Related papers: Three lectures on Fourier analysis and learning th…
We study some features of learning models based on "delayed" and undifferentiated reinforcement and realized by simple algorithms which may be considered of a very elementary nature. We show that a modification of the Hebb-rule works well…
Hamiltonian learning is crucial to the certification of quantum devices and quantum simulators. In this paper, we propose a hybrid quantum-classical Hamiltonian learning algorithm to find the coefficients of the Pauli operator components of…
In this work, we develop a highly efficient representation of functions and differential operators based on Fourier analysis. Using this representation, we create a variational hybrid quantum algorithm to solve static, Schr\"odinger-type,…
The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal…
Let f:{-1,1}^n -> R be a real function on the hypercube, given by its discrete Fourier expansion, or, equivalently, represented as a multilinear polynomial. We say that it is Boolean if its image is in {-1,1}. We show that every function on…
Over the last year significant progress was made in the understanding of the computation of Feynman integrals using differential equations. These lectures give a review of these developments, while not assuming any prior knowledge of the…
Quantum and q-deformed algebras find their application not only in mathematical physics and field theoretical context, but also in phenomenology of particle properties. We describe (i) the use of quantum algebras U_q(su_n) corresponding to…
This paper explores the use of a discrete singular convolution algorithm as a unified approach for numerical integration of the Fokker-Planck equation. The unified features of the discrete singular convolution algorithm are discussed. It is…
Dualities are widely used in quantum field theories and string theory to obtain correlation functions at high accuracy. Here we present examples where dual data representations are useful in supervised classification, linking machine…
Efficient algorithms for computing linear convolutions based on the fast Fourier transform are developed. A hybrid approach is described that combines the conventional practice of explicit dealiasing (explicitly padding the input data with…
Amongst the most remarkable successes of quantum computation are Shor's efficient quantum algorithms for the computational tasks of integer factorisation and the evaluation of discrete logarithms. In this article we review the essential…
The Kirkwood-Dirac distribution, serving as an informationally complete representation of a quantum state, has recently garnered increasing attention. We investigate the Kirkwood-Dirac classicality with respect to mutually unbiased bases.…
The first part of these lecture notes is devoted to an introduction to the theory of exact WKB analysis for second-order Schr\"odinger-type ordinary differential equations. It reviews the construction of the WKB solution, Borel summability,…
We present some results from classical homological algebra using the language of cotorsion theories in abelian categories. The results are a couple of foundational facts about homological dimension, the Kunneth formula and the universal…
We adapt previous research on category theory and topological unsupervised learning to develop a functorial perspective on manifold learning, also known as nonlinear dimensionality reduction. We first characterize manifold learning…
For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have…
The purpose of this note is to discuss several results that have been obtained in the last decade in the context of sharp adjoint Fourier restriction/Strichartz inequalities. Rather than aiming at full generality, we focus on several…
We propose a mathematical framework that we call quantum, higher-order Fourier analysis. This generalizes the classical theory of higher-order Fourier analysis, which led to many advances in number theory and combinatorics. We define a…
We study the deterministic query complexity of Boolean functions on slices of the hypercube. The $k^{th}$ slice $\binom{[n]}{k}$ of the hypercube $\{0,1\}^n$ is the set of all $n$-bit strings with Hamming weight $k$. We show that there…
We consider a wide class of semi linear Hamiltonian partial differential equa- tions and their approximation by time splitting methods. We assume that the nonlinearity is polynomial, and that the numerical tra jectory remains at least uni-…