Search

We consider the problem of approximating Nash equilibria of N functions f 1 , ... , f N of N variables. In particular, we show systems of the form u ̇ j (t) = − ∇ x j f j (u (t)) (j = 1 , ... , N) are well-posed and the large time limits of their solutions u (t) = (u 1 (t) , ... , u N (t)) are Nash equilibria for f 1 , ... , f N provided that these functions satisfy an appropriate monotonicity condition. To this end, we will invoke the theory of maximal monotone operators on a Hilbert space. We will also identify an application of these ideas in game theory and show how to approximate equilibria in some game theoretic problems in function spaces. [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]

In this paper, we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure π σ β , 0 < β ≤ 1 , on the dual of Schwartz test function space ′ . The Hilbert space L 2 (π σ β) of complex-valued functions is described in terms of a system of generalized Appell polynomials ℙ σ , β , α associated to the measure π σ β . The kernels C n σ , β (⋅) , n ∈ ℕ 0 , of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system ℙ σ , β , α , there is a generalized dual Appell system ℚ σ , β , α that is biorthogonal to ℙ σ , β , α . The test and generalized function spaces associated to the measure π σ β are completely characterized using an integral transform as entire functions. [ABSTRACT FROM AUTHOR]

Assume Z F + A D. If is an ordinal and X is a set of ordinals, then [ X ] ∗ is the collection of order-preserving functions f : → X which have uniform cofinality ω and discontinuous everywhere. The weak partition properties on ω 1 and ω 2 yield partition measures on [ ω 1 ] ∗ when < ω 1 and [ ω 2 ] ∗ when < ω 2 . The following almost everywhere continuity properties for functions on partition spaces with respect to these partition measures will be shown. For every < ω 1 and function Φ : [ ω 1 ] → ω 1 , there is a club C ⊆ ω 1 and a ζ < so that for all f , g ∈ [ C ] ∗ , if f ↾ ζ = g ↾ ζ and sup (f) = sup (g) , then Φ (f) = Φ (g). For every < ω 2 and function Φ : [ ω 2 ] → ω 2 , there is an ω -club C ⊆ ω 2 and a ζ < so that for all f , g ∈ [ C ] ∗ , if f ↾ ζ = g ↾ ζ and sup (f) = sup (g) , then Φ (f) = Φ (g). The previous two continuity results will be used to distinguish the cardinalities of some important subsets of (ω 2) : | [ ω 1 ] ω | < | [ ω 1 ] < ω 1 |. | [ ω 2 ] ω | < | [ ω 2 ] < ω 1 | < | [ ω 2 ] ω 1 | < | [ ω 2 ] < ω 2 |. ¬ (| [ ω 1 ] < ω 1 | ≤ | [ ω 2 ] ω |). ¬ (| [ ω 1 ] ω 1 | ≤ | [ ω 2 ] < ω 1 |). It will also be shown that [ ω 1 ] ω has the Jónsson property: For every Φ : < ω ([ ω 1 ] ω) → [ ω 1 ] ω , there is an X ⊆ [ ω 1 ] ω with | X | = | [ ω 1 ] ω | so that Φ [ < ω X ] ≠ [ ω 1 ] ω . [ABSTRACT FROM AUTHOR]

This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains. [ABSTRACT FROM AUTHOR]

This paper aims to explore the optimal nonlinear approximation on the class of functions in Sobolev space in the probabilistic and average case settings. Specifically, we investigate the optimal approximation through the utilization of a set with finite pseudo-dimension, measured by the Kolmogorov probabilistic nonlinear (n , δ) -width (or say, the Kolmogorov probabilistic pseudo- (n , δ) -width). Furthermore, we provide an estimation of the exact asymptotic order of the Kolmogorov probabilistic nonlinear (n , δ) -width and p -average nonlinear n -width on the class of functions in Sobolev space. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of multiple solutions for a Schrödinger logarithmic equation of the form − 2 Δ u + V (x) u = u log u 2 , in ℝ N , u ∈ H 1 (ℝ N) , (P ) where V : ℝ N → ℝ is a continuous function that satisfies some technical conditions and is a positive parameter. We will establish the multiplicity of solution for (P ) by using the notion of Lusternik–Schnirelmann category, by introducing a new function space where the energy functional is C 1 . [ABSTRACT FROM AUTHOR]

We study nonstationary frames of matrix-valued Gabor systems and wavelet systems in the matrix-valued function space L 2 (ℝ , ℂ l × l). First, we show that a diagonal matrix-valued window function constitutes a frame for L 2 (ℝ , ℂ l × l) whenever each diagonal entry constitutes a frame for the space L 2 (ℝ). This is not true for arbitrary nonzero matrix-valued function. Using this, we prove the existence of nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group in terms of density of real numbers. We give a representation of the frame operator of matrix-valued nonstationary Gabor system. A necessary condition with explicit frame bounds for nonstationary matrix-valued Gabor frames associated with the Weyl–Heisenberg group is given. We discuss matrix-valued frame preserving maps in terms of adjointablity, with respect to the matrix-valued inner product, of bounded linear operators acting on L 2 (ℝ , ℂ l × l). It is shown that the image of a matrix-valued Gabor frame under bounded, linear and invertible operator on L 2 (ℝ , ℂ l × l) may not be a frame for L 2 (ℝ , ℂ l × l). In this direction, we give sufficient conditions on bounded linear operators which can preserve frame conditions. Finally, we give necessary and sufficient condition for the existence of nonstationary matrix-valued wavelet frames associated with the extended affine group. [ABSTRACT FROM AUTHOR]

It is known that a tempered exponential dichotomy (which is the version of nonuniform exponential dichotomy for random dynamical systems) can be described by admissibility of a pair of function classes defined in terms of Lyapunov norms. Moreover, for MET-systems (i.e. random dynamical systems which satisfy the assumptions of the Multiplicative Ergodic Theorem) over an ergodic base system, tempered exponential dichotomy can be described by admissibility of a pair of function spaces which are defined in terms of the original norm. In present paper, we find an admissibility description of tempered exponential dichotomies for MET-systems in terms of new output and input classes. These classes can be regarded as a critical case of admissibility classes in the known result. Finally, we use our new pair of admissible spaces to prove the roughness of tempered exponential dichotomies for parametric MET-systems and give a Hölder continuous dependence of the associated projections on the parameter. [ABSTRACT FROM AUTHOR]

We construct solvers for an isogeometric multi-patch discretization, where the patches are coupled via a discontinuous Galerkin approach, which allows for the consideration of discretizations that do not match on the interfaces. We solve the resulting linear system using a Dual-Primal IsogEometric Tearing and Interconnecting (IETI-DP) method. We are interested in solving the arising patch-local problems using iterative solvers since this allows for the reduction of the memory footprint. We solve the patch-local problems approximately using the Fast Diagonalization method, which is known to be robust in the grid size and the spline degree. To obtain the tensor structure needed for the application of the Fast Diagonalization method, we introduce an orthogonal splitting of the local function spaces. We present a convergence theory for two-dimensional problems that confirms that the condition number of the preconditioned system only grows poly-logarithmically with the grid size. The numerical experiments confirm this finding. Moreover, they show that the convergence of the overall solver only mildly depends on the spline degree. We observe a mild reduction of the computational times and a significant reduction of the memory requirements in comparison to standard IETI-DP solvers using sparse direct solvers for the local subproblems. Furthermore, the experiments indicate good scaling behavior on distributed memory machines. Additionally, we present an extension of the solver to three-dimensional problems and provide numerical experiments assessing good performance also in that setting. [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce a class of mixed-norm Herz spaces, Ė q → α → , p → (ℝ n) , which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on Ė q → α → , p → (ℝ n). Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on Ė q → α → , p → (ℝ n). As applications, the authors develop various real-variable theory of Hardy spaces associated with Ė q → α → , p → (ℝ n) by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of Ė q → α → , p → (ℝ n) and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem. [ABSTRACT FROM AUTHOR]

This paper extends the work of Fasshauer and Ye [Reproducing kernels of Sobolev spaces via a Green kernel approach with differential operators and boundary operators, Adv. Comput. Math. 38(4) (2011) 891921] in two different ways, namely, new kernels and associated native spaces are identified as crucial Hilbert spaces in applied mathematics. These spaces include the following spaces defined in bounded domains Ω ⊂ ℝ d with smooth boundary: homogeneous Sobolev–Slobodeckij̆ spaces, denoted by H 0 s (Ω ¯) , and Sobolev–Slobodeckij̆ spaces, denoted by H s (Ω) , where s > d 2 . Our goal is accomplished by obtaining the Green's solutions of equations involving the fractional Laplacian and fractional differential operators defined through interpolation theory. We provide a proof that the Green's kernels satisfying these problems are symmetric and positive definite reproducing kernels of H 0 s (Ω ¯) and H s (Ω) , respectively. Constructing kernels in these two ways enables the characterization of functions in native spaces based on their regularity. The Galerkin/collocation method, based on these kernels, is employed to solve various fractional problems, offering explicit or simplified calculations and efficient solutions. This method yields improved results with reduced computational costs, making it suitable for complex domains. [ABSTRACT FROM AUTHOR]

We study the peeling for the wave equation on the Vaidya spacetime following the approach developed by Mason and Nicolas in Mason–Nicolas 2009. The idea is to encode the regularity at null infinity of the rescaled field, characterized by Sobolev-type norms, in terms of corresponding function spaces of initial data. All function spaces are obtained from energy fluxes associated with an observer constructed from the Morawetz vector field on Minkowski spacetime. We combine conformal techniques and energy estimates to obtain the optimal classes of initial data ensuring a given regularity of the rescaled field. The classes of data are equivalent to those obtained on Minkowski and Schwarzschild spacetimes in that they impose the same decay at infinity and regularity. [ABSTRACT FROM AUTHOR]

Molecular dynamics (MD) simulations have been extensively used to study protein dynamics and subsequently functions. However, MD simulations are often insufficient to explore adequate conformational space for protein functions within reachable timescales. Accordingly, many enhanced sampling methods, including variational autoencoder (VAE) based methods, have been developed to address this issue. The purpose of this study is to evaluate the feasibility of using VAE to assist in the exploration of protein conformational landscapes. Using three modeling systems, we showed that VAE could capture high-level hidden information which distinguishes protein conformations. These models could also be used to generate new physically plausible protein conformations for direct sampling in favorable conformational spaces. We also found that VAE worked better in interpolation than extrapolation and increasing latent space dimension could lead to a trade-off between performances and complexities. Molecular dynamics simulations are widely used to study protein dynamics and functions, but they can be limited in terms of efficiency in exploring protein conformational space. Variational autoencoder (VAE)-based methods could be used to enhance the sampling efficiency of protein conformations. Our study evaluated the feasibility of using VAE to assist in exploring protein conformational landscapes. Results showed that VAE can capture high-level information to distinguish protein conformations and generate new, physically plausible ones for direct sampling. Overall, VAE method performs better in interpolation than extrapolation, and increasing latent space dimension had a trade-off between performance and complexity. [ABSTRACT FROM AUTHOR]

In this paper, we make research on the fractal structure of the space of continuous functions and explore how the fractal dimension of continuous functions under certain operations changes. We prove that any nonzero real power and the logarithm of a positive continuous function can keep the fractal dimensions of bi-Lipschitz invariance closed. For continuous functions having finite zero points, the relationship between its global behavior and the local behavior of its square on zero points has been given. Further, we discuss the fractal dimension of the product of continuous functions and provide the product decomposition of a continuous function in terms of the lower and upper box dimensions. Some special properties of the space of one-dimensional continuous functions have also been shown. [ABSTRACT FROM AUTHOR]

We present an extensive study on boundary layer mesh resolution in flow computation with the Space–Time Variational Multiscale (ST-VMS) method and isogeometric discretization. The study is in the context of 2D flow past a circular cylinder, at Reynolds number ranging from 1 0 2 to 1 0 6 . It was motivated by the need to have in tire aerodynamics a better understanding of the mesh resolution requirements near the tire surface. The focus in the study is on the normal-direction element length for the first layer of elements near the cylinder, with that length varying by a refinement factor ranging from 2 to 40. The evaluation is based mostly on the velocity profile near the cylinder. As the element length for the first layer is varied, the element lengths for the other layers of the disk-shaped inner mesh are adjusted, with no increase in the number of elements for the refinement factors 2, 3, and 4, and with modest increases only in the radial direction for refinement factors beyond that. The computations are performed with quadratic NURBS basis functions in space and linear basis functions in time. The expressions for the stabilization parameters used in the ST-VMS and for the related local lengths scales are those targeting isogeometric discretization, introduced in recent years. The mesh resolution study is based mostly on the strong enforcement of the Dirichlet boundary conditions on the cylinder, but also includes some computations with the weakly-enforced conditions. We expect that the data generated and observations made will be helpful in setting proper near-surface mesh resolution in VMS-based computations with isogeometric discretization, not only for cylindrical shapes but also for comparable geometries. We furthermore expect that although the data generated and observations made are based on computations with nonmoving meshes, they will also be applicable to computations with moving meshes where the mesh around the solid surface rotates with the surface in the framework of the ST Slip Interface method. [ABSTRACT FROM AUTHOR]

In this paper, we explore the concept of dimension preserving approximation of continuous multivariate functions defined on the domain [ 0 , 1 ] q (= [ 0 , 1 ] × ⋯ × [ 0 , 1 ] (q-times) where q is a natural number). We establish a few well-known multivariate constrained approximation results in terms of dimension preserving approximants. In particular, we indicate the construction of multivariate dimension preserving approximants using the concept of α -fractal interpolation functions. We also prove the existence of one-sided approximation of multivariate function using fractal functions. Moreover, we provide an upper bound for the fractal dimension of the graph of the α -fractal function. Further, we study the approximation aspects of α -fractal functions and establish the existence of the Schauder basis consisting of multivariate fractal functions for the space of all real valued continuous functions defined on [ 0 , 1 ] q and prove the existence of multivariate fractal polynomials for the approximation. [ABSTRACT FROM AUTHOR]

We study compactness of embeddings of Sobolev-type spaces of arbitrary integer order into function spaces on domains in ℝ n with respect to upper Ahlfors regular measures ν , that is, Borel measures whose decay on balls is dominated by a power of their radius. Sobolev-type spaces as well as target spaces considered in this paper are built upon general rearrangement-invariant function norms. Several sufficient conditions for compactness are provided and these conditions are shown to be often also necessary, yielding sharp compactness results. It is noteworthy that the only connection between the measure ν and the compactness criteria is how fast the measure decays on balls. Applications to Sobolev-type spaces built upon Lorentz–Zygmund norms are also presented. [ABSTRACT FROM AUTHOR]

In this paper, we establish the validity of the so-called Bounded Approximation Property (BAP) for a comprehensive class of translation and modulation invariant Banach spaces (B, ∥ · ∥B) of tempered distributions on the Euclidean space ℝ d . In fact, such spaces have a double module structure, over some Beurling algebra with respect to convolution, and with respect to pointwise multiplication over some Fourier Beurling algebra. Combining this double module structure with functional analytic arguments which describe the approximation of convolution operators by discrete convolutions we are able to verify the BAP, in fact, for most cases even the Metric Approximation Property (MAP) for such Banach space. The family of spaces under consideration is very rich and contains virtually all the classical function spaces relevant for mathematical analysis, as long as the Schwartz space (ℝ d) is dense in (B, ∥ · ∥B). In particular, all the reflexive spaces in this family are included. Moreover, this family of Banach spaces is closed with respect to intersections, sums and various interpolation methods. [ABSTRACT FROM AUTHOR]

In this paper, we study exponential tractability of multivariate approximation problem for Hermite spaces of analytic functions over ℝ d in the worst case setting. The functions are characterized by the property that their Hermite coefficients decay exponentially fast. The considered algorithms based on tensor products of Gauss–Hermite rules for multivariate integration are constructive and use the class Λ std of only function evaluations. We give matching necessary and sufficient conditions for notions of EC-quasi-polynomial tractability and EC-uniform weak tractability in terms of two weight parameters of the problem. [ABSTRACT FROM AUTHOR]

We construct in a rigorous mathematical way interacting quantum field theories on a p -adic spacetime. The main result is the construction of a measure on a function space which allows a rigorous definition of the partition function. The advantage of the approach presented here is that all the perturbation calculations can be carried out in the standard way using functional derivatives, but in a mathematically rigorous way. [ABSTRACT FROM AUTHOR]

Measles is a highly transmissible disease in children around the world. According to the World Health Organization (WHO), 73% of deaths of children were due to measles in 2018. This study describes the physical solution of the SIQR model for measles spread under the effect of natural delay amongst different compartments. By three different numerical techniques, the efficacies of solutions of the underlying system have been compared and a clear preference of nonstandard finite-difference (NSFD) scheme over the rest has been established. It has also been observed, on principle, that the NSFD formulation recovers all the essential traits of a continuous model namely the boundedness, positivity and stability of equilibriums of populations. The numerical results have also been supported by a very strong classical analysis of the model where the existence of a solution vector in explicit subsets of the function spaces has been guaranteed which leads to optimization of fixed-point methods. [ABSTRACT FROM AUTHOR]

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied. [ABSTRACT FROM AUTHOR]

Using the method of Girsanov's transformation, we investigate Talagrand's quadratic transportation cost inequalities for the laws of the solutions of stochastic partial differential equations (SPDEs) with two reflection walls under the uniform norm on the continuous functions space. These equations are driven by fractional noises. [ABSTRACT FROM AUTHOR]

In this paper, using the method of blow-up analysis, the authors obtain an improved Trudinger–Moser inequality involving L p -norm (p > 1) and prove the existence of its extremal function on a closed Riemann surface (Σ , g) with the action of a finite isometric group G = { σ 1 , σ 2 , ... , σ N }. To be exact, let W 1 , 2 (Σ , g) be the usual Sobolev space, a function space ℋ G = { u ∈ W 1 , 2 (Σ , g) : ∫ Σ u d v g = 0 and u (σ i (x)) = u (x) , ∀ x ∈ Σ , σ i ∈ G } and ℓ = min x ∈ Σ I (x) , where I (x) stands for the number of all distinct points in the set G (x) = { σ 1 (x) , ... , σ N (x) }. Define λ p G = inf u ∈ ℋ G , u ≢ 0 ∥ ∇ g u ∥ 2 2 / ∥ u ∥ p 2 , where ∥ ⋅ ∥ p is the standard L p -norm on (Σ , g). Using blow-up analysis, we prove that if 0 ≤ α < λ p G , the supremum sup u ∈ ℋ G , ∥ ∇ g u ∥ 2 2 − α ∥ u ∥ p 2 ≤ 1 ∫ Σ e 4 π ℓ u 2 d v g < + ∞ , and this supremum can be attained; if α ≥ λ p G , the above supremum is infinite. This kind of inequality will play an important role in the study of prescribing Gaussian curvature problem and mean field equations. In particular, their result generalizes those of Chen [A Trudinger inequality on surfaces with conical singularities, Proc. Amer. Math. Soc. 108 (1990) 821–832], Yang [Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two, J. Differential Equations 258 (2015) 3161–3193] and Fang–Yang [Trudinger–Moser inequalities on a closed Riemannian surface with the action of a finite isometric group, Ann. Sc. Norm. Super. Pisa Cl. Sci. 20 (2020) 1295–1324]. [ABSTRACT FROM AUTHOR]

Let X be a Banach space with a Schauder basis (x m) m = 0 ∞ , and I be the identity operator on X. It is known, at least in essence, that if (T m) m = 0 ∞ is a sequence of bounded linear operators on X such that ∑ m = 0 ∞ ∥ I − T m ∥ < ∞ , then (T m (x m)) m = 0 ∞ is also a basis. The first part of this work acts as an expository note to formally record the aforementioned stability result. In the second part, we apply this stability result to construct a Schauder basis consisting of bivariate fractal functions for the space of continuous functions defined on a rectangle. To this end, fractal perturbations of the elements in the classical bivariate Faber–Schauder system are formulated using a sequence of bounded linear fractal operators close to the identity operator in accordance with the stability result mentioned above. This illustration, although emphasized only for the bivariate case, can easily be extended to higher dimensions. Further, the perturbation technique used here acts as a companion for a few researches on fractal bases in the univariate setting. [ABSTRACT FROM AUTHOR]

In this paper, we make research on the approximation of functions with fractal dimension in continuous functions space. We first investigate fractal dimension of the linear combination of continuous functions with different fractal dimensions. Then, fractal winding of continuous functions has been given. Furthermore, based on Weierstrass theorem and Weierstrass function, we establish a theorem that continuous functions with the non-integer fractal dimension can be approximated by the linear combination of trigonometric polynomials with the same fractal dimension. Condition about continuous functions with fractal dimension one and two has also been discussed elementary. [ABSTRACT FROM AUTHOR]

This work aims to prove a Hardy-type inequality and a trace theorem for a class of function spaces on smooth domains with a nonlocal character. Functions in these spaces are allowed to be as rough as an L p -function inside the domain of definition but as smooth as a W s , p -function near the boundary. This feature is captured by a norm that is characterized by a nonlocal interaction kernel defined heterogeneously with a special localization feature on the boundary. Thus, the trace theorem we obtain here can be viewed as an improvement and refinement of the classical trace theorem for fractional Sobolev spaces W s , p (Ω). Similarly, the Hardy-type inequalities we establish for functions that vanish on the boundary show that functions in this generalized space have the same decay rate to the boundary as functions in the smaller space W s , p (Ω). The results we prove extend existing results shown in the Hilbert space setting with p = 2. A Poincaré-type inequality we establish for the function space under consideration together with the new trace theorem allows formulating and proving well-posedness of a nonlinear nonlocal variational problem with conventional local boundary condition. [ABSTRACT FROM AUTHOR]

In this paper, we construct explicitly the intertwining differential operators for the Jacobi algebra 2 . For the construction, we use the singular vectors of the Verma modules over 2 which we have constructed earlier. We construct the function spaces on which the operators act. We find two versions of the left (representation) action and the right action. These actions are combined with the singular vectors to provide the intertwining differential operators. [ABSTRACT FROM AUTHOR]

Let L be a simplicial complex. In this paper, we study random sub-hypergraphs and random sub-complexes of L. By considering the minimal complex that a sub-hypergraph can be embedded in and the maximal complex that can be embedded in a sub-hypergraph, we define some maps on the space of probability functions on sub-hypergraphs of L. We study the compositions of these maps as well as their actions on the space of probability functions. [ABSTRACT FROM AUTHOR]

This paper addresses the numerical solution of nonlinear time-fractional Fisher equations via local meshless method combined with explicit difference scheme. This procedure uses radial basis functions to compute space derivatives while Caputo derivative scheme utilizes for time-fractional integration to semi-discretize the model equations. To assess the accuracy, maximum error norm is used. In order to validate the proposed method, some non-rectangular domains are also considered. [ABSTRACT FROM AUTHOR]

The topology of the moduli space for Lamé functions of degree m is determined: this is a Riemann surface which consists of two connected components when m ≥ 2 ; we find the Euler characteristics and genera of these components. As a corollary we prove a conjecture of Maier on degrees of Cohn's polynomials. These results are obtained with the help of a geometric description of these Riemann surfaces, as quotients of the moduli spaces for certain singular flat triangles. An application is given to the study of metrics of constant positive curvature with one conic singularity with the angle 2 π (2 m + 1) on a torus. We show that the degeneration locus of such metrics is a union of smooth analytic curves and we enumerate these curves. [ABSTRACT FROM AUTHOR]

We consider a population structured by a space variable and a phenotypical trait, submitted to dispersion, mutations, growth and nonlocal competition. This population is facing an environmental gradient: to survive at location x , an individual must have a trait close to some optimal trait y opt (x). Our main focus is to understand the effect of a nonlinear environmental gradient. We thus consider a nonlocal parabolic equation for the distribution of the population, with y opt (x) = (x) , 0 < | | ≪ 1. We construct steady states solutions and, when is periodic, pulsating fronts. This requires the combination of rigorous perturbation techniques based on a careful application of the implicit function theorem in rather intricate function spaces. To deal with the phenotypic trait variable y we take advantage of a Hilbert basis of L 2 (ℝ) made of eigenfunctions of an underlying Schrödinger operator, whereas to deal with the space variable x we use the Fourier series expansions. Our mathematical analysis reveals, in particular, how both the steady states solutions and the fronts (speed and profile) are distorted by the nonlinear environmental gradient, which are important biological insights. [ABSTRACT FROM AUTHOR]

The algebraic properties of a strict deformation quantization are analyzed on the classical phase space ℝ 2 n . The corresponding quantization maps enable us to take the limit for ℏ → 0 of a suitable sequence of algebraic vector states induced by ℏ -dependent eigenvectors of several quantum models, in which the sequence converges to a probability measure on ℝ 2 n , defining a classical algebraic state. The observables are here represented in terms of a Berezin quantization map which associates classical observables (functions on the phase space) to quantum observables (elements of C ∗ algebras) parametrized by ℏ. The existence of this classical limit is in particular proved for ground states of a wide class of Schrödinger operators, where the classical limiting state is obtained in terms of a Haar integral. The support of the classical state (a probability measure on the phase space) is included in certain orbits in ℝ 2 n depending on the symmetry of the potential. In addition, since this C ∗ -algebraic approach allows for both quantum and classical theories, it is highly suitable to study the theoretical concept of spontaneous symmetry breaking (SSB) as an emergent phenomenon when passing from the quantum realm to the classical world by switching off ℏ. To this end, a detailed mathematical description is outlined and it is shown how this algebraic approach sheds new light on spontaneous symmetry breaking in several physical models. [ABSTRACT FROM AUTHOR]

Let α ∈ (0 , 1 ] , β ∈ [ 0 , n) and T Ω , β be a singular or fractional integral operator with homogeneous kernel Ω. In this paper, a CMO type space CMO α (ℝ n) is introduced and studied. In particular, the relationship between CMO α (ℝ n) and the Lipchitz space Lip α (ℝ n) is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator (T Ω , β) b m on weighted Lebesgue spaces via functions in L i p α (ℝ n) , and an equivalent characterization of the compactness for (T Ω , β) b m via functions in CMO α (ℝ n) are obtained. Some results are new even in the unweighted setting for the first-order commutators. [ABSTRACT FROM AUTHOR]

We introduce a category of fuzzy convergence groups, FCONVGRP a subcategory of the category of fuzzy convergence spaces, FCONV. Viewing I as a complete Heyting algebra, we prove that the category of I -tolerance groups, I -TOLGRP is isomorphic to a subcategory of FCONVGRP. Since FCONV is a topological universe, and thereby possesses function space structure, upon invoking this, we are able, among others, to show that FCONVGRP is topological, and more importantly, it enables us to obtain a compatible fuzzy convergence function space structure on group of homeomorphisms. It is noticeable, however, that the category of fuzzy neighborhood convergence groups, FNCONVGRP — a supercategory of the well-known category FNS, of fuzzy neighborhood spaces, as well as the category of fuzzy neighborhood groups, FNGRP — a subcategory of FNCONVGRP exhibit nice relationships with FCONVGRP. It is important to note that the objects of FCONVGRP are homogeneous, this paves the way to present two pertinent characterization theorems on fuzzy convergence groups. Finally, introducing a category PSTOPGRP, of pseudotopological groups, we reveal the embeddings of FTOPGRP and PSTOPGRP into FCONVGRP. [ABSTRACT FROM AUTHOR]

The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the "static slip models", there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan–Moore–Gibson–Thompson (JMGT) equation arising in acoustics as an alternative model to the well-known Kuznetsov equation. We show a local existence result in appropriate function spaces, and, using the energy method together with a bootstrap argument, we prove a global existence result for small data, without using the linear decay. Finally, polynomial decay rates in time for a norm related to the solution will be obtained. [ABSTRACT FROM AUTHOR]

The classical Aleksandrov–Bakel'man–Pucci estimate (ABP estimate) for a second-order elliptic operator in nondivergence form is one of the fundamental tools for the bounds of subsolutions. Cabre improved the ABP estimate by replacing a constant factor, the diameter of a given domain, with a geometric character, which can be defined and finite for some unbounded domains. In the proof, Cabre used the Krylov–Safonov boundary weak Harnack inequality from Trudinger; thus, it is required that the first-order coefficients belong to a Lebesgue 2 n -integrable function space. Using a growth lemma from Safonov and an approximation method, we improve the result to Lebesgue n -integrable first-order coefficients, which is optimal and coincides with the condition for the original ABP estimate. [ABSTRACT FROM AUTHOR]

Given N ≥ 3 , 1 < p < N , two measurable functions V (r) ≥ 0 and K (r) > 0 , and a continuous function A (r) > 0 (r > 0), we study the quasilinear elliptic equation − div (A (| x |) | ∇ u | p − 2 ∇ u) u + V (| x |) | u | p − 2 u = K (| x |) f (u) in ℝ N. We find existence of nonnegative solutions by the application of variational methods, for which we have to study the compactness of the embedding of a suitable function space X into the sum of Lebesgue spaces L K q 1 + L K q 2 , and thus into L K q (= L K q + L K q ) as a particular case. Our results do not require any compatibility between how the potentials A , V and K behave at the origin and at infinity, and essentially rely on power type estimates of the relative growth of V and K , not of the potentials separately. The nonlinearity f has a double-power behavior, whose standard example is f (t) = min { t q 1 − 1 , t q 2 − 1 } , recovering the usual case of a single-power behavior when q 1 = q 2 . [ABSTRACT FROM AUTHOR]

This work addresses several novel classes of convex function involving arbitrary non-negative function, which is known as approximately generalized (ψ , ℏ) -convex and approximately ψ -quasiconvex function, with respect to Raina's function, which are elaborated in Hilbert space. To ensure the feasibility of the proposed concept and with the discussion of special cases, it is presented that these classes generate other classes of generalized (ψ , ℏ) -convex functions such as higher-order strongly (HOS) generalized (ψ , ℏ) -convex functions and HOS generalized ψ -quasiconvex function. The core of the proposed method is a newly developed Simpson's type of identity in the settings of Riemann–Liouville fractional integral operator. Based on the HOS generalized (ψ , ℏ) -convex function representation, we established several theorems and related novel consequences. The presented results demonstrate better performance for HOS generalized ψ -quasiconvex functions where we can generate several other novel classes for convex functions that exist in the relative literature. Accordingly, the assortment in this study aims at presenting a direction in the related fields. [ABSTRACT FROM AUTHOR]

The expansion of both sides of Einstein's field equations in the weak-field approximation, up to terms of order 1 / c 4 is derived. This new approach leads to an extended form of gravitomagnetism (GEM) properly named as Beyond Gravitomagnetism (BGEM). The metric of BGEM includes a quadratic term in the gravitoelectric potential n the time and also space metric functions in contrast with first post-Newtonian 1 PN approximation where the quadratic term appears only in the time metric function. This nonlinear term does not appear in conventional GEM, but is essential in achieving the exact value of Mercury's perihelion advance as we explicitly show. The new BGEM metric is also applied to the classical problem of light deflection by the Sun, but the contribution of the new nonlinear terms produce higher-order terms in this problem and can be neglected, giving the correct result obtained already in the Lense–Thirring (GEM) approximation. The BGEM approximation also provides new terms that depend on the dynamics of the system, which may bring new insights into galactic and stellar physics. [ABSTRACT FROM AUTHOR]

This paper is concerned with an abstract inf-sup problem generated by a bilinear Lagrangian and convex constraints. We study the conditions that guarantee no gap between the inf-sup and related sup-inf problems. The key assumption introduced in the paper generalizes the well-known Babuška–Brezzi condition. It is based on an inf-sup condition defined for convex cones in function spaces. We also apply a regularization method convenient for solving the inf-sup problem and derive a computable majorant of the critical (inf-sup) value, which can be used in a posteriori error analysis of numerical results. Results obtained for the abstract problem are applied to continuum mechanics. In particular, examples of limit load problems and similar ones arising in classical plasticity, gradient plasticity and delamination are introduced. [ABSTRACT FROM AUTHOR]

Digital signals are often modeled as functions in Banach spaces, such as the ubiquitous Lp spaces. The frame theory in Banach spaces induces flexible representations of signals due to the robustness and redundancy of frames. Nevertheless, the lack of inner product in general Banach spaces limits the direct representations of signals in Banach spaces under a given basis or frame. In this paper, we introduce the concept of semi-inner product (SIP) (p, q)-Bessel multipliers to extend the flexibility of signal representations in separable Banach spaces, where 1/p + 1/q = 1, 1 < p < +∞. These multipliers are defined as composition of analysis operator of an SIP-I Bessel sequence, a multiplication with a fixed sequence and synthesis operator of an SIP-II Bessel sequence. The basic properties of the SIP (p, q)-Bessel multipliers are investigated. Moreover, as special cases, characterizations of q-Riesz bases related to signal representations are given, and the multipliers for q-Riesz bases are discussed. We show that SIP (p, q)-Bessel multipliers for q-Riesz bases are invertible. Finally, the continuity of SIP (p, q)-Bessel multipliers with respect to their parameters is investigated. The results theoretically show that the SIP (p, q)-Bessel multipliers offer a larger range of freedom than frames on signal representations in Banach spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the solution theory of the nonlinear q -fractional differential equation of Caputo type c D q α y (t) = f (t , y (t)) with given initial values D q k y (0 +) = d k , k = 0 , 1 , ... , n − 1 where α > 0 is the order, n = [ α ] and 0 < q < 1 is the scale index. For 0 ≤ β < α − n + 1 , by assuming that function t β f (t , y) is bounded and satisfies the Lipschitz condition on variable y , we prove that this problem admits a unique solution in the q -integrable function space 𝒟 q (n) (0 , b) and this solution is absolutely stable in the L ∞ -norm. This unique existence condition allows that f (t , y) is singular at t = 0 and discontinuous for t ∈ (0 , b ]. Finally, a successive approximation method is presented to find out the analytic approximation solution of this problem. [ABSTRACT FROM AUTHOR]

Based on the distributions space on 𝒮 ℂ ′ (denoted by ℱ 𝜃 ∗ ( 𝒮 ′ ℂ)) which is the topological dual space of the space of entire functions with exponential growth of order 𝜃 and of minimal type, we introduce a new type of differential equations using the Wick derivation operator and the Wick product of elements in ℱ 𝜃 ∗ ( 𝒮 ′ ℂ). These equations are called generalized Bernoulli Wick differential equations which are the analogue of the classical Bernoulli differential equations. We solve these generalized Wick differential equations. The present method is exemplified by several examples. [ABSTRACT FROM AUTHOR]

The special wettability of surfaces has attracted intensive research interests and leads to a rapidly developing field collectively known as the "superwettability" system. As a new member of this system, superwettability-based chemical processes advance the applications of superwettability from single surface functions to a broader space. This review briefly introduces the latest developments in this field, including superwettability-based miniature reactors, highly effective catalytic electrodes, gaseous reactant channels and solid/liquid/gas triphase edge-growth. The reaction types cover electrochemical reaction, polymerization, crystal precipitation and catalytic process. We believe there are abundant unimaginable possibilities along with challenges in this newly emerging promising field. We hope this review will inspire researchers with different backgrounds to use superwettability concept in their reaction systems flexibly and provide a new exploitation for the superwettability system. [ABSTRACT FROM AUTHOR]

We propose and study the framework of dissipative statistical solutions for the incompressible Euler equations. Statistical solutions are time-parameterized probability measures on the space of square-integrable functions, whose time-evolution is determined from the underlying Euler equations. We prove partial well-posedness results for dissipative statistical solutions and propose a Monte Carlo type algorithm, based on spectral viscosity spatial discretizations, to approximate them. Under verifiable hypotheses on the computations, we prove that the approximations converge to a statistical solution in a suitable topology. In particular, multi-point statistical quantities of interest converge on increasing resolution. We present several numerical experiments to illustrate the theory. [ABSTRACT FROM AUTHOR]

In the Dirac theory of the quantum-mechanical interaction of a magnetic monopole and an electric charge, the vector potential is singular from the origin to infinity along a certain direction — the so-called Dirac string. Imposing the famous quantization condition, the singular string attached to the monopole can be rotated arbitrarily by a gauge transformation, and hence is not physically observable. By deriving its analytical expression and analyzing its properties, we show that the gauge function χ (r) which rotates the string to another one is a smooth function everywhere in space, except their respective strings. On the strings, χ (r) is a multi-valued function. Consequently, some misunderstandings in the literature are clarified. [ABSTRACT FROM AUTHOR]

In this paper, regularity properties, Strichartz type estimates for solution of integral problem for linear and nonlinear abstract Schrödinger equations in vector-valued function spaces are obtained. The equation includes a linear operator A defined in a Banach space E , in which by choosing E and A we can obtain numerous classis of initial value problems for Schrödinger equations which occur in a wide variety of physical systems. [ABSTRACT FROM AUTHOR]