Abstract The collective dynamics of interacting dynamical units on a network crucially depends the properties structure. Rather than considering large but finite graphs to capture network, one often resorts graph limits and thereon. We elucidate symmetry systems limits—including graphons graphops—and analyze how shapes dynamics, for example through invariant subspaces. In addition traditional symmetries, can support generalized noninvertible symmetries. Moreover, as asymmetric networks have...

This report describes work performed during SWI 2023 at the University of Groningen in relation with Problem 1 posed by company ASMPT. They have detailed simulation software a machine and they compare results this physical experimental results. There is significant difference between simulated measured data, it goal to study how estimate parameters model using ex-perimentally frequency response. First, two toy models are studied understand challenges pa- rameter estimation domain. Later,...

We prove that the Kuramoto model on a graph can contain infinitely many nonequivalent stable equilibria. More precisely, we for every there is connected such set of equilibria contains manifold dimension . In particular, solve conjecture Delabays, Coletta, and Jacquod about number planar graphs. Our results are based analysis balanced configurations, which correspond to equilateral polygon linkages in topology. order analyze stability manifolds apply topological bifurcation theory.

Abstract Let 𝐺 be a linear algebraic group defined over finite field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi mathvariant="double-struck">F</m:mi> <m:mi>q</m:mi> </m:msub> </m:math> \mathbb{F}_{q} . We present several connections between the isogenies of and groups rational points <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi>G</m:mi> <m:mo>⁢</m:mo> <m:msup> <m:mi>n</m:mi> </m:msup> stretchy="false">)</m:mo> </m:mrow> <m:mo>≥</m:mo> <m:mn>1</m:mn>...

We investigate power series that converge to a bounded function on the real line. First, we establish relations between coefficients of and boundedness resulting function; in particular, show can be prevented by certain Turán inequalities and, case coefficients, sign patterns. Second, set naturally supports three topologies these are inequivalent incomplete. In each case, determine topological completion. Third, study algebra series, revealing key role backward shift operator.

Abstract Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this linearizing system at equilibrium leads a matrix zero every entry. call these completely degenerate. prove that they exist for certain intrinsic frequencies if and only underlying graph bipartite, do not generic frequencies. of frequencies, we has an Euler circuit such number steps between any two...

This paper concerns discrete-time occupancy processes on a finite graph. Our results can be formulated in two theorems, which are stated for vertex processes, but also applied to edge process (e.g., dynamic random graphs). The first theorem shows that concentration of local state averages is controlled by walk the second polynomials states. For graphs, this allows estimate deviations density, triangle and more general subgraph densities. only require Lipschitz continuity hold both dense...

We introduce a broad class of equations that are described by graph, which includes many well-studied systems. For these, we show the number solutions (or dimension solution set) can be bounded studying certain induced subgraphs. As corollaries, obtain novel bounds in spectral graph theory on multiplicities eigenvalues, and nonlinear dynamical system equilibrium set network.

Let A,B be matrices in SL(2,R) having trace greater than or equal to 2. Assume the pair is coherently oriented, that is, can conjugated a nonnegative entries. also either A,B^(-1) oriented as well, have integer Then Lagarias-Wang finiteness conjecture holds for set {A,B}, with optimal product {A,B,AB,A^2B,AB^2}. In particular, it every matrix SL(2,Z>=0).

We analyze graph dynamical systems with odd analytic coupling. The set of equilibria is union manifolds, which can intersect and have different dimension. geometry this depends on the coupling function, as well several properties underlying graph, such homology, coverings, connectivity symmetry. Equilibrium stability change among manifolds within a single manifold equilibria. gradient structure topological bifurcation theory be leveraged to establish Lyapunov, asymptotic, linear stability.

We investigate power series that converge to a bounded function on the real line. First, we establish relations between coefficients of and boundedness resulting function; in particular, show can be prevented by certain Tur\'an inequalities and, case coefficients, sign patterns. Second, set naturally supports three topologies these are inequivalent incomplete. In each case, determine topological completion. Third, study algebra series, revealing key role backward shift operator.

Let G be a linear algebraic group defined over finite field F_q. We present several connections between the isogenies of and groups rational points G(F_q^n). show that an isogeny from G' to F_q gives rise subgroup fixed index in G(F_q^n) for infinitely many n. Conversely, we if is reductive existence k n implies order k. In particular, every infinite sequence subgroups controlled by number isogenies. This result applies classical GLm, SLm, SOm, SUm, Sp2m can extended non-reductive prime...

We prove that the Kuramoto model on a graph can contain infinitely many non-equivalent stable equilibria. More precisely, we for every positive integer d there is connected such set of equilibria contains manifold dimension d. In particular, solve conjecture R. Delabays, T. Coletta and P. Jacquod about number planar graphs. Our results are based analysis balanced configurations, which correspond to equilateral polygon linkages in topology. order analyze stability manifolds apply topological...

Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this linearizing system at equilibrium leads a matrix zero every entry. call these completely degenerate. prove that they exist for certain intrinsic frequencies if and only underlying graph bipartite, do not generic frequencies. of frequencies, we has an Euler circuit such number steps between any two visits same...

The collective dynamics of interacting dynamical units on a network crucially depends the properties structure. Rather than considering large but finite graphs to capture network, one often resorts graph limits and thereon. We elucidate symmetry systems -- including graphons graphops analyze how shape dynamics, for example through invariant subspaces. In addition traditional symmetries, can support generalized noninvertible symmetries. Moreover, as asymmetric networks have symmetric limits,...