Eigenvalues Λ1 Research Articles

Conicting nonlinearities and spectral lacunae bounded on one side by eigenvalues

This paper deals with eigenvalue problems of the form where 0 < σ < τ and V(x) is such that the spectrum of −u″ consists of eigenvalues λ1, λ2,…situated below the continuous spectrum [Λ,+∞[.We analyse the existence of (multiple) solutions for λ < λ1 as well as for λ > λ1 when λ is in a spectral lacuna.The existence of solutions depends on the weight of μ > 0. Moreover, when λ increases (while μ is kept fixed), some solutions are lost when crossing eigenvalues.The above results are derived with the help of an abstract approach based on variational techniques for multiple solutions. This approach can even be applied to a wider class of problems, the one presented herein being only a model problem.

Eigenvalue Analysis for a Frictional Punch Problem of Concave Notch

A numerical solution for the eigenvalue in a frictional punch problem is presented, One edge of a concave notch is assumed to be traction free, and the other edge is in contact with the frictional punch ( Fig, 1), In addition to the normal pressure the friction force is also applied along the contact edge of punch. The friction coefficient δ is assumed within the range – 0.3 ⩾ 6 ⩾ 0,3. The notch angle β between two edges is varying within the range π ⩾ β ⩾ 2π. The target function method is used to evaluate the eigenvalue λ1 in the stress expression σij ≈ O(1/rλ1). Finally, the obtained eigenvalues are expressed in the form of λ1 = s(δ, β). The pressure under punch and the stresses at the corner points are singular in general.

A Robin problem for a class of quasilinear operators and a related minimizing problem

In this paper we establish the existence of multiple radial solutions for a class of quasilinear operators with nonlinear boundary Robin conditions. Besides other conditions, we consider the nonlinearities having a behavior at -∞ at least like a linearity of slope less than first eigenvalue λ1(R). The technical approach is by variational methods, which is mainly based on a version of Mountain Pass Theorem due to Ghoussoub and Preiss.

Upper Bounds for the Laplacian Graph Eigenvalues

We first apply non-negative matrix theory to the matrix K=D+A, where D and A are the degree-diagonal and adjacency matrices of a graph G, respectively, to establish a relation on the largest Laplacian eigenvalue λ1(G) of G and the spectral radius ρ(K) of K. And then by using this relation we present two upper bounds for λ1(G) and determine the extrernal graphs which achieve the upper bounds.

Resolution of the Symmetric Nonnegative Inverse Eigenvalue Problem for Matrices Subordinate to a Bipartite Graph

There is a symmetric nonnegative matrix A, subordinate to a given bipartite graph G on n vertices, with eigenvalues λ1≥λ2≥≥λ n if and only if, λ1 + λ n ≥0, λ2 + λ n-1≥0,...,λ m + λ n - m + 1≥0, λ m + 1≥0,...,λ n - m ≥0, in which m is the matching numberof G. Other observations are also made about the symmetric nonnegative inverse eigenvalue problem with respect to a graph

The spectral representation of Bessel processes with constant drift: applications in queueing and finance

Bessel processes with constant negative drift have recently appeared as heavy-traffic limits in queueing theory. We derive a closed-form expression for the spectral representation of the transition density of the Bessel process of order ν > −1 with constant drift μ ≠ 0. When ν > -½ and μ < 0, the first term of the spectral expansion is the steady-state gamma density corresponding to the zero principal eigenvalue λ0 = 0, followed by an infinite series of terms corresponding to the higher eigenvalues λn, n = 1,2,…, as well as an integral over the continuous spectrum above μ2/2. When −1 < ν < -½ and μ < 0, there is only one eigenvalue λ0 = 0 in addition to the continuous spectrum. As well as applications in queueing, Bessel processes with constant negative drift naturally lead to two new nonaffine analytically tractable specifications for short-term interest rates, credit spreads, and stochastic volatility in finance. The two processes serve as alternatives to the CIR process for modelling mean-reverting positive economic variables and have nonlinear infinitesimal drift and variance. On a historical note, the Sturm–Liouville equation associated with Bessel processes with constant negative drift is closely related to the celebrated Schrödinger equation with Coulomb potential used to describe the hydrogen atom in quantum mechanics. Another connection is with D. G. Kendall's pole-seeking Brownian motion.

Largest eigenvalue of a unicyclic mixed graph

The graphs which maximize and minimize respectively the largest eigenvalue over all unicyclic mixed graphs U on n vertices are determined. The unicyclic mixed graphs U with the largest eigenvalue λ1(U)=n or λ1(U)e(n,n+1) are characterized.

Dispersion of Singularities of Solutions for Schrödinger Equations

We consider the singularities of solutions for the Schrodinger evolution equation associated with where Q is a d×d real symmetric matrix with the eigenvalues λ1,⋯,λ d , and W ∈ C∞(R d ,R) satisfies W(x)=o(|x|2) as |x|→∞. Under additional conditions, we show the dispersion of microlocal singularities of solutions due to the principal symbol in all directions at time and in the nondegenerate directions at t ∈ Σ. We also show the weaker dispersion of microlocal singularities of solutions due to the subprincipal symbol W in the degenerate directions at t ∈ Σ if W satisfies W(x)=O(|x|1+δ δ ) as |x|→∞ for some 0<δ<1 and additional conditions. In particular, we prove the dispersion of microlocal singularities of solutions at resonant times when H is a perturbed harmonic oscillator.

Eigenfunctions and Hardy inequalities for a magnetic Schrödinger operator in ℝ2

AbstractThe zero set {z∈ℝ2:ψ(z)=0} of an eigenfunction ψ of the Schrödinger operator ℒ︁V=(i∇+A)2+V on L2(ℝ2) with an Aharonov–Bohm‐type magnetic potential is investigated. It is shown that, for the first eigenvalue λ1 (the ground state energy), the following two statements are equivalent: (I) the magnetic flux through each singular point of the magnetic potential A is a half‐integer; and (II) a suitable eigenfunction ψ associated with λ1 (a ground state) may be chosen in such a way that the zero set of ψ is the union of a finite number of nodal lines (curves of class C2) which emanate from the singular points of the magnetic potential A and slit the two‐dimensional plane ℝ2. As an auxiliary result, a Hardy‐type inequality near the singular points of A is proved. The C2 differentiability of nodal lines is obtained from an asymptotic analysis combined with the implicit function theorem. Copyright © 2003 John Wiley &amp; Sons, Ltd.

NONRESONANCE BELOW THE SECOND EIGENVALUE FOR A NONLINEAR ELLIPTIC PROBLEM

We study the solvability of the problem −∆pu = g(x, u) + h in Ω; u = 0 on ∂Ω, when the nonlinearity g is assumed to lie asymptotically between 0 and the second eigenvalue λ2 of −∆p. We show that this problem is nonresonant.

Minimal Lagrangian submanifolds in adjoint orbits and upper bounds on the first eigenvalue of the Laplacian

Let G be a compact semisimple Lie group, g its Lie algebra, (,) an AdG-invariant inner product on g, and M an adjoint orbit in 9. In this article, if (M,(,)|M) is Kähler with respect to its canonical complex structure, then we give, for a closed minimal Lagrangian submanifold L⊂M, upper bounds on the first positive eigenvalue λ1(L) of the Laplacian ΔL, which acts on C∞(L), and lower bounds on the volume of L. In particular, when (M,(,)|M) is Kähler-Einstein, (p=cω, where p and ω are Ricci form and Kähler form of (M,(,)|M) with respect to the canonical complex structure respectively, and c is a positive constant,) we prove λ1(L)≤c. Combining with a result of Oh [5], we can see that L is Hamiltonian stable if and only if λ1(L)=c.

On the Number and Structure of Solutions for a Fredholm Alternative with the p-Laplacian

We investigate the existence and multiplicity of weak solutions u∈W01,p(Ω) to the degenerate quasilinear Dirichlet boundary value problem(P) −Δpu=λ1∣u∣p−2u+fT(x)+ζ·ϕ1(x) inΩu=0 on ∂Ω, where ζ∈R is a parameter. It is assumed that 1<p<∞, p≠2, and Ω is a bounded domain in RN. The number λ1 stands for the first (smallest) eigenvalue of the positive p-Laplacian − Δp, where Δpu≡div(∣ ∇u∣p−2∇u). The eigenvalue λ1 being simple, let ϕ1 denote the eigenfunction associated with λ1. Furthermore, fT∈L∞(Ω) is a given function which is assumed to be L2-orthogonal to ϕ1 and fT≢0 in Ω. We show the existence of a solution for problem (P) precisely when the parameter ζ satisfies ζ*⩽ζ⩽ζ*, for some numbers −∞<ζ*<0<ζ*<∞ depending on fT. Otherwise, nonexistence occurs. Moreover, we show that problem (P) possesses at least two distinct solutions provided ζ#<ζ<ζ# and ζ≠0, for some numbers ζ# and ζ# such that ζ*⩽ζ#<0<ζ#⩽ζ*. Finally, given any δ≷0, we show that the set of all weak solutions to problem (P) is bounded in C1(Ω) uniformly for ∣ζ∣⩾δ and for ζ=0 as well. Precise asymptotic behavior (blow-up) of every solution is given as ζ→0 (ζ≠0) using the linearization of equation (P) about ϕ1.

Extremal Chemical Trees

Avariety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W, the largest graph eigenvalue λ1, the connectivity index χ, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W, λ1, E, and Z, whereas the analogous problem for χ was solved earlier. Among chemical trees with 5, 6, 7, and 3k + 2 vertices, k = 2, 3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k + 1 vertices, k = 3, 4..., one tree has minimum W and maximum λ1 and another minimum E and Z.

Asymptotic Behavior of a Nonhomogeneous Linear Recurrence System

Consider the nonhomogeneous linear recurrence systemxn+1=(A+Bn)xn+gn,where A and Bn (n=0,1,…) are square matrices and gn (n=0,1,…) are column vectors. In this paper, we describe, in terms of the initial condition, the asymptotic behavior of the solutions of this equation in the case when A has a simple dominant eigenvalue λ0,∑n=0∞⫫Bn⫫<∞ and ∑n=0∞|λ0|−n⫫gn⫫<∞. The proof is based on the duality between the solutions of the above equation and the solutions of the associated adjoint equation. As a consequence, we obtain a similar result for higher order scalar equations.

Extremal Chemical Trees

A variety of molecular-graph-based structure-descriptors were proposed, in particular the Wiener index W. the largest graph eigenvalue λ1, the connectivity index X, the graph energy E and the Hosoya index Z, capable of measuring the branching of the carbon-atom skeleton of organic compounds, and therefore suitable for describing several of their physico-chemical properties. We now determine the structure of the chemical trees (= the graph representation of acyclic saturated hydrocarbons) that are extremal with respect to W , λ1, E, and Z. whereas the analogous problem for X was solved earlier. Among chemical trees with 5. 6, 7, and 3k + 2 vertices, k = 2,3,..., one and the same tree has maximum λ1 and minimum W, E, Z. Among chemical trees with 3k and 3k +1 vertices, k = 3,4...., one tree has minimum 11 and maximum λ1 and another minimum E and Z .

Constructions of trace zero symmetric stochastic matrices for the inverse eigenvalue problem

In the special case of where the spectrum σ = {λ1 ,λ 2 ,λ 3, 0, 0 ,..., 0} has at most three nonzero eigenvalues λ1 ,λ 2 ,λ 3 with λ1 ≥ 0 ≥ λ2 ≥ λ3 ,a ndλ1 + λ2 + λ3 =0 , the inverse eigenvalue problem for symmetric stochastic n × n matrices is solved. Constructions are provided for the appropriate matrices where they are readily available. It is shown that when n is odd it is not possible to realize the spectrum σ with an n × n symmetric stochastic matrix when λ3 �nd 3 2n−3 > λ2 λ3 ≥ 0, andit is shown that this boundis best possible.

The opial inequality in RN

This paper discusses the minimal eigenvalue λ1(p,q,Ω) and its best estimate of the following nonlinear eigenvalue problem -Δpu=λ|u|q-2u, u|∂Ω=0.Where Ω is a bounded smooth domain in RN, Δp is the p-laplace operator, p≥2, q∈ [p,NpN-p). By this result, the famous Opial inequality in RN are deduced.

Population dynamics and stage structure in a haploid‐diploid red seaweed, Gracilaria gracilis

Summary Many red seaweeds are characterized by a haploid‐diploid life cycle in which populations consist of dioecious haploid (gametophyte) and diploid (tetrasporophyte) individuals as well as an additional diploid zygote‐derived sporangium (carposporophyte) stage. A demographic analysis of Gracilaria gracilis populations was carried out to explore and evaluate the population dynamics and stage structure of a typical haploid‐diploid red seaweed. Four G. gracilis populations were studied at two sites on the French coast of the Strait of Dover. Survival, reproduction and recruitment rates were measured in each population for up to 4 years. Eight two‐sex stage‐based population projection matrices were built to describe their demography. All four populations were characterized by high survival and low recruitment rates. Population growth rates (λ) were similar between populations and between years and ranged from 1.03 to 1.17. In addition, generation times were found to be as long as 42 years. Sex and ploidy ratios were variable across populations and over time. Female frequencies ranged from 0.31 to 0.59 and tetrasporophyte frequencies from 0.44 to 0.63. However, in most cases, the observed population structures were not significantly different from the calculated stage distributions. Eigenvalue elasticity analysis showed that λ was most sensitive to changes in matrix transitions that corresponded to survival of the gametophyte and tetrasporophyte stages. In contrast, the contribution of the fertility elements to λ was small. Eigenvector elasticity analysis also showed that survival elements had the greatest impact on sex and ploidy ratios.

Semilinear hemivariational inequalities at resonance

In this paper we examine a semilinear hemivariational inequality at resonance in the first eigenvalue λ1 of (−Δ,H 0 1 (Z)). We prove two existence theorems for such problems. Our approach is variational and is based on the nonsmooth critical point theory of Chang, which uses the subdifferential calculus of Clarke for locally Lipschitz functions.

On the gap between the first eigenvalues of the Laplacian on functions and 1-forms

We study the first positive eigenvalue λ1(p) of the Laplacian on p-forms for oriented closed Riemannian manifolds. It is known that the inequality λ1(1)≤λ1(0) holds in general. In the present paper, a Riemannian manifold is said to have the gap if the strict inequality λ1(1)<λ1(0) holds. We show that any oriented closed manifold M with the first Betti number b1(M)=0 whose dimension is bigger than two, admits two Riemannian metrics, the one with the gap and the other without the gap.