We prove fractional analogs of the classical Sylvester-Gallai theorem. Our theorems translate local information about collinear triples in a set of points into global bounds on the dimension of the set. Specifically, we show that if for every points v in a finite set V ⊂ C, there are at least δ|V | other points u ∈ V for which the line through v, u contains a third point in V , then V resides i...

A bent function is called self-dual if it is equal to its dual. It is called anti-self-dual if it is equal to the complement of its dual. A spectral characterization in terms of the Rayleigh quotient of the Sylvester Hadamard matrix is derived. Bounds on the Rayleigh quotient are given for Boolean functions in an odd number of variables. An efficient search algorithm based on the spectrum of th...

Using an operator approach we derive Sylvester-Whitworth formulae for sets A'a. By the same token we treat the problem where both sets of A's and B's are involved. Our result extends the Sylvester-Whitworth inclusion and exclusion formula to the resolution of the number of elements in exactly mi sets of A's and mi sets of B's respectively. The formula are applied to the complete graph and compl...

In this paper a method for solving large-scale Sylvester equations is presented. The method is based on the sign function iteration and is particularly effective for Sylvester equations with factorized righthand side. In this case, the solution will be computed in factored form as it is for instance required in model reduction. The hierarchical matrix format and the corresponding formatted arit...

The nonlinear matrix equation X+A∗X−1A = Q can be cast as a linear Sylvester equation subject to unitary constraint. The Sylvester equation can be obtained by means of hermitian eigenvalue computation. The unitary constraint can be satisfied by means of either a straightforward alternating projection method or by a coordinate-free Newton iteration. The idea proposed in this paper originates fro...

We present a nested splitting conjugate gradient iteration method for solving large sparse continuous Sylvester equation, in which both coefficient matrices are (non-Hermitian) positive semi-definite, and at least one of them is positive definite. This method is actually inner/outer iterations, which employs the Sylvester conjugate gradient method as inner iteration to approximate each outer it...

We consider the solution of the ?-Sylvester equation AX±X?B? = C, for ? = T,H and A,B,∈ Cm×n, and some related linear matrix equations (AXB? ± X? = C, AXB? ± CX?D? = E, AX ± X?A? = C, AX ± Y B = C, AXB ± CY D = E, AXA? ± BY B? = C and AXB ± (AXB)? = C). Solvability conditions and stable numerical methods are considered, in terms of the (generalized and periodic) Schur, QR and (generalized) sing...

A necessary and sufficient condition on the support of a generic unmixed bivariate polynomial system is identified such that for polynomial systems with such support, the Dixon resultant formulation produces their resultants. It is shown that Sylvester-type matrices can also be obtained for such polynomial systems. These results are shown to be a generalization of related results recently repor...