Monatshefte für Mathematik最新文献

In this article, we investigate the fine-scale statistics of real-valued arithmetic sequences. In particular, we focus on real-valued vector sequences, generalizing previous works of Boca et al. and the first author on the local statistics of integer-valued and rational-valued vector sequences, respectively. As the main results, we prove the Poissonian behavior of the pair correlation function for certain classes of real-valued vector sequences. This is achieved by extrapolating conditions on the number of solutions of Diophantine inequalities using twisted moments of the Riemann zeta function. Later, we give concrete examples of sequences in this set-up where these conditions are satisfied.

In this paper, we focus on zero-filter limit problem for the Camassa-Holm equation in the more general Besov spaces. We prove that the solution of the Camassa-Holm equation converges strongly in (L^infty (0,T;B^s_{2,r}(mathbb {R}))) to the inviscid Burgers equation as the filter parameter (alpha ) tends to zero with the given initial data (u_0in B^s_{2,r}(mathbb {R})). Moreover, we also show that the zero-filter limit for the Camassa-Holm equation does not converges uniformly with respect to the initial data in (B^s_{2,r}(mathbb {R})).

In the context of the initial data and an amplitude parameter (varepsilon ), we establish a local existence result for a highly nonlinear shallow water equation on the real line. This result holds in the space (H^k) as long as (k>5/2). Additionally, we illustrate that the threshold time for the occurrence of wave breaking in the surging type is on the order of (varepsilon ^{-1},) while plunging breakers do not manifest. Lastly, in accordance with ODE theory, it is demonstrated that there are no exact solitary wave solutions in the form of sech and (sech^2).

We study when Aron–Berner extensions of (separately) almost Dunford–Pettis multilinear operators between Banach lattices are (separately) almost Dunford–Pettis. For instance, for a (sigma )-Dedekind complete Banach lattice F containing a copy of (ell _infty ), we characterize the Banach lattices (E_1, ldots , E_m) for which every continuous m-linear operator from (E_1 times cdots times E_m) to F admits an almost Dunford–Pettis Aron–Berner extension. Illustrative examples are provided.

Let (Phi _i, Psi _i) be Young functions, (omega _i) be weights and (M^{Phi _i,Psi _i}_{omega _i}(mathbb {R} ^{d})) be the corresponding Orlicz modulation spaces for (i=1,2,3). We consider linear (respect. bilinear) multipliers on (mathbb {R} ^{d}), that is bounded measurable functions (m(xi )) (respect. (m(xi ,eta ))) on (mathbb {R} ^{d}) (respect. (mathbb {R} ^{2d})) such that

(respect.

define a bounded linear (respect. bilinear) operator from (M^{Phi _1,Psi _1}_{omega _1}(mathbb {R} ^{d})) to (M^{Phi _2,Psi _2}_{omega _2}(mathbb {R} ^{d})) (respect. (M^{Phi _1,Psi _1}_{omega _1}(mathbb {R} ^{d})times M^{Phi _2,Psi _2}_{omega _2}(mathbb {R} ^{d})) to (M^{Phi _3,Psi _3}_{omega _3}(mathbb {R} ^{d}))). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces.

Abstract

A node of a Sturm–Liouville problem is an interior zero of an eigenfunction. The aim of this paper is to present a simple and new proof of the result on sharp bounds of the node for the Sturm–Liouville equation with the Dirichlet boundary condition when the (L^1) norm of potentials is given. Based on the outer approximation method, we will reduce this infinite-dimensional optimization problem to the finite-dimensional optimization problem.

Abstract

Let (Omega subset mathbb {R}^2) and let (mathcal {L} subset Omega ) be a one-dimensional set with finite length (L =|mathcal {L}|) . We are interested in minimizers of an energy functional that measures the size of a set projected onto itself in all directions: we are thus asking for sets that see themselves as little as possible (suitably interpreted). Obvious minimizers of the functional are subsets of a straight line but this is only possible for (L le text{ diam }(Omega )) . The problem has an equivalent formulation: the expected number of intersections between a random line and (mathcal {L}) depends only on the length of (mathcal {L}) (Crofton’s formula). We are interested in sets (mathcal {L}) that minimize the variance of the expected number of intersections. We solve the problem for convex (Omega ) and slightly less than half of all values of L: there, a minimizing set is the union of copies of the boundary and a line segment.

This paper is concerned with the bounded solutions for a nonlinear second-order differential equation with asymptotic conditions and boundary condition which arise from the study of Arctic gyres. In the case of Lipschitz continuous nonlinearities, we prove the existence, uniqueness and stability of the bounded solution. An existence result for the general nonlinear vorticity term is also obtained.

In this paper, we prove the existence of a nonnegative solution to nonlinear parabolic problems with two absorption terms and a singular lower order term. More precisely, we analyze the interaction between the two absorption terms and the singular term to get a solution for the largest possible class of the data. Also, the regularizing effect of absorption terms on the regularity of the solution of the problem and its gradient is analyzed.