The paper describes the general form of an ordinary differential equation of the second order which allows a nontrivial global transformation consisting of the change of the independent variable and of a nonvanishing factor. A result given by J. Aczél is generalized. A functional equation of the form
\[ f(t,vy,wy+uvz)=f(x,y,z)u^{2}v+g(t,x,u,v,w)vz+h(t,x,u,v,w)y+2uwz \] is solved on $\mathbb R$ for $y\ne 0$, $v\ne 0.$.
Determination of heights with help of GPS in local geodetic networks is still more actual respecting the fact that the GPS technology becames more and more effective with hardware progress, with improvements in measuring and evaluating procedures, and with better modelling of the disturbing influences. In comparison with GPS the employment of classical terrestrial measuring technologies is often more difficult namely in broken mountain environment. In period 1998-2005 authors carried out repeated measurements of GPS baselines of various length and various height differences in local geodynamical network Sněžník and in other experimental areas. On ground of analyses of large GPS data sets the modified procedure for GPS observation was designed. The procedure is based on repetition of shorter static sessions separated by time intervals of optimal length. This technology represents an alternative to the ususal long static sessions, and is offering better effectivity of vertical GPS measurements with minimal loss of accuracy. The paper presents detailed description of the modified procedure together with some statistical analyses of results. The possibilities of elimination or mitigation of some disturbing influences are discussed. Two testing vertical profiles were marked in Sněžník network- longitudinal profile in N-S direction, and transversal profile in E-W direction - which were measured in course of several years by classical method of very precise levelling, and also by modified GPS heighting procedure in repeated sessions. Results obtained contributed to the local quasigeoid model creation., Otakar Švábenský, Josef Weigel and Radovan Machotka., and Obsahuje bibliografii
A proper coloring c : V (G) → {1, 2, . . . , k}, k ≥ 2 of a graph G is called a graceful k-coloring if the induced edge coloring c ′ : E(G) → {1, 2, . . . , k − 1} defined by c ′ (uv) = |c(u) − c(v)| for each edge uv of G is also proper. The minimum integer k for which G has a graceful k-coloring is the graceful chromatic number χg(G). It is known that if T is a tree with maximum degree ∆, then χg(T ) ≤ ⌈ 5⁄3∆⌉ and this bound is best possible. It is shown for each integer ∆ ≥ 2 that there is an infinite class of trees T with maximum degree ∆ such that χg(T ) = ⌈ 5⁄3 ∆⌉. In particular, we investigate for each integer ∆ ≥ 2 a class of rooted trees T∆,h with maximum degree ∆ and height h. The graceful chromatic number of T∆,h is determined for each integer ∆ ≥ 2 when 1 ≤ h ≤ 4. Furthermore, it is shown for each ∆ ≥ 2 that lim h→∞ χg(T∆,h) = ⌈ 5⁄3∆⌉.
Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian index $n$. We prove that for a graph $G$ of order $n \geq 3$ with the largest Laplacian index $n$, $G$ is Hamiltonian if $G$ is regular or its maximum vertex degree is $\triangle (G)=n/2$. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results.
In this paper we introduce and investigate the notion of half cyclically ordered group generalizing the notion of half partially ordered group whose study was begun by Giraudet and Lucas.
In this paper we study Beurling type distributions in the Hankel setting. We consider the space ${\mathcal E}(w)^{\prime }$ of Beurling type distributions on $(0, \infty )$ having upper bounded support. The Hankel transform and the Hankel convolution are studied on the space ${\mathcal E}(w)^{\prime }$. We also establish Paley Wiener type theorems for Hankel transformations of distributions in ${\mathcal E}(w)^{\prime }$.
Some $q$-analysis variants of Hardy type inequalities of the form $$ \int _0^b \bigg (x^{\alpha -1} \int _0^x t^{-\alpha } f(t) {\rm d}_q t \bigg )^{p} {\rm d}_q x \leq C \int _0^b f^p(t) {\rm d}_q t $$ with sharp constant $C$ are proved and discussed. A similar result with the Riemann-Liouville operator involved is also proved. Finally, it is pointed out that by using these techniques we can also obtain some new discrete Hardy and Copson type inequalities in the classical case.
This paper shows that some characterizations of the harmonic majorization of the Martin function for domains having smooth boundaries also hold for cones.