In connection with algebras of unbounded operators, Lassner showed in [4] that, if T is a densely defined, closed linear operator in a Hilbert space such that its domain is contained in the domain of its adjoint T* and is globally invariant under T and T*,then T is bounded. In the case of a Banach space (in particular, a C*-algebra) weshowed in [6] that a densely defined closed derivation in a C*-algebra with domaincontaining its range is automatically bounded (see the references in [6] and [7] for thetheory of derivations in C*-algebras).

The definition of a suitable Jordan analogue of C*-algebras (which we call JB*-algebras in this paper) was recently suggested by Kaplansky (see (26)). The theory of unital JB*-algebras is now comparatively well understood due to the work of Alfsen, Shultz and Størmer (1) from which a Gelfand-Neumark theorem for unital JB*-algebras can be obtained (26). Independently, from work on simply connected symmetric complex Banach manifolds with base point, Kaup introduced the definition of C*-triple systems in (14) and subsequently in (7) it was shown that every unital JB*-algebra is a C*-triple system. In this paper, we wish to extend this result to show that every JB*-algebra is a C*-triple system.

Certain subsemigroups of the full transformation semigroup Tn on a finite set of cardinality n are investigated, namely those subsemigroups S of Tn, which are normalised by the symmetric group on n elements, the group of units of Tn. The Sn-normal closure of an element of Tn is determined, and the structure of the Sn-normal ideals consisting of the members of Tn whose image contains at most r elements is studied.

Let $v$ be a henselian valuation of a field $K$ with value group $G$, let $\bar{v}$ be the (unique) extension of $v$ to a fixed algebraic closure $\bar{K}$ of $K$ and let $(\tilde{K},\tilde{v})$ be a completion of $(K,v)$. For $\alpha\in\bar{K}\setminus K$, let $M(\alpha,K)$ denote the set $\{\bar{v}(\alpha-\beta):\beta\in\bar{K},\ [K(\beta):K] \lt [K(\alpha):K]\}$. It is known that $M(\alpha,K)$ has an upper bound in $\bar{G}$ if and only if $[K(\alpha):K]=[\tilde{K}(\alpha):\tilde{K}]$, and that the supremum of $M(\alpha,K)$, which is denoted by $\delta_{K}(\alpha)$ (usually referred to as the main invariant of $\alpha$), satisfies a principle similar to the Krasner principle. Moreover, each complete discrete rank 1 valued field $(K,v)$ has the property that $\delta_{K}(\alpha)\in M(\alpha,K)$ for every $\alpha\in\bar{K}\setminus K$. In this paper the authors give a characterization of all those henselian valued fields $(K,v)$ which have the property mentioned above.

AMS 2000 Mathematics subject classification: Primary 12J10; 12J25; 13A18

It is shown that any torsion unit of the integral group ring $\mathbb{Z}G$ of a finite group $G$ is rationally conjugate to an element of $\pm G$ if $G=XA$ with $A$ a cyclic normal subgroup of $G$ and $X$ an abelian group (thus confirming a conjecture of Zassenhaus for this particular class of groups, which comprises the class of metacyclic groups).

Semigroup presentations have been studied over a long period, usually as a means of providing examples of semigroups. In 1967 B. H. Neumann introduced an enumeration method for finitely presented semigroups analogous to the Todd–Coxeter coset enumeration process for groups. A proof of Neumann's enumeration method was given by Jura in 1978.

In Section 3 of this paper we describe a machine implementation of a semigroup enumeration algorithm based on that of Neumann. In Section 2 we examine certain semigroup presentations, motivated by the fact that the corresponding group presentation has yielded interesting groups. The theorems, although proved algebraically, were suggested by the semigroup enumeration program.

Through appropriate choices of elements in the underlying iterated function system, the methodology of fractal interpolation enables us to associate a family of continuous self-referential functions with a prescribed real-valued continuous function on a real compact interval. This procedure elicits what is referred to as an α-fractal operator on , the space of all real-valued continuous functions defined on a compact interval I. With an eye towards connecting fractal functions with other branches of mathematics, in this paper we continue to investigate the fractal operator in more general spaces such as the space of all bounded functions and the Lebesgue space , and in some standard spaces of smooth functions such as the space of k-times continuously differentiable functions, Hölder spaces and Sobolev spaces . Using properties of the α-fractal operator, the existence of Schauder bases consisting of self-referential functions for these function spaces is established.

In the Cayley-Klein projective metric it is ordinarily assumed that the measure of angles, plane and dihedral, is always elliptic, i.e. in a sheaf of planes or lines there is no actual plane or line which makes an infinite angle with the others. With this restriction there are only three kinds of geometry—Parabolic, Hyperbolic and Elliptic, i.e. the geometries of Euclid, Lobachevskij and Riemann ; and the form of the absolute is also limited. Thus in plane geometry the only degenerate form of the absolute which is possible is two coincident straight lines and a pair of imaginary points ; in three dimensions the absolute cannot be a ruled quadric, other than two coincident planes. If, however, this restriction as to angular measurement is removed, there are 9 systems of plane geometry and 27 in three dimensions; for the measure of distance, plane angle and dihedral angle may be parabolic, hyperbolic, or elliptic.

Let Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s < 2, . We obtain the existence of infinitely many solutions for the singular critical problem with Dirichlet boundary condition for suitable positive number λ.

The following rather surprising result is noted.

(1) A function f(z) = ∑anzn such that an ↓ 0 (n → ∞) belongs to H1 if and only if ∑(an/(n + 1)) < ∞.

A more subtle analysis is needed to prove that assertion (2) remains true if H1 is replaced by the predual, 1(⊂ H1), of the Bloch space. Assertion (1) extends the Hardy–Littlewood theorem, which says the following.

(2) f belongs to Hp (1 < p < ∞) if and only if ∑(n + 1)p−2anp < ∞.

A new proof of (2) is given and applications of (1) and (2) to the Libera transform of functions with positive coefficients are presented. The fact that the Libera operator does not map H1 to H1 is improved by proving that it does not map 1 into H1.

Train algebras were first introduced by Etherington in (1) and proved very useful in dealing with problems in mathematical genetics. The types of algebras which arose were commutative, non-associative and finite-dimensional. It proved convenient in the general theory to regard them as defined over the complex numbers. We remind the reader of some basic definitions. A baric algebra is one which admits a non-trivial homomorphism into its coefficient field K. A (principal) train algebra is baric and has a rank equation in which the coefficients of a general element x depend only on its baric value, generally called the weight of x. A special train algebra (STA) is a baric algebra in which the nilideal is nilpotent and all its right powers are ideals; the nilideal being the set of elements of A of weight zero. In (2) Etherington showed that in a baric algebra one can always take a very simple basis consisting of a distinguished element of unit weight and all other basis elements of weight zero.

An elementary proposition states that an absolutely convergent series is convergent, i.e. that if

then

In this note we show that two conjectures of George Weiss on admissible and weakly admissible observation operators fail to hold in general for the right shift semigroup in the case that the output space is infinite dimensional.

AMS 2000 Mathematics subject classification: Primary 47D06; 32A37; 93B28

In this paper we prove the existence of at least three classical solutions for the problem

$$ \left\{ \begin{aligned} \amp-(|u'|^{p-2}u')'=\lambda f(t,u)h(u'), \\ \ampu(a)=u(b)=0, \end{aligned} \right. $$

when $\lambda$ lies in an explicitly determined open interval.

Our main tool is a very recent three-critical-points theorem stated in a paper by D. Averna and G. Bonanno (Topolog. Meth. Nonlin. Analysis22 (2003), 93–103).

AMS 2000 Mathematics subject classification: Primary 34B15

Initial–boundary-value problems for the two-dimensional Zakharov–Kuznetsov equation posed on bounded rectangles and on a strip are considered. Spectral properties of a linearized operator and critical sizes of domains are studied. An exponential decay rate of regular solutions for the original nonlinear problems is proved.

We prove two-term supercongruences for generalizations of recently discovered sporadic sequences of Cooper. We also discuss recent progress and future directions concerning other types of supercongruences.

We compute the (generalized) Poincaré series of the multi-index filtration defined by a finite collection of divisorial valuations on the ring $\mathcal{O}_{\mathbb{C}^2,0}$ of germs of functions of two variables. We use the method initially elaborated by the authors and Campillo for computing the similar Poincaré series for the valuations defined by the irreducible components of a plane curve singularity. The method is essentially based on the notions of the so-called extended semigroup and of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions of two variables. The last notion is similar to (and inspired by) the notion of the motivic integration.

AMS 2000 Mathematics subject classification: Primary 14B05; 16W70

We study the interplay between the cohomology of the Koszul complex of the partial derivatives of a homogeneous polynomial f and the pole order filtration P on the cohomology of the open set U = ℙn \ D, with D the hypersurface defined by f = 0. The relation is expressed by some spectral sequences. These sequences may, on the one hand, in many cases be used to determine the filtration P for curves and surfaces and, on the other hand, to obtain information about the syzygies involving the partial derivatives of the polynomial f. The case of a nodal hypersurface D is treated in terms of the defects of linear systems of hypersurfaces of various degrees passing through the nodes of D. When D is a nodal surface in ℙ3, we show that F2H3(U) ≠ P2H3(U) as soon as the degree of D is at least 4.

It is shown that most properties of (bounded) completely hyperexpansive operators remain valid for unbounded $2$-hyperexpansive operators. Powers of closed $2$-hyperexpansive operators are proved to be closed and $2$-hyperexpansive. Various parts of spectra of such operators are calculated. $2$-hyperexpansive weighted shifts are investigated. Examples of unbounded closed $2$-hyperexpansive ($2$-isometric) operators with invariant dense domains are established.

AMS 2000 Mathematics subject classification: Primary 47A05; 47B37. Secondary 47B20; 47B10

The conditions under which the solution of the non-local thermistor problem

\begin{gather*} u_t=\Delta u+\frac{\lambda f(u)}{(\int_{\varOmega}f(u)\,\mathrm{d}x)^{2}},\quad x\in\varOmega\subset\mathbb{R}^N,\ N\geq2,\ t>0, \\ \frac{\partial u(x,t)}{\partial\nu}+\beta(x)u(x,t)=0,\quad x\in\partial\varOmega,\ t>0, \\ u(x,0)=u_0(x),\quad x\in\varOmega, \end{gather*}

blows up are investigated. We assume that $f(s)$ is a decreasing function and that it is integrable in $(0,\infty)$. Considering a suitable functional we prove that for all $\lambda\gt0$ the solution of the Neumann problem blows up in finite time. The same result is obtained for the Robin problem under the assumption that $\lambda$ is sufficiently large $(\lambda\gg 1)$. In the proof of existence of blow-up for the Dirichlet problem we use the subsolution technique. We are able to construct a blowing-up lower solution under the assumption that either $\lambda\gt\lambda^*$ or $0\lt\lambda\lt\lambda^*$, for some critical value $\lambda^*$, and that the initial condition is sufficiently large provided also that $f(s)$ satisfies the decay condition $\int_0^\infty[sf(s)-s^2f'(s)]\,\mathrm{d} s\lt\infty$.