A note on unit and class number of real quadratic fields

Letp be an odd prime withp1 (mod 4) andQ(p) be the real quadratic field. Also let andh denote the fundamental unit and the class number of Q(p), respectively. The main purpose of this paper is to study the explicit expressions of ^h and ^2h , and to discuss the problems related to the conjecture of Ankeny-Artin-Chowla.     

On the construction and calculation of optimal nonlifting critical airfoils

Numerical calculations are carried out in the hodograph plane to construct optimal critical airfoil shapes and the flow about them. These optimal airfoil shapes give the highest free-stream Mach numberM for a given thickness ratio and tail angle _t (nonlifting) for which the flow is nowhere supersonic. A relationship betweenM and for various _t is given. Analytical and numerical solutions to the same problem are found on the basis of transonic small-disturbance theory. These results provide a limiting case asM 1, 0 and agree well with the calculations of the full problem. Using a numerical method to calculate the flow about general (subsonic) airfoils, a comparison is made between the critical free-stream Mach numbers for some standard airfoil shapes and the optimal free stream Mach number of the corresponding and _t . A significant increase in the critical free-stream Mach number is found for the optimal airfoils.     

A condition for nonnegativity of an operator related to the Dirichlet operator

Let L₀ be a positive definite closed linear operator with domain of definition D(L₀) dense in the Hilbert space H; let(, ₁, ₂) be the positive boundary value space of the operator L₀ such that the restriction of L ₀ ^* to ker ₂ is the Friedrichs extension of the operator L₀. We establish a test for nonnegativity of an operator T of the form Ty=L ₀ ^* y+^*(₁–C)y, y D(T)= ker(₂+), where :H and C: are respectively a compact operator and a bounded nonnegative operator.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 32, 1990, pp. 30–33.     

Quadratically constrained least squares and quadratic problems

Summary We consider the following problem: Compute a vectorx such that Ax–b₂=min, subject to the constraint x₂=. A new approach to this problem based on Gauss quadrature is given. The method is especially well suited when the dimensions ofA are large and the matrix is sparse.It is also possible to extend this technique to a constrained quadratic form: For a symmetric matrixA we consider the minimization ofx ^T A x–2b ^T x subject to the constraint x₂=.Some numerical examples are given.This work was in part supported by the National Science Foundation under Grant DCR-8412314 and by the National Institute of Standards and Technology under Grant 60NANB9D0908.     

An intersection property for starshaped sets in the plane

Let be a family of compact starshaped sets in the plane. If every three and every two members of have a union which is connected and simply connected, then {F:F in } is simply connected and nonempty. Of course, if every three and every two members of have a starshaped union, the same result holds.Supported in part by NSF grants DMS-8705336, DMS-8908717 and by a Senior Faculty Summer Research Fellowship, Research Council, University of Oklahoma.     

On the existence of regular conditional probabilities

Summary Let (X, A, P) be a measure space with P(X)=1 and a sub--algebra. It is well known (see e.g. Doob, p. 624) that even if A is separable, a regular conditional probability (r.c.p.) on A, given , does not always exist. All theorems assuring the existence of a r.c.p. known hitherto use in addition to the separability of A conditions of an essentially topological nature such as compact approximation (for references see e.g. Pfanzagl-Pierlo (1966), section 7).It is the purpose of this paper to explore the possibility of sufficient conditions of a purely measure-theoretic nature. Without loss of generality we may assume that (X, A, P) is complete and is complete with respect to P ¦A. For many applications it will be sufficient to have the r.c.p. defined on a sub--algebra A ₀ A with A ₀ ~ A(P|A) rather than on the whole -algebra A. It will be shown that a r.c.p. in this more restricted sense always exists if A is the completion of a separable -algebra and if it contains a P-null set of the power of the continuum. The P-null-set -condition cannot be omitted without any replacements. To be more specific: Even a r.c.p. in the restricted sense does not necessarily exist if A is the completion of a separable -algebra (see Example 1). Furthermore, there are instances, in which a r.c.p. in the restricted sense exists but a r.c.p. in the usual sense does not exist (see Example 2).Even if a r.c.p. exists, it is not necessarily proper. The deep study by Blackwell and Ryll-Nardzewski (1963) reveals that this negative statement even holds under rather restrictive conditions (X = complete separable metric space, A = Borel algebra of X). It therefore seems to be of some interest to state that the r.c.p. in the restricted sense can always be chosen such that it is proper with respect to some sub--algebra +B_o +(+B A ₀ with ₀₀(P¦A).     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

A conditional dilatation equation

Summary The following theorem holds true. Theorem. Let X be a normed real vector space of dimension 3 and let k > 0 be a fixed real number. Suppose that f: X X and g: X × X are functions satisfying x – y = k f(x) – f(y) = g(x, y)(x – y) for all x, y X. Then there exist elements and t X such that f(x) = x + t for all x X and such that g(x, y) = for all x, y X with x – y = k.     

Rank restrictions on the quadratic form in indefinite quadratic programming

Summary A quadratic programming problem, whereq(x) =a ^T x +x ^T Qx is an indefinite objective function, can be solved withSwarup's approach of optimizing (c ^T x + )(d ^T x + ) only if the rank ofQ is two; ifQ is definite, the rank ofQ must be one.

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Zusammenfassung Ein quadratisches Optimierungsproblem, in demq(x) =a ^T x +x ^T Qx eine indefinite Zielfunktion ist, kann mitSwarups Optimierungsansatz (c ^T x + )(d ^T x + ) nur gelöst werden, wenn der Rang vonQ gleich zwei ist; wennQ definit ist, muß der Rang vonQ gleich eins sein.

     

On the divergence of certain Hermite-Fejér interpolation

Summary In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the ebyev roots) not describing the derivative values at exceptional nodes {_n} _n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever –1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [–1, 1].     

Algebraic systems of quadratic forms of number fields and function fields

In this paper we examine for which Witt classes ,..., _n over a number field or a function fieldF there exist a finite extensionL/F and ₂,..., _n L^* such thatT _L/F ()=1 andTr _L/F (_i)=_i fori=2,...n.     

On affine scaling algorithms for nonconvex quadratic programming

We investigate the use of interior algorithms, especially the affine-scaling algorithm, to solve nonconvex — indefinite or negative definite — quadratic programming (QP) problems. Although the nonconvex QP with a polytope constraint is a hard problem, we show that the problem with an ellipsoidal constraint is easy. When the hard QP is solved by successively solving the easy QP, the sequence of points monotonically converge to a feasible point satisfying both the first and the second order optimality conditions.Research supported in part by NSF Grant DDM-8922636 and the College Summer Grant, College of Business Administration, The University of Iowa.     

On quadratic spline interpolation

Using the quadratic spline interpolates(x) fitting the data (x _i,y _i), 0in and satisfying the end conditions_o=y_o, we give formulae approximatingy andy at selected knots by orders up toO(h ⁴).     

Isometries of symmetric operator spaces associated with AFD factors of typeII and symmetric vector-valued spaces

If is a surjective isometry of the separable symmetric operator spaceE(M, ) associated with the approximately finite-dimensional semifinite factorM and if · _E(M,) is not proportional to · _{L 2}, then there exist a unitary operatorUM and a Jordan automorphismJ ofM such that(x)=UJ(x) for allxME(M, ). We characterize also surjective isometries of vector-valued symmetric spacesF((0, 1), E(M, )).Research supported by the Australian Research Council     

The existence of probability measures with specified projections

Let X and Y be locally compact-compact topological spaces, F X×Y is closed, and P(F) is the set of all Borel probability measures on F. For us to find, for the pair of probability measures (_x, _y P (X)×P(Y), a probability measure P(F) such that _X = _X ^–1 , _Y = _Y ^–1 it is necessary and sufficient that, for any pair of Borel sets A X, B Y for which (A× B) F=Ø, the condition _XA+ _YB 1 holds.Translated from Matematicheskie Zametki, Vol. 14, No. 4, pp. 573–576, October, 1973.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

On the convergence in a restricted sense of multiple series

Z ^d — k=(k ₁, ...,k _d) k _j,d1.d- (8), . . a _k s _m= a _k s, >0 N, min (m ₁,...,m _d)N, ¦s _m–s¦. , , >0 N, min (m ₁,...,m _d)N min (n ₁,...,n _d)N, ¦s _m–s _n. . , (8) , >0 N, max (b ₁,...,b _d) N, m — Z ^d , m1, ¦s(b, m)¦ where      

Asymptotic Expansions for the Congestion Period for the M/M/∞ Queue

We consider the M/M/ queue with arrival rate , service rate and traffic intensity =/. We analyze the first passage distribution of the time the number of customers N(t) reaches the level c, starting from N(0)=m>c. If m=c+1 we refer to this time period as the congestion period above the level c. We give detailed asymptotic expansions for the distribution of this first passage time for , various ranges of m and c, and several different time scales. Numerical studies back up the asymptotic results.     

Admissibility of linear estimators of regression coefficients under quadratic loss

For the general fixed effects linear model:Y=X+, N(0,V),V0, we obtain the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS in the class of all estimators under the loss function (d -S)D(d -S), whereD0 is known. For the general random effects linear model: =XV ₁₁ X+XV ₁₂+V ₂₁ X+V ₂₂0, we also get the necessary and sufficient conditions forLY+a to be admissible for a linear estimable functionS+Q in the class of all estimators under the loss function (d -S -Q)D(d -S -Q), whereD0 is known.     

On the concentration of stresses due to a small elliptic inclusion on the neutral axis of a deep beam under constant bending moment

Zusammenfassung Es wird die Spannungsverteilung untersucht, die sich in einem breiten Balken mit konstanter Höhe unter einem konstanten Biegemoment ausbildet, wenn er eine kleine elliptische Einschliessung mit Zentrum auf der Neutralachse enthält. Insbesondere werden die Fälle eines sehr starren Einschlusses sowie eines elliptischen Loches im Detail diskutiert.

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Nomenclature x, y Cartesian coordinates - , elliptic coordinates - u, v (u ,u )=components of displacements - , unit elongations in -and -directions - shearing strain - , normal stress components in elliptical coordinates - shearing stress in elliptic coordinates - _x , _y normal stress in Cartesian coordinates - _xy shearing stress in Cartesian coordinates - E Young's modulus for the beam - v Poisson's ratio for the beam - 1/h ₁, 1/h ₂ stretch ratios - e _x + _y dilatation - 2 rotation - M bending moment