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.     

A Classification of Spectrum-determined Circulant Digraphs

Some researchers have proved that ádám’s conjecture is wrong. However, under special conditions, it is right. Let Z_n be a cyclic group of order n and C_n(S) be the circulant digraph of Z_n with respect to S ? Z_n\{0}. In the literature, some people have used a spectral method to solve the isomorphism for the circulants of prime-power order. In this paper, we also use the spectral method to characterize the circulants of order p^aq^bw^c(where p, q and w are all distinct primes), and to make ádám’s conjecture right.     

The weighted Hardy spaces associated to self-adjoint operators and their duality on product spaces

Let L be a non-negative self-adjoint operator acting on L²(R ⁿ ) satisfying a pointwise Gaussian estimate for its heat kernel. Let w be an A _r weight on R ⁿ × R ⁿ , 1 < r < ∞. In this article we obtain a weighted atomic decomposition for the weighted Hardy space H _L,w ^p (R ⁿ ×R ⁿ ), 0 < p ≤ 1 associated to L. Based on the atomic decomposition, we show the dual relationship between H _L,w ¹ (R ⁿ × R ⁿ ) and BMO_L,w(R ⁿ × R ⁿ ).     

CR deformations for rational homogeneous surface singularities

In this paper, we will present a CR-construction of the versal deformations of the singularitiesV _n ? ?²/? _n ,n ∈ {2,3,4,?} defined by the immersions of ?² into ? ⁿ⁺¹ X ⁿ : (z, w) → (z ⁿ ,z ^n?1 w, ?,zw ^n?1 ,w ⁿ )     

The modular group and words in its two generators<Superscript>*</Superscript>

Consider the full modular group PSL₂(?) with presentation 〈U,?S|U ³,?S ²〉. Motivated by our investigations on quasi-modular forms and the Minkowski question mark function (so that this paper may be considered as a necessary appendix), we are led to the following natural question. Some words in the alphabet {U, S} are equal to the unity; for example, USU ³ SU ² is such a word of length 8, and USU ³ SUSU ³ S ³ U is such a word of length 15. We consider the following integer sequence. For each n ∈ ?₀, let t(n) be the number of words in the alphabet {U, S} that equal the identity in the group. This is the new entry A265434 into the Online Encyclopedia of Integer Sequences. We investigate the generating function of this sequence and prove that it is an algebraic function over ?(x) of degree 3. As an interesting generalization, we formulate the problem of describing all algebraic functions with a Fermat property.     

Economical separability in free groups

Consider the rank n free group F _n with basis X. Bogopol’ski? conjectured in [1, Problem 15.35] that each element w ∈ F _n of length |w| ≥ 2 with respect to X can be separated by a subgroup H ≤ F _n of index at most C log |w| with some constant C. We prove this conjecture for all w outside the commutant of F _n , as well as the separability by a subgroup of index at most |w|/2 + 2 in general.     

Weighted fractional Bernstein’s inequalities and their applications

This paper studies the weighted, fractional Bernstein inequality for spherical polynomials on S^d-1\(\left( {0.1} \right)\;{\left\| {{{\left( { - {\Delta _0}} \right)}^{{\raise0.7ex\hbox{$r$} \!\mathord{\left/ {\vphantom {r 2}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$2$}}}}f} \right\|_{p,w}} \leqslant {C_w}{n^r}{\left\| f \right\|_{p,w}}\;for\;all\;f \in \Pi _n^d\), where Π_n^d denotes the space of all spherical polynomials of degree at most n on S^d-1 and (-Δ₀)^r/2 is the fractional Laplacian-Beltrami operator on S^d-1. A new class of doubling weights with conditions weaker than the A_p condition is introduced and used to characterize completely those doubling weights w on S^d-1 for which the weighted Bernstein inequality (0.1) holds for some 1 ≤ p ≤ 8 and all r > t. It is shown that in the unweighted case, if 0 < p < 8 and r > 0 is not an even integer, (0.1) with w = 1 holds if and only if r > (d - 1)((1/p) - 1). As applications, we show that every function f ∈ L_p(S^d-1) with 0 < p < 1 can be approximated by the de la Vallée Poussin means of a Fourier-Laplace series and establish a sharp Sobolev type embedding theorem for the weighted Besov spaces with respect to general doubling weights.     

Thom-Sebastiani properties of Kohn-Rossi cohomology of compact connected strongly pseudoconvex CR manifolds

Let X_1 and X_2 be two compact connected strongly pseudoconvex embeddable Cauchy-Riemann(CR) manifolds of dimensions 2m-1 and 2n-1 in C~(m+1)and C~(n+1), respectively. We introduce the ThomSebastiani sum X = X_1 ⊕X_2which is a new compact connected strongly pseudoconvex embeddable CR manifold of dimension 2m+2n+1 in C~(m+n+2). Thus the set of all codimension 3 strongly pseudoconvex compact connected CR manifolds in Cn+1for all n 2 forms a semigroup. X is said to be an irreducible element in this semigroup if X cannot be written in the form X_1 ⊕ X_2. It is a natural question to determine when X is an irreducible CR manifold. We use Kohn-Rossi cohomology groups to give a necessary condition of the above question. Explicitly,we show that if X = X_1 ⊕ X_2, then the Kohn-Rossi cohomology of the X is the product of those Kohn-Rossi cohomology coming from X_1 and X_2 provided that X_2 admits a transversal holomorphic S~1-action.     

Nonisometric Domains with the Same Marvizi – Melrose Invariants

For any strictly convex planar domain Ω ? R² with a C^∞ boundary one can associate an infinite sequence of spectral invariants introduced by Marvizi–Merlose [5]. These invariants can generically be determined using the spectrum of the Dirichlet problem of the Laplace operator. A natural question asks if this collection is sufficient to determine Ω up to isometry. In this paper we give a counterexample, namely, we present two nonisometric domains Ω and \(\bar \Omega \) with the same collection of Marvizi–Melrose invariants. Moreover, each domain has countably many periodic orbits {Sⁿ}_n≥1 (resp. \({\left\{ {{{\bar S}^n}} \right\}_{n \geqslant 1}}\)) of period going to infinity such that Sⁿ and \({\bar S^n}\) have the same period and perimeter for each n.     

Castelnuovo-Mumford regularity and projective dimension of a squarefree monomial ideal

Let S = K[x₁; x₂;...; x_n] be the polynomial ring in n variables over a field K; and let I be a squarefree monomial ideal minimally generated by the monomials u₁; u₂;...; u_m: Let w be the smallest number t with the property that for all integers 1 6 i₁ < i₂ <... < i _t 6 m such that \(lcm({u_{{i_1}}},{u_{{i_2}}},...,{u_{{i_t}}}) = lcm({u_1},{u_2},...,{u_m})\) We give an upper bound for Castelnuovo-Mumford regularity of I by the bigsize of I: As a corollary, the projective dimension of I is bounded by the number w.     

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).     

On integral Cayley sum graphs

Let S be a subset of a finite abelian group G. The Cayley sum graph Cay⁺(G, S) of G with respect to S is a graph whose vertex set is G and two vertices g and h are joined by an edge if and only if g + h ∈ S. We call a finite abelian group G a Cayley sum integral group if for every subset S of G, Cay⁺(G, S) is integral i.e., all eigenvalues of its adjacency matrix are integers. In this paper, we prove that all Cayley sum integral groups are represented by Z₃ and Zn₂ ⁿ, n ≥ 1, where Z_k is the group of integers modulo k. Also, we classify simple connected cubic integral Cayley sum graphs.     

Neighbor Connectivity of Two Kinds of Cayley Graphs

In this paper, we determine the neighbor connectivity κNB of two kinds of Cayley graphs: alternating group networks AN _n and star graphs S _n ; and give the exact values of edge neighbor connectivity λ_NB of ANn and Cayley graphs generated by transposition trees Γ _n . Those are κ_NB(AN _n ) = n^?1, λ_NB(AN _n ) = n^?2 and κ_NB(S _n ) = λ_NB(Γ _n ) = n^?1.     

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.     

Verbal Subgroups and Subalgebras in Skew Fields

Let D be an arbitrary skew field and K a central subfield of D. We prove that D can be embedded in a skew field Δ such that w(Δ)=Δ for every nonempty Lie word w on a set of variables y₁,y₂,.?.?. with coefficients in K; moreover, we have for the multiplicative group Δ^* that v(Δ^*)=Δ^* for every nonempty word \(v=x_{1}^{\varepsilon_{1}}x_{2}^{\varepsilon_{2}}\ldots x_{n}^{\varepsilon_{n}}\) (?_i=±1; i=1,2,.?.?.,n).     

Systems of dilated functions: Completeness,minimality, basisness

The completeness, minimality, and basis property in L ²[0, π] and L ^p[0, π], p ≠ 2, are considered for systems of dilated functions u _n (x) = S(nx), n ∈ N, where S is the trigonometric polynomial S(x) = Σ _k=0 ^m a _k sin(kx), a ₀ a _m ≠ 0. A series of results are presented and several unanswered questions are mentioned.     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

Dual submanifolds in rational homology spheres

Let Σ be a simply connected rational homology sphere. A pair of disjoint closed submanifolds M_+, M_-? Σ are called dual to each other if the complement Σ-M_+ strongly homotopy retracts onto M_- or vice-versa. In this paper, we are concerned with the basic problem of which integral triples(n; m_+, m-) ∈ N~3 can appear, where n = dimΣ-1 and m_± = codim M_±-1. The problem is motivated by several fundamental aspects in differential geometry.(i) The theory of isoparametric/Dupin hypersurfaces in the unit sphere S~(n+1) initiated by′Elie Cartan, where M_± are the focal manifolds of the isoparametric/Dupin hypersurface M ? S~(n+1), and m± coincide with the multiplicities of principal curvatures of M.(ii) The Grove-Ziller construction of non-negatively curved Riemannian metrics on the Milnor exotic spheres Σ,i.e., total spaces of smooth S~3-bundles over S~4 homeomorphic but not diffeomorphic to S~7, where M_± =P_±×_(SO(4))S~3, P → S~4 the principal SO(4)-bundle of Σ and P_± the singular orbits of a cohomogeneity one SO(4) × SO(3)-action on P which are both of codimension 2.Based on the important result of Grove-Halperin, we provide a surprisingly simple answer, namely, if and only if one of the following holds true:· m_+ = m_-= n;· m_+ = m_-=1/3n ∈ {1, 2, 4, 8};· m_+ = m_-=1/4n ∈ {1, 2};· m_+ = m_-=1/6n ∈ {1, 2};·n/(m_++m_-)= 1 or 2, and for the latter case, m_+ + m_-is odd if min(m_+, m_-)≥2.In addition, if Σ is a homotopy sphere and the ratio n/(m_++m_-)= 2(for simplicity let us assume 2 m_- m_+),we observe that the work of Stolz on the multiplicities of isoparametric hypersurfaces applies almost identically to conclude that, the pair can be realized if and only if, either(m_+, m_-) =(5, 4) or m_+ + m_-+ 1 is divisible by the integer δ(m_-)(see the table on Page 1551), which is equivalent to the existence of(m_--1) linearly independent vector fields on the sphere S~(m_++m_-)by Adams' celebrated work. In contrast, infinitely many counterexamples are given if Σ is a rational homology sphere.