Continuous approximation of binomial lattices

A systematic analysis of a continuous version of a binomial lattice, containing a real parameter $\gamma$ and covering the Toda field equation as $\gamma\to\infty$, is carried out in the framework of group theory. The symmetry algebra of the equation is derived. Reductions by one-dimensional and two-dimensional subalgebras of the symmetry algebra and their corresponding subgroups, yield notable field equations in lower dimensions whose solutions allow to find exact solutions to the original equation. Some reduced equations turn out to be related to potentials of physical interest, such as the Fermi-Pasta-Ulam and the Killingbeck potentials, and others. An instanton-like approximate solution is also obtained which reproduces the Eguchi-Hanson instanton configuration for $\gamma\to\infty$. Furthermore, the equation under consideration is extended to $(n+1)$--dimensions. A spherically symmetric form of this equation, studied by means of the symmetry approach, provides conformally invariant classes of field equations comprising remarkable special cases. One of these $(n=4)$ enables us to establish a connection with the Euclidean Yang-Mills equations, another appears in the context of Differential Geometry in relation to the socalled Yamabe problem. All the properties of the reduced equations are shared by the spherically symmetric generalized field equation.Comment: 30 pages, LaTeX, no figures. Submitted to Annals of Physic

Functional Integration Over Geometries

The geometric construction of the functional integral over coset spaces ${\cal M}/{\cal G}$ is reviewed. The inner product on the cotangent space of infinitesimal deformations of $\cal M$ defines an invariant distance and volume form, or functional integration measure on the full configuration space. Then, by a simple change of coordinates parameterizing the gauge fiber $\cal G$, the functional measure on the coset space ${\cal M}/{\cal G}$ is deduced. This change of integration variables leads to a Jacobian which is entirely equivalent to the Faddeev-Popov determinant of the more traditional gauge fixed approach in non-abelian gauge theory. If the general construction is applied to the case where $\cal G$ is the group of coordinate reparametrizations of spacetime, the continuum functional integral over geometries, {\it i.e.} metrics modulo coordinate reparameterizations may be defined. The invariant functional integration measure is used to derive the trace anomaly and effective action for the conformal part of the metric in two and four dimensional spacetime. In two dimensions this approach generates the Polyakov-Liouville action of closed bosonic non-critical string theory. In four dimensions the corresponding effective action leads to novel conclusions on the importance of quantum effects in gravity in the far infrared, and in particular, a dramatic modification of the classical Einstein theory at cosmological distance scales, signaled first by the quantum instability of classical de Sitter spacetime. Finite volume scaling relations for the functional integral of quantum gravity in two and four dimensions are derived, and comparison with the discretized dynamical triangulation approach to the integration over geometries are discussed.Comment: 68 pages, Latex document using Revtex Macro package, Contribution to the special issue of the Journal of Mathematical Physics on Functional Integration, to be published July, 1995

Arithmeticity vs. non-linearity for irreducible lattices

We establish an arithmeticity vs. non-linearity alternative for irreducible lattices in suitable product groups, such as for instance products of topologically simple groups. This applies notably to a (large class of) Kac-Moody groups. The alternative relies on a CAT(0) superrigidity theorem, as we follow Margulis' reduction of arithmeticity to superrigidity.Comment: 11 page

Special fast diffusion with slow asymptotics. Entropy method and flow on a Riemannian manifold

We consider the asymptotic behaviour of positive solutions $u(t,x)$ of the fast diffusion equation $u_t=\Delta (u^{m}/m)={\rm div} (u^{m-1}\nabla u)$ posed for x\in\RR^d, $t>0$, with a precise value for the exponent $m=(d-4)/(d-2)$. The space dimension is $d\ge 3$ so that $m<1$, and even $m=-1$ for $d=3$. This case had been left open in the general study \cite{BBDGV} since it requires quite different functional analytic methods, due in particular to the absence of a spectral gap for the operator generating the linearized evolution. The linearization of this flow is interpreted here as the heat flow of the Laplace-Beltrami operator of a suitable Riemannian Manifold (\RR^d,{\bf g}), with a metric ${\bf g}$ which is conformal to the standard \RR^d metric. Studying the pointwise heat kernel behaviour allows to prove {suitable Gagliardo-Nirenberg} inequalities associated to the generator. Such inequalities in turn allow to study the nonlinear evolution as well, and to determine its asymptotics, which is identical to the one satisfied by the linearization. In terms of the rescaled representation, which is a nonlinear Fokker--Planck equation, the convergence rate turns out to be polynomial in time. This result is in contrast with the known exponential decay of such representation for all other values of $m$.Comment: 37 page

The constraint equations for the Einstein-scalar field system on compact manifolds

We study the constraint equations for the Einstein-scalar field system on compact manifolds. Using the conformal method we reformulate these equations as a determined system of nonlinear partial differential equations. By introducing a new conformal invariant, which is sensitive to the presence of the initial data for the scalar field, we are able to divide the set of free conformal data into subclasses depending on the possible signs for the coefficients of terms in the resulting Einstein-scalar field Lichnerowicz equation. For many of these subclasses we determine whether or not a solution exists. In contrast to other well studied field theories, there are certain cases, depending on the mean curvature and the potential of the scalar field, for which we are unable to resolve the question of existence of a solution. We consider this system in such generality so as to include the vacuum constraint equations with an arbitrary cosmological constant, the Yamabe equation and even (all cases of) the prescribed scalar curvature problem as special cases.Comment: Minor changes, final version. To appear: Classical and Quantum Gravit

Structural Instability in Polyacene : A Projector Quantum Monte Carlo Study

We have studied polyacene within the Hubbard model to explore the effect of electron correlations on the Peierls' instability in a system marginally away from one-dimension. We employ the projector quantum Monte Carlo method to obtain ground state estimates of the energy and various correlation functions. We find strong similarities between polyacene and polyacetylene which can be rationalized from the real-space valence-bond arguments of Mazumdar and Dixit. Electron correlations tend to enhance the Peierls' instability in polyacene. This enhancement appears to attain a maximum at $U/t \sim 3.0$ and the maximum shifts to larger values when the alternation parameter is increased. The system shows no tendency to destroy the imposed bond-alternation pattern, as evidenced by the bond-bond correlations. The cis- distortion is seen to be favoured over the trans- distortion. The spin-spin correlations show that undistorted polyacene is susceptible to a SDW distortion for large interaction strength. The charge-charge correlations indicate the absence of a CDW distortion for the parameters studied.Comment: 13 pages, 10 figures available on reques

Quantitative and qualitative relationship between microstructural factors and fatigue lives under load- And strain-controlled conditions of Ti-5Al-2Sn-2Zr-4Cr-4Mo (Ti-17) fabricated using a 1500-ton forging simulator

The fatigue lives of forged Ti-17 using a 1500-ton forging simulator subjected to different solution treatments and a common aging treatment were evaluated under both load- and strain-controlled conditions: high and low cycle fatigue lives, respectively. Then, the tensile properties and microstructures were also examined. Finally, the relationships among fatigue lives and the microstructural factors and tensile properties were examined. The microstructure after solution treatment at 1203 K, which is more than the β transus temperature, and aging treatment exhibits equiaxed prior β grains composed of fine acicular ¡. On the other hand, the microstructures after solution treatment at temperatures of 1063, 1123, and 1143 K, which are less than the β transus temperature, and aging treatment exhibit elongated prior β grains composed of two different microstructural feature regions, which are acicular α and fine spheroidal α phase regions. The 0.2% proof stress, σ₀.₂, and tensile strength, σB, increase with increasing solution treatment temperature up to 1143 K within the (α + β) region, but decrease with further increasing solution treatment temperature to 1203 K within the β region. The elongation (EL) and reduction of area (RA) decrease with increasing solution treatment temperature, and it becomes nearly 0% corresponding to a solution treatment temperature of 1203 K. The high cycle fatigue limit increases with increasing solution treatment temperature up to 1143 K, corresponding to the (α + β) region. However, it decreases with further increase in the solution treatment temperature to 1203 K in the β region. The fatigue ratio in high cycle fatigue life region is increasing with decreasing solution treatment temperature, namely increasing the volume fraction of the primary α phase, and it relates well qualitatively with the volume fraction of the primary α phase when the solution treatment temperature is less than the β transus temperature. The low cycle fatigue life increases with decreasing solution treatment temperature, namely increasing the volume fraction of the primary α phase. The low cycle fatigue life relates well quantitatively with the tensile true strain at breaking of the specimen and the volume fraction of the primary α phase for each total strain range of low cycle fatigue testing.Niinomi M., Akahori T., Nakai M., et al. Quantitative and qualitative relationship between microstructural factors and fatigue lives under load- And strain-controlled conditions of Ti-5Al-2Sn-2Zr-4Cr-4Mo (Ti-17) fabricated using a 1500-ton forging simulator. Materials Transactions 60, 1740 (2019); https://doi.org/10.2320/matertrans.ME201904

A compactness theorem for scalar-flat metrics on manifolds with boundary

Let (M,g) be a compact Riemannian manifold with boundary. This paper is concerned with the set of scalar-flat metrics which are in the conformal class of g and have the boundary as a constant mean curvature hypersurface. We prove that this set is compact for dimensions greater than or equal to 7 under the generic condition that the trace-free 2nd fundamental form of the boundary is nonzero everywhere.Comment: 49 pages. Final version, to appear in Calc. Var. Partial Differential Equation

On the affine group of a normal homogeneous manifold

A very important class of homogeneous Riemannian manifolds are the so-called normal homogeneous spaces, which have associated a canonical connection. In this work we obtain geometrically the (connected component of the) group of affine transformations with respect to the canonical connection for a normal homogeneous space. The naturally reductive case is also treated. This completes the geometric calculation of the isometry group of naturally reductive spaces. In addition, we prove that for normal homogeneous spaces the set of fixed points of the full isotropy is a torus. As an application of our results it follows that the holonomy group of a homogeneous fibration is contained in the group of (canonically) affine transformations of the fibers, in particular this holonomy group is a Lie group (this is a result of Guijarro and Walschap).Comment: Final version to appear in Ann. Global Anal. Geom