Related papers: Computable Integrability. Chapter 5: Factorization…
Several paradigms for declarative problem solving start from a specification in a high-level language, which is then transformed to a low-level language, such as SAT or SMT. Often, this transformation includes a "grounding" step to remove…
Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of…
We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV…
Three kinds of integrable Kondo problems in one-dimensional extended Hubbard models are studied by means of the boundary graded quantum inverse scattering method. The boundary K matrices depending on the local moments of the impurities are…
In the article the problem of the integrable classification of nonlinear lattices depending on one discrete and two continuous variables is studied. By integrability we mean the presence of reductions of a chain to a system of hyperbolic…
We classify order $3$ linear difference operators over $\mathbb{C}(x)$ that are solvable in terms of lower order difference operators. To prove this result, we introduce the notion of absolute irreducibility for difference modules, and…
In this article we present logarithmic methods for solving first order and second order ordinary differential equations. The essence of the method is that we apply the basic properties derivatives and logarithms to reduce the number of…
Nonlinear second-order ordinary differential equations are common in various fields of science, such as physics, mechanics and biology. Here we provide a new family of integrable second-order ordinary differential equations by considering…
We conjecture an integrability and linearizability test for dispersive Z^2-lattice equations by using a discrete multiscale analysis. The lowest order secularity conditions from the multiscale expansion give a partial differential equation…
We study probabilistic complexity classes and questions of derandomisation from a logical point of view. For each logic L we introduce a new logic BPL, bounded error probabilistic L, which is defined from L in a similar way as the…
Quantum algorithms are at the heart of the ongoing efforts to use quantum mechanics to solve computational problems unsolvable on ordinary classical computers. Their common feature is the use of genuine quantum properties such as…
Here we present an algorithm to find elementary first integrals of rational second order ordinary differential equations (SOODEs). In \cite{PS2}, we have presented the first algorithmic way to deal with SOODEs, introducing the basis for the…
Integrable lattice equations arising in the context of singular manifold equations for scalar, multicomponent KP hierarchies and 2D Toda lattice hierarchy are considered. These equation generate the corresponding continuous hierarchy of…
Solving optimization problems is the key to decision making in many real-life analytics applications. However, the coefficients of the optimization problems are often uncertain and dependent on external factors, such as future demand or…
This paper studies first-order algorithms for solving fully composite optimization problems over convex and compact sets. We leverage the structure of the objective by handling its differentiable and non-differentiable components…
A huge family of solvable potentials can be generated by systematically exploiting the factorization (Darboux) method. Starting from the free case, a large class of the known solvable families is thus reproduced, together with new ones. We…
Linear differential equations with polynomial coefficients over a field $K$ of positive characteristic $p$ with local exponents in the prime field have a basis of solutions in the differential extension $\mathcal{R}_p=K(z_1, z_2,…
We investigate the problem of factorization of large numbers on a quantum computer which we imagine to be realized within a linear ion trap. We derive upper bounds on the size of the numbers that can be factorized on such a quantum…
Let Q be a non-singular quadratic form with integer coefficients. When Q is indefinite we provide new upper bounds for the least non-trivial integral solution to the equation Q=0. When Q is positive definite we provide improved upper bounds…
The theory of fractional calculus in the complex plane was not built with a specific application in mind. The main obstacle to application was the difficulty with obtaining analytic continuations of fractional derivatives and integrals. It…