Modulo Equivalence Research Articles

Some results in dynamic model theory

First-order structures over a fixed signature Σ give rise to a family of trace-based and relational Kleene algebras with tests defined in terms of Tarskian frames. A Tarskian frame is a Kripke frame whose states are valuations of program variables and whose atomic actions are state changes effected by variable assignments x≔ e, where e is a Σ-term. The Kleene algebras with tests that arise in this way play a role in dynamic model theory akin to the role played by Lindenbaum algebras in classical first-order model theory. Given a first-order theory T over Σ, we exhibit a Kripke frame U whose trace algebra Tr U is universal for the equational theory of Tarskian trace algebras over Σ satisfying T, although U itself is not Tarskian in general. The corresponding relation algebra Rel U is not universal for the equational theory of relation algebras of Tarskian frames, but it is so modulo observational equivalence.

De Bruijn's syntax and reductional behaviour of λ-terms: the untyped case

In this paper, a notation influenced by de Bruijn's syntax of the λ-calculus is used to describe canonical forms of terms and an equivalence relation which divides terms into classes according to their reductional behaviour. We show that this notation helps describe canonical forms more elegantly than the classical notation. We define reduction modulo equivalence classes of terms up to the permutation of redexes in canonical forms and show that this reduction contains other notions of reductions in the literature including the σ-reduction of Regnier. We establish all the desirable properties of our reduction modulo equivalence classes for the untyped λ-calculus.

Split-2 Bisimilarity has a Finite Axiomatization over CCS with Hennessy's Merge

This note shows that split-2 bisimulation equivalence (also known as timed equivalence) affords a finite equational axiomatization over the process algebra obtained by adding an auxiliary operation proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicating Systems. Thus the addition of a single binary operation, viz. Hennessy's merge, is sufficient for the finite equational axiomatization of parallel composition modulo this non-interleaving equivalence. This result is in sharp contrast to a theorem previously obtained by the same authors to the effect that the same language is not finitely based modulo bisimulation equivalence.

CCS with Hennessy's Merge has no Finite Equational Axiomatization

This paper confirms a conjecture of Bergstra and Klop's from 1984 by establishing that the process algebra obtained by adding an auxiliary operator proposed by Hennessy in 1981 to the recursion free fragment of Milner's Calculus of Communicationg Systems is not finitely based modulo bisimulation equivalence. Thus Hennessy's merge cannot replace the left merge and communication merge operators proposed by Bergstra and Klop, at least if a finite axiomatization of parallel composition is desired.

A note on an expressiveness hierarchy for multi-exit iteration

Multi-exit iteration is a generalization of the standard binary Kleene star operation. The addition of this construct to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse.

Classification of Algebroid Plane Curves with Semigroup ⟨6, 9, 19⟩#

This paper deals with the classification of plane algebroid branches, modulo contact equivalence, within the equisingularity class determined by the semigroup of values generated by the integers 6, 9 and 19. This illustrates how one can use the techniques developed in Hefez and Hernandes (Hefez, A., Hernandes, M. E. (2001). Computational Methods in the Local Theory of Curves. Colóquio Brasileiro de Matemática. Rio de Janeiro: Publicações Matemáticas, IMPA), to make computations in the module of Kähler differentials of the local ring of an algebroid branch. Dedicated to Steven L. Kleiman on the occasion of his 60th birthday.

A Note on an Expressiveness Hierarchy for Multi-exit Iteration

Multi-exit iteration is a generalization of the standard binary Kleene star operation that allows for the specification of agents that, up to bisimulation equivalence, are solutions of systems of recursion equations of the form<br />X_1 = P_1 X_2 + Q_1 <br /><br /> X_n = P_n X_1 + Q_n <br /><br /> where n is a positive integer, and the P_i and the Q_i are process terms. The addition of multi-exit iteration to Basic Process Algebra (BPA) yields a more expressive language than that obtained by augmenting BPA with the standard binary Kleene star. This note offers an expressiveness hierarchy, modulo bisimulation equivalence, for the family of multi-exit iteration operators proposed by Bergstra, Bethke and Ponse.

Milnor K-theory of rings, higher Chow groups and applications

If R is a smooth semi-local algebra of geometric type over an infinite field, we prove that the Milnor K-group K M n (R) surjects onto the higher Chow group CH n (R , n) for all n≥0. Our proof shows moreover that there is an algorithmic way to represent any admissible cycle in CH n (R , n) modulo equivalence as a linear combination of “symbolic elements” defined as graphs of units in R. As a byproduct we get a new and entirely geometric proof of results of Gabber, Kato and Rost, related to the Gersten resolution for the Milnor K-sheaf. Furthermore it is also shown that in the semi-local PID case we have, under some mild assumptions, an isomorphism. Some applications are also given.

Explaining updates by minimal sums

Human reasoning about developments of the world involves always an assumption of inertia. We discuss two approaches for formalizing such an assumption, based on the concept of an explanation: (1) there is a general preference relation ≺ given on the set of all explanations and (2) there is a notion of a distance between models and explanations are preferred if their sum of distances is minimal. Each distance dist naturally induces a preference relation ≺ dist . We show exactly under which conditions the converse is true as well and therefore both approaches are equivalent modulo these conditions. Our main result is a general representation theorem in the spirit of Kraus, Lehmann and Magidor.

Algebroid plane curves whose Milnor and Tjurina numbers differ by one or two

In this paper we describe all irreducible plane algebroid curves, defined over an algebraically closed field of characteristic zero, modulo analytic equivalence, having the property that the difference between their Milnor and Tjurina numbers is 1 or 2. Our work extends a previous result of O. Zariski who described such curves when this difference is zero.

Two-point differentiation for general orders

We establish a Zavadskij-type reduction for orders Λ in a finite-dimensional algebra over a complete field K with respect to a discrete valuation. For a suitable monomorphism u of Λ-lattices, we define a derived order δ u Λ, and a functor ∂ u between Λ- and δ u Λ-lattices which yields an equivalence modulo finitely many indecomposables. The known versions of Zavadskij's differentiation algorithm (for tiled orders [15], representations of posets [14], and vector space categories [10,11]) are unified, and extended in this way to a part of representation theory of general orders.

A Categorical Critical-pair Completion Algorithm

We introduce a general critical-pair/completion algorithm, formulated in the language of category theory. It encompasses the Knuth–Bendix procedure for term rewriting systems (also modulo equivalence relations), the Gröbner basis algorithm for polynomial ideal theory, and the resolution procedure for automated theorem proving. We show how these three procedures fit in the general algorithm, and how our approach relates to other categorical modeling approaches to these algorithms, especially term rewriting.

On a conjecture of Izhboldin on similarity of quadratic forms

In his paper \it Motivic equivalence of quadratic forms, Izhboldin modifies a conjecture of Lam and asks whether two quadratic forms, each of which isomorphic to the product of an Albert form and a k -fold Pfister form, are similar provided they are equivalent modulo I^{k+3} . We relate this conjecture to another conjecture on the dimensions of anisotropic forms in I^{k+3} . As a consequence, we obtain that Izhboldin's conjecture is true for k\leq 1 .

Equivalence between chain categories of representations of affine sl(2) and N=2 superconformal algebras

Highest-weight-type representation theories of the affine sl(2) and N=2 superconformal algebras are shown to be equivalent modulo the respective spectral flows.

Moving algebraic cycles of bounded degree

The Chow Moving Lemma is a theorem which asserts that a given algebraic s-cycle on a smooth algebraic variety X can be moved within its rational equivalence class to intersect properly a given r-cycle on X provided that r + s ≥ dim(X) (cf. [Chow], [S2]). In the past few years, there has been considerable interest in studying spaces of algebraic cycles rather than simply cycles modulo an equivalence relation. With this in mind, it is natural to ask whether one can move a given “bounded family” of s-cycles on the smooth variety X to intersect properly a given “bounded family” of r-cycles. The main point of this paper is to formulate and prove just such a result. In Theorem 3.1, we demonstrate that for any integer e and any smooth projective variety X, one can simultaniously and algebraically “move” all effective s-cycles of degree ≤ e on X so that each such cycle meets every effective r-cycle of degree ≤ e on X in proper dimension. The primary motivation for this Moving Lemma for Cycles of Bounded Degree was the possibility of a duality theorem between cohomology and homology theories defined in terms of homotopy groups of cycle spaces. Using Theorem 3.1, we have proved such a duality theorem for complex quasi-projective varieties in [F-L2]. We prove our Moving Lemma for varieties over an arbitary infinite field, permitting a proof in [F-V] of a duality theorem for “motivic cohomology and homology”. The reader will find that our Moving Lemma has numerous good properties. First of all, the move is given as an algebraic move (parametrized by a punctured projective line) on Chow varietes. Although this move is “good” only for s-cycles of bounded degree, it is defined on all effective s-cycles. Moreover, the move starts at “time 0” by expressing an effective s-cycle Z as a difference of effective s-cycles both of which have intersection properties no worse than Z. Finally, our Moving Lemma is applicable to smooth quasiprojective varieties, for it is stated for a possibly singular projective variety X resulting in a conclusion of proper intersection off the singular locus of X. The classical motivation for the moving lemma was to define an intersection product on algebraic cycles modulo rational equivalence, thereby establishing the Chow ring A∗(X). Some of the classical literature overlooked the question of whether or not intersection of cycles defined via a moving lemma is independent of the move (e.g., [Chow], [R], [S2]; on the other hand, cf. [Chev], [S3])). One direct consequence of our Moving Lemma is a proof for smooth quasi-projective varieties that the intersection product is indeed well defined independent of the choice of move (Theorem 3.4). Of course, the intersection product now has an intrinsic formulation for all smooth algebraic varieties due to Fulton

Representations of Hermitian Kernels by Means of Krein Spaces

Hermitian kernels are studied as generalizations of kernels of positive type. The main tool is the axiomatic concept of induced Krem space. The existence of Kolmogorov decompositions of a hermitian kernel and their uniqueness, modulo unitary equivalence, are characterized. The existence of reproducing kernel Krein spaces is shown to be equivalent to the existence of Kolmogorov decompositions. Applications Applications to the Naimark dilations of Toeplitz hermitian kernels on the set of integers and to the uniqueness of the Krem space completions of nondegenerate inner product spaces are included.

Quantitative assessment of the effects of variability in dietary zinc dose-rate idiorrhythms upon zinc deposition in bone of weanling rats by using a slope-ratio assay

We quantitatively assessed the effect of the idiorrhythmic dose-rate variability in dietary zinc intake on zinc deposition in the femur and incisor of weanling rats by the slope-ratio analysis of analogous idiorrhythms. In idiorrhythmic feeding the dose (x) is viewed over the period of the entire experimental epoch as a dose-time equivalent modulo (Mx) that can be divided into a series of equal products of different doses with different frequency (idiorrhythm, I). Each I is administered in a regularly recurring pattern to deliver the same dose as that from a standard daily regimen of zinc, albeit at a different rate; Ix = [ d nth (Mx)] d nth , where d nth is the sequential number of zinc dosing day separated by 1 to 7 days of feeding the diet without zinc when d nth >1. Four different Mx were tested, M3, M6, M12, and M24 providing 3, 6, 12, and 24 mg Zn · kg −1 diet · d −1, respectively, over a 24 day epoch; each Mx had seven analogous Ix of different dose sizes, but the same dosing frequency ( d nth). The slopes for the M3, M6, and M12 analogous idiorrhythms showed that zinc deposition in the femur and incisor, as a measure of metabolic availability, varied considerably with spacing of the dose with time. Metabolic availability of zinc progressively decreased by 50% from I = Mx 1 to I = 4Mx 4 . Then it rose to approach the initial I = Mx 1 values of animals fed zinc daily for I = 5Mx 5 and I = 6Mx 6 , before it finally dropped again to the bottom level for I = 8Mx 8 . The results showed that the impact of zinc dose-rate in dietary zinc intake is an important determinant of an adequate supply of metabolic zinc.

Compactness and finite equivalence of infinite digraphs

In Bauslaugh (1995) we defined and explored the notion of homomorphic compactness for infinite digraphs. In this paper we show that there are exactly 2 N 0 compact digraphs, up to homomorphic equivalence. We then define the notion of finite equivalence for infinite digraphs. We show that for almost any infinite digraph G, the class of digraphs which are finitely equivalent to G (modulo homomorphic equivalence) is either a proper class or consists of a single homomorphic equivalence class. For undirected graphs we show that this is true in all cases. We also examine some basic properties of a natural partial order we may impose on these classes of digraphs.

Estimates of the number of rational mappings from a fixed variety to varieties of general type

First we find effective bounds for the number of dominant rational maps $f:X \rightarrow Y$ between two fixed smooth projective varieties with ample canonical bundles. The bounds are of the type $\{A \cdot K_X^n\}^{\{B \cdot K_X^n\}^2}$, where $n=dimX$, $K_X$ is the canonical bundle of $X$ and $A,B $ are some constants, depending only on $n$. Then we show that for any variety $X$ there exist numbers $c(X)$ and $C(X)$ with the following properties: For any threefold $Y$ of general type the number of dominant rational maps $f:X \r Y$ is bounded above by $c(X)$. The number of threefolds $Y$, modulo birational equivalence, for which there exist dominant rational maps $f:X \r Y$, is bounded above by $C(X)$. If, moreover, $X$ is a threefold of general type, we prove that $c(X)$ and $C(X)$ only depend on the index $r_{X_c}$ of the canonical model $X_c$ of $X$ and on $K_{X_c}^3$.

Higher Picard varieties and the height pairing

Let X be a smooth projective variety which is defined over a number field. Beilinson and Bloch have defined under suitable asssumptions height pairings between Chow groups of homologically trivial cycles on X. Beilinson has also formulated a hard Lefschetz and a Hodge index conjecture for these Chow groups. We show that the restriction of the height pairing to cycles algebraically equivalent to zero can be computed via Abel-Jacobi maps in terms of the Néron-Tate height pairing on the higher Picard varieties of X . This description is used in the case where X is an abelian variety to prove a consequence of Beilinson conjectures. Namely, we prove a hard Lefschetz and a Hodge index theorem for the groups of cycles algebraically equivalent to zero modulo incidence equivalence.