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In [[mathematics]], a '''Hurwitz polynomial''', named after [[Adolf Hurwitz]], is a [[polynomial]] whose coefficients are positive [[real number]]s and whose [[root of a function|roots]] ([[zero (complex analysis)|zeros]]) are located in the left half-plane of the [[complex number|complex plane]] or on the ''jω'' axis, that is, the real part of every root is zero or negative.<ref name="Kuo">{{cite book 
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  | last = Kuo
  | first = Franklin F.
  | authorlink =
  | coauthors =
  | title = Network Analysis and Synthesis, 2nd Ed.
  | publisher = John Wiley & Sons
  | date = 1966
  | location =
  | pages = 295-296
  | url =
  | doi =
  | id =
  | isbn = 0471511188}}</ref>  The term is sometimes restricted to polynomials whose roots have real parts that are strictly negative, excluding the axis (i.e., a Hurwitz [[stable polynomial]]).<ref name=" Weisstein">{{cite web
  | last =  Weisstein
  | first = Eric W
  | title = Hurwitz polynomial
  | work = Wolfram Mathworld
  | publisher = Wolfram Research
  | date = 1999
  | url = http://mathworld.wolfram.com/HurwitzPolynomial.html
  | format =
  | doi =
  | accessdate = July 3, 2013}}</ref><ref name="Reddy">{{cite conference
  | first = Hari C.
  | last = Reddy
  | title = Theory of two-dimensional Hurwitz polynomials
  | booktitle = The Circuits and Filters Handbook, 2nd Ed.
  | pages = 260-263
  | publisher = CRC Press
  | date = 2002
  | location =
  | url = http://books.google.com/books?id=SmDImt1zHXkC&pg=PA262&dq=hurwitz+polynomial
  | doi =
  | id = ISBN 1420041401
  | accessdate = July 3, 2013}}</ref>
 
A polynomial function ''P''(''s'') of a [[complex variable]] ''s'' is said to be Hurwitz if the following conditions are satisfied:
 
:1. ''P''(''s'') is real when ''s'' is real.
:2. The roots of ''P''(''s'') have real parts which are zero or negative.
 
Hurwitz polynomials are important in [[control system|control systems theory]], because they represent the [[Characteristic polynomial#Characteristic equation|characteristic equations]] of [[Stability theory|stable]] [[linear system]]s. Whether a polynomial is Hurwitz can be determined by solving the equation to find the roots, or from the coefficients without solving the equation by the [[Routh-Hurwitz stability criterion]].
 
== Examples ==
 
A simple example of a Hurwitz polynomial is the following:
 
:<math>x^2 + 2x + 1.</math>
 
The only real solution is &minus;1, as it factors to
 
:<math>(x+1)^2.</math>
 
== Properties ==
 
For a polynomial to be Hurwitz, it is necessary but not sufficient that all of its coefficients be positive. A necessary and sufficient condition that a polynomial is Hurwitz is that it pass the [[Routh-Hurwitz stability criterion]].  A given polynomial can be efficiently tested to be Hurwitz or not by using the Routh continued fraction expansion technique.
 
The properties of Hurwitz polynomials are:
 
# All the [[pole (complex analysis)|poles]] and [[zero (complex analysis)|zeros]] are in the left half plane or on its boundary, the imaginary axis.
# Any poles and zeros on the imaginary axis are simple (have a multiplicity of one).
# Any poles on the imaginary axis have real strictly positive residues, and similarly at any zeros on the imaginary axis, the function has a real strictly positive derivative.
# Over the right half plane, the minimum value of the real part of a PR function occurs on the imaginary axis (because the real part of an analytic function constitutes a harmonic function over the plane, and therefore satisfies the maximum principle).
# The polynomial should not have missing powers of s.
 
== References ==
{{reflist}}
 
{{DEFAULTSORT:Hurwitz Polynomial}}
[[Category:Polynomials]]
 
{{mathanalysis-stub}}

Latest revision as of 17:55, 17 December 2014

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Many "behind the scenes" components greatly contribute to the ultimate success of your roof. Tualatin Roofing Contractors That Offer Guarantees. The price tag of getting numerous appraisals made this practice financially impractical. How do we certainly DETECT and solve the leak and the damage in the most effective, cost powerful way so you will not get nailed with a string of flat roof repairs that will under no circumstances remedy the problem at its core. However, there is one disadvantage of metal roofing, and that is corrosion.

If you are unfamiliar with this process, then hiring a professional is a good idea. EPDM roofing material is popular since it is very simply to apply and is readily available. Indicators of a weak framing are cupping and buckling, so pay extra attention on these. Item number one involves looking not at the roof, but your ceiling. When you decide to undergo any building work on your flat roof you must get several builders to give you a quote.

Is there a specific design they want to have installed. The experienced roofer will provide permanent touches after the temporary repair put up by the home owner. If this is the case then you will simply slide the new tile into place and mortar underneath and all around it to create a concrete seal. Once the fire is established, smaller pieces of wood can be added; controlling the combustion air with the dampers. These may make themselves visible in the attic space, on a ceiling of an upper floor, or even in the basement if water runs down an outer wall or vent stack.

Given below are seven important safety tips regarding to it. It is critical to choose the proper thickness for the type and weight of roofing material being installed. If you choose to hire a roofing contractor who's experienced in flat roof repairs, make certain to discover their company history information just before obtaining the business. If you do have a professional do the job, you will pay more, but on the flip side if you do not replace the flooring, you can be almost guaranteed to reduce your selling price by that much in the end, so you will be out that amount either way. They are much more skilled and know greater than you do.

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