Ähnliche Lösungen bei kompressiblen Grenzschichten mit Wärmeübergang und Absaugen oder Ausblasen

By applying an Illingworth-Stewartson transformation the equations of compressible boundary layers with heat-transfer and pressure gradient can be brought to a form which is similar to the incompressible equations. Special boundary conditions lead here to similar solutions, which are calculated in the present paper also for the case of suction and blowing out.

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Bezeichnungen c Schallgeschwindigkeit - c _p spez. Wärme bei konst. Druck - f dimensionslose Stromfunktion (Gl. (13)) - m Exponent (Gl. (12)) - p Druck - u, v Geschwindigkeitskomponenten - transformierte Geschwindigkeitskomponenten - x, y physikalische Koordinaten - transformierte Koordinaten (Gl. (3) und (4)) - Ma Machsche Zahl - Pr Prandtlzahl - S dimensionslose Temperaturfunktion (Gl. (7)) - S _m Sutherlandsche Temperaturkonstante - T Temperatur - Parameter (Gl. (17)) - dynamische Zähigkeit - kinematische Zähigkeit - Stromfunktion Indices 0 Ruhe - 1 äusserer Rand der Grenzschicht - Wand Mitteilung der Aerodynamischen Versuchsanstalt Göttingen.     

Volume,diameter and the minimal mass of a stationary 1-cycle

In this paper we present upper bounds on the minimal mass of a non-trivial stationary 1-cycle. The results that we obtain are valid for all closed Riemannian manifolds. The first result is that the minimal mass of a stationary 1-cycle on a closed n-dimensional Riemannian manifold Mⁿ is bounded from above by (n + 2)!d/4, where d is the diameter of a manifold Mⁿ. The second result is that the minimal mass of a stationary 1-cycle on a closed Riemannian manifold Mⁿ is bounded from above by where where is the filling radius of a manifold, and where is its volume.     

Approximation by Reciprocals of Polynomials with Positive Coefficients in L p Spaces

We prove that, if f(x) L ^p _[0,1], 1 < p < , f(x) 0, x [0,1], f 0, then there is a polynomial p(x) ⁺ _n such that f - 1/p _LP C(p)(f,n ^-1/2) _LP where ⁺ _n indicates the set of all polynomials of degree n with positive coeficients (see the definition (1) in the text).     

Aperiodische Wahrscheinlichkeitsmaße auf topologischen Gruppen

Several conditions are shown to be equivalent to the aperiodicity of a regular probability measure on a locally compact, separated topological GroupG. In particular, is aperiodic if and only if the sequence ( ( ⁽ⁿ⁾ denoting then-th convolution power of ) is convergent for any nonvoid open subsetU ofG with compact closure. It is always assumed that the support of generatesG as a closed semigroup.     

On the maximum 2-1 matching

Given a simple bipartite graphG=(X,Y, E).M E is called a 2-1 matching ofG if: 1) _x X, either two edges or none inM is incident tox and 2) _y Y, at most one edge inM is incident toy. In this paper, we describe an efficient algorithm for finding a maximum 2-1 matching in a given bipartite graph. We also formulate and prove a duality theorem for 2-1 matching.     

Riemann's Fragment on Limit Values of Elliptic Modular Functions

We give an interpretation different from that of Dedekind to work n° 28 of the complete works of Riemann Fragmente über die Grenzfälle der Elliptischen Modulfunctionen.We prove a theorem of inversion of radial limit and sum in a series of functions. This allows us to justify all of Riemann's reasoning in the fragment to obtain the limit values of modular elliptic functions. In particular we prove the statement of Riemann that for every rational number x we have where denotes the periodic function with period 1, such that (x) = x when |x| < 1/2, and (n + ) = 0 for every n Z.This assertion of Riemann was criticized by Dedekind. We also give the transformation formulae of the logarithms of the classical theta-function ₃(0), giving an alternative form to that obtained by B. C. Berndt [1].     

Oscillation of fourth order nonlinear neutral difference equations-II

Oscillatory and asymptotic behaviour of solutions of a class of fourth order nonlinear neutral difference equations of the form

Contraction of Exterior Powers in Characteristic 2 and the Spin Module

Let K be a field of characteristic 2 and letV be a vector space of dimension 2m over K. Let f be a non-degenerate alternating bilinear form defined on V × V. The symplectic group Sp(2m, K) acts on the exterior powers ^k V for 0 k. 2m There is a contraction map defined on the exterior algebra , which commutes with the Sp(2m, K) action and satisfies ² = 0 and ( ^k V) ^k–1 V We prove that ( ^k V)= ker ^k–1 V except when k=m+2. In the exceptional case, ( ^m+2 V) has codimension 2^m in ker ^m V and we show that the quotient module ker ^m V/ ^m+2 V is a spin module for Sp(2m,K). When K is algebraically closed, we show that this spin module occurs with multiplicity 1 in ^m V and multiplicity 0 in all other components of V.     

A note on the illumination of convex bodies

LetKE ^d be a convex body and letl _r(K) denote the minimum number ofr-dimensional affine subspaces ofE ^d lying outsideK with which it is possible to illuminateK, where 0rd–1. We give a new proof of the theorem thatl _r(K)(d+1)/(r+1) with equality for smoothK.The work was supported by Hung. Nat. Found. for Sci . Research No. 326-0213 and 326-0113.     

Characterization of {v μ+1 + 2v μ,v μ + 2v μ − 1;t,q}-min · hypers and its applications to error-correcting codes

Recently, Hamada [5] characterized all {v ₂ + 2v ₁,v ₁ + 2v ₀;t,q}-min · hypers for any integert 2 and any prime powerq 3 wherev _l = (q ^l – 1)/(q – 1) for any integerl 0. The purpose of this paper is to characterize all {v _{+ 1} + 2v ,v + 2v _{– 1};t,q}-min · hypers for any integerst, and any prime powerq such thatt 3, 2 t – 1 andq 5 and to characterize all (n, k, d; q)-codes meeting the Griesmer bound (1.1) for the casek 3, d = q ^k-1 – (2q ^-1 +q ) andq 5 using the results in Hamada [3, 4, 5].     

The Law of the Logarithm for Weighted Sums of Independent Random Variables

Let X,X _n ;n1 be a sequence of real-valued i.i.d. random variables with E(X)=0. Assume B(u) is positive, strictly increasing and regularly-varying at infinity with index 1/2<1. Set b _n =B(n),n1. If

Global Solutions to the Two Dimensional Quasi-Geostrophic Equation with Critical or Super-Critical Dissipation

The two dimensional quasi-geostrophic (2D QG) equation with critical and super-critical dissipation is studied in Sobolev space H^s(ℝ²). For critical case (α=), existence of global (large) solutions in H^s is proved for s≥ when is small. This generalizes and improves the results of Constantin, D. Cordoba and Wu [4] for s = 1, 2 and the result of A. Cordoba and D. Cordoba [8] for s=. For s≥1, these solutions are also unique. The improvement for pushing s down from 1 to is somewhat surprising and unexpected. For super-critical case (α ∈ (0,)), existence and uniqueness of global (large) solution in H^s is proved when the product is small for suitable s≥2−2α, p ∈ [1,∞] and β ∈ (0,1].     

The average a posteriori error of numerical methods

Summary The definition of the average error of numerical methods (by example of a quadrature formula to approximateS(f)= f d on a function classF) is difficult, because on many important setsF there is no natural probability measure in the sense of an equidistribution. We define the average a posteriori error of an approximation by an averaging process over the set of possible information, which is used by (in the example of a quadrature formula,N(F)={(f(a ₁), ...,f/fF} is the set of posible information). This approach has the practical advantage that the averaging process is related only to finite dimensional sets and uses only the usual Lebesgue measure. As an application of the theory I consider the numerical integration of functions of the classF={f:[0,1]/f(x)–f(y)||x–y|}. For arbitrary (fixed) knotsa _i we determine the optimal coefficientsc _i for the approximation and compute the resulting average error. The latter is minimal for the knots . (It is well known that the maximal error is minimal for the knotsa _i .) Then the adaptive methods for the same problem and methods for seeking the maximum of a Lipschitz function are considered. While adaptive methods are not better when considering the maximal error (this is valid for our examples as well as for many others) this is in general not the case with the average error.     

On squeezing and flow of energy for nonlinear wave equations

We study the nonlinear wave equationü– ²u+f(u)=0 undern-dimensional periodic boundary conditions (n=1,2,3) in a Sobolev phase-space . In [K1] we interpreted the energy transition to higher frequencies problem for this (and similar) equations as a squeezing: time-t flow maps of the equation with larget squeezer-balls inH ³ to narrow cylinders formed by vector-functions such that the norms of their first Fourier coefficients are less then some small . We proved in [K1] a version of Gromov's (non)squeezing theorem: the phenomenon stated above is impossible for <r ifH ³ is chosen to be the symplectic phase space of the equation,H ³=H ^1/2. In this paper we show that in smooth phase spacesH ³,s5, the squeezing is typical if the dispersion is small. We use this result and some related statements to obtain a (rather rough) picture of qualitative behaviour of solutions of the ⁴ ü– ²u+u ³=0.     

Oscillation of fourth order nonlinear neutral difference equations I

Oscillatory and asymptotic behaviour of solutions of a class of nonlinear fourth order neutral difference equations of the form

C 1 changes of variable: Beurling-Helson type theorem and Hörmander conjecture on Fourier multipliers

We prove that if aC ¹ smooth change of variable : generates a bounded composition operatorff° in the spaceA _p()=L ^p ,p2, then is linear (affine).We also prove that for a nonlinearC ¹ mapping , the norms of exponentialse ⁱ as Fourier multipliers inL ^p () tend to infinity (,||). In both results the condition C ¹ is sharp, it cannot be replaced by the Lipschitz condition.     

Linear Discrepancy of Totally Unimodular Matrices*†

  We show that the linear discrepancy of a totally unimodular m×n matrix A is at most

Polynomially convex orbits of compact lie groups

LetV be a finite dimensional complex linear space and letG be a compact subgroup of GL(V). We prove that an orbitG, V, is polynomially convex if and only ifG is closed andG is the real form ofG . For every orbitG which is not polynomially convex we construct an analytic annulus or strip inG with the boundary inG. It is also proved that the group of holomorphic automorphisms ofG which commute withG acts transitively on the set of polynomially convexG-orbits. Further, an analog of the Kempf-Ness criterion is obtained and homogeneous spaces of compact Lie groups which admit only polynomially convex equivariant embeddings are characterized.Supported by Federal program Integratsiya, no. 586.Supported by INTAS grant 97/10170.     

Minimal Blocking Sets in Projective Spaces of Square Order

In this paper minimal m-blocking sets of cardinality at most in projective spaces PG(n,q) of square order q, q 16, are characterized to be (t, 2(m-t-1))-cones for some t with . In particular we will find the smallest m-blocking sets that generate the whole space PG(n,q) for 2m n m.     

Quasi-additive functions

Summary Let 0 < 1 and letX, Y be real normed spaces. In this paper we consider the following functional inequality:f(x + y) – f(x) – f(y) min{f(x + y), f(x) + f(y)} forx, y R, wheref: X Y. Mainly continuous solutions are investigated. In the case whereY = R some necessary and some sufficient conditions for this inequality are given.Let 0 <1. The following functional inequality has been considered in [5]:f(x + y) – f(x) – f(y) min{f(x + y), f(x) + f(y)} forx, y R, wheref: R R. It appeared that the solutions of this inequality have properties very similar to those of additive functions (cf. [1], [2], [3]). The inequality under consideration seems to be interesting also because of its physical interpretation (cf. [5]). In this paper we shall consider this inequality in a more general case, wheref is defined on a real normed space and takes its values in another real normed space.The first part of the paper concerns the general case; in the second part we assume that the range off is inR.