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convex combination - перевод на русский

LINEAR COMBINATION OF POINTS WHOSE COEFFICIENTS ARE NON-NEGATIVE AND SUM TO ONE

Convex sum

$Convex combination of three points <math>v_{0},v_{1},v_{2} \text{ of } 2\text{-simplex} \in \mathbb{R}^{2}</math> in a two dimensional vector space <math> \mathbb{R}^{2}</math> as shown in animation with <math>\alpha^{0}+\alpha^{1}+\alpha^{2}=1</math>, <math>P( \alpha^{0},\alpha^{1},\alpha^{2} )</math> <math>:= \alpha^{0} v_{0} + \alpha^{1} v_{1} + \alpha^{2} v_{2}</math> . When P is inside of the triangle <math>\alpha_{i}\ge 0</math>. Otherwise, when P is outside of the triangle, at least one of the <math>\alpha_{i}</math> is negative.$
$Convex combination of four points <math>A_{1},A_{2},A_{3},A_{4} \in \mathbb{R}^{3}</math> in a three dimensional vector space <math> \mathbb{R}^{3}</math> as animation in [[Geogebra]] with <math>\sum_{i=1}^{4} \alpha_{i}=1</math> and <small><math>\sum_{i=1}^{4} \alpha_{i}\cdot A_{i}=P</math></small>. When P is inside of the tetrahedron <math>\alpha_{i}>=0</math>. Otherwise, when P is outside of the tetrahedron, at least one of the <math>\alpha_{i}</math> is negative.$
$Convex combination of two functions as vectors in a vector space of functions - visualized in Open Source Geogebra with <math>[a,b]=[-4,7]</math> and as the first function <math>f:[a,b]\to \mathbb{R}</math> a polynomial is defined. <math>f(x):= \frac{3}{10} \cdot x^2 - 2 </math> A trigonometric function <math>g:[a,b]\to \mathbb{R}</math> was chosen as the second function. <math>g(x):= 2 \cdot cos(x) + 1</math> The figure illustrates the convex combination <math>K(t):= (1-t)\cdot f + t \cdot g</math> of <math>f</math> and <math>g</math> as graph in red color.$
$Convex combination of two points <math> v_1,v_2 \in \mathbb{R}^2</math> in a two dimensional vector space <math>\mathbb{R}^2</math> as animation in [[Geogebra]] with <math>t \in [0,1]</math> and <math> K(t) := (1-t)\cdot v_1 + t \cdot v_2</math>$

convex combination

математика

выпуклая комбинация

convex combination

мат.
выпуклая комбинация

convex sum

математика

выпуклая сумма

Определение

convex hull

<mathematics, graphics> For a set S in space, the smallest convex set containing S. In the plane, the convex hull can be visualized as the shape assumed by a rubber band that has been stretched around the set S and released to conform as closely as possible to S. (1997-08-03)

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Википедия

Convex combination

In convex geometry and vector algebra, a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-negative and sum to 1. In other words, the operation is equivalent to a standard weighted average, but whose weights are expressed as a percent of the total weight, instead of as a fraction of the count of the weights as in a standard weighted average.

More formally, given a finite number of points $x_{1},x_{2},\dots ,x_{n}$ in a real vector space, a convex combination of these points is a point of the form

\alpha _{1}x_{1}+\alpha _{2}x_{2}+\cdots +\alpha _{n}x_{n}

where the real numbers $\alpha _{i}$ satisfy $\alpha _{i}\geq 0$ and $\alpha _{1}+\alpha _{2}+\cdots +\alpha _{n}=1.$

As a particular example, every convex combination of two points lies on the line segment between the points.

A set is convex if it contains all convex combinations of its points. The convex hull of a given set of points is identical to the set of all their convex combinations.

There exist subsets of a vector space that are not closed under linear combinations but are closed under convex combinations. For example, the interval $[0,1]$ is convex but generates the real-number line under linear combinations. Another example is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one).

https://en.wikipedia.org/wiki/Convex combination

Как переводится convex combination на Русский язык