We study a real hypersurface $M$ in a complex space form ${{M}_{n}}\left( c \right),c\ne 0$, whose shape operator and structure tensor commute each other on the holomorphic distribution of M.

Let $\mathfrak{a}$ be an ideal of a commutative Noetherian ring $R$ and $M$ and $N$ two finitely generated $R$-modules. Our main result asserts that if $R/\mathfrak{a}\,\le \,1$, then all generalized local cohomology modules $H_{\mathfrak{a}}^{i}(M,\,N)$ are $\mathfrak{a}$-cofinite.

A closed convex subset of ${{c}_{0}}$ has the fixed point property (fpp) if every nonexpansive self mapping of it has a fixed point. All nonempty weak compact convex subsets of ${{c}_{0}}$ are known to have the fpp. We show that closed convex subsets with a nonempty interior and nonempty convex subsets which are compact in a topology slightly coarser than the weak topology may fail to have the fpp.

Beineke and Harary gave an example of a family of tournaments Tn such that every subtournament of Tn is irreducible or transitive. We characterize all tournaments with this property.

We give a constructive proof of the carpenter's theorem due to Kadison. Unlike the original proof, our approach also yields the real case of this theorem.

If $A$ is a prime ${{\text{C}}^{*}}$-algebra, $a\,\in \,A$ and $\lambda $ is in the numerical range $W\left( a \right)$ of $a$, then for each $\varepsilon \,>\,0$ there exists an element $h\,\in \,A$ such that $\left\| h \right\|\,=\,1$ and $\left\| {{h}^{*}}(a-\lambda )h \right\|\,<\,\varepsilon $. If $\lambda $ is an extreme point of $W\left( a \right)$, the same conclusion holds without the assumption that $A$ is prime. Given any element $a$ in a von Neumann algebra (or in a general ${{\text{C}}^{*}}$-algebra) $A$, all normal elements in the weak* closure (the norm closure, respectively) of the ${{\text{C}}^{*}}$-convex hull of $a$ are characterized.

The theory of almost everywhere convergence has its roots in the poineering work of A. Kolmogorov, and today it constitutes one of the most captivating and challenging chapters in modern probability theory and analysis. Whereas some modes of convergence for sequences of measurable functions, e.g. convergence in norm, can be readily obtained by an intelligent exploitation of the various properties of the function spaces involved, a.e. convergence invariably requires a rather high, and sometimes surprising, degree of mathematical virtuosity.

We show that the set of values of an additive polynomial in several variables with arguments in a formal Laurent series field over a finite field has the optimal approximation property: every element in the field has a (not necessarily unique) closest approximation in this set of values. The approximation is with respect to the canonical valuation on the field. This property is elementary in the language of valued rings.

Let $c\,=\,\left( {{c}_{1}},\ldots ,{{c}_{n}} \right)$ be such that ${{c}_{1}}\,\ge \,\cdots \,\ge \,{{c}_{n}}$. The $c$-numerical range of an $n\,\times \,n$ matrix $A$ is defined by

$${{W}_{c}}\left( A \right)\,=\,\left\{ \sum\limits_{j=1}^{n}{{{c}_{j}}\left( A{{x}_{j}},\,{{x}_{j}} \right)\,:\,\left\{ {{x}_{1}},\ldots ,{{x}_{n}} \right\}\,\text{an}\,\text{orthonormal basis for }{{\mathbf{C}}^{n}}} \right\}\,,$$

and the $c$-numerical radius of $A$ is defined by ${{r}_{c}}\left( A \right)\,=\,\max \left\{ \left| z \right|\,:\,z\,\in \,{{W}_{c}}\left( A \right) \right\}$. We determine the structure of those linear operators $\phi$ on algebras of block triangular matrices, satisfying

$${{W}_{c}}\left( \phi \left( A \right) \right)={{W}_{c}}\left( A \right)\text{for}\,\,\text{all}\,\,A\,\text{or}\,\,{{r}_{c}}\left( \phi \left( A \right) \right)={{r}_{c}}\left( A \right)\text{for}\,\,\text{all}\,A.$$

Given a positive integer $n$, a finite field ${{\mathbb{F}}_{q}}$ of $q$ elements ($q$ odd), and a non-degenerate symmetric bilinear form $B$ on $\mathbb{F}_{q}^{n}$, we determine the largest possible cardinality of pairwise $B$-orthogonal subsets $\varepsilon \,\subseteq \,\mathbb{F}_{q}^{n}$, that is, for any two vectors $x,\,y\,\in \,\varepsilon $, one has $B(x,\,y)\,=\,0$.

We modify the construction of the mod 2 Dyer-Lashof (co)-algebra to obtain a (co)-algebra W which is (also) unstable over the Steenrod algebra A*. W has canonical sub-coalgebras W[k] such that the hom-dual W[k:]* is isomorphic as an A-algebra to the ring of upper triangular invariants in ℤ/2ℤ [x1, . . . , xk].

We call $\alpha \left( z \right)={{a}_{0}}+{{a}_{1}}z+\cdot \cdot \cdot +{{a}_{n-1}}{{z}^{n-1}}$ a Littlewood polynomial if ${{a}_{j}}=\pm 1$ for all $j$. We call $\alpha \left( z \right)$ self-reciprocal if $\alpha \left( z \right)={{z}^{n-1}}\alpha \left( 1/z \right)$, and call $\alpha \left( z \right)$ skewsymmetric if $n=2m+1$ and ${{a}_{m+j}}={{\left( -1 \right)}^{j}}{{a}_{m-j}}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\mathbb{C}$ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

Does there exist a circulant conference matrix of order > 2? When is there a symmetric Hadamard matrix with constant diagonal? How many pairwise disjoint, amicable weighing matrices of order n can there be? These are questions concerning which the trace function gives a great deal of insight. We offer easy proofs of the known solutions to the first two, the first being new, and develop new results regarding the latter question. It is shown that there are 2t disjoint amicable weighing matrices of order 2tp, where p is odd, and that this is an upper bound for t ≤ 1. An even stronger bound is obtained for certain cases.

We showed in [1] that the Eulerian family of cuboids with integral edges and face diagonals did not have integral inner diagonals. We now show that the derived family does not have integral inner diagonals except possibly when the generators are divisible by 705180. In this case there appears to be no inherent reason why the diagonals cannot be integral.

The Hurwitz zeta function ζ(s, a) is defined by the series

for 0 < a ≤ 1 and σ = Re(s) > 1, and can be continued analytically to the whole complex plane except for a simple pole at s = 1 with residue 1. The integral functions C(s, a) and S(s, a) are defined in terms of the Hurwitz zeta function as follows:

Using integral representations of C(s, a) and S(s, a), we evaluate explicitly a class of improper integrals. For example if 0 < a < 1 we show that

Let R be a prime ring with involution satisfying x/2 ∊ R whenever x ∊ R. Assume that R has two nontrivial symmetric idempotents e1, e2 whose sum is not 1, and that the subrings determined by e1, e2, 1 — (e1 + e2) are not orders in simple rings of dimension at most 4 over their centers. Then if L is a Lie derivation of the skew elements K into R there exists a subring A of R, A ⊆ , a derivation D :A → RC, the central closure of R, and a mapping T:R → C, satisfying L = D + T on K and = 0.

Let (E, E') be a dual pair and G be a collection of σ(E, E') bounded sets in E with the following properties:

1. i) G contains all singletons.

2. ii) If A, B are in G there is a C in G with A ∪B ⊆ C.

3. iii) If A is in G then λ A ε G for any scaler λ.

Then sets of the form the polar of Ai, form base of neighbourhoods in E' for the polar topology TG of G convergence.

An example of the type described in the title is given.

Various generalizations of the Bernstein operator, defined on C[0, 1] by the relation

1.1

where

have been given. Note that bnk(x) is the well-known binomial distribution.