In this paper we introduce a nested family of spaces of continuous functions defined
on the spectrum of a uniform algebra. The smallest space in the family is the
uniform algebra itself. In the ``finite dimensional'' case, from some point on the
spaces will be the space of all continuous complex-valued functions on the
spectrum. These spaces are defined in terms of solutions to the nonlinear
Cauchy--Riemann equations as introduced by the author in 1976, so they are not
generally linear spaces of functions. However, these spaces do shed light on the
higher dimensional properties of a uniform algebra. In particular, these spaces are
directly related to the generalized Shilov boundary of the uniform algebra (as
defined by the author and, independently, by Sibony in the early 1970s).
We provide concise series representations for various
L-series integrals. Different techniques are needed below and above
the abscissa of absolute convergence of the underlying L-series.
Consider a real potential $V$ on
$\RR^d$, $d\geq 2$, and the Schr\"odinger equation:
\begin{equation}
\tag{LS} \label{LS1} i\partial_t u +\Delta u -Vu=0,\quad
u_{\restriction t=0}=u_0\in L^2.
\end{equation}
In this paper, we investigate the minimal local regularity of $V$
needed to get local in time dispersive estimates (such as local in
time Strichartz estimates or local smoothing effect with gain of
$1/2$ derivative) on solutions of \eqref{LS1}. Prior works
show some dispersive properties when $V$ (small at infinity) is in
$L^{d/2}$ or in spaces just a little larger but with a smallness
condition on $V$ (or at least on its negative part).
In this work, we prove the critical character of these results by
constructing a positive potential $V$ which has compact support,
bounded outside $0$ and of the order $(\log|x|)^2/|x|^2$ near $0$.
The lack of dispersiveness comes from the existence of a sequence
of quasimodes for the operator $P:=-\Delta+V$.
The elementary construction of $V$ consists in sticking together
concentrated, truncated potential wells near $0$. This yields a
potential oscillating with infinite speed and amplitude at $0$,
such that the operator $P$ admits a sequence of quasi-modes of
polynomial order whose support concentrates on the pole.
In this paper, we study the tensor product $\pi=\sigma^{\min}\otimes
\sigma^{\min}$ of the minimal representation $\sigma^{\min}$ of $O(p,q)$ with
itself, and decompose $\pi$ into a direct integral of irreducible
representations. The decomposition is given in terms of the Plancherel measure
on a certain real hyperbolic space.
In this paper, we investigate the higher simplicial cohomology
groups of the convolution algebra $\ell^1(S)$ for various semigroups
$S$. The classes of semigroups considered are semilattices, Clifford
semigroups, regular Rees semigroups and the additive semigroups of
integers greater than $a$ for some integer $a$. Our results are of
two types: in some cases, we show that some cohomology groups are $0$,
while in some other cases, we show that some cohomology groups are
Banach spaces.
Fix an integer $m>1$, and set $\zeta_{m}=\exp(2\pi i/m)$. Let ${\bar x}$
denote the multiplicative inverse of $x$ modulo $m$. The Kloosterman
sums $R(d)=\sum_{x} \zeta_{m}^{x + d{\bar x}}$, $1 \leq d \leq
m$, $(d,m)=1$,
satisfy the polynomial
$$f_{m}(x) = \prod_{d} (x-R(d)) = x^{\phi(m)} +c_{1}
x^{\phi(m)-1} + \dots + c_{\phi(m)},$$
where the sum and product are taken over a complete system of reduced residues
modulo $m$. Here we give a natural factorization of $f_{m}(x)$, namely,
$$ f_{m}(x) = \prod_{\sigma} f_{m}^{(\sigma)}(x),$$
where $\sigma$ runs through the square classes of the group ${\bf Z}_{m}^{*}$
of reduced residues modulo $m$. Questions concerning the explicit
determination of the factors $f_{m}^{(\sigma)}(x)$ (or at least their
beginning coefficients), their reducibility over the rational field
${\bf Q}$ and duplication among the factors are studied. The treatment
is similar to what has been done for period polynomials for finite
fields.
Given a finite group $G$, we examine the classification of all
frame representations of $G$ and the classification of all
$G$-frames, i.e., frames induced by group representations of $G$.
We show that the exact number of equivalence classes of $G$-frames
and the exact number of frame representations can be explicitly
calculated. We also discuss how to calculate the largest number
$L$ such that there exists an $L$-tuple of strongly disjoint
$G$-frames.
We study a real hypersurface $M$ in a complex space
form $\mn$, $c \neq 0$, whose shape operator and structure tensor
commute each other on the holomorphic distribution of $M$.
We study natural $*$-valuations, $*$-places and graded $*$-rings
associated with $*$-ordered rings.
We prove that the natural $*$-valuation is always quasi-Ore and is
even quasi-commutative (i.e., the corresponding graded $*$-ring is
commutative), provided the ring contains an imaginary unit.
Furthermore, it is proved that the graded $*$-ring is isomorphic
to a twisted semigroup algebra. Our results are applied to answer a question
of Cimpri\v c regarding $*$-orderability of quantum
groups.
In this paper, we consider Hermitian harmonic maps from
Hermitian manifolds into convex balls. We prove that there exist
no non-trivial Hermitian harmonic maps from closed Hermitian
manifolds into convex balls, and we use the heat flow method to
solve the Dirichlet problem for Hermitian harmonic maps when the
domain is a compact Hermitian manifold with non-empty boundary.
Starting with the Leibniz algebra defined by a $\varphi$-dialgebra, we
construct examples of ``coquecigrues,'' in the sense of Loday, that is to
say, manifolds whose tangent structure at a distinguished point coincides
with that of the Leibniz algebra. We discuss some possible
implications and generalizations of this construction.
We study the structure of the sets of symmetric sequences and
spreading models of an Orlicz sequence space equipped with partial
order with respect to domination of bases. In the cases that these
sets are ``small'', some descriptions of the structure of these posets
are obtained.
We show that there exists a non-archimedean
Fr\'echet-Montel space $W$ with a basis and with a continuous norm
such that any non-archimedean Fr\'echet space of countable type is isomorphic
to a quotient of $W$. We also prove that any non-archimedean nuclear
Fr\'echet space is isomorphic to a quotient of some non-archimedean nuclear
Fr\'echet space with a basis and with a continuous norm.
Let $G(X)$ denote the number of positive composite integers $n$
satisfying $\sum_{j=1}^{n-1}j^{n-1}\equiv -1 \tmod{n}$.
Then $G(X)\ll X^{1/2}\log X$ for sufficiently large $X$.
It is shown that the coniveau filtration on the cohomology
of smooth projective varieties is preserved up to shift
by pushforwards, pullbacks and products.
We show that there exists a closed infinite dimensional subspace
of $H^\infty(B^n)$ such that every function of norm one is
universal for some sequence of automorphisms of $B^n$.
A relation between the anticyclic structure of the dendriform operad
and the Coxeter transformations in the Grothendieck groups of the
derived categories of modules over the Tamari posets is obtained.
We prove that every real
algebraic integer $\alpha$ is expressible by a
difference of two Mahler measures of integer polynomials.
Moreover, these polynomials can be chosen in such a way that they
both have the same degree as that of $\alpha$, say
$d$, one of these two polynomials is irreducible and
another has an irreducible factor of degree $d$, so
that $\alpha=M(P)-bM(Q)$ with irreducible polynomials
$P, Q\in \mathbb Z[X]$ of degree $d$ and a
positive integer $b$. Finally, if $d \leqslant 3$, then one can take $b=1$.
Given an odd surjective Galois representation $\varrho\from \G_\Q\to\PGL_2(\F_3)$ and a
positive integer~$N$, there exists a twisted modular curve $X(N,3)_\varrho$
defined over $\Q$ whose rational points classify the quadratic $\Q$-curves of degree $N$
realizing~$\varrho$. This paper gives a method to provide an explicit plane quartic model for
this curve in the genus-three case $N=5$.
Let $G=({\mathbb Z}/a\rtimes{\mathbb Z}/b) \times
\SL_2(\mathbb{F}_p)$, and let $X(n)$ be an $n$-dimensional
$CW$-complex of the homotopy type of an $n$-sphere. We study the
automorphism group $\Aut (G)$ in order to compute the number of
distinct homotopy types of spherical space forms with respect to free
and cellular $G$-actions on all $CW$-complexes $X(2dn-1)$, where $2d$
is the period of $G$. The groups ${\mathcal E}(X(2dn-1)/\mu)$ of self
homotopy equivalences of space forms $X(2dn-1)/\mu$ associated with
free and cellular $G$-actions $\mu$ on $X(2dn-1)$ are determined as
well.
We prove that the elliptic surface
$y^2=x^3+2(t^8+14t^4+1)x+4t^2(t^8+6t^4+1)$ has geometric Mordell--Weil
rank $15$. This completes a list of Kuwata, who gave explicit examples
of elliptic $K3$-surfaces with geometric Mordell--Weil ranks
$0,1,\dots, 14, 16, 17, 18$.
Let $A$ be a stable, separable, real rank zero $C^{*}$-algebra, and
suppose that $A$ has an AF-skeleton with only finitely many extreme
traces.
Then the corona algebra ${\mathcal M}(A)/A$ is
purely infinite in the sense of Kirchberg and R\o rdam, which implies that
$A$ has the corona factorization property.
The original Sato--Tate Conjecture concerns the angle distribution
of the eigenvalues arising from non-CM elliptic curves. In this paper,
we formulate a modular analogue of the Sato--Tate Conjecture and prove
that the angles arising from non-CM holomorphic Hecke
eigenforms with non-trivial central characters are not distributed
with respect to the Sate--Tate measure
for non-CM elliptic curves. Furthermore, under a reasonable conjecture,
we prove that the expected distribution is uniform.
Dans le papier Beyond Endoscopy une id\'ee pour entamer la
fonctorialit\'e en utilisant la formule des traces a \'et\'e
introduite. Maints probl\`emes, l'existence d'une limite convenable
de la formule des traces, est eqquiss\'ee dans cette note
informelle mais seulement pour $GL(2)$ et les corps des fonctions
rationelles sur un corps fini et en ne pas resolvant
bon nombre de questions.
Using ideas of S. Wassermann on non-exact $C^*$-algebras and
property T groups, we show that one of his examples of non-invertible
$C^*$-extensions is not semi-invertible. To prove this, we
show that a certain element vanishes in the asymptotic tensor
product. We also show that a modification of the example gives
a $C^*$-extension which is not even invertible up to homotopy.
In his last letter to Hardy, Ramanujan
defined 17 functions $F(q)$, where $|q|<1$. He called them mock theta
functions, because as $q$ radially approaches any point $e^{2\pi ir}$
($r$ rational), there is a theta function $F_r(q)$ with $F(q)-F_r(q)=O(1)$.
In this paper we establish the relationship between two families of mock
theta functions.
We prove Beurling's theorem for rank $1$ Riemannian symmetric
spaces and relate its consequences with the characterization of
the heat kernel of the symmetric space.
Let $p$ be a prime greater than or equal to 17 and
congruent to
2 modulo 3. We use results of Beukers and Helou on
Cauchy--Liouville--Mirimanoff
polynomials to show that
the intersection of the Fermat curve of degree $p$ with the
line $X+Y=Z$ in the projective plane
contains no algebraic points of degree
$d$ with $3 \leq d \leq 11$.
We prove a result on
the roots of these polynomials and show that, experimentally,
they seem to satisfy
the conditions of a mild extension of
an irreducibility theorem of P\'{o}lya and Szeg\"{o}.
These conditions are conjecturally
also necessary for irreducibility.
Recently I. Castro and F. Urbano introduced the
Lagrangian catenoid.
Topologically, it is $\mathbb R\times S^{n-1}$ and its induced metric is
conformally flat,
but not cylindrical. Their result is that if a Lagrangian minimal
submanifold in
${\mathbb C}^n$ is foliated by round $(n-1)$-spheres, it is congruent to
a Lagrangian
catenoid. Here we study the question of conformally flat, minimal, Lagrangian
submanifolds in
${\mathbb C}^n$. The general problem is formidable, but we first show that such a
submanifold resembles a Lagrangian catenoid in that its Schouten tensor has an
eigenvalue of multiplicity one. Then, restricting to the case of at most two
eigenvalues, we show that the submanifold is either flat and totally
geodesic or is
homothetic to (a piece of) the Lagrangian catenoid.
We obtain Hauptmoduls of genus zero congruence
subgroups of the type $\Gamma_0^+(p):=\linebreak\Gamma_0(p)+w_p$, where $p$ is
a prime and $w_p$ is the Atkin--Lehner involution. We then use the
Hauptmoduls, along with modular functions on $\Gamma_1(p)$
to construct families of cyclic extensions of quadratic number
fields. Further examples of cyclic extension of bi-quadratic and
tri-quadratic number fields are also given.
In this paper we investigate the existence of positive solutions
for nonlinear elliptic problems driven by the $p$-Laplacian with a
nonsmooth potential (hemivariational inequality). Under asymptotic
conditions that make the Euler functional indefinite and
incorporate in our framework the asymptotically linear problems,
using a variational approach based on nonsmooth critical point
theory, we obtain positive smooth solutions. Our analysis also
leads naturally to multiplicity results.
Let $M$ be a symplectic $4$-dimensional manifold equipped with a
Hamiltonian circle action with isolated fixed points. We describe a
method for computing its integral equivariant cohomology in terms of
fixed point data. We give some examples of these computations.
We study two sufficient conditions that imply global injectivity
for a $C^1$ map $X\colon \R^2\to \R^2$ such that its Jacobian at any
point of $\R^2$ is not zero. One is based on the notion of
half-Reeb component and the other on the Palais--Smale condition.
We improve the first condition using the notion of inseparable
leaves. We provide a new proof of the sufficiency of the second
condition. We prove that both conditions are not equivalent, more
precisely we show that the Palais--Smale condition implies the
nonexistence of inseparable leaves, but the converse is not true.
Finally, we show that the Palais--Smale condition it is not a
necessary condition for the global injectivity of the map $X$.
We extend the notion of linking number of an
ordinary link of two components to that of a singular link
with transverse intersections in which case the linking
number is a half-integer. We then apply it to simplify
the construction of the Seifert matrix, and therefore
the Alexander polynomial, in a natural way.
Beginning with a seminal paper of R\'enyi, expansions in noninteger real bases have been widely
studied in the last
forty years. They turned out to be relevant in
various domains of mathematics, such as the theory of finite
automata, number
theory, fractals or dynamical systems.
Several results were extended by Dar\'oczy and K\'atai
for expansions
in complex bases. We introduce an adaptation of the so-called greedy
algorithm to the complex case, and we
generalize one of their main theorems.
We show that, for most of the elliptic curves $\E$ over a prime finite
field
$\F_p$ of $p$ elements, the discriminant $D(\E)$ of the quadratic number
field containing the endomorphism ring of $\E$ over $\F_p$
is sufficiently large.
We also obtain an asymptotic formula for the number of distinct
quadratic number fields generated by the endomorphism rings
of all elliptic curves over $\F_p$.
Let $X$ be a smooth complex projective curve of genus $g\geq
1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let
$W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$
by $\xi^{-1}$. There is a rational map
$D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$,
where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces
of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable
vector
bundles on $X$ with trivial determinant. We prove that if $n=2$,
then $D_{\xi}$ is
everywhere defined and is injective.
In the present paper, we introduce a modification of the Meyer-K\"{o}nig and
Zeller (MKZ) operators which preserve the test functions $f_{0}(x)=1$ and
$f_{2}(x)=x^{2}$, and we show that this modification provides a better estimation
than the classical MKZ operators on the interval $[\frac{1}{2},1)$ with
respect to the modulus of continuity and the Lipschitz class functionals.
Furthermore, we present the $r-$th order generalization of our operators and
study their approximation properties.
Let $G_1$ and $G_2$ be $p$-adic groups. We describe a decomposition of
${\rm Ext}$-groups in the category of smooth representations of
$G_1 \times G_2$ in terms of ${\rm Ext}$-groups for $G_1$ and $G_2$.
We comment on ${\rm Ext}^1_G(\pi,\pi)$ for a supercuspidal
representation
$\pi$ of a $p$-adic group $G$. We also consider an example of
identifying
the class, in a suitable ${\rm Ext}^1$, of a Jacquet module of certain
representations of $p$-adic ${\rm GL}_{2n}$.
Let $\mathcal{F}$ be a family of vector fields on a manifold or a
subcartesian space spanning a distribution $D$. We prove that an orbit $O$
of $\mathcal{F}$ is an integral manifold of $D$ if $D$ is involutive on $O$
and it has constant rank on $O$. This result implies Frobenius' theorem, and
its various generalizations, on manifolds as well as on subcartesian spaces.
We prove that a separable, nuclear, purely infinite, simple
$C^*$-algebra satisfying the universal coefficient theorem
is weakly semiprojective if and only if
its $K$-groups are direct sums of cyclic groups.
This paper presents some
results on the simple exceptional Jordan algebra over an algebraically
closed field $\Phi$ of characteristic not $2$. Namely an example of
simple decomposition of $H(O_3)$ into the sum of two subalgebras
of the type $H(Q_3)$ is produced, and it is shown that this
decomposition is the only one possible in terms of simple
subalgebras.
Let $M$ be an $m$ dimensional submanifold in the Euclidean space
${\mathbf R}^n$ and $H$ be the mean curvature of $M$. We obtain
some low geometric estimates of the total square mean curvature
$\int_M H^2 d\sigma$. The low bounds are geometric invariants
involving the volume of $M$, the total scalar curvature of $M$,
the Euler characteristic and the circumscribed ball of $M$.
We prove that for a simply connected closed
$4$-dimensional manifold, its embeddings
into the sphere of dimension $6$ are all
concordant to each other.
Let $v \ge k \ge 1$ and $\lam \ge 0$ be integers. A block
design $\BD(v,k,\lambda)$ is a collection $\cA$ of $k$-subsets of a
$v$-set $X$ in which every unordered pair of elements from $X$ is
contained in exactly $\lambda$ elements of $\cA$. More generally, for a
fixed simple graph $G$, a graph design $\GD(v,G,\lambda)$ is a
collection $\cA$ of graphs isomorphic to $G$ with vertices in $X$ such
that every unordered pair of elements from $X$ is an edge of exactly
$\lambda$ elements of $\cA$. A famous result of Wilson says that for a
fixed $G$ and $\lambda$, there exists a $\GD(v,G,\lambda)$ for all
sufficiently large $v$ satisfying certain necessary conditions. A
block (graph) design as above is resolvable if $\cA$ can be
partitioned into partitions of (graphs whose vertex sets partition)
$X$. Lu has shown asymptotic existence in $v$ of resolvable
$\BD(v,k,\lambda)$, yet for over twenty years the analogous problem for
resolvable $\GD(v,G,\lambda)$ has remained open. In this paper, we settle
asymptotic existence of resolvable graph designs.
It is shown that Schatten $p$-classes
of operators between Hilbert spaces of different (infinite)
dimensions have ultrapowers which are (completely) isometric to
non-commutative $L_p$-spaces. On the other hand, these Schatten
classes are not themselves isomorphic to non-commutative $L_p$
spaces. As a consequence, the class of non-commutative $L_p$-spaces
is not axiomatizable in the first-order language developed by
Henson and Iovino for normed space structures, neither in the
signature of Banach spaces, nor in that of operator spaces. Other
examples of the same phenomenon are presented that belong to the
class of corners of non-commutative $L_p$-spaces. For $p=1$ this
last class, which is the same as the class of preduals of ternary
rings of operators, is itself axiomatizable in the signature of
operator spaces.
If $A$ is a subset of the set of reflections of a finite Coxeter
group $W$, we define a sub-$\ZM$-module $\DC_A(W)$ of the group
algebra $\ZM W$. We discuss cases where this submodule is a
subalgebra. This family of subalgebras includes strictly the
Solomon descent algebra, the group algebra and, if $W$ is of type
$B$, the Mantaci--Reutenauer algebra.
The paper offers a study of the inverse Laplace
transforms of the functions $I_n(rs)\{sI_n^{'}(s)\}^{-1}$ where
$I_n$ is the modified Bessel function of the first kind and $r$ is
a parameter. The present study is a continuation of the author's
previous work %[\textit{Canadian Mathematical Bulletin} 45]
on the
singular behavior of the special case of the functions in
question, $r$=1. The general case of $r \in [0,1]$ is addressed,
and it is shown that the inverse Laplace transforms for such $r$
exhibit significantly more complex behavior than their
predecessors, even though they still only have two different types
of points of discontinuity: singularities and finite
discontinuities. The functions studied originate from
non-stationary fluid-structure interaction, and as such are of
interest to researchers working in the area.
In this paper we show that any Frobenius split, smooth, projective
threefold over a perfect field of characteristic $p>0$ is
Hodge--Witt. This is proved by generalizing to the case of
threefolds a well-known criterion due to N.~Nygaard for surfaces to be Hodge-Witt.
We also show that the second crystalline
cohomology of any smooth, projective Frobenius split variety does
not have any exotic torsion. In the last two sections we include
some applications.
For $p>0$ and for a given set $E$ of type $G_{\delta}$ in the boundary
of the unit disc $\partial\mathbb D$ we construct a holomorphic function
$f\in\mathbb O(\mathbb D)$ such that
\[
\int_{\mathbb D\setminus[0,1]E}|ft|^{p}\,d\mathfrak{L}^{2}<\infty\]
and\[
E=E^{p}(f)=\Bigl\{ z\in\partial\mathbb D:\int_{0}^{1}|f(tz)|^{p}\,dt=\infty\Bigr\} .\]
In particular if a set $E$ has a measure equal to zero, then a function
$f$ is constructed as integrable with power $p$ on the unit disc $\mathbb D$.
Let $p$ be a prime and $F$ a field containing a primitive $p$-th
root of unity. Then for $n\in \N$, the cohomological dimension
of the maximal pro-$p$-quotient $G$ of the absolute Galois group
of $F$ is at most $n$ if and only if the corestriction maps
$H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open
subgroups $H$ of index $p$. Using this result, we generalize
Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp}
H^n(H,\Fp)$.
Soit $q$ une puissance d'un nombre premier
$p$. Dans cette note on \'etablit la g\'en\'eralisation suivante
d'un th\'eor\`eme de Wintenberger : tout sous-groupe ab\'elien
ferm\'e du groupe des $\mathbb F_q$-auto\-morphismes continus du corps
des s\'eries formelles $\mathbb F_q(\!(X)\!)$ muni de sa filtration
de ramification est un groupe filtr\'e isomorphe au groupe de Galois
d'une extension ab\'elienne d'un corps local {\`a} corps
r\'esiduel $\mathbb F_q$, filtr\'e par les groupes de ramification
de l'extension en num\'erotation inf\'erieure.
Let $R$ be a commutative Noetherian ring, $\fa$ an ideal
of $R$ and $M$ a finitely generated $R$-module. Let $t$ be a
non-negative integer. It is known that if the local cohomology
module $\H^i_\fa(M)$ is finitely generated for all $i<t$, then
$\Hom_R(R/\fa, \H^t_\fa(M))$ is finitely generated. In this paper it
is shown that if $\H^i_\fa(M)$ is Artinian for all $i<t$, then
$\Hom_R(R/\fa, \H^t_\fa(M))$ need not be Artinian, but it has a
finitely generated submodule $N$ such that
$\Hom_R(R/\fa,\H^t_\fa(M))/N$ is Artinian.
Let $\mathfrak g$ be a semisimple complex Lie algebra and $\k\subset\g$ be
any algebraic subalgebra reductive in $\mathfrak g$. For any simple
finite dimensional $\mathfrak k$-module $V$, we construct simple
$(\mathfrak g,\mathfrak k)$-modules $M$ with finite dimensional $\mathfrak k$-isotypic
components such that $V$ is a $\mathfrak k$-submodule of $M$ and the Vogan
norm of any simple $\k$-submodule $V'\subset M, V'\not\simeq V$, is
greater than the Vogan norm of $V$. The $(\mathfrak g,\mathfrak k)$-modules
$M$ are subquotients of the fundamental series of
$(\mathfrak g,\mathfrak k)$-modules.
We present partial positive results supporting a conjecture that
admitting an equivalent Lipschitz (or uniformly) weak$^*$ Kadec--Klee norm is
a three space property.
It is shown that if a Banach space is saturated with infinite
dimensional subspaces in which all ``special" $n$-tuples of
vectors are equivalent with constants independent of $n$-tuples and
of $n$, then the space contains asymptotic-$l_p$ subspaces
for some $1 \leq p \leq \infty$.
This extends a result by Figiel, Frankiewicz, Komorowski and
Ryll-Nardzewski.
Let $G$ be a compact topological group and let $f\colon G\to G$ be a
continuous transformation of $G$. Define $f^*\colon G\to G$ by
$f^*(x)=f(x^{-1})x$ and let $\mu=\mu_G$ be Haar measure on $G$. Assume
that $H=\Imag f^*$ is a subgroup of $G$ and for every
measurable $C\subseteq H$,
$\mu_G((f^*)^{-1}(C))=\mu_H(C)$. Then for every measurable
$C\subseteq G$, there exist $S\subseteq C$ and $g\in G$ such that
$f(Sg^{-1})\subseteq Cg^{-1}$ and $\mu(S)\ge(\mu(C))^2$.