# Belfer Graduate School of Science Dissertations 1962 - 1978

Permanent URI for this collectionhttps://hdl.handle.net/20.500.12202/8579

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Item Open Access Existence and completeness of wave operators in two Hilbert spaces(Yeshiva University, 1984-06) Altabet, Meryl J.; Schechter, Martin; Lebow, Arnold; Rosenfeld, NormanIn this thesis we consider an unperturbed self-adjoint operator H(,0) on a Hilbert space H(,0), the operators A, B mapping H(,0) to the Hilbert space K, and J a bounded linear operator mapping the Hilbert space H(,0) to the Hilbert space H.;Our first objective is to give conditions under which there exists a perturbed self-adjoint operator H such that R(z)J - JR(,0)(z) = -(BJ('*)R(z))*AR(,0)(z) and HJ(R-HOOK)J(H(,0) + B('*)A). We prove the existence of the operator H by actually constructing its resolvent R(z).(').;Our next objective is to consider two specific operators H(,0), the momentum operator and the kinetic energy operator, and to give examples of A, B, J for which the main conclusions of scattering theory hold. In obtaining conditions for the existence and completeness of the wave operator, we use a combination of time dependent and stationary methods.Item Restricted Difference equation methods for solution of partial differential equations(ProQuest Dissertations & Theses, 1982) Kohn, Meryle Cherrick; Ehrenpreis, Leon; Lebow, Arnold; Rosenfeld, NormanDifference equation techniques are applied to determine sufficient conditions on polynomials P(x,y) for which the Fischer space related differential equation.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;has (a) no non-trivial solutions f(x,y); (b) locally convergent solutions; and (c) formal solutions; {lcub}where P(x,y) is a polynomial with complex coefficients,;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;is the differential operator whose coefficients are the complex conjugates of the coefficients of P, and.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;Solutions of a general difference equation in two dimensions, (SIGMA) K(,i)C(,m+ai, n+bi) = 0 with K(,i) non-vanishing, are analyzed. Particular emphasis is placed on solutions with C(,mn) = 0 in specified regions of the plane.;Difference equations corresponding to the differential equation.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;(where P and Q are polynomials) are examined to determine sufficient conditions on P and Q for which the equation has (a) no non-trivial solution; (b) polynomial solutions; and (c) formal solutions.Item Restricted Two parallel queues created by arrivals with two demands(ProQuest Dissertations & Theses, 1982) Hahn, Susan Ann; Flatto, Leopold; Lebow, Arnold; Schechter, MartinIn this thesis we consider the system of two unbounded single server queues in which a customer upon arrival joins both queues. The arrivals are assumed to form a Poisson process with mean interarrival time 1 and the servers have exponential service time distribution with means 1/(alpha) and 1/(beta) respectively. It is assumed 1 < (alpha) (LESSTHEQ) (beta) and hence the equilibrium probabilities p(,ij) are positive for all i,j and.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;The functional equation for the generating function.;(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI).;is obtained from the equilibrium equations. This equation exhibits a relation between the functions P(z,0), P(0,w) on the algebraic curve S = {lcub}(z,w): (zw)('2) - (1 + (alpha) + (beta))zw + (alpha)w + (beta)z = 0{rcub}. S is parametrized by a pair of elliptic functions z = z(t), w = w(t) and the functional equation is converted into automorphy conditions for A(t) = P(z(t),0) and B(t) = P(0,w(t)), which are then continued analytically to the whole t-plane. A(t) and B(t), and hence P(z,0), P(0,w), P(z,w) are obtained in closed form. Asymptotic formulas for p(,ij) as i,j (--->) (INFIN) are obtained from the expression for P(z,w).Item Restricted ON THE COMPLETENESS OF (F(N THETA)) AND A RECIPROCAL THEOREM FOR ABSOLUTELY CONVERGENT DIRICHLET SERIES(ProQuest Dissertations & Theses, 1980) GOODMAN, ARTHUR; Schechter, Martin; Newman, Donald J.; Rosenfeld, NormanOne of the basic questions in approximation theory is: given a Banach space B, and S a subset of B, under what conditions will S be "rich" enough to approximate any element of B by a finite linear combination of elements of S. The most important result of this type pertains to the case where B is the Banach space of continuous functions on a finite interval with the supremum norm, and S is the set of monomials {l,x,x 2, ••• }. In this case the famous Weierstrass Theorem asserts that any element of B can be uniformly approximated by a finite linear combination of elements of S, that is any continuous function on a finite interval can be uniformly approximated by a polynomial. (from Introduction)Item Restricted The representation ring of some finite simple groups(ProQuest Dissertations & Theses, 1979) Rabinowitz, Aliza DubinThis paper is concerned with some of the representations and representation rings of PSL(n, p). • We show how Frobenius 's original method of character computation can be applied to PSL(2,5), and give derivations for all the characters of PSL(3,p). We find the reprasentation ring of PSL(2,p) and PSL(3,p).Item Restricted Phase transitions in three-component fluid systems(ProQuest Dissertations & Theses, 1978) Sato, Makoto; Lebowitz, Joel L.; Behrends, Ralph E.; Petersen, AageWe prove the existence of phase segregation in a three component Widom-Rowlinson fluid mixture with activities εA=εB=ε≥εc. for sufficiently large ε. Integration over the coordinates of one component in a three-component Widom-Rowlinson Model yields a binary systems with hard core plus attractive many body potentials. We investigate some features of the phase diagram of such a binary mixture using the results of mean field theory and of series expansion for the three-component Widom-Rowlinson model. We also find a general way of obtaining the equation of state of multi-component one-dimensinal systems with non-additive hard core diameters and nearest neighbor soft potentials.Item Restricted Interpretation of molecular absorption in the atmosphere of Jupiter(ProQuest Dissertations & Theses, 1978) SATO, MAKIKO; Frye, Graham; Behrends, Ralph E.; Hansen, James E.Item Restricted Strong Coupling Calculations Of The Nonlinear Sigma Model(ProQuest Dissertations & Theses, 1977, September) Han, Daesoo; Frye, Graham; Behrends, Ralph E.; Mattis, Daniel C.Item Open Access States in separable algebras(ProQuest Dissertations & Theses, 1976) Czarnocha, Bronislaw; Petersen, Aage; Finkelstein, David; Komar, ArthurItem Restricted I. Pyrolysis of Dibenzotropilidene. Ii. Bicyclobutane-coupling constants(ProQuest Dissertations & Theses, 1976) Fink, Rina; Pomerantz, Martin; Beck, Donald R.; Borowitz, Irving J.; Halpern, TeodoroItem Open Access On the kinetics of phase transitions in binary alloys(ProQuest Dissertations & Theses, 1975) Marro, JoaquinItem Restricted A history of finite group theory in the United States: 1888-1916(ProQuest Dissertations & Theses, 1975) Craig, Robert T.; Grosof, Miriam S.; Flatto, Leopold; Newman, Donald J.; Woodruff, Arthur E .The study of finite group theory in the United States at the end of the 19th century and beginning of the 20th century. (from Introduction)Item Restricted Polynomials and harmonic functions on finite fields(ProQuest Dissertations & Theses, 1975) Shiffman, Mary; Ehrenpreis, Leon; Berger, Charles A.; Korányi, Ádám__The aim of this work is to see under what circumstances it is possible to extend the generalized form of the Lagrange Interpolation Formula which states that there is a unique polynomial f of degree n-1 which satisfies the condition: if x1, [see PAI] are points of the real or complex plane respectively, where ,[see PAI] • with multiplicity m1, [ ] ,a are real or complex numbers, then [] [] __In Chapter I this result is extended to the finite fields. Here we work with polynomials as in the classical case. To start, new "derivatives II are defined on the finite field polynomials, and are shown to have properties which can determine the multiplicity of the roots of a polynomial as the ordinary derivative determines multiplicity in the real and complex cases. Using these new operators we prove that a generalized form of the Lagrange Interpolation Formula holds in the finite fields. __Still working in the finite field we try to extend our_ result to complex valued functions. Chapter II deals with the characters as replacements for the xn. We hope that the characters will share with the classical polynomials the property of being harmonic, so we start by looking for a definition of harmonic. The first definition we give is a mean value property where we sum over the units . What seems to be an obvious choice for a definition is shown to be unsatisfactory when we prove that only constant functions are harmonic by this definition. A second definition in which we sum over a smaller group is also tried with the same results. Our final definition alters the first by limiting the group we sum over and the set of radii. Using this definition we show that non-constant, harmonic, complex valued functions do exist. We also show that if the finite field is GF(p n) and the set of radii contains n appropriately chosen elements, the only harmonic functions are the constant functions. Corresponding definitions and results are given for Q, the field of p-adic , and for the field of formal power series. As a note of comparison of these results with the classical case we give a statement of the Two Radius Theorem. __Since the set of radii easily becomes large enough to leave us with only the constant functions for our harmonic functions, it does not seem worthwhile to pursue the subject of harmonic functions any further. Instead, we turn to looking for "derivatives" on the characters to use in defining multiplicity in hopes we can use them in proving an interpolation formula. In the case of the multiplicative characters we show no such operators exist. __Although we have interesting results on harmonicity, none encourages us to use the characters to replace the x . Therefore, we continue in Chapter III to look for some other complex valued functions on the finite field which we can use. The candidates in Chapter III are Legendre polynomials. After looking at the classical Legendre polynomials we give a corresponding definition for Legendre polynomials and associated Legendre polynomials on the finite field. Next we actually compute examples of these polynomials in GF(3), GF(S), GF(9), GF(25) and GF(7). We meet with only limited success in making a correspondence between the new polynomials and the classical ones.Item Open Access Ergodic theory in statistical mechanics(ProQuest Dissertations & Theses, 1975) Aizenman, Michael; Lebowitz, Joel L.; Landford, III, Oscar E.; Penrose, OliverItem Metadata only A NEW METHOD IN THE STUDY OF NONLINEAR FREDHOLM OPERATOR EQUATIONS(ProQuest Dissertations & Theses, 1974) PODOLAK, ESTHERItem Restricted Theoretical isotope effects studies(ProQuest Dissertations & Theses, 1974) Wolf, Gail Ellen; Spindel, William; Borowitz, Irving J.; Goldstein, Martin; Ishida, Takanobu; Stern, Marvin J.Two theoretical isotope effects studies were carried out. The purpose of the first study was to develop a method for determining force fields of molecules from experimental thermodynamic data such as isotope-exchange equilibrium constant measurements or vapor pressure isotope effect measurements instead of from spectroscopic data. The purpose of the second study was to examine how certain structural features of molecules affect the prevalence of temperature-dependence anomalies of isotope-exchange equilibrium. __The technique used in the first study utilized the orthogonal polynomial expansion of the logarithmic form of the isotopic partitions function ratio (ln --6.. f), which is an explicit though non-linear function of the individual force constants. A method using the Gauss-Newton non-linear least squares fitting technique was developed in this study to solve for the unknown force constants. __Molecules for which force fields were available in the s2 literature were examined. The [] f values for these molecules were calculated from the known force fields for a series of temperatures and were used as the "experimental" data. The molecules were then treated as if their force fields were unknown. The non-linear least squares analysis was applied to a set of "guessed" values of the force constants in an effort to regenerate the "original II force constants.Item Open Access Ergodic Theory and Infinite-Systems(ProQuest Dissertations & Theses, 1974) GOLDSTEIN, SHELDON; Lebowitz, Joel L.; Lanford, III, Oscar E.; Newman, Donald E.Item Open Access Solution of the Percus-Yevick Equation(ProQuest Dissertations & Theses, 1973) Ahn, Saeyoung; Lebowitz, Joel L.; Cooper, Fred; Frye, Graham; Mattis, Daniel C.Item Open Access Homeomorphisms, global-univalence and surjectivity of maps between banachspaces.(ProQuest Dissertations & Theses, 1972) Plastock, Roy Alan; Berger, Melvyn S.; Coburn, Lewis; Lebow, ArnoldItem Restricted Monotonicity of quadrature formulas and polynomial representation(ProQuest Dissertations & Theses, 1972) Molluzzo, John Charles; Newman, D. J.; Coburn, L.; Berger, C.It is the purpose of this dissertation to investigate two questions. The first concerns for what functions are certain elementary mechanical quadrature formulas monotonic. The second concerns the possibility of representing polynomials of several variables as sums of superpositions of polynomials of one variable.

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