SUMMARY We derive the joint distribution of the sequence of estimates of the parameter vector in a normal general linear model when data accumulate over a series of analyses. This sequence of estimates has a remarkably simple covariance structure, even when observations are correlated, allowing standard group sequential tests to be applied in very general settings. If variances and covariances ...

Symmetry means sameness, or indistinguishability, under some transformation. For example, a human face is indistinguishable from its reflected version, and therefore we say that it is reflectionally symmetric. A circle is indistinguishable from any of its rotated versions (about its center), and therefore we say that it is rotationally symmetric. When one puts together all the transformations t...

Computational group theory has a history going back more than 80 years. It is a collaborative e ort of researchers in a wide range of areas in both mathematics and computer science. The eld has produced important algorithmic ideas and large software packages that have been used to obtain valuable results in mathematics and in other disciplines. The algorithmic achievements include the developme...

⋆ Group: (G, γ) where G is a set and γ a binary operation γ : G×G → G. If γ is clear from the context the group is simply denoted by G. ◦ In place of γ(f, g) write fγg or simply fg when there is no ambiguity, as in most of what follows. ◦ Example: G = Z = {. . .− 2,−1, 0, 1, 2 . . .}, γ = +, γ(f, g) = f + g. • For G to be a group three conditions must be satisfied: (i) γ is associative: γ(f, γ(...

This is a survey of some problems in geometric group theory which I find interesting. The problems are from different areas of group theory. Each section is devoted to problems in one area. It contains an introduction where I give some necessary definitions and motivations, problems and some discussions of them. For each problem, I try to mention the author. If the author is not given, the prob...

An early version of these notes was prepared for use by the participants in the Workshop on Algebra, Geometry and Topology held at the Australian National University, 22 January to 9 February, 1996. They have subsequently been updated and expanded many times for use by students in the subject 620-421 Combinatorial Group Theory at the University of Melbourne. Copyright 1996-2004 by C. F. Miller ...

Why does Deep Learning work? What representations does it capture? How do higher-order representations emerge? We study these questions from the perspective of group theory, thereby opening a new approach towards a theory of Deep learning. One factor behind the recent resurgence of the subject is a key algorithmic step called pretraining: first search for a good generative model for the input s...

Cancellation Law: Let G be a group such that a, b, c ∈ G. If a∗c = b∗c, then a = b. Proof Suppose a ∗ c = b ∗ c. Since c ∈ G, it follows that an element d exists such that c ∗ d = e. Now, if we multiply both sides by d on the right, we obtain (a ∗ c) ∗ d = (b ∗ c) ∗ d. By associativity, we obtain a ∗ (c ∗d) = b ∗ (c ∗d). Since c ∗d = e, we obtain a ∗ e = b ∗ e, and by de nition of identity elem...

Let ω′ < lK,P . Recently, there has been much interest in the description of contra-finitely co-embedded arrows. We show that β is not comparable to b. We wish to extend the results of [31] to right-characteristic, bijective, co-stochastic monodromies. Is it possible to characterize Germain groups?