The number of points from a random lattice that lie inside a ball

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices.Comment: 29 page

Final report for the REDUS project - Reduced Uncertainty in Stock Assessment

The REDUS project (2016-2020) has been a strategic project at the Institute of Marine Research (IMR) aimed at quantifying and reducing the uncertainty in data-rich and age-structured stock assessments (e.g., cod, herring, haddock, capelin). Work was organized in four topical work-packages: Fisheries-dependent (catch) surveys and assessment modeling (WP1), Fishery-independent (scientific) surveys (WP2), Evaluating and testing of long-term management strategies (WP3), and Communication of uncertainty, dissemination of project results and capacity building (WP4). The Norwegian Computing Center (NR) was contracted in as a strategic partner in statistical modeling and analysis, contributing mainly to WP1 and WP2, but found the research of fundamental interest therefore also allocating internal (NR) funding to develop the statistical science base of several of the methods.publishedVersio

Geometry of numbers, class group statistics and free path lengths

This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies. In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants. In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices. In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures. In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.QC 20151204</p

Geometry of numbers, class group statistics and free path lengths

This thesis contains four papers, where the first two are in the area of geometry of numbers, the third is about class group statistics and the fourth is about free path lengths. A general theme throughout the thesis is lattice points and convex bodies. In Paper A we give an asymptotic expression for the number of integer matrices with primitive row vectors and a given nonzero determinant, such that the Euclidean matrix norm is less than a given large number. We also investigate the density of matrices with primitive rows in the space of matrices with a given determinant, and determine its asymptotics for large determinants. In Paper B we prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot hold if one averages over the space of all lattices. In Paper C, we give a conjectural asymptotic formula for the number of imaginary quadratic fields with class number h, for any odd h, and a conjectural asymptotic formula for the number of imaginary quadratic fields with class group isomorphic to G, for any finite abelian p-group G where p is an odd prime. In support of our conjectures we have computed these quantities, assuming the generalized Riemann hypothesis and with the aid of a supercomputer, for all odd h up to a million and all abelian p-groups of order up to a million, thus producing a large list of “missing class groups.” The numerical evidence matches quite well with our conjectures. In Paper D, we consider the distribution of free path lengths, or the distance between consecutive bounces of random particles in a rectangular box. If each particle travels a distance R, then, as R → ∞ the free path lengths coincides with the distribution of the length of the intersection of a random line with the box (for a natural ensemble of random lines) and we determine the mean value of the path lengths. Moreover, we give an explicit formula for the probability density function in dimension two and three. In dimension two we also consider a closely related model where each particle is allowed to bounce N times, as N → ∞, and give an explicit formula for its probability density function.QC 20151204</p

Final report for the REDUS project - Reduced Uncertainty in Stock Assessment

The REDUS project (2016-2020) has been a strategic project at the Institute of Marine Research (IMR) aimed at quantifying and reducing the uncertainty in data-rich and age-structured stock assessments (e.g., cod, herring, haddock, capelin). Work was organized in four topical work-packages: Fisheries-dependent (catch) surveys and assessment modeling (WP1), Fishery-independent (scientific) surveys (WP2), Evaluating and testing of long-term management strategies (WP3), and Communication of uncertainty, dissemination of project results and capacity building (WP4). The Norwegian Computing Center (NR) was contracted in as a strategic partner in statistical modeling and analysis, contributing mainly to WP1 and WP2, but found the research of fundamental interest therefore also allocating internal (NR) funding to develop the statistical science base of several of the methods