International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
This document discusses the formulation of fractional supersymmetric theories in one dimension. It begins by presenting fractional superspace and fractional supersymmetry of order F=3, including the fractional supersymmetry transformations. It then derives the fractional supercharges and Euler-Lagrange equations for F=3. Finally, it generalizes the formulation to arbitrary fractional order F ≥ 3 by introducing fractional superspace and supersymmetry transformations of order F, as well as an action invariant under such transformations.
Divergence measures are useful for comparing two probability distributions. Depending on the nature of
the problem, different divergence measures are suitable. So it is always desirable to develop a new
divergence measure.
Recently, Jain and Chhabra [6] introduced new series ( ( , ) m x P Q , ( , ) m z P Q and ( , ) m w P Q for
mÎN ) of information divergence measures, defined the properties and characterized, compared with
standard divergences and derived the new series ( ( ) * , m x P Q formÎN ) of metric spaces.
In this work, various important and interesting relations among divergences of these new series and other
well known divergence measures are obtained. Some intra relations among these new divergences are
evaluated as well and bounds of new divergence measure ( ( ) 1 x P,Q ) are obtained by using Csiszar’s
information inequalities. Numerical illustrations (Verification) regarding bounds are done as well.
In this paper we define some new operations in fuzzy soft multi set theory and show that the DeMorgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations in our way. Also some new results along with illustrating examples have been put
forward in our work.
This document contains the marking scheme for the Mathematics exam of class 12 from the year 2017-18. It lists 12 questions from Section A, 7 questions from Section B, and 4 questions from Section C along with the marks assigned to each question. For most questions, the full solution is provided with marks assigned based on the steps shown. The marking scheme provides the question numbers, expected answers, and total marks to evaluate student responses on the Mathematics exam.
This document summarizes research on deficient quartic spline interpolation. It begins by introducing the topic and defining deficient quartic splines. It then proves the existence and uniqueness of a spline interpolation that matches given functional values and derivatives at interior points, with specified boundary conditions. Specifically, it shows there is a unique spline if the mesh size is greater than or equal to the interval length divided by 2. Next, the document derives error bounds for the spline interpolation. It obtains pointwise bounds for the error function and shows the error is bounded above by a function involving the fifth modulus of smoothness of the given function. In conclusion, best possible error bounds are obtained for the deficient quartic spline interpolation method presented.
The document contains a sample mathematics exam with multiple choice and paragraph style questions covering topics such as lines, planes, parabolas, functions, probability, and trigonometric functions. There are 3 sections with a total of 60 questions. Section 1 contains multiple choice questions with one or more correct answers. Section 2 contains paragraph style questions with one correct answer for each related question. Section 3 contains matching questions linking concepts to their values or outcomes.
Decomposition formulas for H B - hypergeometric functions of three variablesinventionjournals
: In this paper we investigate several decomposition formulas associated with hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 5 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell's hypergeometric functions.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
This document discusses the formulation of fractional supersymmetric theories in one dimension. It begins by presenting fractional superspace and fractional supersymmetry of order F=3, including the fractional supersymmetry transformations. It then derives the fractional supercharges and Euler-Lagrange equations for F=3. Finally, it generalizes the formulation to arbitrary fractional order F ≥ 3 by introducing fractional superspace and supersymmetry transformations of order F, as well as an action invariant under such transformations.
Divergence measures are useful for comparing two probability distributions. Depending on the nature of
the problem, different divergence measures are suitable. So it is always desirable to develop a new
divergence measure.
Recently, Jain and Chhabra [6] introduced new series ( ( , ) m x P Q , ( , ) m z P Q and ( , ) m w P Q for
mÎN ) of information divergence measures, defined the properties and characterized, compared with
standard divergences and derived the new series ( ( ) * , m x P Q formÎN ) of metric spaces.
In this work, various important and interesting relations among divergences of these new series and other
well known divergence measures are obtained. Some intra relations among these new divergences are
evaluated as well and bounds of new divergence measure ( ( ) 1 x P,Q ) are obtained by using Csiszar’s
information inequalities. Numerical illustrations (Verification) regarding bounds are done as well.
In this paper we define some new operations in fuzzy soft multi set theory and show that the DeMorgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations in our way. Also some new results along with illustrating examples have been put
forward in our work.
This document contains the marking scheme for the Mathematics exam of class 12 from the year 2017-18. It lists 12 questions from Section A, 7 questions from Section B, and 4 questions from Section C along with the marks assigned to each question. For most questions, the full solution is provided with marks assigned based on the steps shown. The marking scheme provides the question numbers, expected answers, and total marks to evaluate student responses on the Mathematics exam.
This document summarizes research on deficient quartic spline interpolation. It begins by introducing the topic and defining deficient quartic splines. It then proves the existence and uniqueness of a spline interpolation that matches given functional values and derivatives at interior points, with specified boundary conditions. Specifically, it shows there is a unique spline if the mesh size is greater than or equal to the interval length divided by 2. Next, the document derives error bounds for the spline interpolation. It obtains pointwise bounds for the error function and shows the error is bounded above by a function involving the fifth modulus of smoothness of the given function. In conclusion, best possible error bounds are obtained for the deficient quartic spline interpolation method presented.
The document contains a sample mathematics exam with multiple choice and paragraph style questions covering topics such as lines, planes, parabolas, functions, probability, and trigonometric functions. There are 3 sections with a total of 60 questions. Section 1 contains multiple choice questions with one or more correct answers. Section 2 contains paragraph style questions with one correct answer for each related question. Section 3 contains matching questions linking concepts to their values or outcomes.
Decomposition formulas for H B - hypergeometric functions of three variablesinventionjournals
: In this paper we investigate several decomposition formulas associated with hypergeometric functions H B in three variables. Many operator identities involving these pairs of symbolic operators are first constructed for this purpose. By means of these operator identities, as many as 5 decomposition formulas are then found, which express the aforementioned triple hypergeometric functions in terms of such simpler functions as the products of the Gauss and Appell's hypergeometric functions.
On Fuzzy Soft Multi Set and Its Application in Information Systems ijcax
Research on information and communication technologies have been developed rapidly since it can be applied easily to several areas like computer science, medical science, economics, environments, engineering, among other. Applications of soft set theory, especially in information systems have been found paramount importance. Recently, Mukherjee and Das defined some new operations in fuzzy soft multi set theory and show that the De-Morgan’s type of results hold in fuzzy soft multi set theory with respect to these newly defined operations. In this paper, we extend their work and study some more basic properties of their defined operations. Also, we define some basic supporting tools in information system also application of fuzzy soft multi sets in information system are presented and discussed. Here we define the notion of fuzzy multi-valued information system in fuzzy soft multi set theory and show that every fuzzy soft multi set is a fuzzy multi valued information system.
1. This document contains 5 multiple choice questions about functions and their compositions.
2. The questions involve evaluating functions, finding the composition of two functions, and determining the result of adding two functions together.
3. This summary provides a high-level overview of the key elements and purpose contained within the document.
On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACESijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document presents the solution to quadruple Fourier series equations involving heat polynomials. Quadruple series equations are useful for solving four-part boundary value problems in fields like electrostatics and elasticity. The document considers two sets of quadruple series equations, the first kind and second kind, involving heat polynomials of the first and second kind. The solutions are obtained by reducing the problems to simultaneous Fredholm integral equations of the second kind. The specific equations considered and the steps to solve them using operator theory are presented.
This document discusses solving systems of three linear equations (SPLTV) using the elimination method. It provides the general form of three linear equations with three variables (x, y, z), defines the solution set as values of x, y, z that satisfy all three equations, and outlines the 6-step elimination method process. An example problem is shown to demonstrate applying the elimination method to obtain the solution set {(x,y,z)}. Practice problems are provided for students to solve SPLTV systems using the elimination method in their workbooks.
On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
The document provides 5 examples of defining and calculating sequences based on functions. The first two examples show defining sequences by functions f(x)=2x-7 and f(x)=2x^3-4, and calculating the first 5 terms. The third example defines the sequence by f(x)=5x^2-3x+7 and lists the first 3 terms. The fourth example provides 3 infinite sequences defined by patterns in the terms. The fifth example lists 3 finite sequences.
Pedagogy of Mathematics (Part II) - Set language introduction and Ex.1.4, Set Language, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy,
This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
This document discusses multiplication and division of integral expressions. It begins by explaining the rules for multiplying powers of the same base. When powers of the same base are multiplied, the base does not change and the exponents are added. It then explains how to multiply monomials by distributing coefficients and adding exponents of the same variables. Examples are provided to illustrate multiplying powers, monomials, and applying the rules to example problems involving distances and speeds.
The document provides information about calculating marginal gains values for different orderings of elements in a set. It defines the marginal gain function and describes calculating marginal gains for 3 elements in a set under different orderings of those elements. Equations for the marginal gain function are provided along with numeric examples to illustrate the calculations.
Pedagogy of Mathematics (Part II) - Set language introduction and Ex.1.5, Set Language, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy,
X std mathematics - Relations and functions (Ex 1.2), Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, relations, definition of relations, null relation
The document discusses Fourier series and periodic functions. It provides:
- Definitions of Fourier series and periodic functions.
- Examples of periodic functions including trigonometric and other functions.
- Euler's formulae for calculating the coefficients of a Fourier series.
- Integration properties used to solve Fourier series problems.
- Two examples of determining the Fourier series for given periodic functions and using it to deduce mathematical results.
This document introduces two new fractional integration operators associated with the H-function of two variables. The H-function of two variables is defined and some of its key properties are established. These operators generalize previous results given by other authors. The behavior of the H-function for small and large values of the variables is also described. When certain parameters are set to specific values, the H-function reduces to known cases.
The document discusses the neighbor-joining method for reconstructing phylogenetic trees from evolutionary distance data. It was developed by Saitou and Nei in 1987. The method works by finding pairs of operational taxonomic units (OTUs) that minimize the total branch length at each stage of clustering starting with a starlike tree. The input is the number of taxa and the output is an unrooted tree with branches. The document then provides a step-by-step example of applying the neighbor-joining method to calculate and construct a phylogenetic tree from distance data between 6 OTUs.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
The document discusses summation notation and various summation formulas and properties. It defines summation as the sum of all terms in an infinite sequence, represented by ∑. Some key points summarized:
1. The sum of the first n terms of a geometric progression, where the ratio r is between -1 and 1, is Sn = a/(1-r).
2. Common properties of infinite sums include: the sum of two infinite sums is equal to the sum of the individual sums, and the sum of a constant multiplied by terms of an infinite sum is equal to the constant multiplied by the sum.
3. The sum of the first n positive integers can be represented using the formula ∑i=1n
2022 ملزمة الرياضيات للصف السادس الاحيائي - الفصل الخامس - المعادلات التفاضليةanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الخامس المعادلات التفاضلية 2022 anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
This document contains math problems involving ratios, proportions, functions, and sets. It asks the student to:
1) Find missing values in ratios and proportions.
2) Represent sets, relations, and functions using diagrams like arrow diagrams and Cartesian diagrams.
3) Identify properties of functions like their domain, codomain, and range.
4) Graph linear and quadratic functions and find features like the x- and y-intercepts.
5) Solve proportions and equations involving ratios.
So in summary, the document provides exercises to practice fundamental concepts involving ratios, proportions, functions, and sets through problems requiring calculations, diagrammatic representation, graphing, and solving equations.
General Solution of Equations of Motion of Axisymmetric Problem of Micro-Isot...IJERA Editor
In this paper, we obtain the general solution of equations of motion of axisymmetric problem of micro-isotropic,
micro-elastic solid in static case. The equations of motion of axisymmetric problem are converted into vector
matrix differential equations using the Hankel transform. Applying the technique of solving the eigen value
problem, the general solution of the said problem is obtained. The results of the corresponding problem in linear
micropolar elasticity are obtained as a particular case of this paper.
1. This document contains 5 multiple choice questions about functions and their compositions.
2. The questions involve evaluating functions, finding the composition of two functions, and determining the result of adding two functions together.
3. This summary provides a high-level overview of the key elements and purpose contained within the document.
On (1,2)*-πgθ-CLOSED SETS IN BITOPOLOGICAL SPACESijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
This document presents the solution to quadruple Fourier series equations involving heat polynomials. Quadruple series equations are useful for solving four-part boundary value problems in fields like electrostatics and elasticity. The document considers two sets of quadruple series equations, the first kind and second kind, involving heat polynomials of the first and second kind. The solutions are obtained by reducing the problems to simultaneous Fredholm integral equations of the second kind. The specific equations considered and the steps to solve them using operator theory are presented.
This document discusses solving systems of three linear equations (SPLTV) using the elimination method. It provides the general form of three linear equations with three variables (x, y, z), defines the solution set as values of x, y, z that satisfy all three equations, and outlines the 6-step elimination method process. An example problem is shown to demonstrate applying the elimination method to obtain the solution set {(x,y,z)}. Practice problems are provided for students to solve SPLTV systems using the elimination method in their workbooks.
On Some Double Integrals of H -Function of Two Variables and Their ApplicationsIJERA Editor
This paper deals with the evaluation of four integrals of H -function of two variables proposed by Singh and
Mandia [7] and their applications in deriving double half-range Fourier series for the H -function of two
variables. A multiple integral and a multiple half-range Fourier series of the H -function of two variables are
derived analogous to the double integral and double half-range Fourier series of the H -function of two
variables.
The document provides 5 examples of defining and calculating sequences based on functions. The first two examples show defining sequences by functions f(x)=2x-7 and f(x)=2x^3-4, and calculating the first 5 terms. The third example defines the sequence by f(x)=5x^2-3x+7 and lists the first 3 terms. The fourth example provides 3 infinite sequences defined by patterns in the terms. The fifth example lists 3 finite sequences.
Pedagogy of Mathematics (Part II) - Set language introduction and Ex.1.4, Set Language, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy,
This document provides practice problems for additional mathematics Form 4 students in Terengganu, Malaysia. It covers topics on quadratic equations and quadratic functions, with multiple choice and short answer questions. The problems are divided into three sections: quadratic equations, quadratic functions for paper 1, and quadratic functions for paper 2. The document is copyrighted material from the Terengganu State Education Department.
This document discusses multiplication and division of integral expressions. It begins by explaining the rules for multiplying powers of the same base. When powers of the same base are multiplied, the base does not change and the exponents are added. It then explains how to multiply monomials by distributing coefficients and adding exponents of the same variables. Examples are provided to illustrate multiplying powers, monomials, and applying the rules to example problems involving distances and speeds.
The document provides information about calculating marginal gains values for different orderings of elements in a set. It defines the marginal gain function and describes calculating marginal gains for 3 elements in a set under different orderings of those elements. Equations for the marginal gain function are provided along with numeric examples to illustrate the calculations.
Pedagogy of Mathematics (Part II) - Set language introduction and Ex.1.5, Set Language, Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy,
X std mathematics - Relations and functions (Ex 1.2), Maths, IX std Maths, Samacheerkalvi maths, II year B.Ed., Pedagogy, Mathematics, relations, definition of relations, null relation
The document discusses Fourier series and periodic functions. It provides:
- Definitions of Fourier series and periodic functions.
- Examples of periodic functions including trigonometric and other functions.
- Euler's formulae for calculating the coefficients of a Fourier series.
- Integration properties used to solve Fourier series problems.
- Two examples of determining the Fourier series for given periodic functions and using it to deduce mathematical results.
This document introduces two new fractional integration operators associated with the H-function of two variables. The H-function of two variables is defined and some of its key properties are established. These operators generalize previous results given by other authors. The behavior of the H-function for small and large values of the variables is also described. When certain parameters are set to specific values, the H-function reduces to known cases.
The document discusses the neighbor-joining method for reconstructing phylogenetic trees from evolutionary distance data. It was developed by Saitou and Nei in 1987. The method works by finding pairs of operational taxonomic units (OTUs) that minimize the total branch length at each stage of clustering starting with a starlike tree. The input is the number of taxa and the output is an unrooted tree with branches. The document then provides a step-by-step example of applying the neighbor-joining method to calculate and construct a phylogenetic tree from distance data between 6 OTUs.
This document discusses partial ordering in the context of soft sets. It begins with basic definitions of soft sets and soft set operations like complement, Cartesian product, and composition of soft set relations. It then defines what a partial order is in terms of being reflexive, antisymmetric, and transitive. A partially ordered soft set is one where the soft set elements have a partial order defined on them. Linear (total) ordering is also discussed, where all elements in the soft set are comparable. Examples are provided to illustrate these concepts of ordering in soft sets.
The document discusses summation notation and various summation formulas and properties. It defines summation as the sum of all terms in an infinite sequence, represented by ∑. Some key points summarized:
1. The sum of the first n terms of a geometric progression, where the ratio r is between -1 and 1, is Sn = a/(1-r).
2. Common properties of infinite sums include: the sum of two infinite sums is equal to the sum of the individual sums, and the sum of a constant multiplied by terms of an infinite sum is equal to the constant multiplied by the sum.
3. The sum of the first n positive integers can be represented using the formula ∑i=1n
2022 ملزمة الرياضيات للصف السادس الاحيائي - الفصل الخامس - المعادلات التفاضليةanasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
In this paper based on recently introduced approach we formulated some recommendations to optimize
manufacture drift bipolar transistor to decrease their dimensions and to decrease local overheats during
functioning. The approach based on manufacture a heterostructure, doping required parts of the heterostructure
by dopant diffusion or by ion implantation and optimization of annealing of dopant and/or radiation
defects. The optimization gives us possibility to increase homogeneity of distributions of concentrations
of dopants in emitter and collector and specific inhomogenous of concentration of dopant in base and at the
same time to increase sharpness of p-n-junctions, which have been manufactured framework the transistor.
We obtain dependences of optimal annealing time on several parameters. We also introduced an analytical
approach to model nonlinear physical processes (such as mass- and heat transport) in inhomogenous media
with time-varying parameters.
ملزمة الرياضيات للصف السادس التطبيقي الفصل الخامس المعادلات التفاضلية 2022 anasKhalaf4
طبعة جديدة ومنقحة
حل تمارين الكتاب
شرح المواضيع الرياضية بالتفصيل وبأسلوب واضح ومفهوم لجميع المستويات
حلول الاسألة الوزارية
اعداد الدكتور أنس ذياب خلف
email: anasdhyiab@gmail.com
This document contains math problems involving ratios, proportions, functions, and sets. It asks the student to:
1) Find missing values in ratios and proportions.
2) Represent sets, relations, and functions using diagrams like arrow diagrams and Cartesian diagrams.
3) Identify properties of functions like their domain, codomain, and range.
4) Graph linear and quadratic functions and find features like the x- and y-intercepts.
5) Solve proportions and equations involving ratios.
So in summary, the document provides exercises to practice fundamental concepts involving ratios, proportions, functions, and sets through problems requiring calculations, diagrammatic representation, graphing, and solving equations.
General Solution of Equations of Motion of Axisymmetric Problem of Micro-Isot...IJERA Editor
In this paper, we obtain the general solution of equations of motion of axisymmetric problem of micro-isotropic,
micro-elastic solid in static case. The equations of motion of axisymmetric problem are converted into vector
matrix differential equations using the Hankel transform. Applying the technique of solving the eigen value
problem, the general solution of the said problem is obtained. The results of the corresponding problem in linear
micropolar elasticity are obtained as a particular case of this paper.
Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)paperpublications3
Abstract: Ramanujan’s lost notebook contains many results on mock theta functions. In particular, the lost notebook contains eight identities for tenth order mock theta functions. Previously many authors proved the first six of Ramanujan’s tenth order mock theta function identities. It is the purpose of this paper to prove the seventh and eighth identities of Ramanujan’s tenth order mock theta function identities which are expressed by mock theta functions and also a definite integral. The properties of modular forms are used for the proofs of theta function identities and L. J. Mordell’s transformation formula for the definite integral.Keywords: Mock Theta Functions from Ramanujan’s Lost Notebook.
Title: Some Continued Mock Theta Functions from Ramanujan’s Lost Notebook (IV)
Author: MOHAMMADI BEGUM JEELANI SHAIKH
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
The document describes a Prolog program for planning flight routes between locations on a given day. It defines predicates for finding direct and indirect flight routes based on a timetable database. The timetable stores flights between locations with departure and arrival times, flight numbers, and valid days. The route predicate uses the flight predicate to recursively find valid multi-segment routes that ensure a minimum transfer time between connections. Sample queries and timetable facts are provided to demonstrate the program's operation.
The document analyzes a homogeneous cubic equation with six unknowns of the form αxy(x+y) + βzw(z+w) = (α + β)XY(X+Y). It presents three approaches to finding integral solutions to this equation.
The first approach assumes specific forms for the unknowns and reduces the equation to a form involving two new variables, which are then related to special numbers to obtain the integral solutions. Several properties relating the solutions to other numbers are presented.
The second approach uses a linear transformation and factoring to express the two variables as functions involving a parameter, yielding another set of integral solutions. More solution properties are given.
The third approach similarly uses factor
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
A Boundary Value Problem and Expansion Formula of I - Function and General Cl...IJERA Editor
In the present paper, we make a model of a boundary value problem and then obtain its solution involving
products of I -function and a general class of polynomials.
Inversion Theorem for Generalized Fractional Hilbert Transforminventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
This document contains information about a calculus project completed by students of the Mechanical Engineering department at Laxmi Institute of Technology in Sarigam. It includes the names and student IDs of 13 students who participated in the project. The document covers topics in multiple integrals, including double integrals, Fubini's theorem, double integrals in polar coordinates, and triple integrals. Formulas and examples are provided for each topic.
1. The document provides solutions to 5 homework problems involving probability distributions and expectations. It finds probabilities, probability density functions, cumulative distribution functions, and expectations for various random variables.
2. It summarizes the key steps and results for each problem, including defining relevant random variables, identifying their distributions, and calculating requested probabilities, densities, distributions, and expectations through integration.
3. The solutions demonstrate techniques for determining distributions and related metrics of random variables given their definitions and relationships to other random variables.
This document summarizes the key points from a journal article about Ramanujan's formula for an infinite series. It discusses 12 classes of related infinite series and evaluates them in closed form for certain positive integer values. Several theorems are proved relating the different series and evaluating specific cases. The document aims to provide corrected versions and new proofs of some of Ramanujan's formulas for infinite series.
Fixed points theorem on a pair of random generalized non linear contractionsAlexander Decker
1) The document presents a fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of a separable Banach space.
2) It proves that if two random operators A1(w) and A2(w) satisfy a certain inequality involving upper semi-continuous functions, then there exists a unique random variable η(w) that is the common fixed point of A1(w) and A2(w).
3) As an example, the theorem is applied to prove the existence of a solution in a Banach space to a random non-linear integral equation of the form x(t;w) = h(t;w) + integral of k
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Stability of Iteration for Some General Operators in b-MetricKomal Goyal
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1. International Journal of Computational Engineering Research||Vol, 04||Issue, 2||
Fractional Derivative Associated With the Generalized M-Series and
Multivariable Polynomials
1,
Ashok Singh Shekhawat , 2,Jyoti Shaktawat
1,
Department of Mathematics Arya College of Engineering and Information Technology, Jaipur, Rajasthan
2,
Department of Mathematics Kautilya Institute of Technology and Engineering, Jaipur, Rajasthan
ABSTRACT
The aim of present paper is to derive a fractional derivative of the multivariable H-function of Srivastava
and Panda [9], associated with a general class of multivariable polynomials of Srivastava [6] and the
generalized Lauricella functions of Srivastava and Daoust [11] the generalized M-series. Certain special
cases have also been discussed. The results derived here are of a very general nature and hence
encompass several cases of interest hitherto scattered in the literature.
I.
INTRODUCTION
In this paper the H-function of several complex variables introduced and studied by Srivastava and
Panda [9] is an extension of the multivariable G-function and includes Fox’s H-function, Meijer’s G-function of
one and two variables, the generalized Lauricella functions of Srivastava and Daoust [11], Appell functions etc.
In this note we derive a fractional derivative of H-function of several complex variables of Srivastava and Panda
[9], associated with a general polynomials (multivariable) of Srivastava [6] and the generalized Lauricella
functions of Srivastavaand Daoust [11].Generalized M-series extension of the both Mittag-Laffler function and
generalized hypergeometric functions.
II.
DEFINITIONS AND NOTATIONS
By Oldham and Spanner [4] and Srivastava and Goyal [7] the fractional derivative of a function f(t) of
complex order
t
1
1
f(x) dx, Re( 0
0 t x)
D f(t)}
a
t
m
d
m
D
f(t)} 0 Re( m
m a t
dt
Where m is positive integer.
The multivariable H-function is defined by Srivastava and Panda [9] in the following manner
0 u' , v' ) ;...; (u
H [z z H
1
r
1
2 i)
where
r
L
i
(r)
A, C : [B' , D' ] ;...; (B
1
L
1
v
(r)
1
(r)
D
z
1
z
r
(r)
a) : ',...,
(r)
b' ) : ' ] ;...; [b
c) : ',...,
(r)
d' ) : ' ] ;...; [d
r
1
1
r
z
r
1
1
z
r
r
(r)
(r)
(r)
(r)
d d
1
…(2.1)
…(2.2)
r
r
1 .
The general class of multivariable polynomials defined by Srivastava [6] defined as
S
p p
1
s
q q
1
s
q
x x
1
s
||Issn 2250-3005 ||
1
p
1
k 0
1
q
s
p
k
s
s
0
q
1
p k
1 1
k
1
q
s
k
s
p k
s s
||February||2014||
Page 35
2. Fractional Derivative Associated With The…
A [q k q k x
1
where
q
1
s
0 1 2 p
j
j
s
k
1
1
x
k
…(2.3)
s
s
0 j 1,..., s) are
non-zero
arbitrary
positive
integer
the
coefficients
A [q k q k being arbitrary constants, real or complex.
1
1
s
s
The following known result of Srivastava and Panda [10]
Lemma. If ( ≥ 0), 0< x < 1, Re (1+p) > 0, Re(q) > 1, i > 0 and i > 0 or i = 0 and | zi | < , i = 1,2,…,r then
x
z x 1
1
F
z x r
r
. F
1 p q 2M) (
M ! (1 p q M)
0
1
r
2
1 p)
1
M, 1 p q M ;
x
1 p
;
z z F
M
M
1
…(2.4)
where
F
M
E 2 : U' ;...; U
z z F
1
r
p 2 : V' ;...; V
e) : ';...; (r) 1 p 1 r
(r)
g) : ';...; 2 p q M 1 r
(r)
(r)
(r)
x
(r)
M 1; v' ) : t' ] ;...; [(v
1
r
(r)
1 w' ) : x' ] ;...; [(w
r
t
(r)
z z
1
r
…(2.5)
where M ≥ 0,
In this paper, we also use short notations as given
1
1
F
F
(r)
P : V' ,..., V
t
r
denote the generalized Lauricella function of several complex variable.
The special case of the fractional derivative of Oldham and Spanier [4] is
E : U' ,..., U
(r)
1
…(2.7)
Re( 1
1
The generalized M-series is the extension of the both Mittag-Leffler function and generalized hypergeometric
function.
It represent as following
D
t
t
…(2.6)
t
M c c d d z) M z)
1
p, q
k0
p
1
q
c c
1
k
p
(d d
1
k
q
k
k
p, q
z
k
k
III.
z, c, Re( 0
…(2.8)
THE MAIN RESULT
Our main result of this paper is the fractional derivative formula involving the Lauricella functions,
generalized polynomials and the multivariable H-function and generalized M-series as given
||Issn 2250-3005 ||
||February||2014||
Page 36
3. Fractional Derivative Associated With The…
x)
D
M
,m
y
N
N M
1
1
0 k, M 0
z y )}
1
(r)
v
(r)
(r)
D
w x)}
r
1
r
1 y 1
w x)}
1
N
M M
s
S 1
N N
1
s
1
r y r
N
M k
1 1
s
k
w x)
1
w x)}
r
x)
x)
a
b
1 y 1
a
b
s y s
M k
s s
k
1
(r)
z y
r
k 0
s
u' , v' ) ;...; (u
0 3
s
k 0
1
M
s
A 3, C 3 :[B' , D' ] ;...; [B
H
F
x) 1 y 2 H
A[N
1
k N k
1
s
s
s
1 y 1 1
1
1
1
r
r
s
a k k :
i i
1
1
r
i 1
r y r 1
r
s
s
r)
(r)
(r)
a k k :
k b k k : a) : ',...,
b' ) : '; ];...; [(b
1 1
1
1
r
i i
2
1
r
i 1
i 1
s
(r)
(r) (r)
k b k k : c) : ',...,
d' ) : ' ] ,..., [d
i i
2
1
r
1
1
r
r
i 1
…(3.1)
where
1 1 q 2M) (1 p q M)
k ! M ! (1 p q M)
1
k
k 1 p)
s
k
1
k
. x)
. F
M
z z
1
r
k
M
1 p)
1 1
k
s
a k
i i
i 1
k
2
y)
c c
1
R
(d d
1
M)
R
m
R
i 1
b k
i i
t
0 s 0 i 1,2,..., r
i
i
R
and
r
Re(
i 1
d (i)
j
(i)
i
j
1
d (i)
j
Re (
1
(i)
i
i 1
j
Proof. In order to prove (3.1) express the Lauricella function by (2.4) and the multivariable H-function in terms of
Mellin-Barnes type of contour integrals by (2.2) and generalized polynomials given by (2.3) respectively and
r
generalized M-series (2.8) and collecting the power of x) and (y Finally making use of the result
(2.7), we get (3.1).
||Issn 2250-3005 ||
||February||2014||
Page 37
4. Fractional Derivative Associated With The…
IV.
PARTICULAR CASES
With = A = C = 0, the multivariable H-function breaks into product of Fox’s H-function and
consequently there holds the following result
x)
D
M
,m
y
x)
z y )} 1
1
z y r
r
y
1
r
2
H
N
(r)
(r)
3,3 :[B' , D' ] ;...; [B
M
k 0
1
0 3 u' , v' ) ;...; (u
s
s
N
1
v
(r)
(r)
D
M k
1 1
k
k 0
s
u
B
i 1
0 k, M 0
H
N M
1
1
F
M M
s
S 1
N N
1
s
(i)
v
(i)
D
(i)
N
s
M k
s s
k
1
w x)
1
w i x)}
(i)
a
b
x) 1 y 1
x) a s y b s
A[N
1
i
y
i
b
(i)
d
(i)
(i)
(i)
k N k
1
s
s
s
1 y 1 1
1
1
1
r
r
w x)}
r
s
a k k :
i i
1
1
r
i 1
r y r r
r
s
s
(r)
(r)
a k k :
k b k k : b' ) : '; ];...; [(b
1 1
1
1
r
i i
2
1
r
i 1
i 1
s
(r) (r)
k b k k : d' ) : ' ] ,..., [d
i i
2
1
r
1
1
r
r
i 1
…(4.1)
valid under the conditions surrounding (3.1).
If
II.
x)
D
M
(i)
,m
0 k, M 0
H
(i)
1 (i = 1,2,…) equation (4.1) reduces to
y
x)
1
F
z y )} 1
1
z y r
r
y
r
2
G
i 1
N
N M
1
1
k 0
1
0 3 u' , v' ) ;...; (u
3,3 :[B' , D' ] ;...; [B
||Issn 2250-3005 ||
(r)
(r)
s
M
s
N
1
k
k 0
s
v
D
(r)
(r)
w x)
1
M k
1 1
u
B
M M
s
S 1
N N
1
s
(i)
v
(i)
D
1
w x)}
r
w i x)}
(i)
(i)
N
a
b
x) 1 y 1
x) a s y b s
s
M k
s s
k
A[N
1
i
y
i
b
(i)
d
(i)
k N k
1
s
s
s
1 y 1 1
1
1
1
r
r
s
a k k :
i i
1
1
r
i 1
r y) r r
r
||February||2014||
Page 38
5. Fractional Derivative Associated With The…
s
s
(r)
a k k :
k b k k : b' );...; [(b
1 1
1
1
r
i i
2
1
r
i 1
i 1
s
(r)
k b k k : d' ) ,..., [(d
i i
2
1
r
1
1
r
r
i 1
…(4.2)
valid under the conditions as obtainable from (3.1).
III. Let Ni = 0 (i = 1,…,s), the result in (3.1) reduces to the known result given by Sharma and Singh [ ], after a
little simplification.
IV. Replacing N1,…,Ns by N in (3.1) we have a known result recently obtained by Chaurasia and Singh [ ].
V.
ACKNOWLEDGEMENT
The authors are grateful to Professor H.M. Srivastava, University of Victoria, Canada for his kind help
and valuable suggestions in the preparation of this paper.
REFERENCES
[1]
[2]
[3]
[4]
[5]
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[9]
[10]
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(2008), 187-191.
M. Sharma and Jain, R., A note on a generalized series as a special function,n of fractional calculus. J. Fract. Calc. and Appl. Anal.,
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K.B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
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H.M. Srivastava, A multilinear generating function for the Konhauser sets of biorthogonal polynomials suggested by the Laguerre
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Angew. Math. 283/284 (1976), 265-274.
H.M. Srivastava and R. Panda, Certain expansion formulas involving the generalized Lauricella functions, II Comment. Math.Univ.
St. Paul., 24 (1974), 7-14.
H.M. Srivastava and M.C. Daoust, Certain generalized Neuman expansions associated with the Kampé de Fériet function, Nederl.
Akad. Wetensch Indag. Math., 31 (1969), 449-457.
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