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">M</Emphasis> <Subscript> <Emphasis Type="Italic">p</Emphasis> </Subscript>-supplemented subgroups of finite groups

A subgroup K of G is M_p-supplemented in G if there exists a subgroup B of G such that G = KB and TB < G for every maximal subgroup T of K with |K: T| = p^α. In this paper we prove the following: Let p be a prime divisor of |G| and let H be ap-nilpotent subgroup having a Sylow p-subgroup of G. Suppose that H has a subgroup D with D_p ≠ 1 and |H: D| = p_α. Then G is p-nilpotent if and only if every subgroup T of H with |T| = |D| is M_p-supplemented in G and N_G(T_p)/C_G(T_p) is a p-group.     

Solvability of finite groups

H is called an ? _p -embedded subgroup of G, if there exists a p-nilpotent subgroup B of G such that H _p ∈ Syl _p (B) and B is ? _p -supplemented in G. In this paper, by considering prime divisor 3, 5, or 7, we use ? _p -embedded property of primary subgroups to investigate the solvability of finite groups. The main result is follows. Let E be a normal subgroup of G, and let P be a Sylow 5-subgroup of E. Suppose that 1 < d ? |P| and d divides |P|. If every subgroup H of P with |H| = d is ?₅-embedded in G, then every composition factor of E satisfies one of the following conditions: (1) I/C is cyclic of order 5, (2) I/C is 5′-group, (3) I/C ? A₅.     

Tauber-Konstanten für Verschiedene Tauberbedingungen bei den Kreisverfahren der Limitierungstheorie

Till now, we know Tauberian constants for the ‘Kreisverfahren’ with the conditions lim sup |n ^1/2 a _n|<∞ and lim sup |n ¹ a _n|<∞. Now, we obtain constants for the more general condition lim sup |n ^pa_n|<∞ with anyp(=∞<p<+∞). These constants are not always 0 or ∞, even if 1/2<p<1; therefore the Tauberian condition lim sup |n ^pa_n|<∞ is ‘appropriate’ for 1/2≦p≦1.     

Every 2-group with all subgroups normal-by-finite is locally finite

A group G has all of its subgroups normal-by-finite if H/H _G is finite for all subgroups H of G. The Tarski-groups provide examples of p-groups (p a “large” prime) of nonlocally finite groups in which every subgroup is normal-by-finite. The aim of this paper is to prove that a 2-group with every subgroup normal-by-finite is locally finite. We also prove that if |H/H _G | 6 2 for every subgroup H of G, then G contains an Abelian subgroup of index at most 8.     

On curves on K3 surfaces,II

We study the Clifford index c of a smooth irreducible curve X in the linear series |2H| on a special K3 surface S of degree 2n in \({{\mathbb P}}^{n+1}\), with hyperplane section H, and we look for the complete and base point free linear series of S whose restrictions to X compute c. In a more general context, we discuss the features of such series, for an assigned curve on a K3 surface; this discussion is of some independent interest.     

An optimal Berry-Esseen type inequality for expectations of smooth functions

We provide an optimal Berry-Esseen type inequality for Zolotarev’s ideal ζ₃-metric measuring the difference between expectations of sufficiently smooth functions, like |·|³, of a sum of independent random variables X ₁,..., X _n with finite third-order moments and a sum of independent symmetric two-point random variables, isoscedastic to the X _i . In the homoscedastic case of equal variances, and in particular, in case of identically distributed X ₁,..., X _n the approximating law is a standardized symmetric binomial one. As a corollary, we improve an already optimal estimate of the accuracy of the normal approximation due to Tyurin (2009).     

Note: A conjecture on partitions of groups

We conjecture that every infinite group G can be partitioned into countably many cells \(G = \bigcup\limits_{n \in \omega } {A_n }\) such that cov(A _n A _n ^?1 ) = |G| for each n ∈ ω Here cov(A) = min{|X|: X} ? G, G = X A}. We confirm this conjecture for each group of regular cardinality and for some groups (in particular, Abelian) of an arbitrary cardinality.     

Testing for change in the mean via convergence in distribution of sup-functionals of weighted tied-down partial sums processes

Let X ₁,..., X _n, n > 1, be nondegenerate independent chronologically ordered realvalued observables with finite means. Consider the “no-change in the mean” null hypothesis H ₀: X ₁,..., X _n is a randomsample on X with Var X <∞. We revisit the problem of nonparametric testing for H ₀ versus the “at most one change (AMOC) in the mean” alternative hypothesis H _A: there is an integer k*, 1 ≤ k* < n, such that EX ₁ = · · · = EX_k* ≠ EX_k*+1 = ··· = EX _n. A natural way of testing for H ₀ versus H _A is via comparing the sample mean of the first k observables to the sample mean of the last n - k observables, for all possible times k of AMOC in the mean, 1 ≤ k < n. In particular, a number of such tests in the literature are based on test statistics that are maximums in k of the appropriately individually normalized absolute deviations Δ_k = |S _k/k - (S _n - S _k)/(n - k)|, where S _k:= X ₁ + ··· + X _k. Asymptotic distributions of these test statistics under H ₀ as n → ∞ are obtained via establishing convergence in distribution of supfunctionals of respectively weighted |Z _n(t)|, where {Z _n(t), 0 ≤ t ≤ 1}_n≥1 are the tied-down partial sums processes such that

$${Z_n}\left( t \right): = \left( {{S_{\left\lceil {\left( {n + 1} \right)t} \right\rceil }} - \left[ {\left( {n + 1} \right)t} \right]{S_n}/n} \right)/\sqrt n $$

if 0 ≤ t < 1, and Z _n(t):= 0 if t = 1. In the present paper, we propose an alternative route to nonparametric testing for H ₀ versus H _A via sup-functionals of appropriately weighted |Z _n(t)|. Simply considering max_1?k<n Δk as a prototype test statistic leads us to establishing convergence in distribution of special sup-functionals of |Z _n(t)|/(t(1 - t)) under H ₀ and assuming also that E|X|^r < ∞ for some r > 2. We believe the weight function t(1 - t) for sup-functionals of |Z _n(t)| has not been considered before.

     

On the Automorphism Group of a Smooth Schubert Variety

Let G be a simple algebraic group of adjoint type over the field \(\mathbb {C}\) of complex numbers. Let B be a Borel subgroup of G containing a maximal torus T of G. Let w be an element of the Weyl group W and let X(w) be the Schubert variety in G/B corresponding to w. Let α ₀ denote the highest root of G with respect to T and B. Let P be the stabiliser of X(w) in G. In this paper, we prove that if G is simply laced and X(w) is smooth, then the connected component of the automorphism group of X(w) containing the identity automorphism equals P if and only if w ^?1(α ₀) is a negative root (see Theorem 4.2). We prove a partial result in the non simply laced case (see Theorem 6.6).     

Finite <Emphasis Type="Italic">p</Emphasis>-groups all of whose maximal abelian subgroups are soft

A subgroup A of a p-group G is said to be soft in G if C _G (A) = A and |N _G (A/A| = p. In this paper we determined finite p-groups all of whose maximal abelian subgroups are soft; see Theorem A and Proposition 2.4.     

On the absolute Cesàro summability factors of infinite series

The author has established that if {λ _n | is a convex sequence such that the series\(\sum {\frac{{\lambda _n }}{n}} \) is convergent and if Σa _n is bounded [R, logn, 1] with indexk, then\(\sum {a_n \lambda _n } \) is summable |C, 1|k fork>1. The casek=1 of the theorem is due to Pati [3].     

Topological entropy of sets of generic points for actions of amenable groups

Let G be a countable discrete infinite amenable group which acts continuously on a compact metric space X and let μ be an ergodic G-invariant Borel probability measure on X. For a fixed tempered F?lner sequence {Fn} in G with limn→+∞|Fn|/log n= ∞, we prove the following result:h_top~B(G_μ, {F_n}) = h_μ(X, G),where G_μ is the set of generic points for μ with respect to {F_n} and h_top~B(G_μ, {F_n}) is the Bowen topological entropy(along {F_n}) on G_μ. This generalizes the classical result of Bowen(1973).     

Separability and Sequential Separability of the Space <Emphasis Type="Italic">C</Emphasis>(<Emphasis Type="Italic">X</Emphasis>)

The following results are obtained: (1) a criterion for the separability of the space of continuous functions C(X) with the set-open topology; (2) a criterion for the sequential separability of the space C_p(A|X), where A ? X; (3) an answer to Velichko’s question of whether a set-theoretic condition on a metric space X in a criterion for the sequential separability of C_p(X) is necessary.     

A Generalization of Strassen’s Functional LIL

Let X ₁,X ₂,… be a sequence of i.i.d. mean zero random variables and let S _n denote the sum of the first n random variables. We show that whenever we have with probability one, lim?sup? _n→∞|S _n |/c _n =α ₀<∞ for a regular normalizing sequence {c _n }, the corresponding normalized partial sum process sequence is relatively compact in C[0,1] with canonical cluster set. Combining this result with some LIL type results in the infinite variance case, we obtain Strassen type results in this setting.     

Engel-type subgroups and length parameters of finite groups

Let g be an element of a finite group G. For a positive integer n, let E _n (g) be the subgroup generated by all commutators [...[[x, g], g],..., g] over x ∈ G, where g is repeated n times. By Baer’s theorem, if E _n (g) = 1, then g belongs to the Fitting subgroup F(G). We generalize this theorem in terms of certain length parameters of E _n (g). For soluble G we prove that if, for some n, the Fitting height of E _n (g) is equal to k, then g belongs to the (k+1)th Fitting subgroup F_k+1(G). For nonsoluble G the results are in terms of nonsoluble length and generalized Fitting height. The generalized Fitting height h*(H) of a finite group H is the least number h such that F_h* (H) = H, where F₀* (H) = 1, and F_i+1(H)* is the inverse image of the generalized Fitting subgroup F*(H/F*_i (H)). Let m be the number of prime factors of |g| counting multiplicities. It is proved that if, for some n, the generalized Fitting height of E _n (g) is equal to k, then g belongs to F*_f(k,m)(G), where f(k, m) depends only on k and m. The nonsoluble length λ(H) of a finite group H is defined as the minimum number of nonsoluble factors in a normal series each of whose factors either is soluble or is a direct product of nonabelian simple groups. It is proved that if λ(E _n (g)) = k, then g belongs to a normal subgroup whose nonsoluble length is bounded in terms of k and m. We also state conjectures of stronger results independent of m and show that these conjectures reduce to a certain question about automorphisms of direct products of finite simple groups.     

Inversion and subspaces of a finite field

Consider two F _q -subspaces A and B of a finite field, of the same size, and let A ^?1 denote the set of inverses of the nonzero elements of A. The author proved that A ^?1 can only be contained in A if either A is a subfield, or A is the set of trace zero elements in a quadratic extension of a field. Csajbók refined this to the following quantitative statement: if A ^?1 ? B, then the bound |A ^?1∩B| ≤ 2|B|/q ? 2 holds. He also gave examples showing that his bound is sharp for |B| ≤ q ³. Our main result is a proof of the stronger bound |A ^?1 ∩ B| ≤ |B|/q · (1 + O _d (q ^?1/2)), for |B| = q ^d with d > 3. We also classify all examples with |B| ≤ q ³ which attain equality or near-equality in Csajbók’s bound.     

Hyperbolic Chevalley groups on ℂ<Superscript>2</Superscript>

Let Γ ? U (1, 1) be the subgroup generated by the complex reflections. Suppose that Γ acts discretely on the domain K = {(z ₁, z ₂) ∈ ?² ||z ₁|² ? |z ₂|² < 0} and that the projective group PΓ acts on the unit disk B = {|z ₁/z ₂| < 1} as a Fuchsian group of signature (n ₁, ..., n _s ), s ? 3, n _i ? 2. For such groups, we prove a Chevalley type theorem, i.e., find a necessary and sufficient condition for the quotient space K/Γ to be isomorphic to ?² ? {0}.     

Vertex-fault-tolerant cycles embedding on enhanced hypercube networks

In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F _v be the set of faulty vertices in the n-dimensional enhanced hypercube Q _n,k (1 ≤ k ≤ n?1). When |F _v | = 2, we showed that Q _n,k ? F _v contains a fault-free cycle of every even length from 4 to 2 ⁿ ?4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2 ⁿ ? 4, simultaneously, contains a cycle of every odd length from n ? k + 2 to 2 ⁿ ? 3 where n(≥ 3) and k have the different parity. Furthermore, when |F ^v | = f ^v ≤ n ? 2, we proof that there exists the longest fault-free cycle, which is of even length 2 ⁿ ? 2f ^v whether n(n ≥ 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2 ⁿ ? 2f ^v ? 1 in Q ^n,k ? F ^v where n(≥ 3) and k have the different parity.     

On necessary and sufficient conditions for the self-normalized central limit theorem

Let X₁, X₂, … be a sequence of independent random variables and S_n = Σ _i=1 ⁿ X_i and V _n ² = Σ _i=1 ⁿ X _i ² . When the elements of the sequence are i.i.d., it is known that the self-normalized sum S_n=V_n converges to a standard normal distribution if and only if max_1?i?n|X_i|/V_n→0 in probability and the mean of X₁ is zero. In this paper, sufficient conditions for the self-normalized central limit theorem are obtained for general independent random variables. It is also shown that if max_1?i?n|X_i|/V_n→0 in probability, then these sufficient conditions are necessary.