Bulletin of the Belgian Mathematical Society - Simon Stevin Articles (Project Euclid)
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The latest articles from Bulletin of the Belgian Mathematical Society - Simon Stevin on Project Euclid, a site for mathematics and statistics resources.en-usCopyright 2010 Cornell University LibraryEuclid-L@cornell.edu (Project Euclid Team)Thu, 05 Aug 2010 15:41 EDTTue, 07 Jun 2011 09:08 EDThttp://projecteuclid.org/collection/euclid/images/logo_linking_100.gifProject Euclid
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On the Geometry of the Conformal Group in Spacetime
http://projecteuclid.org/euclid.bbms/1274896198
<strong>N. G. Gresnigt</strong>, <strong>P. F. Renaud</strong><p><strong>Source: </strong>Bull. Belg. Math. Soc. Simon Stevin, Volume 17, Number 2, 193--200.</p><p><strong>Abstract:</strong><br/>
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.
</p>projecteuclid.org/euclid.bbms/1274896198_Thu, 05 Aug 2010 15:41 EDTThu, 05 Aug 2010 15:41 EDTOn linear extendability of isometrical embeddingshttp://projecteuclid.org/euclid.bbms/1480993584<strong>Hossein Khodaiemehr</strong>, <strong>Fereshteh Sady</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 515--527.</p><p><strong>Abstract:</strong><br/>
In this paper we first investigate linear extendability of an
isometric embedding $T:\mathcal U \longrightarrow \mathcal Y$ from an open subset
$\mathcal U$ of a real Banach space $\mathcal X$ into a real Banach space $\mathcal Y$
in the case where $\mathcal Y$ is either the space $C_\Bbb R(K)$ of
continuous real-valued functions on a compact space $K$, or is a
strictly convex Banach space. Then we obtain similar results for
the case where $\mathcal Y$ is an arbitrary real Banach space and $T:\mathcal U
\longrightarrow \mathcal Y$ is an isometry whose range satisfies some
additional conditions.
</p>projecteuclid.org/euclid.bbms/1480993584_20161205220633Mon, 05 Dec 2016 22:06 ESTFixed point properties for semigroups of non-expansive mappings in conjugate Banach spaceshttp://projecteuclid.org/euclid.bbms/1480993585<strong>Khadime Salame</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 529--544.</p><p><strong>Abstract:</strong><br/>
In this paper we study common fixed point properties of non-linear actions of semi-topological semigroups on non-void weak* compact convex sets in dual Banach spaces. Among other things, we derive from our main result Theorem 1, the existence of a common fixed point property for semigroups of non-expansive mappings acting on non-empty weakly compact convex sets, generalizing a result of Hsu [13], Mitchell [25].
</p>projecteuclid.org/euclid.bbms/1480993585_20161205220633Mon, 05 Dec 2016 22:06 ESTWeak amenability of Banach algebras with respect to charactershttp://projecteuclid.org/euclid.bbms/1480993586<strong>Rasoul Nasr-Isfahani</strong>, <strong>Somayeh Shahmoradi</strong>, <strong>Sima Soltani Renani</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 545--558.</p><p><strong>Abstract:</strong><br/>
For a Banach algebra ${\cal A}$, we introduce and investigate weak amenability of ${\cal A}$
with respect to a character. We give some necessary conditions for the weak
amenability of $\cal A$ with respect to a character and describe a class of Banach
algebras that are not weakly amenable with respect to characters.
Finally, we give examples of Banach algebras which are
weakly amenable with respect to characters but neither weakly amenable nor amenable
with respect to characters.
</p>projecteuclid.org/euclid.bbms/1480993586_20161205220633Mon, 05 Dec 2016 22:06 ESTFlag-transitive point-primitive non-symmetric $2$-$(v,k,2)$ designs with alternating soclehttp://projecteuclid.org/euclid.bbms/1480993587<strong>Hongxue Liang</strong>, <strong>Shenglin Zhou</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 559--571.</p><p><strong>Abstract:</strong><br/>
We prove that if $\mathcal{D}$ is a non-trivial non-symmetric $2$-$(v,k,2)$ design
admitting a flag-transitive point-primitive automorphism group $G$ with $Soc(G)=A_{n}$ for $n\geq5$,
then $\mathcal{D}$ is a $2$-$(6,3,2)$ or $2$-$(10,4,2)$ design.
</p>projecteuclid.org/euclid.bbms/1480993587_20161205220633Mon, 05 Dec 2016 22:06 ESTOn the Frobenius vector of some simplicial affine semigroupshttp://projecteuclid.org/euclid.bbms/1480993588<strong>Ali Mahdavi</strong>, <strong>Farhad Rahmati</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 573--582.</p><p><strong>Abstract:</strong><br/>
We give a formula for a Frobenius vector of a Gorenstein simplicial affine semigroup $S$, and when the semigroup is Cohen-Macaulay we give an algorithm computing the set of minimal Frobenius vectors of $S$ for a special class of semigroups.
</p>projecteuclid.org/euclid.bbms/1480993588_20161205220633Mon, 05 Dec 2016 22:06 ESTOptimization through dense setshttp://projecteuclid.org/euclid.bbms/1480993589<strong>C. R. Jayanarayanan</strong>, <strong>T. S. S. R. K. Rao</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 583--594.</p><p><strong>Abstract:</strong><br/>
In this paper, we study two optimization problems where solutions on a dense set yield global solution. We study these problems for spaces of Bochner integrable functions and for spaces of continuous functions.
The first one deals with expressing the length of a vector as a sum of the distance to a best approximation and minimal best approximation and the second one relates to approximating a subsequence of a minimizing sequence with a sequence of proximinal vectors.
</p>projecteuclid.org/euclid.bbms/1480993589_20161205220633Mon, 05 Dec 2016 22:06 ESTWeighted composition operators on algebras of differentiable functionshttp://projecteuclid.org/euclid.bbms/1480993590<strong>S. Amiri</strong>, <strong>A. Golbaharan</strong>, <strong>H. Mahyar</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 595--608.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a perfect compact plane set, $n\in \mathbb{N}$ and $D^n(X)$ be
the algebra of complex-valued functions on $X$ with continuous
$n$-th derivative. In this paper we study weighted composition
operators on algebras $D^n(X)$. We give a necessary and sufficient
condition for these operators to be compact. As a consequence, we
characterize power compact composition operators on these
algebras. Then we determine the spectra of Riesz weighted
composition operators on these algebras.
</p>projecteuclid.org/euclid.bbms/1480993590_20161205220633Mon, 05 Dec 2016 22:06 ESTInjective mappings in $\mathbb{R}^\mathbb{R}$ and lineabilityhttp://projecteuclid.org/euclid.bbms/1480993591<strong>P. Jiménez-Rodríguez</strong>, <strong>S. Maghsoudi</strong>, <strong>G.A. Muñoz-Fernández</strong>, <strong>J.B. Seoane-Sepúlveda</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 609--623.</p><p><strong>Abstract:</strong><br/>
It is known that there is not a two dimensional linear space in $\mathbb R^\mathbb R$ every non-zero element of which is an injective function. Here, we generalize this result to arbitrarily large dimensions. We also study the convolution of non-differentiable functions which gives, as a result, a differentiable function. In this latter case, we are able to show the existence of linear spaces of the largest possible dimension formed by functions enjoying the previous property. By doing this we provide both positive and negative results to the recent field of lineability. Some open questions are also provided.
</p>projecteuclid.org/euclid.bbms/1480993591_20161205220633Mon, 05 Dec 2016 22:06 ESTTransitive parallelism of residues in buildingshttp://projecteuclid.org/euclid.bbms/1480993592<strong>Antoine Clais</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 4, 625--640.</p><p><strong>Abstract:</strong><br/>
We study the buildings in which parallelism of residues is an equivalence relation. If the building admits a group action, we describe how parallel residues are related to residues with equal stabilizers. This permits to retrieve the fact that in a Coxeter group or in a graph product, intersections of parabolic subgroups are parabolic.
</p>projecteuclid.org/euclid.bbms/1480993592_20161205220633Mon, 05 Dec 2016 22:06 ESTCoalgebras governing both weighted Hurwitz products and their pointwise transformshttp://projecteuclid.org/euclid.bbms/1483671619<strong>Richard Garner</strong>, <strong>Ross Street</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 643--666.</p><p><strong>Abstract:</strong><br/>
We give further insights into the weighted Hurwitz product and the
weighted tensor product of Joyal species. Our first group of results
relate the Hurwitz product to the pointwise product, including the
interaction with Rota--Baxter operators. Our second group of results
explain the first in terms of convolution with suitable bialgebras,
and show that these bialgebras are in fact obtained in a
particularly straightforward way by freely generating from pointed
coalgebras. Our third group of results extend this from linear
algebra to two-dimensional linear algebra, deriving the existence of
weighted Hurwitz monoidal structures on the category of species
using convolution with freely generated bimonoidales. Our final
group of results relate Hurwitz monoidal structures with
equivalences of Dold--Kan type.
</p>projecteuclid.org/euclid.bbms/1483671619_20170105220101Thu, 05 Jan 2017 22:01 ESTA criterion for reflectiveness of normal extensionshttp://projecteuclid.org/euclid.bbms/1483671620<strong>Andrea Montoli</strong>, <strong>Diana Rodelo</strong>, <strong>Tim Van der Linden</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 667--691.</p><p><strong>Abstract:</strong><br/>
We give a new sufficient condition for the normal extensions in an
admissible Galois structure to be reflective. We then show that
this condition is indeed fulfilled when $\mathbb{X}$ is the (protomodular)
reflective subcategory of $\mathcal{S}$-special objects of a Barr-exact
$\mathcal{S}$-protomodular category $\mathbb{C}$, where $\mathcal{S}$ is the class of
split epimorphic trivial extensions in $\mathbb{C}$. Next to some
concrete examples where the criterion may be applied, we also
study the adjunction between a Barr-exact unital category and its
abelian core, which we prove to be admissible.
</p>projecteuclid.org/euclid.bbms/1483671620_20170105220101Thu, 05 Jan 2017 22:01 ESTThe Projective Class Rings of a family of pointed Hopf algebras of Rank twohttp://projecteuclid.org/euclid.bbms/1483671621<strong>Hui-Xiang Chen</strong>, <strong>Hassen Suleman Esmael Mohammed</strong>, <strong>Weijun Lin</strong>, <strong>Hua Sun</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 693--711.</p><p><strong>Abstract:</strong><br/>
In this paper, we compute the projective class rings of the tensor product
$\mathcal{H}_n(q)=A_n(q)\ot A_n(q^{-1})$ of Taft algebras $A_n(q)$ and $A_n(q^{-1})$, and its
cocycle deformations $H_n(0,q)$ and $H_n(1,q)$, where $n>2$ is a positive integer
and $q$ is a primitive $n$-th root of unity. It is shown that the projective class rings $r_p(\mathcal{H}_n(q))$,
$r_p(H_n(0,q))$ and $r_p(H_n(1,q))$ are commutative rings generated by three elements, three elements and
two elements subject to some relations, respectively.
It turns out that even $\mathcal{H}_n(q)$, $H_n(0,q)$ and $H_n(1,q)$
are cocycle twist-equivalent to each other, they are of different representation types: wild, wild and tame, respectively.
</p>projecteuclid.org/euclid.bbms/1483671621_20170105220101Thu, 05 Jan 2017 22:01 ESTCurved Rota-Baxter systemshttp://projecteuclid.org/euclid.bbms/1483671622<strong>Tomasz Brzeziński</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 713--720.</p><p><strong>Abstract:</strong><br/>
Rota-Baxter systems are modified by the inclusion of a curvature term. It is shown that, subject to specific properties of the curvature form, curved Rota-Baxter systems $(A,R,S,\omega)$ induce associative and (left) pre-Lie products on the algebra $A$. It is also shown that if both Rota-Baxter operators coincide with each other and the curvature is $A$-bilinear, then the (modified by $R$) Hochschild cohomology ring over $A$ is a curved differential graded algebra.
</p>projecteuclid.org/euclid.bbms/1483671622_20170105220101Thu, 05 Jan 2017 22:01 ESTAlgebra depth in tensor categorieshttp://projecteuclid.org/euclid.bbms/1483671623<strong>Lars Kadison</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 721--752.</p><p><strong>Abstract:</strong><br/>
Study of the quotient module of a finite-dimensional Hopf subalgebra pair in order to compute its depth yields a relative Maschke Theorem, in which semisimple extension is characterized as being separable, and is therefore an ordinary Frobenius extension. We study the core Hopf ideal of a Hopf subalgebra, noting that the length of the annihilator chain of tensor powers of the quotient module is linearly related to the depth, if the Hopf algebra is semisimple. A tensor categorical definition of depth is introduced, and a summary from this new point of view of previous results are included. It is shown in a last section that the depth, Bratteli diagram and relative cyclic homology of algebra extensions are Morita invariants.
</p>projecteuclid.org/euclid.bbms/1483671623_20170105220101Thu, 05 Jan 2017 22:01 ESTPicard-Vessiot and categorically normal extensions in differential-difference Galois theoryhttp://projecteuclid.org/euclid.bbms/1483671624<strong>G. Janelidze</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 753--768.</p><p><strong>Abstract:</strong><br/>
The connection between categorical and differential Galois theories established by the author (published in 1989) is extended to the context that includes difference Galois theory.
</p>projecteuclid.org/euclid.bbms/1483671624_20170105220101Thu, 05 Jan 2017 22:01 ESTA note on the restricted universal enveloping algebra of a restricted Lie-Rinehart Algebrahttp://projecteuclid.org/euclid.bbms/1483671625<strong>Peter Schauenburg</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 769--777.</p><p><strong>Abstract:</strong><br/>
Lie-Rinehart algebras, also known as Lie algebroids, give rise to Hopf algebroids by a universal enveloping algebra construction, much as the universal enveloping algebra of an ordinary Lie algebra gives a Hopf algebra, of infinite dimension. In finite characteristic, the universal enveloping algebra of a restricted Lie algebra admits a quotient Hopf algebra which is finite-dimensional if the Lie algebra is. Rumynin has shown that suitably defined restricted Lie algebroids allow to define restricted universal enveloping algebras that are finitely generated projective if the Lie algebroid is. This note presents an alternative proof and possibly fills a gap that might, however, only be a gap in the author's understanding.
</p>projecteuclid.org/euclid.bbms/1483671625_20170105220101Thu, 05 Jan 2017 22:01 ESTRepresentations of the small quasi-quantum group ${\operatorname{Q}}\mathbf{u}_{q}(\mathfrak{sl}_{2})$http://projecteuclid.org/euclid.bbms/1483671626<strong>Gongxiang Liu</strong>, <strong>Fred Van Oystaeyen</strong>, <strong>Yinhuo Zhang</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 23, Number 5, 779--800.</p><p><strong>Abstract:</strong><br/>
The quasi-Frobenius-Lusztig kernel ${\operatorname{Q}}\mathbf{u}_{q}(\mathfrak{sl}_{2})$
associated with $\mathfrak{sl}_{2}$ has been constructed in [9]. In this paper we study the representations of this small quasi-quantum group. We give a complete list of non-isomorphic indecomposables and the tensor product decomposition rules for simples and projectives. A description of the Grothendieck ring is also provided.
</p>projecteuclid.org/euclid.bbms/1483671626_20170105220101Thu, 05 Jan 2017 22:01 ESTFrom finite groups to finite-dimensional Hopf algebrashttp://projecteuclid.org/euclid.bbms/1489888812<strong>Miriam Cohen</strong>, <strong>Sara Westreich</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 1--14.</p>projecteuclid.org/euclid.bbms/1489888812_20170318220020Sat, 18 Mar 2017 22:00 EDTA finite-dimensional Lie algebra arising from a Nichols algebra of diagonal type (rank 2)http://projecteuclid.org/euclid.bbms/1489888813<strong>Nicolás Andruskiewitsch</strong>, <strong>Iván Angiono</strong>, <strong>Fiorela Rossi Bertone</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 15--34.</p><p><strong>Abstract:</strong><br/>
Let ${\mathcal{B}}_{\mathfrak{q}}$ be a finite-dimensional Nichols algebra of diagonal type corresponding to a matrix $\mathfrak{q} \in {\mathbf{k}}^{\theta \times \theta}$.
Let ${\mathcal{L}}_{\mathfrak{q}}$ be the Lusztig algebra associated to ${\mathcal{B}}_{\mathfrak{q}}$.
We present ${\mathcal{L}}_{\mathfrak{q}}$ as an extension (as braided Hopf algebras) of ${\mathcal{B}}_{\mathfrak{q}}$ by ${\mathfrak Z}_{\mathfrak{q}}$ where ${\mathfrak Z}_{\mathfrak{q}}$ is isomorphic to the universal enveloping algebra of a Lie algebra ${\mathfrak{n}}_\mathfrak{q}$. We compute the Lie algebra ${\mathfrak{n}}_{\mathfrak{q}}$ when $\theta = 2$.
</p>projecteuclid.org/euclid.bbms/1489888813_20170318220020Sat, 18 Mar 2017 22:00 EDTPartial actions: what they are and why we carehttp://projecteuclid.org/euclid.bbms/1489888814<strong>Eliezer Batista</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 35--71.</p><p><strong>Abstract:</strong><br/>
We present a survey of recent developments in the theory of partial actions of groups and Hopf algebras.
</p>projecteuclid.org/euclid.bbms/1489888814_20170318220020Sat, 18 Mar 2017 22:00 EDTThree natural subgroups of the Brauer-Picard group of a
Hopf algebra with applicationshttp://projecteuclid.org/euclid.bbms/1489888815<strong>Simon Lentner</strong>, <strong>Jan Priel</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 73--106.</p><p><strong>Abstract:</strong><br/>
In this article we construct three explicit natural subgroups of the
Brauer-Picard group of the category of representations of a
finite-dimensional Hopf algebra. In examples the Brauer Picard group decomposes
into an ordered product of these subgroups, somewhat similar to a Bruhat
decomposition.
Our construction returns for any Hopf algebra three types of braided
autoequivalences and correspondingly three families of invertible bimodule
categories. This gives examples of so-called (2-)Morita
equivalences and defects in topological field theories. We have a closer
look at the case of quantum groups and Nichols algebras and give
interesting applications. Finally, we briefly discuss the three families of
group-theoretic extensions.
</p>projecteuclid.org/euclid.bbms/1489888815_20170318220020Sat, 18 Mar 2017 22:00 EDTA simplicial approach to multiplier bimonoidshttp://projecteuclid.org/euclid.bbms/1489888816<strong>Gabriella Böhm</strong>, <strong>Stephen Lack</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 107--122.</p><p><strong>Abstract:</strong><br/>
Although multiplier bimonoids in general are not known to correspond to
comonoids in any monoidal category, we classify them in terms of maps from the
Catalan simplicial set to another suitable simplicial set; thus they can be
regarded as (co)monoids in something more general than a monoidal
category (namely, the simplicial set itself).
We analyze the particular simplicial maps corresponding to that class of
multiplier bimonoids which can be regarded as comonoids.
</p>projecteuclid.org/euclid.bbms/1489888816_20170318220020Sat, 18 Mar 2017 22:00 EDTA criterion for $p$-henselianity in characteristic $p$http://projecteuclid.org/euclid.bbms/1489888817<strong>Zoé Chatzidakis</strong>, <strong>Milan Perera</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 123--126.</p><p><strong>Abstract:</strong><br/>
Let $p$ be a prime. In this paper we give a proof of the following
result: A valued field $(K,v)$ of characteristic $p > 0$ is
$p$-henselian if and only if every element of strictly positive
valuation is of the form $x^p - x$ for some $x \in K$.
</p>projecteuclid.org/euclid.bbms/1489888817_20170318220020Sat, 18 Mar 2017 22:00 EDTComplex of injective words revisitedhttp://projecteuclid.org/euclid.bbms/1489888818<strong>Wee Liang Gan</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 127--134.</p><p><strong>Abstract:</strong><br/>
We give a simple proof that (a generalization of) the complex of injective words has vanishing homology in all except the top degree.
</p>projecteuclid.org/euclid.bbms/1489888818_20170318220020Sat, 18 Mar 2017 22:00 EDTSolvability of a fractional integral equation with the concept of measure of noncompactnesshttp://projecteuclid.org/euclid.bbms/1489888819<strong>Kishin Sadarangani</strong>, <strong>Bessem Samet</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 135--151.</p><p><strong>Abstract:</strong><br/>
We study the existence of solutions for a new class of integral equations involving a fractional integral with respect to another function. Our techniques are based on the measure of non-compactness concept combined with a generalized version of Darbo's theorem. Some examples are presented to illustrate the obtained results.
</p>projecteuclid.org/euclid.bbms/1489888819_20170318220020Sat, 18 Mar 2017 22:00 EDTVanishing theorems of the basic harmonic forms on a complete foliated Riemannian manifoldhttp://projecteuclid.org/euclid.bbms/1489888820<strong>Seoung Dal Jung</strong>, <strong>Huili Liu</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 1, 153--160.</p><p><strong>Abstract:</strong><br/>
A well-known result by M. Min-Oo et al. states that
there are no nontrivial basic harmonic $r (0<
r<q={\rm codim}\mathcal F)$-forms
on a compact foliated Riemannian manifold $(M,\mathcal F)$. We extend this result to a complete foliated Riemannian
manifold.
</p>projecteuclid.org/euclid.bbms/1489888820_20170318220020Sat, 18 Mar 2017 22:00 EDTCommon fixed point theorems for a pair of multivalued mappings
under weak contractive conditions in ordered metric spaceshttps://projecteuclid.org/euclid.bbms/1503453703<strong>José Eduardo Prado Pires de Campos</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 161--174.</p><p><strong>Abstract:</strong><br/>
We give a direct proof that if two string links have isotopic closures, then there is a braid-special isomorphism between their n-level group diagrams, for every $n \geqslant 2$. In the case of link-homotopy, we give an alternative proof to our previous result that there is a braid-special isomorphism between the group diagrams for the homotopy classes of two string links if and only if they have link-homotopic closures.
</p>projecteuclid.org/euclid.bbms/1503453703_20170822220153Tue, 22 Aug 2017 22:01 EDTOn existence of embeddings for point-line geometrieshttps://projecteuclid.org/euclid.bbms/1503453704<strong>Anderson de Araujo</strong>, <strong>Rafael Abreu</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 175--187.</p><p><strong>Abstract:</strong><br/>
In this work we prove the existence of a classical positive solution for an elliptic equation with a sublinear term. We use Galerkin approximations to show existence of such solution on bounded domains in $\mathbb{R}^N$.
</p>projecteuclid.org/euclid.bbms/1503453704_20170822220153Tue, 22 Aug 2017 22:01 EDTAsymptotic stability of solutions to quasi-geostrophic equationhttps://projecteuclid.org/euclid.bbms/1503453705<strong>Dominika Pilarczyk</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 189--198.</p><p><strong>Abstract:</strong><br/>
We show that sufficiently small mild solutions of the initial value problem to the quasi-geostrophic equation in $\mathbb R^2$ are asymptotically stable under arbitrary large initial $L^2$-perturbations. We obtain also the decay rate.
</p>projecteuclid.org/euclid.bbms/1503453705_20170822220153Tue, 22 Aug 2017 22:01 EDTHölder's inequality: some recent and unexpected applicationshttps://projecteuclid.org/euclid.bbms/1503453706<strong>N. Albuquerque</strong>, <strong>G. Araújo</strong>, <strong>D. Pellegrino</strong>, <strong>J.B. Seoane-Sepúlveda</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 199--225.</p><p><strong>Abstract:</strong><br/>
Hölder's inequality, since its appearance in 1888, has played a fundamental role in Mathematical Analysis and may be considered a milestone in Mathematics. It may seem strange that, nowadays, it keeps resurfacing and bringing new insights to the mathematical community. In this survey we show how a variant of Hölder's inequality (although well-known in PDEs) was essentially overlooked in Functional/Complex Analysis and has had a crucial (and in some sense unexpected) influence in very recent advances in different fields of Mathematics. Some of these recent advances have been appearing since 2012 and include the theory of Dirichlet series, the famous Bohr radius problem, certain classical inequalities (such as Bohnenblust--Hille or Hardy--Littlewood), and Mathematical Physics.
</p>projecteuclid.org/euclid.bbms/1503453706_20170822220153Tue, 22 Aug 2017 22:01 EDTNon-Weakly Supercyclic Classes of Weighted
Composition Operators on Banach Spaces of Analytic Functionshttps://projecteuclid.org/euclid.bbms/1503453707<strong>A. Moradi</strong>, <strong>B. Khani Robati</strong>, <strong>K. Hedayatian</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 227--241.</p><p><strong>Abstract:</strong><br/>
We present a non-weak supercyclicity criterion for
vectors in infinite dimensional Banach spaces.
Also, we give sufficient conditions under which a class
of weighted composition operators on a Banach space of analytic
functions is not weakly supercyclic. In particular, we show that the
semigroup of linear isometries on the spaces $S^p$ ($p>1$), is not weakly supercyclic. Moreover, we observe that every composition operator on some Banach space of analytic functions such as the disc algebra or the analytic Lipschitz space is not weakly supercyclic.
</p>projecteuclid.org/euclid.bbms/1503453707_20170822220153Tue, 22 Aug 2017 22:01 EDTAn ergodic theorem for the quasi-regular representation of the free grouphttps://projecteuclid.org/euclid.bbms/1503453708<strong>Adrien Boyer</strong>, <strong>Antoine Pinochet Lobos</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 243--255.</p><p><strong>Abstract:</strong><br/>
We prove the weak-$*$ convergence of a certain sequence of averages of unitary operators associated to the action of the free group on its Gromov boundary. This result, which can be thought as an ergodic theorem à la von Neumann with coefficients, provides a new proof of the irreducibility of the quasi-regular representation of the free group.
</p>projecteuclid.org/euclid.bbms/1503453708_20170822220153Tue, 22 Aug 2017 22:01 EDTHomogeneous Geodesics in Generalized Wallach Spaceshttps://projecteuclid.org/euclid.bbms/1503453709<strong>Andreas Arvanitoyeorgos</strong>, <strong>Yu Wang</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 257--270.</p><p><strong>Abstract:</strong><br/>
We classify generalized Wallach spaces which are g.o. spaces. We also investigate homogeneous geodesics in generalized Wallach spaces for any given invariant Riemannian metric and we give some examples.
</p>projecteuclid.org/euclid.bbms/1503453709_20170822220153Tue, 22 Aug 2017 22:01 EDTStability constants for weighted composition operators on $L^p(\Sigma)$https://projecteuclid.org/euclid.bbms/1503453710<strong>M. R. Jabbarzadeh</strong>, <strong>M. Jafari Bakhshkandi</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 271--281.</p><p><strong>Abstract:</strong><br/>
In this note we give an explicit formula for the Moore-Penrose
inverse $W^{\dag}$ of a weighted composition operator $W$ on
$L^2(\Sigma)$ and then we obtain the stability constant $K_W$ of
$W$ on $L^p(\Sigma)$, where $1\leq p\leq \infty$. Moreover, we determine,
under certain conditions, the essential norm of $W$ acting on $L^\infty(\Sigma)$.
</p>projecteuclid.org/euclid.bbms/1503453710_20170822220153Tue, 22 Aug 2017 22:01 EDTSome remarks on the structure of Lipschitz-free spaceshttps://projecteuclid.org/euclid.bbms/1503453711<strong>Petr Aizpuru</strong>, <strong>M. Hájek</strong>, <strong>Matěj Novotný</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 283--304.</p><p><strong>Abstract:</strong><br/>
We give several structural results concerning the Lipschitz-free spaces
$\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$
contains a complemented copy of $\ell_1(\Gamma)$, where $\Gamma=\text{dens}(M)$.
If $\mathcal N$ is a net in a finite dimensional Banach space $X$, we
show that $\mathcal F(\mathcal N)$ is isomorphic to its square.
If $X$ contains a complemented copy of $\ell_p, c_0$ then $\f(\mathcal N)$
is isomorphic to its\linebreak $\ell_1$-sum. Finally, we prove that
for all $X\cong C(K)$ spaces, where $K$ is a metrizable
compact, $\f(\mathcal N)$ are mutually isomorphic
spaces with a Schauder basis.
</p>projecteuclid.org/euclid.bbms/1503453711_20170822220153Tue, 22 Aug 2017 22:01 EDTA new analytical technique for solving Lane - Emden type equations arising in astrophysicshttps://projecteuclid.org/euclid.bbms/1503453712<strong>Sinan Deniz</strong>, <strong>Necdet Bildik</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 305--320.</p><p><strong>Abstract:</strong><br/>
Lane - Emden type equations are nonlinear differential equations which represent many scientific phenomena in astrophysics and mathematical physics. In this study, a new analytic approximate technique for addressing nonlinear problems, namely the optimal perturbation iteration method, is introduced and implemented to singular initial value Lane-Emden type problems to test the effectiveness and performance of the method. This technique provides us to adjust the convergence regions when necessary.Comparing different methods reveals that the proposed method is highly accurate and has great potential to be a new kind of powerful analytical tool for Lane-Emden type equations.
</p>projecteuclid.org/euclid.bbms/1503453712_20170822220153Tue, 22 Aug 2017 22:01 EDTBest approximative properties of exposed faces of $l_1$https://projecteuclid.org/euclid.bbms/1506477683<strong>S. Kalaiarasi</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 321--333.</p>projecteuclid.org/euclid.bbms/1506477683_20170926220147Tue, 26 Sep 2017 22:01 EDTAn explicit formula for the cup-length of the rotation grouphttps://projecteuclid.org/euclid.bbms/1506477684<strong>Július Korbaš</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 335--338.</p><p><strong>Abstract:</strong><br/>
This paper gives an explicit formula for the $\mathbb Z_2$-cup-length of
the rotation group $\mathrm{SO}(n)$.
</p>projecteuclid.org/euclid.bbms/1506477684_20170926220147Tue, 26 Sep 2017 22:01 EDTNumerical
treatment of nonlocal boundary value problem with layer behaviourhttps://projecteuclid.org/euclid.bbms/1506477685<strong>Erkan Cimen</strong>, <strong>Musa Cakir</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 339--352.</p><p><strong>Abstract:</strong><br/>
This paper deals with the singularly perturbed nonlocal boundary value
problem for a linear first order differential equation. For the numerical
solution of this problem, we use a fitted difference scheme on a piecewise
uniform Shishkin mesh. An error analysis shows that the method is almost
first order convergent, in the discrete maximum norm, independently of the
perturbation parameter. Numerical results are presented which illustrate the
theoretical results.
</p>projecteuclid.org/euclid.bbms/1506477685_20170926220147Tue, 26 Sep 2017 22:01 EDTGeneralized derivatives and approximation in weighted Lorentz spaceshttps://projecteuclid.org/euclid.bbms/1506477686<strong>Ramazan Akgün</strong>, <strong>Yunus Emre Yildirir</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 353--366.</p><p><strong>Abstract:</strong><br/>
In the present article we prove direct, simultaneous and converse
approximation theorems by trigonometric polynomials for functions $f$ and $
\left( \psi ,\beta \right) $-derivatives of $f$ in weighted Lorentz spaces.
</p>projecteuclid.org/euclid.bbms/1506477686_20170926220147Tue, 26 Sep 2017 22:01 EDTA discrete-time approach in the qualitative theory of skew-product three-parameter semiflowshttps://projecteuclid.org/euclid.bbms/1506477687<strong>Ciprian Preda</strong>, <strong>Adriana-Paula Popiţiu</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 367--379.</p><p><strong>Abstract:</strong><br/>
We obtain a characterization of the uniform exponential stability for the continuous-time skew-product three-parameter semiflows in Banach spaces, using a discrete-time approach. Our technique is based on the classical "test-function" method of O. Perron and Ta Li.
</p>projecteuclid.org/euclid.bbms/1506477687_20170926220147Tue, 26 Sep 2017 22:01 EDTCompact perturbations resulting in hereditarily polaroid operatorshttps://projecteuclid.org/euclid.bbms/1506477688<strong>B.P. Duggal</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 381--392.</p><p><strong>Abstract:</strong><br/>
A Banach space operator $A\in B({\cal X})$ is polaroid,
$A\in(\cal P)$, if the isolated points of the spectrum $\sigma(A)$ are
poles of the operator; $A$ is hereditarily polaroid, $A\in(\cal {HP})$, if
every restriction of $A$ to a closed invariant subspace is polaroid. It is seen that
operators $A\in(\cal {HP})$ have SVEP - the single-valued extension property - on $\Phi_{sf}(A)=\{\lambda: A-\lambda$
is semi Fredholm $\}$. Hence $\Phi^+_{sf}(A)=\{\lambda\in\Phi_{sf}(A), \ind(A-\lambda)>0\}=\emptyset$ for operators
$A\in(\cal {HP})$, and a necessary and sufficient condition for the perturbation $A+K$ of an operator $A\in B({\cal X})$ by a compact operator
$K\inB({\cal X})$ to be hereditarily polaroid is that
$\Phi_{sf}^+(A)=\emptyset$. A sufficient condition for $A\in B({\cal X})$ to
have SVEP on $\Phi_{sf}(A)$ is that its component
$\Omega_a(A)=\{\lambda\in\Phi_{sf}(A): \ind(A-\lambda)\leq 0\}$ is
connected. We prove: If $A\in B({\cal H})$ is a Hilbert space operator, then a
necessary and sufficient condition
for there to exist a compact operator $K\in B({\cal H})$ such that $A+K\in(\cal {HP})$ is that
$\Omega_a(A)$ is connected.
</p>projecteuclid.org/euclid.bbms/1506477688_20170926220147Tue, 26 Sep 2017 22:01 EDTDensity by moduli and Wijsman statistical convergencehttps://projecteuclid.org/euclid.bbms/1506477689<strong>Vinod K. Bhardwaj</strong>, <strong>Shweta Dhawan</strong>, <strong>Oleksiy A. Dovgoshey</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 393--415.</p><p><strong>Abstract:</strong><br/>
In this paper, we have generalized the Wijsman statistical convergence of closed sets in metric space by introducing the $f$-Wijsman statistical convergence of these sets, where $f$ is an unbounded modulus. It is shown that the Wijsman convergent sequences are precisely those sequences which are $f$-Wijsman statistically convergent for every unbounded modulus $f$. We have also introduced a new concept of Wijsman strong Cesàro summability with respect to a modulus $f$, and investigate the relationship between the $f$-Wijsman statistically convergent sequences and the Wijsman strongly Cesàro summable sequences with respect to $f$.
</p>projecteuclid.org/euclid.bbms/1506477689_20170926220147Tue, 26 Sep 2017 22:01 EDTGeneralized CAT(0) spaceshttps://projecteuclid.org/euclid.bbms/1506477690<strong>M. A. Khamsi</strong>, <strong>S. A. Shukri</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 417--426.</p><p><strong>Abstract:</strong><br/>
We extend the Gromov geometric definition of CAT(0) spaces to the case where the comparison triangles are not in the Euclidean plane but belong to a general Banach space. In particular, we study the case where the Banach space is $\ell_p$, for $p > 2$.
</p>projecteuclid.org/euclid.bbms/1506477690_20170926220147Tue, 26 Sep 2017 22:01 EDTCriteria for univalence, Integral means and Dirichlet integral for Meromorphic functionshttps://projecteuclid.org/euclid.bbms/1506477691<strong>Bappaditya Bhowmik</strong>, <strong>Firdoshi Parveen</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 427--438.</p><p><strong>Abstract:</strong><br/>
Let $\mathcal{A}(p)$ be the class consisting of functions $f$ that are holomorphic
in $\mathbb D\setminus \{p\}$, $p\in (0,1)$ possessing a simple pole at the point $z=p$ with nonzero residue and normalized by the condition $f(0)=0=f'(0)-1$.
In this article, we first prove a sufficient condition for univalency for functions in $\mathcal{A}(p)$.
Thereafter, we consider the class denoted by $\Sigma(p)$ that consists of functions $f \in \mathcal{A}(p)$ that are univalent in
$\mathbb D$. We obtain the exact value for $\displaystyle\max_ {f\in \Sigma(p)}\Delta(r,z/f)$, where the Dirichlet integral $\Delta(r,z/f)$ is given by
$$
\Delta(r,z/f)=\displaystyle\iint_{|z|<r} |\left(z/f(z)\right)'|^2 \,dx\, dy, \quad(z=x+iy),~0<r\leq 1.
$$
We also obtain a sharp estimate for $\Delta(r,z/f)$ whenever $f$ belongs to certain subclasses of $\Sigma(p)$.
Furthermore, we obtain sharp estimates of the integral means for the aforementioned classes of functions.
</p>projecteuclid.org/euclid.bbms/1506477691_20170926220147Tue, 26 Sep 2017 22:01 EDTSums of asymptotically midpoint uniformly convex spaceshttps://projecteuclid.org/euclid.bbms/1506477692<strong>S. J. Dilworth</strong>, <strong>Denka Kutzarova</strong>, <strong>N. Lovasoa Randrianarivony</strong>, <strong>Matthew Romney</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 439--446.</p><p><strong>Abstract:</strong><br/>
We study the property of asymptotic midpoint uniform convexity for infinite direct sums of Banach spaces, where the norm of the sum is defined by a Banach space $E$ with a 1-unconditional basis. We show that a sum $(\sum_{n=1}^\infty X_n)_E$ is asymptotically midpoint uniformly convex (AMUC) if and only if the spaces $X_n$ are uniformly AMUC and $E$ is uniformly monotone. We also show that $L_p(X)$ is AMUC if and only if $X$ is uniformly convex.
</p>projecteuclid.org/euclid.bbms/1506477692_20170926220147Tue, 26 Sep 2017 22:01 EDTA remark on the Chow ring of some hyperkähler fourfoldshttps://projecteuclid.org/euclid.bbms/1506477693<strong>Robert Laterveer</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 447--455.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a hyperkähler variety. Voisin has conjectured that the classes of Lagrangian constant cycle subvarieties in the Chow ring of $X$ should lie in a subring injecting into cohomology.
We study this conjecture for the Fano variety of lines on a very general cubic fourfold.
</p>projecteuclid.org/euclid.bbms/1506477693_20170926220147Tue, 26 Sep 2017 22:01 EDTNew Numerical Solution For Solving Nonlinear Singular Thomas-Fermi Differential Equationhttps://projecteuclid.org/euclid.bbms/1506477694<strong>Kourosh Parand</strong>, <strong>Mehdi Delkhosh</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 3, 457--476.</p><p><strong>Abstract:</strong><br/>
In this paper, the nonlinear singular Thomas-Fermi differential equation on a semi-infinite domain for neutral atoms is solved by using the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind. First, this collocation method reduces the solution of this problem to the solution of a system of nonlinear algebraic equations. Second, using solve a system of nonlinear equations, the initial value for the unknown parameter $L$ is calculated, and finally, the value of $L$ to increase the accuracy of the initial slope is improved and the value of $y'(0)=-1.588071022611375312718684509$ is calculated. The comparison with some numerical solutions shows that the present solution is highly accurate.
</p>projecteuclid.org/euclid.bbms/1506477694_20170926220147Tue, 26 Sep 2017 22:01 EDTNielsen-Reidemeister indices for multivalued mapshttps://projecteuclid.org/euclid.bbms/1515035003<strong>M.C. Crabb</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 483--499.</p><p><strong>Abstract:</strong><br/>
A Nielsen-Reidemeister index is constructed for multivalued maps defined by fractions $f/p$ where $p: \tilde X \to X$ is a fibrewise manifold with closed fibres over a compact ENR and $f:\tilde X\to X$ is a continuous map. In the case that $p$ is a finite $n$-fold cover, this index is shown to agree with the index of the $n$-valued map $\tilde X\multimap \tilde X$ associated with $f/p$ by a construction of Brown [4].
</p>projecteuclid.org/euclid.bbms/1515035003_20180103220342Wed, 03 Jan 2018 22:03 ESTLinear systems over $\mathbb Z[Q_{16}]$ and roots of maps of some 3-complexes into $M_{Q_{16}}$https://projecteuclid.org/euclid.bbms/1515035004<strong>Claudemir Aniz</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 501--522.</p><p><strong>Abstract:</strong><br/>
Let $\mathbb Z[Q_{16}]$ be the group ring where $Q_{16}=\langle x,y|x^4=y^2,\,xyx=y\rangle$ is the quaternion group of order 16 and $\varepsilon $ the augmentation map. We show that, if $PX=K(x-1)$ and $PX=K(-xy+1)$ has solution over $\mathbb Z[Q_{16}]$ and all $m\times m$ minors of $\varepsilon (P)$ are relatively prime, then the linear system $PX=K$ has a solution over $\mathbb Z[Q_{16}]$, where $P=[p_{ij}]$ is an $m\times n$ matrix with $m\leq n$. As a consequence of such results, we show that there is no map $f:W\to M_{Q_{16}}$ that is strongly surjective, i.e., such that $MR[f,a]=\min\{\#(g^{-1}(a))|g\in [f]\}\neq 0$. Here, $M_{Q_{16}}$ is the orbit space of the 3-sphere $\mathbb S^3$ with respect to the action of $Q_{16}$ determined by the inclusion $Q_{16}\subseteq \mathbb S^3$ and $W$ is a $CW$-complex of dimension 3 with $H^3(W;\mathbb Z)=0$.
</p>projecteuclid.org/euclid.bbms/1515035004_20180103220342Wed, 03 Jan 2018 22:03 ESTStabilizers of fixed point classes and Nielsen numbers of $n$-valued mapshttps://projecteuclid.org/euclid.bbms/1515035005<strong>Robert F. Brown</strong>, <strong>Junzheng Nan</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 523--535.</p><p><strong>Abstract:</strong><br/>
The stabilizer of a fixed point class of a map is the fixed subgroup of the induced fundamental group homomorphism based at a point in the class. A theorem of Jiang, Wang and Zhang is used to prove that if a map of a graph satisfies a strong remnant condition, then the stabilizers of all its fixed point classes are trivial. Consequently, if $\phi_{p, f}$ is the $n$-valued lift to a covering space $p$ of a map $f$ with strong remnant of a graph, then the Nielsen numbers are related by the equation $N(\phi_{p, f}) = n \cdot N(f)$. Additional information concerning Nielsen numbers is obtained for $n$-valued lifts of maps of graphs with positive Lefschetz numbers and of maps of spaces with abelian fundamental groups and for extensions of $n$-valued maps.
</p>projecteuclid.org/euclid.bbms/1515035005_20180103220342Wed, 03 Jan 2018 22:03 ESTNielsen periodic point theory on infra-nilmanifoldshttps://projecteuclid.org/euclid.bbms/1515035006<strong>Gert-Jan Dugardein</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 537--566.</p><p><strong>Abstract:</strong><br/>
In this paper, we expand certain aspects of Nielsen periodic point theory from tori and nilmanifolds to infra-nilmanifolds. We show that infra-nilmanifolds are essentially reducible to the GCD and essentially toral. With these structural properties in mind, we develop a method to compute the full Nielsen-Jiang number $NF_n(f)$. We also determine for which maps it holds that $NF_n(f)=N(f^n)$, for all $n$.
</p>projecteuclid.org/euclid.bbms/1515035006_20180103220342Wed, 03 Jan 2018 22:03 EST$(H,G)$-coincidence theorems for manifolds and a topological Tverberg type theorem for any natural number $r$https://projecteuclid.org/euclid.bbms/1515035007<strong>Denise de Mattos</strong>, <strong>Edivaldo L. dos Santos</strong>, <strong>Taciana O. Souza</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 567--579.</p><p><strong>Abstract:</strong><br/>
Let $X$ be a paracompact space, let $G$ be a finite group acting freely on $X$ and let $H$ a cyclic subgroup of $G$ of prime order $p$. Let $f:X\rightarrow M$ be a continuous map where $M$ is a connected $m$-manifold (orientable if $p>2$) and $f^* (V_k) = 0$, for $k\geq 1$, where $V_k$ are the $Wu$ classes of $M$. Suppose that $\ind X\geq n> (|G|-r)m$, where $r=\frac{|G|}{p}$. In this work, we estimate the cohomological dimension of the set $A(f,H,G)$ of $(H,G)$-coincidence points of $f$. Also, we estimate the index of a $(H, G)$-coincidence set in the case that $H$ is a $p$-torus subgroup of a particular group $G$ and as application we prove a topological Tverberg type theorem for any natural number $r$. Such result is a weak version of the famous topological Tverberg conjecture, which was proved recently fail for all $r$ that are not prime powers. Moreover, we obtain a generalized Van Kampen-Flores type theorem for any integer $r$.
</p>projecteuclid.org/euclid.bbms/1515035007_20180103220342Wed, 03 Jan 2018 22:03 EST$\mathbb{Z}_2^k$-actions fixing a disjoint union of odd dimensional projective spaceshttps://projecteuclid.org/euclid.bbms/1515035008<strong>Allan E. R. de Andrade</strong>, <strong>Pedro L.Q. Pergher</strong>, <strong>Sérgio T. Urao</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 581--590.</p><p><strong>Abstract:</strong><br/>
Consider the real, complex and quaternionic $n$-dimensional projective spaces, $\mathbb{R}P^n$, $\mathbb{C}P^n$ and $\mathbb{H}P^n$; to unify notation, write $K_dP^n$ for the real ($d=1$), complex ($d=2$) and quaternionic ($d=4$) $n$-dimensional projective space. Consider a pair $(M,\Phi)$, where $M$ is a closed smooth manifold and $\Phi$ is a smooth action of the group $\mathbb{Z}_2^k$ on $M$; here, $\mathbb{Z}_2^k$ is considered as the group generated by $k$ commuting smooth involutions $T_1,T_2,...,T_k$. Write $F$ for the fixed-point set of $\Phi$. In this paper we prove the following two results: i) If $F$ is a disjoint union $F=\mathbb{R}P^{n_1} \sqcup \mathbb{R}P^{n_2} \sqcup ... \sqcup \mathbb{R}P^{n_j}$, where $j \ge 2$, each $n_i$ is odd and $n_i \not=n_t$ if $i \not= t$, then $(M,\Phi)$ bounds equivariantly. ii) If $F= K_dP^n \sqcup K_dP^m$, where $d=1,2$ and $4$ and $n$ and $m$ are odd, then $(M,\Phi)$ bounds equivariantly. These results are found in the literature for $k=1$.
</p>projecteuclid.org/euclid.bbms/1515035008_20180103220342Wed, 03 Jan 2018 22:03 ESTAn averaging formula for the coincidence Reidemeister tracehttps://projecteuclid.org/euclid.bbms/1515035009<strong>Jong Bum Lee</strong>, <strong>P. Christopher Staecker</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 591--612.</p><p><strong>Abstract:</strong><br/>
In the setting of continuous maps between compact orientable manifolds of the same dimension, there is a well known averaging formula for the coincidence Lefschetz number in terms of the Lefschetz numbers of lifts to some finite covering space. We state and prove an analogous averaging formula for the coincidence Reidemeister trace. This generalizes a recent formula in fixed point theory by Liu and Zhao. We give two separate and independent proofs of our main result: one using methods developed by Kim and the first author for averaging Nielsen numbers, and one using an axiomatic approach for the local Reidemeister trace. We also give some examples and state some open questions for the nonorientable case.
</p>projecteuclid.org/euclid.bbms/1515035009_20180103220342Wed, 03 Jan 2018 22:03 ESTThe Nielsen Borsuk-Ulam numberhttps://projecteuclid.org/euclid.bbms/1515035010<strong>Fabiana Santos Cotrim</strong>, <strong>Daniel Vendrúscolo</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 613--619.</p><p><strong>Abstract:</strong><br/>
A Nielsen-Borsuk-Ulam number ($NBU(f,\tau)$) is defined for continuous maps $f:X\to Y$ where $X$ and $Y$ are closed orientable triangulable $n$-mani\-folds and $X$ has a free involution $\tau$. This number is a lower bound, in the homotopy class of $f$, for the number of pairs of points in $X$ satisfying $f(x)=f\circ\tau(x)$. It is proved that $NBU(f,\tau)$ can be realized (Wecken type theorem) when $n\ge 3$.
</p>projecteuclid.org/euclid.bbms/1515035010_20180103220342Wed, 03 Jan 2018 22:03 ESTEquivariant maps between representation sphereshttps://projecteuclid.org/euclid.bbms/1515035011<strong>Zbigniew Błaszczyk</strong>, <strong>Wacław Marzantowicz</strong>, <strong>Mahender Singh</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 621--630.</p><p><strong>Abstract:</strong><br/>
Let $G$ be a compact Lie group. We prove that if $V$ and $W$ are orthogonal $G$-representations such that $V^G=W^G=\{0\}$, then a $G$-equivariant map $S(V) \to S(W)$ exists provided that $\dim V^H \leq \dim W^H$ for any closed subgroup $H\subseteq G$. This result is complemented by a reinterpretation in terms of divisibility of certain Euler classes when $G$ is a torus.
</p>projecteuclid.org/euclid.bbms/1515035011_20180103220342Wed, 03 Jan 2018 22:03 ESTCentral configurations, Morse and fixed point indiceshttps://projecteuclid.org/euclid.bbms/1515035012<strong>D.L. Ferrario</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 631--640.</p><p><strong>Abstract:</strong><br/>
We compute the fixed point index of non-degenerate central configurations for the $n$-body problem in the euclidean space of dimension $d$, relating it to the Morse index of the gravitational potential function $\bar U$ induced on the manifold of all maximal $O(d)$-orbits. In order to do so, we analyze the geometry of maximal orbit type manifolds, and compute Morse indices with respect to the mass-metric bilinear form on configuration spaces.
</p>projecteuclid.org/euclid.bbms/1515035012_20180103220342Wed, 03 Jan 2018 22:03 ESTFixed point sets of equivariant fiber-preserving mapshttps://projecteuclid.org/euclid.bbms/1515035013<strong>Rafael Souza</strong>, <strong>Peter Wong</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 641--655.</p><p><strong>Abstract:</strong><br/>
Given a selfmap $f:X\to X$ on a compact connected polyhedron $X$, H. Schirmer gave necessary and sufficient conditions for a nonempty closed subset $A$ to be the fixed point set of a map in the homotopy class of $f$. R. Brown and C. Soderlund extended Schirmer's result to the category of fiber bundles and fiber-preserving maps. The objective of this paper is to prove an equivariant analogue of Brown-Soderlund theorem result in the category of $G$-spaces and $G$-maps where $G$ is a finite group.
</p>projecteuclid.org/euclid.bbms/1515035013_20180103220342Wed, 03 Jan 2018 22:03 ESTMaps between Sol $3$-manifolds and coincidence Nielsen numbershttps://projecteuclid.org/euclid.bbms/1515035014<strong>Karen Regina Panzarin</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 657--664.</p><p><strong>Abstract:</strong><br/>
Let $M_A$ be the torus bundle over $S^1$ obtained using as gluing map an Anosov matrix $A$. In this paper we discuss maps from $M_{A^r}$ to $M_A$ and compute the coincidence Nielsen numbers for such maps, moreover we use that such manifolds are double covers of torus semi-bundles and compute the coincidence Nielsen number for selfmaps of Sol $3$-manifolds which are torus semi-bundles.
</p>projecteuclid.org/euclid.bbms/1515035014_20180103220342Wed, 03 Jan 2018 22:03 ESTA note on nontrivial intersection for selfmaps of complex Grassmann manifoldshttps://projecteuclid.org/euclid.bbms/1515035015<strong>Thaís F. M. Monis</strong>, <strong>Northon C. L. Penteado</strong>, <strong>Sérgio T. Ura</strong>, <strong>Peter Wong</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 665--672.</p><p><strong>Abstract:</strong><br/>
Let $G(k,n)$ be the complex Grassmann manifold of $k$-planes in $\mathbb C^{k+n}$. In this note, we show that for $1< k<n$ and for any selfmap $f:G(k,n)\to G(k,n)$, there exists a $k$-plane $V^k\in G(k,n)$ such that $f(V^k)\cap V^k\ne \{0\}$.
</p>projecteuclid.org/euclid.bbms/1515035015_20180103220342Wed, 03 Jan 2018 22:03 ESTFixed point index bounds for self-maps on closed surfaceshttps://projecteuclid.org/euclid.bbms/1515035016<strong>D.L. Gonçalves</strong>, <strong>M.R. Kelly</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 673--688.</p><p><strong>Abstract:</strong><br/>
Given a surface with non-positive Euler characteristic and non-empty boundary, and a map which has the least number of fixed points possible within its homotopy class there are known bounds (both upper and lower) regarding the fixed point indices of the map. This paper gives a new proof of this result. In addition, a relative version of the method is developed, which is then used to establish the same index bounds for the case of a closed surface of negative Euler characteristic.
</p>projecteuclid.org/euclid.bbms/1515035016_20180103220342Wed, 03 Jan 2018 22:03 ESTNielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifoldshttps://projecteuclid.org/euclid.bbms/1515035017<strong>Philip R. Heath</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 689--723.</p><p><strong>Abstract:</strong><br/>
In this paper we compute the Nielsen numbers $N(f^m)$ and the Nielsen type numbers $NP_m(f)$ and $N\Phi_m(f)$ {\it for all $m$}, for periodic maps $f$ on tori and nilmanifolds. For fixed $m$, there are known formulas for these numbers for arbitrary maps on tori and nilmanifolds. However when seeking to determine these numbers for all $m$ for periodic maps, fascinating patterns and shortcuts are revealed. Our method has two main thrusts. Firstly we study $N(f^m)$, $NP_m(f)$ and $N\Phi_m(f)$ on primitives (maps whose linearizations consist of primitive roots of unity), and then secondly we employ fibre techniques to give an inductive approach to the general case adding one primitive at a time. This approach is made possible by the eigen structure of the linearizations of the maps involved.
</p>projecteuclid.org/euclid.bbms/1515035017_20180103220342Wed, 03 Jan 2018 22:03 ESTCommon value pairs and their estimationshttps://projecteuclid.org/euclid.bbms/1515035018<strong>Ying Gu</strong>, <strong>Xuezhi Zhao</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 725--739.</p><p><strong>Abstract:</strong><br/>
We shall give a new treatment to intersection points of two maps, named common value pairs. Given two maps $f,g\colon X\to Y$. Instead of considering intersection points on target space $Y$, we focus on the pairs in the domains $X$, the pair $(u,v)$ with $f(u)=g(v)$. The set of all these pairs is exactly the preimage of product $f\times g$ at the diagonal in $Y^2$. We shall apply the idea of Nielsen root theory into such a general case: preimage of a set. Hence, some estimation for common value pairs and therefore for intersection points are obtained.
</p>projecteuclid.org/euclid.bbms/1515035018_20180103220342Wed, 03 Jan 2018 22:03 ESTThe equivalence of two methods: finding representatives of non--empty Nielsen classeshttps://projecteuclid.org/euclid.bbms/1515035019<strong>Evelyn L. Hart</strong>, <strong>Ha T. Vu</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 741--745.</p><p><strong>Abstract:</strong><br/>
Let $f:X\to X$ be a self--map with $X$ a wedge of circles or a compact surface with boundary, so that the fundamental group of $X$ is finitely generated and free. In [3], Wagner presents an algorithm for extracting information from the homomorphism induced by $f$ on the fundamental group. This information involves the fixed point index of $f$ and the Nielsen classes of fixed points of $f$. The step in which the representatives of Nielsen classes, Wagner tails, are calculated is equivalent to a step in the method presented by Fadell and Husseini in [1]. The Fadell--Husseini method was designed for closed two dimensional CW--complexes, but the step in which they use the Fox calculus, to determine terms in the unreduced Reidemeister trace, produces the Wagner tails and their indices. The equivalence of these steps was stated in [2] without proof. Further developments in this area have caused continued interest in the techniques, and a clarification of the equivalence is needed. Here we provide the proof and an example.
</p>projecteuclid.org/euclid.bbms/1515035019_20180103220342Wed, 03 Jan 2018 22:03 ESTPeriodic points on T-fiber bundles over the circlehttps://projecteuclid.org/euclid.bbms/1515035020<strong>Weslem Liberato Silva</strong>, <strong>Rafael Moreira de Souza</strong>. <p><strong>Source: </strong>Bulletin of the Belgian Mathematical Society - Simon Stevin, Volume 24, Number 4, 747--767.</p>projecteuclid.org/euclid.bbms/1515035020_20180103220342Wed, 03 Jan 2018 22:03 EST