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In [[computer algebra]], a '''regular semi-algebraic system''' is a particular kind of triangular system  of multivariate polynomials over a real closed field.
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== Introduction ==
[[Regular chain]]s and [[triangular decomposition]]s are fundamental and well-developed tools for describing the complex solutions of polynomial systems. The notion of a regular semi-algebraic system is an adaptation of the concept of a regular chain focusing on solutions of the real analogue: semi-algebraic systems.
 
Any semi-algebraic system <math>S</math> can be decomposed into finitely many regular semi-algebraic systems <math>S_1, \ldots, S_e</math> such that a point (with real coordinates) is a solution of <math>S</math> if and only if it is a solution of one of the systems <math>S_1, \ldots, S_e</math>.<ref>Changbo Chen, James H. Davenport, John P. May, Marc Moreno-Maza, Bican Xia, Rong Xiao. Triangular decomposition of semi-algebraic systems.  Proceedings of 2010 International Symposium on Symbolic and Algebraic Computation (ISSAC 2010), ACM Press, pp. 187–194, 2010.</ref>
 
== Formal definition ==
 
Let <math>T</math> be a [[regular chain]] of <math>{{\mathbf{k}}}[x_1, \ldots, x_n]</math>
for some ordering of the variables <math>\mathbf{x} = x_1, \ldots, x_n</math> and a [[real closed field]] <math>{{\mathbf{k}}}</math>.
Let <math>\mathbf{u} = u_1, \ldots, u_d</math> and <math>\mathbf{y} = y_1, \ldots, y_{n-d}</math>
designate respectively the variables of <math>\mathbf{x}</math> that are free and algebraic with respect to <math>T</math>. Let <math>P \subset {{\mathbf{k}}}[\mathbf{x}]</math> be finite such that
each polynomial in <math>P</math> is regular w.r.t.\ the saturated ideal of <math>T</math>.
Define <math>P_{>} :=\{p>0\mid p\in P\}</math>. Let <math>\mathcal{Q}</math> be a quantifier-free
formula of <math>{{\mathbf{k}}}[\mathbf{x}]</math> involving only the variables of <math>\mathbf{u}</math>. We say that <math>R := [\mathcal{Q}, T, P_{>}]</math> is a '''regular semi-algebraic system''' if the following three conditions hold.
 
* <math>\mathcal{Q}</math> defines a non-empty open semi-algebraic set <math>S</math> of <math>{{\mathbf{k}}}^d</math>,
* the regular system <math>[T, P] </math> specializes well at every point <math>u</math> of <math>S</math>,
* at each point <math>u</math> of <math>S</math>, the specialized system <math>[T(u), P(u)_{>}</math> has at least one real zero.
 
The zero set of <math>R</math>, denoted by <math>{Z}_{{{\mathbf{k}}}}(R)</math>,
is defined as the set of points <math>(u, y) \in {{{\mathbf{k}}}}^d \times {{{\mathbf{k}}}}^{n-d}</math> such that <math>\mathcal{Q}(u)</math> is true and  <math>t(u, y)=0</math>, <math>p(u, y)>0</math>, for all <math>t\in T</math>and all <math>p\in P</math>.
 
== See also ==
*[[Wu's method of characteristic set|Characteristic set]]
*[[Triangular decomposition]]
*[[Regular chain]]
*[[RegularChains]]
*[[Real algebraic geometry]]
 
== References ==
{{Reflist}}
 
[[Category:Equations]]
[[Category:Algebra]]
[[Category:Polynomials]]
[[Category:Algebraic geometry]]
[[Category:Computer algebra]]

Revision as of 02:42, 4 March 2014

Indoor gardening is a way to extend the expanding season and get pleasure from plants that otherwise would not survive outdoors. These initial generation LED panels are quite sweet and genuinely low-cost as LED develop light technologies goes. On the other hand, there has been fascinating developments in LED technologies recently, with RGB LEDs getting at the front. What puts these above HID lights is that these LEDs are able to make the maximum peak spectral output of a particular colour, some thing a HID light can't receive. As with all LED goods, the price tag is bound to come down as time goes on. Every single year rates are going down, as we‘re nonetheless standing at the very starting of the revolution of light that LEDs will bring.

Hardy deciduous and evergreen trees, such as maples and pines, hardly ever do well indoors and below LED lights because they call for winter situations for their development cycle. The primary advantage to growing bonsai with LED lights is that LED lights use substantially less power than other lighting strategies. LED lights final between 5 and 12 years, requiring less frequent replacement than other forms of lighting. Hydroponic systems normally require develop lights.

In all walks of life to the green, environmental protection, power conservation advocate atmosphere, the use of the hotel LED For the transformation of the lighting method, is displaying growing interest. LED technology just to meet these needs, it also has broad prospects for improvement. Here's more information regarding Growing Autoflower With Led Lights look at the web page. Hotel Lighting LED interior lighting is becoming a marketplace breakthrough. LED develop lights have undergone extensive testing and are verified in the field.

As soon as once again, in an effort to compete with other lights on the market, they have also incorporated undesirable white LEDs All the LEDs are 3W LEDs, which is preferred (more than puny 1W bulbs or the significantly less efficient 5W LEDs) but they never indicate no matter if they are from a trustworthy manufacturer. The absolute max light you want depends on the plant you are increasing. Plants differ and some will modify colour to reflect away undesirable light. The switching of the lights is completed by a timer Not manually.

Environmental awareness, the green energy into the moment with the college of financial improvement, LED low energy there for the truth that gambling, and with the LED technologies continues to evolve,LED LightingIn powerful help of national government policies evolved. No heat also implies LED grow lights are really discreet since there is no thermal footprint. Most LED develop lights are made to operate for more than 50,000 hours.

Also, with indoor farming it may be probable to alter other aspects that may well make plants use light a lot more efficiently such as CO2 concentration, although this is incredibly substantially an it depends” aspect. I reside in an region where a tremendous amount of electricity goes to develop lights. Opinions and comments published on this web page may perhaps not be sanctioned by, and do not necessarily represent the views of Sustainable Enterprises Media, Inc, its owners, sponsors, affiliates, or subsidiaries.

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