DOCSLIB.ORG

23. Dimension Dimension Is Intuitively Obvious but Surprisingly Hard to Define Rigorously and to Work With

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

58 RICHARD BORCHERDS 23. Dimension Dimension is intuitively obvious but surprisingly hard to define rigorously and to work with. There are several different concepts of dimension • It was at first assumed that the dimension was the number or parameters something depended on. This fell apart when Cantor showed that there is a bijective map from R ! R2. The Peano curve is a continuous surjective map from R ! R2. • The Lebesgue covering dimension: a space has Lebesgue covering dimension at most n if every open cover has a refinement such that each point is in at most n + 1 sets. This does not work well for the spectrums of rings. Example: dimension 2 (DIAGRAM) no point in more than 3 sets. Not trivial to prove that n-dim space has dimension n. No good for commutative algebra as A1 has infinite Lebesgue covering dimension, as any finite number of non-empty open sets intersect. • The "classical" definition. Definition 23.1. (Brouwer, Menger, Urysohn) A topological space has dimension ≤ n (n ≥ −1) if all points have arbitrarily small neighborhoods with boundary of dimension < n. The empty set is the only space of dimension −1. This definition is mostly used for separable metric spaces. Rather amazingly it also works for the spectra of Noetherian rings, which are about as far as one can get from separable metric spaces. • Definition 23.2. The Krull dimension of a topological space is the supre- mum of the numbers n for which there is a chain Z0 ⊂ Z1 ⊂ ::: ⊂ Zn of n + 1 irreducible subsets. DIAGRAM pt ⊂ curve ⊂ A2 For Noetherian topological spaces the Krull dimension is the same as the Menger definition, but for non-Noetherian spaces it behaves badly. Any Hausdorff space has dim 0 with this definition, as only irreducible sets are points. • Various forms of (real-valued) Hausdorff dimension of fractals: these mostly require a metric so are of no use for commutative rings. • Transfinite dimensions: we can extend the definition of dimension to allow the dimension to be a transfinite ordinal. This seems to be of little use in ring theory. • The deviation of a poset. A poset has deviation ≤ α (an ordinal) if for any descending chain all but a finite number of the intervals between element of the chain have deviation less than α. For a Noetherian ring the dimension is the deviation of its poset of ideals. This definition has the advantage that it works well for noncommutative Noetherian rings and modules. (Note that the poset of rationals does not have a deviation, so some non-Noetherian rings do not either.) • Algebraic definition: main idea is that higher dimensional spaces have more functions on them. Example 23.3. dim B=k = tr. deg. quotient field of B over k. MATH 250B: COMMUTATIVE ALGEBRA 59 This works fine for "geometric" domains such as coordinate rings of affine varieties (and used to be used as the definition) but does not work well on "arithmetic" rings such as Z: SpecZ should have dimension 1, but its field of fractions has transcendence degree 0. (Even for geometric rings the definition needs some fudging for things like rings with zero divisors or complete local rings.) • The Hilbert polynomial. Example 23.4. Easiest definition to calculate uses Hilbert polynomial of a local ring. Look at dim(A=mk): this is a polynomial in k for large k of degree d, and d is the dimension of the local ring. It is also the same as the dimension of the completion as these have the same Hilbert polynomials. 2 3 For example, k[x1; : : : ; xn] has dimension n. k[x; y]=(y − x − ax − b) has dimension 1. • The Gelfand-Kirillov dimension for possibly non-commutative finitely gen- erated algebras R over a field is (23.1) d = lim sup log dim Rn= log n n where Rn is the subspace generated by monomials in the generators of length at most n. (Informally Rn has dimension very roughly nd). When R is commutative this is the same as the dimension defined using Hilbert polynomials. For non-commutative rings the dimension can be 0, 1, or any real number at least 2, or infinity. • Dimension of a ring: Dimension usually defined as the Krull dimension of its spectrum, in other words sup n such that P0 ⊂ P1 ⊂ ::: ⊂ Pn are distinct prime ideals. Dim R = sup dim Rm, m maximal, so can be reduced to dimension of local rings. Problem: What are irreducible subsets of (say) A4? Hard to describe, and one needs nontrivial commutative algebra to calculate dimension. • The dimension of the tangent space at a point works for non-singular schemes but fails when there are singularities. This is the number of gen- erators of a maximal ideal of a local ring. A variation that gives a good definition of dimension is the minimal number of elements of a system of parameters (generating an ideal containing some power of m). • Homological dimension. There are several ways to define dimension as "the number beyond which certain homology or cohomology groups vanish". These definitions work well, but require developing a lot of homology theory. Most of the topological definitions are "local" in the sense that one can define the dimension of a space at a point, and the dimension of the space is the sup of the dimensions at a point. Theorem 23.5. The Krull dimension is at most equal to the dimension. It is at least the dimension for Noetherian topological spaces. Proof. Suppose the Krull dimension is at least n. Let I0;I1;:::;In be a chain of n + 1 closed irreducible subsets such that Ii is in Ii+1. Then by induction on i we see that Ii is of dimension at least i, because for any point in Ii+1 the set Ii is in the boundary of any neighborhood not containing Ii. So the space has dimension at least n. 60 RICHARD BORCHERDS Now suppose the space is Noetherian. We show that for every closed subset the Krull dimension is at least the dimension. Otherwise there is a minimal closed subset that does not have this property, which we may as well assume is the whole space. The space is irreducible, otherwise we can write it as the union of two proper closed subsets, for which the Krull dimension is at least the dimension by Noetherian induction. Suppose the space has dimension at least n. We can find a point p and a neighborhood of it whose boundary has dimension at least n − 1, so there is a chain of n irreducible subsets not containing p. But then we can add the whole space to get a chain n + 1 irreducible subsets, so the Krull dimension is at least n. In particular for any Noetherian rings, the Krull dimension of the spectrum is the Menger dimension. This also coincides with the dimension defined using Hilbert polynomials, or deviations of posets, or systems of parameters, and for varieties over fields is the same as the definition using transcendence degree. For non-Noetherian rings the dimension does not seem to be such a useful concept: the Krull dimension is usually infinite and behaves in strange ways: for example the dimension of R[x] can be larger than the "expected" value 1 + dim R. There seems to be no really satisfactory concept of dimension for non-Noetherian rings, and in any case many of them tend to be ”infinite dimensional". We want to show the following are equal, for Noetherian local rings: • The Krull dimension • The degree of the Hilbert polynomial • The smallest cardinality of a system of parameters We will prove this by proving 3 inequalities between these numbers. This will take some time and is somewhat technical. The Hilbert polynomial. Suppose that R = ⊕Rn is a graded ring finitely generated over R0 and M = ⊕Mn a finitely generated graded module over it. We want to study the growth of Mn. Usually we will take R0 to be a field, in which case we measure the size of Mn by its dimension lambda(Mn) (as a vector space). It is sometimes useful to allow R0 to be an Artinian ring, in which case we can measure the size of Mn by its length λ(Mn), which generalized the dimension of a vector space. We recall that the dimension (or lenght) is additive for short exact sequences of modules. We put all these lengths together in the POINCARE SERIES X n P (t) = λnt Theorem 23.6. The Poincare series is a rational function of t of the form Y f(t)= (1 − tki ) for some polynomial f. Proof. We take the elements xi to be homogeneous generators of R over R0 of degrees ki. We will prove the theorem by induction on the number of generators. If the number of generators is 0 so the R = R0 then the therom is trivial as Mn = 0 for n large. Now look at the exact sequence ×xs 0 ! Kn ! Mn ! Mn+ks ! Ln+ks ! 0 MATH 250B: COMMUTATIVE ALGEBRA 61 where K and L are both modules where xs acts as 0. This shows that λ(Kn) − λ(Mn) + λ(Mn+ks − λ(Ln+ks ) = 0 so −sk sk PK (t) − PM (t) + t PM (t) − t PL(t) = 0 so ks PL(t) − t PK (t) PM (t) = 1 − tks The result now follows by induction on the number of generators, as both K and L can be regarded as modules over the quotient R=(xs). .
Recommended publications
  • Dimension Theory and Systems of Parameters

    Dimension Theory and Systems of Parameters

    Dimension theory and systems of parameters Krull's principal ideal theorem Our next objective is to study dimension theory in Noetherian rings. There was initially amazement that the results that follow hold in an arbitrary Noetherian ring. Theorem (Krull's principal ideal theorem). Let R be a Noetherian ring, x 2 R, and P a minimal prime of xR. Then the height of P ≤ 1. Before giving the proof, we want to state a consequence that appears much more general. The following result is also frequently referred to as Krull's principal ideal theorem, even though no principal ideals are present. But the heart of the proof is the case n = 1, which is the principal ideal theorem. This result is sometimes called Krull's height theorem. It follows by induction from the principal ideal theorem, although the induction is not quite straightforward, and the converse also needs a result on prime avoidance. Theorem (Krull's principal ideal theorem, strong version, alias Krull's height theorem). Let R be a Noetherian ring and P a minimal prime ideal of an ideal generated by n elements. Then the height of P is at most n. Conversely, if P has height n then it is a minimal prime of an ideal generated by n elements. That is, the height of a prime P is the same as the least number of generators of an ideal I ⊆ P of which P is a minimal prime. In particular, the height of every prime ideal P is at most the number of generators of P , and is therefore finite.
  • Hausdorff Dimension of Singular Vectors

    Hausdorff Dimension of Singular Vectors

    HAUSDORFF DIMENSION OF SINGULAR VECTORS YITWAH CHEUNG AND NICOLAS CHEVALLIER Abstract. We prove that the set of singular vectors in Rd; d ≥ 2; has Hausdorff dimension d2 d d+1 and that the Hausdorff dimension of the set of "-Dirichlet improvable vectors in R d2 d is roughly d+1 plus a power of " between 2 and d. As a corollary, the set of divergent t t −dt trajectories of the flow by diag(e ; : : : ; e ; e ) acting on SLd+1 R= SLd+1 Z has Hausdorff d codimension d+1 . These results extend the work of the first author in [6]. 1. Introduction Singular vectors were introduced by Khintchine in the twenties (see [14]). Recall that d x = (x1; :::; xd) 2 R is singular if for every " > 0, there exists T0 such that for all T > T0 the system of inequalities " (1.1) max jqxi − pij < and 0 < q < T 1≤i≤d T 1=d admits an integer solution (p; q) 2 Zd × Z. In dimension one, only the rational numbers are singular. The existence of singular vectors that do not lie in a rational subspace was proved by Khintchine for all dimensions ≥ 2. Singular vectors exhibit phenomena that cannot occur in dimension one. For instance, when x 2 Rd is singular, the sequence 0; x; 2x; :::; nx; ::: fills the torus Td = Rd=Zd in such a way that there exists a point y whose distance in the torus to the set f0; x; :::; nxg, times n1=d, goes to infinity when n goes to infinity (see [4], chapter V).
  • UC Berkeley UC Berkeley Previously Published Works

    UC Berkeley UC Berkeley Previously Published Works

    UC Berkeley UC Berkeley Previously Published Works Title Operator bases, S-matrices, and their partition functions Permalink https://escholarship.org/uc/item/31n0p4j4 Journal Journal of High Energy Physics, 2017(10) ISSN 1126-6708 Authors Henning, B Lu, X Melia, T et al. Publication Date 2017-10-01 DOI 10.1007/JHEP10(2017)199 Peer reviewed eScholarship.org Powered by the California Digital Library University of California Published for SISSA by Springer Received: July 7, 2017 Accepted: October 6, 2017 Published: October 27, 2017 Operator bases, S-matrices, and their partition functions JHEP10(2017)199 Brian Henning,a Xiaochuan Lu,b Tom Meliac;d;e and Hitoshi Murayamac;d;e aDepartment of Physics, Yale University, New Haven, Connecticut 06511, U.S.A. bDepartment of Physics, University of California, Davis, California 95616, U.S.A. cDepartment of Physics, University of California, Berkeley, California 94720, U.S.A. dTheoretical Physics Group, Lawrence Berkeley National Laboratory, Berkeley, California 94720, U.S.A. eKavli Institute for the Physics and Mathematics of the Universe (WPI), Todai Institutes for Advanced Study, University of Tokyo, Kashiwa 277-8583, Japan E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: Relativistic quantum systems that admit scattering experiments are quan- titatively described by effective field theories, where S-matrix kinematics and symmetry considerations are encoded in the operator spectrum of the EFT. In this paper we use the S-matrix to derive the structure of the EFT operator basis, providing complementary de- scriptions in (i) position space utilizing the conformal algebra and cohomology and (ii) mo- mentum space via an algebraic formulation in terms of a ring of momenta with kinematics implemented as an ideal.
  • Arxiv:1012.0864V3 [Math.AG] 14 Oct 2013 Nbt Emtyadpr Ler.Ltu Mhsz H Olwn Th Following the Emphasize Us H Let Results Cit.: His Algebra

    Arxiv:1012.0864V3 [Math.AG] 14 Oct 2013 Nbt Emtyadpr Ler.Ltu Mhsz H Olwn Th Following the Emphasize Us H Let Results Cit.: His Algebra

    ORLOV SPECTRA: BOUNDS AND GAPS MATTHEW BALLARD, DAVID FAVERO, AND LUDMIL KATZARKOV Abstract. The Orlov spectrum is a new invariant of a triangulated category. It was intro- duced by D. Orlov building on work of A. Bondal-M. van den Bergh and R. Rouquier. The supremum of the Orlov spectrum of a triangulated category is called the ultimate dimension. In this work, we study Orlov spectra of triangulated categories arising in mirror symmetry. We introduce the notion of gaps and outline their geometric significance. We provide the first large class of examples where the ultimate dimension is finite: categories of singular- ities associated to isolated hypersurface singularities. Similarly, given any nonzero object in the bounded derived category of coherent sheaves on a smooth Calabi-Yau hypersurface, we produce a new generator by closing the object under a certain monodromy action and uniformly bound this new generator’s generation time. In addition, we provide new upper bounds on the generation times of exceptional collections and connect generation time to braid group actions to provide a lower bound on the ultimate dimension of the derived Fukaya category of a symplectic surface of genus greater than one. 1. Introduction The spectrum of a triangulated category was introduced by D. Orlov in [39], building on work of A. Bondal, R. Rouquier, and M. van den Bergh, [44] [11]. This categorical invariant, which we shall call the Orlov spectrum, is simply a list of non-negative numbers. Each number is the generation time of an object in the triangulated category. Roughly, the generation time of an object is the necessary number of exact triangles it takes to build the category using this object.
  • Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes

    Commutative Algebra Ii, Spring 2019, A. Kustin, Class Notes

    COMMUTATIVE ALGEBRA II, SPRING 2019, A. KUSTIN, CLASS NOTES 1. REGULAR SEQUENCES This section loosely follows sections 16 and 17 of [6]. Definition 1.1. Let R be a ring and M be a non-zero R-module. (a) The element r of R is regular on M if rm = 0 =) m = 0, for m 2 M. (b) The elements r1; : : : ; rs (of R) form a regular sequence on M, if (i) (r1; : : : ; rs)M 6= M, (ii) r1 is regular on M, r2 is regular on M=(r1)M, ::: , and rs is regular on M=(r1; : : : ; rs−1)M. Example 1.2. The elements x1; : : : ; xn in the polynomial ring R = k[x1; : : : ; xn] form a regular sequence on R. Example 1.3. In general, order matters. Let R = k[x; y; z]. The elements x; y(1 − x); z(1 − x) of R form a regular sequence on R. But the elements y(1 − x); z(1 − x); x do not form a regular sequence on R. Lemma 1.4. If M is a finitely generated module over a Noetherian local ring R, then every regular sequence on M is a regular sequence in any order. Proof. It suffices to show that if x1; x2 is a regular sequence on M, then x2; x1 is a regular sequence on M. Assume x1; x2 is a regular sequence on M. We first show that x2 is regular on M. If x2m = 0, then the hypothesis that x1; x2 is a regular sequence on M guarantees that m 2 x1M; thus m = x1m1 for some m1.
  • A Metric on the Moduli Space of Bodies

    A Metric on the Moduli Space of Bodies

    A METRIC ON THE MODULI SPACE OF BODIES HAJIME FUJITA, KAHO OHASHI Abstract. We construct a metric on the moduli space of bodies in Euclidean space. The moduli space is defined as the quotient space with respect to the action of integral affine transformations. This moduli space contains a subspace, the moduli space of Delzant polytopes, which can be identified with the moduli space of sym- plectic toric manifolds. We also discuss related problems. 1. Introduction An n-dimensional Delzant polytope is a convex polytope in Rn which is simple, rational and smooth1 at each vertex. There exists a natu- ral bijective correspondence between the set of n-dimensional Delzant polytopes and the equivariant isomorphism classes of 2n-dimensional symplectic toric manifolds, which is called the Delzant construction [4]. Motivated by this fact Pelayo-Pires-Ratiu-Sabatini studied in [14] 2 the set of n-dimensional Delzant polytopes Dn from the view point of metric geometry. They constructed a metric on Dn by using the n-dimensional volume of the symmetric difference. They also studied the moduli space of Delzant polytopes Den, which is constructed as a quotient space with respect to natural action of integral affine trans- formations. It is known that Den corresponds to the set of equivalence classes of symplectic toric manifolds with respect to weak isomorphisms [10], and they call the moduli space the moduli space of toric manifolds. In [14] they showed that, the metric space D2 is path connected, the moduli space Den is neither complete nor locally compact. Here we use the metric topology on Dn and the quotient topology on Den.
  • Lectures on Fractal Geometry and Dynamics

    Lectures on Fractal Geometry and Dynamics

    Lectures on fractal geometry and dynamics Michael Hochman∗ June 27, 2012 Contents 1 Introduction 2 2 Preliminaries 3 3 Dimension 4 3.1 A family of examples: Middle-α Cantor sets . 4 3.2 Minkowski dimension . 5 3.3 Hausdorff dimension . 9 4 Using measures to compute dimension 14 4.1 The mass distribution principle . 14 4.2 Billingsley's lemma . 15 4.3 Frostman's lemma . 18 4.4 Product sets . 22 5 Iterated function systems 25 5.1 The Hausdorff metric . 25 5.2 Iterated function systems . 27 5.3 Self-similar sets . 32 5.4 Self-affine sets . 36 6 Geometry of measures 40 6.1 The Besicovitch covering theorem . 40 6.2 Density and differentiation theorems . 45 6.3 Dimension of a measure at a point . 50 6.4 Upper and lower dimension of measures . 52 ∗Send comments to [email protected] 1 6.5 Hausdorff measures and their densities . 55 7 Projections 59 7.1 Marstrand's projection theorem . 60 7.2 Absolute continuity of projections . 63 7.3 Bernoulli convolutions . 65 7.4 Kenyon's theorem . 71 8 Intersections 74 8.1 Marstrand's slice theorem . 74 8.2 The Kakeya problem . 77 9 Local theory of fractals 79 9.1 Microsets and galleries . 80 9.2 Symbolic setup . 81 9.3 Measures, distributions and measure-valued integration . 82 9.4 Markov chains . 83 9.5 CP-processes . 85 9.6 Dimension and CP-distributions . 88 9.7 Constructing CP-distributions supported on galleries . 90 9.8 Spectrum of an Markov chain .
  • On the Hilbert Function of a Graded Cohen-Macaulay Domain

    On the Hilbert Function of a Graded Cohen-Macaulay Domain

    Journal of Pure and Applied Algebra 73 (lW1) 307-314 307 North-Holland Richard P. Stsnley” Department of Mathematics, Massachusetts Institute of Teclmology . Carnhridge. MA 02139. USA Communicated by A. Blass Received 7 May 1990 Revised 6 August 1990 Abstract Stanley, R.P., On the Hilbert function of a graded Cohen-Macaulay domain. Journal of Pure and Applied Algebra 73 (1991) 307-314. A condition is obtained on the Hilbert function of a graded Cohen-Macaulay domain R=R,$f?,G+. - over a field R, = K when R is integral over the subalgebra generated by R,. A result of Eisenbud and Harris leads to a stronger condition when char K = 0 and R is generated as a K-aigebra by R,. An application is given to the Ehrhart polynomial of an :,r____:-&W’ ^ ,,-l -u-i.-I\*..- _- c- --1..+,q,__~“t..‘. ... -_ _ _,_-_. 1. Introduction By a graded algebra over a field K, we mean here a commutative K-algebra R with identity, together with a vector space direct sum decompo&ion & = &_,l R;. such that: (a) RiRi c Ri+i, (b) R, = K (i.e., R is connected), and (c) R is finitely-generated as a K-algebra. R is standard if R is generated as a K-algebra by R,, and semistandard if R is integral over the subalgebra K[ R,] of R generated by R,. The Hilbert function H( R, -) of R is defined by H( R, i) = dim,R,, for i 2 0. while the Hilbert series is given by F(R, A) = 2 H(R, i)A’.
  • Commutative Algebra) Notes, 2Nd Half

    Commutative Algebra) Notes, 2Nd Half

    Math 221 (Commutative Algebra) Notes, 2nd Half Live [Locally] Free or Die. This is Math 221: Commutative Algebra as taught by Prof. Gaitsgory at Harvard during the Fall Term of 2008, and as understood by yours truly. There are probably typos and mistakes (which are all mine); use at your own risk. If you find the aforementioned mistakes or typos, and are feeling in a generous mood, then you may consider sharing this information with me.1 -S. Gong [email protected] 1 CONTENTS Contents 1 Divisors 2 1.1 Discrete Valuation Rings (DVRs), Krull Dimension 1 . .2 1.2 Dedekind Domains . .6 1.3 Divisors and Fractional Ideals . .8 1.4 Dedekind Domains and Integral Closedness . 11 1.5 The Picard Group . 16 2 Dimension via Hilbert Functions 21 2.1 Filtrations and Gradations . 21 2.2 Dimension of Modules over a Polynomial Algebra . 24 2.3 For an arbitrary finitely generated algebra . 27 3 Dimension via Transcendency Degree 31 3.1 Definitions . 31 3.2 Agreement with Hilbert dimension . 32 4 Dimension via Chains of Primes 34 4.1 Definition via chains of primes and Noether Normalization . 34 4.2 Agreement with Hilbert dimension . 35 5 Kahler Differentials 37 6 Completions 42 6.1 Introduction to inverse limits and completions . 42 6.2 Artin Rees Pattern . 46 7 Local Rings and Other Notions of Dimension 50 7.1 Dimension theory for Local Noetherian Rings . 50 8 Smoothness and Regularity 53 8.1 Regular Local Rings . 53 8.2 Smoothness/Regularity for algebraically closed fields . 55 8.3 A taste of smoothness and regularity for non-algebraically closed fields .
  • Expansive Subdynamics

    Expansive Subdynamics

    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 349, Number 1, January 1997, Pages 55–102 S 0002-9947(97)01634-6 EXPANSIVE SUBDYNAMICS MIKE BOYLE AND DOUGLAS LIND Abstract. This paper provides a framework for studying the dynamics of d commuting homeomorphisms. Let α be a continuous action of Z on an infi- d nite compact metric space. For each subspace V of R we introduce a notion of expansiveness for α along V , and show that there are nonexpansive subspaces in every dimension d 1. For each k d the set Ek(α) of expansive k- dimensional subspaces≤ is− open in the Grassmann≤ manifold of all k-dimensional d subspaces of R . Various dynamical properties of α are constant, or vary nicely, within a connected component of Ek(α), but change abruptly when passing from one expansive component to another. We give several examples of this sort of “phase transition,” including the topological and measure-theoretic di- rectional entropies studied by Milnor, zeta functions, and dimension groups. For d = 2 we show that, except for one unresolved case, every open set of directions whose complement is nonempty can arise as an E1(α). The unre- solved case is that of the complement of a single irrational direction. Algebraic examples using commuting automorphisms of compact abelian groups are an important source of phenomena, and we study several instances in detail. We conclude with a set of problems and research directions suggested by our anal- ysis. Contents 1. Introduction 56 2. Definitions and examples 57 3. Nonexpansive subspaces 63 4. Realization 67 5.
  • 2-Dimensional Vertex Decomposable Circulant Graphs

    2-Dimensional Vertex Decomposable Circulant Graphs

    2-DIMENSIONAL VERTEX DECOMPOSABLE CIRCULANT GRAPHS GIANCARLO RINALDO AND FRANCESCO ROMEO n Abstract. Let G be the circulant graph Cn(S) with S ⊆ {1,..., 2 } and let ∆ be its independence complex. We describe the well-covered circulant graphs with 2-dimensional ∆ and construct an infinite family of vertex-decomposable circulant graphs within this family. Key Words: Circulant graphs, Cohen-Macaulay, Vertex decomposability. 2010 Mathematics Subject Classification: Primary 13F55. Secondary 13H10 Introduction N n Let n ∈ and S ⊆ {1, 2,..., 2 }. The circulant graph G := Cn(S) is a graph with vertex set Z = {0,...,n − 1} and edge set E(G) := {{i, j} | n |j − i|n ∈ S} where |k|n = min{|k|,n − |k|}. Let R = K[x0,...,xn−1] be the polynomial ring on n variables over a field K. The edge ideal of G, denoted by I(G), is the ideal of R generated by all square-free monomials xixj such that {i, j} ∈ E(G). Edge ideals of graphs have been introduced by Villarreal [20] in 1990, where he studied the Cohen–Macaulay property of such ideals. Many authors have focused their attention on such ideals (e.g.[10], [12]). A known fact about Cohen- Macaulay edge ideals is that they are well-covered, that is all the maximal independent sets of G have the same cardinality. Despite the nice structure the circulant graphs have, it has been proved that is hard to compute their clique number (see [5]), and hence the Krull dimension of R/I(G). arXiv:1807.05755v1 [math.AC] 16 Jul 2018 In particular, some well-covered circulant graphs have been studied (see [2], [3],[19], [8] and [15]).
  • On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes

    On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes

    Munich Personal RePEc Archive On the Computation of the Hausdorff Dimension of the Walrasian Economy:Further Notes Dominique, C-Rene Independent Research 9 August 2009 Online at https://mpra.ub.uni-muenchen.de/16723/ MPRA Paper No. 16723, posted 10 Aug 2009 10:42 UTC ON THE COMPUTATION OF THE HAUSDORFF DIMENSION OF THE WALRASIAN ECONOMY: FURTHER NOTES C-René Dominique* ABSTRACT: In a recent paper, Dominique (2009) argues that for a Walrasian economy with m consumers and n goods, the equilibrium set of prices becomes a fractal attractor due to continuous destructions and creations of excess demands. The paper also posits that the Hausdorff dimension of the attractor is d = ln (n) / ln (n-1) if there are n copies of sizes (1/(n-1)), but that assumption does not hold. This note revisits the problem, demonstrates that the Walrasian economy is indeed self-similar and recomputes the Hausdorff dimensions of both the attractor and that of a time series of a given market. KEYWORDS: Fractal Attractor, Contractive Mappings, Self-similarity, Hausdorff Dimensions of the Walrasian Economy, the Hausdorff dimension of a time series of a given market. INTRODUCTION Dominique (2009) has shown that the equilibrium price vector of a Walrasian pure exchange economy is a closed invariant set E* Rn-1 (where R is the set of real numbers and n-1 are the number of independent prices) rather than the conventionally assumed stationary fixed point. And that the Hausdorff dimension of the attractor lies between one and two if n self-similar copies of the economy can be made.