We generalize the normalized combinatorial Laplace operator for graphs by defining two Laplace operators for hypergraphs that can be useful in the study of chemical reaction networks. We also ...investigate some properties of their spectra.
In this paper, we explore the relationship between one of the most elementary and important properties of graphs, the presence and relative frequency of triangles, and a combinatorial notion of Ricci ...curvature. We employ a definition of generalized Ricci curvature proposed by Ollivier in a general framework of Markov processes and metric spaces and applied in graph theory by Lin–Yau. In analogy with curvature notions in Riemannian geometry, we interpret this Ricci curvature as a control on the amount of overlap between neighborhoods of two neighboring vertices. It is therefore naturally related to the presence of triangles containing those vertices, or more precisely, the local clustering coefficient, that is, the relative proportion of connected neighbors among all the neighbors of a vertex. This suggests to derive lower Ricci curvature bounds on graphs in terms of such local clustering coefficients. We also study curvature-dimension inequalities on graphs, building upon previous work of several authors.
We first develop a general framework for Laplace operators defined in terms of the combinatorial structure of a simplicial complex. This includes, among others, the graph Laplacian, the combinatorial ...Laplacian on simplicial complexes, the weighted Laplacian, and the normalized graph Laplacian. This framework then allows us to define the normalized Laplace operator Δiup on simplicial complexes which we then systematically investigate. We study the effects of a wedge sum, a join and a duplication of a motif on the spectrum of the normalized Laplace operator and identify some of the combinatorial features of a simplicial complex that are encoded in its spectrum.
As a commutative version of the supersymmetric nonlinear sigma model, Dirac-harmonic maps from Riemann surfaces were introduced fifteen years ago. They are critical points of an unbounded conformally ...invariant functional involving two fields, a map from a Riemann surface into a Riemannian manifold and a section of a Dirac bundle which is the usual spinor bundle twisted with the pull-back of the tangent bundle of the target by the map. As solutions to a coupled nonlinear elliptic system, the existence and regularity theory of Dirac-harmonic maps has already received much attention, while the general uniqueness theory has not been established yet. For uncoupled Dirac-harmonic maps, the map components are harmonic maps. Since the uniqueness theory of harmonic maps from a compact surface with boundary is known, it is sufficient to consider the uniqueness of the spinor components, which are solutions to the corresponding boundary value problems for a nonlinear Dirac equation. In particular, when the map components belong to W1,p with p>2, the spinor components are uniquely determined by boundary values and map components. For coupled Dirac-harmonic maps, the map components are not harmonic maps. So the uniqueness problem is more difficult to solve. In this paper, we study the uniqueness problem on a compact surface with boundary. More precisely, we prove the energy convexity for weakly Dirac-harmonic maps from the unit disk with small energy. This yields the first uniqueness result about Dirac-harmonic maps from a surface conformal to the unit disk with small energy and arbitrary boundary values.
The main thesis developed in this article is that the key feature of biological life is the
a biological process can control and regulate other processes
, and it maintains that ability over time. ...This control can happen hierarchically and/or reciprocally, and it takes place in three-dimensional space. This implies that the information that a biological process has to utilize is only about the control, but not about the content of those processes. Those other processes can be vastly more complex that the controlling process itself, and in fact necessarily so. In particular, each biological process draws upon the complexity of its environment.
We investigate spontaneous critical dynamics of excitatory and inhibitory (EI) sparsely connected populations of spiking leaky integrate-and-fire neurons with conductance-based synapses. We use a ...bottom-up approach to derive a single neuron gain function and a linear Poisson neuron approximation which we use to study mean-field dynamics of the EI population and its bifurcations. In the low firing rate regime, the quiescent state loses stability due to saddle-node or Hopf bifurcations. In particular, at the Bogdanov-Takens (BT) bifurcation point which is the intersection of the Hopf bifurcation and the saddle-node bifurcation lines of the 2D dynamical system, the network shows avalanche dynamics with power-law avalanche size and duration distributions. This matches the characteristics of low firing spontaneous activity in the cortex. By linearizing gain functions and excitatory and inhibitory nullclines, we can approximate the location of the BT bifurcation point. This point in the control parameter phase space corresponds to the internal balance of excitation and inhibition and a slight excess of external excitatory input to the excitatory population. Due to the tight balance of average excitation and inhibition currents, the firing of the individual cells is fluctuation-driven. Around the BT point, the spiking of neurons is a Poisson process and the population average membrane potential of neurons is approximately at the middle of the operating interval
V
Rest
,
V
th
. Moreover, the EI network is close to both oscillatory and active-inactive phase transition regimes.
We develop a new concept of non-positive curvature for metric spaces, based on intersection patterns of closed balls. In contrast to the synthetic approaches of Alexandrov and Busemann, our concept ...also applies to metric spaces that might be discrete. The natural comparison spaces that emerge from our discussion are no longer Euclidean spaces, but rather tripod spaces. These tripod spaces include the hyperconvex spaces which have trivial Čech homology. This suggests a link of our geometrical method to the topological method of persistent homology employed in topological data analysis. We also investigate the geometry of general tripod spaces.
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