On the centers of quantum groups of <Emphasis Type="Italic">A</Emphasis><Subscript><Emphasis Type="Italic">n</Emphasis></Subscript>-type

Let g be the finite dimensional simple Lie algebra of type An, and let U? = U _q (g,Λ) and U = U _q (g,Q) be the quantum groups defined over the weight lattice and over the root lattice, respectively. In this paper, we find two algebraically independent central elements in U? for all n ≥ 2 and give an explicit formula of the Casimir elements for the quantum group U? = U _q (g,Λ), which corresponds to the Casimir element of the enveloping algebra U(g). Moreover, for n = 2 we give explicitly generators of the center subalgebras of the quantum groups U? = U _q (g,Λ) and U = U _q (g,Q).     

Double-bosonization and Majid’s conjecture (IV): Type-crossings from A to BCD

Both in Majid's double-bosonization theory and in Rosso's quantum shuffle theory, the rankinductive and type-crossing construction for U_q(g)'s is still a remaining open question. In this paper, working in Majid's framework, based on the generalized double-bosonization theorem we proved before, we further describe explicitly the type-crossing construction of U_q(g)'s for(BCD)_n series directly from type An-1via adding a pair of dual braided groups determined by a pair of(R, R′)-matrices of type A derived from the respective suitably chosen representations. Combining with our results of the first three papers of this series, this solves Majid's conjecture, i.e., any quantum group U_q(g) associated to a simple Lie algebra g can be grown out of U_q(sl_2)recursively by a series of suitably chosen double-bosonization procedures.     

Quantum quaternion spheres

For the quantum symplectic group SP _q (2n), we describe the C ^?-algebra of continuous functions on the quotient space S P _q (2n)/S P _q (2n?2) as an universal C ^?-algebra given by a finite set of generators and relations. The proof involves a careful analysis of the relations, and use of the branching rules for representations of the symplectic group due to Zhelobenko. We then exhibit a set of generators of the K-groups of this C ^?-algebra in terms of generators of the C ^?-algebra.     

Blow-up of <Emphasis Type="Italic">p</Emphasis>-Laplacian evolution equations with variable source power

We study the blow-up and/or global existence of the following p-Laplacian evolution equation with variable source power

$${s_j} = {\beta _j} + \overline {{\beta _{n - j}}}p$$

where Ω is either a bounded domain or the whole space ? ^N , q(x) is a positive and continuous function defined in Ω with 0 < q _? = inf q(x) ? q(x) ? sup q(x) = q+ < ∞. It is demonstrated that the equation with variable source power has much richer dynamics with interesting phenomena which depends on the interplay of q(x) and the structure of spatial domain Ω, compared with the case of constant source power. For the case that Ω is a bounded domain, the exponent p ? 1 plays a crucial role. If q+ > p ? 1, there exist blow-up solutions, while if q ₊ < p ? 1, all the solutions are global. If q _? > p ? 1, there exist global solutions, while for given q _? < p ? 1 < q ₊, there exist some function q(x) and Ω such that all nontrivial solutions will blow up, which is called the Fujita phenomenon. For the case Ω = ? ^N , the Fujita phenomenon occurs if 1 < q _? ? q ₊ ? p ? 1 + p/N, while if q _? > p ? 1 + p/N, there exist global solutions.

     

Sieve Functions in Arithmetic Bands,II

An arithmetic function f is called a sieve function of range Q if its Eratosthenes transform g = f * μ is supported in [1,Q] ∩ N, where g(q) ? _ε q ^ε (?ε > 0). We continue our study of the distribution of f(n) over short arithmetic bands, n ≡ ar + b (mod q), with n ∈ (N,2N] ∩ N, 1 ≤ a ≤ H = o(N) and r,b ∈ Z such that g:c:d:(r,q) = 1. In particular, the optimality of some results is discussed.     

On Templeman averages and variation functions

Let S be a countable semigroup acting in a measure-preserving fashion (g ? T _g ) on a measure space (Ω, A, µ). For a finite subset A of S, let |A| denote its cardinality. Let (A _k ) _k=1 ^∞ be a sequence of subsets of S satisfying conditions related to those in the ergodic theorem for semi-group actions of A. A. Tempelman. For A-measureable functions f on the measure space (Ω, A, μ) we form for k ≥ 1 the Templeman averages \(\pi _k (f)(x) = \left| {A_k } \right|^{ - 1} \sum\nolimits_{g \in A_k } {T_g f(x)}\) and set V _q f(x) = (Σ _k≥1|π _k+1(f)(x) ? π _k (f)(x)|q)^1/q when q ∈ (1, 2]. We show that there exists C > 0 such that for all f in L ¹(Ω, A, µ) we have µ({x ∈ Ω: V _q f(x) > λ}) ≤ C(∫_Ω | f | dµ/λ). Finally, some concrete examples are constructed.     

Existence and concentration behavior of sign-changing solutions for quasilinear Schrödinger equations

We consider the quasilinear Schrödinger equations of the form ?ε²Δu + V(x)u ? ε²Δ(u²)u = g(u), x∈ R^N, where ε > 0 is a small parameter, the nonlinearity g(u) ∈ C¹(R) is an odd function with subcritical growth and V(x) is a positive Hölder continuous function which is bounded from below, away from zero, and inf_ΛV(x) < inf?_ΛV(x) for some open bounded subset Λ of R^N. We prove that there is an ε₀ > 0 such that for all ε ∈ (0, ε₀], the above mentioned problem possesses a sign-changing solution uε which exhibits concentration profile around the local minimum point of V(x) as ε → 0⁺.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

Joint spectrum and the infinite dihedral group

A subgroup H of a group G is called pronormal if, for any element g ∈ G, the subgroups H and H ^g are conjugate in the subgroup <H,H ^g >. We prove that, if a group G has a normal abelian subgroup V and a subgroup H such that G = HV, then H is pronormal in G if and only if U = N _U (H)[H,U] for any H-invariant subgroup U of V. Using this fact, we prove that the simple symplectic group PSp_6n (q) with q ≡ ±3 (mod 8) contains a nonpronormal subgroup of odd index. Hence, we disprove the conjecture on the pronormality of subgroups of odd indices in finite simple groups, which was formulated in 2012 by E.P. Vdovin and D.O. Revin and verified by the authors in 2015 for many families of simple finite groups.     

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.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Short cycles in repeated exponentiation modulo a prime

Given a prime p, we consider the dynamical system generated by repeated exponentiations modulo p, that is, by the map \({u \mapsto f_g(u)}\), where f _g (u) ≡ g ^u (mod p) and 0 ≤ f _g (u) ≤ p ? 1. This map is in particular used in a number of constructions of cryptographically secure pseudorandom generators. We obtain nontrivial upper bounds on the number of fixed points and short cycles in the above dynamical system.     

Frobenius nonclassicality of Fermat curves with respect to cubics

For Fermat curves F: aX ⁿ + bY ⁿ = Z ⁿ defined over F _q , we establish necessary and sufficient conditions for F to be F _q -Frobenius nonclassical with respect to the linear system of plane cubics. In the new F _q -Frobenius nonclassical cases, we determine explicit formulas for the number N _q (F) of F _q -rational points on F. For the remaining Fermat curves, nice upper bounds for N _q (F) are immediately given by the Stöhr–Voloch Theory.     

Properties of quasi-Boolean function on quasi-Boolean algebra

In this paper, we investigate the following problem: give a quasi-Boolean function Ψ(x ₁, …, x _n ) = (a ∧ C) ∨ (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p ), the term (a ∧ C) can be deleted from Ψ(x ₁, …, x _n )? i.e., (a ∧ C) ∨ (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p ) = (a ₁ ∧ C ₁) ∨ … ∨ (a _p ∧ C _p )? When a = 1: we divide our discussion into two cases. (1) ?₁(Ψ,C) = ø, C can not be deleted; ?₁(Ψ,C) ≠ ø, if S _i ⁰ ≠ ø (1 ≤ i ≤ q), then C can not be deleted, otherwise C can be deleted. When a = m: we prove the following results: (m∧C)∨(a ₁∧C ₁)∨…∨(a _p ∧C _p ) = (a ₁∧C ₁)∨…∨(a _p ∧C _p ) ? (m ∧ C) ∨ C ₁ ∨ … ∨C _p = C ₁ ∨ … ∨C _p . Two possible cases are listed as follows, (1) ?₂(Ψ,C) = ø, the term (m∧C) can not be deleted; (2) ?₂(Ψ,C) ≠ ø, if (?i ₀) such that \(S'_{i_0 } \) = ø, then (m∧C) can be deleted, otherwise ((m∧C)∨C ₁∨…∨C _q )(v ₁, …, v _n ) = (C ₁ ∨ … ∨ C _q )(v ₁, …, v _n )(?(v ₁, …, v _n ) ∈ L ₃ ⁿ ) ? (C ₁ ^′ ∨ … ∨ C _q ^′ )(u ₁, …, u _q ) = 1(?(u ₁, …, u _q ) ∈ B ₂ ⁿ ).     

The Schröder-Bernstein property for weakly minimal theories

For a countable, weakly minimal theory T, we show that the Schröder-Bernstein property (any two elementarily bi-embeddable models are isomorphic) is equivalent to each of the following:

1. 1.
   For any U-rank-1 type q ∈ S(acl ^eq (?)) and any automorphism f of the monster model C, there is some n < ω such that f ⁿ (q) is not almost orthogonal to q ? f(q) ? … ? f ^n?1(q)
    
2. 2.
   T has no infinite collection of models which are pairwise elementarily bi-embeddable but pairwise nonisomorphic.
    

We conclude that for countable, weakly minimal theories, the Schröder-Bernstein property is absolute between transitve models of ZFC.     

Nikol’skii inequality between the uniform norm and <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>-norm with jacobi weight of algebraic polynomials on an interval

We study the Nikol’skii inequality for algebraic polynomials on the interval [?1, 1] between the uniform norm and the norm of the space L _q ^(α,β) , 1 ≤ q < ∞, with the Jacobi weight ?(α,β)(x) = (1 ? x) ^α (1 + x) ^β , α ≥ β > ?1. We prove that, in the case α > β ≥ ?1/2, the polynomial with unit leading coefficient that deviates least from zero in the space L _q ^(α+1,,β) with the Jacobi weight ? ^(α+1,β)(x) = (1?x) ^α+1(1+x) ^β is the unique extremal polynomial in the Nikol’skii inequality. To prove this result, we use the generalized translation operator associated with the Jacobi weight. We describe the set of all functions at which the norm of this operator in the space L _q ^(α,β) for 1 ≤ q < ∞ and α > β ≥ ?1/2 is attained.     

Seifert manifolds and (1, 1)-knots

The aim of this paper is to investigate the relations between Seifert manifolds and (1, 1)-knots. In particular, we prove that each orientable Seifert manifold with invariants

$\{ Oo,0| - 1;\underbrace {(p,q),...,(p,q)}_{n times},(l,l - 1)\} $

has the fundamental group cyclically presented by G _n ((x ₁ ^q ...x _n ^{q l} x _n ^?p ) and, moreover, it is the n-fold strongly-cyclic covering of the lens space L(|nlq ? p|, q) which is branched over the (1, 1)-knot K(q, q(nl ? 2), p ? 2q, p ? q) if p ≥ 2q and over the (1, 1)-knot K(p? q, 2q ? p, q(nl ? 2), p ? q) if p< 2q.

     

Perturbations of strongly continuous operator semigroups,and matrix Muckenhoupt weights

Let A and A ₀ be linear continuously invertible operators on a Hilbert space ? such that A ^?1 ? A ₀ ^?1 has finite rank. Assuming that σ(A ₀) = ? and that the operator semigroup V ₊(t) = exp{iA ₀ t}, t ≥ 0, is of class C ₀, we state criteria under which the semigroups U _±(t) = exp{±iAt}, t ≥ 0, are of class C ₀ as well. The analysis in the paper is based on functional models for nonself-adjoint operators and techniques of matrix Muckenhoupt weights.     

Discrete series of representations for the modular double of the quantum group <Emphasis Type="Italic">U</Emphasis><Subscript><Emphasis Type="Italic">q</Emphasis></Subscript>(sl(2, ℝ))

The modular double of the quantum group U _q (sl(2)) with deformation parameter q = e ^iπτ is a natural object explicitly taking into account the duality τ ? 1/τ. The use of the modular double in conformal field theory allows one to consider the region 1 < c < 25 for the central charge of the Virasoro algebra when |τ| = 1. In this paper, a new discrete series of representations for the modular double of U _q (sl(2, ?)) is found for such τ.     

ON THE COADJOINT ORBITS OF MAXIMAL UNIPOTENT SUBGROUPS OF REDUCTIVE GROUPS

Let G be a simple algebraic group defined over an algebraically closed field of characteristic 0 or a good prime for G. Let U be a maximal unipotent subgroup of G and \( \mathfrak{u} \) its Lie algebra. We prove the separability of orbit maps and the connectedness of centralizers for the coadjoint action of U on (certain quotients of) the dual \( \mathfrak{u} \)* of \( \mathfrak{u} \). This leads to a method to give a parametrization of the coadjoint orbits in terms of so-called minimal representatives which form a disjoint union of quasi-affine varieties. Moreover, we obtain an algorithm to explicitly calculate this parametrization which has been used for G of rank at most 8, except E₈.When G is defined and split over the field of q elements, for q the power of a good prime for G, this algorithmic parametrization is used to calculate the number k(U(q); \( \mathfrak{u} \)*(q)) of coadjoint orbits of U(q) on \( \mathfrak{u} \)*(q). Since k(U(q), \( \mathfrak{u} \)*(q)) coincides with the number k(U(q)) of conjugacy classes in U(q), these calculations can be viewed as an extension of the results obtained in [11]. In each case considered here there is a polynomial h(t) with integer coefficients such that for every such q we have k(U(q)) = h(q). We also explain implications of our results for a parametrization of the irreducible complex characters of U(q).