Efficient adaptive integration of functions with sharp gradients and cusps in n-dimensional parallelepipeds

In this paper, we study the efficient numerical integration of functions with sharp gradients and cusps. An adaptive integration algorithm is presented that systematically improves the accuracy of the integration of a set of functions. The algorithm is based on a divide and conquer strategy and is independent of the location of the sharp gradient or cusp. The error analysis reveals that for a $C^0$ function (derivative-discontinuity at a point), a rate of convergence of $n+1$ is obtained in $R^n$. Two applications of the adaptive integration scheme are studied. First, we use the adaptive quadratures for the integration of the regularized Heaviside function---a strongly localized function that is used for modeling sharp gradients. Then, the adaptive quadratures are employed in the enriched finite element solution of the all-electron Coulomb problem in crystalline diamond. The source term and enrichment functions of this problem have sharp gradients and cusps at the nuclei. We show that the optimal rate of convergence is obtained with only a marginal increase in the number of integration points with respect to the pure finite element solution with the same number of elements. The adaptive integration scheme is simple, robust, and directly applicable to any generalized finite element method employing enrichments with sharp local variations or cusps in $n$-dimensional parallelepiped elements.Comment: 22 page

Majorana spin-flip transitions in a magnetic trap

Atoms confined in a magnetic trap can escape by making spin-flip Majorana transitions due to a breakdown of the adiabatic approximation. Several papers have studied this process for atoms with spin $F = 1/2$ or $F= 1$. The present paper calculates the escape rate for atoms with spin $F > 1$. This problem has new features because the perturbation $\Delta T$ which allows atoms to escape satisfies a selection rule $\Delta F_z = 0, \pm 1, \pm 2$ and multi-step processes contribute in leading order. When the adiabatic approximation is satisfied the leading order terms can be summed to yield a simple expression for the escape rate.Comment: 16page

CONTINUITY OF CONDITION SPECTRUM AND ITS LEVEL SET IN BANACH ALGEBRA

For 0 < � < 1 and a Banach algebra element a, this thesis aims to establish the results related to continuity of condition spectrum and its level set correspondence at (�; a). Here we propose a method of study to achieve the continuity. We first identify the Banach algebras at which the interior of the level set of condition spectrum is empty and then we obtain the continuity results. This thesis consists of four chapters. Chapter 1 contains all the prerequisites which are crucial for the development of the thesis. In particular, this chapter has a quick review of the basic properties of spectrum, condition spectrum, upper and lower hemicontiuous correspondences. We also concentrate on analytic vector valued maps and generalized maximum modulus theorem for them. For an element a in A, Chapter 2 has the results related to interior of the level of set of the condition spectrum of a. At first, we focus on

Generalized Duffy transformation for integrating vertex singularities

For an integrand with a 1/r vertex singularity, the Duffy transformation from a triangle (pyramid) to a square (cube) provides an accurate and efficient technique to evaluate the integral. In this paper, we generalize the Duffy transformation to power singularities of the form p(x)/r α , where p is a trivariate polynomial and α &gt; 0 is the strength of the singularity. We use the map (u, v, w) → (x, y, z) : x = u β , y = x v, z = x w, and judiciously choose β to accurately estimate the integral. For α = 1, the Duffy transformation (β = 1) is optimal, whereas if α ≠ 1, we show that there are other values of β that prove to be substantially better. Numerical tests in two and three dimensions are presented that reveal the improved accuracy of the new transformation. Higher-order partition of unity finite element solutions for the Laplace equation with a derivative singularity at a re-entrant corner are presented to demonstrate the benefits of using the generalized Duffy transformation

PT-symmetric square well and the associated SUSY hierarchies

The PT-symmetric square well problem is considered in a SUSY framework. When the coupling strength $Z$ lies below the critical value $Z_0^{\rm (crit)}$ where PT symmetry becomes spontaneously broken, we find a hierarchy of SUSY partner potentials, depicting an unbroken SUSY situation and reducing to the family of $\sec^2$-like potentials in the $Z \to 0$ limit. For $Z$ above $Z_0^{\rm (crit)}$, there is a rich diversity of SUSY hierarchies, including some with PT-symmetry breaking and some with partial PT-symmetry restoration.Comment: LaTeX, 18 pages, no figure; broken PT-symmetry case added (Sec. 6

An Effective Sentence Ordering Approach For Multi-Document Summarization Using Text Entailment

With the rapid development of modern technology electronically available textual information has increased to a considerable amount. Summarization of textual inform ation manually from unstructured text sources creates overhead to the user, therefore a systematic approach is required. Summarization is an approach that focuses on providing the user with a condensed version of the origina l text but in real time applicat ions extended document summarization is required for summarizing the text from multiple documents. The main focus of multi - document summarization is sentence ordering and ranking that arranges the collected sentences from multiple document in order to gene rate a well - organized summary. The improper order of extracted sentences significantly degrades readability and understandability of the summary. The existing system does multi document summarization by combining several preference measures such as chronology, probabilistic, precedence, succession, topical closeness experts to calculate the preference value between sentences. These approach to sent ence ordering and ranking does not address context based similarity measure between sentences which is very ess ential for effective summarization. The proposed system addresses this issues through textual entailment expert system. This approach builds an entailment model which incorpo rates the cause and effect between sentences in the documents using the symmetric measure such as cosine similarity and non - symmetric measures such as unigram match, bigram match, longest common sub - sequence, skip gram match, stemming. The proposed system is efficient in providing user with a contextual summary which significantly impro ves the readability and understandability of the final coherent summa

Second Order Darboux Displacements

The potentials for a one dimensional Schroedinger equation that are displaced along the x axis under second order Darboux transformations, called 2-SUSY invariant, are characterized in terms of a differential-difference equation. The solutions of the Schroedinger equation with such potentials are given analytically for any value of the energy. The method is illustrated by a two-soliton potential. It is proven that a particular case of the periodic Lame-Ince potential is 2-SUSY invariant. Both Bloch solutions of the corresponding Schroedinger equation equation are found for any value of the energy. A simple analytic expression for a family of two-gap potentials is derived

Alternative Methods to Teach History of Anesthesia

Background: History of Anesthesia [HOA] may be taught through lectures, small group discussions, or by one-on-one teaching. HOA competes for scarce time in a busy didactic schedule and for coverage in mainstream medical journals devoted to anesthesiology. These efforts are hampered by the fact that HOA does not have any direct impact on the delivery of modern anesthesia, and the fact that these topics do not appear in written or oral board examinations. We describe three additional modalities to teach HOA to anesthesia residents that have been successfully employed by the Department of Anesthesiology at the University of Massachusetts: 1) Tours; 2) Historical Narratives and Novels; and 3) Movies and Video Clips. Conclusions: It is difficult to impart information using lectures due to time constraints and a very busy didactic schedule in residency programs. HOA related material does not get adequate coverage in standard textbooks of anesthesiology. We describe successful use of three modalities of imparting interesting information in an informal setting. Such efforts provide a unique experience during residency training. From preliminary reports we are confident that details from such tours, novels and movies remain imprinted in their memory for many years, perhaps permanently. These individuals are likely to remain advocates of history, and may choose to devote a part of their academic career towards exploration of HOA