Constrained von Neumann Inequalities

An equivalent formulation of the von Neumann inequality states that the backward shift S* on ?₂ is extremal, in the sense that if T is a Hilbert space contraction, then ‖p(T)‖?‖p(S*)‖ for each polynomial p. We discuss several results of the following type: if T is a Hilbert space contraction satisfying some constraints, then S* restricted to a suitable invariant subspace is an extremal operator. Several operator radii are used instead of the operator norm. Applications to inequalities of coefficients of rational functions positive on the torus are given.     

The Birman-Schwinger principle in von Neumann algebras of finite type

We introduce a relative index for a pair of dissipative operators in a von Neumann algebra of finite type and prove an analog of the Birman-Schwinger principle in this setting. As an application of this result, revisiting the Birman-Krein formula in the abstract scattering theory, we represent the de la Harpe-Skandalis determinant of the characteristic function of dissipative operators in the algebra in terms of the relative index.     

Free orbit dimension of finite von Neumann algebras

We introduce a new free entropy invariant, which yields improvements of most of the applications of free entropy to finite von Neumann algebras, including those with Cartan subalgebras, simple masas, property T, property Γ, nonprime factors, and thin factors.     

Reversibility of linear cellular automata

In this paper the reversibility problem for linear cellular automata defined by a characteristic matrix of the form of a pentadiagonal matrix is tackled. Specifically, a criterion for the reversibility in terms of the number of cells is stated and, in these cases, the inverse cellular automata are explicitly computed.     

On the notion of relative property (T) for inclusions of von Neumann algebras

We prove that the notion of rigidity (or relative property (T)) for inclusions of finite von Neumann algebras recently defined by the second author is equivalent to a weaker property, in which no “continuity constants” are required. The proof is by contradiction and uses infinite products of completely positive maps, regarded as correspondences.     

A secret sharing scheme based on cellular automata

A new secret sharing scheme based on a particular type of discrete delay dynamical systems: memory cellular automata, is proposed. Specifically, such scheme consists of a (k, n)-threshold scheme where the text to be shared is considered as one of the k initial conditions of the memory cellular automata and the n shares to be distributed are n consecutive configurations of the evolution of such cellular automata. It is also proved to be perfect and ideal.     

A cellular automata model of chemotherapy effects on tumour growth: targeting cancer and immune cells

The effects of therapy on avascular cancer development based on a stochastic cellular automata model are considered. Making the model more compatible with the biology of cancer, the following features are implemented: intrinsic resistance of cancerous cells along with drug-induced resistance, drug-sensitive cells, immune system. Results are reported for no treatment, discontinued treatment after only one cycle of chemotherapy, and periodic drug administration therapy modes. Growth fraction, necrotic fraction, and tumour volume are used as output parameters beside a 2-D graphical growth presentation. Periodic drug administration is more effective to inhibit the growth of tumours. The model has been validated by the verification of the simulation results using in vivo literature data. Considering immune cells makes the model more compatible with the biological realities. Beside targeting cancer cells, the model can also simulate the activation of the immune system to fight against cancer.

Abbreviations CA: cellular automata; DSC: drug sensitive cell; DRC: drug resistant cell; GF: growth fraction; NF: necrotic fraction; ODE: ordinary differential equation; PDE: partial differential equation; SCAM: The proposed stochastic cellular automata model     

Modelling nonlinear sequence generators in terms of linear cellular automata

In this work, a wide family of LFSR-based sequence generators, the so-called clock-controlled shrinking generators (CCSGs), has been analyzed and identified with a subset of linear cellular automata (CA). In fact, a pair of linear models describing the behavior of the CCSGs can be derived. The algorithm that converts a given CCSG into a CA-based linear model is very simple and can be applied to CCSGs in a range of practical interest. The linearity of these cellular models can be advantageously used in two different ways: (a) for the analysis and/or cryptanalysis of the CCSGs and (b) for the reconstruction of the output sequence obtained from this kind of generators.     

The dynamics of expansive invertible onesided cellular automata

Using textile systems, we prove the conjecture of Boyle and Maass that the dynamical system defined by an expansive invertible onesided cellular automaton is topologically conjugate to a topological Markov shift. We also study expansive leftmost-permutive onesided cellular automata and bipermutive endomorphisms of mixing topological Markov shifts.

     

Density of periodic points, invariant measures and almost equicontinuous points of cellular automata

Revisiting the notion of μ-almost equicontinuous cellular automata introduced by R. Gilman, we show that the sequence of image measures of a shift ergodic measure μ by iterations of such automata converges in Cesàro mean to an invariant measure μ_c. Moreover the dynamical system (cellular automaton F, invariant measure μ_c) has still the μ_c-almost equicontinuity property and the set of periodic points is dense in the topological support of the measure μ_c. We also show that the density of periodic points in the topological support of a measure μ occurs for each μ-almost equicontinuous cellular automaton when μ is an invariant and shift ergodic measure. Finally using most of these results we give a non-trivial example of a couple (μ-equicontinuous cellular automaton F, shift and F-invariant measure μ) such that the restriction of F to the topological support of μ has no equicontinuous points.     

Synthesis of cryptographic interleaved sequences by means of linear cellular automata

This work shows that a class of pseudorandom binary sequences, the so-called interleaved sequences, can be generated by means of linear multiplicative polynomial cellular automata. In fact, these linear automata generate all the solutions of a type of linear difference equations with binary coefficients. Interleaved sequences are just particular solutions of such equations. In this way, popular nonlinear sequence generators with cryptographic application can be linearized in terms of simple cellular automata.     

On maximal injective subalgebras of tensor products of von Neumann algebras

Let Mi be a von Neumann algebra, and Bi be a maximal injective von Neumann subalgebra of Mi, i=1,2. If M₁ has separable predual and the center of B₁ is atomic, e.g., B₁ is a factor, then is a maximal injective von Neumann subalgebra of . This partly answers a question of Popa.     

Emergence of density dynamics by surface interpolation in elementary cellular automata

A classic problem in elementary cellular automata (ECAs) is the specification of numerical tools to represent and study their dynamical behaviour. Mean field theory and basins of attraction have been commonly used; however, although the first case gives the long term estimation of density, frequently it does not show an adequate approximation for the step-by-step temporal behaviour; mainly for non-trivial behaviour. In the second case, basins of attraction display a complete representation of the evolution of an ECA, but they are limited up to configurations of 32 cells; and for the same ECA, one can obtain tens of basins to analyse. This paper is devoted to represent the dynamics of density in ECAs for hundreds of cells using only two surfaces calculated by the nearest-neighbour interpolation. A diversity of surfaces emerges in this analysis. Consequently, we propose a surface and histogram based classification for periodic, chaotic and complex ECA.     

Multiplication operators defined by covering maps on the Bergman space: The connection between operator theory and von Neumann algebras

In this paper, we combine methods of complex analysis, operator theory and conformal geometry to construct a class of Type II factors in the theory of von Neumann algebras, which arise essentially from holomorphic coverings of bounded planar domains. One will see how types of such von Neumann algebras are related to algebraic topology of planar domains. As a result, the paper establishes a fascinating connections to one of the long-standing problems in free group factors. An interplay of analytical, geometrical, operator and group theoretical techniques is intrinsic to the paper.     

Persistence of sums of correlated increments and clustering in cellular automata

Let $T$ be the first return time to $(?\infty ,0]$ of sums of increments given by a functional of a stationary Markov chain. We determine the asymptotic behavior of the survival probability, $\mathbb{P}(T\ge t)～C{t}^{?1∕2}$ for an explicit constant $C$. Our analysis is based on a connection between the survival probability and the running maximum of the time-reversed process, and relies on a functional central limit theorem for Markov chains. As applications, we recover known clustering results for the 3-color cyclic cellular automaton and the Greenberg–Hastings model, and we prove a new clustering result for the 3-color firefly cellular automaton.     

A note on cocycle-conjugate endomorphisms of von Neumann algebras

We show that two cocycle-conjugate endomorphisms of an arbitrary von Neumann algebra that satisfy certain stability conditions are conjugate endomorphisms, when restricted to some specific von Neumann subalgebras. As a consequence of this result, we obtain a new criterion for conjugacy of Powers shift endomorphisms acting on factors of type

     

On Von Neumann Regular Rings of Skew Generalized Power Series

In this paper we introduce a construction called the skew generalized power series ring R[[S, ω]] with coefficients in a ring R and exponents in a strictly ordered monoid S which extends Ribenboim's construction of generalized power series rings. In the case when S is totally ordered or commutative aperiodic, and ω(s) is constant on idempotents for some s ∈ S?{1}, we give sufficient and necessary conditions on R and S such that the ring R[[S, ω]] is von Neumann regular, and we show that the von Neumann regularity of the ring R[[S, ω]] is equivalent to its semisimplicity. We also give a characterization of the strong regularity of the ring R[[S, ω]].