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.

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.

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 $G$ .

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.

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.

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.

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 [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].

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.

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.

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.

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.

Throughout this paper k denotes a fixed commutative ground ring. A Cohen–Macaulay complex is a finite simplicial complex satisfying a certain homological vanishing condition. These complexes have been the subject of much research; introductions can be found in, for example, Björner, Garsia and Stanley [6] or Budach, Graw, Meinel and Waack [7]. It is known (see, for example, Cibils [8], Gerstenhaber and Schack [10]) that there is a strong connection between the (co)homology of an arbitrary simplicial complex and that of its incidence algebra. We show how the Cohen–Macaulay property fits into this picture, establishing the following characterization.

A pure finite simplicial complex is Cohen–Macaulay over k if and only if the incidence algebra over k of its augmented face poset, graded in the obvious way by chain lengths, is a Koszul ring.

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 $A$ 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 $(B_A,{\rm weak)\,{\longrightarrow}\,(B_A,{\rm norm})$.

Permutations that have no fixed points have been known for a very long time as ‘derangements’. Under that heading Rouse-Ball [10, p. 46] puts the matter in the following charming way: ‘Suppose you have written a letter to each of n different friends, and addressed the n corresponding envelopes. In how many ways can you make the regrettable mistake of putting every letter into a wrong envelope?’ He traces this problem back to 1713 and since then it has occurred in one form or another in many elementary texts on probability theory (see for example [12, p. 57] and references cited by Rouse-Ball). Let pn be the probability that all the letters are put into the wrong envelope. It is well known that pn→e−1 as n→∞, and that, moreover, convergence is very fast. In fact, pn =[sum ]n0(−1)r/r!, so that [mid ]pn−e−1[mid ]<1/(n+1)!.

A derangement can be thought of as a fixed-point-free element of the symmetric group Sym(n). In this paper we turn our attention to eigenvalue-free elements of finite linear groups. Since an eigenvector of a linear transformation X in a vector space V corresponds to a fixed point of X in the projective space whose points are the 1-dimensional subspaces of V, eigenvalue-free invertible matrices correspond to derangements in projective groups.

The classification theorem of compact surfaces states that every topological orientable compact surface is homeomorphic to a sphere or to a ‘torus’ of genus $g$, with $g\,=\,1,2,\dots$. It is proved in the paper that every hyperbolic Riemann surface except for $\bold D\setminus\{0\}$ can be decomposed into basic pieces of only a few different types: Y-pieces, funnels and half-disks. As a corollary of this result, the generalization of the classification theorem to non-compact surfaces is obtained.

On the basis of some new Liouville theorems, under suitable conditions, a priori estimates are obtained of positive solutions of the problem \[-\Delta _pu=\lambda u^{\alpha }-a(x)u^q\quad \mbox{in}\;\Omega,\qquad u|_{\partial \Omega }=0,\] where $\Omega \subset {\mathbb{R}}^N$ ($N\geq 2$) is a bounded smooth domain, $p>1$ and λ is a parameter, α, q are given constants such that $p-1<\alpha <p^*-1$, $\alpha <q$, $p^*=Np/(N-p)$ if $N > p$ and $p^*=\infty $ when $N\leq p$, and $a(x)$ is a continuous nonnegative function. Making use of the Leray–Schauder degree of a compact mapping and a priori estimates, the paper finds that the problem above possesses at least one positive solution. It also discusses the corresponding perturbed problem, where $a(x)$ is replaced by $a(x)+\epsilon$, $\epsilon>0$. The results are strikingly different from those obtained for the case $\alpha=p-1$.

The relationship between the connections on two Riemannian manifolds that are related by a Riemannian submersion are used to characterise parallel translation on shape space in terms of its behaviour on the pre-shape sphere and so to derive a concrete procedure for the unrolling, or development, of curves in shape space onto tangent spaces. This will enable one to adapt to shape spaces the standard statistical procedures for fitting curves to Euclidean data.