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1. CMB Online first

Wang, Li-an Daniel
 A Multiplier Theorem on Anisotropic Hardy Spaces We present a multiplier theorem on anisotropic Hardy spaces. When $m$ satisfies the anisotropic, pointwise Mihlin condition, we obtain boundedness of the multiplier operator $T_m : H_A^p (\mathbb R^n) \rightarrow H_A^p (\mathbb R^n)$, for the range of $p$ that depends on the eccentricities of the dilation $A$ and the level of regularity of a multiplier symbol $m$. This extends the classical multiplier theorem of Taibleson and Weiss. Keywords:anisotropic Hardy space, multiplier, Fourier transformCategories:42B30, 42B25, 42B35

2. CMB Online first

Bu, Shangquan; Cai, Gang
 HÃ¶lder continuous solutions of degenerate differential equations with finite delay Using known operator-valued Fourier multiplier results on vector-valued HÃ¶lder continuous function spaces $C^\alpha (\mathbb R; X)$, we completely characterize the $C^\alpha$-well-posedness of the first order degenerate differential equations with finite delay $(Mu)'(t) = Au(t) + Fu_t + f(t)$ for $t\in\mathbb R$ by the boundedness of the $(M, F)$-resolvent of $A$ under suitable assumption on the delay operator $F$, where $A, M$ are closed linear operators on a Banach space $X$ satisfying $D(A)\cap D(M) \not=\{0\}$, the delay operator $F$ is a bounded linear operator from $C([-r, 0]; X)$ to $X$ and $r \gt 0$ is fixed. Keywords:well-posedness, degenerate differential equation, $\dot{C}^\alpha$-multiplier, HÃ¶lder continuous function spaceCategories:34N05, 34G10, 47D06, 47A10, 34K30

3. CMB Online first

Buijs, Urtzi; Félix, Yves; Murillo, Aniceto; Tanré, Daniel
 Maurer-Cartan elements in the Lie models of finite simplicial complexes In a previous work, we have associated a complete differential graded Lie algebra to any finite simplicial complex in a functorial way. Similarly, we have also a realization functor from the category of complete differential graded Lie algebras to the category of simplicial sets. We have already interpreted the homology of a Lie algebra in terms of homotopy groups of its realization. In this paper, we begin a dictionary between models and simplicial complexes by establishing a correspondence between the Deligne groupoid of the model and the connected components of the finite simplicial complex. Keywords:complete differential graded Lie algebra, Maurer-Cartan element, rational homotopy theoryCategory:16E45

4. CMB Online first

Lee, Tsiu-Kwen
 Ad-nilpotent elements of semiprime rings with involution Let $R$ be an $n!$-torsion free semiprime ring with involution $*$ and with extended centroid $C$, where $n\gt 1$ is a positive integer. We characterize $a\in K$, the Lie algebra of skew elements in $R$, satisfying $(\operatorname{ad}_a)^n=0$ on $K$. This generalizes both Martindale and Miers' theorem and the theorem of Brox et al. To prove it we first prove that if $a, b\in R$ satisfy $(\operatorname{ad}_a)^n=\operatorname{ad}_b$ on $R$, where either $n$ is even or $b=0$, then $\big(a-\lambda\big)^{[\frac{n+1}{2}]}=0$ for some $\lambda\in C$. Keywords:Semiprime ring, Lie algebra, Jordan algebra, faithful $f$-free, involution, skew (symmetric) element, ad-nilpotent element, Jordan elementCategories:16N60, 16W10, 17B60

5. CMB 2016 (vol 59 pp. 813)

Kaimakamis, George; Panagiotidou, Konstantina; Pérez, Juan de Dios
 A Classification of Three-dimensional Real Hypersurfaces in Non-flat Complex Space Forms in Terms of Their generalized Tanaka-Webster Lie Derivative On a real hypersurface $M$ in a non-flat complex space form there exist the Levi-Civita and the k-th generalized Tanaka-Webster connections. The aim of the present paper is to study three dimensional real hypersurfaces in non-flat complex space forms, whose Lie derivative of the structure Jacobi operator with respect to the Levi-Civita connections coincides with the Lie derivative of it with respect to the k-th generalized Tanaka-Webster connection. The Lie derivatives are considered in direction of the structure vector field and in directions of any vecro field orthogonal to the structure vector field. Keywords:$k$-th generalized Tanaka-Webster connection, non-flat complex space form, real hypersurface, Lie derivative, structure Jacobi operatorCategories:53C15, 53B25

6. CMB 2016 (vol 59 pp. 673)

Bačák, Miroslav; Kovalev, Leonid V.
 Lipschitz Retractions in Hadamard Spaces Via Gradient Flow Semigroups Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped with the Hausdorff metric is called a finite subset space. In this paper we are concerned with the existence of Lipschitz retractions $r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions do not exist if $X$ is the one-dimensional sphere. On the other hand L. Kovalev has recently established their existence in case $X$ is a Hilbert space and he also posed a question as to whether or not such Lipschitz retractions exist for $X$ being a Hadamard space. In the present paper we answer this question in the positive. Keywords:finite subset space, gradient flow, Hadamard space, Lie-Trotter-Kato formula, Lipschitz retractionCategories:53C23, 47H20, 54E40, 58D07

7. CMB 2015 (vol 58 pp. 692)

Anona, F. M.; Randriambololondrantomalala, Princy; Ravelonirina, H. S. G.
 Sur les algÃ¨bres de Lie associÃ©es Ã  une connexion Let $\Gamma$ be a connection on a smooth manifold $M$, in this paper we give some properties of $\Gamma$ by studying the corresponding Lie algebras. In particular, we compute the first Chevalley-Eilenberg cohomology space of the horizontal vector fields Lie algebra on the tangent bundle of $M$, whose the corresponding Lie derivative of $\Gamma$ is null, and of the horizontal nullity curvature space. Keywords:algÃ¨bre de Lie, connexion, cohomologie de Chevalley-Eilenberg, champs dont la dÃ©rivÃ©e de Lie correspondante Ã  une connexion est nulle, espace de nullitÃ© de la courbureCategories:17B66, 53B15

8. CMB 2015 (vol 58 pp. 575)

Martinez-Torres, David
 The Diffeomorphism Type of Canonical Integrations Of Poisson Tensors on Surfaces A surface $\Sigma$ endowed with a Poisson tensor $\pi$ is known to admit canonical integration, $\mathcal{G}(\pi)$, which is a 4-dimensional manifold with a (symplectic) Lie groupoid structure. In this short note we show that if $\pi$ is not an area form on the 2-sphere, then $\mathcal{G}(\pi)$ is diffeomorphic to the cotangent bundle $T^*\Sigma$. This extends results by the author and by Bonechi, Ciccoli, Staffolani, and Tarlini. Keywords:Poisson tensor, Lie groupoid, cotangent bundleCategories:58H05, 55R10, 53D17

9. CMB 2015 (vol 58 pp. 306)

Khoshkhah, Kaveh; Zaker, Manouchehr
 On the Largest Dynamic Monopolies of Graphs with a Given Average Threshold Let $G$ be a graph and $\tau$ be an assignment of nonnegative integer thresholds to the vertices of $G$. A subset of vertices, $D$ is said to be a $\tau$-dynamic monopoly, if $V(G)$ can be partitioned into subsets $D_0, D_1, \ldots, D_k$ such that $D_0=D$ and for any $i\in \{0, \ldots, k-1\}$, each vertex $v$ in $D_{i+1}$ has at least $\tau(v)$ neighbors in $D_0\cup \ldots \cup D_i$. Denote the size of smallest $\tau$-dynamic monopoly by $dyn_{\tau}(G)$ and the average of thresholds in $\tau$ by $\overline{\tau}$. We show that the values of $dyn_{\tau}(G)$ over all assignments $\tau$ with the same average threshold is a continuous set of integers. For any positive number $t$, denote the maximum $dyn_{\tau}(G)$ taken over all threshold assignments $\tau$ with $\overline{\tau}\leq t$, by $Ldyn_t(G)$. In fact, $Ldyn_t(G)$ shows the worst-case value of a dynamic monopoly when the average threshold is a given number $t$. We investigate under what conditions on $t$, there exists an upper bound for $Ldyn_{t}(G)$ of the form $c|G|$, where $c\lt 1$. Next, we show that $Ldyn_t(G)$ is coNP-hard for planar graphs but has polynomial-time solution for forests. Keywords:spread of influence in graphs, irreversible dynamic monopolies, target set selectionCategories:05C69, 05C85

10. CMB 2015 (vol 58 pp. 281)

Kalus, Matthias
 On the Relation of Real and Complex Lie Supergroups A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complex Harish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed. Keywords:Lie supergroup, almost complex structure, Harish-Chandra pair, universal complexificationCategories:32C11, 58A50

11. CMB 2014 (vol 58 pp. 69)

Fulp, Ronald Owen
 Correction to "Infinite Dimensional DeWitt Supergroups and Their Bodies" The Theorem below is a correction to Theorem 3.5 in the article entitled " Infinite Dimensional DeWitt Supergroups and Their Bodies" published in Canad. Math. Bull. Vol. 57 (2) 2014 pp. 283-288. Only part (iii) of that Theorem requires correction. The proof of Theorem 3.5 in the original article failed to separate the proof of (ii) from the proof of (iii). The proof of (ii) is complete once it is established that $ad_a$ is quasi-nilpotent for each $a$ since it immediately follows that $K$ is quasi-nilpotent. The proof of (iii) is not complete in the original article. The revision appears as the proof of (iii) of the revised Theorem below. Keywords:super groups, body of super groups, Banach Lie groupsCategories:58B25, 17B65, 81R10, 57P99

12. CMB 2014 (vol 57 pp. 735)

Cagliero, Leandro; Szechtman, Fernando
 On the Theorem of the Primitive Element with Applications to the Representation Theory of Associative and Lie Algebras We describe of all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,y\in K$. When is $F[x,y]=F[\alpha x+\beta y]$ for some non-zero elements $\alpha,\beta\in F$? Keywords:uniserial module, Lie algebra, associative algebra, primitive elementCategories:17B10, 13C05, 12F10, 12E20

13. CMB 2012 (vol 57 pp. 132)

Mubeena, T.; Sankaran, P.
 Twisted Conjugacy Classes in Abelian Extensions of Certain Linear Groups Given a group automorphism $\phi:\Gamma\longrightarrow \Gamma$, one has an action of $\Gamma$ on itself by $\phi$-twisted conjugacy, namely, $g.x=gx\phi(g^{-1})$. The orbits of this action are called $\phi$-twisted conjugacy classes. One says that $\Gamma$ has the $R_\infty$-property if there are infinitely many $\phi$-twisted conjugacy classes for every automorphism $\phi$ of $\Gamma$. In this paper we show that $\operatorname{SL}(n,\mathbb{Z})$ and its congruence subgroups have the $R_\infty$-property. Further we show that any (countable) abelian extension of $\Gamma$ has the $R_\infty$-property where $\Gamma$ is a torsion free non-elementary hyperbolic group, or $\operatorname{SL}(n,\mathbb{Z}), \operatorname{Sp}(2n,\mathbb{Z})$ or a principal congruence subgroup of $\operatorname{SL}(n,\mathbb{Z})$ or the fundamental group of a complete Riemannian manifold of constant negative curvature. Keywords:twisted conjugacy classes, hyperbolic groups, lattices in Lie groupsCategory:20E45

14. CMB 2012 (vol 56 pp. 606)

Mazorchuk, Volodymyr; Zhao, Kaiming
 Characterization of Simple Highest Weight Modules We prove that for simple complex finite dimensional Lie algebras, affine Kac-Moody Lie algebras, the Virasoro algebra and the Heisenberg-Virasoro algebra, simple highest weight modules are characterized by the property that all positive root elements act on these modules locally nilpotently. We also show that this is not the case for higher rank Virasoro and for Heisenberg algebras. Keywords:Lie algebra, highest weight module, triangular decomposition, locally nilpotent actionCategories:17B20, 17B65, 17B66, 17B68

15. CMB 2011 (vol 55 pp. 870)

Wang, Hui; Deng, Shaoqiang
 Left Invariant Einstein-Randers Metrics on Compact Lie Groups In this paper we study left invariant Einstein-Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics. Keywords:Einstein-Randers metric, compact Lie groups, geodesic, flag curvatureCategories:17B20, 22E46, 53C12

16. CMB 2011 (vol 55 pp. 523)

Iwase, Norio; Mimura, Mamoru; Oda, Nobuyuki; Yoon, Yeon Soo
 The Milnor-Stasheff Filtration on Spaces and Generalized Cyclic Maps The concept of $C_{k}$-spaces is introduced, situated at an intermediate stage between $H$-spaces and $T$-spaces. The $C_{k}$-space corresponds to the $k$-th Milnor-Stasheff filtration on spaces. It is proved that a space $X$ is a $C_{k}$-space if and only if the Gottlieb set $G(Z,X)=[Z,X]$ for any space $Z$ with ${\rm cat}\, Z\le k$, which generalizes the fact that $X$ is a $T$-space if and only if $G(\Sigma B,X)=[\Sigma B,X]$ for any space $B$. Some results on the $C_{k}$-space are generalized to the $C_{k}^{f}$-space for a map $f\colon A \to X$. Projective spaces, lens spaces and spaces with a few cells are studied as examples of $C_{k}$-spaces, and non-$C_{k}$-spaces. Keywords:Gottlieb sets for maps, L-S category, T-spacesCategories:55P45, 55P35

17. CMB 2011 (vol 55 pp. 708)

Demeter, Ciprian
 Improved Range in the Return Times Theorem We prove that the Return Times Theorem holds true for pairs of $L^p-L^q$ functions, whenever $\frac{1}{p}+\frac{1}{q}<\frac{3}{2}$. Keywords:Return Times Theorem, maximal multiplier, maximal inequalityCategories:42B25, 37A45

18. CMB 2011 (vol 55 pp. 579)

Ndogmo, J. C.
 Casimir Operators and Nilpotent Radicals It is shown that a Lie algebra having a nilpotent radical has a fundamental set of invariants consisting of Casimir operators. A different proof is given in the well known special case of an abelian radical. A result relating the number of invariants to the dimension of the Cartan subalgebra is also established. Keywords:nilpotent radical, Casimir operators, algebraic Lie algebras, Cartan subalgebras, number of invariantsCategories:16W25, 17B45, 16S30

19. CMB 2011 (vol 54 pp. 654)

Forrest, Brian E.; Runde, Volker
 Norm One Idempotent $cb$-Multipliers with Applications to the Fourier Algebra in the $cb$-Multiplier Norm For a locally compact group $G$, let $A(G)$ be its Fourier algebra, let $M_{cb}A(G)$ denote the completely bounded multipliers of $A(G)$, and let $A_{\mathit{Mcb}}(G)$ stand for the closure of $A(G)$ in $M_{cb}A(G)$. We characterize the norm one idempotents in $M_{cb}A(G)$: the indicator function of a set $E \subset G$ is a norm one idempotent in $M_{cb}A(G)$ if and only if $E$ is a coset of an open subgroup of $G$. As applications, we describe the closed ideals of $A_{\mathit{Mcb}}(G)$ with an approximate identity bounded by $1$, and we characterize those $G$ for which $A_{\mathit{Mcb}}(G)$ is $1$-amenable in the sense of B. E. Johnson. (We can even slightly relax the norm bounds.) Keywords:amenability, bounded approximate identity, $cb$-multiplier norm, Fourier algebra, norm one idempotentCategories:43A22, 20E05, 43A30, 46J10, 46J40, 46L07, 47L25

20. CMB 2011 (vol 55 pp. 260)

Delvaux, L.; Van Daele, A.; Wang, Shuanhong
 A Note on the Antipode for Algebraic Quantum Groups Recently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra. In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals. We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature. Keywords:multiplier Hopf algebras, algebraic quantum groups, the antipodeCategories:16W30, 46L65

21. CMB 2011 (vol 54 pp. 716)

Okassa, Eugène
 Symplectic Lie-Rinehart-Jacobi Algebras and Contact Manifolds We give a characterization of contact manifolds in terms of symplectic Lie-Rinehart-Jacobi algebras. We also give a sufficient condition for a Jacobi manifold to be a contact manifold. Keywords:Lie-Rinehart algebras, differential operators, Jacobi manifolds, symplectic manifolds, contact manifoldsCategories:13N05, 53D05, 53D10

22. CMB 2011 (vol 54 pp. 442)

García, Esther; Lozano, Miguel Gómez; Neher, Erhard
 Nondegeneracy for Lie Triple Systems and Kantor Pairs We study the transfer of nondegeneracy between Lie triple systems and their standard Lie algebra envelopes as well as between Kantor pairs, their associated Lie triple systems, and their Lie algebra envelopes. We also show that simple Kantor pairs and Lie triple systems in characteristic $0$ are nondegenerate. Keywords:Kantor pairs, Lie triple systems, Lie algebrasCategories:17A40, 17B60, 17B99

23. CMB 2011 (vol 54 pp. 472)

Iacono, Donatella
 A Semiregularity Map Annihilating Obstructions to Deforming Holomorphic Maps We study infinitesimal deformations of holomorphic maps of compact, complex, KÃ¤hler manifolds. In particular, we describe a generalization of Bloch's semiregularity map that annihilates obstructions to deform holomorphic maps with fixed codomain. Keywords:semiregularity map, obstruction theory, functors of Artin rings, differential graded Lie algebrasCategories:13D10, 14D15, 14B10

24. CMB 2010 (vol 54 pp. 527)

Preda, Ciprian; Sipos, Ciprian
 On the Dichotomy of the Evolution Families: A Discrete-Argument Approach We establish a discrete-time criteria guaranteeing the existence of an exponential dichotomy in the continuous-time behavior of an abstract evolution family. We prove that an evolution family ${\cal U}=\{U(t,s)\}_{t \geq s\geq 0}$ acting on a Banach space $X$ is uniformly exponentially dichotomic (with respect to its continuous-time behavior) if and only if the corresponding difference equation with the inhomogeneous term from a vector-valued Orlicz sequence space $l^\Phi(\mathbb{N}, X)$ admits a solution in the same $l^\Phi(\mathbb{N},X)$. The technique of proof effectively eliminates the continuity hypothesis on the evolution family (\emph{i.e.,} we do not assume that $U(\,\cdot\,,s)x$ or $U(t,\,\cdot\,)x$ is continuous on $[s,\infty)$, and respectively $[0,t]$). Thus, some known results given by Coffman and Schaffer, Perron, and Ta Li are extended. Keywords:evolution families, exponential dichotomy, Orlicz sequence spaces, admissibilityCategories:34D05, 47D06, 93D20

25. CMB 2010 (vol 54 pp. 364)

Preda, Ciprian; Preda, Petre
 Lyapunov Theorems for the Asymptotic Behavior of Evolution Families on the Half-Line Two theorems regarding the asymptotic behavior of evolution families are established in terms of the solutions of a certain Lyapunov operator equation. Keywords:evolution families, exponential instability, Lyapunov equationCategories:34D05, 47D06
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