We denote by $K$ the class of all cardinals; put $K^{\prime }= K \cup \lbrace \infty \rbrace $. Let $\mathcal C$ be a class of algebraic systems. A generalized cardinal property $f$ on $\mathcal C$ is defined to be a rule which assings to each $A \in \mathcal C$ an element $f A$ of $K^{\prime }$ such that, whenever $A_1, A_2 \in \mathcal C$ and $A_1 \simeq A_2$, then $f A_1 =f A_2$. In this paper we are interested mainly in the cases when (i) $\mathcal C$ is the class of all bounded lattices $B$ having more than one element, or (ii) $\mathcal C$ is a class of lattice ordered groups.
The problem of observer design for a class of nonlinear discrete-time systems with time-delay is considered. A new approach of nonlinear observer design is proposed for the class of systems. Based on differential mean value theory, the error dynamic is transformed into linear parameter variable system. By using Lyapunov stability theory and Schur complement lemma, the sufficient conditions expressed in terms of matrix inequalities are obtained to guarantee the observer error converges asymptotically to zero. Furthermore, the problem of observer design with affine gain is investigated. The computing method for observer gain matrix is given and it is also demonstrated that the observer error converges asymptotically to zero. Finally, an illustrative example is given to validate the effectiveness of the proposed method.
Recently, the utilization of invariant aggregation operators, i.e., aggregation operators not depending on a given scale of measurement was found as a very current theme. One type of invariantness of aggregation operators is the homogeneity what means that an aggregation operator is invariant with respect to multiplication by a constant. We present here a complete characterization of homogeneous aggregation operators. We discuss a relationship between homogeneity, kernel property and shift-invariance of aggregation operators. Several examples are included.
Homogeneity, as one type of invariantness, means that an aggregation function is invariant with respect to multiplication by a constant, and quasi-homogeneity, as a relaxed version, reflects the original output as well as the constant. In this paper, we characterize all homogeneous/quasi-homogeneous n-ary aggregation functions and present several methods to generate new homogeneous/quasi-homogeneous n-ary aggregation functions by aggregation of given ones.
The main objective of the paper was to propose and evaluate the performance of a regional approach to estimate CN values and to test the impact of different initial abstraction ratios. The curve number (CN) was analyzed for five Slovak and five Polish catchments situated in the Carpathian Mountains. The L-moment based method of Hosking and Wallis and the ANOVA test were combined to delineate the area in two homogenous regions of catchments with similar CN values. The optimization condition enabled the choice of the initial abstraction ratio, which provided the smallest discrepancy between the tabulated and estimated CNs and the antecedent runoff conditions. The homogeneity in the CN within the regions of four Slovak and four Polish catchments was revealed. Finally, the regional CN was proposed to be at the 50% quantile of the regional theoretical distribution function estimated from all the CNs in the region. The approach is applied in a group of Slovak and Polish catchments with physiographic conditions representative for the Carpathian region. The main benefit of introducing a common regional CN is the opportunity to apply this procedure in catchments of similar soil-physiographic characteristics and to verify the existing tabulated CN. The paper could give rise to an alternative way of estimating the CN values in forested catchments and catchments with a lack of data or without observations.