Let G be a permutation group of degree n, and k a positive integer with k\le n. We say that G has the k-existential transversal property , or k-et , if there exists a k-subset A (of the domain \Omega ...) whose orbit under G contains transversals for all k-partitions \mathcal {P} of \Omega . This property is a substantial weakening of the k-universal transversal property, or k-ut, investigated by the first and third author, which required this condition to hold for all k-subsets A of the domain \Omega . Our first task in this paper is to investigate the k-et property and to decide which groups satisfy it. For example, it is known that for k< 6 there are several families of k-transitive groups, but for k\ge 6 the only ones are alternating or symmetric groups; here we show that in the k-et context the threshold is 8, that is, for 8\le k\le n/2, the only transitive groups with k-et are the symmetric and alternating groups; this is best possible since the Mathieu group M_{24} (degree 24) has 7-et. We determine all groups with k-et for 4\le k\le n/2, up to some unresolved cases for k=4,5, and describe the property for k=2,3 in permutation group language. These considerations essentially answer Problem 5 proposed in the paper on k-ut referred to above; we also slightly improve the classification of groups possessing the k-ut property. In that earlier paper, the results were applied to semigroups, in particular, to the question of when the semigroup \langle G,t\rangle is regular, where t is a map of rank k (with k<n/2); this turned out to be equivalent to the k-ut property. The question investigated here is when there is a k-subset A of the domain such that \langle G, t\rangle is regular for all maps t with image A. This turns out to be much more delicate; the k-et property (with A as witnessing set) is a necessary condition, and the combination of k-et and (k-1)-ut is sufficient, but the truth lies somewhere between. Given the knowledge that a group under consideration has the necessary condition of k-et, the regularity question for k\le n/2 is solved except for one sporadic group. The paper ends with a number of problems on combinatorics, permutation groups and transformation semigroups, and their linear analogues.
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms ...that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.
In the present study, we were interested in the appraisal by teachers who have had experience with an individual form of differentiation of the efficiency of grouping learners into homogenous or ...heterogeneous groups. In the study, seven aspects were thus considered: (1) the learning performance and motivation of learners, (2) the learners’ activity, (3) didactic ways of adapting teaching to diverse learners, (4) the impact on the formation of learners’ personality, (5) cooperation among learners, (6) opportunities for professional development, and (7) cooperation with parents. The research has shown that there is a statistically significant difference between the teachers who teach in homogeneous groups and the teachers who teach in heterogeneous groups in all the seven aspects of the appraisal of the efficiency of grouping learners into homogenous or heterogeneous groups. Regarding most of the assertions, teachers belonging to both groups perceive the way of teaching they perform as having the most advantages.
Besides increasing the amount of data that can be used in a fitting process, the Regional Frequency Analysis (RFA) also assesses the quality of weather station networks. This technique assumes that ...it is possible to form homogeneous groups of meteorological series presenting independent and identically distributed data. Based on the hypothesis that such homogeneous groups can be formed under tropical-subtropical conditions, this study applied the RFA to assess the probability of one-day annual maximum rainfall in the State of São Paulo, Brazil. Critical limits used in previous studies to declare a region/group as 'acceptable homogeneous' (H≤1.00) or to select a distribution (|Z|≤1.64) were evaluated through Monte Carlo simulations. While the limit H≤1 is appropriate, the limit |Z|≤1.64 may lead to unacceptably high rates of rejecting a true null hypothesis. This statement is particularly true for the general logistic distribution. A computational algorithm allowing the selection of critical limits corresponding to pre-specified probabilities of rejecting a true null hypothesis is provided. Considering the new critical limits, data from one of the largest weather station networks of the State have been pooled into four homogeneous groups. Both generalized logistic and extreme value distributions are recommended for the probabilistic assessment of such groups.
The purpose of this paper is to advance our knowledge of two of the most classic and popular topics in transformation semigroups: automorphisms and the size of minimal generating sets. In order to do ...this, we examine the k-homogeneous permutation groups (those which act transitively on the subsets of size k of their domain X) where \vert X\vert=n and k<n/2. In the process we obtain, for k-homogeneous groups, results on the minimum numbers of generators, the numbers of orbits on k-partitions, and their normalizers in the symmetric group. As a sample result, we show that every finite 2-homogeneous group is 2-generated. Underlying our investigations on automorphisms of transformation semigroups is the following conjecture: If a transformation semigroup S contains singular maps and its group of units is a primitive group G of permutations, then its automorphisms are all induced (under conjugation) by the elements in the normalizer of G in the symmetric group. For the special case that S contains all constant maps, this conjecture was proved correct more than 40 years ago. In this paper, we prove that the conjecture also holds for the case of semigroups containing a map of rank 3 or less. The effort in establishing this result suggests that further improvements might be a great challenge. This problem and several additional ones on permutation groups, transformation semigroups, and computational algebra are proposed at the end of the paper.
The research aims to detect: - Effectiveness of the competitive style of learning teams with heterogeneous and heterogeneous groups , and the usual method of skillful achievement of some basketball ...skills. The researchers used the experimental approach to suit the nature of the research. The sample included (42) students of the first year in the Faculty of Physical Education and Sports Sciences, University of Mosul, 14 students for the first experimental group, 14 students for the second experimental group, (14) students for the control group. Parity between the two groups was carried out in the basic variables (age, height and mass) and some elements of fitness and skill. The implementation of the educational program lasted 6 weeks from 25/2/2018 to 5/4/2018, 12 educational plans for the first experimental group, 12 educational plans for the second experimental group, 12 educational plans for the control group, And the duration of each educational plan (75) minutes, and after the completion of the program post-test was conducted for the skillful achievement on 11/4/2018 . The researchers used the following statistical means: To obtain the results of the current research, the researcher used the statistical bag (SPSS) using the following statistical means: mean, standard deviation, simple correlation coefficient Pearson, and the t-test of the unrelated samples. - coefficient of easiness and difficulty, the coefficient of power excellence, Spearman equation. - One-way analysis In light of the results of the research and within its limits, the researchers reached the following conclusions: 1- Superiority of the two experimental groups that studied the competitive learning teams method for the homogeneous and heterogeneous groups on the controlling group that studied according to the normal method of post-skillful achievement and for the two experimental groups of some basketball skills. 2- The experimental groups were effective in the skillful achievement for some basketball skills and did not show a difference between the two experimental groups, The researchers recommended a recommendation that is : 1- Emphasize en using the style of competitive science teams with homogeneous and heterogeneous groups in teaching basketball in the first year students . The researchers suggested a proposal that is : 1- Necessity to introduce competition at the end of the educational unit because, it has a positive impact on the achievement of the skill of basketball for the students of the first year students .
Budapest, the capital of Hungary, is famous for its abundant thermal waters. The monitoring network of the thermal karst waters of Budapest consists of 27 sampling locations, from which data for six ...geochemical parameters and temperature were available for the period 1960–2010. Based on this data, the optimal grouping for the sampling locations was sought using combined cluster and discriminant analysis. Furthermore, the homogeneity of the obtained groups, as well as temporal changes in the overall monitoring system were investigated. The seven groups found are in accordance with the previously established hydrogeological conditions, i.e. the grouping can be considered as optimal. The results obtained using combined cluster and discriminant analysis on the geochemical and temperature database of Budapest have important practical implications, since the thermal waters are an intensely used resource. Any artificial intervention will influence locations belonging to the same group, and therefore the optimal grouping of sampling locations will help in the planning of further activities. This case study might serve as an example in other settings where multiple measurements at multiple sampling locations from a monitoring network are available.
•Geochemical and temperature data of the thermal karst waters of Budapest were analysed.•Combined cluster and discriminant analysis (CCDA) can find the optimal grouping of data.•Seven main groups and smaller homogeneous sub-groups were identified.•CCDA can be applied to the investigation of temporal changes.•Considerable temporal changes were shown to have occurred over the decades.
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