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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]

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]

Using the institutional mapping approach, the present research has investigated the important actors of the science, technology and innovation system of Iran's cyber space and their functions in this system. The importance of this issue stems from the fact that the science, technology and innovation system of cyber space has a high number and diversity in actors and functions, and the effort to investigate them can help to better understanding the structural and functional complexities of this system. The data collection of this research has been done through a library study, including the review of previous studies and documents, including the description of the duties of organizations, statutes, laws, and upstream documents, as well as a field study, including holding a panel with key stakeholders. Classification of identified actors based on the theoretical framework in five levels of government, educational and research institutions, intermediate institutions, companies (large companies, small and medium businesses, startups and technological units), and non-governmental organizations (specialized association, trade association, unions, nongovernmental organization) was done. Then, through the actors-functions matrix, it was determined what functions each actor plays in the science, technology and innovation system of cyber space. According to the findings, the main guardian of the cyber space science, technology and innovation system is the Supreme Council of Cyber Space. Also, the Ministry of Communications and Information Technology has the largest role among other players in this system. The role of civil institutions and the private sector also needs to be highlighted in this system. The policy implication of this research is that the governance of science, technology and innovation system of cyber space requires a networked and evidence-based approach that, in addition to considering the views and expectations of all stakeholders, operates based on the facts and objective conditions. [ABSTRACT FROM AUTHOR]

One of the reasons why many neural networks are capable of replicating complicated tasks or functions is their universal approximation property. Though the past few decades have seen tremendous advances in theories of neural networks, a single constructive and elementary framework for neural network universality remains unavailable. This paper is an effort to provide a unified and constructive framework for the universality of a large class of activation functions including most of the existing ones. At the heart of the framework is the concept of neural network approximate identity (nAI). The main result is as follows: any nAI activation function is universal in the space of continuous functions on compacta. It turns out that most of the existing activation functions are nAI, and thus universal. The framework induces several advantages over the contemporary counterparts. First, it is constructive with elementary means from functional analysis, probability theory, and numerical analysis. Second, it is one of the first unified and constructive attempts that is valid for most of the existing activation functions. Third, it provides new proofs for most activation functions. Fourth, for a given activation and error tolerance, the framework provides precisely the architecture of the corresponding one-hidden neural network with a predetermined number of neurons and the values of weights/biases. Fifth, the framework allows us to abstractly present the first universal approximation with a favorable non-asymptotic rate. Sixth, our framework also provides insights into the developments, and hence providing constructive derivations, of some of the existing approaches. [ABSTRACT FROM AUTHOR]

Recent advances in operator learning theory have improved our knowledge about learning maps between infinite dimensional spaces. However, for large-scale engineering problems such as concurrent multiscale simulation for mechanical properties, the training cost for current deep learning methods is high and unscalable. The article presents a thorough analysis on the mathematical underpinnings of the operator learning paradigm and proposes a kernel learning method that maps between Banach spaces of functions. We first provide a survey of modern kernel and operator learning theory, as well as discuss recent results and open problems. From there, the article presents an algorithm to show we can analytically approximate the piecewise constant functions on R for operator learning. This implies the potential feasibility of success of neural operators on clustered functions, at least on the real line. Finally, a k-means clustered domain on the basis of a mechanistic response is considered and the Lippmann-Schwinger equation for micro-mechanical homogenization is solved. The article briefly discusses the mathematics of previous kernel learning methods and some preliminary results with those methods. The proposed kernel operator learning method uses graph kernel networks to come up with a mechanistic reduced order method for multiscale homogenization. [ABSTRACT FROM AUTHOR]

Motivation Advances in omics technologies have revolutionized cancer research by producing massive datasets. Common approaches to deciphering these complex data are by embedding algorithms of molecular interaction networks. These algorithms find a low-dimensional space in which similarities between the network nodes are best preserved. Currently available embedding approaches mine the gene embeddings directly to uncover new cancer-related knowledge. However, these gene-centric approaches produce incomplete knowledge, since they do not account for the functional implications of genomic alterations. We propose a new, function-centric perspective and approach, to complement the knowledge obtained from omic data. Results We introduce our Functional Mapping Matrix (FMM) to explore the functional organization of different tissue-specific and species-specific embedding spaces generated by a Non-negative Matrix Tri-Factorization algorithm. Also, we use our FMM to define the optimal dimensionality of these molecular interaction network embedding spaces. For this optimal dimensionality, we compare the FMMs of the most prevalent cancers in human to FMMs of their corresponding control tissues. We find that cancer alters the positions in the embedding space of cancer-related functions, while it keeps the positions of the noncancer-related ones. We exploit this spacial 'movement' to predict novel cancer-related functions. Finally, we predict novel cancer-related genes that the currently available methods for gene-centric analyses cannot identify; we validate these predictions by literature curation and retrospective analyses of patient survival data. Availability and implementation Data and source code can be accessed at https://github.com/gaiac/FMM. [ABSTRACT FROM AUTHOR]

• We formulated a new class of piecewise impulsive dynamic systems on arbitrary time domain by using the time scales theory. • Study the existence of solutions via fixed point theorems. Also establish the stability results of the considered problem. • Provide some necessary and sufficient conditions for the controllability of the piecewise linear system. • Develop a new class of control function and establish sufficient conditions for the controllability of nonlinear systems. • Some numerical simulations for different time domains are given to verify the proposed theoretical results. The main objective of this article is to establish the existence of solutions, stability, and controllability results for piecewise nonlinear impulsive dynamic systems on an arbitrary time domain. Using the Banach fixed point theorem, we prove the existence of a unique solution while by applying Schauder's fixed point theorem, we prove the existence of at least one solution. Further, we establish the controllability results by converting the controllability problem into a fixed point problem of an operator equation in some suitable function space. Mainly, we used the fixed point theorems, Gramian type matrices, functional analysis, and time scales theory to establish these results. Since the problem is formulated by using the theory of time scales, therefore, the obtained results are true for the continuous-time domain, discrete-time domain, as well as any combination of these two, which shows that the obtained results are non-trivial and generalize the existing ones. In the last, we have given a simulated example for two different time domains to verify the obtained analytical results. [ABSTRACT FROM AUTHOR]

The analysis of linear ill-posed problems often is carried out in function spaces using tools from functional analysis. However, the numerical solution of these problems typically is computed by first discretizing the problem and then applying tools from finite-dimensional linear algebra. The present paper explores the feasibility of applying the Chebfun package to solve ill-posed problems with a regularize-first approach numerically. This allows a user to work with functions instead of vectors and with integral operators instead of matrices. The solution process therefore is much closer to the analysis of ill-posed problems than standard linear algebra-based solution methods. Furthermore, the difficult process of explicitly choosing a suitable discretization is not required. [ABSTRACT FROM AUTHOR]

Linear ill‐posed problems often are analyzed in function spaces using tools from functional analysis, while their numerical solution typically is computed by first discretizing the problem and then applying tools from finite‐dimensional linear algebra. The Chebfun package makes it possible to solve linear ill‐posed problems without explicit discretization. This work discusses the use of Tikhonov regularization with a fairly general linear regularization operator within the Chebfun framework. [ABSTRACT FROM AUTHOR]

The main concern of this manuscript is to investigate some sufficient conditions under which the ABC-fractional neutral dynamic system with non-instantaneous impulsive conditions is controllable. Many physical problems require to control the state of the system not only at the final time of the interval but also at each of the impulses points, i.e., the so-called total controllability. In this regard, this is the first attempt to establish the total controllability results for ABC-fractional system. We also establish these results for the considered system with integral term. The basic technique of our approach is to reduce the controllability problem into a solvability problem of an operator equation in some suitable function space and then we prove the solvability results for the operator equations which in turn imply the controllability of the system. Semi-group theory, functional analysis, measure of non-compactness and Monch fixed point theorem have been used to establish these results. At last, an example is given to validate the obtained analytical outcomes. [ABSTRACT FROM AUTHOR]

Functional analysis aims to decompose the main functions of a product, which can be large and complex, into more manageable subfunctions. Recently, interest in explicitly considering users during functional modeling has grown to enhance analysis completeness, but how this addresses some commonly reported challenges for novice engineers is not clear. This research proposes and assesses a simple way to streamline user considerations in functional analysis as potential mitigation for some challenges (i.e., difficulty in identifying functions to explore design space and expressing the functions with expected syntax and solution neutrality) that novice designers frequently mention at the early design stage. Analysis of the results indicates that embedding requirements into a user workflow supports novice designers generate significantly more functions with correct syntax than starting with requirements only. In addition, the exploration space is prominently broader, especially at the higher levels measured by the geometry of the generated hierarchical models. These results suggest that strategically incorporating user considerations, even in a simple way, positively addresses the common challenges. This initial exploration and assessment could serve as an inspiration for future research on more efficient ways to streamline the functional decomposition process and, in turn, better support subsequent systematic conceptual design. [ABSTRACT FROM AUTHOR]

In this paper, we characterize spaces of L ∞ -functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work is analogous to established results concerning invariant spaces of continuous and measurable functions on a compact Hausdorff space. The case for L ∞ -functions cannot be proved in the same way when endowed with the norm-topology, but a similar argument can be used when the space of L ∞ -functions is given the weak*-topology, as we show in this paper. [ABSTRACT FROM AUTHOR]

In this paper, we would like to study the boundedness of operators of Hardy type on generalized central function spaces, such as the generalized central Hardy space HAφp (ℝn), the generalized central Morrey space Mφp (ℝn), and the generalized central Campanato space CMOφp (ℝn), with p ϵ (1, ∞), and φ (t) : (0, ∞) → (0,∞). We first show that HAφp' (ℝn) is the predual of CMOφp (ℝn). After that, we investigate the boundedness of operators of Hardy type on those spaces. By duality, we obtain the boundedness characterization of function b ϵCMOφp (ℝn) via the Mφp (ℝn) -boundedness of commutator [b, H*]. [ABSTRACT FROM AUTHOR]

We derive three generalized Gagliardo-Nirenberg inequalities in Lorentz, BMO and homogeneous Lipschitz spaces. They have the forms... whose parameters satisfy specific conditions. We use the so-called Muramatu's integral formula as the main approach throughout the paper. [ABSTRACT FROM AUTHOR]

The boundedness, compactness and essential norm of Volterra integral operator Vg from weighted Bergman spaces Aαp to general function spaces F(q, qt -- 2, s) are investigated in this paper. [ABSTRACT FROM AUTHOR]

Paraconvex multimaps are revisited in normed vector space setting. A parallel is provided with the studies conducted for real valued paraconvex functions on generalized convexities and monotonicities. Several characterizations are then obtained. The links with some generalized convexities for multimaps are examined and a first classification is achieved. In addition, two representation results for 2-paraconvex multimaps are given. [ABSTRACT FROM AUTHOR]

Simultaneously with the demographic and territorial expansion of urban areas, the role of big cities is becoming more important, as a result of the services they offer, following the trends of changes in the sphere of the economic sector. In the sphere of first-order administrativeterritorial units which have supported the change of the level of urban morphology, the Municipality of Constanța is considered one of them, being witness to a fluctuating development and strongly influencing the functioning processes of the space, including the dynamics of the urban process, which, in most cases, follow the trends of the changes that appeared as a result of the territorial administration policies. This study aims to analyze the functional character of Constanța Municipality, from the perspective of the main sectors of activity that certainly contribute to maintaining a constant balance, and the ways in which these aspects are reflected in the territory. This paper represents the product of bibliographic research activities as well as methods of graphical and statistical analysis in order to expose. The results of the study capture the evolution of the functional character of the main sectors of activity and how this process manifests itself in the territorial context of Constanța, following the stability of solutions for optimization that feasibly support the socio-economic balance of the area. [ABSTRACT FROM AUTHOR]

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space R n. All continuous functions that vanish at infinity can be uniformly approximated by neural networks with one hidden layer, for all activation functions φ that are continuous, nonpolynomial, and asymptotically polynomial at ± ∞. When φ is moreover bounded, we exactly determine which functions can be uniformly approximated by neural networks, with the following unexpected results. Let N φ l (R n) ¯ denote the vector space of functions that are uniformly approximable by neural networks with l hidden layers and n inputs. For all n and all l ≥ 2 , N φ l (R n) ¯ turns out to be an algebra under the pointwise product. If the left limit of φ differs from its right limit (for instance, when φ is sigmoidal) the algebra N φ l (R n) ¯ (l ≥ 2) is independent of φ and l , and equals the closed span of products of sigmoids composed with one-dimensional projections. If the left limit of φ equals its right limit, N φ l (R n) ¯ (l ≥ 1) equals the (real part of the) commutative resolvent algebra, a C*-algebra which is used in mathematical approaches to quantum theory. In the latter case, the algebra is independent of l ≥ 1 , whereas in the former case N φ 2 (R n) ¯ is strictly bigger than N φ 1 (R n) ¯. [ABSTRACT FROM AUTHOR]

We propose a nonparametric two-sample test procedure based on Maximum Mean Discrepancy (MMD) for testing the hypothesis that two samples of functions have the same underlying distribution, using kernels defined on function spaces. This construction is motivated by a scaling analysis of the efficiency of MMD-based tests for datasets of increasing dimension. Theoretical properties of kernels on function spaces and their associated MMD are established and employed to ascertain the efficacy of the newly proposed test, as well as to assess the effects of using functional reconstructions based on discretised function samples. The theoretical results are demonstrated over a range of synthetic and real world datasets. [ABSTRACT FROM AUTHOR]

The Baire category theorem has numerous applications in real and functional analysis. In this note, we discuss an interesting application in the context of metric spaces. We characterize all subsets of a general metric space that appear as the set of continuity points for functions on that space. Furthermore, we juxtapose our results with the Tietze extension theorem, which beautifully leads us in another direction towards the study of extensions of continuous functions. Most of the results we discuss here already exist scattered in the literature. In this note, we strive to present them in a coherent form. [ABSTRACT FROM AUTHOR]

In this paper, we study spaceability of subsets of generalized Orlicz and Lebesgue spaces associated to a Banach function space. Also, we give some sufficient conditions for spaceability of subsets of a general Banach space which improves an important result on this topic. As an application, it is shown that the set of all bounded linear operators which are not positive semidefinite on a separable Hilbert space is spaceable. [ABSTRACT FROM AUTHOR]

In this paper, we study a different class of bivariate α-fractal functions. First, we introduce the bivariate Bernstein α-fractal functions that are more suitable to approximate both smooth and non-smooth surfaces and investigate their convergence properties. Then, we compute the box-counting dimension of the graph of the bivariate α-fractal functions for equally spaced data set. In regard to the connection of functional analysis and fractal function, we cogitate the bivariate fractal operator in spaces of functions such as k-times continuously differentiable functions space C k (I × J) and the Lebesgue space L p (I × J). Also, we study some approximation properties using bivariate Bernstein α-fractal trigonometric functions. [ABSTRACT FROM AUTHOR]

In their 1976 paper, Nagel and Rudin characterize the closed unitarily and Möbius invariant spaces of continuous and L p -functions on a sphere, for 1 ≤ p < ∞ . In this paper we provide an analogous characterization for the weak*-closed unitarily and Möbius invariant spaces of L ∞ -functions on a sphere. We also investigate the weak*-closed unitarily and Möbius invariant algebras of L ∞ -functions on a sphere. [ABSTRACT FROM AUTHOR]

In this paper, we study a nonlinear integral equation that arises in a model of spatial logistic dynamics. The solvability of this equation is investigated by introducing special spaces of functions that are integrable up to a constant. Sufficient conditions for the biological characteristics and the parameters of the third spatial moment closure are established that guarantee the existence of the solution of the equation described above in some ball centered at zero. In addition, it is shown that this solution is unique in the considered ball and not zero. This means that, under appropriate conditions, the equilibrium state of the population of a certain species exists and does not coincide with the state of extinction. [ABSTRACT FROM AUTHOR]

In this paper, we characterize spaces of L∞-functions on a compact Hausdorff space that are invariant under a transitive and continuous group action. This work is analogous to established results concerning invariant spaces of continuous and measurable functions on a compact Hausdorff space. The case for L∞-functions cannot be proved in the same way when endowed with the norm-topology, but a similar argument can be used when the space of L∞-functions is given the weak*-topology, as we show in this paper.

We laconically describe the great contributions of Professor Francesco Altomare to mathematical research and Ph.D education, and his unique status in the mathematical community. In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory. Moreover, we report on and give concrete examples of his unique way to work together with Ph.D students, both before and sometimes also after their dissertation. Finally, we shortly describe his remarkable "class travel" from "simple" conditions with no academic traditions in his family in the small town Giovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today. [ABSTRACT FROM AUTHOR]

We consider Schmidt’s game on the space of compact subsets of a given metric space equipped with the Hausdorff metric, and the space of continuous functions equipped with the supremum norm. We are interested in determining the generic behavior of objects in a metric space, mostly in the context of fractal dimensions, and the notion of “generic” we adopt is that of being winning for Schmidt’s game.We find properties whose corresponding sets are winning for Schmidt’s game that are starkly different from previously established, and well-known, properties which are generic in other contexts, such as being residual or of full measure. [ABSTRACT FROM AUTHOR]

In this article, we introduce the space D([0; 1];D) of functions defined on [0; 1] with values in the Skorohod space D, which are right-continuous and have left limits with respect to the J1 topology. This space is equipped with the Skorohod-type distance introduced in [13]. Following the classical approach of [4, 5], we give several criteria for tightness of probability measures on this space, by char-acterizing the relatively compact subsets of this space. In particular, one of these criteria has been used in the recent article [1] for proving the existence of a D-valued α-stable Levy motion. Finally, we give a criterion for weak convergence of random elements in D([0; 1];D), and a criterion for the existence of a process with sample paths in D([0; 1];D) based on its finite-dimensional distributions. [ABSTRACT FROM AUTHOR]

• A kind of the petal-inspired space deployable-foldable mechanism is designed. • The dimension of the planar deployable unit is analyzed and the mechanism is constructed. • To realize the space folding function, the spatial deployable unit is designed. • The foldable mechanism configurations are synthesized and selected. • Two engineering application examples for the foldable mechanism are introduced. By applying bionics principles, a petal-inspired space deployable-foldable mechanism for use in space applications is proposed, based on features common to both the flower blooming process and deployable-foldable mechanisms. The mechanism is designed and analyzed according to the geometrical relationships and folded condition of configurations before and after folding of the deployable unit. Functional analysis is then performed to assess, step by step, the total function of the space deployable-foldable mechanism, which can be divided into the driving function, plane folding function, space folding function, and locking and unlocking function. The sub-functions are solved to obtain the morphological matrix through sorting. The deployable unit is carried out configuration synthesis by the metamorphic method of joint change. Using various prismatic pairs, multiple degrees-of-freedom joints, and locking-unlocking joints, the number of configuration of the space deployable-foldable mechanism is increased. Finally, screening methods are used to select five schemes and 36 configurations. The petal-inspired space deployable-foldable mechanism can be applied in space stations, mobile communications, environmental monitoring, and other engineering applications. [ABSTRACT FROM AUTHOR]

We laconically describe the contribution of Lars-Erik Persson to mathematical research and postgraduate studies. In particular, we describe and give examples of his innovative and great contributions to the following areas: Fourier analysis (or more generally harmonic analysis), interpolation theory, homogenization theory, Hardy type inequalities, convexity and inequalities, and applied mathematics together with a brief description of his historical and didactic related contributions. Moreover, we will describe and give concrete examples of his unique way to work together with PhD students both before and sometimes also after their exams. Finally, we will shortly describe his remarkable ''class travel" from a small village in north Sweden to finally become the broad, innovative and powerful mathematician he obviously is regarded to be today. We also refer to the article [A] about Lars-Erik as a great person and great mathematician, which was published in a journal issue dedicated to his 65th anniversary. We essentially update, complement and broaden the information given in [A] especially in the part concerning Lars-Erik Persson's contributions to mathematical research and PhD education. [ABSTRACT FROM AUTHOR]

We consider Hilbert spaces of analytic functions in the disk with a normalized reproducing kernel and such that the backward shift f (z) ↦ f (z) − f (0) z is a contraction on the space. We present a model for this operator and use it to prove the surprising result that functions which extend continuously to the closure of the disk are dense in the space. This has several applications, for example we can answer a question regarding reverse Carleson embeddings for these spaces. We also identify a large class of spaces which are similar to the de Branges–Rovnyak spaces and prove some results which are new even in the classical case. [ABSTRACT FROM AUTHOR]

This paper deals with bilinear operators acting in pairs of Banach function spaces that factor through the pointwise product. We find similar situations in different contexts of the functional analysis, including abstract vector lattices—orthosymmetric maps, C ∗ -algebras—zero product preserving operators, and classical and harmonic analysis—integral bilinear operators. Bringing together the ideas of these areas, we show new factorization theorems and characterizations by means of norm inequalities. The objective of the paper is to apply these tools to provide new descriptions of some classes of bilinear integral operators, and to obtain integral representations for abstract classes of bilinear maps satisfying certain domination properties. [ABSTRACT FROM AUTHOR]

In this paper, we propose a family of nonconforming finite elements for 2mth order partial differential equations in Rn on simplicial grids when m = n + 1. This family of nonconforming elements naturally extends the elements proposed by Wang and Xu [Math. Comp. 82 (2013), pp. 25-43], where m ≤ n is required. We prove the unisolvent property by induction on the dimensions using the similarity properties of both shape function spaces and degrees of freedom. The proposed elements have approximability, pass the generalized patch test, and hence converge. We also establish quasi-optimal error estimates in the broken H³ norm for the 2D nonconforming element. In addition, we propose an H³ nonconforming finite element that is robust for the sixth order singularly perturbed problems in 2D. These theoretical results are further validated by the numerical tests for the 2D triharmonic problem. [ABSTRACT FROM AUTHOR]

Abstract This article is concerned with an initial–boundary-value problem (IBVP) for the sixth order Boussinesq equation on a bounded domain with non-homogeneous boundary conditions, (1) u t t − u x x + β u x x x x − u x x x x x x + (u 2) x x = 0 , x ∈ (0 , 1) , t > 0 , u (x , 0) = φ (x) , u t (x , 0) = ψ (x) , u x (0 , t) = h 1 (t) , u x x x (0 , t) = h 2 (t) , u x x x x x (0 , t) = h 3 (t) , u x (1 , t) = h 4 (t) , u x x x (1 , t) = h 5 (t) , u x x x x x (1 , t) = h 6 (t) , where β = ± 1. For 0 ≤ s ≤ 6 , it is shown that the IBVP is locally well-posed if the initial data lie in the product of L 2 -based Sobolev spaces, H s (0 , 1) × H s − 3 (0 , 1) , provided the boundary data (h 1 , h 2 , h 3) and (h 4 , h 5 , h 6) lie in the product space, H l o c (s + 2) ∕ 3 (R +) × H l o c s ∕ 3 (R +) × H l o c (s − 2) ∕ 3 (R +). [ABSTRACT FROM AUTHOR]

In this paper, we investigate the boundedness and compactness of weighted composition operators from Zygmund-type spaces to weighted-type spaces and little weighted-type spaces in the unit ball of Cn. [ABSTRACT FROM AUTHOR]

In this article, we study an Euler-Bernoulli viscoelastic control system which is dissipative due to the presence of the viscoelastic term. The main feature which distinguishes this paper from other related works lies in the fact that we no longer impose traditional conditions such as complete monotonicity and decay property on the kernel function g. Without loss of generality, we study the system in the case of g ≡ 1. By means of the duality principle and the Hahn-Banach theorem, we show that the system with g = 1 is approximately controllable in the appropriate function space. [ABSTRACT FROM AUTHOR]

In this paper, we study translation invariant function spaces and spectral analysis on KPC-hypergroups and describe a correspondence between ideals in the algebra of compactly supported measures and varieties of continuous functions on a KPC-hypergroup. [ABSTRACT FROM AUTHOR]

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In this paper, we study the existence of infinitely many solutions of the following degenerate magnetic fractional problem involving critical Sobolev‐Hardy nonlinearities: M([u]s,A2)(−△)Asu=λf(x,|u|)u+|u|2s∗(α)−2u|x|α,inRN,where s ∈ (0,1), N > 2s, 2s∗(α)=2(N−α)N−2s is the fractional Hardy‐Sobolev critical exponent with α ∈ [0,2s), λ is a positive real parameter, M:R0+→R0+ is a Kirchhoff function, A:RN→RN is a magnetic potential function, and (−△)As is the fractional magnetic operator. By using the new version of symmetric mountain pass theorem of Kajikiya, we prove that the problem admits infinitely many solutions for the suitable value of λ. [ABSTRACT FROM AUTHOR]

Abstract In this paper we study optimal estimates for two functionals involving the anisotropic p -torsional rigidity T p (Ω) , 1 < p < + ∞. More precisely, we study Φ (Ω) = T p (Ω) | Ω | M (Ω) and Ψ (Ω) = T p (Ω) | Ω | [ R F (Ω) ] p p − 1 , where M (Ω) is the maximum of the torsion function u Ω and R F (Ω) is the anisotropic inradius of Ω. [ABSTRACT FROM AUTHOR]

Abstract We study the complexity of computing the norm ‖ f ‖ L q (Q) , where f is from the unit ball of a Sobolev space W p r (Q) and Q = [ 0 , 1 ] d is the unit cube. The deterministic case is a consequence of a general result by Wasilkowski. We consider the randomized setting with standard information and determine the order of the randomized n th minimal error. Furthermore, we discuss problems related to the arithmetic cost of the proposed algorithms and present modifications of nearly optimal arithmetic cost in the one-dimensional case. [ABSTRACT FROM AUTHOR]

We investigate the Henon type equation involving the critical Sobolev exponent with Dirichret boundary condition ..., in Ω included in a unit ball, under several conditions. Here, Ψ is a non-trivial given function with 0 ≤ Ψ ≤ 1 which may vanish on ∂Ω. Let λ1 be the first eigenvalue of the Dirichret eigenvalue problem -Δφ = λΨφ in Ω. We show that if the dimension N ≥ 4 and 0 < λ < λ1, there exists a positive solution for small α > 0. Our methods include the mountain pass theorem and the Talenti function. [ABSTRACT FROM AUTHOR]

In this paper, the existence of positive solutions in terms of different values of two parameters for a system of conformable-type fractional differential equations with the p-Laplacian operator is obtained via Guo-Krasnosel’skii fixed point theorem. [ABSTRACT FROM AUTHOR]