Automatic Data Layout Transformations in the ExaStencils Code Generator

Performance optimizations should focus not only on the computations of an application, but also on the internal data layout. A well-known problem is whether a struct of arrays or an array of structs results in a higher performance for a particular application. Even though the switch from the one to the other is fairly simple to implement, testing both transformations can become laborious and error-prone. Additionally, there are more complex data layout transformations, such as a color splitting for multi-color kernels in the domain of stencil codes, that are manually difficult. As a remedy, we propose new flexible layout transformation statements for our domain-specific language ExaSlang that support arbitrary affine transformations. Since our code generator applies them automatically to the generated code, these statements enable the simple adaptation of the data layout without the need for any other modifications of the application code. This constitutes a big advance in the ease of testing and evaluating different memory layout schemes in order to identify the best.

Canonical Decompositions of Affine Permutations, Affine Codes, and Split $k$-Schur Functions

We develop a new perspective on the unique maximal decomposition of an arbitrary affine permutation into a product of cyclically decreasing elements, implicit in work of Thomas Lam. This decomposition is closely related to the affine code, which generalizes the $k$-bounded partition associated to Grassmannian elements. We also prove that the affine code readily encodes a number of basic combinatorial properties of an affine permutation. As an application, we prove a new special case of the Littlewood-Richardson Rule for $k$-Schur functions, using the canonical decomposition to control for which permutations appear in the expansion of the $k$-Schur function in noncommuting variables over the affine nil-Coxeter algebra.

The Pseudo-Rigid Rolling Coin

A pseudo-rigid coin is a thin disk that can deform only to the extent of undergoing an arbitrary affine deformation in its own plane. The coupling of the classical rolling problem with this deformability, albeit limited, may shed light on such phenomena as the production of noise by a twirling dish. From the point of view of analytical dynamics, one of the interesting features of this problem is that the rolling constraint turns out to be nonholonomic even in the case of motion on a straight line in a vertical plane. After the analytical formulation of the general problem, explicit solutions are obtained for special shape-preserving motions. For more general motions, numerical studies are carried out for various initial conditions.

HAMILTONIAN FORMALISM OF TWO-LOOP WZNW MODEL

The Hamiltonian canonical formalism of two-dimensional WZNW model based on an arbitrary affine Lie algebra is given under the Chevalley basis. The Poisson brackets of conserved chiral currents are calculated, which turn out to be the classical two-loop Kac-Moody current algebras.

RESTRICTED QUANTUM AFFINE SYMMETRY OF PERTURBED MINIMAL CONFORMAL MODELS

We study the structure of superselection sectors of an arbitrary perturbation of a conformal field theory. We describe how a restriction of the q-deformed sl(2) affine Lie algebra symmetry of the sine-Gordon theory can be used to derive the S-matrices of the Φ(1,3) perturbations of the minimal unitary series. This analysis provides an identification of fields which create the massive kink spectrum. We investigate the ultraviolet limit of the restricted sine-Gordon model, and explain the relation between the restriction and the Fock space cohomology of minimal models. We also comment on the structure of degenerate vacuum states. Deformed Serre relations are proven for arbitrary affine Toda theories, and it is shown in certain cases how relations of the Serre type become fractional spin supersymmetry relations upon restriction.

FEIGIN-FUCHS REPRESENTATIONS OF ARBITRARY AFFINE LIE ALEGBRAS

We develop a systematic method to obtain the Feigin-Fuchs representations for arbitrary affine Lie algebras from a geometrical viewpoint. Choosing canonical coordinates for the flag manifolds associated with the Lie algebras, we get the general formulae for the KacMoody currents. In particular, we use this method to construct explicitly the Feigin-Fuchs representations for the affine Lie algebra of [Formula: see text] type.

Ideals and Higher Derivations in Commutative Rings

In this paper, we wish to generalize the following lemma first proven by O. Zariski [5, Lemma 4]. Let O be a complete local ring containing the rational numbers and let m denote the maximal ideal of O. Assume there exists a derivation δ of O such that δ(x) is a unit in O for some x in m. Then O contains a ring O1 of representatives of the (complete local) ring O/Ox having the following properties: (a) δ is zero O1; (b) x is analytically independent over O1; (c) O is the power series ring O1[[x]]. In [4], A. Seidenberg used Zariski's lemma extensively to study conditions under which an affine algebraic variety V over a base field of characteristic zero is analytically a product along a given subvariety W of V. We should like to generalize Zariski's lemma by removing the condition that O contain the rationals. We could then get some conditions under which an arbitrary affine variety V would be analytically a product along a subvariety W.