In this paper, we study the $L^p$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on $L^p$ provided that their kernels satisfy a size condition much weaker than that for the classical Calder\'{o}n--Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

We search for theorems that, given a $C_i$-field $K$ and a subfield $k$ of $K$, allow us to conclude that $k$ is a $C_j$-field for some $j$. We give appropriate theorems in the case $K=k(t)$ and $K = k\llp t\rrp$. We then consider the more difficult case where $K/k$ is an algebraic extension. Here we are able to prove some results, and make conjectures. We also point out the connection between these questions and Lang's conjecture on nonreal function fields over a real closed field.

We extend the methods of Van der Poorten and Chapman for explicitly evaluating the Dede\-kind eta function at quadratic irrationalities. Via evaluation of Hecke $L$-series we obtain new evaluations at points in imaginary quadratic number fields with class numbers 3 and 4. Further, we overcome the limitations of the earlier methods and via modular equations provide explicit evaluations where the class number is 5 or 7.

Using a modification of Webster's proof of the Newlander--Nirenberg theorem, it is shown that, for a weakly convergent sequence of integrable unitary connections on a complex vector bundle over a complex manifold, there is a subsequence of local holomorphic frames that converges strongly in an appropriate Holder class.

The Lusternik--Schnirelmann category has been described in different ways. Two major ones, the first by Ganea, the second by Whitehead, are presented here with a number of variants. The equivalence of these variants relies on the axioms of Quillen's model category, but also sometimes on an additional axiom, the so-called ``cube axiom''.

In this article, we give an explicit formula to compute the non abelian twisted sign-deter\-mined Reidemeister torsion of the exterior of a fibered knot in terms of its monodromy. As an application, we give explicit formulae for the non abelian Reidemeister torsion of torus knots and of the figure eight knot.

In this note we consider $\overline\partial$-problem in line bundles over complex projective space $\mathbb{CP}^1$ and prove that the equation can be solved for $(0,1)$ forms with compact support. As a consequence, any Cauchy-Riemann function on a compact real hypersurface in such line bundles is a jump of two holomorphic functions defined on the sides of the hypersurface. In particular, the results can be applied to $\mathbb{CP}^2$ since by removing a point from it we get a line bundle over $\mathbb{CP}^1$.

We characterize the weight functions $u,v,w$ on $(0,\infty)$ such that $$ \left(\int_0^\infty f^{*}(t)^ qw(t)\,dt\right)^{1/q} \leq C \sup_{t\in(0,\infty)}f^{**}_u(t)v(t), $$ where $$ f^{**}_u(t):=\left(\int_{0}^{t}u(s)\,ds\right)^{-1} \int_{0}^{t}f^*(s)u(s)\,ds. $$ As an application we present a~new simple characterization of the associate space to the space $\Gamma^ \infty(v)$, determined by the norm $$ \|f\|_{\Gamma^ \infty(v)}=\sup_{t\in(0,\infty)}f^{**}(t)v(t), $$ where $$ f^{**}(t):=\frac1t\int_{0}^{t}f^*(s)\,ds. $$

We introduce roots of indecomposable modules over group algebras of finite groups, and we investigate some of their properties. This allows us to correct an error in Landrock's book which has to do with roots of simple modules.

We give a geometric proof of classical results that characterize Pisot numbers as algebraic $\lambda>1$ for which there is $x\neq0$ with $\lambda^nx \to 0 \mod$ and identify such $x$ as members of $\Z[\lambda^{-1}] \cdot \Z[\lambda]^*$ where $\Z[\lambda]^*$ is the dual module of $\Z[\lambda]$.

We construct a one-parameter generic polynomial for $\PSL(2,2^n)$ over $\mathbb F_{2^n}$.

We consider the w$^*$-closed operator algebra $\cA_+$ generated by the image of the semigroup $SL_2(\R_+)$ under a unitary representation $\rho$ of $SL_2(\R)$ on the Hilbert~space $L_2(\R)$. We show that $\cA_+$ is a reflexive operator algebra and $\cA_+=\Alg\cD$ where $\cD$ is a double triangle subspace lattice. Surprisingly, $\cA_+$ is also generated as a w$^*$-closed algebra by the image under $\rho$ of a strict subsemigroup of $SL_2(\R_+)$.

Given a finite group $G$, we attach to the character degrees of $G$ a graph whose vertex set is the set of primes dividing the degrees of irreducible characters of $G$, and with an edge between $p$ and $q$ if $pq$ divides the degree of some irreducible character of $G$. In this paper, we describe which graphs occur when $G$ is a solvable group of Fitting height $2$.

In this paper we give a characterization of real hypersurfaces of type $A$ in a complex two-plane Grassmannian $G_2(\mathbb{C}^{m+2})$ which are tubes over totally geodesic $G_2(\mathbb{C}^{m+1})$ in $G_2(\mathbb{C}^{m+2})$ in terms of the {\it vanishing Lie derivative\/} of the shape operator $A$ along the direction of the Reeb vector field $\xi$.

Given a non-negative, locally integrable function $V$ on $\RR^n$, we give a necessary and sufficient condition that $-\Delta+V$ have purely discrete spectrum, in terms of the scattering length of $V$ restricted to boxes.

We investigate the geometry of manifolds with bounded Ricci curvature in $L^1$-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem.

We give in this note a weighted version of Brianchon and Gram's decomposition for a simple polytope. We can derive from this decomposition the weighted polar formula of Agapito and a weighted version of Brion's theorem, in a manner similar to Haase, where the unweighted case is worked out. This weighted version of Brianchon and Gram' decomposition is a direct consequence of the ordinary Brianchon--Gram formula.

This paper investigates the relationship between a system of differential equations and the underlying geometry associated with it. The geometry of a surface determines shortest paths, or geodesics connecting nearby points, which are defined as the solutions to a pair of second-order differential equations: the Euler--Lagrange equations of the metric. We ask when the converse holds, that is, when solutions to a system of differential equations reveals an underlying geometry. Specifically, when may the solutions to a given pair of second order ordinary differential equations $d^{2}y^{1}/dt^{2} = f(y,\dot{y},t)$ and $d^{2}y^{2}/dt^{2} = g(y,\dot{y},t)$ be reparameterized by $t\rightarrow T(y,t)$ so as to give locally the geodesics of a Euclidean space? Our approach is based upon Cartan's method of equivalence. In the second part of the paper, the equivalence problem is solved for a generic pair of second order ordinary differential equations of the above form revealing the existence of 24 invariant functions.

Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon.

We give a short proof of Totaro's theorem that every$E_8$-torsor over a field $k$ becomes trivial over a finiteseparable extension of $k$of degree dividing $d(E_8)=2^63^25$.

It is shown that the ergodic Hilbert transform exists for a class of bounded symmetric admissible processes relative to invertible measure preserving transformations. This generalizes the well-known result on the existence of the ergodic Hilbert transform.

An invariant is presented which classifies, up to equivariant isomorphism, $C^*$-dynamical systems arising as limits from inductive systems of elementary $C^*$-algebras on which the Euclidean motion group acts by way of unitary representations that decompose into finite direct sums of irreducibles.

A sharp upper bound on the first $S^{1}$ invariant eigenvalue of the Laplacian for $S^1$ invariant metrics on $S^2$ is used to find obstructions to the existence of $S^1$ equivariant isometric embeddings of such metrics in $(\R^3,\can)$. As a corollary we prove: If the first four distinct eigenvalues have even multiplicities then the metric cannot be equivariantly, isometrically embedded in $(\R^3,\can)$. This leads to generalizations of some classical results in the theory of surfaces.

Continuous mappings defined from compact subsets $K$ of complex Euclidean space $\cc^n$ into complex projective space $\pp^m$ are approximated by rational mappings. The fundamental tool employed is homotopy theory.

Let $X$ be a Polish space. We will prove that $$ \dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to } X\}, $$ where $\dim_L X$ and $\dim_T X$ stand for the concentration dimension and the topological dimension of $X$, respectively.

A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak--Siciak theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a double power series $F(x,y)$\ converges on a set of lines of positive capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero.

If $C=C(R)$ denotes the center of a ring $R$ and $g(x)$ is a polynomial in C[x]$, Camillo and Sim\'{o}n called a ring $g(x)$-clean if every element is the sum of a unit and a root of $g(x)$. If $V$ is a vector space of countable dimension over a division ring $D,$ they showed that $\end {}_{D}V$ is $g(x)$-clean provided that $g(x)$ has two roots in $C(D)$. If $g(x)=x-x^{2}$ this shows that $\end {}_{D}V$ is clean, a result of Nicholson and Varadarajan. In this paper we remove the countable condition, and in fact prove that $\Mend {}_{R}M$ is $g(x)$-clean for any semisimple module $M$ over an arbitrary ring $R$ provided that $g(x)\in (x-a)(x-b)C[x]$ where $a,b\in C$ and both $b$ and $b-a$ are units in $R$.

We give a characterization of products of projective spaces using unsplit covering families of rational curves.

A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence.

We extend a result of Noritzsch, which describes the orbit sizes in the action of a Frobenius group $G$ on a finite vector space $V$ under certain conditions, to a more general class of finite solvable groups $G$. This result has applications in computing irreducible character degrees of finite groups. Another application, proved here, is a result concerning the structure of certain groups with few complex irreducible character degrees.

This paper investigates the modularity of three non-rigid Calabi--Yau threefolds with bad reduction at 11. They are constructed as fibre products of rational elliptic surfaces, involving the modular elliptic surface of level 5. Their middle $\ell$-adic cohomology groups are shown to split into two-dimensional pieces, all but one of which can be interpreted in terms of elliptic curves. The remaining pieces are associated to newforms of weight 4 and level 22 or 55, respectively. For this purpose, we develop a method by Serre to compare the corresponding two-dimensional 2-adic Galois representations with uneven trace. Eventually this method is also applied to a self fibre product of the Hesse-pencil, relating it to a newform of weight 4 and level 27.

We prove a self-adjoint analogue of the Marcus--Pisier inequality, comparing the expected value of convex functionals on randomreflection matrices and on elements of the Gaussian orthogonal (or unitary) ensemble.

We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings.

It is shown that the group of compactly supported, measure-preserving homeomorphisms of a connected, second countable manifold is locally contractible in the direct limit topology. Furthermore, this group is weakly homotopically equivalent to the more general group of compactly supported homeomorphisms.

In this paper we study affine completeness of generalised dihedral groups. We give a formula for the number of unary compatible functions on these groups, and we characterise for every $k \in~\N$ the $k$-affine complete generalised dihedral groups. We find that the direct product of a $1$-affine complete group with itself need not be $1$-affine complete. Finally, we give an example of a nonabelian solvable affine complete group. For nilpotent groups we find a strong necessary condition for $2$-affine completeness.

We show that each point of the principal eigencurve of the nonlinear problem $$ -\Delta_{p}u-\lambda m(x)|u|^{p-2}u=\mu|u|^{p-2}u \quad \text{in } \Omega, $$ is stable (continuous) with respect to the exponent $p$ varying in $(1,\infty)$; we also prove some convergence results of the principal eigenfunctions corresponding.

This paper is concerned with the structure of inner $E_0$-semigroups. We show that any inner $E_0$-semigroup acting on an infinite factor $M$ is completely determined by a continuous tensor product system of Hilbert spaces in $M$ and that the product system associated with an inner $E_0$-semigroup is a complete cocycle conjugacy invariant.

It is known that the derivative of a Blaschke product whose zero sequence lies in a Stolz angle belongs to all the Bergman spaces $A^p$ with $0<p<3/2$. The question of whether this result is best possible remained open. In this paper, for a large class of Blaschke products $B$ with zeros in a Stolz angle, we obtain a number of conditions which are equivalent to the membership of $B'$ in the space $A^p$ ($p>1$). As a consequence, we prove that there exists a Blaschke product $B$ with zeros on a radius such that $B'\notin A^{3/2}$.

Free analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

This note shows that any set of cofibrations containing the standard set of generating projective cofibrations determines a cofibrantly generated proper closed model structure on the category of simplicial presheaves on a small Grothendieck site, for which the weak equivalences are the local weak equivalences in the usual sense.

In this paper, we consider the behavior of the commutators of convolution operators on the Triebel--Lizorkin spaces $\dot{F}^{s, q} _p$.

We construct vector-valued modular forms of weight 2 associated to Jacobi-like forms with respect to a symmetric tensor representation of $\G$ by using the method of Kuga and Shimura as well as the correspondence between Jacobi-like forms and sequences of modular forms. As an application, we obtain vector-valued modular forms determined by theta functions and by pseudodifferential operators.

We call $\alpha(z) = a_0 + a_1 z + \dots + a_{n-1} z^{n-1}$ a Littlewood polynomial if $a_j = \pm 1$ for all $j$. We call $\alpha(z)$ self-reciprocal if $\alpha(z) = z^{n-1}\alpha(1/z)$, and call $\alpha(z)$ skewsymmetric if $n = 2m+1$ and $a_{m+j} = (-1)^j a_{m-j}$ for all $j$. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in $\bC$ 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.

In this paper, we find a lower bound on the number of cyclic function fields of prime degree~$l$ whose class numbers are divisible by a given integer $n$. This generalizes a previous result of D. Cardon and R. Murty which gives a lower bound on the number of quadratic function fields with class numbers divisible by $n$.

The purpose of this note is to show that the homologically trivial cycles contructed by Clemens and their generalisations due to Paranjape can be detected by the technique of spreading out. More precisely, we associate to these cycles (and the ambient varieties in which they live) certain families which arise naturally and which are defined over $\bbC$ and show that these cycles, along with their relations, can be detected in the singular cohomology of the total space of these families.

The number of cyclic cubic fields with a given conductor and a given index is determined.

The aim of this paper is to study sequences of integers for which the second differences between their squares are constant. We show that there are infinitely many nontrivial monotone sextuples having this property and discuss some related problems.

By applying the Cayley--Dickson process to the division algebra of real octonions, one obtains a 16-dimensional real algebra known as (real) sedenions. We denote this algebra by $\bA_4$. It is a flexible quadratic algebra (with unit element 1) but not a division algebra. We classify the subalgebras of $\bA_4$ up to conjugacy (i.e., up to the action of the automorphism group $G$ of $\bA_4$) with one exception: we leave aside the more complicated case of classifying the quaternion subalgebras. Any nonzero subalgebra contains 1 and we show that there are no proper subalgebras of dimension 5, 7 or $>8$. The proper non-division subalgebras have dimensions 3, 6 and 8. We show that in each of these dimensions there is exactly one conjugacy class of such subalgebras. There are infinitely many conjugacy classes of subalgebras in dimensions 2 and 4, but only 4 conjugacy classes in dimension 8.

We consider the growth norm of a measurable function $f$ defined by $$\|f\|_{-\sigma}=\ess\{\delta_D(z)^\sigma|f(z)|:z\in D\},$$ where $\delta_D(z)$ denote the distance from $z$ to $\partial D$. We prove some optimal growth norm estimates for $\bar\partial$ on convex domains of finite type.

Let $\Gamma_0$ be a Fuchsian group of the first kind of genus zero and $\Gamma$ be a subgroup of $\Gamma_0$ of finite index of genus zero. We find universal recursive relations giving the $q_{r}$-series coefficients of $j_0$ by using those of the $q_{h_{s}}$-series of $j$, where $j$ is the canonical Hauptmodul for $\Gamma$ and $j_0$ is a Hauptmodul for $\Gamma_0$ without zeros on the complex upper half plane $\mathfrak{H}$ (here $q_{\ell} := e^{2 \pi i z / \ell}$). We find universal recursive formulas for $q$-series coefficients of any modular form on $\Gamma_0^{+}(p)$ in terms of those of the canonical Hauptmodul $j_p^{+}$.

We prove that convex sets are measure convex and extremal sets are measure extremal provided they are of low Borel complexity. We also present examples showing that the positive results cannot be strengthened.

This paper studies Hausdorff--Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies in particular to yield a Hausdorff--Young inequality for nonunimodular groups.

In this article we will show that there are infinitely many symmetric, integral $3 \times 3$ matrices, with zeros on the diagonal, whose eigenvalues are all integral. We will do this by proving that the rational points on a certain non-Kummer, singular K3 surface are dense. We will also compute the entire N\'eron--Severi group of this surface and find all low degree curves on it.

In this paper we determine the structure of the full lift for the Howe correspondence of $(Sp(n),O(V))$ for rank-one reducibilities.

In this paper we describe six pencils of $K3$-surfaces which have large Picard number ($\rho=19,20$) and each contains precisely five special fibers: four have A-D-E singularities and one is non-reduced. In particular, we characterize these surfaces as cyclic coverings of some $K3$-surfaces described in a recent paper by Barth and the author. In many cases, using 3-divisible sets, resp., 2-divisible sets, of rational curves and lattice theory, we describe explicitly the Picard lattices.

Using the Polyak--Viro Gauss diagram formula for the degree-4 Vassiliev invariant, we extend some previous results on positive knots and the non-triviality of the Jones polynomial of untwisted Whitehead doubles.

For a non-trivial knot in the $3$-sphere, only integral Dehn surgery can create a closed $3$-manifold containing a projective plane. If we restrict ourselves to hyperbolic knots, the corresponding claim for a Klein bottle is still true. In contrast to these, we show that non-integral surgery on a hyperbolic knot can create a closed non-orientable surface of any genus greater than two.

Considering a mapping $g$ holomorphic on a neighbourhood of a rationally convex set $K\subset\cc^n$, and range into the complex projective space $\cc\pp^m$, the main objective of this paper is to show that we can uniformly approximate $g$ on $K$ by rational mappings defined from $\cc^n$ into $\cc\pp^m$. We only need to ask that the second \v{C}ech cohomology group $\check{H}^2(K,\zz)$ vanishes.

No abstract.