence leads to the formation of clusters of agents with similar opinions. However, these clusters remain stable. This result is obtained even though there is always a positive probability of acceptance and even though the opinion replacements are entirely random.\n\nThe intuition is simple. Social influence leads to the formation of clusters, but due to the random opinion replacements, clusters also loose members from time to time. These agents will first adopt a random opinion but will sooner or later join one of the clusters. Thus, all clusters have an input and an output of agents and remain stable when in and output are balanced. To be sure, it is possible that a cluster happens to disappear in this dynamic. However, randomness in tandem with social influence will lead to the formation of a new cluster (Pineda et al. 2009, 2013; Carro et al. 2013) and the collective pattern of opinion clustering will reemerge.\n\nPanel (c) of Figure 2 shows that clusters disappear, however, when random opinion replacements happen too frequently. For this run, we imposed that opinions adopt random values with a probability of 0.5. Under this condition, social influence is too weak to generate stable clustering.", null, "Figure 2. Time evolution of normalized agents density for three different magnitudes of random opinion replacement.\n\nFigures 3 and 4 extend the observations from the three ideal-typical runs to broader parameter spaces, using opinion pattern diagrams to visualize aggregated agent-distributions for quasi-steady states (except for the Panel (a) of Figure 4, as we discuss below) in twenty independent simulation runs per parameter combination. The number of simulation events (Tmax) required for the quasi-steady states greatly depends on parameter settings, hence we adjusted Tmax for each Panel empirically to save computation time.\n\nFigure 3 focuses on the original implementation of confidence bounds, where the probability of acceptance is either zero or one (see Panel (a)). Panel (b), thus, replicates the findings from earlier studies with the original bounded-confidence model (Lorenz 2007), showing that the bounded-confidence model can explain the emergence and stability of opinion clustering when agents’ confidence is sufficiently bounded. When $$\\varepsilon$$ exceeds a value of 0.23, agents arrive at consensus, forming one big cluster close to the center of the opinion space. However, agents form multiple clusters, when they have narrower bounds of confidence ($$\\varepsilon$$ < 0.23). These clusters are stable because the probability that an agent is influenced by an agent from another cluster is zero. The number of clusters forming is roughly $$1/(2\\varepsilon)$$ (e.g., Deffuant et al. 2000; Ben-Naim et al. 2003). Note that Panel (b) fails to visualize the emergent clustering for very small values of $$\\varepsilon$$, as the model generates a large number of opinion clusters with only small (but stable) opinion differences between clusters. When $$\\varepsilon$$ = 0.05, for instance, dynamics typically settle when about eight distinct clusters have formed. Random differences in the initial opinion distributions of the twenty realizations per parameter combination, however, lead to small differences in the exact positions of the clusters between simulation runs. As a consequence, the differences between clusters are not visible in the aggregated opinion distributions shown in Panel (b) of Figure 3.\n\nThe predictions of the standard bounded-confidence model are robust to including random opinion replacements (m = 0.1), as Panel (c) of Figure 3 shows. When agents sometimes happen to adopt a random opinion (drawn from a uniform distribution), clustering can be stable when the bounds of confidence are sufficiently small.\n\nHowever, the ability of the bounded-confidence model to generate stable clustering breaks down when one slightly relaxes the bounded-confidence assumption in that one allows also agents with otherwise to divergent opinions to exert influence on each other with a small probability of $$\\beta$$ = 0.001. Demonstrating this, Panel (d) of Figure 3 shows that the initial opinion diversity decreases and the agent populations arrive at a consensus, even when agents have very narrow bounds of confidence.\n\nStrikingly, the ability of the bounded-confidence model to explain clustering is regained when both forms of randomness are included (see Panel (e) of Figure 3). That is, when agents sometimes adopt random opinion values ($$\\beta$$ = 0.001) and when random violations of the bounded-confidence assumption are included (m = 0.1), opinion clustering remains stable when agents’ confidence in others is sufficiently bounded.", null, "Figure 3. Opinion pattern diagrams for the original Hegselmann-Krause model with bounded confidence. The studied parameter range is 0 ≤ $$\\varepsilon$$ ≤ 0.5 and the resolution is 0.01.\n\nFigure 4 shows the same analyses as Figure 3 but focuses on the less restrictive interpretation of the bounded-confidence assumption, where the probability of acceptance is not modelled as a step function. To generate these figures, we assumed that $$\\varepsilon$$ = 1 and varied parameter f (see Panel (a)). Accordingly, the x-axis represents parameter f rather then $$\\varepsilon$$.\n\nPanel (b) of Figure 4 focuses on the condition without random opinion changes (m = 0) and without random deviations from the bounded-confidence rule ($$\\beta$$ = 0). It is important to understand that the opinion variance shown in the left part of Panel (b) is an artifact of our decision to run simulations for “only” 5000 simulation events and of floating point precision close to zero. Even under small facilitation values the probability of acceptance is always positive, which implies that populations will always reach a consensus, because every possible interaction will continue to occur ad infinitum although quite rarely. However, small f values lead to probabilities of acceptance that are so small that agents with distant opinions hardly influence each other or are even represented by zero due to floating point imprecision. As a consequence, consensus is not reached with in the limit of 5000 simulation events and will perhaps not even do without technical modifications dealing with floating point precision.\n\nIn order to test whether our intuition that the model generates consensus also for small facilitation values, we ran twenty additional simulations with f = 0.05 and without a predefined simulation end (m = 0, $$\\beta$$ = 0). Supporting our intuition, we observed that all twenty populations generated a consensus. On average, it took systems approximately 30,000 iterations to decrease the value range of opinions to less then 10-10.\n\nPanel (d) of Figure 4 is based on the same parameter conditions as Panel (b) but adds a small default acceptance probability of $$\\beta$$ = 0.001, which excludes the artifact observed in Panel (b). Accordingly, Panel (d) shows that all simulation runs generated consensus, independent of the degree f of facilitation. This shows again that the bounded-confidence model fails to generate opinion clustering when the bounded-confidence assumption is implemented in a less restrictive way than in the original contributions.\n\nHowever, system behaviour differs very much when random opinion replacements are included (m = 0.1), as Panels (c) and (e) of Figure 4 show. In particular, Panel (e) shows that the model generates phases of opinion clustering even though there is always a positive probability of acceptance and even though independent opinion changes are added.\n\nIn order to demonstrate that the opinion clustering observed in the simulation runs with random opinion replacements (Panels (c) and (e) of Figure 4) is not an artifact of a limited duration of the simulations and/or floating point issues, we reran the simulation experiment for these experimental treatments assuming that all agents held the same opinion at the outset of the dynamics (xi (0) = 0.5 for all i). In Appendix B we show that the resulting opinion pattern diagrams are virtually identical to those generated in the main simulation exper-iment where simulations departed from a uniform opinion distribution. This shows that the opinion diversity shown in Panels (c) and (e) of Figure 4 also emerges when dynamics start from perfect consensus, showing that these results are not artifacts.\n\nSo far, we focused on the bounded-confidence model developed by Hegselmann & Krause (2002) which assumes that agents consider the opinions of multiple sources when they update their opinions. The alternative bounded-confidence model by Deffuant et al. (2000), in contrast, assumed that influence is dyadic in the sense that agents always consider only the opinion of one other agent for opinion update. We focused our analyses on the Hegselmann-Krause model mainly because it is more similar to the original work of Abelson (1964). We demonstrated in Appendix , however, that our conclusions also hold under the dyadic influence regime proposed by Deffuant et al. (2000).", null, "Figure 4. Opinion pattern diagrams for the generalized Hegselmann-Krause model with a smooth acceptance function. The studied parameter range is f ≥ 0.01 and the resolution is 0.03 in terms of $$log_{10}\\frac{1}{f}$$, which gives finer resolution for smaller f so that we are able to keep better resolution in regions with smaller structures and are able to save the computation cost simultaneously. For example, it corresponds to $$\\sim0.0014$$ in terms of f at 0.02, while $$\\sim0.0067$$ at f = 0.1.\n\nWe provide a NetLogo implementation of our model which allows readers to produce trajectories of all model variations presented here (Lorenz et al. 2016) . The reader can use it to observe and explore our claims on the basis of independent simulation runs. The model also includes “example buttons”, which set the model parameters to the values studied in our examples.\n\n### Summary and conclusions\n\nBounded-confidence models proposed the most prominent solution to Abelson’s puzzle of explaining opinion clustering in connected networks where nodes exert positive influence on each other. However, the models have been criticized for being able to generate opinion clustering only when the assumption that agents’ confidence in others is bound is being interpreted in a maximally strict sense. When one allows agents to sometimes deviate from the bounded-confidence assumption, clustering breaks down even when these deviations are rare and random.\n\nEven though we echoed the criticism that the predictions of the original bounded-confidence models are not robust to random deviations, we showed the models’ ability to explain clustering can be regained if another typical assumption of social-influence models is relaxed. Building on modeling advances from the field of statistical mechanics (Pineda et al. 2009, 2013; Carro et al. 2013), we showed that a bounded-confidence model that takes into account random deviations from the bounded-confidence assumption is able to explain opinion clustering when one includes that agents’ opinion are not only affected by influence from other agents. If, in addition, opinions can change in a random fashion, clustering can emerge and remain stable. Thus, our results demonstrate that the bounded-confidence model does provide an important answer to Abelson’s puzzle, despite the criticism.\n\nOur findings illustrate that in complex systems seemingly innocent events can have profound effects, even when these events are rare and random. One important way to deal with the problem is to base models on empirically validated assumptions. With regard to the bounded-confidence model, however, too little is known about how bounded individuals’ confidence actually is. How is this boundedness distributed in our societies and under what conditions can bounds shift and increase or decrease openness to distant opinions? Under which conditions do individuals deviate from the bounded-confidence assumption? Are these deviations random or do they follow certain patterns? Our finding that deviations have a decisive effect on the predictions of bounded-confidence models shows that empirical answers to these questions are urgently needed.\n\nWhat is more, empirical research is needed to inform models about how to formally incorporate deviations from model assumptions. For instance, we adopted from earlier modeling work (Pineda et al. 2009, 2013; Carro et al. 2013) the assumption that agents sometimes adopt a random opinion and that this random opinion is drawn from a uniform distribution. This is certainly a plausible model of turnover dynamics, where individuals may happen to leave a social group or organization, making space for a replacement with an independent opinion. However, this model of deviations appears to be a less plausible representation of other influences, such as social influences from outside the modelled population or the empirically well documented striving for unique opinions (Mäs et al. 2010). For instance, deviations have also been modelled as “white noise”. That is, rather than assigning a new, random value to the agent’s opinion, a random value drawn from a normal distribution with an average of zero was added to agents’ opinions (Mäs et al. 2010). This “white noise” leads on average to much smaller random opinion changes, which generates very different opinion dynamics, as it does not make agents leave and enter clusters as observed with the uniformly distributed noise. In contrast, white-noise deviations at the level of individual agents aggregate to random opinion shifts of clusters. As a consequence, it is possible that clusters happen to adopt similar opinions and merge, a process that will inevitably generate global consensus and not clustering as observed with uniformly distributed noise (Mäs et al. 2010). In sum, deviations from model assumptions can be incorporated in different ways and model predictions often depend on the exact formal implementation. Theoretical and empirical research is, therefore, needed to identify conditions under which alternative models of deviations are more or less plausible. Empirical research in the field of evolutionary game theory has shown, for instance, that deviations tend to occur more likely when they imply small costs (Mäs & Nax 2016). Similar research should be conducted also in the field of social-influence dynamics.\n\nThe bounded confidence model is yet another example of a theory that makes fundamentally different predictions when its assumptions are implemented probabilistically rather than deterministically (Macy & Tsvetkova 2015). It is, therefore, important that theoretical predictions, independent of whether they have been derived formally or not, are tested for robustness to randomness. To be sure, there is no doubt that deterministic models are insightful, as they allow the researcher to analyze theories in a clean environment without any perturbations. This often simplifies analyses, making it easier to understand the model’s mechanisms and their consequences. Nevertheless, a model prediction that only holds for deterministic settings is of limited scientific value, because it cannot be to put to the empirical test in a non-deterministic world. The notion that individual behaviour is deterministic may be useful but it is not an innocent assumption.\n\nIn sum, we studied a new answer to Abelson’s question of how to generate the empirical finding of bimodal opinion distributions. Abelson’s approach to assume that influence declines with opinion discrepancy is supported by empirical research but turned out to not solve his puzzle. The original bounded-confidence models were able to explain opinion clustering by assuming that the influence network is not fixed, and by further strengthening Abelson’s assumption, implementing a threshold in discrepancy beyond which influence vanishes completely. We showed here that Abelson’s weaker implementation of influence declining in discrepancy does generate opinion clustering when it is combined with random independent opinion replacements.\n\n### Notes\n\n1. Abelson’s diversity puzzle is called “community cleavage problem” by Friedkin (2015).\n2. This observation was later generalized by Davis (1996).\n3. Formally opinion discrepancy is nothing else than distance in opinion and this notion is usually used in the social simulation literature. We use \"discrepancy\" here because it is the original notion used by Abelson and well established in psychological models (see e.g. Hunter et al. 1984). In this paper discrepancy is the same as distance, which is also the case for a large part of the psychological literature.\n4. Another criticism was that there are no “hard” bounds but “smooth” ones. Therefore, already some early variations of bounded confidence models on the evolution of extremism (Deffuant et al. 2002, 2004) introduced different kinds of smoother bounds. Both studies did not analyse the impact of smoothness itself.\n5. Mäs & Bischofberger (2015) use the same function for the definition of similarity and interaction probability. They use a parameter h = 1/f quantifying the degree of homophily in the society. In our parametrization they further used $$\\beta$$ = 0 and $$\\varepsilon$$ = 2 for an opinion space where opinion can take values between -1 and 1.\n6. Based on Ben-Naim et al. (2003) these diagrams are called “bifurcation diagrams” in the physics literature e.g. by (Lorenz 2007; Pineda et al. 2009, 2013; Carro et al. 2013).\n\n### Appendix\n\n#### Appendix A: Model based on Deffuant et al\n\nThe conclusions drawn from our analyses do not only apply to the bounded-confidence model proposed by Hegselmann & Krause (2002) but also hold for the model by Deffuant et al. (2000), as Figures 5 and 6 show. In this alternative model, the agent-interaction scheme is divided into two subprocesses. First, two agents i and j are randomly picked. Second i and j influence each other and adjust their opinions with a given probability of acceptance P if their opinions do not differ too much.\n\nWhen agents i and j are selected for interaction and accept each other, they adjust their opinions towards each other by averaging their current opinions. This leads to the following dynamic equation\n\n $$x_{i}(t+1) = \\begin{cases} \\displaystyle{\\frac{x_{i}(t)+x_{j}(t)}{2}} & \\text{with a probability } P(i,j) \\\\ \\\\ x_{i}(t) & \\text{otherwise.} \\end{cases}$$ (3)", null, "Figure 5. Opinion pattern diagrams for the original Deffuant et al. model with bounded confidence.", null, "Figure 6. Opinion pattern diagrams for the generalized Deffuant et al. model with a smooth acceptance function.\n\n#### Appendix B: Opinion patterns\n\nThe experiments described in the main text cannot exclude the possibility that the opinion patterns shown in Panels (c) and (e) of Figures 3 and 4 also result from the fact that we did not run simulations long enough. To ensure that the diagrams in those Panels are not suffering from insuficient simulation length, we conducted additional simulations that started with a perfect consensus (i.e., all agents have an opinion of 0.5 at the initial state). The additional simulations resulted in very similar general opinion patterns (Figure 7), which shows that our discussion in this study is valid. This holds for both version of the bounded-confidence model.\n\nPanel (a) of Figure 7 should be compared to Panel (c) of Figure 4 and Panel (c) of Figure 7 should be compared to Panel (e) of Figure 4. Likewise, Panel (b) of Figure 7 should be compared to Panel (c) of Figure 3 and Panel (d) of Figure 7 should be compared to Panel (e) of Figure 3.", null, "Figure 7. Opinion patterns emerging when dynamics start from perfect consensus at an opinion value of 0.5.\n\n### References\n\nABELSON, R. P. (1964). Mathematical models of the distribution of attitudes under controversy. In Contributions to mathematical psychology, (pp. 142–160). Holt, Rinehart & Winston, New York.\n\nARONSON, E., Turner, J. A. & Carlsmith, J. M. (1963). Communicator credibility and communication discrepancy as determinants of opinion change. The Journal of Abnormal and Social Psychology, 67(1), 31–36. [doi:10.1037/h0045513]\n\nAXELROD, R. (1997). The dissemination of culture. Journal of Conflict Resolution, 41(2), 203–226. [doi:10.1177/0022002797041002001]\n\nBEN-NAIM, E., Krapivsky, P. L. & Redner, S. (2003). Bifurcation and patterns in compromise processes. Physica D, 183, 190–204. [doi:10.1016/S0167-2789(03)00171-4]\n\nCARRO, A., Toral, R. & San Miguel, M. (2013). The role of noise and initial conditions in the asymptotic solution of a bounded confidence, continuous-opinion model. Journal of Statistical Physics, 151(1), 131–149. [doi:10.1007/s10955-012-0635-2]\n\nDANDEKAR, P., Goel, A. & Lee, D. T. (2013). Biased assimilation, homophily, and the dynamics of polarization. Proceedings of the National Academy of Sciences, 110(15), 5791–5796. [doi:10.1073/pnas.1217220110]\n\nDAVIS, J. H. (1996). Group decision making and quantitative judgments: A consensus model. In E. H. Witte & J. H. D. Mahwah (Eds.), Understanding Group Behavior: Consensual Action by Small Groups, (pp. 35–59). Lawrence Erlbaum.\n\nDEFFUANT, G., Amblard, F. & Weisbuch, G. (2004). Modelling Group Opinion Shift to Extreme: The Smooth Bounded Confidence Model. Arxiv preprint cond-mat/0410199: https://arxiv.org/ftp/cond-mat/papers/0410/0410199.pdf.\n\nDEFFUANT, G., Neau, D., Amblard, F. & Weisbuch, G. (2000). Mixing beliefs among interacting agents. Advances in Complex Systems, 3, 87–98. [doi:10.1142/S0219525900000078]\n\nDEFFUANT, G., Neau, D., Amblard, F. & Weisbuch, G. (2002). How Can Extremism Prevail? A Study Based on the Relative Agreement Interaction Model. Journal of Artificial Societies and Social Simulation, 5(4), 1: http://jasss.soc.surrey.ac.uk/5/4/1.html.\n\nDEGROOT, M. H. (1974). Reaching a consensus. Journal of the American Statistical Association, 69(345), 118–121. [doi:10.1080/01621459.1974.10480137]\n\nFISHBEIN, M. & Ajzen, I. (1975). Belief, attitude, intention and behavior: An introduction to theory and research. Reading, MA: Addison-Wesley.\n\nFISHER, S. & Lubin, A. (1958). Distance as a determinant of influence in a two-person serial interaction situation. The Journal of Abnormal and Social Psychology, 56(2), 230–238. [doi:10.1037/h0044609]\n\nFLACHE, A. & Mäs, M. (2008). How to get the timing right. A computational model of the effects of the timing of contacts on team cohesion in demographically diverse teams. Computational & Mathematical Organization Theory, 14(1), 23–51. [doi:10.1007/s10588-008-9019-1]\n\nFRIEDKIN, N. E. (2015). The problem of social control and coordination of complex systems in sociology: A look at the community cleavage problem. IEEE Control Systems, 35(3), 40–51. [doi:10.1109/MCS.2015.2406655]\n\nHARARY, F. (1959). A criterion for unanimity in French’s theory of social power. Studies in social power, 6, 168.\n\nHEGSELMANN, R. & Krause, U. (2002). Opinion dynamics and bounded confidence, models, analysis and simulation. Journal of Artificial Societies and Social Simulation, 5(3), 2: http://jasss.soc.surrey.ac.uk/5/3/2.html.\n\nHUNTER, J. E., Danes, J. E. & Cohen, S. H. (1984). Mathematical Models of Attitude Change. Human communication research series, Academic Press.\n\nLAZARSFELD, P. F. & Merton, R. K. (1954). Friendship as a social process: A substantive and methodological analysis. In M. Berger & T. Abel (Eds.), Freedom and control in modern society, (pp. 18–66). Van Nostrand.\n\nLEHRER, K. & Wagner, C. (1981). Rational Consensus in Science and Society. D. Reidel Publishing Company, Dordrecht, Holland. [doi:10.1007/978-94-009-8520-9]\n\nLIGGETT, T. M. (2013). Stochastic interacting systems: contact, voter and exclusion processes, vol. 324. Springer Science & Business Media.\n\nLORENZ, J. (2007). Continuous Opinion Dynamics under bounded confidence: A Survey. International Journal of Modern Physics C, 18(12), 1819–1838. [doi:10.1142/S0129183107011789]\n\nLORENZ, J., Kurahashi-Nakamura, T. & Mäs, M. (2016). ContinuousOpinionDynamicsNetlogo: Robust clustering in generalized bounded confidence models. http://doi.org/10.5281/zenodo.61205.\n\nMACY, M. W., Kitts, J. A., Flache, A. & Benard, S. (2003). Polarization in dynamic networks: A Hopfield model of emergent structure. In Dynamic Social Network Modeling and Analysis, (pp. 162–173). National Academies Press.\n\nMACY, M. & Tsvetkova, M. (2015). The signal importance of noise. Sociological Methods & Research, 44, 2, 306-332.\n\nMARK, N. P. (2003). Culture and competition: Homophily and distancing explanations for cultural niches. American Sociological Review, 68(3), 319–345. [doi:10.2307/1519727]\n\nMÄS, M. & Bischofberger, L. (2015). Will the personalization of online social networks foster opinion polarization? SSRN. Available at SSRN: http://ssrn.com/abstract=2553436.\n\nMÄS, M. & Flache, A. (2013). Differentiation without distancing. Explaining bi-polarization of opinions without negative influence. PLoS ONE, 8(11), e74516. [doi:10.1371/journal.pone.0074516]\n\nMÄS, M., Flache, A. & Helbing, D. (2010). Individualization as driving force of clustering phenomena in humans. PLoS Comput Biol, 6(10), e1000959. [doi:10.1371/journal.pcbi.1000959]\n\nMÄS, M., Flache, A., Takacs, K. & Jehn, K. A. (2013). In the short term we divide, in the long term we unite: Demographic crisscrossing and the effects of fault lines on subgroup polarization. Organization science, 24(3), 716–736. [doi:10.1287/orsc.1120.0767]\n\nMÄS, M. & Nax, H. H. (2016). A behavioral study of “noise” in coordination games. Journal of Economic Theory, 162, 195–208. [doi:10.1016/j.jet.2015.12.010]\n\nMCPHERSON, M., Smith-Lovin, L. & Cook, J. M. (2001). Birds of a feather: Homophily in social networks. Annual Review of Sociology, 27, 415–444. [doi:10.1146/annurev.soc.27.1.415]\n\nPINEDA, M., Toral, R. & Hernandez-Garcia, E. (2009). Noisy continuous-opinion dynamics. Journal of Statistical Mechanics: Theory and Experiment, 2009(08), P08001 (18pp).\n\nPINEDA,, M., Toral, R. & Hernández-García, E. (2013). The noisy Hegselmann-Krause model for opinion dynamics. The European Physical Journal B, 86(12), 1–10. [doi:10.1140/epjb/e2013-40777-7]\n\nSALZARULO, L. (2006). A continuous opinion dynamics model based on the principle of meta-contrast. Journal of Artificial Societies and Social Simulation, 9(1) 3: http://jasss.soc.surrey.ac.uk/9/1/13.html.\n\nSHIN, J. K. & Lorenz, J. (2010). Tipping diffusivity in information accumulation systems: More links, less consensus. Journal of Statistical Mechanics: Theory and Experiment, 2010(06), P06005. [doi:10.1088/1742-5468/2010/06/p06005]\n\nWATTS, D. J. & Dodds, P. S. (2007). Influentials, networks, and public opinion formation. Journal of Consumer Research, 34, 441–458. [doi:10.1086/518527]" ]	[ null, "http://jasss.soc.surrey.ac.uk/gifs/pdf-icon.png", null, "http://jasss.soc.surrey.ac.uk/gifs/open-access-logo120.jpg", null, "http://jasss.soc.surrey.ac.uk/gifs/download.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure1.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure2.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure3.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure4.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure5.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure6.png", null, "http://jasss.soc.surrey.ac.uk/19/4/7/Figure7.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8873123,"math_prob":0.89029825,"size":37658,"snap":"2019-51-2020-05","text_gpt3_token_len":8547,"char_repetition_ratio":0.15881447,"word_repetition_ratio":0.039333805,"special_character_ratio":0.23742631,"punctuation_ratio":0.1475882,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9726355,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],"im_url_duplicate_count":[null,null,null,null,null,null,null,1,null,1,null,1,null,1,null,1,null,1,null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-28T00:10:43Z\",\"WARC-Record-ID\":\"<urn:uuid:aca252c3-6f94-41f3-b4ae-c82c08c84929>\",\"Content-Length\":\"72091\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:33aa3df8-5db8-48c5-9a6d-1c729e281cb9>\",\"WARC-Concurrent-To\":\"<urn:uuid:6bb5fded-f125-492a-b4de-a426ffc5475e>\",\"WARC-IP-Address\":\"35.177.28.97\",\"WARC-Target-URI\":\"http://jasss.soc.surrey.ac.uk/19/4/7.html\",\"WARC-Payload-Digest\":\"sha1:4UBNCPVCFBXCWFT6CACGO2JK2CPOKG3O\",\"WARC-Block-Digest\":\"sha1:DQHWOQPVZ6Q7PFEHLGVSCCWAKDZPTTZK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579251737572.61_warc_CC-MAIN-20200127235617-20200128025617-00169.warc.gz\"}"}
https://freakonometrics.hypotheses.org/12081	[ "# Triangle for Parameters of AR(2) Stationary Processes\n\nWe’ve seen yesterday conditions on so that the canonical process, , satisfying\n\nThe condition is rather simple, since should be a triangular region. But the proof is a bit more tricky…\n\nRecall that we want to parametrize the region\n\nSince we have a true process, then . Our polynomial is here\n\nwhere ‘s are the roots – in – of . Consider now some kind of dual version of that polynomial,\n\nHaving the roots of outside the unit circle is the same as having the roots of inside the unit circle. Obserse that we can write\n\nRoots of are then\n\nFrom this point, we should discuss a little bit, depending on the value of .\n\n• if\n\nThen there is one root, and only one. So we need to have or equivalently .\n\n• if\n\nThen we got roots in , and\n\nmeans, equivalently, that\n\n• if\n\nThen we have two (conjugate) roots in , and the square of norm of those roots is . Thus, .\n\nWe get what was mention in the course: the canonical has a stationary solution if, and only if\n\nwhich is a triangular region, see", null, "## 2 thoughts on “Triangle for Parameters of AR(2) Stationary Processes”\n\n1.", null, "dong says:\n\nthere is a little typo of $\\phi_2$ on the parameter set\n\n2.", null, "Adil says:\n\nCan you explain more of how you arrived at “some kind of dual version of that polynomial”?\n\nAlso, should the signs be – instead of + in that dual version? If not, I’m not sure how factoring out \\phi_{2} flips the signs here.\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]	[ null, "http://freakonometrics.hypotheses.org/files/2014/01/AR2-simulation-triangle.gif", null, "https://secure.gravatar.com/avatar/327141b0e2c007396c838966f3a0ddcc", null, "https://secure.gravatar.com/avatar/cfc7e472fb5746df86f749223d83163c", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9260037,"math_prob":0.97573745,"size":1320,"snap":"2023-40-2023-50","text_gpt3_token_len":321,"char_repetition_ratio":0.10106383,"word_repetition_ratio":0.015936255,"special_character_ratio":0.23863636,"punctuation_ratio":0.10989011,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97617835,"pos_list":[0,1,2,3,4,5,6],"im_url_duplicate_count":[null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-04T19:05:11Z\",\"WARC-Record-ID\":\"<urn:uuid:68e8c699-e0f1-498b-b6d2-433d0b8ee607>\",\"Content-Length\":\"200720\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:30e33329-08cb-4543-9540-28ca5884e156>\",\"WARC-Concurrent-To\":\"<urn:uuid:b3b45c24-4a98-4151-864c-aa4381c7bb85>\",\"WARC-IP-Address\":\"134.158.39.132\",\"WARC-Target-URI\":\"https://freakonometrics.hypotheses.org/12081\",\"WARC-Payload-Digest\":\"sha1:CIJHH5T4LIDF47ONMMU5KHQ5IGQ66SIC\",\"WARC-Block-Digest\":\"sha1:7N6XMPIPEB6NSOES6AFLGH33GGIQA7LN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-50/CC-MAIN-2023-50_segments_1700679100534.18_warc_CC-MAIN-20231204182901-20231204212901-00439.warc.gz\"}"}
https://byjus.com/rd-sharma-solutions/class-8-maths-chapter-7-factorization-exercise-7-1/	[ "", null, "# RD Sharma Solutions for Class 8 Chapter - 7 Factorization Exercise 7.1\n\n### RD Sharma Class 8 Solutions Chapter 7 Ex 7.1 PDF Free Download\n\nOur expert team has designed the solutions for RD Sharma Class 8 Maths Chapter 7 to help students prepare for their exams at ease. RD Sharma Solutions is one of the best reference material for CBSE students. Learners can download the pdf from the links provided below. Experts suggest students practice the solutions many numbers of times to yield good results in their exams. In Exercise 7.1 of Chapter 7 Factorization, we shall discuss the basic definitions for factors, factorization and we also find the common factors and greatest common factor of the monomial.\n\n## Download the pdf of RD Sharma For Class 8 Maths Exercise 7.1 Chapter 7 Factorization", null, "", null, "", null, "", null, "### Access other Exercises of RD Sharma Solutions for Class 8 Maths Chapter 7 Factorization\n\nExercise 7.2 Solutions\n\nExercise 7.3 Solutions\n\nExercise 7.4 Solutions\n\nExercise 7.5 Solutions\n\nExercise 7.6 Solutions\n\nExercise 7.7 Solutions\n\nExercise 7.8 Solutions\n\nExercise 7.9 Solutions\n\n### Access Answers to RD Sharma Solutions for Class 8 Maths Exercise 7.1 Chapter 7 Factorization\n\nFind the greatest common factor (GCF/HCF) of the following polynomials: (1-14)\n\n1. 2x2 and 12x2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 2 and 12\n\nThe greatest common factor of 2 and 12 is 2\n\nThe common literals appearing in given monomial is x\n\nThe smallest power of x in two monomials is 2\n\nThe monomial of common literals with smallest power is x2\n\n∴ The greatest common factor = 2x2\n\n2. 6x3y and 18x2y3\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 6 and18\n\nThe greatest common factor of 6 and 18 is 6\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in three monomial is 2\n\nSmallest power of y in three monomial is 1\n\nMonomial of common literals with smallest power is x2y\n\n∴ The greatest common factor = 6x2y\n\n3. 7x, 21x2 and 14xy2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 7, 21 and 14\n\nGreatest common factor of 7, 21 and 14 is 7\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in three monomials is 1\n\nSmallest power of y in three monomials is 0\n\nMonomials of common literals with smallest power is x\n\n∴ The greatest common factor = 7x\n\n4. 42x2yz and 63x3y2z3\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 42 and 63.\n\nGreatest common factor of 42, 63 is 21.\n\nCommon literals appearing in given numerical are x, y and z\n\nSmallest power of x in two monomials is 2\n\nSmallest power of y in two monomials is 1\n\nSmallest power of z in two monomials is 1\n\nMonomials of common literals with smallest power is x2yz\n\n∴ The greatest common factor = 21x2yz\n\n5. 12ax2, 6a2x3 and 2a3x5\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 12, 6 and 2\n\nGreatest common factor of 12, 6 and 2 is 2.\n\nCommon literals appearing in given numerical are a and x\n\nSmallest power of x in three monomials is 2\n\nSmallest power of a in three monomials is 1\n\nMonomials of common literals with smallest power is ax2\n\n∴ The greatest common factor = 2ax2\n\n6. 9x2, 15x2y3, 6xy2 and 21x2y2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 9, 15, 16 and 21\n\nGreatest common factor of 9, 15, 16 and 21 is 3.\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in four monomials is 1\n\nSmallest power of y in four monomials is 0\n\nMonomials of common literals with smallest power is x\n\n∴ The greatest common factor = 3x\n\n7. 4a2b3, -12a3b, 18a4b3\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 4, -12 and 18.\n\nGreatest common factor of 4, -12 and 18 is 2.\n\nCommon literals appearing in given numerical are a and b\n\nSmallest power of a in three monomials is 2\n\nSmallest power of b in three monomials is 1\n\nMonomials of common literals with smallest power is a2b\n\n∴ The greatest common factor = 2a2b\n\n8. 6x2y2, 9xy3, 3x3y2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 6, 9 and 3\n\nGreatest common factor of 6, 9 and 3 is 3.\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in three monomials is 1\n\nSmallest power of y in three monomials is 2\n\nMonomials of common literals with smallest power is xy2\n\n∴ The greatest common factor = 3xy2\n\n9. a2b3, a3b2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 0\n\nCommon literals appearing in given numerical are a and b\n\nSmallest power of a in two monomials = 2\n\nSmallest power of b in two monomials = 2\n\nMonomials of common literals with smallest power is a2b2\n\n∴ The greatest common factor = a2b2\n\n10. 36a2b2c4, 54a5c2, 90a4b2c2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 36, 54 and 90\n\nGreatest common factor of 36, 54 and 90 is 18.\n\nCommon literals appearing in given numerical are a, b and c\n\nSmallest power of a in three monomials is 2\n\nSmallest power of b in three monomials is 0\n\nSmallest power of c in three monomials is 2\n\nMonomials of common literals with smallest power is a2c2\n\n∴ The greatest common factor = 18a2c2\n\n11. x3, -yx2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 0\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in two monomials is 2\n\nSmallest power of y in two monomials is 0\n\nMonomials of common literals with smallest power is x2\n\n∴ The greatest common factor = x2\n\n12. 15a3, -45a2, -150a\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 15, -45 and 150\n\nGreatest common factor of 15, -45 and 150 is 15.\n\nCommon literals appearing in given numerical is a\n\nSmallest power of a in three monomials is 1\n\nMonomials of common literals with smallest power is a\n\n∴ The greatest common factor = 15a\n\n13. 2x3y2, 10x2y3, 14xy\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 2, 10 and 14.\n\nGreatest common factor of 2, 10 and 14 is 2.\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in three monomials is 1\n\nSmallest power of y in three monomials is 1\n\nMonomials of common literals with smallest power is xy\n\n∴ The greatest common factor = 2xy\n\n14. 14x3y5, 10x5y3, 2x2y2\n\nSolution:\n\nWe know that the numerical coefficients of given numerical are 14, 10 and 2.\n\nGreatest common factor of 14, 10 and 2 is 2.\n\nCommon literals appearing in given numerical are x and y\n\nSmallest power of x in three monomials is 2\n\nSmallest power of y in three monomials is 2\n\nMonomials of common literals with smallest power is x2y2\n\n∴ The greatest common factor = 2x2y2\n\nFind the greatest common factor of the terms in each of the following expressions:\n\n15. 5a4 + 10a3 – 15a2\n\nSolution:\n\nThe greatest common factor of the three terms is 5a2\n\n16. 2xyz + 3x2y + 4y2\n\nSolution:\n\nThe greatest common factor of the three terms is y\n\n17. 3a2b2 + 4b2c2 + 12a2b2c2\n\nSolution:\n\nThe greatest common factor of the three terms is b2." ]	[ null, "https://www.facebook.com/tr", null, 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https://www.physicsforums.com/threads/one-to-one-functions.134861/	[ "# One-to-one Functions.\n\n#### AznBoi\n\nHow do you solve one-to-one functions algebraically?\n\nProblem: f(x) (3x+4)/5\n\nWhat are you suppose to substitue for x?\n\nI know that if f(a)=f(b), a=b...\n\n#### Office_Shredder\n\nStaff Emeritus\nGold Member\nWhat do you mean solve?\n\nIf f(x) = (3x+4)/5, then you have your function. If you plug in an x, you can solve for f(x) by doing basic math. Given what f(x) is, you can do something like this:\n\n5*f(x) = 3x + 4\n\n5f(x) - 4=3x\n\n(5f(x)-4)/3=x\n\n#### AznBoi\n\nIt says: Determine algebraically whether the function is one-to-one. How do I do that?\n\n#### Office_Shredder\n\nStaff Emeritus\nGold Member\nOh, I get it. You want to show that if you plug two distinct values, x1 and x2 into f(x), the values that are returned either are not equal to each other, or x1 and x2 are the same (that's the definition of one to one)\n\n#### AznBoi\n\nSo do you plug 2 and -2. Their negatives? or what two numbers do you plug in. I plugged in 2 and -2 and I got 2 and -2/5. What am I doing wrong?\n\n#### Hurkyl\n\nStaff Emeritus\nGold Member\nAznBoi said:\nHow do I do that?\nJust do what it says:\n\nI know that if f(a)=f(b), a=b...\nStart by writing down f(a) = f(b).\nUse algebra to derive a = b.\n\nProof finished.\n\n#### berkeman\n\nMentor\nCan you use calculus? If so, you can show that the slope (derivative) of the function is always positive and bounded. That means that the function cannot double-back on itself to create a second y value for any x value.\n\nEven if you aren't supposed to use calculus, at least for this problem, the equation is the equation of a straight line, right? y = mx + b\n\nEDIT -- Ooo. I like Hurkyl's method better!\n\n#### AznBoi\n\nwhat do you mean? like f(a)=3a+4/5 f(b)=3b+4/5?? a will always equal b if you do it that way wouldn't it? No I can't use calculus, I'm in pre cal xD. Can you show me how to do it?\n\n#### Hurkyl\n\nStaff Emeritus\nGold Member\nAznBoi said:\nwhat do you mean? like f(a)=3a+4/5 f(b)=3b+4/5?? a will always equal b if you do it that way wouldn't it?\nIf you can prove what you just said, then you've proven f is one-to-one. It's that easy.\n\n#### AznBoi\n\nWhat about f(x)=x^2 It's not a one-to-one fucntion even though f(a)=a^2 is equal to f(b)=b^2 a=b in that case but if you put -2 and 2 in f(a)=f(b) but a doesn't equal b.. So I'm confused.\n\n#### AznBoi\n\nCan you give me some examples of functions that aren't one-by-one. I mean if you use the same function for x=a and b. aren't they alwasy equal to each other?\n\n#### Office_Shredder\n\nStaff Emeritus\nGold Member\nFor x^2, if a^2=b^2, then a=-b means x^2 isn't necessarily one to one\n\n#### Hurkyl\n\nStaff Emeritus\nGold Member\nWhat about f(x)=x^2 It's not a one-to-one fucntion even though f(a)=a^2 is equal to f(b)=b^2 a=b in that case\nWhy do you think a=b in that case? You've demonstrated that's not always true... so think hard about why you would (incorrectly) believe a=b must be true here.\n\n#### AznBoi\n\nOkay I know that -2^2 and 2^2 are both equal to 4.. So that means you can't use a and b.. cause a^2 and b^2 would be a=b if you solve it algebrically. What numbers do I need to substitue for x? How do I know that a=b or a doesn't equal b. Thats what I'm trying to figure out. =P\n\n#### Hurkyl\n\nStaff Emeritus\nGold Member\ncause a^2 and b^2 would be a=b if you solve it algebrically.\n\n#### AznBoi\n\na^2=b^2 because you square root both sides and you get a=b?\n\n#### Hurkyl\n\nStaff Emeritus\nGold Member\na^2=b^2 because you square root both sides and you get a=b?\nNope. You get \|a\| = \|b\|.\n\n#### AznBoi\n\nSo basically anything that is to an even power is not a one-to-one function. ok this is weird. lol\n\n#### berkeman\n\nMentor\nAznBoi said:\nSo basically anything that is to an even power is not a one-to-one function. ok this is weird. lol\nJust for my peace of mind, could you please post the textbook definition of a one-to-one function? I think that my practical definition of a one-to-one function (any x maps to only one y) may not match what others are asking you to show.\n\n#### Data\n\nOne-to-one means IF f(x) = f(y) THEN x=y.\n\nSo think about f(x) = x^2. If f(x) = f(y) is it NECESSARILY true that x=y? If not, then f is not 1-1. If so, then f is 1-1.\n\n### The Physics Forums Way\n\nWe Value Quality\n• Topics based on mainstream science\n• Proper English grammar and spelling\nWe Value Civility\n• Positive and compassionate attitudes\n• Patience while debating\nWe Value Productivity\n• Disciplined to remain on-topic\n• Recognition of own weaknesses\n• Solo and co-op problem solving" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95175153,"math_prob":0.94218016,"size":1024,"snap":"2019-13-2019-22","text_gpt3_token_len":300,"char_repetition_ratio":0.12254902,"word_repetition_ratio":0.0,"special_character_ratio":0.28222656,"punctuation_ratio":0.11494253,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99768966,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-03-25T22:05:44Z\",\"WARC-Record-ID\":\"<urn:uuid:a9b78c16-29cc-4d31-8a2a-b7d9a7d87f14>\",\"Content-Length\":\"140101\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:262ed6d6-1d78-4e80-ae55-cdd524d16c0c>\",\"WARC-Concurrent-To\":\"<urn:uuid:74aa3566-bbe0-48de-b213-36d500d752b2>\",\"WARC-IP-Address\":\"23.111.143.85\",\"WARC-Target-URI\":\"https://www.physicsforums.com/threads/one-to-one-functions.134861/\",\"WARC-Payload-Digest\":\"sha1:L57LWRTPS5SEYQA6FKRIANFR5ZU5JF7W\",\"WARC-Block-Digest\":\"sha1:3CLV4XKLNE5SOGRNHGGN772KOL2YFYBY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-13/CC-MAIN-2019-13_segments_1552912204461.23_warc_CC-MAIN-20190325214331-20190326000331-00041.warc.gz\"}"}
https://forces-3-1-0.embotech.com/Documentation/solver_options/index.html	[ "# 12. Solver Options¶\n\nThe default solver options can be loaded when giving a name to the solver with the following command\n\ncodeoptions = getOptions('solvername');\n\n\nIn the documentation below, we assume that you have created this struct and named it codeoptions.\n\n## 12.1. General options¶\n\nWe will first discuss how to change several options that are valid for all the FORCES PRO interfaces.\n\n### 12.1.1. Solver name¶\n\nThe name of the solver will be used to name variables, functions, but also the MEX file and associated help file. This helps you to use multiple solvers generated by FORCES within the same software project or Simulink model. To set the name of the solver use:\n\ncodeoptions.name = 'solvername';\n\n\nAlternatively, you can directly name the solver when generating the options struct by calling:\n\ncodeoptions = getOptions('solvername');\n\n\n### 12.1.3. Maximum number of iterations¶\n\nTo set the maximum number of iterations of the generated solver, use:\n\ncodeoptions.maxit = 200;\n\n\nThe default maximum number of iterations for all solvers provided by FORCES PRO is set to 200.\n\n### 12.1.4. Compiler optimization level¶\n\nThe compiler optimization level can be varied by changing the field optlevel from 0 to 3 (default):\n\ncodeoptions.optlevel = 0;\n\n\nImportant\n\nIt is recommended to set optlevel to 0 during prototyping to evaluate the functionality of the solver without long compilation times. Then set it back to 3 when generating code for deployment or timing measurements.\n\n### 12.1.5. Running solvers in parallel¶\n\nThe generated solver can be run in parallel on different threads by changing the field threadSafeStorage from false to true:\n\ncodeoptions.threadSafeStorage = true;\n\n\n### 12.1.6. Measure Computation time¶\n\nYou can measure the time used for executing the generated code by using:\n\ncodeoptions.timing = 1;\n\n\nBy default the execution time is measured. The execution time can be accessed in the field solvetime of the information structure returned by the solver. In addition, the execution time is printed in the console if the flag printlevel is greater than 0.\n\nImportant\n\nSetting timing on will introduce a dependency on libraries used for accessing the system clock. Timing should be turned off when deploying the code on an autonomous embedded system.\n\nBy default when choosing to generate solvers for target platforms, timing is disabled. You can manually enable timing on embedded platforms by using:\n\ncodeoptions.embedded_timing = 1;\n\n\n### 12.1.7. Datatypes¶\n\nThe type of variables can be changed by setting the field floattype as outlined in Table 12.2.\n\nTable 12.2 Data type options\n\nfloattype\n\nDecimation\n\nWidth (bits)\n\nSupported algorithms\n\n'double' (default)\n\n64 bit\n\nFloating point\n\n'float'\n\n32 bit\n\nFloating point\n\n'int'\n\n32 bit\n\nFixed point\n\n'short'\n\n16 bit\n\nFixed point\n\nImportant\n\nUnless running on a resource-constrained platform, we recommend using double precision floating point arithmetics to avoid problems in the solver. If single precision floating point has to be used, reduce the required tolerances on the solver accordingly by a power of two (i.e. from 1E-6 to 1E-3).\n\n### 12.1.8. Overwriting existing solvers¶\n\nWhen a new solver is generated with the same name as an existing solver one can control the overwriting behaviour by setting the field overwrite as outlined in Table 12.3.\n\nTable 12.3 Overwrite existing solver options\n\noverwrite\n\nResult\n\n0\n\nNever overwrite.\n\n1\n\nAlways overwrite.\n\n2 (default)\n\n### 12.1.10. Code generation server¶\n\nBy default, code generation requests are routed to embotech’s server. To send a code generation request to a local server, for example when FORCES PRO is used in an enterprise setting, set the following field to an appropriate value:\n\ncodeoptions.server = 'http://embotech-server2.com:8114/v1.5.beta';\n\n\n### 12.1.12. Skipping automatic cleanup¶\n\nFORCES PRO automatically cleans up some of the files that it generates during the code generation, but which are usually not needed any more after building the MEX file. In particular, some intermediate CasADi generated files are deleted. If you would like to prevent any cleanup by FORCES, set the option:\n\ncodeoptions.cleanup = 0;\n\n\nThe default value is 1 (true).\n\nImportant\n\nThe library or object files generated by FORCES PRO contain only the solver itself. To retain the CasADi generated files for function evaluations, switch off automatic cleanup as shown above. This is needed if you want to use the solver within another software project, and need to link to it.\n\n### 12.1.13. Target platform¶\n\nAs a default option, FORCES PRO generates code for simulation on the host platform. To obtain code for deployment on a target embedded platform, set the field platform to the appropriate value. The platforms currently supported by FORCES PRO are given in Table 12.4. In order to acquire licenses to use a specific platform, licenses can be requested on the portal by selecting the platform naming stated in the Portal Selection.\n\nTable 12.4 Target platforms supported by FORCES PRO\n\nplatform\n\nDescription\n\nPortal Selection\n\n'Generic' (default)\n\nFor the architecture of the host platform.\n\n'x86_64' (Engineering Node)\n\n'x86_64'\n\nFor x86_64 based 64-bit platforms (detected OS).\n\n'x86_64'\n\n'x86'\n\nFor x86 based 32-bit platforms (detected OS).\n\n'x86'\n\n'Win-x86_64'\n\nFor Windows x86_64 based 64-bit platforms (supports Microsoft/Intel compiler).\n\n'x86_64'\n\n'Win-x86'\n\nFor Windows x86 based 32-bit platforms (supports Microsoft/Intel compiler).\n\n'x86'\n\n'Win-MinGW-x86_64'\n\nFor Windows x86_64 based 64-bit platforms (supports MinGW compiler).\n\n'x86_64'\n\n'Win-MinGW-x86'\n\nFor Windows x86 based 32-bit platforms (supports MinGW compiler).\n\n'x86'\n\n'Mac-x86_64'\n\nFor Mac x86_64 based 64-bit platforms (supports GCC/Clang compiler).\n\n'x86_64'\n\n'Gnu-x86_64'\n\nFor Linux x86_64 based 64-bit platforms (supports GCC compiler).\n\n'x86_64'\n\n'Gnu-x86'\n\nFor Linux x86 based 32-bit platforms (supports GCC compiler).\n\n'x86'\n\n'Docker-Gnu-x86_64'\n\nFor Linux x86_64 based 64-bit platforms on Docker (supports GCC compiler).\n\n'Docker-Gnu-x86_64'\n\n'Docker-Gnu-x86'\n\nFor Linux x86 based 32-bit platforms on Docker (supports GCC compiler).\n\n'Docker-Gnu-x86'\n\n'ARM-Generic'\n\nFor ARM Cortex 32-bit processors (Gnueabih machine type).\n\n'ARM-Generic-Gnu'\n\n'ARM-Generic64'\n\nFor ARM Cortex 64-bit processors (Aarch machine type).\n\n'ARM-Generic64-Gnu'\n\n'Integrity-ARM-x86'\n\nFor ARM Cortex 32-bit processors using the Integrity toolchain.\n\n'Integrity-ARM-x86'\n\n'Integrity-ARM-x64'\n\nFor ARM Cortex 64-bit processors using the Integrity toolchain.\n\n'Integrity-ARM-x64'\n\n'ARM Cortex-M3'\n\nFor ARM Cortex M3 32-bit processors.\n\n'ARM-Cortex-M3'\n\n'ARM-Cortex-M4-NOFPU'\n\nFor the ARM Cortex M4 32-bit processors without a floating-point unit.\n\n'ARM-Cortex-M4'\n\n'ARM-Cortex-M4'\n\nFor the ARM Cortex M4 32-bit processors with a floating-point unit.\n\n'ARM-Cortex-M4'\n\n'ARM-Cortex-A7'\n\nFor the ARM Cortex A7 32-bit processors (Gnueabih machine type).\n\n'ARM-Cortex-A7'\n\n'ARM-Cortex-A8'\n\nFor the ARM Cortex A8 32-bit processors (Gnueabih machine type).\n\n'ARM-Cortex-A8'\n\n'ARM-Cortex-A9'\n\nFor the ARM Cortex A9 32-bit processors (Gnueabih machine type).\n\n'ARM-Cortex-A9'\n\n'ARM-Cortex-A15'\n\nFor the ARM Cortex A15 32-bit processors (Gnueabih machine type).\n\n'ARM-Cortex-A15'\n\n'ARM-Cortex-A53'\n\nFor the ARM Cortex A53 64-bit processors (Gnueabih machine type).\n\n'ARM-Cortex-A53'\n\n'ARM-Cortex-A72'\n\nFor the ARM Cortex A72 64-bit processors (Gnueabih machine type).\n\n'ARM-Cortex-A72'\n\n'TI-Cortex-A15'\n\nFor the ARM Cortex A15 32-bit processors (Gnueabih machine type).\n\n'TI-Cortex-A15'\n\n'NVIDIA-Cortex-A57'\n\nFor the NVIDIA Cortex A57 64-bit processors (Aarch machine type).\n\n'NVIDIA-Cortex-A57'\n\n'AARCH-Cortex-A57'\n\nFor the ARM Cortex A57 64-bit processors (Aarch machine type).\n\n'AARCH-Cortex-A57'\n\n'AARCH-Cortex-A72'\n\nFor the ARM Cortex A72 64-bit processors (Aarch machine type).\n\n'AARCH-Cortex-A72'\n\n'PowerPC'\n\nFor 32-bit PowerPC based platforms (supports GCC compiler).\n\n'PowerPC-Gnu'\n\n'PowerPC64'\n\nFor 64-bit PowerPC based platforms (supports GCC compiler).\n\n'PowerPC64-Gnu'\n\n'MinGW32'\n\nFor Windows x86 based 32-bit platforms (supports MinGW compiler).\n\n'x86'\n\n'MinGW64'\n\nFor Windows x86_64 based 64-bit platforms (supports MinGW compiler).\n\n'x86_64'\n\n'dSPACE-MABII'\n\nFor the dSPACE MicroAutoBox II real-time system (supports Microtec compiler).\n\n'dSPACE-MABII-Microtec'\n\n'dSPACE-MABIII'\n\nFor the dSPACE MicroAutoBox III real-time system (supports Gcc compiler).\n\n'dSPACE-MABIII-Gcc'\n\n'dSPACE-MABXII'\n\nFor the dSPACE MicroAutoBox II real-time system (supports Microtec compiler).\n\n'dSPACE-MABII-Microtec'\n\n'dSPACE-MABXIII'\n\nFor the dSPACE MicroAutoBox III real-time system (supports Gcc compiler).\n\n'dSPACE-MABIII-Gcc'\n\n'Speedgoat-x86'\n\nFor Speedgoat 32-bit real-time platforms (supports Microsoft compiler).\n\n'Speedgoat-x86'\n\n'Speedgoat-x64'\n\nFor Speedgoat 64-bit real-time platforms (supports Microsoft compiler).\n\n'Speedgoat-x64'\n\n'IAtomE680_Bachmann'\n\nFor Bachmann PLC platforms (supports VxWorks compiler).\n\n'IAtomE680-VxWorks'\n\nNote\n\nIf a solver for another platform is requested, FORCES PRO will still provide the simulation interfaces for the 'Generic' host platform to enable users to run simulations.\n\n#### 12.1.13.1. Cross compilation¶\n\nTo generate code for other operating systems different from the host platform, set the appropriate flag from the following list to 1:\n\ncodeoptions.win\ncodeoptions.mac\ncodeoptions.gnu\n\n\nNote that this will only affect the target platform. Interfaces for the host platform will be automatically built.\n\n#### 12.1.13.2. Mac compilation¶\n\nWhen compiling for mac platforms it’s possible to select the compiler to be used for the web compilation. Select from the available values gcc (default) and clang with the following codeoption:\n\ncodeoptions.maccompiler\n\n\n#### 12.1.13.3. SIMD instructions¶\n\nOn x86-based host platforms, one can enable the sse field to accelerate the execution of the solver\n\ncodeoptions.sse = 1;\n\n\nOn x86-based host platforms, one can also add the avx field to significantly accelerate the compilation and execution of the convex solver, from version 1.9.0,\n\ncodeoptions.avx = 1;\n\n\nNote\n\nCurrently when options avx and blckMatrices are enabled simultaneously, blckMatrices is automatically disabled.\n\nNote\n\nWhen sparse parameters are present in the model, the options avx and neon are automatically set to zero.\n\nDepending on the host platform, avx may be automatically enabled. If the machine on which the solver is to be run does not support AVX and the message “Illegal Instruction” is returned at run-time, one can explicitly disable avx by setting:\n\ncodeoptions.avx = -1;\n\n\nIf the host platform supports AVX, but the user prefers not to have AVX intrinsics in the generated code, one can also keep the default option value of the solver:\n\ncodeoptions.avx = 0;\n\n\nOn ‘NVIDIA-Cortex-A57’, ‘AARCH-Cortex-A57’ and ‘AARCH-Cortex-A72’ target platforms, one can also enable the field neon in order to accelerate the execution of the convex solver. From version 1.9.0, the typical behaviour is that the host platform gets vectorized code based on AVX intrinsics when avx = 1, and the target platform gets AVX vectorized code if it supports it when avx = 1 and NEON vectorized code if it is one of the above Cortex platforms and neon = 1.\n\nFor single precision, the options are\n\ncodeoptions.floattype = 'float'\ncodeoptions.neon = 1;\n\n\nFor double precision, the options are\n\ncodeoptions.floattype = 'double'\ncodeoptions.neon = 2;\n\n\nIn case one wants to disable NEON intrinsics in the generated target code, the default value of the neon option is\n\ncodeoptions.neon = 0;\n\n\nIf NEON vectorization is being used and there is a mismatch between float precision and the value of the neon option, the solver is automatically generated with the following options:\n\ncodeoptions.floattype = 'double'\ncodeoptions.neon = 2;\n\n\nand a warning message is raised by the MATLAB client.\n\nNote\n\nFrom version 1.9.0, ARMv8-A NEON instructions are generated. Hence, target platforms based on ARMv7 and previous versions are currently not supported.\n\n### 12.1.14. MISRA 2012 compliance¶\n\nIf your license allows it, add the following field to generate C code that is compliant with the MISRA 2012 rules:\n\ncodeoptions.misra2012_check = 1;\n\n\nThis option makes the generated solver code MISRA compliant. After compilation, the client also downloads a folder whose name terminates with _misra2012_analysis. The folder contains one summary of all MISRA violations for the solver source and header files. Note that the option only produces MISRA compliant code when used with algorithms PDIP and PDIP_NLP.\n\n### 12.1.15. Optimizing code size¶\n\nThe size of the solver libraries generated with code option PDIP_NLP can be reduced by means of the option nlp.compact_code. By setting\n\ncodeoptions.nlp.compact_code = 1;\n\n\nthe user enables the FORCES PRO server to generate smaller code, which results in shorter compilation time and slightly better solve time in some cases. This feature is especially effective on long horizon problems.\n\nThe size of sparse linear algebra routines in the generated code can be reduced by changing the option compactSparse from 0 to 1:\n\ncodeoptions.compactSparse = 1;\n\n\n### 12.1.16. Optimizing Linear Algebra Operations¶\n\nSome linear algebra routines in the generated code have available optimizations that can be enabled by changing the options optimize_<optimization> from 0 to 1. These optimizations change the code in order to make better use of some embedded architectures in which hardware is more limited compared to host PC architectures. Therefore, these optimizations show better results in embedded platforms such as ARM targets rather than during simulations on host PCs. The available optimizations are:\n\n• Cholesky Division: This option performs the divisions included in the Cholesky factorization more efficiently to reduce its computation time.\n\n• Registers: This option attempts to use the architecture’s registers in order to reduce memory operations which can take significant time.\n\n• Use Locals: These options (which are separated into simple/heavy/all in ascending complexity) make better use of data locality in order to reduce memory jumps\n\n• Operations Rearrange: This option rearranges operations in order to make more efficient use of data and reduce memory jumps\n\n• Loop Unrolling: This option unrolls some of the included loops in order to remove their overhead.\n\n• Enable Offset: This option allows the rest of the optimizations to take place in cases where the matrix contains offsets.\n\ncodeoptions.optimize_choleskydivision = 1;\ncodeoptions.optimize_registers = 1;\ncodeoptions.optimize_uselocalsall = 1;\ncodeoptions.optimize_uselocalsheavy = 1; % overriden if uselocalsall is enabled\ncodeoptions.optimize_uselocalssimple = 1; % overriden if uselocalsheavy is enabled\ncodeoptions.optimize_operationsrearrange = 1;\ncodeoptions.optimize_loopunrolling = 1;\ncodeoptions.optimize_enableoffset = 1;\n\n\n### 12.1.17. Dump problem formulation¶\n\nThe MATLAB client of FORCES PRO provides a built-in tool to dump the problem formulation to reproduce the exact same solver for future reference. This tool is explained in detail in Section 13 and can be turned on by using the setting:\n\ncodeoptions.dump_formulation = 1;\n\n\n## 12.2. High-level interface options¶\n\nThe FORCES PRO NLP solver of the high-level interface implements a nonlinear barrier interior-point method. We will now discuss how to change several parameters in the solver.\n\n### 12.2.1. Integrators¶\n\nWhen providing the continuous dynamics the user must select a particular integrator by setting nlp.integrator.type as outlined in Table 12.5.\n\nTable 12.5 Integrators options\n\nnlp.integrator.type\n\nType\n\nOrder\n\n'ForwardEuler'\n\nExplicit Euler Method\n\n1\n\n'ERK2'\n\nExplicit Runge-Kutta\n\n2\n\n'ERK3'\n\nExplicit Runge-Kutta\n\n3\n\n'ERK4' (default)\n\nExplicit Runge-Kutta\n\n4\n\n'BackwardEuler'\n\nImplicit Euler Method\n\n1\n\n'IRK2'\n\nImplicit Euler Method\n\n2\n\n'IRK4'\n\nImplicit Euler Method\n\n4\n\nThe user must also provide the discretization interval (in seconds) and the number of intermediate shooting nodes per interval. For instance:\n\ncodeoptions.nlp.integrator.type = 'ERK2';\ncodeoptions.nlp.integrator.Ts = 0.01;\ncodeoptions.nlp.integrator.nodes = 10;\n\n\nTip\n\nUsually an explicit integrator such as RK4 should suffice for most applications. If you have stiff systems, or suspect inaccurate integration to be the cause of convergence failure of the NLP solver, consider using implicit integrators from the table above.\n\nNote\n\nNote that the implicit integrators BackwardEuler, IRK2 and IRK4 currently rely on the CasADi AD tool to work.\n\n### 12.2.2. Accuracy requirements¶\n\nOne can modify the termination criteria by altering the KKT tolerance with respect to stationarity, equality constraints, inequality constraints and complementarity conditions, respectively, using the following fields:\n\n% default tolerances\ncodeoptions.nlp.TolStat = 1E-5; % inf norm tol. on stationarity\ncodeoptions.nlp.TolEq = 1E-6; % tol. on equality constraints\ncodeoptions.nlp.TolIneq = 1E-6; % tol. on inequality constraints\ncodeoptions.nlp.TolComp = 1E-6; % tol. on complementarity\n\n\nAll tolerances are computed using the infinitiy norm $$\\lVert \\cdot \\rVert_\\infty$$.\n\n### 12.2.3. Barrier strategy¶\n\nThe strategy for updating the barrier parameter is set using the field:\n\ncodeoptions.nlp.BarrStrat = 'loqo';\n\n\nIt can be set to 'loqo' (default) or to 'monotone'. The default settings often leads to faster convergence, while 'monotone' may help convergence for difficult problems.\n\n### 12.2.4. Hessian approximation¶\n\nThe way the Hessian of the Lagrangian function is computed can be set using the field:\n\ncodeoptions.nlp.hessian_approximation = 'bfgs';\n\n\nFORCES PRO currently supports BFGS updates ('bfgs') (default) and Gauss-Newton approximation ('gauss-newton'). Exact Hessians will be supported in a future version. Read the subsequent sections for the corresponding Hessian approximation method of your choice.\n\n#### 12.2.4.1. BFGS options¶\n\nWhen the Hessian is approximated using BFGS updates, the initialization of the estimates can play a critical role in the convergence of the method. The default value is the identity matrix, but the user can modify it using e.g.:\n\ncodeoptions.nlp.bfgs_init = diag([0.1, 10, 4]);\n\n\nNote that BFGS updates are carried out individually per stage in the FORCES NLP solver, so the size of this matrix is the size of the stage variable. Also note that this matrix must be positive definite. When the cost function is positive definite, it often helps to initialize BFGS with the Hessian of the cost function.\n\nThis matrix is also used to restart the BFGS estimates whenever the BFGS updates are skipped several times in a row. The maximum number of updates skipped before the approximation is re-initialized is set using:\n\ncodeoptions.nlp.max_update_skip = 2;\n\n\nThe default value for max_update_skip is 2.\n\n#### 12.2.4.2. Gauss-Newton options¶\n\nFor problems that have a least squares objective, i.e. the cost function can be expressed by a vector-valued function $$r_k : \\mathbb{R}^n \\rightarrow \\mathbb{R}^m$$ which implicitly defines the objective function as:\n\n$f_k(z_k,p_k) = \\frac{1}{2} \\lVert r_k(z_k,p_k) \\rVert_2^2 \\,,$\n\nthe Gauss-Newton approximation of the Hessian is given by:\n\n$\\nabla_{xx}^2 L_k \\approx \\nabla r_k(z_k,p_k) \\nabla r_k(z_k,p_k)^\\top$\n\nand can lead to faster convergence and a more reliable method. When this option is selected, the functions $$r_k$$ have to be provided by the user in the field LSobjective. For example if $$r(z)=\\sqrt{100} z_1^2 + \\sqrt{6} z_2^2$$, i.e. $$f(z) = 50 z_1^2 + 3 z_2^2$$, then the following code defines the least-squares objective (note that $$r$$ is a vector-valued function):\n\nnlp.objective = @(z) 0.1* z(1)^2 + 0.01z(2)^2;\nnlp.LSobjective = @(z) [sqrt(0.2)z(1); sqrt (0.02)z(2)];\n\n\nImportant\n\nThe field LSobjective will have precedence over objective, which need not be defined in this case.\n\nWhen providing your own function evaluations in C, you must populate the Hessian argument with a positive definite Hessian.\n\n### 12.2.5. Line search settings¶\n\nThe line search first computes the maximum step that can be taken while maintaining the iterates inside the feasible region (with respect to the inequality constraints). The maximum distance is then scaled back using the following setting:\n\n % default fraction-to-boundary scaling\ncodeoptions.nlp.ftbr_scaling = 0.9900;\n\n\n### 12.2.6. Regularization¶\n\nTo avoid ill-conditioned saddle point systems, FORCES employs two different types of regularization, static and dynamic regularization.\n\n#### 12.2.6.1. Static regularization¶\n\nStatic regularization of the augmented Hessian by $$\\delta_w I$$, and of the multipliers corresponding to the equality constraints by $$-\\delta_c I$$ helps avoid problems with rank deficiency. The constants $$\\delta_w$$ and $$\\delta_c$$ vary at each iteration according to the following heuristic rule:\n\n$\\begin{split}\\delta_w & = \\eta_w \\min(\\mu,\\lVert c(x) \\rVert))^{\\beta_w} \\cdot (i+1)^{-\\gamma_w} + \\delta_{w,\\min} \\\\ \\delta_c & = \\eta_c \\min(\\mu,\\lVert c(x) \\rVert))^{\\beta_c} \\cdot (i+1)^{-\\gamma_c} + \\delta_{c,\\min} \\\\\\end{split}$\n\nwhere $$\\mu$$ is the barrier parameter and $$i$$ is the number of iterations.\n\nThis rule has been chosen to accommodate two goals: First, make the regularization dependent on the progress of the algorithm - the closer we are to the optimum, the smaller the regularization should be in order not to affect the search directions generated close to the solution, promoting superlinear convergence properties. Second, the amount of regularization employed should decrease with the number of iterations to a certain minimum level, at a certain sublinear rate, in order to prevent stalling due to too large regularization. FORCES NLP does not employ an inertia-correcting linear system solver, and so relies heavily on the parameters of this regularization to be chosen carefully.\n\nYou can change these parameters by using the following settings:\n\n% default static regularization parameters\ncodeoptions.nlp.reg_eta_dw = 1E-4;\ncodeoptions.nlp.reg_beta_dw = 0.8;\ncodeoptions.nlp.reg_min_dw = 1E-9;\ncodeoptions.nlp.reg_gamma_dw = 1.0/3.0;\n\ncodeoptions.nlp.reg_eta_dc = 1E-4;\ncodeoptions.nlp.reg_beta_dc = 0.8;\ncodeoptions.nlp.reg_min_dc = 1E-9;\ncodeoptions.nlp.reg_gamma_dc = 1.0/3.0;\n\n\nNote that by choosing $$\\delta_w=0$$ and $$\\delta_c=0$$, you can turn off the progress and iteration dependent regularization, and rely on a completely static regularization by $$\\delta_{w,\\min}$$ and $$\\delta_{c,\\min}$$, respectively.\n\n#### 12.2.6.2. Dynamic regularization¶\n\nDynamic regularization regularizes the matrix on-the-fly to avoid instabilities due to numerical errors. During the factorization of the saddle point matrix, whenever it encounters a pivot smaller than $$\\epsilon$$, it is replaced by $$\\delta$$. There are two parameter pairs: $$(\\epsilon,\\delta)$$ affects the augmented Hessian and $$(\\epsilon_2,\\delta_2)$$ affects the search direction computation. You can set these parameters by:\n\n% default dynamic regularization parameters\ncodeoptions.regularize.epsilon = 1E-12; % (for Hessian approx.)\ncodeoptions.regularize.delta = 4E-6; % (for Hessian approx.)\ncodeoptions.regularize.epsilon2 = 1E-14; % (for Normal eqs.)\ncodeoptions.regularize.delta2 = 1E-14; % (for Normal eqs.)\n\n\n### 12.2.7. Linear system solver¶\n\nThe interior-point method solves a linear system to find a search direction at every iteration. FORCES NLP offers the following three linear solvers:\n\n• 'normal_eqs' (default): Solving the KKT system in normal equations form.\n\n• 'symm_indefinite_fast': Solving the KKT system in augmented / symmetric indefinite form, using regularization and positive definite Cholesky factorizations only.\n\n• 'symm_indefinite': Solving the KKT system in augmented / symmetric indefinite form, using block-indefinite factorizations.\n\nThe linear system solver can be selected by setting the following field:\n\ncodeoptions.nlp.linear_solver = 'symm_indefinite';\n\n\nIt is recommended to try different linear solvers when experiencing convergence problems. The most stable method is 'symm_indefinite', while the fastest solver is 'symm_indefinite_fast'.\n\nNote\n\nIndependent of the linear system solver choice, the generated code is always library-free and statically allocated, i.e. it can be embedded anywhere.\n\nThe 'normal_eqs' solver is the standard FORCES linear system solver based on a full reduction of the KKT system (the so-called normal equations form). It works well for standard problems, especially convex problems or nonlinear problems where the BFGS or Gauss-Newton approximations of the Hessian are numerically sufficiently well conditioned.\n\nThe 'symm_indefinite' solver is the most robust solver, but still high-speed. It is based on block-wise factorization of the symmetric indefinite form of the KKT system (the so-called augmented form). Each block is handled by symmetric indefinite LDL factorization, with (modified) on-the-fly Bunch-Kaufmann permutations leading to boundedness of lower triangular factors for highest numerical stability. This is our most robust linear system solver, with only a modest performance penalty (about 30% compared to 'symm_indefinite_fast').\n\nThe 'symm_indefinite_fast' solver is robust, but even faster. It is based on block-wise factorization of the symmetric indefinite KKT matrix, where each block is handled by a Cholesky factorization. It uses regularization to increase numerical stability. Currently only used for receding-horizon/MPC-like problems where dimensions of all stages are equal (minus the first and last stage, those are handled separately). It is more robust and faster than the normal equations form. This solver is likely to become the default option in the future.\n\n### 12.2.8. Automatic differentiation tool¶\n\nIf external functions and derivatives are not provided directly as C code by the user, FORCES PRO makes use of an automatic differentiation (AD) tool to generate efficient C code for all the functions (and their derivatives) inside the problem formulation. Currently, three different AD tools are supported that can be chosen by means of the setting nlp.ad_tool as summarized in Table 12.6.\n\nTable 12.6 Automatic differentiation tool options\n\nnlp.ad_tool\n\nTool\n\nURL\n\n'casadi'\n\n'casadi-351'\n\n'symbolic-math-tbx'\n\nMathWorks Symbolic Math Toolbox\n\nMathWorks\n\nNote that MathWorks Symbolic Math Toolbox requires an additional license, which is why the default option is set to\n\ncodeoptions.nlp.ad_tool = 'casadi';\n\n\nAlso note that the use of implicit integrators BackwardEuler, IRK2 and IRK4 (see Section 12.2.1) currently still rely on using the CasADi AD tool.\n\n### 12.2.9. Safety checks¶\n\nBy default, the output of the function evaluations is checked for the presence of NaNs or INFs in order to diagnose potential initialization problems. In order to speed up the solver one can remove these checks by setting:\n\ncodeoptions.nlp.checkFunctions = 0;\n\n\n## 12.3. Convex branch-and-bound options¶\n\nThe settings of the FORCES PRO mixed-integer branch-and-bound convex solver are accessed through the codeoptions.mip struct. It is worthwhile to explore different values for the settings in Table 12.7, as they might have a severe impact on the performance of the branch-and-bound procedure.\n\nNote\n\nAll the options described below are currently not available with the FORCES PRO nonlinear solver. For mixed-integer nonlinear programs and the available options, please have a look at paragraph Mixed-integer nonlinear solver.\n\nTable 12.7 Branch-and-bound options\n\nSetting\n\nValues\n\nDefault\n\nmip.timeout\n\nAny value $$\\geq 0$$\n\n31536000 (1 year)\n\nmip.mipgap\n\nAny value $$\\geq 0$$\n\n0\n\nmip.branchon\n\n'mostAmbiguous', 'leastAmbiguous'\n\n'mostAmbiguous'\n\nmip.stageinorder\n\n0 (OFF), 1 (ON)\n\n1 (ON)\n\nmip.explore\n\n'bestFirst', 'depthFirst'\n\n'bestFirst'\n\nmip.inttol\n\nAny value $$> 0$$\n\n1E-5\n\nmip.queuesize\n\nAny integer value $$\\geq 0$$\n\n1000\n\nA description of each setting is given below:\n\n• mip.timeout: Timeout in seconds, after which the search is stopped and the best solution found so far is returned.\n\n• mip.mipgap: Relative sub-optimality after which the search shall be terminated. For example, a value of 0.01 will search for a feasible solution that is at most 1%-suboptimal. Set to zero if the optimal solution is required.\n\n• mip.branchon: Determines which variable to branch on after having solved the relaxed problem. Options are 'mostAmbiguous' (i.e. the variable closest to 0.5) or 'leastAmbiguous' (i.e. the variable closest to 0 or 1).\n\n• mip.stageinorder: Stage-in-order heuristic: For the branching, determines whether to fix variables in order of the stage number, i.e. first all variables of stage $$i$$ will be fixed before fixing any of the variables of stage $$i+1$$. This is often helpful in multistage problems, where a timeout is expected to occur, and where it is important to fix the early stages first (for example MPC problems). Options are 0 for OFF and 1 for ON.\n\n• mip.explore: Determines the exploration strategy when selecting pending nodes. Options are 'bestFirst', which chooses the node with the lowest lower bound from all pending nodes, or 'depthFirst', which prioritizes nodes with the most number of fixed binaries first to quickly reach a node.\n\n• mip.inttol: Integer tolerance for identifying binary solutions of relaxed problems. A solution of a relaxed problem with variable values that are below inttol away from binary will be declared to be binary.\n\n• mip.queuesize: Maximum number of pending nodes that the branch and bound solver can store. If that number is exceeded during the search, the solver quits with an exitflag value of -2 and returns the best solution found so far.\n\n## 12.4. Solve methods¶\n\nAs a default optimization method the primal-dual interior-point method is used. Several other methods are available. To change the solve method set the solvemethod field as outlined in Table 12.8.\n\nTable 12.8 Solve methods\n\nsolvemethod\n\nMethod\n\nDescription\n\n'PDIP' (default)\n\nPrimal-Dual Interior-Point Method\n\nThe Primal-Dual Interior-Point Method is a stable and robust method for most problems.\n\n'ADMM'\n\nAlternating Direction Methods of Multipliers\n\nFor some problems, ADMM may be faster. The method variant and several algorithm parameters can be tuned in order to improve performance.\n\n'DFG'\n\nFor some problems with simple constraints, our implementation of the dual fast gradient method can be the fastest option. No parameters need to be tuned in this method.\n\n'FG'\n\nFor problems with no equality constraints (only one stage) and simple constraints, the primal fast gradient method can give medium accuracy solutions extremely quickly. The method has several tuning parameters that can significantly affect the performance.\n\n### 12.4.1. Primal-Dual Interior-Point Method¶\n\nThe Primal-Dual Interior-Point Method is the default optimization method. It is a stable and robust method for most of the problems.\n\n#### 12.4.1.1. Solver Initialization¶\n\nThe performance of the solver can be influenced by the way the variables are initialized. The default method (cold start) should work in most cases extremely reliably, so there should be no need in general to try other methods, unless you are experiencing problems with the default initialization scheme. To change the method of initialization in FORCES PRO set the field init to one of the values in Table 12.9.\n\nTable 12.9 PDIP solver initialization\n\ninit\n\nMethod\n\nInitialization method\n\n0 (default)\n\nCold start\n\nSet all primal variables to $$0$$, and all dual variables to the square root of the initial complementarity gap $$\\mu_0: z_i=0, s_i=\\sqrt{\\mu_0}, \\lambda_i=\\sqrt{\\mu_0}$$. The default value is $$\\mu_0=10^6$$.\n\n1\n\nCentered start\n\nSet all primal variables to zero, the slacks to the RHS of the corresponding inequality, and the Lagrange multipliers associated with the inequalities such that the pairwise product between slacks and multipliers is equal to the parameter $$\\mu_0: z_i=0, s_i=b_{\\mathrm{ineq}}$$ and $$s_i \\lambda_i = \\mu_0$$.\n\n2\n\nPrimal warm start\n\nSet all primal variables as provided by the user. Calculate the residuals and set the slacks to the residuals if they are sufficiently positive (larger than $$10^{-4}$$), or to one otherwise. Compute the associated Lagrange multipliers such that $$s_i \\lambda_i = \\mu_0$$.\n\n#### 12.4.1.2. Initial Complementary Slackness¶\n\nThe default value for $$\\mu_0$$ is $$10^6$$. To use a different value, use:\n\ncodeoptions.mu0 = 10;\n\n\n#### 12.4.1.3. Accuracy Requirements¶\n\nThe accuracy for which FORCES PRO returns the OPTIMAL flag can be set as follows:\n\ncodeoptions.accuracy.ineq = 1e-6; % infinity norm of residual for inequalities\ncodeoptions.accuracy.eq = 1e-6; % infinity norm of residual for equalities\ncodeoptions.accuracy.mu = 1e-6; % absolute duality gap\ncodeoptions.accuracy.rdgap = 1e-4; % relative duality gap := (pobj-dobj)/pobj\n\n\n#### 12.4.1.4. Line Search Settings¶\n\nIf FORCES PRO experiences convergence difficulties, you can try selecting different line search parameters. The first two parameters of codeoptions.linesearch, factor_aff and factor_cc are the backtracking factors for the line search (if the current step length is infeasible, then it is reduced by multiplication with these factors) for the affine and combined search direction, respectively.\n\ncodeoptions.linesearch.factor_aff = 0.9;\ncodeoptions.linesearch.factor_cc = 0.95;\n\n\nThe remaining two parameters of the field linesearch determine the minimum (minstep) and maximum step size (maxstep). Choosing minstep too high will cause the generated solver to quit with an exitcode saying that the line search has failed, i.e. no progress could be made along the computed search direction. Choosing maxstep too close to 1 is likely to cause numerical issues, but choosing it too conservatively (too low) is likely to increase the number of iterations needed to solve a problem.\n\ncodeoptions.linesearch.minstep = 1e-8;\ncodeoptions.linesearch.maxstep = 0.995;\n\n\n#### 12.4.1.5. Regularization¶\n\nDuring factorization of supposedly positive definite matrices, FORCES PRO applies a regularization to the $$i$$-th pivot element if it is smaller than $$\\epsilon$$. In this case, it is set to $$\\delta$$, which is the lower bound on the pivot element that FORCES PRO allows to occur.\n\ncodeoptions.regularize.epsilon = 1e-13; % if pivot element < epsilon …\ncodeoptions.regularize.delta = 1e-8; % then set it to delta\n\n\n#### 12.4.1.6. Multicore parallelization¶\n\nFORCES PRO supports the computation on multiple cores, which is particularly useful for large problems and long horizons (the workload is split along the horizon to multiple cores). This is implemented by the use of OpenMP and can be switched on by using\n\ncodeoptions.parallel = 1;\n\n\nBy default multicore computation is switched off.\n\n### 12.4.2. Alternating Directions Method of Multipliers¶\n\nFORCES PRO implements several optimization methods based on the ADMM framework. Different variants can handle different types of constraints and FORCES PRO will automatically choose an ADMM variant that can handle the constraints in a given problem. To manually choose a specific method in FORCES PRO, use the ADMMvariant field of codeoptions:\n\ncodeoptions.ADMMvariant = 1; % can be 1 or 2\n\n\nwhere variant 1 is as follows:\n\n\\begin{align} \\text{minimize} \\quad & \\frac{1}{2} y^\\top H y + f^\\top y \\\\ \\text{subject to} \\quad & Dy=c \\\\ & \\underline{z} \\leq z \\leq \\bar{z} \\\\ & y = z \\end{align}\n\nand variant 2 is as follows:\n\n\\begin{align} \\text{minimize} \\quad & \\frac{1}{2} y^\\top H y + f^\\top y \\\\ \\text{subject to} \\quad & Dy=c \\\\ & A y = z \\\\ & z \\leq b \\end{align*}\n\n#### 12.4.2.1. Accuracy requirements¶\n\nThe accuracy for which FORCES PRO returns the OPTIMAL flag can be set as follows:\n\ncodeoptions.accuracy.consensus = 1e-3; % infinity norm of the consensus equality\ncodeoptions.accuracy.dres = 1e-3; % infinity norm of the dual residual\n\n\nNote that, in contrast to primal-dual interior-point methods, the required number of ADMM iterations varies very significantly depending on the requested accuracy. ADMM typically requires few iterations to compute medium accuracy solutions, but many more iterations to achive the same accuracy as interior-point methods. For feedback applications, medium accuracy solutions are typically sufficient. Also note that the ADMM accuracy requirements have to be changed depending on the problem scaling.\n\n#### 12.4.2.2. Method parameters¶\n\nADMM uses a regularization parameter $$\\rho$$, which also acts as the step size in the gradient step. The convergence speed of ADMM is highly variable in the parameter $$\\rho$$. Its value should satisfy $$\\rho > 0$$. This parameter can be tuned using the following command:\n\ncodeoptions.ADMMrho = 1;\n\n\nIn some cases it may be possible to let FORCES PRO choose the value $$\\rho$$ automatically. To enable this feature set:\n\ncodeoptions.ADMMautorho = 1;\n\n\nPlease note that this does not guarantee that the choice of $$\\rho$$ will be optimal.\n\nADMM can also include an ‘over-relaxation’ step that can improve the convergence speed. This step is typically useful for problems where ADMM exhibits very slow convergence and can be tuned using the parameter $$\\alpha$$. Its value should satisfy $$1 \\leq \\alpha \\leq 2$$. This step using the following command:\n\ncodeoptions.ADMMalpha = 1;\n\n\n#### 12.4.2.3. Precomputations¶\n\nFor problems with time-invariant data, FORCES PRO can compute full matrix inverses at code generation time and then implement matrix solves online by dense matrix-vector multiplication. In some cases, especially when the prediction horizon is long, it may be better to factorize the matrix and implement matrix solves using forward and backward solves with the pre-computed factors. To manually switch on this option, use the ADMMfactorize field of codeoptions.\n\nWhen the data is time-varying, or when the prediction horizon is larger than 15 steps, FORCES PRO automatically switches to a factorization-based method.\n\ncodeoptions.ADMMfactorize = 0;\n\n\n### 12.4.3. Dual Fast Gradient Method¶\n\nFor some problems with simple constraints, our implementation of the dual fast gradient method can be the fastest option. No parameters need to be tuned in this method.\n\n### 12.4.4. Primal Fast Gradient Method¶\n\nFor problems with no equality constraints (only one stage) and simple constraints, the primal fast gradient method can give medium accuracy solutions extremely quickly. The method has several tuning parameters that can significantly affect the performance.\n\n#### 12.4.4.1. Accuracy requirements¶\n\nThe accuracy for which FORCES PRO returns the OPTIMAL flag can be set as follows:\n\ncodeoptions.accuracy.gmap= 1e-5; % infinity norm of the gradient map\n\n\nThe gradient map is related to the difference with respect to the optimal objective value. Just like with other first-order methods, the required number of FG iterations varies very significantly depending on the requested accuracy. Medium accuracy solutions can typically be computed very quickly, but many iterations are needed to achieve the same accuracy as with interior-point methods.\n\n#### 12.4.4.2. Method parameters¶\n\nThe user has to determine the step size in the fast gradient method. The convergence speed of FG is highly variable in this parameter, which should typically be set to be one over the maximum eigenvalue of the quadratic cost function. This parameter can be tuned using the following command:\n\ncodeoptions.FGstep = 1/1000;\n\n\nIn some cases it may be possible to let FORCES PRO choose the step size automatically. To enable this feature set:\n\ncodeoptions.FGautostep = 1;\n\n\n#### 12.4.4.3. Warm starting¶\n\nThe performance of the fast gradient method can be greatly influenced by the way the variables are initialized. Unlike with interior-point methods, fast gradient methods can be very efficiently warm started with a good guess for the optimal solution. To enable this feature set:\n\ncodeoptions.warmstart = 1;\n\n\nWhen the user turns warm start on, a new parameter z_init_0 is automatically added. The user should set it to be a good guess for the solution, which is typically available when solving a sequence of problems." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.7815283,"math_prob":0.92276216,"size":38781,"snap":"2020-45-2020-50","text_gpt3_token_len":9262,"char_repetition_ratio":0.14818062,"word_repetition_ratio":0.12719299,"special_character_ratio":0.22758567,"punctuation_ratio":0.13924411,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9638247,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-11-28T08:29:01Z\",\"WARC-Record-ID\":\"<urn:uuid:282eff7b-823b-4b64-9e0f-f7abbc894e6b>\",\"Content-Length\":\"117821\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1090d724-98af-4f90-bd27-8eef9689286b>\",\"WARC-Concurrent-To\":\"<urn:uuid:65fe33a2-c1b3-4aae-a64e-6ea820e3916b>\",\"WARC-IP-Address\":\"13.94.143.57\",\"WARC-Target-URI\":\"https://forces-3-1-0.embotech.com/Documentation/solver_options/index.html\",\"WARC-Payload-Digest\":\"sha1:MPASK2T5UDFT45PNGP5EW2PPMAL2H2WG\",\"WARC-Block-Digest\":\"sha1:DNHEKOTCTYQV2TBMVFSQO47K445O5JQW\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141195198.31_warc_CC-MAIN-20201128070431-20201128100431-00007.warc.gz\"}"}
https://www.diagramelectric.co/formula-for-finding-total-resistance-in-parallel-circuit/	[ "# Formula For Finding Total Resistance In Parallel Circuit\n\nThe formula for finding the total resistance in a parallel circuit is: ``` Rtotal = 1 / (1 / R1 + 1 / R2 + ... + 1 / Rn) ``` where: \n\n### Rtotal\n\nis the total resistance of the circuit in ohms (Ω). \n\n,\n\n, ...,\n\n### Rn\n\nare the resistances of the individual resistors in the circuit in ohms (Ω). To use this formula, simply add the reciprocals of the individual resistances and then take the reciprocal of the resulting sum. For example, if you have a circuit with three resistors with resistances of 2 Ω, 4 Ω, and 6 Ω, the total resistance would be: ``` Rtotal = 1 / (1 / 2 Ω + 1 / 4 Ω + 1 / 6 Ω) = 1 / (0.5 Ω + 0.25 Ω + 0.167 Ω) = 1 / 0.917 Ω = 1.09 Ω ```\n\nThe total resistance of a parallel circuit is always less than the resistance of any of the individual resistors in the circuit. This is because when resistors are connected in parallel, the current through each resistor is the same, but the voltage across each resistor is different. The total voltage across the circuit is the same as the voltage across any of the individual resistors.\n\nThe formula for finding the total resistance in a parallel circuit can be used to calculate the resistance of a circuit when you know the resistance of the individual resistors. It can also be used to determine the current through a circuit when you know the total resistance and the voltage across the circuit.\n\nParallel circuits are used in a variety of applications, including electrical wiring, electronic circuits, and automotive systems. They are often used to increase the current capacity of a circuit or to reduce the voltage drop across a circuit.\n\nHere are some additional tips for finding the total resistance in a parallel circuit: * If you have a circuit with only two resistors, you can use the following formula to find the total resistance: ``` Rtotal = R1 * R2 / (R1 + R2) ``` * If you have a circuit with a large number of resistors, you can use a spreadsheet or a graphing calculator to find the total resistance. * If you are unsure of the resistance of any of the resistors in a circuit, you can measure it with a multimeter.\n\nBy understanding the formula for finding the total resistance in a parallel circuit, you can calculate the resistance of any circuit and design circuits that meet your specific needs.", null, "How To Find The Percentage Error In Equivalent Resistance When Two Resistors With Values R1 6 0 3 And R2 10 2 Are Connected Parallel Quora", null, "Physics Page Www Thedotcom Com", null, "Simple Parallel Circuits Series And Electronics Textbook", null, "4 Ways To Calculate Total Resistance In Circuits Wikihow", null, "Total Resistance Calculator Of Series Parallel Circuit", null, "Simplified Formulas For Parallel Circuit Resistance Calculations Inst Tools", null, "Series And Parallel Circuits Learn Sparkfun Com", null, "Physics For Kids Resistors In Series And Parallel", null, "4 Ways To Calculate Total Resistance In Circuits Wikihow", null, "How To Find Total Resistance In A Series And Parallel Circuit Brainly", null, "Physics For Kids Resistors In Series And Parallel", null, "How To Solve Parallel Circuits 10 Steps With Pictures Wikihow", null, "Resistors In Series And Parallel Combination Determination Of The Equivalent Resistance Two Procedure Faqs", null, "Lessons In Electric Circuits Volume I Dc Chapter 10", null, "L4 Series And Parallel Resistors Physical Computing", null, "Simple Parallel Circuits Series And Electronics Textbook", null, "How Do You Calculate The Total Resistance Of A Parallel Circuit" ]	[ null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null, "data:image/svg+xml,%3Csvg%20xmlns='http://www.w3.org/2000/svg'%20viewBox='0%200%201%201'%3E%3C/svg%3E", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8348064,"math_prob":0.99280965,"size":3349,"snap":"2023-40-2023-50","text_gpt3_token_len":768,"char_repetition_ratio":0.22242153,"word_repetition_ratio":0.12776831,"special_character_ratio":0.22454464,"punctuation_ratio":0.07628524,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995809,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-12-07T06:18:14Z\",\"WARC-Record-ID\":\"<urn:uuid:a2a3719c-c9e1-4bb0-b4fa-ad45702e2e58>\",\"Content-Length\":\"65351\",\"Content-Type\":\"application/http; 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https://physics.stackexchange.com/questions/92058/is-the-assumption-that-the-two-reference-frames-be-inertial-required-in-the-deri	[ "# Is the assumption that the two reference frames be inertial required in the derivation of transformation equations?\n\nIn the derivation of Galilean transformations the only assumption is that the two frames are moving with some uniform relative velocity $u$.\n\nSuppose with respect to some inertial frame $O$ the two frames $S$ and $S'$ are moving with the same uniform acceleration $a$.\n\nLet $V$ be the velocity of $S$ w.r.t. $O$. Similarly, let $V'$ be the velocity of $S'$ w.r.t. $O$. Furthermore, let $V_0' - V_0 = u$ (const.). Then\n\n$$V = V_0 + at$$ $$V' = V_0' + at$$\n\nThen the relative velocity is $V' - V = u$.\n\nThis is the only result required in deriving the Galilean transformation. So why do people assume that the reference frames be inertial. (I know the point is so that Newton's laws would be valid, but exclusively in the derivation of the transformation equation is this assumption needed?) The same applies in the derivation of Lorentz transformation.\n\n## 2 Answers\n\nI would write a comment but i don't have the privilege so am writing an answer\n\n\"In the derivation of Galilean transformations the only assumption is that the two frames are moving with some uniform relative velocity u.\"\n\nHint: \"In the derivation of Galilean transformations the only assumption is that the two frames are moving with respect to each other with some uniform relative velocity u.\"\nRead this sentecse carefully.\nPS: Since you have marched towards relativity remember every measurement is taken w.r.t some frame of reference(coordinate system). $S$ and $S^{'}$ are inertial w.r.t each other but are non-inertial w.r.t $O$.\n\nWhat do you mean that's the only result required? Are you referring to the assumption that both frames have the same acceleration, that the relative velocity is constant, or what?\n\nThat's true if both frames are accelerating at a uniform velocity - your coordinate transformations would be the same even if Newtonian physical laws don't hold. If they have two separate accelerations then the law \"$\\mathbf{V'}-\\mathbf{V}=u$\" (where $u$ is independent of time) holds, and your equation for $v'$ in the frame of $S$ will look Galilean.\n\nIn the case of two frames with identical accelerations, by measuring the velocity of the other frame and not taking into account any physical observations, yes you can derive the Galilean transformations. But it's easy to construct situations where this is not true. If the accelerations of the frames are different you don't get a galilean transformation, and if the frames have angular velocity (that is, if they are rotating reference frames) things would get even weirder. (In that case I'm not sure what an interesting question to ask would be!)\n\nThe special relativistic version of this would be different. One issue is usually phrased as: If you tie a string between two spaceships accelerating uniformly (with the same acceleration) separated by some distance, the string will break. (Bell's Spaceship Paradox) Since in one frame the other spaceship would look as if it accelerated away, clearly you can't have events transform linearly from one frame to the other, so the Lorentz transform won't hold. I don't know if there's some configuration of reference frames and accelerations that would allow this to hold - that would be something to prove, and you can ask it on stackexchange after phrasing the question precisely! (and giving it a shot yourself. I'd phrase it as something like, \"Given a frame $S_1$ and $S_2$, with positions $X_1(t)$ and $X_2(t)$, with $X_1(0)=x_2(0)$ in inertial reference frame $O$, what restrictions must be placed on $X_1$ and $X_2$ so that events transform in a Lorentz-like way from one frame to another? More specifically, do they have to have zero second derivative always.\" [to avoid lack of clarity, when I say \"frame\" here I mean \"instantaneous rest frame\".] )" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93873906,"math_prob":0.99269485,"size":2240,"snap":"2019-35-2019-39","text_gpt3_token_len":514,"char_repetition_ratio":0.13774598,"word_repetition_ratio":0.0,"special_character_ratio":0.221875,"punctuation_ratio":0.083538085,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996799,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-23T08:51:10Z\",\"WARC-Record-ID\":\"<urn:uuid:b041d38c-1498-493d-85ee-c63a6837db41>\",\"Content-Length\":\"142781\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4f1a817f-269c-4c13-bec4-05051f699304>\",\"WARC-Concurrent-To\":\"<urn:uuid:a34b6764-89bd-4516-81d3-53b255882ee6>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://physics.stackexchange.com/questions/92058/is-the-assumption-that-the-two-reference-frames-be-inertial-required-in-the-deri\",\"WARC-Payload-Digest\":\"sha1:EN3TL7J5HU425BUP4VOPJIRAN2EPZS4H\",\"WARC-Block-Digest\":\"sha1:BUZENOEX3FHCOIYF7PNH52TAPQUS3XMF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027318243.40_warc_CC-MAIN-20190823083811-20190823105811-00099.warc.gz\"}"}
https://www.tutorialspoint.com/java_cryptography/java_cryptography_keys.htm	[ "# Java Cryptography - Keys\n\nAdvertisements\n\nA cryptosystem is an implementation of cryptographic techniques and their accompanying infrastructure to provide information security services. A cryptosystem is also referred to as a cipher system.\n\nThe various components of a basic cryptosystem are Plaintext, Encryption Algorithm, Ciphertext, Decryption Algorithm, Encryption Key and, Decryption Key.\n\nWhere,\n\n• Encryption Key is a value that is known to the sender. The sender inputs the encryption key into the encryption algorithm along with the plaintext in order to compute the cipher text.\n\n• Decryption Key is a value that is known to the receiver. The decryption key is related to the encryption key, but is not always identical to it. The receiver inputs the decryption key into the decryption algorithm along with the cipher text in order to compute the plaintext.\n\nFundamentally there are two types of keys/cryptosystems based on the type of encryption-decryption algorithms.\n\n## Symmetric Key Encryption\n\nThe encryption process where same keys are used for encrypting and decrypting the information is known as Symmetric Key Encryption.\n\nThe study of symmetric cryptosystems is referred to as symmetric cryptography. Symmetric cryptosystems are also sometimes referred to as secret key cryptosystems.\n\nFollowing are a few common examples of symmetric key encryption −\n\n• Digital Encryption Standard (DES)\n• Triple-DES (3DES)\n• IDEA\n• BLOWFISH\n\n## Asymmetric Key Encryption\n\nThe encryption process where different keys are used for encrypting and decrypting the information is known as Asymmetric Key Encryption. Though the keys are different, they are mathematically related and hence, retrieving the plaintext by decrypting cipher text is feasible.\n\nAdvertisements" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9392097,"math_prob":0.875371,"size":1567,"snap":"2021-21-2021-25","text_gpt3_token_len":294,"char_repetition_ratio":0.1887396,"word_repetition_ratio":0.12173913,"special_character_ratio":0.16528398,"punctuation_ratio":0.089147285,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96335316,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-06-21T03:09:03Z\",\"WARC-Record-ID\":\"<urn:uuid:66b79b09-6f7c-40d0-81b4-48893f6fea65>\",\"Content-Length\":\"26327\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:49a3522d-6223-4080-931c-9f2a948b380b>\",\"WARC-Concurrent-To\":\"<urn:uuid:1a1b5bfc-32ea-44c0-8460-cadbe6acb4d4>\",\"WARC-IP-Address\":\"72.21.91.42\",\"WARC-Target-URI\":\"https://www.tutorialspoint.com/java_cryptography/java_cryptography_keys.htm\",\"WARC-Payload-Digest\":\"sha1:FC66PTX5T75RZGO6TFC5SWHVKJLWG6B2\",\"WARC-Block-Digest\":\"sha1:XR4JY6JJV5AYX4JEWK7IZIXEXQSOU724\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-25/CC-MAIN-2021-25_segments_1623488262046.80_warc_CC-MAIN-20210621025359-20210621055359-00445.warc.gz\"}"}
https://rational-equations.com/rational-equations/algebra-formulas/solve-algebra-equation.html	[ "Algebra Tutorials!", null, "Home", null, "Rational Expressions", null, "Graphs of Rational Functions", null, "Solve Two-Step Equations", null, "Multiply, Dividing; Exponents; Square Roots; and Solving Equations", null, "LinearEquations", null, "Solving a Quadratic Equation", null, "Systems of Linear Equations Introduction", null, "Equations and Inequalities", null, "Solving 2nd Degree Equations", null, "Review Solving Quadratic Equations", null, "System of Equations", null, "Solving Equations & Inequalities", null, "Linear Equations Functions Zeros, and Applications", null, "Rational Expressions and Functions", null, "Linear equations in two variables", null, "Lesson Plan for Comparing and Ordering Rational Numbers", null, "LinearEquations", null, "Solving Equations", null, "Radicals and Rational Exponents", null, "Solving Linear Equations", null, "Systems of Linear Equations", null, "Solving Exponential and Logarithmic Equations", null, "Solving Systems of Linear Equations", null, "DISTANCE,CIRCLES,AND QUADRATIC EQUATIONS", null, "Solving Quadratic Equations", null, "Quadratic and Rational Inequalit", null, "Applications of Systems of Linear Equations in Two Variables", null, "Systems of Linear Equations", null, "Test Description for RATIONAL EX", null, "Exponential and Logarithmic Equations", null, "Systems of Linear Equations: Cramer's Rule", null, "Introduction to Systems of Linear Equations", null, "Literal Equations & Formula", null, "Equations and Inequalities with Absolute Value", null, "Rational Expressions", null, "SOLVING LINEAR AND QUADRATIC EQUATIONS", null, "Steepest Descent for Solving Linear Equations", null, "The Quadratic Equation", null, "Linear equations in two variables\nTry the Free Math Solver or Scroll down to Resources!\n\n Depdendent Variable\n\n Number of equations to solve: 23456789\n Equ. #1:\n Equ. #2:\n\n Equ. #3:\n\n Equ. #4:\n\n Equ. #5:\n\n Equ. #6:\n\n Equ. #7:\n\n Equ. #8:\n\n Equ. #9:\n\n Solve for:\n\n Dependent Variable\n\n Number of inequalities to solve: 23456789\n Ineq. #1:\n Ineq. #2:\n\n Ineq. #3:\n\n Ineq. #4:\n\n Ineq. #5:\n\n Ineq. #6:\n\n Ineq. #7:\n\n Ineq. #8:\n\n Ineq. #9:\n\n Solve for:\n\n Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:\n\n### Our users:\n\nI bought Algebrator last year, and now its helping me with my 9th Grade Algebra class, I really like the step by step solving of equations, it's just GREAT !\nKathleen Becker, PA\n\nTo be honest I was a little skeptical at first about how easy Algebrator would be. But it really is the easiest program to get up and running. I was learning algebra within minutes of downloading the software.\nSonya Johnson, TX\n\nGraduating high-school, I was one of the best math students in my class. Entering into college was humbling because suddenly I was barely average. So, my parents helped me pick out Algebrator and, within weeks, I was back again. Your program is not only great for beginners, like my younger brothers in high-school, but it helped me, as a new college student!\nPatrick Ocean, FL\n\nVow! What a teacher! Thanks for making Algebra easy! The software provides amazing ways to deal with complex problems. Anyone caught up there and finds it hard to solve out must buy a copy. Youll get a great tool at a reasonable price.\nAlex Martin, NH\n\nI think this program is one of the most useful learning tools I have purchased (and believe me, I purchased a lot!), It's easy for us as parents- to work with, it saves a lot of our children's precious time.\nM.H., Georgia\n\n### Students struggling with all kinds of algebra problems find out that our software is a life-saver. Here are the search phrases that today's searchers used to find our site. 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lesson\n• mathquizes to solve on line\n• lowest common denominator worksheet\n• green globs graphing equations free trial\n• common denominator with variables\n• free online large equation calculator\n• printable school papers 1st grade\n• convert exponential to decimal excel\n• comparing fractions calculator\n• elementary algebra tutorial or review\n• pizzazz worksheets\n• combining like terms then naming polynomials worksheet\n• positive negative numbers adding subtracting powerpoint presentation\n• notes for permutations and combinations middle school\n• McDougal Little Algebra 2\n• multiplying tenths whole numbers worksheets\n• greatest common factor finder\n• review mathematics grade 10 worksheets\n• solving nonhomogeneous differential equation\n• Scolastic Math\n• quadratic formula in calculator TI84\n• solving equations by completing the square generator\n• multiplacation charts\n• formulae worksheets in maths\n• tricky math questions for grade 2\n• ti-89 laplace\n• root of third order Polynomial expression\n• direct and inverse proportions in algebra\n• whole numbers to decimal\n• Fraction formulas\n• Free Rational Equation Solver\n• Online Polynomial solver\n• basic fractions\n• matlab code solve determinant with variables\n• powerpoint rational exponents algebra\n• holt rinehart winston practical mathematics third edition answers\n• Conceptual Physics third edition answers\n• compound inequalities games\n• year 8 maths papers to do online\n• TI-89 multiple equations\n• McDougal Littell Algebra 2 Florida Edition\n• trigonomic calculators\n• Abstract algebra HW solutions Gallian\n• multiplying fractions with negatives worksheets\n• adding and subtracting negative and positive number worksheets\n• ellipse equation solver\n• free clep guide\n• multiplication lattice worksheets\n• ALGEGBRA HELP\n• rules for adding square roots" ]	[ null, "https://rational-equations.com/images/left_bullet.jpg", null, "https://rational-equations.com/images/left_bullet.jpg", 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http://ballistipedia.com/index.php?title=Ballistic_Accuracy_Classification&oldid=1431	[ "# Ballistic Accuracy Classification\n\n(diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff)\nJump to: navigation, search\n\n## Introduction\n\nBallistic Accuracy Classification™ is a mathematically rigorous system for describing and understanding the precision of ballistic tools like rifles. It provides information that is:\n\n• Useful and easy for consumers to understand\n• Straightforward for enthusiasts and builders to calculate\n• Statistically sound enough for experts to validate\n\nFor more background see Why BAC.\n\nAll firearms and components can be assigned a Ballistic Accuracy Class™ (BAC™), which fully characterizes their accuracy potential. Lower numbers are better. In practice, the lowest possible BAC is 1. (A theoretically perfect rifle system that always puts every shot through the same hole would be BAC 0.)\n\nBehind the scenes, BAC is defined by a single statistical parameter known as sigma (σ). Using the associated statistical model (known as the Rayleigh distribution), statisticians can not only determine the BAC for a particular gun or component, but also compute its expected shooting precision.\n\nBAC™ Sigma (σ) Typical examples Shots within\n¼ MOA radius\nShots within\n½ MOA radius\nClass 1 < 0.1MOA Rail guns 96% 100%\nClass 2 < 0.2MOA Benchrest guns 54% 96%\nClass 3 < 0.3MOA Mil-spec for PSR 29% 75%\nClass 4 < 0.4MOA Competitive auto-loaders 18% 54%\nClass 5 < 0.5MOA Mil-spec for M110 and M24 12% 39%\nClass 6 < 0.6MOA Mil-spec for infantry rifles and ammo 8% 29%\n\nWe can also generate the expected values of more familiar measures, like the extreme spread of a 3- or 5-shot group:\n\nBAC™ 5-shot Groups\n⌀ < 1MOA\nMedian 5-shot\nGroup Spread\n3-shot Groups\n⌀ < 1MOA\nMedian 3-shot\nGroup Spread\nClass 1 100% 0.3MOA 100% 0.2MOA\nClass 2 98% 0.6MOA 99% 0.5MOA\nClass 3 65% 0.9MOA 85% 0.7MOA\nClass 4 26% 1.2MOA 57% 0.9MOA\nClass 5 9% 1.5MOA 35% 1.2MOA\nClass 6 3% 1.8MOA 21% 1.4MOA\n\n### Understanding MOA\n\nAccuracy is described in angular terms, most commonly using the unit \"Minute of Arc\" (MOA). One arc minute spans 1.047\" at 100 yards. Rifle shooters often practice on 100 yard targets, and so they often think in terms of how wide their groups are at 100 yards. People often just round it off and think of 1 MOA as \"one inch at 100 yards.\"\n\nIn the absence of an atmosphere, the angular precision measured at one distance would be valid at all other distances. I.e., a 1\" group at 100 yards would measure 5\" at 500 yards. However, in reality the effects of wind and drag (which, together with gravity, accentuates variations in muzzle velocity) will only increase the angular spread of ballistic groups as distance increases. Therefore, in practice one should expect worse-than-advertised accuracy when shooting at longer distances. The distance at which atmosphere begins to significantly affect precision depends on a bullet’s muzzle velocity and ballistic coefficient. For high-power rifles this is usually beyond 100 yards. Guns shooting subsonic projectiles can begin to suffer after just 25 yards.\n\n### Nomenclature\n\nBAC™ is only meaningful when it conforms to established terms and conventions. BAC must be determined by testing in accordance with the BAC Protocol described in this document.\n\nBallistic Accuracy Classification™ must be supported by the following descriptive parameters:\n\n1. Product tested. (Any component, or group of components, associated with accuracy can be tested.)\n2. Configuration tested. This must include the following details:\n1. Barrel length, material, profile, rifling.\n(E.g., 20\" stainless 1\" bull contour with 6-land 1:10\"-twist cut rifling.)\n2. Receiver, action, and feed mechanism.\n(E.g., AR-10 magazine-fed semi-automatic.)\n3. Ammunition.\n1. If commercial this must include brand, model, and lot.\n(E.g., Federal GM308M Lot#214374H077.)\n2. If custom, this must list component and load formula.\n(E.g., Lapua .308 full-sized brass, WLR primer, 168gr SMK, 44gr Varget, 2.800\" COAL.)\n3. Confidence in BAC. When not conspicuously mentioned, it is assumed that the upper 90% confidence value of sigma is referenced for BAC.\n\nBAC measures are intentionally kept somewhat coarse. Care should be taken to avoid suggesting more than two significant digits of precision. For example, even after shooting 20 rounds through a gun, the 80% confidence interval on the precision estimate typically spans 0.9-1.2 times the estimated value.\n\n### Trademarks\n\nThe following terms are trademarks of Scribe Logistics LLC. They are free to use so long as their use complies with the Nomenclature and Protocol outlined here.\n\n• Ballistic Accuracy Classification™\n• Ballistic Accuracy Class™\n• BAC™\n\nThe trademarks are claimed solely for the purpose of maintaining the integrity of the system and avoiding market confusion.\n\n## Theory\n\n### Statistical Model\n\nBAC™ assumes that the impact of ballistic shots on a target are normally distributed with the same variance along any axis. (Empirical data validate this assumption, and it should be true as long as atmospheric effects are negligible.) Therefore, we use the Rayleigh distribution to model the radius r, or dispersion of each shot, from the center of impact. When the coordinates of the shots have independent $$N(0,\\sigma)$$ distributions along orthogonal axes, the radius of each shot is described by the Rayleigh probability density function:\n\n$$f(r,\\sigma)=\\frac{r}{\\sigma^2}e^{−r^2/2\\sigma^2}$$\n\nThe unbiased estimator for the parameter σ comes from $$\\widehat{\\sigma^2} = \\frac{\\sum r_i^2}{2(n−1)}$$, with confidence $$\\widehat{\\sigma^2} \\sim \\frac{\\sum r^2}{\\chi_{2n-2}^2}$$.\n\n### Simulation\n\nMonte Carlo simulation is adequate for studying and characterizing precision. In fact, many of the results associated with BAC, like the distribution of the extreme spread of a particular number of shots, can only be produced through simulation.\n\nFor simulation purposes random shots should be generated as (x, y) coordinates, where $$X,Y \\sim N(0,\\sigma)$$. It is critical to \"forget\" the known center when using simulated data. When shooting a real gun we never get to know the true center of impact, and instead have to use the sample center. Likewise, Monte Carlo simulations must not reference the known 0 center, and should instead only reference the sample center of whatever group size is being studied.\n\n## Protocol\n\nThe Ballistic Accuracy Classification™ shall be the upper bound of the 90% confidence range on estimated sigma, in units of MOA, multiplied by 10 and rounded to the nearest integer. For example, if the 90% confidence value for sigma on a tested gun is 0.47MOA, then the BAC value is 10 ✕ 0.47 = 4.7, rounded = 5. I.e., in this example we are saying with 90% confidence that the tested gun’s accuracy is no worse than Class 5.\n\n### Classifying a Specimen\n\nIt is not realistic to assign a BAC with fewer than 10 shots. The 90% confidence range with just 10 shots will typically extend to 1.4 times the estimated accuracy value. It will typically take 20 shots to get the outside bound of the 90% confidence interval to within 20% of the estimated value.\n\nYou must not discard data points during testing except for an unrelated failure. (E.g., if you are testing a barrel and you encounter a squib load, that shot may be excluded. But \"fliers\" should not generally be excluded.)\n\nData: Target distance, and (x,y) coordinates of the center of each shot impact. All shots with the same point of aim must be grouped, but multiple groups can be used.\n\nCalculations: The formulas to transform these data into a confidence interval for sigma, and the corresponding BAC, are shown in Media:BallisticAccuracyClassification.xlsx. Given a sample of n shots, over g groups, at a distance d:\n\n1. For each measurement, convert to units of MOA. For example, if measurements are taken in inches, and the target was shot at a distance of d yards, then divide each measurement by 0.01047d\n2. For each group g, calculate the center of the group as $$(\\bar{x}_{i \\in g},\\bar{y}_{i \\in g})$$\n3. For each shot i, find its radius squared relative to the center of the group as $$r_i^2=(x_i−x_g)^2+(y_i−y_g)^2$$\n4. Calculate the upper 90% confidence value for sigma as σU=SQRT[SUM(r2)/CHIINV(0.9,2n-2g)]\n5. Calculate the Ballistic Accuracy Class as =ROUND(σU,0)" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.87972695,"math_prob":0.9622203,"size":9052,"snap":"2019-35-2019-39","text_gpt3_token_len":2273,"char_repetition_ratio":0.10156941,"word_repetition_ratio":0.018220043,"special_character_ratio":0.25375608,"punctuation_ratio":0.12413395,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97014683,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-08-20T16:27:32Z\",\"WARC-Record-ID\":\"<urn:uuid:41701392-d889-4d53-b2f0-0b53fbf1e6d1>\",\"Content-Length\":\"30212\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8f20c793-6394-4486-bf18-a0eb02efc10c>\",\"WARC-Concurrent-To\":\"<urn:uuid:79cdb66d-00bc-4821-bb5a-3c178ebc5347>\",\"WARC-IP-Address\":\"74.208.236.150\",\"WARC-Target-URI\":\"http://ballistipedia.com/index.php?title=Ballistic_Accuracy_Classification&oldid=1431\",\"WARC-Payload-Digest\":\"sha1:WYZMHXAQSVBCLYPED6Y6SKEYSPKNQEDO\",\"WARC-Block-Digest\":\"sha1:ZQBHJ755TVSFXXUCQN5RLSFYDNQYU4OO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-35/CC-MAIN-2019-35_segments_1566027315551.61_warc_CC-MAIN-20190820154633-20190820180633-00060.warc.gz\"}"}
http://hakology.co.uk/tag/objects/	[ "# PyGame : Tutorial PT2 (Drawing objects to the screen)\n\nSo its that time of the week … I apologise if the tutorials are slow in being published as I’m working on a few other projects in my spare time and want these tutorials to be correctly written so I don’t get you in to any bad habits or poor coding situations … (I’ve been reading the documentation for pygame A LOT) so without further a do, here’s pygame tutorial part 2. (Click here for part 1)\n\nIn the last tutorial we covered setting up the basic game window and initialising pygame.\n\nIn this tutorial were going to be drawing some simple shapes and text to the screen and moving them around using the keyboard.\n\nThere are some minor modifications to last weeks code I will explain these as we go along.\n\nnormal = from original tutorial\n\nimport pygame\nfrom pygame.locals import \n\nwWIDTH = 640 # game window width\nwHEIGHT = 480 # game window height\nloc = [0, 0] # location data\n\nThe above code is pretty much the same as last week apart from the loc[0,0] (loc[x,y]) variable this is going to be used to store the xy location of the player.\nHere we set the variables both equal to zero. These numbers represent the players starting position so x=0 and y=0 would result in the player starting in the top left hand corner.\n\ndef main():\n\nr = 1\nwhile(1): # do for a while (game loop)\n\nfor event in pygame.event.get(): # handle events\n\nif event.type == QUIT: # ctrl+c\n\nr = 0 # return 0 = (game is over)\n\nelif event.type == KEYDOWN: # down arrow\n\nif event.key == K_ESCAPE:\n\nr = 0 # return 0 = (game is over)\n\nif event.key == K_q: # q key pressed\n\nr = 0\n\nif event.key == K_LEFT:\n\nloc = loc – 1 # move left\n\nif event.key == K_RIGHT:\n\nloc = loc + 1 # move right\n\nif event.key == K_UP:\n\nloc = loc – 1 # move up\n\nif event.key == K_DOWN:\n\nloc = loc + 1 # move down\n\nSo whats changed above well again most of the code is exactly the same but this time were trapping keys to trigger more events. In the first tutorial we trapped the escape key to make the game exit. Here we are trapping the arrow keys to change the variables held in the (loc) location variable. If the key is pressed down then add or subtract to the loc[x,y] variable depending on which key is being pressed (UP, DOWN, LEFT or RIGHT).\n\nWe will use the loc variable later to draw the objects to the screen.\n\n# keep location visible on the screen\n# check loc is not out of bounds (screen/window size)\n\nif loc < 0:\n\nloc = wWIDTH\n\nif loc > wWIDTH:\n\nloc = 0\n\nif loc < 0:\n\nloc = wHEIGHT\n\nif loc > wHEIGHT:\n\nloc = 0\n\nThe above code checks that the numbers stored in the loc(x,y) variable are within the bounds of the screen. If the bounds are exceeded the code keeps them on the screen. So if the object goes off the left of the screen it will appear on the right and also if the object / player moves off the bottom they will appear at the top and vice versa. The loc positions are compared to the window bounds (the variables we defined at the top of the code. wWIDTH amd wHEIGHT)\n\n#fill the screen with black\nGSURF.fill((0, 0, 0))\n\n#draw stuff here …\n#circle(Surface, color, pos, radius, width=0) -> Rect\n#rect(top,left,width,height)\npygame.draw.circle(GSURF, (0, 255, 0), (loc, loc), 6, 2)\npygame.draw.circle(GSURF, (255, 0, 0), (loc, loc), 4)\n\npygame.draw.rect(GSURF, (0, 0, 255), [loc-10, loc-10, 20, 20], 3)\npygame.draw.ellipse(GSURF, (0,255,0), [loc-30, loc-10, 60, 20], 1)\npygame.draw.arc(GSURF, (255,0,0), [loc-40, loc-40, 80, 80], 3.141, 23.141, 1)\npygame.draw.line(GSURF, (0,0,255), [loc-50, loc-50], [loc+50, loc+50] , 1)\n\nHere’s the good stuff … first of all we fill the canvas / surface black. Its important to draw the background colour first, if we drew the bg colour last we would just end up with a black screen no matter what we’d drawn underneath / first.\n\npygame.draw.circle(GSURF, (0, 255, 0), (loc, loc), 6, 2)\ndraws a green circle to the game surface at the location held in loc[x,y] make the radius 6 pixels and make the stroke / border 2 pixels wide\nso this will not fill the circle\n\npygame.draw.circle(GSURF, (255, 0, 0), (loc, loc), 4)\nThe above code is the same as the last snippet but will fill the circle in red (if the stroke isn’t provided the circle is filled.)\n\npygame.draw.rect(GSURF, (0, 0, 255), [loc-10, loc-10, 20, 20], 3)\nDraws a red rectangle to the screen. Again the last parameter is optional if you omit this parameter the rectangle will be filled red.\n\npygame.draw.ellipse(GSURF, (0,255,0), [loc-30, loc-10, 60, 20], 1)\nDraws an ellipse to the screen. Optional border parameter.\n\npygame.draw.arc(GSURF, (255,0,0), [loc-40, loc-40, 80, 80], 3.141, 23.141, 1)\nDraws an arc to the screen. Last parameter represents line thickness.\nNB. Arc start point and endpoints are measured in radiants.\nThese sound complicated but they are really easy.\nBasically a circle contains 6.28 radiants. (2pi).\nA start point of 0 and endpoint of 3.141 would draw half a circle.\nA start point of 0 and endpoint of 6.282 would draw a whole circle.\nDegrees to radiants = 6.282 / 360 * Degrees\n\npygame.draw.line(GSURF, (0,0,255), [loc-50, loc-50], [loc+50, loc+50] , 1)\nDraws a line from one point to another.\n\nAll the above shapes are drawn @ the location of the player.\n\nmyfont = pygame.font.SysFont(“monospace”, 15)\nlabel = myfont.render(“Frame rate : ” + str(int(GCLOCK.get_fps())), 1, (255,255,0))\nGSURF.blit(label, (loc, loc))\n\n# check the ‘pygame draw’ documentation for more information on other you can draw to the screen\n# above are most of the basics.\npygame.display.flip() # update the screen\nGCLOCK.tick() # update the clock\n\nif r != 1:\n\nbreak\n\nreturn r\n\nAbove we define a font variable as myfont we use the default pygame fonts (which are all available in the documentation.) Next we make a label variable this variable is going to contain the frame rate as this variable isnt a string we have to use str() to convert the fps from a number to a string. (NB. I think I made a school boy error above. Fps is returned as an int, there is no need for the int() function u can remove this. Edit the code so it looks like this str(GCLOCK.get_fps()) ) Then we blit the label object to the screen at the specified location. display.flip() is used for double buffering (which is a whole other topic) ill cover it briefly here if we replaced the last frames image with the current one we were drawing in the loop bad things would happen basically double buffering displays the last drawn image until the next has been drawn and is ready to be displayed. Double buffering cuts out screen flicker.\n\ntick() is then called to update the pygame clock and advance to the next frame. No numbers are passed in here. Before we were passing a number which will restrict or slow the pace of your game by a set amount depending on what number you use. This will also help us to determine accurately how many cycles the game is running at per second accurately.\n\nif __name__ == “__main__”: # main function call\n\nr = 1 # set variable for return value\n\npygame.init() # initialise python pygame\n\nGCLOCK = pygame.time.Clock() # set game clock\n\nGSURF = pygame.display.set_mode((wWIDTH, wHEIGHT)) # main game surface\n\npygame.key.set_repeat(1, 0) # repeat keys delay by 0\n\nwhile r == 1: # quit on 0\n\nr = main() # get return value from main loop (1 == OK … 0 == exit)\n\npygame.quit()\n\nHere is the last bit of code (that initialises the whole game rather important) only one line changed here which is the keyboard repeat rate, this forces the keyboard to repeat key presses if they are held down for a specified period of time.\n\nThat concludes tutorial part two.\n\nHere is a screen shot of what the code does …", null, "Here is a link to the code …\nGIST : https://gist.github.com/caffeinemonster/5953538\nPASTE BIN : http://pastebin.com/sxchRZx7" ]	[ null, "http://hakology.files.wordpress.com/2013/07/screenshot-from-2013-07-09-015841.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8253763,"math_prob":0.91977036,"size":7779,"snap":"2021-31-2021-39","text_gpt3_token_len":2184,"char_repetition_ratio":0.1318328,"word_repetition_ratio":0.047058824,"special_character_ratio":0.30350944,"punctuation_ratio":0.15109573,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.97382116,"pos_list":[0,1,2],"im_url_duplicate_count":[null,7,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-07-28T17:01:48Z\",\"WARC-Record-ID\":\"<urn:uuid:61a9704c-6361-470b-99a7-56ca1ac1baaf>\",\"Content-Length\":\"45325\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:e5a40609-ba98-478a-9727-ccb7c430f975>\",\"WARC-Concurrent-To\":\"<urn:uuid:c6d78a1a-1dd5-4d4c-974a-bf15c622723d>\",\"WARC-IP-Address\":\"85.233.160.139\",\"WARC-Target-URI\":\"http://hakology.co.uk/tag/objects/\",\"WARC-Payload-Digest\":\"sha1:WGR44MIUYRJ43FMYGZ7GGYNDW3A2CVN6\",\"WARC-Block-Digest\":\"sha1:WYAQYO3GMFGZC3ZXSPNENXWPCBWIWE55\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046153739.28_warc_CC-MAIN-20210728154442-20210728184442-00050.warc.gz\"}"}
https://www.unige.ch/math/tggroup/doku.php?id=fables	[ "fables\n\n# Séminaire \"Fables Géométriques\".\n\nThe normal starting time of this seminar is 16.30 on Monday.\n\n2020, Monday, February 17, 16:30, Battelle, Karim Adiprasito\n(University of Copenhagen, Hebrew University of Jerusalem)\n\nAlgebraic geometry of the sphere at infinity, polyhedral de Rham theory and L^2 vanishing conjectures\n\nI will discuss a conjecture of Singer concerning the vanishing of L^2 cohomology on non-positively curved manifolds, and relate it to Hodge theory on a Hilbert space that arises as the limit of Chow rings of certain complex varieties.\n\n2019, Friday, December 6, 15:00, Battelle, Tomasz Pelka (UniBe)\n\nQ-homology planes satisfying the Negativity Conjecture\n\nA smooth complex algebraic surface S is called a Q-homology plane if H_i(S,Q)=0 for i>0. This holds for example if S is a complement of a rational cuspidal curve in P^2. The geometry of such S is understood unless S is of log general type, in which case the log MMP applied to the log smooth completion (X,D) of S is insufficient. The idea of K. Palka was to study the pair (X,(1/2)D) instead. This approach gives much stronger constraints on the shape of D, and leads to the Negativity Conjecture, which asserts that the Kodaira dimension of K_X+(1/2)D is negative. It is a natural generalization e.g. of the Coolidge-Nagata conjecture about rational cuspidal curves, which was recently proved using these methods by M. Koras and K. Palka.\n\nIf this conjecture holds, all Q-homology planes of log general type can be classified. It turns out that, as expected by tom Dieck and Petrie, they are arranged in finitely many discrete families, each obtainable in a uniform way from certain arrangements of lines and conics on P^2. As a consequence, they all satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and their automorphism groups are subgroups of S_3. To illustrate this surprising rigidity, I will show how to construct all rational cuspidal curves (with complements of log general type, satisfying the Negativity Conjecture) inductively, by iterating quadratic Cremona maps. This construction in particular shows that any such curve is uniquely determined, up to a projective equivalence, by the topology of its singular points.\n\n2019, Monday, November 25, 16:30, Battelle, Felix Schlenk (UniNe)\n\n(Real) Lagrangian submanifolds\n\nWe start with describing how Lagrangian submanifolds of symplectic manifolds naturally appear in many ways: In celestial mechanics, integrable systems, symplectic geometry, and algebraic geometry. We then look at real Lagrangians, namely those which are the fixed point set of an anti-symplectic involution. How special is the property of being real? While many of the examples discussed above are real, we explain why the central fibres in toric symplectic manifolds are real only if the moment polytope is centrally symmetric. The talk is based on work of and with Joé Brendel, Yuri Chekanov, and Joontae Kim.\n\n 2019, Friday, November 8, 14:00, Battelle, Johannes Rau (University of Tübingen)\n\nThe dimension of an amoeba\n\nAmoebas are projections of algebraic varieties in logarithmic coordinates and were originally introduced by Gelfand, Kapranov and Zelevinsky in their influential book. Based on some computation, Nisse and Sottile formulated some questions concerning the dimension of amoebas. In a joint work with Jan Draisma and Chi Ho Yuen, we answer these questions by providing a general formula that computes the dimension of amoebas. If time permits, we also discuss the consequences of this formula for matroidal fans.\n\n 2019, Monday, November 4, 16.30, Battelle, Pierrick Bousseau (ETH Zurich)\n\n\nQuasimodular forms from Betti numbers\n\nThis talk will be about refined curve counting on local P2, the noncompact Calabi-Yau 3-fold total space of the canonical line bundle of the projective plane. I will explain how to construct quasimodular forms starting from Betti numbers of moduli spaces of dimension 1 coherent sheaves on P2. This gives a proof of some stringy predictions about the refined topological string theory of local P2 in the Nekrasov-Shatashvili limit. Partly based on work in progress with Honglu Fan, Shuai Guo, and Longting Wu.\n\n\n2019, Monday, October 28, 16.30, Battelle, Ilia Itenberg, (Sorbonne University)\n\n\nPlanes in four-dimensional cubics\n\nWe discuss possible numbers of 2-planes in a smooth cubic hypersurface in the 5-dimensional projective space. We show that, in the complex case, the maximal number of planes is 405, the maximum being realized by the Fermat cubic. In the real case, the maximal number of planes is 357.\n\nThe proofs deal with the period spaces of cubic hypersurfaces in the 5-dimensional complex projective space and are based on the global Torelli theorem and the surjectivity of the period map for these hypersurfaces, as well as on Nikulin's theory of discriminant forms.\n\nJoint work with Alex Degtyarev and John Christian Ottem.\n\n 2019, Monday, October 14, 16:30, Battelle, Igor Krichever (Columbia University)\n\n\nDegenerations of real normalized differentials\n\nThe behavior of real-normalized (RN) meromorphic differentials on Riemann surfaces under degeneration is studied. In particular, it is proved that the residues at the nodes are solutions of a suitable Kirchhoff problem on the dual graph of the curve. It is further shown that the limits of zeroes of RN differentials are the divisor of zeroes of a twisted differential — an explicitly constructed collection of RN differentials on the irreducible components of the stable curve, with higher order poles at some nodes. Our main tool is a new method for constructing differentials on smooth Riemann surfaces, in a plumbing neighborhood of a given stable curve.\n\n 2019, Monday, October 7, 16:30, Battelle, Jérémy Blanc (University of Basel)\n\n\nQuotients of higher dimensional Cremona groups\n\nWe study large groups of birational transformations $\\mathrm{Bir}(X)$, where $X$ is a variety of dimension at least $3$, defined over $\\mathbb{C}$ or a subfield of $\\mathbb{C}$.Two prominent cases are when $X$ is the projective space $\\mathbb{P}^n$, in which case $\\Bir(X)$ is the Cremona group of rank~$n$, or when $X \\subset \\mathbb{P}^{n+1}$ is a smooth cubic hypersurface.In both cases, and more generally when $X$ is birational to a conic bundle, we produce infinitely many distinct group homomorphisms from $\\mathrm{Bir}(X)$ to $\\mathbb{Z}/2$.As a consequence we also obtain that the Cremona group of rank ~$n \\ge 3$ is not generated by linear and Jonquières elements.\n\nJoint work with Stéphane Lamy and Susanna Zimmermann\n\n 2019, Monday, September 30,16:30, Battelle, Roman Golovko (Charles University in Prague)\n\nThe wrapped Fukaya category of a Weinstein manifold is generated by the Lagrangian cocore discs\n\nIn a joint work with B. Chantraine, G. Dimitroglou Rizell and P. Ghiggini, we decompose any object in the wrapped Fukaya category as a twisted complex built from the cocores of the critical (i.e. half-dimensional) handles in a Weinstein handle decomposition. The main tools used are the Floer homology theories of exact Lagrangian immersions, of exact Lagrangian cobordisms in the SFT sense (i.e. between Legendrians), as well as relations between these theories.\n\n 2019, Wednesday, September 25, 11:00, Battelle, Ivan Fesenko, (University of Nottingham)\n\nTwo-dimensional local fields and integration on them\n\nTwo-dimensional local fields include formal loop objects such as $R((t))$, $C((t))$, $Q_p((t))$ and also fields such as $F_p((t_1))((t_2))$, $Q_p\\{\\{t\\}\\}$. They play the fundamental role in two-dimensional number theory, arithmetic geometry, representation theory, algebraic topology and math physics. I will explain basic things about such fields, including their unusual topology and the theory of measure and integration on such fields and Fourier transform which can be viewed as a (rigorous) arithmetic version of the Feynman integral. While one-dimensional local fields show up in tropical geometry of curves, one may expect that two-dimensional local fields should be involved in tropical geometry of surfaces.\n\n 2019, Monday, September 16, 16:30, Battelle, Gleb Smirnov, (ETH Zurich)\n\nFrom flops to diffeomorphism groups\n\nFollowing a short introduction to the flop surgery, I will explain how this surgery can be used to detect non-contractible loops of diffeomorphisms for many algebraic surfaces.\n\n2019, Monday, May 20, 14:00, Battelle, Ziming Ma (Chinese University of Hong Kong)\n\nGeometry of the Maurer-Cartan equation near degenerate Calabi-Yau varieties\n\nIn this talk, we construct a dgBV algebra PV^∗,∗(X) associated to a possibly degenerate Calabi-Yau variety X equipped with local thickening data. This gives a singular version of the (extended) Kodaira-Spencer dgLa which is applicable to both log smooth and maximally degenerated Calabi-Yau. We use this to prove an unobstructedness result about the smoothing of degenerated Log Calabi-Yau varieties X satisfying Hodge-deRham degeneracy property for cohomology of X, in the spirit of Kontsevich-Katzarkov-Pantev. We also demonstrate how our construction can be applied to produce a log Frobenius manifold structure on a formal neighborhood of the extended moduli space using Barannikov’s technique. This is a joint work with Kwokwai Chan and Naichung Conan Leung.\n\n2019, Monday, April 8, 16:00, Battelle, Michele Ancona (Institut Camille Jordan)\n\nRandom sections of line bundles over real Riemann surfaces\n\nGiven a line bundle L over a real Riemann surface, we study the number of real zeros of a random section of L. We prove a rarefaction result for sections whose number of real zeros deviates from the expected one.\n\n2019, Monday, April 1, 16:00, Battelle, Mikhail Shkolnikov (IST Austria)\n\nPSL-tropical limits\n\nThe classical tropical limit is defined for families of varieties in the algebraic torus. One of the ways to generalize this framework is to consider non-commutative groups instead of algebraic tori. We describe tropical limits for subvarieties in PSL(2,C):the result is spelled in terms of floor diagrams and has parallels with symplectic field theory. The talk is based on the work in progress with Grigory Mikhalkin.\n\n2019,Tuesday, March 26, 14:00, Battelle, Enrica Mazzon (Imperial College London)\n\nBerkovich approach to degenerations of hyper-Kähler varieties\n\nTo a degeneration of varieties, we can associate the dual intersection complex, a topological space that encodes the combinatoric of the central fiber and reflects the geometry of the generic fiber. The points of the dual complex can be identified to valuations on the function field of the variety, hence the dual complex can be embedded in the Berkovich space of the variety. In this talk I will explain how this interpretation gives an insight in the study of the dual complexes. I will focus on some degenerations of hyper-Kähler varieties and show that we are able to determine the homeomorphism type of their dual complex using techniques of Berkovich geometry. The results are in accordance with the predictions of mirror symmetry, and the recent work about the rational homology of dual complexes of degenerations of hyper-Kähler varieties, due to Kollár, Laza, Saccà and Voisin. This is joint work with Morgan Brown.\n\n2019, Monday, March 18, 16:00, Battelle, Danilo Lewanski (Max Planck Institut für Mathematik)\n\nRefreshing Tropical Jucys curves\n\nWe derive explicit formulae for the generating series of Grothendieck dessins d’enfant and monotone Hurwitz numbers via the semi-infinite wedge formalism, and from it we obtain bosonic Fock space expressions. This yields to a tropical geometric interpretation involving Gromov-Witten invariants as local multiplicities.\n\n2019, Monday, March 11, 16:00, Battelle, Anton Mellit (University of Vienna)\n\nFive-term relations\n\nI will review how the five term relation for the Fadeev-Kashaev's quantum dilogarithm arises in the Hall algebra context, and sketch a simple proof. Then I will explain how this proof can be transported to the elliptic Hall algebra situation, where the five term relation implies identities between Macdonald polynomials conjectured by Bergeron and Haiman. This is a joint work with Adriano Garsia.\n\n2019, Monday, March 4, 16:00, Battelle, Andras Stipsicz (Budapest University)\n\nKnot Floer homology and double branched covers\n\nWe will review the basic constructions of (various versions of) knot Floer homologies, show some applications and extensions of the definitions to the double branched cover, also using the covering transformation.\n\n2019, Monday, February 25, 16:00, Battelle, Erwan Brugallé (Université de Nantes)\n\nOn the invariance of Welschinger invariants\n\nWelshinger invariants are real analogs of Gromov-Witten invariants for symplectic 4-manifolds X. In this talk, I will strengthen the original Welschinger's invariance result. Our main result is that when X is a real rational algebraic surface, Welschinger invariants eventually only depend on the number of real interpolated points, and some homological data associated to X. This result follows easily from a formula relating Welschinger invariants of two real symplectic manifolds differing by a surgery along a real Lagrangian sphere. As an application, we complete the computation of Welschinger invariants of real rational algebraic surfaces, and obtain vanishing, sign, and sharpness results generalizing previously known statements. If time permits, we will also discuss some hypothetical relations with tropical refined invariants defined by Block-Göttsche and Göttsche-Schroeter.\n\n2018, Monday, December 10, 16:00, Battelle, Arthur Renaudineau (Lille)\n\nLefschetz hyperplane section theorem for tropical hypersurfaces\n\nWe will discuss variants of the Lefschetz hyperplane section theorem for the integral tropical homology groups of non-singular tropical hypersurfaces of toric varieties. As an application, we get that the integral tropical homology groups of non-singular tropical hypersurfaces are torsion free. This is a joint work with Charles Arnal and Kristin Shaw.\n\n2018, Monday, November 26, 16:00, Battelle, Vladimir Fock (Strasbourg)\n\nHigher complex structures on surfaces\n\nWe suggest a definition of a differential geometric structures on surfaces generalizing the notion of complex structure and discuss its properties. The moduli space of such structures share many common features and conjecturally coincide with higher Teichmüller space - the space of positive representations of the fundamental group of the surface into PGL(N) (like moduli of ordinary complex structure give a representation of the fundamental group to PGL(2)). Joint work with A.Thomas.\n\n2018, Monday, November 19, 16:15, Battelle, Stepan Orevkov (Moscow, Toulouse)\n\nOrthogonal polynomials in two variables\n\nA natural generalization of classical systems of (one-variable) orthogonal polynomials is as follows. Let $D$ be a domain in $R^n$ endowed with a Riemannian metric and a mesure. Suppose that the Laplace-Beltrami operator (for the given metric) is symmetric (for the given mesure) and leave invariant the set of polynomials of a given degree. Then its eigenfunctions is a system of orthogonal polynomials.\n\nI present a complete classification of domains in $R2$ for which this construction can be applied. The talk is based on a joint work with D. Bakry and M. Zani.\n\n2018, Monday, October 8, 16:30, Battelle, Sione Ma'u (Auckland)\n\nPolynomial degree via pluripotential theory\n\nGiven a complex polynomial $p$ in one variable, $\\log\|p\|$ is a subharmonic function that grows like $(deg p)\\log\|z\|$ as $\|z\|\\to\\infty$. Such functions are studied using complex potential theory, based on the Laplace operator in the complex plane.\n\nMultivariable polynomials can also be studied using potential theory (more precisely, a non-linear version called pluripotential theory, which is based on the complex Monge-Ampere operator). In this talk I will motivate and define a notion of degree of a polynomial on an affine variety using pluripotential theory (Lelong degree). Using this notion, a straightforward calculation yields a version of Bezout's theorem. I will present some examples and describe how to compute Lelong degree explicitly on an algebraic curve. This is joint work with Jesse Hart.\n\n2018, Monday, October 1, 16:00, Battelle, Mikhail Shkolnikov (Klosterneuburg)\n\nExtended sandpile group and its scaling limit\n\nSince its invention, the sandpile model is believed to be renormalizable due to the presence of power-laws. It appears that, the sandpile group, made of recurrent configurations of the model, approximates a continuous object that we call the extended sandpile group. In fact, this is a tropical abelian variety defined over Z and the subgroup of its integer points is exactly the usual sandpile group. Moreover, the extended sandpile group is naturally a sheaf on discrete domains and, thus, brings an explicit scale renormalization procedure for recurrent configurations. We compute the (projective) scaling limit of sandpile groups along growing convex domains: its is equal to the quotient of real-valued discrete harmonic functions by the subgroup of integer-valued ones. This is a joint work with Moritz Lang.\n\n2018, Wednesday, July 18, 16:30, Battelle, Kristin Shaw (University of Oslo)\n\nChern-Schwartz-MacPherson classes of matroids. Part II\n\nChern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes to singular and non-complete varieties. Matroids are an abstraction of the notion of independence in mathematics. In this talk, I will provide a combinatorial analogue of CSM classes for matroids, motivated by the geometry of hyperplane arrangements. In this setting, CSM classes are polyhedral fans which are Minkowski weights. One goal for defining these classes is to express matroid invariants as invariants from algebraic geometry. The CSM classes can be used to study the complexity of more general objects such as subdivisions of matroid polytopes and tropical manifolds. This is based on joint work with Lucia López de Medrano and Felipe Rincón.\n\n2018, Monday, July 16, 16:00, Battelle, Kristin Shaw (University of Oslo)\n\nChern-Schwartz-MacPherson classes of matroids\n\nChern-Schwarz-Macpherson (CSM) classes are one way to extend the notion of Chern classes to singular and non-complete varieties. Matroids are an abstraction of the notion of independence in mathematics. In this talk, I will provide a combinatorial analogue of CSM classes for matroids, motivated by the geometry of hyperplane arrangements. In this setting, CSM classes are polyhedral fans which are Minkowski weights. One goal for defining these classes is to express matroid invariants as invariants from algebraic geometry. The CSM classes can be used to study the complexity of more general objects such as subdivisions of matroid polytopes and tropical manifolds. This is based on joint work with Lucia López de Medrano and Felipe Rincón.\n\n2018, Monday, July 9, 16:30, Battelle, Ernesto Lupercio (CINVESTAV)\n\nConvex geometry, complex systems and quantum physics\n\nI will speak about our work on sandpiles and quantum integrable systems. Just as in classical mechanics toric manifolds correspond to rational convex polytopes, the irrational case in informed by the theory of sandpiles. Joint with Kalinin, Shkolnikov, Katzarkov, Meersseman and Verjovsky.\n\nWorkshop \"Fables Géométriques\", 2018, June Friday 15 and Saturday 16, Battelle\n\nFriday June 15th\n\n11:00-12:00 Yakov Eliashberg (Stanford)\n\n12:30 lunch\n\n14:30-15:30 Sergey Finashin (METU)\n\n16:00-17:00 Viatcheslav Kharlamov (Strasbourg)\n\nSaturday June 16th\n\n16:00-17:00 Stepan Orevkov (Toulouse)\n\n17:30-18:30 Oleg Viro (Stony Brook)\n\n19:30 dinner\n\n2018, Monday, May 28, 15:00, Battelle, Alexander Esterov (Higher School of Economics - Moscow)\n\nTropical characteristic classes and Plücker formulas\n\nGiven a proper generic map of manifolds, the Thom polynomial counts (in terms of characteristic classes of the manifolds), how many fibers of the map have a prescribed singularity. However, this tool cannot be directly applied to the study of generic polynomial maps $C^m \\to C^n$, because they are not proper. An attempt to extend Thom polynomials in this natural direction leads to what can be called tropical Thom polynomials and tropical characteristic classes.\n\nI will introduce tropical characteristic classes of (very) affine algebraic varieties, compute the tropical version of the simplest Thom polynomials (the Plücker formulas for the number of cusps and nodes of a projectively dual curve), and outline their relation to tropical correspondence theorems and some other possible applications.\n\n2018, Friday, May 25, 10:30, Battelle, Dimitry Kaledin (Steklov & NRU HSE - Moscow)\n\nWitt vectors, commutative and non-commutative, II\n\nWitt vectors were first introduced eighty years ago, but they still come up in different questions of commutative and homological algebra, algebraic geometry, and even algebraic topology. I will try to give a general introduction to this remarkable subject, and show both its classical parts and some recent discoveries. The first talk will be quite elementary, first-year algebra should be enough. In the somewhat more advanced second talk, I will try to explain how the simple constructions of the first talk lead to non-commutative generalization of Grothendieck's cristalline cohomology of smooth algebraic varieties over a finite field.\n\n2018, Tuesday, May 22, 15:30, Battelle, Dimitry Kaledin (Steklov & NRU HSE - Moscow)\n\nWitt vectors, commutative and non-commutative, I\n\nWitt vectors were first introduced eighty years ago, but they still come up in different questions of commutative and homological algebra, algebraic geometry, and even algebraic topology. I will try to give a general introduction to this remarkable subject, and show both its classical parts and some recent discoveries. The first talk will be quite elementary, first-year algebra should be enough. In the somewhat more advanced second talk, I will try to explain how the simple constructions of the first talk lead to non-commutative generalization of Grothendieck's cristalline cohomology of smooth algebraic varieties over a finite field.\n\n2018, Monday, May 14, 16:30, Battelle, Ilia Itenberg (Paris VI - ENS)\n\nFinite real algebraic curves\n\nThe talk is devoted to real plane algebraic curves with finitely many real points. We study the following question: what is the maximal possible number of real points of such a curve provided that it has given (even) degree and given geometric genus? We obtain a complete answer in the case where the degree is sufficiently large with respect to the genus, and prove certain lower and upper bounds for the number in question in the general case. This is a joint work with E. Brugallé, A. Degtyarev and F. Mangolte.\n\n2018, Monday, April 16, 16:30, Battelle, Ludmil Katzarkov (Vienna)\n\nHomological mirror symmetry and the P=W conjecture\n\n2018, Monday, March 5, 16:00, Battelle, Rahul Pandharipande (ETH)\n\nOn Lehn's conjecture for Segre classes on Hilbert schemes of points of surfaces and generalizations\n\nLet L→S be a line bundle on a nonsingular projective surface. I will discuss recent progress concerning the formula conjectured by Lehn in 1999 for the top Segre class of the tautological bundle L^[n] on Hilb(S,n) and the parallel question for vector bundles V→S. Results of Voisin play a crucial role. The talk represents joint work with A. Marian and D. Oprea.\n\n2018, Monday, February 26, 16:30, Battelle, Anton Fonarev (Higher School of Economics)\n\nEmbedding derived categories of curves into derived categories of moduli of stable vector bundles\n\nOne out of many interesting questions about derived categories is the following conjecture by A. Bondal: the bounded derived category of coherent sheaves of a smooth projective variety can be embedded into the bounded derived category of coherent sheaves of a smooth Fano variety. This conjecture is rather nontrivial even for curves. We will show how to embed the derived category of a generic curve of genus g > 1 into the derived category of rank 2 stable vector bundles with a fixed determinant of odd degree. The proof is a nice interplay of algebraic geometry, representation theory and categorical methods. The talk is based on a joint work with A. Kuznetsov.\n\n2017, Monday, November 6, 16:30, Battelle, Jeffrey Giansiracusa (Swansea University)\n\nTropical geometry as a scheme theory\n\nTropical geometry has become a powerful tool set for tackling problems in algebraic geometry, combinatorics, and number theory. The basic objects have traditionally been considered as certain polyhedral sets and heuristically thought of as algebraic objects defined over the real numbers with the max-plus semiring structure. I will explain how to realize this within an extension of scheme theory and describe the particular form of the equations of tropical varieties in terms of matroids.\n\n2017, Monday, October 30, 16:30, Battelle, Diego Matessi (Università degli Studi di Milano)\n\nFrom tropical hypersurfaces to Lagrangian submanifolds\n\nI will explain a construction of Lagrangian submanifolds of $(\\mathbb{C}^)^2$ or $(\\mathbb{C}^)^3$ which lift tropical hypersurfaces in $\\mathbb{R}^2$ or $\\mathbb{R}^3$. The building blocks are what I call Lagrangian pairs of pants. These can be constructed as graphs of the differential of a smooth function defined on a Lagrangian co-ameba. I will also explain some possible generalizations and applications to mirror symmetry.\n\n2017, Monday, October 2, 16:30, Battelle, Dmitry Novikov (Weizmann Institute)\n\nComplex cellular parameterization\n\n(joint work with Gal Binyamini)\n\nWe introduce the notion of a complex cell, a complex analog of the cell decompositions used in real algebraic and analytic geometry. Complex cells defined using holomorphic data admit a natural notion of analytic continuation called $\\delta$-extension, which gives rise to a rich hyperbolic geometric structure absent in the real case. We use this structure to prove that complex cellular decompositions share some interesting features with the classical constructions in the theory of resolution of singularities. Restriction of a complex cellular decomposition to the reals recovers the preparation theorem for subanalytic functions, and can be viewed as an analytic continuation thereof.\n\nA key difference in comparison to the classical resolution of singularities is that the cellular decompositions are intrinsically uniform over (sub)analytic families. We deduce a subanalytic version of the Yomdin-Gromov theorem where $C^k$-smooth maps are replaced by mild maps.\n\n2016, Friday, June 23, 11:00, Battelle, Ernesto Lupercio (CINVESTAV)\n\nQuantum toric varieties\n\nI will describe the theory of quantum toric varieties that generalizes usual toric geometry. Joint with Meersseman, Katzarkov and Verjovsky.\n\n2016, Thursday, June 22, 11:30, Battelle, Conan Leung (CUHK)\n\nInformal introduction to G_2-manifolds III\n\n2016, Wednesday, June 21, 11:30, Battelle, Conan Leung (CUHK)\n\nInformal introduction to G_2-manifolds II\n\n2016, Monday, June 19, 15:00, Battelle, Conan Leung (CUHK)\n\nInformal introduction to G_2-manifolds I\n\n Villa Battelle, May 2, 14:00-15:00; May 3 14:15-15:15; May 5, 14:30-15:30, Aaron Bertram (Utah)\n\nMinicourse: “Moduli Spaces of Complexes in Algebraic Geometry ”\n\nThe ideal of the twisted cubic in projective three-space is completely described by a 2×3 matrix of linear forms in four variables. The space of such matrices (modulo the actions of GL(2) and GL(3)) is a smooth, projective variety compactifying the space of twisted cubics. But the objects parametrized by the points at the boundary of this moduli space are not ideals of curves. They are complexes of line bundles that are stable with respect to a “stability condition on the derived category.” What does this mean? Can this be used to systematically find nice models for moduli and relate them to moduli spaces of coherent sheaves?\n\nDay 1) Introduction to Stability Conditions. Ordinary stability of vector bundles on a Riemann Surface relies on two invariants: the rank and degree (first chern class). A stability condition on the derived category of coherent sheaves on a complex manifold relies on a generalized rank and degree, and also on an exotic t-structure on the derived category, with an abelian category of complexes at its heart. On an algebraic surface, there are stability conditions whose underlying heart can be described by a tilting construction. However, finding a single stability condition on a projective Calabi-Yau threefold (e.g. the quintic in P4) remains open.\n\nDay 2) Models of the Hilbert Schemes of Points on a Surface. As the stability condition varies, the moduli spaces of stable objects (with respect to the stability condition) undergo a series of birational transformations. The particular example of the Hilbert scheme of ideal sheaves on an algebraic surface has been studied for various classes of surfaces. We will survey some results.\n\nDay 3) The Euler Stability Condition on Projective Space. An interesting stability condition on P^n has the Euler characteristic playing the role of the rank. We will use this stability condition to study stratifications of the spaces of symmetric tensors, generalizing the secant varieties to the Veronese embeddings of P^n. This is joint work with Brooke Ullery.\n\n Villa Battelle, Monday, Apr 3, 16:30-17:30, Lionel Lang (Uppsala University)\n\nThe vanishing cycles of curves in toric surfaces : the spin case\n\nIf the interior polygon of a lattice polygon $\\Delta$ is divisible by 2, any generic curve $C$ of the linear system associated to $\\Delta$ admits a spin structure $q$. If a loop in $C$ is a vanishing cycle, then the Dehn twist along the loop has to preserve $q$. As a consequence, the image of the monodromy of the linear system is a subgroup of the mapping class group $MCG(C,q)$ that preserves $q$. The main goal of this talk is to compare the image of the monodromy with $MCG(C,q)$. To this aim, we will show on one side that $MCG(C,q)$ admits a very explicit set of generators. On the other, we will construct elements of the monodromy by tropical means. The conclusion will be that the image of the monodromy is the full group $MCG(C,q)$ if and only if the interior polygon admits no other divisors than 2. (joint with R. Crétois)\n\nVilla Battelle, Wednesday, Mar 8, 12:00, Maksim Karev (PDMI)\n\nMonotone Hurwitz Numbers\n\nUsual Hurwitz numbers count the number of covers over CP^1 with a fixed ramification profile over point \\infty and simply ramified over a specified set of points. They also can be treated as a weighted count of factorizations in the symmetric group. It is known, that Hurwitz numbers can be calculated via intersection indices on the moduli spaces of complex curves by so-called ELSV-formula.\n\nIn my talk, I will discuss monotone Hurwitz numbers, which also arise as factorizations count with restrictions. It turns out, that they also can be related to the intersection indices on the moduli spaces of complex curves. I will give a definition of monotone Hurwitz numbers, and try to explain the origin of the monotone ELSV. If time permits, I will speak about the further development of the subject.\n\nThe talk is based on the joint work with Norman Do (Monash University).\n\nVilla Battelle, Tuesday, Feb 21, 15:30, Yang-Hui He (London, Nankai and Oxford)\n\nCalabi-Yau Varieties: From Quiver Representations to Dessins d'Enfants\n\nWe discuss how bipartite graphs on Riemann surfaces encapture a wealth of information about the physics and the mathematics of gauge theories. The correspondence between the gauge theory, the underlying algebraic geometry of its space of vacua as a quiver variety, the combinatorics of dimers and toric varieties, as well as the number theory of dessin d'enfants becomes particularly intricate under this light.\n\nJoint session of “Fables géométriques” and “Groupes de Lie et espaces des modules” seminars.\n\nVilla Battelle, Monday, Feb 20, 16:30, Yang-Hui He (London, Nankai and Oxford)\n\nThere tends to be exceptional structures in classifications: in geometry, there are the Platonic solids; in algebra, there are the exceptional Lie algebras; in group theory, there are the sporadic groups, to name but a few. Could these exceptional structures be related in some way? A champion for such Correspondences is Prof. John McKay. We take a casual promenade in this land of exceptionology, reviewing some classic results and presenting some new ones based on joint work with Prof. McKay.\n\nSpecial lecture for “Geometry, Topology and Physics” masterclass students.\n\nVilla Battelle, Friday, December 9, 14:30-15:30, Ozgur Ceyhan (Luxembourg)\n\nBackpropagation, its geometry and tropicalisation\n\nThe algorithms that make current artificial neural networks successes possible are decades old. They became applicable only recently as these algorithms demand huge computational power. Any technique which reduces the needs for computation have a potential to make great impact. In this talk, I am going to discuss the basics of backpropagation techniques and tropicalisation of the problem that promises to reduce the time complexity and accelerate computations.\n\n2016,Monday, November 7, 16:30, Battelle, Vladimir Fock.\n\nSeparation of variables in cluster integrable systems\n\nCluster integrable systems can be viewed from five rather different points of view. 1. As a double Bruhat cell of an affine Lie -Poisson group; 2. As a space of pairs (planar algebraic curve, line bundle on it); 3. Space of Abelian connection on a bipartite graph on a torus; 4. Hilbert scheme of points on algebraic torus. 5. Collection of flags in an infinite space invariant under the action of two commuting operators. We will see the relation between all these descriptions and discuss its quantization and possible generalizations.\n\n2016,Friday, Nov 4, 14:30-15:15 part I, 15:30-16:15 part II, Johannes Walcher (Heidelberg).\n\nIdeas of D-branes\n\nAbstract: I will give an introduction to D-branes from the point of view of their origin in the physics of string theory. I will discuss both world-sheet and space-time aspects.\n\n 2016, Monday, 23 mai, 16.30, Battelle, Frédéric Bihan.\n\nUne généralisation de la règle de Descartes pour les systèmes polynomiaux dont le support est un circuit\n\nRésumé : La règle de Descartes borne le nombre de racines positives d'un polynôme réel en une variable par le nombre de changements de signe consécutifs de ses coordonnées dans la base monomiale (ordonnée suivant les puissances croissantes). La borne obtenue est optimale et généraliser la règle de Descartes aux systèmes polynomiaux en plusieurs variables est un problème très difficile. Dans un travail avec Alicia Dickenstein (Université de Buenos Aires), nous avons obtenu une généralisation partielle de la règle de Descartes en plusieurs variables. Notre règle s'applique aux systèmes polynomiaux en un nombre arbitraire n de variables dont le support consiste en n+2 monômes quelconques. Comme pour la règle de Descartes usuelle, notre borne est optimale et s'exprime comme un nombre de changement de signes d'une suite de nombres obtenus en considérant les mineurs maximaux de la matrice des coefficients ainsi que de celle des exposants du système.\n\n(in English)Descartes' Rule of Signs for Polynomial Systems Supported on Circuits\n\nDescartes’ rule of signs bounds the number of positive roots of an univariate polynomial by the number of sign changes between consecutive coefficients. In particular, this produces a sharp bound depending on the number of monomials. Generalizing Descartes’ rule of signs or the corresponding sharp bound to the multivariable case is a challenging problem. In this talk, I will present a generalization of Descartes’ rule of signs for the number of positive solutions of any system of n real polynomial equations in n variables with at most n+2 monomials. This is a joint work with Alicia Dickenstein (Buenos Aires University).\n\n2016, Monday, 9 mai, Battelle, Eugenii Shustin, 18.30-19.15\n\nOn refined tropical invariants of toric surfaces.\n\nWe discuss two examples of refined count of plane tropical curves. One of them is the refined broccoli invariant. It was introduce by Goettsche and Schroeter for genus zero case, and it turns into some descendant invariant or the broccoli invariant according as the parameter takes value 1 or -1. A possible extension of broccoli invariant to positive genera appeared to be rather problematic. However, the refined version turns to be easier to treat. Jointly with F. Schroeter, we have defined a refined broccoli invariant, counting elliptic tropical curves. This can be done for higher genera as well (work in progress). Another example (joint work with L. Blechman) is the refined descendant tropical invariant (involving arbitrary powers of psi-classes). We discuss also the most interesting related question: What is the complex and real enumerative meaning of these invariants?\n\n2016, Monday, 4 april, 16.30, Battelle.\n\nLionel Lang (Uppsala)\n\nThe vanishing cycles of curves in toric surfaces (joint work with Rémi Crétois)\n\nIn [Do], Donaldson addressed the following : Do all Lagrangian spheres in a complex projective manifold arise from the vanishing cycles of a deformation to singular varieties? The answer might depend on the choice of the moduli space in which we are allowed to deform our manifold. Already for curves, it leads to interesting questions. In the Deligne-Mumford moduli space M_g, any loop inside a smooth curve can be contracted along a deformation towards a nodal (stable) curve, provided that the genus g>1. What happens if one restricts to a chosen linear system on a toric surface? Degree d curves in the projective plane, for instance. In the latter, two obstructions occur: the loop should not be separating for d>2 (Bezout), the Dehn twist along the loop should preserve a certain spin structure on the curve for d odd (see [Beau]). In the latter, Beauville proves (in particular) that any non-obstructed loop is homologous to a vanishing cycle. In this talk, we suggest a tropical proof of Beauville's result as well as an extension to any (big enough) linear systems on any smooth toric surface. This problem is directly related to the monodromy group given by the complement of the discriminant in the considered linear system. The proof will involve simple Harnack curves, introduced by Mikhalkin, and monodromy given by partial tropical compactifications of the linear system. If time permits, we will also discuss this problem at the isotopic level, problem that is still open.\n\n[Beau] : Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections complètes. A. Beauville, 1986. [Do] : Polynomials, vanishing cycles and Floer homology. S.K. Donaldson, 2000.\n\n2016, Monday, 21 mars, 16.30, Battelle.\n\nBoris Shapiro (Stockholm)\n\nOn the Waring problem for polynomial rings\n\nWe discuss a natural analog of the classical Waring problem for $C[x_1,…,x_n]$. Namely, we show that a general form p from $C[x_1,…,x_n]$ of degree kd where k>1 can be represented as a sum of at most $k^n$ k-th powers of forms of degree d. Noticeably, $k^n$ coincides with the number obtained by naive dimension count if d is sufficiently large.\n\n2016, Friday, March 18, 14.15, villa Battelle.\n\nSergey Galkin (Moscow)\n\nGamma conjectures and mirror symmetry\n\nI will speak about an exotic integral structure in cohomology of Fano manifolds that conjecturally can be expressed in terms of Euler's gamma-function, how one can observe it by computing asymptotics of a quantum differential equation, and how one can prove the conjectures using mirror symmetry. This is a joint work with Vasily Golyshev and Hiroshi Iritani (1404.6407, 1508.00719).\n\n2016, Thursday, March 17 Colloquium, villa Battelle\n\nVassily Golyshev (Moscow), 16:15\n\nAround the gamma conjectures.\n\nAbstract: We will state the gamma conjectures for Fano manifolds and explain how quantum cohomology makes it possible to enhance the classical Riemann-Roch-Hirzebruch theorem by relating the curve count on a variety to its characteristic classes. We will indicate how the gamma conjectures are proved in the known cases.\n\n2016, Monday, 14 mars, 16.30, Battelle.\n\nE. Abakoumov (Paris-Est)\n\nGrowth of proper holomorphic maps and tropical power series\n\nHow fast a proper holomorphic map, say, from C to C^n can grow? It turns out that the tropical power series appear naturally in answering this question, as well as in some related approximation problems on the complex plane. The talk is based on joint work with E. Dubtsov.\n\n2015, Tuesday, 8 December, 14.30, Battelle. (joint with Séminaire \"Groupes de Lie et espaces des modules”)\n\nBernd Sturmfels (UC Berkeley)\n\nExponential Varieties\n\nExponential varieties arise from exponential families in statistics. These real algebraic varieties have strong positivity and convexity properties, familiar from toric varieties and their moment maps. Another special class, including Gaussian graphical models, are inverses of symmetric matrices satisfying linear constraints. We present a general theory of exponential varieties, with focus on those defined by hyperbolic polynomials. This is joint work with Mateusz Michalek, Caroline Uhler, and Piotr Zwiernik.\n\n2015, Tuesday, December 8, 11:15 -- 12:15, Battelle.\n\nTropical geometry: a graphical interface for the GW/Hurwitz correspondence.\n\nIn their study of the Gromov-Witten theory of curves [OP], Okounkov and Pandharipande used the degeneration formula to express stationary descendant invariants of curves in terms of Hurwitz numbers and one point descendant relative invariants. Then they use operator formalism to organize the combinatorics of the degeneration formula, and the one point invariants into completed cycles. In joint work with Paul Johnson, Hannah Markwig and Dhruv Ranganathan, we revisit their formalism and show that the Feynmann diagrams that are secretly behind the scenes in [OP] are in fact tropical curves. This yields some mild refinements of the Gromov-Witten/Hurwitz correspondence of [OP]. Time permitting we will describe how a generalization of these tecniques should lead to unveiling a similar structure in the stationary/descendant GW theory of sliceable surfaces.\n\n2015, Monday, 7 December, 16.15, Villa Battelle\n\nIsrael Vainsencher (Universidade Federal de Minas Gerais, Brasil)\n\nLegendrian curves\n\nA twisted cubic curve in 3-space is known to define a (non-integrable) distribution of planes. The planes of the distribution osculate the original twc. We show how to define virtual numbers N_d which enumerate the rational curves of degree d which are tangent to that distribution and further meet 2d+1 general lines. (Based on Eden Amorim thesis)\n\nThe next lecture of the course\n\n“Imaginary time in Kaehler geometry, quantization and tropical amoebas” by José Mourão\n\nwill be on Monday 9 November, 17.00 Battelle.\n\n2015, October 27, 15.15 and October 29, 16.15, and November 2, 17.00, Villa Battelle\n\n(Minicourse) Imaginary time in Kaehler geometry, quantization and tropical amoebas.\n\nJosé Mourão, Mathematics Department, Instituto Superior Tecnico, Portugal.\n\nFor a compact Kaehler manifold $M$ and a function $H$ on $M$ we give a simple definition of the continuation of the flow defined by $H$ to complex time, $\\tau$, using the Groebner theory of Lie series. The resulting complexified (or complex time) symplectomorphisms are diffeomorphisms for some $\|\\tau\|< R_H$. For larger values of $\|\\tau\|$ they may correspond e.g. to the collapse of $M$ to a totally real submanifold. Simple examples will be discussed.\n\nKahler geometry applications: Imaginary time symplectomorphisms correspond to Mabuchi geodesics in the infinite dimensional space of Kaehler metrics with fixed cohomology class. We get thus a explicit way of constructing Mabuchi geodesics from Hamiltonian flows.\n\nQuantum theory applications: By lifting the imaginary time symplectomorphisms to the quantum bundle we get generalized coerent state transforms and are able to study the unitary equivalence of quantizations corresponding to nonequivalent polarizations.\n\nTropical geometry applications: For toric varieties the toric geodesics of the Mabuchi metric are straight lines in the space of Guillemin-Abreu symplectic potentials. Taking a strictly convex function $H$ (as a function on the moment polytope) one has that, for large geodesic times s, there is a simple relation between the moment map $\\mu_s$ and the $Log_t$ map of amoeba theory ($t=e^s$) . This relation further simplifies if one takes as $H$ the full symplectic potential, which is continuous but not smooth on $M$ and corresponds to a geodesic of Kaehler metrics with cone angle singularities. The tropical limit corresponds thus, in this setting, to the infinite geodesic time limit corresponding to convex hamiltonians.\n\nLecture 1 (introduction and different definitions of complex time evolution): http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L1.pdf\n\nLecture 2: Kahler tropicalization of C^: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L2.pdf\n\nLecture 3: Kahler tropicalization of C and (strange) actions of G_C on Kahler structures: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L3.pdf\n\nLecture 4: C^infty Kahler tropicalization of toric varieties and of hypersurfaces in toric varieties: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L4.pdf\n\nLecture 5: C^0 Kahler tropicalization of toric varieties and of hypersurfaces in toric varieties: http://www.math.tecnico.ulisboa.pt/~jmourao/talkscourses/Lectures_UG_L5.pdf\n\n2015, October 27, Tuesday, 15.15, Villa Battelle ( together with Séminaire “Groupes de Lie et espaces des modules”)\n\nImaginary time in Kaehler geometry, quantization and tropical amoebas.\n\nJosé Manuel Cidade Mourão, Mathematics Department, Instituto Superior Tecnico, Portugal.\n\nFor a compact Kaehler manifold $M$ and a function $H$ on $M$ we define a continuation of the Hamiltonian flow of $H$ to complex time $\\tau$. The resulting complexified (or complex time) symplectomorphisms are diffeomorphisms for some $\|\\tau\|< R_H$. For larger values of $\|\\tau\|$ they may correspond e.g. to the collapse of $M$ to a totally real submanifold. We'll discuss some simple examples and applications to Kaehler geometry, quantization and tropical geometry. This talk is the first lecture of a mini-course to be given during October-November 2015.\n\n2015, October 5, Monday, 16.20, Villa Battelle\n\nTropicalization of Poisson-Lie groups\n\nAnton Alexeev (UniGe)\n\nIn the first part of the talk, we recall the notion of Poisson-Lie groups and cluster coordinates for some simple examples.\n\nIn the second part, we use the notion of tropicalization to construct completely integrable systems, and for the Poisson-Lie group SU(n)^ match it with the Gelfand-Zeiltin integrable system.\n\nThe talk is based on joint works with I. Davydenkova, M. Podkopaeva and A. Szenes.\n\n2015, September 28, Monday, 16.15, Villa Battelle\n\nWhat is moonshine?\n\nSergey Galkin (HSE, Moscow)\n\nI will describe a few instances of geometric moonshines: surprising appearance of modular forms and sporadic groups as the answers to seemingly unrelated geometric and topological questions.\n\n2015, 21 September, Monday, 16.15, Villa Battelle.\n\nCohomology of superforms on polyhedral complexes and Poincare duality for tropical manifolds\n\nKristin Shaw.\n\nSuperforms introduced by Lagerberg are bigraded differential forms on $\\mathbb R^n$ which can be restricted to polyhedral complexes. We extend these forms to $\\mathbb T^n = [-\\infty, \\infty)^n$ and show that their de Rham cohomology is equivalent to tropical $(p, q)$ cohomology Furthermore, we establish Poincaré duality for cohomology of tropical manifolds. As in the classical theory, the Poincaré pairing can be formulated in terms of integration of superforms.\n\nold page of the seminar http://www.unige.ch/math/folks/langl/fables/\n\nfables.txt · Dernière modification: 2020/01/12 23:23 par weronika\n\n### Outils de la page", null, "" ]	[ null, "https://www.unige.ch/math/tggroup/lib/exe/indexer.php", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8630935,"math_prob":0.9224606,"size":48292,"snap":"2019-51-2020-05","text_gpt3_token_len":11651,"char_repetition_ratio":0.13388419,"word_repetition_ratio":0.097662196,"special_character_ratio":0.20935145,"punctuation_ratio":0.1275551,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.975214,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-26T03:09:32Z\",\"WARC-Record-ID\":\"<urn:uuid:193c65f1-94b3-41d0-96fe-6e490389d793>\",\"Content-Length\":\"68109\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2e67f209-9dea-419b-af40-c006d2547d95>\",\"WARC-Concurrent-To\":\"<urn:uuid:b186a3c0-9672-4be2-8a36-029f2771cbca>\",\"WARC-IP-Address\":\"129.194.6.50\",\"WARC-Target-URI\":\"https://www.unige.ch/math/tggroup/doku.php?id=fables\",\"WARC-Payload-Digest\":\"sha1:3KL4U7HJQDKB2IDB4RYBYBSG6ODWD6OY\",\"WARC-Block-Digest\":\"sha1:VIYFM74VHMB4Z4YBE3HSOS7OLWVHCFBF\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579251684146.65_warc_CC-MAIN-20200126013015-20200126043015-00492.warc.gz\"}"}
http://www.ux1.eiu.edu/~cfadd/1360/23EFields/Continu.html	[ "# Due to Continuous Charge Distributions\n\nCoulomb's Law tells us the force between two point charges. Our variation tells us the Electric field due to a single point charge. What do we do if we have a continuous charge distribution? We can sum up the electric field caused by each tiny, infinitesimal part of the charge distribution. This means an integral over the charge distribution:", null, "For a single point charge Q, we had", null, "where r is the distance from the charge Q. Remember, E is only the magnitude of the electric field; we must take care of its vector nature separately! That's important! Now we have a distribution of charge and we must replace Q by dQ and E by dE -- and take care of the direction of E.", null, "", null, "r, the distance from the tiny, elemental, infinitesimal charge dQ to the point in question, is a function of where that charge dQ is. And, what does it mean to \"integrate over the charge dQ\"? We know how to integrate over a variable like dx, or a plane like dA = dx dy or dA = 2", null, "r dr or dA = r d", null, "dr, or a volume like dV = dx dy dz. So we will need to change from this symbolic charge dQ to a charge density multiplied by some spatial differential,\n\ndQ =", null, "dx\n\ndQ =", null, "dA\n\ndQ =", null, "dV\n\nLook at Example 23.7 in Serway's and Beichner's textbook (p.724): A rod of length", null, "has a uniform charge per unit length", null, "and a total charge Q. Calculate the electric field at a point P along the axis of the rod, a distance d from one end.", null, "What is the electric field at point P because of a little piece of charge", null, "Q located at position x, as shown in the sketch?", null, "E = k", null, "Q / x2\n\nWe will carry out an integration from x = d to x = d +", null, "so we need to change this small amount of charge", null, "Q into a small length", null, "x,", null, "Q =", null, "", null, "x", null, "E = k (", null, "", null, "x) / x2\n\nwhere", null, "= Q /", null, "dE = k (", null, "dx) / x2\n\ndE = k", null, "(dx / x2)", null, "", null, "", null, "", null, "", null, "", null, "", null, "What about other geometries?\n\nLook at Example 23.8, on page 724, of Serway's and Beichner's text. Find the electric field due to a ring of charge: A ring of radius a has a uniform charge density with a total charge Q. Calculate the electric field along the axis of the ring at a point P, a distance x from the center of the ring.", null, "The charge density is", null, "= Q / (2", null, "a)\n\nRemember, our equation for the electric field is for the magnitude of the electric field. Consider a little piece of charge dq as sketched in the diagram. Because that charge dq is there, there is an electric field dE at point P in the direction shown. The component dEx of that electric field along the direction of the axis perpendicular to the plane of the ring is\n\ndEx = dE cos", null, "dEx = dE (x/r)\n\ndEx = [k dq/r2] (x/r)\n\ndEx = [k dq/r3] x\n\ndEx = [k x dq/r3]\n\ndEx = [k x/r3] dq\n\nNotice that, with this geometry, once the radius of the ring a is specified and the position x, that fully specifies r. r and x do not change as we integrate over dq.\n[[ Remember, SQRT() means \"square root of ()\" because that is easier for me to type. ]]\n\nr = SQRT(a2 + x2)\n\nr3 = (a2 + x2)3/2\n\n1/r3 = 1/(a2 + x2)3/2", null, "", null, "", null, "", null, "Remember, x and a are not variables.\n\nWhat about the component of E that is perpendicular to this direction? By symmetry that component is zero. From the diagram, you can see that for each element of charge dq, there is another element of charge dq on the opposite side of the ring that causes an electric field that just cancels the first one -- that is, their components perpendicular to the axis of symmetry just cancel. Notice that their components along the axis do not cancel for they lie in the same direction. Diagrams are very important. Don't start writing equations before you have made good, clear, complete diagrams!\n\nNow that we have looked at the electric field because of a ring of charge, we can build upon that and extend our ideas and look at the electric field due to a disk of charge. Look at Example 23.9, on page 725 of Serway's and Beichner's textbook.\n\nA disk of radius R has a uniform charge per unit area", null, ". Calculate the electric field at a point P that lies along the central axis of the disk and a distance x from its center.", null, "Consider a ring of charge as sketched here. The ring has a radius of r and a thickness of dr and carries a charge of dq. But that charge dq is just proportional to the area,\n\ndq =", null, "dA\n\ndq =", null, "[C dr]\n\ndq =", null, "[ (2", null, "r) dr ]\n\ndq = 2", null, "", null, "r dr\n\nThink back on what we just did in the previous example. For charge Q on a ring of radius r, we found that the electric field at a distance x from the plane of the ring was", null, "That is exactly what we have now -- except we have charge dq instead of Q and the ring has a radius of r instead of a. So we can write", null, "", null, "", null, "", null, "", null, "Be careful; the limits of integration are important.", null, "We could look this up in a table of integrals. But a variable substitution is still fairly direct and straightforward;", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "This result is only valid for x > 0 and must be modified slightly for x < 0." ]	[ null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/Fig23.15.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/EFldEq.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/EFldEq2.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/EFldEq3.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/pi.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/theta.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/lambda.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/sigma.jpg", null, "http://www.ux1.eiu.edu/~cfadd/1360/23EFields/23Images/rho.jpg", null, 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https://whatisconvert.com/234-centimeters-in-meters	[ "# What is 234 Centimeters in Meters?\n\n## Convert 234 Centimeters to Meters\n\nTo calculate 234 Centimeters to the corresponding value in Meters, multiply the quantity in Centimeters by 0.01 (conversion factor). In this case we should multiply 234 Centimeters by 0.01 to get the equivalent result in Meters:\n\n234 Centimeters x 0.01 = 2.34 Meters\n\n234 Centimeters is equivalent to 2.34 Meters.\n\n## How to convert from Centimeters to Meters\n\nThe conversion factor from Centimeters to Meters is 0.01. To find out how many Centimeters in Meters, multiply by the conversion factor or use the Length converter above. Two hundred thirty-four Centimeters is equivalent to two point three four Meters.\n\n## Definition of Centimeter\n\nThe centimeter (symbol: cm) is a unit of length in the metric system. It is also the base unit in the centimeter-gram-second system of units. The centimeter practical unit of length for many everyday measurements. A centimeter is equal to 0.01(or 1E-2) meter.\n\n## Definition of Meter\n\nThe meter (symbol: m) is the fundamental unit of length in the International System of Units (SI). It is defined as \"the length of the path travelled by light in vacuum during a time interval of 1/299,792,458 of a second.\" In 1799, France start using the metric system, and that is the first country using the metric.\n\n## Using the Centimeters to Meters converter you can get answers to questions like the following:\n\n• How many Meters are in 234 Centimeters?\n• 234 Centimeters is equal to how many Meters?\n• How to convert 234 Centimeters to Meters?\n• How many is 234 Centimeters in Meters?\n• What is 234 Centimeters in Meters?\n• How much is 234 Centimeters in Meters?\n• How many m are in 234 cm?\n• 234 cm is equal to how many m?\n• How to convert 234 cm to m?\n• How many is 234 cm in m?\n• What is 234 cm in m?\n• How much is 234 cm in m?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.884468,"math_prob":0.98236406,"size":1793,"snap":"2022-05-2022-21","text_gpt3_token_len":459,"char_repetition_ratio":0.22694242,"word_repetition_ratio":0.07255521,"special_character_ratio":0.26715,"punctuation_ratio":0.11263736,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9995567,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-25T19:29:34Z\",\"WARC-Record-ID\":\"<urn:uuid:a26ae8e2-cc4e-4a18-9cfe-abe172993634>\",\"Content-Length\":\"29681\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b9923e0e-f882-4fb8-990a-2efda133c03a>\",\"WARC-Concurrent-To\":\"<urn:uuid:35cfb23e-8f27-48f5-b7ff-0ec856739126>\",\"WARC-IP-Address\":\"104.21.13.210\",\"WARC-Target-URI\":\"https://whatisconvert.com/234-centimeters-in-meters\",\"WARC-Payload-Digest\":\"sha1:RDN3PUC7RHCA66SVU2X24EZAIBWKC5UE\",\"WARC-Block-Digest\":\"sha1:PCGM4PZML3XSQ3KYZCIOB2WUOMF3II7B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662593428.63_warc_CC-MAIN-20220525182604-20220525212604-00175.warc.gz\"}"}
https://graz.pure.elsevier.com/en/publications/a-tauberian-theorem-for-ideal-statistical-convergence	[ "# A Tauberian theorem for ideal statistical convergence\n\nMarek Balcerzak, Paolo Leonetti\n\nCorresponding author for this work\n\nResearch output: Contribution to journalArticlepeer-review\n\n## Abstract\n\nGiven an ideal I on the positive integers, a real sequence (xn) is said to be I-statistically convergent to ℓ provided that n∈N:[Formula presented]\|{k≤n:xk∉U}\|≥ε∈Ifor all neighborhoods U of ℓ and all ε>0. First, we show that I-statistical convergence coincides with J-convergence, for some unique ideal J=J(I). In addition, J is Borel [analytic, coanalytic, respectively] whenever I is Borel [analytic, coanalytic, resp.]. Then we prove, among others, that if I is the summable ideal {A⊆N:∑a∈A1∕a<∞} or the density zero ideal {A⊆N:limn→∞[Formula presented]\|A∩[1,n]\|=0} then I-statistical convergence coincides with statistical convergence. This can be seen as a Tauberian theorem which extends a classical theorem of Fridy. Lastly, we show that this is never the case if I is maximal.\n\nOriginal language English 83-95 13 Indagationes Mathematicae 31 1 https://doi.org/10.1016/j.indag.2019.10.001 Published - Jan 2020\n\n## Keywords\n\n• Generalized density ideal\n• Ideal statistical convergence\n• Maximal ideals\n• Submeasures\n• Tauberian condition\n\n## ASJC Scopus subject areas\n\n• Mathematics(all)\n\n## Fingerprint\n\nDive into the research topics of 'A Tauberian theorem for ideal statistical convergence'. Together they form a unique fingerprint." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.78846985,"math_prob":0.5342139,"size":1146,"snap":"2022-05-2022-21","text_gpt3_token_len":346,"char_repetition_ratio":0.12434326,"word_repetition_ratio":0.0,"special_character_ratio":0.23734729,"punctuation_ratio":0.10952381,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.979526,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-23T22:14:31Z\",\"WARC-Record-ID\":\"<urn:uuid:23f8a4c9-f8f4-459c-aebf-0f27b93b901e>\",\"Content-Length\":\"43664\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:22b1561f-8040-4f00-87e5-b1d2a91fda4b>\",\"WARC-Concurrent-To\":\"<urn:uuid:dcc4d80e-3f0a-4567-bb15-122d119b1b55>\",\"WARC-IP-Address\":\"34.248.98.230\",\"WARC-Target-URI\":\"https://graz.pure.elsevier.com/en/publications/a-tauberian-theorem-for-ideal-statistical-convergence\",\"WARC-Payload-Digest\":\"sha1:W7GX34X3B2GSVTTXYI5W4IKUE4SRPAMF\",\"WARC-Block-Digest\":\"sha1:ZOLZG6UTEZOVK2SYE4WPM6FPSWYH77UZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662561747.42_warc_CC-MAIN-20220523194013-20220523224013-00703.warc.gz\"}"}