Series-parallel orders are deened as the least class of partial orders containing the one-element order and closed by ordinal sum and disjoint union. From this inductive deenition, it is almost immediate that any series-parallel order may be represented by an algebraic expression, which is unique up to the associativity of ordinal sum and to the associa-tivivity and commutativity of disjoint un...

A novel higher order large-domain hierarchical finite-element technique using curl-conforming vector basis functions constructed from standard Legendre polynomials on generalized curvilinear hexahedral elements is proposed for electromagnetic modeling. The technique combines the inherent modeling flexibility of hierarchical elements with excellent orthogonality and conditioning properties of Le...

Series-parallel orders are deened as the least class of partial orders containing the one-element order and closed by ordinal sum and disjoint union. From this inductive deenition, it is almost immediate that any series-parallel order may be represented by an algebraic expression, which is unique up to the associativity of ordinal sum and to the associa-tivivity and commutativity of disjoint un...

Trotter conjectured that dim P × Q ≥ dimP + dimQ − 2 for all posets P and Q. To shed light on this, we study the k-dimension of products of finite orders. For k ∈ o(ln lnn), the value 2 dimk(P )−dimk(P ×P ) is unbounded when P is an n-element antichain, and 2dim2(mP ) − dim2(mP × mP ) is unbounded when P is a fixed poset with unique maximum and minimum. For products of the “standard” orders Sm ...

Let R be a root system with fixed basis ~' and let W be its Weyl group. For every element w ~ W, there exists a natural correspondence between reduced decompositions of w and some linear orders of the set of inversions of w. We study combinatorial properties of these orders. Furthermore, in the case of a symmetric group, for any permutation w we construct a linear action of the symmetric group ...

A higher order hybrid finite element – method of moments (FEM-MoM) technique for threedimensional analysis and design of radiating and scattering structures is presented. The technique uses generalized parametric hexahedral and quadrilateral elements of arbitrary geometrical orders in conjunction with curland divergence-conforming hierarchical polynomial vector basis functions of arbitrary appr...

In [13] it was demonstrated that the Proper Forcing Axiom implies that there is a five element basis for the class of uncountable linear orders. The assumptions needed in the proof have consistency strength of at least infinitely many Woodin cardinals. In this paper we reduce the upper bound on the consistency strength of such a basis to something less than a Mahlo cardinal, a hypothesis which ...

We make use of binary encoding of a phase wave front for the generation of diffractive elements. Binary encoding enables to introduce a constant phase shift between different diffraction orders. We describe the main features of phase wave front encoding and then suggest to apply them for a design of a diffractive alignment element. The proposed positioning tool is a modified version of the hype...

A finite semifield D is a finite nonassociative ring with identity such that the set D∗ = D \ {0} is closed under the product. From any finite semifield a projective plane can be constructed. In this paper we obtain new semifield planes of orders 64 and 81 by means of computational methods. These computer-assisted results yield to a complete classification (up to isotopy) of 81-element finite s...