This paper presents an adroit utilization of dimensional analysis-based model theory by which the deformation of a structure - however complex - can be elegantly and easily obtained. The structure is loaded by a concentrated lateral load of arbitrary location and magnitude. The relevant technique is outlined in some details; therefore the reader is advised to follow the presented routine closely. By doing so, he will be impressed by the prowees and economy of the described process. In the Preamble, the more important relevant theorems and relations - without proofs - are given in greatly condensed forms. This summary will help the reader to understand the subsequent application presented. Full treatment of the theories and practice of applied dimensional model theory can be found in [1], which the interested and motivated reader is advised to consult. and Obsahuje seznam literatury
We are interested in algorithms for constructing surfaces Γ of possibly small measure that separate a given domain Ω into two regions of equal measure. Using the integral formula for the total gradient variation, we show that such separators can be constructed approximatively by means of sign changing eigenfunctions of the p-Laplacians, p → 1, under homogeneous Neumann boundary conditions. These eigenfunctions turn out to be limits of steepest descent methods applied to suitable norm quotients.
Let $S$ be a non-empty subset of positive integers. A partition of a positive integer $n$ into $S$ is a finite nondecreasing sequence of positive integers $a_1, a_2, \dots , a_r$ in $S$ with repetitions allowed such that $\sum ^r_{i=1} a_i = n$. Here we apply Pólya’s enumeration theorem to find the number $¶(n;S)$ of partitions of $n$ into $S$, and the number ${\mathrm DP}(n;S)$ of distinct partitions of $n$ into $S$. We also present recursive formulas for computing $¶(n;S)$ and ${\mathrm DP}(n;S)$.
We introduce and study some new subclasses of starlike, convex and close-to-convex functions defined by the generalized Bessel operator. Inclusion relations are established and integral operator in these subclasses is discussed.