Hamiltonian type Lie bialgebras

We first prove that,for any generalized Hamiltonian type Lie algebra H,the first co- homology group H~1(H,H(?)H) is trivial.We then show that all Lie bialgebra structures on H are triangular.     

Weighted weak type (1,1) estimate for the Christ-Journé type commutator

Let K be the Calderón-Zygmund convolution kernel on R~d(d≥2).Christ and Journé defined the commutator associated with K and a∈L~∞(R~d)by T_af(x)=p.v.∫_(R~d)K(x-y)m_x,y~a·f(y)dy,which is an extension of the classical Calderón commutator. In this paper, we show that T_a is weighted weak type(1,1) bounded with A,1 weight for d≥2.     

Dennis–Vaserstein type decompositions

A generalization of the Dennis–Vaserstein decomposition is proved for an arbitrary pair of maximal parabolic subgroups P _r and P _s in the general linear group GL(n, R), provided that r − s ≥ sr (R). The usual Dennis–Vaserstein decomposition is the special case where r = n − 1, s =1. Bibliography: 23 titles.     

Isoperimetric type problems and Alexandrov–Fenchel type inequalities in the hyperbolic space

In this paper, we solve various isoperimetric problems for the quermassintegrals and the curvature integrals in the hyperbolic space Hⁿ${\mathbb{H}}^n$, by using quermassintegral preserving curvature flows. As a byproduct, we obtain hyperbolic Alexandrov–Fenchel inequalities.     

Multipliers and Herz type spaces

Hrmander condition for boundedness of multiplier operators will be replaced by a weaker condition described by certain weighted or non-weighted Herz spaces. Some results on boundedness of multiplier operators are then established. As direct corollaries of main theorems in this paper, several celebrated results on boundedness of multiplier operators will be improved or deduced.     

On groups of type Φ

We define a group G to be of type Φ if it has the property that for every -module G, proj. G < ∞ iff proj. H G < ∞ for every finite subgroup H of G. We conjecture that the type Φ is an algebraic characterization of those groups G which admit a finite dimensional model for , the classifying space for the family of the finite subgroups of G. We also conjecture that the type Φ is equivalent to spli being finite, where spli is the supremum of the projective lengths of the injective -modules. Here we prove certain parts of these conjectures. The project is cofounded by the European Social Fund and National Resources–EPEAK II–Pythagoras. Received: 21 June 2006     

Hölder type quasicontinuity

It is proved that a functionuL ^m,p (R ⁿ ) (which coincides with the Sobolev spaceW ^1,p (R ⁿ ) ifm=1) coincides with a Hölder continuous functionw outside a set of smallm,q-capacity, whereq<p. Moreover, ifm=1, then the functionw can be chosen to be close tou in theW ^1,p -norm.     

On weighted Ostrowski type,Trapezoid type,Grüss type and Ostrowski–Grüss like inequalities on time scales

In this paper, we first derive a weighted Montgomery identity on time scales and then establish weighted Ostrowski-type, Trapezoid-type, Grüss-type and Ostrowski–Grüss-like inequalities on time scales, respectively. These results not only provide a generalization of the known results, but also give some other interesting inequalities on time scales as special cases.     

Two Turán type inequalities

Turán’s book [2], in Section 19.4, refers to the following result of Gábor Halász. Let a ₀, a ₁, ..., a _n−1 be complex numbers such that the roots α ₁, ⋯, α _n of the polynomial x ⁿ + a _n−1 x ⁿ⁻¹ + ⋯ + a ₁ x + a ₀ satisfy min _j Re α _j ≧ 0 and let function Y(t) be a solution of the linear differential equation Y ⁽ⁿ⁾ + a _n−1 Y ⁽ⁿ⁻¹⁾ + ⋯ + a ₁ Y′ + a ₀ Y = 0. Then

Lie bialgebras of generalized Witt type

In this paper, all Lie bialgebra structures on the Lie algebras of generalized Witt type are considered. It is proved that, for any Lie algebra W of generalized Witt type, all Lie bialgebras on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H1(W, W(?)W) is trivial.     

Theorems of Denjoy–Wolff type

Using the Kobayashi distance, we first establish a version of the Denjoy–Wolff theorem for a bounded and strictly convex domain in ${{\mathbb{C}}^k}$ . Next, we prove analogous results for semigroups of holomorphic mappings and the resolvents of their generators. Finally, we obtain theorems of Denjoy–Wolff type for families of holomorphic retracts of the open unit ball in a complex, reflexive, and strictly convex Banach space.     

Stein–type shrinkage quantile estimation

The simultaneous asymptotic estimation theory of quantiles is considered for an arbitrary population. The Stein–type estimator and its positive version are considered. The relative merits of the proposed estimators are compared with those of the usual estimator using asymptotic quadratic distributional risk those of the usual estimator using asymptotic quadratic distributional risk under local alternatives. It is shown that both proposed estimators are asymptotically superior to the classical estimator. Further, it is demonstrated that the Stein-type estimator is dominated by its positive part     

New Erdös-Kac type theorems

Assuming a quasi Generalized Riemann Hypothesis (quasi-GRH for short) for Dedekind zeta functions over Kummer fields of the type we prove the following prime analogue of a conjecture of Erd?s & Pomerance (1985) concerning the exponent function f_a(p) (defined to be the minimal exponent e for which a^e ≡ 1 modulo p):

Generalized Baskakov–Durrmeyer type operators

In the present paper, we study some approximation properties of the Durrmeyer type modification of generalized Baskakov operators introduced by Erencin (Appl Math Comput 218(3):4384–4390, 2011). First, we establish a Lorentz-type lemma for the derivatives of the kernel of the generalized Baskakov operators and then obtain a recurrence relation for the moments of their Durrmeyer type modification. Next, we discuss some direct results in simultaneous approximation by these operators e.g. pointwise convergence theorem, Voronovskaja-type theorem and an estimate of error in terms of the modulus of continuity. Finally, we estimate the error in the approximation of functions having derivatives of bounded variation.     

Poincaré series of Drinfeld type

Let K be the rational function field $\mathbb{F}_q (t)$ . We construct Poincaré series on the Bruhat-Tits tree of GL₂ over K _∞ and show that they generate the space of automorphic cusp forms of Drinfeld type.     

Polynomial-time Martin-Löf type theory

Summary Fragments of extensional Martin-Löf type theory without universes,ML ₀, are introduced that conservatively extend S.A. Cook and A. Urquhart'sIPV . A model for these restricted theories is obtained by interpretation in Feferman's theory APP of operators, a natural model of which is the class of partial recursive functions. In conclusion, some examples in group theory are considered.