A new quadrature rule based on a generalized mixed interpolation formula of exponential type

A new method of approximating a function f(x) uniquely by a function fn(x) of the form fn(x) = e1x(aU1(kx) 〈+ bU2(kx) 〈+ ∑ n-2 i=o CiX1), so that fn(x) interpolates f(x) at (n 〈+ 1) equidistant points xo,xo + h,...,xo + nh, with h > O, is derived in a closed-form. Various equivalent forms of the interpolation formula are also derived. A closed-form expression is derived for the error involved in such an approximation. With the aid of the newly derived interpolation formula, a set of Newton Cotes quadrature rules of the closed type are also derived. The total truncation error involved in these quadrature rules are analysed and closed-form expressions for error terms are proposed as conjectures in the two cases when n is odd and when n is even, separately. A more general exponential-type interpolation formula and quadrature rules based upon the generalized mixed interpolation formula are also explained and discussed. A few numerical examples are worked out as illustrations and the results are compared with the results of some of the earlier methods. © 2001 Elsevier Science B.V. All rights reserved

α-convex convergent alternating series .

Various types of convergent alternating series are studied and analysed. Certain important and interesting properties enjoyed by these classes of alternating series, pertaining to the range of the parameter ¬…0 < ¬ < 1†, are discussed. A few illustrations are given, wherever necessary. Further, the generalized ¬-convexity conditions, to be satis®ed by the terms of a series which is l times ¬-convex, are discussed, along with the estimates of the remainders and the term to be added to any partial sum Sk at any lth stage of correction

On modified Gregory rules based on a generalised mixed interpolation formula

AbstractA mixed interpolation function in its generalised form is used to derive the generalised modified Gregory formulae These formulae are expressed in the form of the classical rules along with two correction terms. The error terms are briefly discussed. The newly derived quadrature formulae are tested with certain numerical examples, which shows the efficiency of the generalised modified rules over classical Gregory rules, as well as the modified rules based on the mixed trigonometric interpolation. The importance of the error terms are also discussed

Derivation of the errors involved in interpolation and their application to numerical quadrature formulae

AbstractSimple methods are presented to derive closed-form expressions for the errors involved in the Lagrange interpolation formula. As applications of this formula for the error in the interpolation, the corresponding errors in the quadrature formulae are also taken up. Few examples are considered for numerical experiments

Mathematical approach to analysis of therapeutic properties of four important flowers mentioned in Siribhoovalaya- An ancient Indian multilingual manuscript

Pushpayurveda is considered to be a unique branch of ayurvedic medicine which was first conceived by Jain monks as early as 9th century. Various ancient Indian manuscripts contain notes about the use of flowers for medicinal applications. The petals of flowers could be directly administered orally or can be made into the form of juice, decoction and or by mixing with other active ingredients. ‘Siribhoovalaya’, a multilingual literature written by Jain monk Kumudendu muni mentions the application of eight important flowers for the purification of mercury. In the present study, the antibacterial and antioxidant analysis of extracts of four of the eight mentioned flowers- Panasa, Padari, Kedige, Krishnapushpa was carried out. The results of antioxidant analysis were studied in detail with the help of modern mathematical software MATHEMATICA. Combination of all four flowers was found to be effective against B. cereus, S. aureus and S. marcescenes. The antioxidant activity was found reduced when a combination of the four flowers were used. The extract of Ketaki showed significantly higher antioxidant activity compared to all the other extracts. The whole study is carried out based on the standard tests of hypotheses, using ANOVA and the correlation effect is studied using regression models

Mathematical approach to analysis of therapeutic properties of four important flowers mentioned in Siribhoovalaya- An ancient Indian multilingual manuscript

804-812Pushpayurveda is considered to be a unique branch of ayurvedic medicine which was first conceived by Jain monks as early as 9th century. Various ancient Indian manuscripts contain notes about the use of flowers for medicinal applications. The petals of flowers could be directly administered orally or can be made into the form of juice, decoction and or by mixing with other active ingredients. &lsquo;Siribhoovalaya&rsquo;, a multilingual literature written by Jain monk Kumudendu muni mentions the application of eight important flowers for the purification of mercury. In the present study, the antibacterial and antioxidant analysis of extracts of four of the eight mentioned flowers- Panasa, Padari, Kedige, Krishnapushpa was carried out. The results of antioxidant analysis were studied in detail with the help of modern mathematical software MATHEMATICA. Combination of all four flowers was found to be effective against B. cereus, S. aureus and S. marcescenes. The antioxidant activity was found reduced when a combination of the four flowers were used. The extract of Ketaki showed significantly higher antioxidant activity compared to all the other extracts. The whole study is carried out based on the standard tests of hypotheses, using ANOVA and the correlation effect is studied using regression models

ROLE OF LINEAR ALGEBRA AND ITS APPLICATIONS IN HEALTH CARE MONITORING

Many problems of applications require solving of large system of equations, either under-determined, or determined, or over-determined. The equations may be subjected to constraints. The systems are typically large and sparse systems, wherein the entries of the matrix are predominantly zero. In this review article, we stress upon the applications of solving large systems arising in transmission and emission tomography. Because the measured data is typically insufficient to give a unique solution, optimization techniques such as least-squares method can be used. If the number of equations and the number of variables are small then we can solve the system using Gauss elimination method. It is quite natural in problems of applications, such as medical imaging, to encounter large system of linear equations. Thus, it is common to prefer inexact solutions over exact ones. Even when the number of equations and unknowns is large, there may not be enough information to obtain a unique solution. This is the case of over-determined system of equations and is quite normal in medical tomographic imaging, in which the images are artificially discretized approximations of parts of the interior of the body

WASTE MANAGEMENT IN MATHEMATICAL MODELING

A goal of science is to develop the means for reliable prediction to guidedecision and action. This is accomplished by finding algorithmiccompressions of observations and physical laws. Physical laws arestatements about classes of phenomena, and initial conditions are statementsabout particular systems. Thus, it is the solutions to the equations, and notthe equations themselves, that provide a mathematical description of thephysical phenomena. In constructing and refining mathematical theories, werely heavily on models. At its conception, a model provides the frameworkfor a mathematical interpretation of new phenomena. We apply linearity when we model the behaviour of a device or system that is forced or pushed by a complex set of inputs or excitations. We obtain the response of that device or system to the sum of the individual inputs by adding or superposing the separate responses of the system to each individual input. This important result is called the principle of superposition. Engineers use this principle to predict the response of a system to a complicated input by decomposing or breaking down that input into a set of simpler inputs that produce known system responses or behaviour

On modified Gregory rules based on a generalised mixed interpolation formula

A mixed interpolation function in its generalised form is used to derive the generalised modified Gregory formulae. These formulae are expressed in the form of the classical rules along with two correction terms. The error terms are briefly discussed. The newly derived quadrature formulae are tested with certain numerical examples, which shows the efficiency of the generalised modified rules over classical Gregory rules, as well as the modified rules based on the mixed trigonometric interpolation. The importance of the error terms are also discussed

Derivation of the errors involved in interpolation and their application to numerical quadrature formulae

Simple methods are presented to derive closed-form expressions for the errors involved in the Lagrange interpolation formula. As applications of this formula for the error in the interpolation, the corresponding errors in the quadrature formulae are also taken up. Few examples are considered for numerical experiments