arXiv subjects
Publications and source records attributed to Theodore Kolokolnikov.
The smallest known trivalent graph of girth 14 is constructed. The graph has 372 vertices, improving the old upper bound of 384 \cite{exoo14}. The construction uses methods that may be new to this problem.
We study the mean first-reaction time (MFRT) of a Brownian particle in a bounded three-dimensional domain $Ω$ with a reflecting boundary $\partialΩ$, that contains a thin reactive filament. The filament is a tube of cross-sectional diameter $\varepsilon\ll 1$ around a smooth closed curve $Γ$, and its surface is partially reactive with reactivity $b$. As $\varepsilon\to 0$ the filament reduces to a line source in the outer problem. Its cross-section enters only through a logarithmic reactive capacitance $d_c(b)$, so that the MFRT is O$(-\log[\varepsilon d_c])$. This contrasts with the $O(\varepsilon^{-1})$ scaling for spatially localized traps of radius $\varepsilon$. Using matched asymptotic expansions, we derive a Fredholm integral equation on $Γ$ for the line density of the reactive flux. It sums all powers of $ν\equiv -1/\log[\varepsilon d_c]$ and involves two geometric quantities: the regular part $R$ of the Neumann Green's function, which encodes the interaction of $Γ$ with $\partialΩ$, and an intrinsic self-energy of $Γ$ that depends only on its shape. The framework extends to multiple filaments, splitting probabilities, a spatially heterogeneous diffusivity, and the principal eigenvalue of the Laplacian. For the unit sphere we solve the integral equation with a simple trapezoidal Nyström scheme, and we validate the predictions for rings, linked rings and torus knots against full 3-D finite-element solutions.
We model directed motion of cells that aggregate into a cluster in the presence of chemotaxis or fluid transport. Starting from a stochastic agent-based model (ABM) of Galante \& Levy (2013), Chavy-Waddy \& Kolokolnikov (2016), we consider two model variants, with (I) directed motion of individuals, or (II) implicit individual bias in one direction. In both cases, we derive fourth order nonlinear PDEs representing the continuum limits of the models as a generalization of Chavy-Waddy \& Kolokolnikov (2016). An exact analytic solution then describes cluster density profile as well as the group velocity, and we identify the regimes where the travelling cluster exists. Excellent agreement is observed between the discrete models and the exact solutions of the PDEs in both models, including the existence and speed of the travelling cluster.
We study the problem of optimally placing a large number of small absorbing traps to minimize the mean first-passage time (MFPT) of particles diffusing in a heterogeneous medium with space-dependent diffusivity $D(x)$ and prescribed initial particle distribution $ω(x)$. In two and three dimensions we identify two distinct regimes, depending on how strongly the trap ensemble depletes the particles. In the weak trapping regime (fewer traps) the optimal trap density $ρ$ is the arithmetic average $ρ= \frac12(μ+ω)$ of the normalized inverse diffusivity $μ\propto 1/D$ and the initial particle distribution $ω$. In the strong trapping regime (more traps), it is proportional to their geometric average, $ρ\propto \sqrt{μω}$. We revisit the classical Green's function approach and show that it applies to weak trapping only; in 2D this requires the trap size to be exponentially small in the number of traps, so that most applications of interest fall in the strong trapping regime instead. We therefore develop a homogenization approach that captures both regimes. A special choice of initial particle distribution is $ω\propto 1/D$, which corresponds to the equilibrium particle distribution in the absence of traps according to the Itô interpretation. In this case we show that the optimal trap distribution is the stationary particle distribution itself, regardless of the trapping strength, and this makes the MFPT constant throughout the domain; the same trap distribution also minimizes the worst-case capture time. Finally, in one dimension we find instead $ρ\propto\left(μω\right) ^{1/3}$, and a similar analysis covers space-dependent drift and thin domains of variable cross-section. Direct numerical optimization confirms the analytical results.
We develop a high order asymptotic expansion for the mean first passage time (MFPT) of the capture of Brownian particles by a small elliptical trap in a bounded two dimensional region. This new result describes the effect that trap orientation plays on the capture rate and extends existing results that give information only on the role of trap position on the capture rate. Our results are validated against numerical simulations which confirm the accuracy of the asymptotic approximation. In the case of the unit disk domain, we identify a bifurcation such that the high order correction to the global MFPT (GMFPT) is minimized when the trap is orientated in the radial direction for traps centered at $0<r<r_c :=\sqrt{2-\sqrt{2}}$. When centered at position $r_c<r<1$, the GMFPT correction is minimized by orientating the trap in the angular direction. In the scenario of a general two-dimensional geometry, we identify the orientation that minimizes the GMFPT in terms of the regular part of the Neumann Green's function. This theory is demonstrated on several regular domains such as disks, ellipses and rectangles.
In this study, we provide a detailed analysis of the spike solutions and their stability for theGierer-Meinhardt model on discrete lattices. We explore several phenomena that have no analogues in the continuum limit. For example in the discrete case, the system retains spike patterns even when diffusion of the activator is set to zero. In this limit, we derive a simplified algebraic system to determine the presence of a K-spike solution. The stability of this solution is determined by a K by K matrix. We further delve into the scenarios where K = 2 and K = 3, revealing the existence of stable asymmetric spike patterns. Our stability analysis indicates that the symmetric two-spike solution is the most robust. Furthermore, we demonstrate that symmetric K-spike solutions are locally the most stable configurations. Additionally, we explore spike solutions under conditions where the inhibitor's diffusion rate is not significantly large. In doing so, we uncover zigzag and mesa patterns that do not occur in the continuous system. Our findings reveal that the discrete lattices support a greater variety of stable patterns for the Gierer-Meinhardt model.
We derive attainable upper bounds on the algebraic connectivity (spectral gap) of a regular graph in terms of its diameter and girth. This bound agrees with the well-known Alon-Boppana-Friedman bound for graphs of even diameter, but is an improvement for graphs of odd diameter. For the girth bound, we show that only Moore graphs can attain it, and these only exist for very few possible girths. For diameter bound, we use a combination of stochastic algorithms and exhaustive search to find graphs which attain it. For 3-regular graphs, we find attainable graphs for all diameters $D$ up to and including $D=9$ (the case of $D=10$ is open). These graphs are extremely rare and also have high girth; for example we found exactly 45 distinct cubic graphs on 44 vertices attaining the upper bound when $D=7$; all have girth 8 (out of a total of about $10^{20}$ cubic graphs on 44 vertices, including 266362 having girth 8). We also exhibit families of $d$-regular graphs attaining upper bounds with $D=3$ and $4$, and with $g=6.$ Several conjectures are proposed.
For the Schnakenberg model, we consider a highly symmetric configuration of N spikes whose locations are located at the vertices of a regular N-gon inside either a unit disk or an annulus. We call such configuration a ring of spikes. The ring radius is characterized in terms of the modified Green's function. For a disk, we find that a ring of 9 or more spikes is always unstable with respect to small eigenvalues. Conversely, a ring of 8 or less spikes is stable inside a disk provided that the feed-rate $A$ is sufficiently large. More generally, for sufficiently high feed-rate, a ring of $N$ spikes can be stabilized provided that the annulus is thin enough. As $A$ is decreased, we show that the ring is destabilized due to small eigenvalues first, and then due to large eigenvalues, although both of these thresholds are separated by an asymptotically small amount. For a ring of 8 spikes inside a disk, the instability appears to be supercritical, and deforms the ring into a square-like configuration. For less than 8 spikes, this instability is subcritical and results in spike death.
We study the algebraic connectivity for several classes of random semi-regular graphs. For large random semi-regular bipartite graphs, we explicitly compute both their algebraic connectivity and as well as the full spectrum distribution. For an integer $d\in\left[ 3,7\right] $, we find families of random semi-regular graphs that have higher algebraic connectivity than a random $d$-regular graphs with the same number of vertices and edges. On the other hand, we show that regular graphs beat semi-regular graphs when $d\geq8.$ More generally, we study random semi-regular graphs whose average degree is $d$, not necessary an integer. This provides a natural generalization of a $d$-regular graph in the case of a non-integer $d.$ We characterise their algebraic connectivity in terms of a root of a certain 6th-degree polynomial. Finally, we construct a small-world-type network of average degree 2.5 with a relatively high algebraic connectivity. We also propose some related open problems and conjectures.
In the present work, we explore analytically and numerically the co-existence and interactions of ring dark solitons (RDSs) with other RDSs, as well as with vortices. The azimuthal instabilities of the rings are explored via the so-called filament method. As a result of their nonlinear interaction, the vortices are found to play a stabilizing role on the rings, yet their effect is not sufficient to offer complete stabilization of RDSs. Nevertheless, complete stabilization of the relevant configuration can be achieved by the presence of external ring-shaped barrier potentials. Interactions of multiple rings are also explored, and their equilibrium positions (as a result of their own curvature and their tail-tail interactions) are identified. In this case too, stabilization is achieved via multi-ring external barrier potentials.
We study the spectrum of the Laplace-Beltrami operator on ellipsoids. For ellipsoids that are close to the sphere, we use analytic perturbation theory to estimate the eigenvalues up to two orders. We show that for biaxial ellipsoids sufficiently close to the sphere, the first $N$ eigenvalues have multiplicity at most two, and characterize those that are simple. For the triaxial ellipsoids sufficiently close to the sphere that are not biaxial, we show that at least the first sixteen eigenvalues are all simple. We also give the results of various numerical experiments, including comparisons to our results from the analytic perturbation theory, and approximations for the eigenvalues of ellipsoids that degenerate into infinite cylinders or two-dimensional disks. We propose a conjecture on the exact number of nodal domains of near-sphere ellipsoids.
Precursor gradients in a reaction-diffusion system are spatially varying coefficients in the reaction-kinetics. Such gradients have been used in various applications, such as the head formation in the Hydra, to model the effect of pre-patterns and to localize patterns in various spatial regions. For the 1-D Gierer-Meinhardt (GM) model we show that a simple precursor gradient in the decay rate of the activator can lead to the existence of stable, asymmetric, two-spike patterns, corresponding to localized peaks in the activator of different heights. This is a qualitatively new phenomena for the GM model, in that asymmetric spike patterns are all unstable in the absence of the precursor field. Through a determination of the global bifurcation diagram of two-spike steady-state patterns, we show that asymmetric patterns emerge from a supercritical symmetry-breaking bifurcation along the symmetric two-spike branch as a parameter in the precursor field is varied. Through a combined analytical-numerical approach we analyze the spectrum of the linearization of the GM model around the two-spike steady-state to establish that portions of the asymmetric solution branches are linearly stable. In this linear stability analysis a new class of vector-valued nonlocal eigenvalue problem (NLEP) is derived and analyzed.
Using the Klein-Majda-Damodaran model of nearly-parallel vortex filaments, we construct vortex knots and links on a torus involving periodic boundary conditions and analyze their stability. For a special class of vortex knots -- toroidal knots -- we give a full characterization of both their energetic and dynamical stability. In addition to providing explicit expressions for the relevant waveforms, we derive explicit formulas for their stability boundaries. These include simple links and different realizations of a trefoil knot. It is shown that a ring of more than 7 filaments can potentially be stablized by giving it a slight twist and connecting neighbouring filaments on a torus. In addition to rings, (helical) filament lattice configurations are also considered and are found to be dynamically stable for all rotation frequencies and also energetically stable for sufficiently fast rotations. Numerical simulations are used to compare the Klein-Majda-Damodaran model with the full three-dimensional (3D) Gross-Pitaevskii equations as well as to confirm the analytical theory. Potential differences between the quasi-one-dimensional and the fully 3D description are also discussed.
Synchronization is a universal phenomenon, seen in systems as diverse as superconducting Josephson junctions and discharging pacemaker cells. Here the elements have rhythmic state variables whose mutual influence promotes temporal order. A parallel form of order is seen in swarming systems, such as schools of fish or flocks of birds. Now the degrees of freedom are the individuals' positions, which get redistributed through interactions to form spatial structures. Systems capable of both swarming and synchronizing, dubbed swarmalators, have recently been proposed [O'Keeffe, Kevin P., and Steven H. Strogatz. "Swarmalators: Oscillators that sync and swarm." arXiv preprint arXiv:1701.05670 (2017)] and analyzed in the continuum limit. Here we extend this work by studying finite populations of swarmalators, whose phase similarity affects both their spatial attraction and repulsion. We find ring states, and compute criteria for their existence and stability. Larger populations can form annular distributions, whose density and inner and outer radii we calculate explicitly. These states may be observable in groups of Japanese tree frogs, magnetic colloids, and other systems with an interplay between swarming and synchronization.
We consider vortex dynamics in the context of Bose-Einstein Condensates (BEC) with a rotating trap, with or without anisotropy. Starting with the Gross-Pitaevskii (GP) partial differential equation (PDE), we derive a novel reduced system of ordinary differential equations (ODEs) that describes stable configurations of multiple co-rotating vortices (vortex crystals). This description is found to be quite accurate quantitatively especially in the case of multiple vortices. In the limit of many vortices, BECs are known to form vortex crystal structures, whereby vortices tend to arrange themselves in a hexagonal-like spatial configuration. Using our asymptotic reduction, we derive the effective vortex crystal density and its radius. We also obtain an asymptotic estimate for the maximum number of vortices as a function of rotation rate. We extend considerations to the anisotropic trap case, confirming that a pair of vortices lying on the long (short) axis is linearly stable (unstable), corroborating the ODE reduction results with full PDE simulations. We then further investigate the many-vortex limit in the case of strong anisotropic potential. In this limit, the vortices tend to align themselves along the long axis, and we compute the effective one-dimensional vortex density, as well as the maximum admissible number of vortices. Detailed numerical simulations of the GP equation are used to confirm our analytical predictions.
We consider an aggregation model for two interacting species. The coupling between the species is via their velocities, that incorporate self- and cross-interactions. Our main interest is categorizing the possible steady states of the considered model. Notably, we identify their regions of existence and stability in the parameter space. For assessing the stability we use a combination of variational tools (based on the gradient flow formulation of the model and the associated energy), and linear stability analysis (perturbing the boundaries of the species' supports). We rely on numerical investigations for those steady states that are not analytically tractable. Finally we perform a two-scale expansion to characterize the steady state in the limit of asymptotically weak cross-interactions.
We study the long-time effect of noise on pattern formation for the aggregation model. We consider aggregation kernels that generate patterns consisting of two delta-concentrations. Without noise, there is a one-parameter family of admissible equilibria that consist of two concentrations whose mass is not necessary equal. We show that when a small amount of noise is added, the heavier concentration "leaks" its mass towards the lighter concentration over a very long time scale, eventually resulting in the equilibration of the two masses. We use exponentially small asymptotics to derive the long-time ODE's that quantify this mass exchange. Our theory is validated using full numerical simulations of the original model -- both of the original stochastic particle system and its PDE limit. Our formal computations show that adding noise destroys the degeneracy in the equilibrium solution and leads to a unique symmetric steady state.
For a dissipative variant of the two-dimensional Gross-Pitaevskii equation with a parabolic trap under rotation, we study a symmetry breaking process that leads to the formation of vortices. The first symmetry breaking leads to the formation of many small vortices distributed uniformly near the Thomas-Fermi radius. The instability occurs as a result of a linear instability of a vortex-free steady state as the rotation is increased above a critical threshold. We focus on the second subsequent symmetry breaking, which occurs in the weakly nonlinear regime. At slightly above threshold, we derive a one-dimensional amplitude equation that describes the slow evolution of the envelope of the initial instability. We show that the mechanism responsible for initiating vortex formation is a modulational instability of the amplitude equation. We also illustrate the role of dissipation in the symmetry breaking process. All analyses are confirmed by detailed numerical computations.