A strictly singular non-compact operator on the Gowers and Maurey space GM is constructed.

Let A be an abelian group, F be a field of characteristic 0, and α, β be linearly independent additive maps from A to F, and let δ∈ker(α)\{0}. Then there is a Lie algebra L = L(A, α, β, δ) = [oplus ]x∈AFex under the product

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If, further, β(δ) = 1, and β(A) = Z, there is a subalgebra L+:=L(A+, α, β, δ) = [oplus ]x∈A+Fex, where A+ = {x∈A|β(x)[ges ]0}. The necessary and sufficient conditions are given for L' = [L, L] and L+ to be simple, and all semi-simple elements in L' and L+ are determined. It is shown that L' and L+ cannot be isomorphic to any other known Lie algebras and L' is not isomorphic to any L+ , and all isomorphisms between two L' and all isomorphisms between two L+ are explicitly described.

The paper finds best constants in a class of multiplicative inequalities with one-dimensional spatial variables

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where f is a periodic function with zero mean value, and the norms in the right-hand side of the expression are the L2-norms.

Let g be a primitive root modulo a prime p. It is proved that the triples (gx, gy, gxy), x, y = 1, …, p−1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε>0 be fixed. Then

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uniformly for any integers a, b, c with gcd(a, b, c, p) = 1. Incomplete sums are estimated as well.

The question is motivated by the assumption, often made in cryptography, that the triples (gx, gy, gxy) cannot be distinguished from totally random triples in feasible computation time. The results imply that this is in any case true for a constant fraction of the most significant bits, and for a constant fraction of the least significant bits.

T. Hosokawa, K. Izuchi and D. Zheng recently introduced the concept of asymptotic interpolating sequences (of type 1) in the unit disk for $H^\infty$(${\bb D}$). It is shown that these sequences coincide with sequences that are interpolating for the algebra $QA$. Also a characterization is given of the interpolating sequences of type $1$ for $H^\infty$(${\bb D}$), and asymptotic interpolating sequences in the spectrum of $H^\infty$(${\bb D}$) are studied. The existence of asymptotic interpolating sequences of type $1$ for $H^\infty(\Omega)$ on arbitrary domains is verified. It is shown that any asymptotic interpolating sequence in a uniform algebra eventually is interpolating.

The tangent developables of space curves are classified locally and topologically. It turns out that only seven different tangent developables arise, except for very degenerate cases. An extension of the known classification due to Cleave, Shcherbak and Mond is given.

A random walk that is certain to visit $(0, \infty)$ has associated with it, via a suitable $h$ -transform, a Markov chain called ‘random walk conditioned to stay positive’, which is defined properly below. In continuous time, if the random walk is replaced by Brownian motion then the analogous associated process is Bessel-3. Let $\phi(x) = \log\log x$ . The main result obtained in this paper, which is stated formally in Theorem 1, is that, when the random walk has zero mean and finite variance, the total time for which the random walk conditioned to stay positive is below $x$ ultimately lies between $Lx^2/\phi(x)$ and $Ux^2\phi(x)$ , for suitable (non-random) positive $L$ and finite $U$ , as $x$ goes to infinity. For Bessel-3, the best $L$ and $U$ are identified.

Let $G$ be a compact group and ${\cal C}(G)$ be the $C^{*}$ -algebra of continuous complex-valued functions on $G$ . The paper constructs an imbedding of the Fourier algebra ${\rm A}(G)$ of $G$ into the algebra ${\rm V}(G) = {\cal C}(G)\otimes^h {\cal C}(G)$ (Haagerup tensor product) and deduces results about parallel spectral synthesis, generalizing a result of Varopoulos. It then characterizes which diagonal sets in $G \times G$ are sets of operator synthesis with respect to the Haar measure, using the definition of operator synthesis due to Arveson. This result is applied to obtain an analogue of a result of Froelich: a tensor formula for the algebras associated with the pre-orders defined by closed unital subsemigroups of $G$ .

It is shown that the Schatten–von Neumann p-norm of a Volterra integral operator of the form

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is sandwiched, when 2 [les ] p < ∞, by an integral expression similar to the well-known Hilbert–Schmidt norm.

The $n$ th quantum derivative ${\cal D}_nf(x)$ of the real-valued function $f$ is defined for each real non-zero $x$ as \[ \lim_{q\to 1}\frac{\displaystyle\sum\limits^n_{k=0}(-1)^k\left[\begin{array}{c}n\\k\end{array}\right]_qq^{(k-1)k/2}f(q^{n-k}x)}{q^{(n-1)n/2}(q-1)^nx^n}, \] where $\left[\begin{array}{c}n\\k\end{array}\right]_q$ the $q$ -binomial coefficient. If the $n$ th Peano derivative exists at $x$ , which is to say that if $f$ can be approximated by an $n$ th degree polynomial at the point $x$ , then it is not hard to see that ${\cal D}_nf(x)$ must also exist at that point. Consideration of the function $|1-x|$ at $x=1$ shows that the second quantum derivative is more general than the second Peano derivative. However, it can be shown that the existence of the $n$ th quantum derivative at each point of a set necessarily implies the existence of the $n$ th Peano derivative at almost every point of that set.

In [1] Brauer puts forward a series of questions on group representation theory in order to point out areas which were not well understood. One of these, which we denote by (B1), is the following: what information in addition to the character table determines a (finite) group? In previous papers [5, 7–13], the original work of Frobenius on group characters has been re-examined and has shed light on some of Brauer's questions, in particular an answer to (B1) has been given as follows.

Frobenius defined for each character χ of a group G functions χ(k)[ratio ]G(k) → [Copf ] for k = 1, …, degχ with χ(1) = χ. These functions are called the k-characters (see [10] or [11] for their definition). The 1-, 2- and 3-characters of the irreducible representations determine a group [7, 8] but the 1- and 2-characters do not [12]. Summaries of this work are given in [11] and [13].

Let [Oscr ] be a complete discrete valuation ring with unique maximal ideal J([Oscr ]), let K be its quotient field of characteristic 0, and let k be its residue field [Oscr ]/J([Oscr ]) of prime characteristic p. We fix a finite group G, and we assume that K is big enough for G, that is, K contains all the [mid ]G[mid ]-th roots of unity, where [mid ]G[mid ] is the order of G. In particular, K and k are both splitting fields for all subgroups of G. Suppose that H is an arbitrary subgroup of G. Consider blocks (block ideals) A and B of the group algebras RG and RH, respectively, where R∈{[Oscr ], k}. We consider the following question: when are A and B Morita equivalent? Actually, we deal with ‘naturally Morita equivalent blocks A and B’, which means that A is isomorphic to a full matrix algebra of B, as studied by B. Külshammer. However, Külshammer assumes that H is normal in G, and we do not make this assumption, so we get generalisations of the results of Külshammer. Moreover, in the case H is normal in G, we get the same results as Külshammer; however, he uses the results of E. C. Dade, and we do not.

Simple singularities of functions on space curves are classified. It is shown that their bifurcation sets have properties very similar to those of functions on smooth manifolds and complete intersections: the k(π, 1)-theorem for the bifurcation diagram of functions is true, and both this diagram and the discriminant are Saito's free divisors.

In the existing theory of self-affine tiles, one knows that the Lebesgue measure of any integral self-affine tile corresponding to a standard digit set must be a positive integer and every integral self-affine tile admits some lattice $\varGamma\subseteq\mathbb{Z}^n$ as a translation tiling set of $\mathbb{R}^n$. In this paper, we give algorithms to evaluate the Lebesgue measure of any such integral self-affine tile $K$ and to determine all of the lattice tilings of $\mathbb^n$ by $K$. Moreover, we also propose and determine algorithmically another type of translation tiling of $\mathbb{R}^n$ by $K$, which we call natural tiling. We also provide an algorithm to decide whether or not Lebesgue measure of the set $K\cap (K+j),\ j\in\mathbb{Z}^n$, is strictly positive.

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, the first known family of partial difference sets with negative Latin square type parameters is constructed in nonelementary abelian groups, the groups $\Z_4^{2k}\times \Z_2^{4 \ell-4k}$ for all $k$ when $\ell$ is odd and for all $k < \ell$ when $\ell$ is even. Similarly, partial difference sets with Latin square type parameters are constructed in the same groups for all $k$ when $\ell$ is even and for all $k<\ell$ when $\ell$ is odd. These constructions provide the first example where the non-homomorphic bijection approach outlined by Hagita and Schmidt can produce difference sets in groups that previously had no known constructions. Computer computations indicate that the strongly regular graphs associated to the partial difference sets are not isomorphic to the known graphs, and it is conjectured that the family of strongly regular graphs will be new.

Let $k$ be a field of characteristic zero, $f(X,Y), g(X,Y)\,{\in}\,k[X,Y]$, $g(X,Y)\,{\notin}\,(X,Y)$ and $d\,{:=} g(X,Y)\,{\partial}/ {\partial X}+f(X,Y)\,{\partial}/{\partial Y}$. A connection is established between the $d$-simplicity of the local ring $k[X,Y]_{(X,Y)}$ and the transcendency of the solution in $tk[[t]]$ of the algebraic differential equation $g(t,y(t))\,{\cdot}\, ({\partial}/{\partial t})y(t)=f(t,y(t))$. This connection is used to obtain some interesting results in the theory of the formal power series and to construct new examples of differentially simple rings.

Let $A$ be an infinite-dimensional $C^*$-algebra. It is proved that every nonempty relatively weakly open subset of the closed unit ball $B_A$ of $A$ has diameter equal to 2. This implies that $B_A$ is not dentable, and that there is not any point of continuity for the identity mapping $(B_A,{\rm weak)\,{\longrightarrow}\,(B_A,{\rm norm})$.

the convergence case of a khintchine-type theorem for a large class of hyperplanes is obtained. the approach to the problem is from a dynamical viewpoint, and a method due to kleinbock and margulis is modified to prove the result.

Let $G$ be a finite group scheme operating on an algebraic variety $X$ , both defined over an algebraically closed field $k$ . The paper first investigates the properties of the quotient morphism $X\longrightarrow X/G$ over the open subset of $X$ consisting of points whose stabilizers have maximal index in $G$ . Given a $G$ -linearized coherent sheaf on $X$ , it describes similarly an open subset of $X$ over which the invariants in the sheaf behave nicely in some way. The points in $X$ with linearly reductive stabilizers are characterized in representation theoretic terms. It is shown that the set of such points is nonempty if and only if the field of rational functions $k(X)$ is an injective $G$ -module. Applications of these results to the invariants of a restricted Lie algebra ${\frak g}$ operating on the function ring $k[X]$ by derivations are considered in the final section. Furthermore, conditions are found ensuring that the ring $k[X]^{\frak g}$ is generated over the subring of $p$ th powers in $k[X]$ , where $p={\rm char}\,k>0$ , by a given system of invariant functions and is a locally complete intersection.

The main goals of the paper are to establish a priori bounds for the prime power divisors of the discriminants of spinor regular positive definite primitive integral ternary quadratic lattices, and to describe a procedure for determining all such lattices.