Search Results (5)

Suppose X and Y are Banach spaces, K is a compact Hausdorff space, and $C(K, X)$ is the Banach space of all continuous X-valued functions (with the supremum norm). We will study some strongly bounded operators $T:C(K, X)\to Y$ with representing measures $m:\Sigma \to L(X,Y)$, where $L(X,Y)$ is the Banach space of all operators $T:X\to Y$ and $\Sigma$ is the $\sigma$-algebra of Borel subsets of K. The classes of operators that we will discuss are the Grothendieck, p-limited, p-compact, limited, operators with completely continuous, unconditionally converging, and p-converging adjoints, compact, and absolutely summing. We give a characterization of the limited operators (resp. operators with completely continuous, unconditionally converging, p-convergent adjoints) in terms of their representing measures. Full article

For each $t\in (-1,1)$, the exact value of the least upper bound $H\left(t\right)=sup\left\{\mathbb{E}\right|X{|}^3/\mathbb{E}\left|X-t{|}^3\right\}$ over all the non-degenerate distributions of the random variable X with a fixed normalized first-order moment $\mathbb{E}{X}_1/\sqrt{\mathbb{E}{X}_1^2}=t$, and a finite third-order moment is obtained, yielding the exact value of the unconditional supremum $M:=sup{L}_1\left(X\right)/L_1(X-\mathbb{E}X)=\left(\sqrt{17+7\sqrt{7}}\right)/4$, where $L_1\left(X\right)=\mathbb{E}{\left|X\right|}^3/{\left(\mathbb{E}{X}^2\right)}^{3/2}$ is the non-central Lyapunov ratio, and hence proving S. Shorgin’s (2001) conjecture on the exact value of M. As a corollary, an analog of the Berry–Esseen inequality for the Poisson random sums of independent identically distributed random variables $X_1,X_2,\dots$ is proven in terms of the central Lyapunov ratio $L_1(X_1-\mathbb{E}{X}_1)$ with the constant $0.3031·H\left(t\right){(1-t^2)}^{3/2}\in [0.3031,0.4517)$, $t\in [0,1)$, which depends on the normalized first-moment $t:=\mathbb{E}{X}_1/\sqrt{\mathbb{E}{X}_1^2}$ of random summands and being arbitrarily close to $0.3031$ for small values of t, an almost $1.5$ size improvement from the previously known one. Full article

Pattern recognition is the computerized identification of shapes, designs, and reliabilities in information. It has applications in information compression, machine learning, statistical information analysis, signal processing, image analysis, information retrieval, bioinformatics, and computer graphics. Similarly, a medical diagnosis is a procedure to illustrate or identify diseases or disorders, which would account for a person’s symptoms and signs. Moreover, to illustrate the relationship between any two pieces of intuitionistic hesitant fuzzy (IHF) information, the theory of generalized dice similarity (GDS) measures played an important and valuable role in the field of genuine life dilemmas. The main influence of GDS measures is that we can easily obtain a lot of measures by using different values of parameters, which is the main part of every measure, called DGS measures. The major influence of this theory is to utilize the well-known and valuable theory of dice similarity measures (DSMs) (four different types of DSMs) under the assumption of the IHF set (IHFS), because the IHFS covers the membership grade (MG) and non-membership grade (NMG) in the form of a finite subset of [0, 1], with the rule that the sum of the supremum of the duplet is limited to [0, 1]. Furthermore, we pioneered the main theory of generalized DSMs (GDSMs) computed based on IHFS, called the IHF dice similarity measure, IHF weighted dice similarity measure, IHF GDS measure, and IHF weighted GDS measure, and computed their special cases with the help of parameters. Additionally, to evaluate the proficiency and capability of pioneered measures, we analyzed two different types of applications based on constructed measures, called medical diagnosis and pattern recognition problems, to determine the supremacy and consistency of the presented approaches. Finally, based on practical application, we enhanced the worth of the evaluated measures with the help of a comparative analysis of proposed and existing measures. Full article

Let $\mathrm{Lip}\left(\right[0,1\left]\right)$ be the Banach space of all Lipschitz complex-valued functions f on $[0,1]$, equipped with one of the norms: ${∥f∥}_{\sigma }=\left|f\left(0\right)\right|+{∥f^{\prime }∥}_{L^{\infty }}$ or ${∥f∥}_m=max\left\{\left|f\left(0\right)\right|,{∥f^{\prime }∥}_{L^{\infty }}\right\}$, where ${∥·∥}_{L^{\infty }}$ denotes the essential supremum norm. It is known that the surjective linear isometries of such spaces are integral operators, rather than the more familiar weighted composition operators. In this paper, we describe the topological reflexive closure of the isometry group of $\mathrm{Lip}\left(\right[0,1\left]\right)$. Namely, we prove that every approximate local isometry of $\mathrm{Lip}\left(\right[0,1\left]\right)$ can be represented as a sum of an elementary weighted composition operator and an integral operator. This description allows us to establish the algebraic reflexivity of the sets of surjective linear isometries, isometric reflections, and generalized bi-circular projections of $\mathrm{Lip}\left(\right[0,1\left]\right)$. Additionally, some complete characterizations of such reflections and projections are stated. Full article

In this paper, we find the upper bound for the tail probability $\mathbb{P}\left({sup}_{n⩾0}{\sum }_{\mathrm{I}=1}^n{\xi }_{\mathrm{I}}>x\right)$ with random summands ${\xi }_1,{\xi }_2,\dots$ having light-tailed distributions. We find conditions under which the tail probability of supremum of sums can be estimated by quantity ${\varrho }_{1}exp\{-{\varrho }_{2}x\}$ with some positive constants ${\varrho }_1$ and ${\varrho }_2$. For the proof we use the martingale approach together with the fundamental Wald’s identity. As the application we derive a few Lundberg-type inequalities for the ultimate ruin probability of the inhomogeneous renewal risk model. Full article