On a commutator theorem of robert c. thompson

A very short proof of the following special case of R. C. Thompson's Theorem is presented. If A is an n × n mairix over a field with more than n elements and if det A = 1. then Ais a multiplicative commutator.     

On a theorem of Čebotarev

A new proof is given of the theorem that no submatrix of the p×p matrix S=(ζ^(i-1)(j-1)) is singular, where ζis a primitive pth root of unity and p is a prime. Some related results are also discussed.     

On commutator subgroups

Acta Mathematica Hungarica -     

On the commutator length of a Dehn twist

We show that on a nonorientable surface of genus at least 7 any power of a Dehn twist is equal to a single commutator in the mapping class group and the same is true, under additional assumptions, for the twist subgroup, and also for the extended mapping class group of an orientable surface of genus at least 3.     

On the spread of a hermitian matrix and a conjecture of thompson

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n-1)x(n-1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.     

On the spread of a hermitian matrix and a conjecture of thompson

We prove some inequalities involving the eigenvalues of an nxn Hermitian matrix and the eigenvalues of the (n?1)x(n?1) principal submatrices. We apply this inequality to generalize a known result on the numerical range to the lth numerical range. The method used yields an elegant proof of the converse to the interlacing theorem, which we include. A counterexample to the quardratic spread inequality conjectured by R. C. Thompson is also given.     

On a theorem of E. Cartan

Summary The object of this note is to prove: — Let G be a connected, locally compact subgroup of an analytic group H modelled on a Banach space. Then G itself is a finite dimensional analytic subgroup of H.     

On a theorem of J. Vukman

Summary LetR be a ring. A bi-additive symmetric mappingD:R × R R is called a symmetric bi-derivation if, for any fixedy R, the mappingx D(x, y) is a derivation. J. Vukman [2, Theorem 2] proved that, ifR is a non-commutative prime ring of characteristic not two and three, and ifD:R × R R is a symmetric bi-derivation such that [D(x, x), x] lies in the center ofR for allx R, thenD = 0. This result is in the spirit of the well-known theorem of Posner [1, Theorem 2], which states that the existence of a nonzero derivationd on a prime ringR, such that [d(x), x] lies in the center ofR for allx R, forcesR to be commutative. In this paper we generalize the result of J. Vukman mentioned above to nonzero two-sided ideals of prime rings of characteristic not two and we prove the following. Theorem.Let R be a non-commutative prime ring of characteristic different from two, and I a nonzero two-sided ideal of R. Let D: R × R R be a symmetric bi-derivation. If [D(x, x), x] lies in the center of R for all x I, then D = 0.     

On a theorem of E. Helly

E. Helly's theorem asserts that any bounded sequence of monotone real functions contains a pointwise convergent subsequence. We reprove this theorem in a generalized version in terms of monotone functions on linearly ordered sets. We show that the cardinal number responsible for this generalization is exactly the splitting number. We also show that a positive answer to a problem of S. Saks is obtained under the assumption of the splitting number being strictly greater than the first uncountable cardinal.

     

On a theorem of K. Schmidt

Let Y be a locally compact group, Aut(Y) be the group of topologicalautomorphisms of Y and (Y) be the set of continuous positivedefinite functions on Y which have unit value at the identity.A function (Y²) is said to be of product type if there aresuch functions _j (Y) that (u, v) = ₁(u)₂(v). Define the mappingT: Y² Y² by the formula T(u, v) = (A₁ uA₂ v, A₃ u A₄ v), whereA_j Aut(Y), and assume that T is a one-to-one transform. K.Schmidt proved: (i) if both (u, v) and (T(u, v)) are of producttype, then the functions _j are infinitely divisible; (ii) ifY is Abelian, both (u, v) and (T(u, v)) are of product type,and (u, v) 0, then the functions _j are Gaussian. We show thatstatement (i), generally, is not valid, but K. Schmidt's proofholds true if (u, v) 0. We also give another proof of statement(ii). Our proof uses neither the Levy–Khinchin formulafor a continuous infinitely divisible positive definite functionnor (i) on which K. Schmidt's proof is based.