Peano curves with smooth footprints

We construct Peano curves \(\gamma : [0,\infty ) \rightarrow \mathbb {R}^2\) whose “footprints” \(\gamma ([0,t])\), \(t>0\), have \(C^\infty \) boundaries and are tangent to a common continuous line field on the punctured plane \(\mathbb {R}^2 {\backslash }\{\gamma (0)\}\). Moreover, these boundaries can be taken \(C^\infty \)-close to any prescribed smooth family of nested smooth Jordan curves contracting to a point.     

Invariant Peano curves of expanding Thurston maps

We consider Thurston maps, i.e., branched covering maps f:S ² → S ² that are post-critically finite. In addition, we assume that f is expanding in a suitable sense. It is shown that each sufficiently high iterate F = f ⁿ of f is semi-conjugate to z ^d :S ¹ → S ¹, where d = deg F. More precisely, for such an F we construct a Peano curve γ:S ¹ → S ² (onto), such that F°γ(z) = γ(z ^d ) (for all z∈S ¹).     

Universal space in the Cartesian product of Peano curves without free arcs

We prove that there is the universal space for the class of n-dimensional separable metric spaces in the Cartesian product K₁×?×Kn+1 of Peano curves without free arcs. It is also shown that the set of embeddings of any n-dimensional separable metric space X into this universal space is a residual set in C(X,K₁×?×Kn+1). Other properties of product of Peano curves without free arcs are also proved.     

The maximum ratio clique problem

Computational Management Science - This paper introduces a fractional version of the classical maximum weight clique problem, the maximum ratio clique problem, which is to find a maximal clique...     

Hypersonic airfoils of maximum lift-to-drag ratio

The problem of determining the slender, hypersonic airfoil shape which produces the maximum lift-to-drag ratio for a given profile area, chord, and free-stream conditions is considered. For the estimation of the lift and the drag, the pressure distribution on a surface which sees the flow is approximated by the tangent-wedge relation. On the other hand, for surfaces which do not see the flow, the Prandtl-Meyer relation is used. Finally, base drag is neglected, while the skin-friction coefficient is assumed to be a constant, average value. The method used to determine the optimum upper and lower surfaces is the calculus of variations. Depending on the value of the governing parameter, the optimum airfoil shapes are found to be of three types. For low values of the governing parameter, the optimum shape is a flat plate at an angle of attack followed by slightly concave upper and lower surfaces. The next type of solution has a finite thickness over the entire chord with the upper surface inclined so that the flow is an expansion. Finally, for the last type of solution, the upper surface begins with a portion which sees the flow and is followed by an inclined portion similar to that above. For all of these solutions, the lower surface sees the flow. Results are presented for the optimum dimensionless airfoil shape, its dimensions, and the maximum lift-to-drag ratio. To calculate an actual airfoil shape requires an iteration procedure due to the assumption on the skin-friction coefficient. However, simple results can be obtained by assuming an approximate value for the skin-friction coefficient.This research was supported in part by the Air Force Office of Scientific Research, Office of Aerospace Research, U.S. Air Force, under AFOSR Grant No. 69-1744.     

Boundary conditions on curves for the three-dimensional Laplace operator

Journal of Mathematical Sciences - Boundary conditions on a curve for the three-dimensional Laplace operator are considered in the paper. The result is obtained in terms of a self-adjoint extension...     

Total torsion of curves in three-dimensional manifolds

A classical result in differential geometry assures that the total torsion of a closed spherical curve in the three-dimensional space vanishes. Besides, if a surface is such that the total torsion vanishes for all closed curves, it is part of a sphere or a plane. Here we extend these results to closed curves in three dimensional Riemannian manifolds with constant curvature. We also extend an interesting companion for the total torsion theorem, which was proved for surfaces in by L. A. Santaló, and some results involving the total torsion of lines of curvature. Dedicated to Professor Manfredo P. do Carmo on his 80th birthday.     

Differentiability of Peano derivatives

Peano differentiability is a notion of higher-order Fréchet differentiability. H. W. Oliver gave sufficient conditions for the Peano derivative to be a Fréchet derivative in the case of functions of a real variable. Here we generalize this theorem to functions of several variables.

     

Peano type theorem for abstract parabolic equations

We consider parabolic problems with non-Lipschitz nonlinearity in the different scales of Banach spaces and prove local-in-time existence theorem.     

The completeness of Peano multiplication

It is shown that the set of all theorems of Peano Arithmetic which mention only multiplication is a complete theory in the corresponding restricted language. The notion of a complete decidable covering of a theory is introduced. The author was partially supported by N.S.F. Grant No. MC578-02224.     

Concentration at curves for a singularly perturbed Neumann problem in three-dimensional domains

We prove new concentration phenomena for the equation −ɛ² Δu + u = u^p in a smooth bounded domain and with Neumann boundary conditions. Here p > 1 and ɛ > 0 is small. We show that concentration of solutions occurs at some geodesics of ∂Ω when ɛ → 0. Received: September 2004 Accepted: March 2005     

Lower Bounds for the Square-to-Linear Ratio for Plane Peano Curves

Mathematical Notes - It is proved that, for any map of the unit interval onto the unit square, there exist two points in the interval such that the squared Euclidean distance between their images...     

An injective characterization of Peano spaces

It is shown that a continuous map defined on a closed zero-dimensional subspace S of a compact space T into a Peano space X can be continuously extended over T or, equivalently, X is an AE(0, ∞),and this property precisely characterizes Peano spaces within the class of compact metric spaces. Surjectively, a compact AE(0, ∞) of arbitrary weight is proved to be the continuous image of a Tychonoff cube by a map satisfying the zero-dimensional lifting property.     

Peano Kernel Error Analysis for Quadratic Nodal Spline Interpolation

Using a Peano kernel technique, Jackson-type estimates with respect to the maximum norm are derived for the quadratic nodal spline interpolation error. The explicitly calculated error constants are shown to grow linearly with respect to the local mesh ratio parameter, and are, at least for the important special case of a uniform spline knot sequence, significantly smaller than those previously calculated by different methods.     

Lagrangian flows and the one-dimensional Peano phenomenon for ODEs

We consider the one-dimensional ordinary differential equation with a vector field which is merely continuous and nonnegative, and satisfies a condition on the amount of zeros. Although it is classically known that this problem lacks uniqueness of classical trajectories, we show that there is uniqueness for the so-called regular Lagrangian flow (by now usual notion of flow in nonsmooth situations), as well as uniqueness of distributional solutions for the associated continuity equation. The proof relies on a space reparametrization argument around the zeros of the vector field.