Sequential convergences on <Emphasis Type="Italic">MV</Emphasis>-algebras without Urysohn’s axiom

In a previous author’s paper, sequential convergences on an MV-algebra have been studied; the Urysohn’s axiom was assumed to be valid. The system of all such convergences was denoted by Conv . In the present paper we investigate analogous questions without supposing the validity of the Urysohn’s axiom; the corresponding system of convergences is denoted by conv . Both Conv and conv are partially ordered by the set-theoretical inclusion. We deal with the properties of conv 289-6 and the relations between conv and Conv . We prove that each interval of conv is a distributive lattice. The system conv has the least element, but it does not possess any atom. Hence it is either a singleton set or it is infinite. We consider also the relations between conv and conv G, where (G, u) is a unital lattice-ordered group with = Γ (G, u). This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence — Physics of Information, Grant 1/2/2005.     

<Emphasis Type="Italic">K</Emphasis>-Radical classes and product radical classes of <Emphasis Type="Italic">MV</Emphasis>-algebras

For an MV-algebra let J ₀( ) be the system of all closed ideals of ; this system is partially ordered by the set-theoretical inclusion. A radical class X of MV-algebras will be called a K-radical class iff, whenever ∈ X and is an MV-algebra with J ₀( ) ≅ J ₀( ), then ∈ X. An analogous notation for lattice ordered groups was introduced and studied by Conrad. In the present paper we show that there is a one-to-one correspondence between K-radical classes of MV-algebras and K-radical classes of abelian lattice ordered groups. We also prove an analogous result for product radical classes of MV-algebras; product radical classes of lattice ordered groups were studied by Ton. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information, Grant I/2/2005.     

The commutant of analytic Toeplitz operators on Bergman space

In this paper, using the matrix skills and operator theory techniques we characterize the commutant of analytic Toeplitz operators on Bergman space. For f（z） = z^ng（z） （n ≥1）, g（z） = b0 ＋ b1z^p1 ＋b2z^p2 ＋.. , bk ≠ 0 （k = 0, 1, 2,...）, our main result is =A′（Mf） = A′（Mzn）∩A′（Mg） = A′（Mz^s）, where s = g.c.d.（n,p1,p2,...）. In the last section, we study the relation between strongly irreducible curve and the winding number W（f,f（α））, α ∈ D.     

Estimates for multilinear singular integral operators with nonsmooth kernels

Let (, d, μ) be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, we consider the behavior on × ⋯ × for the m-linear singular integral operators with nonsmooth kernels which were first introduced by Duong, Grafakos and Yan.     

On quantum cluster algebras of finite type

We extend the definition of a quantum analogue of the Caldero-Chapoton map defined by D. Rupel. When Q is a quiver of finite type, we prove that the algebra (Q) generated by all cluster characters is exactly the quantum cluster algebra (Q).     

Direct product decompositions of bounded commutative residuated ℓ-monoids

The notion of bounded commutative residuated ℓ-monoid (BCR ℓ-monoid, in short) generalizes both the notions of MV-algebra and of BL-algebra. Let be a BCR ℓ-monoid; we denote by ℓ( ) the underlying lattice of . In the present paper we show that each direct product decomposition of ℓ( ) determines a direct product decomposition of . This yields that any two direct product decompositions of have isomorphic refinements. We consider also the relations between direct product decompositions of and states on . This work was supported by Slovak Research and Development Agency under the contract No APVV-0071-06. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence-Physics of Information (grant I/2/2005).     

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.     

A superalgebraic interpretation of the quantization maps of Weil algebras

Let G be a Lie group whose Lie algebra g is quadratic. In the paper ＂the non-commutative Weil algebra＂, Alekseev and Meinrenken constructed an explicit G-differential space homomorphism ￡, called the quantization map, between the Well algebra Wg = S（g^＊） χ∧A（g^＊） and Wg= U（g） χ Cl（g） （which they call the noncommutative Weil algebra） for g. They showed that ￡ induces an algebra isomorphism between the basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg）. In this paper, we will interpret the quantization map .~ as the super Duflo map between the symmetric algebra S（Tg[1]） and the universal enveloping algebra U（Tg[1]） of a super Lie algebra T9[1] which is canonically associated with the quadratic Lie algebra g. The basic cohomology rings Hbas^＊（Wg） and Hbas^＊（Wg） correspond exactly to S（Tg[1]）^inv and U（Tg[1]）^inv, respectively. So what they proved is equivalent to the fact that the super Duflo map commutes with the adjoint action of the super Lie algebra, and that the super Duflo map is an algebra homomorphism when restricted to the space of invariants.     

A distribution space for Hilbert transform and its applications

In this paper, a new distribution space is constructed and the definition of the classical Hilbert transform is extended to it. It is shown that is the biggest subspace of on which the extended Hilbert transform is a homeomorphism and both the classical Hilbert transform for L ^p functions and the circular Hilbert transform for periodic functions are special cases of the extension. Some characterizations of the space are given and a class of useful nonlinear phase signals is shown to be in . Finally, the applications of the extended Hilbert transform are discussed. This work was supported by the National Natural Science Foundation of China (Grant Nos. 60475042, 10631080)     

Cohomology of a flag variety as a Bethe algebra

We interpret the equivariant cohomology H_GLn ^*H_{GL_n }^* (ℱ _λ ,ℂ) of a partial flag variety ℱ _λ parametrizing chains of subspaces 0 = F ₀ ⊂ F ₁ ⊂ … ⊂ F _N = ℂ ⁿ , dimF _i /F _i−1 = λ _i , as the Bethe algebra of the -weight subspace of a [t]-module .     

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.     

Nontrivial Solutions of Superquadratic Hamiltonian systems with Lagrangian Boundary Conditions and the L-index Theory

In this paper, the authors study the existence of nontrivial solutions for the Hamiltonian systems z（t） = J△↓H（t, z（t）） with Lagrangian boundary conditions, where ^H（t,z）=1/2（^B（t）z, z） ＋ ^H（t, z）,^B（t） is a semipositive symmetric continuous matrix and ^H（t, z） = satisfies a superquadratic condition at infinity. We also obtain a result about the L-index.     

Adjoint transform of Willmore surfaces in

Using moving frame method, we study the Möbius geometry of a pair of conformally immersed surfaces in . Two new invariants θ and ρ associated with them arise naturally as well as the notion of touch and co-touch. As an application, adjoint transform is defined for any Willmore surface in . It always exists locally, hence generalizes known duality theorems of Willmore surfaces. Finally we characterize a pair of adjoint Willmore surfaces in terms of harmonic map.     

Perturbed Basins of Attraction

Let F be an automorphism of which has an attracting fixed point. It is well known that the basin of attraction is biholomorphically equivalent to . We will show that the basin of attraction of a sequence of automorphisms f ₁, f ₂, . . . is also biholomorphic to if every f _n is a small perturbation of the original map F.     

A note on the schemes of replacement and collection

We derive the schemes of from certain weak forms of the same.     

Ends of graphed equivalence relations,II

Given a graphing of a countable Borel equivalence relation on a Polish space, we show that if there is a Borel way of selecting a non-empty closed set of countably many ends from each -component, then there is a Borel way of selecting an end or line from each -component. Our method yields also Glimm-Effros style dichotomies which characterize the circumstances under which: (1) there is a Borel way of selecting a point or end from each -component; and (2) there is a Borel way of selecting a point, end or line from each -component. The first author was supported in part by NSF Grant DMS-0140503. The second author was supported in part by NSF VIGRE Grant DMS-0502315.     

On <Emphasis Type="Italic">S</Emphasis>-quasinormal and <Emphasis Type="Italic">c</Emphasis>-normal subgroups of a finite group

Let be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are presented: (1) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of H is either c-normal or S-quasinormally embedded in G. (2) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every maximal subgroup of all Sylow subgroups of F*(H), the generalized Fitting subgroup of H, is either c-normal or S-quasinormally embedded in G. (3) G ∈ if and only if there is a normal subgroup H such that G/H ∈ and every cyclic subgroup of F*(H) of prime order or order 4 is either c-normal or S-quasinormally embedded in G. Supported by the Natural Science Foundation of China and the Natural Science Foundation of Guangxi Autonomous Region (No. 0249001). Corresponding author. Supported in part by the Natural Science Foundation of China (10571181), NSF of Guangdong Province (06023728) and ARF(GDEI).     

∞-jets of diffeomorphisms preserving orbits of vector fields

Let F be a C ^∞ vector field defined near the origin O ∈ ℝ ⁿ , F(O) = 0, and (F _t ) be its local flow. Denote by the set of germs of orbit preserving diffeomorphisms h: ℝ ⁿ → ℝ ⁿ at O, and let , (r ≥ 0), be the identity component of with respect to the weak Whitney W ^r topology. Then contains a subset consisting of maps of the form F _α(x)(x), where α: ℝ ⁿ → ℝ runs over the space of all smooth germs at O. It was proved earlier by the author that if F is a linear vector field, then = . In this paper we present a class of examples of vector fields with degenerate singularities at O for which formally coincides with , i.e. on the level of ∞-jets at O. We also establish parameter rigidity of linear vector fields and “reduced” Hamiltonian vector fields of real homogeneous polynomials in two variables.      

Projectability and weak homogeneity of pseudo effect algebras

In this paper we deal with a pseudo effect algebra possessing a certain interpolation property. According to a result of Dvurečenskij and Vettterlein, can be represented as an interval of a unital partially ordered group G. We prove that is projectable (strongly projectable) if and only if G is projectable (strongly projectable). An analogous result concerning weak homogeneity of and of G is shown to be valid. This work has been partially supported by the Slovak Academy of Sciences via the project Center of Excellence—Physics of Information (grant I/2/2005).     

A note on the degree conjecture for the Selberg class

The degree conjecture for the Selberg class of L-functions states that the degree d _F of every F ∈ is an integer. Moreover, it is expected that every F ∈ has polynomial Euler product, and that the degree ∂ _F of such an Euler product coincides with d _F . In this note we prove that a suitable continuity assumption on the degree d _F implies that ∂ _F = d _F for all F ∈ with polynomial Euler product.