Handbook of mathematics for engineers and scienteists part 35 docx

... right-hand sides of the above formulas is called the index of summation (for series) or the index of multiplication (for products). The 1 and n (or the m and n)aretheupper and lower limits of summation ... b 2 ). 1.6. MEAN VALUES AND INEQUALITIES OF GENERAL FORM 13 1.6. Mean Values and Inequalities of General Form 1.6.1. Arithmetic Mean, Geometric Mean, and Other Mea...

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... rank of the matrix A. Properties of the rank of a matrix: 1. For any matrices A and B of the same size the following inequality holds: rank (A + B) ≤ rank (A)+rank(B). 2. For a matrix A of size ... x n–1 n           =  1≤j<i≤n (x i – x j ). 5.2.2-6. Determinant of a sum and a product of matrices. The determinant of the product of two matrices A and B...

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... system of eigenvectors of A and the subspace consisting of vectors orthogonal to all eigenvectors of A.Inthefinite-dimensional case, an orthonormal system of eigenvectors of A is a basis of V. 5. ... =detA. The trace ofthe matrix of a linear operator, Tr(A), isalso independent of the basis. Therefore, one can correctly define the trace Tr(A) of a linear operator as the tra...

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... stated Discriminant D, formula (12.6.1.29) Roots of quadratic equation (12.6.1.27), λ 1 and λ 2 Conditions for coefficients of differential equations (12.6.1.25) Type of equilibrium points or shape of phase ... trajectories of a system of differential equations near an equilibrium point of the center type. 12.6. LINEAR SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 535 By conve...

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... Theorems of stability and instability. In the cases where the theorems of stability and instability by first approximation fail to resolve the issue of stability for a specific system of nonlinear ... solution in the form u k = Ψ(u 3–k ), where k = 1, 2 and Ψ(u) is an arbitrary function of one variable. 2 ◦ . For linear equations (13.1.1.1) with the functions f , g,andh inde...

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... η x  .(13.1.3.13) This is a continuous function of η for fixed x and y. It can be shown that for fixed x and with the exception of a countable set of values of y, function (13.1.3.13) has a unique minimum with ... At the lines of discontinuity, y = y(x), the left and right limit values of w exist: w(x, y–0) and w(x, y + 0). 13.1. LINEAR AND QUASILINEAR EQUATIONS 565 1...

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... ξ, t) for parabolic equations of the form (14.7.2.1) are found by formula (14.7.2.3), and that for hyperbolic equations of the form (14.8.2.1), by formula (14.8.2.3) Type of problem Form of boundary ... Λ 1 (x, t )and 2 (x, t) that occur in the integrands of the last two terms in solution (14.8.2.2) are expressed in terms of the Green’s function of (14.8.2.3). The corresp...

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... values of the functions g(x), h 1 (x), and h 2 (x)onthearc AB, cut off on the given curve (14.8.4.2) by the characteristics x = x 0 and y = y 0 ,and on the values of a(x, y), b(x, y), c(x, y), and ... functions Λ 1 (x, t )and 2 (x, t) that occur in the integrands of the last two terms in solution (14.8.3.2) are expressed in terms of the Green’s function of (14.8.3.3). The c...

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... values of the coefficients s 1 , s 2 , k 1 ,andk 2 in (14.12.1.3) and (14.12.1.4), one can obtain the first, second, third, and mixed boundary value problems for equation (14.12.1.1). The solution of ... x 1 , , x n and t but is independent of z. Let = (x, y, z – ζ, t, τ ) be a fundamental solution of the Cauchy problem for equa- tion (14.11.3.7), i.e., ∂ ∂t = L x,t [ ]+ ∂ 2 ∂...

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... integrals, tables of direct and inverse Laplace transforms can be applied(see Sections T3.1 and T3.2), and, in many cases, to find the inverse transform, methods of the theory of functions of a complex ... case of equation (16.2.3.1) for K(x)=–A sin(λx). We first apply the table of Laplace transforms (see Subsection T3.1.6) and obtain the transform of the kernel of the...

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... 4λ ˆ f(s) 2λ .(16.5.3.19) The inverse transform y(x)=M –1 { ˆy(s)} obtained by means of the Mellin inversion formula (if it exists) is a solution of equation (16.5.3.18). To different signs in the formula for the images ... p n (t) are, for instance, continuous functions of t on the interval [a, b], system (16.5.3.6) becomes a system of nonlinear algebraic equations for A 1 , ,...

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... values of y(x)form + 1 ≤ x ≤ m +2 are determined by the right-hand side of (17.4.1.8) for 0 ≤ x ≤ 1, the quantities y(x+1), , y(x +m–1) are determined by the initial condition (17.4.1.6), and y(x ... ≤ x ≤ m.(17.4.1.6) The values of y(x)form ≤ x ≤ m + 1 are calculated by substituting the initial values (17.4.1.6) into the right-hand side of equation (17.4.1.5) for 0 ≤ x ≤ 1.Weha...

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... multiplication (division) of the equation and the results of its differentiation by suitable functions. Sometimes it is useful to take logarithm of the equation or the results of its transformation (see ... us expand (17.5.1.1) in powers of the small parameter a in a neighborhood of a 0 , taking into account (17.5.1.2), and then divide the resulting equation by a – a 0 and p...

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... equations often arise when themethod of functional separation of variables is used for finding solutions of nonlinear equations of mathematical physics. 1 ◦ . Consider a functional equation of the form f(t)+g(x)+h(x)Q(z)+R(z)=0,wherez ... one of the two functions f(t )and (t) is prescribed and the other is assumed unknown; also one of the functions g(x )and (x) is prescribed...

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... interval. An estimator θ ∗ of a parameter θ of the distribution of an observable random variable X (the estimator is a statistic, i.e., function of a random sample) is itself a random variable with ... value of the population parameter to be estimated; i.e., P (θ ∗ L ≤ θ ≤ θ ∗ R )=γ,(21.2.3.1) where θ ∗ L and θ ∗ R are the lower and upper (left and right) boundaries of the c...

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