Large Codimension Research Articles

Point Spectrum of Periodic Operators on Universal Covering Trees

Abstract For any multi-graph $G$ with edge weights and vertex potential, and its universal covering tree ${\mathcal{T}}$, we completely characterize the point spectrum of operators $A_{{\mathcal{T}}}$ on ${\mathcal{T}}$ arising as pull-backs of local, self-adjoint operators $A_{G}$ on $G$. This builds on work of Aomoto, and includes an alternative proof of the necessary condition for point spectrum derived in [ 5]. Our result gives a finite time algorithm to compute the point spectrum of $A_{{\mathcal{T}}}$ from the graph $G$, and additionally allows us to show that this point spectrum is itself contained in the spectrum of $A_{G}$. Finally, we prove that typical pull-back operators have a spectral delocalization property: the set of edge weight and vertex potential parameters of $A_{G}$ giving rise to $A_{{\mathcal{T}}}$ with purely absolutely continuous spectrum is open, and its complement has large codimension.

JORDAN–KRONECKER INVARIANTS OF LIE ALGEBRA REPRESENTATIONS AND DEGREES OF INVARIANT POLYNOMIALS

For an arbitrary representation ρ of a complex finite-dimensional Lie algebra, we construct a collection of numbers that we call the Jordan–Kronecker invariants of ρ. Among other interesting properties, these numbers provide lower bounds for degrees of polynomial invariants of ρ. Furthermore, we prove that these lower bounds are exact if and only if the invariants are independent outside of a set of large codimension. Finally, we show that under certain additional assumptions our bounds are exact if and only if the algebra of invariants is freely generated.

Robust Degenerate Unfoldings of Cycles and Tangencies

We construct open sets of degenerate unfoldings of heterodimensional cycles of any co-index $c>0$ and homoclinic tangencies of arbitrary codimension $c>0$. These sets are known to be the support of unexpected phenomena in families of diffeomorphisms, such as the Kolmogorov typical co-existence of infinitely many attractors. As a prerequisite we also construct robust homoclinic tangencies of large codimension which cannot be inside a strong partially hyperbolic set.

Capacities, Removable Sets and Lp-Uniqueness on Wiener Spaces

We prove the equivalence of two different types of capacities in abstract Wiener spaces. This yields a criterion for the Lp-uniqueness of the Ornstein-Uhlenbeck operator and its integer powers defined on suitable algebras of functions vanishing in a neighborhood of a given closed set Σ of zero Gaussian measure. To prove the equivalence we show the Wr,p(B,μ)-boundedness of certain smooth nonlinear truncation operators acting on potentials of nonnegative functions. We discuss connections to Gaussian Hausdorff measures. Roughly speaking, if Lp-uniqueness holds then the ‘removed’ set Σ must have sufficiently large codimension, in the case of the Ornstein-Uhlenbeck operator for instance at least 2p. For p = 2 we obtain parallel results on truncations, capacities and essential self-adjointness for Ornstein-Uhlenbeck operators with linear drift. These results apply to the time zero Gaussian free field as a prototype example.

Fano Weighted Complete Intersections of Large Codimension

Fano Weighted Complete Intersections of Large Codimension

Fano weighted complete intersections of large codimension

Fano weighted complete intersections of large codimension

Robust tangencies of large codimension

We construct C2-robust homoclinic and heterodimensional tangencies of large codimension inside transitive partially hyperbolic sets.

On the space of orthogonal multiplications in three and four dimensions and Cayley’s nodal cubic

The study of orthogonal multiplications is more than 100 years old and goes back to the works of Hurwitz and Radon. Yet, apart from the extensive literature on admissibility of domain and range dimensions near the Hurwitz-Radon range (in codimension \(\le 8\)), only sporadic and fragmentary results are known about full classification (in large codimension), more specifically, about the moduli space \(\mathfrak {M}_m\) of orthogonal multiplications \(F:\mathbb {R}^m\times \mathbb {R}^m\rightarrow \mathbb {R}^n\) (for various n), even for \(m\le 4\). In this paper we give an insight to the subtle geometries of \(\mathfrak {M}_3\) and \(\mathfrak {M}_4\). Orthogonal multiplications are intimately connected to quadratic eigenmaps between spheres via the Hopf-Whitehead construction. The 9-dimensional moduli space \(\mathfrak {M}_3\) lies on the boundary of the 84-dimensional moduli of quadratic eigenmaps of \(S^5\) into spheres. Similarly, the 36-dimensional moduli space \(\mathfrak {M}_4\) is on the boundary of the 300-dimensional moduli of quadratic eigenmaps of \(S^7\) into spheres. We will show that \(\mathfrak {M}_3\) is the \(SO(3)\times SO(3)\)-orbit of a 3-dimensional convex body bounded by Cayley’s nodal cubic surface with vertices in a real projective space \(\mathbb {R}P^3\), the latter imbedded equivariantly and minimally in an 8-sphere of the space of quadratic spherical harmonics on \(S^3\). For \(\mathfrak {M}_4\), we show that it possesses two orthogonal 18-dimensional slices each of which is an \(SO(4)\times SO(4)\)-orbit of a 6-dimensional polytope \(\mathcal {P}\subset \mathbb {R}^6\). This polytope itself is the convex hull of two orthogonal regular tetrahedra. The corresponding orthogonal multiplications are explicitly constructible. Finally, we give an algebraic description of the 24-dimensional space of diagonalizable elements in \(\mathfrak {M}_4\). The crucial fact in \(\mathbb {R}^4\) is the splitting of the exterior product \(\Lambda ^2(\mathbb {R}^4)\) into self-dual and anti-self-dual components. The techniques employed here can be traced back to the work of Ziller and the author (Toth and Ziller 1999) in describing the 18-dimensional moduli for quartic spherical minimal immersions of \(S^3\) into spheres. As a new feature, we point out the importance of multiplicity of the zeros of the polynomials that define the boundary \(\partial \,\mathfrak {M}_m\) as a determinantal variety.

On r-periodic orbits of k-periodic maps

In this paper, we analyze r-periodic orbits of k-periodic difference equations, i.e. and their stability. These special orbits were introduced in S. Elaydi and R.J. Sacker (Global stability of periodic orbits of non-autonomous difference equations and population biology, J. Differ. Equ. 208(1) (2005), pp. 258–273). We discuss that, depending on the values of r and k, such orbits generically only occur in finite dimensional systems that depend on sufficiently many parameters, i.e. they have a large codimension in the sense of bifurcation theory. As an example, we consider the periodically forced Beverton–Holt model, for which explicit formulas for the globally attracting periodic orbit, having the minimal period k = r, can be derived. When r factors k the Beverton–Holt model with two time-variant parameters is an example that can be studied explicitly and that exhibits globally attracting r-periodic orbits. For arbitrarily chosen periods r and k, we develop an algorithm for the numerical approximation of an r-periodic orbit and of the associated parameter set, for which this orbit exists. We apply the algorithm to the generalized Beverton–Holt, the 2D stiletto model, and another example that exhibits periodic orbits with r and k relatively prime.

Out-of-core and compressed level set methods

This article presents a generic framework for the representation and deformation of level set surfaces at extreme resolutions. The framework is composed of two modules that each utilize optimized and application specific algorithms: 1) A fastout-of-coredata management scheme that allows for resolutions of the deforming geometry limited only by the available disk space as opposed to memory, and 2) compact and fastcompressionstrategies that reduce both offline storage requirements and online memory footprints during simulation. Out-of-core and compression techniques have been applied to a wide range of computer graphics problems in recent years, but this article is the first to apply it in the context of level set and fluidsimulations. Our framework is generic and flexible in the sense that the two modules can transparently be integrated, separately or in any combination, into existing level set and fluid simulation software based on recently proposed narrow band data structures like the DT-Grid of Nielsen and Museth [2006] and the H-RLE of Houston et al [2006]. The framework can be applied to narrow band signed distances, fluid velocities, scalar fields, particle properties as well as standard graphics attributes like colors, texture coordinates, normals, displacements etc. In fact, our framework is applicable to a large body of computer graphics problems that involve sequential or random access to very large co-dimension one (level set) and zero (e.g. fluid) data sets. We demonstrate this with several applications, including fluid simulations interacting with large boundaries (≈ 15003), surface deformations (≈ 20483), the solution of partial differential equations on large surfaces (≈ 40963) and mesh-to-level set scan conversions of resolutions up to ≈ 350003(7 billion voxels in the narrow band). Our out-of-core framework is shown to be several times faster than current state-of-the-art level set data structures relying on OS paging. In particular we show sustained throughput (grid points/sec) for gigabyte sized level sets as high as 65% of state-of-the-art throughput for in-core simulations. We also demonstrate that our compression techniques out-perform state-of-the-art compression algorithms for narrow bands.

Matching conditions for a brane of arbitrary codimension

We present matching conditions for distributional sources of arbitrary codimension in the context of Lovelock gravity. Then we give examples, treating maximally symmetric distributional p-branes, embedded in flat, de Sitter and anti-de Sitter spacetime. Unlike Einstein theory, distributional defects of locally smooth geometry and codimension greater than 2 are demonstrated to exist in Lovelock theories. The form of the matching conditions depends on the parity of the brane codimension. For odd codimension, the matching conditions involve discontinuities of Chern-Simons forms and are thus similar to junction conditions for hypersurfaces. For even codimension, the bulk Lovelock densities induce intrinsic Lovelock densities on the brane. In particular, this results in the appearance of the induced Einstein tensor for p>2. For the matching conditions we present, the effect of the bulk is reduced to an overall topological solid angle defect which sets the Planck scale on the brane and to extrinsic curvature terms. Moreover, for topological matching conditions and constant solid angle deficit, we find that the equations of motion are obtained from an exact p+1 dimensional action, which reduces to an induced Lovelock theory for large codimension. In essence, this signifies that the distributional part of the Lovelock bulk equations can naturally give rise to induced gravity terms on a brane of even co-dimension. We relate our findings to recent results on codimension 2 branes.

Two-dimensional random tilings of large codimension: new progress

Two-dimensional random tilings of rhombi can be seen as projections of two-dimensional membranes embedded in hypercubic lattices of higher dimensional spaces. Here, we consider tilings projected from a D-dimensional space. We study the limiting case, when the quantity D, and therefore the number of different species of tiles, become large. We had previously demonstrated [M. Widom, N. Destainville, R. Mosseri, F. Bailly, in: Proceedings of the Sixth International Conference on Quasicrystals, World Scientific, Singapore, 1997.] that, in this limit, the thermodynamic properties of the tiling become independent of the boundary conditions. The exact value of the limiting entropy and finite D corrections remain open questions. Here, we develop a mean-field theory, which uses an iterative description of the tilings based on an analogy with avoiding oriented walks on a random tiling. We compare the quantities so-obtained with numerical calculations. We also discuss the role of spatial correlations.

Reconstruction of 3-submanifolds of large codimension in euclidean spaces from their gauss image

We present necessary and sufficient conditions for a regular three-dimensional manifold in the Grassmannian manifoldG(m, m+3) to be the Gauss image of a regular 3-submanifold in (m+3)-dimensional Euclidean space form>4.

Invariant subspaces of the harmonic Dirichlet space with large co-dimension

In this paper, we comment on the complexity of the invariant subspaces (under the bilateral Dirichlet shift f -* (f) of the harmonic Dirichlet space D. Using the sampling theory of Seip and some work on invariant subspaces of Bergman spaces, we will give examples of invariant subspaces .F C D with dim(.F/(F) = n, n E N U foo}. We will also generalize this to the Dirichlet classes D,, 0 < a < 0o, as well as the Besov classes Bp, 1 < p < 00, 0 < a < 1.

Affine Gelfand-Dickey brackets and holomorphic vector bundles

We define the (second) Adler-Gelfand-Dickey Poisson structure on differential operators over an elliptic curve and classify symplectic leaves of this structure. This problem leads to the problem of classification of coadjoint orbits for double loop algebras, conjugacy classes in loop groups, and holomorphic vector bundles over the elliptic curve. We show that symplectic leaves have a finite but (unlike the traditional case of operators on the circle) arbitrarily large codimension, and compute it explicitly.

PROJECTIONS OF ALGEBRAIC VARIETIES

This paper gives conditions which are necessary and sufficient in order that a generic projection of a projective variety not have singularities other than those coming from the singularities of the original variety. In particular it turns out that this is possible only for varieties of large codimension. Bibliography: 5 titles.

On rigged immersions

A self-contained account is given in an efficient formalism of rigged immersions of one manifold-with-connection in another, leading to the analogues of the Gauss, Codazzi and Ricci equations discovered by Schouten. The equations expressing their interdependence are then derived and it is shown that in general one of the two sets of “Codazzi” equations is a consequence of the other set and the Gauss and Ricci equations. The formalism is specialised to the Riemannian case, where it is shown that, for large codimension (specific limits being given), all butn components of the Codazzi equations are determined by the other equations. A local theorem on the existence of rigged immersions is proved.

Diffeomorphic complete intersections with different multidegrees

A complete intersection Xn(d) C CPn+r of hypersurfaces of degrees dx, . . . , dr is determined up to diffeomorphism by the dimension n and the multidegree which is the unordered r-tuple d = (dx, . . . , dr). A general problem is to find invariants which determine the diffeomorphism type of X. In this note we announce partial results which allow us to prove that in any odd dimension there are infinitely many pairs of multidegrees for which the corresponding complete intersections are diffeomorphic. This follows from a result characterizing the homotopy type of X under some restrictions, an estimation of the number of differential structures with prescribed Pontryagin classes on a given homotopy type, and a counting argument. Complete proofs of these results will appear elsewhere. A necessary feature of these examples is a large codimension.

Thickenings of CW complexes of the form 𝑆^{𝑚}∪_{𝛼}𝑒ⁿ

Necessary conditions are given for the existence of a thickening of S m ∪ α e n {S^m}\,{ \cup _\alpha }\,{e^n} in codimension k. I give examples of such complexes requiring arbitrarily large codimension in order to thicken. Sufficient conditions are given for the existence of a tractable thickening in codimension k + 1 k\, + \,1 . The methods used include the study of the reduced product space of a pair of CW complexes.