Related papers: Minimal Potentials with Very Many Minima
The OPE of two N=2 R-symmetry current (short) multiplets is determined by the possible superspace three-point functions that two such multiplets can form with a third, a priori long multiplet. We show that the shortness conditions on the…
We investigate the loss surface of neural networks. We prove that even for one-hidden-layer networks with "slightest" nonlinearity, the empirical risks have spurious local minima in most cases. Our results thus indicate that in general "no…
The relation between random normal matrices and conformal mappings discovered by Wiegmann and Zabrodin is made rigorous by restricting normal matrices to have spectrum in a bounded set. It is shown that for a suitable class of potentials…
We prove some variational analysis of regularity and weak convergence of nonlocal variational principle.
We give a new proof for an equality of certain max-min and min-max approximation problems involving normal matrices. The previously published proofs of this equality apply tools from matrix theory, (analytic) optimization theory and…
The autonomous renormalization of the O(N)-symmetric scalar theory is based on an infinite re-scaling of constant fields, whereas finite-momentum modes remain finite. The natural framework for a detailed analysis of this method is a system…
We show that arbitrarily small antisymmetric perturbations of the zero function are sufficient to produce the stickiness phenomenon for planar nonlocal minimal graphs (with the same quantitative bounds obtained for the case of even…
We perform a global renormalization group study of O(N) symmetric Wess-Zumino theories and their phases in three euclidean dimensions. At infinite N the theory is solved exactly. The phases and phase transitions are worked out for finite…
We consider minimising $p$-harmonic maps from three-dimensional domains to the real projective plane, for $1<p<2$. These maps arise as least-energy configurations in variational models for nematic liquid crystals. We show that the singular…
The renormalization of the Minimal Supersymmetric Standard Model is discussed. In particular we focus on the soft-supersymmetry breaking sector of the MSSM and comment on non-renormalization theorems.
The aim of this work is to establish numerous interrelated gradient estimates in the nonlinear nonlocal setting. First of all, we prove that weak solutions to a class of homogeneous nonlinear nonlocal equations of possibly arbitrarily low…
We consider a nonlocal isoperimetric problem defined in the whole space $\R^N$, whose nonlocal part is given by a Riesz potential with exponent $\alpha\in(0, N-1)$. We show that critical configurations with positive second variation are…
In this paper, we investigate negative eigenvalues of exactly solvable quantum models, particularly one-dimensional Hamiltonians with $\delta'$-like potentials used to represent localized dipoles. These operators arise as norm resolvent…
Working in the time domain, we show that the location of the singularities and the principal symbol of a potential that is conormal to nested submanifolds $S_2 \subset S_1 \subset \mathbb{R}^n$, for $n \geq 3$, can be recovered from the…
A class of exact membrane solutions is quantized.
Let $\Sigma$ be a compact $C^2$ hypersurface in $\R^{2n}$ bounding a convex set with non-empty interior. In this paper it is proved that there always exist at least $n$ geometrically distinct closed characteristics on $\Sigma$ if $\Sigma$…
We explore the asymptotic convergence and nonasymptotic maximal inequalities of supermartingales and backward submartingales in the space of positive semidefinite matrices. These are natural matrix analogs of scalar nonnegative…
We study some divisibility properties of multiperfect numbers. Our main result is: if $N=p_1^{\alpha_1}... p_s^{\alpha_s} q_1^{2\beta_1}... q_t^{2\beta_t}$ with $\beta_1, ..., \beta_t$ in some finite set S satisfies…
We consider the inverse problem of finding a magnitude-symmetric matrix (matrix with opposing off-diagonal entries equal in magnitude) with a prescribed set of principal minors. This problem is closely related to the theory of recognizing…
Let P be the operator $-\Delta+V$ on R^d, where $V$ is a real potential with several inverse square singularities. The usual non-trapping type high-frequency inequality on the truncated resolvent of $P$ is shown, using semi-classical…