<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.     

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 .     

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.     

Toward the theory of ring Q-homeomophisms

We investigate classes of the so-called ring Q-homeomorphisms including, in particular, Q-homeomorphisms, various classes of homeomorphisms with finite length distortion, Sobolev’s classes etc. In terms of the majorant Q(x), we give a series of criteria for normality based on estimates of the distortion of the spherical distance under ring Q-homeomorphisms. In particular, it is shown that the class of all ring Q-homeomorphisms f of a domain D ⊂ ℝ ⁿ into , n ≥ 2, with , forms a normal family, if Q(x) has finite mean oscillation in D. We also prove normality of , for instance, if Q(x) has singularities of logarithmic type whose degrees are not greater than n − 1 at every point x ∈ D. The results are applicable, in particular, to mappings with finite length distortion and Sobolev’s classes.     

Highest weight representations of a family of Lie algebras of Block type

For an additive subgroup G of a field F of characteristic zero, a Lie algebra B（G） of Block type is defined with basis {Lα,i｜ α∈G, i∈Z＋} and relations [Lα,i, Lβ,j] = （β-α）Lα＋β,i＋j＋（αj-βi）Lα＋β,Lα＋β,i＋j-1.It is proved that an irreducible highest weight B（Z）-module is quasifinite if and only if it is a proper quotient of a Verma module. Furthermore, for a total order λ on G and any ∧∈B（G）0^＊（the dual space of B（G）0 = span{L0,i｜i∈Z＋}）, a Verma B（G）-module M（∧,λ） is defined, and the irreducibility of M（A,λ） is completely determined.     

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).     

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 <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).     

Second cohomology group of extended W-algebras

Let \mathbbF\mathbb{F} be a field of characteristic 0, and let G be an additive subgroup of \mathbbF\mathbb{F}. We define a class of infinite-dimensional Lie algebras \mathbbF\mathbb{F}-basis {L _μ, V _μ, W _μ | μ ∈ G}, which are very closely related to W-algebras. In this paper, the second cohomology group of is determined.     

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.     

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).     

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.      

∞-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.      

A quasianalyticity property for monogenic solutions of small divisor problems

We discuss the quasianalytic properties of various spaces of functions suit-able for one-dimensional small divisor problems. These spaces are formed of functions ¹-holomorphic on certain compact sets K _j of the Riemann sphere (in the Whitney sense), as is the solution of a linear or non-linear small divisor problem when viewed as a function of the multiplier (the intersection of K _j with the unit circle is defined by a Diophantine-type condition, so as to avoid the divergence caused by roots of unity). It turns out that a kind of generalized analytic continuation through the unit circle is possible under suitable conditions on the K _j ’s.     

Cotorsion pair extensions

Assume that S is an almost excellent extension of R. Using functors HomR（S,-） and -×R S, we establish some connections between classes of modules lR and lS, cotorsion pairs （AR, BR） and （AS, BS）. If lS is a T-extension or （and） H-extension of lR, we show that lS is a （resp., monomorphic, epimorphic, special） preenveloping class if and only if so is lR. If （AS, BS） is a TH- extension of （AR, BR）, we obtain that （AS, BS） is complete （resp., of finite type, of cofinite type, hereditary, perfect, n-tilting） if and only if so is （AR, BR）.     

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.     

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).     

On modules with <Emphasis Type="Italic">d</Emphasis>-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M ∈ W H J^d（A） if and only if M admits a filtration of submodules： 0 belong to U0 belong to U1 belong to ... belong to Up = M such that all Ui/Ui-1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W H J^d（A） in a special case. Let M ∈ W H J^d（A）. It is proved that the Koszul dual E（M） is Noetherian, Hopfian, of finite dimension in special cases, and E（M） ∈ gr0（E（A））. In particular, we show that M ∈ W H J^d（A） if and only if E（G（M）） ∈ gr0（E（A））, where G is the associated graded functor.     

Construct weak Hopf algebras by using Borcherds matrix

We define a new kind quantized enveloping algebra of a generalized Kac-Moody algebra by adding a new generator J satisfying jm = j for some integer m. We denote this algebra by wUqT（A）. This algebra is a weak Hopf algebra if and only if m = 2,3. In general, it is a bialgebra, and contains a Hopf subalgebra. This Hopf subalgebra is isomorphic to the usual quantum envelope algebra Uq （A） of a generalized Kac-Moody algebra A.     

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.