Connectedness of number theoretical tilings

Digit Set ◽

Number Systems ◽

Arcwise Connected ◽

International Audience ◽

Consecutive Integers ◽

Pisot Unit ◽

International audience Let T=T(A,D) be a self-affine tile in \reals^n defined by an integral expanding matrix A and a digit set D. In connection with canonical number systems, we study connectedness of T when D corresponds to the set of consecutive integers \0,1,..., |det(A)|-1\. It is shown that in \reals^3 and \reals^4, for any integral expanding matrix A, T(A,D) is connected. We also study the connectedness of Pisot dual tilings which play an important role in the study of β -expansion, substitution and symbolic dynamical system. It is shown that each tile generated by a Pisot unit of degree 3 is arcwise connected. This is naturally expected since the digit set consists of consecutive integers as above. However surprisingly, we found families of disconnected Pisot dual tiles of degree 4. Also we give a simple necessary and sufficient condition for the connectedness of the Pisot dual tiles of degree 4. As a byproduct, a complete classification of the β -expansion of 1 for quartic Pisot units is given.

Strong Morita equivalence for the Denjoy C*-Algebras

Canadian Mathematical Bulletin ◽

10.4153/cmb-1988-064-7 ◽

1988 ◽

Vol 31 (4) ◽

pp. 439-447

Author(s):

Ian F. Putnam

Keyword(s):

Periodic Points ◽

Complete Classification ◽

Morita Equivalence ◽

Constant Function ◽

C Algebras ◽

Strong Morita Equivalence ◽

Function Construction

AbstractThe C*-algebras associated with irrational rotations of the circle were classified up to strong Morita equivalence by M. A. Rieffel. As a corollary, he gave a complete classification of the C*-algebras arising from irrational or Kronecker flows on the 2-torus up to *-isomorphism. Here, we extend the result to the socalled Denjoy homeomorphisms. Specifically, we give a necessary and sufficient condition for the strong Morita equivalence of two C*-algebras arising from homeomorphisms of the circle without periodic points. As a corollary, we show that two C*-algebras arising from flows on the 2-torus obtained from such homeomorphisms by the “flow under constant function” construction are *-isomorphic if and only if the flows themselves are topologically conjugate.

Applications of the Sylvester operator in the space of slice semi-regular functions

Concrete Operators ◽

10.1515/conop-2020-0001 ◽

2020 ◽

Vol 7 (1) ◽

pp. 1-12

Author(s):

Altavilla Amedeo ◽

Chiara de Fabritiis

Keyword(s):

Zero Divisor ◽

Complete Classification ◽

Linear Operators ◽

Type Result ◽

Regular Functions ◽

Zero Divisors ◽

Dimension 2 ◽

AbstractIn this paper we apply the results obtained in [3] to establish some outcomes of the study of the behaviour of a class of linear operators, which include the Sylvester ones, acting on slice semi-regular functions. We first present a detailed study of the kernel of the linear operator ℒf,g (when not trivial), showing that it has dimension 2 if exactly one between f and g is a zero divisor, and it has dimension 3 if both f and g are zero divisors. Afterwards, we deepen the analysis of the behaviour of the -product, giving a complete classification of the cases when the functions fv, gv and fvgv are linearly dependent and obtaining, as a by-product, a necessary and sufficient condition on the functions f and g in order their *-product is slice-preserving. At last, we give an Embry-type result which classifies the functions f and g such that for any function h commuting with f + g and f * g, we have that h commutes with f and g, too.

The location of classified edges due to the change in the geometric multiplicity of an eigenvalue in a tree

Special Matrices ◽

10.1515/spma-2019-0019 ◽

2019 ◽

Vol 7 (1) ◽

pp. 257-262

Author(s):

Kenji Toyonaga

Keyword(s):

Symmetric Matrix ◽

Symmetric Matrices ◽

Sufficient Condition ◽

Geometric Multiplicity ◽

Multiplicity Of An Eigenvalue

Abstract Given a combinatorially symmetric matrix A whose graph is a tree T and its eigenvalues, edges in T can be classified in four categories, based upon the change in geometric multiplicity of a particular eigenvalue, when the edge is removed. We investigate a necessary and sufficient condition for each classification of edges. We have similar results as the case for real symmetric matrices whose graph is a tree. We show that a g-2-Parter edge, a g-Parter edge and a g-downer edge are located separately from each other in a tree, and there is a g-neutral edge between them. Furthermore, we show that the distance between a g-downer edge and a g-2-Parter edge or a g-Parter edge is at least 2 in a tree. Lastly we give a combinatorially symmetric matrix whose graph contains all types of edges.

Allee’s Effect Bifurcation in Generalized Logistic Maps

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127419500391 ◽

2019 ◽

Vol 29 (03) ◽

pp. 1950039

Author(s):

J. Leonel Rocha ◽

Abdel-Kaddous Taha

Keyword(s):

Allee Effect ◽

Dynamical Behavior ◽

Complete Classification ◽

Unimodal Maps ◽

New Class ◽

New Type ◽

Local And Global Bifurcations ◽

One Dimensional Maps ◽

This paper concerns the study of the Allee effect on the dynamical behavior of a new class of generalized logistic maps. The fundamentals of the dynamics of this 4-parameter family of one-dimensional maps are presented. A complete classification of the nature and stability of its fixed points is provided. The main results relate to the Allee effect bifurcation: a new type of bifurcation introduced for this class of unimodal maps. A necessary and sufficient condition so that the Allee fixed point is a snap-back repeller is established. In addition, in the parameters space is defined an Allee’s effect region, which determines the existence of an essential extinction for the generalized logistic maps. Local and global bifurcations of generalized logistic maps are investigated.

Finite Eulerian posets which are binomial or Sheffer

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2899 ◽

2011 ◽

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

Author(s):

Hoda Bidkhori

Keyword(s):

Generating Functions ◽

Complete Classification ◽

International Audience ◽

Original Question ◽

Eulerian Posets

International audience In this paper we study finite Eulerian posets which are binomial or Sheffer. These important classes of posets are related to the theory of generating functions and to geometry. The results of this paper are organized as follows: (1) We completely determine the structure of Eulerian binomial posets and, as a conclusion, we are able to classify factorial functions of Eulerian binomial posets; (2) We give an almost complete classification of factorial functions of Eulerian Sheffer posets by dividing the original question into several cases; (3) In most cases above, we completely determine the structure of Eulerian Sheffer posets, a result stronger than just classifying factorial functions of these Eulerian Sheffer posets. We also study Eulerian triangular posets. This paper answers questions posed by R. Ehrenborg and M. Readdy. This research is also motivated by the work of R. Stanley about recognizing the \emphboolean lattice by looking at smaller intervals. Nous étudions les ensembles partiellement ordonnés finis (EPO) qui sont soit binomiaux soit de type Sheffer (deux notions reliées aux séries génératrices et à la géométrie). Nos résultats sont les suivants: (1) nous déterminons la structure des EPO Euleriens et binomiaux; nous classifions ainsi les fonctions factorielles de tous ces EPO; (2) nous donnons une classification presque complète des fonctions factorielles des EPO Euleriens de type Sheffer; (3) dans la plupart de ces cas, nous déterminons complètement la structure des EPO Euleriens et Sheffer, ce qui est plus fort que classifier leurs fonctions factorielles. Nous étudions aussi les EPO Euleriens triangulaires. Cet article répond à des questions de R. Ehrenborg and M. Readdy. Il est aussi motivé par le travail de R. Stanley sur la reconnaissance du treillis booléen via l'étude des petits intervalles.

Some results on the classification of expansive automorphisms of compact abelian groups

Ergodic Theory and Dynamical Systems ◽

10.1017/s0143385700008701 ◽

1996 ◽

Vol 16 (1) ◽

pp. 45-50 ◽

Cited By ~ 10

Author(s):

Fabio Fagnani

Keyword(s):

Finite Group ◽

Topological Entropy ◽

Abelian Groups ◽

Topological Classification ◽

Sufficient Condition ◽

Compact Abelian Groups ◽

A Finite Group ◽

AbstractIn this paper we study expansive automorphisms of compact 0-dimensional abelian groups. Our main result is the complete algebraic and topological classification of the transitive expansive automorpisms for which the maximal order of the elements isp2for a primep. This yields a classification of the transitive expansive automorphisms with topological entropy logp2. Finally, we prove a necessary and sufficient condition for an expansive automorphism to be conjugated, topologically and algebraically, to a shift over a finite group.

Prime Producing Quadratic Polynomials and Class-Numbers of Real Quadratic Fields

Canadian Journal of Mathematics ◽

10.4153/cjm-1990-018-3 ◽

1990 ◽

Vol 42 (2) ◽

pp. 315-341 ◽

Cited By ~ 5

Author(s):

Stéphane Louboutin

Keyword(s):

Class Number ◽

Quadratic Field ◽

Sufficient Condition ◽

Quadratic Polynomials ◽

Real Quadratic Fields ◽

Consecutive Integers ◽

Image Position ◽

Real Quadratic

Frobenius-Rabinowitsch's theorem provides us with a necessary and sufficient condition for the class-number of a complex quadratic field with negative discriminant D to be one in terms of the primality of the values taken by the quadratic polynomial with discriminant Don consecutive integers (See [1], [7]). M. D. Hendy extended Frobenius-Rabinowitsch's result to a necessary and sufficient condition for the class-number of a complex quadratic field with discriminant D to be two in terms of the primality of the values taken by the quadratic polynomials and with discriminant D (see [2], [7]).

Classification of surfaces in three-sphere in Lie sphere geometry

Nagoya Mathematical Journal ◽

10.1017/s0027763000005924 ◽

1996 ◽

Vol 143 ◽

pp. 59-92

Author(s):

Takayoshi Yamazaki ◽

Atsuko Yamada Yoshikawa

Keyword(s):

Plane Curves ◽

Sufficient Condition ◽

Three Sphere ◽

Lie Sphere Geometry ◽

Sphere Geometry ◽

Lie Sphere

We studied plane curves in Lie sphere geometry in [YY]. Especially we constructed Lie frames of curves in S2 and classified them by the Lie equivalence. In this paper we are concerned with surfaces in S3. We construct Lie frames and classify them. We moreover obtain the necessary and sufficient condition that two surfaces are Lie equivalent.

Cuspidal and noncuspidal robot manipulators

Robotica ◽

10.1017/s0263574707003761 ◽

2007 ◽

Vol 25 (6) ◽

pp. 677-689 ◽

Cited By ~ 22

Author(s):

Philippe Wenger

Keyword(s):

Robot Manipulators ◽

Sufficient Condition ◽

Geometric Conditions ◽

Full Classification

SUMMARYThis article synthezises the most important results on the kinematics of cuspidal manipulators i.e. nonredundant manipulators that can change posture without meeting a singularity. The characteristic surfaces, the uniqueness domains and the regions of feasible paths in the workspace are defined. Then, several sufficient geometric conditions for a manipulator to be noncuspidal are enumerated and a general necessary and sufficient condition for a manipulator to be cuspidal is provided. An explicit DH-parameter-based condition for an orthogonal manipulator to be cuspidal is derived. The full classification of 3R orthogonal manipulators is provided and all types of cuspidal and noncuspidal orthogonal manipulators are enumerated. Finally, some facts about cuspidal and noncuspidal 6R manipulators are reported.

The freeness of Ish arrangements

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2485 ◽

2015 ◽

Vol DMTCS Proceedings, 27th... (Proceedings) ◽

Author(s):

Takuro Abe ◽

Daisuke Suyama ◽

Shuhei Tsujie

Keyword(s):

Catalan Numbers ◽

Sufficient Condition ◽

Free Arrangement ◽

International Audience ◽

New Interpretation ◽

Shi Arrangement

International audience The Ish arrangement was introduced by Armstrong to give a new interpretation of the $q; t$-Catalan numbers of Garsia and Haiman. Armstrong and Rhoades showed that there are some striking similarities between the Shi arrangement and the Ish arrangement and posed some problems. One of them is whether the Ish arrangement is a free arrangement or not. In this paper, we verify that the Ish arrangement is supersolvable and hence free. Moreover, we give a necessary and sufficient condition for the deleted Ish arrangement to be free L’arrangement Ish a été introduit par Armstrong pour donner une nouvelle interprétation des nombres $q; t$-Catalan de Garsia et Haiman. Armstrong et Rhoades ont montré qu’il y avait des ressemblances frappantes entre l’arrangement Shi et l’arrangement Ish et ont posé des conjectures. L’une d’elles est de savoir si l’arrangement Ish est un arrangement libre ou pas. Dans cet article, nous vérifions que l’arrangement Ish est supersoluble et donc libre. De plus, on donne une condition nécessaire et suffisante pour que l’arrangement Ish réduit soit libre.