On a theorem of silver

A recent theorem of Silver, in its simplest form, states, that if ω < cf(k) < k and 2^λ=λ⁺ for all λ < k, then 2^k=k⁺. Silver's proof employs Boolean-valued as well as nonstandard models of set theory. In the present note we give an elementary proof of Silver's theorem in its general form.     

A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems

We consider a one-parameter family of Noumerov-type methods for the integration of second order periodic initial-value problems: y″ = f(t, y), y(t₀) = y₀, y′(t₀) = y′₀. By applying these methods to the test equation: y″ + λ²y = 0, we determine the parameter of the family so that the phase-lag (frequency distortion) for the method is minimal. The resulting method has a very small phase-lag of size $\text{1}\text{12096}\text{λ}{}^6\text{h}{}^6$ (h is the step-size); interestingly, this method also possesses an interval of periodicity of size 2.71. Noumerov's method has a phase-lag of size $\text{1}\text{480}\text{λ}{}^4\text{h}{}^4$ and an interval of periodicity of size 2.449. The superiority of our method over Noumerov's method is illustrated by two examples.     

l-Extensions of CM-fields and cyclotomic invariants

The Hurwitz type relation of Iwasawa's λ^?-invariants in l-extensions of CM-fields is given under the assumption of the vanishing of μ^?-invariants.     

Positive solutions of the nonlinear fourth-order beam equation with three parameters

This paper is concerned with the existence and nonexistence of positive solutions of the nonlinear fourth-order beam equation u(4)(t)+ηu^″(t)−ζu(t)=λf(t,u(t)), 0<t<1, u(0)=u(1)=u^″(0)=u^″(1)=0, where is continuous and ζ, η and λ are parameters. We show that there exists a such that the above boundary value problem (BVP) 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 Banach space, we establish a uniqueness criterion for positive solution of the BVP. In particular such a positive solution uλ(t) of the BVP depends continuously on the parameter λ, i.e., uλ(t) is nondecreasing in λ, limλ→⁰⁺‖uλ(t)‖=0 and limλ→+∞‖uλ(t)‖=+∞ for any t∈[0,1].     

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‖=+∞.     

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.     

A second-order nonlinear difference equation: Oscillation and asymptotic behavior

Discrete analogues are investigated for well-known results on oscillation, growth, and asymptotic behavior of solutions of y″ + q(t) y^γ = 0, for q(t) ? 0 and for q(t) ? 0. The analogue of Atkinson's oscillation criterion is shown to be true for Δ²y_{n ? 1} + q_ny_n^γ = 0, but the analogue for Atkinson's nonoscillation criterion is shown to be false.     

Nonnegative (s, t)-potent matrices

This paper contains a study of matrices satisfying A^s = A^t for different positive integers s and t. Representations, similar to Flor's well-known characterization of a nonnegative idempotent matrix, are obtained for nonnegative matrices of this type.     

New classes of complete balanced Howell rotations

Complete balanced Howell rotations (CBHR) owe their origins to duplicate bridge tournaments but have since been shown to possess of deep combinatorial properties. They include many other combinatorial designs as special cases, such as: balanced Howell rotations, weak complete balanced Howell rotations, Room squares, Howell designs, and a class of balanced incomplete block designs.All known CBHR's are for n partnerships such that n = 2^t(p^r + 1), where p^r is an odd prime power and t a natural number. In most cases, p^r ≡ 3(mod 4) is also assumed. Berlekamp and Hwang gave constructions of CBHR's for each such n > 3 with t = 0; Schellenberg gave constructions for each such n with t = 1. In this paper, we construct CBHR for each such n with t arbitrary.     

A construction for PBIBD(2)'s

This note gives a new construction for PBIBD(2)'s that generalizes a construction of Hall's for finite projective planes, and that leads to a new PBIBD(2) with parameters (v, b, k, r, λ₁, λ₂) = (36, 60, 10, 0, 2).     

Cyclotomic Z2-extensions of J-fields

Hurwitz-type relations of Iwasawa's λ₂^?-invariants and the 2-ranks of the “narrow” ideal class groups in the 2-extensions of J-fields are given under the assumption of the vanishing of μ₂-invariants.     

Kostka functions associated to complex reflection groups and a conjecture of Finkelberg-Ionov

Kostka functions K_(λ,μ)~±(t), indexed by r-partitions λ and μ of n, are a generalization of Kostka polynomials K_(λ,μ)(t) indexed by partitions λ,μ of n. It is known that Kostka polynomials have an interpretation in terms of Lusztig's partition function. Finkelberg and Ionov(2016) defined alternate functions K_(λ,μ)(t) by using an analogue of Lusztig's partition function, and showed that K_(λ,μ)(t) ∈Z≥0[t] for generic μ by making use of a coherent realization. They conjectured that K_(λ,μ)(t) coincide with K_(λ,μ)~-(t). In this paper, we show that their conjecture holds. We also discuss the multi-variable version, namely, r-variable Kostka functions K_(λ,μ)~±(t_1,…,t_r).     

Positive and sign-changing solutions for fourth-order BVPs with parameters

This paper is concerned with the existence and multiplicity of positive and sign-changing solutions of the fourth-order boundary value problem u ⁽⁴⁾(t)=λ f(t,u(t),u ^′′(t)), 0<t<1,?u(0)?=?u(1)=u ^′′(0)=u ^′′(1)?=0, where f:[0,1]×?→? is continuous, λ∈? is a parameter. By using the fixed-point index theory of differential operators, it is proved that the above boundary value problem has positive, negative and sign-changing solutions for λ being different intervals. As an example, the boundary value problem u ⁽⁴⁾(t)+?η u ^′′(t)??ζu(t)=?λ f(t,u(t)), ?0<t<1,?u(0)=?u(1)=?u ^′′(0)=?u ^′′(1)=0 is also considered and some obtained results are the complement of the known results.     

A Galerkin Approximation for the Initial-Value Problem for Linear Second-Order Differential Equations

In this paper, a Galerkin type algorithm is given for the numerical solution of L(x)=(r(t)x'(t))'-p(t)x(t)=g(t); x(a)=x_a, x'(a)=x'_a, where r (t)>f0, and Spline hat functions form the approximating basis. Using the related quadratic form, a two-step difference equation is derived for the numerical solutions. A discrete Gronwall type lemma is then used to show that the error at the node points satisfies e_k=0(h²). If e(t) is the error function on a?t?b; it is also shown (in a variety of norms) that e(t)?Ch² and e'(t)?C₁h. Test case runs are also included. A (one step) Richardson or Rhomberg type procedure is used to show that e^R_k=0(h⁴). Thus our results are comparable to Runge-Kutta with half the function evaluations.     

Asymptotic Behavior of Massless Dirac Waves in Schwarzschild Geometry

In this paper, we show that massless Dirac waves in the Schwarzschild geometry decay to zero at a rate t ^?2λ , where λ = 1, 2, . . . is the angular momentum. Our technique is to use Chandrasekhar’s separation of variables whereby the Dirac equations split into two sets of wave equations. For the first set, we show that the wave decays as t ^?2λ . For the second set, in general, the solutions tend to some explicit profile at the rate t ^?2λ . The decay rate of solutions of Dirac equations is achieved by showing that the coefficient of the explicit profile is exactly zero. The key ingredients in the proof of the decay rate of solutions for the first set of wave equations are an energy estimate used to show the absence of bound states and zero energy resonance and the analysis of the spectral representation of the solutions. The proof of asymptotic behavior for the solutions of the second set of wave equations relies on careful analysis of the Green’s functions for time independent Schrödinger equations associated with these wave equations.     

Some mutually disjoint steiner systems

A t-design λ; t-d-n is a system of subsets of size d (called blocks) from an n-set S, such that each t-subset from S is contained in precisely λ blocks. A Steiner system S(l, m, n) is a t-design with parameters 1; l-m-n. Two Steiner systems (or t-designs) are disjoint if they share no blocks. A search has been conducted which resulted in discovering 9 mutually disjoint S(5, 8, 24)'s, 24 mutually disjoint S(4, 7, 23)'s, 60 mutually disjoint S(3, 6, 22)'s, and 197 mutually disjoint S(2, 5, 21)'s. Taking unions of several mutually disjoint Steiner systems will then produce t-designs (with varying λ's) on 21, 22, 23, and 24 points.     

Note on Creasy's confidence limits for the gradient in the linear functional relationship

An apparent gap in the proof of Creasy's t test is filled and some false statements on this topic, found in the literature, are corrected. Creasy's test is shown to be identical with Williams' t test. The latter can be generalized to the multivariate case of a linear functional relationship.     

Potential theory associated with Uhlenbeck-Ornstein process

Some parallel results of Gross' paper (Potential theory on Hilbert space, J. Functional Analysis1 (1967), 123–181) are obtained for Uhlenbeck-Ornstein process U(t) in an abstract Wiener space (H, B, i). Generalized number operator $\text{N}$ is defined by $\text{N}$f(x) = ?lim_∈←0{E[f($\text{U}$(τ^∈_ξ))] ? f(x)}/E[τ^∈_ξ, where τ_x^? is the first exit time of U(t) starting at x from the ball of radius ? with center x. It is shown that $\text{N}$f(x) = ?trace D²f(x)+〈Df(x),x〉 for a large class of functions f. Let r_t(x, dy) be the transition probabilities of U(t). The λ-potential G_λf, λ > 0, and normalized potential Rf of f are defined by G_λf(X) = ∫₀^∞e^?λtr_tf(x) dt and Rf(x) = ∫₀^∞ [r_tf(x) ? r_tf(0)] dt. It is shown that if f is a bounded Lip-1 function then trace D²G_λf(x) ? 〈DG_λf(x), x〉 = ?f(x) + λG_λf(x) and trace D²Rf(x) ? 〈DRf(x), x〉 = ?f(x) + ∫_Bf(y)p₁(dy), where p₁ is the Wiener measure in B with parameter 1. Some approximation theorems are also proved.     

Global branches of periodic solutions for forced delay differential equations on compact manifolds

We prove a global bifurcation result for T-periodic solutions of the T-periodic delay differential equation x^′(t)=λf(t,x(t),x(t−1)) depending on a real parameter λ?0. The approach is based on the fixed point index theory for maps on ANRs.     

Toward a stochastic calculus for several Markov processes

In this paper we investigate a class of harmonic functions associated with a pair xt = (xt11, xt22) of strong Markov processes. In the case where both processes are Brownian motions, a smooth function f is harmonic if Δx1Δx2f(x1,x2) = 0. For these harmonic functions we investigate a certain boundary value problem which is analogous to the Dirichlet problem associated with a single process. One basic tool for this study is a generalization of Dynkin's formula, which can be thought of as a kind of stochastic Green's formula. Another important tool is the use of Markov processes xti?i obtained from xtii by certain random time changes. We call such a process a stochastic wave since it propogates deterministically through a certain family of sets; however its position on a given set is random.