1. CJM 2016 (vol 68 pp. 816)
 Guo, Xiaoli; Hu, Guoen

On the Commutators of Singular Integral Operators with Rough Convolution Kernels
Let $T_{\Omega}$ be the singular integral operator with kernel
$\frac{\Omega(x)}{x^n}$, where $\Omega$ is homogeneous of degree
zero, has mean value zero and belongs to $L^q(S^{n1})$ for
some
$q\in (1,\,\infty]$. In this paper, the authors establish the
compactness on weighted $L^p$ spaces, and the Morrey spaces,
for the commutator generated by $\operatorname{CMO}(\mathbb{R}^n)$ function
and $T_{\Omega}$. The associated maximal operator and the discrete
maximal operator are also considered.
Keywords:commutator, singular integral operator, compact operator, completely continuous operator, maximal operator, Morrey space Categories:42B20, 47B07 

2. CJM 2015 (vol 69 pp. 130)
 Levin, Aaron; Wang, Julie TzuYueh

On NonArchimedean Curves Omitting Few Components and their Arithmetic Analogues
Let $\mathbf{k}$ be an algebraically closed field complete with respect
to a nonArchimedean absolute value of arbitrary characteristic.
Let $D_1,\dots, D_n$ be effective nef divisors intersecting
transversally in an $n$dimensional nonsingular projective variety
$X$.
We study the degeneracy of nonArchimedean analytic maps from
$\mathbf{k}$ into $X\setminus \cup_{i=1}^nD_i$ under various geometric
conditions. When $X$ is a rational ruled surface and $D_1$ and
$D_2$ are ample, we obtain a necessary and sufficient condition
such that
there is no nonArchimedean analytic map from $\mathbf{k}$ into $X\setminus
D_1 \cup D_2$.
Using the dictionary between nonArchimedean Nevanlinna theory
and Diophantine approximation that originated in
earlier work with T. T. H. An, %
we also study arithmetic analogues of these problems, establishing
results on integral points on these varieties over $\mathbb{Z}$
or the ring of integers of an imaginary quadratic field.
Keywords:nonArchimedean Picard theorem, nonArchimedean analytic curves, integral points Categories:11J97, 32P05, 32H25 

3. CJM 2014 (vol 67 pp. 152)
 Lescop, Christine

On Homotopy Invariants of Combings of Threemanifolds
Combings of compact, oriented $3$dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$valued invariant $\theta_G$ of torsion combings
on closed $3$manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$invariant, which is the simplest configuration
space integral invariant of rational homology $3$balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the CassonWalker invariant.
The article also includes a selfcontained presentation of combings
for $3$manifolds.
Keywords:Spin$^c$structure, nowhere zero vector fields, first Pontrjagin class, Euler class, Heegaard Floer homology grading, Gompf invariant, Theta invariant, CassonWalker invariant, perturbative expansion of ChernSimons theory, configuration space integrals Categories:57M27, 57R20, 57N10 

4. CJM 2013 (vol 66 pp. 205)
 Iovanov, Miodrag Cristian

Generalized Frobenius Algebras and Hopf Algebras
"CoFrobenius" coalgebras were introduced as dualizations of
Frobenius algebras.
We previously showed
that they admit
leftright symmetric characterizations analogue to those of Frobenius
algebras. We consider the more general quasicoFrobenius (QcF)
coalgebras; the first main result in this paper is that these also
admit symmetric characterizations: a coalgebra is QcF if it is weakly
isomorphic to its (left, or right) rational dual $Rat(C^*)$, in the
sense that certain coproduct or product powers of these objects are
isomorphic. Fundamental results of Hopf algebras, such as the
equivalent characterizations of Hopf algebras with nonzero integrals
as left (or right) coFrobenius, QcF, semiperfect or with nonzero
rational dual, as well as the uniqueness of integrals and a short
proof of the bijectivity of the antipode for such Hopf algebras all
follow as a consequence of these results. This gives a purely
representation theoretic approach to many of the basic fundamental
results in the theory of Hopf algebras. Furthermore, we introduce a
general concept of Frobenius algebra, which makes sense for infinite
dimensional and for topological algebras, and specializes to the
classical notion in the finite case. This will be a topological
algebra $A$ that is isomorphic to its complete topological dual
$A^\vee$. We show that $A$ is a (quasi)Frobenius algebra if and only
if $A$ is the dual $C^*$ of a (quasi)coFrobenius coalgebra $C$. We
give many examples of coFrobenius coalgebras and Hopf algebras
connected to category theory, homological algebra and the newer
qhomological algebra, topology or graph theory, showing the
importance of the concept.
Keywords:coalgebra, Hopf algebra, integral, Frobenius, QcF, coFrobenius Categories:16T15, 18G35, 16T05, 20N99, 18D10, 05E10 

5. CJM 2012 (vol 65 pp. 544)
 Deitmar, Anton; Horozov, Ivan

Iterated Integrals and Higher Order Invariants
We show that higher order invariants of smooth functions can be
written as linear combinations of full invariants times iterated
integrals.
The nonuniqueness of such a presentation is captured in the kernel of
the ensuing map from the tensor product. This kernel is computed
explicitly.
As a consequence, it turns out that higher order invariants are a free
module of the algebra of full invariants.
Keywords:higher order forms, iterated integrals Categories:14F35, 11F12, 55D35, 58A10 

6. CJM 2011 (vol 64 pp. 1359)
 Nozaki, Hiroshi; Sawa, Masanori

Note on Cubature Formulae and Designs Obtained from Group Orbits
In 1960,
Sobolev proved that for a finite reflection group $G$,
a $G$invariant cubature formula is of degree $t$ if and only if
it is exact for all $G$invariant polynomials of degree at most $t$.
In this paper,
we find some observations on invariant cubature formulas and Euclidean designs
in connection with the Sobolev theorem.
First, we give an alternative proof of
theorems by Xu (1998) on necessary and sufficient conditions
for the existence of cubature formulas with some strong symmetry.
The new proof is shorter and simpler compared to the original one by Xu, and
moreover gives a general interpretation of
the analyticallywritten conditions of Xu's theorems.
Second,
we extend a theorem by Neumaier and Seidel (1988) on
Euclidean designs to invariant Euclidean designs, and thereby
classify tight Euclidean designs obtained from
unions of the orbits of the corner vectors.
This result generalizes a theorem of Bajnok (2007) which classifies
tight Euclidean designs invariant under the Weyl group of type $B$
to other finite reflection groups.
Keywords:cubature formula, Euclidean design, radially symmetric integral, reflection group, Sobolev theorem Categories:65D32, 05E99, 51M99 

7. CJM 2011 (vol 64 pp. 257)
 Chen, Yanping; Ding, Yong; Wang, Xinxia

Compactness of Commutators for Singular Integrals on Morrey Spaces
In this paper we characterize the
compactness of the commutator $[b,T]$ for the singular integral
operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More
precisely, we prove that if
$b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO}
(\mathbb R^n)$closure of $C_c^\infty(\mathbb R^n)$,
then $[b,T]$ is a compact operator on the
Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and
$0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and
$[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$
for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$.
Moreover, the boundedness of a rough singular integral operator $T$
and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also
given. We obtain a sufficient condition for a
subset in Morrey space to be a strongly precompact set,
which has interest in its own right.
Keywords:singular integral, commutators, compactness, VMO, Morrey space Categories:42B20, 42B99 

8. CJM 2011 (vol 63 pp. 1238)
 Bump, Daniel; Nakasuji, Maki

Casselman's Basis of Iwahori Vectors and the Bruhat Order
W. Casselman defined a basis $f_u$ of Iwahori fixed vectors of a spherical
representation $(\pi, V)$ of a split semisimple $p$adic group $G$ over a
nonarchimedean local field $F$ by the condition that it be dual to the
intertwining operators, indexed by elements $u$ of the Weyl group $W$. On
the other hand, there is a natural basis $\psi_u$, and one seeks to find the
transition matrices between the two bases. Thus, let $f_u = \sum_v \tilde{m}
(u, v) \psi_v$ and $\psi_u = \sum_v m (u, v) f_v$. Using the IwahoriHecke
algebra we prove that if a combinatorial condition is satisfied, then $m (u,
v) = \prod_{\alpha} \frac{1  q^{ 1} \mathbf{z}^{\alpha}}{1
\mathbf{z}^{\alpha}}$, where $\mathbf z$ are the Langlands parameters
for the representation and $\alpha$ runs through the set $S (u, v)$ of
positive coroots $\alpha \in \hat{\Phi}$ (the dual root system of $G$) such
that $u \leqslant v r_{\alpha} < v$ with $r_{\alpha}$ the reflection
corresponding to $\alpha$. The condition is conjecturally always satisfied
if $G$ is simplylaced and the KazhdanLusztig polynomial $P_{w_0 v, w_0 u}
= 1$ with $w_0$ the long Weyl group element. There is a similar formula for
$\tilde{m}$ conjecturally satisfied if $P_{u, v} = 1$.
This leads to various combinatorial conjectures.
Keywords:Iwahori fixed vector, Iwahori Hecke algebra, Bruhat order, intertwining integrals Categories:20C08, 20F55, 22E50 

9. CJM 2011 (vol 64 pp. 151)
 Miller, Steven J.; Wong, Siman

Moments of the Rank of Elliptic Curves
Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis
for the $L$function $L(E_D, s)$ for every quadratic twist $E_D$ of
$E$ by $D\in\mathbb{Z}$. We combine Weil's
explicit formula with techniques of HeathBrown to derive an asymptotic
upper bound for the weighted moments of the analytic rank of $E_D$. We
derive from this an upper bound for the density of lowlying zeros of
$L(E_D, s)$ that is compatible with the random matrix models of Katz and
Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$,
the analytic rank and (assuming in addition the Birch and SwinnertonDyer
conjecture)
the number of integral points of $E_D$ are less than $f(D)$
for almost all $D$.
Keywords:elliptic curve, explicit formula, integral point, lowlying zeros, quadratic twist, rank Categories:11G05, 11G40 

10. CJM 2011 (vol 64 pp. 183)
 Nowak, Adam; Stempak, Krzysztof

Negative Powers of Laguerre Operators
We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$.
For these operators we prove twoweight $L^pL^q$ estimates with ranges of $q$ depending
on $p$. The case of the harmonic oscillator (Hermite operator) has recently
been treated by Bongioanni and Torrea by using a straightforward
approach of kernel estimates. Here these results are applied in certain Laguerre settings.
The procedure is fairly direct for Laguerre function expansions of
Hermite type,
due to some monotonicity properties of the kernels involved.
The case of Laguerre function expansions of convolution type is less straightforward.
For halfinteger type indices $\alpha$ we transfer the desired results from the Hermite setting
and then apply an interpolation argument based on a device we call the
convexity principle
to cover the continuous range of $\alpha\in[1/2,\infty)^d$. Finally, we investigate negative powers
of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and
isomorphic to $\mathbb Z^d_2$. The two weight $L^pL^q$ estimates we obtain in this setting are essentially
consequences of those for Laguerre function expansions of convolution type.
Keywords:potential operator, fractional integral, Riesz potential, negative power, harmonic oscillator, Laguerre operator, Dunkl harmonic oscillator Categories:47G40, 31C15, 26A33 

11. CJM 2010 (vol 62 pp. 787)
 Landquist, E.; Rozenhart, P.; Scheidler, R.; Webster, J.; Wu, Q.

An Explicit Treatment of Cubic Function Fields with Applications
We give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for nonsingularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few squarefree polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.
Keywords:cubic function field, discriminant, nonsingularity, integral basis, genus, signature of a place, class number Categories:14H05, 11R58, 14H45, 11G20, 11G30, 11R16, 11R29 

12. CJM 2009 (vol 62 pp. 202)
13. CJM 2008 (vol 60 pp. 241)
 Alexandrova, Ivana

SemiClassical Wavefront Set and Fourier Integral Operators
Here we define and prove some properties of the semiclassical
wavefront set. We also define and study semiclassical Fourier
integral operators and prove a generalization of Egorov's theorem to
manifolds of different dimensions.
Keywords:wavefront set, Fourier integral operators, Egorov theorem, semiclassical analysis Categories:35S30, 35A27, 58J40, 81Q20 

14. CJM 2007 (vol 59 pp. 673)
 Ash, Avner; Friedberg, Solomon

Hecke $L$Functions and the Distribution of Totally Positive Integers
Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 

15. CJM 2006 (vol 58 pp. 897)
 Courtès, François

Distributions invariantes sur les groupes rÃ©ductifs quasidÃ©ployÃ©s
Soit $F$ un corps local non archim\'edien, et $G$ le groupe des
$F$points d'un groupe r\'eductif connexe quasid\'eploy\'e d\'efini sur $F$.
Dans cet article, on s'int\'eresse aux distributions sur $G$ invariantes
par conjugaison, et \`a l'espace de leurs restrictions \`a l'alg\`ebre de
Hecke $\mathcal{H}$ des fonctions sur $G$ \`a support compact
biinvariantes par un sousgroupe d'Iwahori $I$ donn\'e. On montre tout
d'abord que les valeurs d'une telle distribution sur $\mathcal{H}$
sont enti\`erement d\'etermin\'ees par sa restriction au sousespace de
dimension finie des \'el\'ements de $\mathcal{H}$ \`a support dans la
r\'eunion des sousgroupes parahoriques de $G$ contenant $I$. On utilise
ensuite cette propri\'et\'e pour montrer, moyennant certaines conditions
sur $G$, que cet espace est engendr\'e d'une part par certaines
int\'egrales orbitales semisimples, d'autre part par les int\'egrales
orbitales unipotentes, en montrant tout d'abord des r\'esultats
analogues sur les groupes finis.
Keywords:reductive $p$adic groups, orbital integrals, invariant distributions Categories:22E35, 20G40 

16. CJM 2005 (vol 57 pp. 1314)
 Zhitomirskii, M.

Relative Darboux Theorem for Singular Manifolds and Local Contact Algebra
In 1999 V. Arnol'd introduced the local contact algebra: studying the
problem of classification of singular curves in a contact space, he
showed the existence of the ghost of the contact structure (invariants
which are not related to the induced structure on the curve). Our
main result implies that the only reason for existence of the local
contact algebra and the ghost is the difference between the geometric
and (defined in this paper) algebraic restriction of a $1$form to a
singular submanifold. We prove that a germ of any subset $N$ of a
contact manifold is well defined, up to contactomorphisms, by the
algebraic restriction to $N$ of the contact structure. This is a
generalization of the DarbouxGivental' theorem for smooth
submanifolds of a contact manifold. Studying the difference between
the geometric and the algebraic restrictions gives a powerful tool for
classification of stratified submanifolds of a contact manifold. This
is illustrated by complete solution of three classification problems,
including a simple explanation of V.~Arnold's results and further
classification results for singular curves in a contact space. We
also prove several results on the external geometry of a singular
submanifold $N$ in terms of the algebraic restriction of the contact
structure to $N$. In particular, the algebraic restriction is zero if
and only if $N$ is contained in a smooth Legendrian submanifold of
$M$.
Keywords:contact manifold, local contact algebra,, relative Darboux theorem, integral curves Categories:53D10, 14B05, 58K50 

17. CJM 2002 (vol 54 pp. 709)
 Ismail, Mourad E. H.; Stanton, Dennis

$q$Integral and Moment Representations for $q$Orthogonal Polynomials
We develop a method for deriving integral representations of certain
orthogonal polynomials as moments. These moment representations are
applied to find linear and multilinear generating functions for
$q$orthogonal polynomials. As a byproduct we establish new
transformation formulas for combinations of basic hypergeometric
functions, including a new representation of the $q$exponential
function $\mathcal{E}_q$.
Keywords:$q$integral, $q$orthogonal polynomials, associated polynomials, $q$difference equations, generating functions, AlSalamChihara polynomials, continuous $q$ultraspherical polynomials Categories:33D45, 33D20, 33C45, 30E05 

18. CJM 2000 (vol 52 pp. 381)
 Miyachi, Akihiko

Hardy Space Estimate for the Product of Singular Integrals
$H^p$ estimate for the multilinear operators which are finite sums
of pointwise products of singular integrals and fractional
integrals is given. An application to Sobolev space and some
examples are also given.
Keywords:$H^p$ space, multilinear operator, singular integral, fractional integration, Sobolev space Category:42B20 

19. CJM 1998 (vol 50 pp. 1253)
 LópezBautista, Pedro Ricardo; VillaSalvador, Gabriel Daniel

Integral representation of $p$class groups in ${\Bbb Z}_p$extensions and the Jacobian variety
For an arbitrary finite Galois $p$extension $L/K$ of
$\zp$cyclotomic number fields of $\CM$type with Galois group $G =
\Gal(L/K)$ such that the Iwasawa invariants $\mu_K^$, $ \mu_L^$
are zero, we obtain unconditionally and explicitly the Galois
module structure of $\clases$, the minus part of the $p$subgroup
of the class group of $L$. For an arbitrary finite Galois
$p$extension $L/K$ of algebraic function fields of one variable
over an algebraically closed field $k$ of characteristic $p$ as its
exact field of constants with Galois group $G = \Gal(L/K)$ we
obtain unconditionally and explicitly the Galois module structure
of the $p$torsion part of the Jacobian variety $J_L(p)$ associated
to $L/k$.
Keywords:${\Bbb Z}_p$extensions, Iwasawa's theory, class group, integral representation, fields of algebraic functions, Jacobian variety, Galois module structure Categories:11R33, 11R23, 11R58, 14H40 

20. CJM 1997 (vol 49 pp. 1010)
 Lorente, Maria

A characterization of two weight norm inequalities for onesided operators of fractional type
In this paper we give a characterization of the pairs
of weights $(\w,v)$ such that $T$ maps $L^p(v)$ into
$L^q(\w)$, where $T$ is a general onesided operator
that includes as a particular case the Weyl fractional
integral. As an application we solve the following problem:
given a weight $v$, when is there a nontrivial weight
$\w$ such that $T$ maps $L^p(v)$ into $L^q(\w )$?
Keywords:Weyl fractional integral, weights Categories:26A33, 42B25 

21. CJM 1997 (vol 49 pp. 722)
 Elder, G. Griffith; Madan, Manohar L.

Galois module structure of the integers in wildly ramified $C_p\times C_p$ extensions
Let $L/K$ be a finite Galois extension of local fields which are finite
extensions of $\bQ_p$, the field of $p$adic numbers. Let $\Gal (L/K)=G$,
and $\euO_L$ and $\bZ_p$ be the rings of integers in $L$ and $\bQ_p$,
respectively. And let $\euP_L$ denote the maximal ideal of $\euO_L$. We
determine, explicitly in terms of specific indecomposable $\bZ_p[G]$modules,
the $\bZ_p[G]$module structure of $\euO_L$ and $\euP_L$, for $L$, a
composite of two arithmetically disjoint, ramified cyclic extensions of
$K$, one of which is only weakly ramified in the sense of Erez \cite{erez}.
Keywords:Galois module structureintegral representation. Categories:11S15, 20C32 

22. CJM 1997 (vol 49 pp. 543)
 Ismail, Mourad E. H.; Rahman, Mizan; Suslov, Sergei K.

Some summation theorems and transformations for $q$series
We introduce a double sum extension of a very wellpoised series and
extend to this the transformations of Bailey and Sears as well as the
${}_6\f_5$ summation formula of F.~H.~Jackson and the $q$Dixon sum.
We also give $q$integral representations of the double sum.
Generalizations of the NassrallahRahman integral are also found.
Keywords:Basic hypergeometric series, balanced series,, very wellpoised series, integral representations,, AlSalamChihara polynomials. Categories:33D20, 33D60 
