On the number of real critical points of logarithmic derivatives and the Hawaii conjecture

For a given real entire function φ in the class U _2n ^*, n ≥ 0, with finitely many nonreal zeroes, we establish a connection between the number of real zeroes of the functions Q[φ] = (φ′/φ)′ and Q ₁[φ] = (φ″/φ′)′. This connection leads to a proof of the Hawaii Conjecture (T. Craven, G. Csordas, and W. Smith [5]), which states that if φ is a real polynomial, then the number of real zeroes of Q[φ] does not exceed the number of nonreal zeroes of φ.     

Factorization theorems for univariate splines on regular grids

For a given univariate compactly supported distributionφ, we investigate here the spaceS(φ) spanned by its integer translates, the subspaceH(φ) of all exponentials inS(φ) and the kernelK _ϕ of the associated semi-discrete convolutionφ*. The paper addresses a variety of results including a complete structure ofH(φ) andK _ϕ and a characterization of splines of minimal support. The main result shows that eachφ can be expressed asφ = p(∇)τ * M, wherep(∇) is a finite difference operator,τ is a distribution of smaller support satisfyingH(τ) =K _τ = {0}, andM is a spline which depends onH(φ) but not onφ itself, and which in the generic case (termed here “regular”) is the exponential B-spline associated withH(φ). The approach chosen is direct and avoids entirely the Fourier analysis arguments. The fact that a distribution is examined, rather than a function, is essential for the methods employed.     

On a Global Complexity Bound of the Levenberg-Marquardt Method

In this paper, we investigate a global complexity bound of the Levenberg-Marquardt method (LMM) for the nonlinear least squares problem. The global complexity bound for an iterative method solving unconstrained minimization of φ is an upper bound to the number of iterations required to get an approximate solution, such that ‖∇φ(x)‖≤ε. We show that the global complexity bound of the LMM is O(ε ⁻²).     

On the Characterization of Expansion Maps for Self-Affine Tilings

We consider self-affine tilings in ℝ ⁿ with expansion matrix φ and address the question which matrices φ can arise this way. In one dimension, λ is an expansion factor of a self-affine tiling if and only if |λ| is a Perron number, by a result of Lind. In two dimensions, when φ is a similarity, we can speak of a complex expansion factor, and there is an analogous necessary condition, due to Thurston: if a complex λ is an expansion factor of a self-similar tiling, then it is a complex Perron number. We establish a necessary condition for φ to be an expansion matrix for any n, assuming only that φ is diagonalizable over ℂ. We conjecture that this condition on φ is also sufficient for the existence of a self-affine tiling.     

A short proof of the levy continuity theorem in Hilbert space

A short proof of the Levy continuity theorem in Hilbert space. In the theory of the normal distribution on a real Hilbert spaceH, certain functionsφ have been shown by L. Gross to give rise to random variablesφ∼ in a natural way; in particular, this is the case for functions which are “uniformly τ-continuous near zero”. Among such functions are the characteristic functionsφ of probability distributionsm onH, given byφ(y)=∫e ^i(y,x)dm(x). The following analogue of the Levy continuity theorem has been proved by Gross: Letφ _j be the characteristic function of the probability measurem _j onH, Then necessary and sufficient that ∫f dm _j → ∫f dm for some probability measurem and all bounded continuousf, is that there exists a functionφ, uniformly τ-continuous near zero, withφ _j∼ →φ∼ in probability.φ turns out, of course, to be the characteristic function ofm. In the present paper we give a short proof of this theorem. Research supported by National Science Foundation Grant GP-3977.     

Ergodicity of some mappings of the circle and the line

Ifφ is inner and has a fixed point inD, thenφ as a mapping of the circle is exact. Ifφ has a “fixed” point onT, then the condition Σ(1−|φ _n(0)|)=∞ impliesφ _m is weak mixing for allm. These results when transferred to the line by a conformal mapping of the disc onto the upper half plane give a proof of the total weak mixing for the Boole transformation.     

Lip Automorphism Germ and Lip Automorphism

Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.     

Direct and Inverse Estimates for Bernstein Polynomials

Direct estimates for the Bernstein operator are presented by the Ditzian—Totik modulus of smoothness , whereby the step-weight φ is a function such that φ ² is concave. The inverse direction will be established for those step-weights φ for which φ ² and , are concave functions. This combines the classical estimate (φ=1 ) and the estimate developed by Ditzian and Totik ( ). In particular, the cases , λ∈[0,1] , are included. August 2, 1996. Date revised: March 28, 1997.     

Topological rigidity of Hamiltonian loops and quantum homology

This paper studies the question of when a loop φ={φ _t }_{0≤ t ≤1} in the group Symp(M,ω) of symplectomorphisms of a symplectic manifold (M,ω) is isotopic to a loop that is generated by a time-dependent Hamiltonian function. (Loops with this property are said to be Hamiltonian.) Our main result is that Hamiltonian loops are rigid in the following sense: if φ is Hamiltonian with respect to ω, and if φ′ is a small perturbation of φ that preserves another symplectic form ω′, then φ′ is Hamiltonian with respect to ω′. This allows us to get some new information on the structure of the flux group, i.e. the image of π₁(Symp(M,ω)) under the flux homomorphism. We give a complete proof of our result for some manifolds, and sketch the proof in general. The argument uses methods developed by Seidel for studying properties of Hamiltonian loops via the quantum homology of M. Oblatum 31-X-1997 & 20-III-1998 / Published online: 14 October 1998     

Triple automorphisms of simple Lie algebras

An invertible linear map φ on a Lie algebra L is called a triple automorphism of it if φ([x, [y, z]]) = [φ(x), [φ(y), φ(z)]] for ∀ x, y, z ∈ L. Let g be a finite-dimensional simple Lie algebra of rank l defined over an algebraically closed field F of characteristic zero, p an arbitrary parabolic subalgebra of g. It is shown in this paper that an invertible linear map φ on p is a triple automorphism if and only if either φ itself is an automorphism of p or it is the composition of an automorphism of p and an extremal map of order 2.     

Multiplicative Mappings of Rings

Let R and F be arbitrary associative rings. A mapping φ of R onto F is called a multiplicative isomorphism if φ is bijective and satisfies φ（xy） = φ（x）φ（y） for all x, y ∈ R. In this short note, we establish a condition on R, in the case where R may not contain any non-zero idempotents, that assures that φ is additive, which generalizes the famous Martindale＇s result. As an application, we show that under a mild assumption every multiplicative isomorphism from the radical of a nest algebra onto an arbitrary ring is additive.     

Some remarks on the study of good contractions

LetX be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism φ:X→Z onto a normal varietyZ with connected fibers which is given by a (high multiple of a) divisor of the typeK _x+rL, wherer is a positive rational number andL is an ample Cartier divisor. We first prove that the dimension of anu fiberF of φ is bigger or equal to (r-1) and, if φ is birational, thatdimF≥r, with the equalities if and only ifF is the projective space andL the hyperplane bundle (this is a sort of “relative” version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism φ itself in the case in which all fibers have minimal dimension with the respect tor. If φ is a birational divisorial contraction andX has terminal singularities we prove that φ is actually a “blow-up”.     

Quantum Uncertainty and the Spectra of Symmetric Operators

In certain circumstances, the uncertainty, ΔS[φ], of a quantum observable, S, can be bounded from below by a finite overall constant ΔS>0, i.e., ΔS[φ]≥ΔS, for all physical states φ. For example, a finite lower bound to the resolution of distances has been used to model a natural ultraviolet cutoff at the Planck or string scale. In general, the minimum uncertainty of an observable can depend on the expectation value, t=〈φ,S φ〉, through a function ΔS _t of t, i.e., ΔS[φ]≥ΔS _t , for all physical states φ with 〈φ,S φ〉=t. An observable whose uncertainty is finitely bounded from below is necessarily described by an operator that is merely symmetric rather than self-adjoint on the physical domain. Nevertheless, on larger domains, the operator possesses a family of self-adjoint extensions. Here, we prove results on the relationship between the spacing of the eigenvalues of these self-adjoint extensions and the function ΔS _t . We also discuss potential applications in quantum and classical information theory.      

Harmonicity on maps between almost contact metric manifolds

We study (φ,φ′)-holomorphic maps between almost contact metric manifolds, in particular horizontally conformal (φ,φ′)-holomorphic submersions, and obtain some criteria for the harmonicity of such maps.     

Onl p-complemented copies in Orlicz spaces

Let 1<α≦β<∞ andF be an arbitrary closed subset of the interval [α,β]. An Orlicz sequence spacel ^φ (resp. an Orlicz function spaceL ^φ(μ)) with associated indices α and β is found in such a way that the set of valuesp for which thel ^p-space is isomorphic to a complemented subspace ofl ^φ (resp.L ^φ(μ)) is precisely the given setF (resp.F ∪ {2}). Also, a recent result of Hernández and Peirats [1] is extended showing that, even for the case in which the indices satisfy α_φ ^∞<2<β_φ ^∞, there exist minimal Orlicz function spacesL ^φ(μ) with no complemented copy ofl ^p for anyp ≠ 2. Supported in part by CAICYT grant 0338-84.     

Correction to the paper “on the curvature of a generalization of a contact metric manifolds”

We considered in Example 3.1 of the paper [1] an S-structure on R^2n+s . We concluded that when s > 1 this manifold cannot be of constant φ-sectional curvature. Unfortunately this result is wrong. In fact, essentially due to a sign mistake in defining the φ-structure and a consequent transposition of the elements of the φ-basis (3.2), some of the Christoffel’s symbols were incorrect. In the present rectification, using a more slendler tecnique, we prove that our manifold is of constant φ-sectional curvature −3s and then it is η-Einstein.     

The Gleason’s problem for some polyharmonic and hyperbolic harmonic function spaces

Let Ω ⊆ ℝⁿ be a bounded convex domain with C ² boundary. For 0 < p, q ⩽ ∞ and a normal weight φ, the mixed norm space H _k ^p,q,φ (Ω) consists of all polyharmonic functions f of order k for which the mixed norm ∥ · ∥_p,q,φ < ∞. In this paper, we prove that the Gleason’s problem (Ω, a, H _k ^p,q,φ ) is always solvable for any reference point a ∈ Ω. Also, the Gleason’s problem for the polyharmonic φ-Bloch (little φ-Bloch) space is solvable. The parallel results for the hyperbolic harmonic mixed norm space are obtained.     

Uniqueness of Nelsons diffusions

Let ℒ≔Δ/2+(∇φ/φ) ·∇ be a generalized Schr?dinger operator or generator of Nelsons diffusion, defined on C ^∞ ₀(D) where φ is a continuous and strictly positive function on an open domain D⊂ℝ ^d such that ∇φ∈L _loc ²(D). Some results are given about the two questions below: (i) Whether does ℒ generate a unique semigroup in L ¹(D, φ² dx)? (ii) Whether the semigroup determined by ℒ is strong Feller? Received: 21 October 1997 / Revised version: 3 September 1998     

Semifixed Sets of Maps in Hyperspaces with Application to Set Differential Equations

For maps φ on hyperspaces the existence of semifixed sets, i.e., of sets A satisfying one of the relations A ⊂ φ(A), A ⊃ φ(A), A ∩ φ(A) ≠ ∅, is considered. An application to set differential equations is also presented.     

Pseudo-dimension and entropy of manifolds formed by affine-invariant dictionary

We consider the manifolds H _n(φ) formed by all possible linear combinations of n functions from the set {φ(A⋅+b)}, where x→Ax+b is arbitrary affine mapping in the space ℝ^d. For example, neural networks and radial basis functions are the manifolds of type H _n(φ). We obtain estimates for pseudo-dimension of the manifold H _n(φ) for wide collection of the generator function φ. The estimates have the order O(d ² n) in degree scale, that is the order is proportional to number of parameters of the manifold H _n(φ). Moreover the estimates for ɛ-entropy of the manifold H _n(φ) are obtained. Mathematics subject classifications (2000) 41A46, 41A50, 42A61, 42C10 V. Maiorov: Supported by the Center for Absorption in Science, Ministry of Immigrant Absorption, State of Israel.