''surplus variable''
A non-negative variable $y_i$ that is introduced for a (linear) constraint $\sum_ja_{ij}x_j\leq b_i
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$#C+1 = 36 : ~/encyclopedia/old_files/data/F041/F.0401660 Free variable,
''free occurrence of a variable''
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$#C+1 = 15 : ~/encyclopedia/old_files/data/P074/P.0704380 Predicate variable,
''second-order variable''
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...and where it is stressed that only in this context the notion of a random variable becomes meaningful).
...n X(\omega)<x\}$ belongs to the class $\mathcal{A}$. Let $X$ be any random variable and $\mathcal{A}_X$ the class of subsets $C\subset\mathbf{R}^1$ for which $
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''object variable''
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''bound occurrence of a variable''
...nt of a bound variable, where it occurs, by a new (for a given expression) variable leads to an expression with the same meaning. For example, in the expressio
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$#C+1 = 46 : ~/encyclopedia/old_files/data/V096/V.0906080 Variable\AAhdirections method
...sed on operators that are equivalent in spectrum and suitable for use with variable coefficients and non-linear problems.
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$#C+1 = 116 : ~/encyclopedia/old_files/data/A110/A.1100220 Accessible random variable
An accessible random variable will be defined in terms of Borel cylinder functions, a special class of ac
11 KB (1,627 words) - 06:28, 26 March 2023
In applications of the variable-grid method, at each instant the working region is divided into a finite nu
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...s representations of solutions in terms of analytic functions of a complex variable.
...partial, complex-variable methods|Differential equation, partial, complex-variable methods]]),
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-
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...ional vectors $x_1$ and $x_2$. The properties of a convex function of one variable are correspondingly generalized to functions of several variables; for exa
...valign="top"> N. Bourbaki, "Elements of mathematics. Functions of a real variable" , Addison-Wesley (1976) pp. Chapt. 1 Sect. 4 (Translated from French)</T
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...the representation of solutions by way of analytic functions of a complex variable.
of the complex variable $ z = x + iy $
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...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
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...edia/old_files/data/A013/A.0103000 Approximation of functions of a complex variable
...chlet series, etc.). The theory of approximation of functions of a complex variable is intimately connected with other branches of complex analysis, and with m
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...= 73 : ~/encyclopedia/old_files/data/F042/F.0402120 Functions of a complex variable, theory of
...ex plane $ \mathbf C = \mathbf C ^ {1} $ (functions of a single complex variable) or on a set of points $ z = ( z _ {1} \dots z _ {n} ) $
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...heir local and global properties. The modern theory of functions of a real variable typically involves a wide application of set-theoretic methods, along, natu
Thus, the object studied in the theory of functions of a real variable is a function. About this concept N.N. Luzin [[#References|[3]]] wrote "It
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...of]]; [[Functions of a complex variable, theory of|Functions of a complex variable, theory of]]).
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''simple function (of a complex variable)''
...older, names for a [[Univalent function|univalent function]] of a complex variable.
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''bound occurrence of a variable''
...nt of a bound variable, where it occurs, by a new (for a given expression) variable leads to an expression with the same meaning. For example, in the expressio
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...ign to a [[Convex function (of a real variable)|convex function (of a real variable)]].
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...ual variable of the corresponding type (here also it is necessary to avoid variable collisions).
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$#C+1 = 15 : ~/encyclopedia/old_files/data/P074/P.0704380 Predicate variable,
''second-order variable''
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The curve representing the ends of the variable vector $x(t)$ ($t$ is a real variable, such as time) whose origin for all $t$ is a given fixed point $O$.
...ograph. For example, if the velocity of a moving point is represented by a variable vector $\mathbf{v}$, then by drawing the values of $\mathbf{v}$ at differen
1 KB (179 words) - 19:16, 12 April 2017
...ude characterizing the [[Dispersion|dispersion]] of the values of a random variable $ X $.
is the distribution function of the random variable. Another variable which is also occasionally considered is the Gini dispersion coefficient
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''surplus variable''
A non-negative variable $y_i$ that is introduced for a (linear) constraint $\sum_ja_{ij}x_j\leq b_i
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$#C+1 = 36 : ~/encyclopedia/old_files/data/F041/F.0401660 Free variable,
''free occurrence of a variable''
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...and the set of conditional probabilities $\mathrm P_\theta$ of the random variable $X$ given $\Theta=\theta$. According to the [[Bayes formula|Bayes formula]]
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A function $f(z)$ of a complex variable $z$ which is single-valued in this domain and which has a finite derivative
[[Category:Functions of a complex variable]]
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.../> is a [[Convex function (of a real
variable)|convex function (of a real
variable)]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmat
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A [[periodic function]] $f(z)$ of the complex variable $z$ all periods $p$ of which are integer multiples of a single unique funda
...riod $2\omega\in X$. A non-constant continuous periodic function of a real variable is necessarily simply-periodic.
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The argument of a function is the variable (also called the independent variable) on which the value of the function depends.
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...ibly others) in accordance with the following rules: 1) each propositional variable is a propositional formula; and 2) if $A,B$ are propositional formulas, the
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...esis is called the hypothesis distribution. E.g., if one observes a random variable $X$, then the statement "X is subject to the Poisson law with parameter 1"
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...mber for each set of values of the parameters ($x$ in this term is a bound variable). Syntactically, terms are characterized by the fact that they can be subst
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...on a [[probability space]] $(\Omega,\mathcal{F},\mathbb{P})$, to a random variable $X$, defined in the following way: $X_n \stackrel{\mathrm{P}}{\rightarrow}
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''in the theory of functions of a complex variable''
[[Category:Functions of a complex variable]]
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A function that does not change sign when the sign of the independent variable is changed, i.e. satisfying the condition $f(-x)=f(x)$. The graph of an eve
A function that does change sign when the sign of the independent variable is changed, i.e. satisfying the condition $f(x)=-f(-x)$, is called an [[Odd
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...s also defined for discrete distributions: If the values $x_k$ of a random variable $X$ with distribution $p_k = \mathsf{P}(X = x_k)$ are arranged in increasin
...istics)]] the mode acts as a measure of location of the values of a random variable. For distributions which are unimodal and symmetric with respect to some po
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Here $z$ is a complex variable, $m$ is an integer and $n$ is a natural number. The series converge in the
...<TD valign="top"> A.I. Markushevich, "Theory of functions of a complex variable" , '''3''' , Chelsea (1977) pp. Chapt. 8 (Translated from Russian)</TD><
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...f. [[Convex function (of a complex
variable)|Convex function (of a complex
variable)]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.
...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex
variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
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...em is one of the classical results in the theory of functions of a complex variable of the type of [[Distortion theorems|distortion theorems]].
...TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD
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The inverse hyperbolic functions of a real variable $ x $
The inverse hyperbolic functions of a complex variable $ z $
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...i=1,\dotsc,m$; $j=1,\dotsc,m_i$) is either a variable or the negation of a variable, and $\bot$ is the logical symbol denoting falsehood. For each propositiona
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...n of values of the [[Normal distribution|normal distribution]] of a random variable at a distance from its [[Mathematical expectation|mathematical expectation]
Let $X$ be a normally $N(a,\sigma^2)$ distributed random variable. For any $k>0$,
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...ing time (relative to this family of subalgebras) is a mapping (a [[random variable]]) $\tau : \Omega \rightarrow T \cup \{\infty\}$ such that
...iable. This condition has the interpretation that the (time-valued) random variable $\tau$ has no knowledge of the future, since the $\sigma$-algebra $\mathcal
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The probability distribution of the random variable
is a random variable subject to the standard normal law $ N( 0, 1) $
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is a real or complex variable. If $ a _ {0} \neq 0 $,
...lane, that is, it is an [[Entire function|entire function]] of the complex variable $ z $,
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...mula is a formula; and b) if $A$ and $B$ are formulas and $y$ is an object variable, then $(\neg A)$, $(A\mathbin\&B)$, $(A\lor B)$, $(A\supset B)$, $(\forall
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...f Cauchy distributions is closed under linear transformations: If a random variable $ X $
then the random variable $ Y = aX + b $
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cf. [[Random variable|Random variable]]). For $ m < N $,
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...makes it possible to reduce the study of the deformation of surfaces with variable $ K > 0 $
...r-reaching analogy between the properties of deformations of surfaces with variable and constant positive Gaussian curvature.
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be a random variable having a continuous and strictly increasing distribution function $ F $.
Then the random variable $ Y = F ( X) $
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The distribution of a discrete random variable assuming non-negative integral values $ m = 0, 1 \dots $
The random variable equal to the number of independent trials prior to the first successful out
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''conditional expectation, of a random variable''
...ctation of $X$ with respect to $\mathfrak{B}$ is understood to be a random variable $\mathsf{E}(X\, |\, \mathfrak{B})$, measurable with respect to $\mathfrak{B
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...depending on a continuous or discrete argument $\tau$, relative to another variable $\phi(\tau)$ whose behaviour, as a rule, is assumed to be known. In general
...lign="top"> A.F. Timan, "Theory of approximation of functions of a real variable" , Pergamon (1963) (Translated from Russian)</TD></TR><TR><TD valign="top
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A property of the dependence of one random variable on another, defined by the fact that the conditional variance is equal to a
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...valign="top"> G.M. Goluzin, "Geometric theory of functions of a complex variable" , ''Transl. Math. Monogr.'' , '''26''' , Amer. Math. Soc. (1969) (Transl
[[Category:Functions of a complex variable]]
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a random variable defined on $ ( \Omega , {\mathcal A} ) $
is called a (regular) conditional distribution of the random variable $ X $
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...$F(x) = \mathbf{P}\{X \le x\}$ be the distribution function of the random variable $X$. Then the probability distribution for the range is
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...im_i}$ do not contain both the formulas $p$ and $\neg p$, where $p$ is any variable. For any propositional formula $A$ it is possible to construct an equivale
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with respect to the variable $ x _ {1} $
in the single variable $ x _ {1} $,
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be a continuously-distributed random variable for which the probability density $ p( x) $,
...tuation, when instead of the realizations of the initial continuous random variable $ X $,
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A function $f$ that changes sign when the independent variable changes sign, i.e. $f(-x)=-f(x)$ for all values in the domain of $f$. The g
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\Var X$ of the deviation of a [[Random_variable | random variable]] $X$ from its [[Mathematical expectation|mathematical expectation]] $\Ex X
...us functions of a random variable $X$ in terms of the distribution of this variable on the set of real numbers are of importance (cf. [[Mathematical expectatio
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...TD> <TD valign="top"> S. Stoilov, "The theory of functions of a complex variable" , '''1–2''' , Moscow (1962) (In Russian; translated from Rumanian)</TD
...top">[a1]</TD> <TD valign="top"> J.B. Conway, "Functions of one complex variable" , Springer (1978)</TD></TR></table>
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...culation of an area or an [[Integral|integral]] (of a function of a single variable).
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