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Arxiv:1601.07660V1 [Math.AC] 28 Jan 2016 2].Tesemigroup the [26])
INTEGRAL DOMAINS WITH BOOLEAN t-CLASS SEMIGROUP S. KABBAJ AND A. MIMOUNI Abstract. The t-class semigroup of an integral domain is the semigroup of the isomorphy classes of the t-ideals with the operation induced by t- multiplication. This paper investigates integral domains with Boolean t-class semigroup with an emphasis on the GCD and stability conditions. The main results establish t-analogues for well-known results on Pr¨ufer domains and B´ezout domains of finite character. 1. Introduction All rings considered in this paper are integral domains (i.e., commutative with identity and without zero-divisors). The class semigroup of a domain R, denoted S(R), is the semigroup of nonzero fractional ideals modulo its subsemigroup of nonzero principal ideals [11, 41]. The t-class semigroup of R, denoted St(R), is the semigroup of fractional t-ideals modulo its subsemigroup of nonzero principal ideals, that is, the semigroup of the isomorphy classes of the t-ideals of R with the operation induced by ideal t-multiplication. Notice that St(R) is the t-analogue of S(R), as the class group Cl(R) is the t-analogue of the Picard group Pic(R). The following set-theoretic inclusions always hold: Pic(R) ⊆ Cl(R) ⊆ St(R) ⊆ S(R). Note that the first and third inclusions turn into equality for Pr¨ufer domains and the second does so for Krull domains. More details on these objects are provided in the next section. Divisibility properties of a domain R are often reflected in group or semigroup- theoretic properties of Cl(R) or S(R). -
Characterization of Pr¨Ufer-Like Monoids and Domains by Gcd
CHARACTERIZATION OF PRUFER-LIKE¨ MONOIDS AND DOMAINS BY GCD-THEORIES FRANZ HALTER-KOCH Abstract. We combine ideal-theoretic and divisor-theoretic methods to characterize various classes of Pr¨ufer-like monoids and domains by the gcd-properties of certain semi- groups of invertible ideals. 1. Introduction One of the main themes of multiplicative ideal theory during the last decades was the characterization and investigation of various classes of integral domains defined by the invertibility properties of certain classes of ideals. In this field, Pr¨uferdomains form the classical antetype, and there is a wealth of generalizations and variations of this concept in the literature. In this paper, we combine ideal-theoretic and divisor-theoretic methods to arrange some known characterizations of Pr¨ufer-like domains in a new way and to present several new ones. One of the basic ideas in our investigations is to address the gcd-properties of certain semigroups of invertible (integral) ideals and to combine this viewpoint with the concept of gcd-theories. Although the theory of integral domains is our main concern, the paper is written in the language of (commutative cancellative) monoids in order to point out the purely multiplicative character of the theory. The main results are the Theorems 3.4 and 3.5 and the subsequent theorems and corollaries. In Section 2 we gather the necessary results from the theory of monoid homomorphisms and ideal systems. 2. Monoids and homomorphisms Throughout this paper, by a monoid D we mean (deviating from the usual termi- nology) a commutative multiplicative semigroup with unit element 1 ∈ D and a zero element 0 ∈ D (satisfying 1x = x and 0x = 0 for all x ∈ D), and we always assume that D• = D \{0} is cancellative (that is, for all a, b ∈ D and c ∈ D•, if ac = bc, then a = b ). -
Symmetric Coverage of Dynamic Mapping Error for Mobile Sensor Networks
2011 American Control Conference on O'Farrell Street, San Francisco, CA, USA June 29 - July 01, 2011 Symmetric coverage of dynamic mapping error for mobile sensor networks Carlos H. Caicedo-Nu´nez˜ and Naomi Ehrich Leonard Abstract—We present an approach to control design for nonuniform density fields. Related problems include control a mobile sensor network tasked with sampling a scalar field of a mobile sensor network in a noisy, possibly time-varying and providing optimal space-time measurements. The coverage field for minimum-error spatial estimation [9], [10] and metric is derived from the mapping error in objective analysis (OA), an assimilation scheme that provides a linear statistical cooperative exploration of features [11]. In [12] agents are estimation of a sampled field. OA mapping error is an example deployed in a distributed way in order to maximize the of a consumable density field: the error decreases dynamically probability of event detection. The authors of [13] study the at locations where agents move and sample. OA mapping error distributed implementation of maximizing joint entropy in is also a regenerating density field if the sampled field is measurements. However, none of these methods have been time-varying: error increases over time as measurement value decays. The resulting optimal coverage problem presents a chal- designed to specifically address a spatial-temporal density lenge to traditional coverage methods. We prove a symmetric field that changes in response to the motion of the agents. dynamic coverage solution that exploits the symmetry of the In this paper we propose a coverage strategy for a density domain and yields symmetry-preserving coordinated motion field defined by OA mapping error in the case that the of mobile sensors. -
A LOCALIZATION of R[X]
Can. J. Math., Vol. XXXIII, No. 1, 1981, pp. 103-115 A LOCALIZATION OF R[x] JAMES A. HUCKABA AND IRA J. PAPICK 1. Introduction. Throughout this paper, R will be a commutative integral domain with identity and x an indeterminate. If / G R[x], let CR(J) denote the ideal of R generated by the coefficients of/. Define 1 SR = {/ G R[x]: cR(f) = R\ and UR = {/ G lî(s): ^(J)" = *}. For a,b £ R, write (a:6) = {r G R: rb G (a)}. When no confusion may result, we will write c(f),S, U, and (a:b). It follows that both 5 and U are multiplicatively closed sets in R[x] [7, Proposition 33.1], [17, Theorem F], and that R[x]s Q R[x]n. The ring R[x]s, denoted by R(x), has been the object of study of several authors (see for example [1], [2], [3], [12]). An especially inter esting paper concerning R(x) is that of Arnold's [3], where he, among other things, characterizes when R(x) is a Priifer domain. We shall make special use of his results in our work. In § 2 we determine conditions on the ring R so that R(x) = R[x]v. A complete characterization of this property is given for Noetherian domains in Proposition 2.2. In particular, we prove that if R is a Noe therian domain, then R(x) = R[x]v if and only if depth (R) ^ 1. Some sufficient conditions for R(x) = R[x]u are that R be treed (Proposition 2.5), or that SP (R) (see § 2 for définitions) be finite (Proposition 2.9). -
Self-Similar and Self-Affine Sets; Measure of the Intersection of Two
Self-similar and self-affine sets; measure of the intersection of two copies M´arton Elekes , Tam´as Keleti and Andr´as M´ath´e †† ‡† ‡‡ Alfr´ed R´enyi Institute of Mathematics, Hungarian Academy of Sciences, † P.O. Box 127, H-1364, Budapest, Hungary Department of Analysis, E¨otv¨os Lor´and University, P´azm´any P´eter s´et´any 1/c, ‡ H-1117 Budapest, Hungary (e-mail: [email protected], [email protected], [email protected]) (Received 2008 ) Abstract. Let K Rd be a self-similar or self-affine set and let µ be a self-similar or ⊂ self-affine measure on it. Let be the group of affine maps, similitudes, isometries G or translations of Rd. Under various assumptions (such as separation conditions or we assume that the transformations are small perturbations or that K is a so called Sierpi´nski sponge) we prove theorems of the following types, which are closely related to each other; (Non-stability) • There exists a constant c < 1 such that for every g we have either ∈ G µ K g(K) <c µ(K) or K g(K). ∩ · ⊂ (Measure and topology) • For every g we have µ K g(K) > 0 int (K g(K)) = (where ∈ G ∩ ⇐⇒ K ∩ 6 ∅ int is interior relative to K). K (Extension) • The measure µ has a -invariant extension to Rd. G Moreover, in many situations we characterize those g’s for which µ K g(K) > 0. ∩ We also get results about those g’s for which g(K) K or g(K) K. -
Unique Factorization Domains in Commutative Algebra.Pdf (366.6Kb)
Unique Factorization Domains in Commutative Algebra Yongjian Huang Advisor: Prof. Daniel Katz University of Kansas May 20, 2021 1 Introduction In this project, we learn about unique factorization domains in commutative algebra. Most importantly, we explore the relation between unique factorization domains and regular local rings, and prove the main theorem: If R is a regular local ring, so is a unique factorization domain. 2 Prime ideals Before learning the section about unique factorization domains, we first need to know about definition and theorems about prime ideals. Definition 2.1. In a commutative ring R, the ideal I is prime if ab 2 I implies a 2 I or b 2 I. Alternatively, I is prime if R=I is an integral domain. The following theorem tells us another way to define prime ideals. Theorem 2.1. Let S be a multiplicatively closed set in a ring R and let I be an ideal in R maximal with respect to the exclusion of S. Then I is prime. Proof. Given ab 2 I , we want to show a 2 I or b 2 I . We give a proof by contradiction, suppose a2 = I and b2 = I, then the ideal (I; a) generated by I and a is strictly larger than I. So the ideal (I; a) intersects S. Thus, there exists an element s 2 S of the form s1 = i1 + xa, where i1 2 I and x 2 R. Similarly, we have s2 = i2 + yb, where i2 2 I and y 2 R. s1 · s2 = (i1 + xa)(i2 + yb) = i1i2 + i1yb + i2xa + xyab Thus, s1s2 2 I. -
NOTES on UNIQUE FACTORIZATION DOMAINS Alfonso Gracia-Saz, MAT 347
Unique-factorization domains MAT 347 NOTES ON UNIQUE FACTORIZATION DOMAINS Alfonso Gracia-Saz, MAT 347 Note: These notes summarize the approach I will take to Chapter 8. You are welcome to read Chapter 8 in the book instead, which simply uses a different order, and goes in slightly different depth at different points. If you read the book, notice that I will skip any references to universal side divisors and Dedekind-Hasse norms. If you find any typos or mistakes, please let me know. These notes complement, but do not replace lectures. Last updated on January 21, 2016. Note 1. Through this paper, we will assume that all our rings are integral domains. R will always denote an integral domains, even if we don't say it each time. Motivation: We know that every integer number is the product of prime numbers in a unique way. Sort of. We just believed our kindergarden teacher when she told us, and we omitted the fact that it needed to be proven. We want to prove that this is true, that something similar is true in the ring of polynomials over a field. More generally, in which domains is this true? In which domains does this fail? 1 Unique-factorization domains In this section we want to define what it means that \every" element can be written as product of \primes" in a \unique" way (as we normally think of the integers), and we want to see some examples where this fails. It will take us a few definitions. Definition 2. Let a; b 2 R. -
Journal of Computational and Applied Mathematics Numerical Integration
Journal of Computational and Applied Mathematics 233 (2009) 279–292 Contents lists available at ScienceDirect Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam Numerical integration over polygons using an eight-node quadrilateral spline finite elementI Chong-Jun Li a,∗, Paola Lamberti b, Catterina Dagnino b a School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China b Department of Mathematics, University of Torino, via C. Alberto, 10 Torino 10123, Italy article info a b s t r a c t Article history: In this paper, a cubature formula over polygons is proposed and analysed. It is based Received 9 February 2009 on an eight-node quadrilateral spline finite element [C.-J. Li, R.-H. Wang, A new 8-node Received in revised form 8 July 2009 quadrilateral spline finite element, J. Comp. Appl. Math. 195 (2006) 54–65] and is exact for quadratic polynomials on arbitrary convex quadrangulations and for cubic polynomials on MSC: rectangular partitions. The convergence of sequences of the above cubatures is proved for 65D05 continuous integrand functions and error bounds are derived. Some numerical examples 65D07 are given, by comparisons with other known cubatures. 65D30 65D32 ' 2009 Elsevier B.V. All rights reserved. Keywords: Numerical integration Spline finite element method Bivariate splines Triangulated quadrangulation 1. Introduction The problem considered in this paper is the numerical evaluation of Z IΩ .f / D f .x; y/dxdy; (1) Ω 2 where f 2 C.Ω/ and Ω is a polygonal domain in R , i.e. a domain with the boundary composed of piecewise straight lines. -
NOTICES Was Sent to Press, the Meeting Dates Which Fall Rather Far in the Future Are Subject Tr Change
AMERICAN MATHEMATICAL SOCIETY Nottces Edited by GORDON L. WALKER Contents MEETINGS Calendar of Meetings • • • . • • • • • • • • . • • • • • . • • • • • • 4 Program of the February Meeting in Tucson • • • . • . • . • . 5 Abstracts for the Meeting, pp. 80-84 Program of the February Meeting in New York ••••••••• 10 Abstracts for the Meeting, pp. 85-94 PRELIMINARY ANNOUNCEMENTS OF MEETINGS ••••••.••• 15 ACTIVITIES OF OTHER ASSOCIATIONS ••••••••••••••••• 19 NEWS ITEMS AND ANNOUNCEMENTS •••••••••••••••••• 27 FOREIGN SCIENCE INFORMATION •••••••••.•••••••••• 38 PERSONAL ITEMS •••••••.•••••••••.••••••••••••• 47 NEW PUBLICATIONS ••••••••••••••••••••••••••••. 50 CATALOGUE OF LECTURE NOTES - Supplement No.1 . • • • • • 57 ABSTRACTS OF CONTRIBUTED PAPERS • • • • • • . • • • • • • • • 59 RESERVATION FORM . • . • • • • • • • • • • . • • • • • . • . • • • • • • 99 3 MEETINGS CALENDAR OF MEETINGS NOTE: This Calendar lists all of the meetings which have been approved by the Council up to the date at which this issue of the NOTICES was sent to press, The meeting dates which fall rather far in the future are subject tr change. This is particularly true of the meetings to which no numbers have yet been assigned, Meet Deadline ing Date Place for No, Abstracts* 567 April 14-16, 1960 New York, New York Mar. 1 568 April 22-23, 1960 Chicago, Illinois Mar, 1 569 April 22-23, 1960 Berkeley, California Mar, 570 June 18, 1960 Missoula, Montana May 5 571 August 29-September 3, 1960 East Lansing, Michigan July 15 (65th Summer Meeting) 572 October 22, 1960 Worcester, Massachusetts Sept. 8 January 24-27, 1961 Washington, D. C. (67th Annual Meeting) August, 1961 Stillwater, Oklahoma (66th Summer Meeting) January, 1962 Kansas City, Missouri (68th Annual Meeting) August, 1962 Vancouver, British Columbia (67th Summer Meeting) August, 1963 Boulder, Colorado (68th Summer Meeting) *The abstracts of papers to be presented at the meetings must be received i. -
MATHEMATICAL TRIPOS Part IB 2017
MATHEMATICALTRIPOS PartIB 2017 List of Courses Analysis II Complex Analysis Complex Analysis or Complex Methods Complex Methods Electromagnetism Fluid Dynamics Geometry Groups, Rings and Modules Linear Algebra Markov Chains Methods Metric and Topological Spaces Numerical Analysis Optimisation Quantum Mechanics Statistics Variational Principles Part IB, 2017 List of Questions [TURN OVER 2 Paper 3, Section I 2G Analysis II What does it mean to say that a metric space is complete? Which of the following metric spaces are complete? Briefly justify your answers. (i) [0, 1] with the Euclidean metric. (ii) Q with the Euclidean metric. (iii) The subset (0, 0) (x, sin(1/x)) x> 0 R2 { } ∪ { | }⊂ with the metric induced from the Euclidean metric on R2. Write down a metric on R with respect to which R is not complete, justifying your answer. [You may assume throughout that R is complete with respect to the Euclidean metric.] Paper 2, Section I 3G Analysis II Let X R. What does it mean to say that a sequence of real-valued functions on ⊂ X is uniformly convergent? Let f,f (n > 1): R R be functions. n → (a) Show that if each fn is continuous, and (fn) converges uniformly on R to f, then f is also continuous. (b) Suppose that, for every M > 0, (f ) converges uniformly on [ M, M]. Need n − (fn) converge uniformly on R? Justify your answer. Paper 4, Section I 3G Analysis II State the chain rule for the composition of two differentiable functions f : Rm Rn → and g : Rn Rp. → Let f : R2 R be differentiable. -
Conglomerated Filters, Statistical Measures, and Representations by Ultrafilters
CONGLOMERATED FILTERS, STATISTICAL MEASURES, AND REPRESENTATIONS BY ULTRAFILTERS VLADIMIR KADETS AND DMYTRO SELIUTIN Abstract. Using a new concept of conglomerated filter we demonstrate in a purely com- binatorial way that none of Erd¨os-Ulamfilters or summable filters can be generated by a single statistical measure and consequently they cannot be represented as intersections of countable families of ulrafilters. Minimal families of ultrafilters and their intersections are studied and several open questions are discussed. 1. Introduction In 1937, Henri Cartan (1904{2008), one of the founders of the Bourbaki group, introduced the concepts of filter and ultrafilter [3, 4]. These concepts were among the cornerstones of Bourbaki's exposition of General Topology [2]. For non-metrizable spaces, filter convergence is a good substitute for ordinary convergence of sequences, in particular a Hausdorff space X is compact if and only if every filter in X has a cluster point. We refer to [14, Section 16.1] for a brief introduction to filters and compactness. Filters and ultrafiters (or equivalent concepts of ideals and maximal ideals of subsets) are widely used in Topology, Model Theory, and Functional Analysis. Let us recall some definitions. A filter F on a set Ω 6= ; is a non-empty collection of subsets of Ω satisfying the following axioms: (a) ; 2= F; (b) if A; B 2 F then A \ B 2 F; (c) for every A 2 F if B ⊃ A then B 2 F. arXiv:2012.02866v1 [math.FA] 4 Dec 2020 The natural ordering on the set of filters on Ω is defined as follows: F1 F2 if F1 ⊃ F2. -
Complex Analysis
University of Cambridge Mathematics Tripos Part IB Complex Analysis Lent, 2018 Lectures by G. P. Paternain Notes by Qiangru Kuang Contents Contents 1 Basic notions 2 1.1 Complex differentiation ....................... 2 1.2 Power series .............................. 4 1.3 Conformal maps ........................... 8 2 Complex Integration I 10 2.1 Integration along curves ....................... 10 2.2 Cauchy’s Theorem, weak version .................. 15 2.3 Cauchy Integral Formula, weak version ............... 17 2.4 Application of Cauchy Integration Formula ............ 18 2.5 Uniform limits of holomorphic functions .............. 20 2.6 Zeros of holomorphic functions ................... 22 2.7 Analytic continuation ........................ 22 3 Complex Integration II 24 3.1 Winding number ........................... 24 3.2 General form of Cauchy’s theorem ................. 26 4 Laurent expansion, Singularities and the Residue theorem 29 4.1 Laurent expansion .......................... 29 4.2 Isolated singularities ......................... 30 4.3 Application and techniques of integration ............. 34 5 The Argument principle, Local degree, Open mapping theorem & Rouché’s theorem 38 Index 41 1 1 Basic notions 1 Basic notions Some preliminary notations/definitions: Notation. • 퐷(푎, 푟) is the open disc of radius 푟 > 0 and centred at 푎 ∈ C. • 푈 ⊆ C is open if for any 푎 ∈ 푈, there exists 휀 > 0 such that 퐷(푎, 휀) ⊆ 푈. • A curve is a continuous map from a closed interval 휑 ∶ [푎, 푏] → C. It is continuously differentiable, i.e. 퐶1, if 휑′ exists and is continuous on [푎, 푏]. • An open set 푈 ⊆ C is path-connected if for every 푧, 푤 ∈ 푈 there exists a curve 휑 ∶ [0, 1] → 푈 with endpoints 푧, 푤.