Large algebras of singular functions vanishing on prescribed sets

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.Comment: 11 page

Smooth functions with uncountably many zeros

[EN] In this short note we show that there exist uncountably generated alge- bras every non-zero element of which is a smooth function having uncount- ably many zeros. This result complements some recent ones by Enflo et al. [7, 9].The authors would like to thank the anonymous referee, whose thorough analysis and insightful remarks improved the text. The authors were supported by CNPq Grant 401735/2013-3 (PVE - Linha 2), MTM2012-34341, MEC Project MTM2013-47093-P, and Programa de Investigacion y Desarrollo de la UPV, Referencia SP2012070. The third author is also supported by a FPU grant of MEC Project MTM2010-14909.Conejero, JA.; Muñoz-Fernández, GA.; Murillo Arcila, M.; Seoane Sepúlveda, JB. (2015). Smooth functions with uncountably many zeros. Bulletin of the Bengian Mathematical Society. 22:1-5. http://hdl.handle.net/10251/64844152

A hierarchy in the family of real surjective functions

This expository paper focuses on the study of extreme surjective functions in ℝℝ. We present several different types of extreme surjectivity by providing examples and crucial properties. These examples help us to establish a hierarchy within the different classes of surjectivity we deal with. The classes presented here are: everywhere surjective functions, strongly everywhere surjective functions, κ-everywhere surjective functions, perfectly everywhere surjective functions and Jones functions. The algebraic structure of the sets of surjective functions we show here is studied using the concept of lineability. In the final sections of this work we also reveal unexpected connections between the different degrees of extreme surjectivity given above and other interesting sets of functions such as the space of additive mappings, the class of mappings with a dense graph, the class of Darboux functions and the class of Sierpiński-Zygmund functions in ℝℝ

Linear (and non-linear) techniques and its applications

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 11-01-2016La presente tesis est a centrada en dos temas principales: el primero abarca el primer cap tulo y el segundo se divide entre los cap tulos dos y tres. En el primer cap tulo estudio un problema que apareci o como tal hace relativamente poco tiempo (aunque ya en la segunda mitad del pasado siglo se publicaron una serie de resultados que, con la terminolog a adecuada, estar an englobados dentro de esta teor a). Nos interesaremos en la b usqueda de estructuras algebraicas (como espacios vectoriales, algebras, espacios de Banach) contenidas en subconjuntos de funciones cuyos elementos (con la posible excepci on del elemento nulo) veri can ciertas propiedades anti-intuitivas (propiedades de dif cil visualizaci on). Ello nos puede conducir a la idea de c omo la intuci on puede enga~narnos, y sugerir que, aunque se haya dedicado una ingente cantidad de esfuerzo y tiempo para encontrar un unico ejemplo que veri que tales propiedades, y dicho trabajo pueda dar la idea de que no existen muchos m as espec menes de similares caracter sticas, de hecho existen ejemplares su cientes como para construir espacios \grandes" cuyos elementos (salvo el cero) satisfacen las mismas propiedades. M as espec camente, decimos que un subconjunto de un espacio vectorial topol ogico es -lineable (dado un numero cardinal ) si podemos garantizar la existencia de un espacio vectorial de dimensi on contenido en el conjunto (uni on el elemento cero, en caso de que cero no forme parte del conjunto de partida). Si el espacio vectorial es cerrado, nos referiremos a este conjunto como - espaciable (y la propiedad que trataremos ser a la de -espaciabilidad) y si la estructura en cuesti on es un algebra de Banach, entonces diremos que el conjunto es ( ; )-algebrable (donde aqu es la cardinalidad de un conjunto minimal de generadores del algebra)...This thesis will be divided into two topics: the rst one will cover the rst chapter and will deal with a problem that took form little time ago (even though already in the second half of the past century there would be some results). We will be interested on nding algebraic structures (vector spaces, algebras, Banach spaces) contained in subsets of functions whose elements ful ll some anti-intuitive property, union the zero function. Thereby, we can have an idea of how the intuition may mislead us, and hint that, even though we may think that because of having to spend a huge e ort in nding one example of such elements we may not nd many more, in fact there are enough to consider huge spaces all whose elements except from the zero element satisfy the same property. More speci cally, we de ne a subset of a topological vector space to be {u100000}lineable (for a cardinal number ) if we can nd a vector space of dimension contained in the set (union the zero element, in case it is not included). If the vector space is closed, then we will be talking about {u100000}spaceability (and we will say that the set is {u100000}spaceable), and if the structure included is a Banach algebra then we will de ne the set to be ( ; ){u100000}algebrable (where here would be the cardinality of a minimal set of generators of the algebra). If no cardinal number is de ned, then we will assume the structure to be in nite dimensional. This trend was developed as an independent theory in the end of the last Century, in [5], and since its appearance it has resulted in a fruitful eld of study, as the amount of results show (see for example [4], [7], [12], [24], [26] or [54], a very recent and detailed paper giving an exhausting overview of the results published until 2014 can be found in [16]). The sets that will be considered here when studying those anti-intuitive properties will deal with functions de ned over the real line, more concretely results that lie beneath the de nition of di erentiability (for example the relationship between bounds of the di erential and the Lipschitzianity of the function). In particular, we will revisit the famous example given by Weierstrass. There will also be some sections dedicated to the analyticity of real functions and its relation with the in nite di erentiability...Fac. de Ciencias MatemáticasTRUEunpu

(Lineabilidad y propiedades no lineales en el ámbito del Análisis Real)

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Ciencias Matemáticas, leída el 12-09-2022Fac. de Ciencias MatemáticasTRUEunpu

When the identity theorem "seems" to fail

The Identity Theorem states that an analytic function (real or complex) on a connected domain is uniquely determined by its values on a sequence of distinct points that converge to a point of its domain. This result is not true in general in the real setting, if we relax the analytic hypothesis on the function to infinitely many times differentiable. In fact, we construct an algebra of functions A enjoying the following properties: (i) A is uncountably infinitely generated (that is, the cardinality of a minimal system of generators of A is uncountable); (ii) every nonzero element of A is nowhere analytic; (iii) A subset of C-infinity (R); (iv) every element of A has infinitely many zeros in R; and (v) for every f is an element of A\ {0} and n is an element of N, f((n)) (the nth derivative of f) enjoys the same properties as the elements in A\ {0}. This construction complements those made by Cater and by Kim and Kwon, and published in the American Mathematical Monthly in 1984 and 2000, respectively