Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Preface 1. Introduction 2. Graph operations and modifications 3. Spectrum and structure 4. Characterizations by spectra 5. Structure and one eigenvalue 6. Spectral techniques 7. Laplacians 8. Additional topics 9. Applications Appendix Bibliography Index of symbols Index.

An Introduction to the Theory of Graph Spectra: Spectral techniques

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Well Logging For Earth Scientists

Simple singularities and simple algebraic groups

We prove that any crepant resolution of a polyhedral singularity C/G for G ⊂ SO(3) of types Z/nZ, D2n and T is isomorphic to a moduli space of representations of a quiver with relations. Moreover we classify all crepant resolutions explicitly by giving an open cover and find a one-to-one correspondence between them and mutations of the McKay quiver.

https://www.intlpress.com/site/pub/files/_fulltext/journals/ajm/2017/0021/0001/AJM-2017-0021-0001-a001.pdf

Flops and mutations for crepant resolutions of polyhedral singularities

Let G ⊂ GL(2,C) be a finite subgroup acting on the complex plane C2, and consider the following diagram 

C2 -> X <- π:Y
where π is the minimal resolution of singularities. Since Du Val in the 1930s the explicit calculation of Y was made from X by blowing up the singularity at the origin, where we lose any information about the group G in the process. But, is there a direct relation between the resolution Y and the group G?
McKay [McK80] in the late 1970s was the first to realise the link between the group action and the resolution Y , thus giving birth to the so called McKay correspondence. This beautiful correspondence establishes an equivalence between the geometry of the minimal resolution Y of the quotient singularity C2/G, and the G-equivariant geometry of C2.

Dihedral groups and G-Hilbert schemes

Let $G$ be a polyhedral group $G\subset SO(3)$ of types $\mathbb{Z}/n\mathbb{Z}$, $D_{2n}$ and $\mathbb{T}$. We prove that there exists a one-to-one correspondence between flops of $G$-Hilb$\mathbb{C}^3$ and mutations of the McKay quiver with potential which do not mutate the trivial vertex. This correspondence provides two equivalent methods to construct every projective crepant resolution for the singularities $\mathbb{C}^3/G$, which are constructed as moduli spaces $\mathcal{M}_C$ of quivers with potential for some chamber $C$ in the space $\Theta$ of stability conditions. In addition, we study the relation between the exceptional locus in $\mathcal{M}_C$ with the corresponding quiver $Q_C$, and we describe explicitly the part of the chamber structure in $\Theta$ where every such resolution can be found.

For a given finite small binary dihedral group
					$G\subset\mathrm{GL}(2,\mathbf{C})$ we provide an explicit description of the
					minimal resolution $Y$ of the singularity $\mathbf{C}^{2}/G$. The minimal
					resolution $Y$ is known to be either the moduli space of $G$-clusters
					$G$-Hilb$(\mathbf{C}^{2})$, or the equivalent $\mathcal{M}_{\theta}(Q,R)$, the
					moduli space of $\theta$-stable quiver representations of the McKay quiver. We
					use both moduli approaches to give an explicit open cover of $Y$, by assigning
					to every distinguished $G$-graph $\Gamma$ an open set
					$U_{\Gamma}\subset\mathcal{M}_{\theta}(Q,R)$, and calculating the explicit
					equation of $U_{\Gamma}$ using the McKay quiver with relations $(Q,R)$.

/pdf/dihedral-g-hilb-via-representations-of-the-mckay-quiver-2r3jbry31d.pdf

Dihedral $G$-Hilb via representations of the McKay quiver

We give the classification of all possible G-graphs for any small binary dihedral subgroup G in GL(2,C) and use this classification to give the combinatorial description of the special representations of G in terms of its maximal cyclic normal subgroup.

Álvaro Nolla de Celis

Papers

Flops and mutations for crepant resolutions of polyhedral singularities

Dihedral groups and G-Hilbert schemes

Flops and mutations for crepant resolutions of polyhedral singularities

Dihedral $G$-Hilb via representations of the McKay quiver

G-graphs and special representations for binary dihedral groups in GL(2,C)