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This work is a thorough and detailed study on the geometry of the unit sphere of certain Banach spaces of homogeneous polynomials in ${\mathbb{R}}^2$. Specifically, we provide a complete description of the unit spheres, identify the extreme points of the unit balls, derive explicit formulas for the corresponding polynomial norms, and describe the techniques required to tackle these questions. To enhance the comprehensiveness of this work, we complement the results and their proofs with suitable diagrams and figures. The new results presented here settle some open questions posed in the past. For the sake of completeness of this work, we briefly discuss previous known results and provide directions of research and applications of our results., Comment: 46 pages, 18 figures

In this article, we investigate the existence of closed vector subspaces (i.e.spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces $\Lambda_{\varphi,w}$, equipped with the Luxemburg norm. If a family of Orlicz functions $(\varphi_n)_{n=1}^{\infty}$ satisfies certain order relations with respect to a given Orlicz function $\varphi$, the subset of the order-continuous subspace $(\Lambda_{\varphi,w})_a$ whose elements do not belong to $\bigcup_{n=1}^{\infty}\Lambda_{\varphi_n,w}$ is spaceable, and even maximal-spaceable when $\varphi$ satisfies the $\Delta_2$-condition. We also show that this subset is either residual or empty. In addition, sufficient conditions for this subset not being $(\alpha, \beta)$-spaceable are provided. A similar analysis is also performed on the subset $\Lambda_{\varphi,w} \setminus (\Lambda_{\varphi,w})_a$ when $\varphi$ does not satisfy the $\Delta_2$-condition. The comparison between different Orlicz-Lorentz spaces is characterized via the generating pairs $(\varphi,w)$. For a fixed Orlicz function that satisfies the $\Delta_2^{\infty}$-condition, we provide a characterization of disjointly strictly singular inclusion operators between Orlicz-Lorentz spaces with different weights. As a consequence, there are certain subsets of Orlicz-Lorentz spaces on $[0,1]$ for which lineability problem is not valid. Moreover, various types of $(\alpha,\beta)$-lineability and pointwise lineability properties on other nonlinear subsets of Orlicz-Lorentz spaces are examined. These results extend a number of previously known results in Orlicz and Lorentz spaces., Comment: 38 pages

In this paper, using the tools from the lineability theory, we distinguish certain subsets of $p$-adic differentiable functions. Specifically, we show that the following sets of functions are large enough to contain an infinite dimensional algebraic structure: (i) continuously differentiable but not strictly differentiable functions, (ii) strictly differentiable functions of order $r$ but not strictly differentiable of order $r+1$, (iii) strictly differentiable functions with zero derivative that are not Lipschitzian of any order $\alpha >1$, (iv) differentiable functions with unbounded derivative, and (v) continuous functions that are differentiable on a full set with respect to the Haar measure but not differentiable on its complement having cardinality the continuum.

In this note, it is proved the existence of an infinitely generated multiplicative group consisting of entire functions that are, except for the constant function 1, hypercyclic with respect to the convolution operator associated to a given entire function of subexponential type. A certain stability under multiplication is also shown for compositional hypercyclicity on complex domains., Comment: 12 pages

We study some properties convex functions fulfill. Among the conclusions we obtain from such result, we are able to prove some nontrivial inequalities among real numbers, and we give an improvement of the reverse triangle inequality in the particular case where three aligned points are considered in a normed space.

We will give a simple, unified, possible explanation of several debated genetic issues on today's humans, Neandertals and Denisovans. In particular it is shown by means of a simple mathematical model why there is little genetic variation in todays's human population or in Western Neandertal population, why all mtDNA and y-chromosomes in today's humans seem to have African origin with no trace of Neandertal nor Denosovan mtDNA or y-chromosomes, why a big part of the European gene pool is young (from Neolitic time), and why today's East Asians have mode Neandertal genes than today's Europeans.

In this paper, equivalence constants between various polynomial norms are calculated. As an application, we also obtain sharp values of the Hardy--Littlewood constants for $2$-homogeneous polynomials on $\ell_p^2$ spaces, $2

In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust--Hille type inequalities., Comment: 16 pages

We show that a recent interpolative new proof of the Bohnenblust--Hille inequality, when suitably handled, recovers its best known constants. This seems to be unexpectedly surprising since the known interpolative approaches only provide constants having exponential growth. This preprint is no longer an independent submission, it is now contained in the preprint arXiv 1310.2834., Comment: This preprint is no longer an independent submission, it is now fully contained in the preprint arXiv 1310.2834

We introduce the concept of {\em maximal lineability cardinal number}, $\mL(M)$, of a subset $M$ of a topological vector space and study its relation to the cardinal numbers known as: additivity $A(M)$, homogeneous lineability $\HL(M)$, and lineability $\LL(M)$ of $M$. In particular, we will describe, in terms of $\LL$, the lineability and spaceability of the families of the following Darboux-like functions on $\real^n$, $n\ge 1$: extendable, Jones, and almost continuous functions.

Grothendieck's theorem asserts that every continuous linear operator from $\ell_{1}$ to $\ell_{2}$ is absolutely $\left( 1;1\right) $-summing. In this note we prove that the optimal constant $g_{m}$ so that every continuous $m$-linear operator from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}$ is absolutely $\left( g_{m};1\right) $-summing is $\frac{2}{m+1}$. We also show that if $g_{m}<\frac{2}{m+1}$ there is $\mathfrak{c}$ dimensional linear space composed by continuous non absolutely $\left( g_{m};1\right) $-summing $m$-linear operators from $\ell_{1}\times\cdots\times\ell_{1}$ to $\ell_{2}.$ In particular, our result solves (in the positive) a conjecture posed by A.T. Bernardino in 2011.

Let $X$ be a sequence space and denote by $Z(X)$ the subset of $X$ formed by sequences having only a finite number of zero coordinates. We study algebraic properties of $Z(X)$ and show (among other results) that (for $p \in [1,\infty]$) $Z(\ell_p)$ does not contain infinite dimensional closed subspaces. This solves an open question originally posed by R. M. Aron and V. I. Gurariy in 2003 on the linear structure of $Z(\ell_\infty)$. In addition to this, we also give a thorough analysis of the existing algebraic structures within the set $X \setminus Z(X)$ and its algebraic genericity., Comment: 17 pages

Recently it has been proved that, assuming that there is an almost disjoint family of cardinality (2^{\mathfrak c}) in (\mathfrak c) (which is assured, for instance, by either Martin's Axiom, or CH, or even $2^{<\mathfrak c=\mathfrak c$}) one has that the set of Sierpi\'nski-Zygmund functions is (2^{\mathfrak{c}})-strongly algebrable (and, thus, (2^{\mathfrak{c}})-lineable). Here we prove that these two statements are actually equivalent and, moreover, they both are undecidable. This would be the first time in which one encounters an undecidable proposition in the recently coined theory of lineability., Comment: 5 pages

The $n$-linear Bohnenblust-Hille inequality asserts that there is a constant $C_{n}\in\lbrack1,\infty)$ such that the $\ell_{\frac{2n}{n+1}}$-norm of $(U(e_{i_{^{1}}},...,e_{i_{n}}))_{i_{1},...i_{n}=1}^{N}$is bounded above by $C_{n}$ times the supremum norm of $U,$ regardless of the $n$-linear form $U:\mathbb{C}^{N}\times...\times\mathbb{C}^{N}% \rightarrow\mathbb{C}$ and the positive integer $N$ (the same holds for real scalars). The power $2n/(n+1)$ is sharp but the values and asymptotic behavior of the optimal constants remain a mystery. The first estimates for these constants had exponential growth. Very recently, a new panorama emerged and the importance, for many applications, of the knowledge of the optimal constants (denoted by $(K_{n})_{n=1}^{\infty}$) was stressed. The title of this paper is part of our Fundamental Lemma, one of the novelties presented here. It brings surprising new (and precise) information on the optimal constants (for both real and complex scalars). For instance, [K_{n+1}-K_{n}<\frac{0.87}{n^{0.473}}] for infinitely many $n$'s. In the case of complex scalars we present a curious formula, where $\pi,e$ and the famous Euler--Mascheroni constant $\gamma$ appear together: [K_{n}<1+(\frac{4}{\sqrt{\pi}}(1-e^{\gamma/2-1/2}) {\sum\limits_{j=1}^{n-1}}j^{^{\log_{2}(e^{-\gamma/2+1/2}) -1}%})] for all $n\geq2$. Numerically, the above formula shows a surprising low growth, [K_{n}<1.41(n-1)^{0.305}-0.04] for every integer $n \geq2$. We also provide a brief discussion on the interplay between the Kahane-Salem-Zygmund and the Bohnenblust-Hille (polynomial and multilinear) inequalities., Comment: An Appendix was added

This paper has been withdrawn by the authors due to a crucial gap in the estimates for m>=4., Comment: This paper has been withdrawn by the authors due to a crucial gap in the estimates for m>=4

In this note we prove a general version of the Extrapolation Theorem, extending the classical linear extrapolation theorem due to B. Maurey. Our result shows, in particular, that the operators involved do not need to be linear.

The Bohnenblust-Hille inequality was obtained in 1931 and (in the case of real scalars) asserts that for every positive integer $N$ and every $m$-linear mapping $T:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow \mathbb{R}$ one has (\sum\limits_{i_{1},...,i_{m}=1}^{N}|T(e_{i_{^{1}}},...,e_{i_{m}})|^{\frac{2m}{m+1}})^{\frac{m+1}{2m}}\leq C_{m}\VertT\Vert, for some positive constant $C_{m}$. Since then, several authors obtained upper estimates for the values of $C_{m}$. However, the novelty presented in this short note is that we provide lower (and non-trivial) bounds for $C_{m}$.

In the last decades many authors have become interested in the study of multilinear and polynomial generalizations of families of operator ideals (such as, for instance, the ideal of absolutely summing operators). However, these generalizations must keep the essence of the given operator ideal and there seems not to be a universal method to achieve this. The main task of this paper is to discuss, study, and introduce multilinear and polynomial extensions of the aforementioned operator ideals taking into account the already existing methods of evaluating the adequacy of such generalizations. Besides this subject's intrinsic mathematical interest, the main motivation is our belief (based on facts that shall be presented) that some of the already existing approaches are not adequate.

We construct uncountably generated algebras inside the following sets of special functions: Sierpi\'nski-Zygmund functions, perfectly everywhere surjective functions and nowhere continuous Darboux functions. All conclusions obtained in this paper are improvements of some already known results.

In this paper we prove a characterization of continuity for polynomials on a normed space. Namely, we prove that a polynomial is continuous if and only if it maps compact sets into compact sets. We also provide a partial answer to the question as to whether a polynomial is continuous if and only if it transforms connected sets into connected sets. These results motivate the natural question as to how many non-continuous polynomials there are on an infinite dimensional normed space. A problem on the \emph{lineability} of the sets of non-continuous polynomials and multilinear mappings on infinite dimensional normed spaces is answered., Comment: 8 pages

A classical inequality due to Bohnenblust and Hille states that for every positive integer $m$ there is a constant $C_{m}>0$ so that $$(\sum\limits_{i_{1},...,i_{m}=1}^{N}|U(e_{i_{^{1}}},...,e_{i_{m}})| ^{\frac{2m}{m+1}}) ^{\frac{m+1}{2m}}\leq C_{m}| U|$$ for every positive integer $N$ and every $m$-linear mapping $U:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$, where $C_{m}=m^{\frac{m+1}{2m}}2^{\frac{m-1}{2}}.$ The value of $C_{m}$ was improved to $C_{m}=2^{\frac{m-1}{2}}$ by S. Kaijser and more recently H. Qu\'{e}ffelec and A. Defant and P. Sevilla-Peris remarked that $C_{m}=(\frac{2}{\sqrt{\pi}})^{m-1}$ also works. The Bohnenblust--Hille inequality also holds for real Banach spaces with the constants $C_{m}=2^{\frac{m-1}{2}}$. In this note we show that a recent new proof of the Bohnenblust--Hille inequality (due to Defant, Popa and Schwarting) provides, in fact, quite better estimates for $C_{m}$ for all values of $m \in \mathbb{N}$. In particular, we will also show that, for real scalars, if $m$ is even with $2\leq m\leq 24$, then $$C_{\mathbb{R},m}=2^{1/2}C_{\mathbb{R},m/2}.$$ We will mainly work on a paper by Defant, Popa and Schwarting, giving some remarks about their work and explaining how to, numerically, improve the previously mentioned constants., Comment: The present version improves the constants of the previous version

A classical inequality due to Bohnenblust and Hille states that for every $N \in \mathbb{N}$ and every $m$-linear mapping $U:\ell_{\infty}^{N}\times...\times\ell_{\infty}^{N}\rightarrow\mathbb{C}$ we have \[(\sum\limits_{i_{1},...,i_{m}=1}^{N}| U(e_{i_{^{1}}},...,e_{i_{m}})| ^{\frac{2m}{m+1}}) ^{\frac{m+1}{2m}}\leq C_{m}| U|] where $C_{m}=2^{\frac{m-1}{2}}$. The result is also true for real Banach spaces. In this note we show that an adequate use of a recent new proof of Bohnenblust-Hille inequality, due to Defant, Popa and Schwarting, combined with the optimal constants of Khinchine's inequality (due to Haagerup) provides quite better estimates for the constants involved in the real Bohnenblust-Hille inequality. For instance, for $2\leq m\leq 14,$ we show that the constants $C_{m}=2^\frac{m-1}{2}$ can be replaced by $2^{\frac{m^{2}+6m-8}{8m}}$ if $m$ is even and by $2^{\frac{m^{2}+6m-7}{8m}}$ if $m$ is odd, which substantially improve the known values of $C_{m}$. We also show that the new constants present a better asymptotic behavior., Comment: 9 pages. This note is an improvement of a previous version registered on arXiv

In this paper we present two different results in the context of nonlinear analysis. The first one is essentially a nonlinear technique that, in view of its strong generality, may be useful in different practical problems. The second result, more technical, but also connected to the first one, is an extension of the well-known Pietsch Domination Theorem. The last decade witnessed the birth of different families of Pietsch Domination-type results and some attempts of unification. Our result, that we call "full general Pietsch Domination Theorem" is potentially a definitive Pietsch Domination Theorem which unifies the previous versions and delimits what can be proved in this line.The connections to the recent notion of weighted summability are traced., Comment: 24 pages