A Note on Coercivity of Lower Semicontinuous Functions and Nonsmooth Critical Point Theory

The first motivation for this note is to obtain a general version of the following result: let E be a Banach space and f : E → R be a differentiable function, bounded below and satisfying the Palais-Smale condition; then, f is coercive, i.e., f(x) goes to infinity as ||x|| goes to infinity. In recent years, many variants and extensions of this result appeared, see [3], [5], [6], [9], [14], [18], [19] and the references therein. A general result of this type was given in [3, Theorem 5.1] for a lower semicontinuous function defined on a Banach space, through an approach based on an abstract notion of subdifferential operator, and taking into account the “smoothness” of the Banach space. Here, we give (Theorem 1) an extension in a metric setting, based on the notion of slope from [11] and coercivity is considered in a generalized sense, inspired by [9]; our result allows to recover, for example, the coercivity result of [19], where a weakened version of the Palais-Smale condition is used. Our main tool (Proposition 1) is a consequence of Ekeland’s variational principle extending [12, Corollary 3.4], and deals with a function f which is, in some sense, the “uniform” Γ-limit of a sequence of functions.

Perturbations of Critical Values in Nonsmooth Critical Point Theory

* Supported by Ministero dell’Università e della Ricerca Scientifica e Tecnologica (40% – 1993). ** Supported by Ministero dell’Università e della Ricerca Scientifica e Tecnologica (40% – 1993).

Nontrivial Solutions of Quasilinear Equations In BV

The existence of a nontrivial critical point is proved for a functional containing an area-type term. Techniques of nonsmooth critical point theory are applied.

On the critical point structure of eigenfunctions belonging to the first nonzero eigenvalue of a genus two closed hyperbolic surface

We develop a method based on spectral graph theory to approximate the eigenvalues and eigenfunctions of the Laplace-Beltrami operator of a compact riemannian manifold -- The method is applied to a closed hyperbolic surface of genus two -- The results obtained agree with the ones obtained by other authors by different methods, and they serve as experimental evidence supporting the conjectured fact that the generic eigenfunctions belonging to the first nonzero eigenvalue of a closed hyperbolic surface of arbitrary genus are Morse functions having the least possible total number of critical points among all Morse functions admitted by such manifolds

Critical behavior of the susceptible-infected-recovered model on a square lattice

By means of numerical simulations and epidemic analysis, the transition point of the stochastic asynchronous susceptible-infected-recovered model on a square lattice is found to be c(0)=0.176 500 5(10), where c is the probability a chosen infected site spontaneously recovers rather than tries to infect one neighbor. This point corresponds to an infection/recovery rate of lambda(c)=(1-c(0))/c(0)=4.665 71(3) and a net transmissibility of (1-c(0))/(1+3c(0))=0.538 410(2), which falls between the rigorous bounds of the site and bond thresholds. The critical behavior of the model is consistent with the two-dimensional percolation universality class, but local growth probabilities differ from those of dynamic percolation cluster growth, as is demonstrated explicitly.

Precise determination of quantum critical points by the violation of the entropic area law

Finite-size scaling analysis turns out to be a powerful tool to calculate the phase diagram as well as the critical properties of two-dimensional classical statistical mechanics models and quantum Hamiltonians in one dimension. The most used method to locate quantum critical points is the so-called crossing method, where the estimates are obtained by comparing the mass gaps of two distinct lattice sizes. The success of this method is due to its simplicity and the ability to provide accurate results even considering relatively small lattice sizes. In this paper, we introduce an estimator that locates quantum critical points by exploring the known distinct behavior of the entanglement entropy in critical and noncritical systems. As a benchmark test, we use this new estimator to locate the critical point of the quantum Ising chain and the critical line of the spin-1 Blume-Capel quantum chain. The tricritical point of this last model is also obtained. Comparison with the standard crossing method is also presented. The method we propose is simple to implement in practice, particularly in density matrix renormalization group calculations, and provides us, like the crossing method, amazingly accurate results for quite small lattice sizes. Our applications show that the proposed method has several advantages, as compared with the standard crossing method, and we believe it will become popular in future numerical studies.

Out of the shadows: The critical sociology of sport in Australia, 1986 to 1996

The sociology of sport in Australia has reached a key point in its development. A critical tradition in the subdiscipline has been established over the last decade, but its intellectual and institutional progress has been uneven. This article briefly traces the emergence of critical sports sociology in a country outside the major centers in the UK and U.S., its break with functionalist approaches, and its attempts to overcome the neglect of local mainstream sociology. The authors proceed to examine (self-reflexively) the changes of theoretical direction and the new lines of research that are being explored in the field. A recent ''skirmish'' with narrative history over the preferred theories and methods in sports analysis is discussed as illustrative of the difficulties encountered by an energetic but small, dispersed and underorganized scholarly movement in Australia.

The effectiveness of the Media Smart Program on promoting the critical thinking of children towards the use of celebrities in television advertising

A Work Project, presented as part of the requirements for the Award of a Masters Degree in Management from the NOVA – School of Business and Economics

Clarke critical values of subanalytic Lipschitz continuous functions

We prove that any subanalytic locally Lipschitz function has the Sard property. Such functions are typically nonsmooth and their lack of regularity necessitates the choice of some generalized notion of gradient and of critical point. In our framework these notions are defined in terms of the Clarke and of the convex-stable subdifferentials. The main result of this note asserts that for any subanalytic locally Lipschitz function the set of its Clarke critical values is locally finite. The proof relies on Pawlucki's extension of the Puiseuxlemma. In the last section we give an example of a continuous subanalytic function which is not constant on a segment of "broadly critical" points, that is, points for which we can find arbitrarily short convex combinations of gradients at nearby points.

Choline-PET in prostate cancer management: The point of view of the radiation oncologist.

Among PET radiotracers, FDG seems to be quite accepted as an accurate oncology diagnostic tool, frequently helpful also in the evaluation of treatment response and in radiation therapy treatment planning for several cancer sites. To the contrary, the reliability of Choline as a tracer for prostate cancer (PC) still remains an object of debate for clinicians, including radiation oncologists. This review focuses on the available data about the potential impact of Choline-PET in the daily clinical practice of radiation oncologists managing PC patients. In summary, routine Choline-PET is not indicated for initial local T staging, but it seems better than conventional imaging for nodal staging and for all patients with suspected metastases. In these settings, Choline-PET showed the potential to change patient management. A critical limit remains spatial resolution, limiting the accuracy and reliability for small lesions. After a PSA rise, the problem of the trigger PSA value remains crucial. Indeed, the overall detection rate of Choline-PET is significantly increased when the trigger PSA, or the doubling time, increases, but higher PSA levels are often a sign of metastatic spread, a contraindication for potentially curable local treatments such as radiation therapy. Even if several published data seem to be promising, the current role of PET in treatment planning in PC patients to be irradiated still remains under investigation. Based on available literature data, all these issues are addressed and discussed in this review.

Price regulation of plastic money: A critical assessment of Spanish rules

Recent decisions by the Spanish national competition authority (TDC) mandate paymentsystems to include only two costs when setting their domestic multilateral interchange fees(MIF): a fixed processing cost and a variable cost for the risk of fraud. This artificiallowering of MIFs will not lower consumer prices, because of uncompetitive retailing; but itwill however lead to higher cardholders fees and, likely, new prices for point of saleterminals, delaying the development of the immature Spanish card market. Also, to the extent that increased cardholders fees do not offset the fall in MIFs revenue, the task of issuing new cards will be underpaid relatively to the task of acquiring new merchants, causing an imbalance between the two sides of the networks. Moreover, the pricing scheme arising from the decisions will cause unbundling and underprovision of those services whose costs are excluded. Indeed, the payment guarantee and the free funding period will tend to be removed from the package of services currently provided, to be either provided by third parties, by issuers for a separate fee, or not provided at all, especially to smaller and medium-sized merchants. Transaction services will also suffer the consequences that the TDC precludes pricing them in variable terms.

The three strands of intercultural polities: a comprehensive view. A critical review of Bouchard and Cantle recent books on interculturalism

Within the emerging policy debate on interculturalism we critically review two recent books in 2012: Bouchard’s L’interculturalisme: un point de vue quebecois, and Cantle’s Interculturalism: The New Era of Cohesion and Diversity. In my view, both contribute very directly to open a foundational debate on interculturalism. In addressing the point of convergence and the dividing lines of these two contributions, I will claim that in spite of having one core concept of interculturalism, there are, however, at least two basic conceptions that have to be interpreted in complementary ways: Bouchard’s essay represents the contractual strand, Cantle’s book the cohesion strand. At the end I would also suggest that these two strands do not manage to express explicitly that diversity can also be seen as a resource of innovation and creativity, and so can drive individual and social development. This view is based on the diversity advantage literature already informing most of the diversity debate in Europe and elsewhere. This is what I will call the constructivist strand. My ultimate purpose is to defend a comprehensive view, grounded on the argument that no one can have the sole authority to define intercultural policy, since the three strands can be applied at different moments, according to different purposes and policy needs. The challenge now is that policy managers be able to achieve a balance between these three policy drivers.

From caging to rouse dynamics in polymer melts with intramolecular barriers: a critical test of the mode coupling theory

By means of computer simulations and solution of the equations of the mode coupling theory (MCT),we investigate the role of the intramolecular barriers on several dynamic aspects of nonentangled polymers. The investigated dynamic range extends from the caging regime characteristic of glass-formers to the relaxation of the chain Rouse modes. We review our recent work on this question,provide new results, and critically discuss the limitations of the theory. Solutions of the MCT for the structural relaxation reproduce qualitative trends of simulations for weak and moderate barriers. However, a progressive discrepancy is revealed as the limit of stiff chains is approached. This dis-agreement does not seem related with dynamic heterogeneities, which indeed are not enhanced by increasing barrier strength. It is not connected either with the breakdown of the convolution approximation for three-point static correlations, which retains its validity for stiff chains. These findings suggest the need of an improvement of the MCT equations for polymer melts. Concerning the relaxation of the chain degrees of freedom, MCT provides a microscopic basis for time scales from chain reorientation down to the caging regime. It rationalizes, from first principles, the observed deviations from the Rouse model on increasing the barrier strength. These include anomalous scaling of relaxation times, long-time plateaux, and nonmonotonous wavelength dependence of the mode correlators.

Prediction in forensic science: a critical examination of common understandings

In this commentary, we argue that the term 'prediction' is overly used when in fact, referring to foundational writings of de Finetti, the correspondent term should be inference. In particular, we intend (i) to summarize and clarify relevant subject matter on prediction from established statistical theory, and (ii) point out the logic of this understanding with respect practical uses of the term prediction. Written from an interdisciplinary perspective, associating statistics and forensic science as an example, this discussion also connects to related fields such as medical diagnosis and other areas of application where reasoning based on scientific results is practiced in societal relevant contexts. This includes forensic psychology that uses prediction as part of its vocabulary when dealing with matters that arise in the course of legal proceedings.

From caging to rouse dynamics in polymer melts with intramolecular barriers: a critical test of the mode coupling theory

By means of computer simulations and solution of the equations of the mode coupling theory (MCT),we investigate the role of the intramolecular barriers on several dynamic aspects of nonentangled polymers. The investigated dynamic range extends from the caging regime characteristic of glass-formers to the relaxation of the chain Rouse modes. We review our recent work on this question,provide new results, and critically discuss the limitations of the theory. Solutions of the MCT for the structural relaxation reproduce qualitative trends of simulations for weak and moderate barriers. However, a progressive discrepancy is revealed as the limit of stiff chains is approached. This dis-agreement does not seem related with dynamic heterogeneities, which indeed are not enhanced by increasing barrier strength. It is not connected either with the breakdown of the convolution approximation for three-point static correlations, which retains its validity for stiff chains. These findings suggest the need of an improvement of the MCT equations for polymer melts. Concerning the relaxation of the chain degrees of freedom, MCT provides a microscopic basis for time scales from chain reorientation down to the caging regime. It rationalizes, from first principles, the observed deviations from the Rouse model on increasing the barrier strength. These include anomalous scaling of relaxation times, long-time plateaux, and nonmonotonous wavelength dependence of the mode correlators.