Duality for dyadic intervals

2019 ◽  
Vol 29 (01) ◽  
pp. 41-60 ◽  
Author(s):  
K. Matczak ◽  
A. Mućka ◽  
A. B. Romanowska
Keyword(s):  
Convex Sets ◽  
Convex Polytopes ◽  
Arithmetic Mean ◽  
Unit Interval ◽  
Finitely Generated ◽  
One Dimensional ◽  
Additional Constant ◽  
Dyadic Space ◽  
The One ◽  
Dyadic Intervals

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

Duality for dyadic triangles

2019 ◽  
Vol 29 (01) ◽  
pp. 61-83 ◽  
Author(s):  
K. Matczak ◽  
A. Mućka ◽  
A. B. Romanowska
Keyword(s):  
Binary Operation ◽  
Arithmetic Mean ◽  
Unit Cube ◽  
Unit Interval ◽  
Real Plane ◽  
Finitely Generated ◽  
One Dimensional ◽  
Additional Constant ◽  
The One ◽  
Dyadic Intervals

This paper is a direct continuation of the paper “Duality for dyadic intervals” by the same authors, and can be considered as its second part. Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent groupoid under the binary operation of arithmetic mean. The first paper dealt with the structure of finitely generated subgroupoids of the dyadic line, which were shown to be isomorphic to dyadic intervals. Then a duality between the class of dyadic intervals and the class of certain subgroupoids of the dyadic unit square was described. The present paper extends the results of the first paper, provides some characterizations of dyadic triangles, and describes a duality for the class of dyadic triangles. As in the case of intervals, the duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the dyadic unit cube, considered as (commutative, idempotent and entropic) groupoids with additional constant operations.

DYADIC POLYGONS

2011 ◽  
Vol 21 (03) ◽  
pp. 387-408 ◽  
Author(s):  
K. MATCZAK ◽  
A. B. ROMANOWSKA ◽  
J. D. H. SMITH
Keyword(s):  
Binary Operation ◽  
Arithmetic Mean ◽  
Real Plane ◽  
Finitely Generated ◽  
Idempotent Algebra ◽  
One Dimensional ◽  
The Real ◽  
Pythagoras Theorem ◽  
The One ◽  
Dyadic Intervals

Dyadic rationals are rationals whose denominator is a power of 2. Dyadic triangles and dyadic polygons are, respectively, defined as the intersections with the dyadic plane of a triangle or polygon in the real plane whose vertices lie in the dyadic plane. The one-dimensional analogs are dyadic intervals. Algebraically, dyadic polygons carry the structure of a commutative, entropic and idempotent algebra under the binary operation of arithmetic mean. In this paper, dyadic intervals and triangles are classified to within affine or algebraic isomorphism, and dyadic polygons are shown to be finitely generated as algebras. The auxiliary results include a form of Pythagoras' theorem for dyadic affine geometry.

scholarly journals DUALITY FOR CONVEX POLYTOPES

2009 ◽  
Vol 86 (3) ◽  
pp. 399-412 ◽  
Author(s):  
A. B. ROMANOWSKA ◽  
P. ŚLUSARSKI ◽  
J. D. H. SMITH
Keyword(s):  
Convex Sets ◽  
Convex Polytopes ◽  
Finitely Generated ◽  
Additional Constant ◽  
Category Of Polytopes

AbstractThis paper establishes a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations.

DUALITY FOR QUASIPOLYTOPES

2016 ◽  
Vol 101 (1) ◽  
pp. 95-117
Author(s):  
A. MUĆKA ◽  
A. B. ROMANOWSKA
Keyword(s):  
General Class ◽  
Convex Sets ◽  
Finitely Generated ◽  
Dual Spaces ◽  
Additional Constant ◽  
Category Of Polytopes

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. The present paper provides an extension of this duality to a much more general class of so-called quasipolytopes, that is, convex sets with polytopes as closures. The dual spaces of quasipolytopes are Płonka sums of open polytopes, which are considered as barycentric algebras with some additional operations. In constructing this duality, we use several known and new dualities: the Hofmann–Mislove–Stralka duality for semilattices; the Romanowska–Ślusarski–Smith duality for polytopes; a new duality for open polytopes; and a new duality for injective Płonka sums of polytopes.

scholarly journals Comments on open string with ‘massive’ boundary term

2020 ◽  
Vol 476 (2236) ◽  
pp. 20190748
Author(s):  
Arkady A. Tseytlin
Keyword(s):  
Scattering Amplitudes ◽  
Partition Function ◽  
Gauge Field ◽  
Path Integral ◽  
Open String ◽  
One Dimensional ◽  
Conformally Invariant ◽  
Additional Constant ◽  
The One ◽  
Definition Of

We discuss possible definition of open string path integral in the presence of additional boundary couplings corresponding to the presence of masses at the ends of the string. These couplings are not conformally invariant implying that as in a non-critical string case one is to integrate over the one-dimensional metric or reparametrizations of the boundary. We compute the partition function on the disc in the presence of an additional constant gauge field background and comment on the structure of the corresponding scattering amplitudes.

scholarly journals On the measure of the one-skeleton of the sum of convex compact sets

1987 ◽  
Vol 42 (3) ◽  
pp. 385-389
Author(s):  
Leoni Dalla
Keyword(s):  
Convex Sets ◽  
Compact Convex ◽  
Minkowski Inequality ◽  
Convex Compact ◽  
Compact Sets ◽  
One Dimensional ◽  
Special Cases ◽  
Analogous Relation ◽  
The One ◽  
Convex Compact Sets

AbstractFor any two compact convex sets in a Euclidean space, the relation between the volume of the sum of the two sets and the volume of each of them is given by the Brünn-Minkowski inequality. In this note we prove an analogous relation for the one-dimensional Hausdorff measure of the one-skeleton of the above sets. Also, some counterexamples are given which show that the above results are the best possible in some special cases.

Exploring the Multidimensional Facets of Authoritarianism: Authoritarian Aggression and Social Dominance Orientation

2008 ◽  
Vol 67 (1) ◽  
pp. 51-60 ◽  
Author(s):  
Stefano Passini
Keyword(s):  
Social Dominance ◽  
Factor Model ◽  
Three Dimensional ◽  
Social Dominance Orientation ◽  
Model Fit ◽  
Regression Analyses ◽  
One Dimensional ◽  
Confirmatory Factor ◽  
The One ◽  
Better Than

The relation between authoritarianism and social dominance orientation was analyzed, with authoritarianism measured using a three-dimensional scale. The implicit multidimensional structure (authoritarian submission, conventionalism, authoritarian aggression) of Altemeyer’s (1981, 1988) conceptualization of authoritarianism is inconsistent with its one-dimensional methodological operationalization. The dimensionality of authoritarianism was investigated using confirmatory factor analysis in a sample of 713 university students. As hypothesized, the three-factor model fit the data significantly better than the one-factor model. Regression analyses revealed that only authoritarian aggression was related to social dominance orientation. That is, only intolerance of deviance was related to high social dominance, whereas submissiveness was not.

Adiabatic large polarons in anisotropic molecular crystals

2011 ◽  
Vol 35 (1) ◽  
pp. 15-27
Author(s):  
Zoran Ivić ◽  
Željko Pržulj
Keyword(s):  
Energy Transfer ◽  
Effective Mass ◽  
Molecular Crystals ◽  
Single Chain ◽  
Proper Understanding ◽  
One Dimensional ◽  
Interchain Coupling ◽  
Large Polaron ◽  
The One

Adiabatic large polarons in anisotropic molecular crystals We study the large polaron whose motion is confined to a single chain in a system composed of the collection of parallel molecular chains embedded in threedimensional lattice. It is found that the interchain coupling has a significant impact on the large polaron characteristics. In particular, its radius is quite larger while its effective mass is considerably lighter than that estimated within the one-dimensional models. We believe that our findings should be taken into account for the proper understanding of the possible role of large polarons in the charge and energy transfer in quasi-one-dimensional substances.

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