Related papers: A Symbolic Summation Approach to Find Optimal Nest…
Given a Feynman parameter integral, depending on a single discrete variable $N$ and a real parameter $\epsilon$, we discuss a new algorithmic framework to compute the first coefficients of its Laurent series expansion in $\epsilon$. In a…
We give some necessary and sufficient conditions for the possibility to represent a Hermitian operator on an infinite-dimensional Hilbert space (real or complex) in the form $\sum_{i=1}^nQ_iP_i$, where $P_1,\dots,P_n$, $Q_1,\dots,Q_n$ are…
When neural networks are used to solve differential equations, they usually produce solutions in the form of black-box functions that are not directly mathematically interpretable. We introduce a method for generating symbolic expressions…
Representations of nonnegative polynomials as sums of squares are central to real algebraic geometry and the subject of active research. The sum-of-squares representations of a given polynomial are parametrized by the convex body of…
Expansion of higher transcendental functions in a small parameter are needed in many areas of science. For certain classes of functions this can be achieved by algebraic means. These algebraic tools are based on nested sums and can be…
Hyperbolic problems can at times be solved employing symbolic arguments. This is especially true for the construction of forward (and backward) fundamental solutions. We formulate a corresponding abstract scheme and illustrate its…
We present efficient methods for calculating linear recurrences of hypergeometric double sums and, more generally, of multiple sums. In particular, we supplement this approach with the algorithmic theory of contiguous relations, which…
We consider the problem of decomposing a real-valued symmetric tensor as the sum of outer products of real-valued vectors. Algebraic methods exist for computing complex-valued decompositions of symmetric tensors, but here we focus on…
Hyperbolic programming is the problem of computing the infimum of a linear function when restricted to the hyperbolicity cone of a hyperbolic polynomial, a generalization of semidefinite programming. We propose an approach based on symbolic…
We describe a neural-based method for generating exact or approximate solutions to differential equations in the form of mathematical expressions. Unlike other neural methods, our system returns symbolic expressions that can be interpreted…
This paper offers a solution method that allows one to find exact values for a large class of convergent series of rational terms. Sums of this form arise often in problems dealing with Quantum Field Theory.
A general method to express in terms of Gauss sums the number of rational points of subschemes of projective schemes over finite fields is applied to the image of the triple embedding $\mathbb{P}^1\hookrightarrow\mathbb{P}^3$. As a…
We consider nested sums involving the Pochhammer symbol at infinity and rewrite them in terms of a small set of constants, such as powers of $\pi,$ $\log(2)$ or zeta values. In order to perform these simplifications, we view the series as…
We give two general transformations that allows certain quite general basic hypergeometric multi-sums of arbitrary depth (sums that involve an arbitrary sequence $\{g(k)\}$), to be reduced to an infinite $q$-product times a single basic…
This is a review paper of recent results in the perturbative symmetry approach in the symbolic representation.
In this chapter we present the sums of Hermitian squares approach to noncommutative polynomial optimization problems. This is an extension of the sums of squares approach for polynomial optimization arising from real algebraic geometry. We…
In this paper we show that Sum-of-Squares optimization can be used to find optimal semialgebraic representations of sets. These sets may be explicitly defined, as in the case of discrete points or unions of sets; or implicitly defined, as…
We prove a summation formula for a bilateral series whose terms are products of two basic hypergeometric functions. In special cases, series of this type arise as matrix elements of quantum group representations.
Symbolic regression aims to discover human-interpretable equations that explain observational data. However, existing approaches rely heavily on discrete structure search (e.g., genetic programming), which often leads to high computational…
Graph summarization via node grouping is a popular method to build concise graph representations by grouping nodes from the original graph into supernodes and encoding edges into superedges such that the loss of adjacency information is…