Applications of inversion formulas to the joint t-universality of Lerch zeta functions

In this paper, we define λ-joint, a^′-joint, (λ,λ)-joint, (λ,a^′)-joint and (a^′,a^′)-joint t-universality of Lerch zeta functions and consider the relations among those. Next we show the existence of (λ,λ)-joint t-universality. Finally, we also show the existence of λ-joint, a^′-joint, (λ,a^′)-joint and (a^′,a^′)-joint t-universality by using inversion formulas.     

Generalized Christoffel functions for Jacobi-exponential weights

Assume that W=e ^?Q where I:=(a,b), ?∞≦a<0<b≦∞, and Q:?I→[0,∞) is continuous and increasing. Let 0<p<∞, a<t _r <t _r?1<?<t ₁<b, p _i >?1/p, i=1,2,…,r, and $U(x)=\prod_{i=1}^{r} {|x-t_{i}|}^{p_{i}}$ . We give the L _p Christoffel functions for the Jacobi-exponential weight WU. In addition, we obtain restricted range inequalities.     

A multiplication theorem for the Lerch zeta function and explicit representations of the Bernoulli and Euler polynomials

A multiplication theorem for the Lerch zeta function ?(s,a,ξ) is obtained, from which, when evaluating at s=−n for integers n?0, explicit representations for the Bernoulli and Euler polynomials are derived in terms of two arrays of polynomials related to the classical Stirling and Eulerian numbers. As consequences, explicit formulas for some special values of the Bernoulli and Euler polynomials are given.     

Note on forced oscillation of nth-order sublinear differential equations

In the case of oscillatory potentials, we establish an oscillation theorem for the forced sublinear differential equation x(n)+q(t)λ|x|sgnx=e(t), t∈[t₀,∞). No restriction is imposed on the forcing term e(t) to be the nth derivative of an oscillatory function. In particular, we show that all solutions of the equation x^″+tαsintλ|x|sgnx=mtβcost, t?0, 0<λ<1 are oscillatory for all m≠0 if β>(α+2)/(1−λ). This provides an analogue of a result of Nasr [Proc. Amer. Math. Soc. 126 (1998) 123] for the forced superlinear equation and answers a question raised in an earlier paper [J.S.W. Wong, SIAM J. Math. Anal. 19 (1988) 673].     

A cluster of many small holes with negative imaginary surface impedances may generate a negative refraction index

We deal with the scattering of an acoustic medium modeled by an index of refraction n varying in a bounded region Ω of and equal to unity outside Ω. This region is perforated with an extremely large number of small holes D_m's of maximum radius a, a << 1, modeled by surface impedance functions. Precisely, we are in the regime described by the number of holes of the order M:=O(a^{β ? 2}), the minimum distance between the holes is d ～ a^t, and the surface impedance functions of the form λ_m～λ_m,0a^?β with β > 0 and λ_m,0 being constants and eventually complex numbers. Under some natural conditions on the parameters β,t, and λ_m,0, we characterize the equivalent medium generating approximately the same scattered waves as the original perforated acoustic medium. We give an explicit error estimate between the scattered waves generated by the perforated medium and the equivalent one, respectively, as a→0. As applications of these results, we discuss the following findings:

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Any Diophantine quintuple contains a regular Diophantine quadruple

A set {a₁,…,am} of m distinct positive integers is called a Diophantine m-tuple if aiaj+1 is a perfect square for all i, j with 1?i<j?m. It is conjectured that if {a,b,c,d} is a Diophantine quadruple with a<b<c<d, then d=d₊, where d₊=a+b+c+2abc+2rst and , , . In this paper, we show that if {a,b,c,d,e} is a Diophantine quintuple with a<b<c<d<e, then d=d₊.     

Remark on an infinite semipositone problem with indefinite weight and falling zeros

In this work, we consider the positive solutions to the singular problem $$ \left\{\begin{array}{ll} -\Delta u = am(x)u-f(u) - \dfrac{c}{u^{\alpha}} & {\rm in}\;\Omega,\\ u=0 & {\rm on}\; \partial\Omega, \end{array} \right. $$ where 0?<?α?<?1,a?>?0 and c?>?0 are constants, Ω is a bounded domain with smooth boundary $\partial\Omega$ , Δ is a Laplacian operator, and $f:[0,\infty] \longrightarrow{\mathbb R}$ is a continuous function. The weight functions m(x) satisfies m(x)?∈?C(Ω) and m(x)?>?m ₀?>?0 for x?∈?Ω and also ||m||_?∞??=?l?<?∞. We assume that there exist A?>?0, M?>?0, p?>?1 such that alu???M?≤?f(u)?≤?Au ^p for all u?∈?[0,?∞?). We prove the existence of a positive solution via the method of sub-supersolutions when $m_{0}a>\frac{2\lambda_{1} }{1+\alpha}$ and c is small. Here λ ₁ is the first eigenvalue of operator ??Δ with Dirichlet boundary conditions.     

Self-adjusting stochastic processes

Let A_t(i, j) be the transition matrix at time t of a process with n states. Such a process may be called self-adjusting if the occurrence of the transition from state h to state k at time t results in a change in the hth row such that A_t+1(h, k) ? A_t(h, k). If the self-adjustment (due to transition h → kx) is A_{t + 1}(h, j) = λA_t(h, j) + (1 ? λ)δ_jk (0 < λ < 1), then with probability 1 the process is eventually periodic. If A₀(i, j) < 1 for all i, j and if the self-adjustment satisfies A_{t + 1}(h, k) = ?(A_t(h, k)) with ?(x) twice differentiable and increasing, x < ?(x) < 1 for 0 ? x < 1,?(1) = ?′(1) = 1, then, with probability 1, lim A_t does not exist.     

Eigenvalue problems for second-order nonlinear dynamic equations on time scales

This paper is concerned with the existence and nonexistence of positive solutions of the second-order nonlinear dynamic equation uΔΔ(t)+λa(t)f(u(σ(t)))=0, t∈[0,1], satisfying either the conjugate boundary conditions u(0)=u(σ(1))=0 or the right focal boundary conditions u(0)=uΔ(σ(1))=0, where a and f are positive. We show that there exists a λ^∗>0 such that the above boundary value problem has at least two, one and no positive solutions for 0<λ<λ^∗, λ=λ^∗ and λ>λ^∗, respectively. Furthermore, by using the semiorder method on cones of the Banach space, we establish an existence and uniqueness criterion for positive solution of the problem. In particular, such a positive solution uλ(t) of the problem depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ‖=0 and limλ→+∞‖uλ‖=+∞.     

Blowup stability of solutions of the nonlinear heat equation with a large life span

We study the Cauchy problem for the nonlinear heat equation u_t-?u=|u|^p-1u in R^N. The initial data is of the form u₀=λ?, where ?∈C₀(R^N) is fixed and λ>0. We first take 1<p<p_f, where p_f is the Fujita critical exponent, and ?∈C₀(R^N)∩L¹(R^N) with nonzero mean. We show that u(t) blows up for λ small, extending the H. Fujita blowup result for sign-changing solutions. Next, we consider 1<p<p_s, where p_s is the Sobolev critical exponent, and ?(x) decaying as |x|^-σ at infinity, where p<1+2/σ. We also prove that u(t) blows up when λ is small, extending a result of T. Lee and W. Ni. For both cases, the solution enjoys some stable blowup properties. For example, there is single point blowup even if ? is not radial.     

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.     

On the existence of Hadamard matrices

Given any natural number q > 3 we show there exists an integer t ? [2log₂(q ? 3)] such that an Hadamard matrix exists for every order 2^sq where s > t. The Hadamard conjecture is that s = 2.This means that for each q there is a finite number of orders 2^υq for which an Hadamard matrix is not known. This is the first time such a statement could be made for arbitrary q.In particular it is already known that an Hadamard matrix exists for each 2^sq where if q = 2^m ? 1 then s ? m, if q = 2^m + 3 (a prime power) then s ? m, if q = 2^m + 1 (a prime power) then s ? m + 1.It is also shown that all orthogonal designs of types (a, b, m ? a ? b) and (a, b), 0 ? a + b ? m, exist in orders m = 2^t and 2^t+2 · 3, t ? 1 a positive integer.     

Positive solutions to singular semipositone m-point n-order boundary value problems

In this paper, we consider the existence of positive solutions to the following Singular Semipositone m-Point n-order Boundary Value Problems (SBVP): $$\left\{\begin{array}{l@{\quad}l}(-1)^{(n-k)}x^{(n)}(t)=\lambda f(t,x(t)),&0<t<1,\\[4pt]x(1)=\sum_{i=1}^{m-2}a_ix(\eta_i),\qquad x^{(i)}(0)=0,&0\leq i\leq k-1,\\[4pt]x^{(j)}(1)=0,&1\leq j\leq n-k-1,\end{array}\right.$$ where m≥3, λ>0, a _i ∈[0,∞),(i=1,2,…,m?2),0<η ₁<η ₂<???<η _m?2<1 are constants, f:(0,1)×[0,+∞)→R is continuous and may have singularity at t=0 and/or 1. Without making any monotone-type assumption, we obtain the positive solution of the problem for λ lying in some interval, based on fixed-point index theorem in a cone.     

Markov-Nikolskii type inequalities for exponential sums on finite intervals

Let Λn:={λ₀<λ₁<?<λn} be a set of real numbers. The collection of all linear combinations of eλ₀t,eλ₁t,…,eλnt over R will be denoted by

E(Λn):=span{eλ₀t,eλ₁t,…,eλnt}.     

A test for the existence of Gohberg-Krein representations in terms of multiparameter Wiener processes

This paper provides new necessary and sufficient conditions for a Gaussian random field to have a Gohberg-Krein representation in terms of an n-parameter Wiener process (n > 1). As an application, it demonstrates the nonexistence of a Gohberg-Krein representation of W_s,t ? stW_1,1 in terms of the two-parameter Wiener process W_s,t with (s, t) ? [0, a] × [0, b] for 0 < a < 1, 0 < b < 1.     

On diophantine equations of the form {({\boldsymbol x}-{\boldsymbol a}_{\bf 1})({\boldsymbol x}-{\boldsymbol a}_{\bf 2}) \ldots ({\boldsymbol x}-{\boldsymbol a}_{\boldsymbol k}) + {\boldsymbol r} = {\boldsymbol y}^{ \boldsymbol n}}

Erd?s and Selfridge [3] proved that a product of consecutive integers can never be a perfect power. That is, the equation x(x?+?1)(x?+?2)...(x?+?(m???1))?=?y ⁿ has no solutions in positive integers x,m,n where m, n?>?1 and y?∈?Q. We consider the equation $$ (x-a_1)(x-a_2) \ldots (x-a_k) + r = y^n $$ where 0?≤?a ₁?<?a ₂?<???<?a _k are integers and, with r?∈?Q, n?≥?3 and we prove a finiteness theorem for the number of solutions x in Z, y in Q. Following that, we show that, more interestingly, for every nonzero integer n?>?2 and for any nonzero integer r which is not a perfect n-th power for which the equation admits solutions, k is bounded by an effective bound.     

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.     

Asymptotic expansions of the ordered spectrum of symmetric matrices

In this work, we build on ideas of Torki (2001 [6]) and show that if a symmetric matrix-valued map t?A(t) has a one-sided asymptotic expansion at t=0⁺ of order K then so does t?λ_m(A(t)), where λ_m is the mth largest eigenvalue. We derive formulas for computing the coefficients A₀,A₁,…,A_K in the asymptotic expansion. As an application of the approach we give a new proof of a classical result due to Kato (1976 [3]) about the one-sided analyticity of the ordered spectrum under analytic perturbations. Finally, as a demonstration of the derived formulas, we compute the first three terms in the asymptotic expansion of λ_m(A+tE) for any fixed symmetric matrices A and E.     

Multiplier operators and extremal functions related to the dual dunkl-sonine operator

We study the dual Dunkl-Sonine operator ^tS_k,? on ?^d, and give expression of ^tS_k,?, using Dunkl multiplier operators on ?^d. Next, we study the extremal functions f^*_λ, λ >0 related to the Dunkl multiplier operators, and more precisely show that {f^*_λ} λ >0 converges uniformly to ^tS_k,?(f) as λ → 0+. Certain examples based on Dunkl-heat and Dunkl-Poisson kernels are provided to illustrate the results.