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1. CJM Online first

Martins, Luciana de Fátima; Saji, Kentaro
Geometric invariants of cuspidal edges
We give a normal form of the cuspidal edge which uses only diffeomorphisms on the source and isometries on the target. Using this normal form, we study differential geometric invariants of cuspidal edges which determine them up to order three. We also clarify relations between these invariants.

Keywords:cuspidal edge, curvature, wave fronts
Categories:57R45, 53A05, 53A55

2. CJM Online first

an Huef, Astrid; Archbold, Robert John
The C*-algebras of Compact Transformation Groups
We investigate the representation theory of the crossed-product $C^*$-algebra associated to a compact group $G$ acting on a locally compact space $X$ when the stability subgroups vary discontinuously. Our main result applies when $G$ has a principal stability subgroup or $X$ is locally of finite $G$-orbit type. Then the upper multiplicity of the representation of the crossed product induced from an irreducible representation $V$ of a stability subgroup is obtained by restricting $V$ to a certain closed subgroup of the stability subgroup and taking the maximum of the multiplicities of the irreducible summands occurring in the restriction of $V$. As a corollary we obtain that when the trivial subgroup is a principal stability subgroup, the crossed product is a Fell algebra if and only if every stability subgroup is abelian. A second corollary is that the $C^*$-algebra of the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses the classical branching theorem for the special orthogonal group $\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation groups are locally induced from the actions of compact groups, we describe how some of our results can be extended to transformation groups that are locally proper.

Keywords:compact transformation group, proper action, spectrum of a C*-algebra, multiplicity of a representation, crossed-product C*-algebra, continuous-trace C*-algebra, Fell algebra
Categories:46L05, 46L55

3. CJM Online first

Zhang, Tong
Geography of Irregular Gorenstein 3-folds
In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.

Keywords:3-fold, geography, irregular variety

4. CJM Online first

Aluffi, Paolo; Faber, Eleonore
Chern classes of splayed intersections
We generalize the Chern class relation for the transversal intersection of two nonsingular varieties to a relation for possibly singular varieties, under a splayedness assumption. We show that the relation for the Chern-Schwartz-MacPherson classes holds for two splayed hypersurfaces in a nonsingular variety, and under a `strong splayedness' assumption for more general subschemes. Moreover, the relation is shown to hold for the Chern-Fulton classes of any two splayed subschemes. The main tool is a formula for Segre classes of splayed subschemes. We also discuss the Chern class relation under the assumption that one of the varieties is a general very ample divisor.

Keywords:splayed intersection, Chern-Schwartz-MacPherson class, Chern-Fulton class, splayed blowup, Segre class
Categories:14C17, 14J17

5. CJM Online first

Jaffe, Ethan Y.
Pathological phenomena in Denjoy-Carleman classes
Let $\mathcal{C}^M$ denote a Denjoy-Carleman class of $\mathcal{C}^\infty$ functions (for a given logarithmically-convex sequence $M = (M_n)$). We construct: (1) a function in $\mathcal{C}^M((-1,1))$ which is nowhere in any smaller class; (2) a function on $\mathbb{R}$ which is formally $\mathcal{C}^M$ at every point, but not in $\mathcal{C}^M(\mathbb{R})$; (3) (under the assumption of quasianalyticity) a smooth function on $\mathbb{R}^p$ ($p \geq 2$) which is $\mathcal{C}^M$ on every $\mathcal{C}^M$ curve, but not in $\mathcal{C}^M(\mathbb{R}^p)$.

Keywords:Denjoy-Carleman classes, quasianalytic functions, quasianalytic curve, arc-quasianalytic

6. CJM Online first

Carey, Alan L; Gayral, Victor; Phillips, John; Rennie, Adam; Sukochev, Fedor
Spectral flow for nonunital spectral triples
We prove two results about nonunital index theory left open in a previous paper. The first is that the spectral triple arising from an action of the reals on a $C^*$-algebra with invariant trace satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths arising from the odd index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.

Keywords:spectral triple, spectral flow, local index theorem

7. CJM Online first

Drappeau, Sary
Sommes friables d'exponentielles et applications
An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a+b=c$ which is valid unconditionnally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand and Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

Keywords:théorie analytique des nombres, entiers friables, méthode du col
Categories:12N25, 11L07

8. CJM Online first

Kawakami, Yu
Function-theoretic Properties for the Gauss Maps of Various Classes of Surfaces
We elucidate the geometric background of function-theoretic properties for the Gauss maps of several classes of immersed surfaces in three-dimensional space forms, for example, minimal surfaces in Euclidean three-space, improper affine spheres in the affine three-space, and constant mean curvature one surfaces and flat surfaces in hyperbolic three-space. To achieve this purpose, we prove an optimal curvature bound for a specified conformal metric on an open Riemann surface and give some applications. We also provide unicity theorems for the Gauss maps of these classes of surfaces.

Keywords:Gauss map, minimal surface, constant mean curvature surface, front, ramification, omitted value, the Ahlfors island theorem, unicity theorem.
Categories:53C42, 30D35, 30F45, 53A10, 53A15

9. CJM Online first

Zhang, Junqiang; Cao, Jun; Jiang, Renjin; Yang, Dachun
Non-tangential Maximal Function Characterizations of Hardy Spaces Associated with Degenerate Elliptic Operators
Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights or in the class of $QC(\mathbb{R}^n)$ weights, and $L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$, $n\ge 2$. In this article, the authors establish the non-tangential maximal function characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for $p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and $w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$, the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$ is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical Hardy space $H_w^p(\mathbb{R}^n)$.

Keywords:degenerate elliptic operator, Hardy space, square function, maximal function, molecule, Riesz transform
Categories:42B30, 42B35, 35J70

10. CJM Online first

Cojocaru, Alina Carmen; Shulman, Andrew Michael
The Distribution of the First Elementary Divisor of the Reductions of a Generic Drinfeld Module of Arbitrary Rank
Let $\psi$ be a generic Drinfeld module of rank $r \geq 2$. We study the first elementary divisor $d_{1, \wp}(\psi)$ of the reduction of $\psi$ modulo a prime $\wp$, as $\wp$ varies. In particular, we prove the existence of the density of the primes $\wp$ for which $d_{1, \wp} (\psi)$ is fixed. For $r = 2$, we also study the second elementary divisor (the exponent) of the reduction of $\psi$ modulo $\wp$ and prove that, on average, it has a large norm. Our work is motivated by the study of J.-P. Serre of an elliptic curve analogue of Artin's Primitive Root Conjecture, and, moreover, by refinements to Serre's study developed by the first author and M.R. Murty.

Keywords:Drinfeld modules, density theorems
Categories:11R45, 11G09, 11R58

11. CJM Online first

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

12. CJM Online first

Barros, Carlos Braga; Rocha, Victor; Souza, Josiney
Lyapunov Stability and Attraction Under Equivariant Maps
Let $M$ and $N$ be admissible Hausdorff topological spaces endowed with admissible families of open coverings. Assume that $\mathcal{S}$ is a semigroup acting on both $M$ and $N$. In this paper we study the behavior of limit sets, prolongations, prolongational limit sets, attracting sets, attractors and Lyapunov stable sets (all concepts defined for the action of the semigroup $\mathcal{S}$) under equivariant maps and semiconjugations from $M$ to $N$.

Keywords:Lyapunov stability, semigroup actions, generalized flows, equivariant maps, admissible topological spaces
Categories:37B25, 37C75, 34C27, 34D05

13. CJM Online first

Kaniuth, Eberhard
The Bochner-Schoenberg-Eberlein property and spectral synthesis for certain Banach algebra products
Associated with two commutative Banach algebras $A$ and $B$ and a character $\theta$ of $B$ is a certain Banach algebra product $A\times_\theta B$, which is a splitting extension of $B$ by $A$. We investigate two topics for the algebra $A\times_\theta B$ in relation to the corresponding ones of $A$ and $B$. The first one is the Bochner-Schoenberg-Eberlein property and the algebra of Bochner-Schoenberg-Eberlein functions on the spectrum, whereas the second one concerns the wide range of spectral synthesis problems for $A\times_\theta B$.

Keywords:commutative Banach algebra, splitting extension, Gelfand spectrum, set of synthesis, weak spectral set, multiplier algebra, BSE-algebra, BSE-function
Categories:46J10, 46J25, 43A30, 43A45

14. CJM Online first

Sadykov, Rustam
The Weak b-principle: Mumford Conjecture
In this note we introduce and study a new class of maps called oriented colored broken submersions. This is the simplest class of maps that satisfies a version of the b-principle and in dimension $2$ approximates the class of oriented submersions well in the sense that every oriented colored broken submersion of dimension $2$ to a closed simply connected manifold is bordant to a submersion. We show that the Madsen-Weiss theorem (the standard Mumford Conjecture) fits a general setting of the b-principle. Namely, a version of the b-principle for oriented colored broken submersions together with the Harer stability theorem and Miller-Morita theorem implies the Madsen-Weiss theorem.

Keywords:generalized cohomology theories, fold singularities, h-principle, infinite loop spaces
Categories:55N20, 53C23

15. CJM Online first

Carcamo, Cristian; Vidal, Claudio
Stability of Equilibrium Solutions in Planar Hamiltonian Difference Systems
In this paper, we study the stability in the Lyapunov sense of the equilibrium solutions of discrete or difference Hamiltonian systems in the plane. First, we perform a detailed study of linear Hamiltonian systems as a function of the parameters, in particular we analyze the regular and the degenerate cases. Next, we give a detailed study of the normal form associated with the linear Hamiltonian system. At the same time we obtain the conditions under which we can get stability (in linear approximation) of the equilibrium solution, classifying all the possible phase diagrams as a function of the parameters. After that, we study the stability of the equilibrium solutions of the first order difference system in the plane associated to mechanical Hamiltonian system and Hamiltonian system defined by cubic polynomials. Finally, important differences with the continuous case are pointed out.

Keywords:difference equations, Hamiltonian systems, stability in the Lyapunov sense
Categories:34D20, 34E10

16. CJM Online first

Charlesworth, Ian; Nelson, Brent; Skoufranis, Paul
On two-faced families of non-commutative random variables
We demonstrate that the notions of bi-free independence and combinatorial-bi-free independence of two-faced families are equivalent using a diagrammatic view of bi-non-crossing partitions. These diagrams produce an operator model on a Fock space suitable for representing any two-faced family of non-commutative random variables. Furthermore, using a Kreweras complement on bi-non-crossing partitions we establish the expected formulas for the multiplicative convolution of a bi-free pair of two-faced families.

Keywords:free probability, operator algebras, bi-free

17. CJM Online first

Amini, Massoud; Elliott, George A.; Golestani, Nasser
The Category of Bratteli Diagrams
A category structure for Bratteli diagrams is proposed and a functor from the category of AF algebras to the category of Bratteli diagrams is constructed. Since isomorphism of Bratteli diagrams in this category coincides with Bratteli's notion of equivalence, we obtain in particular a functorial formulation of Bratteli's classification of AF algebras (and at the same time, of Glimm's classification of UHF~algebras). It is shown that the three approaches to classification of AF~algebras, namely, through Bratteli diagrams, K-theory, and abstract classifying categories, are essentially the same from a categorical point of view.

Keywords:C$^{*}$-algebra, category, functor, AF algebra, dimension group, Bratteli diagram
Categories:46L05, 46L35, 46M15

18. CJM Online first

Bonfanti, Matteo Alfonso; van Geemen,
Abelian Surfaces with an Automorphism and Quaternionic Multiplication
We construct one dimensional families of Abelian surfaces with quaternionic multiplication which also have an automorphism of order three or four. Using Barth's description of the moduli space of $(2,4)$-polarized Abelian surfaces, we find the Shimura curve parametrizing these Abelian surfaces in a specific case. We explicitly relate these surfaces to the Jacobians of genus two curves studied by Hashimoto and Murabayashi. We also describe a (Humbert) surface in Barth's moduli space which parametrizes Abelian surfaces with real multiplication by $\mathbf{Z}[\sqrt{2}]$.

Keywords:abelian surfaces, moduli, shimura curves
Categories:14K10, 11G10, 14K20

19. CJM Online first

Takeda, Shuichiro
Metaplectic Tensor Products for Automorphic Representation of $\widetilde{GL}(r)$
Let $M=\operatorname{GL}_{r_1}\times\cdots\times\operatorname{GL}_{r_k}\subseteq\operatorname{GL}_r$ be a Levi subgroup of $\operatorname{GL}_r$, where $r=r_1+\cdots+r_k$, and $\widetilde{M}$ its metaplectic preimage in the $n$-fold metaplectic cover $\widetilde{\operatorname{GL}}_r$ of $\operatorname{GL}_r$. For automorphic representations $\pi_1,\dots,\pi_k$ of $\widetilde{\operatorname{GL}}_{r_1}(\mathbb{A}),\dots,\widetilde{\operatorname{GL}}_{r_k}(\mathbb{A})$, we construct (under a certain technical assumption, which is always satisfied when $n=2$) an automorphic representation $\pi$ of $\widetilde{M}(\mathbb{A})$ which can be considered as the ``tensor product'' of the representations $\pi_1,\dots,\pi_k$. This is the global analogue of the metaplectic tensor product defined by P. Mezo in the sense that locally at each place $v$, $\pi_v$ is equivalent to the local metaplectic tensor product of $\pi_{1,v},\dots,\pi_{k,v}$ defined by Mezo. Then we show that if all of $\pi_i$ are cuspidal (resp. square-integrable modulo center), then the metaplectic tensor product is cuspidal (resp. square-integrable modulo center). We also show that (both locally and globally) the metaplectic tensor product behaves in the expected way under the action of a Weyl group element, and show the compatibility with parabolic inductions.

Keywords:automorphic forms, representations of covering groups

20. CJM Online first

Pan, Ivan Edgardo; Simis, Aron
Cremona Maps of de Jonquières Type
This paper is concerned with suitable generalizations of a plane de Jonquières map to higher dimensional space $\mathbb{P}^n$ with $n\geq 3$. For each given point of $\mathbb{P}^n$ there is a subgroup of the entire Cremona group of dimension $n$ consisting of such maps. One studies both geometric and group-theoretical properties of this notion. In the case where $n=3$ one describes an explicit set of generators of the group and gives a homological characterization of a basic subgroup thereof.

Keywords:Cremona map, de Jonquières map, Cremona group, minimal free resolution
Categories:14E05, 13D02, 13H10, 14E07, 14M05, 14M25

21. CJM 2014 (vol 67 pp. 241)

Agler, Jim; McCarthy,
Global Holomorphic Functions in Several Noncommuting Variables
We define a free holomorphic function to be a function that is locally, with respect to the free topology, a bounded nc-function. We prove that free holomorphic functions are the functions that are locally uniformly approximable by free polynomials. We prove a realization formula and an Oka-Weil theorem for free analytic functions.

Keywords:noncommutative analysis, free holomorphic functions

22. CJM 2014 (vol 67 pp. 152)

Lescop, Christine
On Homotopy Invariants of Combings of Three-manifolds
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 Casson-Walker invariant. The article also includes a self-contained 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, Casson-Walker invariant, perturbative expansion of Chern-Simons theory, configuration space integrals
Categories:57M27, 57R20, 57N10

23. CJM 2014 (vol 67 pp. 404)

Hua, Jiajie; Lin, Huaxin
Rotation Algebras and the Exel Trace Formula
We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $\|uv-vu\|\lt 2$ and $uvu^*v^*$ commutes with $u$ and $v,$ then the $C^*$-subalgebra $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of some rotation algebra $A_\theta$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $\theta\in (-1/2, 1/2)$ be a real number. We prove the following: For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following: if $u$ and $v$ are two unitaries in any unital simple $C^*$-algebra $A$ with tracial rank zero such that \[ \|uv-e^{2\pi i\theta}vu\|\lt \delta \text{ and } {1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta, \] for all tracial state $\tau$ of $A,$ then there exists a pair of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$ such that \[ \tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\, \|u-\tilde{u}\|\lt \epsilon \text{ and } \|v-\tilde{v}\|\lt \epsilon. \]

Keywords:rotation algebras, Exel trace formula

24. CJM Online first

Ashraf, Samia; Azam, Haniya; Berceanu, Barbu
Representation stability of power sets and square free polynomials
The symmetric group $\mathcal{S}_n$ acts on the power set $\mathcal{P}(n)$ and also on the set of square free polynomials in $n$ variables. These two related representations are analyzed from the stability point of view. An application is given for the action of the symmetric group on the cohomology of the pure braid group.

Keywords:symmetric group modules, square free polynomials, representation stability, Arnold algebra
Categories:20C30, 13A50, 20F36, 55R80

25. CJM 2014 (vol 67 pp. 369)

Graham, Robert; Pichot, Mikael
A Free Product Formula for the Sofic Dimension
It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability measure preserving $s$-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$ satisfies the equality \[ s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3) \] where $\mathfrak{h}$ is the normalized Haar measure on $G$.

Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy
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