Mostrando recursos 1 - 20 de 1.127

  1. Addendum to: Some new Types of Filter Limit Theorems for Topological group-valued Measures

    Boccuto, Antonio; Dimitriou, Xenofon
    The purpose of this note is to point out some corrections to the paper: A. Boccuto and X. Dimitriou, “Some new types of filter limit theorems for topological group-valued measures,” Real Anal. Exchange 39(1) (2014), 139-174.

  2. Basic Introduction To Exponential and Logarithmic Functions

    Badi, Adel B.
    This article discusses the definitions and properties of exponential and logarithmic functions. The treatment is based on the basic properties of real numbers, sequences and continuous functions. This treatment avoids the use of definite integrals.

  3. Absolute Continuity in Partial Differential Equations

    Farjudian, Amin; Emamizadeh, Behrouz
    In this note, we study a function which frequently appears in partial differential equations. We prove that this function is absolutely continuous; hence it can be written as a definite integral. As a result, we obtain some estimates regarding solutions of the Hamilton-Jacobi systems.

  4. Extreme Results on Certain Generalized Riemann Derivatives

    Georgiou, John C.
    In this paper the following question is investigated. Given a natural number \(r\) and numbers \(\alpha_j,\beta_j\) for \(j=0,1,\dots,r\) satisfying \( \alpha_0 <\alpha_1 < \dots \lt \alpha_r \) and \begin{equation*} \sum_{j=0}^{r} \beta_j \alpha_j^k= \begin{cases} 0 & \text{if \(k=0,1,\dots,r-1\)}\\ r!& \text{if \(k=r\) } \end{cases} , \end{equation*} is there a \( 2\pi\)-periodic, \( r-1\) times continuously differentiable function \( f\) such that \begin{equation*} \limsup_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \limsup_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \infty, \end{equation*} \begin{equation*} \liminf_{h \nearrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+ \alpha_j h)\Big) = \liminf_{h \searrow 0} h^{-r} \Big(\sum_{j=0}^{r} \beta_j f(x+...

  5. A Sufficient Condition for a Bounded Set of Positive Lebesgue Measure in ℝ 2 or ℝ 3 to Contain its Centroid

    Hintikka, Eric A.; Krantz, Steven G.
    In this paper, we give a sufficient condition for a domain in either two- or three-dimensional Euclidean space to contain its centroid. We show that the condition is sharp. The condition is not, however, necessary.

  6. An Integral on a Complete Metric Measure Space

    Bongiorno, Donatella; Corrao, Giuseppa
    We study a Henstock-Kurzweil type integral defined on a complete metric measure space \(X\) endowed with a Radon measure \(\mu\) and with a family of “cells” \(\mathcal{F}\) that satisfies the Vitali covering theorem with respect to \(\mu\). This integral encloses, in particular, the classical Henstock-Kurzweil integral on the real line, the dyadic Henstock-Kurzweil integral, the Mawhin’s integral [19], and the \(s\)-HK integral [4]. The main result of this paper is the extension of the usual descriptive characterizations of the Henstock-Kurzweil integral on the real line, in terms of \(ACG^*\) functions (Main Theorem 1) and in terms of variational measures (Main Theorem...

  7. Generalized Kiesswetter’s Functions

    Li, Delong; Miao, Jie
    In 1966, Kiesswetter found an interesting example of continuous everywhere but differentiable nowhere functions using base-4 expansion of real numbers. In this paper we show how Kiesswetter’s function can be extended to general cases. We also provide an equivalent form for such functions via a recurrence relation.

  8. On Borel Hull Operations

    Filipczak, Tomasz; Rosłanowski, Andrzej; Shelah, Saharon
    We show that some set-theoretic assumptions (for example Martin’s Axiom) imply that there is no translation invariant Borel hull operation on the family of Lebesgue null sets and on the family of meager sets (in \(\mathbb{R}^{n}\)). We also prove that if the meager ideal admits a monotone Borel hull operation, then there is also a monotone Borel hull operation on the \(\sigma\)-algebra of sets with the property of Baire.

  9. Hausdorff and Packing Measures of Balanced Cantor Sets

    Hare, Kathryn; Ng, Ka-Shing
    We estimate the \(h\)-Hausdorff and \(h\)-packing measures of balanced Cantor sets, and characterize the corresponding dimension partitions. This generalizes results known for Cantor sets associated with positive decreasing summable sequences and central Cantor sets.

  10. Bowen’s Formula for Shift-Generated Finite Conformal Constructions

    Ghenciu, Andrei E.; Roy, Mario
    We study shift-generated finite conformal constructions; i.e., conformal constructions generated by a general shift (shift of finite type, sofic shift and non-sofic shift alike) over a finite alphabet. These constructions are not restricted to shifts of finite type or sofic shifts as in the classical limit set constructions. In particular, we prove that the limit sets of such constructions satisfy Bowen’s formula, which gives the Hausdorff dimension of the limit set as the zero of the topological pressure. We look at several examples, including a one-dimensional construction generated by the so-called context-free shift.

  11. The Equality of Mixed Partial Derivatives

    Minguzzi, Ettore
    We review and develop two little-known results on the equality of mixed partial derivatives, which can be considered the best results so far available in their respective domains. The former, due to Mikusiński and his school, deals with equality at a given point, while the latter, due to Tolstov, concerns equality almost everywhere. Applications to differential geometry and General Relativity are commented.

  12. Tangent Measures of Typical Measures

    Sahlsten, Tuomas
    We prove that for a typical Radon measure \(\mu\) in \(\mathbb{R}^d\), every non-zero Radon measure is a tangent measure of \(\mu\) at \(\mu\)-almost every point. This was already shown by T. O’Neil in his Ph.D. thesis from 1994, but we provide a different self-contained proof for this fact. Moreover, we show that this result is sharp: for any non-zero measure we construct a point in its support where the set of tangent measures does not contain all non-zero measures. We also study a concept similar to tangent measures on trees, micromeasures, and show an analogous typical property for them.

  13. Equilateral Weights on the Unit Ball of ℝ n

    Chetcuti, Emmanuel; Muscat, Joseph
    An equilateral set (or regular simplex) in a metric space \(X\) is a set \(A\) such that the distance between any pair of distinct members of \(A\) is constant. An equilateral set is standard if the distance between distinct members is equal to \(1\). Motivated by the notion of frame functions, as introduced and characterized by Gleason in [6], we define an equilateral weight on a metric space \(X\) to be a function \(f:X\longrightarrow \mathbb{R}\) such that \(\sum_{i\in I}f(x_i)=W\) for every maximal standard equilateral set \(\{x_i:i\in I\}\) in \(X\), where \(W\in\mathbb{R}\) is the weight of \(f\). In this paper, we...

  14. The Denjoy-Young-Saks Theorem in Higher Dimensions: A Survey

    Matszangosz, Ákos K.
    The classical theorem of Denjoy, Young and Saks gives a relation between Dini derivatives of a real variable function that holds almost everywhere. We present what is known in the one and two variable case with an emphasis on the latter. Relations that hold a.e. in both the measure and category sense are considered. Classical and approximate derivatives are both discussed.

  15. Strong Derivatives and Integrals

    Thomson, Brian S.
    The strong derivative is not, without some caution, a useful tool in the study of McShaneʼs (i.e., Lebesgueʼs) integral. Even so, the underlying structure of that process of derivation is closely connected to the formulation of the Riemann sums definition that McShane gave for his integral. This article discusses some of the features and traps for the study of those connections.

  16. On Partitions of the Real Line into Continuum Many Thick Subsets

    Kharazishvili, A. B.
    Three classical constructions of Lebesgue nonmeasurable sets on the real line \(\mathbb{R}\) are envisaged from the point of view of the thickness of those sets. It is also shown, within \({\bf ZF}~\&~{\bf DC}\) theory, that the existence of a Lebesgue nonmeasurable subset of \(\mathbb{R}\) implies the existence of a partition of \(\mathbb{R}\) into continuum many thick sets.

  17. Hake’s Theorem on Metric Measure Spaces

    Singh, Surinder Pal; Rana, Inder K.
    In this paper, we extend Hake’s theorem over metric measure spaces. We provide its measure theoretic versions in terms of the Henstock variational measure \(V_F\).

  18. A Simpler Proof for the \(\epsilon\)-\(\delta\) Characterization of Baire Class One Functions

    Fenecios, Jonald P.; Cabral, Emmanuel A.
    We offer a new and simpler proof of a recent \(\epsilon\)-\(\delta\) characterization of Baire class one functions using a theorem by Henri Lebesgue. The proof is more elementary in the sense that it does not use the Baire Category Theorem. Furthermore, the proof requires only that the domain and range be separable metric spaces instead of Polish spaces.

  19. Riemann and Riemann-type Integration in Banach Spaces

    Ali, Sk. Jaker; Mondal, Pratikshan
    Riemann, Riemann-Dunford, Riemann-Pettis and Darboux integrable functions with values in a Banach space and Riemann-Gelfand integrable functions with values in the dual of a Banach space are studied in the light of the work of Graves, Alexiewicz and Orlicz, and Gordon. Various properties of these types of integrals and the interrelation between them are established. The Fundamental Theorem of Integral Calculus for these types of integrals is also studied.

  20. Weighting inequalities for one-sided vector-valued maximal operators with respect to a function

    Corvalán, Álvaro
    The purpose of this paper is to find necessary and sufficient conditions on the weight \(w\) for the weak type \(\left( p,p\right) \) with \(1\leq p<+\infty \) and for the strong type \(\left( p,p\right)\) with \(1

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