Normalized Gompertz wavelets and their applications

We introduce local iterated function systems (IFSs) and present some of their basic properties. A new class of local attractors of local IFSs, namely local fractal functions, is constructed. We derive formulas so that these local fractal functions become elements of various function spaces, such as the Lebesgue spaces L p , the smoothness spaces C n , the homogeneous Hölder spaces $$\dot{C}^{s}$$ , and the Sobolev spaces W m, p .

This paper studies how well computable functions can be approximated by their Fourier series. To this end, we equip the space of Lp-computable functions (computable Lebesgue integrable functions) with a size notion, by introducing Lp-computable Baire categories. We show that Lp-computable Baire categories satisfy the following three basic properties. Singleton sets {f } (where f is Lp-computable) are meager, suitable infinite unions of meager sets are meager, and the whole space of Lp-computable functions is not meager. We give an alternative characterization of meager sets via Banach-Mazur games. We study the convergence of Fourier series for Lp-computable functions and show that whereas for every p > 1, the Fourier series of every Lp-computable function f converges to f in the Lp norm, the set of L1-computable functions whose Fourier series does not diverge almost everywhere is meager (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

For disposing numerous practical application problems involving expert systems, decision-making, image processing, classifications and etc, the investigations on the constructions and basic properties of n-ary aggregation operators (nAAOs) have always been a hot research topic with important research value and significance at theoretical investigations on aggregation operators (AOs). Herein, first, we propose a method for constructing nAAOs on function spaces via a family of known ones defined on a bounded poset, where those function spaces are composed by all fuzzy sets with that bounded poset as the truth values set. This method is different from the existing construction methods of nAAOs on bounded posets and provides a unified way of constructing usual nAAOs (like t-norms, uninorms, overlap functions, etc.) on function spaces via a family of known ones. Second, we present notion of representable nAAOs on function spaces and afford their equivalent characterization. Third, we discuss some vital properties of representable nAAOs on function spaces. Fourth, it is worth noticing that the obtained results cover the cases of nAAOs on function spaces composed of all interval-valued fuzzy sets and type-2 fuzzy sets when underlying bounded poset is taken as the corresponding truth values set, respectively. As a consequence, the theoretical results obtained herein have certain promotion and basic theoretical value for the mining of new potential applications of nAAOs in real problems, especially in expert systems, decision-making, image processing and etc.

The Bargmann-Fock space of slice hyperholomorphic functions is recently introduced by Alpay, Colombo, Sabadini and Salomon. In this paper, we reconsider this space and present a direct proof of its independence of the slice. We also introduce a quaternionic analogue of the classical Segal-Bargmann transform and discuss some of its basic properties. The explicit expression of its inverse is obtained and the connection to the left one-dimensional quaternionic Fourier transform is given.

Shift operators on harmonic Hilbert function spaces on real balls and von Neumann inequality

Emerging trends in soft set theory and related topics.

In this paper we give some basic and important properties of several typical Banach spaces of functions of G-Brownian motion paths induced by a sublinear expectation—G-expectation. Many results can be also applied to more general situations. A generalized version of Kolmogorov’s criterion for continuous modification of a stochastic process is also obtained. The results can be applied in continuous time dynamic and coherent risk measures in finance, in particular for path-dependence risky positions under situations of volatility model uncertainty.

Digital topology is part of the ongoing endeavor to understand and analyze digitized images. With a view to supporting this endeavor, many notions from algebraic topology have been introduced into the setting of digital topology. But some of the most basic notions from homotopy theory remain largely absent from the digital topology literature. We embark on a development of homotopy theory in digital topology, and define such fundamental notions as function spaces, path spaces, and cofibrations in this setting. We establish digital analogues of basic homotopy-theoretic properties such as the homotopy extension property for cofibrations, and the homotopy lifting property for certain evaluation maps that correspond to path fibrations in the topological setting. We indicate that some depth may be achieved by using these homotopy-theoretic notions to give a preliminary treatment of Lusternik–Schnirelmann category in the digital topology setting. This topic provides a connection between digital topology and critical points of functions on manifolds, as well as other topics from topological dynamics.

We study various types of generating functions for higher-order derangement numbers. We present a generating function for a new class of binomial-type polynomials with coefficients given by higher-order derangement numbers. Using this generating function, we derive explicit formulas for these polynomials, as well as derivative and integral formulas involving Stirling numbers and Cauchy numbers of the first kind. By employing this generating function along with other special formulas, we obtain several new results involving higher-order derangement numbers, Bernoulli numbers, Stirling numbers, combinatorial numbers associated with Daehee numbers, Peters-type Simsek numbers, and special finite sums.

Classical Segal–Bargmann theory studies three Hilbert space unitary isomorphisms that describe the wave-particle duality and the configuration space-phase space. In this work, we generalized these concepts to Clifford algebra-valued functions. We establish the unitary isomorphisms among the space of Clifford algebra-valued square-integrable functions on $$\mathbb {R}^n$$ with respect to a Gaussian measure, the space of monogenic square-integrable functions on $$\mathbb {R}^{n+1}$$ with respect to another Gaussian measure and the space of Clifford algebra-valued linear functionals on symmetric tensor elements of $$\mathbb {R}^n$$ .

The main objective of this paper is to compare – analytically as well as numerically – different approaches for obtaining optimal input currents in impedance tomography. Following the approaches described in, e.g. [Cheney and Isaacson, IEEE Trans. Biomed. Eng. 39: 852–860, 1992, Isaacson, IEEE Trans. Med. Imag. 5: 91–95, 1986, Ito and Kunisch, SIAM J. Contr. Optim. 33: 643–666., 1995, Knowles, An optimal current functional for electrical impedance tomography, 2004], we aim at constructing input currents j, which contain the most information about the difference between the unknown physical conductivity σ* and a given approximation σ 0. The differences can be measured by different discrepancy functionals and the optimal input currents which maximize these functionals depend on the function spaces chosen for defining j and on the norm for measuring the discrepancy. Moreover, the definition of the appropriately weighted Sobolev spaces depends on σ and this subsequently influences the iteration for maximizing the functionals. Numerical experiments illustrate features of the optimal input currents obtained for several combinations of function spaces. The reconstructions with these optimal currents are compared with those with standard input currents (sinusoid and dipole). The differences between the optimal currents obtained by different function space settings are significant. Two newly developed optimal currents can yield qualitatively better reconstructions.

The Stirling numbers of the first and second kind enumerate the permutations of a finite set that are decomposed into a given number of cycles and the partitions of a finite set into a given number of subsets, respectively. Basic properties of these numbers and their generalizations of probabilistic and statistical interest are reviewed. Specifically, explicit expressions, generating functions, recurrence relations, and other useful properties of these numbers are presented. Further, the Lah numbers and the generalized factorial coefficients, closely related to the Stirling numbers, are discussed. The noncentral and the associated Stirling and related numbers are briefly presented. The generalized Stirling and Lah numbers and, in particular, the q ‐Stirling and q ‐Lah numbers are only quoted. Probabilistic and statistical applications of the Stirling and related numbers in expressing the probability mass function of certain (a) urn‐model distributions, (b) occupancy distributions, (c) convolutions of truncated discrete distributions and (d) compound discrete distributions are exemplified.

In the present paper, we define the generalized Kwang-Wu Chen matrix. Basic properties of this generalization, such as explicit formulas and generating functions are presented. Moreover, we focus on a new class of generalized Fubini polynomials. Then we discuss their relationship with other polynomials such as Fubini, Bell, Eulerian and Frobenius-Euler polynomials. We have also investigated some basic properties related to the degenerate generalized Fubini polynomials.

Gabor frames and directional time–frequency analysis

The aim of this paper is to introduce and investigative some new function classes of Morrey–Campanato type. The domains \(\Omega \subset {\mathbb {R}}^n\) in this paper are supposed to satisfy the following property: there exists a constant \(A>0\) such that for all \(x_{0}\in \Omega , \rho <\mathrm{diam} \ \Omega \), we have \(|Q(x_{0},\rho )\cap \Omega |\ge A\rho ^n.\) Let \(0<p<\infty \) and \(0\le \lambda \le n+p\). We say that \(f\in \bar {\mathcal{L}}^{p,\lambda }(\Omega )\) if $$\begin{aligned} \sup _{x_{0}\in \Omega ,\rho >0}\rho ^{-\lambda }\int _{\Omega (x_{0},\rho )}\big |f(x)-|f|_{\Omega (x_{0},\rho )}\big |^p {\mathrm{d}}x<\infty , \end{aligned}$$where \(\Omega (x_{0},\rho )=Q(x_{0},\rho )\cap \Omega \) and \(Q(x,\rho )\) is denote the cube of \({\mathbb {R}}^n\). Some basic properties and characterizations of these classes are presented. If \(0\le \lambda <n\), the space is equivalent to related Morrey space under certain conditions. If \(\lambda =n\), then \(f \in \bar {\mathcal{L}}^{p,n}(\Omega )\) if and only if \(f\in {\mathrm{BMO}}(\Omega )\) with \(f^{-}\in L^{\infty }(\Omega )\), where \(f^{-}=-\min \{0,f\}\). If \(n<\lambda \le n+p\), the \(\bar {\mathcal{L}}^{p,\lambda }(\Omega )\) functions establish an integral characterization of the nonnegative Hölder continue functions. As applications, this paper gives unified criteria on the necessity of bounded commutators of maximal functions on ball Banach function spaces.