Using the definition of regular p-group given by M. Hall [1], a new class of finite groups called regular-nilpotent has been defined. The action of these groups as automorphisms of compact Riemann surfaces has been investigated. It is proved that a necessary and sufficient condition for a Fuchsian group to cover a regular-nilpotent group is that its orbit genus be zero and its periods satisfy the least common multiple condition, first defined by Harvey [2] and Maclachlan [4].

Let $X$ be a closed, symplectic 4-manifold. Suppose that there is either a symplectic or an anti-symplectic involution $\sigma : X\,{\to}\, X$ with a 2-dimensional compact, oriented submanifold $\Sigma$ as a fixed point set.

If $\sigma$ is a symplectic involution then the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{\ge}\, 1$ is a symplectic 4-manifold.

If $\sigma$ is an anti-symplectic involution and $\Sigma$ has genus greater than 1 representing non-trivial homology class, we prove a vanishing theorem on Seiberg-Witten invariants of the quotient $X/\sigma$ with $b_2^+(X/\sigma)\,{ >}\,1.$

If $\Sigma$ is a torus with self-intersection number 0, we get a relation between the Seiberg-Witten invariants on $X$ and those of $X/\sigma$ with $b_2^+(X), b_2^+(X/\sigma)\,{ >}\,2$ which was obtained in [21] when the genus $g(\Sigma)\,{ >}\,1$ and $\Sigma\cdot\Sigma\,{=}\,0$.This work was supported by a Korea Research Foundation Grant (No KRF-2002-072-C00010).

We study Gevrey classes of holomorphic functions of several variables on a polysector, and their relation to classes of Gevrey strongly asymptotically developable functions. A new Borel-Ritt-Gevrey interpolation problem is formulated, and its solution is obtained by the construction of adequate linear continuous extension operators. Our results improve those given by Haraoka in this context, and extend to several variables the one-dimensional versions of the Borel-Ritt-Gevrey theorem given by Ramis and Thilliez, respectively. Some rigidity properties for the constructed operators are stated.

In this paper, we study Lagrangian submanifolds $M$ of the nearly Kähler 6-sphere $S^6(1)$. It is well known that such submanifolds, which are 3-dimensional, are always minimal and admit a symmetric cubic form. Following an idea of Bryant, developed in the study of Lagrangian submanifolds of $\mathbb C^3$, we then investigate those Lagrangian submanifolds for which at each point the tangent space admits an isometry preserving this cubic form. We obtain that all such Lagrangian submanifolds can be obtained starting from complex curves in $S^6(1)$ or from holomorphic curves in $\mathbb CP^2(4)$. In the final section we classify the Lagrangian submanifolds which admit a Sasakian structure that is compatible with the induced metric. This last result generalizes theorems obtained by Deshmukh and ElHadi. Note that in this case, the condition that $M$ admits a Sasakian structure implies that $M$ admits a pointwise isometry of the tangent space.

The classical conservation of number principle is an important result in algebraic geometry. We present a version of this principle suitable for the study of topological properties of real algebraic varieties. Our self-contained topological proof does not depend on the intersection theory of algebraic cycles. Some applications are included.

For an arbitrary $K$-algebra $R$, an $R$, $K$-bimodule $M$ is algebraically reflexive if the only $K$-endomorphisms of $M$ leaving invariant every $R$-submodule of $M$ are the scalar multiplications by elements of $R$. Hadwin has shown for an infinite field $K$ and $R = K[x]$ that $R$ is reflexive as an $R$, $K$-bimodule. This paper provides a generalisation by giving a skew polynomial version of his result.

Let B(X) denote the Banach algebra of all bounded linear operators on a Banach space X. We show that B(X) is finite if and only if no proper, complemented subspace of X is isomorphic to X, and we show that B(X) is properly infinite if and only if X contains a complemented subspace isomorphic to X[oplus ]X. We apply these characterizations to find Banach spaces X1, X2, and X3 such that B(X1) is finite, B(X2) is infinite, but not properly infinite, and B(X3) is properly infinite. Moreover, we prove that every unital, properly infinite ring has a continued bisection of the identity, and we give examples of Banach spaces D1 and D2 such that B(D1) and B(D2) are infinite without being properly infinite, B(D1) has a continued bisection of the identity, and B(D2) has no continued bisection of the identity. Finally, we exhibit a unital $C^\ast$-algebra which is finite and has a continued bisection of the identity.

Signed digit representations with base $q$ and digits $-\frac q2,\dots,\frac q2$ (and uniqueness being enforced by applying a special rule which decides whether $-q/2$ or $q/2$ should be taken) are considered with respect to counting the occurrences of a given (contiguous) subblock of length $r$. The average number of occurrences amongst the numbers $0,\dots,n-1$ turns out to be const$\cdot\log_qn+\delta(\log_qn)+\smallOh(1)$, with a constant and a periodic function of period one depending on the given subblock; they are explicitly described. Furthermore, we use probabilistic techniques to prove a central limit theorem for the number of occurrences of a given subblock.

Suppose that $c$ is a linear operator acting on an $n$-dimensional complex Hilbert Space $H$, and let $\tau$ denote the normalized trace on $B(H)$. Set $b_1 = (c+c^*)/2$ and $b_2 = (c-c^*)/2i$, and write $B$ for the spectral scale of $\{b_1, b_2\}$ with respect to $\tau$. We show that $B$ contains full information about $W_k(c)$, the $k$-numerical range of $c$ for each $k = 1,\dots,n$. This is in addition to the matrix pencil information that has been described in previous papers. Thus both types of information are contained in the geometry of a single 3-dimensional compact, convex set. We then use spectral scales to prove a new fact about $W_k(c)$. We show in Theorem 3.4 that the point $\lambda$ is a singular point on the boundary of $W_k(c)$ if and only if $\lambda$ is an isolated extreme point of $W_k(c)$: i.e. it is the end point of two line segments on the boundary of $W_k(c)$. In this case $\lambda = (n/k)\tau(cz)$, where $z$ is a central projection in the algebra generated by $c$ and the identity. In addition we show how the general theory of the spectral scale may be used to derive some other known properties of the $k$-numerical range.

We show that locally supersoluble finitary groups over certain division rings (e.g. fields) have local systems of hypercyclic normal subgroups.

We consider associative algebras $\Lambda$ over a field provided with a direct sum decomposition of a two-sided ideal $M$ and a sub-algebra $A$ – examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of $\Lambda$. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.

We study soluble groups in which every subgroup lying between a characteristic subgroup and its derived subgroup is normal.

Finite groups in which the Frattini subgroup of each proper normal subgroup is trivial, while the group itself has a nontrivial Frattini subgroup, are investigated. A direct result of this study leads to a classification of finite groups in which the Frattini subgroup of each proper subgroup is trivial, while the group itself has a nontrivial Frattini subgroup.

It is shown that each finite translation generalized quadrangle (TGQ) $\mathcal{S}$, which is the translation dual of the point-line dual of a flock generalized quadrangle, has a line $[\infty]$ each point of which is a translation point. This leads to the fact that the full group of automorphisms of $\mathcal{S}$ acts $2$-transitively on the points of $[\infty]$, and the observation applies to the point-line duals of the Kantor flock generalized quadrangles, the Roman generalized quadrangles and the recently discovered Penttila-Williams generalized quadrangle. Moreover, by previous work of the author, the non-classical generalized quadrangles (GQ's) which have two distinct translation points, are precisely the TGQ's of which the translation dual is the point-line dual of a non-classical flock GQ.

We emphasize that, for a long time, it has been thought that every non-classical TGQ which is the translation dual of the point-line dual of a flock GQ has only one translation point. There are important consequences for the theory of generalized ovoids (or eggs) in PG$(4n - 1,q)$, the study of span-symmetric generalized quadrangles, derivation of flocks of the quadratic cone in PG$(3,q)$, subtended ovoids in generalized quadrangles, and the understanding of automorphism groups of certain generalized quadrangles. Several problems on these topics will be solved completely.

In this note, we give a new proof of the fact that an affine semiprime algebra $R$ of Gelfand-Kirillov dimension 1 satisfies a polynomial identity. Our proof uses only the growth properties of the algebra and yields an explicit upper bound for the pi degree of $R$.

We show that every Hilbert C$^*$-module $E$ is a JB$^*$-triple in a canonical way, establish an explicit expression for the holomorphic automorphisms of the unit ball of $E$, discuss the existence of fixed points for these automorphisms and give sufficient conditions for $E$ to have the density property.

Recently, Ballester-Bolinches [1 and 2], Pedraza-Aguilera [2] and Perez-Ramos [2] have studied circumstances under which certain injectors and projectors, which are always pronormal, must be normally embedded. In this note we give a scheme for describing a minimal counterexample to a conjecture of the form: a subnormally embedded subgroup with properties $\alpha_1$, $\alpha_2,{\ldots\,},\alpha_{n}$ is normally embedded, where $\alpha_1$, $\alpha_2,{\ldots\,},\alpha_{n}$ satisfy certain conditions. We then show contradictions in certain cases involving finite solvable groups.

Given a contact form $\eta$, there is a one-to-one correspondence between the Riemannian structures $(\eta,g)$ and the CR-structures $(\eta,L)$. It is interesting to study the interaction between the two associated structures. We approach the geometry of contact Riemannian manifolds in connection with their associated CR-structures. In this context, for a contact Riemannian manifold $(M;\eta,g)$ we consider the Jacobi-type operator $R_{\dot\gamma}=R(\cdot,\dot\gamma)\dot\gamma$ along a self-parallel curve $\gamma$ with respect to the (generalized) Tanaka connection $\hatbnabla$.This work was financially supported by Chonnam National University in the program, 2001.

We prove that up to isomorphism $\langle a,b\,\vert\, ab=0\rangle$ is the unique principal Rees quotient of a free inverse semigroup that is not trivial or monogenic with zero, satisfying a nontrivial identity in signature with involution.

Extending results and methods of Thomson and Trent, we prove the existence of non trivial quasi-invariant subspaces for subnormal families of unbounded operators having sufficiently rich domains. In some special cases, proper invariant subspaces are obtained.