On the zeros of derivatives of Bessel functions

In this paper we are interested in the behaviour respect tov of thekth positive zeroc′ _vk of the derivative of the general Bessel functionC _v(x)=J _v(x)cosα?Y _v(x)sinα, 0≤α<π, whereJ _v(x) andY _v(x) indicate the Bessel functions of first and second kind respectively. It is well known that forc′ _vk>∥v∥,c′ _vk increases asv increases. Here we prove several additional properties forc′ _vk. Our main result is thatc′ _vk is concave as a function ofv, whenc′ _vk>∥v∥>0. This implies the concavity ofc′ _vk for everyk=2,3, ?. In the case of the zerosJ′ _vk of _d ^dx J _v(x) we extend this property tok=1 for everyv≥0.     

A convexity property of zeros of Bessel functions

Fork=1, 2,... letj _vk andc _vk be thek-th positive zeros of the Bessel function $$C_v \left( x \right) = C_v \left( {\alpha ;x} \right) = J_v \left( x \right)\cos \alpha - Y_v \left( x \right)\sin \alpha , 0 \leqslant \alpha< \pi$$ whereY _v (X) is the Bessel function of the second kind. Using the notationj _vκ =C _vk withκ=k?α/π introduced in [3] we show that the functionj _vκ +f(v) is convex with respect toυ≥0 forκ≥0.7070..., wheref(υ) is defined in the theorem of section 2. As an application we find the inequality 0 >j _0κ +j _1κ ? 2κπ > log 8/9, where κ≥0.7070....     

Elementary approximations for zeros of Bessel functions

Let j_vk, y_vk and c_vk denote the kth positive zeros of the Bessel functions J_v(x), Y_v(x) and of the general cylinder function C_v(x) = cos αJ_v(x)?sin αY_v(x), 0 ? α < π, respectively. In this paper we extend to c_vk, k = 2, 3,..., some linear inequalities presently known only for j_vk. In the case of the zeros y_vk we are able to extend these inequalities also to k = 1. Finally in the case of the first positive zero j_v1 we compare the linear enequalities given in [9] with some other known inequalities.     

An asymptotic relation for the zeros of Bessel functions

We define the function j_νκ for all real κ > 0 as follows: for κ = 1, 2, … the j_νκ denotes the kth positive zero of the Bessel function J_ν(z) of first kind and for k ? 1 < κ < k, j_νκ denotes the kth positive zero of the cylinder Bessel function C_ν(z) = cos αJ_ν(z) ? sin αY_ν(Z) with α = (k ? ν)π (see [2]), where Y_ν(x) is the Bessel function of second kind. We introduce the function ι(x) for x > ? 1,

$\text{l(x)=}\text{lim}\text{κ→∞}\text{j}{}_{\mathrm{\kappa ,x,\kappa }}\text{κ}$

. and we prove, among other things, the inequality $\text{j}{}_{\mathrm{\nu \kappa }}\text{< κι(}\text{ν}\text{κ}\text{)}$. Moreover, we find the first three terms of the asymptotic expansion of ι(x), for large values of x and other properties of this function.     

Convergence of solutions of a second order Cauchy problem

Let j_νk, y_νk and c_νk denote the kth positive zeros of the Bessel functions J_ν(x), Y_ν(x) and of the general cylinder function C_ν(x), respectively. We show, among other things, that, for k = 2, 3,… and 0 < ν < ∞, c_νk is a concave function of ν, c_νk > ν + c_0k and $\text{c}{}_{\mathrm{\nu k}}\text{[v + (}\text{2}\text{π}\text{)c}{}_{\mathrm{0k}}\text{]}$ decreases as ν increases. In the cases of j_νk and y_νk, these results hold also for k = 1.     

A note on the conditional moments of a multivariate normal distribution confined to a convex set

Let Y be an N(μ, Σ) random variable on R^m, 1 ≤ m ≤ ∞, where Σ is positive definite. Let C be a nonempty convex set in R^m with closure $\text{C}$. Let (·,-·) be the Eculidean inner product on R^m, and let μ_c be the conditional expected value of Y given Y ∈ C. For v ∈ R^m and s ≥ 0, let β_s(v) be the expected value of |(v, Y) ? (v, μ)|^s and let γ_s(v) be the conditional expected value of |(v, Y) ? (v, μ_c)|^s given Y ∈ C. For s ≥ 1, γ_s(v) < β_s(v) if and only if $\text{C}\text{+ Σ v ≠}\text{C}$, and γ_s(v) < β_s(v) for all v ≠ 0 if and only if $\text{C}\text{+ v ≠}\text{C}$ for any v ∈ R^m such that v ≠ 0.     

Lower semicontinuity and the theorem of Datko and Pazy

Let {T(t)} _t≥0 be aC ₀-semigroup on a real or complex Banach spaceX and letJ:C ⁺[0,∞)→[0,∞] be a lower semicontinuous and nondecreasing functional onC ⁺[0,∞), the positive cone ofC[0,∞), satisfyingJ(c 1)=∞ for allc>0. We prove the following result: if {T(t)} _t≥0 is not uniformly exponentially stable, then the set $\{ x \in X: J(||T( \cdot )x||) = \infty \}$ is residual inX.     

Further results on the existence of nested orthogonal arrays

Nested orthogonal arrays provide an option for designing an experimental setup consisting of two experiments, the expensive one of higher accuracy being nested in a larger and relatively less expensive one of lower accuracy. We denote by OA_(λ, μ)(t, k, (v, w)) (or OA(t, k, (v, w)) if λ = μ = 1) a (symmetric) orthogonal array OA _λ (t, k, v) with a nested OA _μ (t, k, w) (as a subarray). It is proved in this article that an OA(t, t + 1,(v, w)) exists if and only if v ≥ 2w for any positive integers v, w and any strength t ≥ 2. Some constructions of OA_(λ, μ)(t, k, (v, w))′s with λ ≠ μ and k ? t > 1 are also presented.     

Algorithm for the computation of Bessel function integrals

A fortran subroutine is given for the computation of integrals of the form ∫^c₀f(x)J_v(αx)dx, where v = 0, 1,…,10.     

Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities

The paper investigates the structure and properties of the set S of all positive solutions to the singular Dirichlet boundary value problem u″(t) + au′(t)/t ? au(t)/t ² = f(t, u(t),u′(t)), u(0) = 0, u(T) = 0. Here a ∈ (?∞,?1) and f satisfies the local Carathéodory conditions on [0,T]×D, where D = [0,∞)×?. It is shown that S _c = {u ∈ S: u′(T) = ?c} is nonempty and compact for each c ≥ 0 and S = ∪ _c≥0 S _c . The uniqueness of the problem is discussed. Having a special case of the problem, we introduce an ordering in S showing that the difference of any two solutions in S _c ,c≥ 0, keeps its sign on [0,T]. An application to the equation v″(t) + kv′(t)/t = ψ(t)+g(t, v(t)), k ∈ (1,∞), is given.     

On the Nonseparable Subspaces of J(η) and C([1, η])

Let η be a regular cardinal. It is proved, among other things, that: (i) if J(η) is the corresponding long James space, then every closed subspace Y ⊆ J(η), with Dens (Y) = η, has a copy of 𝓁₂(η) complemented in J(η); (ii) if Y is a closed subspace of the space of continuous functions C([1, η]), with Dens (Y) = η, then Y has a copy of c₀(η) complemented in C([1, η]). In particular, every nonseparable closed subspace of J(ω₁) (resp. C([1, ω₁])) contains a complemented copy of 𝓁₂(ω₁) (resp.c₀(ω₁)). As consequence, we give examples (J(ω₁), C([1, ω₁]), C(V), V being the “long segment”) of Banach spaces X with the hereditary density property (HDP) (i.e., for every subspace Y ⊆ X we have that Dens (Y) = w*–Dens (Y*)), in spite of these spaces are not weakly Lindelof determined (WLD).     

Extended Jacobson density theorem for rings with derivations and automorphisms

A deterministic version of the Itô calculus is presented. We consider a modelY _t=H(N _t ,t) with a deterministic Brownian N _t and an unknown functionH. We predictY _c from the observation {Y _t;t ∈ [a, b]}, wherea. We prove that there exists an estimatorY _t based on the observation such thatE[(? _t?Y _c)²]=O((c?b)²) asc ↓b.     

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

In this paper, existence and attractiveness of solutions for quadratic Urysohn fractional integral equations on an unbounded interval are obtained by virtue of Tichonov fixed point theorem and suitable conjunction of the well known measure ω₀(X) and the spaces C(R⁺). Further, three certain solutions sets X_L,γ, X_1,α and X_{1,(1−(α+v))}, which tending to zero at an appropriate rate t^−ν (ν > 0), ν = γ (or α or 1 − (α + v)) as t → ∞, are introduced and stability of solutions for quadratic Urysohn fractional integral equations are obtained based on these solutions sets respectively by applying Schauder fixed point theorem via some easy checked conditions. An example is given to illustrate the results.     

Eventually positive solutions of first order nonlinear differential equations with a deviating argument

The following first order nonlinear differential equation with a deviating argument $ x'(t) + p(t)[x(\tau (t))]^\alpha = 0 $ is considered, where α > 0, α ≠ 1, p ∈ C[t ₀; ∞), p(t) > 0 for t ≧ t ₀, τ ∈ C[t ₀; ∞), lim _t→∞ τ(t) = ∞, τ(t) < t for t ≧ t ₀. Every eventually positive solution x(t) satisfying lim _t→∞ x(t) ≧ 0. The structure of solutions x(t) satisfying lim _t→∞ x(t) > 0 is well known. In this paper we study the existence, nonexistence and asymptotic behavior of eventually positive solutions x(t) satisfying lim _t→∞ x(t) = 0.     

A new proof of a theorem of Harper on the Sperner-Erdös problem

Let P be a finite, partially ordered set and v a weight on P, i.e., a function v: P → R₊/{0. A subset F ? P) is called a k-family, if there are not c₀,…,c_k?F such that c₀ < … <c_k. Let d_k(P, v) = max {Σ_x?Fv(x); F is k-family. It is given a new proof of a theorem of Harper which states that d_k(P, v) = d_k(Q, w), if there is a flow morphism from (P, v) onto (Q, w).     

A Probabilistic Characterization of the Dominance Order on Partitions

We present a probabilistic characterization of the dominance order on partitions. Let ν be a partition and Y _ν its Ferrers diagram, i.e. a stack of rows of cells with row i containing ν _i cells. Let the cells of Y _ν be filled with independent and identically distributed draws from the random variable X = B i n(r, p) with r ≥ 1 and p ∈ (0, 1). Given j, t ≥ 0, let P(ν, j, t) be the probability that the sum of all the entries in Y _ν is j while the sum of the entries in each row of Y _ν is no more than t. It is shown that if ν and μ are two partitions of n, ν dominates μ if and only if P(ν, j, t) ≤ P(μ, j, t) for all j, t ≥ 0. It is shown that the same result holds if X is any log-concave integer valued random variable with {i : P(X = i) > 0} = {0, 1,…,r} for some r ≥ 1.     

On oscillation of solutions of a nonautonomous quasilinear second-order equation

Sufficient conditions are obtained for the initial values of nontrivial oscillating (for t=ω) solutions of the nonautonomous quasilinear equation $$y'' \pm \lambda (t)y = F(t,y,y'),$$ wheret ∈ Δ=[a, ω[,-∞ <a < ω ≤+ ∞, λ(t) > 0, λ(t) ∈ C _Δ ⁽¹⁾ , |F((t,x,y))|≤L(t)(|x|+|y|)^1+α, L(t) ≥-0, α ∈ [0,+∞[, F: Δ × R² →R,F ∈C _Δ×R ²,R is the set of real numbers, and R² is the two-dimensional real Euclidean space.     

Copositive pointwise approximation

We prove that if a functionf ∈C ⁽¹⁾ (I),I: = [?1, 1], changes its signs times (s ∈ ?) within the intervalI, then, for everyn > C, whereC is a constant which depends only on the set of points at which the function changes its sign, andk ∈ ?, there exists an algebraic polynomialP _n =P _n (x) of degree ≤n which locally inherits the sign off(x) and satisfies the inequality $$\left| {f\left( x \right) - P_n \left( x \right)} \right| \leqslant c\left( {s,k} \right)\left( {\frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right)\omega _k \left( {f'; \frac{1}{{n^2 }} + \frac{{\sqrt {1 - x^2 } }}{n}} \right), x \in I$$ , where ω _k (f′;t) is thekth modulus of continuity of the functionf’. It is also shown that iff ∈C (I) andf(x) ≥ 0,x ∈I then, for anyn ≥k ? 1, there exists a polynomialP _n =P _n (x) of degree ≤n such thatP _n (x) ≥ 0,x ∈I, and |f(x) ?P _n (x)| ≤c(k)ω _k (f;n ^?2 +n ^?1 √1 ?x ²),x ∈I.     

Threshold phenomena for a reaction-diffusion system

We consider the pure initial value problem for the system of equations $\text{ν}{}_{\mathrm{t}}\text{= ν}{}_{\mathrm{xx}}\text{+ ?(ν) ? w, w}{}_{\mathrm{t}}\text{= ε(ν ? γw), ε, γ ? 0}$, the initial data being (ν(x, 0), w(x, 0)) = (?(x), 0). Here $\text{?(v) = ?v + H(v ? a)}$, where H is the Heaviside step function and $\text{a ? (0,}\text{1}\text{2}\text{)}$. This system is of the FitzHugh-Nagumo type and has several applications including nerve conduction and distributed chemical/ biochemical systems. It is demonstrated that this system exhibits a threshold phenomenon. This is done by considering the curve s(t) defined by s(t) = sup{x: v(x, t) = a}. The initial datum, ?(x), is said to be superthreshold if lim_{t→∞ s(t) = ∞}. It is proven that the initial datum is superthreshold if ?(x) > a on a sufficiently long interval, ?(x) is sufficiently smooth, and ?(x) decays sufficiently fast to zero as $\text{¦x¦ → ∞}$.     

A note on the evaluation of bounded L2-functionals at B-splines and its application to singular integral equations

An algorithm computing recursively the values of ∫g(t)v(t) dt, whereg is anL ₂-function andv is aB-spline, is presented. For the functionsg _s(t)=log∥s?t∥ the starting values of the recursion formula can be computed analytically. The problem is related to the numerical solution of integral equations with a logarithmic singularity.