Let 2 ≦ s ∊ N and {n1, …, ns) ⊆ N*. In 1884, J. Sylvester [13] published the following well-known result on the singularity degree S of the monomial curve whose corresponding semigroup is S: = 〈n1, …, ns): If s = 2, then

Let K: = –Z\S and

for all 1 ≦ i ≦ s. We introduce the invariant

of S involving a correction term to the Milnor number 2δ [4] of S. As a modified version and extension of Sylvester's result to all monomial space curves, we prove the following theorem: If s = 3, then

We prove similar formulas for s = 4 if S is symmetric.

An invertible operator T on a Hilbert space is a multiplicative commutator if there exist invertible operators A and B on such that T = ABA–1B–1. In this paper we discuss the question of which operators are, and which are not, multiplicative commutators. The analogous question for additive commutators (operators of the form AB — BA) has received considerable attention and has, in fact, been completely settled (2). The present results represent the information we have been able to obtain by carrying over to the multiplicative problem the techniques that proved efficacious in the additive situation. While these results remain incomplete, they suffice, for example, to enable us to determine precisely which normal operators are multiplicative commutators.

The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is ${{C}^{2}}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

1. Introduction and s t a t e m e n t of results. Let F be a field and Mn(F) the n X n matrices over F. Set GL(n, F) to be t he units in Mn(F). Suppose H is a subgroup of GL(n, F) and L is an L-linear operator on Mn(F) mapping H into itself. Can L be neatly characterized?

Marcus [2] answered this question when F is the complex numbers C and H the unitary group. The set of such L is a group and L has the form L(A) = UAV or L(A) = UA'V where U, V are fixed unitary matrices and A’ is t he transpose of A.

In the preceding paper Touchard considers a set of polynomials Qn(x) defined by

1.

Touchard uses (1) to compute Qn(x) for 0 ≤ n ≤ 9 and also finds Qn(-½).

He remarks however “l'expression générale des polynômes Qn(x) nous echappe.”

The object of this note is to derive an explicit expression for Qn(x).

Let and denote complex Banach algebras and let b e a left Banach module and a right Banach -module. If

we define the bounded linear elementary operator R(A, B), acting on , by

For the case , elementary operators were introduced by Lumer and Rosenblum [19], who studied their spectral properties. In this setting many authors subsequently studied spectral, algebraic, metric, and structural properties of elementary operators, with particular attention devoted to the inner derivations δa (δa(x) = ax – xa) [25], generalized derivations τ(a, b) (τ(a, b)(x) = ax – xb) [9, 10], and elementary multiplications S(a, b) (S(a, b)(x) = axb), including left and right multiplications La and Rb [11].

The decreasing rearrangement of a finite sequence a1, a2, … , an of real numbers is a second sequence aπ(1), aπ(2), … , aπ(n), where π(l), π(2), … , π(n) is a permutation of 1, 2, … , n and

(1, p. 260). The kth term of the rearranged sequence will be denoted by . Thus the terms of the rearranged sequence correspond to and are equal to those of the given sequence ak, but are arranged in descending (non-increasing) order.

Let R be an associative ring with unit element. Mod-R and R-Mod will denote the categories of unitary right and left R-modules, respectively, and all modules are assumed to be in Mod-R unless otherwise specified. For all M, N ϵ Mod-R, HomR(M, N) will usually be abbreviated as [M, N]. For the definitions of basic terms, and an exposition on torsion theories in Mod-R, the reader is referred to Lambek [6]. Jans [5] has called a class of modules which is closed under submodules, direct products, homomorphic images, group extensions, and isomorphic images a TTF (torsion-torsionfree) class.

Let n be a positive integer and put N = {1, 2, . . . , n}. A collection {S1, S2, . . . , St} of subsets of N is called determining if, for any T ⊂ N, the cardinalities of the t intersections T ∩ Sj determine T uniquely. Let €1, €2, . . . , €n be n variables with range {0, 1}. It is clear that a determining collection {Sj) has the property that the sums

In this paper, we wish to generalize the following lemma first proven by O. Zariski [5, Lemma 4]. Let O be a complete local ring containing the rational numbers and let m denote the maximal ideal of O. Assume there exists a derivation δ of O such that δ(x) is a unit in O for some x in m. Then O contains a ring O1 of representatives of the (complete local) ring O/Ox having the following properties: (a) δ is zero O1; (b) x is analytically independent over O1; (c) O is the power series ring O1[[x]]. In [4], A. Seidenberg used Zariski's lemma extensively to study conditions under which an affine algebraic variety V over a base field of characteristic zero is analytically a product along a given subvariety W of V. We should like to generalize Zariski's lemma by removing the condition that O contain the rationals. We could then get some conditions under which an arbitrary affine variety V would be analytically a product along a subvariety W.

A classical theorem of Hermite and Joubert asserts that any field extension of degree $n\,=\,5\,\text{or}\,\text{6}$ is generated by an element whose minimal polynomial is of the form ${{\lambda }^{n}}\,+\,{{c}_{1}}{{\lambda }^{n-1}}\,+\,\cdot \cdot \cdot +\,{{c}_{n-1}}\lambda \,+\,{{c}_{n}}$ with ${{c}_{1\,}}\,=\,\,{{c}_{3}}\,=\,0$. We show that this theorem fails for $n\,=\,{{3}^{m}}$ or ${{3}^{m}}+{{3}^{l}}$ (and more generally, for $n={{p}^{m}}$ or ${{p}^{m}}+{{p}^{l}}$, if 3 is replaced by another prime $p$), where $m\,>\,1\,\ge \,0$. We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$.

We also prove a similar result for division algebras and use it to study the structure of the universal division algebra $\text{UD}\left( n \right)$.

Čebyšev centers have been studied extensively. In this paper an alternate concept of center for infinite bounded point sets is introduced. Some of the results in this paper for this new type of center are similar to previous results for Čebyšev centers.

A simple ${{C}^{*}}$-algebra is constructed for which the Murray-von Neumann equivalence classes of projections, with the usual addition—induced by addition of orthogonal projections—form the additive semigroup

$$\left\{ 0,\,2,\,3,\ldots \right\}.$$

(This is a particularly simple instance of the phenomenon of perforation of the ordered ${{K}_{0}}$-group, which has long been known in the commutative case—for instance, in the case of the four-sphere—and was recently observed by the second author in the case of a simple ${{C}^{*}}$-algebra.)

A topology on $\mathbb{Z}$, which gives a nice proof that the set of prime integers is infinite, is characterised and examined. It is found to be homeomorphic to $\mathbb{Q}$, with a compact completion homeomorphic to the Cantor set. It has a natural place in a family of topologies on $\mathbb{Z}$, which includes the $p$-adics, and one in which the set of rational primes $\mathbb{P}$ is dense. Examples from number theory are given, including the primes and squares, Fermat numbers, Fibonacci numbers and $k$-free numbers.

In this paper we shall study the sufficiency of complex jets. Let A (Cn, C) be the set of all analytic functions f : Cn → C with f (0) = 0. We call two functions f and g of A (Cn, C) equivalent of order r at 0 if, at 0, their Taylor expansions up to and including the terms of degree ≦ r are identical.

Let R be a ring with unit element and G a finite group. We denote by RG the group ring of the group G over R and by U(RG) the group of units of this group ring.

The study of the nilpotency of U(RG) has been the subject of several papers.

In this paper, an induction theorem for rearrangements involving w-tuples in Rn is proved, showing that a certain proposition regarding a pair of w-tuples related by the weak spectral order ≪ is true for any integer n ≧ 2 if and only if it is true for n = 2. This theorem contains as particular cases a well-known theorem of Hardy-Littlewood-Pólya [4, Lemma 2, p. 47], a theorem of Pólya [8], a theorem of Rado [9, pp. 1-2], two theorems of Mirsky [6, Theorem 2, p. 232; 7, Theorem 4, p. 90], a result given in [1, Corollary 2.4, p. 1333] and also [2, Proposition 2.1, p. 439].

1. Introduction and statement of results. If pn(z) is a polynomial of degree at most n, then according to a famous result known as Bernstein's inequality (for references see [4])

(1)

Here equality holds if and only if pn(z) has all its zeros at the origin and so it is natural to seek for improvements under appropriate assumptions on the zeros of pn(z). Thus, for example, it was conjectured by P. Erdôs and later proved by Lax [2] that if pn(z) does not vanish in │z│ < 1, then (1) can be replaced by

(2)

On the other hand, Turán [5] showed that if pn(z) is a polynomial of degree n having all its zeros in │z│ ≦ 1, then

(3)

Let Λ be an artin algebra, and denote by mod Λ the category of finitely generated Λ-modules. All modules we consider are finitely generated.

We recall from [6] that a nonsplit exact sequence in mod A is said to be almost split if A and C are indecomposable, and given a map h: X → C which is not an isomorphism and with X indecomposable, there is some t: X → B such that gt = h.

Almost split sequences have turned out to be useful in the study of representation theory of artin algebras. Given a nonprojective indecomposable Λ-module C (or an indecomposable noninjective Λ-module A), we know that

there exists a unique almost split sequence [6, Proposition 4.3], [5, Section 3].

The motion groups of the real Euclidean plane and of the elliptic plane, the group of projectivities of a line, the projective general linear group PGL2(K), some orthogonal groups O3(K, Q) with char K = 2 (see [8]), are all bireflectional (zweispiegelig). There can be no doubt that bireflectional groups are of prime importance in any theory of groups that are generated by involutions. A brief look into F. Bachmann's book [1] gives convincing evidence.