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The present research deals with generalizations of the Salem function with arguments defined in terms of certain alternating expansions of real numbers. Special attention is given to modelling such functions by systems of functional equations. [ABSTRACT FROM AUTHOR]

For a separable rearrangement invariant space X on [0, 1] of fundamental type we identify the set of all p ∈ [ 1 , ∞ ] such that ℓ p is finitely represented in X in such a way that the unit basis vectors of ℓ p ( c 0 if p = ∞ ) correspond to pairwise disjoint and equimeasurable functions. This can be treated as a follow up of a paper by the first-named author related to separable rearrangement invariant spaces on (0 , ∞) . [ABSTRACT FROM AUTHOR]

The paper is focused on the basis property of the system of exponentials and trigonometric systems of sine and cosine functions in a separable subspace of the weighted grand Lebesgue space generated by the shift operator. In this paper, with the help of the shift operator, a separable subspace of the weighted space of the grand Lebesgue space is defined. The density in of the set of infinitely differentiable functions that are finite on is studied. It is proved that if the weight function satisfies the Mackenhoupt condition, then the system of exponentials forms a basis in , and trigonometric systems of sine and cosine functions form bases in . [ABSTRACT FROM AUTHOR]

An operator T acting on a separable complex Banach space B is said to be hypercyclic if there exists f ∈ B such that the orbit { T n f : n ∈ N } is dense in B . Godefroy and Shapiro (J. Funct. Anal., 98(2):229–269, 1991) characterized those elements, which are hypercyclic, in the commutant of the Hardy backward shift. In this paper, we study some dynamical properties of operators X that λ -commute with the Hardy backward shift B, that is, B X = λ X B . [ABSTRACT FROM AUTHOR]

Based on the estimation of the Cauchy matrix of linear impulsive differential equation, by using Banach fixed point theorem and exponential dichotomy, sufficient conditions for the existence of almost periodic solutions in shifts δ± of some nonlinear dynamic equations with impulses on time scales are established. Finally, two impulsive ecosystems defined on some specific time scales are studied to illustrate the effectiveness of the main results. [ABSTRACT FROM AUTHOR]

For a linear autonomous system of neutral type with commensurable delays, an algorithm is given for solving the modal controllability problem (in particular, the finite spectrum assignment problem), which provides a closed-loop system with a given characteristic quasipolynomial. A procedure for editing the finite part of the spectrum is proposed. A criterion for exponential stabilization of the system under study is constructively justified. When the criterion is met, the closed-loop system can be made exponentially stable according to the proposed spectral reduction algorithm. The obtained statements and spectrum assignment algorithms are illustrated with examples. [ABSTRACT FROM AUTHOR]

We extend existing functional relationships for the discrete generating series associated with a single-variable linear polynomial coefficient difference equation to the multivariable case. [ABSTRACT FROM AUTHOR]

As the queue becomes exhausted, different maintenance tasks can be performed according to the fatigue load and wear degree of the service equipment. At the same time, considering the customer's sensitivity to time delay, the service facility will not completely remain inactive during the maintenance period. To describe this objectively existing phenomenon arising in the waiting line system, we consider a hyper-exponential working vacation queue with a batch renewal arrival process. Through the calculation of the well-structured roots of the associated characteristic equation, the shift operator method in the theory of difference equations and the supplementary variable technique for stochastic modeling plays a central role in the queue-length distribution analysis. Comparison with other ways to analyze queueing models, the advantage of our approach is that we can avoid deriving the complex transition probability matrix of the queue-length process embedded at input points. The feasibility of this approach is verified by extensive numerical examples. [ABSTRACT FROM AUTHOR]

The first mixed boundary value problem for a nonlinear functional-differential equation of parabolic type with shifts in the spatial variables is considered. Sufficient conditions are proved under which a nonlinear differential-difference operator is demicontinuous, coercive, and pseudomonotone on the domain of the operator . Based on these properties, existence theorems for a generalized solution are proved. [ABSTRACT FROM AUTHOR]

Let E and F be two complex Hilbert spaces. Let S_{E} and S_{F} be the shift operators on vector-valued Hardy space H_{E}^{2} and H_{F}^{2}, respectively. We show that the cyclic multiplicity of S_{E}\oplus S_{F} ^{\ast } equals 1+\dim E. This result is classical when \dim E=\dim F=1 (see J. A. Deddens [ On A \oplus A^\ast, 1972]; Paul Richard Halmos [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]; Domingo A. Herrero and Warren R. Wogen [Rocky Mountain J. Math. 20 (1990), pp. 445–466]). Our approach is inspired by the elegant and short proof of this classical result attributed to Nikolskii, Peller and Vasunin in Halmos's book [ A Hilbert space problem book , Springer-Verlag, New York-Berlin, 1982]. By using the invariant subspace theorems for S_{E}\oplus S_{F}^{\ast } (see M. C. Câmara and W. T. Ross [Canad. Math. Bull. 64 (2021), pp. 98–111]; Caixing Gu and Shuaibing Luo [J. Funct. Anal. 282 (2022), 31 pp.]; Dan Timotin [Concr. Oper. 7 (2020), pp. 116–123]), we characterize non-cyclic subspaces of S_{E}\oplus S_{F}^{\ast } when \dim E<\infty and \dim F=1. [ABSTRACT FROM AUTHOR]

My book "Eigenvalues and s-Numbers" (1987) deals with classical traces of Banach space operators. Now an extension to singular traces is required. Hopefully, this (mostly) self-contained, comprehensive survey will serve that purpose. The second part is devoted to the future of the theory of traces. My mathematical heritage consists in a collection of open problems, which I was unable to solve. Good luck. [ABSTRACT FROM AUTHOR]

We extend existing functional relationships for the discrete generating series associated with a single-variable linear polynomial coefficient difference equation to the multivariable case.

A class of doubly stochastic graph shift operators (GSO) is proposed, which is shown to exhibit: (i) lower and upper L 2 -boundedness for locally stationary random graph signals, (ii) L 2 -isometry for i.i.d. random graph signals with the asymptotic increase in the incoming neighbourhood size of vertices, and (iii) preservation of the mean of any graph signal – all prerequisites for reliable graph neural networks. These properties are obtained through a statistical consistency analysis of the proposed graph shift operator, and by exploiting the dual role of the doubly stochastic GSO as a Markov (diffusion) matrix and as an unbiased expectation operator. For generality, we consider directed graphs which exhibit asymmetric connectivity matrices. The proposed approach is validated through an example on the estimation of a vector field. [ABSTRACT FROM AUTHOR]

We describe the spectral properties of the dilation operator , with , in separable rearrangement invariant spaces of fundamental type. We show that these properties are determined by the values of the dilation indices of the spaces and apply the results to Orlicz spaces. [ABSTRACT FROM AUTHOR]

This study evaluate the estimated parameters from fitted Beta-ARMA Model on average temperature in North west Nigeria from the years 1990 to 2017 using shift operator factor to combine the estimated parameters of the mixed model. It was observed that maximum temperature contribute to persistence in climate change (Figs. 1, 2) with high degree of randomness in all the 342 months observed. There is presence of serial correlation on average temperature from the year 1990 to 2017 due to variations associated to climatic variables. The relationship between past and current observations based on variability using determinant factor from AR(1) covariance and Equal correlation methods, it shows that the covariance of the two methods are significant but AR(1) is more precise than EC. A reduced form was used to determine the contribution of each estimated coefficients in model from linear combination using shift operator factor, there is an indication that maximum temperature contribute more to persistence in climate change in the region. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the uniqueness problem of generalized shift polynomials of finite order entire functions sharing a small function under the notion of weakly and relaxed weighted sharing environment. Some examples have been exhibited which are relevant to the content of the paper. [ABSTRACT FROM AUTHOR]

Let { h n } be a sequence in R d tending to infinity and let { T h n } be the corresponding sequence of shift operators given by (T h n f) (x) = f (x - h n) for x ∈ R d . We prove that { T h n } converges weakly to the zero operator as n → ∞ on a separable rearrangement-invariant Banach function space X (R d) if and only if its fundamental function φ X satisfies φ X (t) / t → 0 as t → ∞ . On the other hand, we show that { T h n } does not converge weakly to the zero operator as n → ∞ on all Marcinkiewicz endpoint spaces M φ (R d) and on all non-separable Orlicz spaces L Φ (R d) . Finally, we prove that if { h n } is an arithmetic progression: h n = n h , n ∈ N with an arbitrary h ∈ R d \ { 0 } , then { T nh } does not converge weakly to the zero operator on any non-separable rearrangement-invariant Banach function space X (R d) as n → ∞ . [ABSTRACT FROM AUTHOR]

High-performing, real-time pose detection and tracking in real-time will enable computers to develop a finer-grained and more natural understanding of human behavior. However, the implementation of real-time human pose estimation remains a challenge. On the one hand, the performance of semantic keypoint tracking in live video footage requires high computational resources and large parameters, which limiting the accuracy of pose estimation. On the other hand, some transformer-based models were proposed recently with outstanding performance and much fewer parameters and FLOPs. However, the self-attention module in the transformer is not computationally friendly, which makes it difficult to apply these excellent models to real-time jobs. To overcome the above problems, we propose a transformer-like model, named ShiftPose, which is regression-based approach. The ShiftPose does not contain any self-attention module. Instead, we replace the self-attention module with a non-parameter operation called the shift operator. Meanwhile, we adapt the bridge–branch connection, instead of a fully-branched connection, such as HRNet, as our multi-resolution integration scheme. Specifically, the bottom half of our model adds the previous output, as well as the output from the top half of our model, corresponding to its resolution. Finally, the simple, yet promising, disentangled representation (SimDR) was used in our study to make the training process more stable. The experimental results on the MPII datasets were 86.4 PCKH, 29.1PCKH@0.1. On the COCO dataset, the results were 72.2 mAP and 91.5 AP50, 255 fps on GPU, with 10.2M parameters, and 1.6 GFLOPs. In addition, we tested our model for single-stage 3D human pose estimation and draw several useful and exploratory conclusions. The above results show good performance, and this paper provides a new method for high-performance, real-time attitude detection and tracking. [ABSTRACT FROM AUTHOR]

This letter introduces an efficient shift operator discontinuous Galerkin time domain (SO‐DGTD) method with both nonconforming and local time stepping techniques for the simulation of nanophotonic problems. In this method, the constitutive relation of metallic properties is described by introducing the shift operator method (SO) and the relation of electric field and electric displacement can be obtained in the time domain. To make the nanophotonic simulation more efficient, we introduce the non‐conforming technique based on upwind numerical flux to SO‐DGTD method. In addition, we further introduce the local time stepping technique to SO‐DGTD method to improve the computational efficiency for nanophotonic problems with complex structures. Finally, some numerical examples are given to show the accuracy and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

We study the sequence f n of functions that is defined by recursive convolutions as f 1 (x) = Π (x) , f n + 1 (x) = (f n ∗ f 1) (x) , n ∈ N , where Π is the unit rectangle function. We find out the general closed-form of the sequence f n and apply it for the evaluation of the improper integral 2 π ∫ 0 ∞ sin (ξ) ξ n cos (2 x ξ) d ξ , x ∈ R , n ∈ N , n ≥ 2. We also study some interesting features of the numerical coefficients that appear in the closed-form expression of f n . In connection to the numerical coefficients that appear in closed-form expression of f n , we introduce a map F defined on N × N 0 , by the rule, F (n , s) = ∑ i = 0 n i s (- 1) n - i i ! (n - i) ! , and show that its range is in N 0 , where N 0 : = N ∪ { 0 } . [ABSTRACT FROM AUTHOR]

In this paper, we attempt to understand Cowen–Douglas operators by the way of basis theory and shift operator. For a Cowen–Douglas operator T ∈ B n (Ω) and a complex number z 0 ∈ Ω , we show that there is a generalized basis { g k } k = 0 ∞ , such that the adjoint operator (T - z 0) ∗ is the forward shift on { g k } k = 0 ∞ , and if n ≥ 2 , then T - z 0 never is a backward shift on any Markushevich basis. Moreover, we give a characterization of a Cowen–Douglas operator in B 1 (Ω) being a backward shift on some Markushevich basis. Also, we show that a multiplication operator defined on Bergman space with a M o ¨ bius transformation as its multiplier is a forward shift on some Markushevich basis. [ABSTRACT FROM AUTHOR]

For a separable rearrangement invariant space X on (0 , ∞) of fundamental type we identify the set of all p ∈ [ 1 , ∞ ] such that ℓ p is finitely represented in X in such a way that the unit basis vectors of ℓ p ( c 0 if p = ∞ ) correspond to pairwise disjoint and equimeasurable functions. This characterization hinges upon a description of the set of approximate eigenvalues of the doubling operator x (t) ↦ x (t / 2) in X. We prove that this set is surprisingly simple: depending on the values of some dilation indices of such a space, it is either an interval or a union of two intervals. We apply these results to the Lorentz and Orlicz spaces. [ABSTRACT FROM AUTHOR]

Motivated by the fault tolerance for manufacturing, we investigate a renewal input bulk arrival queue with a fault-tolerant server, in which the server can keep working with a low service rate even if the partial failure occurs. Only when there are no customers in the system, the partial failure can be removed. To explore the performance measures of the queue, a more generic and simpler algorithm based on the right shift operator method for solving difference equations is employed to obtain the queue-length distributions at different time epochs. The significant feature of this algorithm lies in that it does not require the derivation of the transition probability matrix for the corresponding embedded Markov chain. Furthermore, we can resort to the queue-length distribution at the pre-arrival epoch to quickly get the expected sojourn time for an arbitrary customer. Finally, with the help of Pad e ´ approximation, several representative numerical examples are illustrated in tables and graphs, under which we show how to verify the correctness of our theoretical results through Little's law. [ABSTRACT FROM AUTHOR]

In this paper, we employ the notion of weighted sharing to study the uniqueness problems of q-c-shift and q-shift monomials of transcendental meromorphic functions of zero order sharing 1-points. The results in this paper extend some previous results. [ABSTRACT FROM AUTHOR]

The quantum calculus has emerged as a connection between mathematics and physics. It has wide applications, particularly in quantum mechanics, analytic number theory, combinatorial analysis, operation theory etc. The q-calculus, which serves as a powerful tool to model quantum computing, has drawn attention of many researchers in the field of special functions and as a result the q-analogues of certain special functions, especially hypergeometric function, 1-variable Hermite polynomials, Appell polynomials etc., are introduced and studied. In this paper, we introduce the 2-variable q-Hermite polynomials by means of generating function. Also, its certain properties like series definition, recurrence relations, q-differential equation and summation formulas are established. The operational definition and some integral representations of these polynomials are obtained. Some examples are also considerd to show the efficacy of the proposed method. Some concluding remarks are given. At the end of this paper, the graphical representations of these polynomials of certain degrees with specified values of q are given.

In this paper, on the basis of constructing a new shift operator S and choosing the Grover coin G as coin operator C , we get the standard evolution operator U on cycles. Using U , we not only got the analytical expression of wavefunction ψ s , j , but also obtained the conclusion that the limit distribution π (ν) of N , which is not uniform distribution, regardless of N is odd or even. [ABSTRACT FROM AUTHOR]

In this paper, we study a specific system of Clifford–Appell polynomials and, in particular, their product. Moreover, we introduce a new family of quaternionic reproducing kernel Hilbert spaces in the framework of Fueter regular functions. The construction is based on a general idea which allows us to obtain various function spaces by specifying a suitable sequence of real numbers. We focus on the Fock and Hardy cases in this setting, and we study the action of the Fueter mapping and its range. [ABSTRACT FROM AUTHOR]

This work is dedicated to Korovkin type theorems in Banach function spaces. The subspace X S of the Banach function space X generated by the shift operator is considered and the density of the set C 0 ∞ in X S is proved. The analogs of the Korovkin theorems in X S are obtained. Also, the analog of the Korovkin theorem for Kantorovich polynomials is derived both in the cases of rearrangement-invariant and general non-rearrangement-invariant Banach function spaces. These results are obtained for Lebesgue spaces, grand-Lebesgue spaces, Morrey-type spaces and their weighted versions, weak Lebesgue spaces, Orlicz spaces. Note that in our case the Korovkin-type theorem for Kantorovich polynomials in Morrey spaces is an only natural analog of the classical L p version of the Korovkin theorem and strengthens the previously known result. [ABSTRACT FROM AUTHOR]

In this article, different standard and nonstandard continualization techniques are applied to a one-dimensional solid consisting in a chain of masses interacting with nearest and next-nearest neighbors through linear springs. The study focuses on the reliability of the different continua in capturing the dispersive behavior of the discrete, on the order of the continuous governing equation because of its effect on the need for including nonclassical boundary conditions, as well as on the physical inconsistencies that appear for short wavelengths. The Regularization method, used by Bacigalupo and Gambarotta for a lattice with nearest interactions, presents advantages over the others. [ABSTRACT FROM AUTHOR]

Let G be a bounded region in the complex plane and H be a Hilbert space of analytic functions on G satisfying (a) z f ∈ H for all f ∈ H , and (b) { z n } n = 0 ∞ forms an orthogonal basis of H with lim n ‖ z n + 1 ‖ / ‖ z n ‖ = 1 . This paper aims to study a random counterpart of the shift operator on H . We replace the weights w n = 1 in the shift operator on H , defined as T z n = w n z n + 1 , by a sequence of i.i.d. random variables { X n } n = 0 ∞ ; that is, w n = X n . We call T a random weighted shift on H with weights { X n } n = 0 ∞ . In this paper we classify the samples of T according to five usual equivalence relations. We compare random weighted shifts acting on various Hilbert spaces of analytic functions, via a binary relation ⊲ borrowed from the representation theory of C ∗ -algebras. Also we discuss the random Hardy space associated with T and, in certain case, determine the spectral picture of its multipliers. [ABSTRACT FROM AUTHOR]

We employ the notion of weighted and truncated sharing to study the uniqueness problems of generalized shift monomial sharing the same 1-points. The corollary deducted from our main results will improve a number of results of recent time. As an application of the main result we will also improve a recent result under the periphery of a more generalized shift operator. Some examples have been exhibited by us relevant to the content of the paper.

The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented. [ABSTRACT FROM AUTHOR]

Let 핋 be a nonempty, closed, and arbitrary set of real numbers, namely a time scale, and consider the following delay dynamical equation xΔ (t) = a (t) x (t) + f (t, x (ϑ (t))), t ∈ 핋, where ϑ stands for the abstract delay function. The main goal of this study is three-fold: obtaining the existence of an equi-bounded solution, proving the asymptotic stability of the zero solution, and showing the existence of a periodic solution based on new periodicity concept on time scales for the given delayed equation under certain conditions. In our analysis, we propose an alternative variation of parameters formulation by using an auxiliary function to invert a mapping for the utilization of fixed point theory. [ABSTRACT FROM AUTHOR]

Riordan matrices are infinite lower triangular matrices corresponding to the certain operators in the space of formal power series. The n th descending diagonal of the ordinary Riordan matrix and the n th descending diagonal of the exponential Riordan matrix have the generating functions respectively g n (φ x) / (1 − φ x) n + 1 and h n (φ x) / (1 − φ x) 2 n + 1 , where g n (x) , h n (x) are polynomials of degree ≤ n. We will call these polynomials the numerator polynomials of Riordan matrices. General properties of these polynomials were considered in separate paper. In this paper, we will consider numerator polynomials of the Riordan matrices associated with the family of series a (β) (x) = a (x (β) a β (x)). The matrices of transformations, in which these polynomials participate, have the form A n E n β A n − 1 , where A n is the certain matrix of order n + 1 , E is the matrix of the shift operator. The main focus is on studying the properties of these matrices. [ABSTRACT FROM AUTHOR]

Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only yields energy levels in a finite range and with a nonzero uncertainty. Within this study we review isospectral potentials from the literature and investigate the applicability of the harmonic oscillator hypothesis to recent measurements. First, we describe experimental results from literature by a harmonic oscillator model, obtaining a realistic size and depth of the resulting quantum well. Second, we use the shift-operator approach to calculate anharmonic non-polynomial potentials producing (partly) equidistant spectra. We discuss different potential types and interpret the possible modeling applications. Finally, by applying nth order perturbation theory we show that exactly equidistant eigenenergies cannot be achieved by polynomial potentials, except by the harmonic oscillator potential. In summary, we aim to give an overview over which conclusions may be drawn from the experimental determination of energy levels and which may not. [ABSTRACT FROM AUTHOR]

High-performing, real-time pose detection and tracking in real-time will enable computers to develop a finer-grained and more natural understanding of human behavior. However, the implementation of real-time human pose estimation remains a challenge. On the one hand, the performance of semantic keypoint tracking in live video footage requires high computational resources and large parameters, which limiting the accuracy of pose estimation. On the other hand, some transformer-based models were proposed recently with outstanding performance and much fewer parameters and FLOPs. However, the self-attention module in the transformer is not computationally friendly, which makes it difficult to apply these excellent models to real-time jobs. To overcome the above problems, we propose a transformer-like model, named ShiftPose, which is regression-based approach. The ShiftPose does not contain any self-attention module. Instead, we replace the self-attention module with a non-parameter operation called the shift operator. Meanwhile, we adapt the bridge–branch connection, instead of a fully-branched connection, such as HRNet, as our multi-resolution integration scheme. Specifically, the bottom half of our model adds the previous output, as well as the output from the top half of our model, corresponding to its resolution. Finally, the simple, yet promising, disentangled representation (SimDR) was used in our study to make the training process more stable. The experimental results on the MPII datasets were 86.4 PCKH, 29.1PCKH@0.1. On the COCO dataset, the results were 72.2 mAP and 91.5 AP50, 255 fps on GPU, with 10.2M parameters, and 1.6 GFLOPs. In addition, we tested our model for single-stage 3D human pose estimation and draw several useful and exploratory conclusions. The above results show good performance, and this paper provides a new method for high-performance, real-time attitude detection and tracking.

A new method, called the SO-DGTD method, for solving time-dependent Maxwell's equations via the discontinuous Galerkin time domain (DGTD) method in dispersive media is presented. The shift operator (SO) method is used to represent the constitutive relations. This method has a universal equation for dispersive media models, i.e. the Debye model, the Drude model, and the Lorentz model. The algorithm is validated by studying the radar cross-section (RCS) of a hybrid dispersive sphere and a suddenly created plasma problem. [ABSTRACT FROM AUTHOR]