Remainder

About: Remainder is a research topic. Over the lifetime, 3443 publications have been published within this topic receiving 35379 citations. The topic is also known as: dividend mod divisor.

The Calculation of Multivariate Polynomial Resultants

Abstract: An efficient algorithm is presented for the exact calculation of resultants of multivariate polynomials with integer coefficients. The algorithm applies modular homomorphisms and the Chinese remainder theorem, evaluation homomorphisms and interpolation, in reducing the problem to resultant calculation for univariate polynomials over GF(p), whereupon a polynomial remainder sequence algorithm is used. The computing time of the algorithm is analyzed theoretically as a function of the degrees and coefficient sizes of its inputs . As a very special case , it is shown that when all degrees are equal and the coefficient size is fixed, its computing time is approximately proportional to λ2r+l , where λ is the common degree and r is the number of variables . Empirically observed computing times of the algorithm are tabulated for a large number of examples, and other algorithms are compared. Potential application of the algorithm to the solution of systems of polynomial equations is briefly discussed.

The product-limit estimator and the bootstrap: Some asymptotic representations

Abstract: The product-limit estimator and its quantile process are represented as i.i.d. mean processes, with a remainder of ordern−3/4(logn)3/4 a.s. Corresponding bootstrap versions of these representations are given, which can help one visualize how the bootstrap procedure operates in this set up.

On dominance and varieties of commuting matrices

Abstract: [4]), a question first raised by Goto. In the same section it is shown that the algebra generated by a pair of commuting n x n matrices can not have dimension greater than n, and that this algebra can always be embedded in a commutative algebra of dimension exactly n, a result derived by introducing the notion of a specialization of an algebra of matrices. The remainder of Chapter II contains related results of independent interest included mainly for later reference. The present paper was originally intended to be the first part of "On nilalgebras and linear varieties of nilpotent matrices, IV" (in preparation), which study continues the program of determining the structure of linear varieties of nilpotent matrices (" nilvarieties "). It is there shown that if V is a nilvariety of n x n matrices, and if c is the dimension of the algebra of all n x n matrices commuting with a generic element of V, then dim V_? 1/2(n2 c). All nilvarieties V for which dim V l/2(X2-c) are determined and shown to be associative nilpotent algebras of a type which we have elected to call anti-semisimple. These may be described

High order symplectic integrators for perturbed Hamiltonian systems

Abstract: We present a class of symplectic integrators adapted for the integration of perturbed Hamiltonian systems of the form $H=A+\epsilon B$. We give a constructive proof that for all integer $p$, there exists an integrator with positive steps with a remainder of order $O(\tau^p\epsilon +\tau^2\epsilon^2)$, where $\tau$ is the stepsize of the integrator. The analytical expressions of the leading terms of the remainders are given at all orders. In many cases, a corrector step can be performed such that the remainder becomes $O(\tau^p\epsilon +\tau^4\epsilon^2)$. The performances of these integrators are compared for the simple pendulum and the planetary 3-Body problem of Sun-Jupiter-Saturn.