<Emphasis Type="Italic">p</Emphasis>-Adic valuation of the Morgan–Voyce sequence and <Emphasis Type="Italic">p</Emphasis>-regularity

We characterize the p-adic valuation of the Morgan–Voyce sequence and its companion sequence. Further, we show that the p-adic valuation of the Morgan–Voyce sequence is a p-regular sequence and we determine its rank explicitly.     

On Analytic Campanato and <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>, <Emphasis Type="Italic">s</Emphasis>) Spaces

We investigate the relation between analytic Campanato spaces \(\mathcal {AL}_{p,s}\) and the spaces F(p, q, s), characterize the bounded and compact Riemann–Stieltjes operators from \(\mathcal {AL}_{p,s}\) to \(F(p,p-s-1,s)\). We also describe the corona theorem and the interpolating sequences for the class \(F(p,p-2,s)\), which is the Möbius invariant subspace of the analytic Besov type spaces \(B_p(s)\).     

On the distribution of <Emphasis Type="Italic">αp</Emphasis> modulo one for primes <Emphasis Type="Italic">p</Emphasis> of a special form

A classical problem in analytic number theory is to study the distribution of αp modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p+2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.     

On the regularity of scalar <Emphasis Type="Italic">p</Emphasis>-Harmonic functions in <Emphasis Type="Italic">n</Emphasis> dimensions with 2 ≤ <Emphasis Type="Italic">p</Emphasis> < <Emphasis Type="Italic">n</Emphasis>

Using Tilli’s technique [Cal Var 25(3):395–401, 2006], we shall give a new proof of the regularity of the local minima of the functional

$J\left( u\right) =\int\limits_{\Omega } \left\vert \partial u\right\vert^{p}\,dx$

with Ω a domain of class C ^{0, 1} in \({\mathbb{R}^{n}}\) and 2 ≤ p < n.

     

Rational approximations of functions of Markov-Stieltjes type in Hardy spaces <Emphasis Type="Italic">H</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript>, 0 < <Emphasis Type="Italic">p</Emphasis> ≤ ∞

Orders of the best approximations for functions of Markov-Stieltjes type in the space H ^p are considered in the case of intersection of the measure support and the boundary of a unit circle. The results of J.-E. Andersson for functions with a singularity at a single point of a unit circumference are extended to the case of two singularities on that circumference.     

Supersymmetry-inspired low-energy <Emphasis Type="Italic">α</Emphasis>–<Emphasis Type="Italic">p</Emphasis> elastic scattering phases

We consider an effective potential model consisting of an electromagnetic part plus a nuclear part as the ground state interaction for an α–p system. The next few higher partial wave interactions are generated using the formalism of supersymmetric quantum mechanics. We adapt the phase function method to compute α–p elastic scattering phases up to 12 MeV, including the effect of the electromagnetic interaction quite rigorously in our phase shift calculation. With the further incorporation of some energy-dependent correction factors to our interactions, we obtain a good agreement with the experimental data.     

Lower semicontinuity of mappings with bounded (<Emphasis Type="Italic">θ</Emphasis>, 1)-weighted (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-distortion

We prove that, under some extra conditions, the locally uniform limit of mappings with bounded (θ, 1)-weighted (p, q)-distortion is a mapping of bounded (θ, 1)-weighted (p, q)-distortion too. Moreover, we obtain the lower semicontinuity of the distortion coefficients.     

Solvability of Mixed Problems for the Klein–Gordon–Fock Equation in the Class <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> for <Emphasis Type="Italic">p</Emphasis> ≥ 1

We prove that the mixed problem for the Klein–Gordon–Fock equation u _tt (x, t) ? u _xx (x, t) + au(x, t) = 0, where a ≥ 0, in the rectangle Q _T = [0 ≤ x ≤ l] × [0 ≤ t ≤ T] with zero initial conditions and with the boundary conditions u(0, t) = μ(t) ∈ L _p [0, T ], u(l, t) = 0, has a unique generalized solution u(x, t) in the class L _p (Q _T ) for p ≥ 1. We construct the solution in explicit analytic form.     

Difference sets with <Emphasis Type="Italic">n</Emphasis> = 5<Emphasis Type="Italic"> p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>

Let D be a (v, k, λ)-difference set in an abelian group G, and (v, 31) = 1. If n = 5p ^r with p a prime not dividing v and r a positive integer, then p is a multiplier of D. In the case 31|v, we get restrictions on the parameters of such difference sets D for which p may not be a multiplier.      

On the <Emphasis Type="Italic">p</Emphasis>-supersolubility of a finite group with a µ-supplemented sylow <Emphasis Type="Italic">p</Emphasis>-subgroup

A subgroup H of a group G is called µ-supplemented in G if there exists a subgroup K such that G = HK and H ₁ K is a proper subgroup in G for every maximal subgroup H ₁ in H. For the initial values of p, we establish the p-supersolubility of a finite group with a μ-supplemented Sylow p-subgroup.     

Poisson brackets of mappings obtained as (<Emphasis Type="Italic">q</Emphasis>,−<Emphasis Type="Italic">p</Emphasis>) reductions of lattice equations

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson brackets are derived from a Lagrangian, using the so-called Ostrogradsky transformation. The (q,?p) reductions are (p + q)-dimensional maps and explicit Poisson brackets for such reductions of the discrete KdV equation, the discrete Lotka–Volterra equation, and the discrete Liouville equation are included. Lax representations of these equations can be used to construct sufficiently many integrals for the reductions. As examples we show that the (3,?2) reductions of the integrable partial difference equations are Liouville integrable in their own right.     

<Emphasis Type="Italic">p</Emphasis>-Adic Brownian motion over ℚ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

In this paper, we generalize the result of Bikulov and Volovich (1997) and construct a p-adic Brownian motion over ℚ _p . First, we construct directly a p-adic white noise over ℚ _p by using a specific complete orthonormal system of (ℚ _p ). A p-adic Brownian motion over ℚ _p is then constructed by the Paley-Wiener method. Finally, we introduce a p-adic random walk and prove a theorem on the approximation of a p-adic Brownian motion by a p-adic random walk.     

The smooth structure set of <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> × <Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">q</Emphasis></Superscript>

We calculate \({\mathcal{S}^{{\it Diff}}(S^p \times S^q)}\), the smooth structure set of S ^p × S ^q , for p, q ≥ 2 and p + q ≥ 5. As a consequence we show that in general \({\mathcal{S}^{Diff}(S^{4j-1}\times S^{4k})}\) cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of the forgetful map \({\mathcal{S}^{Diff}(S^{4j}\times S^{4k}) \rightarrow \mathcal{S}^{Top}(S^{4j}\times S^{4k})}\) is not in general a subgroup of the topological structure set.     

Convergence Theorems for ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>-Norm Minimizers with Respect to <Emphasis Type="Italic">p</Emphasis>

Let C be a fixed compact convex subset of and let x_p be the unique minimal _p-norm element in C for any . In this paper, we study the convergence of x_p as p or , respectively. We characterize also the limit point as the minimal element of C with respect to the lexical minimax order relation or the lexical minitotal order relation, respectively.Communicated by D. G. LuenbergerThe author thanks Professor H. Komiya for valuable advice and an anonymous referee for kind comments.     

Weak-Polynomial Convergence on Spaces ℓ<Subscript><Emphasis Type="Italic">p</Emphasis></Subscript> and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

This paper is concerned with the study of the set P ^-1(0), when P varies over all orthogonally additive polynomials on _p and L _p spaces. We apply our results to obtain characterizations of the weak-polynomial topologies associated to this class of polynomials.     

A note on a class of <Emphasis Type="Italic">p</Emphasis>-valent starlike functions of order <Emphasis Type="Italic">β</Emphasis>

We obtain sharp coefficient bounds for some p-valent starlike functions of order β, 0 ≤ β < p. Initially this problem was handled by Aouf in [1]. We pointed out that the proof by Aouf is incorrect and present a correct proof.     

<Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates for the Schrödinger type operators

Let L _k = (?Δ) ^k + V ^k be a Schrödinger type operator, where k ≥ 1 is a positive integer and V is a nonnegative polynomial. We obtain the L ^p estimates for the operators ?^2k L _k ^?1 and ? ^k L _k ^?1/2 .     

Derivatives for smooth representations of <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℝ) and <Emphasis Type="Italic">GL</Emphasis>(<Emphasis Type="Italic">n</Emphasis>, ℂ)

The notion of derivatives for smooth representations of GL(n, ? _p ) was defined in [BZ77]. In the archimedean case, an analog of the highest derivative was defined for irreducible unitary representations in [Sah89] and called the “adduced” representation. In this paper we define derivatives of all orders for smooth admissible Fréchet representations of moderate growth. The real case is more problematic than the p-adic case; for example, arbitrary derivatives need not be admissible. However, the highest derivative continues being admissible, and for irreducible unitarizable representations coincides with the space of smooth vectors of the adduced representation.In the companion paper [AGS] we prove exactness of the highest derivative functor, and compute highest derivatives of all monomial representations.We apply those results to finish the computation of adduced representations for all irreducible unitary representations and to prove uniqueness of degenerate Whittaker models for unitary representations, thus completing the results of [Sah89, Sah90, SaSt90, GS13a].     

A theorem of the alternatives for the equation |<Emphasis Type="Italic">Ax</Emphasis>| − |<Emphasis Type="Italic">B</Emphasis>||<Emphasis Type="Italic">x</Emphasis>| = <Emphasis Type="Italic">b</Emphasis>

A theorem of the alternatives for the equation \({|Ax|-|B||x|=b\ (A,B\in{\mathbb{R}}^{n\times n},\, b\in{\mathbb{R}}^n)}\) is proved and several consequences are drawn. In particular, a class of matrices A, B is identified for which the equation has exactly 2 ⁿ solutions for each positive right-hand side b.     

Convergence in <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>([0, <Emphasis Type="Italic">T</Emphasis>]) of Wavelet Expansions of <Emphasis Type="Italic">φ</Emphasis>-Sub-Gaussian Random Processes

The article presents new results on convergence in L _p ([0,T]) of wavelet expansions of φ-sub-Gaussian random processes. The convergence rate of the expansions is obtained. Specifications of the obtained results are discussed.