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1 - Chapter
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 1-20
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Preface
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp xxv-xxviii
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Appendix F - Retracts on General NB-Groups (Not Necessarily Simply Connected)
- from Part II - Geometric Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 373-386
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Epilogue
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 582-584
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Appendix A - Semisimple Groups and the Iwasawa Decomposition
- from Part I - Analytic and Algebraic Classification
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 186-196
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Part I - Analytic and Algebraic Classification
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 21-22
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3 - NC-Groups
- from Part I - Analytic and Algebraic Classification
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 65-97
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8 - The Geometric NC-Theorem
- from Part II - Geometric Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 239-266
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4 - The BNB Classification
- from Part I - Analytic and Algebraic Classification
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 98-122
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Index
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 589-596
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Part III - Homology Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 387-388
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14 - Cohomology on Lie Groups
- from Part III - Homology Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 519-561
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Appendix G - Discrete Groups
- from Part III - Homology Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 562-581
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Contents
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp vii-xxiv
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6 - Other Classes of Locally Compact Groups
- from Part I - Analytic and Algebraic Classification
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 168-185
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13 - The Polynomial Homology for Simply Connected Soluble Groups
- from Part III - Homology Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 460-518
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12 - The Homotopy and Homology Classification of Connected Lie Groups
- from Part III - Homology Theory
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- Potential Theory and Geometry on Lie Groups
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- 23 October 2020
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- 22 October 2020, pp 389-459
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Bohr theorems for slice regular functions over octonions
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- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 151 / Issue 5 / October 2021
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- 21 October 2020, pp. 1595-1610
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- October 2021
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In this paper, we mainly investigate two versions of the Bohr theorem for slice regular functions over the largest alternative division algebras of octonions
$\mathbb {O}$. To this end, we establish the coefficient estimates for self-maps of the unit ball of 
$\mathbb {O}$ and the Carathéodory class in this setting. As a further application of the coefficient estimate, the 1/2-covering theorem is also proven for slice regular functions with convex image.
Mixing properties of colourings of the ℤd lattice
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- Combinatorics, Probability and Computing / Volume 30 / Issue 3 / May 2021
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- 19 October 2020, pp. 360-373
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We study and classify proper q-colourings of the ℤd lattice, identifying three regimes where different combinatorial behaviour holds. (1) When
$q\le d+1$, there exist frozen colourings, that is, proper q-colourings of ℤd which cannot be modified on any finite subset. (2) We prove a strong list-colouring property which implies that, when 
$q\ge d+2$, any proper q-colouring of the boundary of a box of side length 
$n \ge d+2$ can be extended to a proper q-colouring of the entire box. (3) When 
$q\geq 2d+1$, the latter holds for any 
$n \ge 1$. Consequently, we classify the space of proper q-colourings of the ℤd lattice by their mixing properties.
Frozen (Δ + 1)-colourings of bounded degree graphs
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- Combinatorics, Probability and Computing / Volume 30 / Issue 3 / May 2021
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- 19 October 2020, pp. 330-343
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Let G be a graph on n vertices and with maximum degree Δ, and let k be an integer. The k-recolouring graph of G is the graph whose vertices are k-colourings of G and where two k-colourings are adjacent if they differ at exactly one vertex. It is well known that the k-recolouring graph is connected for
$k\geq \Delta+2$. Feghali, Johnson and Paulusma (J. Graph Theory 83 (2016) 340–358) showed that the (Δ + 1)-recolouring graph is composed by a unique connected component of size at least 2 and (possibly many) isolated vertices.In this paper, we study the proportion of isolated vertices (also called frozen colourings) in the (Δ + 1)-recolouring graph. Our first contribution is to show that if G is connected, the proportion of frozen colourings of G is exponentially smaller in n than the total number of colourings. This motivates the use of the Glauber dynamics to approximate the number of (Δ + 1)-colourings of a graph. In contrast to the conjectured mixing time of O(nlog n) for
$k\geq \Delta+2$ colours, we show that the mixing time of the Glauber dynamics for (Δ + 1)-colourings restricted to non-frozen colourings can be Ω(n2). Finally, we prove some results about the existence of graphs with large girth and frozen colourings, and study frozen colourings in random regular graphs.
