The Goldberg-Sachs theorem has been very useful in constructing algebraically special exact solutions of Einstein vacuum equation. Most of the physical meaningful vacuum exact solutions are algebraically special. We show that the Goldberg-Sachs theorem is not true in linearized gravity. This is a remarkable result, which gives light on the understanding of the physical meaning of the linearized...

Birkhoff used what has been called ([7]) 'Hilbert's projective metric' or ([9]) the 'Cayley-Hilbert metric'. In each case, it proved possible to obtain sharp estimates for the contraction constant of a positive linear operator with respect to the ' almost' metric. Subsequently, several authors generalized and sharpened the original results and established a close connection between the Birkhoff...

Today two American mathematicians, George David Birkhoff (March 21, 1884 – November 12, 1944) and his son Garrett Birkhoff (January 19, 1911 – November 22, 1996) are featured. The elder Birkhoff was born in Overisel, Michigan, the son of a physician who had come from Holland in 1870. When Birkhoff was two, the family moved to Chicago. From 1896 to 1902, he studied at the Lewis Institute (now th...

We develop and defend the thesis that the Hilbert space formalism of quantum mechanics is a new theory of probability. The theory, like its classical counterpart, consists of an algebra of events, and the probability measures defined on it. The construction proceeds in the following steps: (a) Axioms for the algebra of events are introduced following Birkhoff and von Neumann. All axioms, except...

The effects of multiaccess sources and time varying multipath are considered in analyzing a system in which multiple sources communicate with a fixed base station. We discuss detection and the use of stripping in a multiaccess multipath environment. We finally derive a capacity for these systems. It turns out that CDMA type systems are inherently capable (theoretically) of higher rates than sys...

Methods approaching the problem of the Hermite Birkhoff interpolation of scattered data by combining Shepard operators with local interpolating polynomials are not new in literature [1–4]. In [3] combinations of Shepard operators with bivariate Hermite-Birkhoff local interpolating polynomials are introduced to increase the algebraic degree of precision (polynomial reproduction degree) of Shepar...

We prove the conjecture by Diaconis and Eriksson (2006) that the Markov degree of the Birkhoff model is three. In fact we prove the conjecture in a generalization of the Birkhoff model, where each voter is asked to rank a fixed number, say r, of candidates among all candidates. We also give an exhaustive characterization of Markov bases for small r.

We show that toric ideals of flow polytopes are generated in degree 3. This was conjectured by Diaconis and Eriksson for the special case of the Birkhoff polytope. Our proof uses a hyperplane subdivision method developed by Haase and Paffenholz. It is known that reduced revlex Gröbner bases of the toric ideal of the Birkhoff polytope Bn have at most degree n. We show that this bound is sharp fo...

The Schubert cells eλ are the orbits of the Iwahori subgroup B̃, while the Birkhoff strata Sλ are the orbits of the opposite Iwahori subgroup B̃ −. The cells and the strata are dual in the sense that Sλ ∩ eλ = {λ}, and the intersection is transverse. The closure of eλ is the affine Schubert variety Xλ. It has dimension ` (λ), where ` is the minimal length occuring in the coset λW̃I , and its cells...

Using normal coordinates in a Poincaré-Birkhoff-Witt basis for the Hopf algebra of renormalization in perturbative quantum field theory, we investigate the relation between the twisted antipode axiom in that formalism, the Birkhoff algebraic decomposition and the universal formula of Kontsevich for quantum deformation. PACS: 02.40Gh, 11.10.Gh, 03.70.+k, 03.65.Fd