Some notes on strongly E*-unitary inverse semigroups

In this paper, we give a direct proof that every strongly inverse semigroup can be embedded into a 0-semidirect product of a semilattice with zero by a group. As a corollary, we obtain a new proof of the structure theory of strongly inverse semigroups described in [1]. We also prove that the strongly inverse semigroups are precisely inverse semigroups equipped with a , idempotent pure prehomomorphism to a primitive inverse semigroup.     

Zwei Klassen lokalkompakter maximal fastperiodischer Gruppen

In this paper we study the class of all locally compact groupsG with the property that for each closed subgroupH ofG there exists a pair of homomorphisms into a compact group withH as coincidence set, and the class of all locally compact groupG with the property that finite dimensional unitary representations of subgroups ofG can be extended to finite dimensional representations ofG. It is shown that [MOORE]-groups (every irreducible unitary representation is finite dimensional) have these two properties. A solvable group in is a [MOORE]-group. Moreover, we prove a structure theorem for Lie groups in the class [MOORE], and show that compactly generated Lie groups in [MOORE] have faithful finite dimensional unitary representations.     

Temperature-Dependence of Equilibrium States of a Two-Phase Elastic Medium with the Zero Surface Tension Coefficient

We consider the energy functional of a two-phase elastic medium with quadratic energy densities defined for such that ,where is a measurable characteristic function. Under some natural conditions on the data of the problem, we prove the existence of an interval (t ^-,t ⁺) of the change of temperature such that the energy functional has only a minimizer such that for or such that t^ + $$ " align="middle" border="0"> . The energy functional has no minimizers such that or if . We derive two-sided estimates for the numbers in terms of the characteristics of the two-phase elastic medium and the boundary condition. Bibliography: 3 titles.     

A Theorem Concerning Nets Arising from Generalized Quadrangles with a Regular Point

Suppose is a generalized quadrangle (GQ) of order , with a regular point. Then there is a net which arises from this regular point. We prove that if such a net has a proper subnet with the same degree as the net, then it must be an affine plane of order t. Also, this affine plane induces a proper subquadrangle of order t containing the regular point, and we necessarily have that . This result has many applications, of which we give one example. Suppose is an elation generalized quadrangle (EGQ) of order , with elation point p. Then is called a skew translation generalized quadrangle (STGQ) with base-point p if there is a full group of symmetries about p of order t which is contained in the elation group. We show that a GQ of order s is an STGQ with base-point p if and only if p is an elation point which is regular.     

A Product Formula Related to Quantum Zeno Dynamics

We prove a product formula which involves the unitary group generated by a semibounded self-adjoint operator and an orthogonal projection P on a separable Hilbert space with the convergence in It gives a partial answer to the question about existence of the limit which describes quantum Zeno dynamics in the subspace Ran P. The convergence in is demonstrated in the case of a finite-dimensional P. The main result is illustrated in the example where the projection corresponds to a domain in and the unitary group is the free Schrödinger evolution.submitted 21/06/04, accepted 12/10/04     

Lines of curvature near umbilical points on surfaces immersed in R4

LetM be a compact orientable surface,I:M R⁴ an immersion of classC ^r and a normal unitary smooth vector field. We prove that in the space of pairs (I, ) the family of pairs all whose umbilical points are Darbouxian is generic.     

The number oft-wise balanced designs

We prove that the number oft-wise balanced designs of ordern is asymptotically , provided that blocks of sizet are permitted. In the process, we prove that the number oft-profiles (multisets of block sizes) is bounded below by and above by for constants c₂>c₁>0.     

Fibonacci-Cayley Numbers and Repetition Patterns in Genomic DNA

Let F(k) denote the k-th Fibonacci number in the Fibonacci sequence F(0) := 0, F(1) := 1,..., F(k+1) := F(k-1)+F(k). Motivated by proposals regarding putative mechanisms that may be responsible for producing those often observed long repetitive patterns in genomic DNA, we study in this note the Fibonacci-Cayley index fc_x of positive integers x, i.e., the largest integer for which positive integers a, b with x = aF(k-1)+bF(k) exist and show that holds for the arithmetic mean of the indices of the smallest and the largest Fibonacci numbers occurring in the Zeckendorf decomposition AMS Subject Classification: 11B39, 11A99, 92D20.     

A Remark on the Rank Conjecture

We prove a result about the action of -operations on the homology of linear groups. We use this to give a sharper formulation of the rank conjecture as well as some shorter proofs of various known results. We formulate a conjecture about how the sharper formulation of the rank conjecture together with another conjecture could give rise to a different point of view on the isomorphism between and K_n^{(p)} (F)$ for an infinite field F, and we prove part of this new conjecture.     

Filling Length in Finitely Presentable Groups

We study the filling length function for a finite presentation of a group , and interpret this function as an optimal bound on the length of the boundary loop as a van Kampen diagram is collapsed to the basepoint using a combinatorial notion of a null-homotopy. We prove that filling length is well behaved under change of presentation of . We look at 'AD-pairs' (f,g) for a finite presentation : that is, an isoperimetric function f and an isodiametric function g that can be realised simultaneously. We prove that the filling length admits a bound of the form [g+1][log (f+1)+1] whenever (f,g) is an AD-pair for . Further we show that (up to multiplicative constants) if is an isoperimetric function ( ) for a finite presentation then ( ) is an AD-pair. Also we prove that for all finite presentations filling length is bounded by an exponential of an isodiametric function.Partially supported by NSF grant DMS-9800158Supported by EPSRC Award No. 98001683 and Corpus Christi College, Oxford.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

On the \Gamma-limit of the Mumford-Shah functional

We study by means of -convergence the asymptotics of the rescaled Mumford-Shah functional

Quasisimilar Weak Contractions Have Isomorphic Lattices of Invariant Subspaces

A contraction T acting on a Hilbert space H is called a weak contraction if the spectrum of T does not cover the unit disk and the operator I-T ^* T is of trace class. Operators T₁:H₁ H₁ and T₂:H₂ H₂ are called quasisimilar if there exist operators >X:H₁ H₂ and Y:H₂ H₁ such that T₂X=XT₁, YT₂=T₁Y, and X and Y have zero kernels and dense ranges. It is proved that if two weak contractions T₁ and T₂ acting on separable spaces H₁ and H₂ are quasisimilar, then there exists an operator X:H₁ H₂ such that XT₁=T₂X and the mapping , where E=clos XE for E Lat T₁, is a lattice isomorphism. An example is given of two quasisimilar weak contractions such that for any isomorphism , its inverse is not equal to for a (bounded) operator Y. Bibliography: 4 titles.     

On causality in commutant lifting theorem. I

Two functionals (A) and for an operatorA were introduced in [11] for the study of causality in commutant lifting theory. In this paper we give sufficient and necessary conditions for in a special case. We prove that in this case , and we show by some examples related to nonlinear system control that is the best constant in our inequality.     

Note on a Theorem of Munkres

We prove that given a Riemannian manifold with boundary, having a finite number of compact boundary components, any fat triangulation of the boundary can be extended to the whole manifold. We also show that this result extends to manifolds and to embedded PL manifolds of dimensions 2, 3 and 4. We employ these results to prove that manifolds of the types above admit quasimeromorphic mappings onto As an application we prove the existence of G-automorphic quasimeromorphic mappings, where G is a Kleinian group acting on Dedicated to the memory of Robert BrooksThis paper represents part of the authors Ph.D. thesis written under the supervision of Prof. Uri Srebro.     

The homeomorphism type of unital-like submanifolds in smooth projective planes

We prove that a compact, connected submanifold of the point space of a smooth projective plane is homeomorphic to a sphere provided that certain intersection properties with lines are satisfied. As an application, we show that the set of absolute points of a smooth polarity in a smooth projective plane of dimension 2l is empty or homeomorphic to a sphere of dimension 2l - 1 or .Received: 13 September 2002     

Proof of the Ergodic Hypothesis for Typical Hard Ball Systems

We consider the system of hard balls with masses and radius r in the flat torus of size . We prove the ergodicity (actually, the Bernoulli mixing property) of such systems for almost every selection of the outer geometric parameters. This theorem complements my earlier result that proved the same, almost sure ergodicity for the case . The method of that proof was primarily dynamical-geometric, whereas the present approach is inherently algebraic. Communicated by Eduard ZehnderSubmitted 17/10/02, accepted 01/12/03     

Packing 4-Cycles in Eulerian and Bipartite Eulerian Tournaments with an Application to Distances in Interchange Graphs

We prove that every Eulerian orientation of K_m,n contains arc-disjoint directed 4-cycles, improving earlier lower bounds. Combined with a probabilistic argument, this result is used to prove that every regular tournament with n vertices contains arc-disjoint directed 4-cycles. The result is also used to provide an upper bound for the distance between two antipodal vertices in interchange graphs.Received February 6, 2004