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Theoretical Ecology Seminar from the IITE (https://iite.info)
Digital models and differential equations
Introduction to Fourier's ideas
Differential equations and macroecology: an integration
Digital models and phenology
Fundamentos y estructuras matemáticas
Effects of the mass of the predator on community in WCNRM2020
Notes on Biomathematics, Differential Equations, and Numerics
Modelamiento_Biomatematica_Agricultura_2025__Copy_.pdfThis biomathematics text provides a concise and applied introduction to the use of mathematical models in agricultural sciences, focusing on the formulation and interpretation of models for plant growth, population dynamics, and disease processes. By combining differential equations with biological insight, the book highlights how mathematical modelling can support understanding and decision-making in agricultural systems under environmental variability.
Calculo_Numerico_UCT_2025__Copy_.pdfThis numerical calculus book offers a clear and practice-oriented introduction to computational methods for engineering, covering core topics such as numerical differentiation and integration, root-finding, and the numerical solution of differential equations. Emphasis is placed on algorithmic implementation, accuracy, and stability, making the text a valuable reference for engineers seeking to model and solve real-world problems using reliable numerical methods.
Apuntes_EDO_IEEE_2026.pdfThis differential equations book provides an applied introduction tailored to electrical engineering, focusing on the modelling and analysis of circuits, dynamic systems, and signal behaviour using ordinary differential equations. The text emphasises the connection between mathematical formulation and physical interpretation, illustrating how differential equations describe transient and steady-state responses in electrical systems.
Transportes_en_Plantas_CEAF.pdfThe chapter develops a rigorous synthesis of the principal transport mechanisms in plants from a quantitative and phenomenological perspective. Starting from classical models of diffusion, convection, and active transport, mathematical expressions are derived that describe substrate fluxes between source and sink regions. The text systematically integrates physico-chemical foundations — concentration gradients, osmotic pressure, diffusive resistances, and energetic coupling — with mathematical formulations suitable for modelling at the organ or whole-plant scale.
This chapter corresponds to the Spanish translation of material originally presented in:
Thornley, J. H. M., & Johnson, I. R. (1990). Plant and Crop Modelling: A Mathematical Approach to Plant and Crop Physiology. Oxford: Clarendon Press.
Penman_Monteith_Relaciones_Hidricas_CITRA.pdfThe chapter presents a rigorous synthesis of plant water relations and canopy–atmosphere exchange processes from a quantitative and mechanistic perspective. Building upon the Penman–Monteith equation and soil–plant hydraulic theory, mathematical formulations are derived to describe water fluxes across the soil–plant–atmosphere continuum. The treatment integrates thermodynamic principles — water potential components, vapor pressure gradients, hydraulic conductances, and resistances in series and parallel — with differential equations.
This chapter corresponds to the Spanish translation and adaptation of material originally presented in:
Thornley, J. H. M., & Johnson, I. R. (1990). Plant and Crop Modelling: A Mathematical Approach to Plant and Crop Physiology. Oxford: Clarendon Press.
Apunte_Calculo_Varias_Variables_2026.pdfEsta obra propone una aproximación al cálculo en varias variables inspirada en la tradición intuicionista de la matemática, en la que la construcción de los conceptos, la comprensión geométrica y la resolución de problemas preceden a su formalización.
El recorrido comienza con el teorema del valor medio y las aproximaciones de Taylor en una variable, para extender progresivamente estas ideas a funciones de dos, tres y más variables. A partir de la interpretación geométrica del plano tangente, las derivadas parciales y la diferenciabilidad, el lector desarrolla una comprensión constructiva de las herramientas fundamentales del análisis multivariable.
La obra culmina con el teorema fundamental del cálculo y sus extensiones a varias dimensiones, estableciendo conexiones entre integración, geometría y los teoremas de Green, Stokes y Gauss. Los ejemplos resueltos y las demostraciones detalladas permiten transitar desde la intuición matemática hacia el razonamiento riguroso.
Apuntes_Algebra_Linela_2026.pdfEste libro presenta una introducción sistemática al álgebra lineal, siguiendo la estructura y el rigor característicos de los textos clásicos de la disciplina. Su propósito es desarrollar una comprensión sólida de los espacios vectoriales, las transformaciones lineales y las estructuras algebraicas que fundamentan buena parte de la matemática contemporánea.
A través de ejercicios resueltos paso a paso, el lector estudia sistemas de ecuaciones lineales, matrices, determinantes, bases, dimensión, valores y vectores propios, diagonalización y espacios con producto interno. Cada capítulo combina definiciones precisas, teoremas, demostraciones y problemas de dificultad progresiva.
La obra busca que el estudiante comprenda no solamente cómo realizar un cálculo, sino también por qué funciona un procedimiento y bajo qué condiciones es válido. De esta manera, proporciona una base rigurosa para continuar estudios en análisis matemático, ecuaciones diferenciales, optimización y modelización matemática.
Quantum_Calculus_2026.pdfEsta obra introduce al lector en el cálculo cuántico o Quantum Calculus, una generalización del cálculo diferencial e integral que permite estudiar variaciones y acumulaciones mediante operadores de diferencias sin recurrir necesariamente a los límites tradicionales.
Partiendo de la derivada clásica, se desarrolla la derivada, la integral de Jackson y las principales funciones del análisis qqq, incluyendo las exponenciales y las funciones trigonométricas q-análogas. Los conceptos se presentan mediante ejemplos detallados y demostraciones que permiten comprender cómo el cálculo clásico se recupera, bajo condiciones apropiadas, cuando q→1.
El texto avanza hacia las ecuaciones q-diferenciales, sus propiedades analíticas y sus aplicaciones a problemas matemáticos y modelos dinámicos. Finalmente, introduce conexiones con la teoría de Galois de ecuaciones en diferencias, ofreciendo una perspectiva sobre las estructuras algebraicas que subyacen a esta generalización del análisis.
Ecuaciones_Física_Matemática_2026.pdfFundamentos de Análisis No Lineal y Sistemas Dinámicos es una obra dedicada al estudio de las herramientas matemáticas que permiten comprender la evolución de sistemas cuya dinámica está gobernada por relaciones no lineales.
El libro aborda conceptos fundamentales del análisis matemático, las ecuaciones diferenciales y la teoría de sistemas dinámicos, con especial atención a la existencia y unicidad de soluciones, la estabilidad de los puntos de equilibrio, el comportamiento asintótico y los fenómenos de bifurcación. A través de definiciones rigurosas, demostraciones detalladas y ejemplos, establece conexiones entre la teoría matemática y sus aplicaciones en física, biología, ecología e ingeniería.
Su propósito es proporcionar una base teórica sólida para el estudio, modelamiento y análisis cualitativo de fenómenos dinámicos complejos, integrando el rigor matemático con una perspectiva aplicada.
Libro_Fisica_Matematica_2026.pdfEcuaciones de la Física Matemática: Fundamentos, Demostraciones y Problemas Resueltos es una obra dedicada al estudio de las herramientas matemáticas que permiten formular, analizar y resolver ecuaciones que describen fenómenos físicos.
El libro aborda conceptos fundamentales de la teoría de distribuciones, funciones de Green, convolución, análisis de Fourier, espacios de Sobolev y operadores elípticos, con especial atención a la existencia, unicidad, estabilidad y regularidad de las soluciones. A través de definiciones rigurosas, demostraciones paso a paso y problemas resueltos, establece conexiones entre la teoría matemática y sus aplicaciones en física, ingeniería y modelamiento de fenómenos naturales.
Su propósito es proporcionar una base teórica sólida para comprender y resolver problemas de la física matemática, integrando el rigor de las demostraciones con un enfoque pedagógico y aplicado.
Scientific Dissemination through Dialogue and Teaching
Scientific Dissemination through Dialogue and Teaching
In my early academic years, my formation was strongly shaped by continuous participation in seminars, workshops, and informal academic discussions, where conversations with experienced researchers played a central role in consolidating my interests and perspectives in mathematics. These formative exchanges were fundamental in developing a critical and reflective approach to mathematical thinking, teaching, and interdisciplinary work.
Over the years, this early exposure to academic dialogue evolved into active participation in a wide range of scientific and academic events, where I have shared ideas, presented research, and engaged with diverse communities. These activities reflect a sustained commitment to scientific dissemination, teaching, and the training of new generations, as well as to fostering interdisciplinary dialogue between mathematics and related fields. Together, these experiences highlight the role of communication and collaboration as essential components of mathematical practice and its applications.
Academic Presentations and Publications
The academic trajectory of William Campillay-Llanos began in the early 2000s, with a strong emphasis on mathematics education, mathematical proof, and the epistemological foundations of the discipline.
In 2007, he actively participated as a lecturer at UMCE and the University of Chile, addressing key issues in the teaching of mathematics in Chile. During this period, he delivered talks such as “The Use of Cabri: Is It Always Appropriate? Three Problems for Reflection” and “Proposals to Overcome Shortcomings and Remedy Frequent Errors in Chilean School Mathematics”, both focused on the role of proof in the classroom. That same year, he published the article “A Proposal for Guiding Mathematical Proof in Chilean Curricula and the Idea of Academic Rigour” in the Revista Chilena de Educación Científica.
In 2009, his work gained broader national and international visibility. He was a speaker at the Ibero-American Congress on Mathematics Education (CIBEM) with the presentation “Proportional Calculus: The Multiplicative World”. He also participated in the 23rd Southern Zone Mathematics Conference (University of Magallanes) with “An Introduction to Infinite Calculi: Proportional Calculus”, and presented “To Prove Is to Do Mathematics, and to Do Mathematics Is to Prove” at the meeting of the Chilean Mathematical Society. In the same year, he contributed as a reviewer to the monograph “Algebra as a Quality of the Student”, within the framework of a FONDEF project.
In 2010, he published the article “Mathematical Proof in Chilean Secondary School Curricula” in the Revista Iberoamericana de Educación Matemática.
Between 2011 and 2012, his academic production took on a more interdisciplinary character, integrating mathematics, biology, epistemology, and education. In 2011, he presented “Sensitivity and Rigour in the Study of Mathematics: From Multiplicative Calculus to the Biology of Love” at AEPMAT, and co-authored with L. Espinoza the paper “Didactical Situations Theory in Chile: Does It Work?” published in Acta Latinoamericana de Matemática Educativa.
In 2012, he presented “The Relationship between Galois Theory and the Reflections of Humberto Maturana” at the Southern Zone Mathematics Conference, and “The Autopoiesis of Galois Theory” at the National Mathematics Meeting in Chile.
From 2013 onwards, his academic work increasingly focused on theoretical and applied mathematics, particularly functional analysis and differential equations. That year, he presented “Proportional Differential Equations”, jointly with M. Pinto, at the Annual Meeting of the Group on Evolution Differential Equations and Functional Analysis (USACH).
In 2014, he delivered the talk “Structure and Form” at the Institute of Philosophy and Complexity Sciences (IFICC), and “Question or Conjecture” at the Seminar on Evolution Equations of the University of La Frontera, strengthening the dialogue between mathematics, philosophy, and complexity.
Finally, in 2015, he participated as a speaker in high-level academic settings, delivering the lecture “What Is an Arithmetic?” at the Postgraduate Mathematics Seminar of USACH (August 2015), and presenting, together with F. Guevara-Morales, “Gelfond’s Constant and Proportional Arithmetic” at the Southern Zone Mathematics Conference (April 2015).
Participation and Attendance at Academic Events (2004–2014)
Alongside his work as a speaker, William Campillay-Llanos has maintained continuous participation in schools, conferences, and seminars, strengthening both his academic training and professional networks.
Between 2004 and 2006, he participated in the Colloquium on Challenges and Perspectives in Mathematics Education (UMCE), the First and Second National Schools of Biomathematics (UMCE), the Mathematics Week organised by PUCV, and the Capricorn Mathematics Congress (COMCA 2006).
In 2010, he attended the Seminar on Functional Analysis and Evolution Equations organised by the GAFEVOL research group (USACH).
During 2011, he expanded his international exposure by participating in the IMPA Summer School (Rio de Janeiro) and the First Workshop in Game Theory and Evolutionary Dynamics at the Fluminense Federal University, Brazil.
In 2014, he attended the 7th International Conference on Functional Equations, held at the University of Limoges, France, further consolidating his training in functional equations and mathematical analysis at an international level.
The passage belongs to the poem “Au Lecteur” (“To the Reader”), which opens Les Fleurs du mal by Charles Baudelaire and serves as a true prologue to the book, setting from the outset its moral, aesthetic, and philosophical tone. Written around 1855 and first published in Paris in 1857, Les Fleurs du mal caused a major public and judicial scandal upon its release: Baudelaire was prosecuted for “offence against public morality and good morals,” and several poems were censored. This context underscores the provocative nature of “Au Lecteur”, which introduces the reader to a work that deliberately confronts the anxieties, vices, and spiritual malaise of modern life.
Français (original)
Mais parmi les chacals, les panthères, les lices,
Les singes, les scorpions, les vautours, les serpents,
Les monstres glapissants, hurlants, grognants, rampants,
Dans la ménagerie infâme de nos vices,
Español
Pero entre los chacales, las panteras, las perras,
Los simios, los escorpiones, los buitres, las serpientes,
Los monstruos chillones, aullantes, gruñones, reptantes,
En la infame jauría de nuestros vicios,
English
But among the jackals, panthers, hounds,
Apes, scorpions, vultures, serpents,
The yelping, howling, growling, crawling monsters,
In the infamous menagerie of our vices,
Français (original)
Il en est un plus laid, plus méchant, plus immonde !
Quoiqu’il ne pousse ni grands gestes, ni grands cris,
Il ferait volontiers de la terre un débris
Et dans un bâillement avalerait le monde ;
Español
¡Hay uno más feo, más malo, más inmundo!
Aunque no haga grandes gestos ni lance grandes gritos,
Haría gustoso de la tierra un despojo
Y en un bostezo se tragaría el mundo;
English
There is one more ugly, more wicked, more foul!
Though he makes neither grand gestures nor loud cries,
He would gladly turn the earth into rubble
And in a yawn swallow the world;
Français (original)
C’est l’Ennui ! — l’œil chargé d’un pleur involontaire,
Il rêve d’échafauds en fumant son houka.
Tu le connais, lecteur, ce monstre délicat,
Español
¡Es el Tedio! — con los ojos cargados de un llanto involuntario,
Sueña con patíbulos mientras fuma su pipa.
Tú conoces, lector, a este monstruo delicado,
English
It is Boredom! — eyes heavy with involuntary tears,
He dreams of scaffolds while smoking his pipe.
You know him, reader, this delicate monster,
Français (original)
— Hypocrite lecteur, — mon semblable, — mon frère !
Español
— ¡Hipócrita lector, — mi semejante, — mi hermano!
English
— Hypocritical reader, — my likeness, — my brother!
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