Spectral properties of two-parameter eigenvalue problems

1981 ◽  
Vol 89 (1-2) ◽  
pp. 157-173 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Asymptotic Results ◽  
Solution Sets ◽  
Image Position ◽  
Two Parameters ◽  
Necessary And Sufficient ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe study the self-adjoint eigenvalue problem W(λ)x = 0, (*), in Hilbert space for one equation in two parameters. Hereis bounded below with compact resolvent for each λ = (λ1, λ2). We give necessary and sufficient conditions for the existence of λ so that (*) holds with W(λ)= ≧0 and we investigate the geometry of the set Z0 of such λ. We also discuss higher order solution sets Zi where the ith eigenvalue of W(λ) vanishes, deriving various asymptotic results in a unified fashion.

Existence conditions for two-parameter eigenvalue problems

1981 ◽  
Vol 91 (1-2) ◽  
pp. 15-30 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne ◽  
Lawrence Turyn
Keyword(s):  
Hilbert Spaces ◽  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Compact Resolvent ◽  
Existence Conditions ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe discuss necessary and sufficient conditions for the existence of eigentuples λ=(λl,λ2) and eigenvectors x1≠0, x2≠0 for the problem Wr(λ)xr = 0, Wr(λ)≧0, (*), where Wr(λ)= Tr + λ1Vr2, r=1,2. Here Tr and Vrs are self-adjoint operators on separable Hilbert spaces Hr. We assume the Vrs to be bounded and the Tr bounded below with compact resolvent. Most of our conditions involve the conesWe obtain results under various conditions on the Tr, but the following is typical:THEOREM. If (*) has a solution for all choices ofT1, T2then (a)0∉ V1UV2,(b)V1∩(—V2) =∅ and (c) V1⊂V2∪{0}, V2⊈V1∪{0}. Conversely, if (a) and (b) hold andV1⊈V2∪∩{0}, V2⊈ then (*) has a solution for all choices ofT1, T2.

The embedding problem for Markov chains with three states

1980 ◽  
Vol 87 (2) ◽  
pp. 285-294 ◽  
Author(s):  
Halina Frydman
Keyword(s):  
Markov Chains ◽  
Stochastic Matrix ◽  
Sufficient Conditions ◽  
Parameter Family ◽  
Necessary And Sufficient Conditions ◽  
Embedding Problem ◽  
Stochastic Matrices ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Two Parameter

In this paper we consider the embedding problem for Markov chains with three states. A non-singular stochastic matrix P is called embeddable if there exists a two-parameter family of stochastic matricessatisfyingand such thatThough extensive characterizations of embeddable n × n stochastic matrices have been given in (l), (2), (3), (6), and further characterizations of embeddable 3 × 3 stochastic matrices in (4), they do not provide, except in the case of 2 × 2 stochastic matrices, easily applicable necessary and sufficient conditions for embeddability.

Spectral properties of a two parameter nonlinear Sturm-Liouville problem

1993 ◽  
Vol 123 (6) ◽  
pp. 1041-1058 ◽  
Author(s):  
Tetsutaro Shibata
Keyword(s):  
Level Set ◽  
Spectral Properties ◽  
Liouville Problem ◽  
General Level ◽  
Asymptotic Formulae ◽  
Sturm Liouville ◽  
Image Position ◽  
Two Parameters ◽  
Two Parameter

SynopsisWe consider the nonlinear Sturm–Liouville problem with two parameters on the general level setWe establish asymptotic formulae of the n-th variational eigenvalue λ = λn(μ, α) as α→∞ and α↓(nπ)2.

Spectral properties of two parameter eigenvalue problems II

1987 ◽  
Vol 106 (1-2) ◽  
pp. 39-51 ◽  
Author(s):  
Paul Binding ◽  
Patrick J. Browne
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Asymptotic Properties ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Adjoint Operators ◽  
Bounded Below ◽  
Two Parameter

SynopsisWe consider eigenvalues λ =(λ1, λ2) ∈R2 for the problem W(λ)x = 0, x ≠ 0, x ∈ H, where W(λ) = R + λ1V1 + λ2V2), and R, V1, V2 are self-adjoint operators on a separable Hilbert space H, R being bounded below with compact resolvent and V1, V2 being bounded. The i-th eigencurve Z1 is the set of eigenvalues λ, for which the i-th eigenvalue (counted according to multiplicity and in increasing order) of W(λ) vanishes. We study monotonic and asymptotic properties of Zi, and we give formulae for any asymptotes that exist. Additional results are given in the finite dimensional case.

scholarly journals Topological entropy in totally disconnected locally compact groups

2016 ◽  
Vol 37 (7) ◽  
pp. 2163-2186 ◽  
Author(s):  
ANNA GIORDANO BRUNO ◽  
SIMONE VIRILI
Keyword(s):  
Topological Entropy ◽  
Sufficient Conditions ◽  
Invariant Subgroup ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Totally Disconnected ◽  
Dynamical Interpretation

Let $G$ be a topological group, let $\unicode[STIX]{x1D719}$ be a continuous endomorphism of $G$ and let $H$ be a closed $\unicode[STIX]{x1D719}$-invariant subgroup of $G$. We study whether the topological entropy is an additive invariant, that is, $$\begin{eqnarray}h_{\text{top}}(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719}\restriction _{H})+h_{\text{top}}(\bar{\unicode[STIX]{x1D719}}),\end{eqnarray}$$ where $\bar{\unicode[STIX]{x1D719}}:G/H\rightarrow G/H$ is the map induced by $\unicode[STIX]{x1D719}$. We concentrate on the case when $G$ is totally disconnected locally compact and $H$ is either compact or normal. Under these hypotheses, we show that the above additivity property holds true whenever $\unicode[STIX]{x1D719}H=H$ and $\ker (\unicode[STIX]{x1D719})\leq H$. As an application, we give a dynamical interpretation of the scale $s(\unicode[STIX]{x1D719})$ by showing that $\log s(\unicode[STIX]{x1D719})$ is the topological entropy of a suitable map induced by $\unicode[STIX]{x1D719}$. Finally, we give necessary and sufficient conditions for the equality $\log s(\unicode[STIX]{x1D719})=h_{\text{top}}(\unicode[STIX]{x1D719})$ to hold.

Moving Weighted Averages

1993 ◽  
Vol 45 (3) ◽  
pp. 449-469 ◽  
Author(s):  
M. A. Akcoglu ◽  
Y. Déniel
Keyword(s):  
Weight Function ◽  
Sufficient Conditions ◽  
Nonnegative Function ◽  
Finite Measure ◽  
Weight Functions ◽  
Ergodic Averages ◽  
Fixed Weight ◽  
Finite Measure Space ◽  
Image Position ◽  
Necessary And Sufficient

AbstractLet ℝ denote the real line. Let {Tt}tєℝ be a measure preserving ergodic flow on a non atomic finite measure space (X, ℱ, μ). A nonnegative function φ on ℝ is called a weight function if ∫ℝ φ(t)dt = 1. Consider the weighted ergodic averagesof a function f X —> ℝ, where {θk} is a sequence of weight functions. Some sufficient and some necessary and sufficient conditions are given for the a.e. convergence of Akf, in particular for a special case in whichwhere φ is a fixed weight function and {(ak, rk)} is a sequence of pairs of real numbers such that rk > 0 for all k. These conditions are obtained by a combination of the methods of Bellow-Jones-Rosenblatt, developed to deal with moving ergodic averages, and the methods of Broise-Déniel-Derriennic, developed to deal with unbounded weight functions.

scholarly journals Sufficient conditions for matchings

1972 ◽  
Vol 18 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Ian Anderson
Keyword(s):  
Simple Proof ◽  
Perfect Matching ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Number Of Components ◽  
Image Position ◽  
Necessary And Sufficient

A graph G is said to possess a perfect matching if there is a subgraph of G consisting of disjoint edges which together cover all the vertices of G. Clearly G must then have an even number of vertices. A necessary and sufficient condition for G to possess a perfect matching was obtained by Tutte (3). If S is any set of vertices of G, let p(S) denote the number of components of the graph G – S with an odd number of vertices. Then the conditionis both necessary and sufficient for the existence of a perfect matching. A simple proof of this result is given in (1).

A necessary and sufficient condition for the weak lower semicontinuity of one-dimensional non-local variational integrals

2006 ◽  
Vol 136 (4) ◽  
pp. 701-708 ◽  
Author(s):  
Jonathan Bevan ◽  
Pablo Pedregal
Keyword(s):  
Lower Semicontinuous ◽  
Short Note ◽  
Necessary And Sufficient Condition ◽  
One Dimensional ◽  
Lower Semicontinuous Envelope ◽  
Variational Integrals ◽  
Non Local ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Bounded Below

In this short note we prove that the functional I : W1,p(J;R) → R defined by is sequentially weakly lower semicontinuous in W1,p(J,R) if and only if the symmetric part W+ of W is separately convex. We assume that W is real valued, continuous and bounded below by a constant, and that J is an open subinterval of R. We also show that the lower semicontinuous envelope of I cannot in general be obtained by replacing W by its separately convex hull Wsc.

On the rate of convergence of probabilities of moderate deviations

1970 ◽  
Vol 11 (1) ◽  
pp. 91-94 ◽  
Author(s):  
V. K. Rohatgi
Keyword(s):  
Sufficient Conditions ◽  
Moderate Deviations ◽  
Independent Random Variables ◽  
Standard Normal Distribution ◽  
Positive Real ◽  
Moment Conditions ◽  
Standard Normal ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Asymptotic Forms

Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

Necessary and sufficient conditions for asymptotic decay of oscillations in delayed functional equations

1981 ◽  
Vol 91 (1-2) ◽  
pp. 135-145
Author(s):  
Lu-San Chen ◽  
Cheh-Chih Yeh
Keyword(s):  
Differential Operator ◽  
Functional Equations ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Oscillatory Solutions ◽  
Asymptotic Decay ◽  
Image Position ◽  
Necessary And Sufficient

SynopsisThis paper studies the equationwhere the differential operator Ln is defined byand a necessary and sufficient condition that all oscillatory solutions of the above equation converge to zero asymptotically is presented. The results obtained extend and improve previous ones of Kusano and Onose, and Singh, even in the usual case wherewhere N is an integer with l≦N≦n–1.

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