Computing Integrated Information (Φ) in Discrete Dynamical Systems with Multi-Valued Elements

Integrated information theory (IIT) provides a mathematical framework to characterize the cause-effect structure of a physical system and its amount of integrated information (Φ). An accompanying Python software package (“PyPhi”) was recently introduced to implement this framework for the causal analysis of discrete dynamical systems of binary elements. Here, we present an update to PyPhi that extends its applicability to systems constituted of discrete, but multi-valued elements. This allows us to analyze and compare general causal properties of random networks made up of binary, ternary, quaternary, and mixed nodes. Moreover, we apply the developed tools for causal analysis to a simple non-binary regulatory network model (p53-Mdm2) and discuss commonly used binarization methods in light of their capacity to preserve the causal structure of the original system with multi-valued elements.     

Formal proofs of operator identities by a single formal computation

A formal computation proving a new operator identity from known ones is, in principle, restricted by domains and codomains of linear operators involved, since not any two operators can be added or composed. Algebraically, identities can be modelled by noncommutative polynomials and such a formal computation proves that the polynomial corresponding to the new identity lies in the ideal generated by the polynomials corresponding to the known identities. In order to prove an operator identity, however, just proving membership of the polynomial in the ideal is not enough, since the ring of noncommutative polynomials ignores domains and codomains. We show that it suffices to additionally verify compatibility of this polynomial and of the generators of the ideal with the labelled quiver that encodes which polynomials can be realized as linear operators. Then, for every consistent representation of such a quiver in a linear category, there exists a computation in the category that proves the corresponding instance of the identity. Moreover, by assigning the same label to several edges of the quiver, the algebraic framework developed allows to model different versions of an operator by the same indeterminate in the noncommutative polynomials.     

Upper Bounds for the Expected Maxima of Independent Random Variables Given Known First Four Moments

Mathematical Notes - The paper is devoted to the study of conditional bounds for the expectation of the maximum of independent identically distributed standardized random variables for which the...     

Approximation Properties of the Vallée-Poussin Means of Partial Sums of a Special Series in Laguerre Polynomials

Mathematical Notes - We consider the problem of the approximation of functions, continuous on the semiaxis $$[0,\infty)$$ and for which the derivatives $$f^{(\nu)}(0)$$ , $$\nu=0,\dots,r-1$$ exist...     

On a Ramanujan Identity and Its Generalizations

Mathematical Notes - In the present paper, we propose a new method of derivation of number-theoretic identities which is applied to the proof of the multidimensional analogue of one of the...     

The Existence of Zeros of Multivalued Functionals,Coincidence Points,and Fixed Points in $$f$$ -Quasimetric Spaces

Mathematical Notes - The notion of a $$\lambda$$ -generalized-search multivalued functional on an $$f$$ -quasimetric space is introduced. An existence theorem for zeros of such functionals is...     

Comparison of Arithmetic,Geometric, and Harmonic Means

Mathematical Notes - The main purpose of the paper is to strengthen the results of P. R. Mercer (2003) concerning the comparison of arithmetic, geometric, and harmonic weighted means.     

Correction to: Impact of leg bending in the patient-specific computational fluid dynamics of popliteal stenting

Acta Mechanica Sinica - A Correction to this paper has been published: https://doi.org/10.1007/s10409-021-01066-2     

Sequences of Independent Functions in Rearrangement Invariant Spaces

Siberian Mathematical Journal - We obtain some new estimates that show the extremality of the Rademacher system in the set of sequences of independent functions considered in rearrangement...     

Explicit breather solution of the nonlinear Schrödinger equation

Theoretical and Mathematical Physics - We present a one-line closed-form expression for the three-parameter breather of the nonlinear Schrödinger equation. This provides an analytic proof of...