{"id":5202,"date":"2025-06-25T11:35:40","date_gmt":"2025-06-25T11:35:40","guid":{"rendered":"https:\/\/fs.kibu.ac.ke\/?p=5202"},"modified":"2025-12-26T22:58:14","modified_gmt":"2025-12-26T22:58:14","slug":"circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices","status":"publish","type":"post","link":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/","title":{"rendered":"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices"},"content":{"rendered":"\t\t<div data-elementor-type=\"wp-post\" data-elementor-id=\"5202\" class=\"elementor elementor-5202\" data-elementor-post-type=\"post\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-849d4c6 elementor-section-full_width elementor-section-height-min-height elementor-section-items-stretch elementor-section-height-default\" data-id=\"849d4c6\" data-element_type=\"section\" id=\"ourgoal\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-5d4dbcac\" data-id=\"5d4dbcac\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-4ec79a39 elementor-widget elementor-widget-theme-post-title elementor-page-title elementor-widget-heading\" data-id=\"4ec79a39\" data-element_type=\"widget\" data-widget_type=\"theme-post-title.default\">\n\t\t\t\t\t<h1 class=\"elementor-heading-title elementor-size-default\">Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices<\/h1>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-top-section elementor-element elementor-element-14a7774d elementor-section-height-min-height elementor-section-boxed elementor-section-height-default elementor-section-items-middle\" data-id=\"14a7774d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-no\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-top-column elementor-element elementor-element-65f45fab\" data-id=\"65f45fab\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-6533920d elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"6533920d\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-2f88e6d3 elementor-invisible\" data-id=\"2f88e6d3\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-660698a3 elementor-widget__width-auto elementor-widget elementor-widget-heading\" data-id=\"660698a3\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">2024<\/h2>\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-1d7a5e5 elementor-invisible\" data-id=\"1d7a5e5\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-b1a3c34 elementor-widget elementor-widget-heading\" data-id=\"b1a3c34\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h3 class=\"elementor-heading-title elementor-size-default\">KIBU Authors<\/h3>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-735eecce elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"735eecce\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>Shem Away<\/p><p>Lucy Walingo Chikamai<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t<div class=\"elementor-column elementor-col-33 elementor-inner-column elementor-element elementor-element-58384ed6 elementor-invisible\" data-id=\"58384ed6\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-3d054070 elementor-tablet-align-left elementor-invisible elementor-widget elementor-widget-button\" data-id=\"3d054070\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeInLeft&quot;}\" data-widget_type=\"button.default\">\n\t\t\t\t\t\t\t\t\t\t<a class=\"elementor-button elementor-button-link elementor-size-sm elementor-animation-shrink\" href=\"https:\/\/www.irejournals.com\/paper-details\/1706122\">\n\t\t\t\t\t\t<span class=\"elementor-button-content-wrapper\">\n\t\t\t\t\t\t\t\t\t<span class=\"elementor-button-text\">VIEW ON PUBLISHER SITE<\/span>\n\t\t\t\t\t<\/span>\n\t\t\t\t\t<\/a>\n\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<section class=\"elementor-section elementor-inner-section elementor-element elementor-element-721ee2a3 elementor-section-boxed elementor-section-height-default elementor-section-height-default\" data-id=\"721ee2a3\" data-element_type=\"section\">\n\t\t\t\t\t\t<div class=\"elementor-container elementor-column-gap-default\">\n\t\t\t\t\t<div class=\"elementor-column elementor-col-100 elementor-inner-column elementor-element elementor-element-536e1a6a elementor-invisible\" data-id=\"536e1a6a\" data-element_type=\"column\" data-settings=\"{&quot;background_background&quot;:&quot;classic&quot;,&quot;animation&quot;:&quot;fadeIn&quot;,&quot;animation_mobile&quot;:&quot;none&quot;}\">\n\t\t\t<div class=\"elementor-widget-wrap elementor-element-populated\">\n\t\t\t\t\t\t<div class=\"elementor-element elementor-element-160b01a1 elementor-widget elementor-widget-heading\" data-id=\"160b01a1\" data-element_type=\"widget\" data-widget_type=\"heading.default\">\n\t\t\t\t\t<h2 class=\"elementor-heading-title elementor-size-default\">Abstract<\/h2>\t\t\t\t<\/div>\n\t\t\t\t<div class=\"elementor-element elementor-element-6de57b11 elementor-invisible elementor-widget elementor-widget-text-editor\" data-id=\"6de57b11\" data-element_type=\"widget\" data-settings=\"{&quot;_animation&quot;:&quot;fadeIn&quot;}\" data-widget_type=\"text-editor.default\">\n\t\t\t\t\t\t\t\t\t<p>This study investigates the circularity and polygonal structure of numerical ranges for non-negative matrices. We provide a comprehensive characterization of non-negative matrices with circular numerical ranges and derive conditions for polygonal numerical ranges. Our main results show that a non-negative matrix has a circular numerical range centered at the origin if and only if it is unitarily equivalent to a scalar multiple of a doubly stochastic matrix. Furthermore, we prove that the numerical range of a non-negative matrix is a regular polygon with k vertices if and only if the matrix is unitarily equivalent to a direct sum of k cyclic permutation matrices. We explore the relationships between matrix structure (irreducibility, sparsity, and symmetry) and numerical range geometry, supported by computational methods for visualization and analysis. The study establishes connections between numerical range geometry and applications in Markov chains, quantum information theory, and graph theory. Our findings extend existing theory, provide new matrix characterizations based on numerical range geometry, and offer computational tools for further research. This work contributes to a deeper understanding of non-negative matrices and their properties, with potential implications for various fields in mathematics, physics, and computer science.<\/p>\t\t\t\t\t\t\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t\t<\/div>\n\t\t<\/div>\n\t\t\t\t\t<\/div>\n\t\t<\/section>\n\t\t\t\t<\/div>\n\t\t","protected":false},"excerpt":{"rendered":"<p>2024 KIBU Authors Shem Away Lucy Walingo Chikamai VIEW ON PUBLISHER SITE Abstract This study investigates the circularity and polygonal structure of numerical ranges for non-negative matrices. We provide a comprehensive characterization of non-negative matrices with circular numerical ranges and derive conditions for polygonal numerical ranges. Our main results show that a non-negative matrix has [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"elementor_header_footer","format":"standard","meta":{"footnotes":""},"categories":[9],"tags":[],"class_list":["post-5202","post","type-post","status-publish","format-standard","hentry","category-research-list"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices - Faculty of Science<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices - Faculty of Science\" \/>\n<meta property=\"og:description\" content=\"2024 KIBU Authors Shem Away Lucy Walingo Chikamai VIEW ON PUBLISHER SITE Abstract This study investigates the circularity and polygonal structure of numerical ranges for non-negative matrices. We provide a comprehensive characterization of non-negative matrices with circular numerical ranges and derive conditions for polygonal numerical ranges. Our main results show that a non-negative matrix has [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\" \/>\n<meta property=\"og:site_name\" content=\"Faculty of Science\" \/>\n<meta property=\"article:published_time\" content=\"2025-06-25T11:35:40+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2025-12-26T22:58:14+00:00\" \/>\n<meta name=\"author\" content=\"kibabii\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Written by\" \/>\n\t<meta name=\"twitter:data1\" content=\"kibabii\" \/>\n\t<meta name=\"twitter:label2\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data2\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#article\",\"isPartOf\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\"},\"author\":{\"name\":\"kibabii\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6\"},\"headline\":\"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices\",\"datePublished\":\"2025-06-25T11:35:40+00:00\",\"dateModified\":\"2025-12-26T22:58:14+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\"},\"wordCount\":213,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#organization\"},\"articleSection\":[\"Research List\"],\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\",\"name\":\"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices - Faculty of Science\",\"isPartOf\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#website\"},\"datePublished\":\"2025-06-25T11:35:40+00:00\",\"dateModified\":\"2025-12-26T22:58:14+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/kibu.ac.ke\/fs\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Faculty Publications\",\"item\":\"https:\/\/kibu.ac.ke\/fs\/faculty-publications\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices\"}]},{\"@type\":\"WebSite\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#website\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/\",\"name\":\"Faculty of Science\",\"description\":\"\",\"publisher\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"https:\/\/kibu.ac.ke\/fs\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"en-US\"},{\"@type\":\"Organization\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#organization\",\"name\":\"Faculty of Science\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/logo\/image\/\",\"url\":\"https:\/\/kibu.ac.ke\/fs\/wp-content\/uploads\/2025\/12\/KIbabii-Logo-1024x923-1.png\",\"contentUrl\":\"https:\/\/kibu.ac.ke\/fs\/wp-content\/uploads\/2025\/12\/KIbabii-Logo-1024x923-1.png\",\"width\":1024,\"height\":923,\"caption\":\"Faculty of Science\"},\"image\":{\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/logo\/image\/\"}},{\"@type\":\"Person\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6\",\"name\":\"kibabii\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"en-US\",\"@id\":\"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/image\/\",\"url\":\"https:\/\/secure.gravatar.com\/avatar\/d1290ba6f5d0e2707a88baa0e2f5a00b8214d0c1dce43336da834b265d51f88f?s=96&d=mm&r=g\",\"contentUrl\":\"https:\/\/secure.gravatar.com\/avatar\/d1290ba6f5d0e2707a88baa0e2f5a00b8214d0c1dce43336da834b265d51f88f?s=96&d=mm&r=g\",\"caption\":\"kibabii\"},\"sameAs\":[\"https:\/\/kibu.ac.ke\/fs\"],\"url\":\"https:\/\/kibu.ac.ke\/fs\/author\/kibabii\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices - Faculty of Science","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/","og_locale":"en_US","og_type":"article","og_title":"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices - Faculty of Science","og_description":"2024 KIBU Authors Shem Away Lucy Walingo Chikamai VIEW ON PUBLISHER SITE Abstract This study investigates the circularity and polygonal structure of numerical ranges for non-negative matrices. We provide a comprehensive characterization of non-negative matrices with circular numerical ranges and derive conditions for polygonal numerical ranges. Our main results show that a non-negative matrix has [&hellip;]","og_url":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/","og_site_name":"Faculty of Science","article_published_time":"2025-06-25T11:35:40+00:00","article_modified_time":"2025-12-26T22:58:14+00:00","author":"kibabii","twitter_card":"summary_large_image","twitter_misc":{"Written by":"kibabii","Est. reading time":"1 minute"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#article","isPartOf":{"@id":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/"},"author":{"name":"kibabii","@id":"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6"},"headline":"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices","datePublished":"2025-06-25T11:35:40+00:00","dateModified":"2025-12-26T22:58:14+00:00","mainEntityOfPage":{"@id":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/"},"wordCount":213,"commentCount":0,"publisher":{"@id":"https:\/\/kibu.ac.ke\/fs\/#organization"},"articleSection":["Research List"],"inLanguage":"en-US","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#respond"]}]},{"@type":"WebPage","@id":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/","url":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/","name":"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices - Faculty of Science","isPartOf":{"@id":"https:\/\/kibu.ac.ke\/fs\/#website"},"datePublished":"2025-06-25T11:35:40+00:00","dateModified":"2025-12-26T22:58:14+00:00","breadcrumb":{"@id":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#breadcrumb"},"inLanguage":"en-US","potentialAction":[{"@type":"ReadAction","target":["https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/"]}]},{"@type":"BreadcrumbList","@id":"https:\/\/kibu.ac.ke\/fs\/circularity-and-polygonal-structure-of-numerical-ranges-for-non-negative-matrices\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https:\/\/kibu.ac.ke\/fs\/"},{"@type":"ListItem","position":2,"name":"Faculty Publications","item":"https:\/\/kibu.ac.ke\/fs\/faculty-publications\/"},{"@type":"ListItem","position":3,"name":"Circularity and Polygonal Structure of Numerical Ranges for Non-Negative Matrices"}]},{"@type":"WebSite","@id":"https:\/\/kibu.ac.ke\/fs\/#website","url":"https:\/\/kibu.ac.ke\/fs\/","name":"Faculty of Science","description":"","publisher":{"@id":"https:\/\/kibu.ac.ke\/fs\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https:\/\/kibu.ac.ke\/fs\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"en-US"},{"@type":"Organization","@id":"https:\/\/kibu.ac.ke\/fs\/#organization","name":"Faculty of Science","url":"https:\/\/kibu.ac.ke\/fs\/","logo":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/kibu.ac.ke\/fs\/#\/schema\/logo\/image\/","url":"https:\/\/kibu.ac.ke\/fs\/wp-content\/uploads\/2025\/12\/KIbabii-Logo-1024x923-1.png","contentUrl":"https:\/\/kibu.ac.ke\/fs\/wp-content\/uploads\/2025\/12\/KIbabii-Logo-1024x923-1.png","width":1024,"height":923,"caption":"Faculty of Science"},"image":{"@id":"https:\/\/kibu.ac.ke\/fs\/#\/schema\/logo\/image\/"}},{"@type":"Person","@id":"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/fc9b7f13eeb65cf0be520197c35b17d6","name":"kibabii","image":{"@type":"ImageObject","inLanguage":"en-US","@id":"https:\/\/kibu.ac.ke\/fs\/#\/schema\/person\/image\/","url":"https:\/\/secure.gravatar.com\/avatar\/d1290ba6f5d0e2707a88baa0e2f5a00b8214d0c1dce43336da834b265d51f88f?s=96&d=mm&r=g","contentUrl":"https:\/\/secure.gravatar.com\/avatar\/d1290ba6f5d0e2707a88baa0e2f5a00b8214d0c1dce43336da834b265d51f88f?s=96&d=mm&r=g","caption":"kibabii"},"sameAs":["https:\/\/kibu.ac.ke\/fs"],"url":"https:\/\/kibu.ac.ke\/fs\/author\/kibabii\/"}]}},"_links":{"self":[{"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/posts\/5202","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/comments?post=5202"}],"version-history":[{"count":3,"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/posts\/5202\/revisions"}],"predecessor-version":[{"id":5714,"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/posts\/5202\/revisions\/5714"}],"wp:attachment":[{"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/media?parent=5202"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/categories?post=5202"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/kibu.ac.ke\/fs\/wp-json\/wp\/v2\/tags?post=5202"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}