37,571 research outputs found
Hyperspherical harmonics for tetraatomic systems
A recursion procedure for the analytical generation of hyperspherical harmonics for tetraatomic systems, in terms of row-orthonormal hyperspherical coordinates, is presented. Using this approach and an algebraic Mathematica program, these harmonics were obtained for values of the hyperangular momentum quantum number up to 30 (about 43.8 million of them). Their properties are presented and discussed. Since they are regular at the poles of the tetraatomic kinetic energy operator, are complete, and are not highly oscillatory, they constitute an excellent basis set for performing a partial wave expansion of the wave function of the corresponding Schrödinger equation in the strong interaction region of nuclear configuration space. This basis set is, in addition, numerically very efficient and should permit benchmark-quality calculations of state-to-state differential and integral cross sections for those systems
A clever elimination strategy for efficient minimal solvers
We present a new insight into the systematic generation of minimal solvers in
computer vision, which leads to smaller and faster solvers. Many minimal
problem formulations are coupled sets of linear and polynomial equations where
image measurements enter the linear equations only. We show that it is useful
to solve such systems by first eliminating all the unknowns that do not appear
in the linear equations and then extending solutions to the rest of unknowns.
This can be generalized to fully non-linear systems by linearization via
lifting. We demonstrate that this approach leads to more efficient solvers in
three problems of partially calibrated relative camera pose computation with
unknown focal length and/or radial distortion. Our approach also generates new
interesting constraints on the fundamental matrices of partially calibrated
cameras, which were not known before.Comment: 13 pages, 7 figure
Richardson Extrapolation for Linearly Degenerate Discontinuities
In this paper we investigate the use of Richardson extrapolation to estimate
the convergence rates for numerical solutions to advection problems involving
discontinuities. We use modified equation analysis to describe the expectation
of the approach. In general, the results do not agree with a-priori estimates
of the convergence rates. However, we identify one particular use case where
Richardson extrapolation does yield the proper result. We then demonstrate this
result using a number of numerical examples.Comment: 19 pages, 4 figur
Simplicial ideals, 2-linear ideals and arithmetical rank
In the first part of this paper we study scrollers and linearly joined
varieties.
A particular class of varieties, of important interest in classical Geometry
are Cohen--Macaulay varieties of minimal degree.
They appear naturally studying the fiber cone of of a codimension two toric
ideals.
Let be an ideal defining a linearly joined arrangement of
varieties:
- We compute the depth, and the cohomological dimension. is the connectedness
dimension.
- We characterize sets of generators of , and give an effective algorithm
to find equations, as an application we compute arithmetical rank. in the case
if defines a union of linear spaces, (ara =projective dimension), in
particular this applies to any square free monomial ideal having a linear
resolution.
- In the case where is a union of linear spaces, the ideal , can be
characterized by a tableau, which is an extension of a Ferrer (or Young)
tableau.
- We introduce a new class of ideals called simplicial ideals, ideals
defining linearly-joined varieties are a particular case of simplicial ideals.Comment: 31 pages, 5 figure
Nilpotent Bases for a Class of Non-Integrable Distributions with Applications to Trajectory Generation for Nonholonomic Systems
This paper develops a constructive method for finding a nilpotent basis for a special class of smooth nonholonomic distributions. The main tool is the use of the Goursat normal form theorem which arises in the study of exterior differential systems. The results are applied to the problem of finding a set of nilpotent input vector fields for a nonholonomic control system, which can then used to construct explicit trajectories to drive the system between any two points. A kinematic model of a rolling penny is used to illustrate this approach. The methods presented here extend previous work using "chained form" and cast that work into a coordinate-free setting
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