Mostrando recursos 1 - 20 de 385

  1. On the Derivatives of Functions of Bounded Variation

    Cater, F. S.
    ss{raeq} \usepackage{amsmath,amssymb,amsthm} %\coverauthor{F. S. Cater} %\covertitle{On the Derivatives of Functions of Bounded Variation} \received{November 5, 2000} \firstpagenumber{975} \markboth{F. S. Cater}{Questions on Derivatives of Functions of Bounded Variation} \author{F. S. Cater, Department of Mathematics, Portland State University, Portland, Oregon 97207, USA} \title{ON THE DERIVATIVES OF FUNCTIONS OF BOUNDED VARIATION} \begin{document} \maketitle The following two questions were submitted by F. S. Cater. Let $\cal F$ denote the family of all absolutely continuous , nondecreasing functions on $[0,1]$. Endow $\cal F$ withe the complete metric $d$ defined by $$ d(f,g) = |f(0)-g(0)| + \int_o^1 |f'-g'|$$ Let \begin{align*} {\cal G}=\{g\in {\cal F};& g'(x) =...

  2. The Persistence of ω-Limit Sets under Perturbation of the Generating Function

    Steele, T. H.
    We consider the set valued function $\Omega $ taking $f$ in $\ C(I,I)$ to its collection of $\omega $-limit sets $\Omega (f)=\{\omega (x,f):x\in I\}$, and consider how $\Omega (f)$ is affected by pertubations of $f$. Our main result characterizes those functions $f$ in $C(I,I)$ at which $\Omega $ is upper semicontinuous, so that whenever $g$ is sufficiently close to $f$, every $\omega $-limit set of $g$ is close to some $\omega $-limit set of $f$ in the Hausdorff metric space. We also develop necessary and sufficient conditions for a function $f$ in $C(I,I)$ to be a point of lower semicontinuity...

  3. SAC Property and Approximate Semicontinuity

    Prus-Wiśniowski, Franciszek; Szkibiel, Grzegorz
    In this article we investigate approximate semicontinuity of a function related to Grande's SAC problem.

  4. A Solution to Pfeffer’s Problem

    Tuo-Yeong, Lee
    We give an example of a non-integrable function $f$ on $[0,1]\times [0,1]$ such that $$ \int_{\alpha}^{\beta} (\int_{\gamma}^{\delta} f(x,y)\,dy)\,dx =\int_{\gamma}^{\delta} (\int_{\alpha}^{\beta} f(x,y)\,dx)\,dy$$ for each subinterval $[\alpha,\beta] \times [\gamma, \delta]$ of $[0,1] \times [0,1]$.

  5. On Pointwise, Discrete and Transfinite Limits of Sequences of Closed Graph Functions

    Grande, Zbigniew
    In this article we prove that if a function $f:X \to {\cal R}$ is the pointwise (discrete) [transfinite] limit of a sequence of real functions $f_n$ with closed graphs defined on complete separable metric space $X$ then $f$ is the pointwise (discrete) [transfinite] limit of a sequence of continuous functions. Moreover we show that each Lebesgue measurable function $f:{\cal R} \to {\cal R}$ is the discrete limit of a sequence of functions with closed graphs in the product topology $T_d\times T_e$, where $T_d$ denotes the density topology and $T_e$ the Euclidean topology.

  6. On the Derivatives of Functions of Bounded Variation

    Cater, F. S.
    Using a standard complete metric $w$ on the set $F$ of continuous functions of bounded variation on the interval $[0,1]$, we find that a typical function in $F$ has an infinite derivative at continuum many points in every subinterval of $[0,1]$. Moreover, for a typical function in $F$, there are continuum many points in every subinterval of $[0,1]$ where it has no derivative, finite nor infinite. The restriction of the derivative of a typical function in $F$ to the set of points of differentiability has infinite oscillation at each point of this set.

  7. Measure Zero Sets Whose Algebraic Sum Is Non-Measurable

    Ciesielski, Krzysztof
    In this note we will show that for every natural number $n>0$ there exists an $S\subset[0,1]$ such that its $n$-th algebraic sum $nS=S+\cdots +S$ is a nowhere dense measure zero set, but its $n+1$-st algebraic sum $nS+S$ is neither measurable nor it has the Baire property. In addition, the set $S$ will be also a Hamel base, that is, a linear base of $\mathbb{R}$ over $\mathbb{Q}$.

  8. Functions I-Approximately Continuous in I1 A. E. Direction at Every Point

    Szkopińska, Bożena
    In this paper we shall show that every function $f\colon\Re^2\rightarrow \Re$ having the Baire property and ${\cal I}$-approximately continuous in ${\cal I}_1$-almost everywhere direction at every point is of the first class of Baire.

  9. Small Combinatorial Cardinal Characteristics and Theorems of Egorov and Blumberg

    Ciesielski, Krzysztof; Pawlikowski, Janusz
    We will show that the following set theoretical assumption \begin{quote} $\continuum=\omega_2$, the dominating number ${\mathfrak d}$ equals to $\omega_1$, and there exists an $\omega_1$-generated Ramsey ultrafilter on $\omega$ \end{quote} (which is consistent with ZFC) implies that for an arbitrary sequence $f_n\colon\real\to\real$ of uniformly bounded functions there is a set $P\subset\real$ of cardinality continuum and an infinite $W\subset\omega$ such that $\{f_n\restriction P\colon n\in W\}$ is a monotone uniformly convergent sequence of uniformly continuous functions. Moreover, if functions $f_n$ are measurable or have the Baire property then $P$ can be chosen as a perfect set. We will also show that $\cof(\NN)=\omega_1$ implies...

  10. A Remark on a Maximal Operator for Fourier Multipliers

    Bourgain, J.; Kostyukovsky, S.; Olevskiǐ, A.
    For a finite set $\Lambda$ on the circle we consider a family of the multiplier operators $T_m$ in $l_2 ({\mathbb Z})$ generated by the $2^{-m}$-neighborhoods of $\Lambda$. We show that the norm of the corresponding maximal operator $T$ can not be estimated by an absolute constant.

  11. Graphs of Functions, Regular Sets and S-Straight Sets

    Delaware, R.; Eifler, L.
    A subset $E$ of $\Bbb{R}^p$ is s-straight if $E$ has finite Hausdorff s-dimensional outer measure which equals its Method I s-outer measure. The graph of a continuously differentiable function is shown to be the countable union of closed 1-straight sets together with a set of Hausdorff 1-measure zero. This result is extended to the graphs of absolutely continuous functions and to regular sets.

  12. A Darboux Baire One Fixed Point Problem

    Humke, P. D.; Svetic, R. E.; Weil, C. E.
    K. Ciesielski asked whether the composition of two derivatives from the unit interval to itself always has a fixed point. The question is equivalent to asking if the composition of two Darboux, Baire one maps of $[0,1]$ to $[0,1]$ has a fixed point. The question is answered affirmatively for three subclasses of the Darboux, Baire one maps of $[0,1]$ to $[0,1]$

  13. Some Examples of Meager Sets in Banach Spaces

    Balcerzak, Marek; Wachowicz, Artur
    We show that some sets in the spaces $c_0\times c_0$, $L^1[0,1]\times L^1[0,1]$ and $C[0,1]$, that appear in analysis, are meager.

  14. A Fundamental Theorem of Calculus for the Kurzweil-Henstock Integral in ℝ m

    Cabral, Emmanuel; Lee, Peng-Yee
    In this paper, \ we \ give a characterization of the Kurzweil-Henstock integral in n-dimensional space.

  15. Non-Uniqueness of Composition Square Roots

    Dewsnap, D. J.; Fischer, P.
    In response to a question posed by O.E. Lanford III, it is shown that for each \(\mu>0\) there is a differentiable and non-linearizable interval map $g$ with non-vanishing derivative defined on a neighborhood of a fixed point $0$ with \(g^\prime(0)=\mu\) such that $g$ has infinitely many differentiable and non-linearizable orientation-reversing composition square roots with non-vanishing first derivatives on a neighborhood of $0$.

  16. Henstock-Stieltjes Integrals Not Induced by Measure

    Pal, Suppriya; Ganguly, D. K.; Yee, Lee Peng
    The present paper concerns with the introduction of a new type of generalized Stieltjes integral with an integrator function which depends on multiple points in a division and cannot be induced by a measure. Some properties of this integral were studied.

  17. Packing Measure in General Metric Space

    Edgar, G. A.
    Packing measures are counterparts to Hausdorff measures, used in measuring fractal dimension of sets. C.~Tricot defined them for subsets of finite-di\-men\-sion\-al Euclidean space. We consider here the proper way to phrase the definitions for use in general metric spaces, and for Hausdorff functions other than the simple powers, in particular non-blanketed Hausdorff functions. The question of the Vitali property arises in this context. An example of a metric space due to R.~O. Davies illustrates the concepts.

  18. Note on the Outer Measures of Images of Sets

    Cater, F. S.
    Let $f$ be a real function on ${\mathbb R}$, let $\{I_v\}$ be a family of intervals covering a set $E$ such that $m(E \cap I_v) \ge m\bigl (f(E \cap I_v)\bigr )$ for each $I_v$. We prove that $m\bigl (f(E)\bigr ) \le 2 \cdot m(E)$. No coefficient smaller than $2$ will suffice here in general.

  19. Infinite Peano Derivatives

    Laczkovich, Miklós
    Let $f_{(n)}$ and ${\underline f}{_{(n)}}$ denote the $n^{\rm th}$ Peano derivative and the $n^{\rm th}$ lower Peano derivative of the function $f:[a,b]\to $\mathbb{R} .$ We investigate the validity of the following statements. ¶ $(M_n)$ If the set $H=\{ x\in [a,b]: {\underline f}{_{(n)}} (x)>0\}$ is of positive outer measure, then $f$ is $n$-convex on a subset of $H$ having positive outer measure. ¶ $(Z_n)$ The set $E_n (f)=\{ x\in [a,b] : f_{(n)} (x)=\infty \}$ is of measure zero for every $f:[a,b]\to \mathbb{R} .$ ¶ We prove that ($M_n$) and ($Z_n$) are true for $n=1$ and $n=2,$ but false for $n\ge 3.$ More precisely we show that for every $n\ge 3$ there...

  20. Asymptotics of the Quantization Errors for Self-Similar Probabilities

    Graf, S.; Luschgy, H.
    The formulae for determining the quantization dimensions of self--similar probabilities satisfying the open set condition are proved by a new method. In addition, this method gives the exact order of convergence for the quantization errors.

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