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 \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Symbol <Emphasis Type="Italic">p</Emphasis>-algebras of prime degree and their <Emphasis Type="Italic">p</Emphasis>-central subspaces

We prove that the maximal dimension of a p-central subspace of the generic symbol p-algebra of prime degree p is \({p+1}\). We do it by proving the following number theoretic fact: let \({\{s_1,\dots,s_{p+1}\}}\) be \({p+1}\) distinct nonzero elements in the additive group \({G=(\mathbb{Z}/p \mathbb{Z}) \times (\mathbb{Z}/p \mathbb{Z})}\), then every nonzero element \({g \in G}\) can be expressed as \({d_1 s_1+\dots+d_{p+1} s_{p+1}}\) for some non-negative integers \({d_1,\dots,d_{p+1}}\) with \({d_1+\dots+d_{p+1}\leq p-1}\).     

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 \({r\to 1-}\).     

Some remarks on energy inequalities for harmonic maps with potential

We call the \({\delta}\)-vector of an integral convex polytope of dimension d flat if the \({\delta}\)-vector is of the form \({(1,0,\ldots,0,a,\ldots,a,0,\ldots,0)}\), where \({a \geq 1}\). In this paper, we give the complete characterization of possible flat \({\delta}\)-vectors. Moreover, for an integral convex polytope \({\mathcal{P}\subset \mathbb{R}^N}\) of dimension d, we let \({i(\mathcal{P},n)=|n\mathcal{P}\cap \mathbb{Z}^N|}\) and \({i^*(\mathcal{P},n)=|n(\mathcal{P} {\setminus}\partial \mathcal{P})\cap \mathbb{Z}^N|}\). By this characterization, we show that for any \({d \geq 1}\) and for any \({k,\ell \geq 0}\) with \({k+\ell \leq d-1}\), there exist integral convex polytopes \({\mathcal{P}}\) and \({\mathcal{Q}}\) of dimension d such that (i) For \({t=1,\ldots,k}\), we have \({i(\mathcal{P},t)=i(\mathcal{Q},t),}\) (ii) For \({t=1,\ldots,\ell}\), we have \({i^*(\mathcal{P},t)=i^*(\mathcal{Q},t)}\), and (iii) \({i(\mathcal{P},k+1) \neq i(\mathcal{Q},k+1)}\) and \({i^*(\mathcal{P},\ell+1)\neq i^*(\mathcal{Q},\ell+1)}\).     

Irreducibility criteria for compositions of multivariate polynomials

We provide irreducibility criteria for compositions of multivariate polynomials over a field K, of the form \({f(X_{1},\ldots ,X_{r-1},g(X_{1},\ldots ,X_{r}))}\), with \({f,g\in K[X_{1},\ldots ,X_{r}]}\), for the case that \({f}\) as a polynomial in X_r is irreducible over \({K(X_{1},\ldots ,X_{r-1})}\) and has leading coefficient divisible by a power of an irreducible polynomial \({p(X_{1},\ldots ,X_{r-1})}\) of sufficiently large degree with respect to \({X_{r-1}}\).     

Arens regularity of projective tensor products

For completely contractive Banach algebras A and B (respectively operator algebras A and B), the necessary and sufficient conditions for the operator space projective tensor product \({A\widehat{\otimes}B}\) (respectively the Haagerup tensor product \({A\otimes^{h}B}\)) to be Arens regular are obtained. Using the non-commutative Grothendieck inequality, we show that, for C*-algebras A and B, \({A\otimes^{\gamma} B}\) is Arens regular if \({A\widehat{\otimes}B}\) and \({A\widehat{\otimes}B^{op}}\) are Arens regular whereas \({A\widehat{\otimes}B}\) is Arens regular if and only if \({A\otimes^{h}B}\) and \({B\otimes^{h}A}\) are, where \({\otimes^h}\), \({\otimes^{\gamma}}\), and \({\widehat{\otimes}}\) are the Haagerup, the Banach space projective tensor norm, and the operator space projective tensor norm, respectively.     

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 \({h(\Omega_{a,b})=1}\).     

Minimal tiled orders of finite global dimension

In this paper, we consider the sequence of balancing and Lucas balancing numbers. The balancing numbers \({B_n}\) are given by the recurrence \({B_n = 6 B_{n-1} - B_{n-2}}\) with initial conditions \({B_0 = 0, B_1 = 1}\) and its associated Lucas balancing numbers \({C_n}\) are given by the recurrence \({C_n = 6 C_{n-1} - C_{n-2}}\) with initial conditions \({C_0 = 1, C_1 = 3}\). First we find the perfect powers in the sequence of balancing and Lucas balancing numbers. We also identify those Lucas balancing numbers which are products of a power of 3 and a perfect power. Using this property of Lucas balancing numbers, we solve a conjecture regarding the non-existence of positive integral solution (x, y) for the Diophantine equation \({2x^2 + 1 = 3^b y^m}\) for any even positive integers b and m with \({m > 2}\), given in (Int J Number Theory 11:1259–1274, 2015). Also we prove that the Diophantine equations \({B_n B_{n+d}\ldots B_{n+(k-1)d} = y^m}\) and \({C_n C_{n+d}\ldots C_{n+(k-1)d} = y^m}\) have no solution for any positive integers n, d, k, y, and m with \({m \geq 2, y \geq 2}\) and gcd\({(n,d) = 1}\).     

Neostability-properties of Fraïssé limits of 2-nilpotent groups of exponent {p > 2}

Let \({L(n)}\) be the language of group theory with n additional new constant symbols \({c_1,\ldots,c_n}\). In \({L(n)}\) we consider the class \({{\mathbb{K}}(n)}\) of all finite groups G of exponent \({p > 2}\), where \({G'\subseteq\langle c_1^G,\ldots,c_n^G\rangle \subseteq Z(G)}\) and \({c_1^G,\ldots,c_n^G}\) are linearly independent. Using amalgamation we show the existence of Fraïssé limits \({D(n)}\) of \({{\mathbb{K}}(n)}\). \({D(1)}\) is Felgner’s extra special p-group. The elementary theories of the \({D(n)}\) are supersimple of SU-rank 1. They have the independence property.     

Ratios of harmonic functions with the same zero set

We study the ratio of harmonic functions u,v which have the same zero set Z in the unit ball \({B\subset \mathbb{R}^n}\). The ratio \({f=u/v}\) can be extended to a real analytic nowhere vanishing function in B. We prove the Harnack inequality and the gradient estimate for such ratios in any dimension: for a given compact set \({K\subset B}\) we show that \({\sup_K|f|\le C_1\inf_K|f|}\) and \({\sup_K\left|\nabla f\right|\le C_2 \inf_K|f|}\), where C ₁ and C ₂ depend on K and Z only. In dimension two we specify the dependence of the constants on Z in these inequalities by showing that only the number of nodal domains of u, i.e. the number of connected components of \({B\setminus Z}\), plays a role.     

Goldbach’s conjecture in arithmetic progressions: number and size of exceptional prime moduli

Set \({T=N^{\frac{1}{3}-\epsilon}}\). It is proved that for all but \({\ll TL^{-H},\,H > 0}\), exceptional prime numbers \({k\leq T}\) and almost all integers b ₁, b ₂ co-prime to k, almost all integers \({n\sim N}\) satisfying \({n\equiv b_{1}+b_{2}(mod\,k)}\) can be written as the sum of two primes p ₁ and p ₂ satisfying \({p_{i}\equiv b_{i}(mod\,k),\,i=1,2}\). For prime numbers \({k\leq N^{\frac{5}{24}-\epsilon}}\), this result is even true for all but \({\ll (\log\,N)^{D}}\) primes k and all integers b ₁, b ₂ co-prime to k.     

Solvability near the characteristic set for a special class of complex vector fields

This work deals with the solvability near the characteristic set Σ = {0} × S ¹ of operators of the form \({L=\partial/\partial t + (x^na(x) + ix^mb(x))\partial/\partial x}\), \({b\not\equiv0}\) and a(0) ≠ 0, defined on \({\Omega_\epsilon=(-\epsilon,\epsilon)\times S^1}\), \({\epsilon >0 }\), where a and b are real-valued smooth functions in \({(-\epsilon,\epsilon)}\) and m ≥ 2n. It is shown that given f belonging to a subspace of finite codimension of \({C^\infty(\Omega_\epsilon)}\) there is a solution \({u\in L^\infty}\) of the equation Lu = f in a neighborhood of Σ; moreover, the L ^∞ regularity is sharp.     

On certain functional equations related to Jordan triple $${(\theta, \phi)}$$-derivations on semiprime rings

The main purpose of this paper is to prove the following result. Let R be a 2-torsion free semiprime ring with symmetric Martindale ring of quotients Q _s and let \({\theta}\) and \({\phi}\) be automorphisms of R. Suppose \({T:R\rightarrow R}\) is an additive mapping satisfying the relation \({T(xyx)=T(x)\theta (y)\theta (x)-\phi (x)T(y)\theta (x)+\phi (x)\phi (y)T(x)}\), for all pairs \({x,y\in R}\). In this case T is of the form \({2T(x)=q\theta (x)+\phi (x)q}\), for all \({x\in R}\) and some fixed element \({q\in Q_{s}}\).     

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.     

On generalized Stanley sequences

Let \({\mathbb{N}}\) denote the set of all nonnegative integers. Let \({k \ge 3}\) be an integer and \({A_{0} = \{a_{1}, \dots, a_{t}\} (a_{1} < \cdots < a_{t})}\) be a nonnegative set which does not contain an arithmetic progression of length k. We denote \({A = \{a_{1}, a_{2}, \ldots{}\}}\) defined by the following greedy algorithm: if \({l \ge t}\) and \({a_{1}, \dots{}, a_{l}}\) have already been defined, then \({a_{l+1}}\) is the smallest integer \({a > a_{l}}\) such that \({\{a_{1}, \dots, a_{l}\} \cup \{a\}}\) also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A₀. We prove some results about various generalizations of the Stanley sequence.     

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.     

Continued fraction normality is not preserved along arithmetic progressions

It is well known that if \({0.a_1a_2a_3\ldots}\) is the base-\({b}\) expansion of a number normal to base-\({b}\), then the numbers \({0.a_ka_{m+k}a_{2m+k}\ldots}\) for \({m\ge 2}\), \({k\ge 1}\) are all normal to base-\({b}\) as well. In contrast, given a continued fraction expansion \({\langle a_1,a_2,a_3,\ldots\rangle}\) that is normal (now with respect to the continued fraction expansion), we show that for any integers \({m\ge 2}\), \({k\ge 1}\), the continued fraction \({\langle a_k, a_{m+k},a_{2m+k},a_{3m+k},\ldots\rangle}\) will never be normal.     

A stationary Fleming–Viot type Brownian particle system

We consider a system \({\{X_1,\ldots,X_N\}}\) of N particles in a bounded d-dimensional domain D. During periods in which none of the particles \({X_1,\ldots,X_N}\) hit the boundary \({\partial D}\) , the system behaves like N independent d-dimensional Brownian motions. When one of the particles hits the boundary \({\partial D}\) , then it instantaneously jumps to the site of one of the remaining N ? 1 particles with probability (N ? 1)^?1. For the system \({\{X_1,\ldots,X_N\}}\) , the existence of an invariant measure \({\nu\mskip-12mu \nu}\) has been demonstrated in Burdzy et al. [Comm Math Phys 214(3):679–703, 2000]. We provide a structural formula for this invariant measure \({\nu\mskip-12mu \nu}\) in terms of the invariant measure m of the Markov chain \({\xi}\) which returns the sites the process \({X:=(X_1,\ldots,X_N)}\) jumps to after hitting the boundary \({\partial D^N}\) . In addition, we characterize the asymptotic behavior of the invariant measure m of \({\xi}\) when N → ∞. Using the methods of the paper, we provide a rigorous proof of the fact that the stationary empirical measure processes \({\frac1N\sum_{i=1}^N\delta_{X_i}}\) converge weakly as N → ∞ to a deterministic constant motion. This motion is concentrated on the probability measure whose density with respect to the Lebesgue measure is the first eigenfunction of the Dirichlet Laplacian on D. This result can be regarded as a complement to a previous one in Grigorescu and Kang [Stoch Process Appl 110(1):111–143, 2004].     

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.     

On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation

$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$

where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form

$a_\vartheta = a+b_\vartheta-b,$

where a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods

$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$

as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ _T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form

$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$

where \({H\in[1/2,1]}\) and B ^H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ _T  = T ^?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.