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.

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.

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.

The Unruh effect without space–time

2020 ◽  
Vol 17 (04) ◽  
pp. 2050057
Author(s):  
Michele Arzano
Keyword(s):  
Real Line ◽  
Thermal Effects ◽  
Thermal State ◽  
Space Time ◽  
Unruh Effect ◽  
Affine Transformations ◽  
One Dimensional ◽  
The Real ◽  
Simple Quantum ◽  
The One

We show how the characteristic thermal effects found for a quantum field in space–time geometries admitting a causal horizon can be found in a simple quantum system living on the real line. The analysis we present is essentially group theoretic in nature: a thermal state emerges naturally when comparing representations of the group of affine transformations of the real line. The freedom in the choice of different notions of translation generators is the key to the one-dimensional Unruh effect we describe.

TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A CHEREDNIK TYPE OPERATOR ON THE REAL LINE

2010 ◽  
Vol 08 (04) ◽  
pp. 387-408 ◽  
Author(s):  
MOHAMED ALI MOUROU
Keyword(s):  
Real Line ◽  
Difference Operators ◽  
Generalized Convolution ◽  
One Dimensional ◽  
The Real ◽  
First Order ◽  
Wiener Theorem ◽  
The Fourier Transform ◽  
The One ◽  
Transmutation Operators

We consider a singular differential-difference operator Λ on the real line which generalizes the one-dimensional Cherednik operator. We construct transmutation operators between Λ and first-order regular differential-difference operators on ℝ. We exploit these transmutation operators, firstly to establish a Paley–Wiener theorem for the Fourier transform associated with Λ, and secondly to introduce a generalized convolution on ℝ tied to Λ.

scholarly journals Reconstruction of the One-Dimensional Lebesgue Measure

2020 ◽  
Vol 28 (1) ◽  
pp. 93-104
Author(s):  
Noboru Endou
Keyword(s):  
Lebesgue Measure ◽  
Real Space ◽  
Outer Measure ◽  
Finite Additivity ◽  
One Dimensional ◽  
The Real ◽  
Measurable Set ◽  
The One ◽  
System 1 ◽  
Dimensional Lebesgue Measure

SummaryIn the Mizar system ([1], [2]), Józef Białas has already given the one-dimensional Lebesgue measure [4]. However, the measure introduced by Białas limited the outer measure to a field with finite additivity. So, although it satisfies the nature of the measure, it cannot specify the length of measurable sets and also it cannot determine what kind of set is a measurable set. From the above, the authors first determined the length of the interval by the outer measure. Specifically, we used the compactness of the real space. Next, we constructed the pre-measure by limiting the outer measure to a semialgebra of intervals. Furthermore, by repeating the extension of the previous measure, we reconstructed the one-dimensional Lebesgue measure [7], [3].

scholarly journals The Effects of Filters and Colour on Stellar Occultations and Appropriate Deconvolution Procedures

1971 ◽  
Vol 2 ◽  
pp. 646-661 ◽  
Author(s):  
T. Krishnan
Keyword(s):  
Fresnel Diffraction ◽  
Color Temperature ◽  
System Response ◽  
Total System ◽  
Central Wavelength ◽  
One Dimensional ◽  
The Real ◽  
Stellar Occultations ◽  
The Absolute ◽  
The One

AbstractThe theory of the effect of bandwidth of lunar occultations is reviewed. It is recalled that effective beamshapes can be calculated for symmetrical bandpasses and that their widths are related to the absolute width in wavelength of the bandpasses. Restoration with the second differential of the theoretical Fresnel diffraction curves at the central wavelength, at the correct rates, yield source distributions as viewed by these beamshapes. It is shown that for asymmetric bandpasses, the real and odd parts taken about the centroids lead to equivalent even and odd beams. Assuming an approximate color temperature for the stars, the total system response can be evaluated and hence the even and odd parts. Restoration of the data should then be performed using the second differential of the Fresnel curve at the centroid wavelength to minimize the odd part, adjusting zeroes, rates, and centroids by inspection. The even part should then represent the even theoretical response convolved with the one-dimensional stellar distribution, provided the latter is circularly symmetrical.The technique is applied to the occultation observation of λ-Aquarii by Nather et al. (1970) leading to closely similar results.

Effective aperture of X-ray compound refractive lenses

2017 ◽  
Vol 24 (3) ◽  
pp. 609-614 ◽  
Author(s):  
V. G. Kohn
Keyword(s):  
Integral Intensity ◽  
Two Dimensional ◽  
One Dimensional ◽  
X Ray ◽  
The Real ◽  
A Value ◽  
Effective Aperture ◽  
The One ◽  
Definition Of ◽  
Focusing Properties

A new definition of the effective aperture of the X-ray compound refractive lens (CRL) is proposed. Both linear (one-dimensional) and circular (two-dimensional) CRLs are considered. It is shown that for a strongly absorbing CRL the real aperture does not influence the focusing properties and the effective aperture is determined by absorption. However, there are three ways to determine the effective aperture in terms of transparent CRLs. In the papers by Kohn [(2002). JETP Lett. 76, 600–603; (2003). J. Exp. Theor. Phys. 97, 204–215; (2009). J. Surface Investig. 3, 358–364; (2012). J. Synchrotron Rad. 19, 84–92; Kohn et al. (2003). Opt. Commun. 216, 247–260; (2003). J. Phys. IV Fr, 104, 217–220], the FWHM of the X-ray beam intensity just behind the CRL was used. In the papers by Lengeler et al. [(1999). J. Synchrotron Rad. 6, 1153–1167; (1998). J. Appl. Phys. 84, 5855–5861], the maximum intensity value at the focus was used. Numerically, these two definitions differ by 50%. The new definition is based on the integral intensity of the beam behind the CRL over the real aperture. The integral intensity is the most physical value and is independent of distance. The new definition gives a value that is greater than that of the Kohn definition by 6% and less than that of the Lengeler definition by 41%. A new approximation for the aperture function of a two-dimensional CRL is proposed which allows one to calculate the two-dimensional CRL through the one-dimensional CRL and to obtain an analytical solution for a complex system of many CRLs.

On absolutely nonmeasurable homomorphisms of commutative groups

2013 ◽  
Vol 50 (3) ◽  
pp. 287-295
Author(s):  
A. Kharazishvili
Keyword(s):  
Real Line ◽  
Quotient Group ◽  
Torsion Subgroup ◽  
Dimensional Torus ◽  
One Dimensional ◽  
The Real ◽  
The One

We give a characterization of all those commutative groups which admit at least one absolutely nonmeasurable homomorphism into the real line (or into the one-dimensional torus). These are exactly those commutative groups (G, +) for which the quotient group G/G0 is uncountable, where G0 denotes the torsion subgroup of G.

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