Scientific Papers

The aim of this paper is to explain how to get a complex of smooth representations out of the dual vector space to a smooth representation of a p-adic Lie group, in natural characteristic. The construction does not depend on any finiteness/admissibility assumptions. Imposing such an assumption, one obtains an involutive duality on the derived category of complexes of smooth modules with admissible cohomology modules.

Full information: http://www.heldermann.de/JLT/JLT30/JLT301/jlt30007.htm

The aim of this work is to give a complete picture concerning the asymptotic behaviour of the solutions to fractional Ginzburg-Landau equation. In previous works, we have shown global well-posedness for the past interval in the case where spatial dimension is less than or equal to 3. Moreover, we have also shown blow-up of solutions for the future interval in one dimensional case. In this work, we summarise the asymptotic behaviour in the case where spatial dimension is less than or equal to 3 by proving blow-up of solutions for a future time interval in multidimensional case. The result is obtained via ODE argument by exploiting a new weighted commutator estimate.

The non-existence of global solutions for semirelativistic equations with non-gauge invariant power type nonlinearity with mass is studied in the frame work of weighted L1. In particular, a priori control of weighted integral of solutions is obtained by introducing a pointwise estimate of fractional derivative of some weight functions. Especially, small data blowup with small mass is obtained.

We study the initial value problem for the half Ginzburg-LandauKuramoto (hGLK) equation with the second order elliptic operator having rough coefficients and potential type perturbation. The blow-up of solutions for hGLK equation with non-positive nonlinearity is shown by an ODE argument. The key tools in the proof are appropriate commutator estimates and the essential self-adjointness of the symmetric uniformly elliptic operator with rough metric and potential type perturbation.

In this article we prove the explicit Mordell Conjecture for large families of curves. In addition, we introduce a method, of easy application, to compute all rational points on curves of quite general shape and increasing genus. The method bases on some explicit and sharp estimates for the height of such rational points, and the bounds are small enough to successfully implement a computer search. As an evidence of the simplicity of its application, we present a variety of explicit examples and explain how to produce many others. In the appendix our method is compared in detail to the classical method of Manin–Demjanenko and the analysis of our explicit examples is carried to conclusion.

The harmonic sections of the Kaluza–Klein model can be seen as a variant of harmonic maps with additional gauge symmetry. Geometrically, they are realized as sections of a fiber bundle associated to a principal bundle with a connection. In this paper, we investigate geometric and analytic aspects of a model that combines the Kaluza–Klein model with the Yang–Mills action and a Dirac action for twisted spinors. In dimension two we show that weak solutions of the Euler–Lagrange system are smooth. For a sequence of approximate solutions on surfaces with uniformly bounded energies we obtain compactness modulo bubbles, namely, energy identities and the no-neck property hold.

We study stability times for a family of parameter dependent nonlinear Schr¨odinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.

An explicit lifespan estimate is presented for the derivative Schr¨odinger equations with periodic boundary condition

Recent studies point to the potential storage of a large number of patterns in the celebrated Hopfield associative memory model, well beyond the limits obtained previously. We investigate the properties of new fixed points to discover that they exhibit instabilities for small perturbations and are therefore of limited value as associative memories. Moreover, a large deviations approach also shows that errors introduced to the original patterns induce additional errors and increased corruption with respect to the stored patterns.

View article: https://iopscience.iop.org/article/10.1088/1751-8121/aa8fd7

Restricted Boltzmann machines are key tools in machine learning and are described by the energy function of bipartite spin-glasses. From a statistical mechanical perspective, they share the same Gibbs measure of Hopfield networks for associative memory. In this equivalence, weights in the former play as patterns in the latter. As Boltzmann machines usually require real weights to be trained with gradient-descent-like methods, while Hopfield networks typically store binary patterns to be able to retrieve, the investigation of a mixed Hebbian network, equipped with both real (e.g., Gaussian) and discrete (e.g., Boolean) patterns naturally arises. We prove that, in the challenging regime of a high storage of real patterns, where retrieval is forbidden, an additional load of Boolean patterns can still be retrieved, as long as the ratio between the overall load and the network size does not exceed a critical threshold, that turns out to be the same of the standard Amit–Gutfreund–Sompolinsky theory. Assuming replica symmetry, we study the case of a low load of Boolean patterns combining the stochastic stability and Hamilton-Jacobi interpolating techniques. The result can be extended to the high load by a non rigorous but standard replica computation argument.

In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund type assumptions, and we prove well-posedness in H∞ respectively without loss and with finite loss of derivatives. The key to obtain the results is the construction of a suitable symmetrizer for our system, which allows us to recover energy estimates (with or without loss) for the hyperbolic operator under consideration. This can be achievied, in contrast with the classical case of systems with smooth (say Lipschitz) coefficients, by adding one step in the diagonalization process, and building the symmetrizer up to the second order.

Abstract

Partial regularity for solutions of quasilinear parabolic systems with nonsmooth in time principal matrix is proved in the paper. Only boundedness of this matrix is assumed in time variable. To prove the result, we apply the method of modified A-caloric approximation assuming that the matrix A is not the constant one but depends on the time variable t.

MSC 35K55

Keywords Quasilinear parabolic systems; Nonsmooth in time nonlinearities; Regularity; Morrey; VMO spaces

We prove a superposition principle for Riesz potentials of nonnegative continuous functions on Lie groups of Heisenberg type. More precisely, we show that the Riesz potential
\(R_{\alpha}(\rho)(g) =\int_{\mathbb{G}}N(g^{-1}g')^{\alpha-Q}\rho(g')dg'\)
of a nonnegative function \(\rho \in C_0(\mathbb{G})\) on a group \(\mathbb{G}\) of Heisenberg type is necessarily either p-subharmonic or p-superharmonic, depending on p and α. Here N denotes the anisotropic homogeneous norm on such groups introduced by Kaplan. This result extends to a wide class of nonabelian stratified Lie groups a recent remarkable superposition result of Crandall and Zhang.