62,943 research outputs found
Groebner bases
No full textConsider a polynomial ring k[χ] in one indeterminate over a field k. Given a polynomial in k[χ], one is able to determine whether that polynomial lies in a given ideal I C k[χ] by applying the Division Algorithm in one variable. The condition r = 0 is necessary and sufficient for membership to the ideal. However, this is not the case for the polynomial ring k[χl...,χn] because remainders generated by the Division Algorithm in n variables are not unique.By the Hilbert Basis Theorem, any given ideal J C k[χl...,χn] has a finite generating set, that is, J = 〈gl,...,gt〉. Then gl,...,gt are a basis of J. They are also known as generators of J. A Groebner basis of J is a special basis whereby the remainder on division by the generators is unique with respect to some fixed ordering. We can then determine ideal membership by checking the remainder.Besides studying the basic theory of Groebner bases, we will look at how we can construct them using the Buchberger's Algorithm and S-polynomials.. Groebner bases also have several applications. The two applications that we are going to study are ideal membership and solving systems of polynomial equations
Symbolic Analysis for Boundary Problems: From Rewriting to Parametrized Groebner Bases
No full textWe review our algebraic framework for linear boundary problems (concentrating on ordinary differential equations). Its starting point is an appropriate algebraization of the domain of functions, which we have named integro-differential algebras. The algebraic treatment of boundary problems brings up two new algebraic structures whose symbolic representation and computational realization is based on canonical forms in certain commutative and noncommutative polynomial domains. The first of these, the ring of integro-differential operators, is used for both stating and solving linear boundary problems. The other structure, called integro-differential polynomials, is the key tool for describing extensions of integrodifferential algebras. We use the canonical simplifier for integro-differential polynomials for generating an automated proof establishing a canonical simplifier for integro-differential operators. Our approach is fully implemented in the THEOREMA system; some code fragments and sample computations are included
Letter from J. R. Eakin to Arthur G. Ringland
No full textLetter (copy) from J. R. Eakin to Arthur C. Ringland about the alignment of 40 acres near the Buggeln ranch
Letter from Arno B. Cammerer to J. R. Eakin
No full textLetter from Arno B. Cammerer to J. R. Eakin describing the procedure for purchasing Bright Angel Trail
Letter from J. R. Eakin to Carl Hayden
No full textLetter from J. R. Eakin to Carl T. Hayden concerning access to Rowe Well and the canyon
Letter from J. R. Eakin to Stephen Mather
No full textLetter from J. R. Eakin to Stephen T. Mather about expenses and reconstruction of the Kaibab Trail
Letter from Carl Hayden to J. R. Eakin
No full textLetter from Carl T. Hayden to J. R. Eakin regarding changes to the Grand Canyon National Park boundaries and the purchase of lands from William Randolph Hearst
Letter from J. R. Eakin to Stephen Mather
No full textLetter from J. R. Eaking to the National Park Service director about changes to the Grand Canyon National Park boundaries, and access to water near the Buggeln property on Desert View road
[Letter from J. R. Roberts to Sister, November 24, 1878]
No full textLetter from J. R. Roberts to sister. J. R. thanked his sister for gifts that were sent and went on to update her on what was happening in their families' lives. The letter ended with a mention that people were searching for land claims in the area and the author wanted their mother to not worry about them
Letter from J. R. Eakin, Grand Canyon National Park to Carl Hayden
No full textLetter from J. R. Eakin to Carl Hayden regarding the sale of Bass properties to the Santa Fe Railroad Company
- …
