In Homogeneous permutations, Peter Cameron [Electronic Journal of Combinatorics 2002] classified the homogeneous permutations (homogeneous structures with 2 linear orders), and posed the problem of classifying the homogeneous ndimensional permutation structures (homogeneous structures with n linear orders) for all finite n. We prove here that the lattice of ∅-definable equivalence relations in ...

From symplectic reflection algebras [12], some algebras are naturally introduced. We show that these algebras are non-homogeneous N Koszul algebras. The Koszul property was generalized to homogeneous algebras of degree N > 2 in [6]. In the present paper, the extension of the Koszul property to non-homogeneous algebras is realized through a PBW theorem. This PBW theorem is the generalization to ...

The present work considers (4+1)-dimensional spatially homogeneous vacuum cosmological models. Exact solutions — some already existing in the literature, and others believed to be new — are exhibited. Some of them are the most general for the corresponding Lie group with which each homogeneous slice is endowed, and some others are quite general. The characterization " general " is given based o...

(Almost strongly minimal) generalized n-gons are constructed for all n& 3 for which the automorphism group acts transitively on the set of ordered ordinary (n­1)-gons contained in it, a new class of BN-pairs thus being obtained. Through the construction being modified slightly, 2! many non-isomorphic almost strongly minimal generalized n-gons are obtained for all n& 3, none of which interprets ...

In this paper, we show that there is a family of polynomials {Pn}, where Pn is a polynomial in n variables of degree at most d = O(log2 n), such that • Pn can be computed by linear sized homogeneous depth-5 circuits. • Pn can be computed by poly(n) sized non-homogeneous depth-3 circuits. • Any homogeneous depth-4 circuit computing Pn must have size at least nΩ( √ d). This shows that the paramet...

A homogeneous set is a non-trivial module of a graph, i.e. a nonempty, non-unitary, proper vertex subset such that all its elements present the same outer neighborhood. Given two graphs G1(V, E1) and G2(V,E2), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a graph GS(V, ES), E1 ⊆ ES ⊆ E2, which has a homogeneous set. This paper presents an algorithm that uses the concept ...

Let OTd(n) be the smallest integer N such that every N -element point sequence in R d in general position contains an order-type homogeneous subset of size n, where a set is order-type homogeneous if all (d + 1)-tuples from this set have the same orientation. It is known that a point sequence in R that is order-type homogeneous, forms the vertex set of a convex polytope that is combinatorially ...

It is shown that the Jacobian Conjecture holds for all polynomial maps F : k → k of the form F = x + H , such that JH is nilpotent and symmetric, when n ≤ 4. If H is also homogeneous a similar result is proved for all n ≤ 5. Introduction Let F := (F1, . . . , Fn) : C → C be a polynomial map i.e. each Fi is a polynomial in n variables over C. Denote by JF := (i ∂xj )1≤i,j≤n, the Jacobian matrix ...

The notion of homogeneous tensors is discussed. We show that there is a one-to-one correspondence between multivector fields on a manifold M , homogeneous with respect to a vector field ∆ on M and first-order polydifferential operators on a closed submanifold N of codimension 1 such that ∆ is transversal to N . This correspondence relates the Schouten-Nijenhuis bracket of multivector fields on ...