On orthogonal projectors induced by compact groups and Haar measures

We study the difference of two orthogonal projectors induced by compact groups of linear operators acting on a vector space. An upper bound for the difference is derived using the Haar measures of the groups. A particular attention is paid to finite groups. Some applications are given for complex matrices and unitarily invariant norms. Majorization inequalities of Fan and Hoffmann and of Causey are rediscovered.     

Aspects of compact quantum group theory

We show that if a compact quantum semigroup satisfies certain weak cancellation laws, then it admits a Haar measure, and using this we show that it is a compact quantum group. Thus, we obtain a new characterization of a compact quantum group. We also give a necessary and sufficient algebraic condition for the Haar measure of a compact quantum group to be faithful, in the case that its coordinate -algebra is exact. A representation is given for the linear dual of the Hopf -algebra of a compact quantum group, and a functional calculus for unbounded linear functionals is derived.

     

On the conjugacy classes in the orthogonal and symplectic groups over algebraically closed fields

Let F$\mathbb{F}$ be an algebraically closed field. Let V$\mathbb{V}$ be a vector space equipped with a non-degenerate symmetric or symplectic bilinear form B   over F$\mathbb{F}$. Suppose the characteristic of F$\mathbb{F}$ is sufficiently large  , i.e. either zero or greater than the dimension of V$\mathbb{V}$. Let I(V,B)$I(\mathbb{V},\mathbb{B})$ denote the group of isometries. Using the Jacobson–Morozov lemma we give a new and simple proof of the fact that two elements in I(V,B)$I(\mathbb{V},\mathbb{B})$ are conjugate if and only if they have the same elementary divisors.     

On the group of polynomial functions in a group

Let G be a group and let n be a positive integer. A polynomial function in G is a function from G ⁿ to G of the form , where f(x ₁, . . . , x _n ) is an element of the free product of G and the free group of rank n freely generated by x ₁, . . . , x _n . There is a natural definition for the product of two polynomial functions; equipped with this operation, the set of polynomial functions is a group. We prove that this group is polycyclic if and only if G is finitely generated, soluble, and nilpotent-by-finite. In particular, if the group of polynomial functions is polycyclic, then necessarily it is nilpotent-by-finite. Furthermore, we prove that G itself is polycyclic if and only if the subgroup of polynomial functions which send (1, . . . , 1) to 1 is finitely generated and soluble.      

Quadratic modules and the orthogonal group over polynomial rings

The following conjecture of Parimala is proved: Any quadratic space over a polynomial ring with coefficients from an algebraically closed field of characteristic different from 2 is extended from the coefficient field. In the case of an arbitrary field of characteristic different from 2, an analogous result is obtained for quadratic spaces whose Witt index is at least 2. Also proved are general cancellation theorems for quadratic modules and a stabilization theorem for the orthogonal group over arbitrary polynomial rings.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta, im. V. A. Steklova AN SSSR, Vol. 71, pp. 216–250, 1977.     

On orthogonal polynomials obtained via polynomial mappings

Let (p_n)_n be a given monic orthogonal polynomial sequence (OPS) and k a fixed positive integer number such that k≥2. We discuss conditions under which this OPS originates from a polynomial mapping in the following sense: to find another monic OPS (q_n)_n and two polynomials π_k and θ_m, with degrees k and m (resp.), with 0≤m≤k−1, such that In this work we establish algebraic conditions for the existence of a polynomial mapping in the above sense. Under such conditions, when (p_n)_n is orthogonal in the positive-definite sense, we consider the corresponding inverse problem, giving explicitly the orthogonality measure for the given OPS (p_n)_n in terms of the orthogonality measure for the OPS (q_n)_n. Some applications and examples are presented, recovering several known results in a unified way.     

Asymptotics for polynomials orthogonal over the unit disk with respect to a positive polynomial weight

We derive asymptotics for polynomials orthogonal over the complex unit disk with respect to a weight of the form ²|h(z)|, with h(z) a polynomial without zeros in |z|<1. The behavior of the polynomials is established at every point of the complex plane. The proofs are based on adapting to the unit disk a technique of J. Szabados for the asymptotic analysis of polynomials orthogonal over the unit circle with respect to the same type of weight.     

Constructions for orthogonal designs using signed group orthogonal designs

Craigen introduced and studied signed group Hadamard matrices extensively and eventually provided an asymptotic existence result for Hadamard matrices. Following his lead, Ghaderpour introduced signed group orthogonal designs and showed an asymptotic existence result for orthogonal designs and consequently Hadamard matrices. In this paper, we construct some interesting families of orthogonal designs using signed group orthogonal designs to show the capability of signed group orthogonal designs in generation of different types of orthogonal designs.     

The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles

We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers under fairly weak conditions on the underlying measure μ. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.     

On the divisibility of the discriminant of an integral polynomial by prime powers

The discriminant of an integral polynomial is one of its main characteristics. It influences the distribution of its roots, the structure of the finite extension of the rational field generated by the polynomial's roots. In the paper, we show that, for any given prime power p ^b , there exists an irreducible polynomial with discriminant being a multiple of p ^b .     

On orthogonal generalized equitable rectangles

In this note, we give a complete solution of the existence of orthogonal generalized equitable rectangles, which was raised as an open problem in by Stinson (Des Codes Cryptogr 45:347–357, 2007).      

On orthogonal polynomial approximation with the dimensional expanding technique for precise time integration in transient analysis

We use four orthogonal polynomial series, Legendre, Chebyshev, Hermite and Laguerre series, to approximate the non-homogeneous term for the precise time integration and incorporate them with the dimensional expanding technique. They are applied to various structures subjected to transient dynamic loading together with Fourier and Taylor approximation proposed in previous works. Numerical examples show that all six methods are efficient and have reasonable precision. In particular, Legendre approximation has much higher precision and better convergence; Chebyshev approximation is also good, but only slightly inferior to Legendre approximation. The other four approximation methods usually produce results with errors hundreds of thousands of times larger. Hermite and Laguerre approximation may be useful for some special non-homogeneous terms, but do not work sufficiently well in our numerical examples. Other contributions of this paper include, a Dynamic Programming scheme for computing series coefficients, a general formula to find the assistant matrix for any polynomial series.     

On the level-continuity of fuzzy integrals

In this paper we define the level-convergence of measurable functions on a fuzzy measure space, by using the closure operator in the Moore sense. We study some of the properties of this convergence and give conditions for the continuity of the fuzzy integral in relation to the level-convergence.     

On properties of integrals of the Legendre polynomial

Properties of the integrals $P_{n0} (x) = P_n (x),P_{nk} (x) = \int\limits_{ - 1}^x {P_{n,k - 1} (y)dy} $ of the Legendre polynomials P _n (x) on the base interval ?1 ≤ x ≤ 1 are systematically considered. The generating function $(1 - 2xz + z^2 )^{k - 1/2} = Q_k (x,z) + ( - 1)^k (2k - 1)!!\sum\limits_{n = k}^\infty {P_{nk} (x)z^{n + k} } $ is defined; here, Q ₀ = 0 and Q _k with k > 0 is a polynomial of degree 2k ? 1 in each of the variables x and z. A second-order differential equation is derived, an analogue of Rodrigues’ formula is obtained, and the asymptotic behavior as n → ∞ is determined. It is proved that the representation $P_{nk} (x) = (x^2 - 1)^k f_{nk} (x)$ holds if and only if n ≥ k, where f _nk is a polynomial divisible by neither x ? 1 nor x + 1. The main result is the sharp bound $|P_{nk} (\cos \theta )| < \frac{{A_k }} {{\nu ^{k + 1/2} }}\sin ^{k - 1/2} \theta ,n \geqslant k.$ Here, $\nu ^2 = \left( {n + \frac{1} {2}} \right)^2 - \left( {k^2 - \frac{1} {4}} \right)\left( {1 - \frac{4} {{\pi ^2 }}} \right),A_k = \sqrt t _k J_k (t_k ) \sim \mu _1 k^{1/6} ,\mu _1 = 0.674885, $ where t _k is the maximum of the function $\sqrt t J_k (t)$ on the half-axis t > 0 and J _k (t) is the Bessel function. The first values A _k and differences A _k ? μ₁ k ^1/6 are tabulated below as follows:      

Finding the symmetry group of an LP with equality constraints and its application to classifying orthogonal arrays

For a given linear program (LP) a permutation of its variables that sends feasible points to feasible points and preserves the objective function value of each of its feasible points is a symmetry of the LP. The set of all symmetries of an LP, denoted by $G^{\mathrm{LP}}$, is the symmetry group of the LP. Margot (2010) described a method for computing a subgroup of the symmetry group $G^{\mathrm{LP}}$ of an LP. This method computes $G^{\mathrm{LP}}$ when the LP has only non-redundant inequalities and its feasible set satisfies no equality constraints. However, when the feasible set of the LP satisfies equality constraints this method finds only a subgroup of $G^{\mathrm{LP}}$ and can miss symmetries. We develop a method for finding the symmetry group of a feasible LP whose feasible set satisfies equality constraints. We apply this method to find and exploit the previously unexploited symmetries of an orthogonal array defining integer linear program (ILP) within the branch-and-bound (B&B) with isomorphism pruning algorithm (Margot, 2007). Our method reduced the running time for finding all OD-equivalence classes of OA $(160,8,2,4)$ and OA $(176,8,2,4)$ by factors of $1∕(2.16)$ and $1∕(1.36)$ compared to the fastest known method (Bulutoglu and Ryan, 2018). These were the two bottleneck cases that could not have been solved until the B&B with isomorphism pruning algorithm was applied. Another key finding of this paper is that converting inequalities to equalities by introducing slack variables and exploiting the symmetry group of the resulting ILP’s LP relaxation within the B&B with isomorphism pruning algorithm can reduce the computation time by several orders of magnitude when enumerating a set of all non-isomorphic solutions of an ILP.     

On series by Haar system

The paper considers the series by Haar system \(\sum\limits_{n = 1}^\infty {a_n \chi _n (x)} \), satisfying the conditions \(\sum\limits_{n = 1}^\infty {a_n^2 \chi _n^2 (x)} = \infty \) and a _n χ _n (x) → 0 almost everywhere. Some theorems about correcting a function on sets of arbitrarily small measures are proved.     

Selective approximate identities for orthogonal polynomial sequences

Let be an orthogonal polynomial sequence on the real line with respect to a probability measure π with compact support S. For y∈S, a sequence of polynomials is called a selective approximate identity with respect to y if for all f∈C(S). We prove the existence and give a complete characterization of a selective approximate identity depending on . A Fejér-like construction is performed and is considered in the context of Nevai class M(b,a) and Nevai's G-operator.