On the limit-classifications of even and odd-order formally symmetric differential expressions

In this paper we consider the formally symmetric differential expressionM [.] of any order (odd or even) ≥ 2. We characterise the dimension of the quotient spaceD(T _max)/D(T _min) associated withM[.] in terms of the behaviour of the determinants det [[f _rg_s](∞)] where 1 ≤n ≤ (order of the expression +1); here [fg](∞) = lim [fg](x), where [fg](x) is the sesquilinear form in f andg associated withM. These results generalise the well-known theorem thatM is in the limit-point case at ∞ if and only if [fg](∞) = 0 for everyf, g ε the maximal domain Δ associated withM.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

Continuous functions taking every value a given number of times

We give necessary and sufficient conditions for a function f: [0, 1] → {1,2,...,w, c} under which there exists a continuous function F: [0, 1] → [0, 1] such that for every y ɛ [0, 1], |F ⁻¹ (y)| = f(y).      

<Emphasis Type="Italic">Φ</Emphasis>-Variation of a bifractional Brownian motion

Let B _H,K = {B _H,K (t)} _t⩾0 be a bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1]. For a function Φ: [0, ∞) → [0, ∞) and for a partition κ = {t _i }_n ⁱ⁼⁰ of an interval [0, T] with T > 0, let {ie418-01}. We prove that, for a suitable Φ depending on H and K, {ie418-02} almost surely. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-16/08     

E Φ andh Φ fail to beM-ideals inL Φ andl Φ in the case of the Orlicz norm

It is well known that in the case of the Luxemburg normE ^Φ (resp.h ^Φ) is anM ideal inL ^Φ (resp.l ^Φ), see [1], [9], [15] and [6]; [17] and [18]. It is proved in this paper that in the case of the Orlicz normE ^Φ (resph ^Φ) is anM-ideal inL ^Φ (resp.l ^Φ) iff Φ satisfies the suitable Δ₂ or Φ^*(a(Φ^*)), where a(Φ^*) is linear on the interval [0,u]} and Φ^* denotes the function complementary to Φ in the sense of Young. It is also proved that any linear continuous regular (i.e. order continuous) functional ξ_υ overE ^Φ (resp.h ^Φ) generated byv∈ L(h ^Φ*) (resp.v∈ L(h ^Φ*)) which attains its norm on the unit sphereS(E ^Φ) (resp.S(h ^Φ)), has a unique norm-preserving extension toL ^Φ (resp.l ^Φ). Finally, it is proved thatL ^Φ (resp.l ^Φ) has the property that any linear continuous regular functional ξ_υ overE ^Φ (resp.h ^Φ) has a unique norm-preserving extension toL ^Φ (resp.l ^Φ) iff Φ orE ^Φ satisfies the suitable Δ₂ and in the second case Φ^* attains the value 1.     

A Theorem of Liouville Type for p-Harmonic Morphisms

In this article we prove a Liouville type theorem for p-harmonic morphisms. We show that if : MNis a p-harmonic morphism (p2) from a complete noncompact Riemannian manifold Mof nonnegative Ricci curvature into a Riemannian manifold Nof nonpositive scalar curvature such that the p-energy E _p (), or (2p–2)-energy E _2p–2() is finite, then is constant.     

Nearly flat almost monotone measures are big pieces of lipschitz graphs

A concentrated (ξ, m) almost monotone measure inR ⁿ is a Radon measure Φ satisfying the two following conditions: (1) Θ ^m (Φ,x)≥1 for every x ∈spt (Φ) and (2) for everyx ∈R ⁿ the ratioexp [ξ(r)]r^−mΦ(B(x,r)) is increasing as a function of r>0. Here ξ is an increasing function such thatlim _r→0-ξ(r)=0. We prove that there is a relatively open dense setReg (Φ) ∋spt (Φ) such that at each x∈Reg(Φ) the support of Φ has the following regularity property: given ε>0 and λ>0 there is an m dimensional spaceW ⊂R ⁿ and a λ-Lipschitz function f from x+W into x+W^‖ so that (100-ε)% ofspt(Φ) ∩B (x, r) coincides with the graph of f, at some scale r>0 depending on x, ε, and λ.     

Generalizations of coatomic modules

For a ring R and a right R-module M, a submodule N of M is said to be δ-small in M if, whenever N+X=M with M/X singular, we have X=M. Let ℘ be the class of all singular simple modules. Then δ(M)=Σ{ L≤ M| L is a δ-small submodule of M} = Re _jm(℘)=∩{ N⊂ M: M/N∈℘. We call M δ-coatomic module whenever N≤ M and M/N=δ(M/N) then M/N=0. And R is called right (left) δ-coatomic ring if the right (left) R-module R _R(_RR) is δ-coatomic. In this note, we study δ-coatomic modules and ring. We prove M=⊕ _i=1 ⁿ M_i is δ-coatomic if and only if each M _i (i=1,…, n) is δ-coatomic.     

Non-Zero Degree Maps Between <Emphasis Type="Italic">2n</Emphasis>-Manifolds

Abstract Thom–Pontrjagin constructions are used to give a computable necessary and sufficient condition for a homomorphism ϕ : H ⁿ (L;Z) → H ⁿ (M;Z) to be realized by a map f : M → L of degree k for closed (n − 1)-connected 2n-manifolds M and L, n > 1. A corollary is that each (n − 1)-connected 2n-manifold admits selfmaps of degree larger than 1, n > 1. In the most interesting case of dimension 4, with the additional surgery arguments we give a necessary and sufficient condition for the existence of a degree k map from a closed orientable 4-manifold M to a closed simply connected 4-manifold L in terms of their intersection forms; in particular, there is a map f : M → L of degree 1 if and only if the intersection form of L is isomorphic to a direct summand of that of M. Both authors are supported by MSTC, NSFC. The comments of F. Ding, J. Z. Pan, Y. Su and the referee enhance the quality of the paper     

Regenerative embedding of Markov sets

Summary.  Given a closed Markov (i.e. regenerative) set in [0,∞), we characterize the laws of the Markov sets which are regeneratively embedded into the latter. Typically, let Φ⁽¹⁾ and Φ⁽²⁾ be two Laplace exponents corresponding to two regenerative laws, and M ⁽²⁾ a Markov set with exponent Φ⁽²⁾. There exists a Markov set M ⁽¹⁾ with exponent Φ⁽¹⁾ which is regeneratively embedded into M ⁽²⁾ if and only if Φ⁽¹⁾/Φ⁽²⁾ is a completely monotone function. Several examples and applications are discussed. Received: 12 April 1996 / In revised form: 12 March 1997     

Radon-Nikodym derivatives of finitely additive interval measures taking values in a Banach space with basis

Let X be a Banach space with a Schauder basis { en }, and let Φ( I ) = Σ∞ n=1 en∫I fn(t)dt be a finitely additive interval measure on the unit interval [0, 1], where the integrals are taken in the sense of Henstock-Kurzweil. Necessary and sufficient conditions are given for Φ to be the indefinite integral of a Henstock-Kurzweil-Pettis (or Henstock, or variational Henstock) integrable function f : [0, 1] → X .     

A note about the cyclotomic polynomial Фpq（x） and some related results

Abstract. The expression of cyclotomic polynomial Фpq （x） is concerned for a long time. A simple and explicit expression of Фpq （x） in Z[x] has been showed. The form of the factors of Фpq （x） over F2 and the upper, lower bounds of their Hamming weight are provided.     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.     

On football manifolds of E. Molnár

A closed 3-manifold M is said to be hyperelliptic if it has an involution τ such that the quotient space of M by the action of τ is homeomorphic to the standard 3-sphere. We show that the hyperbolic football manifolds of Emil Molnár [12] are hyperelliptic. Then we determine the isometry groups of such manifolds. Another consequence is that the unique hyperbolic dodecahedral and icosahedral 3-space forms with first homology group ℤ₃₅ (constructed by I. Prok in [16], on the basis of a principal algorithm due to Emil Molnár [13], and by Richardson and Rubinstein in [18]) are also hyperelliptic.     

Uniformly non-l n (1) Musielak-Orlicz sequence spaces

We give a necessary and sufficient condition for the uniformly non-l _n ⁽¹⁾ property of Musielak-Orlicz sequence spacesl ^Φ generated by a sequence Φ=(ϕ_n:n⩾l) of finite Orlicz functions such that for eachn∈ℕ. As a result, forn ₀⩾2, there exist spacesl ^Φ which are only uniformly non-l _n ⁽¹⁾ forn⩾n ₀. Moreover we obtain a characterization of uniformly non-l _n ⁽¹⁾ and reflexive Orlicz sequence spaces over a wide class of purely atomic measures and of uniformly non-l _n ⁽¹⁾ Nakano sequence spaces. This extends a result of Luxemburg in [19]. Submitted in memory of Professor W. Orlicz     

The generalized Boardman homomorphisms

This paper provides universal upper bounds for the exponent of the kernel and of the cokernel of the classical Boardman homomorphism b ⁿ : π ⁿ (X)→H ⁿ (H;ℤ), from the cohomotopy groups to the ordinary integral cohomology groups of a spectrum X, and of its various generalizations π ⁿ (X)→E ⁿ (X), F ⁿ (X)→(E∧F) ⁿ (X), F ⁿ (X)→H ⁿ (X;π ₀ F) and F ⁿ (X)→H ^n+t (X;π _t F) for other cohomology theories E ^*(−) and F ^*(−). These upper bounds do not depend on X and are given in terms of the exponents of the stable homotopy groups of spheres and, for the last three homomorphisms, in terms of the order of the Postnikov invariants of the spectrum F.     

Distribution results for lattices in SL(2,ℚ p )

In this paper we study the ergodic properties of the linear action of lattices Γ of SL(2,ℚ_p) on ℚ_p × ℚ_p and distribution results for orbits of Γ. Following Serre, one can define a “geodesic flow” for an associated tree (actually associated to GL(2,ℚ_p)). The approach we use is based on an extension of this approach to “frame flows” which are a natural compact group extension of the geodesic flow.     

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

Let V be a closed surface, H⊑π₁(V) a subgroup of finite index l and D=[A ₁,...,A _m ] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A _i ∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11]. The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].     

A Note on Edgeworth Expansions for <Emphasis Type="Italic">U</Emphasis>-Statistics under Minimal Conditions

A one-term Edgeworth expansion for U-statistics with kernel h(x, y) was derived by Jing and Wang [3] under optimal moment conditions. In this note, we show that one of the optimal moment conditions E| h(X ₁, X ₂|^5/3 < ∞ can be weakened to lim _t→∞ t ^5/3 P(|h(X ₁, X ₂)| > t) → 0. Printed in Lietuvos Matematikos Rinkinys, Vol. 45, No. 3, pp. 453–440, July–September, 2005.