Boletín Nº 159
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Boletín del IMI
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1) Activities from January 23 to 31, 2025
Seminario de Análisis Matemático y Matemática Aplicada
Title: Finite replicator dynamics as numerical approach of a game with a continuum of pure strategies
Speaker: Mayte Pérez Pérez (Universidad de Sevilla)
Day: January 30, 2024
Place: Seminario Alberto Dou (209)
Hour: 13:00
Organized by: Departamento de Análisis Matemático y Matemática Aplicada and Instituto de Matemática Interdisciplinar (IMI)
Seminario de Doctorandos
Título: Teoría geométrica de invariantes: aprendiendo a cocientar en geometría algebraica
Doctorando: Alejandro Calleja Arroyo (ICMAT - UCM)
Día: 30 de enero, 2025
Lugar: Seminario Alberto Dou (209)
Hora: 17:00
Organizado por: Facultad de Ciencias Matemáticas UCM y Red de Doctorandos UCM, con la colaboración del Instituto de Matemática Interdisciplinar (IMI)
J. I. Díaz, A. V. Podolskiy, T. A. Shaposhnikova. On the corrector term in the homogenization of the nonlinear Poisson-Robin problem giving rise to a strange term: Application to an optimal control problem. Journal of Mathematical Analysis and Applications, 543 (1). 2025. DOI: 10.1016/j.jmaa.2024.128867.
R. Campoamor-Stursberg, M. Reginatto, D. Schuch. Preface. Journal of Physics: Conference Series. 2883 (1). 2024. DOI: 10.1088/1742-6596/2883/1/011001.
3) 1+400. Divulgación con 1 imagen y 400 palabras
Gustavo Adolfo Muñoz Fernández. How a Mendeleev’s study led to a mathematical problem of great interest: The Markov Inequality
Boletín del IMI, Nº 159 (23 enero 2025), Sección "1+400. Divulgación con 1 imagen y 400 palabras."
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En esta sección se publican artículos cortos de divulgación, con una imagen y un máximo de 400 palabras (sin tener en cuenta en estas restricciones los datos de los autores). Las personas que quieran publicar un artículo pueden enviarlo a secreadm.imi@mat.ucm.es
La colección de todos los artículos publicados en esta sección se puede ver en www.ucm.es/imi/1mas400
Gustavo Muñoz defended his doctoral thesis in 1999 under the title: Two Problems in Real Banach Spaces: Complexifications and Bernstein-Markov type Inequalities. One of the main topics addressed in Gustavo’s dissertation concerns the generalization to real Hilbert spaces of the classical Markov inequality for the k-th derivative of a polynomial of degree n (with n≥k), which is closely related to the subject of the following article of the section 1+400. Let us recall that Markov-type inequalities are simply uniform estimates of the derivatives of a polynomial. A refinement of these inequalities is found in Bernstein-type inequalities, which provide pointwise estimates for the derivatives of a polynomial. Nowadays, these inequalities inspire numerous studies in very diverse contexts. The origin of these problems has a history worth telling, and that is precisely the main motivation for this contribution to 1+400.
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How a Mendeleev’s study led to a mathematical problem of great interest: The Markov Inequality
Gustavo Adolfo Muñoz Fernández
In 1887, the author of the periodic table of elements, Dmitri Mendeleev, published a study [1] that highlighted the relationship between the specific gravity of an alcohol-water solution and the percentage by weight of alcohol in the solution. By plotting the specific gravity against the percentage of alcohol, he observed that the resulting graph could be well approximated using arcs of quadratic polynomials. Irregularities emerged at the points where the arcs intersected. Mendeleev was uncertain whether these anomalies were genuine or merely the outcome of measurement errors. Figure 1 shows one example of the kind of graphs obtained by Mendeleev.
Figure 1: Extracted from [2]
To resolve this, Mendeleev posed the following problem:
Given a quadratic polynomial P(x), what is the maximum value of the derivative P'(x) over a given interval [a, b]? The motivation behind this question lies in the following reasoning: If the slope of one arc exceeds at some point the maximum slope of an adjacent arc, it is clear that the two arcs correspond to different quadratic polynomials.
Mendeleev's problem can be normalized as follows: If P is a polynomial of degree at most n and we define ||P|| = max{||P(x)|| : x ∈ [-1, 1]}, what is the best (in the sense of smallest) constant Mn in the inequality ||P'|| ≤ Mn ||P|| for any polynomial P of degree at most n?
Mendeleev himself found that M2 = 4. The problem was later generalized by A. A. Markov in 1889 [3], proving that Mn = n2. A further generalization was carried out by V. A. Markov, the brother of A. A. Markov, publishing in 1892 (see for instance [4]}) that:
where Mnk is the best constant in the inequality ||P(k)|| ≤ M ||P|| for all polynomials P of degree at most n, being P(k) its k-th derivative
(1 ≤ k ≤ n). These inequalities, known today as Markov inequalities, have been studied in more general contexts, showing that they also hold, for example, in real Banach spaces [5].
Interestingly, Markov's inequalities have a completely different form when we consider complex polynomials. Indeed, if ||P||D = max{|Q(z)| : |z| ∈ D} is the norm of a (complex) polynomial Q over the unit disc D in C, the set of complex numbers, then:
||P(k)||D ≤ n (n-1) ··· (n-k+1) ||P||D
for every polynomial P of degree at most n and n(n-1)···(n-k+1) is optimal. The previous result follows straightforwardly from the well-known Bernstein Inequality for trigonometric polynomials.
References
[1] D. Mendeleev, Investigation of Aqueous Solutions Based on Specific Gravity (Russian), St. Petersburg, 1887.
[2] R. P. Boas Jr. Inequalities for the Derivatives of Polynomials, Mathematics Magazine, 42:4 (1969), 165-174.
[3] A. Markov, On a problem of D. I. Mendeleev (Russian), Zapiski Imp. Akad. Nauk, 62 (1889) 1-24.
[3] A. Markov, On a problem of D. I. Mendeleev (Russian), Zapiski Imp. Akad. Nauk, 62 (1889) 1-24.
[4] R. J. Duffin and A. C. Schaeffer, On some inequalities of S. Bernstein and W. Markoff for derivatives of polynomials, Bull. Amer. Math. Soc., 43 (1938) 289-297.
[5] L. A. Harris. A proof of Markov’s theorem for polynomials on Banach spaces. J. Math. Anal. Appl. 368 (2010), no. 1, 374–381.
4) La viñeta matemática
Comic strip sent by Juan Monge, and used with permission.
5) Math Puzzle
Puzzle sent by Kjartan Poskitt.
The solution will be provided in the next issue of Boletin del IMI.
Solution to last issue's Math Puzzle, sent by Rik Tangerman and published on issue No. 158 of the Boletín del IMI:
The blue fraction
The blue fraction is 1/2.
There is a dissection of the regular octagon in two congruent squares and four congruent rhombi from which the fraction of ½ is immediately clear.
6) Math Art
Math Art sent by Sinem Onaran
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