Great time delay in a system with material cycling and delayed biomass growth

In this paper, we consider the effects of time delay on thelinear stability of the positive equilibrium of a reaction–diffusionmodel with time delayed nutrient recycling. We take the twodelay kernels as f(s) = (s), g(s) = p(0) + (1 – p)(r),s (0, +), r [0, +), 0 < p < 1, where is the Dirac deltafunction. When both N₁ and N₂ are space-independent, we canobserve that the increasing time delay destabilizes the positiveequilibrium, and stability-switches from stability to instabilityto stability occur. That is, for some special time delay kernels,the equilibrium is stable for small time delay, bifurcates towardsstable oscillations when the time delay is increased, then regainsstability for larger values. When both N₁ and N₂ are space-dependent,we are concerned with the effects of the delay kernels as wellas diffusion on the dynamics of the system.     

Groups which do not admit ghosts

A ghost in the stable module category of a group is a map between representations of that is invisible to Tate cohomology. We show that the only non-trivial finite -groups whose stable module categories have no non-trivial ghosts are the cyclic groups and . We compare this to the situation in the derived category of a commutative ring. We also determine for which groups the second power of the Jacobson radical of is stably isomorphic to a suspension of .

     

Homotopical localizations of module spectra

We prove that stable -localizations (where is any map of spectra) preserve ring spectrum structures and module spectrum structures, under suitable hypotheses, and we use this fact to describe all possible localizations of the integral Eilenberg-MacLane spectrum . As a consequence of this study, we infer that localizations of stable GEMs are stable GEMs, and it also follows that there is a proper class of nonequivalent stable localizations.

     

Stability of the Picard Bundle

Let X be a non-singular algebraic curve of genus g 2, n 2an integer, a line bundle over X of degree d > 2n(g –1) with (n,d) = 1 and M the moduli space of stable bundles ofrank n and determinant over X. It is proved that the Picardbundle W is stable with respect to the unique polarisation ofM. 2000 Mathematics Subject Classification 14H60, 14J60.     

Rational Gauss-Chebyshev quadrature formulas for complex poles outside

In this paper we provide an extension of the Chebyshev orthogonal rational functions with arbitrary real poles outside to arbitrary complex poles outside . The zeros of these orthogonal rational functions are not necessarily real anymore. By using the related para-orthogonal functions, however, we obtain an expression for the nodes and weights for rational Gauss-Chebyshev quadrature formulas integrating exactly in spaces of rational functions with arbitrary complex poles outside .

     

Local existence of -sets, projective tensor products, and Arens regularity for

Theorem. If are perfect compact subsets of the locally compact metrizable abelian group, then there are pairwise disjoint perfect subsets such that (i) is either a Kronecker set or (ii) for some , is a translate of a -set all of whose elements have order , and (iii) is isomorphic to the projective tensor product .

This extends what was previously known for groups such as or for the case to the general locally compact abelian group. Old results concerning the local existence of Kronecker and -sets are improved.

     

The Banach algebra induced by a double centralizer

Given a Banach algebra , R. Larsen defined, in his book ``An introduction to the theory of multipliers", a Banach algebra by means of a multiplier on , and essentially used it in the case of a commutative semisimple Banach algebra to prove a result on multiplications which preserve regular maximal ideals. Here, we consider the analogue Banach algebra induced by a bounded double centralizer of a Banach algebra . Then, our main concern is devoted to the relationships between , , and the algebras of bounded double centralizers and of and , respectively. By removing the assumption of semisimplicity, we generalize some results proven by Larsen.

     

Does negative type characterize the round sphere?

We discuss the measure-theoretic metric invariants extent, mean distance and symmetry ratio and their relation to the concept of negative type of a metric space. A conjecture stating that a compact Riemannian manifold with symmetry ratio must be a round sphere was put forward by the author in 2004. We resolve this conjecture in the class of Riemannian symmetric spaces by showing that a Riemannian manifold with symmetry ratio must be of negative type and that the only compact Riemannian symmetric spaces of negative type are the round spheres.

     

The modal logic of forcing

A set theoretical assertion is forceable or possible, written , if holds in some forcing extension, and necessary, written , if holds in all forcing extensions. In this forcing interpretation of modal logic, we establish that if is consistent, then the ZFC-provable principles of forcing are exactly those in the modal theory .

     

Higher-order Alexander invariants and filtrations of the knot concordance group

We establish certain ``nontriviality' results for several filtrations of the smooth and topological knot concordance groups. First, as regards the n-solvable filtration of the topological knot concordance group, , defined by K. Orr, P. Teichner and the first author:

we refine the recent nontriviality results of Cochran and Teichner by including information on the Alexander modules. These results also extend those of C. Livingston and the second author. We exhibit similar structure in the closely related symmetric Grope filtration of . We also show that the Grope filtration of the smooth concordance group is nontrivial using examples that cannot be distinguished by the Ozsváth-Szabó -invariant nor by J. Rasmussen's -invariant. Our broader contribution is to establish, in ``the relative case', the key homological results whose analogues Cochran-Orr-Teichner established in ``the absolute case'.

We say two knots and are concordant modulo -solvability if . Our main result is that, for any knot whose classical Alexander polynomial has degree greater than 2, and for any positive integer , there exist infinitely many knots that are concordant to modulo -solvability, but are all distinct modulo -solvability. Moreover, the and share the same classical Seifert matrix and Alexander module as well as sharing the same higher-order Alexander modules and Seifert presentations up to order .

     

Big Cohen-Macaulay algebras and seeds

In this article, we delve into the properties possessed by algebras, which we have termed seeds, that map to big Cohen-Macaulay algebras. We will show that over a complete local domain of positive characteristic any two big Cohen-Macaulay algebras map to a common big Cohen-Macaulay algebra. We will also strengthen Hochster and Huneke's ``weakly functorial" existence result for big Cohen-Macaulay algebras by showing that the seed property is stable under base change between complete local domains of positive characteristic. We also show that every seed over a positive characteristic ring maps to a balanced big Cohen-Macaulay -algebra that is an absolutely integrally closed, -adically separated, quasilocal domain.

     

Factorization in finite-codimensional ideals of group algebras

Let G be a -compact, locally compact group and I be a closed2-sided ideal with finite codimension in L¹(G). It is shownthat there are a closed left ideal L having a right boundedapproximate identity and a closed right ideal R having a leftbounded approximate identity such that I = L + R. The proofuses ideas from the theory of boundaries of random walks ongroups. 2000 Mathematics Subject Classification: primary 43A20;secondary 42A85, 43A07, 46H10, 46H40, 60B11.     

Planar finitely Suslinian compacta

We show that a planar unshielded compact set is finitely Suslinian if and only if there exists a closed set and a lamination of such that is homeomorphic to . If is a continuum, the analogous statement follows from Carathéodory theory and is widely used in polynomial dynamics.

     

Disklikeness of planar self-affine tiles

We consider the disklikeness of the planar self-affine tile generated by an integral expanding matrix and a consecutive collinear digit set . Let be the characteristic polynomial of . We show that the tile is disklike if and only if . Moreover, is a hexagonal tile for all the cases except when , in which case is a square tile. The proof depends on certain special devices to count the numbers of nodal points and neighbors of and a criterion of Bandt and Wang (2001) on disklikeness.

     

Existence of Solutions To Weak Parabolic Equations For Measures

Let A = (a^ij) be a Borel mapping on [0, 1] x R^d with valuesin the space of non-negative operators on R^d and let b = (bⁱ)be a Borel mapping on [0, 1] x R^d with values in R^d. Let Under broad assumptions on A and b, we construct a family µ= (µ_t)_t [0, 1] of probability measures µ_t on R^dwhich solvesthe Cauchy problem L* µ = 0 with initial conditionµ₀ = , where \nu is a probability measure on R^d, in thefollowing weak sense: and Such an equation is satisfied by transition probabilities ofa diffusion process associated with A and b provided such aprocess exists. However, we do not assume the existence of aprocess and allow quite singular coefficients, in particular,b may be locally unbounded or A may be degenerate. An infinite-dimensionalanalogue is discussed as well. Main methods are L^p-analysiswith respect to suitably chosen measures and reduction to theelliptic case (studied previously) by piecewise constant approximationsin time. 2000 Mathematics Subject Classification 35K10, 35K12,60J35, 60J60, 47D07.     

Prime Divisors of Shifted Factorials

For any positive integer n we let P(n) be the largest primefactor of n. We improve and generalize several results of P.Erds and C. Stewart on P(n!+1). In particular, we show thatlim sup_ninfin; P(n!+1)/n2.5, which improves their lower boundof lim sup_ninfin; P(n!+1)/n2. 2000 Mathematics Subject Classification11A05, 11A07, 11J86.     

Self-similar fragmentations derived from the stable tree I

The basic object we consider is a certain model of continuum random tree, called the stable tree. We construct a fragmentation process (F ^– (t),t0) out of this tree by removing the vertices located under height t. Thanks to a self-similarity property of the stable tree, we show that the fragmentation process is also self-similar. The semigroup and other features of the fragmentation are given explicitly. Asymptotic results are given, as well as a couple of related results on continuous-state branching processes. Research supported in part by NSF Grant DMS-0071448. Mathematics Subject Classification (2000):60J25, 60G52, 60J80     

A note on -estimates for stable integrals with drift

Let be of the form where is a symmetric stable process of index with . We obtain various -estimates for the process . In particular, for and any measurable, nonnegative function we derive the inequality

As an application of the obtained estimates, we prove the existence of solutions for the stochastic equation for any initial value .

     

Stable Maps between 4-Manifolds and Elimination of their Singularities

Let f:MN be a stable map between orientable 4-manifolds, whereM is closed and N is stably parallelisable. It is shown thatthe signature of M vanishes if and only if there exists a stablemap g:MN homotopic to f which has only fold and cusp singularities.This together with results of Ando and Èliaberg showsthat, in this situation, the Thom polynomials are the only obstructionsto the elimination of the singularities except for the foldsingularity. Also studied are some topological properties (includingthose of the discriminant set) of stable maps between 4-manifoldswith only A_k-type singularities.