Extreme Contractions Research Articles

Extreme contractions on finite-dimensional Banach spaces

We study extreme contractions in the setting of finite-dimensional polyhedral Banach spaces. Motivated by the famous Krein–Milman Theorem, we prove that a rank $1$ linear operator of unit norm between such spaces can be expressed as a convex combination o

Characterization of extreme contractions through k-smoothness of operators

We characterize extreme contractions defined between finite-dimensional polyhedral Banach spaces using k-smoothness of operators. As application of results obtained, we explicitly compute the number of extreme contractions in some special Banach spaces. Our approach, in this paper, in studying extreme contractions lead to the improvement and generalization of previously known results.

Revealing the role of past and current climate in shaping the distribution of two parapatric European bats, Myotis daubentonii and M. capaccinii

In this contribution, we aim at determining the role of climate in shaping the current and historical (Last Glacial Maximum and Middle Holocene) distributions of two parapatric bat species (Myotis daubentonii and Myotis capaccinii) in Europe, using ensemble Habitat Suitability Modelling (HSM). Model projection to current climatic conditions predicted distributions largely coherent with the ones observed for the two species in the study area. Temperature- and precipitation-linked variables well accounted for the observed parapatry of the two target species. Moreover, areas of co-occurrence turned out to be those where the main ecological needs concerning the most important range-limiting variables are met for both species. Model projections to past scenarios allowed us to hypothesize the effects of climatic oscillations over the distribution of the two species during the Late Pleistocene and Holocene. Extreme range contractions during the Last Glacial Maximum and the subsequent expansions during Middle Holocene were predicted, consistently with general patterns of post-glacial recolonization routes of many temperate bat species in Europe. Our findings are largely coherent with recent phylogeographic studies investigating the two target species, thus corroborating the validity of potential distribution scenarios obtained from the models and, more importantly, confirming the great influence that glacial cycles had in shaping the biogeography of the European fauna.

Characterization of k-smoothness of operators defined between infinite-dimensional spaces

ABSTRACT We characterize k-smoothness of bounded linear operators defined between infinite-dimensional Hilbert spaces. We study the problem in the setting of both finite and infinite-dimensional Banach spaces. We also characterize k-smoothness of operators on some particular spaces, namely where is a finite-dimensional Banach space and is a two-dimensional Banach space. As an application, we characterize extreme contractions on where is a two-dimensional polygonal Banach space.

Extreme contractions on finite-dimensional polygonal Banach spaces. II

We introduce the concept of weak L-P property for a pair of Banach spaces, in the study of extreme contractions. We give examples of pairs of Banach spaces (not) satisfying weak L-P property and apply the concept to compute the exact number of extreme contractions between a particular pair of polygonal Banach spaces. We also study the optimality of our results on the newly introduced weak L-P property for a pair of Banach spaces, by considering appropriate examples.

On extreme contractions between real Banach spaces

We completely characterize extreme contractions between two-dimensional strictly convex and smooth real Banach spaces, perhaps for the very first time. In order to obtain the desired characterization, we introduce the notions of (weakly) compatible point pair (CPP) and μ-compatible point pair (μ-CPP) in the geometry of Banach spaces. As a concrete application of our abstract results, we describe all rank one extreme contractions in L(ℓ42,ℓ42) and L(ℓ42,H), where H is any Hilbert space. We also prove that there does not exist any rank one extreme contractions in L(H,ℓp2), whenever p is even and H is any Hilbert space. Finally, we characterize real Hilbert spaces among real Banach spaces in terms of CPP, that substantiates our motivation behind introducing these new geometric notions.

On extreme contractions and the norm attainment set of a bounded linear operator

In this paper we completely characterize the norm attainment set of a bounded linear operator between Hilbert spaces. In fact, we obtain two different characterizations of the norm attainment set of a bounded linear operator between Hilbert spaces. We further study the extreme contractions on various types of finite-dimensional Banach spaces, namely Euclidean spaces, and strictly convex spaces. In particular, we give an elementary alternative proof of the well-known characterization of extreme contractions on a Euclidean space, which works equally well for both the real and the complex case. As an application of our exploration, we prove that it is possible to characterize real Hilbert spaces among real Banach spaces, in terms of extreme contractions on their 2-dimensional subspaces.

Gene family evolution in green plants with emphasis on the origination and evolution of Arabidopsis thaliana genes

Gene family size variation is an important mechanism that shapes the natural variation for adaptation in various species. Despite its importance, the pattern of gene family size variation in green plants is still not well understood. In particular, the evolutionary pattern of genes and gene families remains unknown in the model plant Arabidopsis thaliana in the context of green plants. In this study, eight representative genomes of green plants are sampled to study gene family evolution and characterize the origination of A.thaliana genes, respectively. Four important insights gained are that: (i) the rate of gene gains and losses is about 0.001359 per gene every million years, similar to the rate in yeast, Drosophila, and mammals; (ii) some gene families evolved rapidly with extreme expansions or contractions, and 2745 gene families present in all the eight species represent the 'core' proteome of green plants; (iii) 70% of A.thaliana genes could be traced back to 450million years ago; and (iv) intriguingly, A.thaliana genes with early origination are under stronger purifying selection and more conserved. In summary, the present study provides genome-wide insights into evolutionary history and mechanisms of genes and gene families in green plants and especially in A.thaliana.

Gene Family Evolution across 12 Drosophila Genomes

Comparison of whole genomes has revealed large and frequent changes in the size of gene families. These changes occur because of high rates of both gene gain (via duplication) and loss (via deletion or pseudogenization), as well as the evolution of entirely new genes. Here we use the genomes of 12 fully sequenced Drosophila species to study the gain and loss of genes at unprecedented resolution. We find large numbers of both gains and losses, with over 40% of all gene families differing in size among the Drosophila. Approximately 17 genes are estimated to be duplicated and fixed in a genome every million years, a rate on par with that previously found in both yeast and mammals. We find many instances of extreme expansions or contractions in the size of gene families, including the expansion of several sex- and spermatogenesis-related families in D. melanogaster that also evolve under positive selection at the nucleotide level. Newly evolved gene families in our dataset are associated with a class of testes-expressed genes known to have evolved de novo in a number of cases. Gene family comparisons also allow us to identify a number of annotated D. melanogaster genes that are unlikely to encode functional proteins, as well as to identify dozens of previously unannotated D. melanogaster genes with conserved homologs in the other Drosophila. Taken together, our results demonstrate that the apparent stasis in total gene number among species has masked rapid turnover in individual gene gain and loss. It is likely that this genomic revolving door has played a large role in shaping the morphological, physiological, and metabolic differences among species.

Peak power of muscles injured by lengthening contractions

Excessive or extreme lengthening contractions have a well-characterized depressive effect on skeletal muscle isometric force. In addition to producing force, active muscles must often shorten in order to meet the power requirements of locomotion and other physical activities. However, the impact of lengthening contractions on muscle power is poorly understood. We evaluated the effect of 20 isometric contractions or 20 lengthening contractions (20% strain at 1.5 fiber lengths/s) on the force-velocity-power relationships of mouse soleus muscles in vitro at 35 degrees C. Pre- and posttreatment data were obtained as the muscles shortened through their optimal length (Lo). The isometric treatment did not alter Lo, the curvature of the force-velocity relationship (a/Po), or soleus maximal shortening velocity (Vmax), whereas peak force (Po) displayed a slow, time-dependent decline of 10% across the experiments. Following the lengthening treatment, Lo increased by 6%, a/Po increased by 22%, and Vmax and Po fell by 24% and 26%, respectively. Under optimal conditions for producing power, muscles damaged by lengthening contractions attained 22% less force and shortened 20% more slowly than before damage. Consequently, soleus peak power fell 37% after lengthening, a 2.5-fold greater decline than noted for the isometric treatment. Under the conditions studied here, the excessive power loss following lengthening contractions was due to force and velocity deficits of approximately equal relative magnitude. Because power represents the ability of the muscle to perform work, reductions in both force and shortening velocity should be considered when evaluating and treating lengthening-induced skeletal muscle injuries.

Extreme contractions on continuous vector-valued function spaces

An old question asks whether extreme contractions on C ( K ) C(K) are necessarily nice; that is, whether the conjugate of such an operator maps extreme points of the dual ball to extreme points. Partial results have been obtained. Determining which operators are extreme seems to be a difficult task, even in the scalar case. Here we consider the case of extreme contractions on C ( K , E ) C(K,E) , where E E itself is a Banach space. We show that every extreme contraction T T on C ( K , E ) C(K,E) to itself which maps extreme points to elements of norm one is nice, where K K is compact and E E is the sequence space c 0 c_{0} .

A class of extreme contractions inL(l p )

In this note we prove that: if T is a contraction in L(lp) that maps elements of disjoint support to elements of disjoint support, then T is an extreme point of the unit ball of L(lp), 1<p < ∞, p ≠ 2, if and only if T is of the form $$T = \sum\limits_{i = 1}^\infty {\delta _i \otimes y_i } $$ , where either (yi) form a p-orthonormal sequence or the nonzero elements of (yi) form a p-orthonormal sequence for which $$\mathop \cup \limits_i $$ supp (yi)=N.

Approximation theorems for positive operators on L p-spaces

Let 1 < r ⩽ p < ∞. Approximation theorems for positive contractions in L ( L p(m), L r(n) ) are presented. The characterization of norm attaining extreme positive contractions is given. Note that these extreme operators are also exposed points of the positive part of the unit ball of L ( l p, l r ).

Bilateral shoulder dislocation: An unusual occurrence

Bilateral shoulder dislocation is an unusual clinical entity. Proposed mechanisms are via various traumatic insults, atraumatic occurrences, and through extreme muscular contractions. A case involving traumatic dislocation as a result of a diving incident is reported. A brief review of this rare clinical condition is presented.

Some Observations on Recent Architectural History

When I was asked to write an essay on recent architectural history, the range of coverage was left open. None of the other contributors to this series had written on architecture in more than a passing way, leaving me the choice of doing anything from my own specialization to the field as a whole. I have opted for the latter extreme, and thus my essay will not be like the others, which have been for the most part true “state-of-research” analyses written by specialists and restricted to single fields, often of quite narrow focus. My comments will be far more selective and exploratory, with extreme contractions and expansions of discussion de-pending on the subject and my knowledge of it and my sense of its relevance to this essay. I will often be skating on thin ice.

Extreme Positive Contractions on Finite Dimensional lp-Spaces

In this paper we give a characterization of the extreme positive contractions on finite dimensional lp-spaces for 1 &lt; p &lt; ∞. This is related to the characterization of the extreme doubly stochastic operators. In Section 2 we present the basic properties of the facial structure of the set of doubly stochastic n × m matrices. In Section 3 we use these facts for description of the facial structure of the set of positive contractions on finite dimensional lp-space. Next is considered stability of the positive part of the unit ball of operators (Section 5). In Section 7 we prove that extreme positive contractions on are strongly exposed.1. Terminology and notation. Let (X, , m) be a a-finite measure space. As usual, we denote by LP(m), 1 &lt; p &lt; ∞, the Banach lattice of all p-summable real-valued functions on X with standard norm and order. If X = {1, 2 , …, n) n &lt; ∞, and m is a counting measure we write instead of LP(m). If X = [0, 1] and m is Lebesgue measure we write briefly LP.

Extreme points in the Hahn-Banach-Kantorovič setting

This paper presents an existence and characterization theorem for the extreme points of the convex set of all extensions of a linear operator from a real vector space into an order complete real vector lattice which are dominated by a sublinear operator. This result is applied to positive extensions, contractions, and dominated invariant extensions. The paper falls into four sections. Section 1 is reserved for preliminaries. In §2 we consider the convex set of all extensions of a linear operator defined on a vector subspace of a real vector space X with values in an order complete real vector lattice 7 which are dominated by a sublinear operator P from X into 7. We present a characterization of the extreme points of this set being useful for applications. This part is related to papers of Kutateladze [7], [8] and Portenier [15]. In §3 we give two applications of the preceding result. The first one yields another proof of an existence and characterization theorem due to Lipecki [10], [11] concerning extreme positive extensions of a linear operator which is defined on a subspace of an ordered vector space. The second one yields a new characterization theorem for extreme contractions from a separable Banach space into the space of real valued continuous functions on a compact extremally disconnected space. In §4 the results of §2 are extended to P-dominated extensions which are positive on a given cone in X, and we apply them to P-dominated extensions which are invariant with respect to a set of mappings from X into X. Furthermore, we obtain a refinement of a dominated extension theorem for positive linear operators due to Luxemburg and Zaanen. 1. Preliminaries. We adhere to the notation of Schaefer's monograph [16]. Throughout X stands for a real vector space, M for its vector subspace and Y for an order complete real vector lattice. P: X-> Y denotes a sublinear mapping, i.e. P is positively homogeneous and subadditive. The space of all linear operators from M into Y is denoted by L(M, 7). Given a vector subspace N of X with M C N and T G L(M, 7), we put

Relaxation after extreme contraction, studied with isotonic technique.

Posttetanic relaxation was studied under nearly isotonic conditions, using an electromagnetic muscle lever and frog gastrocnemius. Greatly prolonged relaxation occurred when, during contraction, the muscle was thrown into slack, reaching lengths less than the minimum length in situ but greater than 40% of initial length. Contraction under such conditions has been called extreme contraction. The relaxing muscle afterward resisted stretching with tensions estimated to average 42% of twitch tension, and a shift in a length-tension characteristic seemed to have taken place. However, adequate stretching or loading overcame this resistance, and appeared to result in complete reversibility. The similarity of relaxation curves after normal and extreme contractions was demonstrated objectively by fitting the curves with the same mathematical relationship between length and time. Possible mechanisms accounting for extreme contraction are discussed.