Handbook of mathematics for engineers and scienteists part 31 docx
Handbook of mathematics for engineers and scienteists part 7 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...
Handbook of mathematics for engineers and scienteists part 31 docx
... 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...
Handbook of mathematics for engineers and scienteists part 35 docx
... 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...
Handbook of mathematics for engineers and scienteists part 82 docx
... 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...
Handbook of mathematics for engineers and scienteists part 84 docx
... 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...
Handbook of mathematics for engineers and scienteists part 86 docx
... 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...
Handbook of mathematics for engineers and scienteists part 94 docx
... 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...
Handbook of mathematics for engineers and scienteists part 95 docx
... 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...
Handbook of mathematics for engineers and scienteists part 97 docx
... 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
∂...
Handbook of mathematics for engineers and scienteists part 121 docx
... 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...
Handbook of mathematics for engineers and scienteists part 129 docx
... 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
, ,...
Handbook of mathematics for engineers and scienteists part 136 docx
... 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...
Handbook of mathematics for engineers and scienteists part 137 docx
... 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...
Handbook of mathematics for engineers and scienteists part 138 docx
... 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...
Handbook of mathematics for engineers and scienteists part 161 docx
... 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...