Towards a maximal extension of Pelczynski`s decomposition method in Banach spaces

l Suppose that X, Y. A and B are Banach spaces such that X is isomorphic to Y E) A and Y is isomorphic to X circle plus B. Are X and Y necessarily isomorphic? In this generality. the answer is no, as proved by W.T. Cowers in 1996. In the present paper, we provide a very simple necessary and sufficient condition on the 10-tuples (k, l, m, n. p, q, r, s, u, v) in N with p+q+u >= 3, r+s+v >= 3, uv >= 1, (p,q)$(0,0), (r,s)not equal(0,0) and u=1 or v=1 or (p. q) = (1, 0) or (r, s) = (0, 1), which guarantees that X is isomorphic to Y whenever these Banach spaces satisfy X(u) similar to X(p)circle plus Y(q), Y(u) similar to X(r)circle plus Y(s), and A(k) circle plus B(l) similar to A(m) circle plus B(n). Namely, delta = +/- 1 or lozenge not equal 0, gcd(lozenge, delta (p + q - u)) divides p + q - u and gcd(lozenge, delta(r + s - v)) divides r + s - v, where 3 = k - I - in + n is the characteristic number of the 4-tuple (k, l, m, n) and lozenge = (p - u)(s - v) - rq is the discriminant of the 6-tuple (p, q, r, s, U, v). We conjecture that this result is in some sense a maximal extension of the classical Pelczynski`s decomposition method in Banach spaces: the case (1, 0. 1, 0, 2. 0, 0, 2. 1. 1). (C) 2009 Elsevier Inc. All rights reserved.

Crossed products by twisted partial actions and graded algebras

For a twisted partial action e of a group G on an (associative non-necessarily unital) algebra A over a commutative unital ring k, the crossed product A x(Theta) G is proved to be associative. Given a G-graded k-algebra B = circle plus(g is an element of G) B-g with the mild restriction of homogeneous non-degeneracy, a criteria is established for B to be isomorphic to the crossed product B-1 x(Theta) G for some twisted partial action of G on B-1. The equality BgBg-1 B-g = B-g (for all g is an element of G) is one of the ingredients of the criteria, and if it holds and, moreover, B has enough local units, then it is shown that B is stably isomorphic to a crossed product by a twisted partial action of G. (c) 2008 Elsevier Inc. All rights reserved.

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171-206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic gamma, we prove the existence of a locally constant integer valued map Lambda(gamma) on the unit circle with the property that the Morse index of the iterated gamma(N) is equal, up to a correction term epsilon(gamma) is an element of {0,1}, to the sum of the values of Lambda(gamma) at the N-th roots of unity. The discontinuities of Lambda(gamma) occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincare map of gamma. We discuss some applications of the theory.

Generalizations of Pelczynski`s decomposition method for Banach spaces containing a complemented copy of their squares

Suppose that X and Y are Banach spaces isomorphic to complemented subspaces of each other. In 1996, W. T. Gowers solved the Schroeder- Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. However, if X-2 is complemented in X with supplement A and Y-2 is complemented in Y with supplement B, that is, { X similar to X-2 circle plus A Y similar to Y-2 circle plus B, then the classical Pelczynski`s decomposition method for Banach spaces shows that X is isomorphic to Y whenever we can assume that A = B = {0}. But unfortunately, this is not always possible. In this paper, we show that it is possible to find all finite relations of isomorphism between A and B which guarantee that X is isomorphic to Y. In order to do this, we say that a quadruple (p, q, r, s) in N is a P-Quadruple for Banach spaces if X is isomorphic to Y whenever the supplements A and B satisfy A(p) circle plus B-q similar to A(r) circle plus B-s . Then we prove that (p, q, r, s) is a P-Quadruple for Banach spaces if and only if p - r = s - q = +/- 1.

The lower central and derived series of the braid groups of the projective plane

In this paper, we determine the lower central and derived series for the braid groups of the projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. The n-string braid groups B(n)(RP(2)) of the projective plane RP(2) were originally studied by Van Buskirk during the 1960s. and are of particular interest due to the fact that they have torsion. The group B(1)(RP(2)) (resp. B(2)(RP(2))) is isomorphic to the cyclic group Z(2) of order 2 (resp. the generalised quaternion group of order 16) and hence their lower central and derived series are known. If n > 2, we first prove that the lower central series of B(n)(RP(2)) is constant from the commutator subgroup onwards. We observe that Gamma(2)(B(3)(RP(2))) is isomorphic to (F(3) X Q(8)) X Z(3), where F(k) denotes the free group of rank k, and Q(8) denotes the quaternion group of order 8, and that Gamma(2)(B(4)(RP(2))) is an extension of an index 2 subgroup K of P(4)(RP(2)) by Z(2) circle plus Z(2). As for the derived series of B(n)(RP(2)), we show that for all n >= 5, it is constant from the derived subgroup onwards. The group B(n)(RP(2)) being finite and soluble for n <= 2, the critical cases are n = 3, 4. We are able to determine completely the derived series of B(3)(RP(2)). The subgroups (B(3)(RP(2)))((1)), (B(3)(RP(2)))((2)) and (B(3)(RP(2)))((3)) are isomorphic respectively to (F(3) x Q(8)) x Z(3), F(3) X Q(8) and F(9) X Z(2), and we compute the derived series quotients of these groups. From (B(3)(RP(2)))((4)) onwards, the derived series of B(3)(RP(2)), as well as its successive derived series quotients, coincide with those of F(9). We analyse the derived series of B(4)(RP(2)) and its quotients up to (B(4)(RP(2)))((4)), and we show that (B(4)(RP(2)))((4)) is a semi-direct product of F(129) by F(17). Finally, we give a presentation of Gamma(2)(B(n)(RP(2))). (C) 2011 Elsevier Inc. All rights reserved.

Periodic geodesics and geometry of compact Lorentzian manifolds with a Killing vector field

We study the geometry and the periodic geodesics of a compact Lorentzian manifold that has a Killing vector field which is timelike somewhere. Using a compactness argument for subgroups of the isometry group, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is nowhere vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics. We also discuss some properties of the topology of such manifolds. In particular, we show that a compact manifold M admits a Lorentzian metric with a nowhere vanishing Killing vector field which is timelike somewhere if and only if M admits a smooth circle action without fixed points.

Cantor-Bernstein Sextuples for Banach Spaces

Let X and Y be Banach spaces isomorphic to complemented subspaces of each other with supplements A and B. In 1996, W. T. Gowers solved the Schroeder-Bernstein (or Cantor-Bernstein) problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain a necessary and sufficient condition on the sextuples (p, q, r, s, u, v) in N with p + q >= 1, r + s >= 1 and u, v is an element of N*, to provide that X is isomorphic to Y, whenever these spaces satisfy the following decomposition scheme A(u) similar to X(P) circle plus Y(q) B(v) similar to X(r) circle plus Y(s). Namely, Phi = (p - u)(s - v) - (q + u)(r + v) is different from zero and Phi divides p + q and r + s. These sextuples are called Cantor-Bernstein sextuples for Banach spaces. The simplest case (1, 0, 0, 1, 1, 1) indicates the well-known Pelczynski`s decomposition method in Banach space. On the other hand, by interchanging some Banach spaces in the above decomposition scheme, refinements of the Schroeder-Bernstein problem become evident.

Nil graded self-similar algebras

In [19], [24] we introduced a family of self-similar nil Lie algebras L over fields of prime characteristic p > 0 whose properties resemble those of Grigorchuk and Gupta-Sidki groups. The Lie algebra L is generated by two derivations v(1) = partial derivative(1) + t(0)(p-1) (partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...))))), v(2) = partial derivative(2) + t(1)(p-1) (partial derivative(3) + t(2)(p-1) (partial derivative(4) + t(3)(p-1) (partial derivative(5) + t(4)(p-1) (partial derivative(6) + ...)))) of the truncated polynomial ring K[t(i), i is an element of N vertical bar t(j)(p) =0, i is an element of N] in countably many variables. The associative algebra A generated by v(1), v(2) is equipped with a natural Z circle plus Z-gradation. In this paper we show that for p, which is not representable as p = m(2) + m + 1, m is an element of Z, the algebra A is graded nil and can be represented as a sum of two locally nilpotent subalgebras. L. Bartholdi [3] andYa. S. Krylyuk [15] proved that for p = m(2) + m + 1 the algebra A is not graded nil. However, we show that the second family of self-similar Lie algebras introduced in [24] and their associative hulls are always Z(p)-graded, graded nil, and are sums of two locally nilpotent subalgebras.

Boutet de Monvel`s calculus and groupoids I

Can Boutet de Monvel`s algebra on a compact manifold with boundary be obtained as the algebra Psi(0)(G) of pseudodifferential operators on some Lie groupoid G? If it could, the kernel G of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra C*(G). While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal I isomorphic to G. In fact, we prove first that G similar or equal to Psi circle times K with the C*-algebra Psi generated by the zero order pseudodifferential operators on the boundary and the algebra K of compact operators. As both Psi circle times K and I are extensions of C(S*Y) circle times K by K (S*Y is the co-sphere bundle over the boundary) we infer from a theorem by Voiculescu that both are isomorphic.

A Contour-based Separation of VerticallyAttached Traffic Signs

This report presents an algorithm for locating the cut points for and separatingvertically attached traffic signs in Sweden. This algorithm provides severaladvanced digital image processing features: binary image which representsvisual object and its complex rectangle background with number one and zerorespectively, improved cross correlation which shows the similarity of 2Dobjects and filters traffic sign candidates, simplified shape decompositionwhich smoothes contour of visual object iteratively in order to reduce whitenoises, flipping point detection which locates black noises candidates, chasmfilling algorithm which eliminates black noises, determines the final cut pointsand separates originally attached traffic signs into individual ones. At each step,the mediate results as well as the efficiency in practice would be presented toshow the advantages and disadvantages of the developed algorithm. Thisreport concentrates on contour-based recognition of Swedish traffic signs. Thegeneral shapes cover upward triangle, downward triangle, circle, rectangle andoctagon. At last, a demonstration program would be presented to show howthe algorithm works in real-time environment.

Varför ska man dölja det? : -en kvalitativ studie av identitet i förhållande till läs-och skrivsvårigheter

In schools today, we expect student performances and achievements to be exceptional. Having good reading and writing skills are essential if students want to excel at their school assignments. Students with reading and writing difficulties have to work much harder than their other classmates. Their having to work harder coupled with being teenagers and facing all the uncertainties which are present at that age, these students face the difficult task of trying to find out who they are and who they want to be. In other words, they try to create their own individual identities. This study investigates the experiences of students with reading and writing difficulties in their interactions with other students and school personnel, in different situations. The collection of data has been done through group interviews. Thirteen, 15 year old students participated in these interviews. Some of the factors which characterise a hermeneutic approach have helped to form the basis on which the study lies. A hermeneutic approach suggests that the data collected is sorted and analysed to enable the identification of differences and patterns. These patterns are arranged to give results that are subjective and which also show an interpretation of the data collected. The results show that students are more comfortable with their identities, when they are diagnosed or made aware that their performances in school are directly affected by their reading and writing difficulties. The study also shows that having reading and writing difficulties tells the students who they are but, at the same time, plays an important role in their interactions with other classmates and adults. The outcomes of these interactions greatly affect the formation of their identities. The way in which school personnel treat students is also shown to be of great importance.

Populärmusik i TV-reklam : En kvalitativ undersökning av musiken som betydelsebärande teckensystem

This paper examines a popular music song (Heartbeats by Jose Gonzalez) as a sign system in television advertising. The study was conducted through qualitative questionnaires in connection to an audio-visual method of analysis called Masking. The method facilitates the analysis of isolated parts in the audio-visual spectrum by masking/hiding parts of the audio-visual totality.The survey had seven respondents where a hermeneutic epistemological approach was used. For the analysis Cooper's theory of brand identity (Practical and Symbolic Attitudes to Buying Brands) was used together with an interaction model for music in audio-visual advertising called "Modes of music-image interaction”. The results showed that the music was associated with values as genuine, honest, responsibility, purity, independence and innovation. The music's symbolic values helped to position the brand in a lifestyle context. The music also helped to express the target group’s identity and attitudes by being innovative and independent. It also enhanced the perception of the visual colour rendition in the film. In general the television advertisement perceived more positive and entertaining when the music was present. In other words the music's social and cultural position contributed to raise the film's credibility. A deeper social and cultural value was created in the movie through resonance between symbolic values of the music and symbolic values of the film.

Concreteness in Business letters : a Corpus-based Analysis of British and Pakistani English

Linguistic features of business letters have been a research target of both linguists and business writers. In this study, the language of British and Pakistani Business letters was compared and contrasted in terms of concreteness and abstractness. A corpus of 100 business letters from Inner Circle and Outer Circle writers were collected for analysis. The findings of the study revealed that British writers use more specific and concrete nouns, definite determiners, numeral, possessive and demonstrative adjectives, cohesive and rhetorical devices than the Pakistani Writers in order to be become concrete and vivid in their communication. The present findings are rather corpus specific since the data include only two countries; however this study may lead to further cross circle research including Expanding Circle research of business letters in terms of concreteness and abstractness. The issue of concreteness in Cross-circle business English can also be studied from psychological, sociological and anthropological perspectives in future Research.

El olvido : Análisis temático de la novela De amor y de sombra

La novela realista De amor y de sombra, por la autora Isabel Allende, describe como dos jóvenes luchan contra la “injusticia legalizada” en un lugar ficticio que alude a Chile bajo la dictadura de Augusto Pinochet. Un tema subyacente pero crucial en la obra es el olvido – individual, colectivo e impuesto. Por consiguiente demostraremos en este análisis cómo se presenta el olvido en la novela De amor y de sombra. La hipótesis que dirige nuestro estudio es que el olvido tiene un propósito mayor – destacar la importancia de recordar el pasado. Haciendo un análisis hermenéutico, nos apoyaremos en tres teorías sobre el olvido y el olvido social, resaltando ejemplos de personajes y acontecimientos de la novela. Por ende, en la conclusión se afirma la hipótesis y que la obra de Isabel Allende hace un llamamiento a recordar el pasado.

Det är värre att inte börja studera än att hoppa av: En fenomenologisk studie om hur icke traditionella studenter upplever studieavhopp från högskoleutbildning

In Sweden we have the highest rate of nontraditional students among Europe. Coincidently, we also have the highest rates of drop outs among OECD. In this particular field, drop out from higher education, there is a need fore more qualitative studies that can support the "statistics". Studies about experiences of drop outs are significant for society, university and individual. The aim of this study is to describe and understand how nontraditional student experiences drop out from higher education. What do they tell regarding their meeting with the university? How is the drop out visible before the decision to quit? How can the way forward to the drop out be described? What does the university mean for them? The theoretical point of departure is the concept lifeworld and the method is phenomenological hermeneutic with narrative interviews with 5 students. The result points out that the meeting with the university implies a meeting with other individuals, and also a "meeting" within the student himself, by inner questioning and reflection. The students’ relationship to the studies is characterized by the experience of understanding. The dropouts are visible through different events. The decision per se has increased over time and the students have also grown personally by the decision. To understand or not understand is crucial for the experience of learning. The drop out can also be seen as a transition and a turning point. The drop out is an effort and takes place without professional staff, therefore needs to be discussed in public.