For a graph property $\mathcal {P}$ and a graph $G$, we define the domination subdivision number with respect to the property $\mathcal {P}$ to be the minimum number of edges that must be subdivided (where each edge in $G$ can be subdivided at most once) in order to change the domination number with respect to the property $\mathcal {P}$. In this paper we obtain upper bounds in terms of maximum degree and orientable/non-orientable genus for the domination subdivision number with respect to an induced-hereditary property, total domination subdivision number, bondage number with respect to an induced-hereditary property, and Roman bondage number of a graph on topological surfaces.
It is one of the fundamental and challenging problems to determine the node numbers of hidden layers in neural networks. Various efforts have been made to study the relations between the approximation ability and the number of hidden nodes of some specific neural networks, such as single-hidden-layer and two-hiddenlayer feedforward neural networks with specific or conditional activation functions. However, for arbitrary feedforward neural networks, there are few theoretical results on such issues. This paper gives an upper bound on the node number of each hidden layer for the most general feedforward neural networks called multilayer perceptrons (MLP), from an algebraic point of view. First, we put forward the method of expansion linear spaces to investigate the algebraic structure and properties of the outputs of MLPs. Then it is proved that given k distinct training samples, for any MLP with k nodes in each hidden layer, if a certain optimization problem has solutions, the approximation error keeps invariant with adding nodes to hidden layers. Furthermore, it is shown that for any MLP whose activation function for the output layer is bounded on R, at most k hidden nodes in each hidden layer are needed to learn k training samples.
The paper deals with the existence of a quasi continuous selection of a multifunction for which upper inverse image of any open set with compact complement contains a set of the form $(G\setminus I)\cup J$, where $G$ is open and $I$, $J$ are from a given ideal. The methods are based on the properties of a minimal multifunction which is generated by a cluster process with respect to a system of subsets of the form $(G\setminus I)\cup J$.
Let $R$ be an integral domain with quotient field $K$ and $f(x)$ a polynomial of positive degree in $K[x]$. In this paper we develop a method for studying almost principal uppers to zero ideals. More precisely, we prove that uppers to zero divisorial ideals of the form $I = f(x)K[x] \cap R[x]$ are almost principal in the following two cases: – $J$, the ideal generated by the leading coefficients of $I$, satisfies $J^{-1} = R$. – $I^{-1}$ as the $R[x]$-submodule of $K(x)$ is of finite type. Furthermore we prove that for $I = f(x)K[x] \cap R[x]$ we have: – $I^{-1}\cap K[x]=(I:_{K(x)}I)$. – If there exists $p/q \in I^{-1}-K[x]$, then $(q,f)\neq 1$ in $K[x]$. If in addition $q$ is irreducible and $I$ is almost principal, then $I' = q(x)K[x] \cap R[x]$ is an almost principal upper to zero. Finally we show that a Schreier domain $R$ is a greatest common divisor domain if and only if every upper to zero in $R[x]$ contains a primitive polynomial.
An article presents technology of grinding and polishing of small glass areas down to 1 mm2 with optical quality. This technology problem relates to grant project solution ''Metastable water and water vapor'' GAČR No. 101/05/2214, where the surface tension of supercooled water has to be measured. We present experimental results of three possible technology approach of capillary tip finishing of final dimension 0.98/0.32 mm. Special focus is attended to capillary mounting appropriate for both machining and later experimental application too. and Příspěvek je zaměřen na technologii broušení a leštění malých skleněných ploch cca do 1mm2 s optickou kvalitou. Zvládnutí této problematiky úzce souvisí s měřením povrchového napětí přechlazené vody v rámci spoluúčasti autorů na řešení grantu ''Metastabilní voda a vodní pára'' GAČR č. 101/05/2214. Jsou uvedeny postupně tři možnosti opracování čela kapiláry s konečným rozměrem 0,98/0,32 mm včetně konkrétních výsledků. Pozornost je věnována i způsobu uchycení kapiláry, vhodným pro její montáž v experimentálním zařízení. To umožňuje sledovat změny tvaru mensiku kapaliny uvnitř kapiláry.
This article deals with modification and renovation of the ellipsometer Gaertner L119 and its using for thin-film structures study. The measurement process is fully automatized. The device works at constant wavelength in PCSA null regime and the data are obtained for various angles of incidence. The system was tested on a set of SiO2 thin-films prepared on Si single crystal wafers by thermal oxidation at 1200°C. The thicknesses obtained from experiment were compared with the results obtained by using of Yamaguchi type spectral ellipsometric system as well as with the data determined from white-light interferometry and spectral reflectometry. and Předložená práce se zabývá popisem úprav a modernizace elipsometru Gaertner L119 a jeho využitím pro studium tenkých vrstev s automatizovaným režimem měřicího procesu. Zařízení se využívá jako nulovací elipsometr v konfiguraci PCSA pro měření na jedné vlnové délce při různých úhlech dopadu. K testování přístroje byla provedena měření tloušťky vrstev SiO2 připravených termickou oxidací na monokrystalických křemíkových deskách při teplotě 1200 °C. Naměřené tloušťky byly porovnány jednak s výsledky měření na spektrálním elipsometru systému Yamaguchi a také s interferometrickými a reflektometrickými měřeními v bílém světle.