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Abstract: Complementing the work of T.-S. Liu and A.C.M. van Rooij we show that the existence of non-zero translation invariant operators between certain function spaces on a locally compact group implies its amenability. [Copyright &y& Elsevier]

In this paper we show that for a set X of real numbers the function space Cp(X) has both a property introduced by Sakai in [Proc. Amer. Math. Soc. 104 (1988) 917–919] and a property introduced by Reznichenko (see [Topology Appl. 104 (2000) 181–190]) if and only if all finite powers of X have a property that was introduced by Gerlits and Nagy in [Topology Appl. 14 (1982) 151–161]. It follows that the minimal cardinality of a set of real numbers for which the function space does not have the properties of Sakai and Reznichenko is equal to the additivity of the ideal of first category sets of real numbers. [Copyright &y& Elsevier]

The article announces that the paper "A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs," by Ian Kopacka and Michael Hintermüller in Volume 50 of the journal has received the "Best Paper Award" from the Computational Optimization and Applications.

This paper presents a series of results about the fuzzy degree of a hypergoup. It is considered a sequence of membership functions and of join spaces, obtained by starting with a hypergroupoid (H, ⊗), see Corsini's paper [Southeast Asian Bull. Math. 27 (2003) 221-229]. The fuzzy grade is the minumum natural number i such that two consecutive associated join spaces, of the above mentioned sequence, Hi and Hi+1 are isomorphic. [ABSTRACT FROM AUTHOR]

The influence of frequency characteristics of devices is complex. The effects of AlN barrier thickness, gate length, gate-drain spacing and work function on the frequency characteristics of AlN/ β-Ga2O3 high electron mobility transistor (HEMT) were studied by Sentaurus TCAD in this paper. The following conclusions are obtained: as the thickness of the AlN barrier layer increases from 10 nm to 25 nm, the cutoff frequency (fT) and maximum oscillation frequency (fmax ) rise by 18 and 17 GHz respectively. The decrease of gate capacitance is the main reason for the increase of fT . Furthermore, it was found that the thinner barrier layer enhanced the gate's ability to control the channel electrons. When the gate length is scaled down from 0. 9 μm to 0. 1 μm, fT and fmax increase by 84 and 98 GHz, respectively, representing a far more profound influence on frequency characteristics than the barrier layer thickness. However, when the gate length fell below 0. 1 μm, short-channel effects emerged. As the gate-drain spacing increase, fT exhibits a slight decrease. Coupled with the concurrent reduction in source resistance, this led to a synchronized trend in fmax and fT only when the gate-source voltage (VGS ) exceeds - 1. 2 V. The work function, on the other hand, has minimal impacts on fT and fmax, but an increase in the work function positively influenced the device' s pinch-off characteristics. In summary, this paper indicates that by shortening the gate length while concurrently augmenting the thickness of the AlN barrier layer, gate-drain spacing, and work function, one can enhance the frequency characteristics while also improving the pinch-off characteristics of the device, which has certain guiding significance for the design of HEMT devices. [ABSTRACT FROM AUTHOR]

This paper intends to analyze the existing problems of remote sensing data from the perspectives of space remote sensing information data capacity and data types. Then, a framework for rapidly analyzing and processing space remote sensing information is constructed. Then, LSTM is used to realize the fault diagnosis of remote sensing data continuity, discrete sample mixing and strong correlation of sample variation. LSTM conducts a multimodal analysis of remote-control commands, which is applied to modeling. The multi-stage LSTM prediction model is established and integrated efficiently to improve its adaptive ability in complex space environments. In this way, the anomaly recognition of remote sensing information is realized. Experiments show that the algorithm can improve the anomaly detection rate of remote sensing data. Experiments show that the algorithm is feasible. It can provide reliable data interpretation function for space remote sensing information control system. [ABSTRACT FROM AUTHOR]

About 100 years ago, N.J. Walsh's fundamental paper [25] was published, in which he introduced the digital version of the trigonometric system. In remembrance of this and the 50th anniversary of Walsh's death, the authors of the paper [2] presented the role of Walsh functions in dyadic analysis and technical applications. 35 years ago, a collaboration between researchers from the Department of Numerical Analysis, Eötvös Loránd University and Professor W.R. Wade (University of Tennessee, USA) resulted in the publication of the first monograph on dyadic analysis. This provided an overview of the significant results in the field before 1990. Since then, several promising results have been achieved that may determine the future direction of research. This paper provides a brief overview of these results. In the commemorations prepared for the anniversary of the department's establishment, we present in detail our contributions to the achievements in the field. Here, the author only highlights the following. Regarding the Vilenkin generalization of the Walsh system, the interpretation of the concept of the conjugate function and the proof of the corresponding fundamental inequalities were significant [21]. The author of the [11] paper introduced the dyadic analogues of Hermite functions as eigenfunctions of the dyadic derivative and pointed out their application possibilities. In harmonic analysis, the examination of multiplier operators and the corresponding filtering procedures in signal processing is a central theme of research. The strong approximation, two-sided Sidon-type inequalities, and Hardy-type spaces related to this have proven to be of fundamental importance in both the trigonometric and dyadic cases [4]. It would be worthwhile to extend these results to Malmquist--Takenaka systems. New, significant results have also been achieved in the extension of multivariable dyadic analysis, traditional stochastic structures, and function spaces [26, 27]. The [6] paper provides insights into the studies related to the direct product of finite, non-commutative groups. [ABSTRACT FROM AUTHOR]

In this paper, we discuss some extensions of the Maligranda–Orlicz lemma. It deals with the problem of constructing a norm in a subspace of the space of bounded functions, for which it becomes a normed algebra so that the norm introduced is equivalent to the initial norm of the subspace. This is done by satisfying some inequality between these norms. We show in this paper how this inequality is relevant to the study of operator equations in Banach algebras. In fact, we study how to equip a subspace of the space of bounded functions with a norm equivalent to a given one so that it is a normed algebra. We give a general condition for the construction of such norms, which allows us to easily check whether a space with a given norm is an algebra with a pointwise product and the consequences of such a choice for measures of noncompactness in such spaces. We also study quasi-normed spaces. We introduce a general property of measures of noncompactness that allows the study of quadratic operator equations, prove a fixed-point theorem suitable for such problems, and complete the whole with examples and applications. [ABSTRACT FROM AUTHOR]

Abstract: In this note, we will modify several gaps in Chidume and Ofoedu [C.E. Chidume, E.U. Ofoedu, A new iteration process for generalized Lipschitz pseudo-contractive and generalized Lipschitz accretive mappings, Nonlinear Anal. (2006), in press (doi:10.1016/j.na.2006.05.012)]. [Copyright &y& Elsevier]

This paper considers a nonlinear fractional-order boundary value problem   H D a , g α 1 , β , μ x (t) + f (t , x (t) , H D a , g α 2 , β , μ x (t)) = 0 , for t ∈ [ a , b ] , α 1 ∈ (1 , 2 ] , α 2 ∈ (0 , 1 ] , β ∈ [ 0 , 1 ] with appropriate integral boundary conditions on the Hölder spaces. Here, f is a real-valued function that satisfies the Hölder condition, and   H D a , g α , β , μ represents the tempered-Hilfer fractional derivative of order α > 0 with parameter μ ∈ R + and type β ∈ [ 0 , 1 ] . The corresponding integral problem is introduced in the study of this issue. This paper addresses a fundamental issue in the field, namely the circumstances under which differential and integral problems are equivalent. This approach enables the study of differential problems using integral operators. In order to achieve this, tempered fractional calculus and the equivalence problem of the studied problems are introduced and studied. The selection of an appropriate function space is of fundamental importance. This paper investigates the applicability of these operators on Hölder spaces and provides a comprehensive rationale for this choice. [ABSTRACT FROM AUTHOR]

This paper delves into fixed point findings within a complete partially ordered b-metric space, focusing on mappings that adhere to weakly contractive conditions in the presence of essential topological characteristics. These findings represent modifications of established results and further extend analogous outcomes in the existing literature. The conclusions are substantiated by illustrative examples that strengthen the conclusion of the paper. [ABSTRACT FROM AUTHOR]

In this paper we consider various subclasses of normalized, analytic functions defined in the open unit disk A = {z = C : z < 1} in the complex plane C and study the Hardy space of the functions in these subclasses. This study provides an analysis of the growth of these functions near the boundary of the open unit disk and the Taylor's coefficients of them. The study is carried out using the methods of integral means and subordination of analytic functions. Determination of explicit indices of the Hardy space and order of the growth rate of the Taylor coefficient of these functions are important results here. The novelty of the work here is an attempt to extend the study of the above mentioned features for functions in standard subclasses of analytic univalent functions which were not considered by researchers in the past. [ABSTRACT FROM AUTHOR]

In this paper, a short review of the calculus of exact finite-differences of integer order is proposed. The finite-difference operators are called the exact finite-differences of integer orders, if these operators satisfy the same characteristic algebraic relations as standard differential operators of the same order on some function space. In this paper, we prove theorem that this property of the exact finite-differences is satisfies for the space of simple entire functions on the real axis (i.e., functions that can be expanded into power series on the real axis). In addition, new results that describe the exact finite-differences beyond the set of entire functions are proposed. A generalized expression of exact finite-differences for non-entire functions is suggested. As an example, the exact finite-differences of the square root function is considered. The use of exact finite-differences for numerical and computer simulations is not discussed in this paper. Exact finite-differences are considered as an algebraic analog of standard derivatives of integer order. [ABSTRACT FROM AUTHOR]

This paper studies the existence of uniform attractors for 3D micropolar equation with damping term. When β>3, with initial data (uτ,ωτ)∈V1×V2 and external forces (f1,f2)H(f01× H(f02), some uniform estimates of the solution in different function spaces are given. Based on these uniform estimates, the ((V1×V2)×(H(f01)×H(f02)),V1×V2)-continuity of the family of processes {U(f1,f2)(t,τ)}t≥τ is demonstrated. Meanwhile, the (V1×V2,H²(Ω)×H²(Ω))-uniform compactness of {U(f1,f2)(t,τ)}t≥τ is proved. Finally, the existence of a (V1×V2,V1×V2)-uniform attractor and a (V1× V2,H²(Ω)×H²(Ω))-uniform attractor are obtained. Furthermore, the (V1×V2,V1×V2)-uniform attractor and the (V1×V2,H²(Ω)×H²(Ω))-uniform attractor are verified to be the same. [ABSTRACT FROM AUTHOR]

We consider optimal control problems for partial differential equations where the controls take binary values but vary over the time horizon; they can thus be seen as dynamic switches. The switching patterns may be sub ject to combinatorial constraints such as, e.g., an upper bound on the total number of switchings or a lower bound on the time between two switchings. In a companion paper [C. Buchheim, A. Gruütering, and C. Meyer, SIAM J. Optim., arXiv:2203.07121, 2024], we describe the Lp -closure of the convex hull of feasible switching patterns as the intersection of convex sets derived from finite-dimensional pro jections. In this paper, the resulting outer description is used for the construction of an outer approximation algorithm in function space, whose iterates are proven to converge strongly in L² to the global minimizer of the convexified optimal control problem. The linear-quadratic subproblems arising in each iteration of the outer approximation algorithm are solved by means of a semismooth Newton method. A numerical example in two spatial dimensions illustrates the efficiency of the overall algorithm. [ABSTRACT FROM AUTHOR]

The direction estimation of the coherent source in a uniform circular array is an essential part of the signal processing area of the array, but the traditional uniform circular array algorithm has a low localization accuracy and a poor localization effect on the coherent source. To solve this problem, this paper proposes a two-dimensional direction of arrival (DOA) estimation for the coherent source in broadband. Firstly, the central frequency of the coherent sound source is estimated using the frequency estimation method of the delayed data, and a real-valued beamformer is constructed using the concept of the multiloop phase mode. Then, the cost function in the beam space is obtained. Finally, the cost function is searched in two dimensions to locate the sound source. In this paper, we simulate the DOA of the sound source at different frequencies and signal-to-noise ratios and analyze the resolution of the circular array. The simulation results show that the proposed algorithm can estimate the direction of arrival with high precision and achieve the desired results. [ABSTRACT FROM AUTHOR]

Shopping malls have replaced traditional public spaces and become an integral part of urban life in many cities. This paper seeks to explore the role of shopping malls as protest sites in Hong Kong during the Anti-Extradition Law Amendment Bill protest movement in 2019. As the protests decentralised and filtered throughout the city, shopping malls became sites of protest and battlegrounds between riot police and protesters. In addition to singing and chanting, organising sit-ins, and exhibiting protest art inside shopping malls, protesters also confronted mall employees as well as disrupted businesses. Based on information gathered through media reports, planning and policy documents, as well as ethnographic observations, this paper aims to examine the role of shopping malls in the urban development of Hong Kong, their function as public spaces during the protest movement, and how the politicisation of shopping malls shaped and sustained the protest movement. This paper contends that the protesters' appropriation of shopping malls not only represented an important first step of reclaiming the right to the city, but also exemplified how such struggle and resistance can be extended beyond traditional protest sites and into different everyday spaces. [ABSTRACT FROM AUTHOR]

Science communication has traditionally been seen as a means of crossing the boundary of science: moving scientific knowledge into the public. This paper presents an alternative understanding. Drawing upon a particular case of social science communication in the form of an interactive installation about the social responsibility of science, it develops the concept of boundary space where phenomena can simultaneously belong to science and nonscience. In addition, the paper describes how the installation functions as a space for interaction between knowledge communication and knowledge production. The paper argues that we should understand science communication as a social practice, which allows scientists and nonscientists to cooperate in performing science as an important part of society. The aim is to add a new kind of analysis to traditional criticisms of deficit-thinking and popularization by asking what can we say more about science communication if we understand it as part of (rather than separated in time and space from) science as a social activity. [ABSTRACT FROM AUTHOR]

In the context of austerity and the COVID‐19 pandemic, this paper draws on 17 interviews conducted with frontline staff and volunteers to explore the use of food banks by older people in a highly deprived North‐West borough. Despite high levels of poverty amongst this age group, older people are infrequent users of food banks and it is their absence from these spaces, as opposed to their use of and experiences within food banks, that has often gained attention. By foregrounding this age group, this paper highlights different circumstances of use, generational dynamics involving heightened feelings of shame, and how food banks function as social spaces for older people. In doing so, this paper adds to literature in gerontology around spaces of ageing, as well as research on food banks, by highlighting how experiences in these spaces are differentiated by age. This paper advances discussions around the impact of austerity on the everyday lives of older people. Due to the timing of this research, it also gives insight into how older people and informal social spaces have been affected by the COVID‐19 pandemic. [ABSTRACT FROM AUTHOR]

Calculus equation is an important tool for mathematical research and plays an important role in most natural science research. Since the beginning of the eighteenth century, people have gradually used differential and integral equations to solve physical problems. In general, several different aspects of differential equations in the field of mathematics are concerned and studied by most scholars. However, this paper studies and establishes the optimal model for numerical solution of differential equations through deep learning and genetic algorithm. In this paper, the solution of ordinary differential equations is solved through the use of polynomial function space, while the linear combination of simple function x and its product nx can obtain multinomial function space. The space function form of polynomial is very simple, and the operation ability is very strong. Almost all functions can be approximated, and the function space can be transformed by a simple function. Through data simulation test results, it can be found that the oscillation of neural network output is stronger and stronger with the increase in depth, that is to say, the deeper depth endows the neural network with stronger oscillation properties, so for the oscillation function, the depth neural network fitting effect is better than the shallow neural network. Therefore, by combining deep learning and genetic algorithm, this paper studies and establishes the optimal model for numerical solution of differential equations, and finds that the deep neural network can largely complete data simulation. [ABSTRACT FROM AUTHOR]

The present paper reformulates and improves a previously proposed determinant-based second-order multi-reference perturbative formalism. Through a rather simple modification of the energy denominators, this formalism takes into account the interactions between the model space determinants, which are repeated in outer space. The method has been shown to be size-consistent when the model space is a complete active space, which is a severe limit. It is shown here that the completeness of the model space is not necessary to keep this property, provided that the zero-order function satisfies some conditions. For instance, size consistency may be obtained from truncated complete active spaces. It may even be satisfied from Singles and Doubles Configuration Interactions, provided that a coupled electron pair approximation is used in the definition of the model space wave function. The physical content of the method is illustrated by a series of model problems, showing its robustness. A major benefit of the fact that the perturbers are single determinants is the possibility to revise with full flexibility the model-space component of the wave function, i.e., to treat the feedback effect of the dynamic correlation on the valence component of the wave function. [ABSTRACT FROM AUTHOR]

This paper is concerned with function reconstruction from samples. The sampling points used in several approaches are (1) structured points connected with fast algorithms or (2) unstructured points coming from, e.g., an initial random draw to achieve an improved information complexity. We connect both approaches and propose a subsampling of structured points in an offline step. In particular, we start with quasi-Monte Carlo (QMC) points with inherent structure and stable L_2 reconstruction properties. The subsampling procedure consists of a computationally inexpensive random step followed by a deterministic procedure to further reduce the number of points while keeping its information. In these points functions (belonging to a reproducing kernel Hilbert space of bounded functions) will be sampled and reconstructed from whilst achieving state of the art error decay. Our method is dimension-independent and is applicable as soon as we know some initial quadrature points. We apply our general findings on the d-dimensional torus to subsample rank-1 lattices, where it is known that full rank-1 lattices lose half the optimal order of convergence (expressed in terms of the size of the lattice). In contrast to that, our subsampled version regains the optimal rate since many of the lattice points are not needed. Moreover, we utilize fast and memory efficient Fourier algorithms in order to compute the approximation. Numerical experiments in several dimensions support our findings. [ABSTRACT FROM AUTHOR]

Sassanid dynasty mainly known as a religious government that tried to develop Zoroastrianism through Iran. Religious structures are among the most outlined archaeological evidence, generally known as "Chahar Taqi". Despite of vast studies about Sassanid religious architecture, during recent half century, there are ambiguities about excavated Sassanid religious sites, including religious function, and relation to the three sacred fires. The site A of Bandian Dargaz, was excavated during late 20th century, is one of the most important Sassanid sites at northeastern Iran, for architectural spaces and modeling. It was suggested as a Bahram V's sanctuary. Later, the suggestion changed to a lord house or a burial complex. The authors attempt to present better understanding of the architectural identity and function of site A, considering comparing the D architectural space of Bandian, known as fire temple, to similar architectures and adaption to Zoroastrian rites. The most significant question is the function of Architecture D of Bandian Site A, in relation to the triad sacred fires, and any relevant application. Consequently, what was the function of Bandian Site A, considering the function of Architecture D? Methodologically, present paper follows descriptive-analytical method, while it has a fundamental nature. The data collected in a bibliographic and field work, which compare the sites in filed, use disseminated reports, and adaptation architectural spaces to Zoroastrian rites. Comparison of "T" form platform of Space D to the features of Space B of Takht-i-Suleiman, and internal features of the modern Zoroastrian Yazišngāh, architectural limitation for keeping fire except Ātaš Dādgāh, and finally conditions relevant to the Zoroastrian Yazišngāh can indicate Space D of Bandian as Yazišngāh, whereas the fire was the third sacred Ātaš Dādgāh. Present paper is significant for a new presentation of the identity of Bandian Dargaz complex, and revelation of a Sassanid family Fire Temple. [ABSTRACT FROM AUTHOR]

The famous Ascoli-Arzelà theorem served as a springboard for research into compactness in function spaces, particularly spaces of continuous functions. This paper investigates compact subsets of spaces of continuous functions endowed with topologies between the topology of pointwise convergence and the topology of uniform convergence. More precisely, this paper studies necessary and sufficient conditions for a subset to be compact in Cλ,u(X) for a locally-λ space X when λ ⊇ 퓕(X), for a hemi- λ λf-space X when λ ⊆ 퓟 퓢(X), and for a k-space X when λ ⊇ 퓚(X). This paper also studies that every bounded subset of Cλ,u(X) has compact closure for some classes of topological spaces X. [ABSTRACT FROM AUTHOR]

The rationale of this work is to introduce the notion of graphical symmetric spaces and some fixed-point results are proved for H - (ϑ , φ) -contractions in this setting. The idea of graphical symmetric spaces generalizes various spaces equipped with a function which characterizes the distance between two points of the space. Some topological properties of graphical symmetric spaces are discussed. Some fixed-point results for the mappings defined on graphical symmetric spaces are proved. The fixed-point results of this paper generalize and extend several fixed-point results in this new setting. The main results of this paper are applied to obtain the positive solutions of fractional periodic boundary value problems. [ABSTRACT FROM AUTHOR]

Characterization of simultaneous similarity for commuting m- tuples of operators is an open problem even in finite-dimensional spaces; known as "A wild problem in linear algebra". In this paper we offer a criterion for simultaneous similarity of m-tuples of k-cyclic commuting operators on an arbitrary Banach space. Moreover, we obtain an additional equivalence condition in the case of finite dimensional Banach spaces, which extends the result found by Shekhtman [Math. Stat. 1 (2013), pp. 157–161] for pairs of cyclic commuting matrices. We also present two applications of our results, one in the case of general multiplication operators on Banach spaces of analytic function, and one for m-tuples of commuting square matrices. [ABSTRACT FROM AUTHOR]

The aim of the paper is to re-visit the 1990 Khamsi-Kozlowski-Reich Fixed Point Theorem, which initiated a flourishing field of fixed point theory in modular function spaces. Our result generalises this theorem as well as other classical fixed point theorems, including celebrated 1965 result of Kirk. As the common setting for our investigation, we choose the modulated LTI-spaces defined as modular spaces equipped with a sequential convergence structure, which allows also to use convergence types not associated with any topology (like convergence almost everywhere). [ABSTRACT FROM AUTHOR]

This paper is an extension of the research on (cartesian) product of c-spaces. This paper demonstrates that the finite (tensor)product of quotients of c-spaces can be represented as a quotient of its (tensor) product. Some properties of the tensor product of c-spaces have been investigated in this context. Properties of the space of c-continuous functions have been probed and the relevance of the standard c-structure on it has been established. [ABSTRACT FROM AUTHOR]

In this paper, we present a result on the existence of asymptotically stable solutions of infinite systems (IS) of quadratic Hammerstein integral equations (IEs). Our study will be conducted in the Banach space of functions, which are continuous and bounded on the half-real axis with values in the classical Banach sequence space consisting of real bounded sequences. The main tool used in our investigations is the technique associated with the measures of noncompactness (MNCs) and a fixed point theorem of Darbo type. The applicability of our result is illustrated by a suitable example at the end of the paper. [ABSTRACT FROM AUTHOR]

The paper aims to propose a two-step iterated method, which is derived from a regularized dynamical system of extragradient-type in terms of time discretizing, for solving an equilibrium problem. We prove that the iterative sequence generated by the method converges strongly to a solution of the equilibrium problem. Some numerical experiments are given to illustrate and compare the behavior of the new method with several other methods. [ABSTRACT FROM AUTHOR]

In this paper, we propose two self-adaptive extragradient-like algorithms for solving pseudomonotone variational inequalities. We consider two cases: the mapping is Lipschitz continuous (with unknown modulus) and is not Lipschitz continuous. The step size in our algorithms can either increase or decrease at each step. This feature makes our algorithms more efficient. We also give some numerical examples to compare the performance of our algorithm with the existing one. [ABSTRACT FROM AUTHOR]

This paper offers a comprehensive analysis of solution representations for ϖ-fractional partial differential equations, specifically focusing on the linear case of the Darboux problem. We exhibit a representation of the solutions for the Darboux problem of ϖ-fractional partial differential equations in the linear case in the space of continuous functions. Through the application of the generalized Gronwall inequality, we establish the Ulam–Hyers–Rassias Mittag–Leffler stability in the space of continuous functions. Three numerical examples are presented to show the effectiveness and the applicability of our results. [ABSTRACT FROM AUTHOR]

In dynamical systems, Hilbert spaces provide a useful framework for analyzing and solving problems because they are able to handle infinitely dimensional spaces. Many dynamical systems are described by linear operators acting on a Hilbert space. Understanding the spectrum, eigenvalues, and eigenvectors of these operators is crucial. Functional analysis typically involves the use of tensors to represent multilinear mappings between Hilbert spaces, which can result in inequality in tensor Hilbert spaces. In this paper, we study two types of function spaces and use convex and harmonic convex mappings to establish various operator inequalities and their bounds. In the first part of the article, we develop the operator Hermite–Hadamard and upper and lower bounds for weighted discrete Jensen-type inequalities in Hilbert spaces using some relational properties and arithmetic operations from the tensor analysis. Furthermore, we use the Riemann–Liouville fractional integral and develop several new identities which are used in operator Milne-type inequalities to develop several new bounds using different types of generalized mappings, including differentiable, quasi-convex, and convex mappings. Furthermore, some examples and consequences for logarithm and exponential functions are also provided. Furthermore, we provide an interesting example of a physics dynamical model for harmonic mean. Lastly, we develop Hermite–Hadamard inequality in variable exponent function spaces, specifically in mixed norm function space ( l q (·) (L p (·)) ). Moreover, it was developed using classical Lebesgue space ( L p ) space, in which the exponent is constant. This inequality not only refines Jensen and triangular inequality in the norm sense, but we also impose specific conditions on exponent functions to show whether this inequality holds true or not. [ABSTRACT FROM AUTHOR]

In this paper, we present explicit expressions for conforming finite element function spaces, basis functions, and degrees of freedom on the pentatope (the 4-simplex) and tetrahedral prism elements. More generally, our objective is to construct high-order finite element function spaces that maintain conformity with infinite-dimensional spaces of a carefully chosen de Rham complex in four dimensions. This paper is a natural extension of the companion paper entitled "Conforming finite element function spaces in four dimensions, part I: Foundational principles and the tesseract" by Nigam and Williams (2024). In contrast to Part I, in this paper we focus on two of the most popular elements which do not possess a full tensor-product structure in all four coordinate directions. We note that these elements appear frequently in existing space-time finite element methods. In order to build our finite element spaces, we utilize powerful techniques from the recently developed 'Finite Element Exterior Calculus'. Subsequently, we translate our results into the well-known language of linear algebra (vectors and matrices) in order to facilitate implementation by scientists and engineers. [ABSTRACT FROM AUTHOR]

Let n ∈ N 0 , ψ be an analytic self-map on D and u be an analytic function on D. The single operator D u , ψ n acting on various spaces of analytic functions has been a subject of investigation for many years. It is defined as (D u , ψ n f) (z) = u (z) f (n) (ψ (z)) , f ∈ H (D) . However, the study of the operator P u → , ψ k , which represents a finite sum of these operators with varying orders, remains incomplete. The boundedness, compactness and essential norm of the operator P u → , ψ k on the Bloch space are investigated in this paper, and several characterizations for these properties are provided. [ABSTRACT FROM AUTHOR]

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of certain set-valued mappings between function spaces. This paper deals with the computational properties of these accretive and (generalized) monotone set-valued operators. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie "non-computational" proofs from the mainstream literature. To this end, we establish logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On the one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches. [ABSTRACT FROM AUTHOR]

A system of fractional differential equations (FDEs) with fractional derivatives of diverse orders is called an incommensurate system of FDEs. In this paper, the well-posedness of the initial value problem for incommensurate systems of FDEs is obtained on the space of continuous functions. Three different methods for this analysis are used and compared. The complexity of such analysis is reduced by new techniques. Strong existence results are obtained by weaker conditions. The uniqueness and the continuous dependency of the solution on initial values are investigated using the Gronwall inequality. [ABSTRACT FROM AUTHOR]

The stability, robustness, accuracy, and efficiency of space-time finite element methods crucially depend on the choice of approximation spaces for test and trial functions. This is especially true for high-order, mixed finite element methods which often must satisfy an inf-sup condition in order to ensure stability. With this in mind, the primary objective of this paper and a companion paper is to provide a wide range of explicitly stated, conforming, finite element spaces in four dimensions. In this paper, we construct explicit high-order conforming finite elements on 4-cubes (tesseracts); our construction uses tools from the recently developed 'Finite Element Exterior Calculus'. With a focus on practical implementation, we provide details including Piola-type transformations, and explicit expressions for the volumetric, facet, face, edge, and vertex degrees of freedom. In addition, we establish important theoretical properties, such as the exactness of the finite element sequences, and the unisolvence of the degrees of freedom. [ABSTRACT FROM AUTHOR]

The goal of the paper is to establish quadrature formulas on combinatorial graphs. Three types of quadrature formulas are developed. Quadrature formulas of the first type are obtained through interpolation by variational splines. This set of formulas is exact on spaces of variational splines on graphs. Since bandlimited functions can be obtained as limits of variational splines we obtain quadrature formulas which are approximately exact on spaces of bandlimited functions. Accuracy of this type of quadrature formulas is given in terms of geometry of the set of nodes of splines and in terms of smoothness of functions which is measured by means of the combinatorial Laplace operator. Quadrature formulas of the second type are obtained through point-wise sampling for bandlimited functions and based on existence of certain frames in appropriate subspaces of bandlimited functions. The third type quadrature formulas are based on the average sampling over subgraphs. Our quadrature formulas which are based on sampling are exact on a relevant subspaces of bandlimited functions. The results of the paper have potential applications to problems that arise in data mining. [ABSTRACT FROM AUTHOR]

This paper demonstrates that the space of piecewise-smooth bivariate functions can be well-approximated by the space of the functions defined by a set of simple (non-linear) operations on smooth uniform tensor product splines. The examples include bivariate functions with jump discontinuities or normal discontinuities across curves, and even across more involved geometries such as a three-corner discontinuity. The provided data may be uniform or non-uniform, and noisy, and the approximation procedure involves non-linear least-squares minimization. Also included is a basic approximation theorem for functions with jump discontinuity across a smooth curve. [ABSTRACT FROM AUTHOR]

In this paper, we consider convolution operators, integral operators with weak singularity, Riesz potentials, in particular, those with kernels Ki (x, y) = xi−yi |x−y|n acting in special classes of Banach function spaces X (Ω) and their subspaces Xs (Ω)), and we prove some representation theorems for the functions from Banach-Sobolev spaces. We also prove the boundedness of Riesz potential in additive-invariant spaces. [ABSTRACT FROM AUTHOR]

In this paper, some concepts of convex analysis on hyperbolic spaces are studied. We first study properties of the intrinsic distance, for instance, we present the spectral decomposition of its Hessian. Next, we study the concept of convex sets and the intrinsic projection onto these sets. We also study the concept of convex functions and present first- and second-order characterizations of these functions, as well as some optimization concepts related to them. An extensive study of the hyperbolically convex quadratic functions is also presented. [ABSTRACT FROM AUTHOR]