On Howard’s conjecture in heterogeneous shear flow problem

Howard’s conjecture, which states that in the linear instability problem of inviscid heterogeneous parallel shear flow growth rate of an arbitrary unstable wave must approach zero as the wave length decreases to zero, is established in a mathematically rigorous fashion for plane parallel heterogeneous shear flows with negligible buoyancy forcegΒ ≪ 1 (Miles J W,J. Fluid Mech. 10 (1961) 496–508), where Β is the basic heterogeneity distribution function). An erratum to this article is available at .     

On short wave stability and sufficient conditions for stability in the extended Rayleigh problem of hydrodynamic stability

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if k > 0 is the wave number of a normal mode then k > k _c (for some critical wave number k _c) implies the stability of the mode for a class of basic flows. Furthermore, if $ K(z) = \frac{{ - (U'_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} $ K(z) = \frac{{ - (U'_0 - T_0 U'_0 )}} {{U_0 - U_{0s} }} , where U ₀ is the basic velocity, T ₀ (a constant) the topography and prime denotes differentiation with respect to vertical coordinate z then we prove that a sufficient condition for the stability of basic flow is $ 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) $ 0 < K(z) \leqslant \left( {\frac{{\pi ^2 }} {{D^2 }} + \frac{{T_0^2 }} {4}} \right) , where the flow domain is 0 ≤ z ≤ D.     

On the construction of minimal skew products

It is shown that under fairly general conditions on a compact metric spaceY there are minimal homeomorphisms onZ×Y of the form(z,y)→(σz, h _z (y)) where (Z, σ) is a arbitrary metric minimal flow andz→h _z is a continuous map fromZ to the space of homeomorphisms ofY. Similar results are obtained for strict ergodicity, topolotical weak mixing and some relativized concepts.     

Stability of a traveling-wave solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation

The asymptotic behavior of the solution to the Cauchy problem for the Korteweg-de Vries-Burgers equation u _t + (f(u)) _x + au _xxx − bu _xx = 0 as t → ∞ is analyzed. Sufficient conditions for the existence and local stability of a traveling-wave solution known in the case of f(u) = u ² are extended to the case of an arbitrary sufficiently smooth convex function f(u).     

Weak local homeomorphisms and <Emphasis Type="Italic">B</Emphasis>-favorable spaces

Let X and Y be topological spaces such that an arbitrary mapping f: X → Y for which every preimage f ⁻¹ (G) of a set G open in Y is an F _σ-set in X can be represented in the form of the pointwise limit of continuous mappings f _n : X → Y. We study the problem of subspaces Z of the space Y for which the mappings f: X → Z possess the same property. Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 60, No. 9, pp. 1189–1195, September, 2008.     

An acyclic extension of the braid group

We relate Artin's braid groupB _℞=lim_→B_n to a certain groupF′ ofpl-homeomorphisms of the interval. Namely, there exists a short exact sequence 1→B _∞→A→F′→1 whereH _kA=0,k≥1.     

A note on the distribution of the sequence (nα) with transcendental α

Let α be an irrational number and let D _N*(α) and D_N(α) denote the star-discrepancy and the discrepancy of the sequence (nα)_{n ≥1} mod 1, resp. We study properties of the maps α→ v *(α) = limsup_{N →∞} N D _N*(α)/log N and α→v(α) = limsup_{N →∞} N D _N(α)/log N where α is transcendental but not a U-number. This revised version was published online in June 2006 with corrections to the Cover Date.     

The Bieberbach-Rademacher problem for the Monge-Ampère operator

Assume that Ω is a bounded, strictly convex, smooth domain in ℝ^N withN≥2. We consider the problem det ((∂ _iju(x)))=f(x,u(x)),u(x)→∞ asx→∂Ω, where (∂ _iju(x)) denotes the Hessian ofu(x) andf meets some natural regularity and growth conditions. We prove that there exists a unique smooth, strictly convex solution of this problem. The boundary-blow-up rate ofu(x) is characterized in terms of the distance ofx from ∂Ω. Partially supported by the Royal Swedish Academy of Sciences, Gustaf Sigurd Magnuson's fund.     

极限跟踪的不变性

It is showed that the LmSP (limit shadowing property) for flows is invariant under topological equivalence, and the suspension flow ϕ _f of a homeomorphism f under a continuous function ϑ: X → R>0 has LmSP₊ if and only if f has LmSP₊. As examples of application, the LmSP₊ of the ergodic flow on torus which has irrational number is discussed, and the characterization of a class of flows on torus with LmSP₊ is given. Supported by the National Natural Science Foundation of China (10371030) and the doctor’s foundation of Hebei Normal Univ. (L2003B05).     

On two classes of permutations with number-theoretic conditions on the lengths of the cycles

Let Λ be an arbitrary set of positive integers andS _n (Λ) the set of all permutations of degreen for which the lengths of all cycles belong to the set Λ. In the paper the asymptotics of the ratio |S _n (Λ)|/n! asn→∞ is studied in the following cases: 1) Λ is the union of finitely many arithmetic progressions, 2) Λ consists of all positive integers that are not divisible by any number from a given finite set of pairwise coprime positive integers. Here |S _n (Λ)| stands for the number of elements in the finite setS _n (Λ). Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 881–891, December, 1997. Translated by A. I. Shtern     

Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ² → ℝ the Cauchy problemy′=f(x,y), y(x₀)=y₀ has a unique almost-everywhere solution in the classAC _t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions. The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114. The research of the second author was partially supported by the Israel Science Foundation. Publication 736.     

Minimal entropy and simplicial volume

We prove that MinEnt (Y) ∥Y∥ = MinEnt(X) ∥X∥, for manifolds Y whose fundamental group is a subexponential extension of the fundamental group of some negatively curved, locally symmetric manifold X. This is a particular case of a more general result holding for an arbitrary representation ρ : π₁ (Y) →π₁ (X), which relates the minimal entropy and the simplicial volume of X to some invariants of the couple (Y, ker (ρ)). Then, we discuss some applications to the minimal volume problem and to Einstein metrics. Received: 23 December 1998     

Algebraically constructible functions and signs of polynomials

LetW be a real algebraic set. We show that the following families of integer-valued functions onW coincide: (i) the functions of the formω →λ(X _ω ), where X _ω are the fibres of a regular morphismf :X →W of real algebraic sets, (ii) the functions of the formω →χ(X _ω ), where X _ω are the fibres of a proper regular morphismf :X →W of real algebraic sets, (iii) the finite sums of signs of polynomials onW. Such functions are called algebraically constructible onW. Using their characterization in terms of signs of polynomials we present new proofs of their basic functorial properties with respect to the link operator and specialization. Research partially supported by an Australian Research Council Small Grant. Second author also partially supported by KBN 610/P3/94.     

Geometric properties of Fréchet spaces and selection of basis sequences

For an arbitrary nuclear Fréchet spaceE possessing the well-known geometric propertiesD ₁ (DN) and Ω, certain sequences of functionals and elements with power-type estimates of the norms are selected. With the help of these objects, two isomorphic spacesK ₁ andK ₂ of power series of infinite type and continuous mapsJ ₂: K₂→E andJ ₁: E→K₁ are defined. Under some auxiliary conditions, it is proved that the_i are isomorphisms and the spaceE possesses a basis. Translated fromMatematicheskie Zametki, Vol. 66, No. 1, pp. 102–111, July, 1999.     

On the deficiency index of a certain class of differential operators

This paper studies problems on the behavior of solutions for a certain class of differential equations of the form l _n y = λy as x → +0, where l _n is a scalar, formally self-adjoint, quasidifferential expression of an arbitrary order with complex coefficients. The results obtained allow us to find the defficiency index of the operator generated by the expression l _n . __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 36, Suzdal Conference-2004, Part 2, 2005.     

Linear ordinary differential equations of the second order with unbounded solutions

We consider three families of equations of the form y″ + (1 + φ(x))y = 0, where the coefficient φ(x) satisfies the condition lim _x→+∞ φ(x) = 0. We obtain solutions of these equations in closed form. We show that the maximum absolute values of solutions grow at the rate of a logarithmic function, a power-law function, and even an exponential function as x → ∞.     

Stability of almost submetries

In this paper, we consider a triple of Gromov-Hausdorff convergence: $ A_i ^{\underrightarrow {d_{GH} }} A,B_i ^{\underrightarrow {d_{GH} }} A $ A_i ^{\underrightarrow {d_{GH} }} A,B_i ^{\underrightarrow {d_{GH} }} A and maps f _i : A _i → B _i converge to a map f: A → B, where A _i are compact Alexandrov n-spaces and B _i are compact Riemannian m-manifolds such that the curvature, diameter and volume are suitably bounded (non-collapsing). When f is a submetry, we give a necessary and sufficient condition for the sequence to be stable, that is, for i large, there are homeomorphisms, Ψ _i : A _i → A, Φ _i : B _i → B such that f ∘ Ψ _i = Φ _i ∘ f _i . When f is an ε-submetry with ε > 0, we obtain a sufficient condition for the stability in the case that A _i are Riemannian manifolds. Our results generalize the stability/finiteness results on fiber bundles by Riemannian submersions and by submetries.     

On the solutions of the translation equation in rings of formal power series

We study in this paper solutions of the translation equation in rings of formal power series K[X] where K ∈R, C (so called one-parameter groups or flows), and even, more generally, homomorphisms Ф from an abelian group (G, +) into the group Г(K) of invertible power series in K[X]. This problem can equivalently be formulated as the question of constructing homomorphisms Ф from (G, +) into the differential group Г¹∞ describing the chain rules of higher order of C∞ functions with fixed point 0. In this paper we present the general form of these homomorphisms Ф : G → Г(K) (or L¹∞),Ф = (f_n ⁿ≤1,forwhich f₁ = l, f₂ = ... = f_p+l =0,f_p+2 ≠ 0 for fixed, but arbitrary p ≤ 0 (see Theorem 5, Corollary 6 and Theorem 6). This representation uses a sequence (w _n ^p )_n≥p+2 of universal polynomials in f_p+2 and a sequence of parameters, which determines the individual one-parameter group. Instead of (w _n ^p )_n≥p+2 we may also use another sequence (L _n ^p )_n≥p+2 of universal polynomials, and we describe the connection between these forms of the solutions.     

Nielsen’s theorem for model aspherical manifolds

The main issue of this paper is the discussion of Nielsen’s realisation-problem for aspherical manifolds arising from (generalised) Seifert fiber space constructions. We present sufficient conditions on such “model” aspherical manifoldsM to have that a finite abstract kernel ψ:G → Out (π₁ (M)) can be (effectively) geometrically realised by a group of fiber preserving homeomorphisms ofM if and only if ψ can be realised by an (admissible) group extension 1 → (π₁ (M)) →E’ →G → 1. Then an algebraic approach to a (partial) study of the symmetry ofM is possible. Our result covers all situations already described in literature and we show with an example that we also deal with other types of Seifert fiber space constructions which were not yet treated before. Research Assistant of the Belgian National Fund for Scientific Research (N.F.W.O.)