On the torsion in K_2 of a field

For a field F,let Gn(F) = {{a,Φn(a)} ∈ K2(F) | a,Φn(a) ∈ F*},where Φn(x) is the n-th cyclotomic polynomial.At first,by using Faltings' theorem on Mordell conjecture it is proved that if F is a number field and if n = 4,8,12 is a positive integer having a square factor then Gn(F) is not a subgroup of K2(F),and then by using the results of Manin,Grauert,Samuel and Li on Mordell conjecture theorem for function fields,a similar result is established for function fields over an algebraically closed field.     

A Characterization of ＊-Automorphism on B（H）

Let H be a Hilbert space and A be a standard ＊-subalgebra of B（H）. We show that a bijective map Ф ： A →A preserves the Lie-skew product AB - BA＊ if and only if there is a unitary or conjugate unitary operator U ∈A（H） such that Ф（A） = UAU＊ for all A ∈ A, that is, Фis a linear ＊ -isomorphism or a conjugate linear ＊-isomorphism.     

Proof of three conjectures on congruences

This paper proves three conjectures on congruences involving central binomial coefficients or Lucas sequences.Let p be an odd prime and let a be a positive integer.It is shown that if p=1(mod 4)or a1then where(—)denotes the Jacobi symbol.This confirms a conjecture of the second author.A conjecture of Tauraso is also confirmed by showing that where the Lucas numbers Lo,L_1,L_2,...are defined by L_0=2,L_1=1 and L_n+1=L_n+L_n-l(n=1,2,3,...).The third theorem states that if p=5 then F_p~a-(p~a/5)mod p~3 can be determined in the following way:which appeared as a conjecture in a paper of Sun and Tauraso in 2010.     

A note on the existence of fractional f-factors in random graphs

Let G=Gn,p be a binomial random graph with n vertices and edge probability p=p(n),and f be a nonnegative integer-valued function defined on V(G) such that 0a≤f(x)≤bnp-2np ㏒n for every x ∈V(G). An fractional f-indicator function is an function h that assigns to each edge of a graph G a number h(e) in [0,1] so that for each vertex x,we have dh G(x)=f(x),where dh G(x) = x∈e h(e) is the fractional degree of x in G. Set Eh = {e:e ∈E(G) and h(e)=0}.If Gh is a spanning subgraph of G such that E(Gh)=Eh,then Gh is called an fractional f-factor of G. In this paper,we prove that for any binomial random graph Gn,p with p≥n-23,almost surely Gn,p contains an fractional f-factor.     

Lie Higher Derivations on Nest Algebras

Let N be a nest on a Banach space X, and Alg N be the associated nest algebra. It is shown that if there exists a non-trivial element in N which is complemented in X, then D = （Ln）n∈N is a Lie higher derivation of AlgAl if and only if each Ln has the form Ln（A） ： Tn（A） ＋ hn（A）I for all A ∈ AlgN, where （Tn）n∈N is a higher derivation and （hn）n∈N is a sequence of additive functionals satisfying hn（[A,B]） = 0 for all A,B ∈ AlgN and all n ∈ N.     

More about the kernel convergence and the ideal convergence

Let I 2N be an ideal and let XI = span{χI ： I ∈ I}, and let pI be the quotient norm of l∞/XI. In this paper, we show first that for each proper ideal I 2N, the ideal convergence deduced by I is equivalent to pI-kernel convergence. In addition, let K = {x＊oχ（·） ： x＊∈ p（e）}, where p（x） = lim supn→∞1/n（∑k=1n｜x（k）｜, and let Iμ = {A N ： μ（A） = 0} for all μ = x＊oχ（·） ∈ K. Then Iμ is a proper ideal. We also show that the ideal convergence deduced by the proper ideal Iμ, the p-kernel convergence and the statistical convergence are also equivalent.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Pseudo-Anosov Mapping Classes and Their Representations by Products of Two Dehn Twists

Let S be a Riemann surface of analytically finite type （p, n） with 3p-3＋n 〉 0. Let a ∈ S and S = S - {a}. In this article, the author studies those pseudo-Anosov maps on S that are isotopic to the identity on S and can be represented by products of Dehn twists. It is also proved that for any pseudo-Anosov map f of S isotopic to the identity on S, there are infinitely many pseudo-Anosov maps F on S - {b} = S - {a, b}, where b is a point on S, such that F is isotopic to f on S as b is filled in.     

Haddad和Helou定理的另一结论

Let K be a finite field of characteristic ≠ 2 and G the additive group of K × K. Let k_1, k_2 be integers not divisible by the characteristic p of K with(k_1, k_2) = 1. In 2004, Haddad and Helou constructed an additive basis B of G for which the number of representations of g ∈ G as a sum b_1+ b_2(b_1, b_2 ∈ B) is bounded by 18. For g ∈ G and B■G, let σk_1,k_2(B, g)be the number of solutions of g = k_1b_1 + k_2b_2, where b_1, b_2 ∈ B. In this paper, we show that there exists a set B ? G such that k_1 B + k2 B = G and σk_1,k_2(B, g)≤16.     

The <Emphasis Type="Italic">k</Emphasis>-point exponent set of primitive digraphs with girth 2

Let D = （V, E） be a primitive digraph. The vertex exponent of D at a vertex v∈ V, denoted by expD（v）, is the least integer p such that there is a v →u walk of length p for each u ∈ V. Following Brualdi and Liu, we order the vertices of D so that exPD（V1） ≤ exPD（V2） …≤ exPD（Vn）. Then exPD（Vk） is called the k- point exponent of D and is denoted by exPD （k）, 1≤ k ≤ n. In this paper we define e（n, k） ：= max{expD （k） ｜ D ∈ PD（n, 2）} and E（n, k） ：= {exPD（k）｜ D ∈ PD（n, 2）}, where PD（n, 2） is the set of all primitive digraphs of order n with girth 2. We completely determine e（n, k） and E（n, k） for all n, k with n ≥ 3 and 1 ≤ k ≤ n.     

On the abelianizations of congruence subgroups of Aut(<Emphasis Type="Italic">F</Emphasis><Subscript>2</Subscript>)

Let F _n be the free group of rank n, and let Aut⁺(F _n ) be its special automorphism group. For an epimorphism π : F _n → G of the free group F _n onto a finite group G we call the standard congruence subgroup of Aut⁺(F _n ) associated to G and π. In the case n = 2 we fully describe the abelianization of Γ⁺(G, π) for finite abelian groups G. Moreover, we show that if G is a finite non-perfect group, then Γ⁺(G, π) ≤ Aut⁺(F ₂) has infinite abelianization.     

Conjugacy classes and finite p-groups

Let G be a finite p-group, where p is a prime number, and a ∈ G. Denote by Cl(a) = {gag⁻¹| g ∈ G} the conjugacy class of a in G. Assume that |Cl(a)| = pⁿ. Then Cl(a) Cl(a⁻¹) = {xy | x ∈ Cl(a), y ∈ Cl(a⁻¹)} is the union of at least n(p − 1) + 1 distinct conjugacy classes of G. Received: 16 December 2004     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

<Emphasis Type="Italic">Φ</Emphasis>-Variation of a bifractional Brownian motion

Let B _H,K = {B _H,K (t)} _t⩾0 be a bifractional Brownian motion with parameters H ∈ (0, 1) and K ∈ (0, 1]. For a function Φ: [0, ∞) → [0, ∞) and for a partition κ = {t _i }_n ⁱ⁼⁰ of an interval [0, T] with T > 0, let {ie418-01}. We prove that, for a suitable Φ depending on H and K, {ie418-02} almost surely. The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No. T-16/08     

Scale functions on $p$-adic Lie groups

Let G be a p-adic Lie group. Then G is a locally compact, totally disconnected group, to which Willis [14] associates its scale function G : G→ℕ. We show that s can be computed on the Lie algebra level. The image of s consists of powers of p. If G is a linear algebraic group over ℚ _p , s(x)=s(h) is determined by the semisimple part h of x∈G. For every finite extension K of ℚ _p , the scale functions of G and H:=G(K) are related by s _H ∣ _G =s _G ^{[ K :ℚ} _p ^]. More generally, we clarify the relations between the scale function of a p-adic Lie group and the scale functions of its closed subgroups and Hausdorff quotients. Received: 20 February 1997; Revised version: 18 May 1998     

Finite solvable groups in which the Sylow <Emphasis Type="Italic">p</Emphasis>-subgroups are either bicyclic or of order <Emphasis Type="Italic">p</Emphasis><Superscript>3</Superscript>

All groups considered in this paper will be finite. Our main result here is the following theorem. Let G be a solvable group in which the Sylow p-subgroups are either bicyclic or of order p ³ for any p ∈ π(G). Then the derived length of G is at most 6. In particular, if G is an A₄-free group, then the following statements are true: (1) G is a dispersive group; (2) if no prime q ∈ π(G) divides p ² + p + 1 for any prime p ∈ π(G), then G is Ore dispersive; (3) the derived length of G is at most 4.     

On hamiltonicity of <Emphasis Type="Italic">P</Emphasis><Subscript>3</Subscript>-dominated graphs

We introduce a new class of graphs which we call P ₃-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P ₃-dominated graph. We prove that G is hamiltonian if α(G ²) ≤ κ(G), with two exceptions: K _2,3 and K _1,1,3. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. This paper was completed when both authors visited the Center for Combinatorics, Nankai University, Tianjin. They gratefully acknowledge the hospitality and support of the Center for Combinatorics and Nankai University. The work of E.Vumar is sponsored by SRF for ROCS, REM.     

On Browkin’s conjecture about the elements of order five in <Emphasis Type="Italic">K</Emphasis><Subscript>2</Subscript>(ℚ)

Based on the work of Lenstra, a succinct proof of Browkin’s conjecture about the elements of order five in K ₂(ℚ) is given. This work was supported by the National Natural Science Foundation of China (Grant No. 10371061).     

Coxeter group actions on <Subscript>4</Subscript><Emphasis Type="Italic">F</Emphasis><Subscript>3</Subscript>(1) hypergeometric series

In this paper we investigate a certain linear combination K([(x)\vec])=K(a;b,c,d;e,f,g)K(\vec{x})=K(a;b,c,d;e,f,g) of two Saalschutzian hypergeometric series of type ₄ F ₃(1). We first show that K([(x)\vec])K(\vec{x}) is invariant under the action of a certain matrix group G _K , isomorphic to the symmetric group S ₆, acting on the affine hyperplane V={(a,b,c,d,e,f,g)∈ℂ⁷:e+f+g−a−b−c−d=1}. We further develop an algebra of three-term relations for K(a;b,c,d;e,f,g). We show that, for any three elements μ ₁,μ ₂,μ ₃ of a certain matrix group M _K , isomorphic to the Coxeter group W(D ₆) (of order 23040) and containing the above group G _K , there is a relation among K(m₁[(x)\vec])K(\mu_{1}\vec{x}), K(m₂[(x)\vec])K(\mu_{2}\vec{x}), and K(m₃[(x)\vec])K(\mu_{3}\vec{x}), provided that no two of the μ _j ’s are in the same right coset of G _K in M _K . The coefficients in these three-term relations are seen to be rational combinations of gamma and sine functions in a,b,c,d,e,f,g.