Unfolding the torus: Oscillator geometry from time delays

Summary We present a simple method of plotting the trajectories of systems of weakly coupled oscillators. Our algorithm uses the time delays between the “firings” of the oscillators. For any system ofn weakly coupled oscillators there is an attracting invariantn-dimensional torus, and the attractor is a subset of this invariant torus. The invariant torus intersects a suitable codimension-1 surface of section at an (n−1)-dimensional torus. The dynamics ofn coupled oscillators can thus be reduced,in principle, to the study of Poincaré maps of the (n−1)-dimensional torus. This paper gives apractical algorithm for measuring then−1 angles on the torus. Since visualization of 3 (or higher) dimensional data is difficult we concentrate onn=3 oscillators. For three oscillators, a standard projection of the Poincaré map onto the plane yields a projection of the 2-torus which is 4-to-1 over most of the torus, making it difficult to observe the structure of the attractor. Our algorithm allows a direct measurement of the 2 angles on the torus, so we can plot a 1-to-1 map from the invariant torus to the “unfolded torus” where opposite edges of a square are identified. In the cases where the attractor is a torus knot, the knot type of the attractor is obvious in our projection.     

A point distal transformation of the torus

A point-distal non-distal homeomorphism of the torus is constructed. By a similar construction, a point-distal homeomorphism of then+1-dimensional torus can be constructed, with any two compact subsets ofR ⁿ among its fibres over some factor.     

Demushkin's Theorem in codimension one

Demushkin's Theorem says that any two toric structures on an affine variety X are conjugate in the automorphism group of X. We provide the following extension: Let an (n–1)-dimensional torus T act effectively on an n-dimensional affine toric variety X. Then T is conjugate in the automorphism group of X to a subtorus of the big torus of X. Mathematics Subject Classification: 13A50, 14L30, 14M25, 14R20.     

Global Hypoellipticity and Simultaneous Approximability

A necessary and sufficient condition is given for a sum of squares operator to be globally hypoelliptic on an N-dimensional torus. This condition is expressed in terms of Diophantine approximation properties of the coefficients. The proof of the Theorem is based on L²-estimates and microlocalization.     

Higher topological cyclic homology and the Segal conjecture for tori

We investigate higher topological cyclic homology as an approach to studying chromatic phenomena in homotopy theory. Higher topological cyclic homology is constructed from the fixed points of a version of topological Hochschild homology based on the n-dimensional torus, and we propose it as a computationally tractable cousin of n-fold iterated algebraic K-theory.The fixed points of toral topological Hochschild homology are related to one another by restriction and Frobenius operators. We introduce two additional families of operators on fixed points, the Verschiebung, indexed on self-isogenies of the n-torus, and the differentials, indexed on n-vectors. We give a detailed analysis of the relations among the restriction, Frobenius, Verschiebung, and differentials, producing a higher analog of the structure Hesselholt and Madsen described for 1-dimensional topological cyclic homology.We calculate two important pieces of higher topological cyclic homology, namely topological restriction homology and topological Frobenius homology, for the sphere spectrum. The latter computation allows us to establish the Segal conjecture for the torus, which is to say to completely compute the cohomotopy type of the classifying space of the torus.     

Note on Weakly n-Dimensional Spaces

 Weakly n-dimensional spaces were first distinguished by Karl Menger. In this note we shall discuss three topics concerning this class of spaces: universal spaces, products, and the sum theorem. We prove that there is a universal space for the class of all weakly n-dimensional spaces, present a simpler proof of Tomaszewski’s result about the dimension of a product of weakly n-dimensional spaces, and show that there is an n-dimensional space which admits a pairwise disjoint countable closed cover by weakly n-dimensional subspaces but is not weakly n-dimensional itself. (Received 17 August 2000)     

Note on Weakly n -Dimensional Spaces

 Weakly n-dimensional spaces were first distinguished by Karl Menger. In this note we shall discuss three topics concerning this class of spaces: universal spaces, products, and the sum theorem. We prove that there is a universal space for the class of all weakly n-dimensional spaces, present a simpler proof of Tomaszewski’s result about the dimension of a product of weakly n-dimensional spaces, and show that there is an n-dimensional space which admits a pairwise disjoint countable closed cover by weakly n-dimensional subspaces but is not weakly n-dimensional itself.     

Simplices and Spectra of Graphs

In this note we show that the (n−2)-dimensional volumes of codimension 2 faces of an n-dimensional simplex are algebraically independent quantities of the volumes of its edge-lengths. The proof involves computation of the eigenvalues of Kneser graphs. We also show examples of families of simplices (of dimension 4 or greater) which show that the set of (n−2)-dimensional volumes of (n−2)-dimensional faces of a simplex do not determine its volume.     

Isometric Embeddings of Families of Special Lagrangian Submanifolds

We prove that certain Riemannian manifolds can be isometrically embedded inside Calabi–Yau manifolds. For example, we prove that given any real-analytic one parameter family of Riemannian metrics g _t on a three-dimensional manifold Y with volume form independent of t and with a real-analytic family of nowhere vanishing harmonic one forms θ _t , then (Y,g _t ) can be realized as a family of special Lagrangian submanifolds of a Calabi–Yau manifold X. We also prove that certain principal torus bundles can be equivariantly and isometrically embedded inside Calabi-Yau manifolds with torus action. We use this to construct examples of n-parameter families of special Lagrangian tori inside n + k-dimensional Calabi–Yau manifolds with torus symmetry. We also compute McLean's metric of 3-dimensional special Lagrangian fibrations with T ²-symmetry. Mathematics Subject Classification (2000): 53-XX, 53C38.Communicated by N. Hitchin (Oxford)     

Loop-Erased Random Walk on Finite Graphs and the Rayleigh Process

Let (G _n ) _n=1^∞ be a sequence of finite graphs, and let Y _t be the length of a loop-erased random walk on G _n after t steps. We show that for a large family of sequences of finite graphs, which includes the case in which G _n is the d-dimensional torus of size-length n for d≥4, the process (Y _t ) _t=0^∞, suitably normalized, converges to the Rayleigh process introduced by Evans, Pitman, and Winter. Our proof relies heavily on ideas of Peres and Revelle, who used loop-erased random walks to show that the uniform spanning tree on large finite graphs converges to the Brownian continuum random tree of Aldous. Supported in part by NSF Grant DMS-0504882.     

Global rigidity results for lattice actions on tori and new examples of volume-preserving actions

Any action of a finite index subgroup in SL(n,ℤ),n≥4 on then-dimensional torus which has a finite orbit and contains an Anosov element which splits as a direct product is smoothly conjugate to an affine action. We also construct first examples of real-analytic volume-preserving actions of SL(n,ℤ) and other higher-rank lattices on compact manifolds which are not conjugate (even topologically) to algebraic models. This work was partially supported by NSF grant DMS9017995.     

On existence of optimal solutions for stochastic differential equations and inclusions with current velocities

For a stochastic differential inclusion given in terms of current velocities (symmetric mean derivatives) on flat n-dimensional torus, we prove the existence of optimal solution minimizing a certain cost criterion. Then this result is applied to the problem of optimal control for equations with current velocities.     

Solving discrete zero point problems

In this paper we present two theorems on the existence of a discrete zero point of a function from the n-dimensional integer lattice ℤⁿ to the n-dimensional Euclidean space ℝⁿ. The theorems differ in their boundary conditions. For both theorems we give a proof using a combinatorial lemma and present a constructive proof based on a simplicial algorithm that finds a discrete zero point within a finite number of steps. This research was carried out when Gerard van der Laan and Zaifu Yang were visiting the CentER of Tilburg University in the summer of 2004. The visit of Zaifu Yang has been made possible by financial support of CentER and the Netherlands Organization for Scientific Research (NWO). The authors gratefully acknowledge the inspiring and helpful remarks of the referees.     

The Upper Bound Conjecture and Cohen-Macaulay Rings

Let Δ be a triangulation of a (d ? 1)-dimensional sphere with n vertices. The Upper Bound Conjecture states that the number of i-dimensional faces of Δ is less than or equal to a certain explicit number c_i(n, d). A proof is given of a more general result. The proof uses the result, proved by G. Reisner, that a certain commutative ring associated with Δ is a Cohen-Macaulay ring.     

Dense Families of Selections and Finite-Dimensional Spaces

A characterization of n-dimensional spaces via continuous selections avoiding Z _n -sets is given, and a selection theorem for strongly countable-dimensional spaces is established. We apply these results to prove a generalized Ostrand's theorem, and to obtain a new alternative proof of the Hurewicz formula. It is also shown that our selection theorem yields an easy proof of a Michael's result.     

On One Method for the Introduction of Local Coordinates in a Neighborhood of an Invariant Toroidal Set

We consider one method for the introduction of local coordinates in a neighborhood of an m-dimensional invariant torus of a dynamical system of differential equations in the Euclidean space R ⁿ in dimensions satisfying the inequalities m + 1 < n 2m.     

Regular simplices and lower volume bounds for hyperbolicn-manifolds

For an-dimensional compact hyperbolic manifoldM ⁿ a new lower volume bound is presented. The estimate depends on the volume of a hyperbolic regularn-simplex of edge length equal to twice the in-radius ofM ⁿ. Its proof relies upon local density bounds for hyperbolic sphere packings.     

On the construction and topological invariance of the Pontryagin classes

We use sheaves and algebraic L-theory to construct the rational Pontryagin classes of fiber bundles with fiber \mathbbRⁿ{\mathbb{R}^n} . This amounts to an alternative proof of Novikov’s theorem on the topological invariance of the rational Pontryagin classes of vector bundles. Transversality arguments and torus tricks are avoided.     

Amalgamation in finite dimensional cylindric algebras

For every finite n > 1, the embedding property fails in the class of all n-dimensional cylindric type algebras which satisfy the following. Their boolean reducts are boolean algebras and two of the cylindrifications are normal, additive and commute. This result also holds for all subclasses containing the representable n-dimensional cylindric algebras. This considerably strengthens a result of S. Comer on CA _n and provides a strong counterexample for interpolation in finite variable fragments of first order logic. We provide a new modern proof, using an argument inspired by modal logic. February 22, 1999.