Example of an -harmonic function which is not on a dense subset.
Examples of functions -extendable for each finite, but not -extendable
In Example 1, we describe a subset X of the plane and a function on X which has a -extension to the whole for each finite, but has no -extension to . In Example 2, we construct a similar example of a subanalytic subset of ; much more sophisticated than the first one. The dimensions given here are smallest possible.
Explicit Construction of Piecewise Affine Mappings with Constraints
We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving...
Extending times differentiable functions of several variables
It is shown that times Peano differentiable functions defined on a closed subset of and satisfying a certain condition on that set can be extended to times Peano differentiable functions defined on if and only if the th order Peano derivatives are Baire class one functions.
Extending o-minimal Fréchet derivatives
We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.
Extension of differentiable functions
Extension of Lipschitz functions defined on metric subspaces of homogeneous type.
If a metric subspace Mº of an arbitrary metric space M carries a doubling measure μ, then there is a simultaneous linear extension of all Lipschitz functions on Mº ranged in a Banach space to those on M. Moreover, the norm of this linear operator is controlled by logarithm of the doubling constant of μ.
Extension of the Two-Variable Pierce-Birkhoff conjecture to generalized polynomials
Let be a continuous, piecewise-polynomial function. The Pierce-Birkhoff conjecture (1956) is that any such is representable in the form , for some finite collection of polynomials . (A simple example is .) In 1984, L. Mahé and, independently, G. Efroymson, proved this for ; it remains open for . In this paper we prove an analogous result for “generalized polynomials” (also known as signomials), i.e., where the exponents are allowed to be arbitrary real numbers, and not just natural numbers;...
Extension to copulas and quasi-copulas as special -Lipschitz aggregation operators
Smallest and greatest -Lipschitz aggregation operators with given diagonal section, opposite diagonal section, and with graphs passing through a single point of the unit cube, respectively, are determined. These results are used to find smallest and greatest copulas and quasi-copulas with these properties (provided they exist).
Extensions of the representation theorems of Riesz and Fréchet
We present two types of representation theorems: one for linear continuous operators on space of Banach valued regulated functions of several real variables and the other for bilinear continuous operators on cartesian products of spaces of regulated functions of a real variable taking values on Banach spaces. We use generalizations of the notions of functions of bounded variation in the sense of Vitali and Fréchet and the Riemann-Stieltjes-Dushnik or interior integral. A few applications using geometry...
Extremal tests for scalar functions of several real variables at degenerate critical points.
Extremal Tests for Scalar Functions of Several Real variables at Degenerate Critical Points. (Short Communication).
Factorization of Lipschitz Functioins and Absolute Convergence of Vilenkin-Fourier Series.
Factorizations of matrices and functions of two variables
Failure of the condition N below .
Fermat’s method of quadrature
The Treatise on Quadratureof Fermat (c. 1659), besides containing the first known proof of the computation of the area under a higher parabola, , or under a higher hyperbola, —with the appropriate limits of integration in each case—has a second part which was mostly unnoticed by Fermat’s contemporaries. This second part of theTreatise is obscure and difficult to read. In it Fermat reduced the quadrature of a great number of algebraic curves in implicit form to the quadrature of known curves: the...
Fields of sets, set functions, set function integrals, and finite additivity.
Filippov's operation and some attributes
Finding new families of rank-one convex polynomials
Finite element approximations of a glaciology problem
In this paper we study a model problem describing the movement of a glacier under Glen’s flow law and investigated by Colinge and Rappaz [Colinge and Rappaz, ESAIM: M2AN 33 (1999) 395–406]. We establish error estimates for finite element approximation using the results of Chow [Chow, SIAM J. Numer. Analysis 29 (1992) 769–780] and Liu and Barrett [Liu and Barrett, SIAM J. Numer. Analysis 33 (1996) 98–106] and give an analysis of the convergence of the successive approximations used in [Colinge and...