Über Tuttes Cages

An undirected graph of valencyd and girth is called a (d, )-cage if its automorphism group acts transitively on the set of alls-paths in ands(+1)/2. We discuss an elementary construction of two known families of cages which allows us to prove easily some facts about their automorphism groups. We give, for example, a new proof of the fact that the automorphism group ofSp ₄(2 ⁿ ) contains elements which are not induced by Sp ₄(2 ⁿ ).     

Computation of total,differential and double-differential cross sections for compound nuclear reactions of the type (a,b γ) and (a,b γ−γ)

Zusammenfassung Zur vollautomatischen Berechnung von totalen, differentiellen und doppeldifferentiellen Wirkungsquerschnitten im Falle von Kernreaktionen des Typus (a, b, ) und (a, b –) wurden Programme in der ALGOL-64-Sprache aufgestellt, die für einen schnellen Computer, wie die CDC 1604-A, geeignet sind. Wie hier beschrieben, liefern sie entsprechende Winkelverteilungen und Winkelkorrelationen in absoluter und normierter Form für Reaktionen, bei denen Spin-1/2-Teilchena (z. B. Nukleonen) mit willkürlichem Bahndrehimpuls von einem Targetkern (dessen Spin im Grundzustand ganz- oder halbzahlig sein darf) aufgefangen werden, unter Bildung eines kurzlebigen Zwischenkernes, dessen Energieniveaus ein statistisches Quasi-Kontinuum bilden, und der unter Emission von Spin-1/2-Teilchenb mit beliebigem Bahndrehimpuls in einen angeregten Zustand eines Restkerns zerfällt. Die Programme ermöglichen es, weitere Ausgangskanäle des Zwischenzustandes sowie gemischte Multipolarität beim -Zerfall des Restkerns zu berücksichtigen; ferner können sie, durch Wahl geeigneter Transmissionskoeffizienten, den Einfluss einer Spin-Bahn-Wechselwirkung miteinschliessen. Sie dienen weiter zur Berechnung des Beitrages weiterer -Kaskaden durch die Behandlung von Fällen der Art (a, b ₁– ₁), bei denen der ₁ unbeobachtet bleibt, während ₂ beobachtet wird. Bei der Winkelverteilung von -übergangsstrahlung gemischter Multipolarität werden normierte, wie auch absolute Entwicklungskoeffizienten der Legendre-Polynomreihe für reine Multipolarität und für gemischte Multipolarität mit einem vorgegebenen Mischungsverhältnis berechnet. Ferner wird ein weiterer Satz von Mischungsverhältnissen behandelt, was zur Bestimmung unbekannter Verhältnisse oder Spin-Paritäts-Zuordnungen eines Kernzustands führen kann.Weitere Möglichkeiten im Falle von Teilchen- und/oder -Verteilungen (wenn die letzteren keinen unbeobachteten -Übergang einschliessen) wurden eingebaut, die automatisch eine systematische Änderung des Spins (mit positiverund negativer Parität) des durch die austretenden Teilchenb bevölkerten Niveaus für irgendeine anfängliche Spin-Sequenz durchführen. Weitere schnelle Computer-Programme, die mit den obigen verwandt sind, werden zusammenfassend diskutiert.     

Über die asymptotische Darstellung der Aufspaltung von Paaren benachbarter Eigenwerte der Differentialgleichung der Sphäroidfunktionen

Summary Let be the difference of a pair of adjacent eigen-values of the differential equation (1) for the spheroidal functions. Until now only the two first terms of the asymptotic expansion (2) of for large ^*2=–² had been known. In this note the next term, i. e. the coefficient of ^*–2, is given, and the way of calculating it is described.

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The preparation of this note containing partial results, was sponsored by the European Office Air Research and Development Command, U. S. Air Force, Project No. AF 61 (514)-443.     

On theA 0-acceptability of rational approximations to the exponential function with only real poles

We consider rational approximations to the exponential function with real poles, ₁ ^–1 ,..., _m ^–1 , that correspond to implicit Runge-Kutta collocation methods. We show that if _i 1/2,i=1,...,m, the rational approximation isA ₀-acceptable.     

Über geschlossene affine Zwangläufe in der Ebene

We consider integral coverings y:{1,2,..,} of an affine plane which occur when is moved under a continuous periodic affine motion(t):. One can distinguish normal points × , i.e. is constant in a certain neighborhood of x, and singular points. If (x) is the number of times x passes through its orbit (t)x all normal points x have (x)=1, and the set of all singular points consists of a number of isolated points and lines. If (x) is the tangent rotation number of the orbit of x all singular points lie on the moving pole curve.     

On generalized gamma functions related to the Laurent coefficients of the Riemann zeta function

Summary We study a class of generalized gamma functions _k (z) which relate to the generalized Euler constants _k (basically the Laurent coefficients of(s)) as (z) does to the Euler constant. A new series expansion for _k is derived, and the constant term in the asymptotic expansion for log _k (z) is studied in detail. These and related constants are numerically computed for 1 k 15.     

Approximate maximum likelihood estimation in linear regression

The application of the ML method in linear regression requires a parametric form for the error density. When this is not available, the density may be parameterized by its cumulants ( _i ) and the ML then applied. Results are obtained when the standardized cumulants ( _i ) satisfy _i = _i+2/ ₂ ^(i+2)/2 =O(v ⁱ ) asv 0 fori>0.Research financed in part by the Research Center of the Athens University of Economics and Business.     

Cycle-pancyclism in tournaments II

LetT be a Hamiltonian tournament withn vertices and a Hamiltonian cycle ofT. In this paper we develope a general method to find cycles of lengthk, n+4/2 < k < n, intersecting in a large number of arcs. In particular we can show that if there does not exist a cycle.C _k intersecting in at leastk – 3 arcs then for any arce of there exists a cycleC _k containinge and intersecting in at leastk – 2(n–3)/n–k+3 – 2 arcs. In a previous paper [3] the case of cycles of lengthk, k n+4/2 was studied.On leave at MIT Laboratory for Computer Science, 545 Technology Square, Cambridge, MA 02139, USA.     

Nonclassical analogs of solitons in quantum field theory

The possible existence of quantum copies of classical soliton solutions that cannot exist if the effective Planck constant of the theory tends to zero is discussed. In the framework of the standard expansion in , such nonclassical solitons have energy of orderO() and thus occupy an intermediate position between purely classical configurations and second-quantized degrees of freedom. Analytic calculations are made for the profile functions, masses, and stability of such configurations in the ⁴ kink model.Physics Faculty, Moscow State University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 94, No. 1, pp. 52–65, January, 1993.     

Generating Singularities of Solutions of Quasilinear Elliptic Equations Using Wolff's Potential

We consider a quasilinear elliptic problem whose left-hand side is a Leray-Lions operator of p-Laplacian type. If p < < N and the right-hand side is a Radon measure with singularity of order at x_0A , then any supersolution in W^1,p() has singularity of order at least (–p)/(p–1) at x₀. In the proof we exploit a pointwise estimate of A-superharmonic solutions, due to Kilpeläinen and Malý, which involves Wolff's potential of Radon's measure.     

On the choice of γ for singly-implicit RK or Rosenbrock methods

Following the slogan a picture is better than thousand data, for restricted Padé approximations with one realm-fold pole atz=1/ of orderm, we plot the error constants, the maximal error for negative realz as functions of, as well as the regions ofA- resp.A(0)-stability.     

On the theory of imploding shocks

This paper is concerned with the non-monotonic distribution of the static pressure behind self-similar imploding shocks as revealed by the numerical solutions of Landau and Stanyukovich. It is found that in terms of the ratio of the reduced gas velocity to the similarity variable, the exponent equation derived from the condition of zero pressure gradient and that derived from the theory of singularity are of identical form. The identicalness is shown to be formal only. It is shown that both equations can be obtained from a unified equation. Using these results, it is shown that within the framework of inviscid self-similar solutions, the zero pressure gradient cannot exist immediately behind the shock for2+3. Applying the unified equation, it is analytically shown that as approaches unity the similarity exponents also approach unity independent of the dimensionality, a result previously obtained by Lazarus and Richtmyer.

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Zusammenfassung Die nicht-monotone statische Druckverteilung hinter selbst-ähnlichen implodierenden Stößen, dargestellt in den numerischen Lösungen von Landau uad Stanyukovich, ist Gegenstand dieser Arbeit. Es wird gezeigt, daß in bezug auf das Verhältnis zwischen reduzierter Gasgeschwindigkeit und der Ähnlichkeits-Veränderlichen, die unter der Null-Druckgradient-Bedingung und der Singularitätstheorie abgeleiteten Exponentengleichungen identisch sind. Des weiteren wird gezeigt, daß die Identität nur formal ist, daß beide Gleichungen von einer Einheitsgleichung abgeleitet werden können und daß im reibungsfreien Bereich der Null-Druckgradient unmittelbar hinter dem Stoß für2+3 nicht bestehen kann. Bei Anwendung der Einheitsgleichung läßt sich zeigen, daß unabhängig von ihrer Dimensionalität, sowohl als auch die Ähnlichkeitsexponenten sich Eins annähern, wie bereits von Lazarus und Richtmyer angegeben wurde.

     

Global solutions describing the collapse of a spherical or cylindrical cavity

The collapse of a spherical (cylindrical) cavity in air is studied analytically. The global solution for the entire domain between the sound front, separating the undisturbed and the disturbed gas, and the vacuum front is constructed in the form of infinite series in time with coefficients depending on an appropriate similarity variable. At timet=0⁺, the exact planar solution for a uniformly moving cavity is assumed to hold. The global analytic solution of this initial boundary value problem is found until the collapse time (=(–1)/2) for 1+(2/(1+v)), wherev=1 for cylindrical geometry, andv=2 for spherical geometry. For higher values of , the solution series diverge at timet — 2(–1)/ (v(1+)+(1–)2) where =2/(–1). A close agreement is found in the prediction of qualitative features of analytic solution and numerical results of Thomaset al. [1].     

Torsion and closed geodesics on complex hyperbolic manifolds

Summary LetX be a compact complex manifold covered by complex hyperbolicn-space with the induced metric. Each stable horocycle has a cocomplex structure preserved by the geodesic flow. To a closed geodesic one can thus associate a piece of the Poincaré map with a holomorphic fixed point. The resulting Atiyah-Bott fixed point indices, together with the length and multiplicity of as a periodic orbit, determine the contribution of to certain zeta functionsR _p(z), 0pn. From the leading coefficient ofR _p atZ=0 and the Hodge numbersh ^ij (X) we calculate the Ray-Singer -torsionT _p (X). This indicates that the known connections between torsion and the dynamical features of closed orbits continue to hold in the holomorphic category.Corresponding results hold for the -torsion of a flat unitary bundle, extending certain formulas of Ray and Singer to the casen>1.Partially supported by the Sloan Foundation and the National Science Foundation     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

The γ-borel conjecture

In this paper we prove that it is consistent that every -set is countable while not every strong measure zero set is countable. We also show that it is consistent that every strong -set is countable while not every -set is countable. On the other hand we show that every strong measure zero set is countable iff every set with the Rothberger property is countable.Thanks to Boise State University for support during the time this paper was written and to Alan Dow for some helpful discussions and to Boaz Tsaban for some suggestions to improve an earlier version.     

Distoriton of area and conditioned Brownian motion

Summary In a simply connected planar domainD the expected lifetime of conditioned Brownian motion may be viewed as a function on the set of hyperbolic geodesics for the domain. We show that each hyperbolic geodesic induces a decomposition ofD into disjoint subregions and that the subregions are obtained in a natural way using Euclidean geometric quantities relating toD. The lifetime associated with on each _j is then shown to be bounded by the product of the diameter of the smallest ball containing _j and the diameter of the largest ball in _j . Because this quantity is never larger than, and in general is much smaller than, the area of the largest ball in _j it leads to finite lifetime estimates in a variety of domains of infinite area.Research of the first author was supported in part by NSF Grant DMS-9100811Research of the second author was supported in part by NSF Grant DMS-9105407     

Optimal pairs of score vectors for positional scoring rules

Letw=(w ₁,,w _m ) andv=(v ₁,,v _m-1 ) be nonincreasing real vectors withw ₁>w _m andv ₁>v _m-1 . With respect to a lista ₁,,a _n of linear orders on a setA ofm3 elements, thew-score ofaA is the sum overi from 1 tom ofw _i times the number of orders in the list that ranka inith place; thev-score ofaA{b} is defined in a similar manner after a designated elementb is removed from everya _j .We are concerned with pairs (w, v) which maximize the probability that anaA with the greatestw-score also has the greatestv-score inA{b} whenb is randomly selected fromA{a}. Our model assumes that linear ordersa _j onA are independently selected according to the uniform distribution over them linear orders onA. It considers the limit probabilityP _m (w, v) forn that the element inA with the greatestw-score also has the greatestv-score inA{b}.It is shown thatP _m (m,v) takes on its maximum value if and only if bothw andv are linear, so thatw _i –w _i+1=w _i+1–w _i+2 forim–2, andv _i –v _i+1 =v _i+1 –v _i+2 forim–3. This general result for allm3 supplements related results for linear score vectors obtained previously form{3,4}.     

Mittelwert der Eulerschen φ-Funktion und des Quadrates der Dirichletschen Teilerfunktion in algebraischen Zahlkörpern

LetK be an algebraic number field, and for every integer K let () andd(), respectively, denote the number of relatively prime residue classes and the number of divisors of the principal ideal (). Asymptotic equalities are proved for the sums () and d ²(), where runs through certain finite sets of integers ofK.     

Universally signable graphs

In a graph, a chordless cycle of length greater than three is called a hole. Let be a {0, 1} vector whose entries are in one-to-one correspondence with the holes of a graphG. We characterize graphs for which, for all choices of the vector , we can pick a subsetF of the edge set ofG such that |F H| H (mod 2), for all holesH ofG and |F T| 1 for all trianglesT ofG. We call these graphsuniversally signable. The subsetF of edges is said to be labelledodd. All other edges are said to be labelledeven. Clearly graphs with no holes (triangulated graphs) are universally signable with a labelling of odd on all edges, for all choices of the vector . We give a decomposition theorem which leads to a good characterization of graphs that are universally signable. This is a generalization of a theorem due to Hajnal and Surányi [3] for triangulated graphs.This work was supported in part by NSF grants DMI-9424348 DMS-9509581 and ONR grant N00014-89-J-1063. Ajai Kapoor was also supported by a grant from Gruppo Nazionale Delle Ricerche-CNR. We also acknowledge the support of Laboratoire ARTEMIS, Université Joseph Fourier, Grenoble.