Dual operator norms and the spectra of matrices

Let ∥·∥ be an operator norm and ∥·∥^D its dual. Then it is shown that ∥A∥^D? ∑|λ_i(A)|, where λ_i(A) are the eigenvalues of A, holds for all matrices A if and only if ∥·∥ is the operator norm subordinate to a Euclidian vector norm.     

A Hard-Core Model on a Cayley Tree: An Example of a Loss Network

The paper is about a nearest-neighbor hard-core model, with fugacity λ>0, on a homogeneous Cayley tree of order k(with k+1 neighbors). This model arises as as a simple example of a loss network with a nearest-neighbor exclusion. We focus on Gibbs measures for the hard core model, in particular on ‘splitting’ Gibbs measures generating a Markov chain along each path on the tree. In this model, ?λ>0 and k≥1, there exists a unique translation-invariant splitting Gibbs measure μ^*. Define λ_c=1/(k?1)×(k/(k?1)) ^k . Then: (i) for λ≤λ_c, the Gibbs measure is unique (and coincides with the above measure μ^*), (ii) for λ>λ_c, in addition to μ^*, there exist two distinct translation-periodic measures, μ₊and μ_?, taken to each other by the unit space shift. Measures μ₊and μ_?are extreme ?λ>λ_c. We also construct a continuum of distinct, extreme, non-translational-invariant, splitting Gibbs measures. For $\lambda >1/(\sqrt k - 1) \times (\sqrt k /\sqrt k - 1))^k $ , measure μ^*is not extreme (this result can be improved). Finally, we consider a model with two fugacities, λ_eand λ_o, for even and odd sites. We discuss open problems and state several related conjectures.     

Asymptotic behavior of solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0

The author discusses the asymptotic behavior of the solutions of the functional differential equation x′(t) = Ax(λt) + Bx(t), λ>0 (1) where x(t) is an n-dimensional column vector and A, B are n × n matrices with complex constant entries. He obtains the following results for the case 0 < λ < 1: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i < 0 for all i, then there is a constant α such that every solution of (1) is O(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that 0 < Re b₁ ? Re b₂ ? ··· ? Re b_n and λ times Re b_n < Re b₁, then every solution of (1) is O(e^bn^t) as t → ∞. For the case λ>1, he has the following results: (i) If B is diagonalizable with eigenvalues b_i such that Re b_i>0 for all i, then there is a constant α such that no solution x(t) of (1), except the identically zero solution, is 0(t^α) as t → ∞. (ii) If B is diagonalizable with eigenvalues b_i such that Re b₁ ? Re b₂ ? ··· ? Re b_n < 0 and λ Re b_n < Re b₁, then no solution x(t) of (1), except the identically zero solution, is 0(e^b1^t) as t → ∞.     

Asymptotic analysis and estimates of blow‐up time for the radial symmetric semilinear heat equation in the open‐spectrum case

We estimate the blow‐up time for the reaction diffusion equation u_t=Δu+ λf(u), for the radial symmetric case, where f is a positive, increasing and convex function growing fast enough at infinity. Here λ>λ^*, where λ^* is the ‘extremal’ (critical) value for λ, such that there exists an ‘extremal’ weak but not a classical steady‐state solution at λ=λ^* with ∥w(?, λ)∥_∞→∞ as 0<λ→λ^*?. Estimates of the blow‐up time are obtained by using comparison methods. Also an asymptotic analysis is applied when f(s)=e^s, for λ?λ^*?1, regarding the form of the solution during blow‐up and an asymptotic estimate of blow‐up time is obtained. Finally, some numerical results are also presented. Copyright © 2007 John Wiley & Sons, Ltd.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

An asymptotic of the negative discrete spectrum of the Schrödinger operator

The Schrödinger operator Hu = -Δu + V(x)u, where V(x) → 0 as ¦x¦ → ∞, is considered in L₂(R^m) for m?3. The asymptotic formula $$N(\lambda ,V) \sim \Upsilon _m \int {(\lambda - V(x))_ + ^{{m \mathord{\left/ {\vphantom {m {2_{dx} }}} \right. \kern-\nulldelimiterspace} {2_{dx} }}} ,} \lambda \to ---0,$$ is established for the number N(λ, V) of the characteristic values of the operator H which are less than λ. It is assumed about the potential V that V = V_o + V₁; V_o < 0, ¦V_o =o (¦V_o¦^3/2) as ¦x¦ → ∞; σ (t/2, V_o) ?cσ (t. V_o) and V₁∈L_m/2,loc, σ(t, V₁) =o (σ (t, V_o)), where σ (t,f)= mes {x:¦f (x) ¦ > t).     

Asymptotical Formulas For Solutions Of Linear Differential Systems Of The First Order

In this paper a system of differential equations y′ ? A(·,λ)y = 0 is considered on the finite interval [a,b] where λ ∈ C, A(·, λ):= λ A₁+ A ₀ +λ _?1A_?1(·,λ) and A ₁,A ₀, A _{? 1} are n × n matrix-functions. The main assumptions: A ₁ is absolutely continuous on the interval [a, b], A ₀ and A _{- 1}(·,λ) are summable on the same interval when ¦λ¦ is sufficiently large; the roots φ₁(x),…,φ_n (x) of the characteristic equation det (φ E — A ₁) = 0 are different for all x ∈ [a,b] and do not vanish; there exists some unlimited set Ω ? C on which the inequalities Re(λφ₁(x)) ≤ … ≤ Re (λφ_n(x)) are fulfilled for all x ∈ [a,b] and for some numeration of the functions φ_j(x). The asymptotic formula of the exponential type for a fundamental matrix of solutions of the system is obtained for sufficiently large ¦λ¦. The remainder term of this formula has a new type dependence on properties of the coefficients A ₁ (x), A _o (x) and A _{- 1} (x).     

Cantor-Lebesgue theorem for double trignometric series

Let ∥·∥ be a norm in R² and let γ be the unit sphere induced by this norm. We call a segment joining points x,y ε R² rational if (x₁ ? y₁)/(x₂ ? y₂) or (x₂ ? y₂)/(x₁ ? y₁) is a rational number. Let γ be a convex curve containing no rational segments. Satisfaction of the condition $$T_\nu (x) = \sum\nolimits_{\parallel n\parallel = \nu } {c_n e^{2\pi i(n_1 x_1 + n_2 x_2 )} } \to 0(\nu \to \infty )$$ in measure on the set e? [- 1/2,1/2)×[- 1/2, 1/2) =T² of positive planar measure implies ∥T _v ∥L₄ (T²) → 0(v → ∞). if, however, γ contains a rational segment, then there exist a sequence of polynomials {T _v } and a set E ? T², ¦E¦ > 0, such that T _v (x) → 0(v → ∞) on E; however, ¦c_n¦ ? 0 for ∥n∥ → ∞.     

Methods of summation and best approximation

Let λ={λ _k ⁿ } be a triangular method of summation,f ε L_p (1 ≤ p ≤ ∞), $$U_n (f,x,\lambda ) = \frac{{a_0 }}{2} + \sum _{k = 1}^n \lambda _k^n (a_k \cos kx + b_k \sin kx).$$ Consideration is given to the problem of estimating the deviations ∥f ? U_n (f, λ) ∥ L_p in terms of a best approximation E_n (f) L_p in abstract form (for a sequence of projectors in a Banach space). Various generalizations of known inequalities are obtained.     

A gap in the energy spectrum of the one-dimensional Dirac operator

One considers the one-dimensional Dirac operator with a slowly oscillating potential (1) $$H = \left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right)\frac{d}{{dx}} + q\left( {\begin{array}{*{20}c} {\cos z(x)} & {\sin z(x)} \\ {\sin z(x)} & { - \cos z(x)} \\ \end{array} } \right)_, x \in ( - \infty ,\infty ),q - const,$$ where . The following statement holds. The double absolutely continuous spectrum of the operator (1) fills the intervals (?∞,?¦q¦), (¦q¦, ∞). The interval (?¦q¦, ¦q¦) is free from spectrum. The operator has a simple eigenvalue only for singn C₊=sign C_?, situated either at the point (under the condition C₊>0) or at the point λ=?¦q¦ (under the condition). The proof is based on the investigation of the coordinate asytnptotics of the corresponding equation.     

Perturbations with several independent parameters

In this paper we discuss the problem of determining a T-periodic solution $\text{x}{}^1\text{(·, λ)}$ of the differential equation x = A(t)x + f(t, x, λ) + b(t), where the perturbation parameter λ is a vector in a parameter-space R^k. The customary approach assumes that λ = λ(?), ??R. One then establishes the existence of an ?₀ > 0 such that the differential equation has a T-periodic solution $\text{x}{}^1\text{(·, λ(?))}$ for all ? satisfying 0 < ? < ?₀. More specifically it is usually assumed that λ(?) has the form λ(?) = ?λ₀ where λ₀ is a fixed vector in R^k. This means that attention is confined in the perturbation procedure to examining the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies along a line segment terminating at the origin in the parameter-space R^k. The results established here generalize this previous work by allowing one to study the dependence of $\text{x}{}^1\text{(·, λ)}$ on λ as λ varies through a “conical-horn” whose vertex rests at the origin in R^k. In the process an implicit-function formula is developed which is of some interest in its own right.     

Bounds for ratios of eigenvalues using traces

Let A be an n × n matrix with real eigenvalues λ₁ ? … ? λ_n, and let 1 ? k < l ? n. Bounds involving trA and trA² are introduced for λ_k/λ_l, (λ_k ? λ_l)/(λ_k + λ_l), and {kλ_k + (n ? l + 1)λ_l}²/{kλ²_k + (n ? l + 1)λ²_l}. Also included are conditions for λ_l >; 0 and for λ_k + λ_l > 0.     

Spectral asymptotic behavior of elliptic systems

One investigates a first-order elliptic self-adjoint pseudodifferential operator A (x,D) acting in sections of a Hermitian vector bundle over a compactn-dimensional manifold x. It is assumed that the principal symbol A(x, ξ) of the operator is locally diagonalizable and that its eigenvaluesaj(x, ξ) have a variable multiplicity and that {a _i,a _k}≠0 whenevera _i=a _k. Under indicated conditions one constructs an expansion of the fundamental solution of the hyperbolic system \(i\frac{{\partial u}}{{\partial t}} = A(x,D)u\) and one investigates the asymptotic properties of the spectrum of the operator A (x,D). For the distribution functionN(λ) of the eigenvalues one establishes that . Under further assumptions on the properties of the bicharacteristic of the symbolsaj(x, ξ) one establishes a stronger estimate of Duistermaat-Guillemin type:N(λ)=Cλ ⁿ +C′λ ^n?i +0(γ ^n?1 )     

The Ostaszewski square and homogeneous Souslin trees

Assume GCH and let λ denote an uncountable cardinal. We prove that if □_λ holds, then this may be witnessed by a coherent sequence 〈C _α|α < λ⁺〉 with the following remarkable guessing property For every sequence 〈A _i | i < λ〉 of unbounded subsets of λ ⁺, and every limit θ < λ, there exists some α < λ ⁺ such that otp(C _α)=θ and the (i + 1) _th -element of C _α is a member of A _i , for all i < θ. As an application, we construct a homogeneous λ ⁺-Souslin tree from □_λ + CH_λ, for every singular cardinal λ. In addition, as a by-product, a theorem of Farah and Veli?kovi?, and a theorem of Abraham, Shelah and Solovay are generalized to cover the case of successors of regulars.     

Local Asymmetry and the Inner Radius of Nodal Domains

Let M be a closed Riemannian manifold of dimension n. Let ?_λ be an eigenfunction of the Laplace–Beltrami operator corresponding to an eigenvalue λ. We show that the volume of {?_λ > 0} ∩ B is ≥C|B|/λ ⁿ , where B is any ball centered at a point of the nodal set. We apply this result to prove that each nodal domain contains a ball of radius ≥C/λ ⁿ . The results in this paper extend previous results of Nazarov, Polterovich, Sodin and of the author.     

On the continuity of the map ϑ → ¦ϑ¦ from the predual of a W1-algebra

Let ?, ψ be elements in the predual of a W¹-algebra. For their absolute value parts ¦?¦, ¦ψ¦, the estimate $\text{∥¦?¦ ? ¦ψ¦∥ ? (2 ∥? + ψ∥ ∥? ? ψ∥)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}$ is obtained.     

Irreducibility Criterion Over Finite Fields

The aim of this article is to prove the irreducibility of the polynomial Λ(Y) = Y ^d  + λ _d?1 Y ^d?1 + … + λ₀ over 𝔽 _q [X] where λ _i ∈ 𝔽 _q [X] and deg λ _d?1 > deg λ _i for each i ≠ d ? 1. We discuss in particular connections between the irreducible polynomials Λ and the number of Pisot elements in the case of formal power series.     

A Functional Equation Of J.A. Baker Involving Integral Means

The functional equation $$f(x)={1\over 2}\int^{x+1}_{x-1}f(t)\ dt\ \ \ {\rm for}\ \ \ x\ \in\ {\rm R}$$ has the linear functions ?(x) = a + bx (a, b ∈ ?) as trivial solutions. It is shown that there are two kinds of nontrivial solutions, (i) ?(x) = eλ_i ^x (i = 1, 2, …), where the λ_i∈ ? are the fixed points of the map z ? sinh z, and (ii) C^∞-solutions ? for which the values in the interval [?1,1] can be prescribed arbitrarily, but with the provision that ?^(j)(? 1) = ?^(j)(0) = ?^(j)(1) = 0 for all j = 0, 1, 2 …     

On the absorption of eigenvalues by continuous spectrum in regular perturbation problems

We consider the family of operators A + λB with A and B self-adjoint and B relatively form bounded. We consider situations where as λ ↓ λ₁, some eigenvalue μ(λ) approaches the continuous spectrum of A + λB. Typical of our results is the following. If B is relatively form compact, and μ(λ) → μ(λ₁), then either $\text{(μ(λ) ? μ(λ}{}_1\text{))}\text{λ}\text{? λ}{}_1\text{→ 0}\text{or}\text{μ(λ}{}_1\text{)}$ is an eigenvalue of A + λ₁B.