of combinatorial designs‚ which are collections of subsets with certain intersection properties. Theory studies various enumeration and asymptotic problems related to integer partitions‚ and is closely related to q-series‚ functions and orthogonal polynomials. Originally a part of number theory and analysis‚ partition theory is now considered a part of combinatorics or an independent field. Order theory is the study of
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Formulas (to differential equations) Math. A3‚ Midterm Test I. sin2 x + cos2 x = 1 sin(x ± y) = sin x cos y ± cos x sin y tan(x ± y) = tan x±tan y 1∓tan x·tan y differentiation rules: (cu) = cu ′ ′ ′ ′ ′ (c is constant) cos(x ± y) = cos x cos y ∓ sin x sin y (u + v) = u + v (uv)′ = u′ v + uv ′ ′ ′ u ′ = u v−uv v v2 df dg d dx f (g(x)) = dg dx sin 2x = 2 sin x cos x tan 2x = sin x = 2 cos 2x = cos2 x − sin2 x 2 tan x 1−tan2 x 1−cos 2x ‚ 2 integration rules: cos x = 2
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TIME SERIES ANALYSIS Chapter Three Univariate Time Series Models Chapter Three Univariate time series models c WISE 1 3.1 Preliminaries We denote the univariate time series of interest as yt. • yt is observed for t = 1‚ 2‚ . . . ‚ T ; • y0‚ y−1‚ . . . ‚ y1−p are available; • Ωt−1 the history or information set at time t − 1. Call such a sequence of random variables a time series. Chapter Three Univariate time series models c WISE 2 Martingales Let {yt} denote
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Diagonally Implicit Block Backward Differentiation Formulas for Solving Ordinary Differential Equations 1.0 Introduction In mathematics‚ if y is a function of x‚ then an equation that involves x‚ y and one or more derivatives of y with respect to x is called an ordinary differential equation (ODE). The ODEs which do not have additive solutions are non-linear‚ and finding the solutions is much more sophisticated because it is rarely possible to represent them by elementary function in close
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IMPORTANT QUESTIONS Physics · How much force is needed to accelerate a trolley of mass 20g through 1 m/s2. · A force of 100N acts on a mass of 25 kg for 5 s .What velocity does it generate? · A bullet leaves a rifle with a velocity of 100m/s and the rifle of mass 2.5 kg recoils with a velocity of 1m/s. find the mass of the bullet? · Certain force acting on a mass of 15kg for 3s‚ gives it a velocity of 2m/s. Find the magnitude of force. · A cricket ball of mass 0.15 kg is moving with
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M.A.M. SCHOOL OF ENGINEERING‚ SIRUGANUR‚ TIRUCHIRAPPALLI – 621 105. M.A.M. SCHOOL OF ENGINEERING‚ SIRUGANUR‚ TIRUCHIRAPPALLI – 621 105. Department of Computer Science and Engineering Department of Computer Science and Engineering LABORATORY MANUAL – CS 2208 – DATA STRUCTURES LABORATORY LABORATORY MANUAL – CS 2208 – DATA STRUCTURES LABORATORY EX: NO: 1 (a) SINGLY LINKED LIST AIM: Step 3:Stop PROGRAM : To write a Program to implement a single linked list ALGORITHM:
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University of Gujrat A World Class University Department of Computer Science COURSE DESCRIPTION Course Code CS-203 Course Title Data Structures Credit Hours 4 Category Core Prerequisite Knowledge and Experience in Programming Fundamentals and Object Oriented Programming Expertise in design‚ implementation‚ testing‚ and strong debugging of object-oriented programs. Inner Classes and Exception Handling Amis and Objectives “An apprentice carpenter may want only hammer and saw‚ but a master craftsman
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Exercise 4.1 2. Establish each of the following for all n ≥ 1 by the Principle of Mathematical Induction. Solution a) S(n): ==‚ S(1): = = =1‚ So S(1) is true. Assume S(k): = Consider S(k+1) = = +=-1+= -1. Hence‚ it follows that S(k)⇒S(k + 1) is true for all n ∈ Z+ by the Principle of Mathematical Induction. b) S( n) for n=1‚ = 2 = 2+(1-1). So S(1) is true. Inductive Step: assume S(k)is true‚ for some (particular) k ∈ Z+—that is‚ assume that =2+(k-1). For n=k+1‚ = + (k+1)
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------------------------------------------------- Real number In mathematics‚ a real number is a value that represents a quantity along a continuum‚ such as 5 (an integer)‚ 3/4 (a rational number that is not an integer)‚ 8.6 (a rational number expressed in decimal representation)‚ and π (3.1415926535...‚ an irrational number). As a subset of the real numbers‚ the integers‚ such as 5‚ express discrete rather than continuous quantities. Complex numbers include real numbers as a special case. Real
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Matrics & Determinants Matrix operations‚ Definition and properties of determinats‚ Cofactors‚ Adjoint‚ Elementry Transformations‚ Rank and inverse of a Matrix‚ Matrix Polynomial‚ Characteristic Equations‚Eigen Values‚ Latent Vectors‚ Caylay Hamilton theorem‚ Linear system of Equations. Theory Of Equations Polynomials and their charcteristics‚ Roots of an equation‚ Relations between Roots and Coefficients‚Transformation of Equations‚ Symmetric function etc. Abstracy Algebra Groups‚ Cyclic
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