Jaroslav Tišer

Fréchet differentiability of Lipschitz functions via a variational principle

We prove a new variational principle which in particular does not assume the complete- ness of the domain. As an application we give a new, more natural, proof of the fact that a real valued Lipschitz function on an Asplund space has points of Fr´ echet differentiability.

Vitali covering theorem in Hilbert space

It is shown that the statement of the Vitali Covering Theorem does not hold for a certain class of measures in a Hilbert space. This class contains all infinite-dimensional Gaussian measures.

On Hadamard powers of polynomials

We introduce the concept of the Hadamard power of a polynomial formed by real powers of its coefficients. We show that the Hadamard power of a Hurwitz polynomial remains Hurwitz.

Collocation and Least Squares Methods

Various methods to adjust given or chosen mathematical models to actual requirements or actual data are presented. Such adjustment, if it exists, means choice of proper values of some parameters, which usually leads to solution of a set of linear or nonlinear equations. When such adjustment is not possible then least squares method gives an alternative.

Maximal and Minimal Values

The well known fact that differentiable functions may have local extremes only at points where their first derivatives vanish is illustrated. In some cases the corresponding equations are not directly accessible or solvable and other considerations have to be applied. Constrained extremal values for real functions of several variables and extremal problems of non-differentiable functions are included.

Center of Mass and Moments

The dynamics of an isolated system of N free particles, each with the mass m i , i=1,2,…,N is completely determined by the collection of their moments (of order 1). Generalizations of this concept opened new ways in applied sciences as well as in mathematics. Interconnections between the moment problem and least squares techniques form the basic idea of orthogonal series, which became an important tool in approximation theory.

Mappings, Composite and Inverse Functions

The chapter deals with with significant special mappings, their inverse mappings,composition, iteration, invariants and fixed points of mappings. It includes examples of functional equations, methods of investigating special functions, examples of mappings f:R 3→R 2.

A generalized

We show that in every Banach space, there is a g-porous set, the complement of which is of ℋ1-measure zero on every C1 curve.

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It is known that every G δ subset E of the plane containing a dense set of lines, even if it has measure zero, has the property that every real-valued Lipschitz function on ℝ 2 has a point of differentiability in E . Here we show that the set of points of differentiability of Lipschitz functions inside such sets may be surprisingly tiny: we construct a G δ set E ⊂ ℝ 2 containing a dense set of lines for which there is a pair of real-valued Lipschitz functions on ℝ 2 having no common point of differentiability in E , and there is a real-valued Lipschitz function on ℝ 2 whose set of points of differentiability in E is uniformly purely unrectifiable.