Class Of Generalized Functions Research Articles

Approximation with fractal radial basis functions

The article reports on the construction of a general class of fractal radial basis functions (RBFs) in the literature. The fractal RBFs is defined through fractal perturbation of a RBF through suitable choice of iterated function system (IFS). A fractal RBF may be smooth depending on the choice of the germ function and the IFS parameters. Characterizations of conditionally strictly positive definite and strictly positive definite fractal functions are studied using the definition of k-times monotonicity. Furthermore, error estimates and shape-preserving properties for the approximants Pfα defined through linear combination of cardinal fractal RBFs are investigated. Several examples are presented to illustrate the convergence of the operator Pfα across various parameters, highlighting the advantages of the fractal approximant Pfα over the corresponding classical operator Pf. Finally, estimates for the box dimension of the graphs of approximants derived from fractal radial basis functions are given.

An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions

An Application of Multiple Erdélyi–Kober Fractional Integral Operators to Establish New Inequalities Involving a General Class of Functions

Simple proofs of certain inequalities with logarithmic coefficients of univalent functions

In this paper, we give simple proofs for the bounds (some of them sharp) of the difference of the moduli of the second and the first logarithmic coefficient for the general class of univalent functions and for the class of convex univalent functions.

The geometrical and physical interpretation of fractional order derivatives for a general class of functions

The aim of this article is to find a geometric and physical interpretation of fractional order derivatives for a general class of functions defined over a bounded or unbounded domain. We show theoretically and geometrically that the absolute value of the fractional derivative value of a function is inversely proportional to the area of the triangle. Further, we prove geometrically that the fractional derivatives are inversely proportional to the classical integration in some sense. The established results are verified numerically for non‐monotonic, trigonometric, and power functions. Further, this article establishes a significant connection between the area of the projected fence and the area of triangles. As the area of triangles decreases, the area of the projected fence increases, and vice versa. We calculate the turning points of the fractional derivative values of different functions with respect to order , including non‐monotonic, trigonometric, and power functions. In particular, we demonstrate that for the power function , with being a positive real number, the value is a turning point when . However, for , the turning point shifts to the left of point and shifts to the right of point for We discuss the physical interpretation of fractional order derivatives in terms of fractional divergence. We present some applications of fractional tangent lines in the field of numerical analysis.

Nonlinear approximation of high-dimensional anisotropic analytic functions

Motivated by nonlinear approximation results for classes of parametric partial differential equations (PDEs), we seek to better understand so-called library approximations to analytic functions of countably infinite number of variables. Rather than approximating a function of interest by a single space, a library approximation uses a collection of spaces and the best space may be chosen for any point in the domain. In the setting of this paper, we use a specific library which consists of local Taylor approximations on sufficiently small rectangular subdomains of the (rescaled) parameter domain Y≔[−1,1]N. When the function of interest is the solution of a certain type of parametric PDE, recent results (Bonito et al., 2021 [4]) prove an upper bound on the number of spaces required to achieve a desired target accuracy. In this work, we prove a similar result for a more general class of functions with anisotropic analyticity, namely the class introduced in Bonito et al. (2021) [5]. In this way we show both where the theory developed in Bonito et al. (2021) [4] depends on being in the setting of parametric PDEs with affine diffusion coefficients, and prove a more general result outside of this setting.

A New Method for Estimating General Coefficients to Classes of Bi-univalent Functions

This study establishes a new method to investigate bounds of ak; k≥n, for certain general classes of bi-univalent functions. The results include a number of improvements and generalizations for well-known estimations. We also discuss bounds of nan2−a2n−1 and consider several corollaries, remarks, and consequences of the results presented in this paper.

On Some Interesting Classes of Analytic Functions Related to Univalent Functions

On Some Interesting Classes of Analytic Functions Related to Univalent Functions

A Note on Monte Carlo Integration in High Dimensions

Monte Carlo integration is a commonly used technique to compute intractable integrals and is typically thought to perform poorly for very high-dimensional integrals. To show that this is not always the case, we examine Monte Carlo integration using techniques from the high-dimensional statistics literature by allowing the dimension of the integral to increase. In doing so, we derive nonasymptotic bounds for the relative and absolute error of the approximation for some general classes of functions through concentration inequalities. We provide concrete examples in which the magnitude of the number of points sampled needed to guarantee a consistent estimate varies between polynomial to exponential, and show that in theory arbitrarily fast or slow rates are possible. This demonstrates that the behavior of Monte Carlo integration in high dimensions is not uniform. Through our methods we also obtain nonasymptotic confidence intervals which are valid regardless of the number of points sampled.

Event-triggered adaptive prescribed performance tracking for nonlinear time-varying systems with unknown control directions

This article deals with the adaptive prescribed performance tracking for nonlinear time-varying systems. To handle multiple unknown control directions, a novel lemma is established by using a general class of Nussbaum functions. To handle unknown time-varying parameters, the congelation of variables method is employed. Different novel functions are introduced to achieve prescribed performance tracking and finite-time prescribed performance tracking, respectively. Meanwhile, the problem of explosion of complexity is avoided by utilizing the command filter and the compensation mechanism. For reasons of saving network resources and reducing communication burden, a new event-triggered mechanism is proposed, which can circumvent the Zeno behavior. Then, an event-triggered adaptive prescribed performance tracking control scheme is constructed and is extended to event-triggered finite-time adaptive prescribed performance tracking, both of which guarantee that all closed-loop signals are bounded and the trajectory of the tracking error can be limited to the prescribed region. Finally, the availability of the control scheme is demonstrated by two examples.

Certain Results on Fuzzy p-Valent Functions Involving the Linear Operator

The idea of fuzzy differential subordination is a generalisation of the traditional idea of differential subordination that evolved in recent years as a result of incorporating the idea of fuzzy set into the field of geometric function theory. In this investigation, we define some general classes of p-valent analytic functions defined by the fuzzy subordination and generalizes the various classical results of the multivalent functions. Our main focus is to define fuzzy multivalent functions and discuss some interesting inclusion results and various other useful properties of some subclasses of fuzzy p-valent functions, which are defined here by means of a certain generalized Srivastava-Attiya operator. Additionally, links between the significant findings of this study and preceding ones are also pointed out.

Phase Spaces, Parity Operators, and the Born–Jordan Distribution

Phase spaces as given by the Wigner distribution function provide a natural description of infinite-dimensional quantum systems. They are an important tool in quantum optics and have been widely applied in the context of time–frequency analysis and pseudo-differential operators. Phase-space distribution functions are usually specified via integral transformations or convolutions which can be averted and subsumed by (displaced) parity operators proposed in this work. Building on earlier work for Wigner distribution functions (Grossmann in Commun Math Phys 48(3):191–194, 1976. https://doi.org/10.1007/BF01617867), parity operators give rise to a general class of distribution functions in the form of quantum-mechanical expectation values. This enables us to precisely characterize the mathematical existence of general phase-space distribution functions. We then relate these distribution functions to the so-called Cohen class (Cohen in J Math Phys 7(5):781–786, 1966. https://doi.org/10.1063/1.1931206) and recover various quantization schemes and distribution functions from the literature. The parity operator approach is also applied to the Born–Jordan distribution which originates from the Born–Jordan quantization (Born and Jordan in Z Phys 34(1):858–888, 1925. https://doi.org/10.1007/BF01328531). The corresponding parity operator is written as a weighted average of both displacements and squeezing operators, and we determine its generalized spectral decomposition. This leads to an efficient computation of the Born–Jordan parity operator in the number-state basis, and example quantum states reveal unique features of the Born–Jordan distribution.

Refinements of generalised Hermite-Hadamard inequality

New insights, improvements and refinements of the well known Hermite-Hadamard inequality are established for a general class of convex functions. Inequalities involving products of two harmonically ϕh−s functions are also obtained.

Optimal consumption, investment and life insurance selection under robust utilities

We study the problem faced by a wage earner with an uncertain lifetime who has access to a Black–Scholes-type financial market consisting of one risk-free security and one risky asset. His preferences relative to consumption, investment and life insurance purchase are described by a robust expected utility. We rewrite this problem in terms of a two-player zero-sum stochastic differential game and we derive the wage earner optimal strategies for a general class of utility functions, studying the case of discounted constant relative risk aversion utility functions with more detail.

On some inequalities for uniformly convex mapping with estimations to normal distributions

In this paper, we introduce notable Jensen–Mercer inequality for a general class of convex functions, namely uniformly convex functions. We explore some interesting properties of such a class of functions along with some examples. As a result, we establish Hermite–Jensen–Mercer inequalities pertaining uniformly convex functions by considering the class of fractional integral operators. Moreover, we establish Mercer–Ostrowski inequalities for conformable integral operator via differentiable uniformly convex functions. Finally, we apply our inequalities to get estimations for normal probability distributions (Gaussian distributions).

Monge solutions and uniqueness in multi-marginal optimal transport via graph theory

We study a multi-marginal optimal transport problem with surplus b(x1,…,xm)=∑{i,j}∈Pxi⋅xj, where P⊆Q:={{i,j}:i,j∈{1,2,...m},i≠j}. We reformulate this problem by associating each surplus of this type with a graph with m vertices whose set of edges is indexed by P. We then establish uniqueness and Monge solution results for two general classes of surplus functions. Among many other examples, these classes encapsulate the Gangbo and Świȩch surplus [15] and the surplus ∑i=1m−1xi⋅xi+1+xm⋅x1 studied in an earlier work by the present authors [26].

Two types of the second Hankel determinant for the class U and the general class S

In this paper we determine the upper bounds of the Hankel determinants of special type H2(3)(f) and H2(4)(f) for the general class of univalent functions and for the class U.

Parameterized transformations and truncation: When is the result a copula?

Our starting point are several general classes of real functions defined on the unit square satisfying some basic properties such as a boundary condition or several types of monotonicity and continuity. Applying to these functions some parameterized transformations and other constructions such as the transpose and flipping (which describe different aspects of symmetry) and truncation, we ask for conditions yielding (again) a bivariate copula. Some of these transformations are involutive (on one or more classes of functions), others are not even injective, and occasionally they induce additional properties, yielding, e.g., a (quasi-)copula. For several typical scenarios we identify the (not necessarily convex) sets of parameters leading to a copula and conditions imposing a minimal set of parameters.

High-order asymptotic expansions of Gaussian quadrature rules with classical and generalized weight functions

Gaussian quadrature rules are a classical tool for the numerical approximation of integrals with smooth integrands and positive weight functions. We derive and explicitly list asymptotic expressions for the points and weights of Gaussian quadrature rules for three general classes of positive weight functions: analytic functions on a bounded interval with algebraic singularities at the endpoints, analytic weight functions on the halfline with exponential decay at infinity and an algebraic singularity at the finite endpoint, and analytic functions on the real line with exponential decay in both directions at infinity. The results include the Gaussian rules of classical orthogonal polynomials (Legendre, Jacobi, Laguerre and Hermite) as special cases. Explicit expressions for these cases are included in the appendix. We present experiments indicating the range of the number of points at which these expressions achieve high precision. We provide an algorithm that can compute arbitrarily many terms in these expansions for the classical cases, and many though not all terms for the generalized cases.

A Generalized Version of Polynomial Convex Functions and Some Interesting Inequalities with Applications

In this article, we introduce a general class of convex functions and proved some of its basic properties. We establish Hermite-Hadamard type inequalities as well as fractional version of Hermite-Hadamard type inequalities by using Riemann-Liouville integral operator. At the end, some application to special means of real numbers are also given. It can be observed from the remarks given in this paper that several exiting results of ligature can be obtained immediacy from our results by taking suitable involved parameters.

Preparing valence-bond-solid states on noisy intermediate-scale quantum computers

Quantum state preparation is a key step in all digital quantum simulation algorithms. Here we propose methods to initialize on a gate-based quantum computer a general class of quantum spin wave functions, the so-called valence-bond-solid (VBS) states, that are important for two reasons. First, VBS states are the exact ground states of a class of interacting quantum spin models introduced by Affleck, Kennedy, Lieb, and Tasaki (AKLT). Second, the two-dimensional VBS states are universal resource states for measurement-based quantum computing. We find that schemes to prepare VBS states based on their tensor-network representations yield quantum circuits that are too deep to be within reach of noisy intermediate-scale quantum (NISQ) computers. We then apply the general nondeterministic method herein proposed to the preparation of the spin-1 and spin-$3/2$ VBS states, the ground states of the AKLT models defined in one dimension and in the honeycomb lattice, respectively. Shallow quantum circuits of depth independent of the lattice size are explicitly derived for both cases, making use of optimization schemes that outperform standard basis gate decomposition methods. The probabilistic nature of the proposed routine translates into an average number of repetitions to successfully prepare the VBS state that scales exponentially with the number of lattice sites $N$. However, two strategies to quadratically reduce this repetition overhead for any bipartite lattice are devised. Our approach should permit to use NISQ processors to explore the AKLT model and variants thereof, outperforming conventional numerical methods in the near future.