Cartesian Closed Extensions of Subcategories of CONT

In this paper we give a uniform way of proving cartesian closedness for many new subcategories of continuous posets. We define C-P to be the category of continuous posets whose D–completions are isomorphic to objects from C, where C is a subcategory of the category CONT of domains. The main result is that if C is a cartesian closed full subcategory of ALG _⊥ or BC, then C-P is also a cartesian closed subcategory of the category CONTP of continuous posets and Scott continuous functions. In particular, we have the following cartesian closed categories : BC-P, LAT-P, aL-P, aBC-P, B-P, aLAT-P, ω -B-P, ω -aLAT-P, etc.     

Grade zero part of forced graded algebras

This paper concerns a certain subcategory of the category of representations for a semisimple algebraic group G in characteristic p, which arises from the semisimple modules for the corresponding quantum group at a p-th root of unity. The subcategory, thus, records the cohomological difference between quantum groups and algebraic groups. We define translation functors in this category and use them to obtain information on the irreducible characters for G when the Lusztig character formula does not hold.     

Frobenius morphisms and stable module categories of repetitive algebras

Let k be the algebraic closure of a finite field F_q and A be a finite dimensional k-algebra with a Frobenius morphism F.In the present paper we establish a relation between the stable module category of the repetitive algebra  of A and that of the repetitive algebra of the fixed-point algebra A~F.As an application,it is shown that the derived category of A~F is equivalent to the subcategory of F-stable objects in the derived category of A when A has a finite global dimension.     

Green’s relations and regularity for semigroups of transformations that preserve reverse direction equivalence

Let T _X denote the full transformation semigroup on a set X. For an equivalence E on X, let

$T_{\exists}(X)=\{\alpha\in T_X:\forall x,y\in X,(x\alpha,y\alpha)\in E\Rightarrow(x,y)\in E\}.$

Then T _?(X) is exactly the semigroup of mappings on the topological space X for which the collection of all E-classes is a basis. In this paper, we discuss regularity of elements and Green’s relations for T _?(X).

     

Differences of Idempotents In <Emphasis Type="Italic">C</Emphasis>*-Algebras and the Quantum Hall Effect

Let ? be a trace on the unital C*-algebra A and M _? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,Q ∈ A are idempotents and P ? Q ∈ M _? , then ?((P ? Q)²ⁿ⁺¹) = ?(P ? Q) ∈ R for all n ∈ N. Let the isometries U ∈ A and A = A*∈ A be such that I+A is invertible and U-A ∈ M _? with ?(U-A) ∈ R. Then I-A, I?U ∈ M _? and ?(I?U) ∈ R. Let n ∈ N, dimH = 2n + 1, the symmetry operators U, V ∈ B(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.     

An algorithm to compute <Emphasis Type="Italic">ω</Emphasis>-primality in a numerical monoid

Let M be a commutative, cancellative, atomic monoid and x a nonunit in M. We define ω(x)=n if n is the smallest positive integer with the property that whenever x∣a ₁???a _t , where each a _i is an atom, there is a T?{1,2,…,t} with |T|≤n such that x∣∏_k∈T a _k . The ω-function measures how far x is from being prime in M. In this paper, we give an algorithm for computing ω(x) in any numerical monoid. Simple formulas for ω(x) are given for numerical monoids of the form 〈n,n+1,…,2n?1〉, where n≥3, and 〈n,n+1,…,2n?2〉, where n≥4. The paper then focuses on the special case of 2-generator numerical monoids. We give a formula for computing ω(x) in this case and also necessary and sufficient conditions for determining when x is an atom. Finally, we analyze the asymptotic behavior of ω(x) by computing \(\lim_{x\rightarrow \infty}\frac{\omega(x)}{x}\).     

Moduli of Continuity of Harmonic Quasiregular Mappings in $\mathbb{B}^n$

We prove that ω _u (δ)?≤?Cω _f (δ), where \(u : \overline{\mathbb{B}^n} \rightarrow \mathbb{R}^n\) is the harmonic extension of a continuous map \(f:\mathbb{S}^{n-1}\rightarrow\mathbb{R}^n\), if u is a K-quasiregular map. Here C is a constant depending only on n, ω _f and K and ω _h denotes the modulus of continuity of h.     

Hamiltonian stability of Lagrangian tori in toric Kähler manifolds

Let (M,J,ω) be a compact toric Kähler manifold of dim_? M=n and L a regular orbit of the T ⁿ-action on M. In the present paper, we investigate Hamiltonian stability of L, which was introduced by Y.-G. Oh (Invent. Math. 101, 501–519 (1990); Math. Z. 212, 175–192) (1993)). As a result, we prove any regular orbit is Hamiltonian stable when (M,ω)=??ⁿ,ω_FS) and (M,ω)=??ⁿ ₁× ??ⁿ ₂,aω_FS⊕ bω_FS), where ω_FS is the Fubini–Study Kähler form and a and b are positive constants. Moreover, they are locally Hamiltonian volume minimizing Lagrangian submanifolds.     

ON ALGEBRAIC VOLUME DENSITY PROPERTY

A smooth affine algebraic variety X equipped with an algebraic volume form ω has the algebraic volume density property (AVDP) if the Lie algebra generated by complete algebraic vector fields of ω-divergence zero coincides with the space of all algebraic vector fields of ω-divergence zero. We develop an effective criterion of verifying whether a given X has AVDP. As an application of this method we establish AVDP for any homogeneous space X = G/R that admits a G-invariant algebraic volume form where G is a linear algebraic group and R is a closed reductive subgroup of G.     

Real elements in groups of type <Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>

Let G be a group. An element x ∈ G is called real if x is conjugate to x ^?1 in G. In this paper we study the structure of real elements in the compact connected Lie group of type F ₄ and algebraic groups of type F ₄ defined over an arbitrary field.     

A new characterization of symmetric group by NSE

Let G be a group and ω(G) be the set of element orders of G. Let k ∈ ω(G) and m _k (G) be the number of elements of order k in G. Let nse(G) = {m _k (G): k ∈ ω(G)}. Assume r is a prime number and let G be a group such that nse(G) = nse(S _r ), where S _r is the symmetric group of degree r. In this paper we prove that G ? S _r , if r divides the order of G and r ² does not divide it. To get the conclusion we make use of some well-known results on the prime graphs of finite simple groups and their components.     

Ranks of matrices with few distinct entries

It is consistent that P(ω ₁) is the union of less than \({2^{{\aleph _1}}}\) parts such that if A ₀,..., A _n?1, B ₀,..., B _m?1 are distinct elements of the same part, then |A ₀ ∩ · · · ∩ A _n?1 ∩ (ω ₁ ? B ₀) ∩ · · ·∩ (ω ₁ ? B _m?1)| = N₁.     

Time symmetry preserving perturbations of systems,and Poincaré mappings

In the present paper, we obtain necessary and sufficient conditions under which two differential systems have the same symmetries described by a reflecting function. Under these conditions, the systems in question have a common shift operator along solutions of these systems on a symmetric time interval [?ω, ω]. Therefore, the mappings over the period [?ω, ω] coincide for such systems provided that these systems are 2ω-periodic.     

The quasi-triangular structures over Hom-<Emphasis Type="Italic">ω</Emphasis>-smash coproduct Hopf algebras

Let (C, α) and (H, β) be Hom-bialgebras and ω: C ?H → H ?C a linear map. We introduce a Hom-ω-smash coproduct (C _ω ? H, γ) and give necessary and sufficient conditions for (C _ω ? H, γ) to be a Hom-bialgebra. We study the quasi-triangular structures over (C _ω ? H, γ) and show the necessary and sufficient conditions for (C _ω ? H, γ, R) to be a quasi-triangular Hom-Hopf algebra. As applications of our results, we introduce the concept of D(H)* and construct quasi-triangular structures over D(H)*.     

Spectra of Automorphic Extensions of Finite Simple Exceptional Groups of Lie Type

The spectrum ω (G) of a finite group G is the set of orders of elements of G. Let S be a simple exceptional group of type E ₆ or E ₇ . We describe all finite groups G such that S ≤ G ≤ Aut S and ω (G) = ω (S) and completes the study of the recognition-by-spectrum problem for all simple exceptional groups of Lie type.     

Span of a DL-algebra

Unless otherswise specified, all objects are defined over a field k of characteristic 0. Let K be a field. The unessentialness of an extension of the algebra Der K by means of a splittable semisimple Lie algebra is established. Let D _K be the category of differential Lie algebras (DL-algebras) (g;K). In this paper for an extension L/K the functor η:D _K → D _L , defining the tensor product L ? _K of vector spaces and the homomorphism of Lie algebras, is constructed. If the extension L/K is algebraic, then η is unique. The results will be required for strengthening the progress on Gelfand–Kirillov problem and weakened conjecture [1, 2].     

A ωn-Related Family of Homotopy Elements in the Stable Homotopy of Spheres

To determine the stable homotopy groups of spheres π_*(S) is one of the central problems in homotopy theory. Let p be a prime greater than 5. The authors make use of the May spectral sequence and the Adams spectral sequence to prove the existence of a ?_n-related family of homotopy elements, β₁ω_nγ_s, in the stable homotopy groups of spheres, where n > 3, 3 ≤ s < p ? 2 and the ?_n-element was detected by X. Liu.     

A characterization of the simple group PSL<Subscript>5</Subscript>(5) by the set of its element orders

Let G be a finite group and let ω(G) denote the set of the element orders of G. For the simple group PSL₅(5) we prove that if G is a finite group with ω(G) = ω(PSL₅(5)), then either G ? PSL₅(5) or G ? PSL₅(5): 〈θ〉 where θ is a graph automorphism of PSL₅(5) of order 2.     

The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S ₃ onto a genus 2 Riemann surface S ₂ by Hol(S ₃, S ₂). Call two mappings f and g in Hol(S ₃, S ₂) equivalent whenever there exist conformal automorphisms α and β of S ₃ and S ₂ respectively with f ? α = β ? g. It is known that Hol(S ₃, S ₂) always consists of at most two equivalence classes.We obtain the following results: If Hol(S ₃, S ₂) consists of two equivalence classes then both S ₃ and S ₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S ₃, S ₂) there exist anticonformal automorphisms α? and β? with f ? α? = β? ? g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S ₃, S ₂) such that Hol(S ₃, S ₂) consists of two equivalence classes.     

Sets of Absolute Continuity for Harmonic Measure in NTA Domains

We show that if Ω is an NTA domain with harmonic measure ω and E??Ω is contained in an Ahlfors regular set, then \(\omega |_{E}\ll \mathcal {H}^{d}|_{E}\). Moreover, this holds quantitatively in the sense that for all τ>0ω obeys an A_∞-type condition with respect to \(\mathcal {H}^{d}|_{E^{\prime }}\), where E^′?E is so that ω(E?E^′)<τω(E), even though ?Ω may not even be locally \(\mathcal {H}^{d}\)-finite. We also show that, for uniform domains with uniform complements, if E??Ω is the Lipschitz image of a subset of \(\mathbb {R}^{d}\), then there is E^′?E with \(\mathcal {H}^{d}(E\backslash E^{\prime })<\tau \mathcal {H}^{d}(E)\) upon which a similar A_∞-type condition holds.