Set Of Codimension Research Articles

Delta‐convex structure of the singular set of distance functions

AbstractFor the distance function from any closed subset of any complete Finsler manifold, we prove that the singular set is equal to a countable union of delta‐convex hypersurfaces up to an exceptional set of codimension two. In addition, in dimension two, the whole singular set is equal to a countable union of delta‐convex Jordan arcs up to isolated points. These results are new even in the standard Euclidean space and shown to be optimal in view of regularity.

Boundary regularity and stability for spaces with Ricci bounded below

This paper studies the structure and stability of boundaries in noncollapsed {{,mathrm{RCD},}}(K,N) spaces, that is, metric-measure spaces (X,{mathsf {d}},{mathscr {H}}^N) with Ricci curvature bounded below. Our main structural result is that the boundary partial X is homeomorphic to a manifold away from a set of codimension 2, and is N-1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits (M_i^N,{mathsf {d}}_{g_i},p_i) rightarrow (X,{mathsf {d}},p) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary partial X. The key local result is an varepsilon -regularity theorem, which tells us that if a ball B_{2}(p)subset X is sufficiently close to a half space B_{2}(0)subset {mathbb {R}}^N_+ in the Gromov–Hausdorff sense, then B_1(p) is biHölder to an open set of {mathbb {R}}^N_+. In particular, partial X is itself homeomorphic to B_1(0^{N-1}) near B_1(p). Further, the boundary partial X is N-1 rectifiable and the boundary measure is Ahlfors regular on B_1(p) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence X_irightarrow X. Specifically, we show a boundary volume convergence which tells us that the N-1 Hausdorff measures on the boundaries converge to the limit Hausdorff measure on partial X. We will see that a consequence of this is that if the X_i are boundary free then so is X.

Codimension-1 simplices in divisible convex domains

Properly embedded simplices in a convex divisible domain $\Omega \subset \mathbb{R} \textrm{P}^d$ behave somewhat like flats in Riemannian manifolds, so we call them flats. We show that the set of codimension-$1$ flats has image which is a finite collection of disjoint virtual $(d-1)$-tori in the compact quotient manifold. If this collection of virtual tori is non-empty, then the components of its complement are cusped convex projective manifolds with type $d$ cusps.

Rough Diffeomorphisms with Basic Sets of Codimension One

The review is devoted to the exposition of results (including those of the authors of the review) obtained from the 2000s until the present, on topological classification of structurally stable cascades defined on a smooth closed manifold M n (n ≥ 3) assuming that their nonwandering sets either contain an orientable expanding (contracting) attractor (repeller) of codimension one or completely consist of basic sets of codimension one. The results presented here are a natural continuation of the topological classification of Anosov diffeomorphisms of codimension one. The review also reflects progress related to construction of the global Lyapunov function and the energy function for dynamical systems on manifolds (in particular, a construction of the energy function for structurally stable 3-cascades with a nonwandering set containing a two-dimensional expanding attractor is described).

A singular radial connection over $$\mathbb B^5$$ B 5 minimizing the Yang-Mills energy

We prove that the pullback of the \(SU(n)\)-soliton of Chern number \(c_2=1\) over \(\mathbb S^4\) via the radial projection \(\pi :\mathbb B^5{\setminus }\{0\}\rightarrow \mathbb S^4\) minimizes the Yang-Mills energy under a topologically fixed boundary trace constraint. In particular this shows that stationary Yang-Mills connections in high dimension can have singular sets of codimension \(5\).

Statistics of Gaussian packets on metric and decorated graphs

We study a semiclassical asymptotics of the Cauchy problem for a time-dependent Schrödinger equation on metric and decorated graphs with a localized initial function. A decorated graph is a topological space obtained from a graph via replacing vertices with smooth Riemannian manifolds. The main term of an asymptotic solution at an arbitrary finite time is a sum of Gaussian packets and generalized Gaussian packets (localized near a certain set of codimension one). We study the number of packets as time tends to infinity. We prove that under certain assumptions this number grows in time as a polynomial and packets fill the graph uniformly. We discuss a simple example of the opposite situation: in this case, a numerical experiment shows a subexponential growth.

Standing ring blowup solutions for cubic nonlinear Schrödinger equations

For all dimensions N≥3 we prove there exist solutions to the focusing cubic nonlinear Schrodinger equations that blow up on a set of codimension two. The blowup set is identified both as the site of L2 concentration and by a bounded supercritical norm outside any neighborhood of the set. In all cases, the global H1 norm grows at the log-log rate.

On the Periodic Orbits of the Static, Spherically Symmetric Einstein-Yang-Mills Equations

In this paper we analyze the existence of the periodic orbits of the static, spherically symmetric Einstein–Yang–Mills Equations by using the qualitative theory of the ordinary differential equation. We prove that there are no periodic orbits restricted to some invariant set of codimension 1. Furthermore if there is a periodic orbit out of this invariant set, then there must be other periodic orbits, which are symmetric to the first one. We also have results on the non–existence of periodic orbits when the cosmological constant is negative.

On Kulikov’s problem

Kulikov has given an etale morphism \(F : X \rightarrow {\mathbb{C}}^n\) of degree d > 1 which is surjective modulo codimension two with X simply connected, settling his generalized jacobian problem. His method reduces the problem to finding a hypersurface \(D \subset {\mathbb{C}}^n\) and a subgroup \(G \subset \pi_{1}({\mathbb{C}}^n - D)\) of index d generated by geometric generators. By contrast we show that if D has simple normal crossings away from a set of codimension three and \(\overline{D} \subset {\mathbb{P}}^n\) meets the hyperplane at infinity transversely, then necessarily d = 1.

Dynamics of local groups of conformal mappings and generic properties of differential equations on ℂ2

This paper is a survey of the present state of the problems related to the generic properties of foliations defined on ℂ2 by algebraic differential equations. We prove that the properties of density, absolute rigidity, and existence of a countable set of complex limit cycles are inherent in all equations except possibly for the union of some real algebraic set and real analytic set of codimension at least two in the space of coefficients.

Path collapse for multidimensional Brownian motion with rebirth

In a bounded open region of the d-dimensional Euclidean space we consider a Brownian motion which is reborn at a fixed interior point as soon as it reaches the boundary. It was shown that in dimension one coupled paths starting at different points but driven by the same Brownian motion either collapse with probability one or never meet. In higher dimensions, for convex or polyhedral regions, the paths with positive probability of collapse differ at start by a vector from a set of codimension one. The problem can be interpreted in terms of the long term mixing properties of the payoff of a portfolio of knock-out barrier options in derivatives markets.

Rotating Singular Perturbations of the Laplacian

We study a system of a quantum particle interacting with a singular time-dependent uniformly rotating potential in 2 and 3 dimensions: in particular we consider an interaction with support on a point (rotating point interaction) and on a set of codimension 1 (rotating blade). We prove the existence of the Hamiltonians of such systems as suitable self-adjoint operators and we give an explicit expression for their unitary semigroups. Moreover we analyze the asymptotic limit of large angular velocity and we prove strong convergence of the time-dependent propagator to some one-parameter unitary group as (\omega \to \infty).

Computing Nash equilibria by iterated polymatrix approximation

This article develops a new algorithm for computing Nash equilibria of N-player games. The algorithm approximates a game by a sequence of polymatrix games in which the players interact bilaterally. We provide sufficient conditions for local convergence to an equilibrium and report computational experience. The algorithm convergences globally and rapidly on test problems, although in theory it is not failsafe because it can stall on a set of codimension 1. But it can stall only at an approximate equilibrium with index +1, thus allowing a switch to the global Newton method, which is slower but can fail only on a set of codimension 2. Thus, the algorithm can be used to obtain a fast start for the more reliable global Newton method.

Two physical applications of the Laplace operator perturbed on a null set

Two physical applications of the Laplace operator perturbed on a set of zero measure are suggested. The approach is based on the theory of self-adjoint extensions of symmetrical operators. The first applicatio is a solvable model of scattering of a plane wave by a perturbed thin cylinder. “Nonlocal” extensions are described. The model parameters can be chosen such that the model solution is an approximation of the corresponding “realistic” solution. The second application is the description of the time evolution of a one-dimensional quasi-Chaplygin medium, which can be reduced using a hodograph transform to the ill-posed problem of the Laplace operator perturbed on a set of codimension two inR3. Stability and instability conditions are obtained.

Perturbations of elliptic operators on high codimension subsets and the extension theory on an indefinite metric space

The spectral aspect of the problem of perturbations supported on thin sets of codimension θ≥2m in ℝn is considered for elliptic operators of order m. The problem of realization of such perturbations is formulated as a problem of self-adjoint extension of a linear symmetric relation in a space with indefinite metric. It is shown how to construct such a relation for a given elliptic operator and a family of distributions. Its functional model is obtained in terms of Q-functions. Self-adjoint extensions and their resolvents are described. The theory developed is applied to quantum models of point interactions in high dimensions and high moments. Bibliography: 35 titles.

Remarks on Singularities, Dimension and Energy Dissipation for Ideal Hydrodynamics and MHD

For weak solutions of the incompressible Euler equations, there is energy conservation if the velocity is in the Besov space B3s with s greater than 1/3. B3s consists of functions that are Lip(s) (i.e., Holder continuous with exponent s) measured in the Lp norm. Here this result is applied to a velocity field that is Lip(α0) except on a set of co-dimension \(\) on which it is Lip($agr;1), with uniformity that will be made precise below. We show that the Frisch-Parisi multifractal formalism is valid (at least in one direction) for such a function, and that there is energy conservation if \(\). Analogous conservation results are derived for the equations of incompressible ideal MHD (i.e., zero viscosity and resistivity) for both energy and helicity . In addition, a necessary condition is derived for singularity development in ideal MHD generalizing the Beale-Kato-Majda condition for ideal hydrodynamics.

Applications of the signs of Melnikov's function

The existence and stability of periodic solutions for the two-dimensional system x′=f(x)+ɛg(x,α), 0<ɛ≪1, α∈R whose unperturbed system is Hamiltonian can be decided by using the signs of Melnikov's function. The results can be applied to the construction of phase portraits in the bifurcation set of codimension two bifurcations of flows with double zero eigenvalues.

Continuation with estimates from analytic sets of codimension 1

Continuation with estimates from analytic sets of codimension 1

Stokes’ phenomenon; smoothing a victorian discontinuity

Small exponentials in asymptotic representations of functionsy(k; X) (k → ∞) can appear and disappear across sets of codimension 1 in the space of variables X. These changes are not discontinuous but happen smoothly and according to a universal law.

Multiplicities of the eigenvalues of the discrete Schrödinger equation in any dimension

The following von Neumann-Wigner type result is proved: The set of potentials a : Γ → R ( Γ ⊆ Z N ) a:\;\Gamma \to {\mathbf {R}}(\Gamma \subseteq {{\mathbf {Z}}^N}) , with the property that the corresponding discrete Schrödinger equation Δ d + a {\Delta _d} + a has multiple eigenvalues when considered with certain boundary conditions, is an algebraic set of codimension ≥ 2 {\text {codimension}} \geq {\text {2}} within R Γ {{\mathbf {R}}^\Gamma } .