On the “Shidov Property” and a Formation Satisfying It

If G is a nonsoluble finite group, the intersection of the maximal subgroups of G which are not nilpotent is the Frattini subgroup of G. This was proved by Shidov (1971 Shidov , L. I. ( 1971 ). On maximal subgroups of finite groups . Sibirsk. Mat. Zh. 12 ( 3 ): 682 – 683 . [Google Scholar]). The authors here present a new formation ? larger than the formation of the nilpotent groups for which the analogous of the theorem of Shidov holds. The theorem makes use of the classification of finite simple groups.     

The Solution of a Problem of Coifman, Meyer, and?Wickerhauser on Wavelet Packets

Wavelet packets provide an algorithm with many applications in signal processing together with a large class of orthonormal bases of L ²(ℝ), each one corresponding to a different splitting of L ²(ℝ) into a direct sum of its closed subspaces. The definition of wavelet packets is due to the work of Coifman, Meyer, and Wickerhauser, as a generalization of the Walsh system. A question has been posed since then: one asks if a (general) wavelet packet system can be an orthonormal basis for L ²(ℝ) whenever a certain set linked to the system, called the “exceptional set” has zero Lebesgue measure. This answer to this question affects the quality of wavelet packet approximation. In this paper we show that the answer to this question is negative by providing an explicit example. In the proof we make use of the “local trace function” by Dutkay and the generalized shift-invariant system machinery developed by Ron and Shen.     

The flow of a supersonic ideal gas with “weak” and “strong” shocks over a wedge

The problem of the flow of a uniform supersonic ideal (inviscid and non-heat-conducting) gas over a wedge is considered. If the turning angle of the flow, which is equal to the angle of inclination of the generatrix of the wedge, is less than the maximum value, the problem has two solutions. In the solution with an oblique low-intensity (“weak”) shock, the uniform flow between the shock and the wedge is almost always supersonic. One exception is a small vicinity of the maximum turning angle. For an ideal gas this vicinity does not exceed a fraction of a degree at all Mach numbers. Behind a high-intensity (“strong”) shock, the flow of an ideal gas is always subsonic. “Weak” shocks are observed in all experiments with finite wedges. Some researchers attribute this preference to the “downstream” boundary conditions (“on the right at infinity” for a flow incident on the wedge from the left), and others attribute it to the instability (“Lyapunov” instability) of a flow with a strong shock when it flows over the wedge and to the stability of flow with a weak shock. The results presented below from calculations of the flows that occur for finite wedges within the two-dimensional unsteady Euler equations, when the parameters behind the strong shock are specified on the right-hand boundary, i.e., on the arc of a circle between the wedge and the shock, demonstrate the correctness of the conclusion of the first group of researchers and the incorrectness of the conclusion of the other group. In these calculations, after both small and fairly large perturbations, the flows investigated (which are, in fact, Lyapunov unstable!) return to the solution with a strong shock. In addition, the problem of steady flow over a wedge was regarded as the limit of the two-dimensional non-steady problems at infinite time. Simplification of one of them leads to the problem of the submerged over-expanded supersonic steady outflow. In the ideal gas model this problem is equivalent to flow over a wedge with both weak and strong shocks. All the solutions considered are stable.     

Volume growth, “a priori” estimates,and geometric applications

We obtain a maximum principle, and a priori upper estimates for solutions of a class of non-linear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumption of volume growth conditions. Various applications of the results obtained are presented.     

The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students’ empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students’ reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.     

“x≥a and x≤2无解”即“a≤x≤2不成立”吗?

众所周知 ,“ ｘ≥ａｘ≤ 2 无解”即“ ｘ≥ａｘ≤ 2 的解集是 ” ,那么“ａ≤ｘ≤ 2不成立”又是什么意思呢 ?拙以为 ,设Ａ =ｘ ｘ≥ａｘ≤ 2 ,则ｘａ≤ｘ≤ 2不成立 =Ａ而Ａ =[ａ ,2 ]　　　 (ａ <2 ){ 2 }　　　　 (ａ=2 ) 　　　　　 (ａ>2 )故Ａ=(-∞ ,ａ) ∪ (2 , ∞ )　　 (ａ<2 )(-∞ ,2 ) ∪ (2 , ∞ )　　 (ａ=2 )　　　　Ｒ　　　　　　 (ａ>2 )再深究下去 ,“ａ≤ｘ≤ 2不成立 /　ａ >2”但“ａ >2 ａ≤ｘ≤ 2不成立”即“不等式ａ≤ｘ≤ 2不成立”是“ａ>2”的必要不充分条件 .文[1 ] 中介绍并解答了这样的问题 :已知集合…     

“I am more of a visual learner”: The disciplinary values and identities in school mathematics learning and group work

This article depicts the disciplinary values and identities that students (re)construct in the realm of school mathematics. Video-based interactional analysis and video-mediated interviews were conducted in the context of group work in a school with a high percentage of immigrant students whose first language is not English. By adopting the framework of figured worlds, we identified figured disciplinary values where the visualization approach tended to be devalued compared to the calculational approach. The group that was not overly constrained by this figured disciplinary value attended more to the contributions made non-verbally and non-symbolically. Some immigrant students in this study identified themselves with the figurative identity as a “visual learner” that constrained their participation in group problem solving and influenced their overall relationship with the discipline of mathematics. The findings underline the importance of attending to the opaque values students project onto ways of engaging in mathematics.     

An overview on “The European school of MCDA: Emergence,basic features and current works”

This tutorial is based on a paper whose full version can be found in Roy and Vanderpooten (1996). In this paper, we were responding to an invitation issued by Simon French, editor of the Journal of Multi-Criteria Decision Analysis. The purpose of this paper has been to trace the development of a way of thinking and type of research which first saw the light of day in France, but which for many years have been influenced and supplemented by work carried out in other European countries. As suggested by Roman Slowinski and Zilla Sinuary-Stem, editors of this feature issue, we provide here a brief overview of our paper.     

The existence of global solution and “blow up” phenomenon for a system of multi dimensional symmetric regularized wave equations

The existence and uniqueness of the global smooth solution to the initial-boundary value problem of a system of multi-dimensions SRWE are proved. The sufficient conditions of blowing up of the solution are given.     

The “Circolo Matematico di Palermo” and the First World War: The crisis of scientific internationalism: a view through the unedited correspondence of De Franchis with Edmund Landau and other mathematicians

In this work the situation of the “Circolo Matematico di Palermo” between 1914 and 1928 is analyzed. It will be observed that during this time the Circolo was among the few European scientific associations with German as well as French associates. During the 1930s, the nationalist politics of Fascism and above all the racial laws dealt a deadly blow to the Circolo as an international scientific association. We will use the rich correspondence in the Circolo's archives to shed some light on this. In particular, the correspondence between M. De Franchis and E. Landau and other recently found documents will figure prominently.