{If A is a subring of a commutative ring B and if n
is a positive integer, a number of sufficient conditions are given for
``A[[X]]is n-root closed in B[[X]]'' to be equivalent to ``A is n-root
closed in B.'' In addition, it is shown that if S is a multiplicative
submonoid of the positive integers ${\bbd P}$ which is generated by
primes, then there exists a one-dimensional quasilocal integral domain
A (resp., a von Neumann regular ring A) such that $S = \{ n \in {\bbd P}\mid
A$ is $n$-root closed$\}$ (resp., $S = \{n \in {\bbd P}\mid A[[X]]$
is $n$-rootclosed$\}$).

The two theorems proved yield simple yet reasonably
general conditions for triangular matrices to be bounded
operators on $l_p$. The theorems are applied to N\"orlund and
weighted mean matrices.

Garrison [3], Forman [2], and Abel and Siebert [1] showed that for all positive integers
$k$ and $N$, there exists a positive integer $\lambda$ such that $n^k+\lambda$ is
prime for at least $N$ positive integers $n$. In other words, there exists $\lambda$
such that $n^k+\lambda$ represents at least $N$ primes.
We give a quantitative version of this result. We show that there exists
$\lambda \leq x^k$ such that $n^k+\lambda$, $1\leq n\leq x$, represents at
least $(\frac 1k+o(1)) \pi(x)$ primes, as $x\rightarrow \infty$. We also give some
related results.

Let $G$ be a finite group, $H$ a copy of its $p$-Sylow
subgroup, and $\kn$ the $n$-th Morava $K$-theory at $p$.
In this paper we prove that the existence of an
isomorphism between $K(n)^\ast(BG)$ and $K(n)^\ast(BH)$ is
a sufficient condition for $G$ to be $p$-nilpotent.

The Gilbert-Pearson characterization of the spectrum is established
for a generalized Sturm-Liouville equation with two singular
endpoints. It is also shown that strong absolute continuity for the
one singular endpoint problem guarantees absolute continuity for the
two singular endpoint problem. As a consequence, we obtain the result
that strong nonsubordinacy, at one singular endpoint, of a particular
solution guarantees the nonexistence of subordinate solutions at both
singular endpoints.

We consider graded connected Gorenstein algebras with respect
to the evaluation map $\ev_G = \Ext_G(k,\varepsilon )=::
\Ext_G(k,G) \longrightarrow \Ext_G(k,k)$. We prove that if
$\ev_G \neq 0$, then the global dimension of $G$ is finite.

Our main result is showing the asymptotic existence of tight
$\OMEP$s. More precisely, for each fixed number $k$ of rows, and with the
exception of $\OMEP$s of the form $2 \times 2 \times \cdots 2 \times 2s\specdiv 4s$
with $s$ odd and with more than three rows, there are only a finite number
of tight $\OMEP$ parameters for which the tight $\OMEP$ does not exist.

Given an integral functional defined on $L_p$, $1 \leq p <\infty$,
under a growth condition we give an upper bound of the Clarke
directional derivative and we obtain a nice inclusion between the
Clarke subdifferential of the integral functional and the set of
selections of the subdifferential of the integrand.

We study the stability of linear filters associated with certain types of
linear difference equations with variable coefficients. We show that
stability is determined by the locations of the poles of a rational transfer
function relative to the spectrum of an associated weighted shift operator.
The known theory for filters associated with constant-coefficient difference
equations is a special case.

In this paper we prove the following:
1.~~Let $m\ge 2$, $n\ge 1$ be integers and let $G$ be a group such
that $(XY)^n = (YX)^n$ for all subsets $X,Y$ of size $m$ in $G$. Then
\item{a)} $G$ is abelian or a $\BFC$-group of finite exponent bounded by
a function of $m$ and $n$.
\item{b)} If $m\ge n$ then $G$ is abelian or $|G|$
is bounded by a function of $m$ and $n$.
2.~~The only non-abelian group $G$ such that $(XY)^2 = (YX)^2$ for
all subsets $X,Y$ of size $2$ in $G$ is the quaternion group of order $8$.
3.~~Let $m$, $n$ be positive integers and $G$ a group such that
$$
X_1\cdots X_n\subseteq \bigcup_{\sigma \in S_n\bs 1} X_{\sigma (1)}
\cdots X_{\sigma (n)}
$$
for all subsets $X_i$ of size $m$ in $G$. Then $G$ is
$n$-permutable or $|G|$ is bounded by a function of $m$
and $n$.

The splitting pattern of a quadratic form $q$ over
a field $k$ consists of all distinct Witt indices that occur for $q$
over extension fields of $k$. In small dimensions, the complete list
of splitting patterns of quadratic forms is known. We show that
{\it all\/} splitting patterns of quadratic forms of dimension at
most nine can be realized by trace forms.

We construct a ring $R$ which is a sum of two subrings
$A$ and $B$ such that the Levitzki radical of $R$ does not
contain any of the hyperannihilators of $A$ and $B$. This
answers an open question asked by Kegel in 1964.

The main result shows that if $R$ is a semiprime ring satisfying
a polynomial identity, and if $Z(R)$ is the center of $R$, then
$\card R \leq 2^{\card Z(R)}$. Examples show that this bound can
be achieved, and that the inequality fails to hold for rings which
are not semiprime.

We discuss the $q$-exponential functions $e_q$, $E_q$ for $q$ on the unit
circle, especially their continuity in $q$, and analogues of the limit
relation $ \lim_{q\rightarrow 1}e_q((1-q)z)=e^z$.

We investigate the problem of determining when $\IA (F_{n}({\bf A}_{m}{\bf A}))$
is finitely generated for all $n$ and $m$, with $n\geq 2$ and $m\neq 1$. If
$m$ is a nonsquare free integer then $\IA(F_{n}({\bf A}_{m}{\bf A}))$ is not
finitely generated for all $n$ and if $m$ is a square free integer then
$\IA(F_{n}({\bf A}_{m}{\bf A}))$ is finitely generated for all $n$, with
$n\neq 3$, and $\IA(F_{3}({\bf A}_{m}{\bf A}))$ is not finitely generated.
In case $m$ is square free, Bachmuth and Mochizuki claimed in ([7],
Problem 4) that $\TR({\bf A}_{m}{\bf A})$ is $1$ or $4$. We correct their
assertion by proving that $\TR({\bf A}_{m}{\bf A})=\infty $.

Quaternionic numerical range is not always a convex set. In this
note, an explicit criterion is given for the convexity of quaternionic
numerical range.

Let $G$ be any group, and $H$ be a normal subgroup of $G$. Then M.~Hartl
identified the subgroup $G \cap(1+\triangle^3(G)+\triangle(G)\triangle(H))$
of $G$. In this note we give an independent proof of the result of Hartl,
and we identify two subgroups
$G\cap(1+\triangle(H)\triangle(G)\triangle(H)+\triangle([H,G])\triangle(H))$,
$G\cap(1+\triangle^2(G)\triangle(H)+\triangle(K)\triangle(H))$ of $G$ for
some subgroup $K$ of $G$ containing $[H,G]$.

Let $M_n(F)$ be the algebra of $n \times n$
matrices over a field $F$ of characteristic $p>2$ and let $\ast$ be an
involution on $M_n(F)$. If $s_1, \ldots, s_r$ are symmetric
variables we determine the smallest $r$ such that the polynomial
$$
P_{r}(s_1, \ldots, s_{r}) = \sum_{\sigma \in {\cal
S}_r}s_{\sigma(1)}\cdots s_{\sigma(r)}
$$
is a $\ast$-polynomial identity of $M_n(F)$ under either the
symplectic or the transpose involution. We also prove an analogous
result for the polynomial
$$
C_r(k_1, \ldots, k_r, k'_1, \ldots, k'_r) = \sum_
{\sigma, \tau \in {\cal S}_r}k_{\sigma(1)}k'_{\tau(1)}\cdots
k_{\sigma(r)}k'_{\tau(r)}
$$
where $k_1, \ldots, k_r, k'_1, \ldots, k'_r$ are skew
variables under the transpose involution.

If $f(x_1,\dots,x_k)$ is a polynomial with complex coefficients, the Mahler measure
of $f$, $M(f)$ is defined to be the geometric mean of $|f|$ over the $k$-torus
$\Bbb T^k$. We construct a sequence of approximations $M_n(f)$ which satisfy
$-d2^{-n}\log 2 + \log M_n(f) \le \log M(f) \le \log M_n(f)$. We use these to prove
that $M(f)$ is a continuous function of the coefficients of $f$ for polynomials
of fixed total degree $d$. Since $M_n(f)$ can be computed in a finite number
of arithmetic operations from the coefficients of $f$ this also demonstrates
an effective (but impractical) method for computing $M(f)$ to arbitrary
accuracy.

A class of Toeplitz type operators acting on the
weighted Bergman spaces of the unit ball in the $n$-dimensional complex
space is considered and two pluriharmonic symbols of commuting
Toeplitz type operators are completely characterized.

In a paper [1], published in 1990, in a (somewhat inaccessible)
conference proceedings, the authors had shown that for the unitary
operators on a separable Hilbert space, endowed with the strong
operator topology, those with singular, continuous, simple spectrum,
with full support, form a dense $G_\delta$. A similar theorem for
bounded self-adjoint operators with a given norm bound (omitting
simplicity) was recently given by Barry Simon [2], [3], with a totally
different proof. In this note we show that a slight modification of
our argument, combined with the Cayley transform, gives a proof of
Simon's result, with simplicity of the spectrum added.

A new class of homology groups associated to a 3-manifold is defined.
The theories measure the syzygies between skein relations in a skein
module. We investigate some of the properties of the homology theory
associated to the Kauffman bracket.

If $F$ and $F^\prime$ are free
$R$-modules, then $M \cong F/H$ and $M \cong F^\prime/H^\prime$ are
viewed as equivalent presentations of the $R$-module $M$ if there is an
isomorphism $F \to F^\prime$ which carries the submodule $H$ onto $H^\prime$.
We study when presentations of modules of projective dimension $1$ over
Pr\"ufer domains of finite character are necessarily equivalent.

In the present paper we consider the problem of finding power
integral bases in number fields which are composits of two
subfields with coprime discriminants. Especially, we consider
imaginary quadratic extensions of totally real cyclic number
fields of prime degree. As an example we solve the index form
equation completely in a two parametric family of fields of degree
$10$ of this type.

In Bernoulli site percolation on Penrose tilings there are
two natural definitions of the critical probability.
This paper shows that they are equal on almost all Penrose tilings.
It also shows that for almost all Penrose tilings the number
of infinite clusters is almost surely~0 or~1.
The results generalize to percolation on a large class of aperiodic
tilings in arbitrary dimension, to percolation on ergodic subgraphs
of $\hbox{\Bbbvii Z}^d$, and to other percolation processes, including
Bernoulli bond percolation.

A theory of minimal realizations of rational matrix functions $W(\lambda)$
in the ``pencil'' form $W(\lambda)=C(\lambda A_1-A_2)^{-1}B$ is developed.
In particular, properties of the pencil $\lambda A_1-A_2$ are discussed when
$W(\lambda)$ is hermitian on the real line, and when $W(\lambda)$ is
hermitian on the unit circle.

The aim of this article is to obtain an upper bound for the exponential sums
$\sum e(f(x)/q)$, where the summation runs from $x=1$ to $x=q$ with $(x,q)=1$
and $e(\alpha)$ denotes $\exp(2\pi i\alpha)$.
We shall show that the upper bound depends only on the values of $q$ and $s$, %% where $s$ is the number of terms in the polynomial $f(x)$.

The spectra of the Toeplitz operators on the weighted Hardy space
$H^2(Wd\th/2\pi)$ and the Hardy space $H^p(d\th/2\pi)$, and the
singular integral operators on the Lebesgue space $L^2(d\th/2\pi)$
are studied. For example, the theorems of Brown-Halmos type and
Hartman-Wintner type are studied.

A new oscillation criterion is given for the delay differential
equation $x'(t)+p(t)x \left(t-\tau(t)\right)=0$, where $p$, $\tau
\in \C \left([0,\infty),[0,\infty)\right)$ and the function
$T$ defined by $T(t)=t-\tau(t)$, $t\ge 0$ is increasing and such
that $\lim_{t\to\infty}T(t)=\infty$. This criterion concerns the
case where $\liminf_{t\to\infty} \int_{T(t)}^{t}p(s)\,ds\le
\frac{1}{e}$.

For two functions $f$ and $g$, define $g\ll f$ to mean that $g$ satisfies
every algebraic differential equation over the constants satisfied by $f$.
The order $\ll$ was introduced in one of a set of problems on algebraic
differential equations given by the late Lee Rubel. Here we characterise
the set of $g$ such that $g\ll f$, when $f$ is a given Liouvillian function.

Various authors have studied when a Banach space can be renormed so
that every weakly compact convex, or less restrictively every
compact convex set is an intersection of balls. We first observe
that each Banach space can be renormed so that every weakly compact
convex set is an intersection of balls, and then we introduce and
study properties that are slightly stronger than the preceding two
properties respectively.

The growth series of compact hyperbolic Coxeter groups with 4 and 5
generators are explicitly calculated. The assertions of J.~Cannon
and Ph.~Wagreich for the 4-generated groups, that the poles of the
growth series lie
on the unit circle, with the exception of a single real reciprocal pair of
poles, are verified. We also verify that for the 5-generated groups, this
phenomenon fails.

It is known that the Toeplitz algebra associated with any flow
which is both minimal and uniquely ergodic always has a trivial
$K_1$-group. We show in this note that if the unique ergodicity is
dropped, then such $K_1$-group can be non-trivial. Therefore, in
the general setting of minimal flows, even the $K$-theoretical
index is not sufficient for the classification of Toeplitz
operators which are invertible modulo the commutator ideal.

It is known that the product $\omega_1 \times X$ of
$\omega_1$ with an $M_1$-space may be nonnormal. In this paper we
prove that the product $\kappa \times X$ of an uncountable regular
cardinal $\kappa$ with a paracompact semi-stratifiable space is normal
if{f} it is countably paracompact. We also give a sufficient
condition under which the product of a normal space with a paracompact
space is normal, from which many theorems involving such a product
with a countably compact factor can be derived.

Let $(X,L)$ be a polarized manifold over the complex number field
with $\dim X=n$. In this paper, we consider a conjecture of
M.~C.~Beltrametti and A.~J.~Sommese and we obtain that this
conjecture is true if $n=3$ and $h^{0}(L)\geq 2$, or $\dim \Bs
|L|\leq 0$ for any $n\geq 3$. Moreover we can generalize the
result of Sommese.

As a consequence of results due to Bourgain and Stegall, on a
separable Banach space whose unit ball is not dentable, the
set of norm attaining functionals has empty interior (in the
norm topology). First we show that any Banach space can be renormed to
fail this property. Then, our main positive result can be stated as
follows: if a separable Banach space $X$ is very smooth or its bidual
satisfies the $w^{\ast }$-Mazur intersection property, then either $X$
is reflexive or the set of norm attaining functionals has empty
interior, hence the same result holds if $X$ has the Mazur
intersection property and so, if the norm of $X$ is Fr\'{e}chet
differentiable. However, we prove that smoothness is not a sufficient
condition for the same conclusion.

It is known that a semigroup of quasinilpotent integral operators,
with positive lower semicontinuous kernels, on $L^2( X, \mu)$,
where $X$ is a locally compact Hausdorff-Lindel\"of space and $\mu$
is a $\sigma$-finite regular Borel measure on $X$, is
triangularizable. In this article we use the Banach lattice version
of triangularizability to establish the ideal-triangularizability
of a semigroup of positive quasinilpotent integral operators on
$C({\cal K})$ where ${\cal K}$ is a compact Hausdorff space.

In this paper we consider solutions to the free Schr\" odinger
equation in $n+1$ dimensions. When we restrict the last variable
to be a smooth function of the first $n$ variables we find that the
solution, so restricted, is locally in $L^2$, when the initial data
is in an appropriate Sobolev space.

We provide more characterizations of varieties with a weak
difference term and of neutral
varieties. We prove that a variety has a (weak)
difference term (is neutral) with respect to the TC-commutator
iff it has a (weak) difference term
(is neutral) with respect to the linear commutator.
We show that a variety \v\ is congruence meet semi-distributive i{f}f
\v\ is neutral,
i{f}f $M_3$ is not a sublattice of \con a, for ${\bf A} \in \v$, i{f}f
there is a positive integer $n$ such that $\v \smc
\a(\b\o\g)\leq\alpha \beta_n$.

Over a decade ago, this author produced class number one criteria for
real quadratic fields in terms of prime-producing quadratic
polynomials. The purpose of this article is to revisit the problem
from a new perspective with new criteria. We look at the more general
situation involving arbitrary real quadratic orders rather than the
more restrictive field case, and use the interplay between the various
orders to provide not only more general results, but also simpler proofs.

The function $\Delta(x,N)$ as defined in the title is closely
associated via $\Delta(N) = \sup_x |\Delta(x,N)|$ to several problems
in the upper bound sieve. It is also known via a classical theorem of
Franel that certain conjectured bounds involving averages of
$\Delta(x,N)$ are equivalent to the Riemann Hypothesis. We improve the
unconditional bounds which have been hitherto obtained for $\Delta(N)$
and show that these are close to being optimal. Several auxiliary
results relating $\Delta(Np)$ to $\Delta(N)$, where $p$ is a prime
with $p \nmid N$, are also obtained and two new conjectures stated.

We study Hausdorff continua in which every set of certain
cardinality contains a subset which disconnects the space. We show
that such continua are rim-finite. We give characterizations of
this class among metric continua. As an application of our
methods, we show that continua in which each countably infinite set
disconnects are generalized graphs. This extends a result of
Nadler for metric continua.

Let $H$ be a faithfully projective Hopf algebra over a commutative
ring $k$. In \cite{CVZ1, CVZ2} we defined the Brauer group
$\BQ(k,H)$ of $H$ and an homomorphism $\pi$ from Hopf automorphism
group $\Aut_{\Hopf}(H)$ to $\BQ(k,H)$. In this paper, we show that
the morphism $\pi$ can be embedded into an exact sequence.

In this article, making use of the second author's criterion for
exponentiality of a connected solvable Lie group, we give a rather
simple necessary and sufficient condition for the semidirect
product of a torus acting on certain connected solvable Lie groups
to be exponential.

We show that normal and stable normal invariants of polarized
homotopy equivalences of lens spaces $M = L(2^m;\r)$ and
$N = L(2^m;\s)$ are determined by certain $\ell$-polynomials
evaluated on the elementary symmetric functions
$\sigma_i(\rsquare)$ and $\sigma_i(\ssquare)$. Each polynomial
$\ell_k$ appears as the homogeneous part of degree $k$ in the
Hirzebruch multiplicative $L$-sequence. When $n = 8$, the
elementary symmetric functions alone determine the relevant normal
invariants.

In this note we further our investigation of Baer invariants of
groups by obtaining, as consequences of an exact sequence of
A.~S.-T.~Lue, some numerical inequalities for their orders,
exponents, and generating sets. An interesting group theoretic
corollary is an explicit bound for $|\gamma_{c+1}(G)|$ given that
$G/Z_c(G)$ is a finite $p$-group with prescribed order and number
of generators.

Kitada and then Onneweer and Quek have investigated multiplier
operators on Hardy spaces over locally compact Vilenkin groups. In
this note, we provide an improvement to their results for the Hardy
space $H^1$ and provide examples showing that our result applies to a
significantly larger group of multipliers.

It is well known that the compactly supported wavelets cannot belong to
the class $C^\infty({\bf R})\cap L^2({\bf R})$. This is also true for
wavelets with exponential decay. We show that one can construct
wavelets in the class $C^\infty({\bf R})\cap L^2({\bf R})$ that are
``almost'' of exponential decay and, moreover, they are
band-limited. We do this by showing that we can adapt the
construction of the Lemari\'e-Meyer wavelets \cite{LM} that
is found in \cite{BSW} so that we obtain band-limited,
$C^\infty$-wavelets on $\bf R$ that have subexponential decay,
that is, for every $0<\varepsilon<1$, there exits $C_\varepsilon>0$
such that $|\psi(x)|\leq C_\varepsilon e^{-|x|^{1-\varepsilon}}$,
$x\in\bf R$. Moreover, all of its derivatives have also
subexponential decay. The proof is constructive and uses the
Gevrey classes of functions.

Let $b(t)$ be an $L^\infty$ function on $\bR$, $\Omega (\,y')$ be
an $H^1$ function on the unit sphere satisfying the mean zero
property (1) and $Q_m(t)$ be a real polynomial on $\bR$ of degree
$m$ satisfying $Q_m(0)=0$. We prove that the singular integral
operator
$$
T_{Q_m,b} (\,f) (x)=p.v. \int_\bR^n b(|y|) \Omega(\,y) |y|^{-n} f
\left( x-Q_m (|y|) y' \right) \,dy
$$
is bounded in $L^p (\bR^n)$ for $1<p<\infty$, and the bound is
independent of the coefficients of $Q_m(t)$.

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 $\fpp$.

Using the theory of $p$-adic Lie groups we give conditions for a
finitely generated group to admit a splitting as a non-trivial
free product with amalgamation. This can be viewed as an extension
of a theorem of Bass.

The aim of this paper is to characterize those linear maps from a
von~Neumann factor $\A$ into itself which preserve the extreme points
of the unit ball of $\A$. For example, we show that if $\A$ is infinite,
then every such linear preserver can be written as a fixed unitary
operator times either a unital $\ast$-homomorphism or a unital
$\ast$-antihomomorphism.

This paper presents an approach to injectivity theorems via the
Mountain Pass Lemma and raises an open question. The main result
of this paper (Theorem~1.1) is proved by means of the Mountain Pass
Lemma and states that if the eigenvalues of $F' (\x)F' (\x)^{T}$
are uniformly bounded away from zero for $\x \in \hbox{\Bbbvii
R}^{n}$, where $F \colon \hbox{\Bbbvii R}^n \rightarrow
\hbox{\Bbbvii R}^n$ is a class $\cC^{1}$ map, then $F$ is
injective. This was discovered in a joint attempt by the authors
to prove a stronger result conjectured by the first author: Namely,
that a sufficient condition for injectivity of class $\cC^{1}$ maps
$F$ of $\hbox{\Bbbvii R}^n$ into itself is that all the eigenvalues
of $F'(\x)$ are bounded away from zero on $\hbox{\Bbbvii
R}^n$. This is stated as Conjecture~2.1. If true, it would imply
(via {\it Reduction-of-Degree}) {\it injectivity of polynomial
maps} $F \colon \hbox{\Bbbvii R}^n \rightarrow \hbox{\Bbbvii R}^n$
{\it satisfying the hypothesis}, $\det F'(\x) \equiv 1$, of the
celebrated Jacobian Conjecture (JC) of Ott-Heinrich Keller. The
paper ends with several examples to illustrate a variety of cases
and known counterexamples to some natural questions.

For each $n\geq 4$ we construct a class of examples of a minimal
$C$-dependent set of $n$ automorphisms of a prime ring $R$, where $C$
is the extended centroid of $R$. For $n=4$ and $n=5$ it is shown that
the preceding examples are completely general, whereas for $n=6$ an
example is given which fails to enjoy any of the nice properties of
the above example.

We show that for certain compact right topological groups,
$\overline{r(G)}$, the strong operator topology closure of
the image of the right regular representation of $G$ in
${\cal L}({\cal H})$, where ${\cal H} = \L2$, is a compact
topological group and introduce a class of representations,
${\cal R}$, which effectively transfers the representation
theory of $\overline{r(G)}$ over to $G$. Amongst the groups
for which this holds is the class of equicontinuous groups
which have been studied by Ruppert in [10]. We use familiar
examples to illustrate these features of the theory and to
provide a counter-example. Finally we remark that every
equicontinuous group which is at the same time a Borel group
is in fact a topological group.

We present a short proof for a classical result on separating
singularities of holomorphic functions. The proof is based on the
open mapping theorem and the fusion lemma of Roth, which is a basic
tool in complex approximation theory. The same method yields
similar separation results for other classes of functions.

Let $R$ be a ring with $1$ and $P(R)$ the periodic radical of $R$.
We obtain necessary and sufficient conditions for $P(\RG) = 0$ when
$\RG$ is the group ring of an $\FC$ group $G$ and $R$ is commutative. We
also obtain a complete description of $P\bigl(I (X, R)\bigr)$ when
$I(X,R)$ is the incidence algebra of a locally finite partially
ordered set $X$ and $R$ is commutative.

We study metaplectic coverings of the adelized group of a split
connected reductive group $G$ over a number field $F$. Assume its
derived group $G'$ is a simply connected simple Chevalley
group. The purpose is to provide some naturally defined sections
for the coverings with good properties which might be helpful when
we carry some explicit calculations in the theory of automorphic
forms on metaplectic groups. Specifically, we
\begin{enumerate}
\item construct metaplectic coverings of $G({\Bbb A})$ from those
of $G'({\Bbb A})$;
\item for any non-archimedean place $v$, show the section for a
covering of $G(F_{v})$ constructed from a Steinberg section is an
isomorphism, both algebraically and topologically in an open
subgroup of $G(F_{v})$;
\item define a global section which is a product of local sections
on a maximal torus, a unipotent subgroup and a set of
representatives for the Weyl group.

We construct Lipschitz functions such that for all $s>0$ they are
$s$-H\"older, and so proximally, subdifferentiable only on dyadic
rationals and nowhere else. As applications we construct Lipschitz
functions with prescribed H\"older and approximate subderivatives.