Effect of suction on the upstream influence in shockwave-boundary layer interactions

Résumé Dans ce travail, on a étudié l'influence qu'exerce l'aspiration sur les perturbations se propageant en amont par l'interaction de l'onde de choc et de la couche limite, dans un coin de compression. Les résultats des expériences montrent que l'aspiration réduit ces perturbations et élimine la séparation. L'augmentation de l'aspiration entraîne une augmentation de la pression à travers le coin, qui approche, en cas de pression adéquate, la valeur de liquides non visqueux. Tous les résultats sont analysés en termes de la distance vers l'amont adimensionnelle définie dans le texte.

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Nomenclature d upstream distance measured from corner to point 0 - ₀ ^* calculated boundary layer displacement thickness at 0 - X _t distance from effective leading edgeE of turbulent boundary layer, to 0 (see Figure 1) - R _et Reynolds number based onX _t - M free stream Mach number - p free stream static pressure - p measured static pressure - h bleed height - wedge angle - Q air flow removed by suction through the bleed - ₁ density in the free stream - U ₁ velocity in the free stream - ₁ momentum thickness of the undisturbed boundary layer     

Sur les difféomorphismes d'Anosov affines à feuilletages stable et instable différentiables

Résumé Nous étudions les difféomorphismes d'Anosov topologiquement transitifs sur une variété compacte dont les feuilletages stable et instable sont de classeC . Nous montrons que si un tel difféomorphisme préserve une structure symplectique ou une connexion, il est alorsC conjugué à un infranilautomorphisme hyperbolique.Oblatum 18-III-1992     

Thermal convection near a partly insulated vertical flat plate

Résumé On obtient une solution numérique pour le developpement d'une convection libre laminaire près d'une plaque plane verticale isolée, chauffée tout le long de sa base par une source calorifique.Les conditions d'isolement réduisent en fait le problème à celui d'une convection libre près d'une plaque plane verticale dont la température varie selonx ^–3/5,x représentant la distance verticale depuis la base. Une solution approximative du type Pohlhausen suggère la forme des paramètres qu'on utilise pour résoudre les équations limite. On a obtenu des solutions numériques pour des valeurs paramétriques du nombre de Prandtl allant de 0,1 à 10.On étudie aussi la convection libre produite en chauffant la longueur, 1, d'une plaque verticale semi-infinie. A une grande hauteur au-dessus de la partie chauffée lavélocité et les distributions de température se comportent comme lorsque la chaleur est fournie par une source rectiligne près de la base de la plaque.Une solution Pohlhausen donnée pour ce cas détermine la position de l'origine effective de la solution de similarité estimée d'après la longueur 1 de la partie chauffée de la plaque.

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Nomenclature f a dimensionless function - g acceleration due to gravity - h local heat transfer - k thermal conductivity - l length of the heated part of the plate - q heat flux rate - u, v longitudinal and transverse velocities - x, y longitudinal and transverse co-ordinates - A scale velocity in the boundary layer - B constant (Equation (44)) - C constant (Equation (19)), dimensional - C ₁ constant=(Ng/4v ²)^1/4, dimensional - Gr local Grashof number based onx - N dimensional constant (Equation (31)) - T temperature - Ta local Takhar number; a Heat flux number based onx - T _w temperature at the wall - T temperature at infinity - coefficient of cubical expansion - boundary layer thickness - non-dimensional length=y/ - ^* non-dimensional length=C ₁ y/x ^2/5 - non-dimensional temperature - thermal diffusivity - coefficient of kinematic viscosity - density - Prandtl number=/ - stream function     

Some Problems for Cubic Nonlinear Equations on a Half-Line

The paper deals with a problem of developing an inverse-scattering based formalism for solving problems for the cubic nonlinear (or the modified Korteweg–de Vries (KdV)) equations: q _t +q _xxx +6q ² q _x =0, 0x<, –<t<,q _t +q _xxx –6q ² q _x =0, with the given initial and boundary conditions: q(x,0)=q(x),q(0,t)=p(t), p(t)L ₁(–,). The relation between the solution of the initial-boundary value problem (1), (3), (4) and that of the KdV equation on the half-line is shown. The Cauchy problem for the cubic nonlinear equation: q _t +q _xxx –6|q|² q _x =0, 0x<, –<t<, with the given initial condition (3) is considered also. Here we solve the above problems on the half-line 0x< but with –<t<.     

Algorithmes de calcul du maximum des formes quadratiques sur la boule unité de la norme du maximum

Résumé Certaines méthodes directes et indirectes pour le calcul de Max {x ^t Ax, (x)1} sont étudiées.Les méthodes directes sont basées sur les propriétés particulières des normes ₁, ₂ et . Ces méthodes sont très simples mais ne s'appliquent qu'à certaines familles de matrices.La méthode indirecte est la méthode autoduale introduite dans [25, 26] avec = ₁. Dans ce cas, le choix du vecteur initial pour qu'il y ait convergence vers une solution optimale est largement discuté.

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Some methods for computing the maximum of quadratic from on the unit ball of the maximum norm

Summary Some direct and indirect methods are studied for computing Max {x ^t Ax, (x)1} whereA is symmetric definite positive.Direct methods are constructed using particular properties of ₁, ₂, norms. These methods are very simple, but uniquely suitable to certains families of matrices.The indirect method is the autodual method, introduced in [25, 26, 29] with = ₁. In this case the problem of choosing an initial vector so that convergence of the iterative sequence occurs to an optimal solution is largely discussed.

     

Remarks on ergodicity of stationary irreducible transient Markov chains

Summary Consider a stationary process {X _n(), – < n < . If the measure of the process is finite (the measure of the whole sample space finite), it is well known that ergodicity of the process {X _n(), - < n < and of each of the subprocesses {X _n(), 0 n < , {X _n(), – < n 0 are equivalent (see [3]). We shall show that this is generally not true for stationary processes with a sigma-finite measure, specifically for stationary irreducible transient Markov chains. An example of a stationary irreducible transient Markov chain {X _n(), - < n <} with {itX_n(), 0 n < < ergodic but {X _n(), < n 0 nonergodic is given. That this can be the case has already been implicitly indicated in the literature [4]. Another example of a stationary irreducible transient Markov chain with both {X _n(), 0 n < and {itX _n(),-< < n 0} ergodic but {X _n(), - < n < nonergodic is presented. In fact, it is shown that all stationary irreducible transient Markov chains {X _n(), - < n < < are nonergodic.This research was supported in part by the Office of Naval Research.John Simon Guggenheim Memorial Fellow.     

A relation between spline approximation and the problem of the approximation of one class by another

An investigation of the approximation in L_q(–, ) of differentiable functions whose k-th derivatives belong to L_p(–, ), by splines S_m (x) with nonfixed nodes, under the extra assumption that the norms in L_s(–, ) of theirl-th derivatives have a common bound. A relation is established with the problem of approximating functions of one class by functions of another class.Translated from Matematicheskie Zametki, Vol. 9, No. 5, pp. 501–510, May, 1971.     

Conservation laws with continuous flux functions

In the present paper we investigate uniqueness problem for generalized entropic solution of stationnary and evolution conservation laws under anisotropic conditions on vector-flux functions in the classesL andL L ¹.This research was performed while visiting the Equipe de Mathématiques in Besançon.     

Embeddings of Banach Spaces Into Banach Lattices and the Gordon–Lewis Property

In this paper we first show that if X is a Banach space and is a left invariant crossnorm on lX, then there is a Banach lattice L and an isometric embedding J of X into L, so that I J becomes an isometry of lX onto l_m J(X). Here I denotes the identity operator on l and l_m J(X) the canonical lattice tensor product. This result is originally due to G. Pisier (unpublished), but our proof is different. We then use this to prove the main results which characterize the Gordon–Lewis property GL and related structures in terms of embeddings into Banach lattices.     

Subsets of a Hilbert space,having finite hausdorff dimension

Let X₂, X₂ be Hilbert spaces, X₂ X₁, X₂ is dense in X₁, the imbedding is compact,m X₂, dim_H ⁱ m and h⁽ⁱ⁾(m) are the Hausdorff dimension and the limit capacity (information dimension) of the setm with respect to the metrics of the spaces X_i (i=1, 2). Two examples are constructed. 1) An example of a setm bounded in X₂, such that: a) h⁽¹⁾(m) < (and, consequently, dim_H ¹ m); b)m cannot be covered by a countable collection of sets, compact in X₂ (and, consequently, dim_H ² m=). 2) an Example of a setm, compact in X₂, such that h⁽¹⁾(m) < and h⁽²⁾(m)=.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 163, pp. 154–165, 1987.     

The calculation of the thermal laminar boundary layer on a rotating sphere

Sommaire La communication expose une méthode permettant de déterminer la couche limite thermique d'une sphère en rotation dans un fluide incompressible à écoulement uniforme. La méthode se base sur l'idée deFrössling, selon laquelle la répartition des températures s'exprime en tant que développement en série de puissances, et utilise les résultats déjà obtenus parHoskin pour la répartition des vitesses dans la couche limite d'une sphère en rotation. L'auteur donne des résultats numériques obtenus sur une machine à calculer IBM 704.

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Nomenclature x distance along surface of body in a meridian plane from forward stagnation point - y distance measured normal to surface - U velocity of main flow - U velocity of undisturbed stream - u component of velocity inx-direction - v component of velocity iny-direction - w transverse component of velocity due to spin - p pressure - density - T temperature - R distance of surface from axis of symmetry measured normal to axis of symmetry - coefficient of viscosity - kinematic viscosity - a radius of sphere - k thermal conductivity - angular velocity about axis of symmetry - Re Reynolds number - Pr Prandtl number - Nu Nusselt number - a constant - S area - Q quantity of heat transferred in unit time across areaS - d a characteristic length     

A property of nonlinear elliptic equations when the right-hand side is a measure

LetA(u)=–diva(x, u, Du) be a Leray-Lions operator defined onW ₀ ^1,p () and be a bounded Radon measure. For anyu SOLA (Solution Obtained as Limit of Approximations) ofA(u)= in ,u=0 on , we prove that the truncationsT _k(u) at heightk satisfyA(T _k(u)) A(u) in the weak * topology of measures whenk + .

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Résumé SoitA(u)=–diva(x, u, Du) un opérateur de Leray-Lions défini surW ₀ ^1,p () et une mesure de Radon bornée. Pour toutu SOLA (Solution Obtenue comme Limite d'Approximations) deA(u)= dans ,u=0 sur , nous démontrons que les troncaturesT _k(u) à la hauteurk vérifientA(T _k(u)) A(u) dans la topologie faible * des mesures quandk + .

     

On a property of functional series

The question of the convergence of functional series everywhere in the segment [0, 1] is considered. Let F=f be the set of such functions in [0, 1] for each of which there is a transposition of the series _k=1 f_k(x), which converges to it everywhere in [0, 1]. An example of a series is constructed such that the set F consists just of an identical zero, but _k=1 |f _k (x ₀)ü=,(x₀ [0,1]) for any point of the segment [0, 1].Translated from Matematicheskie Zametki, Vol. 11, No. 5, pp. 481–490, May, 1972.     

On stationary queue length distributions for G/M/s/r queues

Consider aG/M/s/r queue, where the sequence{A _n } _n=– of nonnegative interarrival times is stationary and ergodic, and the service timesS _n are i.i.d. exponentially distributed. (SinceA _n =0 is possible for somen, batch arrivals are included.) In caser < , a uniquely determined stationary process of the number of customers in the system is constructed. This extends corresponding results by Loynes [12] and Brandt [4] forr= (with=ES₀/EA₀<s) and Franken et al. [9], Borovkov [2] forr=0 ors=. Furthermore, we give a proof of the relation min(i, s)¯p(i)=p(i–1), 1ir + s, between the time- and arrival-stationary probabilities¯p(i) andp(i), respectively. This extends earlier results of Franken [7], Franken et al. [9].     

On the best approximations and Kolmogorov widths of besov classes of periodic functions of many variables

Order estimates are obtained for the best approximations of the classesB _1, ^r in the spaceL _q with 1<q< and classesB _, ^r in a uniform metric. The behavior of Kolmogorov widths of the classesB _p, ^r ,1<p, in the metric of L is studied.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 79–92, January, 1995.     

Absence de la L∞-contractivité pour les semi-groupes associés aux opérateurs elliptiques complexes sous forme divergence

Soit un opérateur elliptique du second ordre à coefficients complexes sur un ouvert de ^N . On obtient des conditions nécessaires et suffisantes sur les coefficients, pour que le semi-groupe qu'il définit, suivant les conditions au bord considérées, contracte L . On montre en particulier que cette propriété est assez spécifique aux coefficients réels. Abstract. Consider a second order elliptic operator with complex coefficients on an open set of ^N . We obtain necessary and sufficient conditions on the coefficients for the contractivity in L of the semigroup defined under different boundary conditions. In particular, we show that this property is closely related to the fact that the coefficients are actually real valued.     

On the Growth of the Maximum of the Modulus of an Entire Function on a Sequence

Let M _f(r) and _f(r) be, respectively, the maximum of the modulus and the maximum term of an entire function f and let be a continuously differentiable function convex on (–, +) and such that x = o((x)) as x +. We establish that, in order that the equality be true for any entire function f, it is necessary and sufficient that ln (x) = o((x)) as x +.     

On extending backwards positive definite sequences

The two-sided Hamburger moment problem¹, also called the strong one [4], has been extensively studied in recent years in connection with rational approximation. We propose to consider the question of when a sequence, say {a _n } _n=0 can be extended backwards so that the resulting sequence {a _n } _n=–N has an integral representation of the Hamburger type. This was settled (without any proof) under different circumstances in [6]. Here we wish to discuss this completely, as well as the possibility of extending {a _n } _n=0 to {a _n } _n– .     

On Infinite Groups with Given Properties of the Norm of Infinite Subgroups

We investigate the relationship between the norm N _G() of infinite subgroups of an infinite group G and the structure of this group. We prove that N _G() is Abelian in the nonperiodic case, and a locally finite group is a finite extension of a quasicyclic subgroup if N _G() is a non-Dedekind group. In both cases, we describe the structure of the group G under the condition that the subgroup N _G() has finite index in G.     

Opérateurs différentiels associés à certaines représentations unitaires d'un groupe de Lie résoluble exponentiel

Soient G = exp g un groupe de Lie résoluble exponentiel et H = exp h un sous-groupe connexe de G. Soient un caractère unitaire de H et = Ind_H ^G. Soit D(G/H) l'algèbre des opérateurs différentiels G-invariants sur G/H. Une question posée par Duflo et Corwin-Greenleaf consiste à voir si la finitude des multiplicités de est équivalente à la commutativité de D (G/H). Nous répondons positivement à cette question quand H est normal dans G. Lorsque H n'est pas normal, nous préparons le terrain pour d'espaces homogènes nilpotents et nous répondons à la question dans différents cas. Nous étudions finalement l'algèbre D (G) ^H , des opérateurs différentiels qui laissent l'espace des vecteurs C de invariant et qui commuttent avec l'action de H sur cet espace.