We present a simple, explicit orthogonal basis of eigenvectors for the Johnson and Kneser graphs, based on Young’s orthogonal representation of the symmetric group. Our basis can also be viewed as an orthogonal basis for the vector space of all functions over a slice of the Boolean hypercube (a set of the form {(x1, . . . , xn) ∈ {0, 1} : ∑ i xi = k}), which refines the eigenspaces of the Johns...

An (r,λ)-design with mutually balanced nested subdesigns ( for brevity, (r,λ)-design with MBN ) is introduced firstly in this paper. It is shown that an (r,λ)design with MBN is equivalent to a balanced array of strength 2 with s symbols. By the use of a nested design and an orthogonal array, a construction of an (r,λ)design with MBN is given. A direct construction of such an (r,λ)-design, based...

Motivated by the recent surge of activity on the subject, we present a brief non-technical survey of numerous classical and new results in discrepancy theory related to Roth’s orthogonal function method. Communicated by Yukio Ohkubo The goal of this paper is to survey the use of Roth’s orthogonal function method in the theory of irregularities of distribution, which, in a broad sense, means app...

L(p) = ∫ 1 −1 p(x)x(1− x2)−1/2eiζxdx, ζ ∈ R. Since the weight function alternates in sign in the interval of orthogonality, the existence of orthogonal polynomials is not assured. A nonconstructive proof of the existence is given. The three-term recurrence relation for such polynomials is investigated and the asymptotic formulae for recursion coefficients are derived. AMS Mathematics Subject Cl...

We bound the number of nearly orthogonal vectors with fixed VCdimension over {−1, 1}n. Our bounds are of interest in machine learning and empirical process theory and improve previous bounds by Haussler. The bounds are based on a simple projection argument and the generalize to other product spaces. Along the way we derive tight bounds on the sum of binomial coefficients in terms of the entropy...

ABSTRACT. The subject of this paper are polynomials in multiple non-commuting variables. For polynomials of this type orthogonal with respect to a state, we prove a Favardtype recursion relation. On the other hand, free Sheffer polynomials are a polynomial family in non-commuting variables with a resolvent-type generating function. Among such families, we describe the ones that are orthogonal. ...

We consider the existence of optimal parameters for generalized logistic model by least orthogonal absolute deviations, and prove the existence of such optimal solution, under the monotonicity condition on the data.

Let fSng denote the sequence of polynomials orthogonal with respect to the Sobolev inner product