Acyclicity Condition Research Articles

Existence of fixed points for mappings of finite sets

We show that the existence theorem for zeros of a vector field (fixed points of a mapping) holds in the case of a “convex” finite set X and a “continuous” vector field (a self-mapping) directed inwards into the convex hull co X of X. The main goal is to give correct definitions of the notions of “continuity” and “convexity”. We formalize both these notions using a reflexive and symmetric binary relation on X, i.e., using a proximity relation. Continuity (we shall say smoothness) is formulated with respect to any proximity relation, and an additional requirement on the proximity (we shall call it the acyclicity condition) transforms X into a “convex” set. If these two requirements are satisfied, then the vector field has a zero (i.e., a fixed point).

Relating Postnikov pieces with the Krull filtration: a spin-off of Serre's theorem

We characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition. When the mod p cohomology of an H-space is finitely generated as an algebra over the Steenrod algebra we prove that its homotopy groups behave like those of a finite complex. In particular, a p-complete infinite loop space has a finite number of non-trivial homotopy groups if and only if its mod p cohomology satisfies this finiteness condition.

A representation theorem for minmax regret policies

Decision making under uncertainty is one of the central tasks of artificial agents. Due to their simplicity and ease of specification, qualitative decision tools are popular in artificial intelligence. Brafman and Tennenholtz [R.I. Brafman, M. Tennenholtz, An axiomatic treatment of three qualitative decision criteria, J. ACM 47 (3) (2000) 452–482] model an agent's uncertain knowledge as her local state, which consists of states of the world that she deems possible. A policy determines for each local state a total preorder of the set of actions, which represents the agent's preference over these actions. It is known that a policy is maximin representable if and only if it is closed under unions and satisfies a certain acyclicity condition. In this paper we show that the above conditions, although necessary, are insufficient for minmax regret and competitive ratio policies. A complete characterization of these policies is obtained by introducing the best-equally strictness.

On domains with regular sections

We prove the generalized convexity of domains satisfying the condition of acyclicity of their sections by a certain continuously parametrized family of two-dimensional planes.

Directed cycles and related structures in random graphs: II—Dynamic properties

We study directed random graphs (random graphs whose edges are directed) as they evolve in discrete time by the addition of nodes and edges. For two distinct evolution strategies, one that forces the graph to a condition of near acyclicity at all times and another that allows the appearance of nontrivial directed cycles, we provide analytic and simulation results related to the distributions of degrees. Within the latter strategy, in particular, we investigate the appearance and behavior of the strong components that were our subject in the first part of this study.

Geometric Form of the Hahn–Banach Theorem for Generalized Convexity

We investigate a class of compact sets convex with respect to a certain family of planes. For compact sets that satisfy the condition of acyclicity of sections by a certain collection of two-dimensional planes, we prove their generalized convexity.

Rationality of fuzzy reciprocal preference relations

We study relationships between different transitivity and acyclicity conditions for fuzzy reciprocal relations. Such conditions may be treated as rationality conditions for preferences described by such kind of relations.

Response-time bounds of EQL rule-based programs under rule priority structure

A key index of the performance of a rule based program used in real time monitoring and control is its response time, defined by the longest program execution time before a fixed point of the program is reached from a start state. Previous work in computing the response time bounds for rule based programs effectively assumes that all rules take the same amount of firing time. It is also assumed that if two rules are enabled, then either one of them may be scheduled first for firing. These assumptions can result in loose bounds, especially in the case programmers choose to impose a priority structure on the set of rules. We remove the uniform firing cost assumption and discuss how to get tighter bounds by taking rule priority information into account. We show that the rule suppression relation we previously introduced can be extended to incorporate rule priority information. A bound derivation algorithm for programs whose potential trigger relations satisfy an acyclicity condition is presented, followed by its correctness proof and an analysis example. >

Acyclicity and the dispersion of the veto power

Blair and Pollak (Econometrica (1982) 50: 931–943) prove that, if there are more alternatives than individuals, then, for every arrovian binary decision rule that is acyclic, there is at least one individual who has a veto power over a critical number of pairs of alternatives. If the number of individuals is larger than the number of alternatives, there need not be single vetoers but there could be small coalitions endowed with a similar power. Kelsey (Soc Choice Welfare (1985) 2: 131–137) states precise results in this respect. In this paper, we first give a new and much simpler proof of the main result of Blair and Pollak and complete proofs of the generalization of this result by Kelsey. Then we give a precise answer as to the minimum size of the coalitions that must have a veto power under any acyclic binary decision rule and the minimum number of pairs of alternatives on which these coalitions may exercise their power. We also show that, if the veto power of the coalitions of the minimal size attainable under the last objective is limited to the minimum number of pairs of alternatives, then all larger coalitions have a veto power on all pairs. All the results are obtained by appealing to an acyclicity condition found by Ferejohn and Fishburn (J Econ Theory (1979) 21: 28–45). In the case of symmetric and monotonic binary decision rules, proofs are even easier and illustrate clearly the reasons for the veto power.

Phytoestrogen influences on the development of behavior and gonadotropin function.

The effect of phytoestrogens on the sexual differentiation of gonadotropin function was examined by neonatal exposure of pups through milk of rat dams fed a coumestrol (100 micrograms/g), control, or chow diet during the "critical period" of the first 10 postnatal days or throughout the 21 days of lactation. In females, exposure to coumestrol throughout the period of lactation produced growth suppression and an acyclic condition in early adulthood resembling the premature anovulatory syndrome. When the period of treatment was restricted to the first 10 postnatal days, however, no effects on vaginal cyclicity were seen. The 10-day exposure period produced more marked effects in males, resulting in transitory reductions in body weight in weanling males and reductions in mount and ejaculation frequency and a prolongation of the latencies to mount and ejaculate. Testicular weights and plasma testosterone levels did not differ among treatment groups, suggesting that the deficits in male sexual behavior were not due to deficits in adult gonadal function. Few effects of chow treatment were observed. However, significant differences from controls were apparent for weight at vaginal opening in females, and mount rate for chow-treated males was intermediate between that of controls and that of the coumestrol-treated group. These data provide evidence that lactational exposure to phytoestrogen diets can alter neuroendocrine development in both female and male rats.

Algebraic torsion, zeta function and Dirichlet series for graph links in homology 3-spheres

AbstractIn this paper, we define the algebraic torsion τ associated with an -constructible sheaf ℱ on a topological space X and a ring homomorphism π1(X) → Z (with an acyclicity condition). If (X, ) has a fiber structure over S1, then τ is equal to the zeta function of the monodromy acting on the hypercohomology of the fiber. If X is the complement of a link in a homology 3-sphere and is given by a link such that admits a fiberable splice diagram, then this zeta function has a product decomposition (corresponding to the Jaco-Shalen-Johansson decomposition). This can be interpreted as a ‘Lefschetz type formula’ for a dynamical system suitably chosen.

The Evans-Griffith Syzygy Theorem and Bass Numbers

Let $(R,\mathfrak {m})$ be a Noetherian local ring containing a field. The syzygy theorem of Evans and Griffith (see The syzygy problem, Ann. of Math. (2) 114 (1981), 323-353) says that a nonfree $m$th syzygy module $M$ over $R$ which has finite projective dimension must have rank $\geq m$. This theorem is an assertion about the ranks of the homomorphisms in certain acyclic complexes. It is the aim of this paper to demonstrate that the condition of acyclicity can be relaxed in a natural way. We shall use the generalization thus obtained to show that the Bass numbers of a module satisfy restrictions analogous to those which the syzygy theorem imposes on Betti numbers.

The Evans-Griffith syzygy theorem and Bass numbers

Let ( R , m ) (R,\mathfrak {m}) be a Noetherian local ring containing a field. The syzygy theorem of Evans and Griffith (see The syzygy problem, Ann. of Math. (2) 114 (1981), 323-353) says that a nonfree m m th syzygy module M M over R R which has finite projective dimension must have rank ≥ m \geq m . This theorem is an assertion about the ranks of the homomorphisms in certain acyclic complexes. It is the aim of this paper to demonstrate that the condition of acyclicity can be relaxed in a natural way. We shall use the generalization thus obtained to show that the Bass numbers of a module satisfy restrictions analogous to those which the syzygy theorem imposes on Betti numbers.

Acyclic logic programs and the completeness of SLDNF-resolution

This paper investigates the class of acyclic programs, programs with the usual hierarchical condition imposed on ground instances of atoms rather than predicate symbols. The acyclic condition seems to naturally capture much of the recursion that occurs in practice and many programs that arise in practical programming satisfy this condition. We prove completeness of SLDNF-resolution for the acyclic programs and discuss several other desirable properties exhibited by programs belonging to this class.

Analytic torsion for group actions

The Reidemeister torsion is a classical topological invariant for nonsimply-connected manifolds [14]. Let M be a closed oriented smooth manifold with the fundamental group πλ(M) nontrivial. Let p:πx{M) -• O(N) be an orthogonal representation of nx(M). If the twisted real cohomology H*(M, Ep) vanishes, then one can define the Reidemeister torsion τp GR , which is a homeomorphism invariant of M. The original interest of τp was that it is not a homotopy invariant, and so can distinguish spaces which are homotopy equivalent but are not homeomorphic. Ray and Singer asked whether, as for many other topological quantities, one can compute τp by analytic methods [17], Given a Riemannian metric g on M, they defined an analytic torsion T' e R as a certain combination of the eigenvalues of the Laplacian acting on twisted differential forms. They showed that under the above acyclicity condition, T is independent of the metric g, and they conjectured that the analytic expression T equals the combinatorial expression τ . This conjecture was proven to be true independently by Cheeger [4] and Mϋller [15]. One can look at the above situation in the following way. The group nχ{M) acts freely on the universal cover M, and so one has an invariant for free group actions. A natural question is whether the Reidemeister torsion can be extended to an invariant for more general group actions. For a finite group acting (not necessarily freely) on a closed oriented PL manifold X, a Reidemeister torsion was defined algebraically by Rothenberg [19]. One can then ask whether there is a corresponding analytic torsion when X is smooth, and whether the analytic torsion equals the combinatorial torsion. In §11 we define the analytic torsion Tp for a finite group action and show that if the relevant cohomology groups vanish, then it is independent of the G-invariant metric used in its definition. (This was shown previously in unpublished work by Cheeger [5].) The analysis involved to

Early short photoperiod effects in female LSH/SsLak Syrian hamsters.

Previous experiments showed that LSH/SsLak female hamsters cease ovarian cyclicity after 2-4 weeks of short photoperiod (SP) treatment. This study was designed to reveal early hormonal and histological changes on day 1 of the estrous cycle (estrus) in SP-treated animals and changes brought about by the SP-induced acyclic condition. Groups of 8-10 animals were killed on day 1 after 0, 4, 8, 12, and 16 days of SP treatment; sera and pituitaries were saved for hormone determinations, and ovaries were prepared for histological analysis. Intraatrial cannulae were inserted into anovulatory animals, and blood samples were withdrawn on the first and second postoperative days; the animals were killed on the third postoperative day. PRL levels were significantly reduced in cycling animals after 16 days of SP exposure and diminished further in acyclic animals. Pituitary PRL did not change in cycling animals, although it was highly depressed in SP-treated acyclic animals. The estrous FSH surge, serum LH levels, and pituitary gonadotropin contents were not affected by SP in cycling animals; in anovulatory animals, pituitary gonadotropin contents were significantly increased, and daily afternoon gonadotropin surges were observed. Uterine weights were reduced in cycling animals and underwent a further reduction after cyclicity ceased. Ovarian analysis revealed that all cycling animals had the same number of recruited follicles, yet significantly fewer corpora lutea and an apparent increase in atretic antral follicles were observed after 16 days of SP exposure. These results suggest that SP-induced cessation of estrous cyclicity occurs abruptly. SP does not alter the secondary FSH surge, but might adversely affect maintenance of follicular growth and depress serum PRL levels.

THE TOPOLOGY OF BASIS SETS FOR SMALE DIFFEOMORPHISMS

Diffeomorphisms of a closed surface are considered which satisfy Smale's axiom A and an acyclicity condition. It is shown that if one of its basis sets is one-dimensional, then there is also a zero-dimensional source or sink. As a preliminary, some auxiliary propositions of general character are established concerning sources and sinks of diffeomorphisms satisfying the axiom and the condition above. Bibliography: 10 entries.