Posin

Posin is an extension of the concept of a polynomial , as the sum of monomials , with the help of an extension of the concept of a monomial. From the properties of such generalized monomials, there follows a restriction of the domain of the function defined by the posinom to strictly positive values.

Content

1 Definition
2 Example
3 Properties
4 Applications
5 notes
6 Literature

Definition

Posin is a generalized polynomial of the form:

g(x)=\sum \limits _{i=1}^{n}u_{i}(x)=\sum \limits _{i=1}^{n}c_{i}\prod \limits _{j=1}^{m}{x_{j}}^{a_{ij}},\quad x_{j}>0,\ c_{i}>0,\ a_{ij}\in \mathbb {R} .

{\ displaystyle g (x) = \ sum \ limits _ {i = 1} ^ {n} u_ {i} (x) = \ sum \ limits _ {i = 1} ^ {n} c_ {i} \ prod \ limits _ {j = 1} ^ {m} {x_ {j}} ^ {a_ {ij}}, \ quad x_ {j}> 0, \ c_ {i}> 0, \ a_ {ij} \ in \ mathbb {R}.}

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Where $u_{i}(x),\ i={\overline {1,n}}$ ${\ displaystyle u_ {i} (x), \ i = {\ overline {1, n}}}$ - monomials .

Example

$g(x)=0.25x^{4}+x^{-1.5}+2x^{9}.$ ${\ displaystyle g (x) = 0.25x ^ {4} + x ^ {- 1.5} + 2x ^ {9}.}$

Properties

if $g(x)$ ${\ displaystyle g (x)}$ - posin, $\lambda >0$ ${\ displaystyle \ lambda> 0}$ Is a constant then $\lambda g(x)$ ${\ displaystyle \ lambda g (x)}$ - posin,
if $f(x),g(x)$ ${\ displaystyle f (x), g (x)}$ - posinomas then $f(x)+g(x)$ ${\ displaystyle f (x) + g (x)}$ - also posin,
if $f(x),g(x)$ ${\ displaystyle f (x), g (x)}$ - posinomas then $f(x)g(x)$ ${\ displaystyle f (x) g (x)}$ - also posin.

Thus, the set of posinom is, like the set of polynomials, a ring .

Since monomials are a special case of pozinom, the set of pozinom is also an algebra over a polynomial ring.

if $g(x)$ ${\ displaystyle g (x)}$ - posin, $u(x)$ ${\ displaystyle u (x)}$ - monom , then $g(x)/u(x)$ ${\ displaystyle g (x) / u (x)}$ - posin,
if $g(x)$ ${\ displaystyle g (x)}$ - posin, then ${g(x)}^{k}\ (k>0,$ ${\ displaystyle {g (x)} ^ {k} \ (k> 0,}$ whole $)$ ${\ displaystyle)}$ - posin.

Applications

Posynomials are a basic concept in geometric programming . With the help of pozinom, problems from a wide range of mathematical problems are described and solved, in particular, it relates to: optimal planning, optimal management, economic problems and risk calculation, coding, etc.

Notes

↑ Geometric programming, 1972 , p. 12.

Literature

R. Duffin, E. Peterson, C. Zener. Geometric programming. - M .: Mir, 1972.- 311 p.

[_7e7fd2caa56aa5fd-1] Geometric programming, 1972 , p. 12.