Axioms for the coincidence index of maps between manifolds of the same dimension

We study the coincidence theory of maps between two manifolds of the same dimension from an axiomatic viewpoint. First we look at coincidences of maps between manifolds where one of the maps is orientation true, and give a set of axioms such that characterizes the local index (which is an integer valued function). Then we consider coincidence theory for arbitrary pairs of maps between two manifolds. Similarly we provide a set of axioms which characterize the local index, which in this case is a function with values in Z circle plus Z(2). We also show in each setting that the group of values for the index (either Z or Z circle plus Z(2)) is determined by the axioms. Finally, for the general case of coincidence theory for arbitrary pairs of maps between two manifolds we provide a set of axioms which characterize the local Reidemeister trace which is an element of an abelian group which depends on the pair of functions. These results extend known results for coincidences between orientable differentiable manifolds. (C) 2012 Elsevier B.V. All rights reserved.

ON THE RADICAL OF A FREE MALCEV ALGEBRA

We prove that the prime radical rad M of the free Malcev algebra M of rank more than two over a field of characteristic not equal 2 coincides with the set of all universally Engelian elements of M. Moreover, let T(M) be the ideal of M consisting of all stable identities of the split simple 7-dimensional Malcev algebra M over F. It is proved that rad M = J(M) boolean AND T(M), where J(M) is the Jacobian ideal of M. Similar results were proved by I. Shestakov and E. Zelmanov for free alternative and free Jordan algebras.

New Optimization Algorithms for Structural Reliability Analysis

Solution of structural reliability problems by the First Order method require optimization algorithms to find the smallest distance between a limit state function and the origin of standard Gaussian space. The Hassofer-Lind-Rackwitz-Fiessler (HLRF) algorithm, developed specifically for this purpose, has been shown to be efficient but not robust, as it fails to converge for a significant number of problems. On the other hand, recent developments in general (augmented Lagrangian) optimization techniques have not been tested in aplication to structural reliability problems. In the present article, three new optimization algorithms for structural reliability analysis are presented. One algorithm is based on the HLRF, but uses a new differentiable merit function with Wolfe conditions to select step length in linear search. It is shown in the article that, under certain assumptions, the proposed algorithm generates a sequence that converges to the local minimizer of the problem. Two new augmented Lagrangian methods are also presented, which use quadratic penalties to solve nonlinear problems with equality constraints. Performance and robustness of the new algorithms is compared to the classic augmented Lagrangian method, to HLRF and to the improved HLRF (iHLRF) algorithms, in the solution of 25 benchmark problems from the literature. The new proposed HLRF algorithm is shown to be more robust than HLRF or iHLRF, and as efficient as the iHLRF algorithm. The two augmented Lagrangian methods proposed herein are shown to be more robust and more efficient than the classical augmented Lagrangian method.

The r-matrix of the Alday-Arutyunov-Frolov model

We investigate the classical integrability of the Alday-Arutyunov-Frolov model, and show that the Lax connection can be reduced to a simpler 2 x 2 representation. Based on this result, we calculate the algebra between the L-operators and find that it has a highly non-ultralocal form. We then employ and make a suitable generalization of the regularization technique proposed by Mail let for a simpler class of non-ultralocal models, and find the corresponding r- and s-matrices. We also make a connection between the operator-regularization method proposed earlier for the quantum case, and the Mail let's symmetric limit regularization prescription used for non-ultralocal algebras in the classical theory.

Counting singularities via fitting ideals

The stable singularities of differential map germs constitute the main source of studying the geometric and topological behavior of these maps. In particular, one interesting problem is to find formulae which allow us to count the isolated stable singularities which appear in the discriminant of a stable deformation of a finitely determined map germ. Mond and Pellikaan showed how the Fitting ideals are related to such singularities and obtain a formula to count the number of ordinary triple points in map germs from C-2 to C-3, in terms of the Fitting ideals associated with the discriminant. In this article we consider map germs from (Cn+m, 0) to (C-m, 0), and obtain results to count the number of isolated singularities by means of the dimension of some associated algebras to the Fitting ideals. First in Corollary 4.5 we provide a way to compute the total sum of these singularities. In Proposition 4.9, for m = 3 we show how to compute the number of ordinary triple points. In Corollary 4.10 and with f of co-rank one, we show a way to compute the number of points formed by the intersection between a germ of a cuspidal edge and a germ of a plane. Furthermore, we show in some examples how to calculate the number of isolated singularities using these results.

delta-Superderivations of Semisimple Finite-Dimensional Jordan Superalgebras

In the paper, a complete description of the delta-derivations and the delta-superderivations of semisimple finite-dimensional Jordan superalgebras over an algebraically closed field of characteristic p not equal 2 is given. In particular, new examples of nontrivial (1/2)-derivations and odd (1/2)-superderivations are given that are not operators of right multiplication by an element of the superalgebra.

Partial projective representations and partial actions II

This paper is a continuation of Dokuchaev and Novikov (2010) [8]. The interaction between partial projective representations and twisted partial actions of groups considered in Dokuchaev and Novikov (2010) [8] is treated now in a categorical language. In the case of a finite group G, a structural result on the domains of factor sets of partial projective representations of G is obtained in terms of elementary partial actions. For arbitrary G we study the component pM'(G) of totally-defined factor sets in the partial Schur multiplier pM(G) using the structure of Exel's semigroup. A complete characterization of the elements of pM'(G) is obtained for algebraically closed fields. (C) 2011 Elsevier B.V. All rights reserved.

Conceptual problem for noncommutative inflation and the new approach for nonrelativistic inflationary equation of state

In a previous paper, we connected the phenomenological noncommutative inflation of Alexander, Brandenberger and Magueijo [ Phys. Rev. D 67 081301 (2003)] and Koh and Brandenberger [ J. Cosmol. Astropart Phys. 2007 21 ()] with the formal representation theory of groups and algebras and analyzed minimal conditions that the deformed dispersion relation should satisfy in order to lead to a successful inflation. In that paper, we showed that elementary tools of algebra allow a group-like procedure in which even Hopf algebras (roughly the symmetries of noncommutative spaces) could lead to the equation of state of inflationary radiation. Nevertheless, in this paper, we show that there exists a conceptual problem with the kind of representation that leads to the fundamental equations of the model. The problem comes from an incompatibility between one of the minimal conditions for successful inflation (the momentum of individual photons being bounded from above) and the Fock-space structure of the representation which leads to the fundamental inflationary equations of state. We show that the Fock structure, although mathematically allowed, would lead to problems with the overall consistency of physics, like leading to a problematic scattering theory, for example. We suggest replacing the Fock space by one of two possible structures that we propose. One of them relates to the general theory of Hopf algebras (here explained at an elementary level) while the other is based on a representation theorem of von Neumann algebras (a generalization of the Clebsch-Gordan coefficients), a proposal already suggested by us to take into account interactions in the inflationary equation of state.

Differentiability of bizonal positive definite kernels on complex spheres

We prove that any continuous function with domain {z ∈ C: |z| ≤ 1} that generates a bizonal positive definite kernel on the unit sphere in 'C POT.Q' , q ⩾ 3, is continuously differentiable in {z ∈ C: |z| < 1} up to order q − 2, with respect to both z and 'Z BARRA'. In particular, the partial derivatives of the function with respect to x = Re z and y = Im z exist and are continuous in {z ∈ C: |z| < 1} up to the same order.

Rigidity for piecewise smooth homeomorphisms on the circle

We find conditions for two piecewise 'C POT.2+V' homeomorphisms f and g of the circle to be 'C POT.1' conjugate. Besides the restrictions on the combinatorics of the maps (we assume that the maps have bounded combinatorics), and necessary conditions on the one-side derivatives of points where f and g are not differentiable, we also assume zero mean-nonlinearity for f and g.

A graded subring of an inverse limit of polynomial rings

We study the power series ring R= K[[x1,x2,x3,...]]on countably infinitely many variables, over a field K, and two particular K-subalgebras of it: the ring S, which is isomorphic to an inverse limit of the polynomial rings in finitely many variables over K, and the ring R', which is the largest graded subalgebra of R. Of particular interest are the homogeneous, finitely generated ideals in R', among them the generic ideals. The definition of S as an inverse limit yields a set of truncation homomorphisms from S to K[x1,...,xn] which restrict to R'. We have that the truncation of a generic I in R' is a generic ideal in K[x1,...,xn]. It is shown in Initial ideals of Truncated Homogeneous Ideals that the initial ideal of such an ideal converge to the initial ideal of the corresponding ideal in R'. This initial ideal need no longer be finitely generated, but it is always locally finitely generated: this is proved in Gröbner Bases in R'. We show in Reverse lexicographic initial ideals of generic ideals are finitely generated that the initial ideal of a generic ideal in R' is finitely generated. This contrast to the lexicographic term order. If I in R' is a homogeneous, locally finitely generated ideal, and if we write the Hilbert series of the truncated algebras K[x1,...,xn] module the truncation of I as qn(t)/(1-t)n, then we show in Generalized Hilbert Numerators that the qn's converge to a power series in t which we call the generalized Hilbert numerator of the algebra R'/I. In Gröbner bases for non-homogeneous ideals in R' we show that the calculations of Gröbner bases and initial ideals in R' can be done also for some non-homogeneous ideals, namely those which have an associated homogeneous ideal which is locally finitely generated. The fact that S is an inverse limit of polynomial rings, which are naturally endowed with the discrete topology, provides S with a topology which makes it into a complete Hausdorff topological ring. The ring R', with the subspace topology, is dense in R, and the latter ring is the Cauchy completion of the former. In Topological properties of R' we show that with respect to this topology, locally finitely generated ideals in R'are closed.

Rappresentazioni lineari di SL3(C)

Espongo i fatti di base della teoria delle rappresentazioni con lo scopo di indagare i possibili modi in cui un dato gruppo di Lie o algebra di Lie agisce su uno spazio vettoriale di dimensione finita. Tali risultati verranno applicati all'algebra di Lie del gruppo speciale lineare.

Expressiveness in biologically inspired languages

A very recent and exciting new area of research is the application of Concurrency Theory tools to formalize and analyze biological systems and one of the most promising approach comes from the process algebras (process calculi). A process calculus is a formal language that allows to describe concurrent systems and comes with well-established techniques for quantitative and qualitative analysis. Biological systems can be regarded as concurrent systems and therefore modeled by means of process calculi. In this thesis we focus on the process calculi approach to the modeling of biological systems and investigate, mostly from a theoretical point of view, several promising bio-inspired formalisms: Brane Calculi and k-calculus family. We provide several expressiveness results mostly by means of comparisons between calculi. We provide a lower bound to the computational power of the non Turing complete MDB Brane Calculi by showing an encoding of a simple P-System into MDB. We address the issue of local implementation within the k-calculus family: whether n-way rewrites can be simulated by binary interactions only. A solution introducing divergence is provided and we prove a deterministic solution preserving the termination property is not possible. We use the symmetric leader election problem to test synchronization capabilities within the k-calculus family. Several fragments of the original k-calculus are considered and we prove an impossibility result about encoding n-way synchronization into (n-1)-way synchronization. A similar impossibility result is obtained in a pure computer science context. We introduce CCSn, an extension of CCS with multiple input prefixes and show, using the dining philosophers problem, that there is no reasonable encoding of CCS(n+1) into CCSn.

Algebraische Beschreibung von Symmetrien: das Modell der Nichtkommutativen Multi-Tori

In der Nichtkommutativen Geometrie werden Räume und Strukturen durch Algebren beschrieben. Insbesondere werden hierbei klassische Symmetrien durch Hopf-Algebren und Quantengruppen ausgedrückt bzw. verallgemeinert. Wir zeigen in dieser Arbeit, daß der bekannte Quantendoppeltorus, der die Summe aus einem kommutativen und einem nichtkommutativen 2-Torus ist, nur den Spezialfall einer allgemeineren Konstruktion darstellt, die der Summe aus einem kommutativen und mehreren nichtkommutativen n-Tori eine Hopf-Algebren-Struktur zuordnet. Diese Konstruktion führt zur Definition der Nichtkommutativen Multi-Tori. Die Duale dieser Multi-Tori ist eine Kreuzproduktalgebra, die als Quantisierung von Gruppenorbits interpretiert werden kann. Für den Fall von Wurzeln der Eins erhält man wichtige Klassen von endlich-dimensionalen Kac-Algebren, insbesondere die 8-dim. Kac-Paljutkin-Algebra. Ebenfalls für Wurzeln der Eins kann man die Nichtkommutativen Multi-Tori als Hopf-Galois-Erweiterungen des kommutativen Torus interpretieren, wobei die Rolle der typischen Faser von einer endlich-dimensionalen Hopf-Algebra gespielt wird. Der Nichtkommutative 2-Torus besitzt bekanntlich eine u(1)xu(1)-Symmetrie. Wir zeigen, daß er eine größere Quantengruppen-Symmetrie besitzt, die allerdings nicht auf die Spektralen Tripel des Nichtkommutativen Torus fortgesetzt werden kann.

Thermal relaxation for particle systems in interaction with several bosonic heat reservoirs

The present thesis is concerned with the study of a quantum physical system composed of a small particle system (such as a spin chain) and several quantized massless boson fields (as photon gasses or phonon fields) at positive temperature. The setup serves as a simplified model for matter in interaction with thermal "radiation" from different sources. Hereby, questions concerning the dynamical and thermodynamic properties of particle-boson configurations far from thermal equilibrium are in the center of interest. We study a specific situation where the particle system is brought in contact with the boson systems (occasionally referred to as heat reservoirs) where the reservoirs are prepared close to thermal equilibrium states, each at a different temperature. We analyze the interacting time evolution of such an initial configuration and we show thermal relaxation of the system into a stationary state, i.e., we prove the existence of a time invariant state which is the unique limit state of the considered initial configurations evolving in time. As long as the reservoirs have been prepared at different temperatures, this stationary state features thermodynamic characteristics as stationary energy fluxes and a positive entropy production rate which distinguishes it from being a thermal equilibrium at any temperature. Therefore, we refer to it as non-equilibrium stationary state or simply NESS. The physical setup is phrased mathematically in the language of C*-algebras. The thesis gives an extended review of the application of operator algebraic theories to quantum statistical mechanics and introduces in detail the mathematical objects to describe matter in interaction with radiation. The C*-theory is adapted to the concrete setup. The algebraic description of the system is lifted into a Hilbert space framework. The appropriate Hilbert space representation is given by a bosonic Fock space over a suitable L2-space. The first part of the present work is concluded by the derivation of a spectral theory which connects the dynamical and thermodynamic features with spectral properties of a suitable generator, say K, of the time evolution in this Hilbert space setting. That way, the question about thermal relaxation becomes a spectral problem. The operator K is of Pauli-Fierz type. The spectral analysis of the generator K follows. This task is the core part of the work and it employs various kinds of functional analytic techniques. The operator K results from a perturbation of an operator L0 which describes the non-interacting particle-boson system. All spectral considerations are done in a perturbative regime, i.e., we assume that the strength of the coupling is sufficiently small. The extraction of dynamical features of the system from properties of K requires, in particular, the knowledge about the spectrum of K in the nearest vicinity of eigenvalues of the unperturbed operator L0. Since convergent Neumann series expansions only qualify to study the perturbed spectrum in the neighborhood of the unperturbed one on a scale of order of the coupling strength we need to apply a more refined tool, the Feshbach map. This technique allows the analysis of the spectrum on a smaller scale by transferring the analysis to a spectral subspace. The need of spectral information on arbitrary scales requires an iteration of the Feshbach map. This procedure leads to an operator-theoretic renormalization group. The reader is introduced to the Feshbach technique and the renormalization procedure based on it is discussed in full detail. Further, it is explained how the spectral information is extracted from the renormalization group flow. The present dissertation is an extension of two kinds of a recent research contribution by Jakšić and Pillet to a similar physical setup. Firstly, we consider the more delicate situation of bosonic heat reservoirs instead of fermionic ones, and secondly, the system can be studied uniformly for small reservoir temperatures. The adaption of the Feshbach map-based renormalization procedure by Bach, Chen, Fröhlich, and Sigal to concrete spectral problems in quantum statistical mechanics is a further novelty of this work.