Limits of integrals involving almost periodic functions

Let Sp ? R⁺ be a discrete countable set, let {α_λ}_λ∈Sp be a sequence in l¹(Sp) and f(x) ? ∑_λ∈Spα_λsin(λx). f is an almost periodic odd function with {λ: ±λ ∈ Sp} as spectrum. We give some conditions about the set S so that $\int _1^{+\infty}\ f(x){\rm sin}(Rx){dx\over x}\rightarrow 0$ whenever R → +∞, R ∈ S.     

On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant.     

Existence and uniqueness of positive periodic solutions of functional differential equations

In this paper we consider the existence and uniqueness of positive periodic solution for the periodic equation y′(t)=−a(t)y(t)+λh(t)f(y(t−τ(t))). By the eigenvalue problems of completely continuous operators and theory of α-concave or −α-convex operators and its eigenvalue, we establish some criteria for existence and uniqueness of positive periodic solution of above functional differential equations with parameter. In particular, the unique solution y_λ(t) of the above equation depends continuously on the parameter λ. Finally, as an application, we obtain sufficient condition for the existence of positive periodic solutions of the Nicholson blowflies model.     

Pressing Down Lemma for λ-trees and its applications

For any ordinal λ of uncountable cofinality, a λ-tree is a tree T of height λ such that |T _α| < cf(λ) for each α < λ, where T _α = {x ∈ T: ht(x) = α}. In this note we get a Pressing Down Lemma for λ-trees and discuss some of its applications. We show that if η is an uncountable ordinal and T is a Hausdorff tree of height η such that |T _α | ? ω for each α < η, then the tree T is collectionwise Hausdorff if and only if for each antichain C ? T and for each limit ordinal α ? η with cf(α) > ω, {ht(c): c ∈ C} ∩ α is not stationary in α. In the last part of this note, we investigate some properties of κ-trees, κ-Suslin trees and almost κ-Suslin trees, where κ is an uncountable regular cardinal.     

Ultrafilter semigroups generated by direct sums

Let λ be an infinite cardinal and for every ordinal α<λ, let A _α be a set with a distinguished element 0 _α ∈A _α . The direct sum of sets A _α , α<λ, is the subset \(X=\bigoplus_{\alpha<\lambda}A_{\alpha}\) of the Cartesian product ∏_α<λ A _α consisting of all x with finite supp?(x)={α<λ:x(α)≠0 _α }. Endow X with a topology by taking as a neighborhood base at x∈X the subsets of the form {y∈X:y(α)=x(α) for all α<γ} where γ<λ. Let Ult?(X) denote the set of all nonprincipal ultrafilters on X converging to 0∈X. There is a natural partial semigroup operation on X which induces a semigroup operation on Ult?(X). We show that if direct sums X and Y are homeomorphic, then the semigroups Ult?(X) and Ult?(Y) are isomorphic.     

Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces

We consider the generalized shift operator, associated with the Laplace-Bessel differential operator . The maximal operator Mγ (B-maximal operator) and the Riesz potential (B-Riesz potential), associated with the generalized shift operator are investigated. At first, we prove that the B-maximal operator Mγ is bounded from the B-Morrey space Lp,λ,γ to Lp,λ,γ for all 1<p<∞ and 0?λ<n+|γ|. We prove that the B-Riesz potential , 0<α<n+|γ| is bounded from the B-Morrey space Lp,λ,γ to Lq,λ,γ if and only if α/(n+|γ|−λ)=1/p−1/q, 1<p<(n+|γ|−λ)/α. Also we prove that the B-Riesz potential is bounded from the B-Morrey space L1,λ,γ to the weak B-Morrey space WLq,λ,γ if and only if α/(n+|γ|−λ)=1−1/q.     

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.     

Continuity of integrated density of states — independent randomness

In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the ν dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly α-Hölder continuous, 0 < α ≤ 1, then the density of states is also uniformly α-Hölder continuous.     

D-spaces and thick covers

The following results are obtained.

-

An open neighbornet U of X has a closed discrete kernel if X has an almost thick cover by countably U-close sets.

-

Every hereditarily thickly covered space is aD and linearly D.

-

Every t-metrizable space is a D-space.

-

X is a D-space if X has a cover {Xα:α<λ} by D-subspaces such that, for each β<λ, the set ?{Xα:α<β} is closed.

     

Cardinal sequences of LCS spaces under GCH

Let C(α) denote the class of all cardinal sequences of length α associated with compact scattered spaces. Also put

C_λ(α)={f∈C(α):f(0)=λ=min[f(β):β<α]}.     

Conditional limit theorems for critical continuous-state branching processes

We study the conditional limit theorems for critical continuous-state branching processes with branching mechanism ψ(λ) = λ1+αL(1/λ), where α∈ [0, 1] and L is slowly varying at ∞. We prove that if α∈(0, 1], there are norming constants Qt→ 0(as t ↑ +∞) such that for every x 0, Px(QtXt∈·| Xt 0)converges weakly to a non-degenerate limit. The converse assertion is also true provided the regularity of ψ at0. We give a conditional limit theorem for the case α = 0. The limit theorems we obtain in this paper allow infinite variance of the branching process.     

On likelihood ratio ordering of parallel systems with exponential components

Let T(λ₁,...,λ _n ) be the lifetime of a parallel system consisting of exponential components with hazard rates λ₁,...,λ _n , respectively. For systems with only two components, Dykstra et. al. (1997) showed that if (λ₁, λ₂) majorizes (γ₁, γ₂), then, T(λ₁, λ₂) is larger than T(γ₁, γ₂) in likelihood ratio order. In this paper, we extend this theorem to general parallel systems. We introduce a new partial order, the so-called d-larger order, and show that if (λ₁,...,λ _n ) is d-larger than (γ₁,...,γ _n ), then T(λ₁,...,λ _n ) is larger than T(γ₁,...,γ _n ) in likelihood ratio order.     

Rings of Formal Power Series in an Infinite Set of Indeterminates

Let α be an infinite cardinal number, Λ be an index set of cardinality > α, and {X _λ}_λ∈Λ be a set of indeterminates over an integral domain D. It is well known that there are three ways of defining the ring of formal power series in {X _λ}_λ∈Λ over D, say, D[[{X _λ}]] _i for i = 1, 2, 3. In this paper, we let D[[{X _λ}]]_α = ∪ {D[[{X _λ}_λ∈Γ]]₃ | Γ ? Λ and |Γ| ≤ α}, and we then show that D[[{X _λ}]]_α is an integral domain such that D[[{X _λ}]]₂ ? D[[{X _λ}]]_α ? D[[{X _λ}]]₃. We also prove that (1) D is a Krull domain if and only if D[[{X _λ}]]_α is a Krull domain and (2) D[[{X _λ}]]_α is a unique factorization domain (UFD) (resp., π-domain) if and only if D[[X ₁,…, X _n ]] is a UFD (resp., π-domain) for every integer n ≥ 1.     

Complex powers of some degenerate differential operators related to the Helmholtz operator

In the L _p -spaces, we study the complex powers of the operator

$G_\lambda = m^2 I + \Delta - i\lambda \frac{{\partial ^2 }}{{\partial x_1^2 }},0 < \lambda < 1,m > 0,$

where δ is the Laplace operator. The complex powers G _λ ^?α/2 , Reα > 0, are realized as potential type operators B _λ ^α with a nonstandard metric. We obtain L _p → L _p + L _s -estimates for the operator B _λ ^α . By using the method of approximate inverse operators, we construct the inversion of the potentials B _λ ^α φ with L _p -densities and describe the range B _λ ^α (L _p ) in terms of the inversion constructions.

     

Symmetry of zeros of Lerch zeta-function for equal parameters

For most values of parameters λ and α, the zeros of the Lerch zeta-function L(λ, α, s) are distributed very chaotically. In this paper, we consider the particular case of equal parameters L(λ, λ, s) and show by calculations that the nontrivial zeros either lie extremely close to the critical line σ = 1/2 or are distributed almost symmetrically with respect to the critical line. We also investigate this phenomenon theoretically.     

Branching one-dimensional periodic diffusion processes

Let X be a nonsingular conservative one-dimensional periodic diffusion process, λ₀ its principal eigenvalue and X a binary splitting branching diffusion process with nonbranching part X. For each α > λ₀ we have two positive martingales Wⁱ_t(α), i = 1, 2, of X attached to the two positive α-harmonic functions of X. The main purpose of this paper is to show that their limit random variables are positive for all α? (λ₀, α_i), where α_i are some constants greater than λ₀.     

On the enumeration of skew Young tableaux

For fixed n and a fixed partition α of k<n we give an explicit formula for the number N(n;α) of standard skew Young tableaux with n squares and shape λ/α for some λ. From this formula the entire asymptotic expansion of N(n;α) as n→∞ can in principle be computed, generalizing recent work of McKay, Morse, and Wilf. We also give asymptotic formulas for the number f^λ/α of standard skew Young tableaux of shape λ/α for α fixed and λ “large.”     

Multivariate Hörmander-Type Multiplier Theorem for the Hankel transform

Let $\mathcal{H}(f)(x)=\int_{(0,\infty)^{d}} f(\lambda) E_{x}(\lambda) d\nu(\lambda )$ , be the multivariate Hankel transform, where $E_{x}(\lambda)=\prod_{k=1}^{d} (x_{k} \lambda_{k})^{-\alpha _{k}+1/2}J_{\alpha_{k}-1/2}(x_{k} \lambda_{k})$ , with dν(λ)=λ ^2α dλ, α=(α ₁,…,α _d ). We give sufficient conditions on a bounded function m(λ) which guarantee that the operator $\mathcal{H}(m\mathcal{H} f)$ is bounded on L ^p (dν) and of weak-type (1,1), or bounded on the Hardy space H ¹((0,∞) ^d ,dν) in the sense of Coifman-Weiss.     

Linear orderings and powers of characterizable cardinals

We say that a countable model M completely characterizes an infinite cardinal κ, if the Scott sentence of M has a model in cardinality κ, but no models in cardinality κ⁺. If a structure M completely characterizes κ, κ is called characterizable. In this paper, we concern ourselves with cardinals that are characterizable by linearly ordered structures (cf. Definition 2.1).Under the assumption of GCH, Malitz completely resolved the problem by showing that κ is characterizable if and only if κ=ℵ_α, for some α<ω₁ (cf. Malitz (1968) [7] and Baumgartner (1974) [1]). Our results concern the case where GCH fails.From Hjorth (2002) [3], we can deduce that if κ is characterizable, then κ⁺ is characterizable by a densely ordered structure (see Theorem 2.4 and Corollary 2.5).We show that if κ is homogeneously characterizable (cf. Definition 2.2), then κ is characterizable by a densely ordered structure, while the converse fails (Theorem 2.3).The main theorems are (1) If κ>2^λ is a characterizable cardinal, λ is characterizable by a densely ordered structure and λ is the least cardinal such that κ^λ>κ, then κ^λ is also characterizable (Theorem 5.4) and (2) if ℵ_α and κ^ℵ_α are characterizable cardinals, then the same is true for κ^ℵ_α+β, for all countable β (Theorem 5.5).Combining these two theorems we get that if κ>2^ℵ_α is a characterizable cardinal, ℵ_α is characterizable by a densely ordered structure and ℵ_α is the least cardinal such that κ^ℵ_α>κ, then for all β<α+ω₁, κ^ℵ_β is characterizable (Theorem 5.7). Also if κ is a characterizable cardinal, then κ^ℵ_α is characterizable, for all countable α (Corollary 5.6). This answers a question of the author in Souldatos (submitted for publication) [8].     

Polynomial almost periodic solutions for a class of Riemann-Hilbert problems with triangular symbols

Let with α,β∈]0,1[ such that α+β<1, αβ⁻¹∉Q and a,b,c∈C?{0}. In this paper the existence of almost-periodic polynomial (APP) solutions to the equation (with and ) is studied. The natural space in which to seek a solution to the above problem is the space of almost periodic functions with spectrum in the group αZ+βZ+Z. Due to the difficulty in dealing with the problem in that generality, solutions are sought with spectrum in the group αZ+βZ. Several interesting and totally new results are obtained. It is shown that, if 1∉αZ+βZ, no polynomial solutions exist, i.e., almost periodic polynomial solutions exist only if αZ+βZ=αZ+βZ+Z. Keeping to this setting, it is shown that APP solutions exist if and only if the function satisfies the simple spectral condition α+β>1/2. The proof of this result is nontrivial and has a number-theoretic flavour. Explicit formulas for the solution to the above problem are given in the final section of the paper. The derivation of these formulas is to some extent a byproduct of the proof of the result on the existence of APP solutions.