Stationary Boltzmann's equation with Maxwell's boundary conditions in a bounded domain

The paper deals with the stationary Boltzmann equation in a bounded convex domain Ω. The boundary ?Ω is assumed to be a piecewise algebraic variety of the C²-class that fulfils Liapunov's conditions. On the boundary we impose the so-called Maxwell boundary conditions, that is a convex combination of specular and diffusive reflections. The non-linear Boltzmann equation is considered with additional volume and boundary source terms and it has been proved that for sufficiently small sources the problem possesses a unique solution in a properly chosen subspace of C(Ω × ?³). The proof is a refined version of the proof delivered by Guiraud for purely diffusive reflection.     

L-Splines

In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL andL ² error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17].Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.This research was supported in part by NSF Grant GP-5553Papers not specifically concerned with splines are referred to in the text by [1, 2], etc.     

Union congruence in a subspace of a Finsler space

Summary The union curves of a Riemannian space were studied bySpringer [8],Misra [2] andUpadhyay [9]. In a Finsler space, these curves have been studied byPrakash-Behari [4],Sinha [7],Mishra-Sinha [3] andSingh [6]. In the present paper, we wish to extend the concept of union curves in the Finsler space and as such the concept of union congruence has been discussed. The two types of union curves of the Finsler subspace are the particular cases of these curves. These are also generalization of union curves and union curvatures of a vector-field of Finsler space. It has also been shown that the λ-geodesics [5] are special case of these curves. Entrata in Redazione il 14 settembre 1970.     

The Newton—Kantorovich method under mild differentiability conditions and the Ptâk error estimates

We study the Newton-Kantorovich method under mild differentiability conditions. Using Zabrejko-Nguen assumptions we extend the results obtained byZabrejko andNguen in [11]. We also derive Ptâk error estimates which compare favorably with the ones obtained previously byKeller [4],Rokne [7], andArgyros in [1].     

A metric on the manifold of immersions and its Riemannian curvature

E. Binz [1] considered two canonical Riemannian metrics on the space of embeddings of a closed (n–1) dimensional manifold into ⁿ , and computed the geodesic sprays. Here we consider the space of immersions Imm (M, N) whereM is without boundary, and we compute the covariant derivative (in the form of its connector) and the Riemannian curvature of one of these metrics, the non trivial one. The setting is close to that used byP. Michor [2], and we refer the reader to this paper for notation.     

Das Taylor-Deansche Stabilitätsproblem für beliebige Spaltbreiten

Summary The stability of a viscous flow between two co-axial circular cylinders, produced by the simultaneous effect of rotation of the interior cylinder and by a constant pressure gradient in peripherical direction, has been examined. (The two single problems had been studied first byTaylor [1] resp.Dean [2].) The perturbations investigated are vortices of the Taylor-Görtler-type and are assumed to be small. It therefore becomes possible to operate on the basis of the linear differential equations for perturbations.The present stability problem had been treated under the same assumptions byDiPrima [5] for small gap-distances by using the series method ofTaylor. In this paper, however, arbitrary gap-distances are admitted, and the integral equation method ofGörtler [3] andHämmerlin [4] has been applied. The calculated dependences of the critical Reynolds number upon the pressure gradient are in good qualitative agreement with the results ofDiPrima. However, our computations performed by more exact methods yielded somewhat different results with regard to the dependence of the critical vortex number upon the pressure gradient.     

About Kacʼs program in kinetic theory

In this Note we present the main results from the recent work of Mischler and Mouhot (2011) [15], which answers several conjectures raised fifty years ago by Kac (1956) [10]. There Kac introduced a many-particle stochastic process (now denoted as Kac?s master equation) which, for chaotic data, converges to the spatially homogeneous Boltzmann equation. We answer the three following questions raised in Kac (1956) [10]: (1) prove the propagation of chaos for realistic microscopic interactions (i.e. in our results: hard spheres and true Maxwell molecules); (2) relate the time scales of relaxation of the stochastic process and of the limit equation by obtaining rates independent of the number of particles; (3) prove the convergence of the many-particle entropy towards the Boltzmann entropy of the solution to the limit equation (microscopic justification of the H-theorem of Boltzmann in this context). These results crucially rely on a new theory of quantitative uniform in time estimates of propagation of chaos.     

Sulle classi filtrali di algebre

Summary Generalizing results of [1], we show that if a classK of algebras is filtral (as defined byMagari [1]), then so is the class of all simple algebras in the varietyW _K generated byK, and these simple objects are precisely all subalgebras of all ultraproducts of copies of algebras inK. All algebras inW _K are semisimple and regular. We then obtain charac terizations of filtral and semifiltral classes of algebras in terms of the existence of certain types of operations in these algebras, and illustrate our results with examples, mainly from ring theory.

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L'autore era parzialmente sostenuto dal contratto GP 9152 della National Science Foundation.     

Einige Eigenschaften monotoner Matrizen

In [5]Ky Fan derived several theorems on real square matrices from well-known covering theorems on simplexes given byE. Sperner [13] andB. Knaster-C. Kuratowski-S. Mazurkiewicz [9]. In this note some theorems on monotone matrices and nonsingularM-matrices are presented which extend some ofFan's results. The proofs will be based on certain topological theorems on simplexes which were recently established by the author in [12] and which include the mentioned covering theorems.     

Propagation of and Maxwellian weighted bounds for derivatives of solutions to the homogeneous elastic Boltzmann equation

We consider the n-dimensional space homogeneous Boltzmann equation for elastic collisions for variable hard potentials with Grad (angular) cutoff. We prove sharp moment inequalities, the propagation of L¹-Maxwellian weighted estimates, and consequently, the propagation L^∞-Maxwellian weighted estimates to all derivatives of the initial value problem associated to the afore mentioned equation. More specifically, we extend to all derivatives of the initial value problem associated to this class of Boltzmann equations corresponding sharp moment (Povzner) inequalities and time propagation of L¹-Maxwellian weighted estimates as originally developed Bobylev [A.V. Bobylev, Moment inequalities for the Boltzmann equation and applications to spatially homogeneous problems, J. Statist. Phys. 88 (1997) 1183–1214] in the case of hard spheres in 3 dimensions. To achieve this goal we implement the program presented in Bobylev–Gamba–Panferov [A.V. Bobylev, I.M. Gamba, V. Panferov, Moment inequalities and high-energy tails for Boltzmann equation with inelastic interactions, J. Statist. Phys. 116 (5–6) (2004) 1651–1682], which includes a full analysis of the moments by means of sharp moment inequalities and the control of L¹-exponential bounds, in the case of stationary states for different inelastic Boltzmann related problems with ‘heating’ sources where high energy tail decay rates depend on the inelasticity coefficient and the type of ‘heating’ source. More recently, this work was extended to variable hard potentials with angular cutoff by Gamba–Panferov–Villani [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press] in the elastic case collision case where the L¹-Maxwellian weighted norm was shown to propagate if initial states have such property. In addition, we also extend to all derivatives the propagation of L^∞-Maxwellian weighted estimates, proven in [I.M. Gamba, V. Panferov, C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, ARMA (2008), in press], to solutions of the initial value problem to the Boltzmann equations for elastic collisions for variable hard potentials with Grad (angular) cutoff.     

Duality for non-linear programming in aBanach space

Summary A constrained maximization problem in a realBanach space is considered, where the objective function is a non-linear pseudo concave functional and constraints are given bym non linear quasi convex functionals. The optimality conditions and converse duality theorem, given byRitter, have been extended to this class of programming problems.

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Zusammenfassung Die Maximierungsaufgabe in einem reellenBanachraum mit nichtlinearerpseudo konkaver Zielfunktion und endlich vielen nichtlinearen quasi konvexen Nebenbedingungen wird untersucht. Die vonRitter aufgestellten Optimalitätskriterien sowie der Dualitätssatz werden für diese Klasse von Programmen als gültig erkannt.

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Vorgel. v.:J. Nitsche.     

Strong convergence towards homogeneous cooling states for dissipative Maxwell models

We show the propagation of regularity, uniformly in time, for the scaled solutions of the inelastic Maxwell model for small inelasticity. This result together with the weak convergence towards the homogeneous cooling state present in the literature implies the strong convergence in Sobolev norms and in the L¹ norm towards it depending on the regularity of the initial data. The strategy of the proof is based on a precise control of the growth of the Fisher information for the inelastic Boltzmann equation. Moreover, as an application we obtain a bound in the L¹ distance between the homogeneous cooling state and the corresponding Maxwellian distribution vanishing as the inelasticity goes to zero.     

Sur l'interpolation des opérateurs maximaux monotones

On the Interpolation of Maximal Monotone Operators. We study here one way to extend to the maximal monotone case the results of linear interpolation, exposed bybalakrishnan in [2]. We obtain a necessary and sufficient condition of convergence for sequences(A ⁿ ) _n of maximal monotone operators on a real Hilbert spaceH.     

Fehlerabschätzung für die numerische Differentiation analytischer Funktionen durch Quadratur

Summary Numerical treatment of the integral in Cauchy's integral formula produces approximations for the derivatives of an analytic functionf; this fact has already been utilized byLyness andMoler [3, 4]. In the present paper this idea is investigated especially in view of the accuracy of these formulas regarded as quadrature formulas. Since the integration can be reduced to the integration of a periodic analytic function, it is possible to continue the considerations ofDavis [2] in order to find bounds for the error of the differentiation rules. For the application of these bounds one essentially needs estimations of the maximum off on a circle inside of its region of analyticity. Examples show the practical use of the bounds.

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Meinem verehrten LehrerH. Görtler zur Vollendung seines 60. Lebensjahres gewidmet     

On the Mean Square Formula of the Error Termfor a Class of Arithmetical Functions

 Based on the method in Meurman [5], we study the mean square formula of the error term for a class of Arithmetical functions whose Dirichlet series satisfies a functional equation with multiple gamma factors. We obtain improvements on some results of Chandrasekharan and Narasimhan [1].     

Generalized Minkowski formulas for closed hypersuperfaces in Riemann space

Summary This paper generalizedMinkowski formulas for a closed orientable hypersurface in aRiemann space with constant curvature which have been introduced byC. C. Hsiung, and studies on some properties of the hypersurface whose the v-th mean curvature is constant, on making use of the generalized formulas. To Enrico Bompiani on his scientific Jubilee     

Die Kleintropfenbewegung in einem homogenen isotropen Turbulenzfeld

Summary On the basis of a numerical calculation using values established byStewart andTownsend [1] it is shown for small droplets with radiusa10^–3 [cm] that they move in a horizontal direction independently of their radius and with the same velocity distribution as the velocity field of the air which moves them. In a vertical direction the same is true except that the fall movement is superimposed on the turbulence movement and is in the same value range as the latter.     

Baire sets in topological spaces

Summary In this paper we study topological properties of Baire sets in various classes of spaces. The main results state that a Baire set in a realcompact space is realcompact; a Baire set in a topologically complete space is topologically complete; and that a pseudocompact Baire set in any topological space is a zero-set. As a consequence, we obtain new characterizations of realcompact and pseudocompact spaces in terms of Baire sets of their Stone-ech compactifications. (Lorch in [3] using a different method has obtained either implicitly or explicitly the same results concerning Baire sets in realcompact spaces.) The basic tools used for these proofs are first, the notions of anr-compactification andr-embedding (see below for definitions), which have also been defined and used independently byMrówka in [4]; second, the idea included in the proof of the theorem: Every compact Baire set is aG as given inHalmos' text on measure theory [2; Section 51, theorem D].The author wishes to thank Professor W. W.Comfort for his valuable advice in the preparation of this paper.     

Das Anfangs-Randwertproblem eines dünnen Profils in kompressibler Unterschallströmung

Summary The initial boundary value problem of a thin profile in a compressible subsonic flow is investigated, being important for the gust problem. Existence and uniqueness of the solution are shown. By a generalizedWiener-Hopf-equation, the solution is reduced to the solution of a system of integro-differential equations for the downwashes in the profile plane. This system can be solved exactly by an iterative calculation of integrals for any value of timet>0. Finally, for smallt>0, the indicial-admittance function for vertical translation is found and is compared with a result given byLomax [6]. The asymptotic case for larget>0 will be investigated in a later paper.     

Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in [7] ofG. H. Meisters and improves the result [5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result [10] ofG. A. Willis.