arXiv subjects

Matthew S. Mizuhara

Publications and source records attributed to Matthew S. Mizuhara.

18 recordsLinked to original sources

arXiv2026

Active matter within flexible boundaries: a novel experimental approach with T. aceti nematodes

We experimentally explore the collective behavior of the nematode T. Aceti inside flexible boundaries showing the emergence of previously unreported states. These nematodes have been previously shown to be able to synchronize their body oscillations in a favorable condition of confined space. In this collective state, they are able to exert a strong pushing force which we exploit to study how their collective motion could deform a pliable boundary. Using a novel experimental technique, we were able to simulate a non-rigid border at a liquid-liquid-gas triple point. We discovered a state with periodic boundary deformations which are due to the synchronized oscillations of the nematodes. More interestingly, we found a novel state where the nematodes are able to form multiple protrusions in the border which oscillate, rotate, merge, and split. We propose a numerical model that is able to reproduce some of the observed parameters of this state as well as offer an analytical insight into protrusions dynamics. Our research opens a pathway for exploring the interaction between active matter and flexible interfaces with a new kinds of active matter agents forming dynamic non-chaotic border deformations.

cond-mat.soft↗

arXiv2026

The Kuramoto model on the Sierpinski Gasket II: Twisted states

We study the Kuramoto model (KM) of coupled phase oscillators on graphs approximating the Sierpinski gasket (SG). As the size of the graph tends to infinity, the limit points of the sequence of stable equilibria in the KM correspond to the minima of the Dirichlet energy, i.e., to harmonic maps from the SG to the circle. We provide a complete description of the stable equilibria of the continuum limit of the KM on graphs approximating the SG, under both Dirichlet and free boundary conditions. We show that there is a unique stable equilibrium in each homotopy class of continuous functions from the SG to the circle. These equilibria serve as generalizations of the classical twisted states on ring networks. Furthermore, we extend the analysis to the KM on post-critically finite fractals. The results of this work reveal the link between self-similar organization and network dynamics.

math-ph↗

arXiv2026

Kuramoto model on Sierpinski Gasket I: Harmonic maps

Motivated by the study of attractors in the Kuramoto model (KM) on graphs approximating the Sierpinski gasket (SG), we revisit the problem of harmonic maps (HMs) from SG to the circle, first considered by Strichartz. We provide a geometric proof of Strichartz's theorem, which states that for a prescribed degree and suitable boundary conditions, there exists a unique HM from SG to the circle. We extend this result to HMs on post-critically finite (p.c.f.) fractals. For continuous functions on SG, we define a degree given by vector of integers of arbitrary finite length. We show that the degree determines a homotopy class on SG with values in the circle. This provides an analog of the Hopf degree theorem on SG. We move on to analyze HMs. At the heart of our method lies an original construction of covering spaces. After lifting continuous functions on SG with values in the unit circle to continuous real-valued functions on the covering space, we use the harmonic extension algorithm to obtain a harmonic function on the covering space. The desired HM is obtained by restricting the domain of the harmonic function to the fundamental domain and projecting the range to the circle. Each covering space is constructed separately for HMs of a given homotopy class, capturing its intrinsic topology. We show that with suitable modifications the method applies to p.c.f. fractals, a large class of self-similar domains. We illustrate our method using numerical examples of HMs from SG to the circle and discuss the construction of covering spaces for several representative p.c.f. fractals, including the 3-level SG, hexagasket, and pentagasket. The results of this paper provide the foundation for a follow-up work where we give a complete description of attractors in the KM on graphs approximating p.c.f. fractals. Specifically, we show that all HMs identified in this paper are stable steady states of the KM.

math-ph↗

arXiv2025

A bounded confidence model to predict how group work affects student math anxiety

Math anxiety is negatively correlated with student performance and can result in avoidance of further math/STEM classes and careers. Cooperative learning (i.e., group work) is a proven strategy that can reduce math anxiety and has additional social and pedagogical benefits. However, depending on the group individuals, some peer interactions can mitigate anxiety while others exacerbate it. We propose a mathematical modeling approach to help untangle and explore this complex dynamic. We introduce a modification of the Hegselmann-Krause bounded confidence model, including both attractive and repulsive interactions to simulate how math anxiety levels are affected by pairwise student interactions. The model is simple but provides interesting qualitative predictions. In particular, Monte Carlo simulations show that there is an optimal group size to minimize average math anxiety, and that switching group members randomly at certain frequencies can dramatically reduce math anxiety levels. The model is easily adaptable to incorporate additional personal and societal factors, making it ripe for future research.

physics.soc-ph↗

arXiv2023

Bifurcations and patterns in the Kuramoto model with inertia

In this work, we analyze the Kuramoto model (KM) with inertia on a convergent family of graphs. It is assumed that the intrinsic frequencies of the individual oscillators are sampled from a probability distribution. In addition, a given graph, which may also be random, assigns network connectivity. As in the original KM, in the model with inertia, the weak coupling regime features mixing, the state of the network when the phases (but not velocities) of all oscillators are distributed uniformly around the unit circle. We study patterns, which emerge when mixing loses stability under the variation of the strength of coupling. We identify a pitchfork (PF) and an Andronov-Hopf (AH) bifurcations in the model with multimodal intrinsic frequency distributions. To this effect, we use a combination of the linear stability analysis and Penrose diagrams, a geometric technique for studying stability of mixing. We show that the type of a bifurcation and a nascent spatiotemporal pattern depend on the interplay of the qualitative properties of the intrinsic frequency distribution and network connectivity.

nlin.PS↗

arXiv2022

A global bifurcation organizing rhythmic activity in a coupled network

We study a system of coupled phase oscillators near a saddle-node on an invariant circle bifurcation and driven by random intrinsic frequencies. Under the variation of control parameters, the system undergoes a phase transition changing the qualitative properties of collective dynamics. Using the Ott-Antonsen reduction and geometric techniques for ordinary differential equations, we identify a heteroclinic bifurcation in a family of vector fields on a cylinder, which explains the change in collective dynamics. Specifically, we show that the heteroclinic bifurcation separates two topologically distinct families of limit cycles: contractible limit cycles before the bifurcation from noncontractibile ones after the bifurcation. Both families are stable for the model at hand.

nlin.AO↗

arXiv2021

Chimeras unfolded

The instability of mixing in the Kuramoto model of coupled phase oscillators is the key to understanding a range of spatiotemporal patterns, which feature prominently in collective dynamics of systems ranging from neuronal networks, to coupled lasers, to power grids. In this paper, we describe a codimension-2 bifurcation of mixing whose unfolding, in addition to the classical scenario of the onset of synchronization, also explains the formation of clusters and chimeras. We use a combination of linear stability analysis and Penrose diagrams to identify and analyze a variety of spatiotemporal patterns including stationary and traveling coherent clusters and twisted states, as well as their combinations with regions of incoherent behavior called chimera states. The linear stability analysis is used to estimate of the velocity distribution within these structures. Penrose diagrams, on the other hand, predict accurately the basins of their existence. Furthermore, we show that network topology can endow chimera states with nontrivial spatial organization. In particular, we present twisted chimera states, whose coherent regions are organized as stationary or traveling twisted states. The analytical results are illustrated with numerical bifurcation diagrams computed for the Kuramoto model with uni-, bi-, and tri-modal frequency distributions and all-to-all and nonlocal nearest-neighbor connectivity.

nlin.CD↗

arXiv2020

Instability of mixing in the Kuramoto model: From bifurcations to patterns

We study patterns observed right after the loss of stability of mixing in the Kuramoto model of coupled phase oscillators with random intrinsic frequencies on large graphs, which can also be random. We show that the emergent patterns are formed via two independent mechanisms determined by the shape of the frequency distribution and the limiting structure of the underlying graph sequence. Specifically, we identify two nested eigenvalue problems whose eigenvectors (unstable modes) determine the structure of the nascent patterns. The analysis is illustrated with the results of the numerical experiments with the Kuramoto model with unimodal and bimodal frequency distributions on certain graphs.

nlin.CD↗

arXiv2020

Dynamics of a cell motility model near the sharp interface limit

Phase-field models have recently had great success in describing the dynamic morphologies and motility of eukaryotic cells. In this work we investigate the minimal phase-field model introduced in [Berlyand, Potomkin, Rybalko (2017)]. Rigorous analysis of its sharp interface limit dynamics was completed in [Mizuhara, Berlyand, Rybalko, Zhang (2016); Mizuhara, Zhang (2019)], where it was observed that persistent cell motion was not stable. In this work we numerically study the pre-limiting phase-field model near the sharp interface limit, to better understand this lack of persistent motion. We find that immobile, persistent, and rotating states are all exhibited in this minimal model, and investigate the loss of persistent motion in the sharp interface limit. In addition we study cell speed as a function of biophysical parameters.

q-bio.CB↗

arXiv2020

Stability of clusters in the second-order Kuramoto model on random graphs

The Kuramoto model of coupled phase oscillators with inertia on Erdos-Renyi graphs is analyzed in this work. For a system with intrinsic frequencies sampled from a bimodal distribution we identify a variety of two cluster patterns and study their stability. To this end, we decompose the description of the cluster dynamics into two systems: one governing the (macro) dynamics of the centers of mass of the two clusters and the second governing the (micro) dynamics of individual oscillators inside each cluster. The former is a low-dimensional ODE whereas the latter is a system of two coupled Vlasov PDEs. Stability of the cluster dynamics depends on the stability of the low-dimensional group motion and on coherence of the oscillators in each group. We show that the loss of coherence in one of the clusters leads to the loss of stability of a two-cluster state and to formation of chimera states. The analysis of this paper can be generalized to cover states with more than two clusters and to coupled systems on W-random graphs. Our results apply to a model of a power grid with fluctuating sources.

nlin.AO↗

arXiv2019

Fluid-Structure Interaction for the Classroom: Speed, Accuracy, Convergence, and Jellyfish!

When is good, good enough? This question lingers in approximation theory and numerical methods as a competition between accuracy and practicality. Numerical Analysis is traditionally where the rubber meets the road: students begin to use numerical algorithms to compute approximate solutions to non-trivial problems. However, it is difficult for students to understand that more accuracy is not always the goal, but rather enough accuracy for practical use and meaningful interpretation. This compromise between accuracy and computational time/resources can be explored through the use of convergence plots. We offer a variety of classroom activities that allow students to discover the usefulness of convergence plots, including a contemporary example involving jellyfish locomotion using fluid-structure interaction modeling. These examples will additionally illustrate subtleties of convergence analysis: the same numerical scheme can exhibit different convergence rates, and the definition of `error' may change the convergence properties. To solve the fluid-structure interaction system, the open source software IB2d is used. All numerical codes, scripts, and movies are provided for streamlined integration into a classroom setting.

math.HO↗

arXiv2018

Bifurcations in the Kuramoto model on graphs

In his classical work, Kuramoto analytically described the onset of synchronization in all-to-all coupled networks of phase oscillators with random intrinsic frequencies. Specifically, he identified a critical value of the coupling strength, at which the incoherent state loses stability and a gradual build-up of coherence begins. Recently, Kuramoto's scenario was shown to hold for a large class of coupled systems on convergent families of deterministic and random graphs. Guided by these results, in the present work, we study several model problems illustrating the link between network topology and synchronization in coupled dynamical systems. First, we identify several families of graphs, for which the transition to synchronization in the Kuramoto model starts at the same critical value of the coupling strength and proceeds in practically the same way. These examples include Erdős-Rényi random graphs, Paley graphs, complete bipartite graphs, and certain stochastic block graphs. These examples illustrate that some rather simple structural properties such as the volume of the graph may determine the onset of synchronization, while finer structural features may affect only higher order statistics of the transition to synchronization. Further, we study the transition to synchronization in the Kuramoto model on power law and small-world random graphs. The former family of graphs endows the Kuramoto model with very good synchronizability: the synchronization threshold can be made arbitrarily low by varying the parameter of the power law degree distribution. For the Kuramoto model on small-world graphs, in addition to the transition to synchronization, we identify a new bifurcation leading to stable random twisted states. The examples analyzed in this work complement the results in [Chiba, Medvedev, The mean field analysis for the Kuramoto model on graphs (parts I and II), arxiv].

math.DS↗

arXiv2017

Minimal Model of Directed Cell Motility on Patterned Substrates

Crawling cell motility is vital to many biological processes such as wound healing and the immune response. Using a minimal model we investigate the effects of patterned substrate adhesiveness and biophysical cell parameters on the direction of cell motion. We show that cells with low adhesion site formation rates may move perpendicular to adhesive stripes while a those with high adhesion site formation rates results in motility only parallel to the substrate stripes. We explore the effects of varying the substrate pattern and the strength of actin polymerization on the directionality of the crawling cell; these results have applications in motile cell sorting and guiding on engineered substrates.

q-bio.CB↗

arXiv2017

Uniqueness and traveling waves in a cell motility model

We study a non-linear and non-local evolution equation for curves obtained as the sharp interface limit of a phase-field model for crawling motion of eukaryotic cells on a substrate. We establish uniqueness of solutions to the sharp interface limit equation in the so-called subcritical parameter regime. The proof relies on a Grönwall estimate for a specially chosen weighted $L^2$ norm. Next, as persistent motion of crawling cells is of central interest to biologists we study the existence of traveling wave solutions. We prove that traveling wave solutions exist in the supercritical parameter regime provided the non-linear term of the sharp interface limit equation possesses certain asymmetry (related, e.g., to myosin contractility). Finally, we numerically investigate traveling wave solutions and simulate their dynamics. Due to non-uniqueness of solutions of the sharp interface limit equation we simulate a related, singularly perturbed PDE system which is uniquely solvable. Our simulations predict instability of traveling wave solutions and capture both bipedal wandering cell motion as well as rotating cell motion; these behaviors qualitatively agree with recent experimental and theoretical fidings.

math.AP↗

arXiv2016

Extension of a proof of the Ramanujan congruences for multipartitions

Recently Lachterman, Schayer, and Younger published an elegant proof of the Ramanujan congruences for the partition function $p(n)$. Their proof uses only the classical theory of modular forms as well as a beautiful result of Choie, Kohnen, and Ono, without need for Hecke operators. In this paper we give a method for generalizing Lachterman, Schayer, and Younger's proof to include Ramanujan congruences for multipartition functions $p_k(n)$, and Ramanujan congruences for $p(n)$ modulo certain prime powers.

math.NT↗

arXiv2015

A periodic approach to plane partition congruences

Ramanujan's celebrated congruences of the partition function $p(n)$ have inspired a vast amount of results on various partition functions. Kwong's work on periodicity of rational polynomial functions yields a general theorem used to establish congruences for restricted plane partitions. This theorem provides a novel proof of several classical congruences and establishes two new congruences. We additionally prove several new congruences which do not fit the scope of the theorem, using only elementary techniques, or a relationship to existing multipartition congruences.

math.NT↗

arXiv2015

On an evolution equation in a cell motility model

This paper deals with the evolution equation of a curve obtained as the sharp interface limit of a non-linear system of two reaction-diffusion PDEs. This system was introduced as a phase-field model of (crawling) motion of eukaryotic cells on a substrate. The key issue is the evolution of the cell membrane (interface curve) which involves shape change and net motion. This issue can be addressed both qualitatively and quantitatively by studying the evolution equation of the sharp interface limit for this system. However, this equation is non-linear and non-local and existence of solutions presents a significant analytical challenge. We establish existence of solutions for a wide class of initial data in the so-called subcritical regime. Existence is proved in a two step procedure. First, for smooth ($H^2$) initial data we use a regularization technique. Second, we consider non-smooth initial data that are more relevant from the application point of view. Here, uniform estimates on the time when solution exists rely on a maximum principle type argument.

math.AP↗

arXiv2013

On groups with a class-preserving outer automorphism

In 1911, Burnside asked whether or not there exist groups that have an outer automorphism which preserves conjugacy classes. Two years later he answered his own question by constructing a family of such groups. Using the small group library in MAGMA we determine all of the groups of order n < 512 that possess such an automorphism. Our investigations led to the discovery of four new infinite families of such groups, all of which are 2-groups of coclass 4.

math.GR↗