Mathematische Annalen最新文献

In this paper we investigate the Fourier coefficients of harmonic Maass forms of negative half-integral weight. We relate the algebraicity of these coefficients to the algebraicity of the coefficients of certain canonical meromorphic modular forms of positive even weight with poles at Heegner divisors. Moreover, we give an explicit formula for the coefficients of harmonic Maass forms in terms of periods of certain meromorphic modular forms with algebraic coefficients.

Recently Pawłucki showed that compact sets that are definable in some o-minimal structure admit triangulations of class ({{mathcal {C}}}^p) for each integer (pge 1). In this work, we make use of these new techniques of triangulation to show that all continuous definable maps between compact definable sets can be approximated by differentiable maps without changing their image. The argument is an interplay between o-minimal geometry and PL geometry and makes use of a ‘surjective definable version’ of the finite simplicial approximation theorem that we prove here.

It is known that the optimal Noether inequality ({text {vol} }(X) ge frac{4}{3}p_g(X) - frac{10}{3}) holds for every 3-fold X of general type with (p_g(X) ge 11). In this paper, we give a complete classification of 3-folds X of general type with (p_g(X) ge 11) satisfying the above equality by giving the explicit structure of a relative canonical model of X. This model coincides with the canonical model of X when (p_g(X) ge 23). We also establish the second and third optimal Noether inequalities for 3-folds X of general type with (p_g(X) ge 11). A novel phenomenon shows that there is a one-to-one correspondence between the three Noether inequalities and three possible residues of (p_g(X)) modulo 3. These results answer two open questions by Chen et al. (Duke Math J 169(9):1603–1164, 2020), and in dimension three an open question by Chen and Lai (Int J Math 31(1):2050005, 2020).

We obtain piecewise regularity results for linear elliptic systems with piecewise regular coefficients arising from composite materials. We find Hölder continuous functions related to the higher-order derivatives by compensating the possible discontinuity due to the geometry and the discontinuity of the coefficients. This leads to the piecewise regularity of the higher-order derivatives and solves an open problem introduced by Li and Vogelius, and Li and Nirenberg for general composite geometry.

Given (h,g in {mathbb {N}}), we write a set (X subset {mathbb {Z}}) to be a (B_{h}^{+}[g]) set if for any (n in {mathbb {Z}}), the number of solutions to the additive equation (n = x_1 + dots + x_h) with (x_1, dots , x_h in X) is at most g, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative (B_{h}^{times }[g]) set analogously. In this paper, we prove, amongst other results, that there exist absolute constants (g in {mathbb {N}}) and (delta >0) such that for any (h in {mathbb {N}}) and for any finite set A of integers, the largest (B_{h}^{+}[g]) set B inside A and the largest (B_{h}^{times }[g]) set C inside A satisfy

In fact, when (h=2), we may set (g = 31), and when h is sufficiently large, we may set (g = 1) and (delta gg (log log h)^{1/2 - o(1)}). The former makes progress towards a recent conjecture of Klurman–Pohoata and quantitatively strengthens previous work of Shkredov.

Bounds for the volume of sections of convex bodies which are in the (L_p) John ellipsoid positions are established. Specifically, when the convex bodies are in the LYZ ellipsoid position, we construct a set of Hanner polytopes attaining the sharp bounds.

This note aim to provide a deeper insight on article dealing with an inverse problem for biharmonic operator with second order perturbation. More precisely, we are referring to the paper An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator by Bhattacharyya and Ghosh An inverse problem on determining second order symmetric tensor for perturbed biharmonic operator, Math Ann (2021). Unfortunately, the paper contains some incorrect part. Indeed, a gap in the proof of the crucial proposition appears, whereas Proposition 3.6 in Bhattacharyya and Ghosh (Math Ann, 2021) is true only in a very specific case.

In this paper we consider a class of evolution operators with coefficients depending on time and space variables ((t,x) in {mathbb {T}}times {mathbb {R}}^n), where ({mathbb {T}}) is the one-dimensional torus, and prove necessary and sufficient conditions for their global solvability in (time-periodic) Gelfand–Shilov spaces. The argument of the proof is based on a characterization of these spaces in terms of the eigenfunction expansions given by a fixed self-adjoint, globally elliptic differential operator on ({mathbb {R}}^n).

It is shown using Schur complement techniques that on dimensional Hilbert spaces, a non-negative operator valued trigonometric polynomial in two variables with degree ((d_1,d_2)) can be written as a sum of hermitian squares of at most (2d_2) analytic polynomials.

Given a self-adjoint operator T on a separable infinite-dimensional Hilbert space we study the problem of characterizing the set ({mathcal {D}}(T)) of all possible diagonals of T. For operators T with at least two points in their essential spectrum (sigma _{ess}(T)), we give a complete characterization of ({mathcal {D}}(T)) for the class of self-adjoint operators sharing the same spectral measure as T with a possible exception of multiplicities of eigenvalues at the extreme points of (sigma _{ess}(T)). We also give a more precise description of ({mathcal {D}}(T)) for a fixed self-adjoint operator T, albeit modulo the kernel problem for special classes of operators. These classes consist of operators T for which an extreme point of the essential spectrum (sigma _{ess}(T)) is also an extreme point of the spectrum (sigma (T)). Our results generalize a characterization of diagonals of orthogonal projections by Kadison [38, 39], Blaschke-type results of Müller and Tomilov [51] and Loreaux and Weiss [48], and a characterization of diagonals of operators with finite spectrum by the authors [15].