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1. CJM Online first

Folha, Abigail; Penafiel, Carlos
Weingarten type surfaces in $\mathbb{H}^2\times\mathbb{R}$ and $\mathbb{S}^2\times\mathbb{R}$
In this article, we study complete surfaces $\Sigma$, isometrically immersed in the product space $\mathbb{H}^2\times\mathbb{R}$ or $\mathbb{S}^2\times\mathbb{R}$ having positive extrinsic curvature $K_e$. Let $K_i$ denote the intrinsic curvature of $\Sigma$. Assume that the equation $aK_i+bK_e=c$ holds for some real constants $a\neq0$, $b\gt 0$ and $c$. The main result of this article state that when such a surface is a topological sphere it is rotational.

Keywords:Weingarten surface, extrinsic curvature, intrinsic curvature, height estimate, rotational Weingarten surface
Categories:53C42, 53C30

2. CJM Online first

Pronk, Dorette; Scull, Laura
Erratum: Translation Groupoids and Orbifold Cohomology
We correct an error in the proof of a lemma in "Translation Groupoids and Orbifold Cohomology", Canadian J. Math Vol 62 (3), pp 614-645 (2010). This error was pointed out to the authors by Li Du of the Georg-August-Universität at Gottingen, who also suggested the outline for the corrected proof.

Keywords:orbifold, equivariant homotopy theory, translation groupoid, bicategory of fractions
Category:57S15

3. CJM Online first

Manon, Christopher
Toric geometry of $SL_2(\mathbb{C})$ free group character varieties from outer space
Culler and Vogtmann defined a simplicial space $O(g)$ called outer space to study the outer automorphism group of the free group $F_g$. Using representation theoretic methods, we give an embedding of $O(g)$ into the analytification of $\mathcal{X}(F_g, SL_2(\mathbb{C})),$ the $SL_2(\mathbb{C})$ character variety of $F_g,$ reproving a result of Morgan and Shalen. Then we show that every point $v$ contained in a maximal cell of $O(g)$ defines a flat degeneration of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ to a toric variety $X(P_{\Gamma})$. We relate $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$ topologically by showing that there is a surjective, continuous, proper map $\Xi_v: \mathcal{X}(F_g, SL_2(\mathbb{C})) \to X(v)$. We then show that this map is a symplectomorphism on a dense, open subset of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ with respect to natural symplectic structures on $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ and $X(v)$. In this way, we construct an integrable Hamiltonian system in $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ for each point in a maximal cell of $O(g)$, and we show that each $v$ defines a topological decomposition of $\mathcal{X}(F_g, SL_2(\mathbb{C}))$ derived from the decomposition of $X(P_{\Gamma})$ by its torus orbits. Finally, we show that the valuations coming from the closure of a maximal cell in $O(g)$ all arise as divisorial valuations built from an associated projective compactification of $\mathcal{X}(F_g, SL_2(\mathbb{C})).$

Keywords:character variety, outer space, analytification, compactification, integrable system
Categories:14M25, 14T05, 14D20

4. CJM Online first

Ghaani Farashahi, Arash
A Class of Abstract Linear Representations for Convolution Function Algebras over Homogeneous Spaces of Compact Groups
This paper introduces a class of abstract linear representations on Banach convolution function algebras over homogeneous spaces of compact groups. Let $G$ be a compact group and $H$ be a closed subgroup of $G$. Let $\mu$ be the normalized $G$-invariant measure over the compact homogeneous space $G/H$ associated to the Weil's formula and $1\le p\lt \infty$. We then present a structured class of abstract linear representations of the Banach convolution function algebras $L^p(G/H,\mu)$.

Keywords:homogeneous space, linear representation, continuous unitary representation, convolution function algebra, compact group, convolution, involution
Categories:43A85, 47A67, 20G05

5. CJM Online first

Hakl, Robert; Zamora, Manuel
Periodic solutions of an indefinite singular equation arising from the Kepler problem on the sphere
We study a second-order ordinary differential equation coming from the Kepler problem on $\mathbb{S}^2$. The forcing term under consideration is a piecewise constant with singular nonlinearity which changes sign. We establish necessary and sufficient conditions to the existence and multiplicity of $T$-periodic solutions.

Keywords:singular differential equation, indefinite singularity, periodic solution, Kepler problem on $\mathbb{S}^1$, degree theory
Categories:34B16, 34C25, 70F05, 70F15

6. CJM Online first

Ng, P. W.; Skoufranis, P.
Closed convex hulls of unitary orbits in certain simple real rank zero C$^*$-algebras
In this paper, we characterize the closures of convex hulls of unitary orbits of self-adjoint operators in unital, separable, simple C$^*$-algebras with non-trivial tracial simplex, real rank zero, stable rank one, and strict comparison of projections with respect to tracial states. In addition, an upper bound for the number of unitary conjugates in a convex combination needed to approximate a self-adjoint are obtained.

Keywords:convex hull of unitary orbits, real rank zero C*-algebras simple, eigenvalue function, majorization
Category:46L05

7. 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

8. CJM Online first

Ciesielski, Krzysztof Chris; Jasinski, Jakub
Fixed point theorems for maps with local and pointwise contraction properties
The paper constitutes a comprehensive study of ten classes of self-maps on metric spaces $\langle X,d\rangle$ with the local and pointwise (a.k.a. local radial) contraction properties. Each of those classes appeared previously in the literature in the context of fixed point theorems. We begin with presenting an overview of these fixed point results, including concise self contained sketches of their proofs. Then, we proceed with a discussion of the relations among the ten classes of self-maps with domains $\langle X,d\rangle$ having various topological properties which often appear in the theory of fixed point theorems: completeness, compactness, (path) connectedness, rectifiable path connectedness, and $d$-convexity. The bulk of the results presented in this part consists of examples of maps that show non-reversibility of the previously established inclusions between theses classes. Among these examples, the most striking is a differentiable auto-homeomorphism $f$ of a compact perfect subset $X$ of $\mathbb R$ with $f'\equiv 0$, which constitutes also a minimal dynamical system. We finish with discussing a few remaining open problems on weather the maps with specific pointwise contraction properties must have the fixed points.

Keywords:fixed point, periodic point, contractive map, locally contractive map, pointwise contractive map, radially contractive map, rectifiably path connected space, d-convex, geodesic, remetrization contraction mapping principle
Categories:54H25, 37C25

9. CJM Online first

Almeida, Víctor; Betancor, Jorge J.; Rodríguez-Mesa, Lourdes
Anisotropic Hardy-Lorentz spaces with variable exponents
In this paper we introduce Hardy-Lorentz spaces with variable exponents associated to dilations in ${\mathbb R}^n$. We establish maximal characterizations and atomic decompositions for our variable exponent anisotropic Hardy-Lorentz spaces.

Keywords:variable exponent Hardy space, Hardy-Lorentz space, anisotropic Hardy space, maximal function, atomic decomposition
Categories:42B30, 42B25, 42B35

10. CJM Online first

Giannopoulos, Apostolos; Koldobsky, Alexander; Valettas, Petros
Inequalities for the surface area of projections of convex bodies
We provide general inequalities that compare the surface area $S(K)$ of a convex body $K$ in ${\mathbb R}^n$ to the minimal, average or maximal surface area of its hyperplane or lower dimensional projections. We discuss the same questions for all the quermassintegrals of $K$. We examine separately the dependence of the constants on the dimension in the case where $K$ is in some of the classical positions or $K$ is a projection body. Our results are in the spirit of the hyperplane problem, with sections replaced by projections and volume by surface area.

Keywords:surface area, convex body, projection
Categories:52A20, 46B05

11. CJM Online first

Bosa, Joan; Petzka, Henning
Comparison Properties of the Cuntz semigroup and applications to C*-algebras
We study comparison properties in the category $\mathrm{Cu}$ aiming to lift results to the C*-algebraic setting. We introduce a new comparison property and relate it to both the CFP and $\omega$-comparison. We show differences of all properties by providing examples, which suggest that the corona factorization for C*-algebras might allow for both finite and infinite projections. In addition, we show that R{\o}rdam's simple, nuclear C*-algebra with a finite and an infinite projection does not have the CFP.

Keywords:classification of C*-algebras, cuntz semigroup
Categories:46L35, 06F05, 46L05, 19K14

12. CJM 2016 (vol 69 pp. 186)

Pan, Shu-Yen
$L$-Functoriality for Local Theta Correspondence of Supercuspidal Representations with Unipotent Reduction
The preservation principle of local theta correspondences of reductive dual pairs over a $p$-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams/Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

Keywords:local theta correspondence, supercuspidal representation, preservation principle, Langlands functoriality
Categories:22E50, 11F27, 20C33

13. CJM Online first

Lee, Jungyun; Lee, Yoonjin
Regulators of an infinite family of the simplest quartic function fields
We explicitly find regulators of an infinite family $\{L_m\}$ of the simplest quartic function fields with a parameter $m$ in a polynomial ring $\mathbb{F}_q [t]$, where $\mathbb{F}_q$ is the finite field of order $q$ with odd characteristic. In fact, this infinite family of the simplest quartic function fields are subfields of maximal real subfields of cyclotomic function fields, where they have the same conductors. We obtain a lower bound on the class numbers of the family $\{L_m\}$ and some result on the divisibility of the divisor class numbers of cyclotomic function fields which contain $\{L_m\}$ as their subfields. Furthermore, we find an explicit criterion for the characterization of splitting types of all the primes of the rational function field $\mathbb{F}_q (t)$ in $\{L_m\}$.

Keywords:regulator, function field, quartic extension, class number
Categories:11R29, 11R58

14. CJM Online first

Lei, Antonio; Loeffler, David; Zerbes, Sarah Livia
On the asymptotic growth of Bloch-Kato--Shafarevich-Tate groups of modular forms over cyclotomic extensions
We study the asymptotic behaviour of the Bloch--Kato--Shafarevich--Tate group of a modular form $f$ over the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$ under the assumption that $f$ is non-ordinary at $p$. In particular, we give upper bounds of these groups in terms of Iwasawa invariants of Selmer groups defined using $p$-adic Hodge Theory. These bounds have the same form as the formulae of Kobayashi, Kurihara and Sprung for supersingular elliptic curves.

Keywords:cyclotomic extension, Shafarevich-Tate group, Bloch-Kato Selmer group, modular form, non-ordinary prime, p-adic Hodge theory
Categories:11R18, 11F11, 11R23, 11F85

15. CJM Online first

Harrison-Trainor, Matthew; Melnikov, Alexander; Miller, Russell
On computable field embeddings and difference closed fields
We investigate when a computable automorphism of a computable field can be effectively extended to a computable automorphism of its (computable) algebraic closure. We then apply our results and techniques to study effective embeddings of computable difference fields into computable difference closed fields.

Keywords:computable algebra, algebraic field, difference field, extension of automorphism
Categories:03D45, 03C57, 12Y05

16. CJM Online first

Diacu, Florin
The classical $N$-body problem in the context of curved space
We provide the differential equations that generalize the Newtonian $N$-body problem of celestial mechanics to spaces of constant Gaussian curvature, $\kappa$, for all $\kappa\in\mathbb R$. In previous studies, the equations of motion made sense only for $\kappa\ne 0$. The system derived here does more than just include the Euclidean case in the limit $\kappa\to 0$: it recovers the classical equations for $\kappa=0$. This new expression of the laws of motion allows the study of the $N$-body problem in the context of constant curvature spaces and thus offers a natural generalization of the Newtonian equations that includes the classical case. We end the paper with remarks about the bifurcations of the first integrals.

Keywords:N-body problem, spaces of constant curvature

17. CJM 2016 (vol 69 pp. 434)

Lee, Hun Hee; Youn, Sang-gyun
New Deformations of Convolution Algebras and Fourier Algebras on Locally Compact Groups
In this paper we introduce a new way of deforming convolution algebras and Fourier algebras on locally compact groups. We demonstrate that this new deformation allows us to reveal some information of the underlying groups by examining Banach algebra properties of deformed algebras. More precisely, we focus on representability as an operator algebra of deformed convolution algebras on compact connected Lie groups with connection to the real dimension of the underlying group. Similarly, we investigate complete representability as an operator algebra of deformed Fourier algebras on some finitely generated discrete groups with connection to the growth rate of the group.

Keywords:Fourier algebra, convolution algebra, operator algebra, Beurling algebra
Categories:43A20, 43A30, 47L30, 47L25

18. CJM Online first

Hartglass, Michael
Free product C*-algebras associated to graphs, free differentials, and laws of loops
We study a canonical C$^*$-algebra, $\mathcal{S}(\Gamma, \mu)$, that arises from a weighted graph $(\Gamma, \mu)$, specific cases of which were previously studied in the context of planar algebras. We discuss necessary and sufficient conditions of the weighting which ensure simplicity and uniqueness of trace of $\mathcal{S}(\Gamma, \mu)$, and study the structure of its positive cone. We then study the $*$-algebra, $\mathcal{A}$, generated by the generators of $\mathcal{S}(\Gamma, \mu)$, and use a free differential calculus and techniques of Charlesworth and Shlyakhtenko, as well as Mai, Speicher, and Weber to show that certain ``loop" elements have no atoms in their spectral measure. After modifying techniques of Shlyakhtenko and Skoufranis to show that self adjoint elements $x \in M_{n}(\mathcal{A})$ have algebraic Cauchy transform, we explore some applications to eigenvalues of polynomials in Wishart matrices and to diagrammatic elements in von Neumann algebras initially considered by Guionnet, Jones, and Shlyakhtenko.

Keywords:free probability, C*-algebra
Category:46L09

19. CJM Online first

Ovchinnikov, Alexey; Wibmer, Michael
Tannakian categories with semigroup actions
Ostrowski's theorem implies that $\log(x),\log(x+1),\dots$ are algebraically independent over $\mathbb{C}(x)$. More generally, for a linear differential or difference equation, it is an important problem to find all algebraic dependencies among a non-zero solution $y$ and particular transformations of $y$, such as derivatives of $y$ with respect to parameters, shifts of the arguments, rescaling, etc. In the present paper, we develop a theory of Tannakian categories with semigroup actions, which will be used to attack such questions in full generality, as each linear differential equation gives rise to a Tannakian category. Deligne studied actions of braid groups on categories and obtained a finite collection of axioms that characterizes such actions to apply it to various geometric constructions. In this paper, we find a finite set of axioms that characterizes actions of semigroups that are finite free products of semigroups of the form $\mathbb{N}^n\times \mathbb{Z}/{n_1}\mathbb{Z}\times\cdots\times\mathbb{Z}/{n_r}\mathbb{Z}$ on Tannakian categories. This is the class of semigroups that appear in many applications.

Keywords:semigroup actions on categories, Tannakian categories, difference algebraic groups, differential and difference equations with parameters
Categories:18D10, 12H10, 20G05, 33C05, 33C80, 34K06

20. CJM 2016 (vol 69 pp. 284)

Chen, Xianghong; Seeger, Andreas
Convolution Powers of Salem Measures with Applications
We study the regularity of convolution powers for measures supported on Salem sets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for $\alpha$ of the form ${d}/{n}$, $n=2,3,\dots$ there exist $\alpha$-Salem measures for which the $L^2$ Fourier restriction theorem holds in the range $p\le \frac{2d}{2d-\alpha}$. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular $\alpha$-Salem measures, with sharp regularity results for $n$-fold convolutions for all $n\in \mathbb{N}$.

Keywords:convolution powers, Fourier restriction, Salem sets, Salem measures, random sparse sets, Fourier multipliers of Bochner-Riesz type
Categories:42A85, 42B99, 42B15, 42A61

21. CJM 2016 (vol 68 pp. 1382)

Zydor, Michał
La Variante infinitésimale de la formule des traces de Jacquet-Rallis pour les groupes unitaires
We establish an infinitesimal version of the Jacquet-Rallis trace formula for unitary groups. Our formula is obtained by integrating a truncated kernel à la Arthur. It has a geometric side which is a sum of distributions $J_{\mathfrak{o}}$ indexed by classes of elements of the Lie algebra of $U(n+1)$ stable by $U(n)$-conjugation as well as the "spectral side" consisting of the Fourier transforms of the aforementioned distributions. We prove that the distributions $J_{\mathfrak{o}}$ are invariant and depend only on the choice of the Haar measure on $U(n)(\mathbb{A})$. For regular semi-simple classes $\mathfrak{o}$, $J_{\mathfrak{o}}$ is a relative orbital integral of Jacquet-Rallis. For classes $\mathfrak{o}$ called relatively regular semi-simple, we express $J_{\mathfrak{o}}$ in terms of relative orbital integrals regularised by means of zêta functions.

Keywords:formule des traces relative
Categories:11F70, 11F72

22. CJM Online first

Nikolidakis, Eleftherios Nikolaos
Extremal sequences for the Bellman function of the dyadic maximal operator and applications to the Hardy operator
We prove that the extremal sequences for the Bellman function of the dyadic maximal operator behave approximately as eigenfunctions of this operator for a specific eigenvalue. We use this result to prove the analogous one with respect to the Hardy operator.

Keywords:Bellman function, dyadic, Hardy operator, maximal
Category:42B25

23. CJM 2016 (vol 68 pp. 1285)

Ehrig, Michael; Stroppel, Catharina
2-row Springer Fibres and Khovanov Diagram Algebras for Type D
We study in detail two row Springer fibres of even orthogonal type from an algebraic as well as topological point of view. We show that the irreducible components and their pairwise intersections are iterated $\mathbb{P}^1$-bundles. Using results of Kumar and Procesi we compute the cohomology ring with its action of the Weyl group. The main tool is a type $\operatorname D$ diagram calculus labelling the irreducible components in a convenient way which relates to a diagrammatical algebra describing the category of perverse sheaves on isotropic Grassmannians based on work of Braden. The diagram calculus generalizes Khovanov's arc algebra to the type $\operatorname D$ setting and should be seen as setting the framework for generalizing well-known connections of these algebras in type $\operatorname A$ to other types.

Keywords:Springer fibers, Khovanov homology, Weyl group type D
Category:17-11

24. CJM Online first

Fischer, Vera; Mejia, Diego Alejandro
Splitting, Bounding, and Almost Disjointness can be quite Different
We prove the consistency of $$ \operatorname{add}(\mathcal{N})\lt \operatorname{cov}(\mathcal{N}) \lt \mathfrak{p}=\mathfrak{s} =\mathfrak{g}\lt \operatorname{add}(\mathcal{M}) = \operatorname{cof}(\mathcal{M}) \lt \mathfrak{a} =\mathfrak{r}=\operatorname{non}(\mathcal{N})=\mathfrak{c} $$ with $\mathrm{ZFC}$, where each of these cardinal invariants assume arbitrary uncountable regular values.

Keywords:cardinal characteristics of the continuum, splitting, bounding number, maximal almost-disjoint families, template forcing iterations, isomorphism-of-names
Categories:03E17, 03E35, 03E40

25. CJM 2016 (vol 68 pp. 1120)

Stange, Katherine E.
Integral Points on Elliptic Curves and Explicit Valuations of Division Polynomials
Assuming Lang's conjectured lower bound on the heights of non-torsion points on an elliptic curve, we show that there exists an absolute constant $C$ such that for any elliptic curve $E/\mathbb{Q}$ and non-torsion point $P \in E(\mathbb{Q})$, there is at most one integral multiple $[n]P$ such that $n \gt C$. The proof is a modification of a proof of Ingram giving an unconditional but not uniform bound. The new ingredient is a collection of explicit formulae for the sequence $v(\Psi_n)$ of valuations of the division polynomials. For $P$ of non-singular reduction, such sequences are already well described in most cases, but for $P$ of singular reduction, we are led to define a new class of sequences called \emph{elliptic troublemaker sequences}, which measure the failure of the Néron local height to be quadratic. As a corollary in the spirit of a conjecture of Lang and Hall, we obtain a uniform upper bound on $\widehat{h}(P)/h(E)$ for integer points having two large integral multiples.

Keywords:elliptic divisibility sequence, Lang's conjecture, height functions
Categories:11G05, 11G07, 11D25, 11B37, 11B39, 11Y55, 11G50, 11H52
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