Bulletin of the Belgian Mathematical Society - Simon Stevin Articles (Project Euclid)

On the Geometry of the Conformal Group in Spacetime

Thu, 05 Aug 2010 15:41 EDT

N. G. Gresnigt, P. F. Renaud

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 2, 193--200.

Abstract:
The study of the conformal group in $R^{p,q}$ usually involves the conformal compactification of $R^{p,q}$. This allows the transformations to be represented by linear transformations in $R^{p+1,q+1}$. So, for example, the conformal group of Minkowski space, $R^{1,3}$ leads to its isomorphism with $SO(2,4)$. This embedding into a higher dimensional space comes at the expense of the geometric properties of the transformations. This is particularly a problem in $R^{1,3}$ where we might well prefer to keep the geometric nature of the various types of transformations in sight. In this note, we show that this linearization procedure can be achieved with no loss of geometric insight, if, instead of using this compactification, we let the conformal transformations act on two copies of the associated Clifford algebra. Although we are mostly concerned with the conformal group of Minkowski space (where the geometry is clearest), generalization to the general case is straightforward.

Subgroup S-commutativity degrees of finite groups

Thu, 24 May 2012 08:57 EDT

Daniele Ettore Otera, Francesco G. Russo

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 2, 373--382.

Abstract:
The so-called subgroup commutativity degree $sd(G)$ of a finite group $G$ is the number of permuting subgroups $(H,K) \in \mathrm{L}(G) \times \mathrm{L}(G)$, where $\mathrm{L}(G)$ is the subgroup lattice of $G$, divided by $|\mathrm{L}(G)|^2$. It allows to measure how $G$ is far from the celebrated classification of quasihamiltonian groups of K. Iwasawa. Here we generalize $sd(G)$, looking at suitable sublattices of $\mathrm{L}(G)$, and show some new lower bounds. More precisely, we define and study the subgroup S-commutativity degree of a group, which measures the probability that subnormal subgroups commute with maximal subgroups..

Permanence properties of amenable, transitive and faithful actions (Erratum)

Thu, 24 May 2012 08:57 EDT

Soyoung Moon

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 2, 383--384.

``Easy" Representations and the {\sc qsf} property for groups

Fri, 14 Sep 2012 13:06 EDT

Daniele Ettore Otera, Valentin Poénaru

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 385--398.

Abstract:
We define the class of \textit{easily-representable groups} as the class of those finitely presented groups $\Gamma $ admitting an \textit{inverse representation} (which, roughly, is a map from some $2$-complex to a certain singular 3-manifold $M^3 (\Gamma )$ associated to $\Gamma$, satisfying several topological properties) for which the set of double points is closed. Our main result is that easily-representable groups are {\sc qsf} (i.e. quasi-simply filtered).

Examples of mixing subalgebras of von Neumann algebras and their normalizers

Fri, 14 Sep 2012 13:06 EDT

Paul Jolissaint

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 399--413.

Abstract:
We discuss different mixing properties for triples of finite von Neumann algebras $B\subset N\subset M$, and we introduce families of triples of groups $H<K<G$ whose associated von Neumann algebras $L(H)\subset L(K)\subset L(G)$ satisfy $\mathcal{N}_{L(G)}(L(H))''=L(K)$. It turns out that the latter equality is implied by two conditions: the equality $\mathcal{N}_G(H)=K$ and the above mentioned mixing properties. Our families of examples also allow us to exhibit examples of pairs $H<G$ such that $L(\mathcal{N}_G(H))\not=\mathcal{N}_{L(G)}(L(H))''$.

On a family of Hopf algebras of dimension 72

Fri, 14 Sep 2012 13:06 EDT

Nicolás Andruskiewitsch, Cristian Vay

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 415--443.

Abstract:
We investigate a family of Hopf algebras of dimension 72 whose coradical is isomorphic to the algebra of functions on $\mathbb S_3$. We determine the lattice of submodules of the so-called Verma modules and as a consequence we classify all simple modules. We show that these Hopf algebras are unimodular (as well as their duals) but not quasitriangular; also, they are cocycle deformations of each other.

Real elliptic surfaces with double section and real tetragonal curves

Fri, 14 Sep 2012 13:06 EDT

Maha Smirani, Mouadh Akriche

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 445--460.

Abstract:
We give an upper bound for the number of nested ovals of a real tetragonal curve $R$ embedded in an Hirzubruch surface in terms of the genus of the curve.

On harmonic combination of univalent functions

Fri, 14 Sep 2012 13:06 EDT

M. Obradović, S. Ponnusamy

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 461--472.

Abstract:
Let ${\mathcal S}$ be the class of all functions $f$ that are analytic and univalent in the unit disk $\mathbb D$ with the normalization $f(0)=f'(0)-1=0$. Let $\mathcal{U} (\lambda)$ denote the set of all $f\in {\mathcal S}$ satisfying the condition $$\left |f'(z)\left (\frac{z}{f(z)} \right )^{2}-1\right | <\lambda ~\mbox{ for $z\in \mathbb D$}, $$ and some $\lambda \in (0,1]$. In this paper, among other things, we study a ``harmonic mean'' of two univalent analytic functions. More precisely, we discuss the properties of the class of functions $F$ of the form $$\frac{z}{F(z)}=\frac{1}{2}\left( \frac{z}{f(z)}+\frac{z}{g(z)} \right), $$ where $f,g\in \mathcal{S}$ or $f,g\in \mathcal{U}(1)$. In particular, we determine the radius of univalency of $F$, and propose two conjectures concerning the univalency of $F$.

$m$-infrabarrelledness and $m$-convexity

Fri, 14 Sep 2012 13:06 EDT

Marina Haralampidou, Mohamed Oudadess

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 473--483.

Abstract:
$m$-infrabarrelledness, in the context of locally convex algebras, is considered to prove results previously obtained for barrelled algebras. Thus, any unital commutative $m$-infrabarrelled advertibly complete and pseudo-complete locally $m$-convex algebra with bounded elements has the $Q$-property; hence, it is functionally continuous (: all characters are continuous). In the framework of commutative $GB^{\ast }$-algebras with jointly continuous multiplication and bounded elements, the notions {\em $m$-infrabarrelled algebra} and {\em $C^{\ast }$-algebra} coincide. In unital uniform locally $m$-convex algebras, $m$-infrabarrelledness is equivalent to the Banach algebra structure, modulo pseudo-completeness. Moreover, $m$-infrabarrelledness for locally $A$-convex algebras (in particular, $A$-normed ones) is also examined.

Abstract Sectional Category

Fri, 14 Sep 2012 13:06 EDT

F.J. Díaz, J.M. García Calcines, P.R. García Díaz, A. Murillo Mas, J. Remedios Gómez

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 485--506.

Abstract:
We study, in an abstract axiomatic setting, the notion of sectional category of a morphism. From this, we unify and generalize known results about this invariant in different settings as well as we deduce new applications.

Topological monomorphisms between free paratopological groups

Fri, 14 Sep 2012 13:06 EDT

Fucai Lin

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 507--521.

Abstract:
Suppose that $X$ is a subspace of a Tychonoff space $Y$. Then the embedding mapping $e_{X, Y}: X\rightarrow Y$ can be extended to a continuous monomorphism $\hat{e}_{X, Y}: AP(X)\rightarrow AP(Y)$, where $AP(X)$ and $AP(Y)$ are the free Abelian paratopological groups over $X$ and $Y$, respectively. In this paper, we mainly discuss when $\hat{e}_{X, Y}$ is a topological monomorphism, that is, when $\hat{e}_{X, Y}$ is a topological embedding of $AP(X)$ to $AP(Y)$.

Rational involutive automorphisms related with standard representations of ${\mathrm{SL}}(2,\mathbb R)$

Fri, 14 Sep 2012 13:06 EDT

Zdeněk Dušek, Oldřich Kowalski

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 523--533.

Abstract:
Standard irreducible representations of the group $\mathrm{SL}(2,\mathbb R)$ on coefficients of homogeneous polynomials in two variables are studied in a new context. It is proved that any standard representation of $\mathrm{SL}(2,\mathbb R)$ on $\mathbb R^{n+1}$ induces an involutive rational mapping of an open dense subset of $\mathbb R^{n+1}$ onto itself. Examples in low dimensions are presented. We also construct formal involutive rational mappings with ``arbitrary complexity''.

Normal families of holomorphic functions and multiple values

Fri, 14 Sep 2012 13:06 EDT

Lijuan Zhao, Xiangzhong Wu

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 535--547.

Abstract:
Let $\mathcal{F}$ be a family of holomorphic functions defined in $D \subset C$, and let $ k, m, n, p $ be four positive integers with $ \frac{k+p+1}{m}+\frac{p+1}{n} < 1 $. Let $\psi (\not \equiv 0, \infty )$ be a meromorphic function in $ D $ and which has zeros only of multiplicities at most $ p $. Suppose that, for every function $ f \in \mathcal{F} $, (i) $ f $ has zeros only of multiplicities at least $ m $; (ii) all zeros of $ f^{(k)}-\psi(z) $ have multiplicities at least $ n $; (iii) all poles of $ \psi $ have multiplicities at most $ k $, and (iv) $ \psi(z) $ and $ f(z) $ have no common zeros, then $\mathcal{F}$ is normal in $ D $.

Extinction phenomenon for Spinor Ginzburg-Landau equations

Fri, 14 Sep 2012 13:06 EDT

Zuhan Liu

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 549--559.

Abstract:
Recent papers in physics literature have introduced spinor Ginzburg-Landau model for complex vector-valued order parameters in order to account for ferromagnetic or antiferromagnetic effects in high-temperature superconductors. In this paper, we study the spatial behavior of interacting components of Spinor Ginzburg-Landau model. We prove the interspecies interaction leads to extinction, that is, configurations where one or more densities are null.

Secondary Cohomology and $k$-invariants

Fri, 14 Sep 2012 13:06 EDT

Mihai D. Staic

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 3, 561--572.

Abstract:
We give a construction that associates to a pointed topological space $(X,x_0)$ a homotopy invariant $\,_2\kappa^4$ which we call the secondary invariant. This construction can be seen a ``3-type" generalization of the classical $k$-invariant.

Common fixed point theorems for a pair of multivalued mappings under weak contractive conditions in ordered metric spaces

Fri, 23 Nov 2012 13:38 EST

Hemant Kumar Nashine, Zoran Kadelburg

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 577--596.

Abstract:
We present common fixed point theorems for a pair of weakly isotone increasing multivalued mappings satisfying general weak contractive conditions, as well as almost contractive conditions in ordered complete metric spaces. Examples are presented to show the usage of these results.

On existence of embeddings for point-line geometries

Fri, 23 Nov 2012 13:38 EST

Anna Kasikova

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 597--632.

Abstract:
We describe a construction of a point-line presheaf on a point-line geometry from a set of presheaves on subspaces of the geometry. Then we combine our construction with theorems of M. Ronan to give a new proof of the fact that all polar spaces of finite rank at least four, and several other Grassmann geometries of spherical buildings, are embeddable in projective spaces.

Uniform stabilization of the Riemannian wave equation with linear lower order term

Fri, 23 Nov 2012 13:38 EST

Ilhem Hamchi

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 633--647.

Abstract:
The Riemannian wave equation with linear lower order term and unspecified behavior of the nonlinear feedback $f$ is considered. Using the method in $ \left[ LT\right] $ we prove that the energy of the solution decays faster than the solution of some associated differential equation. The decay rate of a general second order hyperbolic equation with polynomial growth at the origin of $f$ is also discussed.

On certain results of C. Bereanu and J. Mawhin

Fri, 23 Nov 2012 13:38 EST

Stanisław Sędziwy

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 649--653.

Abstract:
It is shown that the assumption of the singularity of $\varphi$--Laplacian permits to get for the scalar differential equations the existence results of the Dirichlet, Dirichlet--Neumann, Neuman--Steklov or periodic problems using a simple elementary argument.

Occasionally weakly compatible mappings and fixed points

Fri, 23 Nov 2012 13:38 EST

R. P. Pant, R. K. Bisht

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 655--661.

Abstract:
In the present paper we show that contractive condition employed by Al-Thagafi and Shahzad in particular and contractive conditions in general, do not constitute a proper setting for studying common fixed points of occasionally weakly compatible mappings. Further, we improve the results of Al-Thagafi and Shahzad by employing a proper setting.

Double density by moduli and statistical convergence

Fri, 23 Nov 2012 13:38 EST

A. Aizpuru, M. Listán-García, F. Rambla-Barreno

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 663--673.

Abstract:
By using unbounded modulus functions we introduce a new concept of density for sets of pairs of natural numbers. Consequently, we obtain a generalization of the notion of statistical convergence of double sequences which is studied and characterized. As an application, we prove that `Pringsheim convergence' is equivalent to `module statistical convergence for every unbounded modulus function'.

Harmonic Functions in Upper Half Space

Fri, 23 Nov 2012 13:38 EST

Guo-Shuang Pan, Lei Qiao, Guan-Tie Deng

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 675--681.

Abstract:
In this paper, we prove that if the positive part $u^{+}(x)$ of a harmonic function $u(x)$ in the upper half space satisfies a fast growing condition, then its negative part $u^{-}(x)$ can also be dominated by a similar growing condition. Meanwhile, $u(x)$ can be represented in terms of the modified Poisson integral and a harmonic function vanishing on the boundary.

Continuous Gabor transform for a class of non-Abelian groups

Fri, 23 Nov 2012 13:38 EST

Arash Ghaani Farashahi, Rajabali Kamyabi-Gol

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 683--701.

Abstract:
In this article we define the continuous Gabor transform for second countable, non-abelian, unimodular and type I groups and also we investigate a Plancherel formula and an inversion formula for our definition. As an example we show that how these formulas work for the Heisenberg group and also the matrix group ${SL(2,\mathbb{R})}$.

Deficiency of E-valued meromorphic functions

Fri, 23 Nov 2012 13:38 EST

Zhaojun Wu, Zuxing Xuan

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 703--715.

Abstract:
The purpose of this paper is to discuss the deficiency of an E-valued meromorphic function $f$. Results are obtained to extend the related results for meromorphic vector valued function of Lahiri and Ziegler.

Further results on the exponent of convergence of zeros of solutions of certain higher order linear differential equations

Fri, 23 Nov 2012 13:38 EST

Hong-Yan Xu, Jin Tu

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 717--732.

Abstract:
In this paper, we further investigate the exponent of convergence of the zero-sequence of solutions of the differential equation $$ f^{(k)}+a_{k-1}(z)f^{(k-1)}+\cdots+a_1(z)f' +\psi(z)f=0, $$ where $\psi(z)=\sum_{j=1}^\iota Q_j(z)e^{P_j(z)} (\iota\geq 3, \iota\in N_+ )$, $P_j(z)$ are polynomials of degree $n\geq1$, $Q_j(z),a_\Lambda(z)(\Lambda=1,2,\cdots,k-1;j=1,2,\ldots,\iota)$ are entire functions of order less than $n$, and $k\geq2$.

Rigidity theorem for complete spacelike submanifold in $S^{n+p}_q(1)$ with constant scalar curvature

Fri, 23 Nov 2012 13:38 EST

Shicheng Zhang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 733--746.

Abstract:
In this paper, the complete spacelike submanifold with parallel normalized mean curvature vector and constant normalized scalar curvature is discussed in $(n+p)$-dimensional connected semi-Riemannian manifold $S^{n+p}_q(1)$ $(1\leq q\leq p)$ and a rigidity theorem is obtained.

Homoclinic solutions for second order Hamiltonian systems with small forcing terms

Fri, 23 Nov 2012 13:38 EST

Dong-Lun Wu, Xing-Ping Wu, Chun-Lei Tang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 747--761.

Abstract:
The existence of homoclinic solutions is obtained for a class of nonautonomous second order Hamiltonian systems $\ddot{u}(t)+\nabla V(t,u(t))=f(t)$ as the limit of the $2kT$-periodic solutions which are obtained by the Mountain Pass theorem, where $V(t,x)=-K(t,x)+W(t,x)$ is $T$-periodic with respect to $t,T>0$, and $W(t,x)$ satisfies the superquadratic condition: $W(t,x) / |x|^{2} \rightarrow +\infty$ as $|x| \rightarrow \infty$ uniformly in $t$, which needs not to satisfy the global Ambrosetti-Rabinowitz condition.

Radius of convexity for certain multivalent functions with missing coefficients

Fri, 23 Nov 2012 13:38 EST

Ding-Gong Yang, Jin-Lin Liu

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 4, 763--767.

Abstract:
The object of the present paper is to derive the radius of convexity for certain multivalent functions with missing coefficients.

Monoidal categories in, and linking, geometry and algebra

Tue, 27 Nov 2012 10:52 EST

Ross Street

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5, 769--820.

Abstract:
This is a report on aspects of the theory and use of monoidal categories. The first section introduces the main concepts through the example of the category of vector spaces. String notation is explained and shown to lead naturally to a link between knot theory and monoidal categories. The second section reviews the light thrown on aspects of representation theory by the machinery of monoidal category theory, machinery such as braidings and convolution. The category theory of Mackey functors is reviewed in the third section. Some recent material and a conjecture concerning monoidal centres is included. The fourth and final section looks at ways in which monoidal categories are, and might be, used for new invariants of low-dimensional manifolds and for the field theory of theoretical physics.

Aspects of algebraic exponentiation

Tue, 27 Nov 2012 10:52 EST

Dominique Bourn, James R. A. Gray

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5, 821--844.

Abstract:
We analyse some aspects of the notion of \emph{algebraic exponentiation} introduced by the second author [16] and satisfied by the category $Gp$ of groups. We show how this notion provides a new approach to the categorical-algebraic question of the centralization. We explore, in the category $Gp$, the unusual universal properties and constructions determined by this notion, and we show how it is the origin of various properties of this category.

A categorical approach to loops, neardomains and nearfields

Tue, 27 Nov 2012 10:52 EST

Philippe Cara, Rudger Kieboom, Tina Vervloet

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5, 845--857.

Abstract:
In this paper we study loops, neardomains and nearfields from a categorical point of view. By choosing the right kind of morphisms, we can show that the category of neardomains is equivalent to the category of sharply 2-transitive groups. The other categories are also shown to be equivalent with categories whose objects are sets of permutations with suitable extra properties. Up to now the equivalence between neardomains and sharply 2-transitive groups was only known when both categories were equipped with the obvious isomorphisms as morphisms. We thank Hubert Kiechle for this observation.

Grothendieck quantaloids for allegories of enriched categories

Tue, 27 Nov 2012 10:52 EST

Hans Heymans, Isar Stubbe

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5, 859--888.

Abstract:
For any small involutive quantaloid $\cal Q$ we define, in terms of symmetric quantaloid-enriched categories, an involutive quantaloid $\sf Rel(\cal Q)$ of $\cal Q$-sheaves and relations, and a category $\sf Sh(\cal Q)$ of $\cal Q$-sheaves and functions; the latter is equivalent to the category of symmetric maps in the former. We prove that $\sf Rel(\cal Q)$ is the category of relations in a topos if and only if $\cal Q$ is a modular, locally localic and weakly semi-simple quantaloid; in this case we call $\cal Q$ a Grothendieck quantaloid. It follows that $\sf Sh(\cal Q)$ is a Grothendieck topos whenever $\cal Q$ is a Grothendieck quantaloid. Any locale $L$ is a Grothendieck quantale, and $\sf Sh(L)$ is the topos of sheaves on $L$. Any small quantaloid of closed cribles is a Grothendieck quantaloid, and if $\cal Q$ is the quantaloid of closed cribles in a Grothendieck site $(\cal C,J)$ then $\sf Sh(\cal Q)$ is equivalent to the topos $\Sh(\cal C,J)$. Any inverse quantal frame is a Grothendieck quantale, and if $\cal O(G)$ is the inverse quantal frame naturally associated with an étale groupoid $G$ then $\sf Sh(\cal O(G))$ is the classifying topos of $G$.

Calibrated Toposes

Tue, 27 Nov 2012 10:52 EST

Peter Johnstone

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5, 889--907.

Abstract:
We study a particular structure on a topos $\cal E$, related to the notion of a `class of étale maps' due to Joyal and Moerdijk and to Bénabou's notion of `calibration', which corresponds to giving for each object $A$ of $\cal E$ a `natural' comparison between the slice category ${\cal E}/A$ and a smaller `petit topos' associated with $A$. We show that there are many naturally-arising examples of such structures; but rather few of them satisfy the condition that the relation between the `gros' and `petit' toposes of every object is expressed by a local geometric morphism.

Notions of Möbius inversion

Tue, 27 Nov 2012 10:52 EST

Tom Leinster

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 19, Number 5, 909--933.

Abstract:
Möbius inversion, originally a tool in number theory, was generalized to posets for use in group theory and combinatorics. It was later generalized to categories in two different ways, both of which are useful. We provide a unifying abstract framework. This allows us to compare and contrast the two theories of Möbius inversion for categories, and advance each of them. Among several side benefits is an improved understanding of the following fact: the Euler characteristic of the classifying space of a (suitably finite) category depends only on its underlying graph.

Property $(aw)$ and perturbations

Thu, 18 Apr 2013 13:38 EDT

M. H.M. Rashid

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 1--18.

Abstract:
A bounded linear operator $T\in\mathbf{L}(\mathbb{X})$ acting on a Banach space satisfies property $(aw)$, a variant of Weyl's theorem, if the complement in the spectrum $\sigma(T)$ of the Weyl spectrum $\sigma_w(T)$ is the set of all isolated points of the approximate-point spectrum which are eigenvalues of finite multiplicity. In this article we consider the preservation of property $(aw)$ under a finite rank perturbation commuting with $T$, whenever $T$ is polaroid, or $T$ has analytical core $K(T-\lambda_0 I)=\{0\}$ for some $\lambda_0\in \mathbb{C}$. The preservation of property $(aw)$ is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations or under Riesz perturbations. The theory is exemplified in the case of some special classes of operators.

Quadric Veronesean Caps

Thu, 18 Apr 2013 13:38 EDT

J. Schillewaert, H. Van Maldeghem

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 19--25.

Abstract:
In 2008, Ferrara Dentice and Marino provided a characterization theorem for Veronesean caps in $\mathsf{PG}(N,\mathbb{K})$, with $\mathbb{K}$ a skewfield. This result extends the theorem for the finite case proved by J.A. Thas and Van Maldeghem in 2004. However, although the statement of this theorem is correct, the proof given by Ferrara Dentice and Marino is incomplete, as they borrow some lemmas from the paper of J.A. Thas and Van Maldeghem, which are proved using counting arguments and hence require a different approach in the infinite case. In this paper we use the Veblen-Young theorem to fill these gaps. Moreover, we then use this classification of Veronesean caps to provide a further general geometric characterization.

On the hyper-order of solutions of a class of higher order linear differential equations

Thu, 18 Apr 2013 13:38 EDT

Karima Hamani, Benharrat Belaïdi

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 27--39.

Abstract:
In this paper, we investigate the growth of solutions of the linear differential equation \begin{multline*} f^{(k)}+\left( A_{k-1}(z)e^{P_{k-1}(z)}+B_{k-1}\left( z\right) \right) f^{(k-1)}+\cdots +\\\left( A_{1}(z)e^{P_{1}(z)}+B_{1}\left( z\right) \right) f^{\prime } +\left( A_{0}(z)e^{P_{0}(z)}+B_{0}\left( z\right) \right) f=0, \end{multline*} where $k\geq 2$\ is an integer, $P_{j}(z)$ $(j=0,1,\cdots ,k-1)$\ are nonconstant polynomials\ and $A_{j}(z)$ $\left( \not\equiv 0\right) ,$ $ B_{j}\left( z\right) $\ $\left( \not\equiv 0\right) $ $(j=0,1,\cdots ,k-1)$\ are meromorphic functions. Under some conditions, we determine the hyper-order of these solutions.

Note on New General Integral Operators of $p$-valent Functions

Thu, 18 Apr 2013 13:38 EDT

Irina Dorca, Daniel Breaz

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 41--55.

Abstract:
We study new general integral operators of $p$-valent functions by giving sufficient conditions of $p$-valently starlikeness, $p$-valently close-to-convexness, uniformly $p$-valent close-to-convexness and strongly starlikeness of order $\tau \,\,(0<\tau\leq 1)$ in $U$ (open unit disk). We end our investigation with an example from literature and some other references.

An existence result for nonlinear elliptic equations in Musielak-Orlicz-Sobolev spaces

Thu, 18 Apr 2013 13:38 EDT

A. Benkirane, M. Sidi El Vally

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 57--75.

Abstract:
In this paper we prove an existence result for some class of variational boundary value problems for quasilinear elliptic equations in the Musielak-Orlicz spaces $W^mL_\varphi(\Omega)$, under the assumption that the conjugate function of $\varphi$ satisfies the $\Delta_2$ condition. An imbedding theorem has also been provided without assuming this condition.

Conjugation spaces and equivariant Chern classes

Thu, 18 Apr 2013 13:38 EDT

Wolfgang Pitsch, Jérôme Scherer

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 77--90.

Abstract:
Let $\eta$ be a Real bundle, in the sense of Atiyah, over a space $X$. This is a complex vector bundle together with an involution which is compatible with complex conjugation. We use the fact that $BU$ has a canonical structure of a conjugation space, as defined by Hausmann, Holm, and Puppe, to construct equivariant Chern classes in certain equivariant cohomology groups of $X$ with twisted integer coefficients. We show that these classes determine the (non-equivariant) Chern classes of $\eta$, forgetting the involution on $X$, and the Stiefel-Whitney classes of the real bundle of fixed points.

Isometric embeddings of pretangent spaces in $ E^n$

Thu, 18 Apr 2013 13:38 EDT

Viktoriia Bilet, Oleksiy Dovgoshey

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 91--110.

Abstract:
We prove some infinitesimal analogs of classical results of Menger, Schoenberg and Blumenthal giving the existence conditions for isometric embeddings of metric spaces in the finite-dimensional Euclidean spaces.

Growth of solutions of some higher order linear difference equations

Thu, 18 Apr 2013 13:38 EDT

Xiaoguang Qi, Zhenhua Wang, Lianzhong Yang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 111--122.

Abstract:
This paper is devoted to studying the growth of solutions of equations of type $f(z+n)+\sum_{j=0}^{n-1}\{P_{j}(e^{z})+Q_{j}(e^{-z})\}f(z+j)=0$ and $f(z+n)+ \sum_{j=0}^{n-1}\{P_{j}(e^{A(z)})+Q_{j}(e^{-A(z)})\}f(z+j)=0$, where $P_{j}(z)$ and $Q_{j}(z)$ are polynomials in $z$ and $A(z)$ is a transcendental entire function. We prove three theorems of such type, which improve some results in [6,7].

Unbounded analysis operators

Thu, 18 Apr 2013 13:38 EDT

H. Hosseini Giv, M. Radjabalipour, A. Askari Hemmat

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 123--132.

Abstract:
The paper studies bounded or unbounded operators which can act as analysis operators or synthesis operators of various signal processing including generalized frames, semi-frames, discrete frames, Fourier transforms, etc. The paper is concluded by a short discussion of the controllability of the behavior of the processed signals.

Orthogonality of the Meixner-Pollaczek polynomials beyond Favard's theorem

Thu, 18 Apr 2013 13:38 EDT

Samuel G. Moreno, Esther M. García-Caballero

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 133--143.

Abstract:
We extend the family of Meixner-Pollaczek polynomials $\{P_n^{(\lambda)}(\cdot;\phi)\}_{n=0}^{\infty}$, classically defined for $\lambda>0$ and $0<\phi<\pi$, to arbitrary complex values of the parameter $\lambda$, in such a way that both polynomial systems (the classical and the new {\it generalized} ones) share the same three term recurrence relation. The values $\lambda_N=(1-N)/2$, with $N$ a positive integer, are the only ones for which no orthogonality condition can be deduced from Favard's theorem. In this paper we introduce a non-standard discrete-continuous inner product with respect to which the generalized Meixner-Pollaczek polynomials $\{P_n^{(\lambda_N)}(\cdot;\phi)\}_{n=0}^{\infty}$ become orthogonal.

A Characterization of Dupin Hypersurfaces in $\mathbb R^4$

Thu, 18 Apr 2013 13:38 EDT

Carlos M.C. Riveros

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 145--154.

Abstract:
In this paper we study Dupin hypersurfaces in $\mathbb R^4$ parametrized by lines of curvature, with three distinct principal curvatures and $m_{jik}= 0$. We characterize locally a generic family of such hypersurfaces in terms of the principal curvatures and three vector valued functions of one variable, which are invariant under inversions and homotheties.

New existence results on periodic solutions of nonautonomous second order Hamiltonian systems with $(q,p)$-Laplacian

Thu, 18 Apr 2013 13:38 EDT

Daniel Paşca, Zhiyong Wang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 155--166.

Abstract:
Some new existence theorems are obtained for periodic solutions of nonautonomous second order Hamiltonian systems with $(q,p)$-Laplacian by using the least action principle and the minimax methods.

An application of generalized power increasing sequences on factors theorem

Thu, 18 Apr 2013 13:38 EDT

Hûseyin Bor, Dansheng Yu

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 167--174.

Abstract:
In the present paper, by using a new defined -$\left\vert C,\alpha ,\sigma;\alpha_{n}\right\vert _{k}$ summability method and some classes of pairs of sequences, we generalize a result of Bor [5] dealing with $\varphi-\left\vert C,\alpha,\sigma;\beta\right\vert _{k}\ $summability factors.

Chebyshev Upper Estimates for Beurling's Generalized Prime Numbers

Thu, 18 Apr 2013 13:38 EDT

Jasson Vindas

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 175--180.

Abstract:
Let $N$ be the counting function of a Beurling generalized number system and let $\pi$ be the counting function of its primes. We show that the $L^{1}$-condition $$ \int_{1}^{\infty}\left|\frac{N(x)-ax}{x}\right|\frac{\mathrm{d}x}{x}<\infty $$ and the asymptotic behavior $$N(x)=ax+O\left(\frac{x}{\log x}\right)\: ,$$ for some $a>0$, suffice for a Chebyshev upper estimate $$ \frac{\pi(x)\log x}{x}\leq B<\infty\: .$$

A polynomial encoding provability in pure mathematics (outline of an explicit construction)

Thu, 18 Apr 2013 13:38 EDT

M. Carl, B.Z. Moroz

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 1, 181--187.

Infinitely many homoclinic solutions for the second-order discrete $p$-Laplacian systems

Thu, 23 May 2013 09:42 EDT

Peng Chen, X. H. Tang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 193--212.

Abstract:
By using the Symmetric Mountain Pass Theorem, we establish some existence criteria to guarantee the second-order discrete $p$-Laplacian systems $\triangle (\varphi_p(\Delta u(n-1)))-a(n)|u(n)|^{p-2}u(n)+\nabla W(n, u(n))=0$ has infinitely many homoclinic orbits, where $p>1, \ n\in {\mathbb{Z}},\ u\in {\mathbb{R}}^{N}$, $a:{\mathbb{Z}}\rightarrow{\mathbb{R}}$ and $W:{\mathbb{Z}}\times {\mathbb{R}}^{N}\rightarrow {\mathbb{R}}$ are not periodic in $n$. Our conditions on the nonlinear term $W(n,u(n))$ are rather relaxed and we generalize some existing results in the literature.

Some remarks concerning pseudocompactness in pointfree topology

Thu, 23 May 2013 09:42 EDT

Themba Dube

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 213--219.

Abstract:
Recall that a frame $L$ is pseudocompact if $\mathcal{R}L=\mathcal{R}^*L$, where $\mathcal{R}L$ is the $f$-ring of real-valued continuous functions on $L$, and $\mathcal{R}^*L$ its bounded part. Using properties of uniform frames, Walters-Wayland proved that a completely regular frame $L$ is pseudocompact iff the frame homomorphism $\beta L\to L$ is coz-codense. In this note we give a purely ring-theoretic reaffirmation of this characterization by observing that a frame homomorphism $L\to M$ is coz-codense iff the ring homomorphism $\mathcal{R}L\to\mathcal{R}M$ it induces maps non-invertible elements to non-invertible elements, and that $L$ is pseudocompact iff every finitely generated proper ideal of $\mathcal{R}^*L$ is fixed. We also show that if $L$ is not pseudocompact, then $\mathcal{R}^*L$ has a non-maximal free prime ideal -- thus generalizing a 1954 result of Gillman and Henriksen.

Bilinear factorization of algebras

Thu, 23 May 2013 09:42 EDT

Gabriella Böhm, José Gómez-Torrecillas

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 221--244.

Abstract:
We study the (so-called bilinear) factorization problem answered by a weak wreath product (of monads and, more specifically, of algebras over a commutative ring) in the works by Street and by Caenepeel and De Groot. A bilinear factorization of a monad $R$ turns out to be given by monad morphisms $A\to R\leftarrow B$ inducing a split epimorphism of $B$-$A$ bimodules $B\otimes A\rightarrow R$. We prove a biequivalence between the bicategory of weak distributive laws and an appropriately defined bicategory of bilinear factorization structures. As an illustration of the theory, we collect some examples of algebras over commutative rings which admit a bilinear factorization; i.e. which arise as weak wreath products.

Existence of Local Solutions of Nonlinear Wave Equations in $n$-Dimensional Space

Thu, 23 May 2013 09:42 EDT

Yaojun Ye

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 245--252.

Abstract:
In this paper, we investigate the local existence of solutions in $H^s$ for $n$-dimensional nonlinear wave equations with special nonlinear terms, such as $$u_{tt}-\Delta u=u^k|\nabla u|^l,\ \ x\in R^n,\ \ k\in Z^+,\ \ l\geq2.$$ where $\nabla u=(\frac{\partial u}{\partial x_1},\ \frac{\partial u}{\partial x_2},\ \cdots,\ \frac{\partial u}{\partial x_n})$. Meanwhile, we obtain that the regular index $s$ of Sobolev space $H^s$ satisfies $s>\max\{\frac{n+5}4;\ \frac n2+1-\frac1{l-1}\},\ n>3$.

Meromorphic functions sharing a nonzero polynomial with finite weight

Thu, 23 May 2013 09:42 EDT

Pulak Sahoo, Sajahan Seikh

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 253--267.

Abstract:
In the paper, we study the uniqueness theorems of meromorphic function concerning nonlinear differential polynomials sharing a nonzero polynomial with finite weight and obtain two results which improve and generalize the results due to X.M Li and L. Gao.

Strictification of weakly equivariant Hopf algebras

Thu, 23 May 2013 09:42 EDT

Jennifer Maier, Thomas Nikolaus, Christoph Schweigert

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 269--285.

Abstract:
A weakly equivariant Hopf algebra is a Hopf algebra $A$ with an action of a finite group $G$ up to inner automorphisms of $A$. We show that each weakly equivariant Hopf algebra can be replaced by a Morita equivalent algebra $A^{str}$ with a strict action of $G$ and with a coalgebra structure that leads to a tensor equivalent representation category. However, the coproduct of this strictification cannot, in general, be chosen to be unital, so that a strictification of the $G$-action can only be found on a \emph{weak} Hopf algebra $A^{str}$.

On the rotation index of bar billiards and Poncelet's porism

Thu, 23 May 2013 09:42 EDT

W. Cieślak, H. Martini, W. Mozgawa

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 287--300.

Abstract:
We present some new results on the relations between the rotation index of bar billiards of two nested circles $C_R$ and $C_r$, of radii $R$ and $r$ and with distance $d$ between their centers, satisfying Poncelet's porism property. The rational indices correspond to closed Poncelet transverses, without or with self-intersections. We derive an interesting series arising from the theory of special functions. This relates the rotation number $\frac 13$, of a triangle of Poncelet transverses, to a double series involving $R, r$, and $d$. We also provide a Steiner-type formula which gives a necessary condition for a bar billiard to be a pentagon with self-intersections and rotation index $\frac 25$. Finally we show that, close to a pair of circles having Poncelet's porism property for index $\frac{1}{3}$, there exist always circle pairs having indices $\frac{1}{4}$ they and $\frac{1}{6}$; in the case $\frac{1}{4}$ they are even unique.

A note on the lattices $DP(X)$ and $K(X)$

Thu, 23 May 2013 09:42 EDT

Tarun Das, Sejal Shah

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 301--308.

Abstract:
Using the order structure of the lattice $DP(X)$ of density preserving continuous maps on a Hausdorff space $X$ without isolated points, we describe closed nowhere dense subsets of $X$ and, for a subspace $A$ of $X$, we also deduce topological properties of the space $X-A$ from the lattice theoretic properties of $DP(X,A)$. Finally, we use them to obtain Thrivikraman's results concerning $\beta X-X$ and $K(X)$ and, Magill's result concerning the automorphism group of the lattice $K(X)$.

Existence of nontrivial weak solutions for a quasilinear elliptic systems with concave-convex nonlinearities

Thu, 23 May 2013 09:42 EDT

Honghui Yin, Zuodong Yang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 309--328.

Abstract:
In this paper, our main purpose is to establish the existence of nontrivial weak solutions to the following systems: $$\left\{ \begin{array}{ll} -\triangle_p u=\lambda V(x)|u|^{r-2}u+F_u(x,u,v),\;\;\; x\in \Omega,\\ -\triangle_p v=\theta V(x)|v|^{r-2}v+F_v(x,u,v),\;\;\; x\in \Omega,\\ u=v=0,\;\;\; x\in \partial\Omega, \end{array} \right.$$ where $\Omega$ is a bounded domain in ${\bf R}^N$, $\lambda,\theta>0$, $\triangle_s u=\mbox{div}(|\nabla u|^{s-2}\nabla u)$ is the s-Laplacian of u. We obtain the existence results in two cases: (i)$1<r<p<N$; (ii)$1<p<r<p^*$. The existence results of solutions are obtained by variational methods.

Fast vector arithmetic over $\mathbb{F}_3$

Thu, 23 May 2013 09:42 EDT

K. Coolsaet

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 329--344.

Abstract:
We show how binary machine instructions can be used to implement fast vector operations over the finite field $\mathbb{F}_3$. Apart from the standard operations of addition, subtraction and dot product, we also consider combined addition and subtraction, weight, Hamming distance, and iteration over all vectors of a given length. Tests show that our implementation can be as much as 10 times faster than the standard method of using modular arithmetic on arrays of bytes. For computing the Hamming distance even a factor of 33 can sometimes be reached, provided a recent CPU is used.

Some generalizations of Darbo fixed point theorem and applications

Thu, 23 May 2013 09:42 EDT

Asadollah Aghajani, Józef Banaś, Navid Sabzali

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 345--358.

Abstract:
In the paper we provide a few generalizations of Darbo fixed point theorem. Several interconnections among assumptions imposed in the proved theorems are indicated. We also show the applicability of obtained results to the theory of functional integral equations. A concrete example illustrating the mentioned applicability is also included.

The order of the commutator on $SU(3)$ and an application to gauge groups

Thu, 23 May 2013 09:42 EDT

A. Kono, S. Theriault

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 359--370.

Abstract:
We show that the commutator map on $SU(3)$ has order $2^{3}\cdot 3\cdot 5$. As an application, we give an upper bound on the number of homotopy types of gauge groups for principal $SU(3)$-bundles over an $n$-sphere.

Global Existence and Finite Time Blowup for a Nonlocal Parabolic System

Thu, 23 May 2013 09:42 EDT

Zhengqiu Ling, Zejia Wang

Source: Bull. Belg. Math. Soc. Simon Stevin, Volume 20, Number 2, 371--383.

Abstract:
This paper concerns with a parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and finite time blowup of solutions. By using the super- and sub-solution techniques, the critical exponent of the system is determined. Furthermore, the related classification for the parameters in the model is optimal and complete.