3,605 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
Construction of differential operators having Bochner–Krall orthogonal polynomials as eigenfunctions
No full textAbstractSuppose {Qn}n=0∞ is a sequence of polynomials orthogonal with respect to the moment functional τ=σ+ν, where σ is a classical moment functional (Jacobi, Laguerre, Hermite) and ν is a point mass distribution with finite support. In this paper, we develop a new method for constructing a differential equation having {Qn}n=0∞ as eigenfunctions
Differential equations and Sobolev orthogonality
No full textConsider (Sobolev) orthogonal polynomials which are orthogonal relative to a Sobolev bilinear form integral(R) p(x)q(x)d mu(x) + integral(R) p'(x)q'd nu(x), where d mu(x) and d nu(x) are signed Borel measures with finite moments. We give necessary and sufficient conditions under which such orthogonal polynomials satisfy a linear spectral differential equation with polynomial coefficients. We then find a sufficient condition under which such a differential equation is symmetrizable. These results can be applied to Sobolev-Laguerre polynomials found by Koekoek and Meijer
The Sobolev orthogonality and spectral analysis of the Laguerre polynomials {Ln−k} for positive integers k
No full textAbstractFor k∈N, we consider the analysis of the classical Laguerre differential expressionℓ−k[y](x)=1x−ke−x(−(x−k+1e−xy′(x))′+rx−ke−xy(x))(x∈(0,∞)),where r⩾0 is fixed, in several nonisomorphic Hilbert and Hilbert–Sobolev spaces.In one of these spaces, specifically the Hilbert space L2((0,∞);x−ke−x), it is well known that the Glazman–Krein–Naimark theory produces a self-adjoint operator A−k, generated by ℓ−k[·], that is bounded below by rI, where I is the identity operator on L2((0,∞);x−ke−x). Consequently, as a result of a general theory developed by Littlejohn and Wellman, there is a continuum of left-definite Hilbert spaces {Hs,−k=(Vs,−k,(·,·)s,−k)}s>0 and left-definite self-adjoint operators {Bs,−k}s>0 associated with the pair (L2((0,∞);x−ke−x),A−k). For A−k and each of the operators Bs,−k, it is the case that the tail-end sequence {Ln−k}n=k∞ of Laguerre polynomials form a complete set of eigenfunctions in the corresponding Hilbert spaces.In 1995, Kwon and Littlejohn introduced a Hilbert–Sobolev space Wk[0,∞) in which the entire sequence of Laguerre polynomials is orthonormal. In this paper, we construct a self-adjoint operator in this space, generated by the second-order Laguerre differential expression ℓ−k[·], having {Ln−k}n=0∞ as a complete set of eigenfunctions. The key to this construction is in identifying a certain closed subspace of Wk[0,∞) with the kth left-definite vector space Vk,−k
Self-adjoint operators generated from non-Lagrangian symmetric 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
Natalia LL - artystka neoawangardowa
No full textThe paper shows Natalia Lach-Lachowicz (Natalia LL) as a neo avant-garde artist whose works in a specific maximalistic way are very close to the main currents of avant-garde trends: new mediality (photography), minimalism, conceptualism, performance, bodyart, pop-art, and feminist art. The author of the article concentrates mainly on the mutual influences of conceptualism, consumptionism, and feminism in Natalia LL’s works and pays attention to the emancipatory potential of her works of the seventies and the eighties
Energy flux in isotropic turbulence under large variations of external forcing
No full textWe investigate the response of energy flux in isotropic turbulence to step-function like perturbation in external forcing at large length scales. From both physical experiments and direct numerical simulations, we measured the evolution of the Eulerian velocity structure functions, such as , , before and after the perturbation in forcing. In both cases, we observed the cascade of the energy excess at large scales cascade through scales to the dissipative range, which can be used to study the dynamics of the cascade, and in particular, to estimate the relevant time scales
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No full textStructural and functional analysis of the pro-domain of human cathelicidin, LL-37
No full textCathelicidins form a family of small host defense peptides distinct from another class of cationic antimicrobial peptides, the defensins. They are expressed as large precursor molecules with a highly conserved pro-domain known as the cathelin-like domain (CLD). CLDs have high degrees of sequence homology to cathelin, a protein isolated from pig leukocytes and belonging to the cystatin family of cysteine protease inhibitors. In this report, we describe for the first time the X-ray crystal structure of the human CLD (hCLD) of the sole human cathelicidin, LL-37. The structure of hCLD, determined at 1.93 Å resolution, shows the cystatin-like fold and is highly similar to the structure of the CLD of the pig cathelicidin, protegrin-3. We assayed the in vitro antibacterial activities of hCLD, LL-37 and the precursor form, pro-cathelicidin (also known as hCAP18), and we found that the unprocessed protein inhibited the growth of Gramnegative bacteria with efficiencies comparable to the mature peptide, LL-37. In addition, the antibacterial activity of LL-37 was not inhibited by hCLD intermolecularly, since exogenously added hCLD had no effect on the bactericidal activity of the mature peptide. hCLD itself lacked antimicrobial function and did not inhibit the cysteine protease, cathepsin L. Our results contrast with previous reports of hCLD activity. A comparative structural analysis between hCLD and the cysteine protease inhibitor stefin A showed why hCLD is unable to function as an inhibitor of cysteine proteases. In this respect, the cystatin scaffold represents an ancestral structural platform from which proteins evolved divergently, with some losing inhibitory functions
The modulatory effect of TLR2 on LL-37-induced human mast cells activation
No full textThe sole and endogenous anti-microbial peptide LL-37 is a significant effector molecule in the innate host defense system. Apart from its broadly direct anti-microbial activity, the peptide also activates mast cell in respect of allergic diseases and inflammation. On the other hand, mast cell can be activated by Toll-like receptors (TLRs) which are at the center of innate immunity. It was the aim of the study to illustrate the modulatory effect of TLR2 ligands peptidoglycan (PGN) and tripalmitoyl-S-glycero-Cys-(Lys)4 (Pam3CSK4) on LL-37 induced LAD2 cells (a human mast cell line) activation. LL-37 induced LAD2 cells degranulation and the release of IL-8. TLR2 ligands didn't induce LAD2 cells degranulation, but triggered the release of IL-8. Incubation with PGN or Pam3CSK4 significantly suppressed LL-37-induced degranulation through inhibition of calcium mobilization from LAD2 cells. Similarly, the release of IL-8 was inhibited when LAD2 cells were co-stimulated with TLR2 ligands and LL-37. Studies with inhibitors of key enzymes involved in mast cell signaling indicated that the release of IL-8 induced by TLR2 ligands and LL-37 involved the activation of the PI3K, ERK, JNK and calcineurin signaling pathways. In contrast, p38 activation down-regulated the release of IL-8 induced by TLR2 ligands and LL-37. Taken together, these observations suggest that activation of human mast cells by LL-37 could be modified by TLR2 ligands and the function of human mast cells could be switched from allergic reactions to innate immune response. (C) 2016 Elsevier Inc. All rights reserved.National Natural Science Foundation of China [81271755, 81371737]; Guangdong Natural Science Foundation [2014A030313708]; Shenzhen Research Grant [CXZZ20140416144209739, JCYJ20130329110752142, KQCX20120803145850990]SCI(E)[email protected]; [email protected]
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