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We show an efficient algorithm for generating geodesic regular tree structures for periodic hyperbolic and Euclidean tessellations and experimentally verify its performance on tessellations.
Let $G$ be a group and let $X$ be an algebraic variety over an algebraically closed field $k$ of characteristic zero. Denote $A=X(k)$ the set of rational points of $X$. We investigate invertible algebraic cellular automata $τ\colon A^G \to A^G$ whose local defining map is induced by some morphism of algebraic varieties $X^M \to X$ where $M$ is a finite memory. When $G$ is locally embeddable into finite groups (LEF), then we show that the inverses of reversible algebraic cellular automata are automatically algebraic cellular automata. Generalizations are also obtained for finite product Hopfian pointed object alphabets in concrete categories.
In this paper we analyze the curvature corrections to the surface tension in the context of the Shan-Chen (SC) multi-component Lattice Boltzmann method (LBM). We demonstrate that the same techniques recently applied in the context of the Shan-Chen multi-phase model can be applied to multi-component mixtures. We implement, as a new application, the calculation of the surface of tension radius $R_s$ through the minimization of the generalized surface tension $σ[R]$. In turn we are able to estimate the Tolman length, i.e. the first order coefficient of the curvature expansion of the surface tension $σ(R)$, as well as the higher order corrections, i.e. the curvature- and the Gaussian-rigidity coefficients. The SC multi-component model allows to model both fully-symmetric as well as asymmetric interactions among the components. By performing an extensive set of simulations we present a first example of tunable Tolman length in the mesoscopic model, being zero for symmetric interactions and different from zero otherwise. This result paves the way for controlling such interface properties which are paramount in presence of thermal fluctuations. All reported results can be independently reproduced through the "idea.deploy" framework available at https://github.com/lullimat/idea.deploy.
We present a ridership model of individual trajectories of users within a public transport network for which there are several different routes between origin and destiny and for which the automatic fare collection data does not include information about the exit station; only card identification and time of entry into the system. This model is implemented in the case of the Troncal component of TransMilenio, the BRT system that is the backbone of public transport in Bogotá. The granularity of the data allows to identify the occupation of every bus within the system at any particular day. As a validation, we compare the average bus occupation of two particular days, 17 of May of 2020 and the 1 of December of 2020.
We show that the image of a subshift $X$ under various injective morphisms of symbolic algebraic varieties over monoid universes with algebraic variety alphabets is a subshift of finite type, resp. a sofic subshift, if and only if so is $X$. Similarly, let $G$ be a countable monoid and let $A$, $B$ be Artinian modules over a ring. We prove that for every closed subshift submodule $Σ\subset A^G$ and every injective $G$-equivariant uniformly continuous module homomorphism $τ\colon Σ\to B^G$, a subshift $Δ\subset Σ$ is of finite type, resp. sofic, if and only if so is the image $τ(Δ)$. Generalizations for admissible group cellular automata over admissible Artinian group structure alphabets are also obtained.
We discuss initial value problems for time evolution equations in one dimensional space which are expressed by the lattice operators and propose some new equations to which complexity of solutions is of polynomial class. Novel type of expressions using shift operators and binary trees are applied for the derivation of solution.
We discuss a particle system of 5 neighbors with two independent conserved densities. The mean momentum uniquely depends on a pair of the densities and a three-dimensional fundamental diagram is obtained. It shows the phase transition of behavior of asymptotic solution to the system. Moreover, we propose two other systems which have the similar unique dependency obtained numerically.
Magnetohydrodynamic (MHD) flow of liquid metals through conduits play an important role in the proposed systems for harnessing fusion energy, and various other engineering and scientific problems. The interplay between the magnetic fields and the fluid motion gives rise to complex flow physics, which depend on the electrical conductivity of the bounding walls. An effective approach to represent the latter is via the Shercliff boundary condition for thin conducting walls relating the induced magnetic field and its wall normal gradient at the boundary via a parameter referred to as the wall conductance ratio (Shercliff, JA, J. Fluid Mech. 1, 644 (1956)). Within the framework of the highly parallelizable lattice Boltzmann (LB) method, a lattice kinetic scheme for MHD involving a vector distribution function for the magnetic fields was proposed by Dellar (Dellar, PJ, J. Comp. Phys. 179, 95 (2002)). However, the prior LB algorithms only accounted for limiting special cases involving perfectly insulated boundaries and did not consider the finite conductivity effects. In this paper, we present two new boundary schemes that enforce the Shercliff boundary condition in the LB schemes for MHD. It allows for the specification of the wall conductance ratio to any desired value based on the actual conductivities and length scales of the container and the wall thicknesses. One approach is constructed using a link-based formulation involving a weighted combination of the bounce back and anti-bounce back of the distribution function for the magnetic field and the other approach involves an on-node moment-based implementation. Moreover, their extensions to representing moving walls are also presented. Numerical validations of the boundary schemes for body force or shear driven MHD flows for a wide range of the values of the wall conductance ratio and their second order grid convergence are demonstrated.
This paper presents a probabilistic extension of the well-known cellular automaton, Game of Life. In Game of Life, cells are placed in a grid and then watched as they evolve throughout subsequent generations, as dictated by the rules of the game. In our extension, called ProbLife, these rules now have probabilities associated with them. Instead of cells being either dead or alive, they are denoted by their chance to live. After presenting the rules of ProbLife and its underlying characteristics, we show a concrete implementation in ProbLog, a probabilistic logic programming system. We use this to generate different images, as a form of rule-based generative art.
We analyze a stochastic particle system of 5 neighbors. Considering eigenvalue problem of transition matrix, we propose a conjecture that asymptotic distribution of the system is determined by the number of specific local patterns in the asymptotic solution. Based on the conjecture, mean flux which depends of a pair of the conserved quantities is derived theoretically. Moreover, we obtain mean flux in the deterministic case through the limit of stochastic parameter.
Motivated by recent experimental studies in microbiology, we suggest a modification of the classic ballistic deposition model of surface growth, where the memory of a deposition at a site induces more depositions at that site or its neighbors. By studying the statistics of surfaces in this model, we obtain three independent critical exponents: the growth exponent $β=5/4$, the roughening exponent $α= 2$, and the new (size) exponent $γ= 1/2$. The model requires a modification to the Family-Vicsek scaling, resulting in the dynamical exponent $z = \frac{α+γ}β = 2$. This modified scaling collapses the surface width vs time curves for various lattice sizes. This is a previously unobserved universality class of surface growth that could describe surface properties of a wide range of natural systems.
From the perfect radial symmetries of radiolarian mineral skeletons to the broken symmetry of homochirality, the logic of Nature's regularities has fascinated scientists for centuries. Some of Nature's symmetries are clearly visible in morphology and physical structure, whereas others are hidden in the network of interactions among system components. Just as visible symmetries and asymmetries contribute to the system's beauty, might hidden symmetries contribute to the system's functional harmony? And if so, how? Here we demonstrate that the interaction networks of biological systems from cell signaling to cancer display a form of dynamical reflection symmetry that serves to expand their dynamical complexity. The expansion proceeds according to precise rules regarding the lengths of dynamical cycles, made possible by a peculiar imbalance between excitation and inhibition. To probe the conditions under which this reflection symmetry evolves, we use a multi-objective genetic algorithm to produce networks with long dynamical cycles. We find that local structural motifs that break the reflection symmetry are deleted in the evolutionary process, providing evidence for symmetry's causal role in dynamical complexity. Finally, we identify symmetries in real continuous-valued scRNA-seq data of gene expression in drug-resistant cancer cells. Broadly, our work reveals a class of hidden symmetries in biology, present in the network of interactions among system components, and it underscores their functional importance. Mathematically simple and computational inexpensive, our approach is applicable to the types of biological data commonly acquired in modern experiments.
How do cellular automata behave in the limit of a very large number of cells? Is there a continuum limit with simple properties? We attack this problem by mapping certain classes of automata to quantum field theories for which powerful methods exist for this type of problem. Indeed, many cellular automata admit an interpretation in terms of fermionic particles. Reversible automata on space-lattices with a local updating rule can be described by a partition function or Grassmann functional integral for interacting fermions moving in this space. We discuss large classes of automata that are equivalent to discretized fermionic quantum field theories with various types of interactions. Two-dimensional models include relativistic Thirring or Gross-Neveu type models with abelian or non-abelian continuous global symmetries, models with local gauge symmetries, and spinor gravity with local Lorentz symmetry as well as diffeomorphism invariance in the (naive) continuum limit. The limit of a very large number of cells needs a probabilistic description. Probabilistic cellular automata are characterized by a probability distribution over initial bit-configurations. They can be described by the quantum formalism with wave functions, density matrix and non-commuting operators associated to observables, which are the same for the automata and associated fermionic quantum theories. This formalism is crucial for a discussion of concepts as vacuum states, spontaneous symmetry breaking, coarse graining and the continuum limit for probabilistic cellular automata. In particular, we perform explicitly the continuum limit for an automaton that describes a quantum particle in a potential for one space dimension.
We describe a cellular-automaton based, two-dimensional (2D) lattice model which generates global oscillations in the EEG spectrum as well as time series of local field potentials resembling those observed during slow wave sleep. This is made possible by the presence of pacemakers (local oscillators) which can be spontaneously ignited and whose activity propagates through space to synchronize all connected nodes.
The prediction of the behavior of the disease, the level of affectation in a population and the ways to control it are the most important aspects studied by epidemiology using tools such as historical data and mathematical models. So, our objective is (1) to provide a tool capable of analyzing epidemiological phenomena starting from the most common social interactions within a group of individuals. (2) To provide a methodology to build epidemiological models from patterns and logical rules. (3) Determine the impact of social interactions on the spread of a disease. This paper describes the logical construction of two epidemiological models in cellular automata together with two of their variations based on topological and dynamical principles.
The exponential growth of the power of modern digital computers is based upon the miniaturisation of vast nanoscale arrays of electronic switches, but this will be eventually constrained by fabrication limits and power dissipation. Chemical processes have the potential to scale beyond these limits performing computations through chemical reactions, yet the lack of well-defined programmability limits their scalability and performance. We present a hybrid digitally programmable chemical array as a probabilistic computational machine that uses chemical oscillators partitioned in interconnected cells as a computational substrate. This hybrid architecture performs efficient computation by distributing between chemical and digital domains together with error correction. The efficiency is gained by combining digital with probabilistic chemical logic based on nearest neighbour interactions and hysteresis effects. We demonstrated the implementation of one- and two- dimensional Chemical Cellular Automata and solutions to combinatorial optimization problems.
In this essay I aim to investigate and discuss the process through which bundles of things "self-organize" into other things. In particular, I engage in such investigation by trying to apply a framework of analysis of structural stability of morphological forms to the context of a continuous cellular automata known as $\textit{Lenia}$, which displays great diversity of distinguishable, individuated complex subsystems, who behave autonomously and in lifelike manner (such subsystems are often known as $\textit{solitons}$). I do so in order to suggest a path to the development of methods to formally distinguish said subsystems from their environment.
Recent work with Lenia, a continuously-valued cellular automata (CA) framework, has yielded $\sim$100s of compelling, bioreminiscent and mobile patterns. Lenia can be viewed as a continuously-valued generalization of the Game of Life, a seminal cellular automaton developed by John Conway that exhibits complex and universal behavior based on simple birth and survival rules. Life's framework of totalistic CA based on the Moore neighborhood includes many other interesting, Life-like, CA. A simplification introduced in Lenia limits the types of Life-like CA that are expressible in Lenia to a specific subset. This work recovers the ability to easily implement any Life-like CA by splitting Lenia's growth function into genesis and persistence functions, analogous to Life's birth and survival rules. We demonstrate the capabilities of this new CA variant by implementing a puffer pattern from Life-like CA Morley/Move, and examine differences between related CA in Lenia and Glaberish frameworks: Hydrogeminium natans and s613, respectively. These CA exhibit marked differences in dynamics and character based on spatial entropy over time, and both support several persistent mobile patterns. The CA s613, implemented in the Glaberish framework, is more dynamic than the Hydrogeminium CA in terms of a consistently high variance in spatial entropy over time. These results suggest there may be a wide variety of interesting CA that can be implemented in the Glaberish variant of the Lenia framework, analogous to the many interesting Life-like CA outside of Conway's Life.