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class="vector-toc-link" href="#Classifications"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Classifications</span> </div> </a> <ul id="toc-Classifications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Construction" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Construction"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Construction</span> </div> </a> <button aria-controls="toc-Construction-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Construction subsection</span> </button> <ul id="toc-Construction-sublist" class="vector-toc-list"> <li id="toc-A_priori_information" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#A_priori_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span><i>A priori</i> information</span> </div> </a> <ul id="toc-A_priori_information-sublist" class="vector-toc-list"> <li id="toc-Subjective_information" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Subjective_information"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1.1</span> <span>Subjective information</span> </div> </a> <ul id="toc-Subjective_information-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Complexity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complexity"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Complexity</span> </div> </a> <ul id="toc-Complexity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Training,_tuning,_and_fitting" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Training,_tuning,_and_fitting"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.3</span> <span>Training, tuning, and fitting</span> </div> </a> <ul id="toc-Training,_tuning,_and_fitting-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Evaluation_and_assessment" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Evaluation_and_assessment"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4</span> <span>Evaluation and assessment</span> </div> </a> <ul id="toc-Evaluation_and_assessment-sublist" class="vector-toc-list"> <li id="toc-Prediction_of_empirical_data" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Prediction_of_empirical_data"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.1</span> <span>Prediction of empirical data</span> </div> </a> <ul id="toc-Prediction_of_empirical_data-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Scope_of_the_model" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Scope_of_the_model"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.2</span> <span>Scope of the model</span> </div> </a> <ul id="toc-Scope_of_the_model-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Philosophical_considerations" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Philosophical_considerations"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.4.3</span> <span>Philosophical considerations</span> </div> </a> <ul id="toc-Philosophical_considerations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Significance_in_the_natural_sciences" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Significance_in_the_natural_sciences"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Significance in the natural sciences</span> </div> </a> <ul id="toc-Significance_in_the_natural_sciences-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Some_applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Some_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Some applications</span> </div> </a> <ul id="toc-Some_applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Examples" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Examples"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Examples</span> </div> </a> <ul id="toc-Examples-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Further reading</span> </div> </a> <button aria-controls="toc-Further_reading-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>Toggle Further reading subsection</span> </button> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> <li id="toc-Books" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Books"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.1</span> <span>Books</span> </div> </a> <ul id="toc-Books-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Specific_applications" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Specific_applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">9.2</span> <span>Specific applications</span> </div> </a> <ul id="toc-Specific_applications-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="Toggle the table of contents" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">Toggle the table of contents</span> </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Mathematical model</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. Available in 56 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-56" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">56 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%86%D9%85%D9%88%D8%B0%D8%AC_%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A" title="نموذج رياضي – Arabic" lang="ar" hreflang="ar" data-title="نموذج رياضي" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-az mw-list-item"><a href="https://az.wikipedia.org/wiki/Riyazi_modell%C9%99%C5%9Fdirm%C9%99" title="Riyazi modelləşdirmə – Azerbaijani" lang="az" hreflang="az" data-title="Riyazi modelləşdirmə" data-language-autonym="Azərbaycanca" data-language-local-name="Azerbaijani" class="interlanguage-link-target"><span>Azərbaycanca</span></a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="Математически модел – Bulgarian" lang="bg" hreflang="bg" data-title="Математически модел" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Model_matem%C3%A0tic" title="Model matemàtic – Catalan" lang="ca" hreflang="ca" data-title="Model matemàtic" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cv mw-list-item"><a href="https://cv.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%C4%83%D0%BB%D0%BB%C4%83_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C" title="Математикăллă модель – Chuvash" lang="cv" hreflang="cv" data-title="Математикăллă модель" data-language-autonym="Чӑвашла" data-language-local-name="Chuvash" class="interlanguage-link-target"><span>Чӑвашла</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Matematick%C3%BD_model" title="Matematický model – Czech" lang="cs" hreflang="cs" data-title="Matematický model" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Model_(matematik)" title="Model (matematik) – Danish" lang="da" hreflang="da" data-title="Model (matematik)" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Mathematisches_Modell" title="Mathematisches Modell – German" lang="de" hreflang="de" data-title="Mathematisches Modell" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Matemaatiline_mudel" title="Matemaatiline mudel – Estonian" lang="et" hreflang="et" data-title="Matemaatiline mudel" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Modelo_matem%C3%A1tico" title="Modelo matemático – Spanish" lang="es" hreflang="es" data-title="Modelo matemático" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Matematika_modelo" title="Matematika modelo – Esperanto" lang="eo" hreflang="eo" data-title="Matematika modelo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Eredu_matematiko" title="Eredu matematiko – Basque" lang="eu" hreflang="eu" data-title="Eredu matematiko" data-language-autonym="Euskara" data-language-local-name="Basque" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%AF%D9%84_%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C" title="مدل ریاضی – Persian" lang="fa" hreflang="fa" data-title="مدل ریاضی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Mod%C3%A8le_math%C3%A9matique" title="Modèle mathématique – French" lang="fr" hreflang="fr" data-title="Modèle mathématique" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ga mw-list-item"><a href="https://ga.wikipedia.org/wiki/Samhail_mhatamaitici%C3%BAil" title="Samhail mhatamaiticiúil – Irish" lang="ga" hreflang="ga" data-title="Samhail mhatamaiticiúil" data-language-autonym="Gaeilge" data-language-local-name="Irish" class="interlanguage-link-target"><span>Gaeilge</span></a></li><li class="interlanguage-link interwiki-gl mw-list-item"><a href="https://gl.wikipedia.org/wiki/Modelo_matem%C3%A1tico" title="Modelo matemático – Galician" lang="gl" hreflang="gl" data-title="Modelo matemático" data-language-autonym="Galego" data-language-local-name="Galician" class="interlanguage-link-target"><span>Galego</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%88%98%ED%95%99%EC%A0%81_%EB%AA%A8%EB%8D%B8" title="수학적 모델 – Korean" lang="ko" hreflang="ko" data-title="수학적 모델" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hy mw-list-item"><a href="https://hy.wikipedia.org/wiki/%D5%84%D5%A1%D5%A9%D5%A5%D5%B4%D5%A1%D5%BF%D5%AB%D5%AF%D5%A1%D5%AF%D5%A1%D5%B6_%D5%B4%D5%B8%D5%A4%D5%A5%D5%AC" title="Մաթեմատիկական մոդել – Armenian" lang="hy" hreflang="hy" data-title="Մաթեմատիկական մոդել" data-language-autonym="Հայերեն" data-language-local-name="Armenian" class="interlanguage-link-target"><span>Հայերեն</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%97%E0%A4%A3%E0%A4%BF%E0%A4%A4%E0%A5%80%E0%A4%AF_%E0%A4%AE%E0%A5%89%E0%A4%A1%E0%A4%B2" title="गणितीय मॉडल – Hindi" lang="hi" hreflang="hi" data-title="गणितीय मॉडल" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Matemati%C4%8Dki_model" title="Matematički model – Croatian" lang="hr" hreflang="hr" data-title="Matematički model" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target"><span>Hrvatski</span></a></li><li class="interlanguage-link interwiki-io mw-list-item"><a href="https://io.wikipedia.org/wiki/Matematikala_modelo" title="Matematikala modelo – Ido" lang="io" hreflang="io" data-title="Matematikala modelo" data-language-autonym="Ido" data-language-local-name="Ido" class="interlanguage-link-target"><span>Ido</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Model_matematika" title="Model matematika – Indonesian" lang="id" hreflang="id" data-title="Model matematika" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Modello_matematico" title="Modello matematico – Italian" lang="it" hreflang="it" data-title="Modello matematico" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%9E%D7%95%D7%93%D7%9C_%D7%9E%D7%AA%D7%9E%D7%98%D7%99" title="מודל מתמטי – Hebrew" lang="he" hreflang="he" data-title="מודל מתמטי" data-language-autonym="עברית" data-language-local-name="Hebrew" class="interlanguage-link-target"><span>עברית</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA%D0%B0%D0%BB%D1%8B%D2%9B_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C%D0%B4%D0%B5%D1%83" title="Математикалық модельдеу – Kazakh" lang="kk" hreflang="kk" data-title="Математикалық модельдеу" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-lo mw-list-item"><a href="https://lo.wikipedia.org/wiki/%E0%BB%81%E0%BA%9A%E0%BA%9A%E0%BA%88%E0%BA%B3%E0%BA%A5%E0%BA%AD%E0%BA%87%E0%BA%97%E0%BA%B2%E0%BA%87%E0%BA%84%E0%BA%B0%E0%BA%99%E0%BA%B4%E0%BA%94%E0%BA%AA%E0%BA%B2%E0%BA%94" title="ແບບຈຳລອງທາງຄະນິດສາດ – Lao" lang="lo" hreflang="lo" data-title="ແບບຈຳລອງທາງຄະນິດສາດ" data-language-autonym="ລາວ" data-language-local-name="Lao" class="interlanguage-link-target"><span>ລາວ</span></a></li><li class="interlanguage-link interwiki-lt badge-Q70894304 mw-list-item" title=""><a href="https://lt.wikipedia.org/wiki/Matematinis_modelis" title="Matematinis modelis – Lithuanian" lang="lt" hreflang="lt" data-title="Matematinis modelis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target"><span>Lietuvių</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Matematikai_modell" title="Matematikai modell – Hungarian" lang="hu" hreflang="hu" data-title="Matematikai modell" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Model_matematik" title="Model matematik – Malay" lang="ms" hreflang="ms" data-title="Model matematik" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-mn mw-list-item"><a href="https://mn.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D0%BA_%D0%B7%D0%B0%D0%B3%D0%B2%D0%B0%D1%80" title="Математик загвар – Mongolian" lang="mn" hreflang="mn" data-title="Математик загвар" data-language-autonym="Монгол" data-language-local-name="Mongolian" class="interlanguage-link-target"><span>Монгол</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiskundig_model" title="Wiskundig model – Dutch" lang="nl" hreflang="nl" data-title="Wiskundig model" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E6%95%B0%E7%90%86%E3%83%A2%E3%83%87%E3%83%AB" title="数理モデル – Japanese" lang="ja" hreflang="ja" data-title="数理モデル" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Matematisk_modell" title="Matematisk modell – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Matematisk modell" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Matematisk_modell" title="Matematisk modell – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Matematisk modell" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target"><span>Norsk nynorsk</span></a></li><li class="interlanguage-link interwiki-uz mw-list-item"><a href="https://uz.wikipedia.org/wiki/Matematik_model" title="Matematik model – Uzbek" lang="uz" hreflang="uz" data-title="Matematik model" data-language-autonym="Oʻzbekcha / ўзбекча" data-language-local-name="Uzbek" class="interlanguage-link-target"><span>Oʻzbekcha / ўзбекча</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Modelowanie_matematyczne" title="Modelowanie matematyczne – Polish" lang="pl" hreflang="pl" data-title="Modelowanie matematyczne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Modelo_(matem%C3%A1tica)" title="Modelo (matemática) – Portuguese" lang="pt" hreflang="pt" data-title="Modelo (matemática)" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-kaa mw-list-item"><a href="https://kaa.wikipedia.org/wiki/Matematikal%C4%B1q_modellestiriw" title="Matematikalıq modellestiriw – Kara-Kalpak" lang="kaa" hreflang="kaa" data-title="Matematikalıq modellestiriw" data-language-autonym="Qaraqalpaqsha" data-language-local-name="Kara-Kalpak" class="interlanguage-link-target"><span>Qaraqalpaqsha</span></a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Modelare_matematic%C4%83" title="Modelare matematică – Romanian" lang="ro" hreflang="ro" data-title="Modelare matematică" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target"><span>Română</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B0%D1%8F_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C" title="Математическая модель – Russian" lang="ru" hreflang="ru" data-title="Математическая модель" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sco mw-list-item"><a href="https://sco.wikipedia.org/wiki/Mathematical_model" title="Mathematical model – Scots" lang="sco" hreflang="sco" data-title="Mathematical model" data-language-autonym="Scots" data-language-local-name="Scots" class="interlanguage-link-target"><span>Scots</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Mathematical_model" title="Mathematical model – Simple English" lang="en-simple" hreflang="en-simple" data-title="Mathematical model" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Matematick%C3%BD_model" title="Matematický model – Slovak" lang="sk" hreflang="sk" data-title="Matematický model" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BA%D0%B8_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB" title="Математички модел – Serbian" lang="sr" hreflang="sr" data-title="Математички модел" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Matemati%C4%8Dki_model" title="Matematički model – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Matematički model" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target"><span>Srpskohrvatski / српскохрватски</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Matemaattinen_malli" title="Matemaattinen malli – Finnish" lang="fi" hreflang="fi" data-title="Matemaattinen malli" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Matematisk_modell" title="Matematisk modell – Swedish" lang="sv" hreflang="sv" data-title="Matematisk modell" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Modelong_matematikal" title="Modelong matematikal – Tagalog" lang="tl" hreflang="tl" data-title="Modelong matematikal" data-language-autonym="Tagalog" data-language-local-name="Tagalog" class="interlanguage-link-target"><span>Tagalog</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%95%E0%AE%A3%E0%AE%BF%E0%AE%A4_%E0%AE%AE%E0%AE%BE%E0%AE%A4%E0%AE%BF%E0%AE%B0%E0%AE%BF" title="கணித மாதிரி – Tamil" lang="ta" hreflang="ta" data-title="கணித மாதிரி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%81%E0%B8%9A%E0%B8%9A%E0%B8%88%E0%B8%B3%E0%B8%A5%E0%B8%AD%E0%B8%87%E0%B8%97%E0%B8%B2%E0%B8%87%E0%B8%84%E0%B8%93%E0%B8%B4%E0%B8%95%E0%B8%A8%E0%B8%B2%E0%B8%AA%E0%B8%95%E0%B8%A3%E0%B9%8C" title="แบบจำลองทางคณิตศาสตร์ – Thai" lang="th" hreflang="th" data-title="แบบจำลองทางคณิตศาสตร์" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Matematiksel_model" title="Matematiksel model – Turkish" lang="tr" hreflang="tr" data-title="Matematiksel model" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%9C%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B0_%D0%BC%D0%BE%D0%B4%D0%B5%D0%BB%D1%8C" title="Математична модель – Ukrainian" lang="uk" hreflang="uk" data-title="Математична модель" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D8%B1%DB%8C%D8%A7%D8%B6%DB%8C%D8%A7%D8%AA%DB%8C_%D9%86%D9%85%D9%88%D9%86%DB%81" title="ریاضیاتی نمونہ – Urdu" lang="ur" hreflang="ur" data-title="ریاضیاتی نمونہ" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target"><span>اردو</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%C3%B4_h%C3%ACnh_to%C3%A1n_h%E1%BB%8Dc" title="Mô hình toán học – Vietnamese" lang="vi" hreflang="vi" data-title="Mô hình toán học" 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href="/w/index.php?title=Mathematical_modelling&redirect=no" class="mw-redirect" title="Mathematical modelling">Mathematical modelling</a>)</span></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Description of a system using mathematical concepts and language</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For other uses, see <a href="/wiki/Mathematical_model_(disambiguation)" class="mw-disambig" title="Mathematical model (disambiguation)">Mathematical model (disambiguation)</a>.</div> <p> A <b>mathematical model</b> is an <a href="/wiki/Abstract_and_concrete" title="Abstract and concrete">abstract</a> description of a concrete <a href="/wiki/System" title="System">system</a> using <a href="/wiki/Mathematics" title="Mathematics">mathematical</a> concepts and <a href="/wiki/Language_of_mathematics" title="Language of mathematics">language</a>. The process of developing a mathematical <a href="/wiki/Model" title="Model">model</a> is termed <i><b>mathematical modeling</b></i>. Mathematical models are used in <a href="/wiki/Applied_mathematics" title="Applied mathematics">applied mathematics</a> and in the <a href="/wiki/Natural_science" title="Natural science">natural sciences</a> (such as <a href="/wiki/Physics" title="Physics">physics</a>, <a href="/wiki/Biology" title="Biology">biology</a>, <a href="/wiki/Earth_science" title="Earth science">earth science</a>, <a href="/wiki/Chemistry" title="Chemistry">chemistry</a>) and <a href="/wiki/Engineering" title="Engineering">engineering</a> disciplines (such as <a href="/wiki/Computer_science" title="Computer science">computer science</a>, <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>), as well as in non-physical systems such as the <a href="/wiki/Social_science" title="Social science">social sciences</a><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> (such as <a href="/wiki/Economics" title="Economics">economics</a>, <a href="/wiki/Psychology" title="Psychology">psychology</a>, <a href="/wiki/Sociology" title="Sociology">sociology</a>, <a href="/wiki/Political_science" title="Political science">political science</a>). It can also be taught as a subject in its own right.<sup id="cite_ref-Edwards_2-0" class="reference"><a href="#cite_note-Edwards-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>The use of mathematical models to solve problems in business or military operations is a large part of the field of <a href="/wiki/Operations_research" title="Operations research">operations research</a>. Mathematical models are also used in <a href="/wiki/Music" title="Music">music</a>,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Linguistics" title="Linguistics">linguistics</a>,<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> and <a href="/wiki/Philosophy" title="Philosophy">philosophy</a> (for example, intensively in <a href="/wiki/Analytic_philosophy" title="Analytic philosophy">analytic philosophy</a>). A model may help to explain a system and to study the effects of different components, and to make predictions about behavior. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Elements_of_a_mathematical_model">Elements of a mathematical model</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=1" title="Edit section: Elements of a mathematical model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical models can take many forms, including <a href="/wiki/Dynamical_systems" class="mw-redirect" title="Dynamical systems">dynamical systems</a>, <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a>, <a href="/wiki/Differential_equations" class="mw-redirect" title="Differential equations">differential equations</a>, or <a href="/wiki/Game_theory" title="Game theory">game theoretic models</a>. These and other types of models can overlap, with a given model involving a variety of abstract structures. In general, mathematical models may include <a href="/wiki/Model_theory" title="Model theory">logical models</a>. In many cases, the quality of a scientific field depends on how well the mathematical models developed on the theoretical side agree with results of repeatable experiments. Lack of agreement between theoretical mathematical models and experimental measurements often leads to important advances as better theories are developed. In the <a href="/wiki/Physical_sciences" class="mw-redirect" title="Physical sciences">physical sciences</a>, a traditional mathematical model contains most of the following elements: </p> <ol><li><a href="/wiki/Governing_equation" title="Governing equation">Governing equations</a></li> <li>Supplementary sub-models <ol><li>Defining equations</li> <li><a href="/wiki/Constitutive_equation" title="Constitutive equation">Constitutive equations</a></li></ol></li> <li>Assumptions and constraints <ol><li><a href="/wiki/Initial_condition" title="Initial condition">Initial</a> and <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a></li> <li><a href="/wiki/Constraint_(classical_mechanics)" class="mw-redirect" title="Constraint (classical mechanics)">Classical constraints</a> and <a href="/wiki/Kinematics" title="Kinematics">kinematic equations</a></li></ol></li></ol> <div class="mw-heading mw-heading2"><h2 id="Classifications">Classifications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=2" title="Edit section: Classifications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical models are of different types: </p> <ul><li>Linear vs. nonlinear. If all the operators in a mathematical model exhibit <a href="/wiki/Linear" class="mw-redirect" title="Linear">linearity</a>, the resulting mathematical model is defined as linear. A model is considered to be nonlinear otherwise. The definition of linearity and nonlinearity is dependent on context, and linear models may have nonlinear expressions in them. For example, in a <a href="/wiki/Linear_model" title="Linear model">statistical linear model</a>, it is assumed that a relationship is linear in the parameters, but it may be nonlinear in the predictor variables. Similarly, a differential equation is said to be linear if it can be written with linear <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a>, but it can still have nonlinear expressions in it. In a <a href="/wiki/Optimization_(mathematics)" class="mw-redirect" title="Optimization (mathematics)">mathematical programming</a> model, if the objective functions and constraints are represented entirely by <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a>, then the model is regarded as a linear model. If one or more of the objective functions or constraints are represented with a <a href="/wiki/Nonlinearity" class="mw-redirect" title="Nonlinearity">nonlinear</a> equation, then the model is known as a nonlinear model.<br />Linear structure implies that a problem can be decomposed into simpler parts that can be treated independently and/or analyzed at a different scale and the results obtained will remain valid for the initial problem when recomposed and rescaled.<br />Nonlinearity, even in fairly simple systems, is often associated with phenomena such as <a href="/wiki/Chaos_theory" title="Chaos theory">chaos</a> and <a href="/wiki/Irreversibility" class="mw-redirect" title="Irreversibility">irreversibility</a>. Although there are exceptions, nonlinear systems and models tend to be more difficult to study than linear ones. A common approach to nonlinear problems is <a href="/wiki/Linearization" title="Linearization">linearization</a>, but this can be problematic if one is trying to study aspects such as irreversibility, which are strongly tied to nonlinearity.</li> <li>Static vs. dynamic. A <i>dynamic</i> model accounts for time-dependent changes in the state of the system, while a <i>static</i> (or steady-state) model calculates the system in equilibrium, and thus is time-invariant. Dynamic models typically are represented by <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> or <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a>.</li> <li>Explicit vs. implicit. If all of the input parameters of the overall model are known, and the output parameters can be calculated by a finite series of computations, the model is said to be <i>explicit</i>. But sometimes it is the <i>output</i> parameters which are known, and the corresponding inputs must be solved for by an iterative procedure, such as <a href="/wiki/Newton%27s_method" title="Newton's method">Newton's method</a> or <a href="/wiki/Broyden%27s_method" title="Broyden's method">Broyden's method</a>. In such a case the model is said to be <i>implicit</i>. For example, a <a href="/wiki/Jet_engine" title="Jet engine">jet engine</a>'s physical properties such as turbine and nozzle throat areas can be explicitly calculated given a design <a href="/wiki/Thermodynamic_cycle" title="Thermodynamic cycle">thermodynamic cycle</a> (air and fuel flow rates, pressures, and temperatures) at a specific flight condition and power setting, but the engine's operating cycles at other flight conditions and power settings cannot be explicitly calculated from the constant physical properties.</li> <li>Discrete vs. continuous. A <a href="/wiki/Discrete_modeling" class="mw-redirect" title="Discrete modeling">discrete model</a> treats objects as discrete, such as the particles in a <a href="/wiki/Molecular_model" title="Molecular model">molecular model</a> or the states in a <a href="/wiki/Statistical_model" title="Statistical model">statistical model</a>; while a <a href="/wiki/Continuous_model" class="mw-redirect" title="Continuous model">continuous model</a> represents the objects in a continuous manner, such as the velocity field of fluid in pipe flows, temperatures and stresses in a solid, and electric field that applies continuously over the entire model due to a point charge.</li> <li>Deterministic vs. probabilistic (stochastic). A <a href="/wiki/Deterministic_system" title="Deterministic system">deterministic</a> model is one in which every set of variable states is uniquely determined by parameters in the model and by sets of previous states of these variables; therefore, a deterministic model always performs the same way for a given set of initial conditions. Conversely, in a stochastic model—usually called a "<a href="/wiki/Statistical_model" title="Statistical model">statistical model</a>"—randomness is present, and variable states are not described by unique values, but rather by <a href="/wiki/Probability" title="Probability">probability</a> distributions.</li> <li>Deductive, inductive, or floating. A <style data-mw-deduplicate="TemplateStyles:r1238216509">.mw-parser-output .vanchor>:target~.vanchor-text{background-color:#b1d2ff}@media screen{html.skin-theme-clientpref-night .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .vanchor>:target~.vanchor-text{background-color:#0f4dc9}}</style><span class="vanchor"><span id="deductive_model"></span><span class="vanchor-text">deductive model</span></span> is a logical structure based on a theory. An inductive model arises from empirical findings and generalization from them. The floating model rests on neither theory nor observation, but is merely the invocation of expected structure. Application of mathematics in social sciences outside of economics has been criticized for unfounded models.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> Application of <a href="/wiki/Catastrophe_theory" title="Catastrophe theory">catastrophe theory</a> in science has been characterized as a floating model.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>Strategic vs. non-strategic. Models used in <a href="/wiki/Game_theory" title="Game theory">game theory</a> are different in a sense that they model agents with incompatible incentives, such as competing species or bidders in an auction. Strategic models assume that players are autonomous decision makers who rationally choose actions that maximize their objective function. A key challenge of using strategic models is defining and computing <a href="/wiki/Solution_concept" title="Solution concept">solution concepts</a> such as <a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a>. An interesting property of strategic models is that they separate reasoning about rules of the game from reasoning about behavior of the players.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li></ul> <div class="mw-heading mw-heading2"><h2 id="Construction">Construction</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=3" title="Edit section: Construction"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Business" title="Business">business</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>, mathematical models may be used to maximize a certain output. The system under consideration will require certain inputs. The system relating inputs to outputs depends on other variables too: <a href="/wiki/Decision_theory" title="Decision theory">decision variables</a>, <a href="/wiki/State_variable" title="State variable">state variables</a>, <a href="/wiki/Exogeny" title="Exogeny">exogenous</a> variables, and <a href="/wiki/Random_variable" title="Random variable">random variables</a>. Decision variables are sometimes known as independent variables. Exogenous variables are sometimes known as <a href="/wiki/Parameter" title="Parameter">parameters</a> or <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constants</a>. The variables are not independent of each other as the state variables are dependent on the decision, input, random, and exogenous variables. Furthermore, the output variables are dependent on the state of the system (represented by the state variables). </p><p><a href="/wiki/Goal" title="Goal">Objectives</a> and <a href="/wiki/Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a> of the system and its users can be represented as <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">functions</a> of the output variables or state variables. The <a href="/wiki/Objective_function" class="mw-redirect" title="Objective function">objective functions</a> will depend on the perspective of the model's user. Depending on the context, an objective function is also known as an <i>index of performance</i>, as it is some measure of interest to the user. Although there is no limit to the number of objective functions and constraints a model can have, using or optimizing the model becomes more involved (computationally) as the number increases. For example, <a href="/wiki/Economist" title="Economist">economists</a> often apply <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> when using <a href="/wiki/Input%E2%80%93output_model" title="Input–output model">input–output models</a>. Complicated mathematical models that have many variables may be consolidated by use of <a href="/wiki/Vector_space" title="Vector space">vectors</a> where one symbol represents several variables. </p> <div class="mw-heading mw-heading3"><h3 id="A_priori_information"><i>A priori</i> information</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=4" title="Edit section: A priori information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Blackbox3D-withGraphs.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Blackbox3D-withGraphs.svg/220px-Blackbox3D-withGraphs.svg.png" decoding="async" width="220" height="63" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Blackbox3D-withGraphs.svg/330px-Blackbox3D-withGraphs.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5b/Blackbox3D-withGraphs.svg/440px-Blackbox3D-withGraphs.svg.png 2x" data-file-width="1423" data-file-height="407" /></a><figcaption>To analyse something with a typical "black box approach", only the behavior of the stimulus/response will be accounted for, to infer the (unknown) <i>box</i>. The usual representation of this <i>black box system</i> is a <a href="/wiki/Data_flow_diagram" class="mw-redirect" title="Data flow diagram">data flow diagram</a> centered in the box.</figcaption></figure> <p>Mathematical modeling problems are often classified into <a href="/wiki/Black_box" title="Black box">black box</a> or <a href="/wiki/White_box_(software_engineering)" title="White box (software engineering)">white box</a> models, according to how much <a href="/wiki/A_priori_(philosophy)" class="mw-redirect" title="A priori (philosophy)">a priori</a> information on the system is available. A black-box model is a system of which there is no a priori information available. A white-box model (also called glass box or clear box) is a system where all necessary information is available. Practically all systems are somewhere between the black-box and white-box models, so this concept is useful only as an intuitive guide for deciding which approach to take. </p><p>Usually, it is preferable to use as much a priori information as possible to make the model more accurate. Therefore, the white-box models are usually considered easier, because if you have used the information correctly, then the model will behave correctly. Often the a priori information comes in forms of knowing the type of functions relating different variables. For example, if we make a model of how a medicine works in a human system, we know that usually the amount of medicine in the blood is an <a href="/wiki/Exponential_decay" title="Exponential decay">exponentially decaying</a> function, but we are still left with several unknown parameters; how rapidly does the medicine amount decay, and what is the initial amount of medicine in blood? This example is therefore not a completely white-box model. These parameters have to be estimated through some means before one can use the model. </p><p>In black-box models, one tries to estimate both the functional form of relations between variables and the numerical parameters in those functions. Using a priori information we could end up, for example, with a set of functions that probably could describe the system adequately. If there is no a priori information we would try to use functions as general as possible to cover all different models. An often used approach for black-box models are <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">neural networks</a> which usually do not make assumptions about incoming data. Alternatively, the NARMAX (Nonlinear AutoRegressive Moving Average model with eXogenous inputs) algorithms which were developed as part of <a href="/wiki/Nonlinear_system_identification" title="Nonlinear system identification">nonlinear system identification</a><sup id="cite_ref-SAB1_8-0" class="reference"><a href="#cite_note-SAB1-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> can be used to select the model terms, determine the model structure, and estimate the unknown parameters in the presence of correlated and nonlinear noise. The advantage of NARMAX models compared to neural networks is that NARMAX produces models that can be written down and related to the underlying process, whereas neural networks produce an approximation that is opaque. </p> <div class="mw-heading mw-heading4"><h4 id="Subjective_information">Subjective information</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=5" title="Edit section: Subjective information"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Sometimes it is useful to incorporate subjective information into a mathematical model. This can be done based on <a href="/wiki/Intuition_(knowledge)" class="mw-redirect" title="Intuition (knowledge)">intuition</a>, <a href="/wiki/Experience" title="Experience">experience</a>, or <a href="/wiki/Expert_opinion" class="mw-redirect" title="Expert opinion">expert opinion</a>, or based on convenience of mathematical form. <a href="/wiki/Bayesian_statistics" title="Bayesian statistics">Bayesian statistics</a> provides a theoretical framework for incorporating such subjectivity into a rigorous analysis: we specify a <a href="/wiki/Prior_probability_distribution" class="mw-redirect" title="Prior probability distribution">prior probability distribution</a> (which can be subjective), and then update this distribution based on empirical data. </p><p>An example of when such approach would be necessary is a situation in which an experimenter bends a coin slightly and tosses it once, recording whether it comes up heads, and is then given the task of predicting the probability that the next flip comes up heads. After bending the coin, the true probability that the coin will come up heads is unknown; so the experimenter would need to make a decision (perhaps by looking at the shape of the coin) about what prior distribution to use. Incorporation of such subjective information might be important to get an accurate estimate of the probability. </p> <div class="mw-heading mw-heading3"><h3 id="Complexity">Complexity</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=6" title="Edit section: Complexity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In general, model complexity involves a trade-off between simplicity and accuracy of the model. <a href="/wiki/Occam%27s_razor" title="Occam's razor">Occam's razor</a> is a principle particularly relevant to modeling, its essential idea being that among models with roughly equal predictive power, the simplest one is the most desirable. While added complexity usually improves the realism of a model, it can make the model difficult to understand and analyze, and can also pose computational problems, including <a href="/wiki/Numerical_instability" class="mw-redirect" title="Numerical instability">numerical instability</a>. <a href="/wiki/Thomas_Kuhn" title="Thomas Kuhn">Thomas Kuhn</a> argues that as science progresses, explanations tend to become more complex before a <a href="/wiki/Paradigm_shift" title="Paradigm shift">paradigm shift</a> offers radical simplification.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> </p><p>For example, when modeling the flight of an aircraft, we could embed each mechanical part of the aircraft into our model and would thus acquire an almost white-box model of the system. However, the computational cost of adding such a huge amount of detail would effectively inhibit the usage of such a model. Additionally, the uncertainty would increase due to an overly complex system, because each separate part induces some amount of variance into the model. It is therefore usually appropriate to make some approximations to reduce the model to a sensible size. Engineers often can accept some approximations in order to get a more robust and simple model. For example, <a href="/wiki/Isaac_Newton" title="Isaac Newton">Newton's</a> <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a> is an approximated model of the real world. Still, Newton's model is quite sufficient for most ordinary-life situations, that is, as long as particle speeds are well below the <a href="/wiki/Speed_of_light" title="Speed of light">speed of light</a>, and we study macro-particles only. Note that better accuracy does not necessarily mean a better model. <a href="/wiki/Statistical_model" title="Statistical model">Statistical models</a> are prone to <a href="/wiki/Overfitting" title="Overfitting">overfitting</a> which means that a model is fitted to data too much and it has lost its ability to generalize to new events that were not observed before. </p> <div class="mw-heading mw-heading3"><h3 id="Training,_tuning,_and_fitting"><span id="Training.2C_tuning.2C_and_fitting"></span>Training, tuning, and fitting</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=7" title="Edit section: Training, tuning, and fitting"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Any model which is not pure white-box contains some <a href="/wiki/Parameter" title="Parameter">parameters</a> that can be used to <a href="/wiki/Model_fitting" class="mw-redirect" title="Model fitting">fit the model</a> to the system it is intended to describe. If the modeling is done by an <a href="/wiki/Artificial_neural_network" class="mw-redirect" title="Artificial neural network">artificial neural network</a> or other <a href="/wiki/Machine_learning" title="Machine learning">machine learning</a>, the optimization of parameters is called <i>training</i>, while the optimization of model hyperparameters is called <i>tuning</i> and often uses <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">cross-validation</a>.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> In more conventional modeling through explicitly given mathematical functions, parameters are often determined by <i><a href="/wiki/Curve_fitting" title="Curve fitting">curve fitting</a>.</i><sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (September 2017)">citation needed</span></a></i>]</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Evaluation_and_assessment">Evaluation and assessment</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=8" title="Edit section: Evaluation and assessment"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A crucial part of the modeling process is the evaluation of whether or not a given mathematical model describes a system accurately. This question can be difficult to answer as it involves several different types of evaluation. </p> <div class="mw-heading mw-heading4"><h4 id="Prediction_of_empirical_data">Prediction of empirical data</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=9" title="Edit section: Prediction of empirical data"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Usually, the easiest part of model evaluation is checking whether a model predicts experimental measurements or other empirical data not used in the model development. In models with parameters, a common approach is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters. This practice is referred to as <a href="/wiki/Cross-validation_(statistics)" title="Cross-validation (statistics)">cross-validation</a> in statistics. </p><p>Defining a <a href="/wiki/Metric_(mathematics)" class="mw-redirect" title="Metric (mathematics)">metric</a> to measure distances between observed and predicted data is a useful tool for assessing model fit. In statistics, decision theory, and some <a href="/wiki/Economic_model" title="Economic model">economic models</a>, a <a href="/wiki/Loss_function" title="Loss function">loss function</a> plays a similar role. While it is rather straightforward to test the appropriateness of parameters, it can be more difficult to test the validity of the general mathematical form of a model. In general, more mathematical tools have been developed to test the fit of <a href="/wiki/Statistical_model" title="Statistical model">statistical models</a> than models involving <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. Tools from <a href="/wiki/Nonparametric_statistics" title="Nonparametric statistics">nonparametric statistics</a> can sometimes be used to evaluate how well the data fit a known distribution or to come up with a general model that makes only minimal assumptions about the model's mathematical form. </p> <div class="mw-heading mw-heading4"><h4 id="Scope_of_the_model">Scope of the model</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=10" title="Edit section: Scope of the model"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Assessing the scope of a model, that is, determining what situations the model is applicable to, can be less straightforward. If the model was constructed based on a set of data, one must determine for which systems or situations the known data is a "typical" set of data. The question of whether the model describes well the properties of the system between data points is called <a href="/wiki/Interpolation" title="Interpolation">interpolation</a>, and the same question for events or data points outside the observed data is called <a href="/wiki/Extrapolation" title="Extrapolation">extrapolation</a>. </p><p>As an example of the typical limitations of the scope of a model, in evaluating Newtonian <a href="/wiki/Classical_mechanics" title="Classical mechanics">classical mechanics</a>, we can note that Newton made his measurements without advanced equipment, so he could not measure properties of particles traveling at speeds close to the speed of light. Likewise, he did not measure the movements of molecules and other small particles, but macro particles only. It is then not surprising that his model does not extrapolate well into these domains, even though his model is quite sufficient for ordinary life physics. </p> <div class="mw-heading mw-heading4"><h4 id="Philosophical_considerations">Philosophical considerations</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=11" title="Edit section: Philosophical considerations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many types of modeling implicitly involve claims about <a href="/wiki/Causality" title="Causality">causality</a>. This is usually (but not always) true of models involving differential equations. As the purpose of modeling is to increase our understanding of the world, the validity of a model rests not only on its fit to empirical observations, but also on its ability to extrapolate to situations or data beyond those originally described in the model. One can think of this as the differentiation between qualitative and quantitative predictions. One can also argue that a model is worthless unless it provides some insight which goes beyond what is already known from direct investigation of the phenomenon being studied. </p><p>An example of such criticism is the argument that the mathematical models of <a href="/wiki/Optimal_foraging_theory" title="Optimal foraging theory">optimal foraging theory</a> do not offer insight that goes beyond the common-sense conclusions of <a href="/wiki/Evolution" title="Evolution">evolution</a> and other basic principles of ecology.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> It should also be noted that while mathematical modeling uses mathematical concepts and language, it is not itself a branch of mathematics and does not necessarily conform to any <a href="/wiki/Mathematical_logic" title="Mathematical logic">mathematical logic</a>, but is typically a branch of some science or other technical subject, with corresponding concepts and standards of argumentation.<sup id="cite_ref-Edwards_2-1" class="reference"><a href="#cite_note-Edwards-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Significance_in_the_natural_sciences">Significance in the natural sciences</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=12" title="Edit section: Significance in the natural sciences"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Mathematical models are of great importance in the natural sciences, particularly in <a href="/wiki/Physics" title="Physics">physics</a>. Physical <a href="/wiki/Theory" title="Theory">theories</a> are almost invariably expressed using mathematical models. Throughout history, more and more accurate mathematical models have been developed. <a href="/wiki/Newton%27s_laws_of_motion" title="Newton's laws of motion">Newton's laws</a> accurately describe many everyday phenomena, but at certain limits <a href="/wiki/Theory_of_relativity" title="Theory of relativity">theory of relativity</a> and <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a> must be used. </p><p>It is common to use idealized models in physics to simplify things. Massless ropes, point particles, <a href="/wiki/Ideal_gases" class="mw-redirect" title="Ideal gases">ideal gases</a> and the <a href="/wiki/Particle_in_a_box" title="Particle in a box">particle in a box</a> are among the many simplified models used in physics. The laws of physics are represented with simple equations such as Newton's laws, <a href="/wiki/Maxwell%27s_equations" title="Maxwell's equations">Maxwell's equations</a> and the <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a>. These laws are a basis for making mathematical models of real situations. Many real situations are very complex and thus modeled approximately on a computer, a model that is computationally feasible to compute is made from the basic laws or from approximate models made from the basic laws. For example, molecules can be modeled by <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular orbital</a> models that are approximate solutions to the Schrödinger equation. In <a href="/wiki/Engineering" title="Engineering">engineering</a>, physics models are often made by mathematical methods such as <a href="/wiki/Finite_element_analysis" class="mw-redirect" title="Finite element analysis">finite element analysis</a>. </p><p>Different mathematical models use different geometries that are not necessarily accurate descriptions of the geometry of the universe. <a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a> is much used in classical physics, while <a href="/wiki/Special_relativity" title="Special relativity">special relativity</a> and <a href="/wiki/General_relativity" title="General relativity">general relativity</a> are examples of theories that use <a href="/wiki/Geometry" title="Geometry">geometries</a> which are not Euclidean. </p> <div class="mw-heading mw-heading2"><h2 id="Some_applications">Some applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=13" title="Edit section: Some applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Often when engineers analyze a system to be controlled or optimized, they use a mathematical model. In analysis, engineers can build a descriptive model of the system as a hypothesis of how the system could work, or try to estimate how an unforeseeable event could affect the system. Similarly, in control of a system, engineers can try out different control approaches in <a href="/wiki/Simulation" title="Simulation">simulations</a>. </p><p>A mathematical model usually describes a system by a <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> of variables and a set of equations that establish relationships between the variables. Variables may be of many types; <a href="/wiki/Real_number" title="Real number">real</a> or <a href="/wiki/Integer" title="Integer">integer</a> numbers, <a href="/wiki/Boolean_data_type" title="Boolean data type">Boolean</a> values or <a href="/wiki/String_(computing)" class="mw-redirect" title="String (computing)">strings</a>, for example. The variables represent some properties of the system, for example, the measured system outputs often in the form of <a href="/wiki/Signal_(electronics)" class="mw-redirect" title="Signal (electronics)">signals</a>, <a href="/wiki/Chronometry" title="Chronometry">timing data</a>, counters, and event occurrence. The actual model is the set of functions that describe the relations between the different variables. </p> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=14" title="Edit section: Examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>One of the popular examples in <a href="/wiki/Computer_science" title="Computer science">computer science</a> is the mathematical models of various machines, an example is the <a href="/wiki/Deterministic_finite_automaton" title="Deterministic finite automaton">deterministic finite automaton</a> (DFA) which is defined as an abstract mathematical concept, but due to the deterministic nature of a DFA, it is implementable in hardware and software for solving various specific problems. For example, the following is a DFA M with a binary alphabet, which requires that the input contains an even number of 0s:</li></ul> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:DFAexample.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/DFAexample.svg/250px-DFAexample.svg.png" decoding="async" width="250" height="150" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9d/DFAexample.svg/375px-DFAexample.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9d/DFAexample.svg/500px-DFAexample.svg.png 2x" data-file-width="500" data-file-height="299" /></a><figcaption>The <a href="/wiki/State_diagram" title="State diagram">state diagram</a> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span></figcaption></figure> <dl><dd><dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=(Q,\Sigma ,\delta ,q_{0},F)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>Q</mi> <mo>,</mo> <mi mathvariant="normal">Σ</mi> <mo>,</mo> <mi>δ</mi> <mo>,</mo> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mi>F</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=(Q,\Sigma ,\delta ,q_{0},F)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fbd4a0841c154d4f448f8001ce4ea0c33f1816cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.883ex; height:2.843ex;" alt="{\displaystyle M=(Q,\Sigma ,\delta ,q_{0},F)}"></span> where <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q=\{S_{1},S_{2}\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q=\{S_{1},S_{2}\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c4f3cc6c73e186ee8138df7876a984af211e3f8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.901ex; height:2.843ex;" alt="{\displaystyle Q=\{S_{1},S_{2}\},}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Sigma =\{0,1\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Σ</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma =\{0,1\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f84d07a170f94e1ca28488304bdedd6c61afa1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.107ex; height:2.843ex;" alt="{\displaystyle \Sigma =\{0,1\},}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q_{0}=S_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>q</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle q_{0}=S_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc6549c49de187aed92feed4d6ac45fcfe4b1827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.316ex; height:2.509ex;" alt="{\displaystyle q_{0}=S_{1},}"></span></li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F=\{S_{1}\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F=\{S_{1}\},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9cb1cec3e74481b84effbc81f31d9f91a5c9ed45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.29ex; height:2.843ex;" alt="{\displaystyle F=\{S_{1}\},}"></span> and</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c5321cfa797202b3e1f8620663ff43c4660ea03a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }"></span> is defined by the following <a href="/wiki/State-transition_table" title="State-transition table">state-transition table</a>:</li></ul></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><dl><dd><table border="1"> <tbody><tr> <td></td> <td><div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><b>0</b></div></td> <td><div class="center" style="width:auto; margin-left:auto; margin-right:auto;"><b>1</b></div> </td></tr> <tr> <td><b><i>S</i><sub>1</sub></b></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span> </td></tr> <tr> <td><b><i>S</i><sub>2</sub></b></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span> </td></tr></tbody></table></dd></dl></dd></dl></dd></dl></dd></dl> <dl><dd>The state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf84e7fd4fb8259a9b37f956afdf83ee2a020f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{1}}"></span> represents that there has been an even number of 0s in the input so far, while <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1143e284d5f25cef778ab482edf6617a523ddd9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.479ex; height:2.509ex;" alt="{\displaystyle S_{2}}"></span> signifies an odd number. A 1 in the input does not change the state of the automaton. When the input ends, the state will show whether the input contained an even number of 0s or not. If the input did contain an even number of 0s, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> will finish in state <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{1},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5825a2c55b7c98a3eaaaa8d48a7e8aff00a49aff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.126ex; height:2.509ex;" alt="{\displaystyle S_{1},}"></span> an accepting state, so the input string will be accepted.</dd></dl> <dl><dd>The language recognized by <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> is the <a href="/wiki/Regular_language" title="Regular language">regular language</a> given by the <a href="/wiki/Regular_expression" title="Regular expression">regular expression</a> 1*( 0 (1*) 0 (1*) )*, where "*" is the <a href="/wiki/Kleene_star" title="Kleene star">Kleene star</a>, e.g., 1* denotes any non-negative number (possibly zero) of symbols "1".</dd></dl> <ul><li>Many everyday activities carried out without a thought are uses of mathematical models. A geographical <a href="/wiki/Map_projection" title="Map projection">map projection</a> of a region of the earth onto a small, plane surface is a model which can be used for many purposes such as planning travel.<sup id="cite_ref-LAND_INFO_Worldwide_Mapping_12-0" class="reference"><a href="#cite_note-LAND_INFO_Worldwide_Mapping-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup></li> <li>Another simple activity is predicting the position of a vehicle from its initial position, direction and speed of travel, using the equation that distance traveled is the product of time and speed. This is known as <a href="/wiki/Dead_reckoning" title="Dead reckoning">dead reckoning</a> when used more formally. Mathematical modeling in this way does not necessarily require formal mathematics; animals have been shown to use dead reckoning.<sup id="cite_ref-13" class="reference"><a href="#cite_note-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup></li> <li><i><a href="/wiki/Population" title="Population">Population</a> Growth</i>. A simple (though approximate) model of population growth is the <a href="/wiki/Malthusian_growth_model" title="Malthusian growth model">Malthusian growth model</a>. A slightly more realistic and largely used population growth model is the <a href="/wiki/Logistic_function" title="Logistic function">logistic function</a>, and its extensions.</li> <li><i>Model of a particle in a potential-field</i>. In this model we consider a particle as being a point of mass which describes a trajectory in space which is modeled by a function giving its coordinates in space as a function of time. The potential field is given by a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\!:\mathbb {R} ^{3}\!\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mspace width="negativethinmathspace" /> <mo>:</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="negativethinmathspace" /> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V\!:\mathbb {R} ^{3}\!\to \mathbb {R} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3a6edae3390e6d31cda601ecb6443d4b4ad76e1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.975ex; height:2.676ex;" alt="{\displaystyle V\!:\mathbb {R} ^{3}\!\to \mathbb {R} }"></span> and the trajectory, that is a function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {r} \!:\mathbb {R} \to \mathbb {R} ^{3},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mspace width="negativethinmathspace" /> <mo>:</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo stretchy="false">→</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {r} \!:\mathbb {R} \to \mathbb {R} ^{3},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/280612aad2a8d72c77b8fa6dd01425e7337b9c6b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.323ex; height:3.009ex;" alt="{\displaystyle \mathbf {r} \!:\mathbb {R} \to \mathbb {R} ^{3},}"></span> is the solution of the differential equation: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mi>m</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">x</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">y</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>V</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold">z</mi> <mo mathvariant="bold" stretchy="false">^</mo> </mover> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd8f22ac8b8b4f56b4f541adcd524f759b5ed380" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:51.325ex; height:6.676ex;" alt="{\displaystyle -{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}m={\frac {\partial V[\mathbf {r} (t)]}{\partial x}}\mathbf {\hat {x}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial y}}\mathbf {\hat {y}} +{\frac {\partial V[\mathbf {r} (t)]}{\partial z}}\mathbf {\hat {z}} ,}"></span> that can be written also as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∇</mi> <mi>V</mi> <mo stretchy="false">[</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">r</mi> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc987d59f3eb6aea08a9def1a21e4441af4456ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:23.295ex; height:6.343ex;" alt="{\displaystyle m{\frac {\mathrm {d} ^{2}\mathbf {r} (t)}{\mathrm {d} t^{2}}}=-\nabla V[\mathbf {r} (t)].}"></span></li></ul> <dl><dd>Note this model assumes the particle is a point mass, which is certainly known to be false in many cases in which we use this model; for example, as a model of planetary motion.</dd></dl> <ul><li><i>Model of rational behavior for a consumer</i>. In this model we assume a consumer faces a choice of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"></span> commodities labeled <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1,2,\dots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1,2,\dots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784519df154bf219c605efe867b35e0c6f5b040c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.932ex; height:2.509ex;" alt="{\displaystyle 1,2,\dots ,n}"></span> each with a market price <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{1},p_{2},\dots ,p_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{1},p_{2},\dots ,p_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0749d85c05792d900bc29e3ab54a3100ba790d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:13.784ex; height:2.009ex;" alt="{\displaystyle p_{1},p_{2},\dots ,p_{n}.}"></span> The consumer is assumed to have an <a href="/wiki/Ordinal_utility" title="Ordinal utility">ordinal utility</a> function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"></span> (ordinal in the sense that only the sign of the differences between two utilities, and not the level of each utility, is meaningful), depending on the amounts of commodities <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c2d357bc1b965979bf171b5ba3bac0f68961c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots ,x_{n}}"></span> consumed. The model further assumes that the consumer has a budget <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}"></span> which is used to purchase a vector <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},\dots ,x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},\dots ,x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75c2d357bc1b965979bf171b5ba3bac0f68961c2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.528ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},\dots ,x_{n}}"></span> in such a way as to maximize <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U(x_{1},x_{2},\dots ,x_{n}).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U(x_{1},x_{2},\dots ,x_{n}).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/682ec2d5f93ac20e067807b731f83c8b19b596d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.767ex; height:2.843ex;" alt="{\displaystyle U(x_{1},x_{2},\dots ,x_{n}).}"></span> The problem of rational behavior in this model then becomes a <a href="/wiki/Mathematical_optimization" title="Mathematical optimization">mathematical optimization</a> problem, that is: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \max \,U(x_{1},x_{2},\ldots ,x_{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">max</mo> <mspace width="thinmathspace" /> <mi>U</mi> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \max \,U(x_{1},x_{2},\ldots ,x_{n})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6781d587998bfd61f080339db4ebdf61da198178" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.22ex; height:2.843ex;" alt="{\displaystyle \max \,U(x_{1},x_{2},\ldots ,x_{n})}"></span> subject to: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq M,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≤</mo> <mi>M</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq M,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/065ffa7ef8ac48cb74a46cbb9881076a4a18c94e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:14.028ex; height:6.843ex;" alt="{\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq M,}"></span><span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}\geq 0\;\;\;{\text{ for all }}i=1,2,\dots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>≥</mo> <mn>0</mn> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mspace width="thickmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> for all </mtext> </mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}\geq 0\;\;\;{\text{ for all }}i=1,2,\dots ,n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d6a3bfdcd890a552df30388782a81c9fbcd7202" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:29.789ex; height:2.509ex;" alt="{\displaystyle x_{i}\geq 0\;\;\;{\text{ for all }}i=1,2,\dots ,n.}"></span> This model has been used in a wide variety of economic contexts, such as in <a href="/wiki/General_equilibrium_theory" title="General equilibrium theory">general equilibrium theory</a> to show existence and <a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a> of economic equilibria.</li> <li><i><a href="/wiki/Neighbour-sensing_model" title="Neighbour-sensing model">Neighbour-sensing model</a></i> is a model that explains the <a href="/wiki/Mushroom" title="Mushroom">mushroom</a> formation from the initially chaotic <a href="/wiki/Fungus" title="Fungus">fungal</a> network.</li> <li>In <a href="/wiki/Computer_science" title="Computer science">computer science</a>, mathematical models may be used to simulate computer networks.</li> <li>In <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, mathematical models may be used to analyze the movement of a rocket model.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=15" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=16" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSaltelli2020" class="citation journal cs1">Saltelli, Andrea; et al. (June 2020). "Five ways to ensure that models serve society: a manifesto". <i>Nature</i>. <b>582</b> (7813): 482–484. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2020Natur.582..482S">2020Natur.582..482S</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1038%2Fd41586-020-01812-9">10.1038/d41586-020-01812-9</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/1885%2F219031">1885/219031</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/32581374">32581374</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Nature&rft.atitle=Five+ways+to+ensure+that+models+serve+society%3A+a+manifesto&rft.volume=582&rft.issue=7813&rft.pages=482-484&rft.date=2020-06&rft_id=info%3Ahdl%2F1885%2F219031&rft_id=info%3Apmid%2F32581374&rft_id=info%3Adoi%2F10.1038%2Fd41586-020-01812-9&rft_id=info%3Abibcode%2F2020Natur.582..482S&rft.aulast=Saltelli&rft.aufirst=Andrea&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-Edwards-2"><span class="mw-cite-backlink">^ <a href="#cite_ref-Edwards_2-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Edwards_2-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFEdwardsHamson2007" class="citation book cs1">Edwards, Dilwyn; Hamson, Mike (2007). <i>Guide to Mathematical Modelling</i> (2 ed.). New York: Industrial Press Inc. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8311-3337-5" title="Special:BookSources/978-0-8311-3337-5"><bdi>978-0-8311-3337-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Guide+to+Mathematical+Modelling&rft.place=New+York&rft.edition=2&rft.pub=Industrial+Press+Inc.&rft.date=2007&rft.isbn=978-0-8311-3337-5&rft.aulast=Edwards&rft.aufirst=Dilwyn&rft.au=Hamson%2C+Mike&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text">D. Tymoczko, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice (Oxford Studies in Music Theory), Oxford University Press; Illustrated Edition (March 21, 2011), <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0195336672" title="Special:BookSources/978-0195336672">978-0195336672</a></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Andras Kornai, Mathematical Linguistics (Advanced Information and Knowledge Processing), Springer, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1849966948" title="Special:BookSources/978-1849966948">978-1849966948</a></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAndreski1972" class="citation book cs1"><a href="/wiki/Stanislav_Andreski" title="Stanislav Andreski">Andreski, Stanislav</a> (1972). <i>Social Sciences as Sorcery</i>. <a href="/wiki/St._Martin%E2%80%99s_Press" class="mw-redirect" title="St. Martin’s Press">St. Martin’s Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-14-021816-5" title="Special:BookSources/0-14-021816-5"><bdi>0-14-021816-5</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Social+Sciences+as+Sorcery&rft.pub=St.+Martin%E2%80%99s+Press&rft.date=1972&rft.isbn=0-14-021816-5&rft.aulast=Andreski&rft.aufirst=Stanislav&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTruesdell1984" class="citation book cs1"><a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, Clifford</a> (1984). <i>An Idiot's Fugitive Essays on Science</i>. Springer. pp. 121–7. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-90703-3" title="Special:BookSources/3-540-90703-3"><bdi>3-540-90703-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Idiot%27s+Fugitive+Essays+on+Science&rft.pages=121-7&rft.pub=Springer&rft.date=1984&rft.isbn=3-540-90703-3&rft.aulast=Truesdell&rft.aufirst=Clifford&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Li, C., Xing, Y., He, F., & Cheng, D. (2018). A Strategic Learning Algorithm for State-based Games. ArXiv.</span> </li> <li id="cite_note-SAB1-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-SAB1_8-0">^</a></b></span> <span class="reference-text">Billings S.A. (2013), <i>Nonlinear System Identification: NARMAX Methods in the Time, Frequency, and Spatio-Temporal Domains</i>, Wiley.</span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://plato.stanford.edu/entries/thomas-kuhn/">"Thomas Kuhn"</a>. <i>Stanford Encyclopedia of Philosophy</i>. August 13, 2004<span class="reference-accessdate">. Retrieved <span class="nowrap">January 15,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Stanford+Encyclopedia+of+Philosophy&rft.atitle=Thomas+Kuhn&rft.date=2004-08-13&rft_id=https%3A%2F%2Fplato.stanford.edu%2Fentries%2Fthomas-kuhn%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFThornton" class="citation web cs1">Thornton, Chris. <a rel="nofollow" class="external text" href="http://users.sussex.ac.uk/~christ/crs/ml/lec03a.html">"Machine Learning Lecture"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">February 6,</span> 2019</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Machine+Learning+Lecture&rft.aulast=Thornton&rft.aufirst=Chris&rft_id=http%3A%2F%2Fusers.sussex.ac.uk%2F~christ%2Fcrs%2Fml%2Flec03a.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPyke1984" class="citation journal cs1">Pyke, G. H. (1984). "Optimal Foraging Theory: A Critical Review". <i>Annual Review of Ecology and Systematics</i>. <b>15</b>: 523–575. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.es.15.110184.002515">10.1146/annurev.es.15.110184.002515</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Ecology+and+Systematics&rft.atitle=Optimal+Foraging+Theory%3A+A+Critical+Review&rft.volume=15&rft.pages=523-575&rft.date=1984&rft_id=info%3Adoi%2F10.1146%2Fannurev.es.15.110184.002515&rft.aulast=Pyke&rft.aufirst=G.+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-LAND_INFO_Worldwide_Mapping-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-LAND_INFO_Worldwide_Mapping_12-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.landinfo.com/resources_dictionaryMP.htm">"GIS Definitions of Terminology M-P"</a>. <i>LAND INFO Worldwide Mapping</i><span class="reference-accessdate">. Retrieved <span class="nowrap">January 27,</span> 2020</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=LAND+INFO+Worldwide+Mapping&rft.atitle=GIS+Definitions+of+Terminology+M-P&rft_id=https%3A%2F%2Fwww.landinfo.com%2Fresources_dictionaryMP.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-13">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGallistel1990" class="citation book cs1">Gallistel (1990). <i>The Organization of Learning</i>. Cambridge: The MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-262-07113-4" title="Special:BookSources/0-262-07113-4"><bdi>0-262-07113-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Organization+of+Learning&rft.place=Cambridge&rft.pub=The+MIT+Press&rft.date=1990&rft.isbn=0-262-07113-4&rft.au=Gallistel&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhishawHinesWallace2001" class="citation journal cs1">Whishaw, I. Q.; Hines, D. J.; Wallace, D. G. (2001). "Dead reckoning (path integration) requires the hippocampal formation: Evidence from spontaneous exploration and spatial learning tasks in light (allothetic) and dark (idiothetic) tests". <i>Behavioural Brain Research</i>. <b>127</b> (1–2): 49–69. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0166-4328%2801%2900359-X">10.1016/S0166-4328(01)00359-X</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/11718884">11718884</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:7897256">7897256</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Behavioural+Brain+Research&rft.atitle=Dead+reckoning+%28path+integration%29+requires+the+hippocampal+formation%3A+Evidence+from+spontaneous+exploration+and+spatial+learning+tasks+in+light+%28allothetic%29+and+dark+%28idiothetic%29+tests&rft.volume=127&rft.issue=1%E2%80%932&rft.pages=49-69&rft.date=2001&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A7897256%23id-name%3DS2CID&rft_id=info%3Apmid%2F11718884&rft_id=info%3Adoi%2F10.1016%2FS0166-4328%2801%2900359-X&rft.aulast=Whishaw&rft.aufirst=I.+Q.&rft.au=Hines%2C+D.+J.&rft.au=Wallace%2C+D.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=17" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Books">Books</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=18" title="Edit section: Books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Aris, Rutherford [ 1978 ] ( 1994 ). <i>Mathematical Modelling Techniques</i>, New York: Dover. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-68131-9" title="Special:BookSources/0-486-68131-9">0-486-68131-9</a></li> <li>Bender, E.A. [ 1978 ] ( 2000 ). <i>An Introduction to Mathematical Modeling</i>, New York: Dover. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-41180-X" title="Special:BookSources/0-486-41180-X">0-486-41180-X</a></li> <li><a href="/wiki/Gary_Chartrand" title="Gary Chartrand">Gary Chartrand</a> (1977) <i>Graphs as Mathematical Models</i>, Prindle, Webber & Schmidt <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0871502364" title="Special:BookSources/0871502364">0871502364</a></li> <li>Dubois, G. (2018) <a rel="nofollow" class="external text" href="https://www.taylorfrancis.com/books/9781351241120/">"Modeling and Simulation"</a>, Taylor & Francis, CRC Press.</li> <li>Gershenfeld, N. (1998) <i>The Nature of Mathematical Modeling</i>, <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-57095-6" title="Special:BookSources/0-521-57095-6">0-521-57095-6</a> .</li> <li>Lin, C.C. & Segel, L.A. ( 1988 ). <i>Mathematics Applied to Deterministic Problems in the Natural Sciences</i>, Philadelphia: SIAM. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-89871-229-7" title="Special:BookSources/0-89871-229-7">0-89871-229-7</a></li></ul> <div class="mw-heading mw-heading3"><h3 id="Specific_applications">Specific applications</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=19" title="Edit section: Specific applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Papadimitriou, Fivos. (2010). Mathematical Modelling of Spatial-Ecological Complex Systems: an Evaluation. Geography, Environment, Sustainability 1(3), 67-80. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.24057%2F2071-9388-2010-3-1-67-80">10.24057/2071-9388-2010-3-1-67-80</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPeierls1980" class="citation journal cs1">Peierls, R. (1980). "Model-making in physics". <i>Contemporary Physics</i>. <b>21</b>: 3–17. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980ConPh..21....3P">1980ConPh..21....3P</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F00107518008210938">10.1080/00107518008210938</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Contemporary+Physics&rft.atitle=Model-making+in+physics&rft.volume=21&rft.pages=3-17&rft.date=1980&rft_id=info%3Adoi%2F10.1080%2F00107518008210938&rft_id=info%3Abibcode%2F1980ConPh..21....3P&rft.aulast=Peierls&rft.aufirst=R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AMathematical+model" class="Z3988"></span></li> <li><i><a rel="nofollow" class="external text" href="http://anintroductiontoinfectiousdiseasemodelling.com/">An Introduction to Infectious Disease Modelling</a></i> by Emilia Vynnycky and Richard G White.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Mathematical_model&action=edit&section=20" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><b>General reference</b> </p> <ul><li>Patrone, F. <a rel="nofollow" class="external text" href="http://www.fioravante.patrone.name/mat/u-u/en/differential_equations_intro.htm">Introduction to modeling via differential equations</a>, with critical remarks.</li> <li><a rel="nofollow" class="external text" href="http://plus.maths.org/issue44/package/index.html">Plus teacher and student package: Mathematical Modelling.</a> Brings together all articles on mathematical modeling from <i><a href="/wiki/Plus_Magazine" title="Plus Magazine">Plus Magazine</a></i>, the online mathematics magazine produced by the Millennium Mathematics Project at the University of Cambridge.</li></ul> <p><b>Philosophical</b> </p> <ul><li>Frigg, R. and S. Hartmann, <a rel="nofollow" class="external text" href="http://plato.stanford.edu/entries/models-science/">Models in Science</a>, in: The Stanford Encyclopedia of Philosophy, (Spring 2006 Edition)</li> <li>Griffiths, E. C. (2010) <a rel="nofollow" class="external text" href="https://sites.google.com/a/ncsu.edu/emily-griffiths/whatisamodel.pdf">What is a model?</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output 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