Remainder Sets Research Articles

Symmetrization of bounded remainder sets

Symmetrization of bounded remainder sets

Bounded Remainder Sets

The paper considers the category ( $$ \mathcal{T} $$ , S, X) consisting of mappings S : $$ \mathcal{T} $$ −→ $$ \mathcal{T} $$ of spaces $$ \mathcal{T} $$ with distinguished subsets X ⊂ $$ \mathcal{T} $$ . Let rX (i, x0) be the distribution function of points of an S-orbit x0, x1 = S(x0), . . . , xi−1 = Si−1(x0) getting into X, and let δX (i, x0) be the deviation defined by the equation rX (i, x0) = aX i + δX (i, x0), where aX i is the average value. If δX (i, x0) = O(1), then such sets X are called bounded remainder sets. In the paper, bounded remainder sets X are constructed in the following cases: (1) the space $$ \mathcal{T} $$ is the circle, torus, or the Klein bottle; (2) the map S is a rotation of the circle, a shift or an exchange mapping of the torus; (3) X is a fixed subset X ⊂ $$ \mathcal{T} $$ or a sequence of subsets depending on the iteration number i = 0, 1, 2, . . .. Bibliography: 27 titles.

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices, II

Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be obtained from a geometric argument which was previously employed by Duneau and Oguey in the study of deformation properties of mathematical models for quasicrystals.

Dividing Toric Tilings and Bounded Remainder Sets

With the use of differentiation of exchanged toric developments, an infinite sequence of twodimensional toric tilings is constructed. The karyon of these tilings is proved to be a bounded remainder set.

Bounded remainder sets under torus exchange transformations

Bounded remainder sets under torus exchange transformations

Constructing bounded remainder sets and cut-and-project sets which are bounded distance to lattices

For any irrational cut-and-project setup, we demonstrate a natural infinite family of windows which gives rise to separated nets that are each bounded distance to a lattice. Our proof provides a new construction, using a sufficient condition of Rauzy, of an infinite family of non-trivial bounded remainder sets for any totally irrational toral rotation in any dimension.

Exchanged toric tilings, Rauzy substitution, and bounded remainder sets

This paper is devoted to the two-dimensional problem of the distribution of the fractional parts of a linear function. A new class of tilings of the two-dimensional torus into bounded remainder sets with an effective estimate of the remainder is introduced. It is shown that examples of the tilings under consideration can be obtained by using the geometric version of the Rauzy substitution.

Multi-colour dynamical tilings of tori into bounded remainder sets

We use tilings of multi-dimensional tori to construct bounded remainder sets that are finite unions of convex polyhedra. For the deviations of the distribution of points in the orbits with respect to translations of the torus over these sets, we prove a multi-dimensional version of Hecke's theorem on the distribution of fractional parts on a circle.

Rational evaluation in belief revision

We introduce a new operator, called rational evaluation, in belief change. The operator evaluates new information according to the agent’s core beliefs, and then exports the plausible part of the new information. It belongs to the decision module in belief change. We characterize rational evaluation by axiomatic postulates and propose two functional constructions for it, based on the well-known constructions of kernel sets and remainder sets, respectively. The main results of the paper are two representation theorems with respect to the two functional constructions. We also compare and connect the evaluation operator with some related operators in belief revision, including contraction, set contraction, and prioritized multiple revision.

Bounded Remainder Sets on a Double Covering of the Klein Bottle

A shift $$ \tilde{\mathbb{S}}\kern0.5em :\kern0.5em {\tilde{\mathbb{K}}}^2\to {\tilde{\mathbb{K}}}^2 $$ on the double covering of the Klein bottle $$ {\tilde{\mathbb{K}}}^2={\mathbb{K}}^2\times \left\{\pm 1\right\} $$ is considered. This shift $$ \tilde{\mathbb{S}} $$ generates a tiling $$ {\tilde{\mathbb{K}}}^2={\tilde{\mathbb{K}}}_0^2\bigsqcup {\tilde{\mathbb{K}}}_1^2 $$ by two bounded remainder sets $$ {\tilde{\mathbb{K}}}_0^2 $$ and $$ {\tilde{\mathbb{K}}}_1^2 $$ with respect to the shift $$ \tilde{\mathbb{S}} $$ . Two-sided bounds for the deviation functions of these sets are proved. Bibliography: 16 titles.

Sets of bounded discrepancy for multi-dimensional irrational rotation

We study bounded remainder sets with respect to an irrational rotation of the d-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one.

Multi-colour dynamical tilings of tori into bounded remainder sets

С помощью разбиений многомерных торов строятся множества ограниченного остатка, представляющие собою конечные объединения выпуклых многогранников. Для отклонений распределения точек орбит относительно сдвигов тора по указанным множествам доказывается многомерный вариант теоремы Гекке о распределении дробных долей на окружности. Библиография: 14 наименований.

Exchanged Toric Tilings, Rauzy Substitution, and Bounded Remainder Sets

Работа посвящена двумерной проблеме распределения дробных долей линейной функции. Введен новый класс разбиений двумерного тора на множества ограниченного остатка с эффективной оценкой остаточного члена. Показано, что примеры рассматриваемых разбиений могут быть получены при помощи геометрического варианта подстановки Рози. Библиография: 25 названий.

Universal sampling, quasicrystals and bounded remainder sets

We examine the result due to Matei and Meyer that simple quasicrystals are universal sampling sets, in the critical case when the density of the sampling set is equal to the measure of the spectrum. We show that in this case, an arithmetical condition on the quasicrystal determines whether it is a universal set of “stable and non-redundant” sampling.

Horn Clause Contraction Functions

In classical, AGM-style belief change, it is assumed that the underlying logic contains classical propositional logic. This is clearly a limiting assumption, particularly in Artificial Intelligence. Consequently there has been recent interest in studying belief change in approaches where the full expressivity of classical propositional logic is not obtained. In this paper we investigate belief contraction in Horn knowledge bases. We point out that the obvious extension to the Horn case, involving Horn remainder sets as a starting point, is problematic. Not only do Horn remainder sets have undesirable properties, but also some desirable Horn contraction functions are not captured by this approach. For Horn belief set contraction, we develop an account in terms of a model-theoretic characterisation involving weak remainder sets. Maxichoice and partial meet Horn contraction is specified, and we show that the problems arising with earlier work are resolved by these approaches. As well, constructions of the specific operators and sets of postulates are provided, and representation results are obtained. We also examine Horn package contraction, or contraction by a set of formulas. Again, we give a construction and postulate set, linking them via a representation result. Last, we investigate the closely-related notion of forgetting in Horn clauses. This work is arguably interesting since Horn clauses have found widespread use in AI; as well, the results given here may potentially be extended to other areas which make use of Horn-like reasoning, such as logic programming, rule-based systems, and description logics. Finally, since Horn reasoning is weaker than classical reasoning, this work sheds light on the foundations of belief change

A New Compression Method for Bayer Color Filter Array Images

In order to improve the compression performance of Bayer CFA images exposed continuously, a new high performance remainder set near-lossless compression method is presented. Based on channel-separated-filtering, several typical Bayer CFA image compression methods are compared with the proposed remainder set algorithm. It is proved that the remainder set algorithm has not only the better compression performance, i.e., the lower bits per pixel (average about 2.16bpp), but also the better reconstructed CFA image PSNR (average about 52.31dB). Finally, the proposed method is employed in a multiple channel CMOS image sampling system and some test results are given.

Moduli of toric tilings into bounded remainder sets and balanced words

Moduli of toric tilings into bounded remainder sets and balanced words

Exchanged toric developments and bounded remainder sets

Using exchanged toric developments, we construct tilings of toric by bounded remainder sets. To this end, two special methods for stretching the unit cubes and a general method for multiplying toric developments are used. A multidimensional analog of Hecke’s theorem on the distribution of fractional parts is proved. Bibliography: 7 titles.

A Remainder Set Near-Lossless Compression Method for Bayer Color Filter Array Images

Abstract In order to improve the compression performance of Bayer CFA images exposed continuously, a new high performance remainder set near-lossless compression method is presented. Based on channel-separated-filtering, several typical Bayer CFA image compression methods are compared with the proposed remainder set algorithm. It is proved that the remainder set algorithm has not only the better compression performance, i.e., the lower bits per pixel (average about 2.16bpp), but also the better reconstructed CFA image PSNR (average about 52.31 dB).

The Critique of Trauma and the Afterlife of the Novel in Tom McCarthy’s Remainder

In the last few years, Tom Mcarthy has launched a sustained campaign against the pieties of the traditional British novel. His debut novel Remainder sets out to debunk the conventions of trauma fiction, that, despite its emphasis on fragmentation, repetition, and temporal dislocation, often continues to rely on the conventions of psychological realism. In Remainder , trauma emerges not as a psychological event, but rather as an intractable, dysphoric, subjectless affect. Showing how trauma operates across the domains of the social, the somatic, and the psychological, the novel inaugurates a different account of the linkage of literature and trauma.