Lattice (group)

Summary

In geometry and group theory, a lattice in the real coordinate space \mathbb{R}^n is an infinite set of points in this space with the properties that coordinate-wise addition or subtraction of two points in the lattice produces another lattice point, that the lattice points are all separated by some minimum distance, and that every point in the space is within some maximum distance of a lattice point. Closure under addition and subtraction means that a lattice must be a subgroup of the additive group of the points in the space, and the requirements of minimum and maximum distance can be summarized by saying that a lattice is a Delone set. More abstractly, a lattice can be described as a free abelian group of dimension n which spans the vector space \mathbb{R}^n. For any basis of \mathbb{R}^n, the subgroup of all linear combinations with integer coefficients of the basis vectors forms a lattice, and every lattice can be formed from

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https://en.wikipedia.org/wiki/Lattice_(group)

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On the number of lattice points in a small sphere and a recursive lattice decoding algorithm

Annika Meyer

Let L be a lattice in . This paper provides two methods to obtain upper bounds on the number of points of L contained in a small sphere centered anywhere in . The first method is based on the observation that if the sphere is sufficiently small then the lattice points contained in the sphere give rise to a spherical code with a certain minimum angle. The second method involves Gaussian measures on L in the sense of Banaszczyk (Math Ann 296:625-635, 1993). Examples where the obtained bounds are optimal include some root lattices in small dimensions and the Leech lattice. We also present a natural decoding algorithm for lattices constructed from lattices of smaller dimension, and apply our results on the number of lattice points in a small sphere to conclude on the performance of this algorithm.

Springer US2013

We show that the Chow covectors of a linkage matching field define a bijection of lattice points and we demonstrate how one can recover the linkage matching field from this bijection. This resolves two open questions from Sturmfels & Zelevinsky (1993) on linkage matching fields. For this, we give an explicit construction that associates a bipartite incidence graph of an ordered partition of a common set to all lattice points in a dilated simplex. Given a triangulation of a product of two simplices encoded by a set of bipartite trees, we similarly prove that the bijection from left to right degree vectors of the trees is enough to recover the triangulation. As additional results, we show a cryptomorphic description of linkage matching fields and characterise the flip graph of a linkage matching field in terms of its prodsimplicial flag complex. Finally, we relate our findings to transversal matroids through the tropical Stiefel map.

2018

Pin-by-Pin Treatment of LWR Cores with Cross Section Equivalence Procedures and Higher-Order Transport Methods

Petra Mala

This thesis presents a systematic study on the merits and limitations on using pin-by-pin resolution and transport theory based approaches for nuclear core design calculations. Starting from the lattice codes and an optimal cross section generation scheme, it compares different methods, transport approximations, and spatial discretizations used in pin-by-pin homogenized codes. It is in the interest of nuclear power plant operators to employ more heterogeneous core loadings in order to improve the fuel utilization and decrease the amount of spent fuel. This necessarily increases the requirements on the accuracy of the computation tools used for the core design and safety analysis. One possibility is employing 3D core solvers with higher spatial resolution, e.g. pin-cell wise. The comparison of several lattice codes indicates that already the proper generation of diffusion coefficients and higher-order scattering moments for pin-cell geometry is not straightforward. Out of three available lattice codes, two generated unphysical diffusion coefficient when using the inscatter approximation, while the last code was not able to provide the higher scattering moments. Several few-assembly test cases with either high neutron leakage effects or MOX/uranium interfaces showed that the quality of the results of diffusion and SP3 solvers depends critically on the choice of the diffusion coefficient: in the outscatter approximation, it can cause major deviations, while the inscatter seems overall more adequate. Regarding the transport solvers, results obtained with the SN solver DORT showed very good performance for MOX/uranium interfaces, but major deviations occurred for problems with large power gradients. On the other hand, the MOC solver nTRACER showed in all cases small average error, but 2 - 3 % error around the interface of different assemblies. A study on spatial discretization indicated that the finite difference method applied on pin-cells does not properly capture the big flux changes between MOX and uranium fuel, while the nodal expansion method is more accurate but too slow. It was suggested to use the finite difference method with finer mesh in the outer assembly pin-cells, which increases the required computation time by only 50 % and decreases the pin power errors below 1 % with respect to lattice code results. Due to some problems which were observed with the available diffusion/SP3 solvers, a new SP3 solver was implemented in the DORT-TD platform. Several core tests showed that the SP3 pin-by-pin solver can significantly outperform the state-of-the-art nodal solver SIMULATE-5, in particular for reactor cores with inserted control rods. Finally, the pin-by-pin solvers were coupled to a depletion solver. For that, an accurate and fast interpolation routine had to be implemented. The obtained results of full-cycle depletion with complex core loading showed very good performance in comparison to a heterogeneous transport-based fine-group calculations.

EPFL2018

Pin-by-Pin Treatment of LWR Cores with Cross Section Equivalence Procedures and Higher-Order Transport Methods

Petra Mala

EPFL2018