Handbook of algorithms for physical design automation part 28 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 ... Products 1.5.2-1. Notations for finite series and products. A finite series is just the sum of a finite number of terms and a finite product is the product of a finite number...

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... root of equation (5.1.5.1) with integer coefficients a k is an irreducible fraction of the form p/q ,where p is a divisor of a 0 and q is a divisor of a n .If a n = 1 , then all rational roots of ... equation belong to the interval [–5, 5]. Formula (5.1.5.4) for ρ = 2, n = 4, |a n | = 9, A 1 = 9 yields a better estimate for the bounds of the roots of this polynomial, N 1 = 3....

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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 ... zero. The adjugate (classical adjoint) for a matrix A ≡ [a ij ]ofsizen × n is a matrix C ≡ [c ij ] of size n×n whose entries coincide with the cofactors of the entries of the...

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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. ... A =det  A. Therefore, one can correctly define the determinant det A of a linear operator as the determinant of its matrix in any basis: det A =detA. The trace ofthe matrix of...

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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 convent...

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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 ... function of two variables. With equation (13.1.1.4) solved for u 1 or u 2 , we often specify the general solution in the form u k = Ψ(u 3–k ), where k = 1, 2 and Ψ(u) is an arbitrary...

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... hold for any y 1 and y 2 . It has a simple physical interpretation: the rate of change of the total value of w distributed over the interval (y 1 , y 2 ) is compensated for by the “flux” of the function ... characteristics of equa- tion (13.1.4.1) can intersect in some domain, which results in the nonuniqueness of the solution and the absence of a physical interpretation...

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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 ... 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 corresponding formulas for Λ m (x, t) are given in...

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... 0 1 σ n  e –σ n y sinh(σ n η)fory > η, e –σ n η sinh(σ n y )for > y 0 ≤ y < ∞ ∂ y w = 0 for y = 0 1 σ n  e –σ n y cosh(σ n η)fory > η, e –σ n η cosh(σ n y )for > y 0 ≤ y < ∞ ∂ y w – β 3 w = 0 for y ... at the segment AN (which is part of the characteristic x = x 1 ), the values of h(x) at the segment BN (which is part of the characteristic y = y 1 ), and the...

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... problems for equation (14.11.3.7) via the fundamental solution of the Cauchy problem. TABLE 14.13 Representation of the Green’s functions of some nonstationary boundary value problems in terms of the ... 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 ∂z 2 for...

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