Some extensions of the principles of idealization transfer and choice in the relative internal set theory

The results established in this paper are in connection with the Relative Internal Set Theory (R.I.S.T.). The main result is the general principle of choice: Let be a level and let (x, y) be anexternalbounded formula of the language of R.I.S.T.. Suppose that to each elementx, dominated by , corresponds an elementy _x such that (x, y _x ) holds, then there exists a function of choice such that, which is a very general principle of choice, for everyx dominated by , (x, (x)) holds. More than that, we establish that if all the elementsy _x are uniformly dominated by a level then we can prescribe that the function of choice is also dominated by .     

Singular Eigenvalue Problems for the Equation y (n) + λp(x)y = 0

The paper studies singular eigenvalue problems for the equation y ⁽ⁿ⁾ +p(x)y=0 with boundary conditions imposed on the derivatives y ⁽ⁱ⁾ at the points x=a and x=. We look for singular problems which are analogous to regular problems on a finite interval. It is characterized when each eigenfunction has a finite number of zeros and when the spectrum is discrete or continuous, respectively.     

Two-sided polynomial approximation of functions

Conditions are established when the collocation polynomials P_m(x) and P_M(x), m M, constructed respectively using the system of nodes x_j of multiplicities a_j 1, j = O,, n, and the system of nodes x_-r,,x_o,,x_n,,x_n+r1, r O, r₁ O, of multiplicities a_-r,,(a_o + y_o),,(a_n + y_n),,a_n+r1, a_j + y_j 1, are two sided-approximations of the function f on the intervals , x_j[, j = O,...,n + 1, and on unions of any number of these intervals. In this case, the polynomials P_m (x), P_M ^(l) (x) with l a_j are two-sided approximations of the function f⁽¹⁾ in the neighborhood of the node x_j and the integrals of the polynomials P_m(x), P_M(x) over D_j are two-sided approximations of the integral of the function f (over D_j). If the multiplicities a_j a_j + y_j of the nodes x_j are even, then this is also true for integrals over the set _j= _µ ^k D_j µ 1, k n. It is shown that noncollocation polynomials (Fourier polynomials, etc.) do not have these properties.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 67, pp. 31–37, 1989.     

A Class of Iterative Algorithms and Solvability of Nonlinear Variational Inequalities Involving Multivalued Mappings

The solvability of the following class of nonlinear variational inequality (NVI) problems based on a class of iterative procedures, which possess an equivalence to a class of projection formulas, is presented.Determine an element x ^* K and u ^* T(x ^*) such that u ^*, x – x ^* 0 for all x K where T: K P(H) is a multivalued mapping from a real Hilbert space H into P(H), the power set of H, and K is a nonempty closed convex subset of H. The iterative procedure adopted here is represented by a nonlinear variational inequality: for arbitrarily chosen initial points x ⁰, y ⁰ K, u ⁰ T(y ⁰) and v ⁰ T(x ⁰), we have u ^k + x ^k+1 – y ^k , x – x ^k+1 0, x K, for u ^k T(y ^k ) and for k 0where v ^k + y ^k – x ^k , x – y ^k 0, x K and for v ^k T(x ^k ).     

A decomposition for the exponential dispersion model generated by the invariant measure on the hyperboloid

For = ₀, ₁, ₂) andx=(x₀, x₁, x₂) in R³, define [,x] = ₀ x ₀ – ₁ x ₁ – ₂ x ₂,C = {x³:x ₀ > 0 and [x, x]>0},R(x)=([x, x]) ^1/2 forx inC andH ₁={xC: x₀>0,R(x)=1}. Define the measure onH ₁ such that if is inC and =R(), then exp (–[,x])(dx = ( exp )^–1. Therefore, is invariant under the action ofSO (1, 2), the connected component ofO(1, 2) containing the identity. We first prove that there exists a positive measure in ³ such that its Laplace transform is ( exp )^– if and only if >1. Finally, for 1 and inC, denotingP(,)(dx) = ( exp )^– exp (–[,x])(dx, we show that ifY ₀,...,Y _n aren+1 independent variables with densityP(,),j=0,...,n and ifS _k =X ₀ + ... +X _k andQ _k =R(S _k) –R(S _k–1) –R(Y _k),k=1,...,n, then then+1 statisticsD _n = [/,S _k ] –R _{k – 1} ),Q ₁,...,Q _n are independent random variables with the exponential () or gamma (1,1/) distribution.This research has been partially funded by NSERC Grant A8947.     

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).     

Superstable implicit Runge-Kutta methods for second order initial value problems

Using the well known properties of thes-stage implicit Runge-Kutta methods for first order differential equations, single step methods of arbitrary order can be obtained for the direct integration of the general second order initial value problemsy=f(x, y, y),y(x _o)=y _o,y(x _o)=y _o. These methods when applied to the test equationy+2y+ ² y=0, ,0, +>0, are superstable with the exception of a finite number of isolated values ofh. These methods can be successfully used for solving singular perturbation problems for which f/y and/or f/y are negative and large. Numerical results demonstrate the efficiency of these methods.     

Upper and lower bounds for the frequencies of rectangular free plates

Zusammenfassung Eine neuentwickelte Methode für untere Schranken und das Rayleigh-Ritzverfahren für obere Schranken werden von den Verfassern dazu angewandt, die Eigenfrequenzen der Schwingungen von dünnen gleichförmigen, rechteckigen Platten mit freien Rändern abzuschätzen. Die Anwendbarkeit des Verfahrens für untere Schranken wird hervorgehoben, und es werden Berechnungen für eine Symmetrieklasse von rechteckigen sowie für eine Unterklasse von quadratischen Platten angegeben. Die dadurch entstehenden Schranken führen zu einer Verbesserung bisher veröffentlichter Resultate und weisen auf die Brauchbarkeit der Methode für untere Schranken hin.

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Nomenclature The following symbols are not defined in the text ² Laplacian differential operator (= ²/x ² + ²/y ²) - ⁴ Biharmonic differential operator (= ⁴/x ⁴ + 2 ⁴/x ² y ² + ⁴/y ⁴) - u Generic name for displacement functions - Generic name for eigenvalues - a,b Plate side lengths - Circular frequency of free vibration - Mass density of plate material - h Plate thickness - D Plate flexural rigidity - First variation - d Element of surface area - _ij Kronecker's delta - {} Matrix or vector of the included elements The research reported in this article has been sponsored by the Department of the Navy under Contract NOw-62-0604-c with the Bureau of Naval Weapons.     

A general minimax theorem

This paper is concerned with minimax theorems for two-person zero-sum games (X, Y, f) with payofff and as main result the minimax equality inf supf (x, y)=sup inff (x, y) is obtained under a new condition onf. This condition is based on the concept of averaging functions, i.e. real-valued functions defined on some subset of the plane with min {x, y}< (x, y)x, y} forx y and (x, x)=x. After establishing some simple facts on averaging functions, we prove a minimax theorem for payoffsf with the following property: Forf there exist averaging functions and such that for any x₁, x₂ X, > 0 there exists x₀ X withf (x₀, y) > f (x₁,y),f (x₂,y))– for ally Y, and for any y₁, y₂ Y, > 0 there exists y₀ Y withf (x, y₀) (f (x, y₁),f (x, y₂))+. This result contains as a special case the Fan-König result for concave-convex-like payoffs in a general version, when we take linear averaging with (x, y)=x+(1–)y, (x, y)=x+(1–)y, 0 <, < 1.Then a class of hide-and-seek games is introduced, and we derive conditions for applying the minimax result of this paper.

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Zusammenfassung In dieser Arbeit werden Minimaxsätze für Zwei-Personen-Nullsummenspiele (X, Y,f) mit Auszahlungsfunktionf behandelt, und als Hauptresultat wird die Gültigkeit der Minimaxgleichung inf supf (x, y)=sup inff (x, y) unter einer neuen Bedingung an f nachgewiesen. Diese Bedingung basiert auf dem Konzept mittelnder Funktionen, d.h. reellwertiger Funktionen, welche auf einer Teilmenge der Ebene definiert sind und dort der Eigenschaft min {x, y} < < (x, y)x, y} fürx y, (x, x)=x, genügen. Nach der Herleitung einiger einfacher Aussagen über mittelnde Funktionen beweisen wir einen Minimaxsatz für Auszahlungsfunktionenf mit folgender Eigenschaft: Zuf existieren mittelnde Funktionen und, so daß zu beliebigen x₁, x₂ X, > 0 mindestens ein x₀ X existiert mitf (x₀,y) (f (x ₁,y),f (x₂,y)) – für alley Y und zu beliebigen y₁, y₂ Y, > 0 mindestens ein y₀ Y existiert mitf (x, y₀) (f (x, y₁),f (x, y ₂))+ für allex X. Dieses Resultat enthält als Spezialfall den Fan-König'schen Minimaxsatz für konkav-konvev-ähnliche Auszahlungsfunktionen in einer allgemeinen Version, wenn wir lineare Mittelung mit (x, y)=x+(1–)y, (x, y)= x+(1–)y, 0 <, < 1, betrachten.Es wird eine Klasse von Suchspielen eingeführt, welche mit dem vorstehenden Resultat behandelt werden können.

     

Intuitionistic weak arithmetic

We construct -framed Kripke models of i₁ and i₁ non of whose worlds satisfies xy(x=2yx=2y+1) and x,yzExp(x, y, z) respectively. This will enable us to show that i₁ does not prove ¬¬xy(x=2yx=2y+1) and i₁ does not prove ¬¬x, yzExp(x, y, z). Therefore, i₁¬¬lop and i₁¬¬i₁. We also prove that HAl₁ and present some remarks about i₂. Mathematics Subject Classification (2000):03F30, 03F55, 03H15.     

Über gewisse diophantische Gleichungen mit unendlich vielen Lösungen

Let be a real number with 1<<12/11. Denote by [A] the integer part of the real number A. Using van der Corput-type estimates for trigonometrical sums, we prove that there exist infinitely many squares of the form [x] with natural x; actually, we give an asymptotic formula for the number of natural numbers x]=y² for some natural y. Furthermore, for every natural number d, the (Pell-like) equation [x]–dy²=1 has infinitely many solutions in natural numbers x and y · Also, every natural number n>n() can be written (in many ways) as n=[x]+y² with natural x and y. Instead of squares we study integer-valued polynomials of degree at least two.

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Unterstützt durch NSF-grant 9038     

Convergence of the Gradient Projection Method for Generalized Convex Minimization

This paper develops convergence theory of the gradient projection method by Calamai and Moré (Math. Programming, vol. 39, 93–116, 1987) which, for minimizing a continuously differentiable optimization problem min{f(x) : x } where is a nonempty closed convex set, generates a sequence x_k+1 = P(x_k – _k f(x_k)) where the stepsize _k > 0 is chosen suitably. It is shown that, when f(x) is a pseudo-convex (quasi-convex) function, this method has strong convergence results: either x_k x^* and x^* is a minimizer (stationary point); or x_k arg min{f(x) : x } = , and f(x_k) inf{f(x) : x }.     

Tritronquée Solutions of Perturbed First Painlevé Equations

We consider solutions of the class of ODEs y=6y ²–x , which contains the first Painlevé equation (PI) for =1. It is well known that PI has a unique real solution (called a tritronquée solution) asymptotic to and decaying monotonically on the positive real line. We prove the existence and uniqueness of a corresponding solution for each real nonnegative 1.     

Superconvergence for galerkin methods for the two point boundary problem via local projections

The two point boundary problemy'-a(x)y–b(x)y=-f(x), o<x<1,y(0)=y(1)=0, is first solved approximately by the standard Galerkin method, (Y, ) + (aY+bY, )=(f, ), ₁ ⁰ (r, ), for a function Y ₁ ⁰ (r, ), the space ofC ¹-piecewise--degree-polynomials vanishing atx=0 andx=1 and having knots at {x ₀ ,x ₁ , ...,x _M }=. ThenY is projected locally into a polynomial of higher degree by means of one of several projections. It is then shown that higher-order convergence results locally, provided thaty is locally smooth and is quasi-uniform.This research was supported in part by the National Science Foundation.     

A multiple recursive non-linear congruential pseudo random number generator

On-linear multiple recursive congruential pseudo random number generator with prime modulus p is introduced. Let x, n0, be the sequence generated by a usual linear (r+1)-step recursive congruential generator with prime modulus p and denote by N(n), n0, the sequence of non-negative integers with x_N(n)0 (mod p). The non-linear generator is defined by z_nx_N(n)+1·x _N(n) ^–1 (mod p), n0, where x _N(n) ^–1 denotes the inverse element of x_N(n) in the Galois field GF(p). A condition is given which ensures that the generated sequence is purely periodic with period length p^r and all (p–1)^r r-tupels (y₁,...,y_r) with 1y₁,...,y_rp are generated once per period when r-tupels of consecutive numbers of the generated sequence are formed. For r=1 this generator coincides with the generator introduced by Eichenauer and Lehn [2].     

Measure properties of the set of initial data yielding non uniqueness for a class of differential inclusions

We study uniqueness property for the Cauchy problemxV(x), x(0)=, whereVR ⁿR is a locally Lipschitz continuous, quasiconvex function (i.e. the sublevel sets {Vc} are convex) and V(x) is the generalized gradient ofV atx. We prove that if 0V(x) forV(x)b, then the set of initial data {V=b} yielding non uniqueness of solution in a geometric sense has (n–1)-dimensional Hausdorff measure zero in {V=b}.     

On the Embedding of an Affine Space into a Projective Space

Let k, K be fields, and assume that |k| 4 and n, m 2, or |k| = 3 and n 3, m 2. Then, for any embedding of AG(n, k) into PG(m, K), there exists an isomorphism from k into K and an (n+1) × (m+1) matrix B with entries in K such that can be expressed as (x₁,x₂,...,x_n) = [(1,x₁ ,x₂ ,...,x_n )B], where the right-hand side is the equivalence class of (1,x₁ ,x₂ ,...,x_n )B. Moreover, in this expression, is uniquely determined, and B is uniquely determined up to a multiplication of element of K*. Let l 1, and suppose that there exists an embedding of AG(m+l, k) into PG(m, K) which has the above expression. If we put r = dim _k K, then we have r 3 and m > 2 l-1)/(r-2). Conversely, there exists an embedding of AG(l+m, k) into PG(m, K) with the above expression if K is a cyclic extension of k with dim _k K=r 3, and if m 2l/(r-2) with m even or if m 2l/(r-2) +1 with m odd.     

On properties of the probabilistic constrained linear programming problem and its dual

In this paper, the two problems inf{inf{cx:x R ⁿ,A ₁ xy,A ₂ xb}:y suppF R ^m,F(y)p} and sup{inf{uy:y suppF R ^m,F(y)p}+vb:uA ₁+vA ₂=c, (u,v0} are investigated, whereA ₁,A ₂,b,c are given matrices and vectors of finite dimension,F is the joint probability distribution of the random variables ₁,..., _m, and 0<p<1. The first problem was introduced as the deterministic equivalent and the second problem was introduced as the dual of the probabilistic constrained linear programming problem inf{cx:P(A ₁ x)p,A ₂ xb}.b}. Properties of the sets and the functions involved in the two problems and regularity conditions of optimality are discussed.     

Monotonicity properties of zeros of generalized Airy functions

Summary We show, among other things, that the positive zeros of a solution ofy +x y=0,y(0)=0 decrease to 1 as increases, 0.

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Sommario Si dimostra, tra l'altro, che gli zeri positivi d'una soiuzione diy +x y=0,y(0)=0 decrescono al limite 1, quando cresce, 0.

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To the memory of Milo Háik

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This research was supported by grants from the Natural Sciences and Engineering Research Council (Canada) and Consiglio Nazionale delle Ricerche (Italy). Some of the work was done while the second-named author was visiting the Department of Mathematics, University of Torino.     

Bounds for the nonhomogeneous GASPT equation

Summary Weighted a priori bounds for the equation u+(/y)u_y=f(>0), in the halfplane y>0, are proved. If p>1, 0<+p^–1<1+, u has bounded support and y^µu_y0 (as y0₊), then the L^p norms of uu and yD²u are bounded by the L^p norm of yf. A boundary value problem in a rectangle is also studied in the appropriate weighted Sobolev class.Lavoro eseguito nell'ambito dell'Istituto di Analisi Globale ed Applicazioni del C.N.R.Lavoro eseguito nell'ambito del Gruppo Nazionale di Analisi Funzionale ed Applicazioni del C.N.R.