We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).

Let W denote the intersection with the pseudovariety of completely regular semigroups of the Mal'cev product of the pseudovariety of bands with a pseudovariety V of completely regular semigroups. It is shown that the (pseudo)word problem for W is reduced to that for V in such a way that decidability is preserved in the case in which terms involving only multiplication and weak inversion are considered. It is also shown that, if V is a hyperdecidable (respectively canonically reducible) pseudovariety of groups, then so is W.

Hirota representations of soliton equations have proved veryuseful. They produced many of the known families of multisoliton solutions, andhave often led to a disclosure of the underlying Lax systems and infinite sets ofconserved quantities.

A striking feature is the ease with which direct insight can be gained into thenature of the eigenvalue problem associated with soliton equations derivable from aquadratic Hirota equation (for a single Hirota function), such as the KdV equationor the Boussinesq equation. A key element is the bilinear Bäcklund transformation(BT) which can be obtained straight away from the Hirota representation of theseequations, through decoupling of a related “two field condition” by means of anappropriate constraint of minimal weight. Details of this procedure have beenreported elsewhere. The main point is that bilinear BT's are obtained systematically,without the need of tricky “exchange formulas”. They arise in the formof “Y-systems”, each equation of which belongs to a linear space spanned by a basisof binary Bell polynomials (Y-polynomials).

An integer n\geq2 is said to be a genus of a finite group G if there is a compact Riemann surface of genus n on which G acts as a group of automorphisms. In this paper, formulae are given for the minimum genus, minimum stable genus and the gap sequence, i.e., the (finite) set of non-genera, for a split metacyclic group of order pq, where p and q are primes. This information completely determines the genus spectrum for such groups.

For a nontrivial additive character \lambda and a multiplicative character \chi of the finite field with q elements (q a power of an odd prime), and for each positive integer r, the exponential sums \sum \lambda ((\tr w)^r) over w\in {SO}(2n+1,q) and \sum \chi (\det w)\lambda ((\tr w)^r) over {O}(2n+1,q) are considered. We show that both of them can be expressed as polynomials in q involving certain exponential sums. Also, from these expressions we derive the formulas for the number of elements w in {SO}(2n+1,q) and {O}(2n+1,q) with (\tr w)^r=\beta , for each \beta in the finite field with q elements.

In 1997, M. Cho, M. Ito and S. Oshiro showed that Weyl's theorem holds for p-hyponormal operators, for any p>0. In this note we give another proof of this result. Also, it is shown that Weyl's theorem holds for M-hyponormal operators. Further, in 1962, Stampfli showed that if T is hyponormal and its Weyl spectrum is {0} then T is compact and normal. We show that this result remains true if the hypothesis of hyponormality is replaced by either (a) p-hyponormality or (b) M-hyponormality.

In this paper the authors consider the class of locally nilpotent groups that have the maximum condition on non-nilpotent subgroups.

The N-soliton solution of a generalised Vakhnenko equation is found, where N is an arbitrary positive integer. The solution, which is obtained by using a blend of transformations of the independent variables and Hirota's method, is expressed in terms of a Moloney & Hodnett (1989) type decomposition. Different types of soliton are possible, namely loops, humps or cusps. Details of the different types of interactions between solitons, including resonant soliton interactions, are discussed in detail for the case N=2. A proof of the ‘N-soliton condition’ is given in the Appendix.

The radical of a module over a commutative ring is the intersection of all prime submodules. It is proved that if R is a commutative domain which is either Noetherian or a UFD then R is one-dimensional if and only if every (finitely generated) primary R-module has prime radical, and this holds precisely when every (finitely generated) R-module satisfies the radical formula for primary submodules.

The extension E of degree n over the Galois field F={\text GF}(q)} is called regular over F, if {\text ord}_r(q) and n have greatest common divisor 1 for all prime divisors r of n which are different from the characteristic p of F (here, \order{r}{q} denotes the multiplicative order of q modulo r). Under the assumption that E is regular over F and that q-1 is divisible by 4 if q is odd and n is even, we prove the existence of a primitive element w \in E which is also completely normal over F (the latter means that w simultaneously generates a normal basis for E over every intermediate field K of E/F). Our result achieves, for the class of extensions under consideration, a common generalization of the theorem of Lenstra and Schoof on the existence of primitive normal bases [12] and the theorem of Blessenohl and Johnsen on the existence of complete normal bases [1].

In 1953, Pjateckii˘-S˘apiro has proved that there are infinitely many primes of the form [n^c] for 1 [less than] c [less than] {12\over11} (with an asymptotic result). This range, which measures our progress in the technique of exponential sums, has been improved by many authors. In this paper we obtain 1 [less than] c\leq{243\over205}.

Conserved quantities of box and ball system(BBS) are presented from the hungry Toda molecule equation, an inverse ultra-discrete limit of the BBS.

By using the bilinear technique of soliton theory, a pfaffian version of the SU(2) self-dual Yang-Mills equation and its solution is constructed.

We give algebraic conditions characterizing chain-finite operators and locally chain-finite operators on Banach spaces. For instance, it is shown that T is a chain-finite operator if and only if some power of T is relatively regular and commutes with some generalized inverse; that is there are a bounded linear operator B and a positive integer k such that

TkBTk =Tk and TkB=BTk.

Moreover, we obtain an algebraic characterization of locally chain-finite operators similar to (1).

We obtain the complex orbifold structure of the moduli space for one parameter equisymmetric Riemann surfaces of genus two. For each family, by using the orbifold structure, we obtain the points in the moduli corresponding to real algebraic curves and a special form for the period matrices of Riemann surfaces that admit an anticonformal involution. We describe the topological type of anti-conformal involutions admitted by surfaces of the families depending on the type of period matrix.

Let G be a finite group. The symmetric crosscap number \tilde \sigma (G) is the minimum topological genus of any compact non-orientable surface (with empty boundary) on which G acts effectively. We first survey some of the basic facts about the symmetric crosscap number; this includes relationships between this parameter and others. We obtain formulas for the symmetric crosscap number for three families of groups, the dicyclic groups, the abelian groups with most factors in the canonical form isomorphic to Z_(2), and the hamiltonian groups with no odd order part. We also determine \tilde \sigma (G) for each group G with order less than 16. The groups with symmetric crosscap numbers 1 and 2 have been classified. We show here that there are no groups with \tilde\sigma=3; this affirms a conjecture of Tucker.

In this note we use the Hopf map to construct a family of metrics in the 3-sphere parametrized on the space of positive smooth functions in the 2-sphere. All these metrics make the Hopf map a Riemannian submersion. Also, the fibres are all geodesics if and only if the metric comes from a constant function and so, we have a Berger 3-sphere. Every geodesic in a 3-dimensional Riemannian manifold is a minimum for each elastic energy functional. Therefore, we characterize those functions on the 2-sphere that locally give metrics which have all the fibres being elastica, i.e., critical points of those functionals. Some applications are given including one to the Willmore-Chen variational problem.

In this paper, we show that if T=S+N, where S is similar to a hyponormal operator, S and N commute and N is a nilpotent operator of order m (i.e., N^m=0), then T is a subscalar operator of order 2m. As a corollary, we get that such a T has a nontrivial invariant subspace if its spectrum \sigma(T\hskip1) has the property that there exists some non-empty open set U such that \sigma(T\hskip1)\capU is dominating for U.

We give an example of an element r of a free group F, and an element s of minimal length in the normal closure of r in F, such that s is not conjugate to r±1 or to [r±1f], for any element f of F.

We present a general scheme to derive higher-order members of the Painlevé VI (PVI) hierarchy of ODE's as well as their difference analogues. The derivation is based on a discrete structure that sits on the background of the PVI equation and that consists of a system of partial difference equations on a multidimensional lattice. The connection with the isomonodromic Garnier systems is discussed.