Sturm's Theorem Research Articles

Nodal domain theorems for $p$-Laplacians on signed graphs

We establish various nodal domain theorems for p -Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph p -Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of p -Laplacians and Atay–Liu's multi-way Cheeger constants on signed graphs. In the particular case of p=1 , this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph 1 -Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of p -Laplacians with p>1 on connected bipartite graphs.

Sturm's Theorem for Min matrices

<abstract><p>In the present paper, we study Min matrix $ \mathcal{A}_{min} = \left[a_{min\left(i, j\right)}\right]_{i, j = 1}^n $, where $ a_s $'s are the elements of a real sequence $ \left\lbrace a_s\right\rbrace $. We first obtain a recurrence relation for the characteristic polynomial for matrix $ \mathcal{A}_{min} $, and some relations between the coefficients of its characteristic polynomial. Next, we show that the sequence of the characteristic polynomials of the $ i \times i \left(i \leq n\right) $ Min matrices satisfies the Sturm sequence properties according to different required conditions of the sequence $ \left\lbrace a_s\right\rbrace $. Using Sturm's Theorem, we get some results about the eigenvalues, such as the number of eigenvalues in an interval. Thus, we obtain the number of positive and negative eigenvalues of Min matrix $ \mathcal{A}_{min} $. Finally, we give an example to illustrate our results.</p></abstract>

A refined Gallai-Edmonds structure theorem for weighted matching polynomials

In this work, we prove a refinement of the Gallai-Edmonds structure theorem for weighted matching polynomials by Ku and Wong. Our proof uses a connection between matching polynomials and branched continued fractions. We also show how this is related to a modification by Sylvester of the classical Sturm's theorem on the number of zeros of a real polynomial in an interval. In addition, we obtain some other results about zeros of matching polynomials.

Analysis of Equilibria for a Class of Recurrent Neural Networks With Two Subnetworks.

This article is concerned with the problem of the number and dynamical properties of equilibria for a class of connected recurrent networks with two switching subnetworks. In this network model, parameters serve as switches that allow two subnetworks to be turned ON or OFF among different dynamic states. The two subnetworks are described by a nonlinear coupled equation with a complicated relation among network parameters. Thus, the number and dynamical properties of equilibria have been very hard to investigate. By using Sturm's theorem, together with the geometrical properties of the network equation, we give a complete analysis of equilibria, including the existence, number, and dynamical properties. Necessary and sufficient conditions for the existence and exact number of equilibria are established. Moreover, the dynamical property of each equilibrium point is discussed without prior assumption of their locations. Finally, simulation examples are given to illustrate the theoretical results in this article.

CONDITIONS OF MONOTONE APPROXIMATION OF RAMSEY CURVES AND THEIR MODIFICATIONS

The algorithm for approximating the experimental data of the Ramsey curve and its modifications has been developed, which provides a monotonic increase of the approximating function in the interval [0;\infty) and an existence of a given number of inflection points. The Ramsey curve belongs to the family of logistic curves that are widely used in modeling of limited increasing processes in various subject fields. The classical Ramsey curve has two parameters and has a left constant asymmetry. It is also known that its three-parameter modification provides the possibility of displacement along the axes of ordinate. The extensive practical use of the Ramsey curve with both two and more parameters for approximating experimental dependences is restrained by the frequent loss by this curve of the logistic shape when approximating without additional restrictions on the relationships between its parameters. The article discusses modifications of the Ramsey curve with three and five parameters. The first and second derivatives of the studied modifications of the Ramsey function have a special structure. They are products of polynomial and exponential functions. This allows using Sturm's theorem on the number of polynomial roots in a given interval to control the shape of the approximating curve. It has been shown that with an increase in the number of parameters for the modified curve, the number of possible combinations of restrictions on the values of the parameters ensuring the preservation of its like shape increases significantly. The solution to the approximation problem in this case consists of solving a sequence of conditional global optimization problems with different constraints and choosing a solution that provides the smallest approximation error. Also, the studies of the accuracy of estimating the parameters of the Ramsey curve in accordance with the accuracy of the experimental data have been carried out. In order to simulate the presence of measurement errors, the values of a normally distributed random variable with a mathematical expectation equal to zero and different values of the standard deviation for different series of computational experiments were added to the values of the deterministic sequence. Computational experiments have shown a significant sensitivity of the values of the Ramsey function parameters to the measurement accuracy of experimental data.

Sturm theorem for the generalized Frank matrix

One of the popular test matrices for eigenvalue routines is the Frank matrix due to its well-conditioned and poorly conditioned eigenvalues. All the eigenvalues of the Frank matrix are real, positive and different. Sturm Theorem is a very useful tool for computing the eigenvalues of tridiagonal symmetric matrices. In this paper, we apply Sturm Theorem to the generalized Frank matrix which is a special form of the Hessenberg matrix and examine its eigenvalues by using Sturm property. Moreover, we illustrate our results with an example.

Hörmander's index and oscillation theory

One of the obstacles that arises in the generalization of Sturm's oscillation theorem to the case of general linear Hamiltonian systems is the need to associate a sign with each crossing point, necessitating a signed count rather than a direct count of such points. It's known, however, that this difficulty does not arise for all system/boundary-condition combinations, and so it can be overcome in some cases by exchanging one boundary condition for another while also keeping track of any ancillary counts that accompany the exchange. The primary tool for making such an exchange is Hörmander's index, and in this analysis we develop a straightforward method for computing Hörmander's index, and employ our method to formulate oscillation-type theorems for linear Hamiltonian systems on both bounded and unbounded intervals.

Sturm theory with applications in geometry and classical mechanics

Classical Sturm non-oscillation and comparison theorems as well as the Sturm theorem on zeros for solutions of second order differential equations have a natural symplectic version, since they describe the rotation of a line in the phase plane of the equation. In the higher dimensional symplectic version of these theorems, lines are replaced by Lagrangian subspaces and intersections with a given line are replaced by non-transversality instants with a distinguished Lagrangian subspace. Thus the symplectic Sturm theorems describe some properties of the Maslov index. Starting from the celebrated paper of Arnol’d on symplectic Sturm theory for optical Hamiltonians, we provide a generalization of his results to general Hamiltonians. We finally apply these results for detecting some geometrical information about the distribution of conjugate and focal points on semi-Riemannian manifolds and for studying the geometrical properties of the solutions space of singular Lagrangian systems arising in Celestial Mechanics.

A converse of Sturm's separation theorem

We show that Sturm's classical separation theorem on the interlacing of the zeros of linearly independent solutions of real second order two-term ordinary differential equations necessarily fails in the presence of a turning point in the principal part of the equation. Related results are discussed.

A hyperbolic two-fluid model for compressible flows with arbitrary material-density ratios

Abstract

Sturm's Theorem on the Zeros of Sums of Eigenfunctions: Gelfand's Strategy Implemented

In the second section ``Courant-Gelfand theorem'' of his last published paper (Topological properties of eigenoscillations in mathematical physics, Proc. Steklov Institute Math. 273 (2011) 25--34), Arnold recounts Gelfand's strategy to prove that the zeros of any linear combination of the $n$ first eigenfunctions of the Sturm-Liouville problem $$-\, y''(s) + q(x)\, y(x) = \lambda\, y(x) \mbox{ in } ]0,1[\,, \mbox{ with } y(0)=y(1)=0\,,$$ divide the interval into at most $n$ connected components, and concludes that ``the lack of a published formal text with a rigorous proof \dots is still distressing.''\\ Inspired by Quantum mechanics, Gelfand's strategy consists in replacing the ana\-lysis of linear combinations of the $n$ first eigenfunctions by that of their Slater determinant which is the first eigenfunction of the associated $n$-particle operator acting on Fermions.\\ In the present paper, we implement Gelfand's strategy, and give a complete proof of the above assertion. As a matter of fact, refining Gelfand's strategy, we prove a stronger property taking the multiplicity of zeros into account, a result which actually goes back to Sturm (1836).

Real-Time Switching Angle Computation for Selective Harmonic Control

A two-stage algorithm is proposed to compute switching angles for selective harmonic elimination or selective harmonic control. In the first offline computing stage, the multivariate nonlinear equations are equivalently converted to a group of univariate linear equations by using the theories of symmetric polynomials and Grobner bases, which are then stored into the target microcontrollers (MCUs) and solved online. In the second online computing stage, the linear equations are solved, and their solutions are used to construct a polynomial whose real roots are actually the final solutions; then, an algebraic-numerical hybrid method is developed to solve the constructed polynomial, which uses the Sturm's theorem to bracket each real root into an individual interval, and then, the bisection method is used to find all the exact real roots. Compared to the traditional numerical methods, the executing efficiency is increased significantly. This method is implemented on several commonly used MCUs and shows very good real-time performance. Experiments verify the correctness of the solved switching angles and demonstrate the feasibility of applying this method in the real control system.

Determining the limits of bivariate rational functions by Sturm's theorem

In this paper, we present an algorithm for determining the limits of real rational functions in two variables, based on Sturm's familiar theorem and the general Sturm–Tarski theorem for counting certain roots of univariate polynomials in a real closed field. Let R[x,y] be the ring of polynomials with real coefficients in two variables x, y, and let u(x,y), v(x,y)∈R[x,y] be two non-zero polynomials such that u(a,b)=v(a,b)=0 for a, b∈R. The purpose of this paper is to decide the existence of lim(x,y)→(a,b)⁡u(x,y)v(x,y) and compute the limit if it exists. Our algorithm needs no assumption on the denominators and does not involve the computation of Puiseux series.

Dynamic Picone's identity and its applications

We prove some Picone-type identities and inequalities for a class of first-order nonlinear dynamic systems and derive various weighted inequalities of Wirtinger type and Hardy type on time scales. As applications we study oscillatory and related properties of these systems including Reid's roundabout theorem on disconjugacy, Sturm's separation and comparison theorems, as well as a variational method in the oscillation theory.

A Decision Procedure for Univariate Polynomial Systems Based on Root Counting and Interval Subdivision.

This paper presents a formally verified decision procedure for determinining the satisfiability of a system of univariate polynomial relations over the real line. The procedure combines a root counting function, based on Sturm's theorem, with an interval subdivision algorithm. Given a system of polynomial relations over the same variable, the decision procedure progressively subdivides the real interval into smaller intervals. The subdivision continues until the satisfiability of the system can be determined on each subinterval using Sturm's theorem on a subset of the system's polynomials. The decision procedure has been formally verified in the Prototype Verification System (PVS). In PVS, the decision procedure is specified as a computable boolean function on a deep embedding of polynomial relations. This function is used to define a proof producing strategy for automatically proving existential and universal statements on polynomial systems. The soundness of the strategy solely depends on the internal logic of PVS.

On the application of Sturm's theorem to analysis of dynamic pull-in for a graphene-based MEMS model

A novel procedure based on the Sturm’s theorem for real-valued polynomials is developed to predict and identify periodic and non-periodic solutions for a graphene-based MEMS lumped parameter model with general initial conditions. It is demonstrated that under specific conditions on the lumped parameters and the initial conditions, the model has certain periodic solutions and otherwise there are no such solutions. This theoretical procedure is made practical by numerical implementations with Python scripts to verify the predicted behaviour of the solutions. Numerical simulations are performed with sample data to justify by this procedure the analytically predicted existence of periodic solutions.

Multiple-rank modification of symmetric eigenvalue problem

Rank-1 modifications applied k-times (k?>?1) often are performed to achieve a rank-k modification. We propose a rank- k modification for enhancing computational efficiency. As the first step toward a rank- k modification, an algorithm to perform a rank-2 modification is proposed and tested. The computation cost of our proposed algorithm is in O(n2) where n is the cardinality of the matrix of interest. We also propose a general rank-k update algorithm based on the Sturm Theorem, and compare our results to those of the direct eigenvalue decomposition and of a perturbation method.

Sturm theorems and distance between adjacent zeros for second order integro-differential equations

Sturm theorems and distance between adjacent zeros for second order integro-differential equations

Much ado about Mathieu

Eguchi, Ooguri and Tachikawa observed that the coefficients of the elliptic genus of type II string theory on K3 surfaces appear to be dimensions of representations of the largest Mathieu group. Subsequent work by several people established a candidate for the elliptic genus twisted by each element of M24. In this paper we prove that the resulting sequence of class functions are true characters of M24, proving the Eguchi–Ooguri–Tachikawa ‘Mathieu Moonshine’ conjecture. The integrality of multiplicities is proved using a small generalisation of Sturm's Theorem, while positivity involves a modification of a method of Hooley, for finding an effective bound on a family of Selberg–Kloosterman zeta functions at s=3/4. We also prove the evenness property of the multiplicities, as conjectured by several authors, and use that to investigate the proposal of Gaberdiel–Hohenegger–Volpato that Mathieu Moonshine lifts to the Conway groups Co0 and Co1. We identify the role group cohomology plays in both Mathieu Moonshine and Monstrous Moonshine; in particular this gives a cohomological interpretation for the non-Fricke elements in Norton's Generalised Monstrous Moonshine conjecture, and gives a lower bound for H3(BCo1;C×).

Application of Sturm's theorem to marginal stable circular orbits of a test body in spherically symmetric and static spacetimes

In terms of Sturm's theorem, we reexamine a marginal stable circular orbit (MSCO) such as the innermost stable circular orbit (ISCO) of a time-like geodesic in any spherically symmetric and static spacetime. MSCOs for some of the exact solutions to Einstein's equation are discussed. Sturm's theorem is explicitly applied to the Kottler (often called Schwarzschild-de Sitter) spacetime. Moreover, we analyze MSCOs for a spherically symmetric, static and vacuum solution in Weyl conformal gravity.