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"Discrete geometry"
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Frameworks, tensegrities, and symmetry
\"This introduction to the theory of rigid structures explains how to analyze the performance of built and natural structures under loads, paying special attention to the role of geometry. The book unifies the engineering and mathematical literatures by exploring different notions of rigidity -- local, global, and universal -- and how they are interrelated. Important results are stated formally, but also clarified with a wide range of revealing examples. An important generalization is to tensegrities, where fixed distances are replaced with \"cables\" not allowed to increase in length and \"struts\" not allowed to decrease in length. A special feature is the analysis of symmetric tensegrities, where the symmetry of the structure is used to simplify matters and allows the theory of group representations to be applied. Written for researchers and graduate students in structural engineering and mathematics, this work is also of interest to computer scientists and physicists\"-- Provided by publisher.
Book
Cell complexes, poset topology and the representation theory of algebras arising in algebraic combinatorics and discrete geometry
In recent years it has been noted that a number of combinatorial structures such as real and complex hyperplane arrangements,
interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the
monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left
regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such
structures. In a recent paper, the authors established a close connection between algebraic and combinatorial invariants of a left
regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order
complexes of posets naturally associated to the left regular band.
The purpose of the present monograph is to further develop and
deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all
simple modules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left
regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the
examples that have arisen in the literature belong to this class. A new and important class of examples is a left regular band structure
on the face poset of a CAT(0) cube complex. Also, the recently introduced notion of a COM (complex of oriented matroids or conditional
oriented matroid) fits nicely into our setting and includes CAT(0) cube complexes and certain more general CAT(0) zonotopal complexes. A
fairly complete picture of the representation theory for CW left regular bands is obtained.
eBook
Asymptotic Counting in Conformal Dynamical Systems
In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic
subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the
former.
We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic
orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features
of the distribution of their weights.
These results have direct applications to a wide variety of examples, including the case of
Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions,
Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known
results and proves new results.
Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius
operators and Tauberian theorems as used in classical problems of prime number theory.
eBook
Isoperimetric inequalities in unbounded convex bodies
We consider the problem of minimizing the relative perimeter under a volume constraint in an unbounded convex body
eBook
Convexity of Singular Affine Structures and Toric-Focus Integrable Hamiltonian Systems
This work is devoted to a systematic study of symplectic convexity for integrable Hamiltonian systems with elliptic and focus-focus
singularities. A distinctive feature of these systems is that their base spaces are still smooth manifolds (with boundary and corners),
analogous to the toric case, but their associated integral affine structures are singular, with non-trivial monodromy, due to focus
singularities. We obtain a series of convexity results, both positive and negative, for such singular integral affine base spaces. In
particular, near a focus singular point, they are locally convex and the local-global convexity principle still applies. They are also
globally convex under some natural additional conditions. However, when the monodromy is sufficiently large, the local-global convexity
principle breaks down and the base spaces can be globally non-convex, even for compact manifolds. As a surprising example, we construct
a 2-dimensional “integral affine black hole”, which is locally convex but for which a straight ray from the center can never escape.
eBook
Existence of unimodular triangulations — positive results
Unimodular triangulations of lattice polytopes arise in algebraic geometry, commutative algebra, integer programming and, of course,
combinatorics.
In this article, we review several classes of polytopes that do have unimodular triangulations and constructions
that preserve their existence.
We include, in particular, the first effective proof of the classical result by
Knudsen-Mumford-Waterman stating that every lattice polytope has a dilation that admits a unimodular triangulation. Our proof yields an
explicit (although doubly exponential) bound for the dilation factor.
eBook
The Bounded and Precise Word Problems for Presentations of Groups
We introduce and study the bounded word problem and the precise word problem for groups given by means of generators and defining
relations. For example, for every finitely presented group, the bounded word problem is in
eBook
The Brunn-Minkowski Inequality and A Minkowski Problem for Nonlinear Capacity
In this article we study two classical potential-theoretic problems in convex geometry. The first problem is an inequality of
Brunn-Minkowski type for a nonlinear capacity,
In the first part of this article, we prove the Brunn-Minkowski inequality for this
capacity:
In the second part of this article we study a Minkowski problem for a certain measure associated with a compact
convex set
eBook
Horizons of fractal geometry and complex dimensions : 2016 Summer School, Fractal Geometry and Complex Dimensions, in celebration of the 60th birthday of Michel Lapidus, June 21-29, 2016, California Polytechnic State University, San Luis Obispo, California
This volume contains the proceedings of the 2016 Summer School on Fractal Geometry and Complex Dimensions, in celebration of Michel L. Lapidus's 60th birthday, held from June 21-29, 2016, at California Polytechnic State University, San Luis Obispo, California. The theme of the contributions is fractals and dynamics and content is split into four parts, centered around the following themes: Dimension gaps and the mass transfer principle, fractal strings and complex dimensions, Laplacians on fractal domains and SDEs with fractal noise, and aperiodic order (Delone sets and tilings).
eBook