Combinatorics of Nonnegative Matrices

Sachkov, Vladimir Nikolaevich; Tarakanov, V. E.

doi:10.1090/mmono/213

Cited by 42 publications

(30 citation statements)

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“…Scattering amplitudes for bosons instead involve the permanent [42,72,73], a complex-valued function of matrices similar to the determinant, but that does not feature equivalent symmetry properties. The number of effective parameters for bosons is therefore larger than for fermions.…”

Section: A Fermionic and Bosonic Boundsmentioning

confidence: 99%

Limits to multipartite entanglement generation with bosons and fermions

2013

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Many-photon interference in linear-optics setups can be exploited to generate and detect multipartite entanglement. Without recurring to any inter-particle interaction, many entangled states have been created experimentally, and a panoply of theoretical schemes for the generation of various classes of entangled states is available. Here, we present a unifying framework that accommodates the present experiments and theoretical protocols for the creation of multiparticle entanglement via interference. A general representation of the states that can be created is provided for bosons and fermions, for any particle number, and for any dimensionality of the entangled degree of freedom. Using this framework, we derive an upper bound on the generalized Schmidt number of the states that can be generated, and we establish bounds on the dimensionality of the manifold of these states. We show that -at the expense of a smaller success probability -more states can be created with bosons than with fermions, and give an intuitive interpretation of the state representation and of the established bounds in terms of superimposed many-particle paths.

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Section: A Fermionic and Bosonic Boundsmentioning

confidence: 99%

Limits to multipartite entanglement generation with bosons and fermions

2013

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“….} has been studied extensively; see, for example, Brualdi and Ryser (1991), Sachkov and Tarakanov (2002). Next, we use the Boolean operators AND and OR to examine the sequence of powers of A and infer results about the conjunctive Boolean network that corresponds to the adjacency matrix A.…”

Section: The Adjacency Matrixmentioning

confidence: 99%

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

2010

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For many biological networks, the topology of the network constrains its dynamics. In particular, feedback loops play a crucial role. The results in this paper quantify the constraints that (unsigned) feedback loops exert on the dynamics of a class of discrete models for gene regulatory networks. Conjunctive (resp. disjunctive) Boolean networks, obtained by using only the AND (resp. OR) operator, comprise a subclass of networks that consist of canalyzing functions, used to describe many published gene regulation mechanisms. For the study of feedback loops, it is common to decompose the wiring diagram into linked components each of which is strongly connected. It is shown that for conjunctive Boolean networks with strongly connected wiring diagram, the feedback loop structure completely determines the long-term dynamics of the network. A formula is established for the precise number of limit cycles of a given length, and it is determined which limit cycle lengths can appear. For general wiring diagrams, the situation is much more complicated, as feedback loops in one strongly connected component can influence the feedback loops in other components. This paper provides a sharp lower bound and an upper bound on the number of limit cycles of a given length, in terms of properties of the partially ordered set of strongly connected components.

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“…It is obvious, that Q n  B n . About the necessary definitions and denotes in the theory of matrices we refer to [7] and [10]. It is not difficult to see, that (1) If A = (a i j )  B n , then A = (a j i ) , 1  i, j  n denotes the transposed matrix A .…”

Section: Introductionmentioning

confidence: 99%

Mathematical Modeling in the Textile Industry

Yordzhev

Kostadinova

2012

BMSA

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A mathematical model, describing some different weaving structures, is made in this article. The terms self-mirror and rotation-stable weaving structure are initiated here. There are used the properties and operations in the set of the binary matrices and an equivalence relation in this set. Some combinatorial problems about finding the cardinal number and the elements of the factor set according to this relation is discussed. We propose an algorithm, which solves these problems. The presentation of an arbitrary binary matrix using sequence of nonnegative integers is discussed. It is shown that the presentation of binary matrices using ordered n-tuples of natural numbers makes the algorithms faster and saves a lot of memory. Implementing these ideas a computer program, which receives all of the examined objects, is created. In the paper we use object-oriented programming using the syntax and the semantic of C++ programming language. Some advantages in the use of bitwise operations are shown. The results we have received are used to describe the topology of the different weaving structures.

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Combinatorics of Nonnegative Matrices

Cited by 42 publications

References 0 publications

Limits to multipartite entanglement generation with bosons and fermions

Limits to multipartite entanglement generation with bosons and fermions

The Dynamics of Conjunctive and Disjunctive Boolean Network Models

Mathematical Modeling in the Textile Industry

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