Closed Simplicial Model Structures for Exterior and Proper Homotopy Theory

The notion of exterior space consists of a topological space together with a certain nonempty family of open subsets that is thought of as a system of open neighbourhoods at infinity while an exterior map is a continuous map which is continuous at infinity. The category of spaces and proper maps is a subcategory of the category of exterior spaces.In this paper we show that the category of exterior spaces has a family of closed simplicial model structures, in the sense of Quillen, depending on a pair {T,T} of suitable exterior spaces. For this goal, for a given exterior space T, we construct the exterior T-homotopy groups of an exterior space under T. Using different spaces T we have as particular cases the main proper homotopy groups: the Brown–Grossman, erin–Steenrod, p-cylindrical, Baues–Quintero and Farrell–Taylor–Wagoner groups, as well as the standard (Hurewicz) homotopy groups.The existence of this model structure in the category of exterior spaces has interesting applications. For instance, using different pairs {T,T}, it is possible to study the standard homotopy type, the homotopy type at infinity and the global proper homotopy type.     

A generalization of the relative conditional expectatio—Further properties of Pitman'sT * and their applications to statistics

Summary This paper is concerned with the mappingT ^* which is a generalization of the relative conditional expectation. It has been introduced by E.J.G. Pitman (1979,Some Basic Theory for Statistical Inference, Chapman and Hall). First we extend the definition of the mappingT ^* and describe its fundamental properties. Moreover, we establish inequalities for convex functions with respect toT ^*. The mappingT ^* is very useful in analysing quantities associated with the distribution of a statisticT. The application of the mappingT ^* to statistics is another interest of this paper. The Institute of Statistical Mathematics     

Classification of disordered tilings

A tilingT is a disordered realization of a periodic tilingP with symmetry group Γ if we can map the complement of a compact set ofT onto the quotientP/Γ in such a way that this map respects the features of the tilingT andP. We show that the global type of a 2-dimensional tilingT is determined by the periodic tilingP it is a disordered realization of, a conjugacy class of Γ which can be associated toT and a winding number. In some cases, we need in addition some kind of orientation. For higher-dimensional tilings of spaces which are simply connected at infinity, e.g. ℝ ⁿ withn≥3, the associated periodic tiling alone is sufficient.     

Local and Global LIPSCHITZian Mappings on Ordered Metric Spaces

We consider the set ?? of nonhomogeneous Markov fields on T = N or T = Z with finite state spaces E_n, n ? T , with fixed local characteristics. For T = N we show that ?? has at most \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_\infty = \mathop {\lim \inf}\limits_{n \to \infty} \left| {\mathop E\nolimits_n} \right| $\end{document} phases. If T = Z , ?? has at most N_-∞ · N_∞; phases, where \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop N\nolimits_{-\infty} = \mathop {\lim \inf}\limits_{n \to -\infty} \left| {\mathop E\nolimits_n} \right| $\end{document}. We give examples, that for T = N for any number k, 1 ≦ k ≦ N_∞, there are local characteristics with k phases, whereas for T = Z every number l · k, 1 ≦ l ≦ N_-∞, 1 ≦ k ≦ N_∞ occurs. We describe the inner structure of ??, the behaviour at infinity and the connection between the one-sided and the two-sided tail-fields. Simple examples of Markov fields which are no Markov processes are given.     

Translations of Logical Formulas and the Equiconsistency Problem

A translation of formulas in a language L₁ to formulas in a language L₂ is a mapping which preserves the parameters and commutes with the substitution prefix, the propositional connectives and the quantifiers. Every translation generates a corresponding transformation of theories in L₁ to theories in L₂. We formulate the equiconsistency problem for such transformations and propose a variant of its solution. First, for a transformation F we find the least theory A(F) in L₁ such that its inclusion in a theory T (in L₁) guarantees equiconsistency of F and F(T), then we propose axiomatizations of A(F) for some F's. Mathematics Subject Classification: 03B10, 03B15, 03F25.     

Standard basis in the <Emphasis Type="Italic">T</Emphasis>-ideal formed by polynomial identities of triangular matrices

We give the definition of a standard basis of a T-ideal of the free associative algebra over a field of zero characteristic and indicate some basis called canonical in the linear space of n-linear forms. Using this basis, we construct a standard basis in the T-ideal of identities satisfied by the algebra of upper triangular (n × n)-matrices.     

Lagrangian submanifold landscapes of necessary conditions for maxima in optimal control: Global parameterizations and generalized solutions

We construct global generating functions of the initial and of the evolution Lagrangian submanifolds related to a Hamiltonian flow. These global parameterizations are realized by means of Amann—Conley—Zehnder reduction. In some cases, we have to to face generating functions that are weakly quadratic at infinity; more precisely, degeneracy points can occurs. Therefore, we develop a theory which allows us to treat possibly degenerate cases in order to define a Chaperon—Sikorav—Viterbo weak solution of a time-dependent Hamilton-Jacobi equation with a Cauchy condition given at time t = T (T > 0). The starting motivation is to study some aspects of Mayer problems in optimal control theory. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 21, Geometric Problems in Control Theory, 2004. An erratum to this article is available at .     

Multiple blowup of solutions for a semilinear heat equation

The present paper is concerned with a Cauchy problem for a semilinear heat equation with u₀L(R^N). A solution u of (P) is said to blow up at t=T<+ if lim sup_tT|u(t)|=+ with the supremum norm |·| in R^N. We show that if and N11, then there exists a proper solution u of (P) which blows up at t=T₁, becomes a regular solution for t(T₁,T₂) and blows up again at t=T₂ for some T₁,T₂ with 0<T₁<T₂<+.Mathematics Subject Classification (2000): 35K20, 35K55, 58K57Revised version: 20 July 2004Acknowledgment The author expresses her gratitude to Professor Marek Fila for useful discussion.     

Ramsey Numbers for Trees of Small Maximum Degree

For a tree T we write and , , for the sizes of the vertex classes of T as a bipartite graph. It is shown that for T with maximum degree , the obvious lower bound for the Ramsey number R(T,T) of is asymptotically the correct value for R(T,T). Received December 15, 1999 RID=" " ID=" " The first and third authors were partially supported by NSERC. The second author was partially supported by KBN grant 2 P03A 021 17.     

Behavior of Distant Maximal Geodesics in Finitely Connected Complete Two-Dimensional Riemannian Manifolds II

We study the behavior of maximal geodesics in a finitely connected complete two-dimensional Riemannian manifold M admitting curvature at infinity. In the case where M is homeomorphic to ² the Cohn–Vossen theorem states that the total curvature of M, say c(M), is 2. We already studied the case c(M)<2 in our previous paper. So we study the behavior of geodesics in M with total curvature 2 in this paper. Next we consider the case where M has nonempty boundary. In order to know the behavior of distant geodesics in M with boundary, it is useful to investigate the 'visual image' of the boundary of M. The latter half of this paper will be spent to study the asymptotic behavior of the visual image of a subset of M with located point tending to infinity.     

Structure Theorem for (d, g, h)-Maps

The (3x + 1)-Map, T, acts on the set, Π, of positive integers not divisible by 2 or 3. It is defined by , where k is the largest integer for which T (x) is an integer. The (3x + 1)-Conjecture asks if for every x ε Π there exists an integer, n, such that T ⁿ (x) = 1. The Statistical (3x + 1)-Conjecture asks the same question, except for a subset of Π of density 1. The Structure Theorem proven in [𝕊] shows that infinity is in a sense a repelling point, giving some reasons to expect that the (3x + 1)-Conjecture may be true. In this paper, we present the analogous theorem for some generalizations of the (3x + 1)-Map, and expand on the consequences derived in [𝕊]. The generalizations we consider are determined by positive coprime integers, d and g, with g > d ≥ 2, and a periodic function, h (x). The map T is defined by the formula , where k is again the largest integer for which T (x) is an integer. We prove an analogous Structure Theorem for (d, g, h)-Maps, and that the probability distribution corresponding to the density converges to the Wiener measure with the drift and positive diffusion constant. This shows that it is natural to expect that typical trajectories return to the origin if and escape to infinity otherwise. Received: 18 April 2002     

On the topology of a generic fibre of a polynomial function

In this work we study the topologies of the fibres of some families of complex polynomial functions with isolated critical points. We consider polynomials with some transversality conditions at infinity and compute explicitly its global Milnor number μ(f). the invariant λ(f) and therefore the Euler characteristic of its generic fibre. We show that under some mild ransversality condition (transversal at infinity) the behavior of f at infinity is good and the topology of the generic fibre is determined by the two homogeneous parts of higher degree of f Finally we study families of polynomials, called two-term polynomials. This polynomials may have atypical values at infinity. Given such a two-term polynomial f we characterize its atypical values by some invariants of f. These polynomials are a source of interesting examples.     

Multiplicity Results for Asymptotically Linear Equations, Using the Rotation Number Approach

By using a topological approach and the relation between rotation numbers and weighted eigenvalues, we give some multiplicity results for the boundary value problem u′′ + f(t, u) = 0, u(0) = u(T) = 0, under suitable assumptions on f(t, x)/x at zero and infinity. Solutions are characterized by their nodal properties. Supported by MIUR, GNAMPA and FCT.     

The algebraic structure of the isomorphic types of tally,polynomial time computable sets

 We investigate the polynomial time isomorphic type structure of (the class of tally, polynomial time computable sets). We partition P _T into six parts: D ⁻, D^ ⁻ , C, S, F, F^, and study their p-isomorphic properties separately. The structures of , , and are obvious, where F, F^, and C are the class of tally finite sets, the class of tally co-finite sets, and the class of tally bi-dense sets respectively. The following results for the structures of and will be proved, where D^ is the class of tally, co-dense, polynomial time computable sets and S is the class of tally, scatted (i.e., neither dense nor co-dense), polynomial time computable sets. 1. is a countable distributive lattice with the greatest element. 2. Infinitely many intervals in can be distinguished by first order formulas. 3. There exist infinitely many nontrivial automorphisms for . 4. is not distributive, but any interval in is a countable distributive lattice. RID="ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID="ID=" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID="ID=" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies. Received: 23 July 1999 / Published online: 27 March 2002 RID=" ID=" <E5>Mathematics Subject Classification (2000):</E5> 03D15, 03D25, 03D30, 03D35, 06A06, 06B20 RID=" ID=" <E5>Key words or phrases:</E5> Computational complexity &ndash; Polynomial time &ndash; Degree structure &ndash; Lattice &ndash; Isomorphism RID=" ID=" Part of this work was done when the author was a PhD student at the University of Heidelberg under the direction of Professor Ambos-Spies.     

Leibniz algebras with pseudo-Riemannian bilinear forms

M. Bordemann has studied non-associative algebras with nondegenerate associative bilinear forms. In this paper, we focus on pseudo-Riemannian bilinear forms and study pseudo-Riemannian Leibniz algebras, i.e., Leibniz algebras with pseudo-Riemannian non-degenerate symmetric bilinear forms. We give the notion and some properties of T*-extensions of Leibniz algebras. In addition, we introduce the definition of equivalence and isometrical equivalence for two T*-extensions of a Leibniz algebra, and give a sufficient and necessary condition for the equivalence and isometrical equivalence.     

Nonlinear models of diffusion on a finite space

Summary The basic convergence theorems for finite state Markov chains are extended to the nonlinear case. An operatorT inl ₁ of a finite space with counting measure is called nonexpansive if Tf-Tg₁f-g₁ holds for allf, g. It is shown that, for anyf, there exists an integer p>=1 such thatT ^pnf converges. Sufficient conditions forp=1 are given. In the case of continuous parameter nonexpansive semigrous {T _t, t>=0},T _tf converges fort.The main tool is a geometric theorem on isometriesS of compact subsets of the abovel ₁: It is shown that any orbit underS is finite.The exponential speed of convergence does not extend from the Markov chain case to nonlinearT.This research has been done during a visit of M.A.A. to the University of Göttingen. The principal results were announced in C.R. Math. Rep. Acad. Sci. Canada Vol. VIII, 1. Feb. 1986The research of this author is supported in part by an N.S.E.R.C.-Grant     

Quantum Dynamics of a Resonator Coupled to a SET

We study the quantum dynamics of a resonator driven by a superconducting single-electron transistor. We prove the existence of the Minimal Quantum Dynamical Semigroup T ^(min){\mathcal {T}^{\,({\rm min})}} solving the evolution equation for the density matrix, then we prove that T ^(min){\mathcal {T}^{\,({\rm min})}} is Markov. Moreover we prove the existence and uniqueness of a faithful normal stationary state and show that the dynamics converges towards this stationary state when time goes to infinity.     

On the number of points of bounded height¶on arithmetic projective spaces

Let K be a number field and its ring of integers. Let be a Hermitian vector bundle over . In the first part of this paper we estimate the number of points of bounded height in (generalizing a result by Schanuel). We give then some applications: we estimate the number of hyperplanes and hypersurfaces of degree d>1 in of bounded height and containing a fixed linear subvariety and we estimate the number of points of height, with respect to the anticanonical line bundle, less then T (when T goes to infinity) of ℙ ^N _K blown up at a linear subspace of codimension two. Received: 20 February 1998 / Revised version: 9 November 1998     

Spectral properties of cosine operator functions

LetA be the generator of a cosine functionC _t ,t R in a Banach spaceX; we shall connect the existence and uniqueness of aT-periodic mild solution of the equationu = Au + f with the spectral property 1 (C _T ) and, in caseX is a Hilbert space, also with spectral properties ofA. This research was supported in part by DAAD, West Germany.