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Abstract: Abstract We give a proof of the Lieb–Thirring inequality on the kinetic energy of orthonormal functions by using a microlocal technique, in which the uncertainty and exclusion principles are combined through the use of the Besicovitch covering lemma. PubDate: 2023-01-04

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Abstract: Abstract This paper is concerned with primal and dual second-order optimality conditions for the second-order strict efficiency of nonsmooth vector equilibrium problem with set, cone and equality conditions. First, we propose some second-order constraint qualifications via the second-order tangent sets. Second, we establish necessary optimality conditions of order two in terms of second-order contingent derivatives and second-order Shi sets for a second-order strict local Pareto minima to such problem under suitable assumptions on the second-order constraint qualifications. An application of the result for the twice Fréchet differentiable functions for the second-order local strict efficiency of that problem is also presented. Some illustrative examples are also provided for our findings. PubDate: 2022-12-17

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Abstract: Abstract Endpoint weak-type bounds and non-endpoint strong type bounds are obtained for the spherical maximal function in the setting of Choquet spaces with respect to certain Hausdorff contents or Sobolev capacities. PubDate: 2022-12-11

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Abstract: Abstract Partially saturated flow in a porous medium is typically modeled by the Richards equation, which is nonlinear, parabolic and possibly degenerated. This paper presents domain decomposition-based numerical schemes for the Richards equation, in which different time steps can be used in different subdomains. Two global-in-time domain decomposition methods are derived in mixed formulations: the first method is based on the physical transmission conditions and the second method is based on equivalent Robin transmission conditions. For each method, we use substructuring techniques to rewrite the original problem as a nonlinear problem defined on the space-time interfaces between the subdomains. Such a space-time interface problem is linearized using Newton’s method and then solved iteratively by GMRES; each GMRES iteration involves parallel solution of time-dependent problems in the subdomains. Numerical experiments in two dimensions are carried out to verify and compare the convergence and accuracy of the proposed methods with local time stepping. PubDate: 2022-12-07

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Abstract: Abstract The aim of this paper is to give some properties of Hilbert genus fields and construct the Hilbert genus fields of the fields \(L_{m,d}:=\mathbb {Q}(\zeta _{2^{m}},\sqrt {d})\) , where m ≥ 3 is a positive integer and d is a square-free integer whose prime divisors are congruent to ± 3 (mod 8) or 9 (mod 16). PubDate: 2022-12-02

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Abstract: Abstract Let R be a prime ring of char(R)≠ 2, U its Utumi ring of quotients and center C = Z(U) its extended centroid, I a both sided ideal of R, f(x1,…,xn) a multilinear polynomial over C, that is noncentral-valued on R, F, G be two generalized derivations of R and d be a derivation of R. Let f(I) be the set of all evaluations of the multilinear polynomial f(x1,…,xn) in I. If @@@ for all u ∈ f(I), then all possible forms of the maps are determined. As an application of this result, we also study the commutator identity [F2(u)u,G2(v)v] = 0 for all u,v ∈ f(I), where F and G are two generalized derivations of R. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00471-w

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Abstract: Abstract The aim of the paper is to investigate all transcendental entire solutions of the following general nonlinear equation of the form \(f^{n}+P_{d}(z,f)=p_{1}e^{\alpha _{1}z}+p_{2}e^{\alpha _{2}z}\) , where Pd(z,f) is a differential-difference polynomial in f of degree d. Our result is a generalization and complement of known results obtained by Liu-Mao, L \({\ddot {\mathrm {u}}}\) et al. and the references therein. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00464-9

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Abstract: Abstract In this paper, basing on the forward-backward method and inertial techniques, we introduce a new algorithm for solving a variational inequality problem over the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is established under strongly monotone and Lipschitz continuous assumptions imposed on the cost mapping. As an application, we also apply and analyze our algorithm to solve a convex minimization problem of the sum of two convex functions. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00467-6

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Abstract: Abstract Consider the ideal \((x_{1} , \dotsc , x_{n})^{d} \subseteq k[x_{1} , \dotsc , x_{n}]\) , where k is any field. This ideal can be resolved by both the L-complexes of Buchsbaum and Eisenbud, and the Eliahou-Kervaire resolution. Both of these complexes admit the structure of an associative DG algebra, and it is a question of Peeva as to whether these DG structures coincide in general. In this paper, we construct an isomorphism of complexes between the aforementioned complexes that is also an isomorphism of algebras with their respective products, thus giving an affirmative answer to Peeva’s question. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00474-7

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Abstract: Abstract The purpose of this paper is to introduce a new kind of q −Stancu-Kantorovich type operators and study its various approximation properties. We establish some local direct theorems, e.g., Voronovskaja type asymptotic theorem, global approximation and an estimate of error by means of the Lipschitz type maximal function and the Peetre K-functional. We also consider a n th-order generalization of these operators and study its approximation properties. Next, we define a bivariate case of these operators and investigate the order of convergence by means of moduli of continuity and the elements of Lipschitz class. Furthermore, we consider the associated Generalized Boolean Sum (GBS) operators and examine the approximation degree for functions in a Bögel space. Some numerical examples to illustrate the convergence of these operators to certain functions are also given. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00472-9

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Abstract: Abstract In 1980, White conjectured that the toric ideal of a matroid is generated by quadratic binomials corresponding to a symmetric exchange. In this paper, we compute Gröbner bases of toric ideals associated with matroids and show that, for every matroid on ground sets of size at most seven except for two matroids, Gröbner bases of toric ideals consist of quadratic binomials corresponding to a symmetric exchange. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00468-5

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Abstract: Abstract We give a proof of the combinatorial Brill-Noether conjecture for cactus graphs. This conjecture was formulated by Baker in 2008 when studying the interaction between algebraic curves theory and graph theory. By analyzing the treelike structure of cactus graphs, we produce a construction proof that is based on the Chip Firing Game theory. PubDate: 2022-12-01 DOI: 10.1007/s40306-021-00475-6

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Abstract: Abstract The corresponding Herz-type Hardy spaces to new weighted Herz spaces \(HK^{\beta ,p}_{\alpha ,q}\) associated with the Dunkl operator on \(\mathbb {R}\) have been characterized by atomic decompositions. Later a new characterization of \(HK^{\beta ,p}_{\alpha ,q}\) on the real line is introduced. This helped us in the work to characterize that the norms of the Herz-type Hardy spaces for the Dunkl Operator can be achieved by finite central atomic decomposition in some dense subspaces of them. Secondly, as an application we prove that a sublinear operator satisfying many conditions can be uniquely extended to a bounded operator in the Herz-type Hardy spaces for the Dunkl Operator. PubDate: 2022-11-05 DOI: 10.1007/s40306-022-00483-0

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Abstract: Abstract We consider the Dirichlet-to-Neumann operator in strip-like and half-space domains with Lipschitz boundary. It is shown that the quadratic form generated by the Dirichlet-to-Neumann operator controls some sharp homogeneous fractional Sobolev norms. As an application, we prove that the global Lipschitz solutions constructed in Dong et al. (2021) for the one-phase Muskat problem decays exponentially in time in any Hölder norm Cα, α ∈ (0,1). PubDate: 2022-11-03 DOI: 10.1007/s40306-022-00484-z

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Abstract: Abstract Inspired by the definition of balanced neighborly spheres, we define balanced neighborly polynomials and study the existence of these polynomials. The goal of this article is to construct balanced neighborly polynomials of type (k,k,k,k) over any field K for all k≠ 2, and show that a balanced neighborly polynomial of type (2,2,2,2) exists if and only if char(K)≠ 2. Besides, we also discuss a relation between balanced neighborly polynomials and balanced neighborly simplicial spheres. PubDate: 2022-11-03 DOI: 10.1007/s40306-022-00482-1

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Abstract: Abstract In this work, we show the existence and multiplicity for the nonlocal Lane-Emden system of the form $$ \begin{array}{@{}rcl@{}} \left\{ \begin{aligned} \mathbb L u &= v^{p} + \rho \nu \quad &&\text{in } {\varOmega}, \\ \mathbb L v &= u^{q} + \sigma \tau \quad &&\text{in } {\varOmega},\\ u&=v = 0 \quad &&\text{on } \partial {\varOmega} \text{ or in } {\varOmega}^{c} \text{ if applicable}, \end{aligned} \right. \end{array} $$ where \({\varOmega } \subset \mathbb {R}^{N}\) is a C2 bounded domain, \(\mathbb L\) is a nonlocal operator, ν,τ are Radon measures on Ω, p,q are positive exponents, and ρ,σ > 0 are positive parameters. Based on a fine analysis of the interaction between the Green kernel associated with \(\mathbb L\) , the source terms uq,vp and the measure data, we prove the existence of a positive minimal solution. Furthermore, by analyzing the geometry of Palais-Smale sequences in finite dimensional spaces given by the Galerkin type approximations and their appropriate uniform estimates, we establish the existence of a second positive solution, under a smallness condition on the positive parameters ρ,σ and superlinear growth conditions on source terms. The contribution of the paper lies on our unifying technique that is applicable to various types of local and nonlocal operators. PubDate: 2022-10-25 DOI: 10.1007/s40306-022-00485-y

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Abstract: Abstract We prove a regularity theorem for the solutions of the Donaldson geometric flow equation on the space of symplectic forms on a closed smooth four-manifold, representing a fixed cohomology class. The minimal initial conditions lay in the Besov space \(B^{1,p}_{2}(M, {\varLambda }^{2})\) for p > 4. The Donaldson geometric flow was introduced by Simon Donaldson in Donaldson (Asian J. Math. 3, 1–16 1999). For a detailed exposition see Krom and Salamon (J. Symplectic Geom. 17, 381–417 2019). PubDate: 2022-09-01 DOI: 10.1007/s40306-021-00454-x

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Abstract: Abstract Suppose F is a field with valuation v and valuation domain Ov, E/F is a finite-dimensional field extension, and R is an Ov-subalgebra of E such that F ⋅ R = E and R ∩ F = Ov. It is known that R satisfies LO, INC, GD and SGB over Ov; it is also known that under certain conditions R satisfies GU over Ov. In this paper, we present a necessary and sufficient condition for the existence of such R that does not satisfy GU over Ov. We also present an explicit example of such R that does not satisfy GU over Ov. PubDate: 2022-09-01 DOI: 10.1007/s40306-021-00461-y

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Abstract: Abstract We give an explicit description of all irreducible components and their dimensions of mixed commuting varieties over nilpotent 3 × 3 matrices, hence describing the varieties of 3-dimensional modules for certain quotients of polynomial algebras over an algebraically closed field. Our results also provide insights on support varieties of simple modules over Frobenius kernels of SL3. PubDate: 2022-09-01 DOI: 10.1007/s40306-021-00457-8

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Abstract: Abstract In this paper, we propose two new proximal point methods involving quasi-pseudocontractive mappings in Hadamard spaces. We prove that the first method converges strongly to a common solution of a finite family of minimization problems and fixed point problem for a finite family of quasi-pseudocontractive mappings in an Hadamard space. We then extend this method to a more general method involving multivalued monotone operators to approximate the solution of monotone inclusion problem, which is an important optimization problem. We establish that this method converges strongly to a common zero of a finite family of multivalued monotone operators which is also a common fixed point of a finite family of quasi-pseudocontractive mappings in an Hadamard space. Furthermore, we provide various nontrivial numerical implementations of our method in Hadamard spaces (which are non-Hilbert) and compare them with some other recent methods in the literature. PubDate: 2022-08-05 DOI: 10.1007/s40306-022-00480-3