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1. CJM Online first

Dow, Alan; Tall, Franklin D.
Normality versus paracompactness in locally compact spaces
This note provides a correct proof of the result claimed by the second author that locally compact normal spaces are collectionwise Hausdorff in certain models obtained by forcing with a coherent Souslin tree. A novel feature of the proof is the use of saturation of the non-stationary ideal on $\omega_1$, as well as of a strong form of Chang's Conjecture. Together with other improvements, this enables the consistent characterization of locally compact hereditarily paracompact spaces as those locally compact, hereditarily normal spaces that do not include a copy of $\omega_1$.

Keywords:normal, paracompact, locally compact, countably tight, collectionwise Hausdorff, forcing with a coherent Souslin tree, Martin's Maximum, PFA(S)[S], Axiom R, moving off property
Categories:54A35, 54D20, 54D45, 03E35, 03E50, 03E55, 03E57

2. CJM Online first

Fricain, Emmanuel; Rupam, Rishika
On asymptotically orthonormal sequences
An asymptotically orthonormal sequence is a sequence which is "nearly" orthonormal in the sense that it satisfies the Parseval equality up to two constants close to one. In this paper, we explore such sequences formed by normalized reproducing kernels for model spaces and de Branges-Rovnyak spaces.

Keywords:function space, de Branges-Rovnyak and model space, reproducing kernel, asymptotically orthonormal sequence
Categories:30J05, 30H10, 46E22

3. CJM Online first

Dantas, Sheldon; García, Domingo; Maestre, Manuel; Martín, Miguel
The Bishop-Phelps-Bollobás property for compact operators
We study the Bishop-Phelps-Bollobás property (BPBp for short) for compact operators. We present some abstract techniques which allows to carry the BPBp for compact operators from sequence spaces to function spaces. As main applications, we prove the following results. Let $X$, $Y$ be Banach spaces. If $(c_0,Y)$ has the BPBp for compact operators, then so do $(C_0(L),Y)$ for every locally compact Hausdorff topological space $L$ and $(X,Y)$ whenever $X^*$ is isometrically isomorphic to $\ell_1$. If $X^*$ has the Radon-Nikodým property and $(\ell_1(X),Y)$ has the BPBp for compact operators, then so does $(L_1(\mu,X),Y)$ for every positive measure $\mu$; as a consequence, $(L_1(\mu,X),Y)$ has the the BPBp for compact operators when $X$ and $Y$ are finite-dimensional or $Y$ is a Hilbert space and $X=c_0$ or $X=L_p(\nu)$ for any positive measure $\nu$ and $1\lt p\lt \infty$. For $1\leq p \lt \infty$, if $(X,\ell_p(Y))$ has the BPBp for compact operators, then so does $(X,L_p(\mu,Y))$ for every positive measure $\mu$ such that $L_1(\mu)$ is infinite-dimensional. If $(X,Y)$ has the BPBp for compact operators, then so do $(X,L_\infty(\mu,Y))$ for every $\sigma$-finite positive measure $\mu$ and $(X,C(K,Y))$ for every compact Hausdorff topological space $K$.

Keywords:Bishop-Phelps theorem, Bishop-Phelps-Bollobás property, norm attaining operator, compact operator
Categories:46B04, 46B20, 46B28, 46B25, 46E40

4. CJM Online first

Günther, Christian; Schmidt, Kai-Uwe
$L^q$ norms of Fekete and related polynomials
A Littlewood polynomial is a polynomial in $\mathbb{C}[z]$ having all of its coefficients in $\{-1,1\}$. There are various old unsolved problems, mostly due to Littlewood and Erdős, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small $L^q$ norm on the complex unit circle. We consider the Fekete polynomials \[ f_p(z)=\sum_{j=1}^{p-1}(j\,|\,p)\,z^j, \] where $p$ is an odd prime and $(\,\cdot\,|\,p)$ is the Legendre symbol (so that $z^{-1}f_p(z)$ is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of $L^q$ and $L^2$ norm of $f_p$ when $q$ is an even positive integer and $p\to\infty$. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many $q$. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the $L^4$ norm of these polynomials.

Keywords:character polynomial, Fekete polynomial, $L^q$ norm, Littlewood polynomial
Categories:11B83, 42A05, 30C10

5. CJM 2016 (vol 68 pp. 1257)

Cascante, Carme; Fàbrega, Joan; Ortega, Joaquín M.
Sharp Norm Estimates for the Bergman Operator from Weighted Mixed-norm Spaces to Weighted Hardy Spaces
In this paper we give sharp norm estimates for the Bergman operator acting from weighted mixed-norm spaces to weighted Hardy spaces in the ball, endowed with natural norms.

Keywords:weighted Hardy space, Bergman operator, sharp norm estimate
Categories:47B38, 32A35, 42B25, 32A37

6. CJM 2016 (vol 69 pp. 321)

De Bernardi, Carlo Alberto; Veselý, Libor
Tilings of Normed Spaces
By a tiling of a topological linear space $X$ we mean a covering of $X$ by at least two closed convex sets, called tiles, whose nonempty interiors are pairwise disjoint. Study of tilings of infinite-dimensional spaces initiated in the 1980's with pioneer papers by V. Klee. We prove some general properties of tilings of locally convex spaces, and then apply these results to study existence of tilings of normed and Banach spaces by tiles possessing certain smoothness or rotundity properties. For a Banach space $X$, our main results are the following. 1. $X$ admits no tiling by Fréchet smooth bounded tiles. 2. If $X$ is locally uniformly rotund (LUR), it does not admit any tiling by balls. 3. On the other hand, some $\ell_1(\Gamma)$ spaces, $\Gamma$ uncountable, do admit a tiling by pairwise disjoint LUR bounded tiles.

Keywords:tiling of normed space, Fréchet smooth body, locally uniformly rotund body, $\ell_1(\Gamma)$-space
Categories:46B20, 52A99, 46A45

7. CJM 2015 (vol 68 pp. 334)

Demchenko, Oleg; Gurevich, Alexander
Kernels in the Category of Formal Group Laws
Fontaine described the category of formal groups over the ring of Witt vectors over a finite field of characteristic $p$ with the aid of triples consisting of the module of logarithms, the Dieudonné module and the morphism from the former to the latter. We propose an explicit construction for the kernels in this category in term of Fontaine's triples. The construction is applied to the formal norm homomorphism in the case of an unramified extension of $\mathbb{Q}_p$ and of a totally ramified extension of degree less or equal than $p$. A similar consideration applied to a global extension allows us to establish the existence of a strict isomorphism between the formal norm torus and a formal group law coming from $L$-series.

Keywords:formal groups, $p$-divisible groups, Dieudonne modules, norm tori

8. CJM 2013 (vol 66 pp. 373)

Kim, Sun Kwang; Lee, Han Ju
Uniform Convexity and Bishop-Phelps-Bollobás Property
A new characterization of the uniform convexity of Banach space is obtained in the sense of Bishop-Phelps-Bollobás theorem. It is also proved that the couple of Banach spaces $(X,Y)$ has the bishop-phelps-bollobás property for every banach space $y$ when $X$ is uniformly convex. As a corollary, we show that the Bishop-Phelps-Bollobás theorem holds for bilinear forms on $\ell_p\times \ell_q$ ($1\lt p, q\lt \infty$).

Keywords:Bishop-Phelps-Bollobás property, Bishop-Phelps-Bollobás theorem, norm attaining, uniformly convex
Categories:46B20, 46B22

9. CJM 2012 (vol 66 pp. 102)

Birth, Lidia; Glöckner, Helge
Continuity of convolution of test functions on Lie groups
For a Lie group $G$, we show that the map $C^\infty_c(G)\times C^\infty_c(G)\to C^\infty_c(G)$, $(\gamma,\eta)\mapsto \gamma*\eta$ taking a pair of test functions to their convolution is continuous if and only if $G$ is $\sigma$-compact. More generally, consider $r,s,t \in \mathbb{N}_0\cup\{\infty\}$ with $t\leq r+s$, locally convex spaces $E_1$, $E_2$ and a continuous bilinear map $b\colon E_1\times E_2\to F$ to a complete locally convex space $F$. Let $\beta\colon C^r_c(G,E_1)\times C^s_c(G,E_2)\to C^t_c(G,F)$, $(\gamma,\eta)\mapsto \gamma *_b\eta$ be the associated convolution map. The main result is a characterization of those $(G,r,s,t,b)$ for which $\beta$ is continuous. Convolution of compactly supported continuous functions on a locally compact group is also discussed, as well as convolution of compactly supported $L^1$-functions and convolution of compactly supported Radon measures.

Keywords:Lie group, locally compact group, smooth function, compact support, test function, second countability, countable basis, sigma-compactness, convolution, continuity, seminorm, product estimates
Categories:22E30, 46F05, 22D15, 42A85, 43A10, 43A15, 46A03, 46A13, 46E25

10. CJM 2012 (vol 64 pp. 1222)

Bobiński, Grzegorz
Normality of Maximal Orbit Closures for Euclidean Quivers
Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

Keywords:normal variety, complete intersection, Euclidean quiver, concealed-canonical algebra
Categories:16G20, 14L30

11. CJM 2012 (vol 64 pp. 1182)

Tall, Franklin D.
PFA$(S)[S]$: More Mutually Consistent Topological Consequences of $PFA$ and $V=L$
Extending the work of Larson and Todorcevic, we show there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$ is a coherent Souslin tree.

Keywords:PFA$(S)[S]$, proper forcing, coherent Souslin tree, locally compact, normal, collectionwise Hausdorff, supercompact cardinal
Categories:54A35, 54D15, 54D20, 54D45, 03E35, 03E57, 03E65

12. CJM 2010 (vol 63 pp. 200)

Rahman, Mizan
An Explicit Polynomial Expression for a $q$-Analogue of the 9-$j$ Symbols
Using standard transformation and summation formulas for basic hypergeometric series we obtain an explicit polynomial form of the $q$-analogue of the 9-$j$ symbols, introduced by the author in a recent publication. We also consider a limiting case in which the 9-$j$ symbol factors into two Hahn polynomials. The same factorization occurs in another limit case of the corresponding $q$-analogue.

Keywords:6-$j$ and 9-$j$ symbols, $q$-analogues, balanced and very-well-poised basic hypergeometric series, orthonormal polynomials in one and two variables, Racah and $q$-Racah polynomials and their extensions
Categories:33D45, 33D50

13. CJM 2009 (vol 61 pp. 1118)

Pontreau, Corentin
Petits points d'une surface
Pour toute sous-vari\'et\'e g\'eom\'etriquement irr\'eductible $V$ du grou\-pe multiplicatif $\mathbb{G}_m^n$, on sait qu'en dehors d'un nombre fini de translat\'es de tores exceptionnels inclus dans $V$, tous les points sont de hauteur minor\'ee par une certaine quantit\'e $q(V)^{-1}>0$. On conna\^it de plus une borne sup\'erieure pour la somme des degr\'es de ces translat\'es de tores pour des valeurs de $q(V)$ polynomiales en le degr\'e de $V$. Ceci n'est pas le cas si l'on exige une minoration quasi-optimale pour la hauteur des points de $V$, essentiellement lin\'eaire en l'inverse du degr\'e. Nous apportons ici une r\'eponse partielle \`a ce probl\`eme\,: nous donnons une majoration de la somme des degr\'es de ces translat\'es de sous-tores de codimension $1$ d'une hypersurface $V$. Les r\'esultats, obtenus dans le cas de $\mathbb{G}_m^3$, mais compl\`etement explicites, peuvent toutefois s'\'etendre \`a $\mathbb{G}_m^n$, moyennant quelques petites complications inh\'erentes \`a la dimension $n$.

Keywords:Hauteur normalisée, groupe multiplicatif, problème de Lehmer, petits points
Categories:11G50, 11J81, 14G40

14. CJM 2008 (vol 60 pp. 520)

Chen, Chang-Pao; Huang, Hao-Wei; Shen, Chun-Yen
Matrices Whose Norms Are Determined by Their Actions on Decreasing Sequences
Let $A=(a_{j,k})_{j,k \ge 1}$ be a non-negative matrix. In this paper, we characterize those $A$ for which $\|A\|_{E, F}$ are determined by their actions on decreasing sequences, where $E$ and $F$ are suitable normed Riesz spaces of sequences. In particular, our results can apply to the following spaces: $\ell_p$, $d(w,p)$, and $\ell_p(w)$. The results established here generalize ones given by Bennett; Chen, Luor, and Ou; Jameson; and Jameson and Lashkaripour.

Keywords:norms of matrices, normed Riesz spaces, weighted mean matrices, Nörlund mean matrices, summability matrices, matrices with row decreasing
Categories:15A60, 40G05, 47A30, 47B37, 46B42

15. CJM 2007 (vol 59 pp. 966)

Forrest, Brian E.; Runde, Volker; Spronk, Nico
Operator Amenability of the Fourier Algebra in the $\cb$-Multiplier Norm
Let $G$ be a locally compact group, and let $A_{\cb}(G)$ denote the closure of $A(G)$, the Fourier algebra of $G$, in the space of completely bounded multipliers of $A(G)$. If $G$ is a weakly amenable, discrete group such that $\cstar(G)$ is residually finite-dimensional, we show that $A_{\cb}(G)$ is operator amenable. In particular, $A_{\cb}(\free_2)$ is operator amenable even though $\free_2$, the free group in two generators, is not an amenable group. Moreover, we show that if $G$ is a discrete group such that $A_{\cb}(G)$ is operator amenable, a closed ideal of $A(G)$ is weakly completely complemented in $A(G)$ if and only if it has an approximate identity bounded in the $\cb$-multiplier norm.

Keywords:$\cb$-multiplier norm, Fourier algebra, operator amenability, weak amenability
Categories:43A22, 43A30, 46H25, 46J10, 46J40, 46L07, 47L25

16. CJM 2005 (vol 57 pp. 750)

Sabourin, Hervé
Sur la structure transverse à une orbite nilpotente adjointe
We are interested in Poisson structures to transverse nilpotent adjoint orbits in a complex semi-simple Lie algebra, and we study their polynomial nature. Furthermore, in the case of $sl_n$, we construct some families of nilpotent orbits with quadratic transverse structures.

Keywords:nilpotent adjoint orbits, conormal orbits, Poisson transverse structure
Categories:22E, 53D

17. 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

18. CJM 2001 (vol 53 pp. 944)

Ludwig, J.; Molitor-Braun, C.
Représentations irréductibles bornées des groupes de Lie exponentiels
Let $G$ be a solvable exponential Lie group. We characterize all the continuous topologically irreducible bounded representations $(T, \calU)$ of $G$ on a Banach space $\calU$ by giving a $G$-orbit in $\frn^*$ ($\frn$ being the nilradical of $\frg$), a topologically irreducible representation of $L^1(\RR^n, \o)$, for a certain weight $\o$ and a certain $n \in \NN$, and a topologically simple extension norm. If $G$ is not symmetric, \ie, if the weight $\o$ is exponential, we get a new type of representations which are fundamentally different from the induced representations. Soit $G$ un groupe de Lie r\'esoluble exponentiel. Nous caract\'erisons toutes les repr\'esentations $(T, \calU)$ continues born\'ees topologiquement irr\'eductibles de $G$ dans un espace de Banach $\calU$ \`a l'aide d'une $G$-orbite dans $\frn^*$ ($\frn$ \'etant le radical nilpotent de $\frg$), d'une repr\'esentation topologiquement irr\'eductible de $L^1(\RR^n, \o)$, pour un certain poids $\o$ et un certain $n \in \NN$, d'une norme d'extension topologiquement simple. Si $G$ n'est pas sym\'etrique, c. \`a d. si le poids $\o$ est exponentiel, nous obtenons un nouveau type de repr\'esentations qui sont fondamentalement diff\'erentes des repr\'esentations induites.

Keywords:groupe de Lie résoluble exponentiel, représentation bornée topologiquement irréductible, orbite, norme d'extension, sous-espace invariant, idéal premier, idéal primitif

19. CJM 2000 (vol 52 pp. 897)

Christiansen, T. J.; Joshi, M. S.
Higher Order Scattering on Asymptotically Euclidean Manifolds
We develop a scattering theory for perturbations of powers of the Laplacian on asymptotically Euclidean manifolds. The (absolute) scattering matrix is shown to be a Fourier integral operator associated to the geodesic flow at time $\pi$ on the boundary. Furthermore, it is shown that on $\Real^n$ the asymptotics of certain short-range perturbations of $\Delta^k$ can be recovered from the scattering matrix at a finite number of energies.

Keywords:scattering theory, conormal, Lagrangian

20. CJM 1999 (vol 51 pp. 26)

Fabian, Marián; Mordukhovich, Boris S.
Separable Reduction and Supporting Properties of Fréchet-Like Normals in Banach Spaces
We develop a method of separable reduction for Fr\'{e}chet-like normals and $\epsilon$-normals to arbitrary sets in general Banach spaces. This method allows us to reduce certain problems involving such normals in nonseparable spaces to the separable case. It is particularly helpful in Asplund spaces where every separable subspace admits a Fr\'{e}chet smooth renorm. As an applicaton of the separable reduction method in Asplund spaces, we provide a new direct proof of a nonconvex extension of the celebrated Bishop-Phelps density theorem. Moreover, in this way we establish new characterizations of Asplund spaces in terms of $\epsilon$-normals.

Keywords:nonsmooth analysis, Banach spaces, separable reduction, Fréchet-like normals and subdifferentials, supporting properties, Asplund spaces
Categories:49J52, 58C20, 46B20

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