Semidirect products of regular semigroups
Peter
R.
Jones;
Peter
G.
Trotter
4265-4310
Abstract: Within the usual semidirect product $S*T$ of regular semigroups $S$ and $T$ lies the set $\text {Reg}\,% (S*T)$ of its regular elements. Whenever $S$ or $T$ is completely simple, $\text {Reg}\,% (S*T)$ is a (regular) subsemigroup. It is this `product' that is the theme of the paper. It is best studied within the framework of existence (or e-) varieties of regular semigroups. Given two such classes, ${\mathbf U}%$ and ${\mathbf V}%$, the e-variety ${\mathbf U}*{\mathbf V}%$ generated by $\{\text {Reg}\,% (S*T) : S \in {\mathbf U}% , T \in {\mathbf V}% \}$ is well defined if and only if either ${\mathbf U} %$ or ${\mathbf V}%$ is contained within the e-variety ${\mathbf {CS}}%$ of completely simple semigroups. General properties of this product, together with decompositions of many important e-varieties, are obtained. For instance, as special cases of general results the e-variety $L{\mathbf I}%$ of locally inverse semigroups is decomposed as ${\mathbf I}% * {\mathbf {RZ}}%$, where ${\mathbf I}%$ is the variety of inverse semigroups and ${\mathbf {RZ}}%$ is that of right zero semigroups; and the e-variety ${\mathbf {ES}}%$ of $E$-solid semigroups is decomposed as ${\mathbf {CR}}*{\mathbf G}%$, where ${\mathbf {CR}}%$ is the variety of completely regular semigroups and ${\mathbf G}%$ is the variety of groups. In the second half of the paper, a general construction is given for the e-free semigroups (the analogues of free semigroups in this context) in a wide class of semidirect products ${\mathbf U}% * {\mathbf V}%$ of the above type, as a semidirect product of e-free semigroups from ${\mathbf U}%$ and ${\mathbf V}%$, ``cut down to regular generators''. Included as special cases are the e-free semigroups in almost all the known important e-varieties, together with a host of new instances. For example, the e-free locally inverse semigroups, $E$-solid semigroups, orthodox semigroups and inverse semigroups are included, as are the e-free semigroups in such sub-e-varieties as strict regular semigroups, $E$-solid semigroups for which the subgroups of its self-conjugate core lie in some given group variety, and certain important varieties of completely regular semigroups. Graphical techniques play an important role, both in obtaining decompositions and in refining the descriptions of the e-free semigroups in some e-varieties. Similar techniques are also applied to describe the e-free semigroups in a different `semidirect' product of e-varieties, recently introduced by Auinger and Polák. The two products are then compared.
Doi-Hopf modules, Yetter-Drinfel'd modules and Frobenius type properties
S.
Caenepeel;
G.
Militaru;
Shenglin
Zhu
4311-4342
Abstract: We study the following question: when is the right adjoint of the forgetful functor from the category of $(H,A,C)$-Doi-Hopf modules to the category of $A$-modules also a left adjoint? We can give some necessary and sufficient conditions; one of the equivalent conditions is that $C\otimes A$ and the smash product $A\# C^*$ are isomorphic as $(A, A\# C^*)$-bimodules. The isomorphism can be described using a generalized type of integral. Our results may be applied to some specific cases. In particular, we study the case $A=H$, and this leads to the notion of $k$-Frobenius $H$-module coalgebra. In the special case of Yetter-Drinfel'd modules over a field, the right adjoint is also a left adjoint of the forgetful functor if and only if $H$ is finite dimensional and unimodular.
Essential embedding of cyclic modules in projectives
José
L. Gómez
Pardo;
Pedro
A. Guil
Asensio
4343-4353
Abstract: Let $R$ be a ring and $E = E(R_R)$ its injective envelope. We show that if every simple right $R$-module embeds in $R_R$ and every cyclic submodule of $E_R$ is essentially embeddable in a projective module, then $R_R$ has finite essential socle. As a consequence, we prove that if each finitely generated right $R$-module is essentially embeddable in a projective module, then $R$ is a quasi-Frobenius ring. We also obtain several other applications and, among them: a) we answer affirmatively a question of Al-Huzali, Jain, and López-Permouth, by showing that a right CEP ring (i.e., a ring $R$ such that every cyclic right module is essentially embeddable in a projective module) is always right artinian; b) we prove that if $R$ is right FGF (i.e., any finitely generated right $R$-module embeds in a free module) and right CS, then $R$ is quasi-Frobenius.
$\beta$-expansions with deleted digits for Pisot numbers $\beta$
Steven
P.
Lalley
4355-4365
Abstract: An algorithm is given for computing the Hausdorff dimension of the set(s) $\Lambda =\Lambda (\beta ,D)$ of real numbers with representations $x=\sum _{n=1}^\infty d_n \beta ^{-n}$, where each $d_n \in D$, a finite set of ``digits'', and $\beta >0$ is a Pisot number. The Hausdorff dimension is shown to be $\log \lambda /\log \beta$, where $\lambda$ is the top eigenvalue of a finite 0-1 matrix $A$, and a simple algorithm for generating $A$ from the data $\beta ,D$ is given.
Some ramifications of a theorem of Boas and Pollard concerning the completion of a set of functions in $L^2$
K.
S.
Kazarian;
Robert
E.
Zink
4367-4383
Abstract: About fifty years ago, R. P. Boas and Harry Pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed, bounded, nonnegative measurable function. In subsequent years, several variations and extensions of this theorem have been given by a number of other investigators, and this program is continued here. A mildly surprising corollary of one of the results is that the trigonometric and Walsh systems can be multiplicatively transformed into quasibases for $L^{1}[0,1]$.
The transfer and symplectic cobordism
Malkhaz
Bakuradze
4385-4399
Abstract: The main result of this paper is the nilpotency fomula $\phi _{i}^{4} =0$, $\forall i\geq 1$ for N. Ray classes $\phi _{i}$ in the torsion of the symplectic bordism ring $MSp_{*}$
Order evaluation of products of subsets in finite groups and its applications. II
Z.
Arad;
M.
Muzychuk
4401-4414
Abstract: In this paper we give a new estimate of the cardinality of the product of subsets $AB$ in a finite non-abelian simple group, where $A$ is normal and $B$ is arbitrary. This estimate improves the one given in J. Algebra 182 (1996), 577-603.
Approximation by harmonic functions
Evgeny
A.
Poletsky
4415-4427
Abstract: For a compact set $X\subset \mathbb R^n$ we construct a restoring covering for the space $h(X)$ of real-valued functions on $X$ which can be uniformly approximated by harmonic functions. Functions from $h(X)$ restricted to an element $Y$ of this covering possess some analytic properties. In particular, every nonnegative function $f\in h(Y)$, equal to 0 on an open non-void set, is equal to 0 on $Y$. Moreover, when $n=2$, the algebra $H(Y)$ of complex-valued functions on $Y$ which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set $X\subset \mathbb C$ has a nontrivial Jensen measure, then $X$ contains a nontrivial compact set $Y$ with analytic algebra $H(Y)$.
On composite twisted unknots
Chaim
Goodman-Strauss
4429-4463
Abstract: Following Mathieu, Motegi and others, we consider the class of possible composite twisted unknots as well as pairs of composite knots related by twisting. At most one composite knot can arise from a particular $V$-twisting of an unknot; moreover a twisting of the unknot cannot be composite if we have applied more than a single full twist. A pair of composite knots can be related through at most one full twist for a particular $V$-twisting, or one summand was unaffected by the twist, or the knots were the right and left handed granny knots. Finally a conjectured characterization of all composite twisted unknots that do arise is given.
Only single twists on unknots can produce composite knots
Chuichiro
Hayashi;
Kimihiko
Motegi
4465-4479
Abstract: Let $K$ be a knot in the $3$-sphere $S^{3}$, and $D$ a disc in $S^{3}$ meeting $K$ transversely more than once in the interior. For non-triviality we assume that $\vert K \cap D \vert \ge 2$ over all isotopy of $K$. Let $K_{n}$($\subset S^{3}$) be a knot obtained from $K$ by cutting and $n$-twisting along the disc $D$ (or equivalently, performing $1/n$-Dehn surgery on $\partial D$). Then we prove the following: (1) If $K$ is a trivial knot and $K_{n}$ is a composite knot, then $\vert n \vert \le 1$; (2) if $K$ is a composite knot without locally knotted arc in $S^{3} - \partial D$ and $K_{n}$ is also a composite knot, then $\vert n \vert \le 2$. We exhibit some examples which demonstrate that both results are sharp. Independently Chaim Goodman-Strauss has obtained similar results in a quite different method.
Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string
Marianna
A.
Shubov
4481-4499
Abstract: We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter $h$. Corresponding to different values of $h$, the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of non-selfadjoint operators $A_h$ in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators $A_h$ are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator $A_1 (h=1)$ coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators $A_h$ and to establishing the exact and approximate controllability of the system governed by the damped wave equation.
Nonsymmetric systems on nonsmooth planar domains
G.
C.
Verchota;
A.
L.
Vogel
4501-4535
Abstract: We study boundary value problems, in the sense of Dahlberg, for second order constant coefficient strongly elliptic systems. In this class are systems without a variational formulation, viz. the nonsymmetric systems. Various similarities and differences between this subclass and the symmetrizable systems are examined in nonsmooth domains.
Compact groups and fixed point sets
Alex
Chigogidze;
Karl
H.
Hofmann;
John
R.
Martin
4537-4554
Abstract: Some structure theorems for compact abelian groups are derived and used to show that every closed subset of an infinite compact metrizable group is the fixed point set of an autohomeomorphism. It is also shown that any metrizable product containing a positive-dimensional compact group as a factor has the property that every closed subset is the fixed point set of an autohomeomorphism.
Coloring graphs with fixed genus and girth
John
Gimbel;
Carsten
Thomassen
4555-4564
Abstract: It is well known that the maximum chromatic number of a graph on the orientable surface $S_g$ is $\theta (g^{1/2})$. We prove that there are positive constants $c_1,c_2$ such that every triangle-free graph on $S_g$ has chromatic number less than $c_2(g/\log (g))^{1/3}$ and that some triangle-free graph on $S_g$ has chromatic number at least $c_1\frac {g^{1/3}}{\log (g)}$. We obtain similar results for graphs with restricted clique number or girth on $S_g$ or $N_k$. As an application, we prove that an $S_g$-polytope has chromatic number at most $O(g^{3/7})$. For specific surfaces we prove that every graph on the double torus and of girth at least six is 3-colorable and we characterize completely those triangle-free projective graphs that are not 3-colorable.
Virtually free groups with finitely many outer automorphisms
Martin
R.
Pettet
4565-4587
Abstract: Let $G$ be a finitely generated virtually free group. From a presentation of $G$ as the fundamental group of a finite graph of finite-by-cyclic groups, necessary and sufficient conditions are derived for the outer automorphism group of $G$ to be finite. Two versions of the characterization are given, both effectively verifiable from the graph of groups. The more purely group theoretical criterion is expressed in terms of the structure of the normalizers of the edge groups (Theorem 5.10); the other version involves certain finiteness conditions on the associated $G$-tree (Theorem 5.16). Coupled with an earlier result, this completes a description of the finitely generated groups whose full automorphism groups are virtually free.
Curves of maximum genus in range A and stick-figures
Edoardo
Ballico;
Giorgio
Bolondi;
Philippe
Ellia;
Rosa
Maria
Mirò-Roig
4589-4608
Abstract: In this paper we show the existence of smooth connected space curves not contained in a surface of degree less than $m$, with genus maximal with respect to the degree, in half of the so-called range A. The main tool is a technique of deformation of stick-figures due to G. Fløystad.
Existence of conservation laws and characterization of recursion operators for completely integrable systems
Joseph
Grifone;
Mohamad
Mehdi
4609-4633
Abstract: Using the Spencer-Goldschmidt version of the Cartan-Kähler theorem, we give conditions for (local) existence of conservation laws for analytical quasi-linear systems of two independent variables. This result is applied to characterize the recursion operator (in the sense of Magri) of completely integrable systems.
On the Kolyvagin cup product
Amnon
Besser
4635-4657
Abstract: We define a new cohomological operation, which we call the Kolyvagin cup product, that is a generalization of the derivative operator introduced by Kolyvagin in his work on Euler systems. We show some of the basic properties of this operation. We also define a higher dimensional derivative in certain cases and a dual operation which we call the Kolyvagin cap product and which generalizes a computation of Rubin.
Cohen-Macaulay Section Rings
Zhou
Caijun
4659-4667
Abstract: In this paper, we study the section rings of sheaves of Cohen-Macaulay algebras (over a field $F$) on a ranked poset. A necessary and sufficient condition for these rings to be Cohen-Macaulay will be given. This is a further generalization of a result of Yuzvinsky, which generalizes Reisner's theorem concerning Stanley-Reisner rings.
Contractions on a manifold polarized by an ample vector bundle
Marco
Andreatta;
Massimiliano
Mella
4669-4683
Abstract: A complex manifold $X$ of dimension $n$ together with an ample vector bundle $E$ on it will be called a generalized polarized variety. The adjoint bundle of the pair $(X,E)$ is the line bundle $K_X + det(E)$. We study the positivity (the nefness or ampleness) of the adjoint bundle in the case $r := rank (E) = (n-2)$. If $r\geq (n-1)$ this was previously done in a series of papers by Ye and Zhang, by Fujita, and by Andreatta, Ballico and Wisniewski. If $K_X+detE$ is nef then, by the Kawamata-Shokurov base point free theorem, it supports a contraction; i.e. a map $\pi :X \longrightarrow W$ from $X$ onto a normal projective variety $W$ with connected fiber and such that $K_X + det(E) = \pi^*H$, for some ample line bundle $H$ on $W$. We describe those contractions for which $dimF \leq (r-1)$. We extend this result to the case in which $X$ has log terminal singularities. In particular this gives Mukai's conjecture 1 for singular varieties. We consider also the case in which $dimF = r$ for every fiber and $\pi$ is birational.
Correction and extension of ``Measurable quotients of unipotent translations on homogeneous spaces''
Dave
Witte
4685-4688
Abstract: The statements of Main Theorem 1.1 and Theorem 2.1 of the author's paper [Trans. Amer. Math. Soc. 345 (1994), 577-594] should assume that $\Gamma$ is discrete and $G$ is connected. (Corollaries 1.3, 5.6, and 5.8 are affected similarly.) These restrictions can be removed if the conclusions of the results are weakened to allow for the possible existence of transitive, proper subgroups of $G$. In this form, the results can be extended to the setting where $G$ is a product of real and $p$-adic Lie groups.
Correction to ``Bifurcation of minimal surfaces in Riemannian manifolds''
Jürgen
Jost;
Xianqing
Li-Jost;
Xiao-Wei
Peng
4689-4690
Erratum to ``On a quadratic-trigonometric functional equation and some applications"
J.
K.
Chung;
B.
R.
Ebanks;
C.
T.
Ng;
P.
K.
Sahoo
4691 - 4691