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The dimension of a variety V of algebras of a given type was introduced by E. Graczynska and D. Schweigert in [7] as the cardinality of the set of all derived varieties of V which are properly contained in V. In this paper, we characterize all solid varieties of dimensions 0, 1, and 2; prove that the dimension of a variety of finite type is at most N-0; give an example of a variety which has infinite dimension; and show that for every n is an element of N there is a variety with dimension n. Finally, we show that the dimension of a variety is related to the concept of the semantical kernel of a hypersubstitution and apply this connection to calculate the dimension of the class of all algebras of type tau = (n).
An n-ary cooperation is a mapping from a nonempty set A to the nth copower of A. A clone of cooperations is a set of cooperations which is closed under superposition and contains all injections. Coalgebras are pairs consisting of a set and a set of cooperations defined on this set. We define terms for coalgebras, coidentities and cohyperidentities. These concepts will be applied to give a new solution of the completeness problem for clones of cooperations defined on a two-element set and to separate clones of cooperations by coidentities.
Arenes with various alkyl side-chains were synthesized in high yields and excellent regioselectivities. Starting from toluic and naphthoic acids, the carboxylate group was conveniently substituted by alkyl halides by Birch reduction and subsequent decarbonylation. The method is characterized by inexpensive starting materials and reagents, and methylation of arenes was realized. Besides simple alkyl substituents, the scope of arene functionalization was extended by benzyl, fluoro, amino, and ester groups. We were able to control the alkylation of 1-naphthoic acid during Birch reduction by the addition of tert-butanol. This allowed the regioselective synthesis of mono and bis-substituted naphthalenes from the same starting material.
We discuss the relaxation of a class of nonlinear elliptic Cauchy problems with data on a piece S of the boundary surface by means of a variational approach known in the optimal control literature as "equation error method". By the Cauchy problem is meant any boundary value problem for an unknown function y in a domain X with the property that the data on S, if combined with the differential equations in X, allow one to determine all derivatives of y on S by means of functional equations. In the case of real analytic data of the Cauchy problem, the existence of a local solution near S is guaranteed by the Cauchy-Kovalevskaya theorem. We also admit overdetermined elliptic systems, in which case the set of those Cauchy data on S for which the Cauchy problem is solvable is very "thin". For this reason we discuss a variational setting of the Cauchy problem which always possesses a generalised solution.
Precision farming overcomes the paradigm of uniform field treatment by site-specific data acquisition and treatment to cope with within-field variability. Precision farming heavily relies on spatially dense information about soil and crop status. While it is often difficult and expensive to obtain precise soil information by traditional soil sampling and laboratory analysis some geophysical methods offer means to obtain subsidiary data in an efficient way. In particular, geoelectrical soil mapping has become widely accepted in precision farming. At present it is the most successful geophysical method providing the spatial distribution of relevant agronomic information that enables us to determine management zones for precision farming. Much work has been done to test the applicability of existing geoelectrical methods and to develop measurement systems applicable in the context of precision farming. Therefore, the aim of this paper was to introduce the basic ideas of precision farming, to discuss current precision farming applied geoelectrical methods and instruments and to give an overview about our corresponding activities during recent years. Different experiments were performed both in the laboratory and in the field to estimate first, electrical conductivity affecting factors, second, relationships between direct push and surface measurements, third, the seasonal stability of electrical conductivity patterns and fourth, the relationship between plant yield and electrical conductivity. From the results of these experiments, we concluded that soil texture is a very dominant factor in electrical conductivity mapping. Soil moisture affects both the level and the dynamic range of electrical conductivity readings. Nevertheless, electrical conductivity measurements can be principally performed independent of season. However, electrical conductivity field mapping does not produce reliable maps of spatial particle size distribution of soils, e.g., necessary to generate input parameters for water and nutrient transport models. The missing step to achieve this aim may be to develop multi-sensor systems that allow adjusting the electrical conductivity measurement from the influence of different soil water contents.