Size ramsey numbers of stars versus 4‐chromatic graphs

We investigate the asymptotics of the size Ramsey number î(K_1,nF), where K_1,n is the n‐star and F is a fixed graph. The author 11 has recently proved that r?(K_1,n,F)=(1+o(1))n² for any F with chromatic number χ(F)=3. Here we show that r?(K_1,n,F)≤ n²+o(n²), if χ (F) ≥ 4 and conjecture that this is sharp. We prove the case χ(F)=4 of the conjecture, that is, that r?(K_1,n,F)=(4+o(1))n² for any 4‐chromatic graph F. Also, some general lower bounds are obtained. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 220–233, 2003     

A new approach to Matsumoto's theorem

We give a proof of Matsumoto's theorem on K ₂ of a field using techniques from homological algebra. By considering a complex associated to the action of GL(2, F) on P ₁(F) (F a field), we derive the Matsumoto presentation for H ₀ (F ^., H ₂(SL(2, F))) and, by considering the action of GL(n + 1, F) on P ⁿ (F), we prove the stability part of the theorem; namely, that H ₀(F ^., H ₂(SL(2, F))) is isomorphic to H ₂(SL(F)) = K ₂(F).     

Some involutory similarities

Matrices A,B over an arbitrary field F, when given to be similar to each other, are shown to be involutorily similar (over F) to each other (i.e.B = CAC^-1 for some C = C^-1 over F) in the following cases: (1)B= aI ? Afor some a ε F and (2) B = A^-1 . Result (2) for the cases where char F ≠ 2 is essentially a 1966 result of Wonenburger.     

The dynamics of marginal equilibria

Suppose F is a real-valued function defined on ^m. The dynamics of the system which is obtained by minimizing F by one (real) component at a time is studied. Two distinct cases arise depending upon whether F is C¹ or not. Thus when F ε C², the dynamical system obtained may be studied in terms of iterations of a C² local diffeomorphism. In particular when F is a Morse function, the system will converge to a minimum of F. However, when F is not C¹, but convex and suitably approximable, then the system exhibits a notion of turbulence which leads to orbits terminating at trapping points. These are not minima of F but are fixed points of the system.     

Trialitarian groups and the Hasse principle

Let F be a field of characteristic ≠ 2 such that is of cohomological 2- and 3-dimension ≤ 2. For G a simply connected group of type ³ D ₄ or ⁶ D ₄ over F, we show that the natural map

The Operator Factorization Method in Inverse Obstacle Scattering

The standard factorization method from inverse scattering theory allows to reconstruct an obstacle pointwise from the normal far field operator F. The kernel of this method is the study of the first kind Fredholm integral equation (F^* F)^1/4 f = Φ_z with the right-hand part In this paper we extend the factorization method to cover some kinds of boundary conditions which leads to non-normal far field operators. We visualize the scatterer explicitly in terms of the singular system of the selfadjoint positive operator F_# = [(ReF)^* (ReF)]^1/2 + ImF. The following characterization criterium holds: a given point z is inside the obstacle if and only if the function Φ_z belongs to the range of F_#^1/2. Our operator approach provides the tool for treatment of a wide class of inverse elliptic problems.     

Unique and faithful embeddings of projective-planar graphs

A graph G is uniquely embeddable in a surface F² if for any two embeddings f₁,f₂: G → F², there exists an isomorphism σ: G → G and a homeomorphism h: F² → F² for which h → f₁ = f₂ σ. A graph G is faithfully embeddable in a surface F² if G admits an embedding f: G → F² such that for any isomorphism σ: G → G, there is a homeomorphism h: F² → F² with h → f = f → σ. It will be shown that if a projective-planar graph G is 5-connected and contains a subdivision of the complete graph K₆ as its subgraph, then G is uniquely embeddable in a projective plane, and that moreover if G is not isomorphic to K₆, then G is faithfully embeddable in a projective plane.     

Algebraic relations for reciprocal sums of odd terms in Fibonacci numbers

In this paper, we prove the algebraic independence of the reciprocal sums of odd terms in Fibonacci numbers ∑ _n=1^∞ F _2n−1⁻¹, ∑ _n=1^∞ F _2n−1⁻², ∑ _n=1^∞ F _2n−1⁻³ and write each ∑ _n=1^∞ F _2n−1^−s (s≥4) as an explicit rational function of these three numbers over ℚ. Similar results are obtained for various series including the reciprocal sums of odd terms in Lucas numbers.      

Quasirecognition by prime graph of the simple group <Superscript>2</Superscript><Emphasis Type="Italic">F</Emphasis><Subscript>4</Subscript>(<Emphasis Type="Italic">q</Emphasis>)

As the main result, we show that if G is a finite group such that Γ(G) = Γ(² F ₄(q)), where q = 2^2m+1 for some m ≧ 1, then G has a unique nonabelian composition factor isomorphic to ² F ₄(q). We also show that if G is a finite group satisfying |G| =|² F ₄(q)| and Γ(G) = Γ(² F ₄(q)), then G ≅ ² F ₄(q). As a consequence of our result we give a new proof for a conjecture of W. Shi and J. Bi for ² F ₄(q). The third author was supported in part by a grant from IPM (No. 87200022).     

Spectral self-concordant functions in the space of two-by-two symmetric matrices

We show that given a two-variable, symmetric, ?-self-concordant function f, the spectral function F = f ○ λ is 2(1 + 3?)-self-concordant. Correspondingly, if f is ?-self-concordant barrier, then 4(1 + 3?)² F is a 4(1 + 3?)²?-self-concordant barrier.     

Potenzieren und Wurzelziehen in rationalen Quaternionenalgebren

LetF be a field not of characteristic 2 andQ =F +F i +F j +F k the quaternion algebra overF whereij = -ji =k andi ² = α andj ² = β with 0 ≠ α, β ∈F fixed. (IfF = ℝ and α = β = - 1 thenQ is the division algebra of the Hamilton quaternions.) IfF = ℚ and Q is a division algebra then by embedding certain quadratic number fields inQ we derive an efficient formula to compute the powers of any quaternion. This formula is even true in general and reads as follows. If a, a₁, a₂, a₃ ∈F andn ∈ ℕ then where ω ig a square root of αa₁ ² + βa ₂ ² - αβa ₃ ² in or overF and andA ₀ =na ^n-1. With the help of this formula and related ones we are able to solve the equationX ⁿ =q for arbitrary quaternionsq and positive integers n in case ofF = ℝ and hence in case ofF ⊂ ℝ as well. IfF = ℝ then the total number of all solutions equals 0, 1, 2, 4,n or ∞. (4 is possible even whenn < 4.) In case ofF = ℚ, which we are primarily interested in, there are always either at most six or infinitely many solutions. Further, for everyq ≠ 0 there is at most one solution provided thatn is odd and not divisible by 3. The questions when there are infinitely many solutions and when there are none can always be decided by checking simple conditions on the radicandq ifF = ℝ. ForF = ℚ the two questions are comprehensively investigatet in a natural connection with ternary and quaternary quadratic rational forms. Finally, by applying some of our theorems on powers and roots of quate-rions we also obtain several nice results in matrix theory. For example, for every k ∈ ℤ the mappingA ↦A ^k on the group of all nonsingular 2-by-2 matrices over ℚ is injective if and only ifk is odd and not divisible by 3.

     

A New Proof of a Theorem of Suslin

In this paper we compute the group H₂(SL₂(F)), for F an infinite field. In particular, using some techniques from homological algebra developed by Hutchinson [Hutchinson, K: K-Theory 4 (1990), 181–200], we give a new proof of the following theorem obtained by [Su2]: The group H₂(SL₂, (F)) is the fiber product of λ_*:K₂(F)→ I²(F)/I³(F) and σ: I²(F) → I²(F)/I³(F) where λ_* and σ map onto I²(F)/I³(F). (Received: February 2003)     

First Cohomology Group from the Virasoro-Like Algebra to its Larsson Functor Module

Let 𝒲 be the Virasoro-like algebra, and d ₁, d ₂ be the degree derivations of 𝒲. Set ? = W⊕C d ₁⊕C d ₂. Let F ^α(V) be the ?-module defined by Larsson's functor applied to a finite dimensional sl ₂-module V. In this article, the derivations from ? to ?-modules F ^α(V) and the first cohomology group H ¹(?, F ^α(V)) are given.     

Unique Ergodicity of Harmonic Currents On Singular Foliations of {\mathbb{P}^2}

Let F{\mathcal{F}} be a holomorphic foliation of \mathbbP²{\mathbb{P}^2} by Riemann surfaces. Assume all the singular points of F{\mathcal{F}} are hyperbolic. If F{\mathcal{F}} has no algebraic leaf, then there is a unique positive harmonic (1, 1) current T of mass one, directed by F{\mathcal{F}}. This implies strong ergodic properties for the foliation F{\mathcal{F}}. We also study the harmonic flow associated to the current T.     

Dimensions of fixed point sets of involutions

Suppose the fixed point set F of a smooth involution T:M → M on a smooth, closed and connected manifold M decomposes into two components Fⁿ and F² of dimensions n and 2, respectively, with n > 2 odd. We show that the codimension k of Fⁿ is small if the normal bundle of F² does not bound; specifically, we show that k≦ 3 in this case. In the more general situation where F is not a boundary, n (not necessarily odd) is the dimension of a component of F of maximal dimension and k is the codimension of this component, and fixed components of all dimensions j, 0≦ j≦ n, may occur, a theorem of Boardman gives that . In addition, we show that this bound can be improved to k≦ 1 (hence k = 1) for some specific values of n and some fixed stable cobordism classes of the normal bundle of F² in M; further, we determine in these cases the equivariant cobordism class of (M, T). Received: 25 August 2005     

Combined perturbation bounds: Ⅱ. Polar decompositions

In this paper, we study the perturbation bounds for the polar decomposition A= QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ2r||△Q||2F ≤ ||△A||2F,1/2||△H||2F ≤ ||△A||2F and ||△∑||2F ≤ ||△A||2F, respectively, where ∑ = diag(σ1, σ2,..., σr, 0,..., 0) is the singular value matrix of A and σr denotes the smallest nonzero singular value. Here we present some new combined (asymptotic)perturbation bounds σ2r ||△Q||2F 1/2||△H||2F≤ ||△A||2F and σ2r||△Q||2F ||△∑ ||2F ≤||△A||2F which are optimal for each factor. Some corresponding absolute perturbation bounds are also given.     

A Banach-lattice version of the Josefson-Nissenzweig theorem

Let E, F be two Banach lattices with E order continuous. If F can be mapped positively onto E then the dual F^* contains a weak^* -null sequence of positive and norm-one elements (Theorem 1). This is a Banach-lattice version of the classical Josefson-Nissenzweig theorem. It is an immediate consequence of the dual characterization of order continuity: E is order continuous iff E is Dedekind complete and every norm-one and pairwise disjoint sequence in E^* is weak^*-null (Theorem 2).     

Ternary codes of steiner triple systems

The code over a finite field F^q of a design ?? is the space spanned by the incidence vectors of the blocks. It is shown here that if ?? is a Steiner triple system on v points, and if the integer d is such that 3^d ≤ v < 3^d+1, then the ternary code C of ?? contains a subcode that can be shortened to the ternary generalized Reed-Muller code ?_F3(2(d ? 1),d) of length 3^d. If v = 3^d and d ≥ 2, then C_? ? ?_F3(1,d)? ? _F3(2(d ? 1),d) ? C. © 1994 John Wiley & Sons, Inc.     

Some estimations for the least degree of identities of subspaces M1(m,k) (F) of the matrix superalgebra M(m,k)(F)

We estimate the least degree of identities of subspaces M ₁^(m,k) (F) of the matrix superalgebra M ^(m,k)(F) over the field F for arbitrary m and k. For subspaces M ₁^(m,1) (F) (m≥1) and M ₁^(2,2) (F) we obtain concrete minimal identities.     

On the kernel of the Gassner representation

We study the Gassner representation of the pure braid group P_n by considering its restriction to a free subgroup F. The kernel of the restriction is shown to lie in the subgroup [Γ³F, Γ²F], sharpening a result of Lipschutz.Received: 25 August 2004; revised: 1 November 2004