Partitions of an Integer into Powers

International audience In this paper, we use a simple discrete dynamical model to study partitions of integers into powers of another integer. We extend and generalize some known results about their enumeration and counting, and we give new structural results. In particular, we show that the set of these partitions can be ordered in a natural way which gives the distributive lattice structure to this set. We also give a tree structure which allow efficient and simple enumeration of the partitions of an integer.

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Finitely generated pseudocomplemented distributive lattices

Journal of the Australian Mathematical Society ◽

10.1017/s1446788700029530 ◽

1975 ◽

Vol 19 (2) ◽

pp. 238-246 ◽

Cited By ~ 8

Author(s):

J. Berman ◽

ph. Dwinger

Keyword(s):

Distributive Lattice ◽

Partially Ordered Set ◽

Ordered Set ◽

Distributive Lattices ◽

Basic Fact ◽

Finitely Generated ◽

Partially Ordered ◽

Finite Set ◽

Natural Way

If L is a pseudocomplemented distributive lattice which is generated by a finite set X, then we will show that there exists a subset G of L which is associated with X in a natural way that ¦G¦ ≦ ¦X¦ + 2¦x¦ and whose structure as a partially ordered set characterizes the structure of L to a great extent. We first prove in Section 2 as a basic fact that each element of L can be obtained by forming sums (joins) and products (meets) of elements of G only. Thus, L considered as a distributive lattice with 0,1 (the operation of pseudocomplementation deleted), is generated by G. We apply this to characterize for example, the maximal homomorphic images of L in each of the equational subclasses of the class Bω of pseudocomplemented distributive lattices, and also to find the conditions which have to be satisfied by G in order that X freely generates L.

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Recognition of human hand activity to convey series of numerical digits in a natural way using hand shape tree structure

2017 International Conference on Computing, Communication, Control and Automation (ICCUBEA) ◽

10.1109/iccubea.2017.8463783 ◽

2017 ◽

Author(s):

H. Chidananda ◽

T. Hanumantha Reddy

Keyword(s):

Tree Structure ◽

Human Hand ◽

Hand Shape ◽

Natural Way

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Generalized intervals in partially ordered groups

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100033843 ◽

1959 ◽

Vol 55 (2) ◽

pp. 165-171 ◽

Cited By ~ 1

Author(s):

D. C. J. Burgess

Keyword(s):

Distributive Lattice ◽

Lattice Structure ◽

Ordered Group ◽

Partially Ordered ◽

Partially Ordered Group ◽

Ordered Groups ◽

Normal Sense

1. Introduction. The present paper is chiefly concerned with a generalization, to be known as a ‘D-interval’, of the notion of interval or segment in an arbitrary partially ordered group. This idea is originally due to Duthie (2), but was developed by him only in a lattice. In analogy with the use of the interval in the normal sense, notions of ‘D-distributivity’ and ‘D-modularity’ are defined in terms of the D-interval, and analogues of known properties of lattice-groups or ‘l– groups’ can be formulated which might be valid when a lattice structure is no longer assumed to exist; in particular, an attempt is made to provide such a generalization of the result of Freudenthal (3) that every Z-group is a distributive lattice, but, for an arbitrary partially ordered group, it is shown that only an ‘approximation’ (in terms of non-Archimedean elements) to the desired result actually holds, although any Archimedean partially ordered group is necessarily D-distributive.

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The Inheritance Procedure: Multiple Testing of Tree-structured Hypotheses

Statistical Applications in Genetics and Molecular Biology ◽

10.1515/1544-6115.1554 ◽

2012 ◽

Vol 11 (1) ◽

Cited By ~ 25

Author(s):

Jelle J. Goeman ◽

Livio Finos

Keyword(s):

Multiple Testing ◽

Error Control ◽

Testing Procedure ◽

Tree Structure ◽

Graph Structure ◽

Multiple Testing Procedure ◽

Natural Way ◽

Chromosome Level

Hypotheses tests in bioinformatics can often be set in a tree structure in a very natural way, e.g. when tests are performed at probe, gene, and chromosome level. Exploiting this graph structure in a multiple testing procedure may result in a gain in power or increased interpretability of the results.We present the inheritance procedure, a method of familywise error control for hypotheses structured in a tree. The method starts testing at the top of the tree, following up on those branches in which it finds significant results, and following up on leaf nodes in the neighborhood of those leaves. The method is a uniform improvement over a recently proposed method by Meinshausen. The inheritance procedure has been implemented in the globaltest package which is available on www.bioconductor.org.

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A generalization of Bochner's formula

10.46298/hrj.2004.150 ◽

2004 ◽

Vol Volume 27 ◽

Author(s):

S Kanemitsu ◽

Y Tanigawa ◽

H Tsukada

Keyword(s):

Functional Equation ◽

Dirichlet Series ◽

Hierarchy Theorems ◽

International Audience ◽

Natural Way

International audience In this note we expound our general hierarchy theorems by the example of a Ramified-Type Functional Equarion H, which gives all possbile forms, in terms of se-ries with H-function coefficients, of the functional equation of higher hierarchy arising from the original ramified one satisfied by the Dirichlet series. Then by sepcifying the parameters, we shall deduce a few concrete examples scattered in the literature in the most natural way.

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Lattice Structure of D, T, and SR Fuzzy Flip-Flops Under Max-Min Logic

Journal of Advanced Computational Intelligence and Intelligent Informatics ◽

10.20965/jaciii.2005.p0661 ◽

2005 ◽

Vol 9 (6) ◽

pp. 661-668 ◽

Cited By ~ 1

Author(s):

Shinichi Yoshida ◽

◽

Kaoru Hirota ◽

Keyword(s):

Fuzzy Logic ◽

Distributive Lattice ◽

System Design ◽

Lattice Structure ◽

Design Method ◽

Boolean Lattice ◽

Lattice Structures ◽

Flip Flop ◽

Different Types ◽

Fuzzy Logical

Lattice structures of fuzzy flip-flops are described. A binary flip-flop (e.g. D, T, set-type SR, or reset-type SR flip-flop) can be extended to a fuzzy flip-flop in various ways. Under max-min fuzzy logic, there are 4 types of D fuzzy flip-flops extended from a binary D flip-flop, 136 types of SR fuzzy flip-flops extended from a binary SR flip-flop, and only one T fuzzy flip-flop. There is a lattice structure among different types of fuzzy flip-flops extended from a same binary flip-flop in terms of the order of ambiguity and the order of fuzzy logical value. These results show that fuzzy flip-flops under max-min fuzzy logic construct distributive lattice structures. Moreover D and T fuzzy flip-flops constructs Boolean lattice. And there exists a order monotone between two lattices of same fuzzy flip-flop under the order of ambiguity and the order of fuzzy logical value. Proposed analysis and results have potential to establish a fuzzy sequential system design method.

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The Sorting Order on a Coxeter Group

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3602 ◽

2008 ◽

Vol DMTCS Proceedings vol. AJ,... (Proceedings) ◽

Author(s):

Drew Armstrong

Keyword(s):

Distributive Lattice ◽

Partial Order ◽

Coxeter Group ◽

Bruhat Order ◽

Distributive Lattices ◽

Coxeter System ◽

Maximal Lattice ◽

International Audience ◽

The Way

International audience Let $(W,S)$ be an arbitrary Coxeter system. For each sequence $\omega =(\omega_1,\omega_2,\ldots) \in S^{\ast}$ in the generators we define a partial order― called the $\omega \mathsf{-sorting order}$ ―on the set of group elements $W_{\omega} \subseteq W$ that occur as finite subwords of $\omega$ . We show that the $\omega$-sorting order is a supersolvable join-distributive lattice and that it is strictly between the weak and strong Bruhat orders on the group. Moreover, the $\omega$-sorting order is a "maximal lattice'' in the sense that the addition of any collection of edges from the Bruhat order results in a nonlattice. Along the way we define a class of structures called $\mathsf{supersolvable}$ $\mathsf{antimatroids}$ and we show that these are equivalent to the class of supersolvable join-distributive lattices.

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Beyond Rescorla-Wagner: the ups and downs of learning

10.31234/osf.io/ngqyx ◽

2020 ◽

Author(s):

Gianluca Calcagni ◽

Ricardo Pellón ◽

Justin Harris

Keyword(s):

Learning Curve ◽

Dynamical Model ◽

Associative Process ◽

Damped Oscillations ◽

Oscillatory Pattern ◽

Trial Outcomes ◽

Pavlovian Learning ◽

Oscillatory Model ◽

Natural Way

We check the robustness of a recently proposed dynamical model of associative Pavlovian learning that extends the Rescorla-Wagner (RW) model in a natural way and predicts progressively damped oscillations in the response of the subjects. Using the data of two experiments, we compare the dynamical oscillatory model (DOM) with a non-associative oscillatory model (NAOM) made of the superposition of the RW learning curve and oscillations. Not only do data clearly show an oscillatory pattern, but they also favour the DOM over the NAOM, thus pointing out that these oscillations are the manifestation of an associative process. The latter is interpreted as the fact that subjects make predictions on trial outcomes more extended in time than in the RW model, but with more uncertainty.

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Rate of Escape of Random Walks on Regular Languages and Free Products by Amalgamation of Finite Groups

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.3580 ◽

2008 ◽

Vol DMTCS Proceedings vol. AI,... (Proceedings) ◽

Author(s):

Lorenz A. Gilch

Keyword(s):

Random Walks ◽

Finite Groups ◽

Transition Probabilities ◽

Finite Alphabet ◽

Free Products ◽

Rate Of Escape ◽

International Audience ◽

Special Case ◽

Current Word ◽

Natural Way

International audience We consider random walks on the set of all words over a finite alphabet such that in each step only the last two letters of the current word may be modified and only one letter may be adjoined or deleted. We assume that the transition probabilities depend only on the last two letters of the current word. Furthermore, we consider also the special case of random walks on free products by amalgamation of finite groups which arise in a natural way from random walks on the single factors. The aim of this paper is to compute several equivalent formulas for the rate of escape with respect to natural length functions for these random walks using different techniques.

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