On the curvature of warped product Finsler spaces and the Laplacian of the Sasaki–Finsler metrics

As an opening, we prove that a warped product Finsler space F=_F1×fF2$F=F_1{\times }_f{F}_2$ is of constant curvature c$c$ if and only if the base space _F1$F_1$ is also of constant curvature c$c$, the fiber space _F2$F_2$ is of some constant curvature α$\alpha$, and five other partial differential equations are satisfied. A rather similar result is proved for the case of warped product Finsler spaces of scalar curvature. Close relationships between the geometry of the warped product Finsler spaces of constant curvature and the spectral theory of the Laplacian (Laplace–Beltrami operator) of the well-known Sasaki–Finsler metrics of the base space _F1$F_1$ is established by detailed investigation of the above mentioned PDEs. We also define a new tensor for warped product Finsler spaces, which we call a warped-Cartan tensor. Using the tensor we define a new class of warped product Finsler spaces, calling them C-Warped spaces, which contain Landsberg, Berwald, locally Minkowski and Riemannian spaces, but not necessarily all of the constant curvature Finsler spaces of warped product type. Several results are obtained and special cases, for example the case of Riemannian, C-Warped and projectively flat spaces are also considered.     

On a generalized 1-harmonic equation and the inverse mean curvature flow

We introduce and study generalized 1-harmonic equations (1.1). Using some ideas and techniques in studying 1-harmonic functions from Wei (2007)  [1], and in studying nonhomogeneous 1-harmonic functions on a cocompact set from Wei (2008)  [2, (9.1)], we find an analytic   quantity w$w$ in the generalized 1-harmonic equations  (1.1) on a domain in a Riemannian n$n$-manifold that affects the behavior of weak solutions of  (1.1), and establish its link with the geometry   of the domain. We obtain, as applications, some gradient bounds and nonexistence results for the inverse mean curvature flow, Liouville theorems for p$p$-subharmonic functions of constant p$p$-tension field, p≥n$p\ge n$, and nonexistence results for solutions of the initial value problem of inverse mean curvature flow.     

Invariant fourth root Finsler metrics on the Grassmannian manifolds

Fourth root metrics are a special and important class of Finsler metrics, which have been applied to physics. In this paper, we study invariant fourth root Finsler metrics on the Grassmannian manifolds SO(p+q)/SO(p)×SO(q)$SO(p+q)/SO\left(p\right)\times SO\left(q\right)$. By using the results from the theory of invariant polynomials of Lie groups, we obtain a complete classification of such metrics. Further, some invariant 2m$2m$-th root Finsler metrics are also given.     

Curve flows in Lagrange–Finsler geometry,bi-Hamiltonian structures and solitons

Methods in Riemann–Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N$N$-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces G/SO(n)$G/SO\left(n\right)$ provided with an N$N$-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle.     

The first Chevalley–Eilenberg Cohomology group of the Lie algebra on the transverse bundle of a decreasing family of foliations

In [L. Lebtahi, Lie algebra on the transverse bundle of a decreasing family of foliations, J. Geom. Phys. 60 (2010), 122–133], we defined the transverse bundle V^k$V^k$ to a decreasing family of k$k$ foliations _Fi$F_i$ on a manifold M$M$. We have shown that there exists a (1,1)$(1,1)$ tensor J$J$ of V^k$V^k$ such that J^k≠0$J^k\ne 0$, J^k+1=0$J^{k+1}=0$ and we defined by _LJ(V^k)$L_J\left(V^k\right)$ the Lie Algebra of vector fields X$X$ on V^k$V^k$ such that, for each vector field Y$Y$ on V^k$V^k$, [X,JY]=J[X,Y]$[X,JY]=J[X,Y]$.     

The role of curvature in the slowing down acceleration scenario

We introduce the curvature _Ωk${\Omega }_k$ as a new free parameter in the Bayesian analysis using SNIa, BAO and CMB data, in a model with variable equation of state parameter w(z)$w(z)$. We compare the results using both the Constitution and Union 2 data sets, and also study possible low redshift transitions in the deceleration parameter q(z)$q(z)$. We found that, incorporating _Ωk${\Omega }_k$ in the analysis, it is possible to make all the three observational probes consistent using both SNIa data sets. Our results support dark energy evolution at small redshift, and show that the tension between small and large redshift probes is ameliorated. However, although the tension decreases, it is still not possible to find a consensus set of parameters that fit all the three data set using the Chevalier–Polarski–Linder CPL parametrization.     

Magnetic strings in Einstein–Born–Infeld-dilaton gravity

A class of spinning magnetic string in 4-dimensional Einstein-dilaton gravity with Liouville type potential which produces a longitudinal nonlinear electromagnetic field is presented. These solutions have no curvature singularity and no horizon, but have a conic geometry. In these spacetimes, when the rotation parameter does not vanish, there exists an electric field, and therefore the spinning string has a net electric charge which is proportional to the rotation parameter. Although the asymptotic behavior of these solutions are neither flat nor (A)dS, we calculate the conserved quantities of these solutions by using the counterterm method. We also generalize these four-dimensional solutions to the case of (n+1)$(n+1)$-dimensional rotating solutions with k?[n/2]$k?[n/2]$ rotation parameters, and calculate the conserved quantities and electric charge of them.     

Scalar curvature and holomorphy potentials

A holomorphy potential is a complex valued function whose complex gradient, with respect to some Kähler metric, is a holomorphic vector field. Given k$k$ holomorphic vector fields on a compact complex manifold, form, for a given Kähler metric, a product of the following type: a function of the scalar curvature multiplied by functions of the holomorphy potentials of each of the vector fields. It is shown that the stipulation that such a product be itself a holomorphy potential for yet another vector field singles out critical metrics for a particular functional. This may be regarded as a generalization of the extremal metric variation of Calabi, where k=0$k=0$ and the functional is the square of the L²$L^2$-norm of the scalar curvature. The existence question for such metrics is examined in a number of special cases. Examples are constructed in the case of certain multifactored product manifolds. For the SKR metrics investigated by Derdzinski and Maschler and residing in the complex projective space, it is shown that only one type of nontrivial criticality holds in dimension three and above.     

Random matrix theory for transition strength densities in finite quantum systems: Results from embedded unitary ensembles

Embedded random matrix ensembles are generic models for describing statistical properties of finite isolated interacting quantum many-particle systems. For the simplest spinless fermion (or boson) systems, with say m$m$ fermions (or bosons) in N$N$ single particle states and interacting via k$k$-body interactions, we have EGUE(k$k$) [embedded GUE of k$k$-body interactions] with GUE embedding and the embedding algebra is U(N)$U\left(N\right)$. A finite quantum system, induced by a transition operator, makes transitions from its states to the states of the same system or to those of another system. Examples are electromagnetic transitions (then the initial and final systems are same), nuclear beta and double beta decay (then the initial and final systems are different), particle addition to or removal from a given system and so on. Towards developing a complete statistical theory for transition strength densities (transition strengths multiplied by the density of states at the initial and final energies), we have derived formulas for the lower order bivariate moments of the strength densities generated by a variety of transition operators. Firstly, for a spinless fermion system, using EGUE(k$k$) representation for a Hamiltonian that is k$k$-body and an independent EGUE(t$t$) representation for a transition operator that is t$t$-body and employing the embedding U(N)$U\left(N\right)$ algebra, finite-N$N$ formulas for moments up to order four are derived, for the first time, for the transition strength densities. Secondly, formulas for the moments up to order four are also derived for systems with two types of spinless fermions and a transition operator similar to beta decay and neutrinoless beta decay operators. In addition, moments formulas are also derived for a transition operator that removes k₀$k_0$ number of particles from a system of m$m$ spinless fermions. In the dilute limit, these formulas are shown to reduce to those for the EGOE version derived using the asymptotic limit theory of Mon and French (1975). Numerical results obtained using the exact formulas for two-body (k=2$k=2$) Hamiltonians (in some examples for k=3$k=3$ and 4$4$) and the asymptotic formulas clearly establish that in general the smoothed (with respect to energy) form of the bivariate transition strength densities take bivariate Gaussian form for isolated finite quantum systems. Extensions of these results to bosonic systems and EGUE ensembles with further symmetries are discussed.     

On Randers spaces of constant flag curvature

This paper concerns a ubiquitous class of Finsler metrics on smooth manifolds of dimension n. These are the Randers metrics. They figure prominently in both the theory and the applications of Finsler geometry. For n ≥ 3, we consider only those with constant flag curvature. For n = 2, we focus on those whose flag curvature is a (possibly constant) function of position only. We characterize such metrics by three efficient conditions. With the help of examples in 2 and 3 dimensions, we deduce that the Yasuda-Shimada classification of Randers space forms actually addresses only a special case. The corrected classification for that special case is sharp, holds for n ≥ 2, and follows readily from our three necessary and sufficient conditions.     

Percolation processes in monomer-polyatomic mixtures

In this paper, the percolation of mixtures of monomers and polyatomic species (k$k$-mers) on a square lattice is studied. By means of a finite-size scaling analysis, the critical exponents and the scaling collapsing of the fraction of percolating lattices are found. A phase diagram separating a percolating from a non-percolating region is determined. The main features of the phase diagram are discussed in order to predict its evolution for larger k$k$-mer sizes.     

Ranking the spreading influence in complex networks

Identifying the node spreading influence in networks is an important task to optimally use the network structure and ensure the more efficient spreading in information. In this paper, by taking into account the shortest distance between a target node and the node set with the highest k$k$-core value, we present an improved method to generate the ranking list to evaluate the node spreading influence. Comparing with the epidemic process results for four real networks and the Barabási–Albert network, the parameterless method could identify the node spreading influence more accurately than the ones generated by the degree k$k$, closeness centrality, k$k$-shell and mixed degree decomposition methods. This work would be helpful for deeply understanding the node importance of a network.     

Topology and phase transitions II. Theorem on a necessary relation

In this second paper, we prove a necessity theorem about the topological origin of phase transitions. We consider physical systems described by smooth microscopic interaction potentials VN(q)$V_N(q)$, among N   degrees of freedom, and the associated family of configuration space submanifolds _{Mv}v∈R${\{M_v\}}_{v\in \mathbb{R}}$, with Mv={q∈RN|VN(q)?v}$M_v=\{q\in {\mathbb{R}}^N|V_N(q)?v\}$. On the basis of an analytic relationship between a suitably weighed sum of the Morse indexes of the manifolds _{Mv}v∈R${\{M_v\}}_{v\in \mathbb{R}}$ and thermodynamic entropy, the theorem states that any possible unbound growth with N   of one of the following derivatives of the configurational entropy S(−)(v)=(1/N)logMv_∫dNq$S^{(-)}(v)=(1/N)\log {\int }_{M_v}{d}^{N}q$, that is of |k^∂S(−)(v)/∂vk|$|{\partial }^k{S}^{(-)}(v)/\partial v^k|$, for k=3,4$k=3,4$, can be entailed only by the weighed sum of Morse indexes. Since the unbound growth with N of one of these derivatives corresponds to the occurrence of a first- or of a second-order phase transition, and since the variation of the Morse indexes of a manifold is in one-to-one correspondence with a change of its topology, the Main Theorem of the present paper states that a phase transition necessarily stems from a topological transition in configuration space. The proof of the theorem given in the present paper cannot be done without Main Theorem of paper I.