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We motivate and explain the system introduced by Conway and Sloane for working with quadratic forms over the 2-adic integers, and prove its validity. Their system is far better for actual calculations than earlier methods, and has been used…
In this article we present first an algorithm for calculating the determining equations associated with so-called ``nonclassical method'' of symmetry reductions (a la Bluman and Cole) for systems of partial differentail equations. This…
In this paper, ellipsoid method for linear programming is derived using only minimal knowledge of algebra and matrices. Unfortunately, most authors first describe the algorithm, then later prove its correctness, which requires a good…
In this paper we present a unified method for solving general polynomial equations of degree less than five.
We explain how to encode an algebraic series by finite data and how to do effective arithmetic on the level of these encodings. The reasoning is based on the Newton-Puiseux algorithm and an effective equality test for algebraic series.…
We present a quantum algorithm which identifies with certainty a hidden subgroup of an arbitrary finite group G in only a polynomial (in log |G|) number of calls to the oracle. This is exponentially better than the best classical algorithm.…
Considering the difficult problem under classical computing model can be solved by the quantum algorithm in polynomial time, t-multiple discrete logarithm problems presented. The problem is non-degeneracy and unique solution. We talk about…
We present a family of non-abelian groups for which the hidden subgroup problem can be solved efficiently on a quantum computer.
We prove that the Eulerian polynomial satisfies certain polynomial congruences. Furthermore, these congruences characterize the Eulerian polynomial.
In this paper we presents an algorithm for finding a solution of the linear nonhomogeneous quaternionic-valued differential equations. Moveover, several examples shows the feasibility of our algorithm.
We develop an algorithm for computing affine Kazhdan-Lusztig polynomials, for all Lie types. This generalizes our previously published algorithm for type A, which in turn is a faster version of an algorithm due to Lascouz, Leclerc and…
Vladimir Shpilrain and Jie-Tai Yu have asked for an effective algorithm to decide if two elements of C[x,y] are related by an automorphism of C[x,y]. We describe here an efficient algorithm that decides this question and finds the…
The scheme of divided differences is widely used in many approximation and interpolation problems. Computing the Newton coefficients of the interpolating polynomial is the first step of the Bj\"{o}rck and Pereyra algorithm for solving…
We derive a collection of identities for bivariate Fibonacci and Lucas polynomials using essentially a matrix approach as well as properties of such polynomials when the variables $x$ and $y$ are replaced by polynomials. A wealth of…
We show that with a few modifications the Adomian's method for solving second order differential equations can be used to obtain the known results of the special functions of mathematical physics. The modifications are necessary in order to…
We present an efficient quantum algorithm to simulate nonlinear differential equations with polynomial vector fields of arbitrary degree on quantum platforms. Models of physical systems that are governed by ordinary differential equations…
In this article we implement a method for the computation of a nonlinear elliptic problem with nonstandard growth driven by the $p(x)-$Laplacian operator. Our implementation is based in the {\em decomposition--coordination} method that…
A symbolic computational algorithm which detects " linear "` solutions of nonlinear polynomial differential equations of single functions, is developed throughout this paper.
Computing the determinant of a matrix with the univariate and multivariate polynomial entries arises frequently in the scientific computing and engineering fields. In this paper, an effective algorithm is presented for computing the…
We consider a natural generalization of an abelian Hidden Subgroup Problem where the subgroups and their cosets correspond to graphs of linear functions over a finite field F with d elements. The hidden functions of the generalized problem…