Shift Equivalence Research Articles

A complete invariant for shift equivalence for Boolean matrices and finite relations

We give a complete invariant for shift equivalence for Boolean matrices (equivalently finite relations), in terms of the period, the induced partial order on recurrent components, and the cohomology class of the relation on those components.

On strong shift equivalence for row-finite graphs and C*-algebras

On strong shift equivalence for row-finite graphs and C*-algebras

Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence

Recasting the Hazrat Conjecture: Relating Shift Equivalence to Graded Morita Equivalence

Splittings for $C^*$-correspondences and strong shift equivalence

We present an extension of the notion of in-splits from symbolic dynamics to topological graphs and, more generally, to $C^*$-correspondences. We demonstrate that in-splits provide examples of strong shift equivalences of $C^*$-correspondences. Furthermore, we provide a streamlined treatment of Muhly, Pask, and Tomforde's proof that any strong shift equivalence of regular $C^*$-correspondences induces a (gauge-equivariant) Morita equivalence between Cuntz-Pimsner algebras. For topological graphs, we prove that in-splits induce diagonal-preserving gauge-equivariant $*$-isomorphisms in analogy with the results for Cuntz-Krieger algebras. Additionally, we examine the notion of out-splits for $C^*$-correspondences.

The Szymczak Functor and Shift Equivalence on the Category of Finite Sets and Finite Relations

AbstractThe construction of the Conley index for dynamical systems with discrete time requires an equivalence relation between morphisms induced on index pairs. It follows from the features of the Szymczak functor that shift equivalence, whose equivalence classes are the isomorphism classes in the Szymczak category, is the most general equivalence available. In the case of dynamics modeled from data, the morphisms induced on index pairs are relations. We present an algorithmizable classification of shift equivalence classes for the category of finite sets with arbitrary relations as morphisms. The research is the first step towards the construction of a Conley theory for relations.

Investigating Shift Equivalence of Convolutional Neural Networks in Industrial Defect Segmentation

Investigating Shift Equivalence of Convolutional Neural Networks in Industrial Defect Segmentation

A Low-Complexity SLM PAPR Reduction Scheme for OFDMA

In orthogonal frequency division multiplexing (OFDM) systems, selected mapping (SLM) techniques are widely used to minimize the peak to average power ratio (PAPR). The candidate signals are generated in the time domain by linearly mixing the original time-domain transmitted signal with numerous cyclic shift equivalents to reduce the amount of Inverse Fast Fourier Transform (IFFT) operations in typical SLM systems. The weighting factors and number of cyclic shifts, on the other hand, should be carefully chosen to guarantee that the elements of the appropriate frequency domain phase rotation vectors are of equal magnitude. A low-complexity expression is chosen from among these options to create the proposed low-complexity scheme, which only requires one IFFT. In comparison to the existing SLM technique, the new SLM scheme achieves equivalent PAPR reduction performance with significantly less computing complexity. MATLAB tool is used for simulating the proposed work.

Flip signatures

AbstractA $D_{\infty }$ -topological Markov chain is a topological Markov chain provided with an action of the infinite dihedral group $D_{\infty }$ . It is defined by two zero-one square matrices A and J satisfying $AJ=JA^{\textsf {T}}$ and $J^2=I$ . A flip signature is obtained from symmetric bilinear forms with respect to J on the eventual kernel of A. We modify Williams’ decomposition theorem to prove the flip signature is a $D_{\infty }$ -conjugacy invariant. We introduce natural $D_{\infty }$ -actions on Ashley’s eight-by-eight and the full two-shift. The flip signatures show that Ashley’s eight-by-eight and the full two-shift equipped with the natural $D_{\infty }$ -actions are not $D_{\infty }$ -conjugate. We also discuss the notion of $D_{\infty }$ -shift equivalence and the Lind zeta function.

Some Results and Examples on The Relation of Leray Equivalent and (Strong) Shift Equivalent

Using the concept of Leray functor and the way it is used to define the Cohomological Conley index, we define Leray equivalent, Leray shift equivalent, Leray elementary strong shift equivalent and Leray strong shift equivalent. We established their relationship with shift equivalent, strong shift equivalent and elementary strong shift equivalent. Then, we show that in the setting of Artinian and Noetherian module Leray equivalent implies strong shift equivalent.

Balanced strong shift equivalence, balanced in-splits, and eventual conjugacy

AbstractWe introduce the notion of balanced strong shift equivalence between square non-negative integer matrices, and show that two finite graphs with no sinks are one-sided eventually conjugate if and only if their adjacency matrices are conjugate to balanced strong shift equivalent matrices. Moreover, we show that such graphs are eventually conjugate if and only if one can be reached by the other via a sequence of out-splits and balanced in-splits, the latter move being a variation of the classical in-split move introduced by Williams in his study of shifts of finite type. We also relate one-sided eventual conjugacies to certain block maps on the finite paths of the graphs. These characterizations emphasize that eventual conjugacy is the one-sided analog of two-sided conjugacy.

Design of Cyclic Shifted PAM-DMT Signals in HACO-OFDM Visible Light Communication

This letter proposed a novel design of the hybrid asymmetrically clipped optical orthogonal frequency division multiplexing (HACO-OFDM) system with cyclic shifted PAM-DMT signals called CSPAM-HACO-OFDM for visible light communication. The transmitted signal generated by the CSPAM-HACO-OFDM system combines the PAM-DMT signals with cyclic shift equivalents by using even subcarriers and the ACO-OFDM signals by using odd subcarriers. Peak-to-average power ratio (PAPR) reduction with remarkably low computational complexity is achieved, and side information is not required. A near-optimal side information detector, which has a similar detection error probability as the perfect-side information detector, is used at the receiver side. Simulation results demonstrate the effectiveness of the CSPAM-HACO-OFDM system in terms of PAPR reduction and average bit error rate.

Subshifts, λ-graph bisystems and C⁎-algebras

We introduce a notion of λ-graph bisystem that consists of a pair (L−,L+) of two labeled Bratteli diagrams L−,L+ satisfying certain compatibility condition for labeling their edges. It is a two-sided extension of λ-graph system, that has been previously introduced by the author. Its matrix presentation is called a symbolic matrix bisystem. We first show that any λ-graph bisystem presents subshifts and conversely any subshift is presented by a λ-graph bisystem, called the canonical λ-graph bisystem for the subshift. We introduce an algebraically defined relation on symbolic matrix bisystems called properly strong shift equivalence and show that two subshifts are topologically conjugate if and only if their canonical symbolic matrix bisystems are properly strong shift equivalent. A λ-graph bisystem (L−,L+) yields a pair of C⁎-algebras written OL−+,OL+− that are first defined as the C⁎-algebras of certain étale groupoids constructed from (L−,L+). We study structure of the C⁎-algebras, and show that they are universal unital unique C⁎-algebras subject to certain operator relations among canonical generators of partial isometries and projections encoded by the structure of the λ-graph bisystem (L−,L+). If a λ-graph bisystem comes from a λ-graph system of a finite directed graph, then the associated subshift is the two-sided topological Markov shift (ΛA,σA) by its transition matrix A of the graph, and the associated C⁎-algebra OL−+ is isomorphic to the Cuntz–Krieger algebra OA, whereas the other C⁎-algebra OL+− is isomorphic to the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z of the commutative C⁎-algebra C(ΛA) of continuous functions on the shift space ΛA of the two-sided topological Markov shift by the automorphism σA⁎ induced by the homeomorphism of the shift σA. This phenomenon shows a duality between Cuntz–Krieger algebra OA and the crossed product C⁎-algebra C(ΛA)⋊σA⁎Z.

Design of a Cyclic Shifted LACO-OFDM for Optical Wireless Communication

This paper proposes a cyclic shifted layer asymmetrically clipped optical orthogonal frequency division multiplexing system (CS-LACO-OFDM) for optical wireless communications. The transmitted signal in CS-LACO-OFDM is generated by combining the signal of the first layer and signals of the remaining $L$ -1 layers with cyclic shift equivalents, where $L$ is the number of the CS-LACO-OFDM layers. In particular, the sets of cyclic shift value in CS-LACO-OFDM are modulated on the odd subcarriers of the first layer with complex-valued symbols. Information on cyclic shift sets can be easily detected with the help of modulated symbols conveyed in the first layer without increasing the receiver complexity. Simulation results show that the CS-LACO-OFDM has a similar peak-to-average power ratio performance as LACO-OFDM with separate selective mapping scheme when the number of candidate signals is the same but at remarkably low computational complexity. Furthermore, the average bit error rate of CS-LACO-OFDM over all layers is minimized by the proposed optimal optical power allocation.

Shifts in Translation of English Noun Phrases into Indonesian with Reference to the Short Story A Scandal in Bohemia

Shifts in translation of English noun phrases into Indonesian with reference to a short story A Scandal in Bohemia is a study of how to transfer a message, information from one to another language to overcome its language barrier. Due to the systematical difference of every language, the occurrence of the translation shift is unavoidable in order to make the translation result natural. Translation shift and meaning equivalent are two core elements in translation that are correlated and inseparable, this study is conducted to find out the types of translation shifts and their implication towards the meaning equivalents found in the short story “A Scandal in Bohemia”. The data was taken from a short story entitled “A Scandal of Bohemia” using documentation and note-taking technique. In addition, the data was analyzed qualitatively and comparatively. It was classified based on the types of shifts and the analysis of its implication towards the meaning equivalent under the theory of Translation Shift proposed by Catford (1965) about translation shift and Theory of Meaning-Based Translation proposed by Larson (1998).This study revealed that the occurrence of the translation shift can be classified into four types of shift namely structure shift, class shift, unit shift and intra-system shift in which all of them belong to the category shift. None of the data shown level shift occurrence. This study also revelaed that as what Catford (1965) has stated in his theory that the most frequent shift is the structure shift, meanwhile the least one is the class shift. Multiple shifts are also occurred in some of the NPs, there are a few pairs namely unit and class shifts, class and intra-system shifts, unit and intra-system shifts and structure and intra-system shifts are occurred. Furthermore the implication of translation shift towards the meaning equivalent is also the core aspect in this study; it is shown that the occurrence of the shift is obligatory in order to make an equivalent and natural translation which the meaning that is carried from the SL to the TL is held constant and relevant to the feature of substance. 
 

Canonical mixed-polarity multi-target Toffoli circuits: Shift and removal

In this paper, we study reversible circuits as cascades of multi-target Toffoli gates. This type of gates allows to shift parts of a gate to the preceding gate within a circuit provided that a certain independence condition holds. It turns out that shifts decrease the so-called waiting degree such that shifting as long as possible always terminates and yields shift-reduced circuits. As the main result, we show that shift-reduced circuits are unique canonical representatives of their shift equivalence classes. Canonical circuits are optimal with respect to maximal and as-early-as-possible parallelism of targets within gates. Further, we discuss how successive equal subgates can be removed in order to reduce the waiting degree even more. Interestingly after applying shifts and removals as long as possible again a unique normal form is obtained.

Strong topological shift equivalence for dynamical systems

In this paper, we introduce some new equivalence relations for topological dynamical systems named strong topological shift equivalence and topological shift equivalence, which are similar to the strong shift equivalence and shift equivalence for subshifts of finite type. We study the relations between the new equivalences and other equivalences such as topological conjugacy, mutually topological semi-conjugacy and canonical homeomorphism extensions being topologically conjugate. Some properties and examples are shown. In particular, mean topological dimension is an invariant for topological shift equivalence but not for mutually topologically semi-conjugate equivalence. In this topic, linear operators are also considered.

The Conley index for discrete dynamical systems and the mapping torus

The classical Conley index for flows is defined as a certain homotopy type. In the case of a discrete dynamical system, one usually considers the shift equivalence class of the so-called index map. This equivalence relation is rarely used in other contexts and not well understood in general. Here we propose using a topological invariant of the shift equivalence definition: The homotopy type of the mapping torus of the index map. Using a homotopy type offers new ways for comparing Conley indices–theoretically and numerically. We present some basic properties and examples, compare it to the definition via shift equivalence and sketch an idea for its construction using rigorous numerics.

State splitting, strong shift equivalence and stable isomorphism of Cuntz–Krieger algebras

ABSTRACTWe prove that if two nonnegative matrices are strong shift equivalent, the associated stable Cuntz–Krieger algebras with generalized gauge actions are conjugate. The proof is done by a purely functional analytic method and based on constructing imprimitivity bimodule from bipartite directed graphs through strong shift equivalent matrices, so that we may clarify K-theoretic behaviour of the stable isomorphism between the associated stable Cuntz–Krieger algebras. We also examine our machinery for the matrices obtained by state splitting graphs, so that topological conjugacy of the topological Markov shifts is described in terms of some equivalence relation of the Cuntz–Krieger algebras with canonical masas and the gauge actions without stabilization.

Shift equivalence in the generalized factor order

We provide a geometric condition that guarantees strong Wilf equivalence in the generalized factor order. This provides a powerful tool for proving specific and general Wilf equivalence results, and several such examples are given.

Topological Conjugacy of Topological Markov Shifts and Cuntz–Krieger Algebras

For an irreducible non-permutation matrix A , the triplet (\mathcal{O}_A,\mathcal{D}_A,\rho^A) for the Cuntz–Krieger algebra \mathcal{O}_A , its canonical maximal abelian C^\ast -subalgebra \mathcal{D}_A , and its gauge action \rho^A is called the Cuntz–Krieger triplet. We introduce a notion of strong Morita equivalence in the Cuntz–Krieger triplets, and prove that two Cuntz–Krieger triplets (\mathcal{O}_A,\mathcal{D}_A,\rho^A) and (\mathcal{O}_B,\mathcal{D}_B,\rho^B) are strong Morita equivalent if and only if A and B are strong shift equivalent. We also show that the generalized gauge actions on the stabilized Cuntz–Krieger algebras are cocycle conjugate if the underlying matrices are strong shift equivalent. By clarifying K-theoretic behavior of the cocycle conjugacy, we investigate a relationship between cocycle conjugacy of the gauge actions on the stabilized Cuntz–Krieger algebras and topological conjugacy of the underlying topological Markov shifts.