scholarly journals A remark on projective modules

1987 ◽  
Vol 36 (3) ◽  
pp. 515-520
Author(s):  
Wojciech Kucharz
Keyword(s):  
Projective Modules ◽  
Real Numbers ◽  
Algebraic Subset ◽  
Regular Functions ◽  
The Ideal

Let R denote the field of real numbers and let A be the ring of regular functions on Rn, that is the localization of R[T1 …, Tn] with respect to the set of all polynomials vanishing nowhere on Rn. Let X be an algebraic subset of Rn and let I(X) be the ideal of A of all functions vanishing on X. Assume that X is compact and nonsingular and k = codim X = 1, 2, 4 or 8. we prove here that if the A/I(X)-module I(X)/I(X)2 can be generated by k elements, then there exist a projective A-module P of rank k and a homomorphism from P onto I(X).

A Note on the Dubois-Efroymson Dimension Theorem

1989 ◽  
Vol 32 (1) ◽  
pp. 24-29 ◽  
Author(s):  
Wojciech Kucharz
Keyword(s):  
Algebraic Subset ◽  
Algebraic Set ◽  
Regular Functions ◽  
The Ideal

AbstractLet X ⊂ Rnn be an irreducible nonsingular algebraic set and let Z be an algebraic subset of X with dim Z ≦ dim X — 2. In this paper it is shown that there exists an irreducible algebraic subset Y of X satisfying the following conditions: dim Y = dim X — 1, Z ⊂ Y and that the ideal of regular functions on X vanishing on Y is principal.

K-theory of real algebraic surfaces and threefolds

1989 ◽  
Vol 106 (3) ◽  
pp. 471-480 ◽  
Author(s):  
J. Bochnak ◽  
W. Kucharz
Keyword(s):  
Vector Bundles ◽  
Grothendieck Group ◽  
Algebraic Surfaces ◽  
Real Algebraic Variety ◽  
Projective Modules ◽  
Algebraic Subset ◽  
Witt Group ◽  
Regular Functions ◽  
Real Algebraic Surfaces ◽  
Continuous Complex

LetXbe an affine real algebraic variety, i.e., up to biregular isomorphism an algebraic subset of ℝn. (For definitions and notions of real algebraic geometry we refer the reader to the book [6].) Letdenote the ring of regular functions onX([6], chapter 3). (IfXis an algebraic subset of ℝnthenis comprised of all functions of the formf/g, whereg, f: X→ ℝ are polynomial functions withg−1(O) = Ø.) In this paper, assuming thatXis compact, non-singular, and that dimX≤ 3, we compute the Grothendieck groupof projective modules over(cf. Section 1), and the Grothendieck groupand the Witt groupof symplectic spaces over(cf. Section 2), in terms of the algebraic cohomology groupsandgenerated by the cohomology classes associated with the algebraic subvarieties ofX. We also relate the groupto the Grothendieck groupKO(X) of continuous real vector bundles overX, and the groupsandto the Grothendieck groupK(X)of continuous complex vector bundles overX.

Equations and Complexity for the Dubois–Efroymson Dimension Theorem

2009 ◽  
Vol 52 (2) ◽  
pp. 224-236
Author(s):  
Riccardo Ghiloni
Keyword(s):  
Real Closed Field ◽  
Closed Field ◽  
Algebraic Subset ◽  
Algebraically Closed Field ◽  
Algebraic Set ◽  
Real Algebraic Set ◽  
Nonsingular Point ◽  
Minimum Integer ◽  
The Ideal

AbstractLetRbe a real closed field, letX⊂Rnbe an irreducible real algebraic set and letZbe an algebraic subset ofXof codimension ≥ 2. Dubois and Efroymson proved the existence of an irreducible algebraic subset ofXof codimension 1 containingZ. We improve this dimension theorem as follows. Indicate by μ the minimum integer such that the ideal of polynomials inR[x1, … ,xn] vanishing onZcan be generated by polynomials of degree ≤ μ. We prove the following two results: (1) There exists a polynomialP∈R[x1, … ,xn] of degree≤ μ+1 such thatX∩P–1(0) is an irreducible algebraic subset ofXof codimension 1 containingZ. (2) LetFbe a polynomial inR[x1, … ,xn] of degreedvanishing onZ. Suppose there exists a nonsingular pointxofXsuch thatF(x) = 0 and the differential atxof the restriction ofFtoXis nonzero. Then there exists a polynomialG∈R[x1, … ,xn] of degree ≤ max﹛d, μ + 1﹜ such that, for eacht∈ (–1, 1) \ ﹛0﹜, the set ﹛x∈X|F(x) +tG(x) = 0﹜ is an irreducible algebraic subset ofXof codimension 1 containingZ. Result (1) and a slightly different version of result (2) are valid over any algebraically closed field also.

The Gray World of Practical Data Analysis: An Introduction to Part 2

2011 ◽  
Author(s):  
Gidon Eshel
Keyword(s):  
Data Analysis ◽  
Linear System ◽  
Linear Algebra ◽  
Actual Data ◽  
Distinguishing Characteristic ◽  
Real Numbers ◽  
Ideal World ◽  
Main Distinguishing Characteristic ◽  
Realistic Data ◽  
The Ideal

This chapter provides an overview of the second part of the book. This part is the crux of the matter: how to analyze actual data. While this part builds on Part 1, especially on linear algebra fundamentals covered in Part 1, the two are not redundant. The main distinguishing characteristic of Part 2 is its nuanced grayness. In the ideal world of algebra (and thus in most of part 1), things are black or white: two vectors are either mutually orthogonal or not, real numbers are either zero or not, a vector either solves a linear system or does not. By contrast, realistic data analysis, the province of Part 2, is always gray, always involves subjective decisions.

scholarly journals Normalité de certains anneaux déterminantiels quantiques

1999 ◽  
Vol 42 (3) ◽  
pp. 621-640 ◽  
Author(s):  
Laurent Rigal
Keyword(s):  
Residue Class ◽  
Quantum Algebras ◽  
Regular Functions ◽  
Maximal Orders ◽  
Ring Of Fractions ◽  
Prime Factors ◽  
The Matrix ◽  
Residue Class Ring ◽  
Class Ring ◽  
The Ideal

Let Kq[X] = Oq(M(m, n)) be the quantization of the ring of regular functions on m × n matrices and Iq (X) be the ideal generated by the 2 × 2 quantum minors of the matrix X=(Xij)l≤i≤m, I≤j≤n of generators of Kq[X]. The residue class ring Rq(X) = Kq[X]/Iq(X) (a quantum analogue of determinantal rings) is shown to be an integral domain and a maximal order in its divisionring of fractions. For the proof we use a general lemma concerning maximalorders that we first establish. This lemma actually applies widely to prime factors of quantum algebras. We also prove that, if the parameter isnot a root of unity, all the prime factors of the uniparameter quantum space are maximal orders in their division ring of fractions.

On α-Polynomial Regular Functions, with Applications to Ordinary Differential Equations

2014 ◽  
Vol 57 (2) ◽  
pp. 405-421 ◽  
Author(s):  
Peter Fenton ◽  
Janne Grohn ◽  
Janne Heittokangas ◽  
John Rossi ◽  
Jouni Rattya
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Unit Disc ◽  
Maximum Modulus ◽  
Linear Differential Equations ◽  
Real Numbers ◽  
Regular Functions ◽  
Function Classes ◽  
Finite Set ◽  
Linear Differential

AbstractThis research deals with properties of polynomial regular functions, which were introduced in a recent study concerning Wiman-Valiron theory in the unit disc. The relation of polynomial regular functions to a number of function classes is investigated. Of particular interest is the connection to the growth class Gα, which is closely associated with the theory of linear differential equations with analytic coefficients in the unit disc. If the coefficients are polynomial regular functions, then it turns out that a finite set of real numbers containing all possible maximum modulus orders of solutions can be found. This is in contrast to what is known about the case when the coefficients belong to Gα.

scholarly journals A Data Pre-Processing Model for the Topsis Method

2016 ◽  
Vol 16 (2) ◽  
pp. 219-235 ◽  
Author(s):  
Andrzej Kobryń ◽  
Joanna Prystrom
Keyword(s):  
Decision Making ◽  
Fuzzy Numbers ◽  
Interval Data ◽  
Decision Making Process ◽  
Multi Criteria Decision Making ◽  
Topsis Method ◽  
Real Numbers ◽  
Calculation Algorithms ◽  
Negative Ideal Solution ◽  
The Ideal

AbstractTOPSIS is one of the most popular methods of multi-criteria decision making (MCDM). Its fundamental role is the establishment of chosen alternatives ranking based on their distance from the ideal and negative-ideal solution. There are three primary versions of the TOPSIS method distinguished: classical, interval and fuzzy, where calculation algorithms are adjusted to the character of input rating decision-making alternatives (real numbers, interval data or fuzzy numbers). Various, specialist publications present descriptions on the use of particular versions of the TOPSIS method in the decision-making process, particularly popular is the fuzzy version. However, it should be noticed, that depending on the character of accepted criteria – rating of alternatives can have a heterogeneous character. The present paper suggests the means of proceeding in the situation when the set of criteria covers characteristic criteria for each of the mentioned versions of TOPSIS, as a result of which the rating of the alternatives is vague. The calculation procedure has been illustrated by an adequate numerical example.

scholarly journals A monadic, functional implementation of real numbers

2007 ◽  
Vol 17 (1) ◽  
pp. 129-159 ◽  
Author(s):  
RUSSELL ÓCONNOR
Keyword(s):  
Real Number ◽  
Large Scale ◽  
Metric Spaces ◽  
Continuous Functions ◽  
Proof Assistants ◽  
Elementary Functions ◽  
Real Numbers ◽  
Mathematical Proofs ◽  
Regular Functions ◽  
Regular Sequences

Large scale real number computation is an essential ingredient in several modern mathematical proofs. Because such lengthy computations cannot be verified by hand, some mathematicians want to use software proof assistants to verify the correctness of these proofs. This paper develops a new implementation of the constructive real numbers and elementary functions for such proofs by using the monad properties of the completion operation on metric spaces. Bishop and Bridges's notion (Bishop and Bridges 1985) of regular sequences is generalised to what I call regular functions, which form the completion of any metric space. Using the monad operations, continuous functions on length spaces (which are a common subclass of metric spaces) are created by lifting continuous functions on the original space. A prototype Haskell implementation has been created. I believe that this approach yields a real number library that is reasonably efficient for computation, and still simple enough to verify its correctness easily.

Projective Modules Over Central Separable Algebras

1969 ◽  
Vol 21 ◽  
pp. 39-43 ◽  
Author(s):  
F. R. DeMeyer
Keyword(s):  
Finite Number ◽  
Commutative Rings ◽  
Projective Modules ◽  
Principal Ideal Domain ◽  
Ideal Structure ◽  
Perfect Field ◽  
Maximal Ideals ◽  
Ring Of Polynomials ◽  
Separable Algebras ◽  
The Ideal

In (2), M. Auslander and O. Goldman laid the foundations for the study of central separable algebras. For unexplained terminology and notation, see (2). Here we are interested in projective modules and the ideal structure of a central separable algebra A over some special commutative rings K. When K is a field, one consequence of Wedderburn's Theorem is that there is a unique (up to isomorphism) irreducible A-module. We show here that if K is a commutative ring with a finite number of maximal ideals (semi-local) and with no idempotents other than 0 and 1 or if K is the ring of polynomials in one variable over a perfect field, then there is a unique (up to isomorphism) indecomposable finitely generated projective A-module. An example in (3) shows that this result fails if one only assumes that K is a principal ideal domain.

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