Advances in Mathematics最新文献

In this paper we prove that a totally integrable strictly-convex symplectic billiard table, whose boundary has everywhere strictly positive curvature, must be an ellipse. The proof, inspired by the analogous result of Bialy for Birkhoff billiards, uses the affine equivariance of the symplectic billiard map.

Denote the virtual cohomological dimension of ${\mathrm{SL}}_n\left(Z\right)$ by ${\nu }_n=n(n-1)/2$. Let St denote the Steinberg module of ${\mathrm{SL}}_n\left(Q\right)$ tensored with $Q$. Let $S{h}_{•}\to St$ denote the sharbly resolution of the Steinberg module. By Borel-Serre duality, $H^{{\nu }_n-i}\left({\mathrm{SL}}_n\right(Z),Q)$ is isomorphic to $H_i\left({\mathrm{SL}}_n\right(Z),St)$. The latter is isomorphic to the sharbly homology $H_i\left({\left(S{h}_{•}\right)}_{{\mathrm{SL}}_n\left(Z\right)}\right)$. We produce nonzero classes in $H_i\left({\mathrm{SL}}_n\right(Z),St)$, for certain small i, in terms of sharbly cycles and cosharbly cocycles.

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved $H^1$ and $L^1$ quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space $H^3$. Furthermore, the global well-posedness of the solution is established for certain initial data in $H^s$ with $s\ge 3$.

In 2012 Paule and Radu proved a difficult family of congruences modulo powers of 5 for Andrews' 2-colored generalized Frobenius partition function. The family is associated with the classical modular curve of level 20. We demonstrate the existence of a congruence family for a related generalized Frobenius partition function associated with the same curve. We construct an isomorphism between this new family and the original family of congruences via a mapping on the associated rings of modular functions. The pairing of the congruence families provides a new strategy for future work on congruences associated with modular curves of composite level. We show how a similar approach can be made to multiple other recent examples in the literature. We also give some important insights into the behavior of these congruence families with respect to the Atkin–Lehner involution which proved very important in Paule and Radu's original proof.

We introduce the Pythagorean dimension: a natural number (or infinity) for all representations of the Cuntz algebra and certain unitary representations of the Richard Thompson groups called Pythagorean. For each finite Pythagorean dimension d we completely classify (in a functorial manner) all such representations using finite dimensional linear algebra. Their irreducible classes form a nice moduli space: a real manifold of dimension $2d^2+1$. Apart from a finite disjoint union of circles, each point of the manifold corresponds to an irreducible unitary representation of Thompson's group F (which extends to the other Thompson groups and the Cuntz algebra) that is not monomial. The remaining circles provide monomial representations which we previously fully described and classified. We translate in our language a large number of previous results in the literature. We explain how our techniques extend them.

In this paper, we prove that if a solution to the Muskat problem with different densities and the same viscosity is sufficiently smooth, the solution is analytic in a region that degenerates at the turnover points, provided some additional conditions are satisfied. This paper studies the analyticity of the solution near turnover points, complementing the result in [38].

In [23], the first author and the third author introduced and studied the horospherical p-Minkowski problem for smooth horospherically convex domains in hyperbolic space. In this paper, we introduce and solve the discrete horospherical p-Minkowski problem in hyperbolic space for all $p\in (-\infty ,+\infty )$ when the given measure is even on the unit sphere.

We prove that the category $M$ of abelian groups with a Polish cover introduced in collaboration with Bergfalk and Panagiotopoulos is the left heart of (the derived category of) the quasi-abelian category $A$ of abelian Polish groups in the sense of Beilinson–Bernstein–Deligne and Schneiders. Thus, $M$ is an abelian category containing $A$ as a full subcategory such that the inclusion functor $A\to M$ is exact and finitely continuous. Furthermore, $M$ is uniquely characterized up to equivalence by the following universal property: for every abelian category $B$, a functor $A\to B$ is exact and finitely continuous if and only if it extends to an exact and finitely continuous functor $M\to B$. In particular, this provides a description of the left heart of $A$ as a concrete category.

We provide similar descriptions of the left heart of a number of categories of algebraic structures endowed with a topology, including: non-Archimedean abelian Polish groups; locally compact abelian Polish groups; totally disconnected locally compact abelian Polish groups; Polish R-modules, for a given Polish group or Polish ring R; and separable Banach spaces and separable Fréchet spaces over a separable complete non-Archimedean valued field.

In this paper we prove that there are no families of cyclic $Z_n$-covers of elliptic curves which generate non-compact Shimura (special) curves that lie generically in the Torelli locus $T_g$ of abelian varieties with $g\ge 8$ when n has a proper prime factor $p\ge 7$. This non-existence is also shown for families of $Z_n$-covers of curves of any genus s provided that n has a large enough prime factor p (depending on s). We achieve these results by applying the theory of Higgs bundles and the Viehweg-Zuo characterization of Shimura curves in the moduli space of principally polarized abelian varieties.

In this article, we develop a linear profile decomposition for the $L^p\to L^q$ adjoint Fourier restriction operator associated to the sphere, valid for exponent pairs $p<q$ for which this operator is bounded. Such theorems are new when $p\ne 2$. We apply these methods to prove new results regarding the existence of extremizers and the behavior of extremizing sequences for the spherical extension operator. Namely, assuming boundedness, extremizers exist if $q>\max \{p,\frac{d+2}{d}{p}^{\prime }\}$, or if $q=\frac{d+2}{d}{p}^{\prime }$ and the operator norm exceeds a certain constant times the operator norm of the parabolic extension operator.