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Diagonalizability and symmetrizability of Sobolev-type bilinear forms: A combinatorial approach
Kim, H.K.,Kwon, K.H.,Littlejohn, L.L.,Yoon, G.J. North Holland [etc.] 2014 Linear algebra and its applications Vol.460 No.-
In an earlier paper, Kwon, Littlejohn and Yoon characterized symmetric Sobolev bilinear forms and showed that they have, like symmetric matrices, a diagonal representation. In this paper, we present a new proof of one of their main results by interpreting the coefficients in the diagonal representation of a Sobolev-type bilinear form from a combinatorial point of view. We view this as an improvement over the original proof which relied on mathematical induction.
The effecal of irradiance during leaf development on photoinhibition in Panag ginseng C. A. Meyer
Parmenter, Graeme,Littlejohn, Roger The Korean Society of Ginseng 1998 Journal of Ginseng Research Vol.22 No.2
This experiment used chlorophyll fluorescence techniques to assess the effect of irradiant during leaf development on photoinhibition of photosynthesis in Panax ginseng. Seedlings of p. ginseng were grown in the 91asshouse at four shade levels. The maximum mid-day irradiant in each treatment between emergence (January 4) and completion of the experiment (February 25) was 1220, 485, 235, 125 $\mu$mol/$\textrm{m}^2$/s. To assess the rapidity of photosynthetic readaptation to changes in light levels, fluorescence parameters (Fo, F, Fm, Fm', AF/Fm;, Fv/Fm) were measured for three days before and after transfer of plants (on February 21) from each light treatment into each of the other light treatments. Before transfer, dark adapted values of Fv/Fm in the 1220 (0.699) and 485 (0.739) treatments were different from each other and lower than values in the 235 (0.764) and 125 (0.768) treatments, indicating mild photoinhibition. Patterns of change in F during the day also differed between treatments, with low light treatments tracking irradiant levels, but F in the high light treatment (1220) declined in the morning, presumably due to fluorescence quenching. Although plants grown at high irradiant had relatively low photosynthetic efficiency, relative electron transport rate was greater than in lower irradiant treatments. After transfer, plants adopted the daily pattern of change in F of the treatment to which they were moved with little change in absolute levels of F, except in plants transferred from the highest (1220) to the lowest light level (125), where F increased over the course of the three days following transfer. After plants were transferred, Fm' converged on values similar to those in plants raised in the treatments to which they were moved. Values of Fv/Fm in plants moved from low to high light declined dramatically, but there was no decline in plants from 485 moved to 1220. Values of Pv/Fm in plants that were moved from high light to lower light increased to values above those recorded in plants raised in the lower light treatments. Reductions in quantum efficiency caused by photoinhibition at high irradiant may be more than compensated for by higher electron transport rates, although evidence suggests that under high irradiant this tends to be balanced by reduced leaf area and earlier senescence. Chlorophyll fluorescence techniques appear capable of indicating effects of irradiant induced stress in ginseng, yielding results comparable to those obtained with gas exchange techniques but in less time and with greater replication.
Sobolev orthogonal polynomials in two variables and second order partial differential equations
Lee, Jeong Keun,Littlejohn, L.L. Elsevier 2006 Journal of mathematical analysis and applications Vol.322 No.2
<P><B>Abstract</B></P><P>We consider polynomials in two variables which satisfy an admissible second order partial differential equation of the form<ce:label>(∗)</ce:label>A<SUB>uxx</SUB>+2B<SUB>uxy</SUB>+C<SUB>uyy</SUB>+D<SUB>ux</SUB>+E<SUB>uy</SUB>=λu, and are orthogonal relative to a symmetric bilinear form defined byϕ(p,q)=〈σ,pq〉+〈τ,<SUB>px</SUB><SUB>qx</SUB>〉, where A,…,E are polynomials in <I>x</I> and <I>y</I>, <I>λ</I> is an eigenvalue parameter, <I>σ</I> and <I>τ</I> are linear functionals on polynomials. We find a condition for the partial differential equation <ce:cross-ref refid='fd001'>(∗)</ce:cross-ref> to have polynomial solutions which are orthogonal relative to a symmetric bilinear form ϕ(⋅,⋅). Also examples are provided.</P>
Construction of differential operators having Bochner–Krall orthogonal polynomials as eigenfunctions
Kwon, K.H.,Littlejohn, L.L.,Yoon, G.J. Elsevier 2006 Journal of mathematical analysis and applications Vol.324 No.1
<P><B>Abstract</B></P><P>Suppose {<SUB>Qn</SUB>}n=0∞ is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where <I>σ</I> is a classical moment functional (Jacobi, Laguerre, Hermite) and <I>ν</I> is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having {<SUB>Qn</SUB>}n=0∞ as eigenfunctions.</P>
Ghost matrices and a characterization of symmetric Sobolev bilinear forms
Kwon, K.H.,Littlejohn, Lance L.,Yoon, G.J. Elsevier 2009 Linear algebra and its applications Vol.431 No.1
<P><B>Abstract</B></P><P>In this paper, we characterize symmetric Sobolev bilinear forms defined on P×P, where P is the space of polynomials. More specifically we show that symmetric Sobolev bilinear forms, like symmetric matrices, can be re-written with a diagonal representation. As an application, we introduce the notion of a ghost matrix, extending some classic work of Stieltjes.</P>
ORTHOGONAL POLYNOMIALS SATISFYING PARTIAL DIFFERENTIAL EQUATIONS BELONGING TO THE BASIC CLASS
Lee, J.K.,L.L. Littlejohn,Yoo, B.H. Korean Mathematical Society 2004 대한수학회지 Vol.41 No.6
We classify all partial differential equations with polynomial coefficients in $\chi$ and y of the form A($\chi$) $u_{{\chi}{\chi}}$ + 2B($\chi$, y) $u_{{\chi}y}$ + C(y) $u_{yy}$ + D($\chi$) $u_{{\chi}}$ + E(y) $u_{y}$ = λu, which has weak orthogonal polynomials as solutions and show that partial derivatives of all orders are orthogonal. Also, we construct orthogonal polynomials in d-variables satisfying second order partial differential equations in d-variables.s.
Everitt, W.N.,Kwon, K.H.,Littlejohn, L.L.,Wellman, R.,Yoon, G.J. Koninklijke Vlaamse Ingenieursvereniging 2007 Journal of computational and applied mathematics Vol.208 No.1
<P><B>Abstract</B></P><P>We develop the left-definite analysis associated with the self-adjoint Jacobi operator Ak(α,β), generated from the classical second-order Jacobi differential expression<SUB>ℓα,β,k</SUB>[y](t)=1<SUB>wα,β</SUB>(t)((-(1-t<SUP>)α+1</SUP>(1+t<SUP>)β+1</SUP><SUP>y′</SUP>(t)<SUP>)′</SUP>+k(1-t<SUP>)α</SUP>(1+t<SUP>)β</SUP>y(t))(t∈(-1,1)),in the Hilbert space Lα,β2(-1,1)≔<SUP>L2</SUP>((-1,1);<SUB>wα,β</SUB>(t)), where <SUB>wα,β</SUB>(t)=(1-t<SUP>)α</SUP>(1+t<SUP>)β</SUP>, that has the Jacobi polynomials {Pm(α,β)}m=0∞ as eigenfunctions; here, α,β>-1 and <I>k</I> 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 <SUB>ℓα,β,k</SUB>[·]. In turn, the key to determining these powers is a double sequence of numbers which we introduce in this paper as the <I>Jacobi–Stirling numbers</I>. 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).</P>
Molecular and clinical characteristics of hepatitis B virus in Korea
Ahn, Sang Hoon,Yuen, Lilly,Han, Kwang-Hyub,Littlejohn, Margaret,Chang, Hye Young,Damerow, Hans,Ayres, Anna,Heo, Jeong,Locarnini, Stephen,Revill, Peter A. Wiley Subscription Services, Inc., A Wiley Company 2010 Journal of Medical Virology Vol.82 No.7
<P>Korea is an endemic area of hepatitis B virus (HBV) infection but very little is known about the molecular characteristics of HBV isolates from Korean patients or the association with disease progression. The complete HBV genome sequences from 53 Korean patients with chronic hepatitis B, advanced cirrhosis, or hepatocellular carcinoma (HCC) were analyzed to identify (i) subgenotype distribution and genetic diversity and (ii) signature mutations associated with liver disease progression. With the exception of 1 patient infected with HBV/B, all 52 patients (98.1%) were infected with HBV/C, subgenotype C2. These strains were 98.4% identical and the frequency of amino acid substitutions occurring within key immunological epitopes increased with disease severity. A number of amino acid/nucleotide substitutions were associated with HCC, namely sR24K (HBsAg), SI126T (HBsAg), and pcA1846T (precore gene) mutations (P = 0.029, 0.001, and 0.008, respectively). HBV harboring deletions in the pre-S region were also associated with increased liver disease severity (chronic hepatitis B vs. cirrhosis, P = 0.040; chronic hepatitis B vs. HCC, P = 0.040). Despite the high degree of sequence conservation, several key HBV mutations were associated with disease progression. Prospective studies with larger cohorts of patients are required to evaluate further the clinical manifestation of HBV/C2 in Korea. J. Med. Virol. 82: 1126–1134, 2010. © 2010 Wiley-Liss, Inc.</P>