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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]

We show that the results of the paper mentioned in the title are known or follow from the known results. [ABSTRACT FROM AUTHOR]

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]

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]

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]

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 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]

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]

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]

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 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 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]

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]

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]

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]

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]

In this paper we characterize some basic properties of composition operators on the spaces of harmonic Bloch functions. First we provide some equivalent conditions for boundedness and compactness of composition operators. In the sequel we investigate closed range composition operators. These results extends the similar results that were proven for composition operators on the Bloch spaces. [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 response to the increasing building demands in Turkey, particularly in the metropolitan area of Istanbul, followed by other major cities such as Ankara and Izmir, the expansion of construction zones has led to the emergence of tall towers as a pragmatic solution. The design and implementation of tall buildings require newer technologies and interdisciplinary collaboration in aspects such as facade installation, vertical circulation solutions, and fire systems, compared to low-rise buildings. In spite of the proliferation of skyscrapers, there is a noticeable lack of thorough study on space efficiency in Turkey's tall buildings. This article aims to fill this significant gap in the literature. The research method employed in this study focuses on a case study of 54 modern towers constructed in Turkey between 2010 and 2023, ranging in height from 147 to 284 m. Key findings are as follows: (1) residential use, central core, and prismatic forms are the most prevalent architectural preferences; (2) the most preferred structural material and system are concrete and the shear-walled frame system, respectively; (3) average space efficiency and the percentage of core-to-gross-floor area (GFA) were 78% and 19%, respectively, with measurement ranges varying from a minimum of 64% and 9% to a maximum of 86% and 34%. This paper will provide insight for construction stakeholders, especially architects, for sound planning decisions in the development of Turkish tall buildings. [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]

Let (M , g) be a smooth compact Riemannian manifold of dimension n ≥ 3 . Let also A be a smooth symmetrical positive (0 , 2) -tensor field in M. By the Sobolev embedding theorem, we can write that there exist K , B > 0 such that for any u ∈ H 1 ⁢ (M) , (0.1) ∥ u ∥ L 2 ⋆ 2 ≤ K ⁢ ∥ ∇ A ⁡ u ∥ L 2 2 + B ⁢ ∥ u ∥ L 1 2 where H 1 ⁢ (M) is the standard Sobolev space of functions in L 2 with one derivative in L 2 , | ∇ A ⁡ u | 2 = A ⁢ (∇ ⁡ u , ∇ ⁡ u) and 2 ⋆ is the critical Sobolev exponent for H 1 . We compute in this paper the value of the best possible K in (0.1) and investigate the validity of the corresponding sharp inequality. [ABSTRACT FROM AUTHOR]

Some fundamental properties of the Moritoh wavelet are discussed in this paper. The Moritoh transform is approximated for ultra-distributions in generalised Sobolev space. The adjoint formula of the Fourier transform is extended to the Moritoh transform. The convolution for quaternion-valued functions is defined for a modified representation of quaternions. Furthermore, the quaternionic Moritoh transform is defined with the help of convolution. The inner product relation and the uncertainty principle are also established for the quaternionic Moritoh wavelet transform. [ABSTRACT FROM AUTHOR]

This paper focuses on exploring the existence of solutions for a specific class of FDEs by leveraging fixed point theorem. The equation in question features the Caputo fractional derivative of order 3 < u ˆ ≤ 4 and includes a term Θ (β , Z (β)) alongside boundary conditions. Through the application of a fixed point theorem in appropriate function spaces, we consider nonlocal conditions along with necessary assumptions under which solutions to the given FDE exist. Furthermore, we offer an example to illustrate the results. [ABSTRACT FROM AUTHOR]

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In this paper, we introduce a reconstructed weak Galerkin (RWG) finite element method to solve the Poisson equation, capitalizing on the weak Galerkin (WG) finite element method and the least squares reconstruction technique. We construct a piecewise high-degree polynomial function space using the least squares reconstruction technique. The reconstructed space has only one degree of freedom per element while achieving high-order approximation accuracy. The RWG method presents notable efficiency advantages over standard WG methods, attaining the same numerical accuracy with fewer degrees of freedom. Additionally, we have developed an efficient preconditioner based on piecewise constant functions for the RWG method and demonstrated that the condition number of the preconditioned system is independent of the mesh size. The accuracy and efficiency of the proposed method are validated through numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper, we characterize the boundedness and compactness of product type operators, including Stević-Sharma operator ..., from weak vector valued derivative Besov space wEβp X) into weak vector-valued Besov space wBβp(X). As an application, we obtain the boundedness and compactness characterizations of the weighted composition operator on the weak vector valued derivative Besov space. [ABSTRACT FROM AUTHOR]

It it well known that a Hilbert space V of functions defined on U is a reproducing kernel Hilbert space if and only if for any z ∈ U , in the set V z := { f ∈ V ∣ f ⁢ (z) = 1 } , if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality K ⁢ (z , w) = K ⁢ (w , z) ¯ does not have to be true without assumptions that K ⁢ (z , w) ≠ 0 , K ⁢ (w , z) ≠ 0 . We give sufficient and necessary conditions for a Banach space of functions to be a reproducing kernel Banach space. At the end, we give some examples including ones which show how reproducing kernel Banach spaces can be used to solve extremal problems of Partial Differential Equations Theory. [ABSTRACT FROM AUTHOR]

In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented. [ABSTRACT FROM AUTHOR]