Analysis of heat kernel highlights the strongly modular and heat-preserving structure of proteins

In this paper, we study the structure and dynamical properties of protein contact networks with respect to other biological networks, together with simulated archetypal models acting as probes. We consider both classical topological descriptors, such as the modularity and statistics of the shortest paths, and different interpretations in terms of diffusion provided by the discrete heat kernel, which is elaborated from the normalized graph Laplacians. A principal component analysis shows high discrimination among the network types, either by considering the topological and heat kernel based vector characterizations. Furthermore, a canonical correlation analysis demonstrates the strong agreement among those two characterizations, providing thus an important justification in terms of interpretability for the heat kernel. Finally, and most importantly, the focused analysis of the heat kernel provides a way to yield insights on the fact that proteins have to satisfy specific structural design constraints that the other considered networks do not need to obey. Notably, the heat trace decay of an ensemble of varying-size proteins denotes subdiffusion, a peculiar property of proteins

Lagrangian Description, Symplectic Structure, and Invariants of 3D Fluid Flow

Three dimensional unsteady flow of fluids in the Lagrangian description is considered as an autonomous dynamical system in four dimensions. The condition for the existence of a symplectic structure on the extended space is the frozen field equations of the Eulerian description of motion. Integral invariants of symplectic flow are related to conservation laws of the dynamical equation. A scheme generating infinite families of symmetries and invariants is presented. For the Euler equations these invariants are shown to have a geometric origin in the description of flow as geodesic motion; they are also interpreted in connection with the particle relabelling symmetry.Comment: Plain Latex, 15 page

Algorithmic recognition of infinite cyclic extensions

We prove that one cannot algorithmically decide whether a finitely presented $\mathbb{Z}$-extension admits a finitely generated base group, and we use this fact to prove the undecidability of the BNS invariant. Furthermore, we show the equivalence between the isomorphism problem within the subclass of unique $\mathbb{Z}$-extensions, and the semi-conjugacy problem for deranged outer automorphisms.Comment: 24 page

A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition

In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for the primary visual cortex of mammals. This model is neurophysiologically justified. Further developments of this theory lead to efficient algorithms for image reconstruction, based upon the consideration of an associated hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or certain of its improvements) is a left-invariant structure over the group $SE(2)$ of rototranslations of the plane. Here, we propose a semi-discrete version of this theory, leading to a left-invariant structure over the group $SE(2,N)$, restricting to a finite number of rotations. This apparently very simple group is in fact quite atypical: it is maximally almost periodic, which leads to much simpler harmonic analysis compared to $SE(2).$ Based upon this semi-discrete model, we improve on previous image-reconstruction algorithms and we develop a pattern-recognition theory that leads also to very efficient algorithms in practice.Comment: 123 pages, revised versio

An Algorithm to Simplify Tensor Expressions

The problem of simplifying tensor expressions is addressed in two parts. The first part presents an algorithm designed to put tensor expressions into a canonical form, taking into account the symmetries with respect to index permutations and the renaming of dummy indices. The tensor indices are split into classes and a natural place for them is defined. The canonical form is the closest configuration to the natural configuration. In the second part, the Groebner basis method is used to simplify tensor expressions which obey the linear identities that come from cyclic symmetries (or more general tensor identities, including non-linear identities). The algorithm is suitable for implementation in general purpose computer algebra systems. Some timings of an experimental implementation over the Riemann package are shown.Comment: 15 pages, Latex2e, submitted to Computer Physics Communications: Thematic Issue on "Computer Algebra in Physics Research