Lie jets and symmetries of prolongations of geometric objects

The Lie jet L _θ λ of a field of geometric objects λ on a smooth manifold M with respect to a field θ of Weil A-velocities is a generalization of the Lie derivative L _v λ of a field λ with respect to a vector field v. In this paper, Lie jets L _θ λ are applied to the study of A-smooth diffeomorphisms on a Weil bundle T ^A M of a smooth manifold M, which are symmetries of prolongations of geometric objects from M to T ^A M. It is shown that vanishing of a Lie jet L _θ λ is a necessary and sufficient condition for the prolongation λ ^A of a field of geometric objects λ to be invariant with respect to the transformation of the Weil bundle T ^A M induced by the field θ. The case of symmetries of prolongations of fields of geometric objects to the second-order tangent bundle T ² M are considered in more detail.     

Weak mixing disc and annulus diffeomorphisms with arbitrary Liouville rotation number on the boundary

Let M be an m-dimensional differentiable manifold with a nontrivial circle action S={St_}t∈R, St+1=St, preserving a smooth volume μ. For any Liouville number α we construct a sequence of area-preserving diffeomorphisms Hn such that the sequence converges to a smooth weak mixing diffeomorphism of M. The method is a quantitative version of the approximation by conjugations construction introduced in [Trans. Moscow Math. Soc. 23 (1970) 1].For m=2 and M equal to the unit disc D²={x²+y²?1} or the closed annulus A=T×[0,1] this result proves the following dichotomy: α∈R?Q is Diophantine if and only if there is no ergodic diffeomorphism of M whose rotation number on the boundary equals α (on at least one of the boundaries in the case of A). One part of the dichotomy follows from our constructions, the other is an unpublished result of Michael Herman asserting that if α is Diophantine, then any area preserving diffeomorphism with rotation number α on the boundary (on at least one of the boundaries in the case of A) displays smooth invariant curves arbitrarily close to the boundary which clearly precludes ergodicity or even topological transitivity.     

Exact functions on manifolds

It is proved that the property of a manifold Mⁿ possessing a smooth function with given numbers of critical points of each index is homotopic invariant if Wh( ₁ (Mⁿ)) = 0 and every Z( ₁ (Mⁿ))-stable free module is free.Translated from Matematicheskie Zametki, Vol. 8, No. 1, pp. 77–83, July, 1970.     

On the ideal (<Emphasis Type="Italic">v</Emphasis><Superscript>0</Superscript>)

The σ-ideal (v ⁰) is associated with the Silver forcing, see [5]. Also, it constitutes the family of all completely doughnut null sets, see [9]. We introduce segment topologies to state some resemblances of (v ⁰) to the family of Ramsey null sets. To describe add(v ⁰) we adopt a proof of Base Matrix Lemma. Consistent results are stated, too. Halbeisen’s conjecture cov(v ⁰) = add(v ⁰) is confirmed under the hypothesis t = min{cf(c), r}. The hypothesis cov(v ⁰) = ω ₁ implies that (v ⁰) has the ideal type (c, ω ₁, c).      

Entire functions decaying rapidly on strips

It has been known for many years that there exist non-constant entire functions f which decay to 0 along every infinite line. Recently it has been shown that if 0 < α < 1/2, then there exist entire functions f such that exp(|z| ^α )f(z) → 0 (z → ∞, z ? S) for every strip S; moreover there is a vector space M which consists of such functions and which is dense in the space of all entire functions with the topology of local uniform convergence. In this note the result is shown to hold for every α > 0. The proof depends upon a theorem about tangential harmonic approximation on unbounded sets. As a corollary, a new result is proved about the class of entire functions which have zero integral on every doubly infinite line.     

Automatic Smoothing Spline Projection Pursuit

Abstract

A highly flexible nonparametric regression model for predicting a response y given covariates {x_k}^d _k=1 is the projection pursuit regression (PPR) model ? = h(x) = β₀ + Σ_jβ_jf_j(α^T _jx) where the f_j , are general smooth functions with mean 0 and norm 1, and Σ^d _k=1α² _kj=1. The standard PPR algorithm of Friedman and Stuetzle (1981) estimates the smooth functions f_j using the supersmoother nonparametric scatterplot smoother. Friedman's algorithm constructs a model with M _max linear combinations, then prunes back to a simpler model of size M ≤ M _max, where M and M _max are specified by the user. This article discusses an alternative algorithm in which the smooth functions are estimated using smoothing splines. The direction coefficients α_j, the amount of smoothing in each direction, and the number of terms M and M _max are determined to optimize a single generalized cross-validation measure.     

Two-Dimensional Graphs Moving by Mean Curvature Flow

A surface Σ is a graph in ?⁴ if there is a unit constant 2-form ω on ?⁴ such that <e ₁∧e ₂, ω≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if $ \vartheta _{0} \geqslant \frac{1} {{{\sqrt 2 }}} A surface Σ is a graph in ℝ⁴ if there is a unit constant 2-form ω on ℝ⁴ such that <e ₁∧e ₂, ω>≥v ₀>0 where {e ₁, e ₂} is an orthonormal frame on Σ. We prove that, if v ₀≥ on the initial surface, then the mean curvature flow has a global solution and the scaled surfaces converge to a self-similar solution. A surface Σ is a graph in M ₁×M ₂ where M ₁ and M ₂ are Riemann surfaces, if <e ₁∧e ₂, ω₁>≥v ₀>0 where ω₁ is a K?hler form on M ₁. We prove that, if M is a K?hler-Einstein surface with scalar curvature R, v ₀≥ on the initial surface, then the mean curvature flow has a global solution and it sub-converges to a minimal surface, if, in addition, R≥0 it converges to a totally geodesic surface which is holomorphic. Received July 25, 2001, Accepted October 11, 2001     

On the Space of Weakly Almost Periodic Functionals and Its Amenability

Let S be a locally compact semitopological semigroup with measure algebra M(S), M₀(S) the set of all probability measures in M(S) and WF(S) the space of weakly almost periodic functionals on M(S)^*. Assuming that M₀(S) has the semiright invariant isometry property, it is shown that WF(S) has a topological left invariant mean (TLIM) whenever the center of M₀(S) is nonempty; in particular if either the center of S is nonempty or S has a left identity, then WF(S) has a TLIM. Finally if, for each M₀(S), the mapping v v * of M₀(S) into itself is surjective and the center of M₀(S) is nonempty, then WF(S) has a TLIM. We also generalize some results from discrete case to topological one.AMS Subject Classification (1991): 43A07     

An integral formula for the heat kernel of tubular neighborhoods of complete (connected) Riemannian manifolds

Let M be a complete connected Riemannian manifold and let N be a submanifold of M. Let v: E _v»N be the normal bundle of N and exp _v : E _v»M its exponential map.Let (exp _infv ^/sup-1 , M ₀) be the Fermi chart relative to the submanifold N. Then, by using the Fermi coordinates we obtain an integral formula for the Dirichlet heat kernel p _t ^m (-,-). That is, we obtain a probabilistic representation for the integral _N f(y)p _t ^M (x,y) dywhere f is any measurable function of compact support in M ₀. This representation involves a submanifold semi-classical Brownian Riemannian bridge process. Then applying the integral formula via a Riemannian submersion in [5], we obtain heat kernel formulae for the complex projective space cP ⁿ, the quaternionic projective space QP ⁿ and the Caley line CaP ¹. The case of the Caley plane CaP ² eludes us due to the lack of a submersion theorem.This work is part of a Ph.D. Thesis which was undertaken under Professor K. D. Elworthy, Mathematics Institute, Warwick University, Coventry CV47AL, England, Great Britain.     

Counterexamples to the Kneser conjecture in dimension four

We construct a connected closed orientable smooth four-manifold whose fundamental group is the free product of two non-trivial groups such that it is not homotopy equivalent toM ₀#M ₁ unlessM ₀ orM ₁ is homeomorphic toS ⁴. LetN be the nucleus of the minimal elliptic Enrique surfaceV ₁(2, 2) and putM=N∪ _∂NN. The fundamental group ofM splits as ℤ/2 * ℤ/2. We prove thatM#k(S ²×S²) is diffeomorphic toM ₀#M ₁ for non-simply connected closed smooth four-manifoldsM ₀ andM ₁ if and only ifk≥8. On the other hand we show thatM is homeomorphic toM ₀#M ₁ for closed topological four-manifoldsM ₀ andM ₁ withπ ₁(M_i)=ℤ/2.     

Universal entire functions with gap power series

Let be the family of all compact sets in which have connected complement. For K ε M we denote by A(K) the set of all functions which are continuous on K and holomorphic in its interior.Suppose that {z_n} is any unbounded sequence of complex numbers and let Q be a given sub-sequence of ₀.If Q has density Δ(Q) = 1 then there exists a universal entire function with lacunary power series

1. (1) (z) = ε^∞_{v = 0 v}Z^v, _v = 0 for v Q, which has for all K ε M the following properties simultaneously

2. (2) the sequence {(Z + Z_n)} is dense in A(K)

3. (3) the sequence { (ZZ_n)} is dense in A(K) if 0 K.

Also a converse result is proved: If is an entire function of the form (1) which satisfies (3), then Q must have maximal density Δ_max(Q) = 1.     

Instability of the Rossby–Haurwitz wave in the invariant sets of perturbations

Stability of the Rossby–Haurwitz (RH) wave of subspace H₁H_n in an ideal incompressible fluid on a rotating sphere is analytically studied (H_n is the subspace of homogeneous spherical polynomials of degree n). It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets M₋ⁿ, M₊ⁿ, H_n and M₀ⁿ−H_n depending on the value of their mean spectral number χ(t)=η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M₋ⁿ and M₊ⁿ due to a hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced using the invariant subspace H_n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H_n, the factor norm controls the perturbation part orthogonal to H_n. It is shown that in the set M₋ⁿ (χ(t)<n(n+1)), any nonzonal RH wave of subspace H₁H_n (n2) is Liapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M₀ⁿ−H_n. A necessary condition for this instability is given. The condition states that the spectral number χ(t) of the amplitude of each unstable mode must be equal to n(n+1), where n is the RH-wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave is shown. The instability in the invariant set M₊ⁿ of small-scale perturbations (χ(t)>n(n+1)) is still open problem.     

Existence and uniqueness of packings with specified combinatorics

Generalizations of the Andreev-Thurston circle packing theorem are proved. One such result is the following. Let G=G(V) be a planar graph, and for each vertex v ∈ V, let ℱ _v be a proper 3-manifold of smooth topological disks in S ²,with the property that the pattern of intersection of any two sets A, B ∈ ℱ _v is topologically the pattern of intersection of two circles (i.e., there is a homeomorphism h:S ²→S ² taking A and B to circles). Then there is a packing P=(P _v :v ∈V)whose nerve is G, and which satisfies P _v ∈ ℱ _ν for v ∈ V. (‘The nerve is G’ means that two sets, P _v ,P _u ,touch, if, and only if, u ↔ v is an edge in G.) In the case whereG is the 1-skeleton of a triangulation, we also give a precise uniqueness statement. Various examples and applications are discussed.     

On the Rao modules of minimal curves in $${\mathbb{P}^N}$$

We study the Hartshorne-Rao modules M _C of minimal curves C in \mathbbP^N{\mathbb{P}^N} , with N ≥ 4, lying in the same liaison class of curves on a smooth rational scroll surface. We get a free minimal resolution of M _C for some of such curves and an upper bound for Betti numbers of M _C , for any C.     

A Characterization of Quadric Constant Mean Curvature Hypersurfaces of Spheres

Let be an immersion of a complete n-dimensional oriented manifold. For any v∈ℝ ⁿ⁺², let us denote by ℓ _v :M→ℝ the function given by ℓ _v (x)=〈φ(x),v〉 and by f _v :M→ℝ, the function given by f _v (x)=〈ν(x),v〉, where is a Gauss map. We will prove that if M has constant mean curvature, and, for some v≠0 and some real number λ, we have that ℓ _v =λ f _v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M ⁿ in which is neither totally umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n+4. A. Brasil Jr. was partially supported by CNPq, Brazil, 306626/2007-1.     

Nonlinear and linear instability of the Rossby-Haurwitz wave

The dynamics of perturbations to the Rossby-Haurwitz (RH) wave is analytically analyzed. These waves, being of great meteorological importance, are exact solutions to the nonlinear vorticity equation describing the motion of an ideal incompressible fluid on a rotating sphere. Each RH wave belongs to a space H ₁ ⊕ H _n , where H _n is the subspace of homogeneous spherical polynomials of degree n. It is shown that any perturbation of the RH wave evolves in such a way that its energy K(t) and enstrophy η(t) decrease, remain constant, or increase simultaneously. A geometric interpretation of variations in the perturbation energy is given. A conservation law for arbitrary perturbations is obtained and used to classify all the RH-wave perturbations in four invariant sets, M ₋ ⁿ , M ₊ ⁿ , H _n , and M ₀ ⁿ − H _n , depending on the value of their mean spectral number χ(t) = η(t)/K(t). The energy cascade of growing (or decaying) perturbations has opposite directions in the sets M ₋ ⁿ and M ₊ ⁿ due to the hyperbolic dependence between K(t) and χ(t). A factor space with a factor norm of the perturbations is introduced, using the invariant subspace H _n of neutral perturbations as the zero factor class. While the energy norm controls the perturbation part belonging to H _n , the factor norm controls the perturbation part orthogonal to H _n . It is shown that in the set M ₋ ⁿ (χ(t) < n(n + 1)), any nonzonal RH wave of subspace H ₁ ⊕ H _n (n ≥ 2) is Lyapunov unstable in the energy norm. This instability has nothing in common with the orbital (Poincaré) instability and is caused by asynchronous oscillations of two almost coinciding RH-wave solutions. It is also shown that the exponential instability is possible only in the invariant set M ₀ ⁿ − H _n . A necessary condition for this instability is given. The condition states that the spectral number η(t) of the amplitude of each unstable mode must be equal to n(n + 1), where n is the RH wave degree. The growth rate is estimated and the orthogonality of the unstable normal modes to the RH wave are shown in two Hilbert spaces. The instability in the invariant set M ₊ ⁿ of small-scale perturbations (χ(t) > n(n + 1)) is still an open problem. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Indivisible plexes in latin squares

In this paper, we present two constructions of divisible difference sets based on skew Hadamard difference sets. A special class of Hadamard difference sets, which can be derived from a skew Hadamard difference set and a Paley type regular partial difference set respectively in two groups of orders v ₁ and v ₂ with |v ₁ − v ₂| = 2, is contained in these constructions. Some result on inequivalence of skew Hadamard difference sets is also given in the paper. As a consequence of Delsarte’s theorem, the dual set of skew Hadamard difference set is also a skew Hadamard difference set in an abelian group. We show that there are seven pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁵ or 3⁷, and also at least four pairwisely inequivalent skew Hadamard difference sets in the elementary abelian group of order 3⁹. Furthermore, the skew Hadamard difference sets deduced by Ree-Tits slice symplectic spreads are the dual sets of each other when q ≤ 3¹¹.      

Aubry‐Mather theory and periodic solutions of the forced Burgers equation

Consider a Hamiltonian system with Hamiltonian of the form H(x, t, p) where H is convex in p and periodic in x, and t and x ∈ ℝ¹. It is well‐known that its smooth invariant curves correspond to smooth Z²‐periodic solutions of the PDE u_t + H(x, t, u)_x = 0. In this paper, we establish a connection between the Aubry‐Mather theory of invariant sets of the Hamiltonian system and Z²‐periodic weak solutions of this PDE by realizing the Aubry‐Mather sets as closed subsets of the graphs of these weak solutions. We show that the complement of the Aubry‐Mather set on the graph can be viewed as a subset of the generalized unstable manifold of the Aubry‐Mather set, defined in (2.24). The graph itself is a backward‐invariant set of the Hamiltonian system. The basic idea is to embed the globally minimizing orbits used in the Aubry‐Mather theory into the characteristic fields of the above PDE. This is done by making use of one‐ and two‐sided minimizers, a notion introduced in [12] and inspired by the work of Morse on geodesics of type A [26]. The asymptotic slope of the minimizers, also known as the rotation number, is given by the derivative of the homogenized Hamiltonian, defined in [21]. As an application, we prove that the Z²‐periodic weak solution of the above PDE with given irrational asymptotic slope is unique. A similar connection also exists in multidimensional problems with the convex Hamiltonian, except that in higher dimensions, two‐sided minimizers with a specified asymptotic slope may not exist. © 1999 John Wiley & Sons, Inc.     

Existence, attractiveness and stability of solutions for quadratic Urysohn fractional integral equations

In this paper, existence and attractiveness of solutions for quadratic Urysohn fractional integral equations on an unbounded interval are obtained by virtue of Tichonov fixed point theorem and suitable conjunction of the well known measure ω₀(X) and the spaces C(R⁺). Further, three certain solutions sets X_L,γ, X_1,α and X_{1,(1−(α+v))}, which tending to zero at an appropriate rate t^−ν (ν > 0), ν = γ (or α or 1 − (α + v)) as t → ∞, are introduced and stability of solutions for quadratic Urysohn fractional integral equations are obtained based on these solutions sets respectively by applying Schauder fixed point theorem via some easy checked conditions. An example is given to illustrate the results.