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.     

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 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.      

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).     

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.     

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.     

Using as a Semantic Markup Format

One of the great problems of Mathematical Knowledge Management (MKM) systems is to obtain access to a sufficiently large corpus of mathematical knowledge to allow the management/search/navigation techniques developed by the community to display their strength. Such systems usually expect the mathematical knowledge they operate on in the form of semantically enhanced documents, but mathematicians and publishers in Mathematics have heavily invested into the format and workflow. We analyze the current practice of semi-semantic markup in documents and extend it by a markup infrastructure that allows to embed semantic annotations into documents without changing their visual appearance. This collection of macro packages is called (semantic ) as it allows to markup documents semantically without leaving the time-tried workflow, essentially turning into an MKM format. At the heart of is a definition mechanism for semantic macros for mathematical objects and a non-standard scoping construct for them, which is oriented at the semantic dependency relation rather than the document structure. We evaluate the macro collection on a large case study: the course materials of a two-semester course in Computer Science was annotated semantically and converted to the OMDoc MKM format by Bruce Miller’s LaTeXML system.      

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.     

Nielsen equivalence classes of free abelianized extensions of groups

Let be a presentation of a group G. We obtain information about the Nielsen equivalence classes of F _n /R′ in terms of the group K ₁(ℤG) that arises in algebraic K-theory. We deduce some results about Nielsen equivalence classes and T-systems of polycyclic groups.     

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.     

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.     

∞-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.      

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.     

Diameter preserving surjection on alternate matrices

Let F be a field with ｜F｜ ≥ 3, Km be the set of all m × m （m ≥ 4） alternate matrices over F. The arithmetic distance of A, B ∈ Km is d（A, B） ：= rank（A - B）. If d（A, B） = 2, then A and B are said to be adjacent. The diameter of Km is max{d（A, B） ： A, B ∈ km}. Assume that φ ： Km→Km is a map. We prove the following are equivalent： （a） φ is a diameter preserving surjection in both directions, （b） φ is both an adjacency preserving surjection and a diameter preserving map, （c） φ is a bijective map which preserves the arithmetic distance.     

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).     

On stably -monotone Banach couples

The $ \vec E $ \vec E = (E, E(2^-k )) is a stably $ \vec E $ \vec E is $ \vec E $ \vec E is $ \vec E $ \vec E is -monotone, then either E = l _p (1 ≤ p < ∞) or E = c ₀.     

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).     

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.     

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.     

Systems of correlation functions, coinvariants, and the Verlinde Algebra

We study the Gaberdiel-Goddard spaces of systems of correlation functions attached to affine Kac-Moody Lie algebras