Article

Preliminary Version

June 2004

Authors:

Assuming the Generalized Riemann Hypothesis, we prove the following: If b is an integer greater than one, then the multiplicative order of b modulo N is larger than N for all N in a density one subset of the integers. If A is a hyperbolic unimodular matrix with integer coecients, then the order of A modulo p is greater than p for all p in a density one subset of the primes.

Grid Graphs, Gorenstein Polytopes, and Domino Stackings

Article

Full-text available

Nov 2007

We examine domino tilings of rectangular boards, which are in natural bijection with perfect matchings of grid graphs. This leads to the study of their associated perfect matching polytopes, and we present some of their properties, in particular, when these polytopes are Gorenstein. We also introduce the notion of domino stackings and present some results and several open questions. Our techniques use results from graph theory, polyhedral geometry, and enumerative combinatorics. Comment: 14 pages, 6 figures, uses graphs package

ResearchGate has not been able to resolve any references for this publication.

Preprint

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Infinite matrix of odd natural numbers. A bit about Sophie Germain prime numbers

January 2025

Gennady Eremin

In this paper we work with number sequences from the On-Line Encyclopedia of Integer Sequences (OEIS). Using the Pepis-Kalmar pairing function, we obtain an infinite matrix of natural numbers in which odd natural numbers are separated from even ones; such a matrix simplifies working with prime numbers. With point's shell numbers and shell lines we give another proof of the infinity of primes, ... [Show full abstract] based on Bertrand's postulate. We have shown that the asymptotic density of Mersenne numbers (OEIS A000225) is positive and equal to an infinitesimal value. It is also proven that Sophie Germain prime numbers (OEIS A005384) are located in the set of natural numbers with asymptotic density of 1/6.

Article

A Fast Algorithm for Expansion over Spherical Harmonics

November 1995 · Applicable Algebra in Engineering Communication and Computing

Michael A. Frumkin

Algorithms for expansion over spherical harmonics are often used in electrostatic field calculation, calculation of the density functions in quantum chemistry and calculation of molecular surfaces. It usually includes expansion over spherical harmonics of degrees to several dozens. The usual method is to use an integration method over some grid on the unit sphere and in fact is a multiplication ... [Show full abstract] of the matrix of values of spherical harmonics in the grid points by a vector of values of the expanding function in the set of points. This algorithm executes O(NL2) operations whereN is the number of the grid points andL is the maximal degree of the spherical harmonics involved. We provide an algorithm of complexity O(NLlog2 L) for multiplication of the matrix of values of spherical harmonics in points of an arbitrary grid on the unit sphere. The algorithm is based on interrelation between spherical harmonics and Legendre polynomials and on a fast algorithm for expansion over Legendre polynomials.

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The lower bound on the HK multiplicities of quadric hypersurfaces

January 2021

Vijaylaxmi Trivedi

Here we prove that the Hilbert-Kunz mulitiplicity of a quadric hypersurface of dimension d and odd characteristic

p\geq 2d-4

is bounded below by

1+m_d

, where

m_d

is the

d^{th}

coefficient in the expansion of \mbox{sec}+\mbox{tan}. This proves a part of the long standing conjecture of Watanabe-Yoshida. We also give an upper bound on the HK multiplicity of such a hypersurface. We ... [Show full abstract] approach the question using the HK density function and the classification of ACM bundles on the smooth quadrics via matrix factorizations.

Article

Density estimation on the spaces of symmetric and rectangular matrices

October 1997 · Linear Algebra and its Applications

Yasuko Chikuse

This paper develops the theory of density estimation on the space Sm of all m × m symmetric matrices and on the space Rm, p of all m × p rectangular matrices. We consider the two methods, kernel density estimation and density estimation by orthogonal series, and illustrate them with examples for the normal kernel density functions and for the Hermite orthonormal bases with symmetric and ... [Show full abstract] rectangular matrix arguments on Sm and Rm, p, respectively. The problem of estimating unknown joint density functions of multiple random matrices is also discussed.

Article

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The Reduced Density Matrix Method: Applications of the T 20 N-Representability Condition and Develop...

January 1970

We are interested in realizing the variational calculation with the 2-order Reduced Den- sity Matrix (2-RDM) for fermionic systems. The known necessary N-representability conditions P , Q, G, T 1, and T 20 are im- posed, resulting in an optimization prob- lem called semidenite programming (SDP) problem. The ground state energies for var- ious small atoms and molecules were calcu- lated (3). ... [Show full abstract] Additionally, we show some results of the one-dimensional Hubbard model for high correlation limit using multiple preci- sion arithmetic version of the solver (3). 1. The Reduced Density Matrix Method The Hamiltonian of an N-particle fermionic system in the second quantized form is: H = X ij vi ja y iaj + 1 2 X i1i2j1j2 wi1i2 j1j2a y i1a y i2aj2aj1: Then, the total energy of some N-particle state j i is given by: E = h jHj i = X

Article

Separability of rank two quantum states on multiple quantum spaces with different dimensions

June 2003 · International Journal of Quantum Information

We consider the separability of rank two quantum states on multiple quantum spaces with different dimensions. The sufficient and necessary conditions for separability of these multiparty quantum states are explicitly presented. A nonseparability inequality is also given, for the case where one of the eigenvectors corresponding to nonzero eigenvalues of the density matrix is maximally entangled.

Article

Universality in the Two Matrix Model with a Monomial Quartic and a General Even Polynomial Potential

December 2008 · Communications in Mathematical Physics

Man Yue Mo

In this paper we studied the asymptotic eigenvalue statistics of the 2 matrix model with a quartic monomial and a general even polynomial potential. We studied the correlation kernel for the eigenvalues of one of the matrices in asymptotic limit. We extended the results of Duits and Kuijlaars to the case when the limiting eigenvalue density for one of the matrices is supported on multiple ... [Show full abstract] intervals. The results are achieved by constructing the parametrix to a Riemann-Hilbert problem obtained by Duits and Kuijlaars with theta functions and then showing that this parametrix is well-defined by studying the theta divisor.

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Multiple-Input Multiple-Output (MIMO) Linear Systems Extreme Inputs/Outputs

January 2007 · Shock and Vibration

David O. Smallwood

A linear structure is excited at multiple points with a stationary normal random process. The response of the structure is measured at multiple outputs. If the autospectral densities of the inputs are specified, the phase relationships between the inputs are derived that will minimize or maximize the trace of the autospectral density matrix of the outputs. If the autospectral densities of the ... [Show full abstract] outputs are specified, the phase relationships between the outputs that will minimize or maximize the trace of the input autospectral density matrix are derived. It is shown that other phase relationships and ordinary coherence less than one will result in a trace intermediate between these extremes. Least favorable response and some classes of critical response are special cases of the development. It is shown that the derivation for stationary random waveforms can also be applied to nonstationary random, transients, and deterministic waveforms.

Conference Paper

The (Minimum) Rank of Typical Fooling-Set Matrices

May 2017 · Lecture Notes in Computer Science

A fooling-set matrix is a square matrix with nonzero diagonal, but at least one in every pair of diagonally opposite entries is 0. Dietzfelbinger et al. ’96 proved that the rank of such a matrix is at least

\sqrt{n}

, for a matrix of order n. It is known that the bound is tight (up to a multiplicative constant). We ask for the typical minimum rank of a fooling-set matrix: For a fooling-set ... [Show full abstract] zero-nonzero pattern chosen at random, is the minimum rank of a matrix with that zero-nonzero pattern over a field

{\mathbb {F}}

closer to its lower bound

\sqrt{n}

or to its upper bound n? We study random patterns with a given density p, and prove an

\varOmega (n)

bound for the cases when(a)p tends to 0 quickly enough; (b)p tends to 0 slowly, and

{\left|{{\mathbb {F}}}\right|}=O(1)

; (c)

p\in \left]0,1\right]

is a constant. We have to leave open the case when

p\rightarrow 0

slowly and

{\mathbb {F}}

is a large or infinite field (e.g.,

{\mathbb {F}}={{\mathrm{GF}}}(2^n)

,

{\mathbb {F}}={\mathbb {R}}

).

Article

Full-text available

Non-Identifiability of Simultaneous Spatial Autoregressive Model and Singularity of Fisher Informati...

June 2017 · International Journal of Statistics and Probability

Simultaneous spatial autoregressive model is widely used for spatial data analysis, observed at a set of grid points in a space. However a problem, not so well known, is that there exists no unique model unlike time series AR model for given autocovariances or spectral density. We show that such a non-identifiability of the model implies existence of multiple maximum likelihood estimates under ... [Show full abstract] Gaussianity and causes non-estimability of parameters and the singularity of Fisher information matrix. Several types of necessary and sufficient conditions for the singularity are given.

Preprint

Exact non-Markovian evolution with multiple reservoirs

December 2019

Alexander E. Teretenkov

The model of a multi-level system interacting with several reservoirs is considered. The exact reduced density matrix evolution could be obtained for this model without Markov approximation. Namely, this evolution is fully defined by the finite set of linear differential equations. In this work the results which were obtained previously for only one Lorentz peak in the spectral density are ... [Show full abstract] generalized to the case of an arbitrary number of such peaks. The case of the Ohmic contribution in the spectral density is also taken here into account.