Bulletin of the London Mathematical Society最新文献

It is known that a small spherical cap (rigorously its surface measure) admits Fourier frames, while the whole sphere does not. In this paper, we prove more general results. Consequences include that a small spherical cap in $�R^d$ near the north pole cannot have a frame spectrum near the $�x_d$-axis, and $�S$ does not admit any Fourier frame if its interior contains a closed hemisphere. We also resolve the endpoint case, that is, a hemisphere does not admit any Fourier frame. This answers a question of Kolountzakis and Lai. Our results also hold on more general smooth surfaces with nonvanishing Gaussian curvature. In particular, any compact $��(�d�-�1�)�$-dimensional smooth submanifold immersed in $�R^d$ with nonvanishing Gaussian curvature does not admit any Fourier frame. This generalizes a previous result of Iosevich, Lai, Wyman, and the second author on the boundary of convex bodies, as well as improves a recent result of Kolountzakis and Lai from tight frame to frame.

Denote the symmetric group of degree $�n$ by $�S_n$. Let $�\rho$ be an irreducible representation of $�S_n$ over the field of complex numbers and $��\sigma �\in �S_n�$. In this paper, we describe the set of eigenvalues of $��\rho �\left(�\sigma �\right)�$. Based on this result, we also obtain a description in the case of alternating groups.

We give a sharp lower bound for energy in homotopy classes of maps from real projective space to Riemannian manifolds, together with an upper bound for the infimum of the energy in such a homotopy class. We characterize the maps attaining this lower bound for energy, and we explain how the infimum of the energy in a homotopy class of maps of real projective $�n$-space is determined by an associated class of maps of the real projective plane.

We analyze a Fokker–Planck like equation, driven by a scalar parameter in order to reach an integral constraint. We exhibit criteria guaranteeing existence-uniqueness of a solution. We also provide counter-examples. This problem is motivated by an application to the immune control of tumor growth.

We study the maximal Steklov eigenvalues of trees with given number of boundary vertices and total number of vertices. Trees can be regarded as discrete analogue of Hadamard manifolds, namely simply connected Riemannian manifolds of nonpositive sectional curvature. Let $��{\sigma }_{�k�,�\max �}��(�b�,�n�)��$ be the maximal of $�k$th Steklov eigenvalue of trees with $�b$ leaves as boundary and $�n$ vertices. We determine that

Plank problems concern the covering of convex bodies by planks in Euclidean space and are related to famous open problems in convex geometry. In this survey, we introduce plank problems and present surprising applications of plank theorems in various areas of mathematics.

Let $�G$ be a group and let $�F$ be a family of subgroups of $�G$. The generalised Lusternik–Schnirelmann category $��{cat}_F��\left(�G�\right)��$ is the minimal cardinality of covers of $��B�G�$ by open subsets with fundamental group in $�F$. We prove a combination theorem for $��{cat}_F��\left(�G�\right)��$ in terms of the stabilisers of contractible $�G$-CW-complexes. As applications for the amenable category, we obtain vanishing results for the simplicial volume of gluings of manifolds (along not necessarily amenable boundaries) and of cyclic branched coverings. Moreover, we deduce an upper bound for Farber's topological complexity, generalising an estimate for amalgamated products of Dranishnikov–Sadykov.

The homogeneous Kuramoto model on a graph $��G�=�(�V�,�E�)�$ is a network of $��\left|�V�\right|�$ identical oscillators, one at each vertex, where every oscillator is coupled bidirectionally (with unit strength) to its neighbors in the graph. A graph $�G$ is said to be globally synchronizing if, for almost every initial condition, the homogeneous Kuramoto model converges to the all-in-phase synchronous state. Confirming a conjecture of Abdalla, Bandeira, Kassabov, Souza, Strogatz, and Townsend, we show that with high probability, the random graph process becomes globally synchronizing as soon as it is connected. This is best possible, since connectivity is a necessary condition for global synchronization.