Let l a prime number and Phi(l)(X, Y) the modular polynomial of level l. Since this polynomial has integer coefficients one may compute it modulo primes. Petr Lisonek and Yung-Jung Kim compute the polynomial Phi(l)(X,...
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ISBN:
(纸本)9783319150819;9783319150802
Let l a prime number and Phi(l)(X, Y) the modular polynomial of level l. Since this polynomial has integer coefficients one may compute it modulo primes. Petr Lisonek and Yung-Jung Kim compute the polynomial Phi(l)(X, Y) modulo 2 explicitly for several l and they conjecture that the coefficients under the diagonal vanish. In this note we prove their conjecture and that the same property holds modulo the primes 3 and 5.
In the first part of this paper we present a short survey on the problem of the representation of rational normal curves as set-theoretic complete intersections. In the second part we use a method, introduced by Robbi...
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ISBN:
(纸本)9783319150819;9783319150802
In the first part of this paper we present a short survey on the problem of the representation of rational normal curves as set-theoretic complete intersections. In the second part we use a method, introduced by Robbiano and Valla, to prove that the rational normal quartic is set-theoretically complete intersection of quadrics: it is an original proof of a classical result of Perron, and Gallarati-Rollero.
This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties...
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This paper introduces Riordan-Krylov matrices. These matrices naturally generalize Riordan matrices by using Krylov matrices and a more general class of algebras in place of formal power series. Fundamental properties of RiordanKrylov matrices that are analogous to those of Riordan matrices are developed. These results and techniques are used to study incidence algebras of a poset as well as the chain and zeta-multichain polynomials of the poset. Throughout the paper applications to enumeration problems in combinatorics and numbertheory are provided.(c) 2021 Elsevier Inc. All rights reserved.
We use computeralgebra to-expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter epsilon, with coefficients that are polynomials in the n...
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We use computeralgebra to-expand the Pekeris secular determinant for two-electron atoms symbolically, to produce an explicit polynomial in the energy parameter epsilon, with coefficients that are polynomials in the nuclear charge Z. Repeated differentiation of the polynomial, followed by a simple transformation, gives a series for epsilon in decreasing powers of Z. The leading term is linear, consistent with well-known behavior that corresponds to the approximate quadratic dependence of ionization potential on atomic number (Moseley's law). Evaluating the 12-term series for individual Z gives the roots to a precision of 10 or more digits for Z greater than or equal to 2. This suggests the use of similar tactics to construct formulas for roots vs atomic, molecular, and variational parameters in other eigenvalue problems, in accordance with the general objectives of gradient theory. Matrix elements can be represented by symbols in the secular determinants, enabling the use of analytical expressions for the molecular integrals in the differentiation of the explicit polynomials. The mathematical and computational techniques include modular arithmetic to handle matrix and polynomial operations, and unrestricted precision arithmetic to overcome severe digital erosion. These are likely to find many further applications in computational chemistry. (C) 2001 American Institute of Physics.
We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the ortho...
We establish a new perturbation theory for orthogonal polynomials using a Riemann-Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with respect to two measures can be effectively compared using the difference of their Stieltjes transforms on a suitably chosen contour. Moreover, when two measures are close and satisfy some regularity conditions, we use the theta functions of a hyperelliptic Riemann surface to derive explicit and accurate expansion formulae for the perturbed orthogonal polynomials. In contrast to other approaches, a key strength of the methodology is that estimates can remain valid as the degree of the polynomial grows. The results are applied to analyze several numerical algorithms from linear algebra, including the Lanczos tridiagonalization procedure, the Cholesky factorization, and the conjugate gradient algorithm. As a case study, we investigate these algorithms applied to a general spiked sample covariance matrix model by considering the eigenvector empirical spectral distribution and its limits. For the first time, we give precise estimates on the output of the algorithms, applied to this wide class of random matrices, as the number of iterations diverges. In this setting, beyond the first order expansion, we also derive a new mesoscopic central limit theorem for the associated orthogonal polynomials and other quantities relevant to numerical algorithms.
A systematic and algorithmic approach for the theory of bifurcations with symmetry is presented. Using the computeralgebra as a tool, an algorithm is developed for arbitrary group actions. This includes that instead ...
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A systematic and algorithmic approach for the theory of bifurcations with symmetry is presented. Using the computeralgebra as a tool, an algorithm is developed for arbitrary group actions. This includes that instead of germs, polynomials which creates new problems but nevertheless seems to be a reasonable approach are handled.
In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the computer Aided Geometric Design Community. The analysis of the validity o...
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ISBN:
(纸本)9783319150819;9783319150802
In the nineties, several methods for dealing in a more efficient way with the implicitization of rational parametrizations were explored in the computer Aided Geometric Design Community. The analysis of the validity of these techniques has been a fruitful ground for Commutative algebraists and algebraic Geometers, and several results have been obtained so far. Yet, a lot of research is still being done currently around this topic. In this note we present these methods, show their mathematical formulation, and survey current results and open questions.
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric mo...
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ISBN:
(纸本)9783319150819;9783319150802
In this expository article we give an introduction to Ehrhart theory, i.e., the theory of integer points in polyhedra, and take a tour through its applications in enumerative combinatorics. Topics include geometric modeling in combinatorics, Ehrhart's method for proving that a counting function is a polynomial, the connection between polyhedral cones, rational functions and quasisymmetric functions, methods for bounding coefficients, combinatorial reciprocity theorems, algorithms for counting integer points in polyhedra and computing rational function representations, as well as visualizations of the greatest common divisor and the Euclidean algorithm.
We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use m...
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We present applications of matrix methods to the analytic theory of polynomials. We first show how matrix analysis can be used to give new proofs of a number of classical results on roots of polynomials. Then we use matrix methods to establish a new log-majorization result on roots of polynomials. The theory of multiplier sequences gives the common link between the applications. (C) 2011 Elsevier Inc. All rights reserved.
A commutative ring A is said to be binomial if A is torsion-free (as a Z-module) and the element a(a - 1)(a - 2)center dot center dot center dot(a - n + 1)/n! of A circle times(Z) Q lies in A for every a epsilon A and...
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A commutative ring A is said to be binomial if A is torsion-free (as a Z-module) and the element a(a - 1)(a - 2)center dot center dot center dot(a - n + 1)/n! of A circle times(Z) Q lies in A for every a epsilon A and every positive integer n. Binomial rings were first defined circa 1969 by Philip Hall in connection with his groundbreaking work in the theory of nilpotent groups. They have since had further applications to integer-valued polynomials, Witt vectors, and lambda-rings. For any set X, the ring of integer-valued polynomials in Q[X] is the free binomial ring on the set X. Thus the binomial property provides a universal property for rings of integer-valued polynomials. We give several characterizations of binomial rings and their homomorphic images. For example, we prove that a binomial ring is equivalently lambda-ring A whose Adams operations are all the identity on A. This allows us to construct a right adjoint Bin(U) for the inclusion from binomial rings to rings which has several applications in commutative algebra and numbertheory. For example, there is a natural Bin(U)(A)-algebra structure on the universal lambda-ring Lambda(A), and likewise on the abelian group of multiplicative A-arithmetic functions. Similarly, there is a natural BinU (A)module structure on the abelian group 1 + a for any ideal a in A with respect to which A is complete. (c) 2005 Elsevier B.V. All rights reserved.
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