Hole probability for nodal sets of the cut-off Gaussian Free Field

Let ($\Sigma$, g) be a closed connected surface equipped with a riemannian metric. Let ($\lambda$ n) n$\in$N and ($\psi$ n) n$\in$N be the increasing sequence of eigenvalues and the sequence of corresponding L 2-normalized eigenfunctions of the laplacian on $\Sigma$. For each L \textgreater{} 0, we consider $\phi$ L = 0\textless{}$\lambda$n$\le$L $\xi$n \sqrt $\lambda$n $\psi$ n where the $\xi$ n are i.i.d centered gaussians with variance 1. As L $\rightarrow$ $\infty$, $\phi$ L converges a.s. to the Gaussian Free Field on $\Sigma$ in the sense of distributions. We first compute the asymptotic behavior of the covariance function for this family of fields as L $\rightarrow$ $\infty$. We then use this result to obtain the asymptotics of the probability that $\phi$ L is positive on a given open proper subset with smooth boundary. In doing so, we also prove the concentration of the supremum of $\phi$ L around 1 \sqrt 2$\pi$ ln L

The critical threshold for Bargmann-Fock percolation

In this article, we study the excursions sets $\mathcal{D}\_p=f^{-1}([-p,+\infty[)$ where $f$ is a natural real-analytic planar Gaussian field called the Bargmann-Fock field. More precisely, $f$ is the centered Gaussian field on $\mathbb{R}^2$ with covariance $(x,y) \mapsto \exp(-\frac{1}{2}|x-y|^2)$. In [BG16], Beffara and Gayet prove that, if $p \leq 0$, then a.s. $\mathcal{D}\_p$ has no unbounded component. We show that conversely, if $p>0$, then a.s. $\mathcal{D}\_p$ has a unique unbounded component. As a result, the critical level of this percolation model is $0$. We also prove exponential decay of crossing probabilities under the critical level. To show these results, we develop several tools including a KKL-type result for biased Gaussian vectors (based on the analogous result for product Gaussian vectors by Keller, Mossel and Sen in [KMS12]) and a sprinkling inspired discretization procedure. These intermediate results hold for more general Gaussian fields, for which we prove a discrete version of our main result.Comment: 49 pages, 6 figures, minor changes introduce

The Second Welfare Theorem with public goods in general economies

In this paper we prove a general version of the Second Welfare Theorem for a non-convex and non-transitive economy, with public goods and other externalities in consumption. For this purpose we use the sub-gradient to the distance function (normal cone) to define the pricing rule in this general context.Non-convex separation, Second Welfare Theorem, public goods, externalities.

About the phase space of SL(3) Black Holes

In this note we address some issues of recent interest, related to the asymptotic symmetry algebra of higher spin black holes in $sl(3,\mathbb{R})\times sl(3,\mathbb{R})$ Chern Simons (CS) formulation. We compute the fixed time Dirac bracket algebra that acts on two different phase spaces. Both of these spaces contain black holes as zero modes. The result for one of these phase spaces is explicitly shown to be isomorphic to $W^{(2)}_3\times W^{(2)}_3$ in first order perturbations.Comment: improved presentatio

Expected number of nodal components for cut-off fractional Gaussian fields

Let $({\mathcal{X}},g)$ be a closed Riemmanian manifold of dimension $n>0$. Let $\Delta$ be the Laplacian on ${\mathcal{X}}$, and let $(e\_k)\_k$ be an $L^2$-orthonormal and dense family of Laplace eigenfunctions with respective eigenvalues $(\lambda\_k)\_k$. We assume that $(\lambda\_k)\_k$ is non-decreasing and that the $e\_k$ are real-valued. Let $(\xi\_k)\_k$ be a sequence of iid $\mathcal{N}(0,1)$ random variables. For each $L>0$ and $s\in{\mathbb{R}}$, possibly negative, set$$f^s\_L=\sum\_{0<\lambda\_j\leq L}\lambda\_j^{-\frac{s}{2}}\xi\_je\_j\, .$$Then, $f\_L^s$ is almost surely regular on its zero set. Let $N\_L$ be the number of connected components of its zero set. If $s<\frac{n}{2}$, then we deduce that there exists $\nu=\nu(n,s)>0$ such that $N\_L\sim \nu {Vol}\_g({\mathcal{X}})L^{n/2}$ in $L^1$ and almost surely. In particular, ${\mathbb{E}}[N\_L]\asymp L^{n/2}$. On the other hand, we prove that if $s=\frac{n}{2}$ then$${\mathbb{E}}[N\_L]\asymp \frac{L^{n/2}}{\sqrt{\ln\left(L^{1/2}\right)}}\, .$$In the latter case, we also obtain an upper bound for the expected Euler characteristic of the zero set of $f\_L^s$ and for its Betti numbers. In the case $s>n/2$, the pointwise variance of $f\_L^s$ converges so it is not expected to have universal behavior as $L\rightarrow+\infty$.Comment: 31 pages, corrections of typographical mistakes, correction in Theorem 1.3 and details added in proof

Lesson Plan, Social Studies, 6th Grade

TEKS (Texas Essential Knowledge and Skills): (1) History. The student understands that historical events influence contemporary events. The student is expected to: (A) trace characteristics of various contemporary societies in regions that resulted from historical events or factors such as colonization, immigration, and trade; and (B) analyze the historical background of various contemporary societies to evaluate relationships between past conflicts and current conditions. (2) History. The student understands the influences of individuals and groups from various cultures on various historical and contemporary societies. The student is expected to: (A) identify and describe the historical influence of individuals or groups on various contemporary societies; and (B) describe the social, political, economic, and cultural contributions of individuals and groups from various societies, past and present. (3) Geography. The student understands the factors that influence the locations and characteristics of locations of various contemporary societies on maps and/or globes. The student is expected to: (A) identify and explain the geographic factors responsible for patterns of population in places and regions; (B) explain ways in which human migration influences the character of places and regions (4) Geography. The student understands how geographic factors influence the economic development and political relationships of societies. The student is expected to: (A) explain the geographic factors responsible for the location of economic activities in places and regions; and (B) identify geographic factors such as location, physical features, transportation corridors and barriers, and distribution of natural resources that influence a society\u27s political relationships. (5) Geography. The student understands the impact of interactions between people and the physical environment on the development and conditions of places and regions. The student is expected to: (B) identify and analyze ways people have adapted to the physical environment in various places and regions; and (C) identify and analyze ways people have modified the physical environment such as mining, irrigation, and transportation infrastructure. (6) Economics. The student understands the factors of production in a society\u27s economy. The student is expected to: (A) describe ways in which the factors of production (natural resources, labor, capital, and entrepreneurs influence the economies of various contemporary societies; (B) identify problems that may arise when one or more of the factors of production is in relatively short supply. Lesson objective(s): 1.Students will be able to apply main ideas, key terms, and places by completing questions, analysis and discussion related to Nuevo Santander 2. Students will be able to retell, define compare and summarize main ideas, key terms and place related to Nuevo Santander 3.Students will be able to locate Nuevo Santander Geographically. Differentiation strategies to meet diverse learner needs: Differentiation provided for all students include the following repeated instruction, check for understanding, extended time; visual support, monitorin

Migraine management

Migraines are common illnesses. Studies conducted in 12 Latin American cities, including two in Mexico, have found that its prevalence in our country is 15%. The rate in gender is 3:1 (Women/Men) worldwide. The diagnostic criteria for migraines were published for the first time in 1988, in the first edition of the International Headache Classification, promoted by the International Headache Society, with its second edition in 2003, and a third beta version that will probably be published in 2015. Diagnostic criteria for the different forms of migraines were first described in this document, which has simplified communication among doctors and made possible comparisons between studies. The current migraine criteria (with and without aura) are shown in Table 1. As in all primary headaches, paraclinic and imaging studies are normal and rarely necessary, exception made for the cases where there is clinical doubt or whenever the patient is too anxious and wishes to be ‘‘as certain as you can be’’. This may be a valid reason; however it may increase the cost of medical attention and can be problematic in institutions with a high volume of patients

HAWC response to atmospheric electricity activity

The HAWC Gamma Ray observatory consists of 300 water Cherenkov detectors (WCD) instrumented with four photo multipliers tubes (PMT) per WCD. HAWC is located between two of the highest mountains in Mexico. The high altitude (4100 m asl), the relatively short distance to the Gulf of Mexico (~100 km), the large detecting area (22 000 m$^2$) and its high sensitivity, make HAWC a good instrument to explore the acceleration of particles due to the electric fields existing inside storm clouds. In particular, the scaler system of HAWC records the output of each one of the 1200 PMTs as well as the 2, 3, and 4-fold multiplicities (logic AND in a time window of 30 ns) of each WCD with a sampling rate of 40 Hz. Using the scaler data, we have identified 20 enhancements of the observed rate during periods when storm clouds were over HAWC but without cloud-earth discharges. These enhancements can be produced by electrons with energy of tens of MeV, accelerated by the electric fields of tens of kV/m measured at the site during the storm periods. In this work, we present the recorded data, the method of analysis and our preliminary conclusions on the electron acceleration by the electric fields inside the clouds.Comment: Presented at the 35th International Cosmic Ray Conference (ICRC2017), Bexco, Busan, Korea. See arXiv:1708.02572 for all HAWC contribution

Covering THz Gap using photomixers techonologies: arrays and new antenna topologies

Mención Internacional en el título de doctorEl objetivo de esta tesis es el diseño y construcción de generadores de señal de terahercios basados en fotomezcladores de bajo coste y que funcionen a temperatura ambiente. Como se explica en la introducción, existen muchas aplicaciones potenciales a frecuencias comprendidas entre los 100 GHz y 1.000 GHz. Propuestas en el campo de la seguridad, radioastronomía y medicina, entre otros, han sido consideradas y evaluadas. Algunas de ellas han comenzado a ser explotadas comercialmente, por ejemplo, en la detección de armas y explosivos ocultos en puestos de control y seguridad. No obstante, todavía existe un nicho, el de la electrónica de consumo (comunicaciones, entretenimiento, etc), en el que las frecuencias de esta banda no son utilizadas por su alto coste. En una primera parte, se analizan distintos tipos de antenas planares que son especialmente atractivas para ser integradas con dispositivos fotomezcladores generadores de terahercios. El efecto de los sustratos semiconductores en sus propiedades de radiación es analizado en profundidad. Así mismo, se estudian las principales familias de dispositivos fotomezcladores. A continuación, se propone el uso de antenas dieléctricas combinadas con este tipo de fuentes. Con esto, se pretende evitar el uso de lentes dieléctricas de alto costo sin reducir drásticamente la cantidad de potencia de terahercios radiada por la fuente. Se propone un diseño de antena dieléctrica que permitiría aprovechar el ancho de banda de este tipo de generadores, comenzando en los 50-60 GHz hasta por encima del terahercio. Este tipo de antenas dieléctricas ofrece la posibilidad de combinar varios generadores a fin de cumplir dos objetivos: incrementar la potencia disponible y escanear electrónicamente en uno o dos planos. Se propone el uso de nanotubos de carbono para la realización de desfasadores controlados electrónicamente, componente imprescidible para arrays de escaneo electrónico. Finalmente, se estudia la capacidad de este tipo de antenas de actuar como acoplos de campo cercano en resonadores. Como ejemplo de aplicación, se propone su uso en un receptor de terahercios cuya sensibilidad, no limitada por el ruido térmico, llega al límite cuántico, esto es, la detección de fotones de terahercios.The objective of this thesis is the design and manufacturing of roomtemperature cost-affordable terahertz sources based on photomixing. As explained in the preface, there are many potential applications for frequencies between 100 GHz and 1 THz. Many of them, specially in the fields of security, radioastronomy, and medicine have been deeply studied and evaluated. Some of them have led to already available commercial products, e.g. in hidden weapons and explosives detection in security controls. Nevertheless, there are many consumer-oriented applications related with communication and entertainment industry that cannot be implemented because of the actual costs of terahertz systems. Firstly, some planar antenna topologies appealing for photomixing sources are shown. The effect of the semiconductor substrates on its radiation properties is theoretically analyzed. Also, the main families of terahertz photomixing sources are described. Secondly, the use of dielectric antennas in combination with photomixing sources is proposed. This allows to reduce costs without a drastic decrement of the radiated power. An utra-wideband design that could allow us to take advantage of the photomixers bandwidth (from 50-60 GHz up to beyond the terahertz) is presented. Thirdly, the combination of many sources in array configurations is proposed. It allows us to accomplish two objectives: increasing the available terahertz power and 1D or 2D beam steering. A novel phase shifter made of carbon nanotubes is presented, since such component is critical for the development of a electronically steerable array. Finally, the use of this family of dielectric antennas as a near field coupler for dielectric resonators is studied. A room temperature photoncounting teraherz receiver is proposed as an example of application. This receiver is limited only by a quantum limit, which gives the theoretical maximum achievable sensitivity for a receiver.Programa Oficial de Doctorado en Multimedia y ComunicacionesPresidente: Zoya Popovic.- Secretario: Luis Emilio García Castillo.- Vocal: Cyril Luxe