Smallest Modulus Research Articles

Certain Geometric Properties of the Canonical Weierstrass Product of an Entire Function Associated with Conic Domains

In this paper, we determine the radius of λ -uniform convexity, λ -starlikeness, and α -convexity of order δ for the Weierstrass canonical product of an entire function as a root having smallest modulus and argument ϕ of a functional equation. As special cases, we also determine the radius of λ -uniform convexity, λ -starlikeness, and α -convexity of order δ for the entire function 1 / Γ .

On the extreme value statistics of normal random matrices and 2D Coulomb gases: Universality and finite N corrections

In this paper we extend the orthogonal polynomials approach for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar (J. Stat. Mech. P04001 arXiv:1102.0738), to normal random matrices and 2D Coulomb gases in general. Firstly, we show that this approach provides an alternative derivation of results in the literature. More precisely, we show convergence of the rescaled eigenvalue with largest modulus of a normal Gaussian ensemble to a Gumbel distribution, as well as universality for an arbitrary radially symmetric potential. Secondly, it is shown that this approach can be generalised to obtain convergence of the eigenvalue with smallest modulus and its universality for ring distributions. Most interestingly, the here presented techniques are used to compute all slowly varying finite N correction of the above distributions, which is important for practical applications, given the slow convergence. Another interesting aspect of this work is the fact that we can use standard techniques from Hermitian random matrices to obtain the extreme value statistics of non-Hermitian random matrices resembling the large N expansion used in context of the double scaling limit of Hermitian matrix models in string theory.

The second Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of Schur

Schur's [20] Markov-type extremal problem is to determine (i) \(M_n= \sup_{-1\leq \xi\leq 1}\sup_{P_n\in\mathbf{B}_{n,\xi,2}}(|P_n^{(1)}(\xi)| / n^2)\), where \(\mathbf{B}_{n,\xi,2}=\{P_n\in\mathbf{B}_n:P_n^{(2)}(\xi)=0\}\subset \mathbf{B}_n=\{P_n:|P_n(x)|\leq 1 \;\textrm{for}\; |x| \leq 1\}\) and \(P_n\) is an algebraic polynomial of degree \(\leq n\). Erdos and Szego [4] found that for \(n\geq 4\) this maximum is attained if \(\xi=\pm 1\) and \(P_n\in\mathbf{B}_{n,\pm 1,2}\) is a (unspecified) member of the one-parameter family of hard-core Zolotarev polynomials. An extremal such polynomial as well as the constant \(M_n\) we have explicitly specified for \(n=4\) in [17], and in this paper we strive to obtain an analogous amendment to the Erdos-Szego solution for \(n = 5\). The cases \(n>5\) still remain arcane.Our approach is based on the quite recently discovered explicit algebraic power form representation [6], [7] of the quintic hard-core Zolotarev polynomial, \(Z_{5,t}\), to which we add here explicit descriptions of its critical points, the explicit form of Pell's (aka: Abel's) equation, as well as an alternative proof for the range of the parameter, \(t\). The optimal \(t=t^*\) which yields \(M_5 = |Z_{5,t^*}^{(1)}(1)|/25\) we identify as the negative zero with smallest modulus of a minimal \(P_{10}\). We then turn to an extension of (i), to higher derivatives as proposed by Shadrin [22], and we provide an analogous solution for \(n=5\). Finally, we describe, again for \(n = 5\), two new algebraic approaches towards a solution to Zolotarev's so-called first problem [2], [24] which was originally solved by means of elliptic functions.

Closed-loop identification of unstable systems using noncausal FIR models

ABSTRACTNoncausal finite impulse response (FIR) models are used for closed-loop identification of unstable multi-input, multi-output plants. These models are shown to approximate the Laurent series inside the annulus between the asymptotically stable pole of the largest modulus and the unstable pole of the smallest modulus. By delaying the measured output relative to the measured input, the identified FIR model is a noncausal approximation of the unstable plant. We present examples to compare the accuracy of the identified model obtained using least squares, instrumental variables methods, and prediction error methods for both infinite impulse response (IIR) and noncausal FIR models under arbitrary noise that is fed back into the loop. Finally, we reconstruct an IIR model of the system from its stable and unstable parts using the eigensystem realisation algorithm.

Estimates on the Lower Bound of the Eigenvalue of the Smallest Modulus Associated with a General Weighted Sturm-Liouville Problem

We obtain a lower bound on the eigenvalue of smallest modulus associated with a Dirichlet problem in the general case of a regular Sturm-Liouville problem. The main motivation for this study is the result obtained by Mingarelli (1988).

Spectral asymmetry of the massless Dirac operator on a 3-torus

Consider the massless Dirac operator on a 3-torus equipped with Euclidean metric and standard spin structure. It is known that the eigenvalues can be calculated explicitly: the spectrum is symmetric about zero and zero itself is a double eigenvalue. The aim of the paper is to develop a perturbation theory for the eigenvalue with smallest modulus with respect to perturbations of the metric. Here the application of perturbation techniques is hindered by the fact that eigenvalues of the massless Dirac operator have even multiplicity, which is a consequence of this operator commuting with the antilinear operator of charge conjugation (a peculiar feature of dimension 3). We derive an asymptotic formula for the eigenvalue with smallest modulus for arbitrary perturbations of the metric and present two particular families of Riemannian metrics for which the eigenvalue with smallest modulus can be evaluated explicitly. We also establish a relation between our asymptotic formula and the eta invariant.

Neural-Network-Based Approach for Extracting Eigenvectors and Eigenvalues of Real Normal Matrices and Some Extension to Real Matrices

This paper introduces a novel neural-network-based approach for extracting some eigenpairs of real normal matrices of ordern. Based on the proposed algorithm, the eigenvalues that have the largest and smallest modulus, real parts, or absolute values of imaginary parts can be extracted, respectively, as well as the corresponding eigenvectors. Although the ordinary differential equation on which our proposed algorithm is built is onlyn-dimensional, it can succeed to extractn-dimensional complex eigenvectors that are indeed 2n-dimensional real vectors. Moreover, we show that extracting eigen-pairs of general real matrices can be reduced to those of real normal matrices by employing the norm-reducing skill. Numerical experiments verified the computational capability of the proposed algorithm.

Note on the Smallest Root of the Independence Polynomial

One can define the independence polynomial of a graph G as follows. Let ik(G) denote the number of independent sets of size k of G, where i0(G)=1. Then the independence polynomial of G is I(G,x)=∑k=0n(−1)kik(G)xk. In this paper we give a new proof of the fact that the root of I(G,x) having the smallest modulus is unique and is real.

A covering system whose smallest modulus is 40

Text Paul Erdős, in 1950, asked whether for each positive integer N there exists a finite set of congruence classes, with distinct moduli, covering the integers, whose smallest modulus is N. In this vein, we construct a covering system of the integers with smallest modulus N = 40 . Video For a video summary of this paper, please visit http://www.youtube.com/watch?v=3ev1YjVl0RY.

Effect of oligomer length on the buckling of long and high aspect ratio microwalls UV embossed from oligomer/monomer mixtures

Abstract High aspect ratio microwalls separating long and deep microchannels are important for high sensitivity and high throughput microfluidic devices. Ultraviolet (UV) embossing is a feasible replication method for such microstructures. But buckling easily happens with thin and tall UV embossed microwalls with some formulations due to shrinkage stress. In this paper oligomer/monomer mixtures containing principally (68 wt.%) bisphenol A diacrylate with different degrees of ethoxylation (0, 3, 10 and 30) were used to study microwall buckling; these mixtures were denoted by E0, E3, E10 and E30, respectively. The effect of oligomer length on shrinkage, shrinkage stress, cross-linking kinetics and modulus and their inter-relations was studied. The fugitive multi-acrylated monomers present (trimethylolpropane triacrylate (20%) and dipropylene glycol diacrylate (10%)) caused the four mixtures to exhibit similar ultimate shrinkage stresses; this is contrary to the observation that long oligomers develop low shrinkage stress in neat oligomer systems. However, the E30 mixture network has the smallest modulus. The high shrinkage stress and low modulus of E30 mixture cross-linked network explains the buckling of long and high aspect ratio UV microwalls UV embossed from it. The in-plane shrinkage stress in the cured thin film was found to be the integrated effect of modulus and shrinkage rate built-up: only at high conversions did the stress developed drastically. Almost 100% conversion could be reached with the more flexible and mobile E10 and E30. E10 and E30 also have higher shrinkage values than E0 and E3.

The Mechanical Properties of Common Interlevel Dielectric Films and Their Influences on Aluminum Interconnect Extrusions

AbstractKnowledge of the mechanical properties of interlevel dielectric films and their impact on sub-micron interconnect reliability is becoming more and more important as critical dimensions in ULSI circuits are scaled down. For example, lateral aluminum (Al) extrusions into spaces between metal lines, which become a more of a concern as the pitches shrink, appear to depend partially on properties of SiO2 underlayers. In this paper, the mechanical properties of several common interlevel dielectric SiO2 films such as undoped silica glass using a silane (SiH4) precursor, undoped silica glass using a tetraethylorthosilicate (TEOS) precursor, phosphosilicate glass (PSG) deposited by plasma-enhanced chemical vapor deposition (PECVD) and borophosphosilicate glass (BPSG) deposited by sub-atmosphere chemical vapor deposition (SACVD) were studied. Among the four common interlevel layers, BPSG exhibits the smallest modulus (E), hardness (H) and the highest the coefficients of thermal expansion (CTE). BPSG again has the lowest as-deposited compressive stress and the lowest local Si-O-Si strain before annealing. Stress interactions between the various SiO2 underlayers and the Al metal film are further investigated. The impact of dielectric elastic properties on interconnect reliability during thermal cycles is proposed.

Matrix-geometric invariant measures for G/M/l type Markov chains

Necessary and sufficient conditions for an irreducible Markov chain of G/M/l type to have an invariant measure that is matrix-geometric are given. For example, it is shown that such an invariant measure exists when a(z), the generating function corresponding to transitions in the homogeneous part of the chain, is either an entire function or a rational function. This generalizes a recent result of Latouche, Pearce and Taylor, who showed that a matrix-geometric invariant measure always exists for level-independent quasi-birth-and-death processes. Conditions ensuring the uniqueness of such an invariant measure up to multiplication by a positive constant are also given. Examples of G/M/l type Markov chains with no matrix-geometric invariant measure and with more than one distinct matrix-geometric invariant measure are presented. As a byproduct of our work, it is shown in the transient case that if det has a solution in the exterior of the closed unit disk, then the solution of smallest modulus there is real and positive.

An inexact inverse iteration for large sparse eigenvalue problems

In this paper, we propose an inverse inexact iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the linear convergence of inverse iteration. We also propose two practical formulas for the accuracy bound which are used in actual implementation. © 1997 John Wiley & Sons, Ltd.

An inexact inverse iteration for large sparse eigenvalue problems

In this paper, we propose an inverse inexact iteration method for the computation of the eigenvalue with the smallest modulus and its associated eigenvector for a large sparse matrix. The linear systems of the traditional inverse iteration are solved with accuracy that depends on the eigenvalue with the second smallest modulus and iteration numbers. We prove that this approach preserves the linear convergence of inverse iteration. We also propose two practical formulas for the accuracy bound which are used in actual implementation. © 1997 John Wiley & Sons, Ltd.

On natural exactly covering systems of congruences having moduli occuring at most M times

In this paper we generalize to NECS's( M), certain results obtained in [2] for NECS's(2), namely the boundedness of (a) the greatest prime divisor of any modulus, (b) the number of disjoint systems and of (c) the smallest modulus.

Real dominant zero of a Hurwitz polynomial

A method presenting a very close approximation to a real dominant zero of a Hurwitz polynomial is given. The procedure is based on the Ostrowski method for calculating the zero of smallest modulus of a polynomial.

The remainder in Watson's lemma

Estimates are found for the remainder at the term of smallest modulus in the asymptotic expansions of integrals of a general type. When there is approximately a constant shift of phase between consecutive terms the remainder is easily estimated; in particular, in the case where the terms alternate in sign. The case where all term s have the same sign corresponds to a singularity on the positive real axis, and the same asymptotic expansion corresponds to different functions according as the integral is a principal value (in a somewhat extended sense) or on a loop. The two definitions may differ by a considerable multiple of the smallest term.