Initial value formulation (general relativity)

Template:More footnotes Wenjun Wu's method is an algorithm for solving multivariate polynomial equations introduced in the late 1970s by the Chinese mathematician Wen-Tsun Wu. This method is based on the mathematical concept of characteristic set introduced in the late 1940s by J.F. Ritt. It is fully independent of Gröbner basis method, introduced by Bruno Buchberger (1965), even if Gröbner bases may be used to compute characteristic sets.

Wu's method is powerful for mechanical theorem proving in elementary geometry, and provides a complete decision process for certain classes of problem. It has been used in research in his laboratory (KLMM, Key Laboratory of Mathematics Mechanization in Chinese Academy of Science) and around the world. The main trends of research on Wu's method concern systems of polynomial equations of positive dimension and differential algebra where Ritt's results have been made effective. Wu's method has been applied in various scientific fields, like biology, computer vision, robot kinematics and especially automatic proofs in geometry

Informal description

Wu's method uses polynomial division to solve problems of the form:

$\mathrm{\forall }x,y,z,\dots I(x,y,z,\dots )⟹f(x,y,z,\dots )$

where f is a polynomial equation and I is a conjunction of polynomial equations. The algorithm is complete for such problems over the complex domain.

The core idea of the algorithm is that you can divide one polynomial by another to give a remainder. Repeated division results in either the remainder vanishing (in which case the I implies f statement is true), or an irreducible remainder is left behind (in which case the statement is false).

More specifically, for an ideal I in the ring k[x₁, ..., x_n] over a field k, a (Ritt) characteristic set C of I is composed of a set of polynomials in I, which is in triangular shape: polynomials in C have distinct main variables (see the formal definition below). Given a characteristic set C of I, one can decide if a polynomial f is zero modulo I. That is, the membership test is checkable for I, provided a characteristic set of I.

Ritt characteristic set

A Ritt characteristic set is a finite set of polynomials in triangular form of an ideal. This triangular set satisfies certain minimal condition with respect to the Ritt ordering, and it preserves many interesting geometrical properties of the ideal. However it may not be its system of generators.

• Notation

Let R be the multivariate polynomial ring k[x₁, ..., x_n] over a field k. The variables are ordered linearly according to their subscript: x₁ < ... < x_n. For a non-constant polynomial p in R, the greatest variable effectively presenting in p, called main variable or class, plays a particular role: p can be naturally regarded as a univariate polynomial in its main variable x_k with coefficients in k[x₁, ..., x_k−1]. The degree of p as a univariate polynomial in its main variable is also called its main degree.

• Triangular set

A set T of non-constant polynomials is called a triangular set if all polynomials in T have distinct main variables. This generalizes triangular systems of linear equations in a natural way.

• Ritt ordering

For two non-constant polynomials p and q, we say p is smaller than q with respect to Ritt ordering and written as p <_r q, if one of the following assertions holds:

(1) the main variable of p is smaller than the main variable of q, that is, mvar(p) < mvar(q),

(2) p and q have the same main variable, and the main degree of p is less than the main degree of q, that is, mvar(p) = mvar(q) and mdeg(p) < mdeg(q).

In this way, (k[x₁, ..., x_n],<_r) forms a well partial order. However, the Ritt ordering is not a total order: there exist polynomials p and q such that neither p <_r q nor p _r q. In this case, we say that p and q are not comparable. Note that the Ritt ordering is comparing the rank of p and q. The rank, denoted by rank(p), of a non-constant polynomial p is defined to be a power of its main variable: mvar(p)^mdeg(p) and ranks are compared by comparing first the variables and then, in case of equality of the variables, the degrees.

• Ritt ordering on triangular sets

A crucial generalization on Ritt ordering is to compare triangular sets. Let T = { t₁, ..., t_u} and S = { s₁, ..., s_v} be two triangular sets such that polynomials in T and S are sorted increasingly according to their main variables. We say T is smaller than U w.r.t. Ritt ordering if one of the following assertions holds

(1) there exists k ≤ min(u, v) such that rank(t_i) = rank(s_i) for 1 ≤ i < k and t_k <_r s_k,

(2) u > v and rank(t_i) = rank(s_i) for 1 ≤ i ≤ v.

Also, there exists incomparable triangular sets w.r.t Ritt ordering.

• Ritt characteristic set

Let I be a non-zero ideal of k[x₁, ..., x_n]. A subset T of I is a Ritt characteristic set of I if one of the following conditions holds:

(1) T consists of a single nonzero constant of k,

(2) T is a triangular set and T is minimal w.r.t Ritt ordering in the set of all fine triangular sets contained in I.

A polynomial ideal may possess (infinitely) many characteristic sets, since Ritt ordering is a partial order.

Wu characteristic set

The Ritt–Wu process, first devised by Ritt, subsequently modified by Wu, computes not a Ritt characteristic but an extended one, called Wu characteristic set or ascending chain.

A non-empty subset T of the ideal <F> generated by F is a Wu characteristic set of F if one of the following condition holds

(1) T = {a} with a being a nonzero constant,

(2) T is a triangular set and there exists a subset G of <F> such that <F> = <G> and every polynomial in G is pseudo-reduced to zero with respect to T.

Note that Wu characteristic set is defined to the set F of polynomials, rather to the ideal <F> generated by F. Also it can be shown that a Ritt characteristic set T of <F> is a Wu characteristic set of F. Wu characteristic sets can be computed by Wu's algorithm CHRST-REM, which only requires pseudo-remainder computations and no factorizations are needed.

Wu's characteristic set method has exponential complexity; improvements in computing efficiency by weak chains, regular chains, saturated chain were introduced

Decomposing algebraic varieties

An application is an algorithm for solving systems of algebraic equations by means of characteristic sets. More precisely, given a finite subset F of polynomials, there is an algorithm to compute characteristic sets T₁, ..., T_e such that:

$V(F)=W(T_1)\cup \cdots \cup W(T_e),$

where W(T_i) is the difference of V(T_i) and V(h_i), here h_i is the product of initials of the polynomials in T_i.

See also

• Groebner basis
• Regular chain
• Mathematics-Mechanization Platform
• RegularChains

References

• P. Aubry, M. Moreno Maza (1999) Triangular Sets for Solving Polynomial Systems: a Comparative Implementation of Four Methods. J. Symb. Comput. 28(1–2): 125–154
• David A. Cox, John B. Little, Donal O'Shea. Ideals, Varieties, and Algorithms. 2007.
• Template:Cite web
• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534
• Ritt, J. (1966). Differential Algebra. New York, Dover Publications.
• Dongming Wang (1998). Elimination Methods. Springer-Verlag, Wien, Springer-Verlag
• Dongming Wang (2004). Elimination Practice, Imperial College Press, London ISBN 1-86094-438-8
• Wu, W. T. (1984). Basic principles of mechanical theorem proving in elementary geometries. J. Syst. Sci. Math. Sci., 4, 207–35
• Wu, W. T. (1987). A zero structure theorem for polynomial equations solving. MM Research Preprints, 1, 2–12
• One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
  
  In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
  
  Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
  
  Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
  
  A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
  
  The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
  
  There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

External links

• wsolve Maple package
• The Characteristic Set Method