Interpolation Whitney constants

We obtain new estimates for interpolation Whitney constants.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1038–1043, August, 1995.     

Piecewise Coapproximation and the Whitney Inequality

All cases of the validity of the piecewise q-monotone analog of the Whitney inequality are clarified. For similar analogs of the Jackson inequality negative results are proved.     

Holomorphic Extensions of Whitney Jets

Let φ be a Whitney jet on a closed set F ? ?. By Whitney’s extension theorem φ can be extended to an infinitely differentiable function f on ? which is real analytic on ? F. The main purpose of this article is to show that f can be chosen in such a way that f¦_?F has a holomorphic continuation to the open set (? F) × i? ? ?. In the special case that F is a compact interval or a single point we can even achieve that f¦_?F has a holomorphic continuation to all of $\hat {\rm C}\setminus F$ . In particular, this implies an improvement of the well-known theorem of E. Borel. We also investigate the question when such extensions are given by a so-called extension operator.     

Stiefel—Whitney currents

A canonically defined mod 2 linear dependency current is associated to each collection v of sections, v₁,…,v_m, of a real rank n vector bundle. This current is supported on the linear dependency set of v. It is defined whenever the collection v satisfies a weak measure theoretic condition called “atomicity.” Essentially any reasonable collection of sections satisfies this condition, vastly extending the usual general position hypothesis. This current is a mod 2 d-closed locally integrally flat current of degree q = n −m + 1 and hence determines a ℤ₂-cohomology class. This class is shown to be well defined independent of the collection of sections. Moreover, it is the qth Stiefel-Whitney class of the vector bundle. More is true if q is odd or q = n. In this case a linear dependency current which is twisted by the orientation of the bundle can be associated to the collection v. The mod 2 reduction of this current is the mod 2 linear dependency current. The cohomology class of the linear dependency current is 2-torsion and is the qth twisted integral Stiefel-Whitney class of the bundle. In addition, higher dependency and general degeneracy currents of bundle maps are studied, together with applications to singularities of projections and maps. These results rely on a theorem of Federer which states that the complex of integrally flat currents mod p computes cohomology mod p. An alternate approach to Federer’s theorem is offered in an appendix. This approach is simpler and is via sheaf theory.     

Whitney集与图递归弧

通过构造具有有向图结构的迭代函数系通的子系通,证明了HausdorfF维数大于1的图递归弧均为Whitney集,该结果不需要有向图满足传递条件．     

自相似曲线与Whitney型临界集

本文讨论一类（包括Ｋｏｃｈ曲线在内的）自相似曲线，证明了其上的简单曲线是Ｗｈｉｔｎｅｙ型临界集，并在微小扰动下仍是简单曲线．     

Whitney Homology of Semipure Shellable Posets

We generalize results of Calderbank, Hanlon and Robinson on the representation of the symmetric group on the homology of posets of partitions with restricted block size. Calderbank, Hanlon and Robinson consider the cases of block sizes that are congruent to 0 mod d and 1 mod d for fixed d. We derive a general formula for the representation of the symmetric group on the homology of posets of partitions whose block sizes are congruent to k mod d for any k and d. This formula reduces to the Calderbank-Hanlon-Robinson formulas when k = 0, 1 and to formulas of Sundaram for the virtual representation on the alternating sum of homology. Our results apply to restricted block size partition posets even more general than the k mod d partition posets. These posets include the lattice of partitions whose block sizes are bounded from below by some fixed k. Our main tools involve the new theory of nonpure shellability developed by Björner and Wachs and a generalization of a technique of Sundaram which uses Whitney homology to compute homology representations of Cohen-Macaulay posets. An application to subspace arrangements is also discussed.     

Bounds from a card trick

We describe a new variation of a mathematical card trick, whose analysis leads to new lower bounds for data compression and estimating the entropy of a Markov source.     

Boundedness by 3 of the Whitney Interpolation Constant

Let the function fC[0,1] satisfy f( . We prove the estimate

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On the theorem and constants of H. Whitney

Whitney's famous theorem shows that the error of approximation to a functionf by algebraic polynomials of degree <n can be estimated by thenth order modulus of smoothness off. We show that the constants in this theorem can be taken independent ofn.     

On the Lagrangian form of the variational equations of Lagrangian dynamical systems

An extremal principle for obtaining the variational equations of a Lagrangian system is reviewed and formalized. Formalization is accomplished by relating the new Lagrangian function γ needed in such scheme to a prolongation of the original Lagrangian L. This formalization may be regarded as a necessary step before using the approach for stablishing nonintegrability of dynamical systems, or before applying it to analyse chaos-producing perturbations of integrable Lagrangian systems. The configuration manifold in which γ is defined is the double tangent bundle T(TQ) of the original configuration manifold Q modulo a flip mapping in such manifold. Our main result establishes that both the Euler–Lagrange equations and the corresponding variational equations of the original system can be viewed as the Lagrangian vector field associated with the composition of the first prolongation of the original Lagrangian with a flip mapping. Some applications of the approach to chaos and integrability issues are discussed.     

On the Sendov problem on the Whitney interpolation constant

For a function ƒ continuous on [0, 1] and satisfying the equalities we prove that where ω₄(t,ƒ) is the fourth modulus of smoothness of the function ƒ. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 5, pp. 732–734, May, 1998.     

Extension of a theorem of Whitney

It is shown that every planar graph with no separating triangles is a subgraph of a Hamiltonian planar graph; that is, Whitney’s theorem holds without the assumption of a triangulation.