On the Gosper’s <Emphasis Type="Italic">q</Emphasis>-constant Π<Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>

Gosper introduced the functions sin_qz and cos_qz as q-analogues for the trigonometric functions sin z and cos z respectively. He stated a variety of identities involving these two q-trigonometric functions along with certain constants denoted by \({\Pi _{{q^n}}}\) (n ∈ N). Gosper noticed that all his formulas on these constants have more than two of the \({\Pi _{{q^n}}}\). So, it is natural to raise the question of establishing identities involving only two of the \({\Pi _{{q^n}}}\). In this paper, our main goal is to give examples of such formulas in only two \({\Pi _{{q^n}}}\).     

On the towers of torsion Bertrandias and Payan modules

For an odd prime p, let K/k be a Galois p-extension and S be a set of primes of k containing the primes lying over p. For the p ^r th roots \({\mu _{{p^r}}}\left( K \right)\) of unity in K, we describe the so-called Sha group Sha _S (G(K/k), \({\mu _{{p^r}}}\left( K \right)\)) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields \({\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0\) where \({k_{{T^{i + 1}}}}\) is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of \({k_{{T^i}}}\) over \({k_{{T^i}}}\). This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees \({\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}\) over the towers. Using this formula we investigate whether the towers are stationary or not.     

A note on standard modules and Vogan L-packets

Let F be a non-Archimedean local field of characteristic 0, let G be the group of F-rational points of a connected reductive group defined over F and let \({G\prime}\) be the group of F-rational points of its quasi-split inner form. Given standard modules \({I(\tau, \nu )}\) and \({I(\tau\prime, \nu\prime)}\) for G and \({G\prime}\) respectively with \({\tau\prime}\) a generic tempered representation, such that the Harish-Chandra \({\mu}\)-function of a representation in the supercuspidal support of \({\tau}\) agrees with the one of a generic essentially square-integral representation in some Jacquet module of \({\tau\prime}\) (after a suitable identification of the underlying spaces under which \({\nu = \nu\prime}\)), we show that \({I(\tau, \nu)}\) is irreducible whenever \({I(\tau\prime, \nu\prime)}\) is. The conditions are satisfied if the Langlands quotients \({J(\tau, \nu})\) and \({J(\tau\prime, \nu\prime)}\) of respectively \({I(\tau, \nu)}\) and \({I(\tau\prime, \nu\prime)}\) lie in the same Vogan L-packet (whenever this Vogan L-packet is defined), proving that, for any Vogan L-packet, all the standard modules with Langlands quotient in a given Vogan L-packet are irreducible, if and only if this Vogan L-packet contains a generic representation. This result for generic Vogan L-packets was proven for quasi-split orthogonal and symplectic groups by Moeglin-Waldspurger and used in their proof of the general case of the local Gan-Gross-Prasad conjectures for these groups.     

A Supplement to Hölder’s Inequality. The Resonance Case. I

Suppose that m ≥ 2, numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ₁ ∈ \({L^{{p_1}}}\)(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L ^p (?¹), p ∈ (1, +∞], were introduced by the author in his previous papers.     

The Representation of a C0-Semigroup of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

For a C₀-semigroup \({\{U(t)\}_{t \geq 0}}\) of linear operators in a Banach space \({{\mathfrak{B}}}\) with generator A, we describe the set of elements \({x \in {\mathfrak{B}}}\) whose orbits U(t)x can be extended to entire \({{\mathfrak{B}}}\)-valued functions of a finite order and a finite type, and establish the conditions under which this set is dense in \({{\mathfrak{B}}}\). The Hille problem of finding vectors \({x \in {\mathfrak{B}}}\) such that there exists the limit \({\lim\limits_{n \to \infty}\left(I + \frac{tA}{n}\right)^{n}x}\) is also solved in the paper. We prove that this limit exists if and only if x is an entire vector of the operator A, and if this is the case, then it coincides with U(t)x.     

Solutions of elliptic equations with a level surface parallel to the boundary: Stability of the radial configuration

A positive solution of a homogeneous Dirichlet boundary value problem or initial-value problems for certain elliptic or parabolic equations must be radially symmetric and monotone in the radial direction if just one of its level surfaces is parallel to the boundary of the domain. Here, for the elliptic case, we prove the stability counterpart of that result. We show that if the solution is almost constant on a surface at a fixed distance from the boundary, then the domain is almost radially symmetric, in the sense that is contained in and contains two concentric balls \({B_{{r_e}}}\) and \({B_{{r_i}}}\), with the difference r_e-r_i (linearly) controlled by a suitable norm of the deviation of the solution from a constant. The proof relies on and elaborates arguments developed by Aftalion, Busca, and Reichel.     

Supercyclicity in Spaces of Operators

A sufficient criterion for the map \({C_{A, B}(S) = ASB}\) to be supercyclic on certain algebras of operators on Banach spaces is given. If T is an operator satisfying the Supercyclicity Criterion on a Hilbert space H, then the linear map \({C_{T}(V) = TVT^*}\) is shown to be norm-supercyclic on the algebra \({\mathcal{K}(H)}\) of all compact operators, COT-supercyclic on the real subspace \({\mathcal{S}(H)}\) of all self-adjoint operators and weak*-supercyclic on \({\mathcal{L}(H)}\) of all bounded operators on H. Examples including operators of the form \({C_{B_w, F_\mu}}\) are provided, where B_w and \({F_\mu}\) are respectively backward and forward shifts on Banach sequence spaces.     

Theta characteristics of hyperelliptic graphs

We study theta characteristics of hyperelliptic metric graphs of genus g with no bridge edges. These graphs have a harmonic morphism of degree two to a metric tree that can be lifted to a morphism of degree two of a hyperelliptic curve X over K to the projective line, with K an algebraically closed field of char\({(K) \not =2}\), complete with respect to a non-Archimedean valuation, with residue field k of char\({(k)\not=2}\). The hyperelliptic curve has \({2^{2g}}\) theta characteristics. We show that for each effective theta characteristic on the graph, \({2^{g-1}}\) even and \({2^{g-1}}\) odd theta characteristics on the curve specialize to it; and \({2^g}\) even theta characteristics on the curve specialize to the unique not effective theta characteristics on the graph.     

<Emphasis Type="Italic">p</Emphasis>-adic valuation of exponential sums associated to trinomials and some consequences

In this paper, we compute the p-adic valuation of exponential sums associated to trinomials \(F\left( X \right) = a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over Fp. As a byproduct of our results, we obtain restrictions for permutation polynomials of type \(a{X^{{d_1}}} + b{X^{{d_2}}} + c{X^{{d_3}}}\) over F_p.     

A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B′in separable complex Hilbert spaces H and H′, respectively. It is found that if the mapping f ∈ C~(1+α)at z_0∈ ?B with f(z_0) = w_0∈ ?B′, then the Fr′echet derivative operator Df(z_0) maps the tangent space Tz_0(?B~n) to Tw_0(?B′), the holomorphic tangent space T_(z_0)~(1,0)(?B~n) to T_(w_0)~(1,0)(?B′),respectively.     

On a theorem of Rigby

Let \({\Delta = BAG(2, q)}\) denote the classical biaffine plane of order q, that is, the symmetric \({((q^2 - 1)_q)}\) configuration obtained from the classical affine plane \({\Sigma = AG(2, q)}\) of order q by omitting a point of \({\Sigma}\) together with all lines through this point. Now let \({q \geq 4}\) be a power of a prime p and assume that \({\Delta}\) admits an embedding into the projective plane \({\Pi = PG(2, F)}\), where F is a (not necessarily commutative) field. Then this embedding extends to a projective subplane \({\Pi_0 \cong PG(2, q)}\) of \({\Pi}\); in particular, F has characteristic p. Consequently, \({BAG(2, q)}\) with \({q\geq 4}\) admits an embedding into \({PG(2, q')}\) if only if q′ is a power of q. This strengthens a result of Rigby (Canad J Math 17:977–1009, 1965) in a special case while simultaneously providing a more elegant proof.     

A sufficient condition for a rational differential operator to generate an integrable system

For a rational differential operator \({L=AB^{-1}}\), the Lenard–Magri scheme of integrability is a sequence of functions \({F_n, n \geq 0}\), such that (1) \({B(F_{n+1})=A(F_n)}\) for all \({n \geq 0}\) and (2) the functions \({B(F_n)}\) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of \({B(F_n)}\) is unbounded, property (2) holds if and only if L belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator L is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence (F _n ) starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.     

Remarks on D-integral complete multipartite graphs

A graph is called distance integral (or D-integral) if all eigenvalues of its distance matrix are integers. In their study of D-integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on D-integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs \({K_{{p_1},{p_2},{p_3}}}\) with p₁ < p₂ < p₃, and \({K_{{p_1},{p_2},{p_3},{p_4}}}\) with p₁ < p₂ < p₃ < p₄, as well as the infinite classes of distance integral complete multipartite graphs \({K_{{a_1}{p_1},{a_2}{p_2},...,{a_s}{p_s}}}\) with s = 5, 6.     

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}}\neq\emptyset}\) for every infinite set \({{\mathcal{H}}\subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X^X. Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.     

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q ₁(r) + q ₂(r) such that q ₁(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane.     

A Nonlocal <Emphasis Type="Italic">C</Emphasis>*-Algebra of Singular Integral Operators with Shifts Having Periodic Points

Representations on Hilbert spaces for a nonlocal C*-algebra \({{\mathfrak {B}}}\) of singular integral operators with piecewise slowly oscillating coefficients and unitary shift operators are constructed. The group of unitary shift operators U _g of the C*-algebra \({{\mathfrak {B}}}\) is associated with an amenable discrete group of homeomorphisms \({g:{\mathbb{T}}\to{\mathbb{T}}}\) that have piecewise continuous derivatives and the same nonempty set of periodic points. An isometric C*-algebra homomorphism of the quotient C*-algebra \({{\mathfrak {B}}^\pi={\mathfrak {B}}/{\mathcal {K}}}\), where \({{\mathcal {K}}}\) is the ideal of compact operators, into an n × n matrix algebra associated to a C*-algebra \({{\mathfrak {B}}_0}\) of singular integral operators with shifts having only fixed points is established making use of a spectral measure. Based on this generalization of the Litvinchuk–Gohberg–Krupnik reduction scheme, a symbol calculus for the C*-algebra \({{\mathfrak {B}}}\) as well as a Fredholm criterion for the operators in \({{\mathfrak {B}}}\) are obtained.     

Improvements of the Weil bound for Artin–Schreier curves

For the Artin–Schreier curve y ^q ? y = f(x) defined over a finite field \({{\mathbb F}_q}\) of q elements, the celebrated Weil bound for the number of \({{\mathbb F}_{q^r}}\)-rational points can be sharp, especially in super-singular cases and when r is divisible. In this paper, we show how the Weil bound can be significantly improved, using ideas from moment L-functions and Katz’s work on ?-adic monodromy calculations. Roughly speaking, we show that in favorable cases (which happens quite often), one can remove an extra \({\sqrt{q}}\) factor in the error term.     

Toeplitz Operators from One Fock Space to Another

In this paper, we study Toeplitz operators T _μ from one Fock space \({F^{p}_{\alpha}}\) to another \({F^{q}_{\alpha}}\) for 1 < p, q < ∞ with positive Borel measures μ as symbols. We characterize the boundedness (and compactness) of \({T_\mu: F^{p}_{\alpha} \to F^{q}_{\alpha}}\) in terms of the averaging function \({\widehat{\mu}_r}\) and the t-Berezin transform \({\widetilde{\mu}_t}\) respectively. Quite differently from the Bergman space case, we show that T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for some p ≤ q if and only if T _μ is bounded (or compact) from \({F^{p}_{\alpha}}\) to \({F^{q}_{\alpha}}\) for all p ≤ q. In order to prove our main results on T _μ , we introduce and characterize (vanishing) (p, q)-Fock Carleson measures on C ⁿ .     

The special L-value of the winding quotient of square-free level

Let X be a smooth projective curve of genus \({g \geq 2}\) over an algebraically closed field k of characteristic \({p > 0}\). Let \({F_{X/k} : X \rightarrow X_{1}}\) be the relative Frobenius morphism, and E be a semistable vector bundle on X. Mehta and Pauly asked that whether the length of the Harder–Narasimhan filtration of \({(F_{X/k})^*E}\) is at most p. In this article, we answer the above question negatively by constructing an example.     

On countably {\mu}-paracompact spaces

A topological space X is countably paracompact if and only if X satisfies the condition (A): For any decreasing sequence {F_i} of non-empty closed sets with \({\bigcap_{i=1}^{\infty} F_{i} = \emptyset}\) there exists a sequence {G_i} of open sets such that \({\bigcap_{i=1}^{\infty}\overline{G_{i}}=\emptyset}\) and \({F_{i} \subset G_{i}}\) for every i. We will show, by an example, that this is not true in generalized topological spaces. In fact there is a \({\mu}\)-normal generalized topological space satisfying the analogue of A which is not even countably \({\mu}\)-metacompact. Then we study the relationships between countably \({\mu}\)-paracompactness, countably \({\mu}\)-metacompactness and the condition corresponding to condition A in generalized topological spaces.