Certain Classes Of Functions Research Articles

Generalized Technique for constructing the closed form of the generating functions of certain classes of sequences

We develop generalized techniques for constructing the closed form of the generating functions of sequences of the form𝑓𝑚+1 = 𝑘(𝑓𝑚 + 𝑚𝑘 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 0, 𝑘 > 0 𝑎𝑛𝑑 𝑓0 = 0 𝑎𝑛𝑑 𝑓𝑚+1 = 𝑘(𝑓𝑚 + (𝑚 − 1) 𝑘 )𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 1, 𝑘 > 0 𝑎𝑛𝑑 𝑓1 = 0 using method of differencing in generating function. Also, the techniques of generating functions were applied to solve some important problems of recurrence relations. The findings of this study provide generalized technique and fast method of obtaining the closed form of the generating functions of sequences of the form 𝑓𝑚+1 = 𝑘(𝑓𝑚 + 𝑚𝑘 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 0, 𝑘 > 0 𝑎𝑛𝑑 𝑓0 = 0 and 𝑓𝑚+1 = 𝑘(𝑓𝑚 + (𝑚 − 1) 𝑘 ) 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑚 ≥ 1, 𝑘 > 0 𝑎𝑛𝑑 𝑓1 = 0.

Об операторах типа свертки с ядрами из модифицированных пространств Морри

We investigate operators of the form MaH, HMa, and Hb, where Ma is the multiplication operator on the function a, H is the integral convolution operator with the kernel h, and Hb is the integral operator with the bounded characteristic b(x,y). It is assumed that the kernel h belongs to the intersection of the Lebesgue and Morrey spaces, and the operators themselves act from the Lebesgue space to the Morrey space. First, using conditions for the precompactness of a set in the Morrey space, we prove the compactness of the operator Ma H, wherein it is assumed that the function a approaches zero at infinity. Next, the commutator of the operators Ma and H is considered. It is shown that if the function a belongs to a certain class of functions with a given behavior at infinity, then the commutator is the compact operator. This, in turn, allows us to establish the compactness of the operator HMa. In particular, we prove that the operators PX H and HPX are compact, where PX is the multiplication operator on the characteristic function of a bounded measurable set X. Finally, the integral operator Hb is considered. It is shown that if characteristic b has a given behavior at infinity, then the operator Hb is compact.

CFLOBDDs: Context-Free-Language Ordered Binary Decision Diagrams

This article presents a new compressed representation of Boolean functions, called CFLOBDDs (for Context-Free-Language Ordered Binary Decision Diagrams). They are essentially a plug-compatible alternative to BDDs (Binary Decision Diagrams), and hence are useful for representing certain classes of functions, matrices, graphs, relations, and so forth in a highly compressed fashion. CFLOBDDs share many of the good properties of BDDs, but—in the best case—the CFLOBDD for a Boolean function can be exponentially smaller than any BDD for that function . Compared with the size of the decision tree for a function, a CFLOBDD—again, in the best case—can give a double-exponential reduction in size . They have the potential to permit applications to (i) execute much faster and (ii) handle much larger problem instances than has been possible heretofore. We applied CFLOBDDs in quantum-circuit simulation and found that for several standard problems, the improvement in scalability, compared to BDDs, is quite dramatic. With a 15-minute timeout, the number of qubits that CFLOBDDs can handle are 65,536 for Greenberger-Horne-Zellinger, 524,288 for Bernstein-Vazirani, 4,194,304 for Deutsch-Jozsa, and 4,096 for Grover’s algorithm, besting BDDs by factors of 128×, 1,024×, 8,192×, and 128×, respectively.

Distortion, Radius of Concavity and Several Other Radii Results for Certain Classes of Functions

Distortion, Radius of Concavity and Several Other Radii Results for Certain Classes of Functions

Central limit theorem for bifurcating Markov chains: the mother-daughters triangles case

The main objective of this article is to establish a central limit theorem for additive three-variable functionals of bifurcating Markov chains. We thus extend the central limit theorem under point-wise ergodic conditions studied in S.V. Bitseki Penda and J.-F. Delmas [Central limit theorem for bifurcating Markov chains under pointwise ergodic conditions, Ann. Appl. Probab. 32(5) (2022), pp. 3817–3849] and to a lesser extent, the results of S.V. Bitseki Penda and J.-F. Delmas [Central limit theorem for bifurcating Markov chains under L 2 -ergodic conditions, Adv. Appl. Probab. 54(4) (2022), pp. 999–1031] on central limit theorem under L 2 ergodic conditions. Our results also extend and complement those of J. Guyon [Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging, Ann. Appl. Probab. 17(5–6) (2007), pp. 1538–1569] and J.-F. Delmas and L. Marsalle [Detection of cellular aging in a Galton-Watson process, Stochastic Process. Appl. 120(12) (2010), pp. 2495–2519]. In particular, when the ergodic rate of convergence is greater than 1 / 2 , we have, for certain class of functions, that the asymptotic variance is non-zero at a speed faster than the usual central limit theorem studied by Guyon and Delmas-Marsalle.

Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

Second Hankel determinant of logarithmic coefficients of inverse functions in certain classes of univalent functions

Further Geometric Properties of the Barnes–Mittag-Leffler Function

In this paper, we find sufficient conditions to be imposed on the parameters of a class of functions related to the Barnes–Mittag-Leffler function that allow us to conclude that it possesses certain geometric properties (such as starlikeness, uniformly starlike (convex), strongly starlike (convex), convexity, and close-to-convexity) in the unit disk. The key tools in some of our proofs are the monotonicity properties of a certain class of functions related to the gamma function.

Structured Condition Number for a Certain Class of Functions of Non-commuting Matrices

Structured Condition Number for a Certain Class of Functions of Non-commuting Matrices

Bohr–Rogosinski-Type Inequalities for Certain Classes of Functions: Analytic, Univalent, and Convex

Bohr–Rogosinski-Type Inequalities for Certain Classes of Functions: Analytic, Univalent, and Convex

Refined Bohr inequalities for certain classes of functions: analytic, univalent, and convex

AbstractIn this article, we prove several refined versions of the classical Bohr inequality for the class of analytic self-mappings on the unit disk $ \mathbb {D} $ , class of analytic functions $ f $ defined on $ \mathbb {D} $ such that $\mathrm {Re}\left (f(z)\right )<1 $ , and class of subordination to a function g in $ \mathbb {D} $ . Consequently, the main results of this article are established as certainly improved versions of several existing results. All the results are proved to be sharp.

Efficient Spectral Collocation Method for Tempered Fractional Differential Equations

Transient anomalous diffusion may be modeled by a tempered fractional diffusion equation. In this paper, we present a spectral collocation method with tempered fractional Jacobi functions (TFJFs) as basis functions and obtain an efficient algorithm to solve tempered-type fractional differential equations. We set up the approximation error as O(Nμ−ν) for projection and interpolation by the TFJFs, which shows “spectral accuracy” for a certain class of functions. We derive a recurrence relation to evaluate the collocation differentiation matrix for implementing the spectral collocation algorithm. We demonstrate the effectiveness of the new method for the nonlinear initial and boundary problems, i.e., the fractional Helmholtz equation, and the fractional Burgers equation.

О моментах функций от случайных величин с 𝜌-перемешиванием

Estimates for the moments for a certain class of functions of random variables from a stationary sequence with 𝜌-mixing is obtained in this article. This estimates that cannot be improved in order.

Hyperbolic summation for functions of the GCD and LCM of several integers

Let kge 2 be a fixed integer. We consider sums of type sum _{n_1cdots n_kle x} F(n_1,ldots ,n_k), taken over the hyperbolic region {(n_1,ldots ,n_k)in {mathbb {N}}^k: n_1cdots n_kle x}, where F:{mathbb {N}}^krightarrow {mathbb {C}} is a given function. In particular, we deduce asymptotic formulas with remainder terms for the hyperbolic summations sum _{n_1cdots n_kle x} f((n_1,ldots ,n_k)) and sum _{n_1cdots n_kle x} f([n_1,ldots ,n_k]), involving the GCD and LCM of the integers n_1,ldots ,n_k, where f:{mathbb {N}}rightarrow {mathbb {C}} belongs to certain classes of functions. Some of our results generalize those obtained by the authors (Heyman and Tóth in Results Math 76(1): 22, 2021) for k=2.

The system of ordinary differential questions with rectangular matrices that is partially solved to the derivatives near the pole.

In modern theory of ordinary differential equations and systems of equations with an unknown complex-valued function of a complex variable, a prominent place is occupied by systems of equations that are either unsolved or partially solved with respect to derivatives. The system of ordinary differential equations, which is partially solved with respect to the derivatives, with rectangular matrices around the pole has been studied. The article presents conditions for transforming a system of ordinary differential equations, which is partially solvable with respect to derivatives, to a system of ordinary differential equations with a special form. The theorem with sufficient conditions of the existence at least one solution of the Cauchy problem is proven, some components of the solution are analytic functions in domains with the fixed singularity on the boundary, and the remaining components are functions chosen from a certain class of functions.

On On the approximation of functions by Jacobi – Dunkl expansion in the weighted space 𝕃 2 ( α , β )

UDC 517.5 We prove some new estimates useful in applications for the approximation of certain classes of functions characterized by the generalized continuity modulus from the space 𝕃 2 ( α , β ) by partial sums of the Jacobi – Dunkl series. For this purpose, we use the generalized Jacobi – Dunkl translation operator obtained by Vinogradov in the monograph [Theory of approximation of functions of real variable, Fizmatgiz, Moscow (1960) (in Russian)].

Monotonic self-stabilization and its application to robust and adaptive pattern formation

We introduce, as an enhancement of self-stabilization, the concept of monotonic self-stabilization for distributed systems that ensures that a certain measure of quality of state monotonically improves when the system in an illegitimate state progresses toward a legitimate state. In the concept, the quality is measured by a real-valued function that can be chosen from a certain class of functions. The concept is applied to a multi-robot pattern formation problem in which a group of autonomous mobile robots move from their respective initial positions to the goal positions like a marching band. We solve the problem by presenting two monotonic self-stabilizing pattern formation algorithms, one of which is for FSYNC model and the other is for SSYNC model. The considered real-valued functions to measure the quality take into account both the distance to the goal location and the accuracy of the formation. We present a formal proof of the algorithms' correctness and monotonic self-stability.

Coset-wise affine functions and cycle types of complete mappings

Let K be a finite field of characteristic p. We study a certain class of functions K→K that agree with an Fp-affine function K→K on each coset of a given additive subgroup W of K – we call them W-coset-wiseFp-affine functions of K. We show that these functions form a permutation group on K with the structure of an imprimitive wreath product and characterize which of them are complete mappings of K. As a consequence, we are able to provide various new examples of cycle types of complete mappings of K – for instance, if p>2, then all cycle types where each cycle has length a power of p are achieved by complete mappings of K.

Generalization of Titchmarsh's theorems for the Minkowski algebra

The Minkowski space–time algebra is a powerful tool for studying the theory of electromagnetism and special relativity. In this paper, we intend to establish generalizations of Titchmarsh theorems for certain classes of functions, namely Lipschitz functions in the space for .

An integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions

While an integration by parts formula for the bilinear form of the hypersingular boundary integral operator for the transient heat equation in three spatial dimensions is available in the literature, a proof of this formula seems to be missing. Moreover, the available formula contains an integral term including the time derivative of the fundamental solution of the heat equation, whose interpretation is difficult at second glance. To fill these gaps, we provide a rigorous proof of a general version of the integration by parts formula and an alternative representation of the mentioned integral term, which is valid for a certain class of functions including the typical tensor-product discretization spaces.

On third order Hankel determinant for inverse functions of certain classes of univalent functions

On third order Hankel determinant for inverse functions of certain classes of univalent functions