Search results for "Set theory"
showing 10 items of 751 documents
Construction of 3D Triangles on Dupin Cyclides
2011
This paper considers the conversion of the parametric Bézier surfaces, classically used in CAD-CAM, into patched of a class of non-spherical degree 4 algebraic surfaces called Dupin cyclides, and the definition of 3D triangle with circular edges on Dupin cyclides. Dupin cyclides was discovered by the French mathematician Pierre-Charles Dupin at the beginning of the 19th century. A Dupin cyclide has one parametric equation, two implicit equations, and a set of circular lines of curvature. The authors use the properties of these surfaces to prove that three families of circles (meridian arcs, parallel arcs, and Villarceau circles) can be computed on every Dupin cyclide. A geometric algorithm …
On generalized covering subgroups and a characterisation of ?pronormal?
1983
Introduction. The context of this note is the theory of Schunck classes and formations of finite soluble groups. In a 1972 manuscript Fischer [4] generalized the concept of an ~-covering subgroup of a group G to a (P, ~)-covering subgroup, where P is some pronormal subgroup of G, and proved universal existence (for P satisfying a stronger embedding property) in case the class ~ is a saturated formation. The fact tha t the Schunck classes are the classes ~ with the property that every group has an ~-projector [9, 4.3, 4.4; 6] (which coincides with an ~-covering subgroup in the soluble universe | [6, II.15]) raises the question whether it is possible to determine the whole range of universal …
Extremal Frobenius numbers in a class of sets
1998
For given $ A_k=\{ a_1,\ldots ,a_k \}, a_1 \le \ldots \le a_k $ coprime the Frobenius number $ {g}(A_k) $ is defined as the greatest integer ${g}$ with no representation¶¶ ${g}=\sum \limits ^k_{i=1}\,x_i\,a_i,\;x_i\in {\Bbb N}_0 $ . ¶¶A class $ {\bf A}^*_k $ is given, such that ¶¶ $ {\overline {g}}^*(k,y):= \max \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_k\le y \} $ ¶¶has the same asymptotic behaviour as the general function¶¶ $ {\overline {g}}(k,y):= \max \{ {g}(A_k)| a_k\le y \}\, {\rm for} \, y\to \infty $ .¶¶ Furthermore, ¶¶ $ {\underline {g}}^*(k,x):= \min \{ {g}(A_k)|A_k\in {\bf A}^*_k,\, a_1\ge x \} $ ¶¶is shown to have the same order of magnitude as the general function¶¶ $ {\underline {g}…
Quasianalytic Denjoy-Carleman classes and o-minimality
2003
We show that the expansion of the real field generated by the functions of a quasianalytic Denjoy-Carleman class is model complete and o-minimal, provided that the class satisfies certain closure conditions. Some of these structures do not admit analytic cell decomposition, and they show that there is no largest o-minimal expansion of the real field.
Two Questions of L. A. Shemetkov on Critical Groups
1996
Throughout the paper we consider only finite groups. Let X be a class of groups. A group G is called s-critical for X , or simply X-critical, if G is not in X but all proper subgroups of G are in X. w Ž .x Ž . Following Doerk and Hawkes 3, VII, 6.1 , we denote Crit X the class s of all X-critical groups. Knowledge of the structure of the groups in Ž . Crit X for a class of groups X can often help one to obtain detailed s information for the structure of the groups belonging to X. Ž w Ž .x. O. J. Schmidt see 5, III, 5.2 studied the N-critical groups, where N is the formation of the nilpotent groups. These groups are also called w x Schmidt groups. In 2 , answering to a question posed by Shem…
LR(k) Parsing
1990
In this chapter we shall generalize the notion of strong LL(k) parsing presented in Chapter 5 and consider a method for deterministic left parsing that applies to a slightly wider class of context-free grammars than does the strong LL(k) parsing method. This method will be called “canonical LL(k) parsing”. As in strong LL(k) parsing, the acronym “LL(k)” means that the input string is parsed (1) in a single Left-to-right scan, (2) producing a Left parse, and (3) using lookahead of length k.
A local approach to a class of locally finite groups
2003
This paper is devoted to the study of a class of generalised P-nilpotent groups in the universe cℒ̄ of all radical locally finite groups satisfying min-q for every prime q. Some results of finite groups are extended and a characterisation of the injectors associated with this class is given.
Lp-Spaces
1998
For (X, ℜ, μ) a positive measure space, it has already been noted that μ - a.e. equality is an equivalence relation, and the relation ≤ μ-a.e. a preorder, on.This section studies the structure of the equivalence classes into which μ-a,e. equality partitions.Since the set X/X( ℜ) is always u-null (2.7.7 a)), only the function values on the set X(ℜ) have any significance when equivalence classes are formed: whether we form equivalence classes by partitioning or by partitioningX(ℜ) the resulting structures will be isomorphic. Nevertheless, it is natural to allow functions on an arbitrary X ⊃ X(ℜ). Our choice is to form μ-equivalence classes by partitioning the set X(ℜ). For arbitrary X ⊃ X(ℜ),…
Analytic extension of non quasi-analytic Whitney jets of Roumieu type
1997
Let (Mr)r∈ℕ0 be a logarithmically convex sequence of positive numbers which verifies M0 = 1 as well as Mr≥ 1 for every r ∈ ℕ and defines a non quasi-analytic class. Let moreover F be a closed proper subset of ℝn. Then for every function ƒ on ℝn belonging to the non quasi-analytic (Mr)-class of Roumieu type, there is an element g of the same class which is analytic on ℝnF and such that Dα ƒ(x) = Dαg(x) for every σ ∈ ƒ0n SBAP and x ∈ F.
Generators of Random Processes in Ultrametric Spaces and Their Spectra
2009
The L 2(\( \mathbb{S} \)) space of square integrable functions on an ultrametric space \( \mathbb{S} \) has rather specific structure. As a consequence in a natural way there appear in L 2(\( \mathbb{S} \)) the operators of which unitary counterparts in L 2(ℝn) would be difficult to construct. Such class of self-adjoint operators emerge from theory of random processes on ultrametric spaces. In this paper we collect known material on spectral properties of the generators of random processes on \( \mathbb{S}_B \) an ultrametric space of sequences. (The set of p-adic numbers is a subset of \( \mathbb{S}_B \).) Then we discuss structure of the eigenspaces of the generators.