On independent permutation separability criteria

2006 ◽  
Vol 6 (3) ◽  
pp. 277-288
Author(s):  
L. Clarisse ◽  
P. Wocjan
Keyword(s):  
Complete Classification ◽  
Necessary Condition ◽  
Numerical Verification ◽  
Separability Criteria

Recently, P.\ Wocjan and M.\ Horodecki [Open Syst.\ Inf.\ Dyn.\ 12, 331 (2005)] gave a characterization of combinatorially independent permutation separability criteria. Combinatorial independence is a necessary condition for permutations to yield truly independent criteria meaning that no criterion is strictly stronger that any other. In this paper we observe that some of these criteria are still dependent and analyze why these dependencies occur. To remove them we introduce an improved necessary condition and give a complete classification of the remaining permutations. We conjecture that the remaining class of criteria only contains truly independent permutation separability criteria. Our conjecture is based on the proof that for two, three and four parties all these criteria are truly independent and on numerical verification of their independence for up to 8 parties. It was commonly believed that for three parties there were 9 independent criteria, here we prove that there are exactly 6 independent criteria for three parties and 22 for four parties.

scholarly journals CONVEXLY ORDERABLE GROUPS AND VALUED FIELDS

2014 ◽  
Vol 79 (01) ◽  
pp. 154-170
Author(s):  
JOSEPH FLENNER ◽  
VINCENT GUINGONA
Keyword(s):  
Abelian Groups ◽  
Complete Classification ◽  
Necessary Condition ◽  
Valued Fields ◽  
Ordered Groups ◽  
Algebraic Theories ◽  
Convexly Orderable

Abstract We consider the model theoretic notion of convex orderability, which fits strictly between the notions of VC-minimality and dp-minimality. In some classes of algebraic theories, however, we show that convex orderability and VC-minimality are equivalent, and use this to give a complete classification of VC-minimal theories of ordered groups and abelian groups. Consequences for fields are also considered, including a necessary condition for a theory of valued fields to be quasi-VC-minimal. For example, the p-adics are not quasi-VC-minimal.

Periods for Continuous Self-Maps of the Figure-Eight Space

2003 ◽  
Vol 13 (07) ◽  
pp. 1743-1754 ◽  
Author(s):  
Jaume Llibre ◽  
José Paraños ◽  
J. Ángel Rodríguez
Keyword(s):  
Euler Characteristic ◽  
Connected Graph ◽  
Complete Classification ◽  
Branching Points ◽  
Branching Point

Let 8 be the graph shaped like the number 8. This paper contains a characterization of all possible sets of periods for all continuous self-maps of 8 with the branching point fixed. We remark that this characterization is the first complete classification of the sets of periods for all continuous self-maps on a connected graph with negative Euler characteristic with fixed branching points.

scholarly journals Characterization of Combinatorially Independent Permutation Separability Criteria

2005 ◽  
Vol 12 (04) ◽  
pp. 331-345 ◽  
Author(s):  
Paweł Wocjan ◽  
Michał Horodecki
Keyword(s):  
Density Matrix ◽  
The State ◽  
Necessary Condition ◽  
Trace Norm ◽  
Mixed States ◽  
Graphical Notation ◽  
Operational Conditions ◽  
Separability Criteria ◽  
Partial Transpose

The so-called permutation separability criteria are simple operational conditions that are necessary for separability of mixed states of multipartite systems: (1) permute the indices of the density matrix and (2) check if the trace norm of at least one of the resulting operators is greater than one. If it is greater than one then the state is necessarily entangled. A shortcoming of the permutation separability criteria is that many permutations give rise to equivalent separability criteria. Therefore, we introduce a necessary condition for two permutations to yield independent criteria called combinatorial independence. This condition basically means that the map corresponding to one permutation cannot be obtained by concatenating the map corresponding to the second permutation with a norm-preserving map. We characterize completely combinatorially independent criteria, and determine simple permutations that represent all independent criteria. The representatives can be visualized by means of a simple graphical notation. They are composed of three basic operations: partial transpose, and two types of so-called reshufflings. In particular, for a four-partite system all criteria except one are composed of partial transpose and only one type of reshuffling; the exceptional one requires the second type of reshuffling. Furthermore, we show how to obtain efficiently a simple representative for every permutation. This method allows to check easily if two permutations are combinatorially equivalent or not.

Unstable K1-group and homotopy type of certain gauge groups

2006 ◽  
Vol 136 (1) ◽  
pp. 149-155 ◽  
Author(s):  
Hiroaki Hamanaka ◽  
Akira Kono
Keyword(s):  
Gauge Group ◽  
Homotopy Type ◽  
Complete Classification ◽  
Necessary Condition ◽  
Gauge Groups ◽  
Homotopy Types

We denote the group of homotopy set [X, U(n)] by the unstable K1-group of X. In this paper, using the unstable K1-group of the multi-suspended CP2, we give a necessary condition for two principal SU(n)-bundles over §4 to have the associated gauge group of the same homotopy type, which is an improvement of the result of Sutherland and, particularly, show the complete classification of homotopy types of SU(3)-gauge groups over S4.

Inequalities in Discrete Subgroups of PSL(2, R)

1988 ◽  
Vol 40 (1) ◽  
pp. 115-130 ◽  
Author(s):  
Jane Gilman
Keyword(s):  
Discrete Group ◽  
Sufficient Conditions ◽  
Complete Classification ◽  
Necessary Condition ◽  
Discrete Subgroups ◽  
Elementary Subgroup ◽  
Van Vleck ◽  
Necessary And Sufficient ◽  
Number Of Authors

Conditions for a subgroup, F, of PSL(2, R) to be discrete have been investigated by a number of authors. Jørgensen's inequality [5] gives an elegant necessary condition for discreteness for subgroups of PSL(2, C). Purzitsky, Rosenberger, Matelski, Knapp, and Van Vleck, among others [12, 13, 14, 9, 16, 17, 18, 19, 20, 7, 21] studied two generator discrete subgroups of PSL(2, R) in a long series of papers. The complete classification of two generator subgroups was surprisingly complicated and elusive. The most complete result appears in [20].In this paper we use the results of [20] to prove that a nonelementary subgroup F of PSL(2, R) is discrete if and only if every non-elementary subgroup, G, generated by two hyperbolics is discrete (Theorem 5.2) and that F contains no elliptics if and only if each such G is free (Theorem 5.1). Thus, we produce necessary and sufficient conditions for a non-elementary subgroup F of PSL(2, R) to be a discrete group without elliptic elements (Theorem 6.1) or a discrete group containing only hyperbolic elements (Theorem 7.1).

scholarly journals Phase retrieval in frame theory

2020 ◽  
Author(s):  
◽  
Dorsa Ghoreishi
Keyword(s):  
Phase Retrieval ◽  
Complete Classification ◽  
Frame Theory ◽  
Weak Phase ◽  
The Relationship

This dissertation is the study of phase retrieval in frame theory. The first part is concerned with the analysis of phase retrieval and the complete classification of norm retrieval. Norm retrieval is essential to transfer the properties of phase retrieval to the complement space. The first section includes the results regarding projections and also the characterization of phase retrieval and norm retrieval for subspaces. The second part is the study of weak phase retrieval which was motivated by the idea of reducing the number of vectors satisfying the properties close to phase retrieval. The last section provides the correlation between weak phase retrieval and phase retrieval properties along with the examples illustrating the relationship between weak phase retrieval and the related concepts.

scholarly journals A criterion for discrete branching laws for Klein four symmetric pairs and its application to E6(−14)

2020 ◽  
Vol 31 (06) ◽  
pp. 2050049
Author(s):  
Haian He
Keyword(s):  
Lie Group ◽  
Simple Formula ◽  
Complete Classification ◽  
Necessary Condition ◽  
Symmetric Pair ◽  
Branching Laws ◽  
Nonidentity Element

Let [Formula: see text] be a noncompact connected simple Lie group, and [Formula: see text] a Klein four-symmetric pair. In this paper, we show a necessary condition for the discrete decomposability of unitarizable simple [Formula: see text]-modules for Klein for symmetric pairs. Precisely, if certain conditions hold for [Formula: see text], there does not exist a unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module. As an application, for [Formula: see text], we obtain a complete classification of Klein four symmetric pairs [Formula: see text], with [Formula: see text] noncompact, such that there exists at least one nontrivial unitarizable simple [Formula: see text]-module that is discretely decomposable as a [Formula: see text]-module and is also discretely decomposable as a [Formula: see text]-module for some nonidentity element [Formula: see text].

scholarly journals CONJUGACY IN ARTIN GROUPS AND APPLICATION TO THE CLASSIFICATION OF SURFACES

2006 ◽  
Vol 05 (05) ◽  
pp. 563-570
Author(s):  
GODELLE EDDY ◽  
KAPLAN SHMUEL ◽  
TEICHER MINA
Keyword(s):  
Algebraic Geometry ◽  
Word Problem ◽  
Braid Group ◽  
Necessary Condition ◽  
Fast Method ◽  
Artin Group ◽  
Spherical Type ◽  
Potential Applications

We show that the double reversing algorithm proposed by Dehornoy in [3] for solving the word problem in the braid group can also be used to recognize the conjugates of powers of the generators in an Artin group of spherical type. The proof uses a characterization of these powers in terms of their fractional decomposition. This algorithm could have potential applications to braid-based cryptography; it also provides a fast method for testing a necessary condition in the classification of surfaces in algebraic geometry.

scholarly journals Geometric Classification of Graph C*-algebras over Finite Graphs

2018 ◽  
Vol 70 (2) ◽  
pp. 294-353 ◽  
Author(s):  
Søren Eilers ◽  
Gunnar Restorff ◽  
Efren Ruiz ◽  
Adam P.W. Sørensen
Keyword(s):  
Standard Condition ◽  
Classification Problem ◽  
Complete Classification ◽  
Type I ◽  
Simple Graphs ◽  
Finite Graphs ◽  
C Algebras ◽  
Special Case

AbstractWe address the classification problem for graph C*-algebras of finite graphs (finitely many edges and vertices), containing the class of Cuntz-Krieger algebras as a prominent special case. Contrasting earlier work, we do not assume that the graphs satisfy the standard condition (K), so that the graph C*-algebras may come with uncountably many ideals.We find that in this generality, stable isomorphism of graph C*-algebras does not coincide with the geometric notion of Cuntz move equivalence. However, adding a modest condition on the graphs, the two notions are proved to be mutually equivalent and equivalent to the C*-algebras having isomorphicK-theories. This proves in turn that under this condition, the graph C*-algebras are in fact classifiable byK-theory, providing, in particular, complete classification when the C* - algebras in question are either of real rank zero or type I/postliminal. The key ingredient in obtaining these results is a characterization of Cuntz move equivalence using the adjacency matrices of the graphs.Our results are applied to discuss the classification problem for the quantumlens spaces defined by Hong and Szymański, and to complete the classification of graph C*-algebras associated with all simple graphs with four vertices or less.

scholarly journals Multiplicative automatic sequences

2021 ◽  
Author(s):  
Jakub Konieczny ◽  
Mariusz Lemańczyk ◽  
Clemens Müllner
Keyword(s):  
Complete Classification ◽  
Automatic Sequences ◽  
Complex Valued

AbstractWe obtain a complete classification of complex-valued sequences which are both multiplicative and automatic.

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