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

On weighted zero-sum sequences

Let G be a finite additive abelian group with exponent exp(G)=n>1 and let A be a nonempty subset of {1,...,n-1}. In this paper, we investigate the smallest positive integer $m$, denoted by s_A(G), such that any sequence {c_i}_{i=1}^m with terms from G has a length n=exp(G) subsequence {c_{i_j}}_{j=1}^n for which there are a_1,...,a_n in A such that sum_{j=1}^na_ic_{i_j}=0. When G is a p-group, A contains no multiples of p and any two distinct elements of A are incongruent mod p, we show that s_A(G) is at most $\lceil D(G)/|A|\rceil+exp(G)-1$ if |A| is at least (D(G)-1)/(exp(G)-1), where D(G) is the Davenport constant of G and this upper bound for s_A(G)in terms of |A| is essentially best possible. In the case A={1,-1}, we determine the asymptotic behavior of s_{{1,-1}}(G) when exp(G) is even, showing that, for finite abelian groups of even exponent and fixed rank, s_{{1,-1}}(G)=exp(G)+log_2|G|+O(log_2log_2|G|) as exp(G) tends to the infinity. Combined with a lower bound of $exp(G)+sum{i=1}{r}\lfloor\log_2 n_i\rfloor$, where $G=\Z_{n_1}\oplus...\oplus \Z_{n_r}$ with 1<n_1|... |n_r, this determines s_{{1,-1}}(G), for even exponent groups, up to a small order error term. Our method makes use of the theory of L-intersecting set systems. Some additional more specific values and results related to s_{{1,-1}}(G) are also computed.Comment: 24 pages. Accepted version for publication in Adv. in Appl. Mat

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

Training fisherwomen in fish processing

A project on training fisherwomen for their participation in rural development, sponsored by Ford Foundation (U.S.A), has been started by the Centre for Agricultural and Rural Development Studies, T.N AU.,at the Fisheries College, Tuticorin. The project alms to select a few literate rural women with leadership qualities for Imparting to them a training In the organisational and managerial aspects of a viable fish processing enterprise. It also alms to assist the trained fisherwomen in organising and operating cottage industries by continued technical backing and thus making the production units demonstration centres for the benefit of other women In the region. The preliminary survey helped in identifying 5 candidates from each of the 3 selected villages. The pre-survey revealed the respondents' choice of subject-areas to undergo training and their enthusiasm to learn techniques for the preparation of fish products like fish pickle and MasI Meen. It also revealed their desire to be exposed to new products like fish wafers, fish oil, fish meal, shark fin-rays etc. The pre- and post evaluations of the training programme helped In I) identifying training needs In the fields of marketing and financial management; II) Identifying some low-cost technological substitutes for some of the commercial products (eg: 'Gadi' for vineger); ill) identifying the products or techniques appreciated by the trainees and the products or techniques that received lukewarm response with reasons for such a response: Iv) Identifying the level of managerial efficiency gained by the trainees and the kind of support required for each Individual to start cottage industries. The programme Is being followed up by Interpersonal contacts and the co-ordinate efforts of the development departments

Multi-Channel Inverse Scattering Problem on the Line: Thresholds and Bound States

We consider the multi-channel inverse scattering problem in one-dimension in the presence of thresholds and bound states for a potential of finite support. Utilizing the Levin representation, we derive the general Marchenko integral equation for N-coupled channels and show that, unlike to the case of the radial inverse scattering problem, the information on the bound state energies and asymptotic normalization constants can be inferred from the reflection coefficient matrix alone. Thus, given this matrix, the Marchenko inverse scattering procedure can provide us with a unique multi-channel potential. The relationship to supersymmetric partner potentials as well as possible applications are discussed. The integral equation has been implemented numerically and applied to several schematic examples showing the characteristic features of multi-channel systems. A possible application of the formalism to technological problems is briefly discussed.Comment: 19 pages, 5 figure

Prediction of Cardiovascular disease using machine learning algorithms on healthcare data

Cardiovascular Disease (CVD) is a leading cause of death worldwide, with the potential to cause serious conditions such as heart attacks and strokes. Early assessment of CVD can significantly reduce mortality rates. In recent studies, machine learning algorithms have been applied to Electronic Health Records (EHR) to estimate risk factors for myocardial infarction. This article explores the use of various machine learning techniques on a healthcare dataset to predict a 10-year risk of future coronary heart disease (CHD). The dataset used in this study was obtained from the Framingham and Massachusetts cardiovascular study. We found that our models achieved varying levels of accuracy: 64% for logistic regression, 83% for Naïve Bayes classifier, 42% for Support Vector Machine (SVM), 65% for Random Forest, 78% for KNN classifier, and 70% for XGBOOST classifier. It is revealed that a patient with no history of heart disease may benefit from an algorithm such as&nbsp; Naive Bayes Classifier, while an older patient with a history of heart disease may require an algorithm such as Support Vector Machine. These factors can help guide the physician in selecting the most appropriate algorithm for each individual patient, ensuring that the diagnosis is as accurate as possible and that the treatment plan is tailored to meet the patient's unique needs

Phase shift effective range expansion from supersymmetric quantum mechanics

Supersymmetric or Darboux transformations are used to construct local phase equivalent deep and shallow potentials for $\ell \neq 0$ partial waves. We associate the value of the orbital angular momentum with the asymptotic form of the potential at infinity which allows us to introduce adequate long-distance transformations. The approach is shown to be effective in getting the correct phase shift effective range expansion. Applications are considered for the $^1P_1$ and $^1D_2$ partial waves of the neutron-proton scattering.Comment: 6 pages, 3 figures, Revtex4, version to be publised in Physical Review

Toward a Spin- and Parity-Independent Nucleon-Nucleon Potential

A supersymmetric inversion method is applied to the singlet $^1S_0$ and $^1P_1$ neutron-proton elastic phase shifts. The resulting central potential has a one-pion-exchange (OPE) long-range behavior and a parity-independent short-range part; it fits inverted data well. Adding a regularized OPE tensor term also allows the reproduction of the triplet $^3P_0$, $^3P_1$ and $^3S_1$ phase shifts as well as of the deuteron binding energy. The potential is thus also spin-independent (except for the OPE part) and contains no spin-orbit term. These important simplifications of the neutron-proton interaction are shown to be possible only if the potential possesses Pauli forbidden bound states, as proposed in the Moscow nucleon-nucleon model.Comment: 9 pages, RevTeX, 5 ps figure