Equivalence and symmetries of second-order differential equations

In this article we investigate the equivalence of underdetermined differential equations and differential equations with deviations of second order with respect to the pseudogroup of transformations = φ(x), ȳ = ȳ() = L(x) + y(x), = () = M(x) + z(x). Our main aim is to determine such equations that admit a large pseudogroup of symmetries. Instead the common direct calculations, we use some more advanced tools from differential geometry, however, our exposition is self-contained and only the most fundamental properties of differential forms are employed. This research has been conducted at the Department of Mathematics as part of the research project CEZ: Progressive reliable and durable structures, MSM 0021630519.     

On $$ \mathcal{C} $$*-sets and decompositions of continuous and ηζ-continuous functions

The main purpose of this paper is to introduce the concepts of *-sets, *-continuous functions and to obtain new decompositions of continuous and ηζ-continuous functions. Moreover, properties of *-sets and some properties of -sets are discussed.      

Some functional equations on standard operator algebras

The main purpose of this paper is to prove the following result. Let H be a complex Hilbert space, let (H) be the algebra of all bounded linear operators on H, and let (H) ⊂ (H) be a standard operator algebra which is closed under the adjoint operation. Suppose that T: (H) → (H) is a linear mapping satisfying T(AA* A) = T(A)A* A − AT(A*)A + AA*T(A) for all A ∈ (H). Then T is of the form T(A) = AB + BA for all A ∈ (H), where B is a fixed operator from (H). A result concerning functional equations related to bicircular projections is proved      

Approximately isometric lifting in quasidiagonal extensions

Let 0 → I → A → A/I → 0 be a short exact sequence of C*-algebras with A unital. Suppose that the extension 0 → I → A → A/I → 0 is quasidiagonal, then it is shown that any positive element (projection, partial isometry, unitary element, respectively) in A/I has a lifting with the same form which commutes with some quasicentral approximate unit of I consisting of projections. Furthermore, it is shown that for any given positive number ε, two positive elements (projections, partial isometries, unitary elements, respectively) in A/I, and a positive element (projection, partial isometry, unitary element, respectively) a which is a lifting of , there is a positive element (projection, partial isometry, unitary element, respectively) b in A which is a lifting of such that ∥a−b∥ < . As an application, it is shown that for any positive numbers ε and in U(A/I) ₀ , there exists u in U(A)₀ which is a lifting of such that cel(u) < cel. This work was supported by National Natural Science Foundation of China (Grant No. 10771161)     

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let be a separable Hilbert space, an open convex subset, and f: a smooth map. Let Ω be an open convex set in with , where denotes the closure of Ω in . We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ , where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L ²-space under the flow of a class of kinetic transport equations so called BGK model.      

Distribution of the maxima of random Takagi functions

This paper concerns the maximum value and the set of maximum points of a random version of Takagi’s continuous, nowhere differentiable function. Let F(x):=∑ _n=1^∞ ε _n ϕ(2 ⁿ⁻¹ x), x ∈ R, where ɛ ₁, ɛ ₂, ... are independent, identically distributed random variables taking values in {−1, 1}, and ϕ is the “tent map” defined by ϕ(x) = 2 dist (x, Z). Let p:= P (ɛ ₁ = 1), M:= max {F(x): x ∈ R}, and := {x ∈ [0, 1): F(x) = M}. An explicit expression for M is given in terms of the sequence {ɛ _n }, and it is shown that the probability distribution μ of M is purely atomic if p < , and is singular continuous if p ≧ . In the latter case, the Hausdorff dimension and the multifractal spectrum of μ are determined. It is shown further that the set is finite almost surely if p < , and is topologically equivalent to a Cantor set almost surely if p ≧ . The distribution of the cardinality of is determined in the first case, and the almost-sure Hausdorff dimension of is shown to be (2p − 1)/2p in the second case. The distribution of the leftmost point of is also given. Finally, some of the results are extended to the more general functions Σa ^{n − 1} ɛ _n ϕ(2 ^{n − 1} x), where 0 < a < 1.      

Stability of homogeneity almost everywhere

We consider approximately ϕ-homogeneous mappings almost everywhere, that is functions F such that the difference F(αx) − ϕ(α)F(x) is in some sense bounded almost everywhere in a product space. We will prove, under some assumptions, that then either we have some kind of boundedness of ϕ and F, or there exist a homomorphism and a -homogeneous function , which are almost everywhere equal to ϕ and F, respectively. From this result we derive the superstability effect for the multiplicativity almost everywhere.      

Measurable orthogonally additive functions modulo a discrete subgroup

Under appropriate conditions on Abelian topological groups G and H, an orthogonality ⊥ ⊂ G ² and a σ-algebra of subsets of G we decompose an -measurable function f: G → H which is orthogonally additive modulo a discrete subgroup K of H into its continuous additive and continuous quadratic part (modulo K). Research supported by the Silesian University Mathematics Department (Functional Equations on Abstract Structures program — the first author, and Iterative Functional Equations and Real Analysis program — the second author).     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

The \bar \partial -Neumann operator and commutators of the Bergman projection and multiplication operators

We prove that compactness of the canonical solution operator to restricted to (0, 1)-forms with holomorphic coefficients is equivalent to compactness of the commutator defined on the whole L _(0,1)²(Ω), where is the multiplication by and is the orthogonal projection of L _(0,1)²(Ω) to the subspace of (0, 1) forms with holomorphic coefficients. Further we derive a formula for the -Neumann operator restricted to (0, 1) forms with holomorphic coefficients expressed by commutators of the Bergman projection and the multiplications operators by z and . Partially supported by the FWF grant P19147-N13.     

Projectively condensed semigroups,generalized completely regular semigroups and projective orthomonoids

The class of projectively condensed semigroups is a quasivariety of unary semigroups, the class of projective orthomonoids is a subquasivariety of . Some well-known classes of generalized completely regular semigroups will be regarded as subquasivarieties of . We give the structure semilattice composition and the standard representation of projective orthomonoids, and then obtain the structure theorems of various generalized orthogroups. Partially supported by a UGC (HK) grant #2060123 (04-05).     

On a problem of D.H. Lehmer and pseudorandom binary sequences

Let p be an odd prime, and f(x), g(x) ∈ [x]. Define

The hardy and bellman operators in spaces connected with H( \mathbb{T} ) and BMO( \mathbb{T} )

Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series $ \sum\limits_{n = 1}^\infty {a_n } Assume that 1 ≤ p < ∞ and a function f ∈ L ^p [0, π] has the Fourier series cos nx. According to one result of G.H. Hardy, the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. But if 1 < p ≤ ∞ and f ∈ L ^p [0, π], then the series cos nx is the Fourier series for a certain function (f) ∈ L ^p [0, π]. Similar assertions are true for sine series. This allows one to define the Hardy operator on L ^p (), 1 ≤ p < ∞, and to define the Bellman operator on L ^p (), 1 < p ≤ ∞. In this paper we prove that the Bellman operator boundedly acts in VMO(), and the Hardy operator also maps a certain subspace C() onto VMO(). We also prove the invariance of certain classes of functions with given majorants of modules of continuity or best approximations in the spaces H(), L(), VMO() with respect to the Hardy and Bellman operators. Original Russian Text ? S.S. Volosivets and B.I. Golubov, 2008, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2008, No. 5, pp. 4–13.     

A note on the crossed product $$ \mathcal{R}(\mathcal{A},\alpha ) $$ associated with PSL2(ℝ)

Given the hyperbolic measure dxdy/y ² on the upper half plane ℍ, the rational actions of PSL₂(ℝ) on ℍ induces a continuous unitary representation α of this group on the Hilbert space L ²(ℍ, dxdy/y ²). Supposing that = {M _f : f ∈ L ^∞ (ℍ, dxdy/y ²)}, we show that the crossed product is of type I. In fact, the crossed product is *-isomorphic to the von Neumann algebra , where is the abelian group von Neumann algebra generated by the left regular representation of K. This work was supported by the Youth Foundation of Sichuan Education Department of China (Grant No. 2003B017)     

Nonparametric inference in a simple change-point model

In this paper, we consider a change point model allowing at most one change, X() = f() + e(), where f(t) = α + θ (t), 0 ≤ t ≤ 1, {e(), ..., e()} is a sequence of i.i.d. random variables distributed as e with 0 being the median of e. For this change point model, hypothesis test problem about the change-point t0 is studied and a test statistic is constructed. Furthermore, a nonparametric estimator of t0 is proposed and shown to be strongly consistent. Finally, we give an estimator of jump θ and obtain it’s asymptotic property. Performance of the proposed approach is investigated by extensive simulation studies. Research partially supported by National Natural Science Foundation of China (Grant No. 10471136), Ph.D. Program Foundation of the Ministry of Education of China, and Special Foundations of the Chinese Academy of Sciences and USTC     

On some new criteria for infinite differentiability of periodic functions

We study the set of infinitely differentiable periodic functions in terms of generalized -derivatives defined by a pair of sequences ψ ₁ and ψ ₂. It is shown that every function ƒ from the set has at least one derivative whose parameters ψ ₁ and ψ ₂ decrease faster than any power function and, at the same time, for any function ƒ ∈ different from a trigonometric polynomial, there exists a pair ψ whose parameters ψ ₁ and ψ ₂ have the same rate of decrease and for which the -derivative no longer exists. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 10, pp. 1399–1409, October, 2007.     

On the <Emphasis Type="Italic">C</Emphasis>*-valued triangle equality and inequality in Hilbert <Emphasis Type="Italic">C</Emphasis>*-modules

We prove that for two elements x, y in a Hilbert C*-module V over a C*-algebra the C*-valued triangle equality |x + y| = |x| + |y| holds if and only if 〈x, y〉 = |x| |y|. In addition, if has a unit e, then for every x, y ∊ V and every ɛ > 0 there are contractions u, υ ∊ such that |x + y| ≦ u|x|u* + υ|y|υ* + ɛe.      

A formula for the Bloch norm of a C 1-function on the unit ball of ℂ n

For a C ¹-function f on the unit ball ⊂ ℂ ⁿ we define the Bloch norm by , where is the invariant derivative of f, and then show that . Supported by MNZŽS Serbia, Project No. 144010.     

Finite quasihypermetric spaces

Let (X, d) be a compact metric space and let (X) denote the space of all finite signed Borel measures on X. Define I: (X) → ℝ by I(μ) = ∫ _X ∫ _X d(x, y)dμ(x)dμ(y), and set M(X) = sup I(μ), where μ ranges over the collection of measures in (X) of total mass 1. The space (X, d) is quasihypermetric if I(μ) ≦ 0 for all measures μ in (X) of total mass 0 and is strictly quasihypermetric if in addition the equality I(μ) = 0 holds amongst measures μ of mass 0 only for the zero measure. This paper explores the constant M(X) and other geometric aspects of X in the case when the space X is finite, focusing first on the significance of the maximal strictly quasihypermetric subspaces of a given finite quasihypermetric space and second on the class of finite metric spaces which are L ¹-embeddable. While most of the results are for finite spaces, several apply also in the general compact case. The analysis builds upon earlier more general work of the authors [11] [13].      

Some remarks on trigonometric sums

Let