Engineering Mathematics for 4th Semester.

•

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In these notes, you will get all the best and simple explanations of topics for 4th Semester Engineering aspirants. This includes examples and the best ways to solve all the tricky questions. Topics covered in this notes are:- 1. Introduction to Graph Theory. 2. Planner and Trees. 3. Recurrence Relations. 4. Joint Random Variables. 5. Numerical Methods.

Education

Page 0 of 158
MATHEMATICS NOTES
Hand – Written Notes
By

Page 1 of 158
Index
S.NO. Topic Name Page Number
1 Introduction to Graph
Theory.
03 – 32
2 Planner and Trees 33 – 65
3 Recurrence Relation 66 - 105
4 Joint Random Variables 139 - 159
5 Numerical Methods 106 - 138
*NOTE: - Important topics, formulas and
questions are marked by green colures pen
whereas most important topics, Formulas and
questions are marked by red pen.

The document provides information about numerical methods topics including: 1) Lagrange's interpolation formula for finding a polynomial that passes through given data points, either equally or unequally spaced. The formula uses divided differences to find the coefficients. 2) Newton's divided difference interpolation formula for unequal intervals that also uses divided differences. 3) The nature of divided differences - for a polynomial of degree n, the nth divided difference is constant. 4) Examples of evaluating divided differences and constructing divided difference tables are given.

Grovers Algorithm

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Interpolation functions

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The document discusses interpolation, which involves using a function to approximate values between known data points. It provides examples of Lagrange interpolation, which finds a polynomial passing through all data points, and Newton's interpolation, which uses divided differences to determine coefficients for approximating between points. The examples demonstrate constructing Lagrange and Newton interpolation polynomials using given data sets.

Gaussian Quadrature Formula

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Problemas, rectas y planos en el espacio

Tensor

Numerical Methods - Oridnary Differential Equations - 1

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The document discusses solving ordinary differential equations using Taylor's series method. It presents the Taylor's series for the first order differential equation dy/dx = f(x,y) and gives an example of solving the equation y = x + y, y(0) = 1 using this method. The solution is obtained by taking the Taylor's series expansion and determining the derivatives of y evaluated at x0 = 0. The values of y are computed at x = 0.1 and x = 0.2. A second example solves the differential equation dy/dx = 3x + y^2 using the same approach.

Higher Order Differential Equation

Shrey Patel

The document provides an overview of advanced engineering mathematics concepts for differential equations. It covers topics such as homogeneous and non-homogeneous linear differential equations with constant and variable coefficients. Methods for solving differential equations are discussed, including finding the auxiliary equation, complementary function, and particular integral. Specific solving techniques like the Cauchy-Euler and Legendre methods for variable coefficient equations are also mentioned. Examples of different types of differential equations are provided throughout.

08 interpolation lagrange

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This document contains lecture slides from an ENEM602 engineering mathematics course taught by Dr. Eng. Mohammad Tawfik in Spring 2007. It discusses Lagrange interpolation, which is a method for interpolating a polynomial that passes through a set of points. The document provides examples of using Lagrange interpolation for 2, 3 and 4 points. It derives the general Lagrange interpolation formula and shows an example of applying it to find a 3rd order polynomial to interpolate 4 given data points. Students are assigned homework problems applying Lagrange interpolation.

The document discusses least-square optimization and sparse linear systems. It introduces least-square optimization as a technique to find approximate solutions when exact solutions do not exist. It provides an example of using least-squares to find the line of best fit through three points. The objective is to minimize the sum of squared distances between the line and points. Solving the optimization problem yields a set of linear equations that can be solved using techniques like pseudo-inverse or conjugate gradient. Sparse linear systems with many zero entries can be solved more efficiently than dense systems.

Fuzzy logic member functions

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This document provides an overview of fuzzy logic concepts for a course on soft computing. It discusses key fuzzy logic topics like membership functions, fuzzy sets, linguistic variables, fuzzy rules, fuzzy inference, and neuro-fuzzy systems. The document also provides examples of commonly used membership functions like triangular, trapezoidal, and Gaussian functions. It explains how fuzzy logic allows for approximate reasoning using natural language terms and multivalent logic with membership values between 0 and 1.

Convex optmization in communications

Deepshika Reddy

This document provides an overview of optimization techniques. It begins with an introduction to optimization and examples of optimization problems. It then discusses the historical development of optimization methods. The document categorizes optimization problems into convex, concave, and subclasses like linear programming, quadratic programming, and semidefinite programming. It also covers advanced optimization methods like interior point methods. Finally, the document discusses applications of convex optimization techniques in fields such as engineering, finance, and wireless communications.

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The false position method is a root-finding algorithm that uses linear interpolation to estimate the root of a function. It improves upon the bisection method by using the function values at the endpoints of the interval rather than just their signs. The method chooses the intercept of the secant line through the two endpoints as the next approximation of the root, and continues iteratively narrowing the interval until the root is found.

1. introduction to complex numbers

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This document provides information about the Engineering Mathematics 1 course taught by Dr. Ir. Arman Djohan Diponegoro. It includes details about the class code, credits, schedule, topics to be covered including complex variables and vector analysis. It also outlines the grading breakdown and test schedule. Several concepts from complex numbers are then defined and explained, such as the complex plane, polar coordinates, trigonometric form, and operations like addition, subtraction and multiplication of complex numbers. Examples are provided to demonstrate converting between rectangular and trigonometric forms.

Circular convolution

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The document discusses different types of convolution operations on signals. Convolution involves taking one signal, flipping and shifting it, then multiplying it sample-by-sample with another signal and summing the results. Linear convolution produces a output signal that is the sum of the two input signal lengths minus one. Circular convolution treats the signals as if they repeat periodically, producing a periodic output signal of the same period. It can approximate linear convolution if the period is long enough. The discrete Fourier transform (DFT) and its fast version, the FFT, can also be used to efficiently compute circular convolution by multiplying the signals' Fourier transforms.

Grover's algorithm simplified.pptx

SundarappanKathiresa

Grover's algorithm is a quantum algorithm that allows finding an item in an unstructured database with fewer queries than classical algorithms. It works by amplifying the probability of measuring the target item. The key steps are: (1) initialize a superposition of all possible inputs, (2) apply an oracle to mark the target item, (3) apply Grover's diffusion operator to amplify the target amplitude, and (4) repeat steps 2-3 until the target can be measured with high probability. An example is using Grover's algorithm to find a phone number in an unsorted database with only sqrt(N) queries, compared to N/2 queries classically.

Cryptography using probability

Prankit Mishra

This document discusses the use of probability in cryptography. It begins with introductions to cryptography and probability. Key probability terms and concepts like events, sample spaces, and Markov models are defined. Public key cryptography using Fermat's Little Theorem is explained. Applications of probability in cryptography are explored, including checksums and the birthday problem, pseudo-random number generators, and code breaking using the Metropolis-Hastings algorithm. The document concludes that probability and cryptography are important fields that help secure communications and protect society from cyber attacks.

Mc culloch pitts neuron

Siksha 'O' Anusandhan (Deemed to be University )

Here is a MATLAB program to implement logic functions using a McCulloch-Pitts neuron: % McCulloch-Pitts neuron for logic functions % Inputs x1 = 1; x2 = 0; % Weights w1 = 1; w2 = 1; % Threshold theta = 2; % Net input net = x1*w1 + x2*w2; % Activation function if net >= theta y = 1; else y = 0; end % Output disp(y) This implements a basic AND logic gate using a McCulloch-Pitts neuron.

QR Algorithm Presentation

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Z transform

ayushagrawal464

Z-transforms can be used to evaluate discrete functions, similar to how Laplace transforms are used for continuous functions. The z-transform of a discrete function f(n) is defined as the sum of f(n) multiplied by z to the power of -n, from n=0 to infinity. Some standard z-transform results include formulas for exponential, sinusoidal, and polynomial functions. Z-transforms have properties of linearity and shifting, and can be used to solve differential equations with constant coefficients and in applications of signal processing.

Unit vi

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This document provides an overview of numerical methods for solving ordinary differential equations. It outlines several numerical methods including Taylor's series method, Picard's method of successive approximation, Euler's method, modified Euler's method, Runge-Kutta methods, and predictor-corrector methods like Milne's method and Adams-Moulton method. Examples of the formulas used in each method are given. The document also lists references and provides context about the course and unit.

Fuzzylogic

Manju Rajput

The document discusses fuzzy rule-based systems and fuzzy reasoning. It defines fuzzy if-then rules which have antecedents and consequents that are fuzzy sets rather than binary variables. Fuzzy reasoning can involve single or multiple rules with single or multiple antecedents. Graphical representations are used to illustrate the fuzzy reasoning process. The types of fuzzy inference systems including Mamdani and Sugeno systems are described along with their components and working. Applications of fuzzy logic in various domains are also mentioned.

numerical differentiation&integration

8laddu8

This document contains the syllabus for a course on Mathematical Methods taught according to the JNTU-Hyderabad new syllabus. It covers topics like matrices and linear systems, eigenvalues and eigenvectors, linear transformations, solution of nonlinear systems, curve fitting, numerical integration, Fourier series, and partial differential equations. The specific section summarized discusses numerical differentiation using forward, backward, and central differences. It also covers numerical integration techniques like the trapezoidal rule, Simpson's 1/3 rule, and Simpson's 3/8 rule.

Mediumet e rrjeteve

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Methods of successive over relaxation

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Linear algebra-Basis & Dimension

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Analytic function 1

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- The document discusses complex analytic functions and their derivatives. - A complex function is analytic if it has a complex derivative everywhere in some open region of the complex plane. This allows drawing richer conclusions than for real differentiable functions. - The derivative of a complex function f at z0 is defined as the limit of (f(z)-f(z0))/(z-z0) as z approaches z0. If this limit exists, f is said to be differentiable or analytic at z0.

NUMERICAL METHODS -Iterative methods(indirect method)

krishnapriya R

The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.

Euler's and picard's

Manikanta satyala

- The document discusses numerical methods for solving first order differential equations, namely Picard's method and Euler's method. - Picard's method involves iteratively replacing y with the previous approximation in the differential equation to obtain better approximations that converge to the solution. - Euler's method approximates the solution at the next point by the current value plus the rate of change times the change in x. This provides a first order approximation to the solution.

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Circular convolution

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Grover's algorithm simplified.pptx

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Cryptography using probability

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QR Algorithm Presentation

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Engineering Mathematics for 4th Semester.

1. Page 0 of 158 MATHEMATICS NOTES Hand – Written Notes By

2. Page 1 of 158 Index S.NO. Topic Name Page Number 1 Introduction to Graph Theory. 03 – 32 2 Planner and Trees 33 – 65 3 Recurrence Relation 66 - 105 4 Joint Random Variables 139 - 159 5 Numerical Methods 106 - 138 *NOTE: - Important topics, formulas and questions are marked by green colures pen whereas most important topics, Formulas and questions are marked by red pen.

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