Notes on the Kernels of Locally Finite Higher Derivations in Polynomial Rings

Let A = k^[3] be the polynomial ring in three variables over a field k, and let D be a nontrivial locally finite iterative higher derivation on A. Let A^D denote the kernel of D. In this note, we prove that, if chark > 0 and ML(A^D) ≠ A^D, then A^D ? k^[2]. As a consequence of this result, we give another proof of the cancellation theorem for k^[2] over any field k of positive characteristic.     

Continuous Linear Right Inverses for Partial Differential Operators on Non - Quasianalytic Classes and on Ultradistributions

Characterizations are given of those linear partial differential operators with constant coefficients which admit a continuous linear right inverse on ε_(ω()ω) )resp. ε_[ω](Ω) and/or D′_(ω)(Ω) (resp. D′_[ω](Ω), where Ω is an open set in ?ⁿ. The characterisations are in the same spirit as in the previous results of the authors on the existence of right inverses on C^∞ and/or D′(Ω).     

Quadratically normal and congruence-normal matrices

A matrix A ∈ C ^n×n is unitarily quasidiagonalizable if A can be brought by a unitary similarity transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. In particular, the square roots of normal matrices, i.e., the so-called quadratically normal matrices are unitarily quasidiagonalizable. A matrix A ∈ C ^n×n is congruence-normal if B = A[`(A)] B = A\overline A is a conventional normal matrix. We show that every congruence-normal matrix A can be brought by a unitary congruence transformation to a block diagonal form with 1 × 1 and 2 × 2 diagonal blocks. Our proof emphasizes andexploitsalikenessbetween theequations X <sup>2</sup> = B and X[`(X)] = B X\overline X = B for a normal matrix B. Bibliography: 13 titles.     

Parallel interval multisplittings

Summary We introduce interval multisplittings to enclose the setS={A^–1b|A[A], b[b]}, where [A] denotes an interval matrix and [b] an interval vector. The resulting iterative multisplitting methods have a natural parallelism. We investigate these methods with respect to convergence, speed of convergence and quality of the resulting enclosure forS.Dedicated to the memory of Peter Henrici     

Multi-variable weighted geometric means of positive definite matrices

We define a family of weighted geometric means {G(t;ω;A)}_t∈[0,1]ⁿ where ω and A vary over all positive probability vectors in Rⁿ and n-tuples of positive definite matrices resp. Each of these weighted geometric means interpolates between the weighted ALM (t=0_n) and BMP (t=1_n) geometric means (ALM and BMP geometric means have been defined by Ando-Li-Mathias and Bini-Meini-Poloni, respectively.) We show that the weighted geometric means satisfy multidimensional versions of all properties that one would expect for a two-variable weighted geometric mean.     

The cup product of Hilbert schemes for K3 surfaces

To any graded Frobenius algebra A we associate a sequence of graded Frobenius algebras A ^[n] so that there is canonical isomorphism of rings (H ^*(X;ℚ)[2]) ^[n] ≅H ^*(X ^[n] ;ℚ)[2n] for the Hilbert scheme X ^[n] of generalised n-tuples of any smooth projective surface X with numerically trivial canonical bundle. Oblatum 25-I-2001 & 18-IX-2002?Published online: 24 February 2003     

Continuous Linear Extension of Ultradifferentiable Functions of Beurling Type

For a weight function ω and a closed set A ? ?^N let ?_(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?^N there exists an ω-extension operator EA: ?_(ω)(A) → ?_(ω)(?^N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?^N there exists an ω-extension operator E_K: ?_(ω)(K) → ?_(ω)(?^N) if and only if the Fréchet space ?_(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.).     

On free quadratic extensions of rings

The quaternion algebraB[j] over a commutative ringB with 1 defined byS. Parimala andR. Sridharan is generalized in two directions: (1) the ringB may be non-commutative with 1, and (2)j ² may be any invertible element (not necessarily –1). LetG={} be an automorphism group ofB of order 2, andA={b inB| (b)=b}. LetB[j] be a generalized quaternion algebra such thataj (a) for eacha inB. It will be shown thatB is Galois (for non-commutative ring extensions) overA which is contained in the center ofB if and only ifB[j] is Azumaya overA. Also,A[j] is a splitting ring forB[j] such thatA[j] is Galois overA. Moreover, we shall determine which automorphism group ofA[j] is a Galois group.     

REPRESENTATIONS OF A CLASS OF DRINFELD'S DOUBLES

Let k be a field and A_n(ω) be the Taft's n²-dimensional Hopf algebra. When n is odd, the Drinfeld quantum double D(A_n(ω)) of A_n(ω) is a ribbon Hopf algebra. In the previous articles, we constructed an n₄-dimensional Hopf algebra H_n(p, q) which is isomorphic to D(A_n(ω)) if p ≠ 0 and q = ω^?1 , and studied the irreducible representations of H_n(1, q) and the finite dimensional representations of H₃(1, q). In this article, we examine the finite-dimensional representations of H_n(l q), equivalently, of D(A_n(ω)) for any n ≥ 2. We investigate the indecomposable left H_n(1, q)-module, and describe the structures and properties of all indecomposable modules and classify them when k is algebraically closed. We also give all almost split sequences in mod H_n(1, q), and the Auslander-Reiten-quiver of H_n(1 q).     

Some more primitive group rings

In this paper, we show that the group rings of several families of groups are primitive. LetA andB be two groups with 1<|A|≦|B| andB infinite. Then the main result is that ifK is a field for whichK[A ^ω] is semiprimitive, thenK[A∫B] is primitive. In addition, the field may be replaced by a subdomian in caseA is not torsion orA is not locally finite andK has characteristic 0. Certain other wreath products and free products are discussed. Partially supported by the N.S.F.     

Some results on unstable polynomial algebras over the steenrod algebra

In this paper we give a condition under which an unstable polynomialA _p -algebra can be obtained from the modular polynomial algebra. From this we car deduce both the Aguades theorem^[2] and Clark's theorem^[4]. Project supported by the Science Fund of the Chinese Academy of Sciences.     

PSL（2，F）的一个嵌入定理及其应用

设Ｆ是任意域，Ｇ代表ＳＬ（２，Ｆ）或ＰＳＬ（２，Ｆ）．本文的主要结果是：设Ｋ是Ｆ的子域，则Ｇ中同构于ＳＬ（２，Ｋ）或ＰＳＬ（２，Ｋ）的子群在Ｇ的自同构的作用下彼此共轭，利用这一结果，本文明确确定了A₁^[1]型的仿射Ｋａｃ－Ｍｏｏｄｙ群的一类极大正规子群．     

Sufficient conditions for permutation equivalence to a WHS-matrix

Square matrices with positive leading principal minors, called WHS-matrices (weak Hawkins–Simon), are considered in economics. Some sufficient conditions for a matrix to be a WHS-matrix after suitable row and/or column permutations have recently appeared in the literature. New and unified proofs and generalizations of some results to rectangular matrices are given. In particular, it is shown that if left multiplication of a rectangular matrix A by some nonnegative matrix is upper triangular with positive diagonal, then some row pemutation of A is a WHS-matrix. For a nonsingular A with either the first nonzero entry of each of its rows positive or the last nonzero entry of each column of A ^?1 positive, again some row permutation of A is a WHS-matrix. In addition, any rectangular full rank semipositive matrix is shown to be permutation equivalent to a WHS-matrix.     

On the tournament matrices with smallest rank

A tournament matrix is a square zero-one matrix A satisfying the equation A+A^t = J ? I, where J is the all-ones matrix. In [1] it was proved that if A is an n × n tournament matrix, then the rank of A is at least (n - 1)/2, over any field; and in characteristic zero rank (A) equals n - 1 or n. Michael [3] has constructed examples having rank (n - 1)/2; they are double borderings of Hadamard tournaments of order n - 2, and so must satisfy n ≡ 1 (mod 4). In this note, we supplement this result by showing that an analogous construction is sometimes impossible when n ≡ 3 (mod 4).     

On power-boundedness of interval matrices

In a recent paper by G. Mayer [8] the convergence of the sequence {[A]^k} of the powers of an interval matrix [A] to the nullmatrix was investigated. In this note we give some conditions for the boundedness of the sequence {k^−α[A]^k}, where α is a nonnegative number. The connection to the α-stability of the set [A] is discussed.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Generic Modules Over a Class of Drinfeld's Quantum Doubles

Let k be a field and A _n (ω) be the Taft's n ²-dimensional Hopf algebras. When n is odd, the Drinfeld quantum double D(A _n (ω)) of A _n (ω) is a Ribbon Hopf algebra. In the previous articles, we constructed an n ⁴-dimensional Hopf algebra H _n (p, q) which is isomorphic to D(A _n (ω)) if p ≠ 0 and q = ω^?1, and studied the finite dimensional representations of H _n (1, q). We showed that the basic algebra of any nonsimple block of H _n (1, q) is independent of n. In this article, we examine the infinite representations of H ₂(1, ? 1), or equivalently of H _n (1, q)?D(A _n (ω)) for any n ≥ 2. We investigate the indecomposable and algebraically compact modules over H ₂(1, ? 1), describe the structures of these modules and classify them under the elementary equivalence.     

Circularity of numerical ranges and block-shift matrices

Let A be a square complex matrix. Several characterizations are found for A to be permutationally similar to a block-shift matrix. One interesting equivalent condition is that the numerical range of every matrix with the same zero pattern as A is a circular disk. Equivalent conditions for the characteristic polynomial of the hermitian part of the matrix e ^iθA to be the same for all real values θ are also obtained.Complex matrices of order four that are unitarily similar to a block-shift matrix are identified. A result of Marcus and Pesce [6] is extended, and an open question of Li and Tsing [4] is also answered partially.