An Algebraic Structure of Orthogonal Wavelet Space

In this paper we study the algebraic structure of the space of compactly supported orthonormal wavelets over real numbers. Based on the parameterization of wavelet space, one can define a parameter mapping from the wavelet space of rank 2 (or 2-band, scale factor of 2) and genus gto the (g−1) dimensional real torus (the products of unit circles). By the uniqueness and exactness of factorization, this mapping is well defined and one-to-one. Thus we can equip the rank 2 orthogonal wavelet space with an algebraic structure of the torus. Because of the degenerate phenomenon of the paraunitary matrix, the parameterization map is not onto. However, there exists an onto mapping from the torus to the closure of the wavelet space. And with such mapping, a more complete parameterization is obtained. By utilizing the factorization theory, we present a fast implementation of discrete wavelet transform (DWT). In general, the computational complexity of a rank morthogonal DWT is O(m²g). In this paper we start with a given scaling filter and construct additional (m−1) wavelet filters so that the DWT can be implemented in O(mg). With a fixed scaling filter, the approximation order, the orthogonality, and the smoothness remain unchanged; thus our fast DWT implementation is quite general.     

Degree-regular triangulations of torus and Klein bottle

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices. In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists ann-vertex degree-regular triangulation of the Klein bottle if and only if n is a composite number ≥ 9. We have constructed two distinctn-vertex weakly regular triangulations of the torus for eachn ≥ 12 and a (4m + 2)-vertex weakly regular triangulation of the Klein bottle for eachm ≥ 2. For 12 ≤n ≤ 15, we have classified all then-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.     

Weil-Petersson Areas of The Moduli Spaces Of Tori

We study the Teichmüller spaces of torus with one branch point of order v and of torus with a totally geodesic boundary curve of length m, respectively. Applying the obtained results for the corresponding moduli spaces we find that the Weil-Petersson area of the moduli space of torus with one conical point of order v is (π²/6)(1 - l/v²) and that of the moduli space of torus with a totally geodesic boundary curve of length m is π²/6 + m²/24.     

Torus Actions, Equivariant Moment-Angle Complexes, and Coordinate Subspace Arrangements

We show that the cohomology algebra of the complement of a coordinate subspace arrangement in the m-dimensional complex space is isomorphic to the cohomology algebra of the StanleyReisner face ring of a certain simplicial complex on m vertices. (The face ring is regarded as a module over the polynomial ring on m generators.) After that we calculate the latter cohomology algebra by means of the standard Koszul resolution of a polynomial ring. To prove these facts, we construct a homotopy equivalence (equivariant with respect to the torus action) between the complement of a coordinate subspace arrangement and the moment-angle complex defined by a simplicial complex. The moment-angle complex is a certain subset of the unit polydisk in the m-dimensional complex space invariant with respect to the action of the m-dimensional torus. This complex is a smooth manifold provided that the simplicial complex is a simplicial sphere; otherwise, the complex has a more complicated structure. Then we investigate the equivariant topology of the moment-angle complex and apply the EilenbergMoore spectral sequence. We also relate our results with well-known facts in the theory of toric varieties and symplectic geometry. Bibliography: 23 titles.     

Bifurcations of resonant solutions and chaos in physical pendulum equation with suspension axis vibrations

This paper is a continuation of ＂Complex Dynamics in Physical Pendulum Equation with Suspension Axis Vibrations＂[1].In this paper,we investigate the existence and the bifurcations of resonant solution for ω0：ω：Ω ≈ 1：1：n,1：2：n,1：3：n,2：1：n and 3：1：n by using second-order averaging method,give a criterion for the existence of resonant solution for ω0：ω：Ω ≈ 1：m：n by using Melnikov＇s method and verify the theoretical analysis by numerical simulations.By numerical simulation,we expose some other interesting dynamical behaviors including the entire invariant torus region,the cascade of invariant torus behaviors,the entire chaos region without periodic windows,chaotic region with complex periodic windows,the entire period-one orbits region;the jumping behaviors including invariant torus behaviors converting to period-one orbits,from chaos to invariant torus behaviors or from invariant torus behaviors to chaos,from period-one to chaos,from invariant torus behaviors to another invariant torus behaviors;the interior crisis;and the different nice invariant torus attractors and chaotic attractors.The numerical results show the difference of dynamical behaviors for the physical pendulum equation with suspension axis vibrations between the cases under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations.It exhibits many invariant torus behaviors under the resonant conditions.We find a lot of chaotic behaviors which are different from those under the periodic/quasi-periodic perturbations.However,we did not find the cascades of period-doubling bifurcation.     

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.     

Torus Knots and Dunwoody Manifolds

We obtain an explicit representation as Dunwoody manifolds of all cyclic branched coverings of torus knots of type (p,mp±1), with p > 1 and m > 0.     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

A characterization of the Clifford torus

We provide a characterization of the Clifford torus via a Ricci type condition among minimal surfaces in S⁴. More precisely, we prove that a compact minimal surface in S⁴, with induced metric ds² and Gaussian curvature K, for which the metric is flat away from points where K = 1, is the Clifford torus, provided that m is an integer with m > 2.Received: 8 September 2004     

Completeness of quasi-filiform Lie algebras

It is proved that for any quasi-filiform of non-zero rank the solvable Lie algebra obtained by adjoining a maximal torus of outer derivations is complete. Further, for any positive integer m, it is shown that there exist solvable complete Lie algebras with the second Chevalley–Eilenberg cohomology group of arbitrary dimension.     

Conservation and bifurcation of an invariant torus of a vector field

We consider small perturbations with respect to a small parameter ε≥0 of a smooth vector field in ℝ^n+m possessing an invariant torusT ^m. The flow on the torusT ^m is assumed to be quasiperiodic withm basic frequencies satisfying certain conditions of Diophantine type; the matrix Ω of the variational equation with respect to the invariant torus is assumed to be constant. We investigate the existence problem for invariant tori of different dimensions for the case in which Ω is a nonsingular matrix that can have purely imaginary eigenvalues. Translated fromMatematicheskie Zametki, Vol. 61, No. 1, pp. 34–44, January, 1997. Translated by S. K. Lando     

Sufficient condition for the hyperbolicity of mappings of the torus

We consider mappings of the m-dimensional torus T^m (m ≥ 2) that are C ¹-perturbations of linear hyperbolic automorphisms. We obtain sufficient conditions for such mappings to be one-to-one hyperbolic mappings (i.e., Anosov diffeomorphisms). These results are used to study the blue-sky catastrophe related to the vanishing of a saddle-node invariant torus with a quasiperiodic winding in a system of ordinary differential equations.     

On hyper-Kähler manifolds of typeA ∞

In this paper we shall construct new families of 4m dimensional non-compact complete hyper-Kähler manifolds on whichm dimensional torus acts. In the 4 dimensional case our manifolds should be considered as hyper-Kähler manifolds which correspond to the extended Dynkin diagram of typeA .     

Scheduling independent jobs for torus connected networks with/without link contention

In this paper, we investigate the problem of how to schedule n independent jobs on an m × m torus based network. We develop a model to to quantify the effect of contention for communication links on the dilation of job execution time when multiple jobs share communication links; we then design an efficient algorithm to schedule a set of n independent jobs with different torus size requirements on a given torus with an objective to minimize the total schedule length. We also develop a feasibility algorithm for pre-emptively scheduling a given set of jobs on a torus of given size with a given deadline. We provide analysis for both the algorithms.     

Local torus actions modeled on the standard representation

We introduce the notion of a local torus action modeled on the standard representation (for simplicity, we call it a local torus action). It is a generalization of a locally standard torus action and also an underlying structure of a locally toric Lagrangian fibration. For a local torus action, we define two invariants called a characteristic pair and an Euler class of the orbit map, and prove that local torus actions are classified topologically by them. As a corollary, we obtain a topological classification of locally standard torus actions, which includes the topological classifications of quasi-toric manifolds by Davis and Januszkiewicz and of effective T²-actions on four-dimensional manifolds without nontrivial finite stabilizers by Orlik and Raymond. We discuss locally toric Lagrangian fibrations from the viewpoint of local torus actions. We also investigate the topology of a manifold equipped with a local torus action when the Euler class of the orbit map vanishes.     

Deterministic symmetric rendezvous with tokens in a synchronous torus

In the rendezvous problem, the goal for two mobile agents is to meet whenever this is possible. In the rendezvous with detection problem, an additional goal for the agents is to detect the impossibility of a rendezvous (e.g., due to symmetrical initial positions of the agents) and stop. We consider the rendezvous problem with and without detection for identical anonymous mobile agents (i.e., running the same deterministic algorithm) with tokens in an anonymous synchronous torus with a sense of direction, and show that there is a striking computational difference between one and more tokens. Specifically, we show that (1) two agents with a constant number of unmovable tokens, or with one movable token each, cannot rendezvous in an n×n torus if they have o(logn) memory, while they can solve the rendezvous with detection problem in an n×m torus as long as they have one unmovable token and O(logn+logm) memory; in contrast, (2) when two agents have two movable tokens each then the rendezvous problem (respectively, rendezvous with detection problem) is solvable with constant memory in an arbitrary n×m (respectively, n×n) torus; and finally, (3) two agents with three movable tokens each and constant memory can solve the rendezvous with detection problem in an n×m torus. This is the first publication in the literature that studies tradeoffs between the number of tokens, memory and knowledge the agents need in order to meet in a torus.     

On spectral characterizations of Willmore hypersurfaces in a sphere

Let M be a closed Willmore hypersurface in the sphere S^n＋1（1） （n ≥ 2） with the same mean curvature of the Willmore torus Wm,n-m, if SpecP（M） = Spec^P（Wm,n-m ） （p = 0, 1,2）, then M is Wm,n-m.     

On the toroidal thickness of graphs

The toroidal thickness t₁(G) of a graph G is the minimum value of k such that G is the union of k graphs each of which is embeddable on a torus. We find t₁(G_m), where G_m is the graph obtained from the complete graph K_m by removing a Hamiltonian cycle, and we show that t₁(K_n(3)) = [1/2n] for many values of n. The method of approach involves the construction of sets of triples related to Skolem triples.     

Torus actions on homotopy complex projective spaces

We prove that an effective action of a torus T on a homotopy P^m is linear if m < 4 rk(T) – 1. Examples show that the bound is optimal. Combining this with a theorem of Hattori we conclude that the total Pontrjagin class of such a manifold is given by the usual formula (1 + x²)^{m + 1}.The second named author is an Alfred P. Sloan Research Fellow and was partly supported by an NSF grant.in final form: 10 May 2003