I. Exact solution of some nonlinear evolution equations. II. The similarity solution for the Korteweg-De Vries equation and the related Painleve Transcendent

<p>In Part I, a method for finding solutions of certain diffusive dispersive nonlinear evolution equations is introduced. The method consists of a straightforward iteration procedure, applied to the equation as it stands (in most cases), which can be carried out to all terms, followed by a summation of the resulting infinite series, sometimes directly and other times in terms of traces of inverses of operators in an appropriate space.</p> <p>We first illustrate our method with Burgers' and Thomas' equations, and show how it quickly leads to the Cole-Hopft transformation, which is known to linearize these equations.</p> <p>We also apply this method to the Korteweg and de Vries, nonlinear (cubic) Schr��dinger, Sine-Gordon, modified KdV and Boussinesq equations. In all these cases the multisoliton solutions are easily obtained and new expressions for some of them follow. More generally we show that the Marcenko integral equations, together with the inverse problem that originates them, follow naturally from our expressions.</p> <p>Only solutions that are small in some sense (i.e., they tend to zero as the independent variable goes to ���) are covered by our methods. However, by the study of the effect of writing the initial iterate u_1 = u_(1)(x,t) as a sum u_1 = ^���/u_1 + ^���/u_1 when we know the solution which results if u_1 = ^���/u_1, we are led to expressions that describe the interaction of two arbitrary solutions, only one of which is small. This should not be confused with Backlund transformations and is more in the direction of performing the inverse scattering over an arbitrary ���base��� solution. Thus we are able to write expressions for the interaction of a cnoidal wave with a multisoliton in the case of the KdV equation; these expressions are somewhat different from the ones obtained by Wahlquist (1976). Similarly, we find multi-dark-pulse solutions and solutions describing the interaction of envelope-solitons with a uniform wave train in the case of the Schrodinger equation.</p> <p>Other equations tractable by our method are presented. These include the following equations: Self-induced transparency, reduced Maxwell-Bloch, and a two-dimensional nonlinear Schrodinger. Higher order and matrix-valued equations with nonscalar dispersion functions are also presented.</p> <p>In Part II, the second Painleve transcendent is treated in conjunction with the similarity solutions of the Korteweg-de Vries equat ion and the modified Korteweg-de Vries equation.</p>

Negative regulators of a growth factor-mediated signaling pathway in the nematode Caenorhabditis elegans

<p>Vulval differentiation in C. elegans is mediated by an Epidermal growth factor (EGF)- EGF receptor (EGFR) signaling pathway. I have cloned unc-101, a negative regulator of vulval differentiation of the nematode C. elegans. unc-101 encodes a homolog of AP47, the medium chain of the trans-Golgi clathrin-associated protein complex. This identity was confirmed by cloning and comparing sequence of a C. elegans homolog of AP50, the medium chain of the plasma membrane clathrin-associated protein complex. I provided the first genetic evidence that the trans-Golgi clathrin-coated vesicles are involved in regulation of an EGF signaling pathway. Most of the unc-101 alleles are deletions or nonsense mutations, suggesting that these alleles severely reduce the unc-101 activity. A hybrid gene that contains parts of unc-101 and mouse AP4 7 rescued at least two phenotypes of unc-101 mutations, the Unc and the suppression of vulvaless phenotype of let-23(sy1) mutation. Therefore, the functions of AP47 are conserved between nematodes and mammals.</p> <p>unc-101 mutations can cause a greater than wild-type vulval differentiation in combination with certain mutations in sli-1, another negative regulator of the vulval induction pathway. A mutation in a new gene, rok-1, causes no defect by itself, but causes a greater than wild-type vulval differentiation in the presence of a sli-1 mutation. The unc-101; rok-1; sli-1 triple mutants display a greater extent of vulval differentiation than any double mutant combinations of unc-101, rok-1 and sli-1. Therefore, rok-1 locus defines another negative regulator of the vulval induction pathway.</p> <p>I analyzed a second gene encoding an AP47 homolog in C. elegans. This gene, CEAP47, encodes a protein 72% identical to both unc-101 and mammalian AP47. A hybrid gene containing parts of unc-101 and CEAP47 sequences can rescue phenotypes of unc-101 mutants, indicating that UNC- 101 and CEAP47 proteins can be redundant if expressed in the same set of cells.</p>

Chains of Non-regular de Branges Spaces

<p>We consider canonical systems with singular left endpoints, and discuss the concept of a scalar spectral measure and the corresponding generalized Fourier transform associated with a canonical system with a singular left endpoint. We use the framework of de Branges��� theory of Hilbert spaces of entire functions to study the correspondence between chains of non-regular de Branges spaces, canonical systems with singular left endpoints, and spectral measures.</p> <p>We find sufficient integrability conditions on a Hamiltonian H which ensure the existence of a chain of de Branges functions in the first generalized P��lya class with Hamiltonian H. This result generalizes de Branges��� Theorem 41, which showed the sufficiency of stronger integrability conditions on H for the existence of a chain in the P��lya class. We show the conditions that de Branges came up with are also necessary. In the case of Krein���s strings, namely when the Hamiltonian is diagonal, we show our proposed conditions are also necessary.</p> <p>We also investigate the asymptotic conditions on chains of de Branges functions as t approaches its left endpoint. We show there is a one-to-one correspondence between chains of de Branges functions satisfying certain asymptotic conditions and chains in the P��lya class. In the case of Krein���s strings, we also establish the one-to-one correspondence between chains satisfying certain asymptotic conditions and chains in the generalized P��lya class.</p>

Stationary absolute distributions for chains of infinite order

<p>Let {��_n}^���_{n = -���} be a stochastic process with state space S₁ = {0, 1, ���, D ��� 1}. Such a process is called a chain of infinite order. The transitions of the chain are described by the functions</p> <p>Q_i(i⁽⁰⁾) = ��(��_n = i | ��_{n - 1} = i ⁽⁰⁾₁, ��_{n - 2} = i ⁽⁰⁾₂, ���) (i �� S₁), where i⁽⁰⁾ = (i⁽⁰⁾₁, i⁽⁰⁾₂, ���) ranges over infinite sequences from S₁. If i⁽ⁿ⁾ = (i⁽ⁿ⁾₁, i⁽ⁿ⁾₂, ���) for n = 1, 2,���, then i⁽ⁿ⁾ ��� i⁽⁰⁾ means that for each k, i⁽ⁿ⁾_k = i⁽⁰⁾_k for all n sufficiently large.</p> <p>Given functions Q_i(i⁽⁰⁾) such that </p> <p> (i) 0 ��� Q_i(i⁽⁰) ��� �� �� 1</p> <p>(ii)D ��� 1/��/i = 0 Q_i(i⁽⁰⁾) �� 1 </p> <p>(iii) Q_i(i⁽ⁿ⁾) ��� Q_i(i⁽⁰⁾) whenever i⁽ⁿ⁾ ��� i⁽⁰⁾,</p> <p>we prove the existence of a stationary chain of infinite order {��_n} whose transitions are given by</p> <p>�� (��_n = i | ��_{n - 1}, ��_{n - 2}, ���) = Q_i(��_{n - 1}, ��_{n - 2}, ���)</p> <p>With probability 1. The method also yields stationary chains {��_n} for which (iii) does not hold but whose transition probabilities are, in a sense, ���locally Markovian.��� These and similar results extend a paper by T.E. Harris [<u>Pac. J. Math.,</u> 5 (1955), 707-724].</p> <p>Included is a new proof of the existence and uniqueness of a stationary absolute distribution for an Nth order Markov chain in which all transitions are possible. This proof allows us to achieve our main results without the use of limit theorem techniques.</p>

Ortho-positronium annihilation: steps toward computing the first order radiative corrections

<p>The 0.2% experimental accuracy of the 1968 Beers and Hughes measurement of the annihilation lifetime of ortho-positronium motivates the attempt to compute the first order quantum electrodynamic corrections to this lifetime. The theoretical problems arising in this computation are here studied in detail up to the point of preparing the necessary computer programs and using them to carry out some of the less demanding steps -- but the computation has not yet been completed. Analytic evaluation of the contributing Feynman diagrams is superior to numerical evaluation, and for this process can be carried out with the aid of the Reduce algebra manipulation computer program. </p> <p>The relation of the positronium decay rate to the electronpositron annihilation-in-flight amplitude is derived in detail, and it is shown that at threshold annihilation-in-flight, Coulomb divergences appear while infrared divergences vanish. The threshold Coulomb divergences in the amplitude cancel against like divergences in the modulating continuum wave function. </p> <p>Using the lowest order diagrams of electron-positron annihilation into three photons as a test case, various pitfalls of computer algebraic manipulation are discussed along with ways of avoiding them. The computer manipulation of artificial polynomial expressions is preferable to the direct treatment of rational expressions, even though redundant variables may have to be introduced. </p> <p>Special properties of the contributing Feynman diagrams are discussed, including the need to restore gauge invariance to the sum of the virtual photon-photon scattering box diagrams by means of a finite subtraction. </p> <p>A systematic approach to the Feynman-Brown method of Decomposition of single loop diagram integrals with spin-related tensor numerators is developed in detail. This approach allows the Feynman-Brown method to be straightforwardly programmed in the Reduce algebra manipulation language. </p> <p>The fundamental integrals needed in the wake of the application of the Feynman-Brown decomposition are exhibited and the methods which were used to evaluate them -- primarily dis persion techniques are briefly discussed. </p> <p>Finally, it is pointed out that while the techniques discussed have permitted the computation of a fair number of the simpler integrals and diagrams contributing to the first order correction of the ortho-positronium annihilation rate, further progress with the more complicated diagrams and with the evaluation of traces is heavily contingent on obtaining access to adequate computer time and core capacity. </p>

Studies on rabbit reticulocyte ribosomes and polyribosomes and their relation to hemoglobin synthesis

<p>This dissertation is divided into three parts.</p> <p>The first section is concerned with protein synthesis in cellfree systems from reticulocytes. The sub-cellular reticulocyte fractions, reagents, etc. have been examined for the presence of traces of ribonuclease, using. an assay based upon the loss of infectivity of RNA fran bacteriophage MS2. This assay is sensitive to 5 x 10^-7 �� RNase/ml. In addition, the loss of synthetic capacity of an 80S ribosome on dissociation has been studied, and can be attributed to loss of messenger RNA when the monomer is separated into subunits. The presence of ribonuclease has been shown to be a major cause of polyribosome disintegration during cell-free protein synthesis.</p> <p>The second section concerns the changes in ribosomes and polyribosomes which occur during the maturation of a reticulocyte into an erythrocyte. With increasing age, the cells lose a large proportion of the ribonucleoprotein, but the percentage of ribosomes present as polyribosomes is only slightly altered. The loss of hemoglobin synthesis on maturation is probably due to both the loss of total ribosomes and to the lessened specific activity of the polyribosomes.</p> <p>The third section contains analytical ultracentrifugation data on 80S ribosomes, polyribosomes, and ribosomal RNA from reticulocytes. The 60s and 40s subunits, obtained by dissociation of the 80s particle with inorganic pyrophosphate, were also studied. The RNA from reticulocyte ribosomes has been examined under a variety of denaturing conditions, including dimethyl sulfoxide treatment, formaldehyde reaction and thermal denaturation. From these studies we can conclude that the 28S and 16S RNA's are single polynucleotide chains and are not made up of smaller RNA subunits hydrogen-bonded together.</p>