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Different definitions of integrability, as a rule, use linearization of initial equation and/or expansion on some basic functions which are themselves solutions of some linear differential equation. Important fact here is that linearization…
An explicit formula is given for a fundamental solution for a class of semielliptic operators. The fundamental solution is used to investigate properties of these operators as mappings between weighted function spaces. Necessary and…
Polynomial optimization problems are infinite-dimensional, nonconvex, NP-hard, and are often handled in practice with the moment-sums of squares hierarchy of semidefinite programming bounds. We consider problems where the objective function…
Many supervised learning problems involve high-dimensional data such as images, text, or graphs. In order to make efficient use of data, it is often useful to leverage certain geometric priors in the problem at hand, such as invariance to…
A subset of Q^n is called semilinear (or piecewise linear) if it is Boolean combination of linear half-spaces. We study the computational complexity of the constraint satisfaction problem (CSP) over the rationals when all the constraints…
It is known that point searching in basic semialgebraic sets and the search for globally minimal points in polynomial optimization tasks can be carried out using $(s\,d)^{O(n)}$ arithmetic operations, where $n$ and $s$ are the numbers of…
Theoretical complexity is a vital subfield of computer science that enables us to mathematically investigate computation and answer many interesting queries about the nature of computational problems. It provides theoretical tools to assess…
Symmetry is conventionally described in a contrariety manner that the system is either completely symmetric or completely asymmetric. Using group theoretical approach to overcome this dichotomous problem, we introduce the degree of symmetry…
Motivated by the recent rapid development of complexity theory applied to quantum mechanical processes we present the complete derivation of Nielsen's complexity of unitaries belonging to the representations of oscillator group. Our…
Higher-order WKB methods are used to investigate the border between the solvable and insolvable portions of the spectrum of quasi-exactly solvable quantum-mechanical potentials. The analysis reveals scaling and factorization properties that…
In complexity theory, there exists a famous unsolved problem whether NP can be P or not. In this paper, we discuss this aspect in SAT (satisfiability) problem, and it is shown that the SAT can be solved in plynomial time by means of quantum…
The probabilistic satisfiability of a logical expression is a fundamental concept known as the partition function in statistical physics and field theory, an evaluation of a related graph's Tutte polynomial in mathematics, and the…
We describe a new incomplete but terminating method for real root finding for large multivariate polynomials. We take an abstract view of the polynomial as the set of exponent vectors associated with sign information on the coefficients.…
There is a subset of computational problems that are computable in polynomial time for which an existing algorithm may not complete due to a lack of high performance technology on a mission field. We define a subclass of deterministic…
Recently, Peter Doyle and Curt McMullen devised an iterative solution to the fifth degree polynomial. At the method's core is a rational mapping of the Riemann sphere with the icosahedral symmetry of a general quintic. Moreover, this map…
We propose a framework to study models of computation of indeterministic data, represented by abstract "distributions". In these distributions, probabilities are replaced by "amplitudes" drawn from a fixed semi-ring $S$, of which the…
Polynomial factorization is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity…
These notes contain, among others, a proof that the average running time of an easy solution to the satisfiability problem for propositional calculus is, under some reasonable assumptions, linear (with constant 2) in the size of the input.…
Answering a question of Haugland, we show that the pooling problem with one pool and a bounded number of inputs can be solved in polynomial time by solving a polynomial number of linear programs of polynomial size. We also give an overview…
People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…