Annali di Matematica Pura ed Applicata最新文献

We prove various estimates for the asymptotics of counting functions associated to point sets of coherent frames and Riesz sequences. The obtained results recover the necessary density conditions for coherent frames and Riesz sequences for general unimodular amenable groups, while providing more precise estimates under additional localization conditions on the coherent system for groups of polynomial growth.

A tensor—meaning here a tensor field (,Theta ) of any type (p, q) on a manifold—may be called integrable if it is parallel relative to some torsion-free connection. We provide analytical and geometric characterizations of integrability for differential q-forms, (q=0,1,2,n-1,n) (in dimension n), vectors, bivectors, symmetric (,(2,0),) and (,(0,2),) tensors, as well as complex-diagonalizable and nilpotent tensors of type (,(1,1)). In most cases, integrability is equivalent to algebraic constancy of (,Theta ,) coupled with the vanishing of one or more suitably defined Nijenhuis-type tensors, depending on (,Theta ,) via a quasilinear first-order differential operator. For (,(p,q)=(1,1)), they include the ordinary Nijenhuis tensor.

We consider a unitarily invariant complete Nevanlinna–Pick kernel denoted by s and a commuting d-tuple of bounded operators (varvec{T}= (T_{1}, dots , T_{d})) satisfying a natural contractivity condition with respect to s. We associate with (varvec{T}) its curvature invariant which is a non-negative real number bounded above by the dimension of a defect space of (varvec{T}). The instrument that makes this possible is the characteristic function developed in Adv Math 426:109089, 2023, https://doi.org/10.1016/j.aim.2023.109089. We present an asymptotic formula for the curvature invariant. In the special case when (varvec{T}) is pure, we provide a notably simpler formula, revealing that in this instance, the curvature invariant is an integer. We further investigate its connection with an algebraic invariant known as fiber dimension. Moreover, we obtain a refined and simplified asymptotic formula for the curvature invariant of (varvec{T}) specifically when its characteristic function is a polynomial.

We consider the problem of sampling random supersingular elliptic curves over finite fields of cryptographic size (SRS problem). The currently best-known method combines the reduction of a suitable complex multiplication (CM) elliptic curve and a random walk over some supersingular isogeny graph. Unfortunately, this method is not suitable when the endomorphism ring of the generated curve needs to be hidden, like in some cryptographic applications. This motivates a stricter version of the SRS problem, requiring that the sampling algorithm gives no information about the endomorphism ring of the output curve (cSRS problem). In this work we formally define the SRS and cSRS problems, which are both of theoretical interest. We discuss the relevance of the two problems for cryptographic applications, and we provide a self-contained survey of the known approaches to solve them. Those for the cSRS problem have exponential complexity in the characteristic of the base finite field (since they require computing and finding roots of polynomials of large degree), leaving the problem open. In the second part of the paper, we propose and analyse some alternative techniques—based either on the Hasse invariant or division polynomials—and we explain the reasons why they do not readily lead to efficient cSRS algorithms, but they may open promising research directions.

In this paper, we investigate the dynamics of geophysically stratified water waves with constant vorticity near the equator. Firstly, we prove that the bounded solutions of the three-dimensional inviscid gravity water wave governed by the f-plane approximation equation are two-dimensional. Moreover, we establish the two-dimensional characteristics of bounded solutions for gravity-capillary water waves and gravity water waves with rigid lids. Secondly, we focus on rigid lid case of gravity and gravity-capillary stratified water waves controlled by the (beta )-plane approximation. Here, we find that the only flow with a constant vorticity vector is a stationary flow with a flat interface and a vanishing velocity field.

We study the boundary-value problem

which has a degeneracy or singularity arising from the nonnegative matrix weight (mathbb {M}(x)). A global Calderón-Zygmund estimate for the relative weight is established under minimal regularity requirements on the associated operator by proving that ( |nabla u(x)|^{p(x)}) is as integrable as ( |F(x)|^{p(x)}) in (L^{gamma }left( Omega , |mathbb {M}(x)|^{ gamma p(x) }dxright) ) for every (1<gamma <infty ), under the assumptions that the variable exponent p(x) has a small log-Hölder constant, (mathbb {M}(x)) has a small log-BMO semi-norm and that the boundary (partial Omega ) of the nonsmooth bounded domain (Omega ) is flat in the Reifenberg sense. Our work is a natural extension and outgrowth of the uniformly elliptic problem when the matrix (mathbb {M}(x)) is a constant matrix as in [1, 7] to the degenerate or singular one when a coefficient of the nonlinearity might goes to zero or (infty ).

Biconservative hypersurfaces are hypersurfaces which have conservative stress-energy tensor with respect to the bienergy, containing all minimal and constant mean curvature hypersurfaces. The purpose of this paper is to study biconservative hypersurfaces (M^n) with constant scalar curvature in a space form (N^{n+1}(c)). We prove that every biconservative hypersurface with constant scalar curvature in (N^4(c)) has constant mean curvature. Moreover, we prove that any biconservative hypersurface with constant scalar curvature in (N^5(c)) is ether an open part of a certain rotational hypersurface or a constant mean curvature hypersurface. These solve an open problem proposed recently by D. Fetcu and C. Oniciuc for (nle 4).

This paper is concerned with the following Hamilton system for the curl–curl operator

in a simply connected bounded Lipschitz domain (Omega subset {mathbb {R}}^3) with connected boundary, where (nabla times ) denotes the curl operator in ({mathbb {R}}^3) and (nu :partial Omega rightarrow {mathbb {R}}^3) is the exterior normal. By using some variational approaches inspired by Szulkin and Weth (J Funct Anal 257(12):3802–3822, 2009) and Bartsch and Mederski (Arch Ration Mech Anal 215(1):283–306, 2015), we show that there exists a ground state solution for the above system if (f_1) and (f_2) are both subcritical and satisfy some other growth conditions and convexity conditions. Furthermore, if the nonlinearities are both even, we establish the existence of infinitely many solutions. Finally, we prove the existence of two types of cylindrically symmetric solutions under some symmetry conditions on the domain and the nonlinearities.

We prove that the depth formula holds for two finitely generated Tor-independent modules over Cohen–Macaulay local rings if one of the modules considered has finite reducing projective dimension (for example, if it has finite projective dimension, or the ring is a complete intersection). This generalizes a result of Bergh–Jorgensen which shows that the depth formula holds for two finitely generated Tor-independent modules over Cohen–Macaulay local rings if one of the modules considered has reducible complexity and certain additional conditions hold. Each module that has reducible complexity also has finite complexity and finite reducing projective dimension, but not necessarily vice versa. So a new advantage we have is that, unlike modules of reducible complexity, Betti numbers of modules of finite reducing projective dimension can grow exponentially.

One of the most celebrated problems in Euclidean Harmonic analysis is the Carleson’s problem: determining the optimal regularity of the initial condition f of the Schrödinger equation given by

in terms of the index (alpha ) such that f belongs to the inhomogeneous Sobolev space (H^alpha ({mathbb {R}}^n)), so that the solution of the Schrödinger operator u converges pointwise to f, (displaystyle lim _{t rightarrow 0+} u(x,t)=f(x)), almost everywhere. In this article, we consider the Carleson’s problem for the Schrödinger equation with radial initial data on Damek-Ricci spaces and obtain the sharp bound up to the endpoint (alpha ge 1/4), which agrees with the classical Euclidean case.