Results 1  10
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15
A Poincaré section for the horocycle flow on the space of lattices
, 2012
"... We construct a Poincare section for the horocycle flow on the modular surface SL(2,R)/SL(2,Z), and study the associated first return map, which coincides with a transformation (the BCZ map) defined by BocaCobeliZaharescu [8]. We classify ergodic invariant measures for this map and prove equidistri ..."
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Cited by 7 (3 self)
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We construct a Poincare section for the horocycle flow on the modular surface SL(2,R)/SL(2,Z), and study the associated first return map, which coincides with a transformation (the BCZ map) defined by BocaCobeliZaharescu [8]. We classify ergodic invariant measures for this map and prove equidistribution of periodic orbits. As corollaries, we obtain results on the average depth of cusp excursions and on the distribution of gaps for Farey sequences and slopes of lattice vectors.
FEASIBILITY OF INTEGER KNAPSACKS
, 2010
"... Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider the set F(A) of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ Rn≥0: Ax = b} contains an integer point. When m = 1 the set F(A) is known to contain all consecutive integers greater than th ..."
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Cited by 5 (2 self)
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Given a matrix A ∈ Zm×n satisfying certain regularity assumptions, we consider the set F(A) of all vectors b ∈ Zm such that the associated knapsack polytope P (A, b) = {x ∈ Rn≥0: Ax = b} contains an integer point. When m = 1 the set F(A) is known to contain all consecutive integers greater than the Frobenius number associated with A. In this paper we introduce the diagonal Frobenius number g(A) which reflects in an analogous way feasibility properties of the problem and the structure of F(A) in the general case. We give an optimal upper bound for g(A) and also estimate the asymptotic growth of the diagonal Frobenius number on average.
GENERALIZED FROBENIUS NUMBERS: BOUNDS AND AVERAGE BEHAVIOR
"... Let n ≥ 2 and s ≥ 1 be integers and a = (a1,..., an) be a relatively prime integer ntuple. The sFrobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as ∑n i=1 aixi in at least s different ways, where x1,..., xn are nonnegative integer ..."
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Cited by 3 (2 self)
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Let n ≥ 2 and s ≥ 1 be integers and a = (a1,..., an) be a relatively prime integer ntuple. The sFrobenius number of this ntuple, Fs(a), is defined to be the largest positive integer that cannot be represented as ∑n i=1 aixi in at least s different ways, where x1,..., xn are nonnegative integers. This natural generalization of the classical Frobenius number, F1(a), has been studied recently by a number of authors. We produce new upper and lower bounds for the sFrobenius number by relating it to the so called scovering radius of a certain convex body with respect to a certain lattice; this generalizes a wellknown theorem of R. Kannan for the classical Frobenius number. Using these bounds, we obtain results on the average behavior of the sFrobenius number, extending analogous recent investigations for the classical Frobenius number by a variety of authors. We also derive bounds on the scovering radius, an interesting geometric quantity in its own right.
Diameters of random circulant graphs
, 2013
"... The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many smallworld networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters incre ..."
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Cited by 3 (1 self)
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The diameter of a graph measures the maximal distance between any pair of vertices. The diameters of many smallworld networks, as well as a variety of other random graph models, grow logarithmically in the number of nodes. In contrast, the worst connected networks are cycles whose diameters increase linearly in the number of nodes. In the present study we consider an intermediate class of examples: Cayley graphs of cyclic groups, also known as circulant graphs or multiloop networks. We show that the diameter of a random circulant 2kregular graph with n vertices scales as n1/k, and establish a limit theorem for the distribution of their diameters. We obtain analogous results for the distribution of the average distance and higher moments.
Finescale statistics for the multidimensional Farey sequence
, 2012
"... We generalize classical results on the gap distribution (and other finescale statistics) for the onedimensional Farey sequence to arbitrary dimension. This is achieved by exploiting the equidistribution of horospheres in the space of lattices, and the equidistribution of Farey points in a certain ..."
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Cited by 3 (0 self)
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We generalize classical results on the gap distribution (and other finescale statistics) for the onedimensional Farey sequence to arbitrary dimension. This is achieved by exploiting the equidistribution of horospheres in the space of lattices, and the equidistribution of Farey points in a certain subspace of the space of lattices. The argument follows closely the general approach developed by A. Strömbergsson and the author [Annals of Math. 172 (2010) 1949–2033].
ON THE DISTRIBUTION OF ANGLES BETWEEN THE N SHORTEST VECTORS IN A RANDOM LATTICE
, 2010
"... We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random ndimensional lattice as n → ∞. Moreover we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally we discuss the limit distribution ..."
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Cited by 2 (0 self)
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We determine the joint distribution of the lengths of, and angles between, the N shortest lattice vectors in a random ndimensional lattice as n → ∞. Moreover we interpret the result in terms of eigenvalues and eigenfunctions of the Laplacian on flat tori. Finally we discuss the limit distribution of any finite number of successive minima of a random ndimensional lattice as n → ∞.
Diffusion in the Lorentz gas
, 2014
"... The Lorentz gas, a point particle making mirrorlike reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite ..."
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The Lorentz gas, a point particle making mirrorlike reflections from an extended collection of scatterers, has been a useful model of deterministic diffusion and related statistical properties for over a century. This survey summarises recent results, including periodic and aperiodic models, finite and infinite horizon, external fields, smooth or polygonal obstacles, and in the BoltzmannGrad limit. New results are given for several moving