Finite List Research Articles

An algorithm to detect full irreducibility by bounding the volume of periodic free factors

We provide an effective algorithm for determining whether an element φ of the outer automorphism group of a free group is fully irreducible. Our method produces a finite list that can be checked for periodic proper free factors.

Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P3-free) and B induces a triangle-free graph (i.e., G[B] is K3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs.

Almost polynomial growth: Classifying varieties of graded algebras

Let G be a finite group, V a variety of associative G-graded algebras and c (V), n = 1, 2, …, its sequence of graded codimensions. It was recently shown by Valenti that such a sequence is polynomially bounded if and only if V does not contain a finite list of G-graded algebras. The list consists of group algebras of groups of order a prime number, the infinite-dimensional Grassmann algebra and the algebra of 2 × 2 upper triangular matrices with suitable gradings. Such algebras generate the only varieties of G-graded algebras of almost polynomial growth, i.e., varieties of exponential growth such that any proper subvariety is polynomially bounded. In this paper we completely classify all subvarieties of the G-graded varieties of almost polynomial growth by giving a complete list of finite-dimensional G-graded algebras generating them.

The Kauffman bracket ideal for genus-1 tangles

Given a compact oriented 3-manifold M in S3 with boundary, an (M, 2n)-tangle [Formula: see text] is a 1-manifold with 2n boundary components properly embedded in M. We say that [Formula: see text] embeds in a link L in S3 if [Formula: see text] can be completed to L by a 1-manifold with 2n boundary components exterior to M. The link L is called a closure of [Formula: see text]. We define the Kauffman bracket ideal of [Formula: see text] to be the ideal [Formula: see text] of ℤ[A, A-1] generated by the reduced Kauffman bracket polynomials of all closures of [Formula: see text]. If this ideal is non-trivial, then [Formula: see text] does not embed in the unknot. We give an algorithm for computing a finite list of generators for the Kauffman bracket ideal of any (S1 × D2, 2)-tangle, also called a genus-1 tangle, and give an example of a genus-1 tangle with non-trivial Kauffman bracket ideal. Furthermore, we show that if a single-component genus-1 tangle [Formula: see text] can be obtained as the partial closure of a (B3, 4)-tangle [Formula: see text], then [Formula: see text].

Linguistic Features of NATO’s Media Image (the Case Study of British Online Publications)

The paper investigates linguistic specific means of NATO's image making in online British leading newspapers. The case study involves news reports and analytical articles that represent explicit and implicit evaluation of the North Atlantic Treaty Organization. The investigation is based on the principles of the media linguistic approach and suggests the structural peculiarities of a media image that can be analyzed as a three-sided unit including positive, negative and neutral structural elements of image introduction into a media text. The NATO's image can be viewed as that of a geopolitical actor, creating its Natoland. Certain media topics are analyzed as creating positive and negative images of the NATO as a military organization. A finite list of positive and negative constituents to the Alliance's image is presented in the paper. It is assumed that the simplistic writer-reader model of interaction shapes certain elements of a media image that can be interpreted by a reader. The author distinguishes in the paper the principle constituents of an international organization in general, and the NATO in particular. Findings of the study demonstrate that the NATO's media image is interpreted by the readers as the conceptual entity of two key images: a peacemaker and a hawk that are restored on lexical and intertextual levels

Maker‐breaker games on random geometric graphs

AbstractIn a Maker‐Breaker game on a graph G, Breaker and Maker alternately claim edges of G. Maker wins if, after all edges have been claimed, the graph induced by his edges has some desired property. We consider four Maker‐Breaker games played on random geometric graphs. For each of our four games we show that if we add edges between n points chosen uniformly at random in the unit square by order of increasing edge‐length then, with probability tending to one as n →∞, the graph becomes Maker‐win the very moment it satisfies a simple necessary condition. In particular, with high probability, Maker wins the connectivity game as soon as the minimum degree is at least two; Maker wins the Hamilton cycle game as soon as the minimum degree is at least four; Maker wins the perfect matching game as soon as the minimum degree is at least two and every edge has at least three neighbouring vertices; and Maker wins the H‐game as soon as there is a subgraph from a finite list of “minimal graphs.” These results also allow us to give precise expressions for the limiting probability that G(n, r) is Maker‐win in each case, where G(n, r) is the graph on n points chosen uniformly at random on the unit square with an edge between two points if and only if their distance is at most r. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 45, 553–607, 2014

Mapping tori of small dilatation expanding train-track maps

An expanding train-track map on a graph of rank n is P-small if its dilatation is bounded above by Pn. We prove that for every P there is a finite list of mapping tori X1,…,XA, with A depending only on P and not n, so that the mapping torus associated with every P-small expanding train-track map can be obtained by surgery on some Xi. We also show that, given an integer P>0, there is a bound M depending only on P and not n, so that the fundamental group of the mapping torus of any P-small expanding train-track map has a presentation with less than M generators and M relations. We also provide some bounds for the smallest possible dilatation.

Maximally positive polynomial systems supported on circuits

A real polynomial system with support W⊂Zn is called maximally positive if all its complex solutions are positive solutions. A support W having n+2 elements is called a circuit. We previously showed that the number of non-degenerate positive solutions of a system supported on a circuit W⊂Zn is at most m(W)+1, where m(W)≤n is the degeneracy index of W. We prove that if a circuit W⊂Zn supports a maximally positive system with the maximal number m(W)+1 of non-degenerate positive solutions, then it is unique up to the obvious action of the group of invertible integer affine transformations of Zn. In the general case, we prove that any maximally positive system supported on a circuit can be obtained from another one having the maximal number of positive solutions by means of some elementary transformations. As a consequence, we get for each n and up to the above action a finite list of circuits W⊂Zn which can support maximally positive polynomial systems. We observe that the coefficients of the primitive affine relation of such circuit have absolute value 1 or 2 and make a conjecture in the general case for supports of maximally positive systems.

Use of the Drug Burden Index to identify and reduce potentially inappropriate prescribing of anticholinergic and sedative agents in elderly patients in institutionalized care: a systematic review protocol

Review question/objective The aim of this review is to critically appraise, synthesize and present evidence on the use of the Drug Burden Index (DBI) to identify and reduce potentially inappropriate prescribing of anticholinergic and sedative agents in elderly patients in institutionalized care. More specifically, this review seeks to answer the following questions: 1.In which specific settings and patient groups has the Drug Burden Index been applied? 2.Which outcomes have been studied? (E.g. occurrence and incidence of adverse drug reactions, physical functioning, mental functioning, cause of admission to hospital, etc.) 3.Has the use of the Drug Burden Index impacted upon inappropriate prescribing of anticholinergic and sedative agents to elderly patients in institutionalized care? (E.g. cessation of therapy, prescribing altered to other agents, reduction in adverse drug reactions, etc.) Inclusion criteria Types of participants The focus of this review will specifically be on ‘elderly patients’ as described within the studies. If no classification is given within studies, then only those studies reporting on patients aged 65 years and over will be included in the review. Furthermore, patients receiving care within either hospital or care home settings (institutionalized care) will be included in the review. The care of these patients and hence the use of the Drug Burden Index is likely to be markedly different to home dwelling patients. Types of intervention(s) Use of the Drug Burden Index tool to identify and reduce potentially inappropriate prescribing of anticholinergic or sedative medications. Types of comparisons Patients with a score of zero on the Drug Burden Index (i.e. no prescription of anticholinergic or sedative medicines) compared to the Drug Burden Index score; or different levels of Drug Burden Index scores between sub-samples of patients. Most studies will have no comparison. Types of outcomes This review will consider studies that include the following outcome measures: 1.Drug Burden Index scores 2.Physical and mental functioning; adverse effects of anticholinergic and sedative medicines 3.Changes to therapy following application of the Drug Burden Index as a tool to identify potentially inappropriate prescribing

The Uniformization Problem for Σ-Predicates in a Hereditarily Finite List Superstructure over the Real Exponential Field

We prove a uniformization theorem for Σ-predicates and show that there exists a universal function for Σ-definable functions in a hereditarily finite list superstructure over the real exponential field.

Power theories for multi-choice organizations and political rules: Rank-order equivalence

Voting power theories measure the ability of voters to influence the outcome of an election under a given voting rule. In general, each theory gives a different evaluation of power, raising the question of their appropriateness, and calling for the need to identify classes of rules for which different theories agree. We study the ordinal equivalence of the generalizations of the classical power concepts–the influence relation, the Banzhaf power index, and the Shapley–Shubik power index–to multi-choice organizations and political rules. Under such rules, each voter chooses a level of support for a social goal from a finite list of options, and these individual choices are aggregated to determine the collective level of support for this goal. We show that the power theories analyzed do not always yield the same power relationships among voters. Thanks to necessary and/or sufficient conditions, we identify a large class of rules for which ordinal equivalence obtains. Furthermore, we prove that ordinal equivalence obtains for all linear rules allowing a fixed number of individual approval levels if and only if that number does not exceed three. Our findings generalize all the previous results on the ordinal equivalence of the classical power theories, and show that the condition of linearity found to be necessary and sufficient for ordinal equivalence to obtain when voters have at most three options to choose from is no longer sufficient when they can choose from a list of four or more options.

Constraints on counterexamples to the Casas-Alvero conjecture and a verification in degree 12

In the first (theoretical) part of this paper, we prove a number of constraints on hypothetical counterexamples to the Casas-Alvero conjecture, building on ideas of Graf von Bothmer, Labs, Schicho and van de Woestijne that were recently reinterpreted by Draisma and de Jong in terms of p p -adic valuations. In the second (computational) part, we present ideas improving upon Diaz-Toca and Gonzalez-Vega’s Gröbner basis approach to the Casas-Alvero conjecture. One application is an extension of the proof of Graf von Bothmer et al. to the cases 5 p k 5p^k , 6 p k 6p^k and 7 p k 7p^k (that is, for each of these cases, we determine the finite list of primes p p to which their proof is not applicable). Finally, by combining both parts, we settle the Casas-Alvero conjecture in degree 12 12 (the smallest open case).

Development and evaluation of a short 24-h food list as part of a blended dietary assessment strategy in large-scale cohort studies

The validity of dietary assessment in large-scale cohort studies has been questioned. Combining data sources for the estimation of usual intake in a blended approach may enhance the validity of dietary measurement. Our objective was to develop a web-based 24-h food list for Germany to identify foods consumed during the previous 24 h and to evaluate the performance of the new questionnaire in a feasibility study. Available data from the German National Nutrition Survey II were used to develop a finite list of food items. A total of 508 individuals were invited to fill in the 24-h food list via the Internet up to three times during a 3-6-month time period. In addition, participants were asked to evaluate the questionnaire using a brief online evaluation form. In total, 246 food items were identified for the 24-h food list, reflecting >75% variation in intake of 27 nutrients and four major food groups. Among the individuals invited, 64% participated in the feasibility study. Of these, 100%, 85% and 68% of participants completed the 24-h food list one, two or three times, respectively. The average time needed to complete the questionnaire was 9 min, and its acceptability by participants was rated as high. The 24-h food list represents a promising new dietary assessment tool that can be used as part of a blended approach combining multiple data sources for valid estimation of usual dietary intake in large-scale cohort studies.

Taking Gender into Account in Occupational Health Research: Continuing Tensions

AbstractThinking about women’s occupational health reveals tensions that are involved in doing all occupational health research. By ‘tensions’ we mean conflicting pressures that are not easily or perhaps ever resolved. Recognising such tensions can lead to better science, even when the underlying issues persist. Based on research about women’s occupational health, this paper identifies a series of tensions that have become sources of conflicting pressures within this specific field and which have more general implications for occupational health and, indeed, public health research: sex vs. gender; universal patterns vs. context-specific knowledge; women as a group vs. particular groups of women; quantitative studies vs. qualitative studies; male-female comparisons vs. the study of women; short-term health effects vs. long-term health effects. The tensions identified here are not intended as a finite list but rather as a starting point for the explicit recognition of tensions within specific research proje...

The thick-thin decomposition and the bilipschitz classification of normal surface singularities

We describe a natural decomposition of a normal complex surface singularity $(X,0)$ into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical; the link of the singularity is then Seifert fibered. In general the thin part will not be empty, in which case it always carries essential topology. Our decomposition has some analogy with the Margulis thick-thin decomposition for a negatively curved manifold. However, the geometric behavior is very different; for example, often most of the topology of a normal surface singularity is concentrated in the thin parts. By refining the thick-thin decomposition, we then give a complete description of the intrinsic bilipschitz geometry of $(X,0)$ in terms of its topology and a finite list of numerical bilipschitz invariants.

Solving Rota's Conjecture

I n 1970, Gian-Carlo Rota posed a conjecture predicting a beautiful combinatorial characterization of linear dependence in vector spaces over any given finite field. We have recently completed a fifteen-year research program that culminated in a solution to Rota’s Conjecture. In this article we discuss the conjecture and give an overview of the proof. Matroids are a combinatorial abstraction of linear independence among vectors; given a finite collection of vectors in a vector space, each subset is either dependent or independent. A matroid consists of a finite ground set together with a collection of subsets that we call independent; the independent sets satisfy natural combinatorial axioms coming from linear algebra. Not all matroids can be represented by a collection of vectors and, ever since their introduction by Hassler Whitney [26] in 1935, mathematicians have sought ways to characterize those matroids that are. Rota’s Conjecture asserts that representability over any given finite field is characterized by a finite list of obstructions. We will formalize these notions, and the conjecture, in the next section. In the remainder of this introduction, we will describe the journey that led us to a solution. In the late 1990s, Rota’s Conjecture was already known to hold for fields of size two, three, and

On geodesic homotopies of controlled width and conjugacies in isometry groups

We give an analytical proof of the Poincaré-type inequalities for widths of geodesic homotopies between equivariant maps valued in Hadamard metric spaces. As an application we obtain a linear bound for the length of an element conjugating two finite lists in a group acting on an Hadamard space.

Signal Reconstruction From the Magnitude of Subspace Components

We consider signal reconstruction from the norms of subspace components generalizing standard phase retrieval problems. In the deterministic setting, a closed reconstruction formula is derived when the subspaces satisfy certain cubature conditions, that require at least a quadratic number of subspaces. Moreover, we address reconstruction under the erasure of a subset of the norms; using the concepts of $p$-fusion frames and list decoding, we propose an algorithm that outputs a finite list of candidate signals, one of which is the correct one. In the random setting, we show that a set of subspaces chosen at random and of cardinality scaling linearly in the ambient dimension allows for exact reconstruction with high probability by solving the feasibility problem of a semidefinite program.

Complete solution to the TP2 completion problem

For any pattern of the specified entries, an explicit finite list of polynomial inequalities in the specified entries is given that characterizes the completability of a partial positive matrix to a TP matrix. Each polynomial happens to be a monomial, and an algorithm, whose input is the pattern, is given for finding the polynomials. The method uses some new partial orders on matrices and the logarithmic method to reduce the question to determining the generators of a certain finitely generated, pointed cone.

Productive coprogramming with guarded recursion

Total functional programming offers the beguiling vision that, just by virtue of the compiler accepting a program, we are guaranteed that it will always terminate. In the case of programs that are not intended to terminate, e.g., servers, we are guaranteed that programs will always be productive . Productivity means that, even if a program generates an infinite amount of data, each piece will be generated in finite time. The theoretical underpinning for productive programming with infinite output is provided by the category theoretic notion of final coalgebras. Hence, we speak of co programming with non-well-founded co data, as a dual to programming with well-founded data like finite lists and trees. Systems that offer facilities for productive coprogramming, such as the proof assistants Coq and Agda, currently do so through syntactic guardedness checkers. Syntactic guardedness checkers ensure that all self-recursive calls are guarded by a use of a constructor. Such a check ensures productivity. Unfortunately, these syntactic checks are not compositional, and severely complicate coprogramming. Guarded recursion, originally due to Nakano, is tantalising as a basis for a flexible and compositional type-based approach to coprogramming. However, as we show, by itself, guarded recursion is not suitable for coprogramming due to the fact that there is no way to make finite observations on pieces of infinite data. In this paper, we introduce the concept of clock variables that index Nakano's guarded recursion. Clock variables allow us to "close over" the generation of infinite data, and to make finite observations, something that is not possible with guarded recursion alone.