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...

In this paper an integral representation for the Hermite matrix polynomials is given. By means of the exact computation of certain matrix integrals and the integral representation of Hermite matrix polynomials, a formula for the generating function of the product of Her-mite matrix polynomials is obtained. Both the integral representation of the Hermite matrix polynomials and the formula for th...

in this paper we establish several polynomials similar to bernstein's polynomials and several refinements of  hermite-hadamard inequality for convex functions.

In the paper, the authors consider the generating functions of the Hermite polynomials and their squares, present explicit formulas for higher order derivatives of the generating functions of the Hermite polynomials and their squares, which can be viewed as ordinary differential equations or derivative polynomials, find differential equations that the generating functions of the Hermite polynom...

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 introduce a class of stationary 1-D interpolating subdivision schemes, denoted by Hermite(m, L, k), which classifies all stationary Lagrange or Hermite interpolating subdivision schemes with prescribed multiplicity, support and polynomial reproduction property: Given m > 0, L > 0 and 0 ≤ k ≤ 2mL − 1, Hermite(m, L, k) is a family parametrized by m2L− m(k +1)/2 (in the symmetric case) or 2m2L−...

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...

In this work, we have theoretically analyzed and numerically evaluated the accuracy of high-order lattice Boltzmann (LB) models for capturing non-equilibrium effects in rarefied gas flows. In the incompressible limit, the LB equation is proved to be equivalent to the linearized Bhatnagar-Gross-Krook (BGK) equation. Therefore, when the same Gauss-Hermite quadrature is used, LB method closely ass...

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...