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For first order systems, we obtain an efficient bound on the exponential decay of an eigenfunction in terms of the distance between the corresponding eigenvalue and the essential spectrum. As an example, the Dirac operator is considered.
In this paper we describe the well studied process of renormalization of quadratic polynomials from the point of view of their natural extensions. In particular, we describe the topology of the inverse limit of infinitely renormalizable…
PCA-Net is a recently proposed neural operator architecture which combines principal component analysis (PCA) with neural networks to approximate operators between infinite-dimensional function spaces. The present work develops…
We proved the so called complex bounds for multimodal, infinitely renormalizable analytic maps with bounded combinatorics: deep renormalizations have polynomial-like extensions with definite modulus. The complex bounds is the first step to…
This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…
We introduce different notions of polynomial convexity with bounds on degrees of polynomials in $\mathbb C^n$. We provide some examples in higher dimensions and show necessary and sufficient conditions for polynomial convexity with degree…
We study the high-dimensional inference of a rank-one signal corrupted by sparse noise. The noise is modelled as the adjacency matrix of a weighted undirected graph with finite average connectivity in the large size limit. Using the replica…
Given a polynomial P in several variables over an algebraically closed field, we show that except in some special cases that we fully describe, if one coefficient is allowed to vary, then the polynomial is irreducible for all but at most…
In this paper, we investigate the impact of high-dimensional Principal Component (PC) adjustments on inferring the effects of variables on outcomes, with a focus on applications in genetic association studies where PC adjustment is commonly…
We develop a theory of probabilistic coherence spaces equipped with an additional extensional structure and apply it to approximating probability of convergence of ground type programs of probabilistic PCF whose free variables are of ground…
In this paper, we establish sufficient conditions for the existence of error bounds at infinity for lower semicontinuous inequality systems. We also show that the existence of an error bound at infinity of constraint systems plays an…
In this paper we present a new perspective on error analysis of Legendre approximations for differentiable functions. We start by introducing a sequence of Legendre-Gauss-Lobatto polynomials and prove their theoretical properties, such as…
We propose a novel a posteriori error estimator for conforming finite element discretizations of two- and three-dimensional Helmholtz problems. The estimator is based on an equilibrated flux that is computed by solving patchwise mixed…
Probabilities of causation (PoCs) are fundamental quantities for counterfactual analysis and personalized decision making. However, existing analytical results are largely confined to binary settings. This paper extends PoCs to multi-valued…
There is a known best possible upper bound on the probability of undetected error for linear codes. The $[n,k;q]$ codes with probability of undetected error meeting the bound have support of size $k$ only. In this note, linear codes of full…
This paper presents an achievability bound that evaluates the exact probability of error of an ensemble of random codes that are decoded by a minimum distance decoder. Compared to the state-of-the-art which demands exponential computation…
Upper and lower bounds on the error probability of linear codes under maximum-likelihood (ML) decoding are shortly surveyed and applied to ensembles of codes on graphs. For upper bounds, focus is put on Gallager bounding techniques and…
Let $(M,g)$ be a compact Riemannian manifold and $P_1:=-h^2\Delta_g+V(x)-E_1$ so that $dp_1\neq 0$ on $p_1=0$. We assume that $P_1$ is quantum completely integrable in the sense that there exist functionally independent pseuodifferential…
The success of algorithms in the analysis of high-dimensional data is often attributed to the manifold hypothesis, which supposes that this data lie on or near a manifold of much lower dimension. It is often useful to determine or estimate…
This paper develops upper and lower bounds for the probability of Boolean functions by treating multiple occurrences of variables as independent and assigning them new individual probabilities. We call this approach dissociation and give an…