In the study of locally finite varieties of groups it has often been illuminating to consider when a group A is a factor (i.e. quotient of a subgroup) of a group B. We write A ⋨ B to express this and say that A is involved in B. It follows from elementary isomorphism theorems that the relation ⋨ is a partial order on any set of finite groups. The conjecture that we consider in this paper (and to which we only give the beginning of an answer) is the following:

To assess the mathematical work of the late M. L. Urquhart is, paradoxically, both easy and extremely difficult. It is easy, in that he never published a mathematical paper in any of the journals. Thus one does not have to read a large volume of published work. On the other hand, Urquhart was far from being mathematically inactive, but he communicated his ideas verbally to his associates. At this point of time it is well nigh impossible to recall all the ideas that he discussed so freely during his lifetime. Conversations have long since been forgotten and ideas are now only vaguely remembered. Consequently any objective assessment of Urquhart's mathematical work is very difficult.

We show that the family of all holomorphic functions $f$ in a domain $D$ satisfying

$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$

$k$

$C>0$

$\unicode[STIX]{x1D6FC}>1$

$k\geq 2$

$$\begin{eqnarray}\frac{|f^{(k)}|}{1+|f|^{\unicode[STIX]{x1D6FC}}}(z)\leq C\quad \text{for all }z\in D\end{eqnarray}$$

We give new presentations of the five Mathieu groups, the simple groups J1, J2, HS, McL, Co3, and some other simple and related groups. All generators in these presentations are involutions. Our presentations are simpler than the known presentations of this type for the groups mentioned above.

Hardy space theory has been studied on manifolds or metric measure spaces equipped with either Gaussian or sub-Gaussian heat kernel behaviour. However, there are natural examples where one finds a mix of both behaviours (locally Gaussian and at infinity sub-Gaussian), in which case the previous theory does not apply. Still we define molecular and square function Hardy spaces using appropriate scaling, and we show that they agree with Lebesgue spaces in some range. Besides, counterexamples are given in this setting that the $H^{p}$ space corresponding to Gaussian estimates may not coincide with $L^{p}$ . As a motivation for this theory, we show that the Riesz transform maps our Hardy space $H^{1}$ into $L^{1}$ .

The Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

Szekeres defined a continuous analogue of the additive ordinary continued fraction expansion, which iterates a map T on a domain which can be identified with the unit square [0, 1]2. Associated to it are continuous analogues of the Lagrange and Markoff spectrum. Our main result is that these are identical with the usual Lagrange and Markoff spectra, respectively; thus providing an alternative characterization of them.

Szekeres also described a multi-dimensional analogue of T, which iterates a map Td on a higherdimensional domain; he proposed using it to bound d-dimensional Diophantine approximation constants. We formulate several open problems concerning the Diophantine approximation properties of the map Td.

In this paper we show that, for analytic composition operators between weighted Bergman spaces (including Hardy spaces) and as far as boundedness, compactness, order boundedness and certain summing properties of the adjoint are concerned, it is possible to modify domain spaces in a systematic fashion: there is a space of analytic functions which embeds continuously into each of the spaces under consideration and on which the above properties of the operator are decided.

A remarkable consequence is that, in the setting of composition operators between weighted Bergman spaces, the properties in question can be identified as properties of the operator as a map between appropriately chosen Hilbert spaces.

In this paper, we construct from any given good (3,1)-dimensional manifold pair finitely many almost identical imitations of it whose exteriors are mutative hyperbolic 3-manifolds. The equivariant versions with the mutative reduction property on the isometry group are also established. As a corollary, we have finitely many hyperbolic 3-manifolds with the same volume and the same isometry group.

Bieberbach's conjecture, proposed in 1916 and still unsolved, states that if ƒ(z) = z+a2z2+… is holomorphic and univalent in the disc ∣z∣ < 1 then ∣an∣ ≦ n for each n ≧ 2, with equality for some n only if ƒ(z) is the Koebe function of is obtained from this function by a rotation. Very recently Bombieri has succeeded in showing that if ƒ(z) is sufficiently close to the Koebe function, then with equality only if ƒ(z) = k(z). This had previously been proved by Garabedian, Ross and Schiffer [3] for even values of n.

The first paper (written jointly with L. G. Kovács) of this three-part series reduced the problem of determining all varieties of the title to the study of the varieties of nilpotent groups of class (at most) four whose free groups have no nontrivial elements of odd order. The present paper deals with these under the additional assumption that the variety contains all nilpotent groups of class three. We label each such variety by a vector of eleven parameters, each parameter a nonnegative integer or ∞, subject to numerous but simple conditions. Each vector satisfying these conditions is in fact used, and matches directly a (finite) defining set of laws for the variety it labels. Moreover, one can readily recognize from the parameters whether one variety is contained in another. The third paper will complete the determination of all varieties of nilpotent groups of class four.

We consider positive linear operators of probabilistic type L1f acting on real functions f defined on the positive semi-axis. We deal with the problem of uniform convergence of L1f to f, both in the usual sup-norm and in a uniform Lp type of norm. In both cases, we obtain direct and converse inequalities in terms of a suitable weighted first modulus of smoothness of f. These results are applied to the Baskakov operator and to a gamma operator connected with real Laplace transforms, Poisson mixtures and Weyl fractional derivatives of Laplace transforms.

We characterize the generalized ordered topological spaces X for which the uniformity (X) is convex. Moreover, we show that a uniform ordered space for which every compatible convex uniformity is totally bounded, need not be pseudocompact.

We consider the cosine functional equation (see [1, 2, 3]) , where f(z) is an entire function of a complex variable z and x, y are complex variables.

In this note we characterize the abelian groups G which have two different proper subgroups N and M such that the subgroup lattice L(G)=[0, M]∪ [N, G] is the union of these intervals.

These notes provide a concise introduction to the representation theory of reductive algebraic groups in positive characteristic, with an emphasis on Lusztig's character formula and geometric representation theory. They are based on the first author's notes from a lecture series delivered by the second author at the Simons Centre for Geometry and Physics in August 2019. We intend them to complement more detailed treatments.

We will show the existence, uniqueness and regularity of global solutions for the Cauchy problem for nonlinear evolution equations with the damping term .

As an application of our results, we give the global solvability and regularity of the mixed problem with Dirichiet boundary conditions:

Using an elementary counting procedure on biquadratic polynomials over Zp it is shown that the probability distribution of odd, unramified rational primes according to decomposition type in a fixed dihedral numberfield is identical to the probility of separable quartic polynomials (mod p) whose roots generate numberfields with normal closure having Galois group isomorphic to D4, as p → ∞. This verifies a conjecture about a converse to the Tschebotarev density theorem. Further evidence in support of this conjecture is provided in quadratic and coubic numberfields.

An analogue in a solid of the well known Pascal's theorem (Baker, [1], p. 219) for a conic is established by Baker ([2], pp. 53–54, Ex. 15) after Chasles [6] and by Salmon ([2], p. 142). The same is discussed in detail by Court [8]. The purpose of this paper is to extend it to a projective space of n dimensions or briefly to an n-space Sn. To prove it, we introduce here once again the idea of a set of n+1 associated lines in Sn as indicated in an earlier work (Mandan, [12]) in analogy with a set of 5 associated lines in S4 (Baker, [4], p. 122), and make use of the method of induction.

In a previous paper, ‘The concept of “largeness” in group theory’, a partial order was defined on the class of infinite groups, and this partial order was seen to give some precision to our intuitive notions of what it means for one infinite group to be ‘larger’ than another. The aim of this paper is to look more closely at groups which are ‘low down’ in this partial order, and to examine the interplay between properties of groups and finiteness conditions in group theory.

Subject classification (Amer. Math. Soc. (MOS) 1970): primary 20 E 15, 20 F 15, 20 K 10, 20 K 25; secondary 20 E 10.