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The analysis of observable phenomena (for instance, in biology or physics) allows the detection of dynamical behaviors and, conversely, starting from a desired behavior allows the design of objects exhibiting that behavior in engineering.…
Algorithms for computing rational generating functions of solutions of one-dimensional difference equations are well-known and easy to implement. We propose an algorithm for computing rational generating functions of solutions of…
Modify the Blum-Shub-Smale model of computation replacing the permitted computational primitives (the real field operations) with any finite set $B$ of real functions semialgebraic over the rationals. Consider the class of boolean decision…
We show that multiplication can be done in polynomial time on a three counter machine that receives its input as the contents of two counters. The technique is generalized to functions of two variables computable by deterministic Turing…
Some properties and relations satisfied by the polynomial solutions of the bispectral problem are studied. Given a differential operator, under certain restrictions its polynomial eigenfunctions are explicitly obtained, as well as the…
The problem of calculating real-time correlation functions is formulated in terms of an imaginary-time partial differential equation. The latter is solved analytically for the perturbed harmonic oscillator and compared with the known exact…
In this note we provide an algorithm for computing the fractional integrals of orthogonal polynomials, which is more stable than that using the expression of the polynomials w.r.t. the canonical basis. This algorithm is aimed at solving…
Polynomial factoring has famous practical algorithms over fields-- finite, rational \& $p$-adic. However, modulo prime powers it gets hard as there is non-unique factorization and a combinatorial blowup ensues. For example, $x^2+p \bmod…
A function that is analytic on a domain of $\mathbb{C}^n$ is holonomic if it is the solution to a holonomic system of linear homogeneous differential equations with polynomial coefficients. We define and study the Bernstein-Sato polynomial…
Runtime efficiency and termination are crucial properties in the studies of program verification. Instead of dealing with these issues in an ad hoc manner, it would be useful to develop a robust framework in which such properties are…
A major open problem in computational complexity is the existence of a one-way function, namely a function from strings to strings which is computationally easy to compute but hard to invert. Levin (2023) formulated the notion of one-way…
In this sequence of work we investigate polynomial equations of additive functions. We consider the solutions of equation \[ \sum_{i=1}^{n}f_{i}(x^{p_{i}})g_{i}(x)^{q_{i}}= 0 \qquad \left(x\in \mathbb{F}\right), \] where $n$ is a positive…
The present article introduces ptarithmetic (short for "polynomial time arithmetic") -- a formal number theory similar to the well known Peano arithmetic, but based on the recently born computability logic (see…
Fractional derivative can be defined as a fractional power of derivative. The commutator (i/h)[H, ], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule.…
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
The analysis of solutions to algebraic equations is further simplified. A couple of functions and their analytic continuation or root findings are required.
We give deterministic polynomial-time algorithms that, given an order, compute the primitive idempotents and determine a set of generators for the group of roots of unity in the order. Also, we show that the discrete logarithm problem in…
A method is suggested for the computation of the generalized dimensions of fractal attractors at the period-doubling transition to chaos. The approach is based on an eigenvalue problem formulated in terms of functional equations, with a…
Partition functions of certain classes of "spin glass" models in statistical physics show strong connections to combinatorial graph invariants. Also known as homomorphism functions they allow for the representation of many such invariants,…
We present a polynomial time algorithm to compute any fixed number of the highest coefficients of the Ehrhart quasi-polynomial of a rational simplex. Previously such algorithms were known for integer simplices and for rational polytopes of…