In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even selfdual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow ha...

Figure 1. (a) Example of loop nest. (b)Bounded iteration space (BIS) for this loop nest. Figure 2. (a) TIS obtained through a unimodular transformation. (b) TIS obtained through a non-unimodular transformation. White and black dots represent points of the TIS with integer antiimage. The black dots represent the points of the BTIS with integer antiimage. Figure 3. Parameters which characterize t...

Definition 1.1. A matrix A ∈ Zm×n is totally unimodular if the determinant of every square submatrix B ∈ Zk×k equals either −1, 0, or +1. Alternatively, by Cramer’s rule, A ∈ Zm×n is totally unimodular if every nonsingular submatrix B ∈ Zk×k has an integral inverse B−1 ∈ Zk×k. Recall: B−1 = 1 det B B ∗, where B∗ is the adjugate matrix (transpose of the matrix of cofactors) of B. One important c...

A majority of loop restructuring techniques developed so far assume that loops are perfectly nested. The unimodular approach unifies three individual transformations – loop interchange, skewing and reversal – but is still limited to perfect loop nests. This paper outlines a framework that enables the use of unimodular transformations to restructure imperfect loop nests. The concepts previously ...

1 We show that, for any regular Poisson manifold, there is an injective natural linear map from the first leafwise cohomology space into the first Poisson cohomology space which maps the Reeb class of the symplectic foliation to the modular class of the Poisson manifold. The Riemannian interpretation of those classes will permit us to show that a regular Poisson manifold whose symplectic foliat...

The paper narrows down the possible automorphisms of extremal even unimodular lattices of dimension 72. With extensive computations in Magma using the very sophisticated algorithm for computing class groups of algebraic number fields written by Steve Donnelly it is shown that the extremal even unimodular lattice Γ72 from [17] is the unique extremal even unimodular lattice of dimension 72 that a...

The random interlacement point process (introduced by Sznitman, generalized Teixeira) is a Poisson on the space of labeled doubly infinite nearest neighbour trajectories modulo time-shift transient graph $G$. We show that any transitive $G$ factor i.i.d., i.e., it can be constructed from family i.i.d. variables indexed vertices via an equivariant measurable map. Our proof uses variant soft loca...

It is shown that a lossless first-order optical system whose real symplectic ray transformation matrix can be diagonalized and has only unimodular eigenvalues is similar to a separable fractional Fourier transformer in the sense that the ray transformation matrices of the unimodular system and the separable fractional Fourier transformer are related by means of a similarity transformation. More...