Some Schroeder-Bernstein type theorems for Banach spaces

We first introduce the notion of (p, q, r)-complemented subspaces in Banach spaces, where p, q, r is an element of N. Then, given a couple of triples {(p, q, r), (s, t, u)} in N and putting Lambda = (q + r - p)(t + u - s) - ru, we prove partially the following conjecture: For every pair of Banach spaces X and Y such that X is (p, q, r)-complemented in Y and Y is (s, t, u)-complemented in X, we have that X is isomorphic Y if and only if one of the following conditions holds: (a) Lambda not equal 0, Lambda divides p - q and s - t, p = 1 or q = 1 or s = 1 or t = 1. (b) p = q = s = t = 1 and gcd(r, u) = 1. The case {(2, 1, 1), (2, 1,1)} is the well-known Pelczynski`s decomposition method. Our result leads naturally to some generalizations of the Schroeder-B em stein problem for Banach spaces solved by W.T. Gowers in 1996. (C) 2007 Elsevier Inc. All rights reserved.

On the extent of star countable spaces

For a topological property P, we say that a space X is star Pif for every open cover Uof the space X there exists Y aS, X such that St(Y,U) = X and Y has P. We consider star countable and star Lindelof spaces establishing, among other things, that there exists first countable pseudocompact spaces which are not star Lindelof. We also describe some classes of spaces in which star countability is equivalent to countable extent and show that a star countable space with a dense sigma-compact subspace can have arbitrary extent. It is proved that for any omega (1)-monolithic compact space X, if C (p) (X)is star countable then it is Lindelof.

Spaces of compact operators on C(2(m) circle plus [0, alpha]) spaces

We classify up to isomorphism the spaces of compact operators K(E, F), where E and F are Banach spaces of all continuous functions defined on the compact spaces 2(m) circle plus [0, alpha], the topological sum of Cantor cubes 2(m) and the intervals of ordinal numbers [0, alpha]. More precisely, we prove that if 2(m) and aleph(gamma) are not real-valued measurable cardinals and n >= aleph(0) is not sequential cardinal, then for every ordinals xi, eta, lambda and mu with xi >= omega(1), eta >= omega(1), lambda = mu < omega or lambda, mu is an element of [omega(gamma), omega(gamma+1)[, the following statements are equivalent: (a) K(C(2(m) circle plus [0, lambda]), C(2(n) circle plus [0, xi])) and K(C(2(m) circle plus [0, mu]), C(2(n) circle plus [0, eta]) are isomorphic. (b) Either C([0, xi]) is isomorphic to C([0, eta] or C([0, xi]) is isomorphic to C([0, alpha p]) and C([0, eta]) is isomorphic to C([0,alpha q]) for some regular cardinal alpha and finite ordinals p not equal q. Thus, it is relatively consistent with ZFC that this result furnishes a complete isomorphic classification of these spaces of compact operators. (C) 2010 Elsevier Inc. All rights reserved.

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.

A BROADER CONTEXT FOR MONOTONICALLY MONOLITHIC SPACES

This is a sequel of the work done on (strongly) monotonically monolithic spaces and their generalizations. We introduce the notion of monotonically kappa-monolithic space for any infinite cardinal kappa and present the relevant results. We show, among other things, that any sigma-product of monotonically kappa-monolithic spaces is monotonically kappa-monolithic for any infinite cardinal kappa; besides, it is consistent that any strongly monotonically omega-monolithic space with caliber omega(1) is second countable. We also study (strong) monotone kappa-monolithicity in linearly ordered spaces and subspaces of ordinals.

Ether-Bond-Containing Ionic Liquids and the Relevance of the Ether Bond Position to Transport Properties

The ionic liquids (ILs) 1-ethoxyethyl-2,3-dimethylimidazolium bis(trifluoromethanesulfonyl)imide, [EtO-(CH(2))(2)MMI][Tf(2)N], and N-(ethoxyethyl)-N-methylmorpholinium bis(trifluoromethanesulfonyl)imide, [EtO(CH(2))(2)MMor][Tf(2)N] were synthesized, and relevant properties, such as thermal stability, density, viscosity, electrochemical behavior, ionic conductivity, and self-diffusion coefficients for both ionic species, were measured and compared with those of their alkyl counterparts, 1-n-butyl-2,3-dimethylimidazolium bis(trifluoromethanesulfonyl)imide, [BMMI][Tf(2)N], and N-n-butyl-N-methylpiperidinium bis(trifluoromethanesulfonyl)imide,[BMP][Tf(2)N] and N-n-butyl-N-methylmorpholinium bis(trilfuoromethanesulfonyl)imide [BMMor][Tf(2)N][. This comparison was done to evaluate the effects caused by the presence of the ether bond in either the side chain or in the organic cation ring. The salt, LiTf(2)N, was added to the systems to estimate IL behavior with regard to lithium cation transport. Pure [EtO(CH(2))(2)MMI][Tf(2)N] and their LiTf(2)N solutions showed low viscosity and the highest conductivity among the ILs studied. The H(R) (AC conductivity/NMR calculated conductivity ratio) values showed that, after addition of LiTf(2)N, ILs containing the ether bond seemed to have a greater number of charged species. Structural reasons could explain these high observed HR values for [EtO(CH(2))(2)MMor][Tf(2)N].

Solar position diagram

Program SOLVEJ är ett användarvänligt program som visar solens vandring över himlavalvet vid upp till fem valfria datum och vid valfri ort. Programmet är utvecklat av två skäl. För det första, att demonstreras för en intresserad allmänhet som del av vandringsutställning om solenergi, vilken är initierad och utarbetad av SERC. För det andra, att användas av solenerglintressenter för att snabbt få en uppfattning om solinstrålningen på en ort vid olika tidpunkter på året.Indata till programmet ges från tangentbordet. Som svar på frågor skrivs för vilken ort diagrammet skall gälla, max fem datum, ortens latitud och longitud, som anges positiv i västlig riktning, samt tidszonen. Varje uppgift avslutas med tryck på tangenten ENTER. Programmet kommer nu att rita ett koordinatsystem på skärmen. Första axeln visar vädersträcken, norr, öster, söder, väster och norr, varje delstreck utgör 10 grader. För södra halvklotet byter norr och söder plats. Andra axeln visar höjden över horisonten i grader, 0 till 90 grader och 10 grader för varje delstreck. Efter några sekunder ritas diagrammet upp med solhöjden som funktion av väderstrecket och varje hel timme markerad. Se fig. 1-4. Slutligen frågas efter om diagrammet skall ritas ut på printer. SOLVEJ avbrytes med att trycka CTRL+BREAK.SOLVEJ är skrivet i Quick-BASIC (se App. 1) och leveras både som källkod och körklar version. Lämplig dator är IBM-kompatibel AT med EGA- eller VGA-skärmkort (ej Herkules Lämplig printer är IBM Proprinter eller liknande matrisskrivare, kopplad till LPT1 på kommunikationskortet.Till grund för beräkningarna har använts artikeln On Calculating the Position of the Sun, publicerad i nr. 1 1988 av The International Journal of Ambient Energy. Fem empiriska ekvationer beträffande beräkningar av solens position har studerats för att undersöka deras tillförlitlighet. Felaktigheter på fem grader eller mer kan uppträda om man använder sig av de enkla ekvationer som kan hittas solenergi-böcker och som inte kräver tillgång till dator. FORTRAN-rutinen SUNAE2 (se App. 2) beräknar solpositionen med noggrannast kända metod. Program SOLVEJ är en utveckling av SUNAE2.

The Role of Gay Spaces for a Gay Destination : Gay Tourism in Gran Canaria

This thesis has the aim to evaluate the role of gay spaces in Gran Canaria as a gay tourists destination with particular reference to gay exclusive resorts. The validation of the gay identity is a key motivation for homosexuals to travel, in order to connect with other homosexuals and experience the gay life that they might not be able to experience at home. Gay spaces have been defined both as liberated areas as well as ghettos, where the homosexuals are, in a way, restrained. The method chosen, a small number of semi- structured interviews with managers of gay exclusive resorts in Maspalomas, the hub of gay life in Gran Canaria, where major LGBT events are held, there is a gay friendly environment, a thriving gay scene and many gay exclusive resorts. In the case of Gran Canaria the gay-specific offer is complementary to the ‘regular’ tourism offer, as they coexist, complement and at times overlap. Nevertheless the gay centric holiday is still predominant amongst gay men, and it is likely to continue to be according to the informants. This is because gay tourists seek freedom and a sense of inclusion that they would not be able to find in mixed environments.

Barnfri : en stigmatiserad position

Childfree: a stigmatized position International research has addressed the subject but in Sweden voluntary childlessness has until now been overlooked. This article draws on qualitative, semi-structured interviews with 21 Swedish childfree women. The interviews focused their decision not to have children and attitudes they faced due to their rejection of motherhood. They all had encountered pressure to conform to a pronatalistic norm, proclaiming parenthood to be self-evident in an adult normal life. The results highlight different strategies used by the women to avoid instigating the dislike of those around them. The article argues that understanding childfree as a stigmatized position helps providing new insights to what conditions the social relations between the childfree and ‘the normals’, i.e. persons who advocate having children. Further, viewing the childfree as a stigmatized group has theoretical implications that contribute to developing Goffman’s classical theory of social stigma.

After work : en longitudinell studie i arbetsmiljöns långsiktiga inverkan på sambandet mellan socioekonomisk position och äldres ohälsa

After Work. The long-term effects of work environment on the association between adult socioeconomic position and ill health among the elderly The aim of this study is to analyse the long-term effects of work environment on the association between adult socioeconomic position (SEP) and six ill health outcomes among the elderly. Data was drawn from the longitudinal Level of Living and the SWEOLD-surveys. The individuals are followed from 1968 to 1992 and from 1981 to 2002 and 2004, combining baseline information regarding SEP and work environment during the period of occupational activity with the ill health outcomes from the follow-up studies. Strongest effects where revealed when controlling for the physical work environment on the association between both measures of SEP and two of the ill health outcomes: musculoskeleta lpain and physical performance. The psychological work environment, however, explained very small parts of the associations. The results, even controlled for SEP, exposed strong direct relations between ill health and psychological work environment and rather strong relations to physical work environment. As socioeconomic position indicates a particular structural position within society it may determine the likelihood of health damaging exposures during the period of occupational activity. This study shows that the effect of work environment significantly affects ill health among the elderly. Hence, the results indicate the importance of taking measures in improving work environment during the labour-market participation period, especially since policy-makers attempt to convince workers to stay longer in the workforce.

Languaging in virtual learning sites : studies of online encounters in the language-focused classroom

This thesis focuses upon a series of empirical studies which examine communication and learning in online glocal communities within higher education in Sweden. A recurring theme in the theoretical framework deals with issues of languaging in virtual multimodal environments as well as the making of identity and negotiation of meaning in these settings; analyzing the activity, what people do, in contraposition to the study of how people talk about their activity. The studies arise from netnographic work during two online Italian for Beginners courses offered by a Swedish university. Microanalyses of the interactions occurring through multimodal video-conferencing software are amplified by the study of the courses’ organisation of space and time and have allowed for the identification of communicative strategies and interactional patterns in virtual learning sites when participants communicate in a language variety with which they have a limited experience. The findings from the four studies included in the thesis indicate that students who are part of institutional virtual higher educational settings make use of several resources in order to perform their identity positions inside the group as a way to enrich and nurture the process of communication and learning in this online glocal community. The sociocultural dialogical analyses also shed light on the ways in which participants gathering in discursive technological spaces benefit from the opportunity to go to class without commuting to the physical building of the institution providing the course. This identity position is, thus, both experienced by participants in interaction, and also afforded by the ‘spaceless’ nature of the online environment.