Space Rn Research Articles

Characterizations of infinitesimal relative boundedness for higher order Schrödinger operators

Let m∈N and H=(−Δ)m/2+V be a higher order Schrödinger operators in the Euclidean space Rn with V∈Lloc1(Rn). In this paper, the authors characterize the infinitesimal relative boundedness and Trudinger's subordination for H on Lp(Rn) with p∈[1,∞), via the limit behavior of the family of operators {V(λ2−Δ)−m/2}λ∈(0,∞) and a generalized Kato-type class condition. The latter is weaker than the classical Kato class condition corresponding to the case p=1. All these characterizations are new even when H=−Δ+V is a second order Schrödinger operator.

Equivariant insight into hyperspaces of convex bodies

Let cc(Rn) denote the hyperspace of all nonempty compact convex subsets of the Euclidean space Rn endowed with the Hausdorff metric, and let cc(Bn)={A∈cc(Rn)|A⊂Bn}, where Bn is the closed unit ball of Rn. Let cb(Rn) be the subset of cc(Rn) consisting of all convex bodies, i.e., of sets A∈cc(Rn) with non-empty interior. In this survey article we reveal fundamental properties of the natural action Aff(n)↷cb(Rn) of the affine group that leads to structure results for cb(Rn) and for several important subsets of it. Orbit spaces of cb(Rn) and its subsets by some appropriate subgroups G<Aff(n) are homeomorphic to such important objects like the Hilbert cube or the Banach-Mazur compacta. Among other hyperspaces studied here are cw(Rn) - the hyperspace of convex bodies of constant width, Bp - the hyperspace of convex bodies with smooth boundary of positive Gaussian curvature, L(n) - the hyperspace of convex bodies for which the Euclidean unit ball is the Löwner ellipsoid, cˇ(n) - the hyperspace of convex sets for which the Euclidean unit ball is the Chebyshev ball, etc. The paper contains also several new results concerning equivariant extension properties of the hyperspaces in question, as well as the homeomorphism type of the hyperspace cb(Bn)=cb(Rn)∩cc(Bn) and its orbit spaces. Equivariant extension properties of these hyperspaces are applied to provide a very short proof of the classical Grünbaum Conjecture from Convex Geometry. Along the paper seventeen open problems are raised.

An invariant for affine maximal type equations

Let y:M→Rn+1 be a locally strongly convex hypersurface immersion of a smooth, connected manifold into the real affine space Rn+1, given as the graph of a smooth, strictly convex function xn+1=f(x1,...,xn) defined on a domain Ω⊂Rn. Considering the α-relative normalization of the graph of the convex function f, we will prove a Bernstein theorem for a class of nonlinear, fourth order partial differential equations of affine maximal type. As applications, we define an invariant of the equations and prove a rigidity result of the complete Tn-invariant Kähler metric on complex torus (C⁎)n with vanishing scalar curvature for n≤5.

Optimal Dynamic Production Planning for Supply Network with Random External and Internal Demands

This paper focuses on joint production/inventory optimization in single and multiple horizons, respectively, within a complicated supply network (CSN) consisting of firm nodes with coupled demands and firm nodes with coupled demands. We first formulate the single-epoch joint optimal output model by allowing the production of extra quantity for stock underage, considering the fixed costs incurred by having inventory over demand and shortfalls. Then, the multi-temporal dynamic joint production model is further investigated to deal with stochastic demand fluctuations among CSN nodes by constructing a dynamic input–output model. The K-convexity defined in Rn space is proved to obtain the optimal control strategy. According to physical flow links, all demands associated to the nodes of CSN are categorized into the inter-node demand inside CSN (intermediate demand) and external demand outside CSN (final demand). We exploit the meliorated input–output matrix to describe demand relations, building dynamic input–output models where demand fluctuates randomly in single-cycle CSN and finite multi-cycle CSN. The novel monocyclic and multicyclic dynamic models have been developed to minimize system-wide operational costs. Unlike existent literature, we consider fixed costs incurred by overdemand and underdemand inventory into system operational cost functions and then demonstrate the convexity of objective functions. The cost function with two fixed penalty costs due to excess and shortage of inventory is developed in a multicycle model, and the K-convexity defined in Rn is proved to find out the optimal strategy for joint dynamic production of CSNs in the case of multi-products and multicycles.

Results of existence and uniqueness for the Cauchy problem of semilinear heat equations on stratified Lie groups

The aim of this paper is to give existence and uniqueness results for solutions of the Cauchy problem for semilinear heat equations on stratified Lie groups G with the homogeneous dimension N. We consider the nonlinear function behaves like |u|α or |u|α−1u(α>1) and the initial data u0 belongs to the Sobolev spaces Lsp(G) for 1<p<∞ and 0<s<N/p. Since stratified Lie groups G include the Euclidean space Rn as an example, our results are an extension of the existence and uniqueness results obtained by F. Ribaud on Rn to G. It should be noted that our proof is very different from it given by Ribaud on Rn. We adopt the generalized fractional chain rule on G to obtain the estimate for the nonlinear term, which is very different from the paracomposition technique adopted by Ribaud on Rn. By using the generalized fractional chain rule on G, we can avoid the discussion of Fourier analysis on G and make the proof more simple.

Large Deviations of Invariant Measures of Stochastic Reaction–Diffusion Equations on Unbounded Domains

This paper is concerned with the large deviation principle of invariant measures of the stochastic reaction–diffusion equation with polynomial drift driven by additive noise defined on the entire space Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document}. Since the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not compact and the spectrum of the Laplace operator on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} are not discrete, there are many issues for proving the large deviations of invariant measures in the case of unbounded domains, including the difficulties for proving the compactness of the level sets of rate functions, the uniform Dembo–Zeitouni large deviations on compact sets as well as the exponential tightness on compact sets. Currently, there is no result available in the literature on the large deviations of invariant measures for stochastic PDEs on unbounded domains, and this paper is the first one to deal with this issue. The non-compactness of the standard Sobolev embeddings on Rn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^n$$\\end{document} is circumvented by the idea of uniform tail-ends estimates together with the arguments of weighted spaces.

The Approximate Integrals by the Reproducing Kernel Method on the Elliptical Region Eₙ In the Space Rⁿ

In this paper, we applied the reproducing kernel method on elliptical region, we establish the formulas of the reproducing kernel from several degrees and generalized those formulas whatever the dimension of space, we obtained all surfaces, points, and their constants for compensation in the cubature formula, tables are included to show our results and examples to show the approximation solution in this method.

Three-dimensional quasi-umbilical locally strongly convex homogeneous affine hypersurfaces

The study of strongly convex, locally homogeneous hypersurfaces in the affine space ℝn+1 has a rich history. In dimension 2, these hypersurfaces were first examined in Guggenheimer's 1977 book “Differential Geometry” [4], and the classification was completed by Nomizu and Sasaki in their 1991 paper “A new model of unimodular-affinely homogeneous surfaces” [6]. Apart from the work of Magid and Vrancken in their 1995 paper [5] and Ooguri’s 2013 paper [9], most studies have focused on locally strongly convex affine hypersurfaces. These are characterized by an induced affine metric that is positive definite, and have been extensively studied in Sasaki’s 1980 paper “Hyperbolic affine hyperspheres” [10]. When the hypersurface is an affine sphere, where the shape operator has all eigenvalues equal to zero, Dillen and Vrancken provided a complete classification in a series of papers. In dimension 3, their 1993 paper “The classification of 3-dimensional locally strongly convex homogeneous affine hypersurfaces” [3] offers a comprehensive classification for any dimensions of eigenvalues of the shape operator. In dimension 4, the quasi-umbilical case was explored by Dillen and Vrancken in their 1994 paper “Quasi-umbilical, locally strongly convex homogeneous affine hypersurfaces” [2]. The case where the affine shape operator has two distinct eigenvalues, each with a multiplicity of two, was investigated by Chikh-Salah and Vrancken in their 2017 paper “Four-dimensional locally strongly convex homogeneous affine hypersurfaces” [1]. In this paper, we provide a different proof of the final theorem in Dillen and Vrancken’s [3], utilizing Mathematica software to calculate all Christoffel symbol functions.

Local and global existence for the Ericksen - Leslie problem in unbounded domains

The work deals with the Ericksen-Leslie System for nematic liquid crystals on the space RN with N≥3, on R+3 and on Ω⊆R3 exterior domain with sufficiently smooth boundary. The crystal orientation is described by unit vector v that is a small perturbation of a fixed constant vector η. We prove through a combination of energy method with dispersive a priori estimates a local existence and global existence for small initial data by a contraction argument. In particular, we obtain the following regularity of the liquid velocity u and of the crystal orientation vu∈L∞((0,T);Hs(Ω)),∇u∈L2((0,T);Hs(Ω)),∇v∈L∞((0,T);Hs(Ω)),∇2v∈L2((0,T);Hs(Ω)) for s>N2−1 if Ω=RN and s∈(12,1] if Ω=R+3 or in the exterior case, asking low regularity assumptions on u0 and v0.

Asymptotic behavior for Kirchhoff type stochastic plate equations on unbounded domains

This paper is concerned with the asymptotic behaviour of solutions to a class of Kirchhoff type stochastic nonlinear plate equations with dispersive and viscosity dissipative terms driven additive noise defined on the entire space Rn. The existence, uniqueness as well as upper semi-continuity of pullback random attractors are proved in H2(Rn)×H1(Rn).

Multiple standing waves of matrix nonlinear Schrödinger equations with mixed growth nonlinearities in [formula omitted]

In the whole space RN, we study the existence of two standing waves for a class of matrix nonlinear Schrödinger equations with potentials by using variational methods, where the nonlinearities are sublinear or asymptotically linear at infinity. The novelties are as follows. (1) The matrix nonlinear equations are defined in the whole space RN. (2) The nonlinearities are composed of two mixed nonlinear terms with different growth conditions. (3) The weight function may be sign-changing. (4) Our results can be applied to many examples.

Existence Results Related to a Singular Fractional Double-Phase Problem in the Whole Space

In this paper, we will study a singular problem involving the fractional (q1(x,.)-q2(x,.))-Laplacian operator in the whole space RN,(N≥2). More precisely, we combine the variational method with monotonicity arguments to prove that the associated functional energy admits a critical point, which is a weak solution for such a problem.

A higher dimensional Marcinkiewicz exponent and the Riemann boundary value problems for polymonogenic functions on fractals domains

We use a high-dimensional version of the Marcinkiewicz exponent, a metric characteristic for non-rectifiable plane curves, to present a direct application to the solution of some kind of Riemann boundary value problems on fractal domains of Euclidean space Rn+1,n≥2 for Clifford algebra-valued polymonogenic functions with boundary data in classes of higher order Lipschitz functions. Sufficient conditions to guarantee the existence and uniqueness of solution to the problems are proved. To illustrate the delicate nature of this theory we described a class of hypersurfaces where the results are more refined than those that exist in literature.

Pursuit and Evasion Linear Differential Game Problems with Generalized Integral Constraints

In this paper, we study pursuit and evasion differential game problems of one pursuer/one evader and many pursuers/one evader, respectively, in the space Rn. In both problems, we obtain sufficient conditions that guarantee the completion of a pursuit and an evasion. We construct the players’ optimal strategies in both problems, and we estimate the possible distance that an evader can preserve from pursuers. Lastly, we illustrate our results via some numerical examples.

A characterisation of linear repetitivity for cut and project sets with general polytopal windows

The cut and project method is a central construction in the theory of Aperiodic Order for generating quasicrystals with pure point diffraction. Linear repetitivity (LR) is a form of ideal regularity of aperiodic patterns. Recently, Koivusalo and the present author characterised LR for cut and project sets with convex polytopal windows whose supporting hyperplanes are commensurate with the lattice, the weak homogeneity property. For such cut and project sets, we show that LR is equivalent to two properties. One is a low complexity condition, which may be determined from the cut and project data by calculating the ranks of the intersections of the projection of the lattice to the internal space with the subspaces parallel to the supporting hyperplanes of the window. The second condition is that the projection of the lattice to the internal space is Diophantine (or ‘badly approximable’), which loosely speaking means that the lattice points in the total space stay far from the physical space, relative to their norm. We review then extend these results to non-convex and disconnected polytopal windows, as well as windows with polytopal partitions producing cut and project sets of labelled points. Moreover, we obtain a complete characterisation of LR in the fully general case, where weak homogeneity is not assumed. Here, the Diophantine property must be replaced with an inhomogeneous analogue. We show that cut and project schemes with internal space isomorphic to Rn⊕G⊕Zr, for G finite Abelian, can, up to MLD equivalence, be reduced to ones with internal space Rn, so our results also cover cut and project sets of this form, such as the (generalised) Penrose tilings.

Periodic solutions for Boussinesq systems in weak-Morrey spaces

We prove the existence and polynomial stability of periodic mild solutions for Boussinesq systems in critical weak-Morrey spaces for dimension n⩾3. Those systems are derived via the Boussinesq approximation and describe the movement of an incompressible viscous fluid under natural convection filling the whole space Rn. Using certain dispersive and smoothing properties of heat semigroups on Morrey-Lorentz spaces as well as Yamazaki-type estimate on block spaces, we prove the existence of bounded mild solutions for the linear systems corresponding to the Boussinesq systems. Then, we establish a Massera-type theorem to obtain the existence and uniqueness of periodic solutions to corresponding linear systems on the half line time-axis by using a mean-ergodic method. Next, using fixed point arguments, we can pass from linear systems to prove the existence uniqueness and polynomial stability of such solutions for Boussinesq systems. Finally, we apply the results to Navier-Stokes equations.

Noncompact uniform universal approximation

The universal approximation theorem is generalised to uniform convergence on the (noncompact) input space Rn. 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 Nφl(Rn)¯ 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, Nφl(Rn)¯ 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 Nφl(Rn)¯ (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, Nφl(Rn)¯ (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 Nφ2(Rn)¯ is strictly bigger than Nφ1(Rn)¯.

A Liouville type theorem for a scaling invariant parabolic system with no gradient structure

We prove a Liouville-type theorem for semilinear parabolic systems of the form∂tui−Δui=∑j=1mβijuirujp−r,i=1,2,...,m in the whole space RN×R, where r>0, p>r+1 and (βij) is a real (m,m) symmetric matrix with nonnegative entries, positive on the diagonal. Due to the lack of gradient structure, a nontrivial modifications of the techniques of Gidas and Spruck and of Bidaut-Véron is introduced. We expect to see more applications of this method to other problems with no gradient structure.

Effect of local fractional derivatives on Riemann curvature tensor

In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated M− and V−fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space Rn that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of Rn endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.

Об одном приведении системы булевых уравнений к эквивалентной системе полиномиальных уравнений

In this paper, we study the problem of constructing a special continuation of a Boolean function to the entire space R𝑛, thanks to which, without adding any restrictions, asystem of Boolean equations is transformed into an equivalent system of polynomial equations. As a result of the study, for any Boolean function (𝑥1, 𝑥2, ..., 𝑥𝑛), a corresponding infinitely differentiable rational function (𝑥1, 𝑥2, ..., 𝑥𝑛) is constructed such that𝑓𝑠 (𝑥1, 𝑥2, ..., 𝑥𝑛) ∈ {0, 1} ⇐⇒𝑓𝑠 (𝑥1, 𝑥2, ..., 𝑥𝑛) ∈ {0, 1} ⇐⇒.).)𝑏𝑏1122𝑠𝑠1122{ (𝑥1, 𝑥2, ..., 𝑥𝑛) ∈ {0, 1}𝑛𝑓 (𝑥 , , ..., 𝑥𝑛) = (𝑥 , , ..., (𝑥 , , ..., 𝑥𝑛) = (𝑥 , , ..., 𝑓 (𝑥 , , ..., 𝑥𝑛) = (𝑥 , , ..., (𝑥 , , ..., 𝑥𝑛) = (𝑥 , , ..., Thanks to the constructed function (𝑥1, 𝑥2, ..., 𝑥𝑛), firstly, without adding any restrictions, an arbitrary system of Boolean equations is transformed into an equivalent system of rational equations, and secondly, the solution of the transformed an equivalent system of rational equations is reduced to the problem of numerical minimization of some infinitely differentiable target function, solved by optimization methods, and to an equivalent system of polynomial equations, solved and analyzed by the F4 algorithm.