Notion Of Duality Research Articles

On the Dual Notion of Multiplication Modules

This paper deals with some results concerning the dual notion of multiplication modules (i.e., comultiplication modules) over a commutative ring.

Optimal Design for Synchronization of Cooperative Systems: State Feedback, Observer and Output Feedback

This technical note studies synchronization of identical general linear systems on a digraph containing a spanning tree. A leader node or command generator is considered, which generates the desired tracking trajectory. A framework for cooperative tracking control is proposed, including full state feedback control, observer design and dynamic output feedback control. The classical system theory notion of duality is extended to networked systems. It is shown that unbounded synchronization regions that achieve synchronization on arbitrary digraphs containing a spanning tree can be guaranteed by using linear quadratic regulator based optimal control and observer design methods at each node.

The Dual Notions of Some Generalizations of Prime Submodules

In this article, we introduce the dual notions of strongly prime, weakly prime, semiprime, and primal submodules, that is, strongly second, weakly second, semisecond, and secondal submodules. Furthermore, we investigate the first properties of these classes of modules and obtain some related useful characterizations.

On $q$-analogs of Steiner systems and covering designs

The $q$-analogs of covering designs, Steiner systems, and Turan designs are studied. It is shown that $q$-covering designs and $q$-Turan designs are dual notions. A strong necessary condition for the existence of Steiner structures (the $q$-analogs of Steiner systems) over $\mathbb F$2 is given. No Steiner structures of strength $2$ or more are currently known, and our condition shows that their existence would imply the existence of new Steiner systems of strength $3$. The exact values of the $q$-covering numbers $\mathcal C$q$(n,k,1)$ and $\mathcal C$q$(n,n-1,r)$ are determined for all $q,n,k,r$. Furthermore, recursive upper and lower bounds on the size of general $q$-covering designs and $q$-Turan designs are presented. Finally, it is proved that $\mathcal C$2$(5,3,2) = 27$ and $\mathcal C$2$(7,3,2) \leq 399$. Tables of upper and lower bounds on $\mathcal C$2$(n,k,r)$ are given for all $n \leq 8$.

On the Dual Notion of Prime Submodules (II)

Let R be a commutative ring and let M be an R-module. In this paper, we study the dual notion of prime submodules (that is, second submodules of M). Also we introduce the dual notion of weak multiplication R-modules (that is, weak comultiplication modules) in terms of second submodules and investigate some related results.

A Remark on Twists and the Notion of Torsion-free Discrete Quantum Groups

In this paper twists of reduced locally compact quantum groups are studied. Twists of the dual coaction on a reduced crossed product are introduced and the twisted dual coactions are proved to satisfy a type of Takesaki-Takai duality. The twisted Takesaki-Takai duality implies that twists of discrete, torsion-free quantum groups are torsion-free. Cocycle twists of duals of semisimple, compact Lie are studied leading to a locally compact quantum group contained in the Drinfeld-Jimbo algebra which gives a dual notion of Woronowicz deformations for semisimple, compact Lie groups. These cocycle twists are proven to be torsion-free whenever the Lie group is simply connected.

Fleury's spanning dimension and chain conditions on non-essential elements in modular lattices

Based on a lattice theoretical approach, we give a complete characterization of modules with Fleury’s spanning dimension. An example of a non-Artinian, non-hollow module satisfying this finiteness condition is constructed. Furthermore we introduce and characterize the dual notion of Fleury’s spanning dimension.

Output-to-state stability for hybrid systems

Output-to-state stability (OSS) is a dual notion of input-to-state stability for dynamical systems. This paper presents Lyapunov and asymptotic characterizations of OSS for hybrid dynamical systems, emphasizing that a globally detectable (i.e. nonuniformly OSS) hybrid system admits a smooth OSS-Lyapunov function.

TIGHT CLOSURE OF IDEALS RELATIVE TO MODULES

In this paper the dual notion of tight closure of ideals relative to modules is introduced and some related results are ob- tained.

Toward a Phonetic Representation of Signs: Sequentiality and Contrast

In this paper we examine the theory of the structure of signs that grew from Stokoe’s (1965) proposals. We begin by examining argument for the structural simultaneity of signs by examining claims about how signs contrast and how cheremes function. Historically, such discussions have involved three claims: (1) that signs are composed of a single handshape, a single location, and a sequence of movements, and (2) that these structural aspects account for contrast between signs in the same way that phonemes account for contrast in spoken languages, and (3) that this is evidence of dual patterning. Using a number of different examples, we evaluate these claims from the perspective of the standard definitions of contrast and duality of pattern, showing that discussions drawn from this reasoning are not consistent with standard notions of the phoneme or of double articulation and duality of pattern. We suggest that, if signed languages in fact are structured in the way Stokoe proposed—an approach maintained even in recent work (Meir et al. 2007, 537–39)—then the phoneme and notions of duality of pattern must be redefined for signed languages. We demonstrate, however, that the putative simultaneity proposed by this model does not adequately represent actual observations about the structure of signing and conclude that a model with inherent sequentiality of segments provides a more precise description of signs. We evaluate several such models, finding each to be inadequate, and end with the claim that an adequate descriptive system for signed languages will employ sequential segments. Such a system would be consistent with notions of contrast and duality of pattern developed for other languages.

Tolerant, Classical, Strict

In this paper we investigate a semantics for first-order logic originally proposed by R. van Rooij to account for the idea that vague predicates are tolerant, that is, for the principle that if x is P, then y should be P whenever y is similar enough to x. The semantics, which makes use of indifference relations to model similarity, rests on the interaction of three notions of truth: the classical notion, and two dual notions simultaneously defined in terms of it, which we call tolerant truth and strict truth. We characterize the space of consequence relations definable in terms of those and discuss the kind of solution this gives to the sorites paradox. We discuss some applications of the framework to the pragmatics and psycholinguistics of vague predicates, in particular regarding judgments about borderline cases.

Duality and subdifferential for convex functions on complete [formula omitted] metric spaces

Thanks to the recent concept of quasilinearization of Berg and Nikolaev, we have introduced the notion of duality and subdifferential on complete C A T ( 0 ) (Hadamard) spaces. For a Hadamard space X , its dual is a metric space X ∗ which strictly separates non-empty, disjoint, convex closed subsets of X , provided that one of them is compact. If f : X → ( − ∞ , + ∞ ] is a proper, lower semicontinuous, convex function, then the subdifferential ∂ f : X ⇉ X ∗ is defined as a multivalued monotone operator such that, for any y ∈ X there exists some x ∈ X with x y ⃗ ∈ ∂ f ( x ) . When X is a Hilbert space, it is a classical fact that R ( I + ∂ f ) = X . Using a Fenchel conjugacy-like concept, we show that the approximate subdifferential ∂ ϵ f ( x ) is non-empty, for any ϵ > 0 and any x in efficient domain of f . Our results generalize duality and subdifferential of convex functions in Hilbert spaces.

Accounting Research Homogeneity and the Possibilities of Structural Change

Concerns about the homogenization of accounting research have been expressed over an extended period of time, especially in the U.S. context. Numerous researchers have documented the homogeneity problem and examined the factors that have contributed to research homogeneity. One weakness in the current literature is that limited attention has been paid to understanding the process by which change might be effected. Although researchers have suggested specific changes in accounting academe that are warranted, there is no widely accepted theoretical framework within which proposals for reform can be analyzed. This may explain in part why there has been so little success in reversing the well documented homogeneity trend. In this paper, we apply the theoretical framework developed by Dillard, Rigsby and Goodman (2004) in the managerial accounting context to understand the current research homogeneity problem and to demonstrate the changes that are necessary for lasting reform to occur. At the heart of the model is Giddens’ (1976, 1979, 1981, 1984) notion of duality of structure. This duality of structure implies that the underlying rules that govern social behavior not only create resources but are themselves directly sustained by those same resources. Ultimately then, we suggest that if reform is to be successful, it will almost certainly necessitate a resource shift. In order to effect reform, the policy makers that control the resources by which the current structures shaping accounting research are sustained must act intentionally. There is evidence that such a resource shift precipitated the development of the current structure and it is unlikely that significant structural change can be achieved without a similar resource realignment.

EXTENDED DISTRIBUTIVE LAW: COWREATH OVER CORINGS

We introduce and study comodules over cowreath defined over a given coring. If our coring arises from an entwining structure, we then give a procedure to construct a cowreath from a given cowreath over the factor coalgebra. We also include the dual notions, that is, wreaths over ring extension and their modules. In particular, we show that the study of twisted algebras and twisted bimodules already introduced in [A. Čap, H. Schichl and J. Vanžura, On twisted tensor products of algebras, Commun. Algebra23 (1995) 4701–4735] has its origin in the study of wreath and their bimodules.

A self-dual induction for three-interval exchange transformations

We describe completely a new induction process on three-interval exchange transformations, under its additive and multiplicative forms; we describe the natural extension of the induction map, show that it is self-dual for the notion of duality which links the Rauzy/Zorich and da Rocha inductions, and compute a finite ergodic invariant measure (for the multiplicative form). Then we explain the connection with the published negative slope algorithm of Ferenczi, Holton and Zamboni [S. Ferenczi, C. Holton, and L. Zamboni, The structure of three-interval exchange transformations I: An arithmetic study, Ann. Inst. Fourier (Grenoble), 51 (2001), pp. 861–901], and use our measure-theoretic study to get new results on three-interval exchange transformations.

A MacWilliams Identity for Convolutional Codes: The General Case

A MacWilliams identity for convolutional codes will be established. It makes use of the weight adjacency matrices of the code and its dual, based on state space realizations (the controller canonical form) of the codes in question. The MacWilliams identity applies to various notions of duality appearing in the literature on convolutional coding theory.

Fault Detection for Linear Periodic Systems Using a Geometric Approach

Using the geometric approach, a solution is given to the problem of detecting faults in linear periodic discrete-time systems. The solution consists on developing an unknown input observer. The existing conditions are provided in terms of new geometric notion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">outer</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">observable</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">subspace</i> that is the dual notion of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">inner</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">reachable</i> <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">subspace</i> . Making use of the new notion, a residual generator is developed, such that the residual is made independent of unknown input disturbance, and its output-free response goes asymptotically to zero.

CPS-translation as adjoint

We show that there exist translations between polymorphic λ -calculus and a subsystem of minimal logic with existential types, which form a Galois insertion (embedding). The translation from polymorphic λ -calculus into the existential type system is the so-called call-by-name CPS-translation that can be expounded as an adjoint from the neat connection. The construction of an inverse translation is investigated from a viewpoint of residuated mappings. The duality appears not only in the reduction relations but also in the proof structures, such as paths between the source and the target calculi. From a programming point of view, this result means that abstract data types can interpret polymorphic functions under the CPS-translation. We may regard abstract data types as a dual notion of polymorphic functions.

The anatomy of learning anatomy

The experience of clinical teachers as well as research results about senior medical students' understanding of basic science concepts has much been debated. To gain a better understanding about how this knowledge-transformation is managed by medical students, this work aims at investigating their ways of setting about learning anatomy. Second-year medical students were interviewed with a focus on their approach to learning and their way of organizing their studies in anatomy. Phenomenographic analysis of the interviews was performed in 2007 to explore the complex field of learning anatomy. Subjects were found to hold conceptions of a dual notion of the field of anatomy and the interplay between details and wholes permeated their ways of studying with an obvious endeavor of understanding anatomy in terms of connectedness and meaning. The students' ways of approaching the learning task was characterized by three categories of description; the subjects experienced their anatomy studies as memorizing, contextualizing or experiencing. The study reveals aspects of learning anatomy indicating a deficit in meaningfulness. Variation in approach to learning and contextualization of anatomy are suggested as key-elements in how the students arrive at understanding. This should be acknowledged through careful variation of the integration of anatomy in future design of medical curricula.

Submodular partition functions

Adapting the method introduced in Graph Minors X, we propose a new proof of the duality between the bramble number of a graph and its tree-width. Our approach is based on a new definition of submodularity on partition functions which naturally extends the usual one on set functions. The proof does not rely on Menger’s theorem, and thus generalises the original one. It thus provides a dual for matroid tree-width. One can also derive all known dual notions of other classical width-parameters from it.