139 research outputs found
Self-adjoint operators generated from non-Lagrangian symmetric differential equations having orthogonal polynomial eigenfunctions
No full textWe discuss the self-adjoint spectral theory associated with a certain fourth-order non-Lagrangian symmetrizable ordinary differential equation t(4)[y] = lambday that has a sequence of orthogonal polynomial solutions. This example was first discovered by Jung, Kwon, and Lee. In their paper, they derive the remarkable formula for these polynomials {Q(n)(x)}(n=0)infinity : Q(n)(x) = n integral(1)(x) PLn-1(t)dt, n is an element of N, where {PLn(x)}(n=0)(infinity) are the left Legendre type polynomials. The left Legendre type polynomials and the spectral analysis of the associated symmetric fourth-order differential equation that they satisfy have been extensively studied previously by Krall, Loveland, Everitt, and Littlejohn
Jacobi–Stirling numbers, Jacobi polynomials, and the left-definite analysis of the classical Jacobi differential expression
No full textAbstractWe develop the left-definite analysis associated with the self-adjoint Jacobi operator Ak(α,β), generated from the classical second-order Jacobi differential expressionℓα,β,k[y](t)=1wα,β(t)((-(1-t)α+1(1+t)β+1y′(t))′+k(1-t)α(1+t)βy(t))(t∈(-1,1)),in the Hilbert space Lα,β2(-1,1)≔L2((-1,1);wα,β(t)), where wα,β(t)=(1-t)α(1+t)β, that has the Jacobi polynomials {Pm(α,β)}m=0∞ as eigenfunctions; here, α,β>-1 and k is a fixed, non-negative constant. More specifically, for each n∈N, we explicitly determine the unique left-definite Hilbert–Sobolev space Wn,k(α,β)(-1,1) and the corresponding unique left-definite self-adjoint operator Bn,k(α,β) in Wn,k(α,β)(-1,1) associated with the pair (Lα,β2(-1,1),Ak(α,β)). The Jacobi polynomials {Pm(α,β)}m=0∞ form a complete orthogonal set in each left-definite space Wn,k(α,β)(-1,1) and are the eigenfunctions of each Bn,k(α,β). Moreover, in this paper, we explicitly determine the domain of each Bn,k(α,β) as well as each integral power of Ak(α,β). The key to determining these spaces and operators is in finding the explicit Lagrangian symmetric form of the integral composite powers of ℓα,β,k[·]. In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the Jacobi–Stirling numbers. Some properties of these numbers, which in some ways behave like the classical Stirling numbers of the second kind, are established including a remarkable, and yet somewhat mysterious, identity involving these numbers and the eigenvalues of Ak(α,β)
The Oak House, West Bromwich, Staffordshire (Proof)
No full textOriginal sketch of a house with three large chimney stacks depicting a man holding two horses for the hunt by Allen Edward Everitt (1824-1882). Published in London July 1, 1845 by Chapman & Hall. Artist proof. Originally produced for "The Baronial Halls, and Picturesque Edifices of England", London 1848, author; Sameul Carter HallMr JA van Tilburg bequeathed his "prentenkabinet" of over 10 000 graphic works to the University of PretoriaJacob van Tilburgab201
q-Differential equations for q-classical polynomials and q-Jacobi-Stirling numbers
No full textWe introduce, characterise and provide a combinatorial interpretation for the so-called q-Jacobi–Stirling numbers.
This study is motivated by their key role in the (reciprocal) expansion of any power of a second order
q-differential operator having the q-classical polynomials as eigenfunctions in terms of other even order operators,
which we explicitly construct in this work. The results here obtained can be viewed as the q-version of
those given by Everitt et al. and by the first author, whilst the combinatorics of this new set of numbers is a
q-version of the Jacobi–Stirling numbers given by Gelineau and the second author
Singular point-like perturbations of the Laguerre operator in a Pontryagin space
No full textThe spectral problem for the Laguerre equation on (0, infinity) with real parameter a in the case 0 </alpha/ <1 is closely related to the Nevanlinna functionQ(alpha)(z) -piGamma(-z)/(sinpialpha)Gamma(-z -alpha ).If /alpha/ > 1 and /alpha/ not equal 2, 3,..., this function belongs to the generalized Nevanlinna class N-m, m = [/alpha/+1/2]. A natural question appears: to what spectral problem does this function correspond? For alpha <-1, alpha not equal -2, -3...., an answer was given by Derkach [D]. He obtained an operator representation for the function m(alpha)(Z) = -Q(alpha)(-z)/Gamma(2)(1 + alpha) in terms of a self-adjoint operator in a Pontyragin space. and an interpretation, of m. (z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That ail indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for alpha > 1, alpha not equal 2, 3... we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Q (z) and compare them with the models for -alpha. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Q(alpha)(z) and -Q(alpha)(z)(-1). Finally, we describe the connection between the self-adjoint linear relations in the representations of Q(alpha)(z) and Q(-alpha)(z+alpha) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions.</p
50 A qualitative study of public online discussion forums: exploring parents’ concerns about children’s sleep problems and views about online, community and primary care support
No full textIntroduction Chronic insomnia is common in children. Behavioural interventions are effective.1 A systematic review (pending publication) revealed UK research about primary healthcare (PC) management is limited. Parents seek advice online,2 however, no published research to date has explored parents’ discussions online about PC management. This qualitative study explored (in online discussions) parents’ concerns/expectations about children’s sleep problems, awareness of online, PC, and community management resources, and perceptions of management within PC.Methods Two public online discussion forums were searched for parents’ discussions about children’s sleep problems. Eligible threads were analysed with Braun and Clarke’s reflexive thematic analysis.Results Ninety-three threads were included.Five main themes were developed. Parents had many ‘concerns about children’s sleep problems’ and were emotional/practical support for one another: ‘parents experiences or sharing advice online as a resource’. Parents expressed little regarding PC but had ‘mixed experiences and perceptions of community-based PC professionals’ and ‘limited experiences and perceptions of general practice’. They often discussed ‘other resources for supporting parents with child sleep problems’ (e.g. apps, private sleep consultants).Discussion Parents may have unmet management needs, act as resources for one another, and use non-healthcare resources, however the accuracy of these resources must be explored. The management of chronic insomnia within PC specifically must be further explored.This study/project is funded by the National Institute for Health Research (NIHR) School for Primary Care Research. The views expressed are those of the author(s) and not necessarily those of the NIHR or the Department of Health and Social Care
Singular point-like perturbations of the Laguerre operator in a Pontryagin space
No full textThe spectral problem for the Laguerre equation on (0, infinity) with real parameter a in the case 0 </alpha/ <1 is closely related to the Nevanlinna functionQ(alpha)(z) -piGamma(-z)/(sinpialpha)Gamma(-z -alpha ).If /alpha/ > 1 and /alpha/ not equal 2, 3,..., this function belongs to the generalized Nevanlinna class N-m, m = [/alpha/+1/2]. A natural question appears: to what spectral problem does this function correspond? For alpha <-1, alpha not equal -2, -3...., an answer was given by Derkach [D]. He obtained an operator representation for the function m(alpha)(Z) = -Q(alpha)(-z)/Gamma(2)(1 + alpha) in terms of a self-adjoint operator in a Pontyragin space. and an interpretation, of m. (z) as the Titchmarsh-Weyl function of some boundary value problem related to the Laguerre equation. That ail indefinite metric was needed was made clear earlier by Morton and Krall [MK]. In this note for alpha > 1, alpha not equal 2, 3... we answer this and related questions by using Pontryagin space operator realizations of suitable singular point-like perturbations of the Laguerre operator. We describe the operator models for Q (z) and compare them with the models for -alpha. Also we discuss the spectral properties of the self-adjoint linear relations in the representation of the functions Q(alpha)(z) and -Q(alpha)(z)(-1). Finally, we describe the connection between the self-adjoint linear relations in the representations of Q(alpha)(z) and Q(-alpha)(z+alpha) and show that this connection can be viewed as an operator implementation of the Kummer transform for confluent hypergeometric functions.</p
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