Dyson's Theorem Research Articles

A symmetric function generalization of the Zeilberger–Bressoud $q$-Dyson theorem

In 2000, Kadell gave an orthogonality conjecture for a symmetric function generalization of the Zeilberger--Bressoud $q$-Dyson theorem or the $q$-Dyson constant term identity. This conjecture was proved by Karolyi, Lascoux and Warnaar in 2015. In this paper, by slightly changing the variables of Kadell's conjecture, we obtain another symmetric function generalization of the $q$-Dyson constant term identity. This new generalized constant term admits a simple product-form expression.

A combinatorial analog of a theorem of F.J. Dyson

Tucker's lemma is a combinatorial analog of the Borsuk–Ulam theorem and the case n = 2 was proposed by Tucker in 1945. Numerous generalizations and applications of the lemma have appeared since then. In 2006 Meunier proved the lemma in its full generality in his PhD thesis. There are generalizations and extensions of the Borsuk–Ulam theorem that do not yet have combinatorial analogs. In this note, we give a combinatorial analog of a result of Freeman J. Dyson and show that our result is equivalent to Dyson's theorem. As with Tucker's lemma, we hope that this will lead to generalizations and applications and ultimately a combinatorial analog of Yang's theorem of which both Borsuk–Ulam and Dyson are special cases.

Dyson's theorem for curves

Let K be a number field and X 1 and X 2 two smooth projective curves defined over it. In this paper we prove an analogue of the Dyson theorem for the product X 1 × X 2 . If X i = P 1 we find the classical Dyson theorem. In general, it will imply a self contained and easy proof of Siegel theorem on integral points on hyperbolic curves and it will give some insight on effectiveness. This proof is new and avoids the use of Roth and Mordell–Weil theorems, the theory of Linear Forms in Logarithms and the Schmidt subspace theorem.

A short proof of the Zeilberger-Bressoud $q$-Dyson theorem

We give a formal Laurent series proof of Andrews's q-Dyson Conjecture, first proved by Zeilberger and Bressoud.

Mode-I Elliptical Crack in an Infinite Elastic Body. Part 1. Crack-Face Displacement for the Polynomial Law of Loading

For a mode-I embedded elliptical crack in an infinite elastic body, we propose a modified procedure for the calculation of the weight function and stress intensity factor based on our earlier development. Analytical and numerical values of the stress intensity factor along the crack front have been obtained for different cases of the polynomial law of loading. We propose an approach to the determination of the crack-face displacements from the stress intensity factor values, in which the Rice energy-balance equation, Dyson's theorem, and the theory of crack translation in a nonuniform stress field are used. An expression of closed form for the elliptical crack-face displacement for a polynomial law of loading of any degree has been derived, which can be employed in solving three-dimensional problems of the elasticity theory for cracked bodies.

Method of translations for a mode I elliptic crack in an infinite body. Part I: polynomial loading

A method has been proposed for determining the displacement of the elliptic crack faces in an infinite body and consequently stress intensity factors under the action of polynomial loading. The method is based: on the Rice integral formula which relates the stress and displacement fields for two different states of a body; on Dyson's theorem which defines the form of the displacement field for the prescribed law of the action of the polynomial loading; on the theory of the elliptic crack translations in a nonuniform stress field developed in the present study, and finally on the known solution for a uniform loading.The method proposed does not require the solution of boundary problem and actually represents itself the recurrent procedure for step by step determination of the displacement field for higher and higher degrees of polynomial loading. In its structure, objectives and complexity the method corresponds to the weight function methods known in the literature whose main feature is the use of the known particular solutions for the given body in order to obtain new solutions.

Method of translations for elliptic mode I cracks in infinite bodies. Part 1. Polynomial loads

We propose a method for evaluation of the displacement of the lips of an elliptic crack in an infinite body and, hence, for the determination of the stress intensity factors under the action of a polynomial load. The method is based on the Rice integral formula for the fields of stresses and displacements in two different states of the body, the Dyson theorem for the form of the field of displacements for a given polynomial load, the author's theory of translations of an elliptic crack in a nonuniform stress field, and the use of the existing solution for the case of uniform loading. The structure and complexity of the proposed method are similar to those of the well-known methods of weight functions based on the application of known particular solutions of the problem for a given body in constructing new more general solutions. For the special case of elliptic cracks in infinite bodies, the proposed method realizes the prediction of Fett concerning integration of the method of weight functions with general approaches of the theory of elasticity.

Extreme level repulsion for chaotic quantum Hamiltonians

We study the repulsion of energy levels of a fermion shell-model Hamiltonian, the classical limit of which corresponds to ergodic motion. We show that when the (triaxial) symmetries of the Hamiltonian are broken the level repulsion is modified. It becomes extreme (i.e. the nearest-neighbour distribution is quartic at the origin) when the Hamiltonian does not possess any symmetry other than total parity and time-reversal. This is in perfect agreement with the predictions of Dyson's theorem.

A Note on Dyson's Theorem

A Note on Dyson's Theorem

CVII. The electromagnetic field in a Lagrangian quantum theory

Abstract The graphical rules of Feynman and Dyson are stated in a very general form, which depends only on the field Lagrangian, and does not explicitly require either the use of field operators or the construction of a Hamiltonian scheme. It is shown that by choosing these rules as a starting-point one is able to quantize the electromagnetic field in a simpler way, since there is no longer any need to modify the classical Lagrangian 1/4Fμν Fμν (which does not lead to a Hamiltonian as it stands); to impose a supplementary condition; or to employ an indefinite metric. Dyson's theorem on photon propagation functions is established, and the gauge-invariance of the S-matrix is discussed.

On the Nuclear Forces

Summary. Generalizing the Lagrangian of the interaction as has been done by Schoen­ berg, and assuming the relations between coupling constants in the symmetrical and charged spin 0 meson fields, nuclear forces which have the same character as the Bethe s potential by the neutral vector meson theory does will be given. There is, however, no difference between the usual scalar-pseudoscalar mixed theory and our generalized interaction one in the approximation adopted here. § 1. Introduction. In studying the interaction of mesons with nucleons the pseudoscalar meson theory has been considered by several authors. For many years the central problem of the meson theory has been to look for a satisfactory type of the meson field to explain the properties of nuclei by the customary perturbation method. As is well-known, definite advances towards the understanding of the divergence problem have been made. Recently, using the newly developed method easel) ~nd others!) showed that the additional nuclear magnetic moments calculated by the pseudoscalar meson theory in the Tomonaga-Schwinger formalism could not fit the existing experimental data in all cases, namely, neutral, charged and symmetrical theories. This point is the serious difficulties for the present meson theory. It must be, however, noted that, although it is hardly possible for us to decide the type of the meson which interacts with nucleons by investigating only the nuclear magnetic moment, still, the difficulty is the important disproof for the present meson theory. On the other band, considering that many results obtained by the pseudoscalar meson field have been in agreement with various experimental data, one could not discard it at once. In the present situation there seems to be another pro­ blem on the pseudo scalar meson theory. That is the Dyson's theorem!> concerning the interactions between mesons and nucleons. Dyson has shown that the pseudoscalar interaction and the pseudovector one are completely equivalent for the first order radiative process. It is suggested from his argument that we * Preliminary report of the results appeared in Prog. Theor. Phys. 4 (1949), 572. 102