Mostrando recursos 1 - 20 de 1.174

  1. On Filter Convergence of Series

    Leonov, Alexander; Orhan, Cihan
    A series $\sum x_k$ is $\mathcal{F}$-convergent to $s$ if the sequence $(\sum_{k=1}^n x_k)$ of its partial sums is $\mathcal{F}$-convergent to $s$. We describe filters $\mathcal{F}$ for which $\mathcal{F}$-convergence of a series $\sum x_k$ implies $\mathcal{F}$-convergence to $0$ of the series terms $x_k$. If $(x_k)$ is small enough with respect to a given filter $\mathcal{F}$, then there is an $\mathcal{F}$-subseries $\sum_{k\in I} x_k$ which is absolutely convergent in the usual sense. Filters corresponding to summable ideals, Erdős-Ulam ideals, matrix summability ideals, lacunary ideals and Louveau-Veličković ideals are considered.

  2. Weakly Symmetric Functions and Weakly Symmetrically Continuous Functions

    Muthuvel, Kandasamy
    We prove that there exists a nowhere weakly symmetric function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is everywhere weakly symmetrically continuous and everywhere weakly continuous. Existence of a nowhere weakly symmetrically continuous function $f: \mathbb{R} \rightarrow \mathbb{R}$ that is everywhere weakly symmetric remains open.

  3. A Simple Example Concerning the Upper Box-Counting Dimension of a Cartesian Product

    Olson, Eric J.; Robinson, James C.
    We give a simple example of two countable sets $X$ and $Y$ of real numbers such that their upper box-counting dimension satisfies the strict inequality $\dim_B(X\times Y)\lt\dim_B(X)+\dim_B(Y)$.

  4. Special Maximal Operator and $A_p^+$ Weights

    Chen, Liwei
    We introduce a special maximal operator associated to a special variant of Muckenhoupt’s weights. By using this special maximal operator, we can construct the special weights. We also prove a weighted weak-type estimate of the special maximal operator.

  5. Packing and Hausdorff Measures of Cantor Sets Associated with Series

    Hare, Kathryn; Mendivil, Franklin; Zuberman, Leandro
    We study a generalization of Morán’s sum sets, obtaining information about the $h$-Hausdorff and $h$-packing measures of these sets and certain of their subsets.

  6. New Definitions of Continuity

    Ash, Arlene; Ash, J. Marshall; Catoiu, Stefan
    We classify all generalized $\mathcal{A}$-differences of any order $n\geq0$ for which $\mathcal{A}$-continuity at $x$ implies ordinary continuity at $x$. We show that the only $\mathcal{A}$-continuities that are equivalent to ordinary continuity at $x$ correspond to the limits of the form \[ \lim_{h\rightarrow0}A\left[ f(x+rh) +f\left( x-rh\right) -2f(x)\right] +B\left[ f\left( x+sh\right) -f\left( x-sh\right) \right] , \] with $ABrs\neq0$. All other $\mathcal{A}$-continuities truly generalize ordinary continuity.

  7. The Fourier Dimension is not Finitely Stable

    Ekström, Fredrik
    The Fourier dimension is not, in general, stable under finite unions of sets. Moreover, the stability of the Fourier dimension on particular pairs of sets is independent from the stability of the compact Fourier dimension.

  8. Best $L^p$-Approximant Pair on Small Intervals

    Levis, Fabián E.; Rodriguez, Claudia N.
    In this paper, we study the behavior of best $L^p$-approximations by algebraic polynomial pairs on unions of intervals when the measure of those intervals tends to zero.

  9. On Baire Classification of Strongly Separately Continuous Functions

    Karlova, Olena
    We investigate strongly separately continuous functions on a product of topological spaces and prove that if $X$ is a countable product of real lines, then there exists a strongly separately continuous function $f:X\to\mathbb{R}$ which is not Baire measurable. We show that if $X$ is a product of normed spaces $X_n$, $a\in X$ and $\sigma(a)=\{x\in X:|\{n\in\mathbb{N}: x_n\ne a_n\}|\lt\aleph_0\}$ is a subspace of $X$ equipped with the Tychonoff topology, then for any open set $G\subseteq \sigma(a)$, there is a strongly separately continuous function $f:\sigma(a)\to \mathbb{R}$ such that the discontinuity point set of $f$ is equal to $G$.

  10. Analogues of Transitiveness and Decomposition of Continuity

    Volodymyr, Maslyuchenko; Vasyl’, Nesterenko
    We consider two conditions that weaken the closed graph condition and we study their properties. We show that if $X$ is a locally connected Baire space, $Y$ is a separable metrizable space and $f:X \to Y$ is a $w^*$-quasi-continuous, almost continuous and weakly Darboux function, then $f$ is continuous.

  11. A New Unification of Continuous, Discrete and Impulsive Calculus through Stieltjes Derivatives

    Pouso, Rodrigo López; Rodríguez, Adrián
    We study a simple notion of derivative with respect to a function which we assume to be nondecreasing and continuous from the left everywhere. Derivatives of this type were already considered by Young in 1917 and Daniell in 1918, in connection with the fundamental theorem of calculus for Stieltjes integrals. We show that our definition contains as a particular case the delta derivative in time scales, thus providing a new unification of the continuous and the discrete calculus. Moreover, we can consider differential equations in the new sense, and we show that not only dynamic equations on time scales, but...

  12. On Closed Subsets of $\mathbb{R}$ and of $\mathbb{R}^2$ Admitting Peano Functions

    Ciesielski, Krzysztof Chris; Jasinski, Jakub
    In this note we describe closed subsets of the real line $P\subset {\mathbb R}$ for which there exists a continuous function from $P$ onto $P^2$, called a Peano function. Our characterization of those sets is based on the number of connected components of $P$. We also include a few remarks on compact subsets of $\mathbb{R}^2$ admitting Peano functions, expressed in terms of connectedness and local connectedness.

  13. A Stieltjes Type Extension of the $L^{r}$-Perron Integral

    Massarwi, Eyad; Musial, Paul
    We explore properties of $L^{r}$-derivates with respect to a monotone increasing Lipschitz function. We then define $L^{r}$-ex-major and $L^{r}$-ex-minor functions with respect to a monotone increasing Lipschitz function and use these to define a Perron-Stieltjes-type integral which extends the integral of L. Gordon.

  14. Homogeneous Multifractal Measures with Disjoint Spectrum and Monohölder Monotone Functions

    Buczolich, Zoltán; Seuret, Stéphane
    We proved in an earlier paper that the support of the multifractal spectrum of a homogeneously multifractal (HM) measure within $[0,1]$ must be an interval. In this paper we construct a homogeneously multifractal measure with spectrum supported by $[0,1] \cup \{ 2\}$. This shows that there can be a different behaviour for exponents exceeding one. ¶ We also provide details of the construction of a strictly monotone increasing monohölder (and hence HM) function which has exact Hölder exponent one at each point. This function was also used in our paper about measures and functions with prescribed homogeneous multifractal spectrum.

  15. On Functions of Bounded Semivariation

    Monteiro, Giselle A.
    The concept of bounded variation has been generalized in many ways. In the frame of functions taking values in Banach space, the concept of bounded semivariation is an important generalization. The aim of this paper is to provide an accessible summary of this notion, to illustrate it with an appropriate body of examples, and to outline its connection with the integration theory due to Kurzweil.

  16. Approximately Continuous Functions Have Approximate Extrema, a New Proof

    Freiling, Chris; Humke, Paul D.; O'Malley, Richard J.
    In 1975, Richard O’Malley proved that every approximately continuous function has approximate extrema, and this result provides an immediate solution to SB 157. The purpose of this paper is to provide an additional proof of O’Malley's result.

  17. An Alternate Solution to Scottish Book 157

    Freiling, Chris; Humke, Paul D.; O'Malley, Richard~J.
    In 1971, D. Ornstein proved a theorem that completely solved Problem 157 of the Scottish Book. The purpose of this paper is to give an independent proof.

  18. A Nice Example of Lebesgue Integration

    Gerver, Joseph L.
    We explore the properties of an interesting new example of a function which is Lebesgue integrable but not Riemann integrable.

  19. The Implicit Function Theorem when the Partial Jacobian Matrix is only Continuous at the Base Point

    de Oliveira, Oswaldo
    This article presents an elementary proof of the Implicit Function Theorem for differentiable maps $F(x,y)$, defined on a finite-dimensional Euclidean space, with $\frac{\partial F}{\partial y}(x,y)$ only continuous at the base point. In the case of a single scalar equation this continuity hypothesis is not required. A stronger than usual version of the Inverse Function Theorem is also shown. The proofs rely on the mean-value and the intermediate-value theorems and Darboux’s property (the intermediate-value property for derivatives). These proofs avoid compactness arguments, fixed-point theorems, and integration theory.

  20. Sobolev Subspaces of Nowhere Bounded Functions

    Lamberti, Pier Domenico; Stefani, Giorgio
    We prove that in any Sobolev space which is subcritical with respect to the Sobolev Embedding Theorem there exists a closed infinite dimensional linear subspace whose non zero elements are nowhere bounded functions. We also prove the existence of a closed infinite dimensional linear subspace whose non zero elements are nowhere $L^q$ functions for suitable values of $q$ larger than the Sobolev exponent.

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