Complex vector measure and integral over manifolds with locally finite variations

It is well known that any compactly supported continuous complex differential n-form can be integrated over real n-dimensional C¹ manifolds in C^m (m ≥ n). For n = 1, the integral along any locally rectifiable curve is defined. Another generalization is the theory of currents (linear functionals on the space of compactly supported C^∞ differential forms). The topic of the article is the integration of measurable complex differential (n, 0)-forms (containing no \(d{\bar z_j}\)) over real n-dimensional C⁰ manifolds in C^m with locally finite n-dimensional variations (a generalization of locally rectifiable curves to dimensions n > 1). The last result is that a real n-dimensional manifold C¹ embedded in C^m has locally finite variations, and the integral of a measurable complex differential (n, 0)-form defined in the article can be calculated by a well-known formula.     

On finite alperin <Emphasis Type="Italic">p</Emphasis>-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number d(G) of generators of a finite Alperin p-group G is n ≥ 3, then d(G′) ≤ C _n ² for p≠ 3 and d(G′) ≤ C _n ² + C _n ³ for p = 3. The first section of the paper deals with finite Alperin p-groups G with p≠ 3 and d(G) = n ≥ 3 that have a homocyclic commutator subgroup of rank C _n ² . In addition, a corollary is deduced for infinite Alperin p-groups. In the second section, we prove that, if G is a finite Alperin 3-group with homocyclic commutator subgroup G- of rank C _n ² + C _n ³ , then G″ is an elementary abelian group.     

The Bergman kernel: Explicit formulas,deflation, Lu Qi-Keng problem and Jacobi polynomials

We investigate the Bergman kernel function for the intersection of two complex ellipsoids {(z,w ₁,w ₂) ∈ C ⁿ⁺²: |z ₁|²+...+|z _n |²+|w ₁| ^q < 1, |z ₁|²+...+|z _n |²+|w ₂| ^r < 1}. We also compute the kernel function for {(z ₁,w ₁,w ₂) ∈ C³: |z ₁|^2/n + |w ₁| ^q < 1, |z ₁|^2/n + |w ₂| ^r < 1} and show deflation type identity between these two domains. Moreover in the case that q = r = 2 we express the Bergman kernel in terms of the Jacobi polynomials. The explicit formulas of the Bergman kernel function for these domains enables us to investigate whether the Bergman kernel has zeros or not. This kind of problem is called a Lu Qi-Keng problem.     

<Emphasis Type="Italic">C</Emphasis>*-simplicity of <Emphasis Type="Italic">n</Emphasis>-periodic products

The C*-simplicity of n-periodic products is proved for a large class of groups. In particular, the n-periodic products of any finite or cyclic groups (including the free Burnside groups) are C*-simple. Continuum-many nonisomorphic 3-generated nonsimple C*-simple groups are constructed in each of which the identity xⁿ = 1 holds, where n ≥ 1003 is any odd number. The problem of the existence of C*-simple groups without free subgroups of rank 2 was posed by de la Harpe in 2007.     

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.     

Über das Schwarzsche Lemma und Verwandte Sätze

Upper bounds for the Jacobian determinant by holomorphic mappings of bounded domainsD into itself were given first more then thirty years ago by Stefan Bergman by means of his theory of the kernel function ofD. In this paper a different method shall be developed and distortion theorems for holomorphic mappings of bounded domains of a Kähler manifoldM ⁿ into a Kähler manifoldM ₀ ⁿ shall be proved. The special casesM ⁿ =C _n (unit sphere of C ⁿ ) andM ⁿ =M ₀ ⁿ =|C _n shall also be considered. The proof depends essentially on the two Hermitian quadratic forms corresponding to the metric and to the Ricci tensor. The manifolds must be of negative Ricci curvature and fulfil two conditions given in section 4.     

On the summability factors of infinite series

The author has established that if [λ_n] is a convex sequence such that the series Σn ^-1λ_n is convergent and the sequence {K _n} satisfies the condition |K _n|=O[log(n+1)]^k(C, 1),k?0, whereK _n denotes the (R, logn, 1) mean of the sequence {n log (n+1)a _n}, then the series Σlog(n+1)^1-kλ_n a _n is summable |R, logn, 1|. The result obtained for the particular casek=0 generalises a previous result of the author [1].     

Expansive homoclinic classes of generic <Emphasis Type="Italic">C</Emphasis><Superscript>1</Superscript>-vector fields

Let γ be a hyperbolic closed orbit of a C ¹ vector field X on a compact C ^∞ manifold M of dimension n ≥ 3, and let H _X(γ) be the homoclinic class of X containing γ. In this paper, we prove that C ¹-generically, if H _X(γ) is expansive and isolated, then it is hyperbolic.     

Holomorphic factorization of polynomials

We give necessary and sufficient conditions for a holomorphic factorization of an irreducible polynomial P(s, λ), s ∈ Cⁿ, λ ∈ C, in a domain Ω ? Cⁿ which is connected with the ordering of the real part of the roots of the equation P(s, λ) = 0, s ∈ Ω.     

On the sharp Jackson-Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere

The best constant C _n,m in the Jackson-Nikol’skii inequality between the uniform and integral norms of algebraic polynomials of a given total degree n ≥ 0 on the unit sphere \(\mathbb{S}^{m - 1} \) of the Euclidean space ? ^m is studied. Two-sided estimates for the constant C _n,m are obtained, which, in particular, give the order n ^m?1 of its behavior with respect to n as n → +∞ for a fixed m.     

Implicit differential equations with impasse singularities and singular perturbation problems

This paper concerns families of constrained differential systems having the form {fx1-1} where h: ? ⁿ → ? is a C ^r -function, (α ₁, …, α _n ) is a C ^r vector field on ? ⁿ , r ≥ 1, x ∈ ? ⁿ and λ is a real parameter. Using singular perturbation techniques we present some results on the dynamics of the system around a point (x, λ) of the impasse set λ = h(x). In addition, planar constrained systems with non-regular impasse surfaces are considered.     

Augmentation quotients for burnside rings of generalized dihedral groups

Let H be a finite abelian group of odd order, D be its generalized dihedral group, i.e., the semidirect product of C₂ acting on H by inverting elements, where C₂ is the cyclic group of order two. Let Ω (D) be the Burnside ring of D, Δ(D) be the augmentation ideal of Ω (D). Denote by Δⁿ(D) and Q_n(D) the nth power of Δ(D) and the nth consecutive quotient group Δⁿ(D)/Δⁿ⁺¹(D), respectively. This paper provides an explicit Z-basis for Δⁿ(D) and determines the isomorphism class of Q_n(D) for each positive integer n.     

Spectral (isotropic) manifolds and their dimension

A set of n × n symmetric matrices whose ordered vector of eigenvalues belongs to a fixed set in ?ⁿ is called spectral or isotropic. In this paper, we establish that every locally symmetric C^k submanifoldMof ?ⁿ gives rise to a C^k spectral manifold for k ∈ {2, 3, …,∞,ω}. An explicit formula for the dimension of the spectral manifold in terms of the dimension and the intrinsic properties of M is derived. This work builds upon the results of Sylvester and ?ilhavý and uses characteristic properties of locally symmetric submanifolds established in recent works by the authors.     

Ck-regularity for the $$\bar \partial $$ -equation with a support condition

Let D be a C ^d q-convex intersection, d ≥ 2, 0 ≤ q ≤ n ? 1, in a complex manifold X of complex dimension n, n ≥ 2, and let E be a holomorphic vector bundle of rank N over X. In this paper, C ^k -estimates, k = 2, 3,...,∞, for solutions to the \(\bar \partial \)-equation with small loss of smoothness are obtained for E-valued (0, s)-forms on D when n ? q ≤ s ≤ n. In addition, we solve the \(\bar \partial \)-equation with a support condition in C ^k -spaces. More precisely, we prove that for a \(\bar \partial \)-closed form f in C _0,q ^k (X D,E), 1 ≤ q ≤ n ? 2, n ≥ 3, with compact support and for ε with 0 < ε < 1 there exists a form u in C _0,q?1 ^k?ε (X D,E) with compact support such that \(\bar \partial u = f\) in \(X\backslash \bar D\). Applications are given for a separation theorem of Andreotti-Vesentini type in C ^k -setting and for the solvability of the \(\bar \partial \)-equation for currents.     

Separately radial and radial Toeplitz operators on the projective space and representation theory

We consider separately radial (with corresponding group T ⁿ ) and radial (with corresponding group U(n)) symbols on the projective space P ⁿ (C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between T ⁿ and U(n).     

Slow entropy for some smooth flows on surfaces

We study slow entropy in some classes of smooth mixing flows on surfaces. The flows we study can be represented as special flows over irrational rotations and under roof functions which are C² everywhere except one point (singularity). If the singularity is logarithmic asymmetric (Arnol’d flows), we show that in the scale a_n(t) = n(log n)^t slow entropy equals 1 (the speed of orbit growth is n log n) for a.e. irrational α. If the singularity is of power type (x^?γ, γ ∈ (0, 1)) (Kochergin flows), we show that in the scale a_n(t) = n^t slow entropy equals 1 + γ for a.e. α.We show moreover that for local rank one flows, slow entropy equals 0 in the n(log n)^t scale and is at most 1 for scale n^t. As a consequence we get that a.e. Arnol’d and a.e Kochergin flow is never of local rank one.     

Removable Singularities of Solutions of the Minimal Surface Equation

Suppose that G is a bounded domain in ? ⁿ (n ? 2), E ≠ G is a relatively closed set in G, and 0 < α < 1. We prove that E is removable for solutions of the minimal surface equation in the class C ^1,α(G)_loc if and only if the (n ? 1 + α)-dimensional Hausdorff measure of E is zero.     

Homogenization of the Dirichlet problem for elliptic and parabolic systems with periodic coefficients

Let O ? R ^d be a bounded domain of class C ^1,1. Let 0 < ε - 1. In L ₂(O;C ⁿ ) we consider a positive definite strongly elliptic second-order operator B _D,ε with Dirichlet boundary condition. Its coefficients are periodic and depend on x/ε. The principal part of the operator is given in factorized form, and the operator has lower order terms. We study the behavior of the generalized resolvent (B _D,ε ? ζQ ₀(·/ε))^?1 as ε → 0. Here the matrix-valued function Q ₀ is periodic, bounded, and positive definite; ζ is a complex-valued parameter. We find approximations of the generalized resolvent in the L ₂(O;C ⁿ )-operator norm and in the norm of operators acting from L ₂(O;C ⁿ ) to the Sobolev space H ¹(O;C ⁿ ) with two-parameter error estimates (depending on ε and ζ). Approximations of the generalized resolvent are applied to the homogenization of the solution of the first initial-boundary value problem for the parabolic equation Q ₀(x/ε)? _t v _ε (x, t) = ?(B _D,ε v _ε )(x, t).     

A new characterization of RBMO(<Emphasis Type="Italic">μ</Emphasis>) by John-Strömberg sharp maximal functions

Let μ be a nonnegative Radon measure on ? ^d which only satisfies μ (B(x, r)) ? C ₀ r ⁿ for all x ∈ ? ^d , r > 0, with some fixed constants C ₀ > 0 and n ∈ (0, d]. In this paper, a new characterization for the space RBMO(μ) of Tolsa in terms of the John-Strömberg sharp maximal function is established.     

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 ).