Systematic Recipe Research Articles

Legendrian fronts for affine varieties

In this article we study Weinstein structures endowed with a Lefschetz fibration in terms of the Legendrian front projection. First we provide a systematic recipe for translating from a Weinstein Lefschetz bifibration to a Legendrian handlebody. Then we present several applications of this technique to symplectic topology. This includes the detection of flexibility and rigidity for several families of Weinstein manifolds and the existence of closed exact Lagrangian submanifolds. In addition, we prove that the Koras--Russell cubic is Stein deformation equivalent to affine complex 3-space and verify the affine parts of the algebraic mirrors of two Weinstein 4-manifolds.

Mesh arrangements for solid geometry

Many high-level geometry processing tasks rely on low-level constructive solid geometry operations. Though trivial for implicit representations, boolean operations are notoriously difficult to execute robustly for explicit boundary representations. Existing methods for 3D triangle meshes fall short in one way or another. Some methods are fast but fail to produce closed, self-intersection free output. Other methods are robust but place prohibitively strict assumptions on the input, e.g., no hollow cavities, non-manifold edges or self-intersections. We propose a systematic recipe for conducting a family of exact constructive solid geometry operations. The two-stage method makes no general position assumptions and does not resort to numerical perturbation. The method is variadic , operating on any number of input meshes. This generalizes unary mesh-repair operations, classic binary boolean differencing, and n -ary operations such as finding all regions inside at least k out of n inputs. We demonstrate the superior effectiveness and robustness of our method on a dataset of 10,000 "real-world" meshes from a popular online repository. To encourage development, validation, and comparison, we release both our code and dataset to the public.

Semiclassical description of resonance-assisted tunneling in one-dimensional integrable models

Resonance-assisted tunneling is investigated within the framework of one-dimensional integrable systems. We present a systematic recipe, based on Hamiltonian normal forms, to construct one-dimensional integrable models that exhibit resonance island chain structures with accurately controlled sizes and positions of the islands. Using complex classical trajectories that evolve along suitably defined paths in the complex time domain, we construct a semiclassical theory of the resonance-assisted tunneling process. This semiclassical approach yields a compact analytical expression for tunnelling-induced level splittings which is found to be in very good agreement with the exact splittings obtained through numerical diagonalization.

A derivation of the master equation from path entropy maximization

The master equation and, more generally, Markov processes are routinely used as models for stochastic processes. They are often justified on the basis of randomization and coarse-graining assumptions. Here instead, we derive nth-order Markov processes and the master equation as unique solutions to an inverse problem. We find that when constraints are not enough to uniquely determine the stochastic model, an nth-order Markov process emerges as the unique maximum entropy solution to this otherwise underdetermined problem. This gives a rigorous alternative for justifying such models while providing a systematic recipe for generalizing widely accepted stochastic models usually assumed to follow from the first principles.

The worker/wrapper transformation

AbstractThe worker/wrapper transformation is a technique for changing the type of a computation, usually with the aim of improving its performance. It has been used by compiler writers for many years, but the technique is little known in the wider functional programming community, and has never been described precisely. In this article we explain, formalise and explore the generality of the worker/wrapper transformation. We also provide a systematic recipe for its use as an equational reasoning technique for improving the performance of programs, and illustrate the power of this recipe using a range of examples.

How to eliminate self-reference: a précis

We provide a systematic recipe for eliminating self-reference from a simple language in which semantic paradoxes (whether purely logical or empirical) can be expressed. We start from a non-quantificational language L which contains a truth predicate and sentence names, and we associate to each sentence F of L an infinite series of translations h0(F), h1(F), ..., stated in a quantificational language L*. Under certain conditions, we show that none of the translations is self-referential, but that any one of them perfectly mirrors the semantic behavior of the original. The result, which can be seen as a generalization of recent work by Yablo (1993, Analysis, 53, 251–252; 2004, Self-reference, CSLI) and Cook (2004, Journal of Symbolic Logic, 69(3), 767–774), shows that under certain conditions self-reference is not essential to any of the semantic phenomena that can be obtained in a simple language.

Representation of pure fluid coexistence curves by series expansion

We give a systematic recipe for constructing the coexistence curve associated with a pure fluid described by an analytical equation of state. The novelty of our result lies not in its theoretical basis, which is well known, but in the high degree of accuracy that it will routinely yield over a broad range of density and temperature for a relatively modest expenditure of computing effort. Our results are quite general: they may be applied to any analytical equation of state. The effectiveness of our method is illustrated by applying our general solution to four widely known model equations of state: van der Waals, Dieterici, Soave-Redlich-Kwong, and Carnahan-Starling with van der Waals tail term. These models show a considerable diversity of coexistence curve shapes, and our method yields accurate representations for each model. An investigation is also initiated of the representation of experimental coexistence curves that exhibit pronounced “non-classical” behavior over an appreciable range of densities.

Skew-selfadjoint form for systems of conservation laws

Hyperbolic systems of conservation laws augumented with an entropy inequality are studied. It is shown that such systems can be written in a (quasilinear) skew-selfadjoint form. Centered differencing of such a form under the smooth regime ends up with a systematic recipe for constructing quasiconservative schemes where the global entropy conservation is recovered. Employing the above formulation in bounded regions under the nonsmooth regime as well, a local entropy decay estimate is also concluded. Examples of the shallow-water and the full gasdynamics equations are explicitly treated.

SU(3) ⊃ R(3) Wigner coefficients in the 2s-1d shell

Explicit algebraic expressions are given for SU(3) ⊃ R(3) Wigner coefficients, which are of particular interest in p-shell and 2s-1d shell-model calculations. An orthogonal basis tied closely to Elliott's basis is chosen according to a systematic recipe. The Wigner coefficients are those involving the Kronecker products ( λ 1 μ 1) × ( λ 2 μ 2) → ( λ 3 μ 3) in the following cases: 1. (i) ( λ 2 μ 2) = (10), (01) and μ 1 ≦ 4 (arbitrary λ i ), i = 1, 3. (For the case μ i = 4, only auxillary coefficients are tabulated whenever the angular momentum multiplicity is 3. From these the final Wigner coefficients can be calculated for any specific case of interest.) 2. (ii) ( λ 2 μ 2) = (20), (02) and μ i ≦ 3 (arbitrary λ i ), or λ i ≦ 3 (arbitrary μ i ) i = 1, 3. 3. (iii) ( λ 2 μ 2 = (11) and μ i ≦ 3 (arbitrary λ i ), or λ i ≦ 3 (arbitrary μ i ) i = 1, 3 except for ( λ3) × (11) → ( λ3) which is not given. A method is also given whereby coefficients with more complicated (λ 2μ 2) can be calculated from coefficients of type (i) by a build-up process. Some symmetry properties are discussed.