Bounds for the spectral abscissa of an element in a Banach algebra

For an arbitrary element x with spectrum sp(x) in a Banach algebra with identity e ≠ 0 we define the upper (lower) spectral abscissa \(\mathop {\sigma + (x)}\limits_{( - )} = \mathop {\max }\limits_{(\min )} \operatorname{Re} \lambda ,\lambda \in sp(x)\) . With the aid of the spectral radius \(\rho (x) = \mathop {\max }\limits_{\lambda \in sp(x)} \left| \lambda \right| = \mathop {\lim }\limits_{n \to + \infty } \parallel x^n {{1 - } \mathord{\left/ {\vphantom {{1 - } n}} \right. \kern-0em} n}\) we prove the following bounds: γ_?(x)?σ_?(x)?Γ_?(x)?₊(x)?σ₊(x)?γ₊(x), Γ_(±)(x)=(2δ_(±))^?1 (ρ _δ ² )_(±)?δ _(±) ² ?ρ ₀ ² )(δ_(±)≠0), γ_(±)(x)= (±)ρ_δ(±)?δ_(±), δ₊?0, δ_??0 ρ _(±) ^δ = ρ(x+eδ_(±)). We mention a case where equality is achieved, some corollaries,and discuss the sharpness of the bounds: for every ? > 0 there is a δ: ¦δ¦ ≥ρ ₀ ² /2?, such that Δ: = ¦γ_(±) x?Γ_(±) x¦?ε and conversely, if the bounds are computed for some δ ≠ 0, then △ ≤ρ ₀ ² /2 ¦δ¦. An example is considered.     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

Multipliers for the Weyl transform and Laguerre expansions

Let P_k denote the projection of L²(R ^R ) onto the kth eigenspace of the operator (-δ+?x?² andS _N ^α =(1/A _N ^α )Σ _k ^N =0A _N?k ^α P _k . We study the multiplier transformT _N ^α for the Weyl transform W defined byW(T _N ^αf )=S _n ^αW(f) . Applications to Laguerre expansions are given.     

A remark concerning Pincherle bases

In this note we find sufficient conditions for uniqueness of expansion of any two functionsf(z) and g(z) which are analytic in the circle ¦ z ¦ < R (0 < R <∞) in series $$f(z) = \sum\nolimits_{n = 0}^\infty {(a_n f_2 (z) + b_n g_n (z))}$$ and $$g_i (z) = \sum\nolimits_{n = 0}^\infty {a_n \lambda _n f_n (z)} + b_n \mu _n f_n (x)),$$ which are convergent in the compact topology, where (f _n {z} _n=0 ^∞ and {g} _n=0 ^∞ are given sequences of functions which are analytic in the same circle while {λ _n } _n=0 ^∞ and {μ _n } _n=0 ^∞ are fixed sequences of complex numbers. The assertion obtained here complements a previously known result of M. G. Khaplanov and Kh. R. Rakhmatov.     

On the relation of generalized Valiron summability to Cesàro summability

A family (V _a ^k ) of summability methods, called generalized Valiron summability, is defined. The well-known summability methods (B_α,γ), (E _ρ, (T_α), (S _β) and (V _a) are members of this family. In §3 some properties of the (B _α,γ) and (V _a ^k ) transforms are established. Following Satz II of Faulhaber (1956) it is proved that the members of the (V _a ^k ) family are all equivalent for sequences of finite order. This paper is a good illustration of the use of generalized Boral summability. The following theorem is established: Theorem.If s _n (n ≥ 0) isa real sequence satisfying $$\mathop {lim}\limits_{ \in \to 0 + } \mathop {lim inf}\limits_{m \to \infty } \mathop {min}\limits_{m \leqslant n \leqslant m \in \sqrt m } \left( {\frac{{S_n - S_m }}{{m^p }}} \right) \geqslant 0(\rho \geqslant 0)$$ , and if s_n →s (V _a ^k ) thens _n → s (C, 2ρ).     

A new modification of the Hermite-Fejér interpolation

пУсть жАДАНы Ужлы $$ - \infty< x_1< x_2< ...< x_k< x_{k + 1}< ...< x_n< + \infty ,$$ , И пУстьx ₁ ^* <x ₂ ^* <...<x _n-1 ^* — кОРНИ МНОгО ЧлЕНА Ω′(х). гДЕ $$\omega (x) = \prod\limits_{k = 1}^n {(x - x_k ).} $$ В РАБОтЕ ИсслЕДУЕтсь жАДАЧА: кАк ОпРЕДЕлИт ь МНОгОЧлЕНР(х) МИНИМАльНОИ стЕп ЕНИ, Дль кОтОРОгО ВыпОлНь Утсь слЕДУУЩИЕ ИНтЕР пОльцИОННыЕ УслОВИь гДЕ {y _k И {y _k′}-жАДАННы Е сИстЕМы жНАЧЕНИИ.     

On the eigenvalues of the Sturm-Liouville operator with potentials from Sobolev spaces

We study the asymptotic behavior of the eigenvalues the Sturm-Liouville operator Ly = ?y″ + q(x)y with potentials from the Sobolev space W ₂ ^θ?1 , θ ≥ 0, including the nonclassical case θ ∈ [0, 1) in which the potential is a distribution. The results are obtained in new terms. Let s _2k (q) = λ _k ^1/2 (q) ? k, s _2k?1(q) = μ _k ^1/2 (q) ? k ? 1/2, where {λ _k } ₁ ^∞ and {μ _k } ₁ ^∞ are the sequences of eigenvalues of the operator L generated by the Dirichlet and Dirichlet-Neumann boundary conditions, respectively,. We construct special Hilbert spaces t ₂ ^θ such that the mapping F:W ₂ ^θ?1 →t ₂ ^θ defined by the equality F(q) = {s _n } ₁ ^∞ is well defined for all θ ≥ 0. The main result is as follows: for θ > 0, the mapping F is weakly nonlinear, i.e., can be expressed as F(q) = Uq + Φ(q), where U is the isomorphism of the spaces W ₂ ^θ?1 and t ₂ ^θ , and Φ(q) is a compact mapping. Moreover, we prove the estimate ∥Ф(q)∥_τ ≤ C∥q∥_θ?1, where the exact value of τ = τ(θ) > θ ? 1 is given and the constant C depends only on the radius of the ball ∥q∥_θ? ≤ R, but is independent of the function q varying in this ball.     

Asymptotics of the eigenvalues of the Sturm-Liouville problem with discrete self-similar weight

We study the asymptotics of the spectrum of the boundary-value problem $$ - y'' - \lambda \rho y = 0,y(0) = y(1) = 0 $$ for the case in which the weight ρ ∈ W? ₂ ^?1 [0, 1] is the generalized (in the sense of distributions) derivative of a self-similar function P ∈ L ₂[0, 1] of zero spectral order.     

On Surface Attractors and Repellers in 3-Manifolds

We show that if f: M ³ → M ³ is an A diffeomorphism with a surface two-dimensional attractor or repeller $\mathcal{B}$ with support $M_\mathcal{B}^2$ , then $\mathcal{B} = M_\mathcal{B}^2$ and there exists a k ≥ 1 such that (1) $M_\mathcal{B}^2$ is the disjoint union M ₁ ² ? ? ? M _k ² of tame surfaces such that each surface M _i ² is homeomorphic to the 2-torus T ²; (2) the restriction of f ^k to M _i ² , i ∈ {1,..., k}, is conjugate to an Anosov diffeomorphism of the torus T ².     

On the chromatic numbers of spheres in ℝ n

Let χ(S _r ^n?1 )) be the minimum number of colours needed to colour the points of a sphere S _r ^n?1 of radius $r \geqslant \tfrac{1} {2}$ in ? ⁿ so that any two points at the distance 1 apart receive different colours. In 1981 P. Erd?s conjectured that χ(S _r ^n?1 )→∞ for all $r \geqslant \tfrac{1} {2}$ . This conjecture was proved in 1983 by L. Lovász who showed in [11] that χ(S _r ^n?1 ) ≥ n. In the same paper, Lovász claimed that if $r < \sqrt {\frac{n} {{2n + 2}}}$ , then χ(S _r ^n?1 ) ≤ n+1, and he conjectured that χ(S _r ^n?1 ) grows exponentially, provided $r \geqslant \sqrt {\frac{n} {{2n + 2}}}$ . In this paper, we show that Lovász’ claim is wrong and his conjecture is true: actually we prove that the quantity χ(S _r ^n?1 ) grows exponentially for any $r > \tfrac{1} {2}$ .     

Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy

We consider Riccati foliations ?_ρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π ₁(S) → PSL(2,?) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ⁺ and µ^? for positive and negative times, respectively. These measures have a common projection to a harmonic measure μ_ρ for the Riccati foliation. We describe μ _ρ ⁺ , μ _ρ ^- and μ_ρ in terms of the Patterson-Sullivan construction, and we show that the measures μ_ρ provide examples of the conformal harmonic measures introduced by M. Brunella.     

Interpolation in certain Hilbert spaces of analytic functions

In this article we study, for a Hilbert spaceB of analytic functions in the open unit disk, the dependence of the structure of the space of sequencesB(Z)={{f(z_k)} _k=1 ^∞ :f∈B} on the choice of the sequence Z={z_k} _k=1 ^∞ of distinct points of the unit disk [6].     

Multi-neighboring grids schemes for solving PDE eigen-problems

Instead of most existing postprocessing schemes,a new preprocessing approach,called multineighboring grids(MNG),is proposed for solving PDE eigen-problems on an existing grid G(Δ).The linear or multi-linear element,based on box-splines,are taken as the frst stage Kh1Uh=λh1Mh1Uh.In this paper,the j-th stage neighboring-grid scheme is defned asKh jUh=λh j Mh jUh,where Kh j:=Mh j 1Kh1and Mh jUh is to be found as a better mass distribution over the j-th stage neighboring-gridG(Δ),and Kh jcan be seen as an expansion of Kh1on the j-th neighboring-grid with respect to the(j 1)-th mass distribution Mh j 1.It is shown that for an ODE model eigen-problem,the j-th stage scheme with 2j-th order B-spline basis can reach2j-th order accuracy and even(2j+2)-th order accuracy by perturbing the mass matrix.The argument can be extended to high dimensions with separable variable cases.For Laplace eigen-problems with some 2-D and 3-D structured uniform grids,some 2j-th order schemes are presented for j 3.     

A moment theorem for completely hyperexpansive operators

Given a family $ \{ A_m^x \} _{\mathop {m \in \mathbb{Z}_ + ^d }\limits_{x \in X} } $ (X is a non-empty set) of bounded linear operators between the complex inner product space $ \mathcal{D} $ and the complex Hilbert space ? we characterize the existence of completely hyperexpansive d-tuples T = (T ₁, … , T _d ) on ? such that A _m ^x = T ^m A ₀ ^x for all m ? ? ₊ ^d and x ? X.     

Singular integrals along Lipschitz curves with holomorphic kernels

If γ(x)=x+iA(x),tan ^?1‖A′‖_∞<ω<π/2,S _ω ⁰ ={z∈C}| |argz|<ω, or, |arg(-z)|<ω} We have proved that if φ is a holomorphic function in S _ω ⁰ and \(\left| {\varphi (z)} \right| \leqslant \frac{C}{{\left| z \right|}}\) , denotingT f (z)= ∫?(z-ζ)f(ζ)dζ, ?f∈C ₀(γ), ?z∈suppf, where C_c(γ) denotes the class of continuous functions with compact supports, then the following two conditions are equivalent:

1. T can be extended to be a bounded operator on L²(γ);
2. there exists a function ?₁ ∈H ^∞(S _ω ⁰ ) such that ?′₁(z)=?(z)+?(-z), ?z∈S _ω ⁰ ?z∈S _w ⁰ .

     

A quadratic approximation for protein sequence to structure mapping

The existence and representations of some generalized inverses, includingA _{T, *} ⁽²⁾ ,A _{T, *} ^(1,2) ,A _{T, *} ^(2,3) ,A _*,S ⁽²⁾ ,A _*,S ^(1,2) andA _*,S ^(2,4) , are showed. As applications, the perturbation theory for the generalized inverseA _T,S ⁽²⁾ and the perturbation bound for unique solution of the general restricted systemAx=b (dim (AT)=dimT,b∈AT andx∈T) are studied. Moreover, a characterization and representation of the generalized inverseA _{T, *} ^Emphasis>(2) is obtained.     

Γ-inverses of Bounded Linear Operators

Let B(H) be the algebra of all the bounded linear operators on a Hilbert space H.For A,P and Q in B(H),if there exists an operator X∈ B(H) such thatAP X QA=A,X QAP X=X,(QAP X)*=QAP X and(X QAP)*=X QAP,then X is said to be the Γ-inverse of A associated with P and Q,and denoted by AP,Q+.In this note,we present some necessary and su?cient conditions for which A+P,Qexists,and give an explicit representation of AP,Q+(if AP,Q+exists).     

The reconstruction problem for certain infinite graphs

We are concerned with the notion of the degree-type (D _G ⁱ )_i∈ω of a graphG, whereD _G ⁱ is defined to be the number of vertices inG with degreei. In the first section the following results are proven:

1. IfG is a connected, locally finite, countably infinite graph such that there exists ani so thatD _G ⁱ andD _G ⁱ⁺¹ are both finite and different from 0, thenG is reconstructible.
2. Locally finite, countably infinite graphsG, for which infinitely manyD _G ⁱ are different from 0 but only finitely manyD _G ⁱ are infinite, are reconstructible.

In the second section we give some results about the reconstructibility of certain locally finite countably infinite interval graphs and show that a reconstruction of a planar, infinite graph has to be planar too.