Related papers: Zeilberger to the rescue
The syntactic nature of logic and computation separates them from other fields of mathematics. Nevertheless, syntax has been the only way to adequately capture the dynamics of proofs and programs such as cut-elimination, and the finiteness…
By Kolmogorov Complexity,two number-theoretic problems are solved in different way than before,one problem is Maxim Kontsevich and Don Bernard Zagier's Problem 3 \emph{Exhibit at least one number which does not belong to} $ \mathcal{P}$…
This essay considers ways that recent uses of computers in mathematics challenge contemporary views on the nature of mathematical understanding. It also puts these challenges in a historical perspective and offers speculation as to a…
In this methodological article on experimental-yet-rigorous enumerative combinatorics, we use two instructive case studies, to show that often, just like Alexander the Great before us, the simple, "cheating" solution to a hard problem is…
Using the methodology of (rigorous) {\it experimental mathematics}, we give a simple and motivated solution to Zudilin's question concerning a $q$-analog of a problem posed by Asmus Schmidt about a certain binomial coefficients sum. Our…
Many NP-complete problems take integers as part of their input instances. These input integers are generally binarized, that is, provided in the form of the "binary" numeral representation, and the lengths of such binary forms are used as a…
From the matrix point of view, we use the recursion to discuss four combinatorial numbers in terms of the integer lattice paths, this is different from Andr\'a's method (Andra). We give four tables and matrices, and their relations, and…
Many real life problems can be reduced to the solution of a complex exponentials approximation problem which is usually ill posed. Recently a new transform for solving this problem, formulated as a specific moments problem in the plane, has…
Traditional formulations of geometric problems from the Schubert calculus, either in Plucker coordinates or in local coordinates provided by Schubert cells, yield systems of polynomials that are typically far from complete intersections and…
In a recent paper by L. Fel two new identities for the degree of syzygies are given. We present an algebraic proof of them, using only basic homological algebra tools. We also extend these results.
We introduce a new -as far as we know- problem, according to which we are asked to match sequences of two digits in matrices having entries among those two digits (but others too) and prove that this problem is NP-complete
In the last quarter of a century, algebraic statistics has established itself as an expanding field which uses multilinear algebra, commutative algebra, computational algebra, geometry, and combinatorics to tackle problems in mathematical…
This short expository paper outlines applications of computer algebra to the implication problem of conditional independence for Gaussian random variables. We touch on certificates for validity and invalidity of inference rules from the…
We use the arithmetic of the Kummer surface associated to the Jacobian of a hyperelliptic curve to study the primality of integers of the form $4m^2 5^n-1$. We provide an algorithm capable of proving the primality or compositeness of most…
Every semigroup which is a finite disjoint union of copies of the free mono- genic semigroup (natural numbers under addition) has soluble word prob- lem and soluble membership problem. Efficient algorithms are given for both problems.
In the recent literature of Artificial Intelligence, an intensive research effort has been spent, for various algebras of qualitative relations used in the representation of temporal and spatial knowledge, on the problem of classifying the…
The vast use of computers on scientific numerical computation makes the awareness of the limited precision that these machines are able to provide us an essential matter. A limited and insufficient precision allied to the truncation and…
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…
Numerical relativity has made big strides over the last decade. A number of problems that have plagued the field for years have now been mostly solved. This progress has transformed numerical relativity into a powerful tool to explore…
Mechanized reasoning uses computers to verify proofs and to help discover new theorems. Computer scientists have applied mechanized reasoning to economic problems but -- to date -- this work has not yet been properly presented in economics…