On the q-Bernstein polynomials of rational functions with real poles

The paper aims to investigate the convergence of the q  -Bernstein polynomials _Bn,q(f;x)$B_{n,q}(f;x)$ attached to rational functions in the case q>1$q>1$. The problem reduces to that for the partial fractions (x−α)^−j${(x-\alpha )}^{-j}$, j∈N$j\in \mathbb{N}$. The already available results deal with cases, where either the pole α   is simple or α≠q^−m$\alpha \ne q^{-m}$, m∈_N0$m\in {\mathbb{N}}_0$. Consequently, the present work is focused on the polynomials _Bn,q(f;x)$B_{n,q}(f;x)$ for the functions of the form f(x)=(x−q^−m)^−j$f(x)={(x-q^{-m})}^{-j}$ with j?2$j?2$. For such functions, it is proved that the interval of convergence of {_Bn,q(f;x)}$\{B_{n,q}(f;x)\}$ depends not only on the location, but also on the multiplicity of the pole – a phenomenon which has not been considered previously.     

Invertible bimodule categories over the representation category of a Hopf algebra

For any finite-dimensional Hopf algebra H   we construct a group homomorphism BiGal(H)→BrPic(Rep(H))$\mathrm{BiGal}(H)\to \text{BrPic}(\mathrm{Rep}(H))$, from the group of equivalence classes of H  -biGalois objects to the group of equivalence classes of invertible exact Rep(H)$\mathrm{Rep}(H)$-bimodule categories. We discuss the injectivity of this map. We exemplify in the case H=_Tq$H=T_q$ is a Taft Hopf algebra and for this we classify all exact indecomposable Rep(_Tq)$\mathrm{Rep}(T_q)$-bimodule categories.     

Hypergroups and invariant complemented subspaces

Let K   be a hypergroup with a Haar measure. The purpose of the present paper is to initiate a systematic approach to the study of the class of invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$, the class of left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$ and finally the class of non-zero left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$ in the hypergroup context with the goal of finding some relations between these function spaces. Among other results, we construct two correspondences: one, between closed Weil subhypergroups and certain left translation invariant w^?$w^{?}$-subalgebras of _L∞(K)$L_{\infty }(K)$, and another, between compact subhypergroups and a specific subclass of the class of left translation invariant C^?$C^{?}$-subalgebras of _C0(K)$C_0(K)$. By the help of these two characterizations, we extract some results about invariant complemented subspaces of _L∞(K)$L_{\infty }(K)$ and _C0(K)$C_0(K)$.     

Spectral problems of a class of non-self-adjoint one-dimensional Schrodinger operators

In this paper we investigate the one-dimensional Schrodinger operator L(q)$L(q)$ with complex-valued periodic potential q   when q∈_L1[0,1]$q\in L_1[0,1]$ and _qn=0$q_n=0$ for n=0,−1,−2,...$n=0,-1,-2,...$, where _qn$q_n$ are the Fourier coefficients of q   with respect to the system {e^i2πnx}$\{e^{i2\pi nx}\}$. We prove that the Bloch eigenvalues are (2πn+t)²${(2\pi n+t)}^2$ for n∈Z$n\in \mathbb{Z}$, t∈C$t\in \mathbb{C}$ and find explicit formulas for the Bloch functions. Then we consider the inverse problem for this operator.     

A note on p-completely bounded homomorphisms of the Figà-Talamanca–Herz algebras

For a locally compact group G   and 1<p<∞$1<p<\infty$ let _Ap(G)$A_p(G)$ be the Figà-Talamanca–Herz algebras, which include in particular the Fourier algebra of G  , A(G)$A(G)$ (p=2$p=2$). It is shown that for any amenable group H  , a proper affine map α:Y⊂H→G$\alpha :Y\subset H\to G$ induces a p  -completely contractive algebra homomorphism _?α:_Ap(G)→_Ap(H)${?}_{\alpha }:A_p(G)\to A_p(H)$ by setting _?α(u)=u°α${?}_{\alpha }(u)=u°\alpha$ on Y   and _?α(u)=0${?}_{\alpha }(u)=0$ off of Y. Moreover, we show that if both G and H are amenable then any p  -completely contractive algebra homomorphism ?:_Ap(G)→_Ap(H)$?:A_p(G)\to A_p(H)$ is of this form. These results are the analogs in the context of the Figà-Talamanca–Herz algebras of the ones in the Fourier algebra setting (p=2$p=2$) initiated by the author and continued with N. Spronk, which in turn generalize results of P.J. Cohen and B. Host from abelian group algebra setting.     

Note on a Korovkin-type theorem

A new Korovkin-type theorem and its converse theorem are established. We compare our direct result with the Korovkin-type theorem given by Wang [15]. As applications we obtain quantitative estimates for q  -Bernstein-type operators which preserve the functions _e0(x)=1$e_0(x)=1$ and _ej(x)=x^j$e_j(x)=x^j$.     

A core-free semicovering of the Hawaiian Earring

The connected covering spaces of a connected and locally path-connected topological space X   can be classified by the conjugacy classes of those subgroups of _π1(X,x)${\pi }_1(X,x)$ which contain an open normal subgroup of _π1(X,x)${\pi }_1(X,x)$, when endowed with the natural quotient topology of the compact-open topology on based loops. There are known examples of semicoverings (in the sense of Brazas) that correspond to open subgroups which do not contain an open normal subgroup. We present an example of a semicovering of the Hawaiian Earring H$\mathbb{H}$ with corresponding open subgroup of _π1(H)${\pi }_1(\mathbb{H})$ which does not contain any   nontrivial normal subgroup of _π1(H)${\pi }_1(\mathbb{H})$.     

On the norming constants for normal maxima

Given n   independent standard normal random variables, it is well known that their maxima _Mn$M_n$ can be normalized such that their distribution converges to the Gumbel law. In a remarkable study, Hall proved that the Kolmogorov distance _dn$d_n$ between the normalized _Mn$M_n$ and its associated limit distribution is less than 3/log?n$3/\log ?n$. In the present study, we propose a different set of norming constants that allow this upper bound to be decreased with _dn≤C(m)/log?n$d_n\le C(m)/\log ?n$ for n≥m≥5$n\ge m\ge 5$. Furthermore, the function C(m)$C(m)$ is computed explicitly, which satisfies C(m)≤1$C(m)\le 1$ and _limm→∞?C(m)=1/3${\lim }_{m\to \infty }?C(m)=1/3$. As a consequence, some new and effective norming constants are provided using the asymptotic expansion of a Lambert W type function.     

Asymptotic expansions for elliptic-like regularizations of semilinear evolution equations

Consider in a real Hilbert space H the Cauchy problem (_P0$P_0$): u^′(t)+Au(t)+Bu(t)=f(t)$u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, where −A   is the infinitesimal generator of a _C0$C_0$-semigroup of contractions, B is a nonlinear monotone operator, and f is a given H-valued function. Inspired by the excellent book on singular perturbations by J.L. Lions, we associate with problem (_P0$P_0$) the following regularization (_Pε$P_{\epsilon }$): −εu^″(t)+u^′(t)+Au(t)+Bu(t)=f(t)$-\epsilon u^{″}(t)+u^{\prime }(t)+Au(t)+Bu(t)=f(t)$, 0≤t≤T$0\le t\le T$; u(0)=_u0$u(0)=u_0$, u^′(T)=_uT$u^{\prime }(T)=u_T$, where ε>0$\epsilon >0$ is a small parameter. We investigate existence, uniqueness and higher regularity for problem (_Pε$P_{\epsilon }$). Then we establish asymptotic expansions of order zero, and of order one, for the solution of (_Pε$P_{\epsilon }$). Problem (_Pε$P_{\epsilon }$) turns out to be regularly perturbed of order zero, and singularly perturbed of order one, with respect to the norm of C([0,T];H)$C([0,T];H)$. However, the boundary layer of order one is not visible through the norm of L²(0,T;H)$L^2(0,T;H)$.     

Minimal time of controllability of two parabolic equations with disjoint control and coupling domains

We consider two parabolic equations coupled by a matrix A(x)=q(x)A₀$A(x)=q(x)A_0$, where A₀$A_0$ is a Jordan block of order 1, and controlled by a single localized function, or by a single boundary control. The support of the coupling coefficient, q  , and the control domain may be disjoint. We exhibit an explicit minimal time of null-controllability, T₀(q)∈[0,+∞]$T_0(q)\in [0,+\infty ]$.     

On the existence of representations of finitely presented groups in compact Lie groups

Given a finite, connected 2-complex X   such that _b2(X)?1$b_2(X)?1$ we establish two existence results for representations of the fundamental group of X into compact connected Lie groups G  , with prescribed values on certain loops. If _b2(X)=1$b_2(X)=1$ we assume G=SO(3)$G=\mathrm{SO}(3)$ and that the cup product on H¹(X;Q)$H^1(X;\mathbb{Q})$ is non-zero.     

Decomposing a graph into bistars

Bárat and the present author conjectured that, for each tree T  , there exists a natural number _kT$k_T$ such that the following holds: If G   is a _kT$k_T$-edge-connected graph such that |E(T)|$|E(T)|$ divides |E(G)|$|E(G)|$, then G has a T-decomposition, that is, a decomposition of the edge set into trees each of which is isomorphic to T  . The conjecture has been verified for infinitely many paths and for each star. In this paper we verify the conjecture for an infinite family of trees that are neither paths nor stars, namely all the bistars S(k,k+1)$S(k,k+1)$.     

Compact covers and function spaces

For a Tychonoff space X  , we denote by _Cp(X)$C_p(X)$ and _Cc(X)$C_c(X)$ the space of continuous real-valued functions on X equipped with the topology of pointwise convergence and the compact-open topology respectively. Providing a characterization of the Lindelöf Σ-property of X   in terms of _Cp(X)$C_p(X)$, we extend Okunev?s results by showing that if there exists a surjection from _Cp(X)$C_p(X)$ onto _Cp(Y)$C_p(Y)$ (resp. from _Lp(X)$L_p(X)$ onto _Lp(Y)$L_p(Y)$) that takes bounded sequences to bounded sequences, then υY is a Lindelöf Σ-space (respectively K-analytic) if υX has this property. In the second part, applying Christensen?s theorem, we extend Pelant?s result by proving that if X is a separable completely metrizable space and Y   is first countable, and there is a quotient linear map from _Cc(X)$C_c(X)$ onto _Cc(Y)$C_c(Y)$, then Y   is a separable completely metrizable space. We study also a non-separable case, and consider a different approach to the result of J. Baars, J. de Groot, J. Pelant and V. Valov, which is based on the combination of two facts: Complete metrizability is preserved by _?p${?}_p$-equivalence in the class of metric spaces (J. Baars, J. de Groot, J. Pelant). If X   is completely metrizable and _?p${?}_p$-equivalent to a first-countable Y, then Y is metrizable (V. Valov). Some additional results are presented.