Finite Configurations Research Articles

A statistical mechanical approach to the Payne effect in filled rubbers

In this paper we apply and discuss some new aspects to the steric interaction of filler particles in reinforced elastomers under dynamic mechanical loading conditions. At certain concentration the filler particles (for example, carbon black, or silica) form loose clusters which themselves interact with each other and form a filler network with a significant contribution to the dynamic modulus of the rubber material. The filler concentration is relatively high, so that it is likely that the clusters undergo a ‘jamming transition’. With increasing strain amplitude under periodic mechanical deformation the disruption of the filler network resp. of finite filler cluster configurations leads to dejamming observed as softening of the rubber. As a theoretical approach we map the problem on a simple one dimensional Ising model. We present here a static model of this jamming (dejamming) and discuss the consequences on the mechanical and deformation properties of the filled rubber.

Branching processes for the fragmentation equation

We investigate branching properties of the solution of a fragmentation equation for the mass distribution and we properly associate a continuous time càdlàg Markov process on the space S↓ of all fragmentation sizes, introduced by J. Bertoin. A binary fragmentation kernel induces a specific class of integral type branching kernels and taking as base process the solution of the initial fragmentation equation for the mass distribution, we construct a branching process corresponding to a rate of loss of mass greater than a given strictly positive threshold d. It turns out that this branching process takes values in the set of all finite configurations of sizes greater than d. The process on S↓ is then obtained by letting d tend to zero. A key argument for the convergence of the branching processes is given by the Bochner–Kolmogorov theorem. The construction and the proof of the path regularity of the Markov processes are based on several newly developed potential theoretical tools, in terms of excessive functions and measures, compact Lyapunov functions, and some appropriate absorbing sets.

Three-state von Neumann cellular automata and pattern generation

In this study, we consider the theoretical structure and classification of two-dimensional (2D) three-state uniform cellular automata (CA) based on their visual behavior. The basic features of a CA comprise their discrete dynamic structure and the fact that they are modeled locally, but their behavior can be similar to a continuous system at large time and spatial scales. The geometrical structures of the patterns produced by iterating CA can be considered according to some basic properties. After applying the rules iteratively, it has been shown that CA are capable of producing complex behaviors. Some types of CA exhibit remarkably regular behavior with finite configurations. With simple initial configurations, the patterns generated may be self-replicating (SR), self-similar (SS), or mixed. In this study, we consider the theory of 2D uniform periodic boundary (PB), adiabatic boundary (AB), and reflexive boundary (RB) CA with a von Neumann neighborhood, and applications of image analysis for generating patterns. We investigate a 2D CA under these boundary restrictions for three-state cases, i.e., the ternary field Z3. We also study the applications of SR and SS patterns that correspond to the linear CA rules of 2D uniform CA with different boundary cases over Z3. The von Neumann neighborhood CA rule (i.e., Rule 2460) is classified into SR and SS types depending on the non-zero boundary values a,b,c,d of the neighboring cells that influence the cells under consideration. Based on the visual appearance of the patterns, we also show that Rule 2460 is sometimes sensitive, depending on the boundary conditions and chaotic behavior. Finally, we conclude by analyzing some results in detail for CA defined by Rules 2460NB, 2460PB, 2460AB, and 2460RB for non-symmetric figures.

Noncolliding system of continuous-time random walks

The continuous-time random walk is defined as a Poissonization of discrete-time random walk. We study the noncolliding system of continuous-time simple and symmetric random walks on . We show that the system is determinantal for any finite initial configuration without multiple point. The spatio-temporal correlation kernel is expressed by using the modified Bessel functions. We extend the system to the noncolliding process with an infinite number of particles, when the initial configuration has equidistant spacing of particles, and show a relaxation phenomenon to the equilibrium determinantal point process with the sine kernel.

A Chern–Simons–Yang–Mills–Higgs system in 3+1 dimensions

We study spherically symmetric solutions of an SO(5) Chern–Simons–Yang–Mills–Higgs system in 3+1 dimensions. The Chern–Simons densities are defined in terms of both Yang–Mills fields and a five-component isomultiplet Higgs. The SO(3) × SO(2) solutions are analyzed in a systematic way, by employing numerical methods. These finite energy configurations possess both electric and magnetic global charges, differing radically, however, from Julia–Zee dyons. When two or more of these Chern–Simons densities are present in the Lagrangian, solutions with vanishing electric charge but nonvanishing electrostatic potential may exist.

Numerical analysis of canister movements in an engineered barrier system

The mid-term safety of deep geological nuclear waste repositories is based in part on the presence of a buffer, the main role of which is to isolate the environment from radionuclides. A design evaluation of such a repository is necessary to assess the potential vertical canister movement inside the drift that could reduce the buffer efficiency. A thermo-hydro-mechanical (THM) simulation is performed in a vertical cross-section of the drift. The THM couplings are described, and their influences on the mid-term (300 years) response of the engineered barrier system (EBS) are revealed. This study uses an advanced constitutive model to simulate the THM processes that occur in a specific EBS design case. The near-field simulations of the nuclear waste canister are performed in a two-dimensional finite element configuration that considers the effect of gravity. The focus of this study is on the mechanical behaviour of the buffer, which consists of two different forms of bentonite. Such an approach allows realistic consideration of the effect of the wetting and drying of the buffer material in non-isothermal conditions. Due to a specific design that includes bentonite blocks and pellets, the canister is observed to heave slightly during the re-saturation period, which extends up to 100 years.

Push-Forward Measures on Configuration Spaces: Integration by Parts and Log-Sobolev Inequality

We study a class of measures on the space \(\Gamma _{X}\) of locally finite configurations in \(X=\mathbb {R}^{d}\), obtained as images of “lattice” Gibbs measures on \(X^{\mathbb {Z} ^{d}}\) with respect to an embedding \(\mathbb {Z}^{d}\subset \mathbb {R}^{d}\). For these measures, we prove the integration by parts formula and log-Sobolev inequality.

On the Asymptotic Behavior of a Dynamic Version of the Neyman Contagious Point Process

We consider a dynamic version of the Neyman contagious point process that can be used for modeling the spacial dynamics of biological populations, including species invasion scenarios. Starting with an arbitrary finite initial configuration of points in with nonnegative weights, at each time step a point is chosen at random from the process according to the distribution with probabilities proportional to the points’ weights. Then a finite random number of new points is added to the process, each displaced from the location of the chosen “mother” point by a random vector and assigned a random weight. Under broad conditions on the sequences of the numbers of newly added points, their weights and displacement vectors (which include a random environments setup), we derive the asymptotic behavior of the locations of the points added to the process at time step n and also that of the scaled mean measure of the point process after time step n → ∞.

Bounded and [formula omitted]-weak solutions for nonlinear equations of measure-valued branching processes

We study the nonlinear Schrödinger type equation (Δ−c)u+c∑k=1∞qkuk=f, where c is a bounded positive function. We may replace the Laplace operator with the generator of a Borel right Markov process with state space E and consider a nonlinear term induced by a perturbing kernel. We solve this type of equation in an Lp-weak sense, involving positive definite extensions of the solution from E to the set of all finite positive measures on E. The extensions to the set of measures are related to the classical connection (from the works of M. Nagasawa, N. Ikeda, S. Watanabe, and M.L. Silverstein) between this nonlinear equation and the discrete branching Markov processes. We also investigate the existence of the bounded solutions for the Dirichlet problem associated to the above equation, with bounded boundary data. Our approach is based on probabilistic and analytic potential theoretical methods, used on both spaces E and the set of all finite configurations of E, like the Revuz formula for the continuous additive functionals and the perturbation with kernels of the transition functions and resolvents.

Quantification and control of non-Markovian evolution in finite quantum systems via feedback

We consider the unitary time evolution of continuous quantum mechanical systems confined to a cavity in contact with a finite bath of variable size. Measures for Markovianity for such finite system-bath configurations are developed in terms of Hilbert-Schmidt distances of time evolving wave packets. The relevant time scales are identified, which characterize pseudo-Markovian transient behavior, boundary scattering induced non-Markovian oscillations at intermediate times, and non-Markovian rephasing events at long time scales. It is shown how these time scales can be controlled by tunable parameters such as the bath size and the strength of the system-bath coupling.

Fermi liquids from D-branes

We discuss finite density configurations on probe D-branes, in the presence of worldvolume fermions. To this end we consider a phenomenological model whose bosonic sector is governed by the DBI action, and whose charged sector is purely fermionic. In this model, we demonstrate the existence of a compact worldvolume embedding, stabilized by a Fermi surface on the D- brane. The finite density state in the boundary QFT is a Fermi-like liquid. We comment on the possibility of realizing non-Fermi liquids in this setup.

A low-dimensional analogue of holographic baryons

Baryons in holographic QCD correspond to topological solitons in the bulk. The most prominent example is the Sakai–Sugimoto model, where the bulk soliton in the five-dimensional spacetime of AdS-type can be approximated by the flat space self-dual Yang–Mills instanton with a small size. Recently, the validity of this approximation has been verified by comparison with the numerical field theory solution. However, multi-solitons and solitons with finite density are currently beyond numerical field theory computations. Various approximations have been applied to investigate these important issues and have led to proposals for finite density configurations that include dyonic salt and baryonic popcorn. Here we introduce and investigate a low-dimensional analogue of the Sakai–Sugimoto model, in which the bulk soliton can be approximated by a flat space sigma model instanton. The bulk theory is a baby Skyrme model in a three-dimensional spacetime with negative curvature. The advantage of the lower-dimensional theory is that numerical simulations of multi-solitons and finite density solutions can be performed and compared with flat space instanton approximations. In particular, analogues of dyonic salt and baryonic popcorn configurations are found and analysed.

Deformed self-dual magnetic monopoles

We develop a deformation method for attaining new magnetic monopole analytical solutions consistent with generalized Yang–Mills–Higgs model introduced recently. The new solutions fulfill the usual radially symmetric ansatz and the boundary conditions suitable to assure finite energy configurations. We verify our prescription by studying some particular cases involving both exactly and partially analytical initial configurations whose deformation leads to new analytic BPS monopoles. The results show consistency among the models, the deformation procedure and the profile of the new solutions.

A general mathematical framework for the analysis of spatiotemporal point processes

Spatial and stochastic models are often straightforward to simulate but difficult to analyze mathematically. Most of the mathematical methods available for nonlinear stochastic and spatial models are based on heuristic rather than mathematically justified assumptions, so that, e.g., the choice of the moment closure can be considered more of an art than a science. In this paper, we build on recent developments in specific branch of probability theory, Markov evolutions in the space of locally finite configurations, to develop a mathematically rigorous and practical framework that we expect to be widely applicable for theoretical ecology. In particular, we show how spatial moment equations of all orders can be systematically derived from the underlying individual-based assumptions. Further, as a new mathematical development, we go beyond mean-field theory by discussing how spatial moment equations can be perturbatively expanded around the mean-field model. While we have suggested such a perturbation expansion in our previous research, the present paper gives a rigorous mathematical justification. In addition to bringing mathematical rigor, the application of the mathematically well-established framework of Markov evolutions allows one to derive perturbation expansions in a transparent and systematic manner, which we hope will facilitate the application of the methods in theoretical ecology.

Multilinear generalized Radon transforms and point configurations

Abstract We study multilinear generalized Radon transforms using a graph-theoretic paradigm that includes the widely studied linear case. These provide a general mechanism to study Falconer-type problems involving (k+1)-point configurations in geometric measure theory, with k ≥ 2, including the distribution of simplices, volumes and angles determined by the points of fractal subsets E ⊂ ℝ d , d ≥ 2. If Tk (E) denotes the set of noncongruent ( k + 1 ) ${(k+1)}$ -point configurations determined by E, we show that if the Hausdorff dimension of E is greater than d - (d-1)/(2k), then the k + 1 2 ${\binom{k+1}{2}}$ -dimensional Lebesgue measure of Tk (E) is positive. This complements previous work on the Falconer conjecture ([Int. Math. Res. Not. IMRN 23 (2005), 1411–1425] and the references there), as well as work on finite point configurations [Recent Advances in Harmonic Analysis and Applications, Springer-Verlag, New York (2013), 93–103; Anal. PDE 5 (2012), no. 2, 397–409]. We also give applications to Erdős-type problems in discrete geometry and a fractal regular value theorem, providing a multilinear framework for the results in [Adv. Math. 228 (2011), 2385–2402].

On the Equivalence of Cellular Automata and the Tile Assembly Model

In this paper, we explore relationships between two models of systems which are governed by only the local interactions of large collections of simple components: cellular automata (CA) and the abstract Tile Assembly Model (aTAM). While sharing several similarities, the models have fundamental differences, most notably the dynamic nature of CA (in which every cell location is allowed to change state an infinite number of times) versus the static nature of the aTAM (in which tiles are static components that can never change or be removed once they attach to a growing assembly). We work with 2-dimensional systems in both models, and for our results we first define what it means for CA systems to simulate aTAM systems, and then for aTAM systems to simulate CA systems. We use notions of simulate which are similar to those used in the study of intrinsic universality since they are in some sense strict, but also intuitively natural notions of simulation. We then demonstrate a particular nondeterministic CA which can be configured so that it can simulate any arbitrary aTAM system, and finally an aTAM tile set which can be configured so that it can be used to simulate any arbitrary nondeterministic CA system which begins with a finite initial configuration.

Configurations of limit cycles in Liénard equations

We show that every finite configuration of disjoint simple closed curves in the plane is topologically realizable as the set of limit cycles of a polynomial Liénard equation. The related vector field X is Morse–Smale. Moreover it has the minimum number of singularities required for realizing the configuration in a Liénard equation. We provide an explicit upper bound on the degree of X, which is lower than the results obtained before, obtained in the context of general polynomial vector fields.

When is a quantum cellular automaton (QCA) a quantum lattice gas automaton (QLGA)?

Quantum cellular automata (QCA) are models of quantum computation of particular interest from the point of view of quantum simulation. Quantum lattice gas automata (QLGA - equivalently partitioned quantum cellular automata) represent an interesting subclass of QCA. QLGA have been more deeply analyzed than QCA, whereas general QCA are likely to capture a wider range of quantum behavior. Discriminating between QLGA and QCA is therefore an important question. In spite of much prior work, classifying which QCA are QLGA has remained an open problem. In the present paper we establish necessary and sufficient conditions for unbounded, finite QCA (finitely many active cells in a quiescent background) to be QLGA. We define a local condition that classifies those QCA that are QLGA, and we show that there are QCA that are not QLGA. We use a number of tools from functional analysis of separable Hilbert spaces and representation theory of associative algebras that enable us to treat QCA on finite but unbounded configurations in full detail.

Add Isotropic Gaussian Kernels at Own Risk: More and More Resilient Modes in Higher Dimensions

The fact that a sum of isotropic Gaussian kernels can have more modes than kernels is surprising. Extra (ghost) modes do not exist in $${{\mathbb{R }}}^1$$R1 and are generally not well studied in higher dimensions. We study a configuration of $$n+1$$n+1 Gaussian kernels for which there are exactly $$n+2$$n+2 modes. We show that all modes lie on a finite set of lines, which we call axes, and study the restriction of the Gaussian mixture to these axes in order to discover that there are an exponential number of critical points in this configuration. Although the existence of ghost modes remained unknown due to the difficulty of finding examples in $${{\mathbb{R }}}^2$$R2, we show that the resilience of ghost modes grows like the square root of the dimension. In addition, we exhibit finite configurations of isotropic Gaussian kernels with superlinearly many modes.

On convolutions on configuration spaces. II. spaces of locally finite configurations

We consider the convolution of probability measures on spaces of locally finite configurations (subsets of a phase space) and their connection with the convolution of the corresponding correlation measures and functionals. In particular, the convolution of Gibbs measures is studied. We also describe a relationship between invariant measures with respect to some operator and properties of the corresponding image of this operator on correlation functions.