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1. CJM Online first

Henniart, Guy; Sécherre, Vincent
Types et contragrédientes
Soit $\mathrm{G}$ un groupe réductif $p$-adique, et soit $\mathrm{R}$ un corps algébriquement clos. Soit $\pi$ une représentation lisse de $\mathrm{G}$ dans un espace vectoriel $\mathrm{V}$ sur $\mathrm{R}$. Fixons un sous-groupe ouvert et compact $\mathrm{K}$ de $\mathrm{G}$ et une représentation lisse irréductible $\tau$ de $\mathrm{K}$ dans un espace vectoriel $\mathrm{W}$ de dimension finie sur $\mathrm{R}$. Sur l'espace $\mathrm{Hom}_{\mathrm{K}(\mathrm{W},\mathrm{V})}$ agit l'algèbre d'entrelacement $\mathscr{H}(\mathrm{G},\mathrm{K},\mathrm{W})$. Nous examinons la compatibilité de ces constructions avec le passage aux représentations contragrédientes $\mathrm{V}^ėe$ et $\mathrm{W}^ėe$, et donnons en particulier des conditions sur $\mathrm{W}$ ou sur la caractéristique de $\mathrm{R}$ pour que le comportement soit semblable au cas des représentations complexes. Nous prenons un point de vue abstrait, n'utilisant que des propriétés générales de $\mathrm{G}$. Nous terminons par une application à la théorie des types pour le groupe $\mathrm{GL}_n$ et ses formes intérieures sur un corps local non archimédien.

Keywords:modular representations of p-adic reductive groups, types, contragredient, intertwining
Category:22E50

2. CJM Online first

Zhang, Xi; Zhang, Xiangwen
Generalized Kähler--Einstein Metrics and Energy Functionals
In this paper, we consider a generalized Kähler-Einstein equation on Kähler manifold $M$. Using the twisted $\mathcal K$-energy introduced by Song and Tian, we show that the existence of generalized Kähler-Einstein metrics with semi-positive twisting $(1, 1)$-form $\theta $ is also closely related to the properness of the twisted $\mathcal K$-energy functional. Under the condition that the twisting form $\theta $ is strictly positive at a point or $M$ admits no nontrivial Hamiltonian holomorphic vector field, we prove that the existence of generalized Kähler-Einstein metric implies a Moser-Trudinger type inequality.

Keywords:complex Monge--Ampère equation, energy functional, generalized Kähler--Einstein metric, Moser--Trudinger type inequality
Categories:53C55, 32W20

3. CJM 2013 (vol 66 pp. 284)

Eikrem, Kjersti Solberg
Random Harmonic Functions in Growth Spaces and Bloch-type Spaces
Let $h^\infty_v(\mathbf D)$ and $h^\infty_v(\mathbf B)$ be the spaces of harmonic functions in the unit disk and multi-dimensional unit ball which admit a two-sided radial majorant $v(r)$. We consider functions $v $ that fulfill a doubling condition. In the two-dimensional case let $u (re^{i\theta},\xi) = \sum_{j=0}^\infty (a_{j0} \xi_{j0} r^j \cos j\theta +a_{j1} \xi_{j1} r^j \sin j\theta)$ where $\xi =\{\xi_{ji}\}$ is a sequence of random subnormal variables and $a_{ji}$ are real; in higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients $a_{ji} $ which imply that $u$ is in $h^\infty_v(\mathbf B)$ almost surely. Our estimate improves previous results by Bennett, Stegenga and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces, which generalizes results by Anderson, Clunie and Pommerenke and by Guo and Liu.

Keywords:harmonic functions, random series, growth space, Bloch-type space
Categories:30B20, 31B05, 30H30, 42B05

4. CJM 2012 (vol 66 pp. 170)

Guitart, Xavier; Quer, Jordi
Modular Abelian Varieties Over Number Fields
The main result of this paper is a characterization of the abelian varieties $B/K$ defined over Galois number fields with the property that the $L$-function $L(B/K;s)$ is a product of $L$-functions of non-CM newforms over $\mathbb Q$ for congruence subgroups of the form $\Gamma_1(N)$. The characterization involves the structure of $\operatorname{End}(B)$, isogenies between the Galois conjugates of $B$, and a Galois cohomology class attached to $B/K$. We call the varieties having this property strongly modular. The last section is devoted to the study of a family of abelian surfaces with quaternionic multiplication. As an illustration of the ways in which the general results of the paper can be applied we prove the strong modularity of some particular abelian surfaces belonging to that family, and we show how to find nontrivial examples of strongly modular varieties by twisting.

Keywords:Modular abelian varieties, $GL_2$-type varieties, modular forms
Categories:11G10, 11G18, 11F11

5. CJM 2012 (vol 65 pp. 721)

Adamus, Janusz; Randriambololona, Serge; Shafikov, Rasul
Tameness of Complex Dimension in a Real Analytic Set
Given a real analytic set $X$ in a complex manifold and a positive integer $d$, denote by $\mathcal A^d$ the set of points $p$ in $X$ at which there exists a germ of a complex analytic set of dimension $d$ contained in $X$. It is proved that $\mathcal A^d$ is a closed semianalytic subset of $X$.

Keywords:complex dimension, finite type, semianalytic set, tameness
Categories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15

6. CJM 2012 (vol 65 pp. 195)

Penegini, Matteo; Polizzi, Francesco
Surfaces with $p_g=q=2$, $K^2=6$, and Albanese Map of Degree $2$
We classify minimal surfaces of general type with $p_g=q=2$ and $K^2=6$ whose Albanese map is a generically finite double cover. We show that the corresponding moduli space is the disjoint union of three generically smooth irreducible components $\mathcal{M}_{Ia}$, $\mathcal{M}_{Ib}$, $\mathcal{M}_{II}$ of dimension $4$, $4$, $3$, respectively.

Keywords:surface of general type, abelian surface, Albanese map
Categories:14J29, 14J10

7. CJM 2012 (vol 65 pp. 52)

Christensen, Erik; Sinclair, Allan M.; Smith, Roger R.; White, Stuart
C$^*$-algebras Nearly Contained in Type $\mathrm{I}$ Algebras
In this paper we consider near inclusions $A\subseteq_\gamma B$ of C$^*$-algebras. We show that if $B$ is a separable type $\mathrm{I}$ C$^*$-algebra and $A$ satisfies Kadison's similarity problem, then $A$ is also type $\mathrm{I}$ and use this to obtain an embedding of $A$ into $B$.

Keywords:C$^*$-algebras, near inclusions, perturbations, type I C$^*$-algebras, similarity problem
Category:46L05

8. CJM 2011 (vol 64 pp. 1201)

Aistleitner, Christoph; Elsholtz, Christian
The Central Limit Theorem for Subsequences in Probabilistic Number Theory
Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Keywords:central limit theorem, lacunary sequences, linear Diophantine equations, Ramsey type theorem
Categories:60F05, 42A55, 11D04, 05C55, 11K06

9. CJM 2011 (vol 64 pp. 892)

Hytönen, Tuomas; Liu, Suile; Yang, Dachun; Yang, Dongyong
Boundedness of Calderón-Zygmund Operators on Non-homogeneous Metric Measure Spaces
Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

Keywords:upper doubling, geometrical doubling, dominating function, weak type $(1,1)$ estimate, Calderón-Zygmund operator, maximal operator
Categories:42B20, 42B25, 30L99

10. CJM 2011 (vol 63 pp. 500)

Dadarlat, Marius; Elliott, George A.; Niu, Zhuang
One-Parameter Continuous Fields of Kirchberg Algebras. II
Parallel to the first two authors' earlier classification of separable, unita one-parameter, continuous fields of Kirchberg algebras with torsion free $\mathrm{K}$-groups supported in one dimension, one-parameter, separable, uni continuous fields of AF-algebras are classified by their ordered $\mathrm{K}_0$-sheaves. Effros-Handelman-Shen type theorems are pr separable unital one-parameter continuous fields of AF-algebras and Kirchberg algebras.

Keywords:continuous fields of C$^*$-algebras, $\mathrm{K}_0$-presheaves, Effros--Handeen type theorem
Category:46L35

11. CJM 2010 (vol 62 pp. 961)

Aleman, Alexandru; Duren, Peter; Martín, María J.; Vukotić, Dragan
Multiplicative Isometries and Isometric Zero-Divisors
For some Banach spaces of analytic functions in the unit disk (weighted Bergman spaces, Bloch space, Dirichlet-type spaces), the isometric pointwise multipliers are found to be unimodular constants. As a consequence, it is shown that none of those spaces have isometric zero-divisors. Isometric coefficient multipliers are also investigated.

Keywords:Banach spaces of analytic functions, Hardy spaces, Bergman spaces, Bloch space, Dirichlet space, Dirichlet-type spaces, pointwise multipliers, coefficient multipliers, isometries, isometric zero-divisors
Categories:30H05, 46E15

12. CJM 2010 (vol 62 pp. 827)

Ouyang, Caiheng; Xu, Quanhua
BMO Functions and Carleson Measures with Values in Uniformly Convex Spaces
This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $\mathbf{T}$, respectively. For $1< q<\infty$ and a Banach space $B$, we prove that there exists a positive constant $c$ such that $$\sup_{z_0\in D}\int_{D}(1-|z|)^{q-1}\|\nabla f(z)\|^q P_{z_0}(z) dA(z) \le c^q\sup_{z_0\in D}\int_{\mathbf{T}}\|f(z)-f(z_0)\|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ if and only if $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=\frac{1-|z_0|^2}{|1-\bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.

Keywords:BMO, Carleson measures, Lusin type, Lusin cotype, uniformly convex spaces, uniformly smooth spaces
Categories:46E40, 42B25, 46B20

13. CJM 2010 (vol 62 pp. 1116)

Jin, Yongyang; Zhang, Genkai
Degenerate p-Laplacian Operators and Hardy Type Inequalities on H-Type Groups
Let $\mathbb G$ be a step-two nilpotent group of H-type with Lie algebra $\mathfrak G=V\oplus \mathfrak t$. We define a class of vector fields $X=\{X_j\}$ on $\mathbb G$ depending on a real parameter $k\ge 1$, and we consider the corresponding $p$-Laplacian operator $L_{p,k} u= \operatorname{div}_X (|\nabla_{X} u|^{p-2} \nabla_X u)$. For $k=1$ the vector fields $X=\{X_j\}$ are the left invariant vector fields corresponding to an orthonormal basis of $V$; for $\mathbb G$ being the Heisenberg group the vector fields are the Greiner fields. In this paper we obtain the fundamental solution for the operator $L_{p,k}$ and as an application, we get a Hardy type inequality associated with $X$.

Keywords:fundamental solutions, degenerate Laplacians, Hardy inequality, H-type groups
Categories:35H30, 26D10, 22E25

14. CJM 2008 (vol 60 pp. 1067)

Kariyama, Kazutoshi
On Types for Unramified $p$-Adic Unitary Groups
Let $F$ be a non-archimedean local field of residue characteristic neither 2 nor 3 equipped with a galois involution with fixed field $F_0$, and let $G$ be a symplectic group over $F$ or an unramified unitary group over $F_0$. Following the methods of Bushnell--Kutzko for $\GL(N,F)$, we define an analogue of a simple type attached to a certain skew simple stratum, and realize a type in $G$. In particular, we obtain an irreducible supercuspidal representation of $G$ like $\GL(N,F)$.

Keywords:$p$-adic unitary group, type, supercuspidal representation, Hecke algebra
Categories:22E50, 22D99

15. CJM 2008 (vol 60 pp. 391)

Migliore, Juan C.
The Geometry of the Weak Lefschetz Property and Level Sets of Points
In a recent paper, F. Zanello showed that level Artinian algebras in 3 variables can fail to have the Weak Lefschetz Property (WLP), and can even fail to have unimodal Hilbert function. We show that the same is true for the Artinian reduction of reduced, level sets of points in projective 3-space. Our main goal is to begin an understanding of how the geometry of a set of points can prevent its Artinian reduction from having WLP, which in itself is a very algebraic notion. More precisely, we produce level sets of points whose Artinian reductions have socle types 3 and 4 and arbitrary socle degree $\geq 12$ (in the worst case), but fail to have WLP. We also produce a level set of points whose Artinian reduction fails to have unimodal Hilbert function; our example is based on Zanello's example. Finally, we show that a level set of points can have Artinian reduction that has WLP but fails to have the Strong Lefschetz Property. While our constructions are all based on basic double G-linkage, the implementations use very different methods.

Keywords:Weak Lefschetz Property, Strong Lefschetz Property, basic double G-linkage, level, arithmetically Gorenstein, arithmetically Cohen--Macaulay, socle type, socle degree, Artinian reduction
Categories:13D40, 13D02, 14C20, 13C40, 13C13, 14M05

16. CJM 2008 (vol 60 pp. 379)

rgensen, Peter J\o
Finite Cohen--Macaulay Type and Smooth Non-Commutative Schemes
A commutative local Cohen--Macaulay ring $R$ of finite Cohen--Macaulay type is known to be an isolated singularity; that is, $\Spec(R) \setminus \{ \mathfrak {m} \}$ is smooth. This paper proves a non-commutative analogue. Namely, if $A$ is a (non-commutative) graded Artin--Schelter \CM\ algebra which is fully bounded Noetherian and has finite Cohen--Macaulay type, then the non-commutative projective scheme determined by $A$ is smooth.

Keywords:Artin--Schelter Cohen--Macaulay algebra, Artin--Schelter Gorenstein algebra, Auslander's theorem on finite Cohen--Macaulay type, Cohen--Macaulay ring, fully bounded Noetherian algebra, isolated singularity, maximal Cohen--Macaulay module, non-commutative
Categories:14A22, 16E65, 16W50

17. CJM 2005 (vol 57 pp. 648)

Nevins, Monica
Branching Rules for Principal Series Representations of $SL(2)$ over a $p$-adic Field
We explicitly describe the decomposition into irreducibles of the restriction of the principal series representations of $SL(2,k)$, for $k$ a $p$-adic field, to each of its two maximal compact subgroups (up to conjugacy). We identify these irreducible subrepresentations in the Kirillov-type classification of Shalika. We go on to explicitly describe the decomposition of the reducible principal series of $SL(2,k)$ in terms of the restrictions of its irreducible constituents to a maximal compact subgroup.

Keywords:representations of $p$-adic groups, $p$-adic integers, orbit method, $K$-types
Categories:20G25, 22E35, 20H25

18. CJM 2004 (vol 56 pp. 897)

Borwein, Jonathan M.; Borwein, David; Galway, William F.
Finding and Excluding $b$-ary Machin-Type Individual Digit Formulae
Constants with formulae of the form treated by D.~Bailey, P.~Borwein, and S.~Plouffe (\emph{BBP formulae} to a given base $b$) have interesting computational properties, such as allowing single digits in their base $b$ expansion to be independently computed, and there are hints that they should be \emph{normal} numbers, {\em i.e.,} that their base $b$ digits are randomly distributed. We study a formally limited subset of BBP formulae, which we call \emph{Machin-type BBP formulae}, for which it is relatively easy to determine whether or not a given constant $\kappa$ has a Machin-type BBP formula. In particular, given $b \in \mathbb{N}$, $b>2$, $b$ not a proper power, a $b$-ary Machin-type BBP arctangent formula for $\kappa$ is a formula of the form $\kappa = \sum_{m} a_m \arctan(-b^{-m})$, $a_m \in \mathbb{Q}$, while when $b=2$, we also allow terms of the form $a_m \arctan(1/(1-2^m))$. Of particular interest, we show that $\pi$ has no Machin-type BBP arctangent formula when $b \neq 2$. To the best of our knowledge, when there is no Machin-type BBP formula for a constant then no BBP formula of any form is known for that constant.

Keywords:BBP formulae, Machin-type formulae, arctangents,, logarithms, normality, Mersenne primes, Bang's theorem,, Zsigmondy's theorem, primitive prime factors, $p$-adic analysis
Categories:11Y99, 11A51, 11Y50, 11K36, 33B10

19. CJM 1998 (vol 50 pp. 152)

Min, G.
Inequalities for rational functions with prescribed poles
This paper considers the rational system ${\cal P}_n (a_1,a_2,\ldots,a_n):= \bigl\{ {P(x) \over \prod_{k=1}^n (x-a_k)}, P\in {\cal P}_n\bigr\}$ with nonreal elements in $\{a_k\}_{k=1}^{n}\subset\Bbb{C}\setminus [-1,1]$ paired by complex conjugation. It gives a sharp (to constant) Markov-type inequality for real rational functions in ${\cal P}_n (a_1,a_2,\ldots,a_n)$. The corresponding Markov-type inequality for high derivatives is established, as well as Nikolskii-type inequalities. Some sharp Markov- and Bernstein-type inequalities with curved majorants for rational functions in ${\cal P}_n(a_1,a_2,\ldots,a_n)$ are obtained, which generalize some results for the classical polynomials. A sharp Schur-type inequality is also proved and plays a key role in the proofs of our main results.

Keywords:Markov-type inequality, Bernstein-type inequality, Nikolskii-type inequality, Schur-type inequality, rational functions with prescribed poles, curved majorants, Chebyshev polynomials
Categories:41A17, 26D07, 26C15

20. CJM 1998 (vol 50 pp. 210)

Zhao, Kaiming
Isomorphisms between generalized Cartan type $W$ Lie algebras in characteristic $0$
In this paper, we determine when two simple generalized Cartan type $W$ Lie algebras $W_d (A, T, \varphi)$ are isomorphic, and discuss the relationship between the Jacobian conjecture and the generalized Cartan type $W$ Lie algebras.

Keywords:Simple Lie algebras, the general Lie algebra, generalized Cartan type $W$ Lie algebras, isomorphism, Jacobian conjecture
Categories:17B40, 17B65, 17B56, 17B68

21. CJM 1997 (vol 49 pp. 675)

de Cataldo, Mark Andrea A.
Some adjunction-theoretic properties of codimension two non-singular subvarities of quadrics
We make precise the structure of the first two reduction morphisms associated with codimension two non-singular subvarieties of non-singular quadrics $\Q^n$, $n\geq 5$. We give a coarse classification of the same class of subvarieties when they are assumed not to be of log-general-type.}

Keywords:Adjunction Theory, classification, codimension two, conic bundles,, low codimension, non log-general-type, quadric, reduction, special, variety.
Categories:14C05, 14E05, 14E25, 14E30, 14E35, 14J10

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