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Log-Concavity in Planar Random Walks
We prove log-concavity of exit probabilities of lattice random walks in certain planar regions.
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Lamps in slim rectangular planar semimodular lattices
A planar (upper) semimodular lattice L is slim if the five-element nondistributive modular lattice M3 does not occur among its sublattices. (Planar...
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The First Syzygy of Hibi Rings Associated with Planar Distributive Lattices
In this article, we give explicit minimal generators of the first syzygy of the Hibi ring for a planar distributive lattice in terms of sublattices....
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The density of planar sets avoiding unit distances
By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set...
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From wave functions to tau-functions for the Volterra lattice hierarchy
For an arbitrary solution to the Volterra lattice hierarchy, the logarithmic derivatives of the tau-function of the solution can be computed by the...
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The Matching Lattice and Optimal Ear Decompositions
The objective of this chapter is to present a characterization of the matching lattice of a matching covered graph. Our approach to this...
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Typology of Planar Graphs
From a theoretical point of view, an important problem amounts to understand the structure of random planar graphs and eventually to propose a...
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Tropically planar graphs
We study tropically planar graphs, which are the graphs that appear in smooth tropical plane curves. We develop necessary conditions for graphs to be...
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Planar Graphs
A curveCurve in the plane is the image of a continuous map from [0, 1] to $$\mathbb...
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Coarse-Graining of a Discrete Model for Edge Dislocations in the Regular Triangular Lattice
We consider a discrete model of planar elasticity where the particles, in the reference configuration, sit on a regular triangular lattice and...
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Global planar dynamics with a star node and contracting nonlinearity
This is a complete study of the dynamics of polynomial planar vector fields whose linear part is a multiple of the identity and whose nonlinear part...
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Feynman checkers: lattice quantum field theory with real time
We present a new completely elementary model that describes the creation, annihilation, and motion of non-interacting electrons and positrons along a...
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Atom-generated planar lattices
In this note, we discuss planar lattices generated by their atoms. We prove that if L is a planar lattice generated by n atoms, then both the left...
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Exact Algorithm for Generating H-Cores in Simplified Lattice-Based Protein Model
Modeling protein folding, which is the process by which a protein obtains its spacial shape, still remains a challenging problem. Protein geometry...
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Length-preserving Extensions of a Semimodular Lattice by Lowering a Join-irreducible Element
We extend the bijective correspondence between finite semimodular lattices and Faigle geometries to an analogous correspondence between semimodular...
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Discrete-to-Continuum Limits of Planar Lattice Energies
Before dealing with their evolution, in this chapter we examine the ‘static’ limit of families of energies on lattices with vanishing spacing. This...
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Lattice Coverage of Cuboid with Minimum Number of Hemispheres*
The problem of partial lattice coverage of a cuboid of given dimensions with a minimum number of identical hemispheres with a given coverage factor...