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# Abbreviations

llitre

mmetre

 cm centimetre km kilometre s second EUR euro LPP linear programming problem s.t. subject to bv basic variables of a system of linear equations nbv non-basic variables of a system of linear equations ## Notations

A

A B

A B

A = B

A B

A(x)

x

A(x)

x

a A b / A

|A|

P(A) A B A B A B A \ B A × B

n

Ai

i=1

An

n!

n

k

n = 1, 2, . . . , k

N

N0

Z

negation of proposition A conjunction of propositions A and B disjunction of propositions A and B implication (if A then B) equivalence of propositions A and B universal proposition

existential proposition

a is an element of set A

b is not an element of set A empty set

cardinality of a set A (if A is a finite set, then |A| is equal to the number of elements in set A), the same notation is used for the determinant of a square matrix A

power set of set A

set A is a subset of set B union of sets A and B intersection of sets A and B difference of sets A and B

Cartesian product of sets A and B

Cartesian product of sets A1, A2, . . . , An

n

Cartesian product Ai , where A1 = A2 = . . . = An = A

 i=1 n factorial: n! = 1 · 2 · . . . · (n − 1) · n binomial coefficient: k = k (n ! k) n n ! · − !

equalities n = 1, n = 2, . . . , n = k

set of all natural numbers: N = {1, 2, 3, . . .} union of set N with number zero: N0 = N {0}

union of set N0 with the set of all negative integers

 xvi List of notations Q set of all rational numbers, i.e. set of all fractions p/q with p Z and q N R set of all real numbers R+ set of all non-negative real numbers (a, b) open interval between a and b [a, b] closed interval between a and b ± denotes two cases of a mathematical term: the first one with sign + and the second one with sign − denotes two cases of a mathematical term: the first one with sign − |a| and the second one with sign + absolute value of number a R sign of approximate equality, e.g. √ ≈ 1.41 ≈ 2 = sign ‘not equal’ π irrational number equal to the circle length divided by the diameter length: π ≈ 3.14159... e Euler’s number: e ≈ 2.71828... ∞ infinity √ a square root of a e: y exp(x) ex exp = = notation used for the exponential function with base y log notation used for the logarithm: if y = loga x, then a = x lg notation used for the logarithm with base 10: lg x = log10 x ln notation used for the logarithm with base e: ln a = loge x n summation sign: ai = a1 + a2 + · · · + an i=1 n product sign: 2 ai = a1 · a2 · . . . · an i=1 = −1 i imaginary unit: i C set of all complex numbers: z = a + bi, where a and b are real numbers |z| modulus of number z C {an} sequence: {an} = a1, a2, a3, . . . , an, . . . {sn} series, i.e. the sequence of partial sums of a sequence {an} lim limit sign aRb a is related to b by the binary relation R aRb a is not related to b by the binary relation R R−1 inverse relation of R S ◦ R composite relation of R and S f : A → B mapping or function f A × B: f is a binary relation b = f (a) which assigns to a A exactly one b B b is the image of a assigned by mapping f f −1 inverse mapping or function of f g ◦ f composite mapping or function of f and g Df domain of a function f of n ≥ 1 real variables Rf range of a function f of n ≥ 1 real variables y = f (x) y R is the function value of x R, i.e. the value of

function f at point x

deg P

x x0

x x0 + 0 x x0 0

f(x), y (x)

f(x), y (x)

dy, df

ρf (x0) εf (x0)

Rn Rn+ a

aT

|a|

|a b| a b

dim V

Am,n

AT An

det A, (or |A|)

x1, x2, . . . , xn 0

Ri {≤, =, ≥}

zmin!

zmax!

fx (x0, y0)

fxi (x0)

ρf ,xi (x0) εf ,xi (x0)

Hf (x0)

List of notations xvii

degree of polynomial P x tends to x0

x tends to x0 from the right side x tends to x0 from the left side derivative of function f

derivative of function f with y = f (x) at point x second derivative of function f with y = f (x) at point x differential of function f with y = f (x)

sign of identical equality, e.g. f (x) 0 means that equality f (x) = 0 holds for any value x

proportional rate of change of function f at point x0 elasticity of function f at point x0

integral sign

n-dimensional Euclidean space, i.e. set of all real n-tuples set of all non-negative real n-tuples

vector: ordered n-tuple of real numbers a1, a2, . . . , an corresponding to a matrix with one column transposed vector of vector a

Euclidean length or norm of vector a

Euclidean distance between vectors a Rn and b Rn means that vectors a and b are orthogonal

dimension of the vector space V matrix of order (dimension) m × n transpose of matrix A

nth power of a square matrix A determinant of a matrix A inverse matrix of matrix A adjoint of matrix A

rank of matrix A

denotes the inequalities x1 0, x2 0, . . . , xn 0 means that one of these signs hold in the ith constraint of a system of linear inequalities

indicates that the value of function z should become minimal for the desired solution

indicates that the value of function z should become maximal for the desired solution

partial derivative of function f with z = f (x, y) with respect to x at point (x0, y0)

partial derivative of function f with z = f (x1, x2, . . . , xn) with respect to xi at point x0 = (x10, x20, . . . , xn0)

gradient of function f at point x0

partial rate of change of function f with respect to xi at point x0 partial elasticity of function f with respect to xi at point x0 Hessian matrix of function f at point x0

Q.E.D. (quod erat demonstrandum

– ‘that which was to be demonstrated’)

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