Problems and results on additive properties of general sequences,V

A very special case of one of the theorems of the authors states as follows: Let 1a ₁a ₂... be an infinite sequence of integers for which all the sumsa _i +a _j , 1ij, are distinct. Then there are infinitely many integersk for which 2k can be represented in the forma _i +a _j but 2k+1 cannot be represented in this form. Several unsolved problems are stated.Dedicated to our friend Professor E. Hlawka on the occasion of his seventieth birthday     

On the multiplicative group generated by a dense sequence of integers

The subgroup of the positive rationals, generated by a sequence of upper density , has index essentially not exceeding ^–1.In honor of the seventieth birthday of Professor E. HlawkaPartially supported by N.S.F. Contract No. DMS-8500949.     

Limiting Lagrangians: A primal approach

In this paper, we consider a convex program with either a finite or an infinite number of constraints and its formal Lagrangian dual. We show that either the primal program satisfies a general condition which implies there is no duality gap or that there is a nonzero vectord with the following properties: First, wheneverd is added to the objective function, where is a positive number not greater than one, the resulting program satisfies the general sufficient condition cited above for no duality gap. Second, the optimal value of this perturbed program is attained and tends to the optimal value of the original program as tends to zero. Third, the optimal solutions of the perturbed programs form a minimizing sequence of the original program. As a consequence of the above, we derive the limiting Lagrangian theory of Borwein, Duffin, and Jeroslow.The authors are indebted to an unknown referee who suggested the very short and elegant proofs of Lemma 2.3 and Theorem 2.3.This work was completed while the first author was a member of the College of Management, Georgia Institute of Technology, Atlanta, Georgia.     

Observables on hypergraphs

Observables on hypergraphs are described by event-valued measures. We first distinguish between finitely additive observables and countably additive ones. We then study the spectrum, compatibility, and functions of observables. Next a relationship between observables and certain functionals on the set of measures M(H) of a hypergraph H is established. We characterize hypergraphs for which every linear functional on M(H) is determined by an observable. We define the concept of an effect and show that observables are related to effect-valued measures. Finally, we define operational transformations from M(H) to itself and show that they can be described as a certain combination of effects.On leave from University of Berne, Institute of Mathematical Statistics, Sidlerstrasse 5, CH-3012 Berne, Switzerland.     

Hamiltonian structures of nonlinear evolution equations connected with a polynomial pencil

For a generalized Zakharov-Shabat system in which the matrix potential is a polynomial in the spectral parameter a generating operator is constructed which makes it possible to compactly write out the nonlinear evolution equations (NEE) connected with the system. The eigenfunctions of the generating operator — the squares of solutions of the original system — are found. The Hamiltonian property of the NEE and the existence of a hierarchy of Hamiltonian structures are established.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 120, pp. 55–68, 1982.     

k-convexn-person games and their cores

For any natural numbersk andn, the subclass ofk-convexn-person games is introduced. In casek=n, the subclass consists of the convexn-person games. Ak-convexn-person game is characterized in several ways in terms of the core and certain marginal worth vectors. The marginal worth vectors of a game are described in terms of an upper bound for the core and the corresponding gap function. It is shown that thek-convexity of ann-person gamev is equivalent to

1. all marginal worth vectors ofv belong to the core ofv; or
2. the core ofv is the convex hull of the set consisting of all marginal worth vectors ofv; or
3. the extreme points of the core ofv are exactly the marginal worth vectors ofv.

Examples ofk-convexn-person games are also treated.     

The constant-volume-velocity nature of hearing aids: conclusions based on computer simulations

In the literature there are several references which imply that various parts of a hearing aid are sources of constant volume velocity. Reported herein are the findings of an investigation of the validity of such statements. A computer scheme, referenced elsewhere, for modeling in situ hearing aids was utilized to test the constant-volume-velocity hypothesis. In particular, capabilities of the receiver, ear hook, and earmold tip to deliver constant volume velocity were investigated via a computer. To facilitate such an investigation, a universal receiver/earmold model was created. This model was broken down into "source" and "load"at three locations: the receive output, output of the ear hook, and medial tip of the earmold. At each location comparisons were made between computed values of source and load impedance. The constant-volume-velocity hypothesis was assumed to be valid for those cases where source impedance was much, much greater than load impedance. Plots of such impedances show that, for the cases investigated, this rarely occurred, except over certain frequency bands. With the exception of in-the-ear hearing aids, these results appear to contradict inferences made in the literature about the constant-volume-velocity nature of hearing aids.     

Joint production of ϕ mesons andπ ±,π 0,p,\bar p,K s 0 andK ± in hadronic interactions

The joint production of ? mesons andπ ^±,π ⁰,p, \(\bar p\) ,K _s ⁰ andK ^± is investigated using a sample of 600,000 inclusive ? meson events obtained in hadron Be interactions with incidentπ ^±,p, \(\bar p\) andK ^± beams. Evidence is presented for the joint production of ? mesons and strange particles produced with non-strange incident beams. With incidentK ^± beam the number of additional strange particles is suppressed. The results are found to be in agreement with the qualitative predictions of a parton fusion model. The comparison with the Lund model for lowp _T processes is fair.     

High-p T αα andαp interactions

Properties of secondaries associated with a high-p _T charged trigger particle (3<p _T <5 GeV/c) were studied for αp and αα interactions at c.m. energies \(\sqrt s = 88\) GeV and 125 GeV, respectively. Thep _T distributions of secondaries in the away-jet and trigger-jet regions were compared with those for high-p _{T pp} interactions. No statistically significant differences were seen, except at lowp _T . Momentum and angular distributions of spectator and leading protons were studied as a function of charge and rapidity of the trigger hadron. The observed correlations between trigger charge and number of spectator protons provide evidence of valence quark contributions to the trigger jet.     

The high resolution infrared spectrum and laser Stark spectroscopy of thev8 band of propyne

The perpendicularv ₈ band lying in the 1000–1100 cm^–1 region has been studied from infrared and laser Stark, spectra. We were interested in the part of spectrum corresponding to the spectral range of the 9 m CO₂ laser lines. Assignments of rovibrational lines with J'<40 and K'<6 have been made. About 100 Stark resonances have been assigned to 12 rovibrational transitions. Effective molecular constants and dipole moment have been determined with high accuracy. A list of close resonances with CO₂ laser lines is given and may be used for optical pumping experiments.