Fully-connected networks with local connections

Fully-connected mesh networks with local connections are described. Each connector links only nearest neighbors of the node lattice and carries enough passive pass-through vias to provide direct one-to-one links between all the nodes. If the nodes form a one-dimensional ring, then each connector must contain at least N(N−1)/2 physical channels. However, if the nodes are arranged in a d-dimensional hyper-torus, the number of channels per connector drops to N(N ^1/d −1)/2, which scales much more favorably at large N. Such arrangements can provide fully-meshed connectivity when parts of the network are physically inaccessible or when the network needs to be scaled up in a modular fashion.     

On the Renormalized Volume of Hyperbolic 3-Manifolds

The renormalized volume of hyperbolic manifolds is a quantity motivated by the AdS/CFT correspondence of string theory and computed via a certain regularization procedure. The main aim of the present paper is to elucidate its geometrical meaning. We use another regularization procedure based on surfaces equidistant to a given convex surface ?N. The renormalized volume computed via this procedure is equal to what we call the W-volume of the convex region N given by the usual volume of N minus the quarter of the integral of the mean curvature over ?N. The W-volume satisfies some remarkable properties. First, this quantity is self-dual in the sense explained in the paper. Second, it verifies some simple variational formulas analogous to the classical geometrical Schläfli identities. These variational formulas are invariant under a certain transformation that replaces the data at ?N by those at infinity of M. We use the variational formulas in terms of the data at infinity to give a simple geometrical proof of results of Takhtajan et al on the Kähler potential on various moduli spaces.     

Cooperation in N-person evolutionary snowdrift game in scale-free Barabási-Albert networks

Cooperation in the N-person evolutionary snowdrift game (NESG) is studied in scale-free Barabási-Albert (BA) networks. Due to the inhomogeneity of the network, two versions of NESG are proposed and studied. In a model where the size of the competing group varies from agent to agent, the fraction of cooperators drops as a function of the payoff parameter. The networking effect is studied via the fraction of cooperative agents for nodes with a particular degree. For small payoff parameters, it is found that the small-k agents are dominantly cooperators, while large-k agents are of non-cooperators. Studying the spatial correlation reveals that cooperative agents will avoid to be nearest neighbors and the correlation disappears beyond the next-nearest neighbors. The behavior can be explained in terms of the networking effect and payoffs. In another model with a fixed size of competing groups, the fraction of cooperators could show a non-monotonic behavior in the regime of small payoff parameters. This non-trivial behavior is found to be a combined effect of the many agents with the smallest degree in the BA network and the increasing fraction of cooperators among these agents with the payoff for small payoffs.     

Dynamics of a neural network model with finite connectivity and cycle stored patterns

The spatiotemporal evolution and memory retrieval properties of a Hopfield-like neural network with cycle-stored patterns and finite connectivity are studied. The analytical studies on a mean-field version show that, given the number of stored patterns p, there is a critical connectivity k_c such that the retrieval states are stable fixed points if and only if k > k_c. The dependence of k_c on the number of stored patterns is also present. The numerical simulations are applied to the short-ranged model with local interaction. It is revealed that, given p, the memory retrieval function is kept if the connectivity is high enough while the dynamics of the system is in the frozen phase. However when the connectivity k is less than a critical value k_c the system is in the chaotic phase and loses its memory retrieval ability. The critical points of both the dynamical phase transition and memory-loss phase transition are obtained by simulation data.     

Ground magnetic state of an assembly of single-domain particles confined in a spherical layer

The ground magnetic state of systems of finite number N of single-domain particles confined in a spherical monolayer is investigated by numeric simulations. Two model situations are considered. In the first, the particle positions are imposed and fixed, in the second, the particles are able to move freely within the layer; in the latter case the excluded volume effect is taken into account. It is found that in all the range studied (N?200) the ground state of the system retains a considerable extent of magnetic vorticity (toroid moment). Moreover, the magnetic and toroid moments are correlated: they are approximately perpendicular to one another.     

Gauge Theoretical Construction of Non-compact Calabi-Yau Manifolds

We construct the non-compact Calabi-Yau manifolds interpreted as the complex line bundles over the Hermitian symmetric spaces. These manifolds are the various generalizations of the complex line bundle over CP^N−1. Imposing an F-term constraint on the line bundle over CP^N−1, we obtain the line bundle over the complex quadric surface Q^N−2. On the other hand, when we promote the U(1) gauge symmetry in CP^N−1 to the non-abelian gauge group U(M), the line bundle over the Grassmann manifold is obtained. We construct the non-compact Calabi-Yau manifolds with isometries of exceptional groups, which we have not discussed in the previous papers. Each of these manifolds contains the resolution parameter which controls the size of the base manifold, and the conical singularity appears when the parameter vanishes.     

Wealth condensation in a Barabasi-Albert network

We study the flow of money among agents in a Barabasi-Albert (BA) scale free network, where each network node represents an agent and money exchange interactions are established through links. The system allows money trade between two agents at a time, betting a fraction f of the poorer’s agent wealth. We also allow for the bet to be biased, giving the poorer agent a winning probability p. In the no network case there is a phase transition involving a relationship between p and f. In the networked case, we also found a condensation interface, however, this is not a complete condensation due to the presence of clusters in the network and its topology. As can be expected, the winner is always a well-connected agent, but we also found that the mean wealth decreases with the agents’ connectivity.     

Renormalization group for network models of quantum Hall transitions

We analyze in detail the renormalization group flows which follow from the recently proposed all orders β functions for the Chalker–Coddington network model. The flows in the physical regime reach a true singularity after a finite scale transformation. Other flows are regular and we identify the asymptotic directions. One direction is in the same universality class as the disordered XY model. The all orders β function is computed for the network model of the spin quantum Hall transition and the flows are shown to have similar properties. It is argued that fixed points of general current–current interactions in 2d should correspond to solutions of the Virasoro master equation. Based on this we identify two coset conformal field theories osp(2N|2N)₁/u(1)₀ and osp(4N|4N)₁/su(2)₀ as possible fixed points and study the resulting multifractal properties. We also obtain a scaling relation between the typical amplitude exponent α₀ and the typical point contact conductance exponent X_t which is expected to hold when the density of states is constant.     

Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N → 0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D → 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low- and high-temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in d = 3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.     

On the number of solutions for the two-point boundary value problem on Riemannian manifolds

We study the solutions joining two fixed points of a time-independent dynamical system on a Riemannian manifold (M,g) from an enumerative point of view. We prove a finiteness result for solutions joining two points p,q∈M that are non-conjugate in a suitable sense, under the assumption that (M,g) admits a non-trivial convex function. We discuss in some detail the notion of conjugacy induced by a general dynamical system on a Riemannian manifold. Using techniques of infinite dimensional Morse theory on Hilbert manifolds we also prove that, under generic circumstances, the number of solutions joining two fixed points is odd. We present some examples where our theory applies.     

Two-loop superstrings VII. Cohomology of chiral amplitudes

The relation between superholomorphicity and holomorphicity of chiral superstring N-point amplitudes for NS bosons on a genus 2 Riemann surface is shown to be encoded in a hybrid cohomology theory, incorporating elements of both de Rham and Dolbeault cohomologies. A constructive algorithm is provided which shows that, for arbitrary N and for each fixed even spin structure, the hybrid cohomology classes of the chiral amplitudes of the N-point function on a surface of genus 2 always admit a holomorphic representative. Three key ingredients in the derivation are a classification of all kinematic invariants for the N-point function, a new type of 3-point Green's function, and a recursive construction by monodromies of certain sections of vector bundles over the moduli space of Riemann surfaces, holomorphic in all but exactly one or two insertion points.     

On the application of normal forms near attracting fixed points of dynamical systems

It is shown on an integrable example in the plane, that normal form solutions need not converge over the full basin of attraction of fixed points of dissipative dynamical systems. Their convergence breaks down at a singularity in the complex time plane of the exact solutions of the problem. However, as is demonstrated on a nonintegrable example with 3-dimensional phase space, the region of convergence of normal forms can be large enough to extend almost to a nearby hyperbolic fixed point, whose invariant manifolds “embrace” the attracting fixed point forming a complicated basin boundary. Thus, in such problems, normal forms are shown to be useful in practice, as a tool for finding large regions of initial conditions for which the solutions are necessarily attracted to the fixed point at t → ∞.     

Relaxation on the energy method for the transient Rayleigh-Bénard convection

The onset of buoyancy-driven instability in initially quiescent fluid layers having the various boundary conditions is analyzed by using the energy method. New energy stability equation is derived under the Boussinesq approximation and the relative stability concept. The predicted critical conditions are compared with the previous results based on the conventional energy method. The stability limits which are related to the onset time of instabilities are presented as a function of the Rayleigh number Ra and the Prandtl number Pr. The present stability results predict that the onset time of convective instability decreases with increasing Ra and Pr. For the case of high Ra, the onset time of the instability is relatively insensitive to the boundary conditions of the upper boundaries.     

A local-world node deleting evolving network model

A new type network growth rule which comprises node addition with the concept of local-world connectivity and node deleting is studied. A series of theoretical analysis and numerical simulation to the LWD network are conducted in this Letter. Firstly, the degree distribution p(k) of this network changes no longer pure scale free but truncates by an exponential tail and the truncation in p(k) increases as pa decreases. Secondly, the connectivity is tighter, as the local-world size M increases. Thirdly, the average path length L increases and the clustering coefficient 〈C〉 decreases as generally node deleting increases. Finally, 〈C〉 trends up when the local-world size M increases, so as to kmax. Hence, the expanding local-world can compensate the infection of the node deleting.     

Time interval photoelectron statistics of non-gaussian scattered light

Time interval distributions are evaluated for coherent light scattered by a small collection of independent, but polydispensive, particles. Two situations are considered: when the number of scatterers within the scattering volume, N, is fixed, and when N is allowed to fluctuate randomly with a Poisson distribution.     

Compactifications of F-theory on Calabi-Yau threefolds (II)

We continue our study of compactifications of F-theory on Calabi-Yau threefolds. We gain more insight into F-theory duals of heterotic strings and provide a recipe for building F-theory duals for arbitrary heterotic compactifications on elliptically fibered manifolds. As a byproduct we find that string/string duality in six dimensions gets mapped to fiber/base exchange in F-theory. We also construct a number of new N = 1, d = 6 examples of F-theory vacua and study transitions among them. We find that some of these transition points correspond upon further compactification to 4 dimensions to transitions through analogues of Argyres-Douglas points of N = 2 moduli. A key idea in these transitions is the notion of classifying (0,4) fivebranes of heterotic strings.     

Asymptotically Hyperbolic Manifolds with Small Mass

For asymptotically hyperbolic manifolds of dimension n with scalar curvature at least equal to ?n(n ? 1) the conjectured positive mass theorem states that the mass is non-negative, and vanishes only if the manifold is isometric to hyperbolic space. In this paper we study asymptotically hyperbolic manifolds which are also conformally hyperbolic outside a ball of fixed radius, and for which the positive mass theorem holds. For such manifolds we show that the conformal factor tends to one as the mass tends to zero.