Abstract: Abstract Let G be a group and \(\omega (G)=\{o(g) g\in G\}\) be the set of element orders of G. Let \(k\in \omega (G)\) and \(s_k= \{g\in G o(g)=k\} \) . Let \(nse(G)=\{s_k k\in \omega (G) \}\) . In this paper, we prove that if G is a group and \(G_2 (4)\) is the Chevalley group such that \(nse(G)=nse(G_2 (4))\) , then \(G\cong G_2 (4)\) . PubDate: 2017-09-22

Abstract: Abstract Sub-Riemannian spaces are spaces whose metric structure may be viewed as a constrained geometry, where motion is only possible along a given set of directions, changing from point to point. The simplest example of such spaces is given by the so-called Heisenberg group. The characteristic constrained motion of sub-Riemannian spaces has numerous applications in robotic control in engineering and neurobiology where it arises naturally in the functional magnetic resonance imaging (FMRI). It also arises naturally in other branches of pure mathematics as Cauchy Riemann geometry, complex hyperbolic spaces, and jet spaces. In this paper, we review the use of the relationship between Heisenberg geometry and Cauchy Riemann (CR) geometry. More precisely, we focus on the problem of the prescription of the scalar curvature using techniques related to the theory of critical points at infinity. These techniques were first introduced by Bahri, Bahri and Brezis for the Yamabe conjecture in the Riemannian settings. PubDate: 2017-09-11

Abstract: Abstract A well-known result, due to Dirichlet and later generalized by de la Vallée–Poussin, expresses a relationship between the sum of fractional parts and the Euler–Mascheroni constant. In this paper, we prove an asymptotic relationship between the summation of the products of fractional parts with powers of integers on the one hand, and the values of the Riemann zeta function, on the other hand. Dirichlet’s classical result falls as a particular case of this more general theorem. PubDate: 2017-09-06

Abstract: Abstract For a sequence of positive numbers \(\beta =\{\beta _{n}\}_{n\in \mathbb {Z}}\) , the space \(L^2(\beta )\) consists of all \(f(z)=\sum _{-\infty }^\infty a_nz^n\) , \(a_n\in \mathbb {C}\) for which \(\sum _{-\infty }^\infty a_n ^2\beta _n^2<\infty \) . For a bounded function \(\varphi (z)=\sum _{-\infty }^\infty a_nz^n\) , the slant weighted Toeplitz operator \(A_\varphi ^{(\beta )}\) is an operator on \(L^2(\beta )\) defined as \(A_\varphi ^{(\beta )}=WM_\varphi ^{(\beta )}\) , where \(M_\varphi ^{(\beta )}\) is the weighted multiplication operator on \(L^2(\beta )\) and W is an operator on \(L^2(\beta )\) such that \(Wz^{2n}=z^n\) , \(Wz^{2n-1}=0\) for all \(n\in \mathbb {Z}\) . In this paper we show that for a trigonometric polynomial \(\varphi (z)=\sum _{n=-p}^q a_nz^n\) , \(A_\varphi ^{(\beta )}\) cannot be hyponormal unless \(\varphi \equiv 0\) . We also show that, for \(k \ge 2 \) the \(k^{th}\) order slant weighted Toeplitz operator \( U_{k,\varphi }^{(\beta )}\) cannot be hyponormal unless \(\phi \equiv 0 \) . Also the compression of \( U_{k,\varphi }^{(\beta )}\) to \(H^2(\beta )\) , denoted by \( V_{k,\varphi }^{(\beta )}\) , cannot be hyponormal unless \(\phi \equiv 0 \) . PubDate: 2017-09-05

Abstract: Abstract In this paper, we consider the problem of prescribing scalar curvature under minimal boundary conditions on the standard four-dimensional half sphere. We describe the lack of compactness of the associated variational problem and we give new existence and multiplicity results. PubDate: 2017-09-01

Abstract: Abstract We study the nonexistence of nontrivial solutions for the nonlinear elliptic system $$\begin{aligned} \left\{ \begin{array}{lll} (-\Delta _x)^{\alpha /2}u+ x ^{2\delta } (-\Delta _y)^{\beta /2}u+ x ^{2\eta } y ^{2\theta } (-\Delta _z)^{\gamma /2}u&{}=&{} v^p,\\ \\ (-\Delta _x)^{\mu /2}v+ x ^{2\delta } (-\Delta _y)^{\nu /2}v+ x ^{2\eta } y ^{2\theta } (-\Delta _z)^{\sigma /2}v&{}=&{} u^q, \\ \end{array} \right. \end{aligned}$$ where \((x,y,z)\in \mathbb {R}^{N_1}\times \mathbb {R}^{N_2}\times \mathbb {R}^{N_3}\) , \(0<\alpha ,\beta ,\gamma ,\mu , \nu , \sigma \le 2\) , \(\delta , \eta ,\theta \ge 0\) , and \(p,q>1\) . Here, \((-\Delta _x)^{\alpha /2}\) , \(0<\alpha <2\) , is the fractional Laplacian operator of order \(\alpha /2\) with respect to the variable \(x\in \mathbb {R}^{N_1}\) , \((-\Delta _y)^{\beta /2}\) , \(0<\beta <2\) , is the fractional Laplacian operator of order \(\beta /2\) with respect to the variable \(y\in \mathbb {R}^{N_2}\) , and \((-\Delta _z)^{\gamma /2}\) , \(0<\gamma <2\) , is the fractional Laplacian operator of order \(\gamma /2\) with respect to the variable \(z\in \mathbb {R}^{N_3}\) . Using a weak formulation approach, sufficient conditions are provided in terms of space dimension and system parameters. PubDate: 2017-09-01

Abstract: Abstract This paper is a survey on bubbling phenomena occurring in some geometric problems. We present here a few problems from conformal geometry, gauge theory and contact geometry and we give the main ideas of the proofs and important results. We focus in particular on the Yamabe type problems and the Weinstein conjecture, where A. Bahri made a huge contribution by introducing new methods in variational theory. PubDate: 2017-09-01

Abstract: Abstract Let X be a simply connected CW-complex of finite type and \({\mathbb {K}}\) an arbitrary field. In this paper, we use the Eilenberg–Moore spectral sequence of \(C_*(\Omega (X), \mathbb K)\) to introduce a new homotopical invariant \(\textsc {r}(X, {\mathbb {K}})\) . If X is a Gorenstein space with nonzero evaluation map, then \(\textsc {r}(X, {\mathbb {K}})\) turns out to interpolate \(\mathrm {depth}(H_*(\Omega (X), {\mathbb {K}}))\) and \(\mathrm {e}_{{\mathbb {K}}}(X)\) . We also define for any minimal Sullivan algebra \((\Lambda V,d)\) a new spectral sequence and make use of it to associate to any 1-connected commutative differential graded algebra (A, d) a similar invariant \(\textsc {r}(A,d)\) . When \((\Lambda V,d)\) is a minimal Sullivan model of X, this invariant fulfills the relation \(\textsc {r}(X, {\mathbb {K}}) = \textsc {r}(\Lambda V,d)\) . PubDate: 2017-09-01

Abstract: Abstract In this paper, we characterize spaces such that their one-point compactification (resp., Herrlich compactification) is weakly submaximal. We also establish a necessary and sufficient condition on \(T_{0}\) -spaces in order to get their one-point compactification (resp., Herrlich compactification) \(T_{D}\) -spaces. PubDate: 2017-08-30

Abstract: Abstract In this paper, our aim is to deduce some sharp Turán type inequalities for the remainder q-exponential functions. Our results are shown to be generalizations of results which were obtained by Alzer (Arch Math 55, 462–464, 1990). PubDate: 2017-08-21

Abstract: Abstract In this paper, we prove the existence of at least one periodic solution for some nonlinear parabolic boundary value problems associated with Leray–Lions’s operators with variable exponents under the hypothesis of existence of well-ordered sub- and supersolutions. PubDate: 2017-08-19

Abstract: Abstract We obtain minimal dimension matrix representations for each of the Lie algebras of dimensions five, six, seven and eight obtained by Turkowski that have a non-trivial Levi decomposition. The key technique involves using the invariant subspaces associated to a particular representation of a semi-simple Lie algebra to help in the construction of the radical in the putative Levi decomposition. PubDate: 2017-08-17

Abstract: Abstract We investigate some new results concerning the m-stability property. We show in particular under the martingale representation property with respect to a bounded martingale S that an m-stable set of probability measures is the set of supermartingale measures for a family of discrete integral processes with respect to S. PubDate: 2017-06-01

Abstract: Abstract This paper gives the complete proof of the Conjecture given by Hazarika and this author jointly which deals with a necessary and sufficient condition for the hyponormality of Toeplitz operator, \(T_\varphi \) on the weighted Bergman space with certain polynomial symbols under some assumptions about the Fourier coefficients of the symbol \(\varphi \) . PubDate: 2017-05-16

Abstract: Abstract In this paper, we consider the problem of existence and multiplicity of conformal metrics on a Riemannian compact 4-dimensional manifold \((M^4,g_0)\) with positive scalar curvature. We prove a new existence criterium which provides existence results for a dense subset of positive functions and generalizes Bahri–Coron Euler–Poincaré type criterium. Our argument gives estimates of the Morse index of the founded solutions and has the advantage to extend known existence results. Moreover, it provides, for generic K Morse Inequalities at Infinity, which give a lower bound on the number of metrics with prescribed scalar curvature in terms of the topological contribution of its critical points at Infinity to the difference of topology between the level sets of the associated Euler–Lagrange functional. PubDate: 2017-05-02

Abstract: Abstract This paper is concerned with prescribing the fractional Q-curvature on the unit sphere \(\mathbb {S}^{n}\) endowed with its standard conformal structure \(g_0\) , \(n\ge 4\) . Since the associated variational problem is noncompact, we approach this issue with techniques passed by Abbas Bahri, as the well known theory of critical points at infinity, as well as some lesser known topological invariants that appear here as criteria for existence results. PubDate: 2017-04-18

Abstract: Abstract In the paper, the authors establish explicit formulas for asymptotic and power series expansions of the exponential and the logarithm of asymptotic and power series expansions. The explicit formulas for the power series expansions of the exponential and the logarithm of a power series expansion are applied to find explicit formulas for the Bell numbers and logarithmic polynomials in combinatorics and number theory. PubDate: 2017-04-17

Abstract: Abstract In this article, we investigate the direct problem of approximation theory in the variable exponent Smirnov classes of analytic functions, defined on a doubly connected domain bounded by two Dini-smooth curves. PubDate: 2017-04-11