The C(K, X) spaces for compact metric spaces K and X with a uniformly convex maximal factor

In this paper, we prove that if a Banach space X contains some uniformly convex subspace in certain geometric position, then the C(K, X) spaces of all X-valued continuous functions defined on the compact metric spaces K have exactly the same isomorphism classes that the C(K) spaces. This provides a vector-valued extension of classical results of Bessaga and Pelczynski (1960) [2] and Milutin (1966) [13] on the isomorphic classification of the separable C(K) spaces. As a consequence, we show that if 1 < p < q < infinity then for every infinite countable compact metric spaces K(1), K(2), K(3) and K(4) are equivalent: (a) C(K(1), l(p)) circle plus C(K(2), l(q)) is isomorphic to C(K(3), l(p)) circle plus (K(4), l(q)). (b) C(K(1)) is isomorphic to C(K(3)) and C(K(2)) is isomorphic to C(K(4)). (C) 2011 Elsevier Inc. All rights reserved.

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.

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.

Visualization of spirography-based objective measures in Parkinson's disease

Objective: To investigate whether advanced visualizations of spirography-based objective measures are useful in differentiating drug-related motor dysfunctions between Off and dyskinesia in Parkinson’s disease (PD). Background: During the course of a 3 year longitudinal clinical study, in total 65 patients (43 males and 22 females with mean age of 65) with advanced PD and 10 healthy elderly (HE) subjects (5 males and 5 females with mean age of 61) were assessed. Both patients and HE subjects performed repeated and time-stamped assessments of their objective health indicators using a test battery implemented on a telemetry touch screen handheld computer, in their home environment settings. Among other tasks, the subjects were asked to trace a pre-drawn Archimedes spiral using the dominant hand and repeat the test three times per test occasion. Methods: A web-based framework was developed to enable a visual exploration of relevant spirography-based kinematic features by clinicians so they can in turn evaluate the motor states of the patients i.e. Off and dyskinesia. The system uses different visualization techniques such as time series plots, animation, and interaction and organizes them into different views to aid clinicians in measuring spatial and time-dependent irregularities that could be associated with the motor states. Along with the animation view, the system displays two time series plots for representing drawing speed (blue line) and displacement from ideal trajectory (orange line). The views are coordinated and linked i.e. user interactions in one of the views will be reflected in other views. For instance, when the user points in one of the pixels in the spiral view, the circle size of the underlying pixel increases and a vertical line appears in the time series views to depict the corresponding position. In addition, in order to enable clinicians to observe erratic movements more clearly and thus improve the detection of irregularities, the system displays a color-map which gives an idea of the longevity of the spirography task. Figure 2 shows single randomly selected spirals drawn by a: A) patient who experienced dyskinesias, B) HE subject, and C) patient in Off state. Results: According to a domain expert (DN), the spirals drawn in the Off and dyskinesia motor states are characterized by different spatial and time features. For instance, the spiral shown in Fig. 2A was drawn by a patient who showed symptoms of dyskinesia; the drawing speed was relatively high (cf. blue-colored time series plot and the short timestamp scale in the x axis) and the spatial displacement was high (cf. orange-colored time series plot) associated with smooth deviations as a result of uncontrollable movements. The patient also exhibited low amount of hesitation which could be reflected both in the animation of the spiral as well as time series plots. In contrast, the patient who was in the Off state exhibited different kinematic features, as shown in Fig. 2C. In the case of spirals drawn by a HE subject, there was a great precision during the drawing process as well as unchanging levels of time-dependent features over the test trial, as seen in Fig. 2B. Conclusions: Visualizing spirography-based objective measures enables identification of trends and patterns of drug-related motor dysfunctions at the patient’s individual level. Dynamic access of visualized motor tests may be useful during the evaluation of drug-related complications such as under- and over-medications, providing decision support to clinicians during evaluation of treatment effects as well as improve the quality of life of patients and their caregivers. In future, we plan to evaluate the proposed approach by assessing within- and between-clinician variability in ratings in order to determine its actual usefulness and then use these ratings as target outcomes in supervised machine learning, similarly as it was previously done in the study performed by Memedi et al. (2013).

Dokumentationens dilemman : Förskollärare samtalar om pedagogisk dokumentation

Abstract  Pedagogical documentation is a certain procedure for documenting that, in recent years, has been embraced in several Swedish preschools. Teachers document children’s actions and conversations usually by photos or video recordings. This documentation is to be used for a pedagogical purpose. However, studies and governmental inspections have shown that pedagogical documentation gives rise to many questions among preschool teachers. The purpose of this study is to gain insight into what is being expressed when preschool teachers discuss pedagogical documentation, focusing on themes of content and on the participants’ expressions of their points of view. The data is comprised of transcriptions from audio recordings of discussions conducted in a research circle. The participants are eight preschool teachers that met over the course of one year. Each meeting focused on the documentation provided by a different participant. In that way the contents of the discussions were framed by the teachers own questions and narratives. Theoretically, the study departs from Social Constructionism and Discursive Psychology. The preschool teachers’ utterances have been analyzed using concepts of interpretative repertoires and ideological dilemmas. The results show the main themes to be: Knowledge content in a preschool setting, children’s learning, the teacher’s role and implementation of pedagogical documentation. The participants’ joint position is that the knowledge content at the preschool level is defined by the curriculum for the preschool. Concerning children’s learning and the teacher’s role, two main standpoints are disclosed. Ideologically those standpoints derive from two opposing theories of education. Based on how the standpoints have been expressed I have called them ”predetermined learning” versus ”non-predetermined learning”. One main distinction between the standpoints is that predetermined learning emphasizes the results of learning, while non-predetermined learning emphasizes the processes of learning. The participants’ utterances show that teachers tend to subscribe to the idea that there is only one acceptable way of working with pedagogical documentation. This sometimes creates performance anxiety and feelings of not succeeding and has led to arguments advocating an alternate approach; pedagogical documentation can be done in many ways. The ideological dilemmas within the discourse can be perceived as resources by which the participants argue about knowledge, learning, teaching and about the implementation of pedagogical documentation.

Capability outcomes from educational and ICT capability inputs : an analysis of ICT use in informal education in Kenya

In the field of Information and Communication Technologies for Development (ICT4D) ICT use in education is well studied. Education is often seen as a pre-requisite for development and ICTs are believed to aid in education, e.g. to make it more accessible and to increase its quality. In this paper we study the access and use of ICT in a study circle (SC) education program in the south coast of Kenya. The study is qualitative reporting results based on interviews and observations with SC participants, government officers and SC coordinators and teachers. The study builds on the capability approach perspective of development where individuals’ opportunities and ability to live a life that they value are focused. The aim of the study is to investigate the capability outcomes enabled through the capability inputs access and use of ICT in education as well as the factors that enabled and/or restricted the outcomes. Findings show that many opportunities have been enabled such as an increase in the ability to generate an income, learning benefits, community development and basic human development (e.g. literacy and self-confidence). However, conversion factors such as a poorly developed infrastructure and poor IT literacy prevent many of the individuals from taking full advantage of the ICT and the opportunities it enables.