We introduce and study the type I-, II-, and III-Λ-complete continuity property of Banach spaces, where Λ is a subset of the dual group of a compact metrizable abelian group G.

Let T be a dominant operator that is a quasi-affine transform of an M-hyponormal operator. In this paper we show that if f is a function analytic on a neighborhood of the spectrum of T, then Weyl's theorem holds for f(T{\hskip1}).

The Kadets path distance between Banach spaces X and Y is defined to be the infimum of the lengths with respect to the Kadets distance of all curves joining X and Y. If there is no curve joining X and Y, the Kadets path distance between X and Y is defined to be infty .

Some approaches to estimates of the Kadets path distance from above and from below are developed. In particular, the Kadets path distances between the spaces l_p^n,\ p\in [1,+\rm (inf)ty ], n\in {\b (N)} are estimated.

We prove that the module categories of Noether algebras (i.e., algebras module finite over a noetherian center) and affine noetherian PI algebras over a field enjoy the following product property: whenever a direct product \prod _(n \in ℕ) M_n of finitely generated indecomposable modules M_n is a direct sum of finitely generated objects, there are repeats among the isomorphism types of the M_n. The rings with this property satisfy the pure semisimplicity conjecture which stipulates that vanishing one-sided pure global dimension entails finite representation type.

It is shown that an immersion of n dimensional compact oriented manifold without boundary into the n+1 dimensional Euclidean space, hyperbolic space or open half sphere is a totally umbilic immersion if one of the mean curvature function H_l does not vanish and the ratio H_k/H_l is constant, 1\leq k, l \leq n, k\ne l.

Every compact well-bounded operator has a representation as a linear combination of disjoint projections reminiscent of the representation of compact self-adjoint operators. In this note we show that the converse of this result holds, thus characterizing compact well-bounded operators. We also apply this result to study compact well-bounded operators on some special classes of Banach spaces such as hereditarily indecomposable spaces and certain spaces constructed by G. Pisier.

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.

It is shown that the truncated multidimensional moment problem is more general than the full multidimensional moment 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.

Methods of calculating discriminant and image Milnor numbers in terms of the Milnor numbers of multiple point spaces are described for cases f:{ℂ}^3 \to {ℂ}^4 and F:{ℂ}^3 \to {ℂ}^3.

In this paper we prove that if M_C=\pmatrix {A&#TAB;C\cr0&#TAB;B} is a 2\times 2 upper triangular operator matrix on the Hilbert space H\bigoplus K and if \sigma (A)\cap \sigma (B)=\emptyset , then \sigma is continuous at A and B if and only if \sigma is continuous at M_C, for every C\in B(K,H{\hskip1}).

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.

Since its elaboration in 1983 by Weiss, Tabor and Carnevale, the method to explicitly build the Bäcklund transformation of a partial differential equation (PDE) from singularity analysis only has been improved in several complementary directions, and at the present time it succeeds for practically all PDEs in 1+1-dimensions. The current state of the art is presented, and the emphasis is put on understanding the method. There are two important stages: first, the definition (identified with a Darboux transformation) of a resummation variable to make the Laurent series a finite one as requested by the definition of the word integrability; second, the link (identified with a linearizing formula to be taken from the classification of Painlevé and Gambier) between this resummation variable and the Lax pair to be found.

Two coupled bilinear equations are considered, and then two new coupled differential-difference systems are found. Also two special reductions of these two systems are studied. By using Hirota's method, Bäcklund transformation and superposition formulae, soliton solutions to these equations are presented.

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

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.

Soliton equations whose solutions are expressed by Pfaffians are briefly discussed. Included are a discrete-time Toda equation of BKP type, a modified Toda equation of BKP type, a coupled modified KdV equation and a coupled modified KdV equation of derivative type.

In a recent paper we presented a truncation-type method of deriving Bäcklund transformations for ordinary differential equations. This method is based on a consideration of truncation as a mapping that preserves the locations of a natural subset of the movable poles that the equation possesses. Here we apply this approach to the third and fifth Painlevé equations. For the third Painlevé equation we are able to obtain all fundamental Bäcklund transformations for the case where the parameters satisfy \gamma \delta \neq 0. For the fifth Painlevé equation our approach yields what appears to be all known Bäcklund transformations.

It is shown that the Hirota derivative can be used to construct the plethysm for tensor products of representations of {sl}_2(k).

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).