217 research outputs found
Studying Wythoff and Zometool Constructions using Maple
We describe a Maple package that serves at least four purposes. First, one
can use it to compute whether or not a given polyhedral structure is Zometool
constructible. Second, one can use it to manipulate Zometool objects, for
example to determine how to best build a given structure. Third, the package
allows for an easy computation of the polytopes obtained by the kaleiodoscopic
construction called the Wythoff construction. This feature provides a source of
multiple examples. Fourth, the package allows the projection on Coxeter planesComment: 11 pages, 11 figure
Regular Incidence Complexes, Polytopes, and C-Groups
Regular incidence complexes are combinatorial incidence structures
generalizing regular convex polytopes, regular complex polytopes, various types
of incidence geometries, and many other highly symmetric objects. The special
case of abstract regular polytopes has been well-studied. The paper describes
the combinatorial structure of a regular incidence complex in terms of a system
of distinguished generating subgroups of its automorphism group or a
flag-transitive subgroup. Then the groups admitting a flag-transitive action on
an incidence complex are characterized as generalized string C-groups. Further,
extensions of regular incidence complexes are studied, and certain incidence
complexes particularly close to abstract polytopes, called abstract polytope
complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder,
A. Deza, and A. Ivic Weiss (eds), Springe
Ultrametric spaces of branches on arborescent singularities
Let be a normal complex analytic surface singularity. We say that is
arborescent if the dual graph of any resolution of it is a tree. Whenever
are distinct branches on , we denote by their intersection
number in the sense of Mumford. If is a fixed branch, we define when and
otherwise. We generalize a theorem of P{\l}oski concerning smooth germs of
surfaces, by proving that whenever is arborescent, then is an
ultrametric on the set of branches of different from . We compute the
maximum of , which gives an analog of a theorem of Teissier. We show that
encodes topological information about the structure of the embedded
resolutions of any finite set of branches. This generalizes a theorem of Favre
and Jonsson concerning the case when both and are smooth. We generalize
also from smooth germs to arbitrary arborescent ones their valuative
interpretation of the dual trees of the resolutions of . Our proofs are
based in an essential way on a determinantal identity of Eisenbud and Neumann.Comment: 37 pages, 16 figures. Compared to the first version on Arxiv, il has
a new section 4.3, accompanied by 2 new figures. Several passages were
clarified and the typos discovered in the meantime were correcte
Categorification of a linear algebra identity and factorization of Serre functors
We provide a categorical interpretation of a well-known identity from linear
algebra as an isomorphism of certain functors between triangulated categories
arising from finite dimensional algebras.
As a consequence, we deduce that the Serre functor of a finite dimensional
triangular algebra A has always a lift, up to shift, to a product of suitably
defined reflection functors in the category of perfect complexes over the
trivial extension algebra of A.Comment: 18 pages; Minor changes, references added, new Section 2.
Frieze Patterns of Integers
The famous theorem of Conway and Coxeter on frieze patterns gave a geometric
interpretation to integral friezes via triangulations of polygons. In this
article, we review this result and show some of the development it has led to.
The last decade has seen a lot of activities on friezes. One reason behind this
is the connection to cluster combinatorics.Comment: 9 pages, 6 figure
An algorithm to compute the polar decomposition of a 3 × 3 matrix
We propose an algorithm for computing the polar decomposition of a 3 × 3 real matrix that is based on the connection between orthogonal matrices and quaternions. An important application is to 3D transformations in the level 3 Cascading Style Sheets specification used in web browsers. Our algorithm is numerically reliable and requires fewer arithmetic operations than the alternative of computing the polar decomposition via the singular value decomposition
On the Decomposition of Clifford Algebras of Arbitrary Bilinear Form
Clifford algebras are naturally associated with quadratic forms. These
algebras are Z_2-graded by construction. However, only a Z_n-gradation induced
by a choice of a basis, or even better, by a Chevalley vector space isomorphism
Cl(V) \bigwedge V and an ordering, guarantees a multi-vector decomposition
into scalars, vectors, tensors, and so on, mandatory in physics. We show that
the Chevalley isomorphism theorem cannot be generalized to algebras if the
Z_n-grading or other structures are added, e.g., a linear form. We work with
pairs consisting of a Clifford algebra and a linear form or a Z_n-grading which
we now call 'Clifford algebras of multi-vectors' or 'quantum Clifford
algebras'. It turns out, that in this sense, all multi-vector Clifford algebras
of the same quadratic but different bilinear forms are non-isomorphic. The
usefulness of such algebras in quantum field theory and superconductivity was
shown elsewhere. Allowing for arbitrary bilinear forms however spoils their
diagonalizability which has a considerable effect on the tensor decomposition
of the Clifford algebras governed by the periodicity theorems, including the
Atiyah-Bott-Shapiro mod 8 periodicity. We consider real algebras Cl_{p,q} which
can be decomposed in the symmetric case into a tensor product Cl_{p-1,q-1}
\otimes Cl_{1,1}. The general case used in quantum field theory lacks this
feature. Theories with non-symmetric bilinear forms are however needed in the
analysis of multi-particle states in interacting theories. A connection to
q-deformed structures through nontrivial vacuum states in quantum theories is
outlined.Comment: 25 pages, 1 figure, LaTeX, {Paper presented at the 5th International
Conference on Clifford Algebras and their Applications in Mathematical
Physics, Ixtapa, Mexico, June 27 - July 4, 199
Conway–Coxeter Friezes and Mutation: A Survey
In this survey chapter, we explain the intricate links between Conway–Coxeter friezes and cluster combinatorics. More precisely, we provide a formula, relying solely on the shape of the frieze, describing how each individual entry in the frieze changes under cluster mutation. Moreover, we provide a combinatorial formula for the number of submodules of a string module, and with that a simple way to compute the frieze associated to a fixed cluster-tilting object in a cluster category of Dynkin type A in the sense of Caldero and Chapoton
VASP: A Volumetric Analysis of Surface Properties Yields Insights into Protein-Ligand Binding Specificity
Many algorithms that compare protein structures can reveal similarities that suggest related biological functions, even at great evolutionary distances. Proteins with related function often exhibit differences in binding specificity, but few algorithms identify structural variations that effect specificity. To address this problem, we describe the Volumetric Analysis of Surface Properties (VASP), a novel volumetric analysis tool for the comparison of binding sites in aligned protein structures. VASP uses solid volumes to represent protein shape and the shape of surface cavities, clefts and tunnels that are defined with other methods. Our approach, inspired by techniques from constructive solid geometry, enables the isolation of volumetrically conserved and variable regions within three dimensionally superposed volumes. We applied VASP to compute a comparative volumetric analysis of the ligand binding sites formed by members of the steroidogenic acute regulatory protein (StAR)-related lipid transfer (START) domains and the serine proteases. Within both families, VASP isolated individual amino acids that create structural differences between ligand binding cavities that are known to influence differences in binding specificity. Also, VASP isolated cavity subregions that differ between ligand binding cavities which are essential for differences in binding specificity. As such, VASP should prove a valuable tool in the study of protein-ligand binding specificity
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