Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

Non-Gaussian distribution of collective operators in quantum spin chains

We numerically analyse the behavior of the full distribution of collective observables in quantum spin chains. While most of previous studies of quantum critical phenomena are limited to the first moments, here we demonstrate how quantum fluctuations at criticality lead to highly non-Gaussian distributions. Interestingly, we show that the distributions for different system sizes collapse on thesame curve after scaling for a wide range of transitions: first and second order quantum transitions and transitions of the Berezinskii–Kosterlitz–Thouless type. We propose and analyse the feasibility of an experimental reconstruction of the distribution using light–matter interfaces for atoms in optical lattices or in optical resonators.

Existence of disjoint weakly mixing operators that fail to satisfy the Disjoint Hypercyclicity Criterion

Recently, Bès, Martin, and Sanders [11] provided examples of disjoint hypercyclic operators which fail to satisfy the Disjoint Hypercyclicity Criterion. However, their operators also fail to be disjoint weakly mixing. We show that every separable, infinite dimensional Banach space admits operators T1,T2,…,TN with N⩾2 which are disjoint weakly mixing, and still fail to satisfy the Disjoint Hypercyclicity Criterion, answering a question posed in [11]. Moreover, we provide examples of disjoint hypercyclic operators T1, T2 whose corresponding set of disjoint hypercyclic vectors is nowhere dense, answering another question posed in [11]. In fact, we explicitly describe their set of disjoint hypercyclic vectors. Those same disjoint hypercyclic operators fail to be disjoint topologically transitive. Lastly, we create examples of two families of d-hypercyclic operators which fail to have any d-hypercyclic vectors in common.

Eigenfunctions and fundamental solutions of the fractional Laplace and Dirac operators: the Riemann-Liouville case

In this paper we study eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator $\Delta_+^{(\alpha,\beta,\gamma)}:= D_{x_0^+}^{1+\alpha} +D_{y_0^+}^{1+\beta} +D_{z_0^+}^{1+\gamma},$ where $(\alpha, \beta, \gamma) \in \,]0,1]^3$, and the fractional derivatives $D_{x_0^+}^{1+\alpha}$, $D_{y_0^+}^{1+\beta}$, $D_{z_0^+}^{1+\gamma}$ are in the Riemann-Liouville sense. Applying operational techniques via two-dimensional Laplace transform we describe a complete family of eigenfunctions and fundamental solutions of the operator $\Delta_+^{(\alpha,\beta,\gamma)}$ in classes of functions admitting a summable fractional derivative. Making use of the Mittag-Leffler function, a symbolic operational form of the solutions is presented. From the obtained family of fundamental solutions we deduce a family of fundamental solutions of the fractional Dirac operator, which factorizes the fractional Laplace operator. We apply also the method of separation of variables to obtain eigenfunctions and fundamental solutions.

Eigenfunctions and fundamental solutions of the Caputo fractional Laplace and Dirac operators

In this paper, by using the method of separation of variables, we obtain eigenfunctions and fundamental solutions for the three parameter fractional Laplace operator defined via fractional Caputo derivatives. The solutions are expressed using the Mittag-Leffler function and we show some graphical representations for some parameters. A family of fundamental solutions of the corresponding fractional Dirac operator is also obtained. Particular cases are considered in both cases.

Solutions of Tikhonov functional equations and applications to multiplication operators on Szegö spaces

We consider a natural representation of solutions for Tikhonov functional equations. This will be done by applying the theory of reproducing kernels to the approximate solutions of general bounded linear operator equations (when defined from reproducing kernel Hilbert spaces into general Hilbert spaces), by using the Hilbert-Schmidt property and tensor product of Hilbert spaces. As a concrete case, we shall consider generalized fractional functions formed by the quotient of Bergman functions by Szegö functions considered from the multiplication operators on the Szegö spaces.

Invertibility characterization of Wiener-Hopf plus Hankel operators on variable exponent Lebesgue spaces via even asymmetric factorization

We obtain invertibility and Fredholm criteria for the Wiener-Hopf plus Hankel operators acting between variable exponent Lebesgue spaces on the real line. Such characterizations are obtained via the so-called even asymmetric factorization which is applied to the Fourier symbols of the operators under study.

Homogenization of the p-Laplacian with nonlinear boundary condition on critical size particles: Identifying the strange term for the some non smooth and multivalued operators

We extend previous papers in the literature concerning the homogenization of Robin type boundary conditions for quasilinear equations, in the case of microscopic obstacles of critical size: here we consider nonlinear boundary conditions involving some maximal monotone graphs which may correspond to discontinuous or non-Lipschitz functions arising in some catalysis problems.

Strictly singular operators in pairs of L (p) space

Let E and F be Banach spaces. A linear operator from E to F is said to be strictly singular if, for any subspace Q aS, E, the restriction of A to Q is not an isomorphism. A compactness criterion for any strictly singular operator from L (p) to L (q) is found. There exists a strictly singular but not superstrictly singular operator on L (p) , provided that p not equal 2.

The significant role of third-party logistics (3pl) operators as fosterers of organizational performance in colombian companies

La eliminación de barreras entre países es una consecuencia que llega con la globalización y con los acuerdos de TLC firmados en los últimos años. Esto implica un crecimiento significativo del comercio exterior, lo cual se ve reflejado en un aumento de la complejidad de la cadena de suministro de las empresas. Debido a lo anterior, se hace necesaria la búsqueda de alternativas para obtener altos niveles de productividad y competitividad dentro de las empresas en Colombia, ya que el entorno se ha vuelto cada vez más complejo, saturado de competencia no sólo nacional, sino también internacional. Para mantenerse en una posición competitiva favorable, las compañías deben enfocarse en las actividades que le agregan valor a su negocio, por lo cual una de las alternativas que se están adoptando hoy en día es la tercerización de funciones logísticas a empresas especializadas en el manejo de estos servicios. Tales empresas son los Proveedores de servicios logísticos (LSP), quienes actúan como agentes externos a la organización al gestionar, controlar y proporcionar actividades logísticas en nombre de un contratante. Las actividades realizadas pueden incluir todas o parte de las actividades logísticas, pero como mínimo la gestión y ejecución del transporte y almacenamiento deben estar incluidos (Berglund, 2000). El propósito del documento es analizar el papel de los Operadores Logísticos de Tercer nivel (3PL) como promotores del desempeño organizacional en las empresas colombianas, con el fin de informar a las MIPYMES acerca de los beneficios que se obtienen al trabajar con LSP como un medio para mejorar la posición competitiva del país.

Comparasion of two minimally invasive methods on the longevity of glass ionomer cement restorations: short-term results of a pilot study

The purpose of this study was to evaluate the clinical performance of glass ionomer cement (GIC) restorations comparing two minimally invasive methods in permanent teeth after 12 months. Fifty pregnant women (second trimester of pregnancy), mean age 22 ± 5.30 years, were treated by two previously trained operators. The treatment approaches tested were: chemomechanical method (CarisolvTM; MediTeam) and atraumatic restorative treatment (ART). A split-mouth study design was used in which the two treatments were randomly placed in 50 matched pairs of permanent teeth. The chemomechanical method (CM) was the test group and the ART was the control group. The treatments were performed in Public Health Centers. The tested restorative material was a high-strength GIC (Ketac Molar; 3M/ESPE). The restorations were placed according to the ART guidelines. Two calibrated independent examiners evaluated the restorations in accordance with ART criteria. The inter-examiner kappa was 0.97. Data were analyzed using 95% confidence interval on the binomial distribution and Fisher's exact test at 5% significance level. In a 12-month follow-up, 86% of the restorations were evaluated. In the test group (CM), 100% (CI=93.3-100%) of the restorations were considered successful. In the control group (ART) 97.6% (CI=87.4-99.9%) of the restorations were considered successful and 2.4% unsuccessful (marginal defect >0.5 mm). There was no statistically significant difference between the 12-mounth success rate for both groups (Fisher's exact test: P=0.49) and between the two operators (Fisher's exact test: P=1.00). Both minimally invasive methods, chemomechanical method and ART, showed a similar clinical performance after 12 months of follow up.

Use of flowable composite as intermediary layer in non-carious cervical lesions restored with composite resin: 48-month follow-up

PURPOSE: In this case report, the clinical performance of a microhybrid resin composite placed with or without a flowable resin composite was compared, over a 48-month period. CASE DESCRIPTION: The patient of this case report presented 2 pairs of equivalent cervical abfraction lesions, under occlusion. Four restorations were placed in teeth 34, 35, 44 and 45. The restorations were divided into groups (Single Bond + Filtek-Flow + Filtek Z250 or Single Bond + Filtek Z250) and the materials were applied according to the manufactures instructions. Two previously calibrated operators placed the restorations and two other independent examiners evaluated the restorations at baseline and after 48 months, according to the USPHS criteria and modified criteria for color match. CONCLUSION: After 48 months of evaluation the lesions restored with Filtek-Flow as a liner under Filtek Z250 did not show better clinical performance than the restorations without Filtek-Flow. All restorations showed a trend toward dark yellowing after 48 months.

Spatial patterns of malaria in a land reform colonization project, Juruena municipality, Mato Grosso, Brazil

Background: In Brazil, 99% of malaria cases are concentrated in the Amazon, and malaria's spatial distribution is commonly associated with socio-environmental conditions on a fine landscape scale. In this study, the spatial patterns of malaria and its determinants in a rural settlement of the Brazilian agricultural reform programme called ""Vale do Amanhecer"" in the northern Mato Grosso state were analysed. Methods: In a fine-scaled, exploratory ecological study, geocoded notification forms corresponding to malaria cases from 2005 were compared with spectral indices, such as the Normalized Difference Vegetation Index (NDVI) and the third component of the Tasseled Cap Transformation (TC_3) and thematic layers, derived from the visual interpretation of multispectral TM-Landsat 5 imagery and the application of GIS distance operators. Results: Of a total of 336 malaria cases, 102 (30.36%) were caused by Plasmodium falciparum and 174 (51.79%) by Plasmodium vivax. Of all the cases, 37.6% (133 cases) were from residents of a unique road. In total, 276 cases were reported for the southern part of the settlement, where the population density is higher, with notification rates higher than 10 cases per household. The local landscape mostly consists of open areas (38.79 km(2)). Training forest occupied 27.34 km(2) and midsize vegetation 7.01 km(2). Most domiciles with more than five notified malaria cases were located near areas with high NDVI values. Most domiciles (41.78%) and malaria cases (44.94%) were concentrated in areas with intermediate values of the TC_3, a spectral index representing surface and vegetation humidity. Conclusions: Environmental factors and their alteration are associated with the occurrence and spatial distribution of malaria cases in rural settlements.

Numerical investigation of the quantum fluctuations of optical fields transmitted through an atomic medium

We have numerically solved the Heisenberg-Langevin equations describing the propagation of quantized fields through an optically thick sample of atoms. Two orthogonal polarization components are considered for the field, and the complete Zeeman sublevel structure of the atomic transition is taken into account. Quantum fluctuations of atomic operators are included through appropriate Langevin forces. We have considered an incident field in a linearly polarized coherent state (driving field) and vacuum in the perpendicular polarization and calculated the noise spectra of the amplitude and phase quadratures of the output field for two orthogonal polarizations. We analyze different configurations depending on the total angular momentum of the ground and excited atomic states. We examine the generation of squeezing for the driving-field polarization component and vacuum squeezing of the orthogonal polarization. Entanglement of orthogonally polarized modes is predicted. Noise spectral features specific to (Zeeman) multilevel configurations are identified.