Convex Manifolds Research Articles

The Lie group of real analytic diffeomorphisms is not real analytic

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense.

From <i>E</i> = <i>mc</i><sup>2</sup> to <i>E</i> = <i>mc</i><sup>2</sup>/22—A Short Account of the Most Famous Equation in Physics and Its Hidden Quantum Entanglement Origin

Einstein’s energy mass formula is shown to consist of two basically quantum components E(O) = mc2/22 and E(D) = mc2(21/22). We give various arguments and derivations to expose the quantum entanglement physics residing inside a deceptively simple expression E = mc2. The true surprising aspect of the present work is however the realization that all the involved “physics” in deriving the new quantum dissection of Einstein’s famous formula of special relativity is actually a pure mathematical necessity anchored in the phenomena of volume concentration of convex manifold in high dimensional quasi Banach spaces. Only an endophysical experiment encompassing the entire universe such as COBE, WMAP, Planck and supernova analysis could have discovered dark energy and our present dissection of Einstein’s marvelous formula.

The Hidden Quantum Entanglement Roots of E = mc2 and Its Genesis to E = (mc2/22) Plus mc2 (21/22) Confirming Einstein’s Mass-Energy Formula

Einstein’s energy mass formula is shown to consist of two basically quantum components E (O) = mc2/22 and E(D) = mc2(21/22). We give various arguments and derivations to expose the quantum entanglement physics residing inside a deceptively simple expression E = mc2. The true surprising aspect of the present work is however the realization that all the involved ‘physics’ in deriving the new quantum dissection of Einstein’s famous formula of special relativity is actually a pure mathematical necessity anchored in the phenomena of volume concentration of convex manifold in high dimensional quasi Banach spaces. Thus the present derivation plus the measurements of accelerated cosmic expansion are endophysical theoretical and experimental confirmation of Einstein but with a new quantum relativistic interpretation.

A semigroup approach to functional differential evolution equations

We study functional differential evolution equations of the form , where − A is infinitesimal generator of an analytic semigroup in a Banach space E (with or without order) and the given right‐hand side modelling delay. In many cases, E is a Banach space of sections, such as vector field or differential forms of a (real or complex) vector bundle Ω (of possibly infinite dimension) over a locally convex manifold, for example, a Carathéodory–Finsler manifold. The operator A may in particular be generated by a Dirichlet form acting on an ordered Hilbert space. As an application, we consider a problem from thermo‐magnetohydrodynamics. Copyright © 2013 John Wiley & Sons, Ltd.

A View on Optimal Transport from Noncommutative Geometry

We discuss the relation between the Wasserstein distance of order 1 between probability distributions on a metric space, arising in the study of Monge-Kantorovich transport problem, and the spectral distance of noncommutative geometry. Starting from a remark of Rieffel on compact manifolds, we first show that on any - i.e. non-necessary compact - complete Riemannian spin manifolds, the two distances coincide. Then, on convex manifolds in the sense of Nash embedding, we provide some natural upper and lower bounds to the distance between any two probability distributions. Specializing to the Euclidean space $R^n$, we explicitly compute the distance for a particular class of distributions generalizing Gaussian wave packet. Finally we explore the analogy between the spectral and the Wasserstein distances in the noncommutative case, focusing on the standard model and the Moyal plane. In particular we point out that in the two-sheet space of the standard model, an optimal-transport interpretation of the metric requires a cost function that does not vanish on the diagonal. The latest is similar to the cost function occurring in the relativistic heat equation.

A matrix-free cone complementarity approach for solving large-scale, nonsmooth, rigid body dynamics

This paper proposes an iterative method that can simulate mechanical systems featuring a large number of contacts and joints between rigid bodies. The numerical method behaves as a contractive mapping that converges to the solution of a cone complementarity problem by means of iterated fixed-point steps with separable projections onto convex manifolds. Since computational speed and robustness are important issues when dealing with a large number of frictional contacts, we have performed special algorithmic optimizations in order to translate the numerical scheme into a matrix-free algorithm with O( n) space complexity and easy implementation. A modified version, that can run on parallel computers is discussed. A multithreaded version of the method has been used to simulate systems with more than a million contacts with friction.

Log-Sobolev inequality on non-convex Riemannian manifolds

Counterexamples are constructed to show that when the second fundamental form of the boundary is bounded below by a negative constant, any curvature lower bound is not enough to imply the log-Sobolev inequality. This indicates that in the study of functional inequalities on non-convex manifolds, the concavity of the boundary cannot be compensated by the positivity of the curvature. Next, when the boundary is merely concave on a bounded domain, a criterion on the log-Sobolev inequality known for convex manifolds is proved. Finally, when the concave part of the boundary is unbounded, a Sobolev inequality for a weighted volume measure is established, which implies an explicit sufficient condition for the log-Sobolev inequality to hold on non-convex manifolds.

The Levy-Gromov Isoperimetric Inequality in Convex Manifolds with Boundary

We observe after Bayle and Rosales that the Levy-Gromov isoperimetric inequality generalizes to convex manifolds with boundary and certain singularities.

Correction to the article Covering of a holomorphically convex manifold carrying a positive line bundle

Correction to the article Covering of a holomorphically convex manifold carrying a positive line bundle

Locally Stein domains over holomorphically convex manifolds

Let $\pi : Y \longrightarrow X$ be a domain over a complex space $X$. Assume that $\pi$ is locally Stein. Then we show that $Y$ is Stein provided that $X$ is Stein and either there is an open set $W$ containing $X_{\mathrm{sing}}$ with $\pi^{-1}(W)$ Stein or $\pi$ is locally hyperconvex over any point in $X_{\mathrm{sing}}$. In the same vein we show that, if $X$ is $q$-complete and $X$ has isolated singularities, then $Y$ results $q$-complete.

Center vortices and the Gribov horizon

We show how the infinite color-Coulomb energy of color-charged states is related to enhanced density of near-zero modes of the Faddeev-Popov operator, and calculate this density numerically for both pure Yang-Mills and gauge-Higgs systems at zero temperature, and for pure gauge theory in the deconfined phase. We find that the enhancement of the eigenvalue density is tied to the presence of percolating center vortex configurations, and that this property disappears when center vortices are either removed from the lattice configurations, or cease to percolate. We further demonstrate that thin center vortices have a special geometrical status in gauge-field configuration space: Thin vortices are located at conical or wedge singularities on the Gribov horizon. We show that the Gribov region is itself a convex manifold in lattice configuration space. The Coulomb gauge condition also has a special status; it is shown to be an attractive fixed point of a more general gauge condition, interpolating between the Coulomb and Landau gauges.

REAL-TIME FACIAL POSE IDENTIFICATION WITH HIERARCHICALLY STRUCTURED ML POSE CLASSIFIER

Since pose-varying face images form nonlinear convex manifold in high dimensional image space, it is difficult to model their pose distribution in terms of a simple probabilistic density function. To solve this difficulty, we divide the pose space into many constituent pose classes and treat the continuous pose estimation problem as a discrete pose-class identification problem. We propose to use a hierarchically structured ML (Maximum Likelihood) pose classifiers in the reduced feature space to decrease the computation time for pose identification, where pose space is divided into several pose groups and each group consists of a number of similar neighboring poses. We use the CONDENSATION algorithm to find a newly appearing face and track the face with a variety of poses in real-time. Simulation results show that our proposed pose identification using the hierarchically structured ML pose classifiers can perform a faster pose identification than conventional pose identification using the flat structured ML pose classifiers. A real-time facial pose tracking system is built with high speed hierarchically structured ML pose classifiers.

The cohomology of Banach space bundles over 1–convex manifolds is not always Hausdorff

AbstractA holomorphic Hilbert space bundle E over ℙ1 is constructed such that H1(ℙ1, E) is not Hausdorff. Lifting this bundle to 𝒪 (–1) it is shown that such bundles exist also on non–compact 1–convex spaces.

Le problème de Neumann pour certaines équations du type de Monge-Ampère sur une variété riemannienne

AbstractLet (Mn, g) be a strictly convex riemannian manifold with C∞ boundary. We prove the existence of classical solution for the nonlinear elliptic partial differential equation of Monge-Ampère: det in M with a Neumann condition on the boundary of the form , where is an everywhere strictly positive function satisfying some assumptions, ν stands for the unit normal vector field and is a non-decreasing function in u.

Necessary and sufficient conditions for the symmetrizability of dierential operators over innite dimensional state spaces

In this paper, we establish some necessary and sufficient conditions for the symmetrizability of second-order differential operators over infinite dimensional state spaces including locally convex spaces and separable metrizable Banach manifolds. In particular, we provide an abstract and simple criterion which can be applied to verify many important known examples.

Co)bordism Groups in Quantum PDEs

In this paper we formulate a theory of noncommutative manifolds (quantum manifolds) and for such manifolds we develop a geometric theory of quantum PDEs (QPDEs). In particular, a criterion of formal integrability is given that extends to QPDEs previously obtained by D. C. Spencer and H. Goldschmidt for PDEs for commutative manifolds, and by Prastaro for super PDEs. Quantum manifolds are seen as locally convex manifolds where the model has the structure A m 1 1×···×A m s s , with A≡A 1×···×A s a noncommutative algebra that satisfies some particular axioms (quantum algebra). A general theory of integral (co)bordism for QPDEs is developed that extends our previous for PDEs. Then, noncommutative Hopf algebras (full quantum p-Hopf algebras, 0≤p≤m−1) are canonically associated to any QPDE, Ek ⊂ Ĵk m(W) whose elements represent all the possible invariants that can be recognized for such a structure. Many examples of QPDEs are considered where we apply our theory. In particular, we carefully study QPDEs for quantum field theory and quantum supergravity. We show that the corresponding regular solutions, observed by means of quantum relativistic frames, give curvature, torsion, gravitino and electromagnetic fields as A-valued distributions on spacetime, where A is a quantum algebra. For such equations, canonical quantizations are obtained and the quantum and integral bordism groups and the full quantum p-Hopf algebras, 0≤p≤3, are explicitly calculated. Then, the existence of (quantum) tunnel effects for quantum superstrings in supergravity is proved.

An Isoperimetric Inequality and the First Steklov Eigenvalue

Let (Mn, g) be a compact Riemannian manifold with boundary. In this paper we give upper and lower estimates for the first nonzero Steklov eigenvalueΔϕ=0inM,∂ϕ∂η=ν1ϕon∂M,where ν1 is a positive real number. The estimate from below is for a star-shaped domain on a manifold whose Ricci curvature is bounded from below. The upper estimate is for a convex manifold with nonnegative Ricci curvature and is given in terms of the first nonzero eigenvalue for the Laplacian on the boundary. We prove a comparison theorem for simply connected domains in a simply connected manifold. We exhibit annuli domains for which the comparison theorem fails to be true. In (J. F. Escobar, J. Funct. Anal.60 (1997), 544–556) we introduced the isoperimetric constant I(M) defined asI(M)=infΩ⊂MVol(Σ)min{Vol(Ω1), Vol(Ω2)}, where Ω1=Ω∩∂M is a nonempty domain with boundary in the manifold ∂M, Ω2=∂M−Ω1, and Σ=∂Ω∩int(M), where int(M) is the interior of M. We proved a Cheeger's type inequality for ν1 using the constant I(M). In this paper we give upper and lower estimates for the constant I in terms of isoperimetric constants of the boundary of M.

Covering of a holomorphically convex manifold carrying a positive line bundle

A classical theorem of Siegel [S] asserts that a bounded domain in C covering a compact complex manifold is a domain of holomorphy. In [Wa] Watanabe showed that if a complex manifold D covering a projective manifold M and satisfies H1(D,O∗) = 0, then D is a Stein manifold with H(D,Z) = 0, where O∗ is the sheaf of germs of nowhere-vanishing holomorphic functions and Z is the additive group of integers. The purpose of this paper is to study the case where the base of covering is a holomorphically convex complex manifold carrying a positive holomorphic line bundle. Recall that a complex manifold M is holomorphically convex if, for every infinite subset S of M without limit points, there is a holomorphic function f on M which is unbounded on S. Our main result is the following:

On the maximalC *-algebra of zeros of completely positive mapping and on the boundary of a dynamic semigroup

In this paper, we study the parametrization of conservative extensions of minimal dynamic semigroups used in the quantum theory of open systems for describing irreversible processes. The particular case of extensions parametrized by normal states is discussed in [1, 2]. The construction used below is similar to that of self-ajoint extensions for densely-defined, bounded below, symmetric operators in a Hilbert space. The dynamic semigroup, unlike the unitary operator group, acts not on elements of a Hilbert space 7/but on elements of a topological operator algebra in a Hilbert space B(H); moreover, this semigroup is completely positive (see [3]). For such semigroups, direct methods of describing domains of generating mappings are absent; this complicates the parametrization of extensions of the minimal generating mapping. It turns out that the range of the resolvent of the minimal dynamic semigroup belongs to the complement of the zero set (the kernel) of the limit mapping QOO(.) associated with the formal generating mapping of the dynamic semigroup. For this reason, we first consider general properties of the zero set for a completely positive mapping (see Sec. 1). In what follows, the C*-algebra generated by positive roots of a mapping Q(-) plays the central role. In Sec. 2, we prove that this algebra has the following property: it is maximal in the class of algebras that contain an element X if and only if X* X and X X* are elements of the algebra. In Sec. 3, the mentioned property is extended to the set of positive operators that are zeros of the limit mapping QOO(.) _- s-lira Qn (.). We prove that the domain of the infinitesimal mapping of the minimal dynamic semigroup is contained in the maximal C*-Mgebra generated by positive roots of the mapping Q~(-) associated with the semigroup. In Sec. 4, we give a construction of conservative extensions of the minimal dynamic semigroup, which is based on limit properties of the resolvent mapping. In Sec. 5, we prove the characteristic theorem for completely positive mappings with given ranges, which enables us to consider the set of conservative extensions as a *-weakly compact convex manifold in/3+(/2(7-~)), whose points admit an expansion in the set of extreme elements. The barycentric expansion of elements T in extreme points of E C T establishes a one-to-one correspondence between classes of equivalent normed measures on E and conservative extensions of the minimal dynamic semigroup; in this case, E plays the role of boundary and the class of equivalent measures plays the role of boundary condition for a conservative extension.

Some invariants of convex manifolds

Some invariants of convex manifolds