Characterization of Dirichlet Forms by their Excessive and Co–Excessive Elements

It is proved that if S, T are two elliptic Dirichlet operators on an ordered Hilbert space such that the excessive (resp. coexcessive) elements with respect to S and T are the same then there exists > 0 with T = S. Particularly if , are two elliptic Dirichlet forms on L² ( ) having the same domain of definition and the same -excessive (resp. -coexcessive) elements for any > 0 then = .     

On the solvability of a complete second-order differential equation in Banach space

The paper deals with the Cauchy problem for a complete second-order differential equation with unbounded operator coefficientsu+A(t)u+B(t)u=f, u(0)=u0, u(0)=u ₁ . By using the commutant method, we construct a coercive solution of this problem in Holder space in the case where the operatorB is as strong as the operator A².Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 10, pp. 1449–1454, October, 1993.     

Bilateral difference method for solving the boundary value problem for an ordinary differential equation

A method is proposed for calculating the bilateral approximations of the solution of the boundary value problem on [0, 1] for the equation y+p(x)y-q(x)y=f(x) and the derivative of the solution having the maximum deviation O(h² (h)+h³) on {kh} _k ^N =0, where(t) is the sum of the continuity moduli of the functions p, q,f, on the set of points {kh} _k ^N =0, h=1/N by means of O(N) operations. The data obtained for fairly smooth p, q,f allow interpolation to be used for calculating the bilateral approximations of the solution and its higher derivatives having the maximum deviation O(h³) on [0, 1].Translated from Matematicheskie Zametkii, Vol. 11, No. 4, pp. 421–430, April, 1972.     

The best one-sided approximation of the classes WrHω

In this paper we calculate the upper bounds of the best one-sided approximations, by trigonometric polynomials and splines of minimal defect in the metric of the space L, of the classes W^rH (r = 2, 4, 6, ...) of all 2-periodic functions f(x) that are continuous together with their r-th derivative f^r(x) and such that for any points x and x we have ¦f ^r (x) f^r (x) ¦ (x–x¦), where (t) is a modulus of continuity that is convex upwards.Translated from Matematicheskie Zametki, Vol. 21, No. 3, 313–327, March, 1977.     

Necessary and sufficient condition of the existence of global Hopf bifurcation for Hamiltonian systems

ForH C ² (,R) where 0 R ²ⁿ ,H (0)=0 and detH(0)0, the paper proves that there is a global Hopf bifurcation fromx=0 for Hamiltonian systemx=JH(x) iffJH(0)possesses purely imaginary eigenvalues. The work improves the corresponding result of J.C.Alexander and J. Yorke (Amer. J. Math., 100 (1978), 263–292).     

Über einen Typ von injektiven Klassen

If u is a morphism, A an object of some category, we say, A is injective relative to u, iff Hom (u, A) is epi. In this paper we consider classesU of monomorphisms in categories of modules with the following property: if u, u are monos, such that uu exists and is inU, then u is inU. We show, that the closure a(U) ofU is an injective class in the sence of S. Eilenberg and J. C. Moore, that is, for each module A there is a mono a:AB, such that B is injective relative to u for all uU and that each module C, that is injective relative to u for all uU is injective relative to a.     

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.     

Nonresonant acoustic absorption of Ising magnet

Zubarev's nonequilibrium statistical operator method is used to construct a theory of the nonresonant acoustic absorption of the linear Ising magnet, the basic Hamiltonian of which is of the two-particle kind and is treated without approximations. The frequency dependence of the absorption is determined and found to differ analytically from the well-known Debye curve _D () but for reasonable values of the physical parameters gives an envelope that agrees qualitatively with _D ().Kazan Physicotechnical Institute, Russian Academy of Sciences; Moscow Institute of Instrument Manufacture. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 98, No. 1, pp. 149–159, January, 1994.     

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.     

Orderh 2 method for a singular two-point boundary value problem

A method is described based on auniform mesh for the singular two-point boundary value problem:y+(/x)y+f(x, y)=0, 0<x1,y(0)=0,y(1)=A, and it is shown to be orderh ² convergent forall 1.     

On the zeros of generalized Airy functions

We investigate the behaviour with respect to the parameter > 0 of the zeros of the solutions of the differential equation y+y=0. We show that under appropriate restrictions such zeros are logconvex.Work sponsored by CNR, Gruppo Nazionale per l'Informatica Matematica, of Italy.     

The imaginary zeros of a mixed Bessel function

In the present work we study the existence and monotonicity properties of the imaginary zeros of the mixed Bessel functionM _v(z)=(z²+)J_v(z)+zJ_v(z). Such a function includes as particular cases the functionsJ _v(z)(==0), J_v(z)(=–v²,=1)x andH _v(z)=J_v(z)+zJ_v(z), whereJ _v(z) is the Bessel function of the first kind and of orderv>–1 andJ _v(z), J_v(z) are the first two derivatives ofJ _v(z). Upper and lower bounds found for the imaginary zeros of the functionsJ _v(z), J_v(z) andH _v(z) improve previously known bounds.

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Zusammenfassung Dieser Artikel betrifft die Existenz und Monotonie von Eigenschaften imaginärer Nullen der gemischten BesselfunktionM _v(z)=(z²+)J_v(z)+zJ_v(z). Eine solche Funktion enthält als Spezialfall die FunktionenJ _v(z)(==0), J_v(z)(=–v²,=1) undH _v(z)=J_v(z)+zJ_v(z), woJ _v(z)die Besselfunktion von erster Art und Ordnungv>–1 andJ _v(z), J_v(z) sind die erste und zweite Ableitung vonJ _v(z). Untere und obere Schranken, die für die imaginären Nullen der FunktionenJ _v(z), J_v(z) undH _v(z) gefunden wurden, verbessern früher bekannte Resultate.

     

Description of Hyperinvariant Subspaces of a Contraction in Terms of Its Characteristic Function

If T is a completely nonunitary contraction on a Hilbert space and L is its invariant subspace corresponding to a regular factorization of its characteristic function = , then L is hyperinvariant if and only if the following two conditions are fulfilled: (1) supp _* supp is of Lebesgue measure zero; (2) for every pair A H (E E) and A _* H (E _* E _*) intertwining by , i.e., such that A =A _*, there exists a function A _F H (F F) intertwining with A by and with A _* by , i.e., such that A = A _F and A _F = A _*. Bibliography: 4 titles.     

On a variational problem with lack of compactness: the topological effect of the critical points at infinity

We study the subcritical problemsP :–u=u ^p–,u>0 on;u=0 on , being a smooth and bounded domain in ^N,N–3,p+1=2N/N–2 the critical Sobolev exponent and >0 going to zero — in order to compute the difference of topology that the critical points at infinity induce between the level sets of the functional corresponding to the limit case (P₀).

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Résumé Nous étudions les problèmes sous-critiquesP :–u=u ^p–,u > 0 sur;u=0 sur –où est un domaine borné et régulier de ^N,N–3,p + 1=2N/N –2 est l'exposant critique de Sobolev, et >0 tend vers zéro, afin de calculer la différence de toplogie induite par les points critiques à l'infini entre les ensembles de niveau de la fonctionnelle correspondant au cas limite (P₀).

     

Large deflections of eccentrically loaded arches

Zusammenfassung Auf Grund der Hypothesen von Ebenbleiben und Normalität der Querschnitte werden die Differentialgleichungen der nichtlinearen Theorie der Bogenträger abgeleitet und im Falle des schlanken, durch Einzellasten belasteten Kreisbogenträgers mit undehnbarer Mittellinie auf die Form der Pendelgleichung gebracht. Diese Gleichung wird dann benutzt, um die grossen Durchbiegungen und die Spannungsresultierenden eines Zweigelenkkreisbogens, der durch eine lotrechte exzentrische Einzellast belastet wird, zu berechnen. In der Nähe der kritischen Last bewirken kleine Exzentrizitäten bedeutende Grössenänderungen der Spannungsresultierenden und der Durchbiegungen.

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Notation A cross-sectional area of curved beam - a radius of centroidal circle - E modulus of elasticity - e eccentricity of the load (Fig. 2) - F an arbitrary function - H horizontal component of the internal forceR acting on a cross section of the arch rib (Fig. 2) - h _P horizontal displacement of the loadP (Fig. 2) - I moment of inertia of the cross-sectional area - k ² =4p ²/(1+4p ² sin²₀) - L span (distance between supports),L=2a sin - M internal bending couple (Figs. 1 and 2) - N internal normal tensile force (Figs. 1 and 2) - n distributed tangential load (Fig. 1) - P downward point load (Fig. 2) - p ² –R a ² /E I - Q internal shearing force (Figs. 1 and 2) - q distributed normal load (Fig. 1) - R internal resultant force (Fig. 2);R ²=H ²+V ²=N ²+Q ² - radius of curvature of the undeformed centroidal curve - s length along the unextended centroidal curve measured from the left support - length along the unextended centroidal curve measured from the right support - u tangential displacement component of the centroidal curve (Fig. 1) - V vertical component ofR (Fig. 2) - v _P vertical displacement of the loadP (Fig. 2) - w normal displacement component (Fig. 1) - x, y rectangular coordinates of the deformed left portion of the centroidal curve (Fig. 2) - Z - z normal distance (positive inward) from centroidal curve (Fig. 1) - half subtending angle of the arch (Fig. 2) - angle of rotation of the centroidal curve (Fig. 1) - extensional strain of the centroidal curve - ^z extensional strain of the linez=constant - y cos–x sin - angle between the tangent to the formed left portion of the centroidal curve and the horizontal (Fig. 2) - (u–w)/r, whereu=du/dø - angle betweenH andR - x cos+y sin - normal stress along the centroidal curve - ^z normal stress along the linez=constant - angle measured from the radius at the left support of the undeformed arch - (–)/2 (Fig. 2) - (+u)/r, where =d/dø A bar over a letter indicates that the entity pertains to the right portion of the arch. Asterisk indicates the deformed configuration. Primes indicate derivatives with respect to ø.     

Some results about the spectrum of commutative Banach algebras under the weak topology and applications

LetA be a commutative Banach algebra with a nonempty spectrum A. By weak we denote the relative weak topology induced on A by (A ^*,A ^**). In this note we study some properties of the topological space (A, weak) and present some applications of the results obtained and tools used to amenability, weakly compact homomorphisms, weakly compact subsets of the spectrum of the uniform algebras and to a characterization of the synthesizable ideals of the algebraA.     

Lubin-Hensel factorization for Laurent series

Summary Let K be a complete ultrametric algebraically closed field. Let D be a bounded closed strongly infraconnected set in K with no T-filter, and let H(D) be the Banach algebra of the analytic elements in D. Let r, r be functions from D toR with bounds a, b such that 0 (D,r,r) be the Banach algebra of the Laurent series with coefficients a_s in H(D) such that , provided with a suitable norm. In (D, r, r) we give a kind of Hensel Factorization for series whose dominating coefficients at r(x) and at r(x) conserve the same rank. We take advantage of this method to correcting a mistake that happened in our previous article on the Hensel Factorization for Taylor series.And Erratum to «Maximum principle for analytic elements and Lubin-Hensel's Theorem inH(D)Y»,135, pp. 265–278 of this Journal.     

Zur Aufllösung linearer homogener Differentialgleichungen 2. Ordnung

Summary By the transformationy(x)=v(u),u = exp (–G(x) dx) dx the differential equationDy+G(x)y+H(x)y=0 turns toT(u) ² v ^**+H v=0, wherev ^** signifiesd ² v/du ², andu=du/dx andH=H(x) should be expressed as functions ofu.From the solutionv(u) ofT follows immediately the solutiony(x) ofD, and vice versa.In this paper there are treated some of the types of differential equations, that may be solved by this method.     

Peak sets for analytic Hölder classes

A closed subset E of the unit circumference T is said to be a peak set for the analytic Hölder class A, 0 < < 1 there exists a functionf,fA such that f¦_E1 and ¦f(z)¦<1 for. It is shown that the set E is a peak set of the algebra A if and only if there exists a nonnegative Borel measure on T such that the function coincides almost everywhere with a function of the Hölder class , equal to zero on E. A sufficient condition in order that a closed set E should belong to the family of peak sets is obtained.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 129–136, 1987.     

Topology of Definable Hausdorff Limits

Let A ^n+r be a set definable in an o-minimal expansion S of the real field, let A ^r be its projection, and assume that the non-empty fibers A_a ⁿ are compact for all a A and uniformly bounded, i.e. all fibers are contained in a ball of fixed radius B(0,R). If L is the Hausdorff limit of a sequence of fibers A_ai, we give an upper-bound for the Betti numbers b_k(L) in terms of definable sets explicitly constructed from a fiber A_a. In particular, this allows us to establish effective complexity bounds in the semialgebraic case and in the Pfaffian case. In the Pfaffian setting, Gabrielov introduced the relative closure to construct the o-minimal structure S_Pfaff generated by Pfaffian functions in a way that is adapted to complexity problems. Our results can be used to estimate the Betti numbers of a relative closure (X,Y)₀ in the special case where Y=.