To an integral lattice L in the euclidean space (R, (, )), one associates the set of characteristic vectors v ∈ R with (v, x) ≡ (x, x) mod 2Z for all x ∈ L. They form a coset modulo 2L, where L = {v ∈ R | (v, x) ∈ Z ∀x ∈ L} is the dual lattice of L. Recall that L is called integral, if L ⊂ L and unimodular, if L = L. For a unimodular lattice, the square length of a characteristic vector is cong...

Let G be a simple graph, and consider an orientation of the edges of G. Where V is the vertex-set of G, let A = (A,, : u, w E V) be the integral antisymmetric (0, +_ 1)-matrix such that A,,, = 1 if and only if VW is an edge of G directed from u to w. The orientation of G is said to be unimodular if for every WC V the determinant of the submatrix (A,, : u, w E W) is 0 or 1. For example, it is kn...

Random mechanisms have been used in real-life situations for reasons such as fairness. Voting and matching are two examples of such situations. We investigate whether desirable properties of a random mechanism survive decomposition of the mechanism as a lottery over deterministic mechanisms that also hold such properties. To this end, we represent properties of mechanisms—such as ordinal strate...

In this paper, we discuss several relations between the existence of invariant volume forms for Hamiltonian systems on Poisson-Lie groups and unimodularity structure. particular, prove that vector fields a Lie group endowed with unimodular structure preserve multiple any left-invariant group. Conversely, also if there exists function such identity element is nondegenerate singularity associated...

We consider unimodular random rooted trees (URTs) and invariant forests in Cayley graphs. We show that URTs of bounded degree are the same as the law of the component of the root in an invariant percolation on a regular tree. We use this to give a new proof that URTs are sofic, a result of Elek. We show that ends of invariant forests in the hyperbolic plane converge to ideal boundary points. We...

A unimodular complex surface is a complex 2-manifoldX endowed with a holomorphic volume form Υ. A strictly pseudoconvex real hypersurface in X inherits not only a CR-structure but a canonical coframing as well. In this article, this canonical coframing is defined, its invariants are discussed and interpreted geometrically, and its basic properties are studied. A natural evolution equation for s...

We use conference matrices to define an action of the complex numbers on the real Euclidean vector space R. In certain cases, the lattice D n becomes a module over a ring of quadratic integers. We can then obtain new unimodular lattices, essentially by multiplying the lattice D n by a non-principal ideal in this ring. We show that lattices constructed via quadratic residue codes, including the ...

in this paper we introduce two-wavelet constants for square integrable representations of homogeneous spaces. we establishthe orthogonality relations for square integrable representationsof homogeneous spaces which give rise to the existence of aunique self adjoint positive operator on the set of admissiblewavelets. finally, we show that this operator is a constant multiple of identity operator...