<?xml version="1.0" encoding="ISO-8859-1"?><article xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance">
<front>
<journal-meta>
<journal-id>1409-2433</journal-id>
<journal-title><![CDATA[Revista de Matemática Teoría y Aplicaciones]]></journal-title>
<abbrev-journal-title><![CDATA[Rev. Mat]]></abbrev-journal-title>
<issn>1409-2433</issn>
<publisher>
<publisher-name><![CDATA[Centro de Investigaciones en Matemática Pura y Aplicada (CIMPA) y Escuela de Matemática, San José, Costa Rica.]]></publisher-name>
</publisher>
</journal-meta>
<article-meta>
<article-id>S1409-24332025000200091</article-id>
<article-id pub-id-type="doi">10.15517/3fczkg02</article-id>
<title-group>
<article-title xml:lang="en"><![CDATA[Transparent spheres as gravitational lenses]]></article-title>
<article-title xml:lang="es"><![CDATA[Esferas transparentes como lentes gravitacionales]]></article-title>
</title-group>
<contrib-group>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Santiago-Leandro]]></surname>
<given-names><![CDATA[Edwin]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Mora-Chaverri]]></surname>
<given-names><![CDATA[Alexander]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
<contrib contrib-type="author">
<name>
<surname><![CDATA[Frutos-Alfaro]]></surname>
<given-names><![CDATA[Francisco]]></given-names>
</name>
<xref ref-type="aff" rid="Aff"/>
</contrib>
</contrib-group>
<aff id="Af1">
<institution><![CDATA[,Universidad de Costa Rica Centro de Investigaciones Espaciales ]]></institution>
<addr-line><![CDATA[ San José]]></addr-line>
<country>Costa Rica</country>
</aff>
<aff id="Af2">
<institution><![CDATA[,Universidad de Costa Rica Centro de Investigaciones Espaciales ]]></institution>
<addr-line><![CDATA[ San José]]></addr-line>
<country>Costa Rica</country>
</aff>
<aff id="Af3">
<institution><![CDATA[,Universidad de Costa Rica Centro de Investigaciones Espaciales ]]></institution>
<addr-line><![CDATA[ San José]]></addr-line>
<country>Costa Rica</country>
</aff>
<pub-date pub-type="pub">
<day>00</day>
<month>12</month>
<year>2025</year>
</pub-date>
<pub-date pub-type="epub">
<day>00</day>
<month>12</month>
<year>2025</year>
</pub-date>
<volume>32</volume>
<numero>2</numero>
<fpage>91</fpage>
<lpage>115</lpage>
<copyright-statement/>
<copyright-year/>
<self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_arttext&amp;pid=S1409-24332025000200091&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_abstract&amp;pid=S1409-24332025000200091&amp;lng=en&amp;nrm=iso"></self-uri><self-uri xlink:href="http://www.scielo.sa.cr/scielo.php?script=sci_pdf&amp;pid=S1409-24332025000200091&amp;lng=en&amp;nrm=iso"></self-uri><abstract abstract-type="short" xml:lang="en"><p><![CDATA[Abstract In this work, we present a concise and educational study of gravitational lensing by transparent matter distributions. We focus on the calculation of image properties for several idealized mass profiles, including the uniform transparent sphere, the isothermal gas sphere, the non singular isothermal sphere, and the transparent King profile. Using numerical techniques and the XFGLenses software, we compute and visualize the resulting lensed im-ages, along with the associated critical curves and caustics. The results are consistent with established theoretical predictions for transparent lenses, for example, the occurrence of an odd number of images, and the reduction of two images as the source crosses a caustic. The caustic geometries observed include diamond-shaped, elliptical, and lemniscate-like structures. Among the critical curves, ellipses were most commonly observed, while lemniscate-like forms emerged specifically in the transparent non-singular isothermal sphere case, in agreement with known behaviors in gravitational lensing by smooth matter distributions.]]></p></abstract>
<abstract abstract-type="short" xml:lang="es"><p><![CDATA[Resumen En esta contribución, presentamos un estudio conciso y educativo de lentes gravitacionales debido a distribuciones de masa transparentes. Nos enfocamos en los cálculos de las propiedades de la imagen para perfiles de masa idealizados, incluyendo la esfera transparente uniforme, la esfera de gas isotérmico, la esfera de gas isotérmico no singular, y el perfil de King transparente. Utilizando técnicas numéricas y el software XFGLenses, se calculan y se visualizan las imágenes resultantes, junto con las curvas críticas y cáusticas asociadas. Los resultados son consistentes con las predicciones teóricas de los lentes transparentes, como lo son un número impar de imágenes, y la reducción del número de imágenes en dos cuando la fuente atraviesa la cáustica. Las geometrías que presentan las curvas cáusticas encontradas incluyen la forma de diamante, forma elíptica, y tipo lemniscata. Entre las curvas críticas, formas elípticas fueron las más encontradas, y la forma tipo lemniscata aparecieron específicamente en el case de la esfera isotérmica no singular, lo cual es esperado de lo conocido de lentes gravitacionales debido a distribuciones ideales.]]></p></abstract>
<kwd-group>
<kwd lng="en"><![CDATA[gravitational lensing]]></kwd>
<kwd lng="en"><![CDATA[numerical simulations]]></kwd>
<kwd lng="en"><![CDATA[general relativity]]></kwd>
<kwd lng="es"><![CDATA[lentes gravitacionales]]></kwd>
<kwd lng="es"><![CDATA[simulaciones numéricas]]></kwd>
<kwd lng="es"><![CDATA[relatividad general]]></kwd>
</kwd-group>
</article-meta>
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