On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Uniform in bandwidth exact rates for a class of kernel estimators

Given an i.i.d sample (Y _i , Z _i ), taking values in \({\mathbb{R}^{d'}\times\mathbb{R}^d}\), we consider a collection Nadarya–Watson kernel estimators of the conditional expectations \({\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}\), where z belongs to a compact set \({H\subset \mathbb{R}^d}\), g a Borel function on \({\mathbb{R}^{d'}}\) and c _g (·), d _g (·) are continuous functions on \({\mathbb{R}^d}\). Given two bandwidth sequences \({h_n<\mathfrak{h}_n}\) fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in \({g\in\mathcal{G},\;z\in H}\) and \({h_n\le h\le \mathfrak{h}_n}\) under mild conditions on the density f _Z , the class \({\mathcal{G}}\), the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities \({\mathbb{P}( Y\in C\mid Z=z)}\), that hold uniformly in \({z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}\). Here \({\mathcal{C}}\) is a Vapnik–Chervonenkis class of sets.     

Nonarchimedean Green functions and dynamics on projective space

Let \({\varphi: \mathbb{P}^N_K\to\mathbb{P}^N_K}\) be a morphism of degree d ≥ 2 defined over a field K that is algebraically closed field and complete with respect to a nonarchimedean absolute value. We prove that a modified Green function \({\hat{g}_\varphi}\) associated to \({\varphi}\) is Hölder continuous on \({\mathbb{P}^N(K)}\) and that the Fatou set \({\mathcal{F}(\varphi)}\) of \({\varphi}\) is equal to the set of points at which \({\hat{g}_\Phi}\) is locally constant. Further, \({\hat{g}_\varphi}\) vanishes precisely on the set of points P such that \({\varphi}\) has good reduction at every point in the forward orbit \({\mathcal{O}_\varphi(P)}\) of P. We also prove that the iterates of \({\varphi}\) are locally uniformly Lipschitz on \({\mathcal{F}(\varphi)}\) .     

Complex homomorphisms on self-conjugate algebras of functions

Let X be a non-void set and A be a subalgebra of \({\mathbb{C}^{X}}\) . We call a \({\mathbb{C}}\) -linear functional \({\varphi}\) on A a 1-evaluation if \({\varphi(f) \in f(X) }\) for all \({f\in A}\) . From the classical Gleason–Kahane–?elazko theorem, it follows that if X in addition is a compact Hausdorff space then a mapping \({\varphi}\) of \({C_{\mathbb{C}}(X) }\) into \({\mathbb{C}}\) is a 1-evaluation if and only if \({\varphi}\) is a \({\mathbb{C}}\) -homomorphism. In this paper, we aim to investigate the extent to which this equivalence between 1-evaluations and \({\mathbb{C}}\) -homomorphisms can be generalized to a wider class of self-conjugate subalgebras of \({\mathbb{C}^{X}}\) . In this regards, we prove that a \({\mathbb{C}}\) -linear functional on a self-conjugate subalgebra A of \({\mathbb{C}^{X}}\) is a positive \({\mathbb{C}}\) -homomorphism if and only if \({\varphi}\) is a \({\overline{1}}\) -evaluation, that is, \({\varphi(f) \in\overline{f\left(X\right)}}\) for all \({f\in A}\) . As consequences of our general study, we prove that 1-evaluations and \({\mathbb{C}}\) -homomorphisms on \({C_{\mathbb{C}}\left( X\right)}\) coincide for any topological space X and we get a new characterization of realcompact topological spaces.     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Diagonalization of the Finite Hilbert Transform on Two Adjacent Intervals

We continue the study of stability of solving the interior problem of tomography. The starting point is the Gelfand–Graev formula, which converts the tomographic data into the finite Hilbert transform (FHT) of an unknown function f along a collection of lines. Pick one such line, call it the x-axis, and assume that the function to be reconstructed depends on a one-dimensional argument by restricting f to the x-axis. Let \(I_1\) be the interval where f is supported, and \(I_2\) be the interval where the Hilbert transform of f can be computed using the Gelfand–Graev formula. The equation to be solved is \(\left. {\mathcal {H}}_1 f=g\right| _{I_2}\), where \({\mathcal {H}}_1\) is the FHT that integrates over \(I_1\) and gives the result on \(I_2\), i.e. \({\mathcal {H}}_1: L^2(I_1)\rightarrow L^2(I_2)\). In the case of complete data, \(I_1\subset I_2\), and the classical FHT inversion formula reconstructs f in a stable fashion. In the case of interior problem (i.e., when the tomographic data are truncated), \(I_1\) is no longer a subset of \(I_2\), and the inversion problems becomes severely unstable. By using a differential operator L that commutes with \({\mathcal {H}}_1\), one can obtain the singular value decomposition of \({\mathcal {H}}_1\). Then the rate of decay of singular values of \({\mathcal {H}}_1\) is the measure of instability of finding f. Depending on the available tomographic data, different relative positions of the intervals \(I_{1,2}\) are possible. The cases when \(I_1\) and \(I_2\) are at a positive distance from each other or when they overlap have been investigated already. It was shown that in both cases the spectrum of the operator \({\mathcal {H}}_1^*{\mathcal {H}}_1\) is discrete, and the asymptotics of its eigenvalues \(\sigma _n\) as \(n\rightarrow \infty \) has been obtained. In this paper we consider the case when the intervals \(I_1=(a_1,0)\) and \(I_2=(0,a_2)\) are adjacent. Here \(a_1 < 0 < a_2\). Using recent developments in the Titchmarsh–Weyl theory, we show that the operator L corresponding to two touching intervals has only continuous spectrum and obtain two isometric transformations \(U_1\), \(U_2\), such that \(U_2{\mathcal {H}}_1 U_1^*\) is the multiplication operator with the function \(\sigma (\lambda )\), \(\lambda \ge (a_1^2+a_2^2)/8\). Here \(\lambda \) is the spectral parameter. Then we show that \(\sigma (\lambda )\rightarrow 0\) as \(\lambda \rightarrow \infty \) exponentially fast. This implies that the problem of finding f is severely ill-posed. We also obtain the leading asymptotic behavior of the kernels involved in the integral operators \(U_1\), \(U_2\) as \(\lambda \rightarrow \infty \). When the intervals are symmetric, i.e. \(-a_1=a_2\), the operators \(U_1\), \(U_2\) are obtained explicitly in terms of hypergeometric functions.     

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.     

Lattice homomorphisms between Sobolev spaces

We show in Theorem 4.4 that every vector lattice homomorphism T from \({\mathsf{W}^{1,p}_0(\Omega_1)}\) into \({\mathsf{W}^{1,q}(\Omega_2)}\) for \({p,q\in (1,\infty)}\) and open sets \({\Omega_1,\Omega_2\subset\mathbb{R}^N}\) has a representation of the form \({T\mathsf{u}=(\mathsf{u}\circ\xi)g}\) (Cap _q -quasi everywhere on Ω₂) with mappings ξ : Ω₂ → Ω₁ and g : Ω₂ → [0, ∞). This representation follows as an application of an abstract and more general representation theorem (Theorem 3.5). Other applications are also given.     

Note on Bernoulli numbers associated to some Dirichlet character of prime conductor

We fix an integer \({n \geq 1}\) and a divisor m of n such that n/m is odd. Let p be a prime number of the form \({p=2n\ell+1}\) for some odd prime number \({\ell}\) with \({\ell \nmid m}\). Let \({S=pB_{1,2m\ell}}\) be the p times of the generalised Bernoulli number associated to an odd Dirichlet character of conductor p and order \({2m\ell}\), which is an algebraic integer of the \({2m\ell}\)th cyclotomic field. It is known that \({S \neq 0}\). More strongly, we show that when \({\ell}\) is sufficiently large, the trace of \({\zeta^{-1}S}\) to the \({2m}\)th cyclotomic field does not vanish for any\({\ell}\)th root \({\zeta}\) of unity. We also show a related result on indivisibility of relative class numbers.     

Convergence of Martingale and Moderate Deviations for a Branching Random Walk with a Random Environment in Time

We consider a branching random walk on \({\mathbb {R}}\) with a stationary and ergodic environment \(\xi =(\xi _n)\) indexed by time \(n\in {\mathbb {N}}\). Let \(Z_n\) be the counting measure of particles of generation n and \(\tilde{Z}_n(t)=\int \mathrm{e}^{tx}Z_n(\mathrm{d}x)\) be its Laplace transform. We show the \(L^p\) convergence rate and the uniform convergence of the martingale \(\tilde{Z}_n(t)/{\mathbb {E}}[\tilde{Z}_n(t)|\xi ]\), and establish a moderate deviation principle for the measures \(Z_n\).     

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.     

Reducing Subspaces of Multiplication Operators on the Dirichlet Space

In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product \({\phi}\) on the Dirichlet space D. We prove that any two distinct nontrivial minimal reducing subspaces of \({M_\phi}\) are orthogonal. When the order n of \({\phi}\) is 2 or 3, we show that \({M_\phi}\) is reducible on D if and only if \({\phi}\) is equivalent to \({z^n}\). When the order of \({\phi}\) is 4, we determine the reducing subspaces for \({M_\phi}\), and we see that in this case \({M_\phi}\) can be reducible on D when \({\phi}\) is not equivalent to \({z^4}\). The same phenomenon happens when the order n of \({\phi}\) is not a prime number. Furthermore, we show that \({M_\phi}\) is unitarily equivalent to \({M_{z^n} (n > 1)}\) on D if and only if \({\phi = az^n}\) for some unimodular constant a.     

On the additivity of block designs

We show that symmetric block designs \({\mathcal {D}}=({\mathcal {P}},{\mathcal {B}})\) can be embedded in a suitable commutative group \({\mathfrak {G}}_{\mathcal {D}}\) in such a way that the sum of the elements in each block is zero, whereas the only Steiner triple systems with this property are the point-line designs of \({\mathrm {PG}}(d,2)\) and \({\mathrm {AG}}(d,3)\). In both cases, the blocks can be characterized as the only k-subsets of \(\mathcal {P}\) whose elements sum to zero. It follows that the group of automorphisms of any such design \(\mathcal {D}\) is the group of automorphisms of \({\mathfrak {G}}_\mathcal {D}\) that leave \(\mathcal {P}\) invariant. In some special cases, the group \({\mathfrak {G}}_\mathcal {D}\) can be determined uniquely by the parameters of \(\mathcal {D}\). For instance, if \(\mathcal {D}\) is a 2-\((v,k,\lambda )\) symmetric design of prime order p not dividing k, then \({\mathfrak {G}}_\mathcal {D}\) is (essentially) isomorphic to \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\), and the embedding of the design in the group can be described explicitly. Moreover, in this case, the blocks of \(\mathcal {B}\) can be characterized also as the v intersections of \(\mathcal {P}\) with v suitable hyperplanes of \(({\mathbb {Z}}/p{\mathbb {Z}})^{\frac{v-1}{2}}\).     

Integral means of weighted Hardy–Bloch functions

Let \({\mathcal{B}^\omega(p, q, B_d)}\) denote the \({\omega}\)-weighted Hardy–Bloch space on the unit ball B _d of \({\mathbb{C}^d}\), \({d\ge 1}\). For \({2< p,q < \infty}\) and \({f\in \mathcal{B}^\omega(p, q, B_d)}\), we obtain sharp estimates on the growth of the p-integral means M _p (f, r) as \({r\to 1-}\).     

Intersection of Hamming codes avoiding Hamming subcodes

We prove that given a binary Hamming code \({{\mathcal{H}}^n}\) of length n = 2 ^m ? 1, m ≥ 3, or equivalently a projective geometry PG(m ? 1, 2), there exist permutations \({\pi \in \mathcal{S}_n}\) , such that \({{\mathcal{H}}^n}\) and \({\pi({\mathcal{H}}^n)}\) do not have any Hamming subcode with the same support, or equivalently the corresponding projective geometries do not have any common flat. The introduced permutations are called AF permutations. We study some properties of these permutations and their relation with the well known APN functions.     

On Modular Edge-Graceful Graphs

Let G be a connected graph of order \({n\ge 3}\) and size m and \({f:E(G)\to \mathbb{Z}_n}\) an edge labeling of G. Define a vertex labeling \({f': V(G)\to \mathbb{Z}_n}\) by \({f'(v)= \sum_{u\in N(v)}f(uv)}\) where the sum is computed in \({\mathbb{Z}_n}\) . If f′ is one-to-one, then f is called a modular edge-graceful labeling and G is a modular edge-graceful graph. A graph G is modular edge-graceful if G contains a modular edge-graceful spanning tree. Several classes of modular edge-graceful trees are determined. For a tree T of order n where \({n\not\equiv 2 \pmod 4}\) , it is shown that if T contains at most two even vertices or the set of even vertices of T induces a path, then T is modular edge-graceful. It is also shown that every tree of order n where \({n\not\equiv 2\pmod 4}\) having diameter at most 5 is modular edge-graceful.     

Semigroups of rectangular matrices under a sandwich operation

Let \({\mathcal {M}}_{mn}={\mathcal {M}}_{mn}({\mathbb {F}})\) denote the set of all \(m\times n\) matrices over a field \({\mathbb {F}}\), and fix some \(n\times m\) matrix \(A\in {\mathcal {M}}_{nm}\). An associative operation \(\star \) may be defined on \({\mathcal {M}}_{mn}\) by \(X\star Y=XAY\) for all \(X,Y\in {\mathcal {M}}_{mn}\), and the resulting sandwich semigroup is denoted \({\mathcal {M}}_{mn}^A={\mathcal {M}}_{mn}^A({\mathbb {F}})\). These semigroups are closely related to Munn rings, which are fundamental tools in the representation theory of finite semigroups. We study \({\mathcal {M}}_{mn}^A\) as well as its subsemigroups \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\) and \({\mathcal {E}}_{mn}^A\) (consisting of all regular elements and products of idempotents, respectively), and the ideals of \(\hbox {Reg}({\mathcal {M}}_{mn}^A)\). Among other results, we characterise the regular elements; determine Green’s relations and preorders; calculate the minimal number of matrices (or idempotent matrices, if applicable) required to generate each semigroup we consider; and classify the isomorphisms between finite sandwich semigroups \({\mathcal {M}}_{mn}^A({\mathbb {F}}_1)\) and \({\mathcal {M}}_{kl}^B({\mathbb {F}}_2)\). Along the way, we develop a general theory of sandwich semigroups in a suitably defined class of partial semigroups related to Ehresmann-style “arrows only” categories; we hope this framework will be useful in studies of sandwich semigroups in other categories. We note that all our results have applications to the variants \({\mathcal {M}}_n^A\) of the full linear monoid \({\mathcal {M}}_n\) (in the case \(m=n\)), and to certain semigroups of linear transformations of restricted range or kernel (in the case that \(\hbox {rank}(A)\) is equal to one of m, n).     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).