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1. CJM 2015 (vol 68 pp. 258)

Calixto, Lucas; Moura, Adriano; Savage, Alistair
 Equivariant Map Queer Lie Superalgebras An equivariant map queer Lie superalgebra is the Lie superalgebra of regular maps from an algebraic variety (or scheme) $X$ to a queer Lie superalgebra $\mathfrak{q}$ that are equivariant with respect to the action of a finite group $\Gamma$ acting on $X$ and $\mathfrak{q}$. In this paper, we classify all irreducible finite-dimensional representations of the equivariant map queer Lie superalgebras under the assumption that $\Gamma$ is abelian and acts freely on $X$. We show that such representations are parameterized by a certain set of $\Gamma$-equivariant finitely supported maps from $X$ to the set of isomorphism classes of irreducible finite-dimensional representations of $\mathfrak{q}$. In the special case where $X$ is the torus, we obtain a classification of the irreducible finite-dimensional representations of the twisted loop queer superalgebra. Keywords:Lie superalgebra, queer Lie superalgebra, loop superalgebra, equivariant map superalgebra, finite-dimensional representation, finite-dimensional moduleCategories:17B65, 17B10

2. CJM 2015 (vol 67 pp. 990)

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 diagramCategories:46L05, 46L35, 46M15

3. CJM 2013 (vol 65 pp. 1384)

Wright, Paul
 Estimates of Hausdorff Dimension for Non-wandering Sets of Higher Dimensional Open Billiards This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set $M_0$ of the billiard ball map is a topological Cantor set and its Hausdorff dimension has been previously estimated for billiards in $\mathbb{R}^2$, using well-known techniques. We extend these estimates to billiards in $\mathbb{R}^n$, and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of $\mathbb{R}^n$ formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension. Keywords:dynamical systems, billiards, dimension, HausdorffCategories:37D20, 37D40

4. CJM 2013 (vol 66 pp. 625)

Giambruno, Antonio; Mattina, Daniela La; Zaicev, Mikhail
 Classifying the Minimal Varieties of Polynomial Growth Let $\mathcal{V}$ be a variety of associative algebras generated by an algebra with $1$ over a field of characteristic zero. This paper is devoted to the classification of the varieties $\mathcal{V}$ which are minimal of polynomial growth (i.e., their sequence of codimensions growth like $n^k$ but any proper subvariety grows like $n^t$ with $t\lt k$). These varieties are the building blocks of general varieties of polynomial growth. It turns out that for $k\le 4$ there are only a finite number of varieties of polynomial growth $n^k$, but for each $k \gt 4$, the number of minimal varieties is at least $|F|$, the cardinality of the base field and we give a recipe of how to construct them. Keywords:T-ideal, polynomial identity, codimension, polynomial growth,Categories:16R10, 16P90

5. CJM 2013 (vol 66 pp. 303)

Elekes, Márton; Steprāns, Juris
 Haar Null Sets and the Consistent Reflection of Non-meagreness A subset $X$ of a Polish group $G$ is called Haar null if there exists a Borel set $B \supset X$ and Borel probability measure $\mu$ on $G$ such that $\mu(gBh)=0$ for every $g,h \in G$. We prove that there exist a set $X \subset \mathbb R$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu(X + t) = 0$ for every $t \in \mathbb R$. This answers a question from David Fremlin's problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.) This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C \subset G$ such that for every non-meagre set $X \subset G$ there exists a $t \in G$ such that $C \cap (X + t)$ is relatively non-meagre in $C$. This essentially generalises results of BartoszyÅski and Burke-Miller. Keywords:Haar null, Christensen, non-locally compact Polish group, packing dimension, Problem FC on Fremlin's list, forcing, generic realCategories:28C10, 03E35, 03E17, , , , , 22C05, 28A78

6. 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, tamenessCategories:32B10, 32B20, 32C07, 32C25, 32V15, 32V40, 14P15

7. CJM 2011 (vol 64 pp. 755)

Brown, Lawrence G.; Lee, Hyun Ho
 Homotopy Classification of Projections in the Corona Algebra of a Non-simple $C^*$-algebra We study projections in the corona algebra of $C(X)\otimes K$, where K is the $C^*$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$, or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in $K_0$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct. Keywords:essential codimension, continuous field of Hilbert spaces, Corona algebraCategories:46L05, 46L80

8. CJM 2011 (vol 64 pp. 573)

Nawata, Norio
 Fundamental Group of Simple $C^*$-algebras with Unique Trace III We introduce the fundamental group ${\mathcal F}(A)$ of a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of Fundamental Group of Simple $C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian. Keywords:fundamental group, Picard group, Hilbert module, countable basis, stably projectionless algebra, dimension functionCategories:46L05, 46L08, 46L35

9. CJM 2011 (vol 63 pp. 551)

Hadwin, Don; Li, Qihui; Shen, Junhao
 Topological Free Entropy Dimensions in Nuclear C$^*$-algebras and in Full Free Products of Unital C$^*$-algebras In the paper, we introduce a new concept, topological orbit dimension of an $n$-tuple of elements in a unital C$^{\ast}$-algebra. Using this concept, we conclude that Voiculescu's topological free entropy dimension of every finite family of self-adjoint generators of a nuclear C$^{\ast}$-algebra is less than or equal to $1$. We also show that the Voiculescu's topological free entropy dimension is additive in the full free product of some unital C$^{\ast}$-algebras. We show that the unital full free product of Blackadar and Kirchberg's unital MF algebras is also an MF algebra. As an application, we obtain that $\mathop{\textrm{Ext}}(C_{r}^{\ast}(F_{2})\ast_{\mathbb{C}}C_{r}^{\ast}(F_{2}))$ is not a group. Keywords: topological free entropy dimension, unital C$^{*}$-algebraCategories:46L10, 46L54

10. CJM 2011 (vol 63 pp. 616)

Lee, Edward
 A Modular Quintic Calabi-Yau Threefold of Level 55 In this note we search the parameter space of Horrocks-Mumford quintic threefolds and locate a Calabi-Yau threefold that is modular, in the sense that the $L$-function of its middle-dimensional cohomology is associated with a classical modular form of weight 4 and level 55. Keywords: Calabi-Yau threefold, non-rigid Calabi-Yau threefold, two-dimensional Galois representation, modular variety, Horrocks-Mumford vector bundleCategories:14J15, 11F23, 14J32, 11G40

11. CJM 2011 (vol 63 pp. 648)

Ngai, Sze-Man
 Spectral Asymptotics of Laplacians Associated with One-dimensional Iterated Function Systems with Overlaps We set up a framework for computing the spectral dimension of a class of one-dimensional self-similar measures that are defined by iterated function systems with overlaps and satisfy a family of second-order self-similar identities. As applications of our result we obtain the spectral dimension of important measures such as the infinite Bernoulli convolution associated with the golden ratio and convolutions of Cantor-type measures. The main novelty of our result is that the iterated function systems we consider are not post-critically finite and do not satisfy the well-known open set condition. Keywords:spectral dimension, fractal, Laplacian, self-similar measure, iterated function system with overlaps, second-order self-similar identitiesCategories:28A80, , , , 35P20, 35J05, 43A05, 47A75

12. CJM 2011 (vol 63 pp. 481)

Baragar, Arthur
 The Ample Cone for a K3 Surface In this paper, we give several pictorial fractal representations of the ample or KÃ¤hler cone for surfaces in a certain class of $K3$ surfaces. The class includes surfaces described by smooth $(2,2,2)$ forms in ${\mathbb P^1\times\mathbb P^1\times \mathbb P^1}$ defined over a sufficiently large number field $K$ that have a line parallel to one of the axes and have Picard number four. We relate the Hausdorff dimension of this fractal to the asymptotic growth of orbits of curves under the action of the surface's group of automorphisms. We experimentally estimate the Hausdorff dimension of the fractal to be $1.296 \pm .010$. Keywords:Fractal, Hausdorff dimension, K3 surface, Kleinian groups, dynamicsCategories:14J28, , , , 14J50, 11D41, 11D72, 11H56, 11G10, 37F35, 37D05

13. CJM 2010 (vol 63 pp. 436)

Mine, Kotaro; Sakai, Katsuro
 Simplicial Complexes and Open Subsets of Non-Separable LF-Spaces Let $F$ be a non-separable LF-space homeomorphic to the direct sum $\sum_{n\in\mathbb{N}} \ell_2(\tau_n)$, where $\aleph_0 < \tau_1 < \tau_2 < \cdots$. It is proved that every open subset $U$ of $F$ is homeomorphic to the product $|K| \times F$ for some locally finite-dimensional simplicial complex $K$ such that every vertex $v \in K^{(0)}$ has the star $\operatorname{St}(v,K)$ with $\operatorname{card} \operatorname{St}(v,K)^{(0)} < \tau = \sup\tau_n$ (and $\operatorname{card} K^{(0)} \le \tau$), and, conversely, if $K$ is such a simplicial complex, then the product $|K| \times F$ can be embedded in $F$ as an open set, where $|K|$ is the polyhedron of $K$ with the metric topology. Keywords:LF-space, open set, simplicial complex, metric topology, locally finite-dimensional, star, small box product, ANR, $\ell_2(\tau)$, $\ell_2(\tau)$-manifold, open embedding, $\sum_{i\in\mathbb{N}}\ell_2(\tau_i)$Categories:57N20, 46A13, 46T05, 57N17, 57Q05, 57Q40

14. CJM 2010 (vol 62 pp. 1182)

Yue, Hong
 A Fractal Function Related to the John-Nirenberg Inequality for $Q_{\alpha}({\mathbb R^n})$ A borderline case function $f$ for $Q_{\alpha}({\mathbb R^n})$ spaces is defined as a Haar wavelet decomposition, with the coefficients depending on a fixed parameter $\beta>0$. On its support $I_0=[0, 1]^n$, $f(x)$ can be expressed by the binary expansions of the coordinates of $x$. In particular, $f=f_{\beta}\in Q_{\alpha}({\mathbb R^n})$ if and only if $\alpha<\beta<\frac{n}{2}$, while for $\beta=\alpha$, it was shown by Yue and Dafni that $f$ satisfies a John--Nirenberg inequality for $Q_{\alpha}({\mathbb R^n})$. When $\beta\neq 1$, $f$ is a self-affine function. It is continuous almost everywhere and discontinuous at all dyadic points inside $I_0$. In addition, it is not monotone along any coordinate direction in any small cube. When the parameter $\beta\in (0, 1)$, $f$ is onto from $I_0$ to $[-\frac{1}{1-2^{-\beta}}, \frac{1}{1-2^{-\beta}}]$, and the graph of $f$ has a non-integer fractal dimension $n+1-\beta$. Keywords:Haar wavelets, Q spaces, John-Nirenberg inequality, Greedy expansion, self-affine, fractal, Box dimensionCategories:42B35, 42C10, 30D50, 28A80

15. CJM 2009 (vol 61 pp. 124)

Dijkstra, Jan J.; Mill, Jan van
 Characterizing Complete Erd\H os Space The space now known as {\em complete Erd\H os space\/} $\cerdos$ was introduced by Paul Erd\H os in 1940 as the closed subspace of the Hilbert space $\ell^2$ consisting of all vectors such that every coordinate is in the convergent sequence $\{0\}\cup\{1/n:n\in\N\}$. In a solution to a problem posed by Lex G. Oversteegen we present simple and useful topological characterizations of $\cerdos$. As an application we determine the class of factors of $\cerdos$. In another application we determine precisely which of the spaces that can be constructed in the Banach spaces $\ell^p$ according to the Erd\H os method' are homeomorphic to $\cerdos$. A novel application states that if $I$ is a Polishable $F_\sigma$-ideal on $\omega$, then $I$ with the Polish topology is homeomorphic to either $\Z$, the Cantor set $2^\omega$, $\Z\times2^\omega$, or $\cerdos$. This last result answers a question that was asked by Stevo Todor{\v{c}}evi{\'c}. Keywords:Complete Erd\H os space, Lelek fan, almost zero-dimensional, nowhere zero-dimensional, Polishable ideals, submeasures on $\omega$, $\R$-trees, line-free groups in Banach spacesCategories:28C10, 46B20, 54F65

16. CJM 2009 (vol 61 pp. 76)

Christensen, Lars Winther; Holm, Henrik
 Ascent Properties of Auslander Categories Let $R$ be a homomorphic image of a Gorenstein local ring. Recent work has shown that there is a bridge between Auslander categories and modules of finite Gorenstein homological dimensions over $R$. We use Gorenstein dimensions to prove new results about Auslander categories and vice versa. For example, we establish base change relations between the Auslander categories of the source and target rings of a homomorphism $\varphi \colon R \to S$ of finite flat dimension. Keywords:Auslander categories, Gorenstein dimensions, ascent properties, Auslander--Buchsbaum formulasCategories:13D05, 13D07, 13D25

17. CJM 2008 (vol 60 pp. 658)

Mihailescu, Eugen; Urba\'nski, Mariusz
 Inverse Pressure Estimates and the Independence of Stable Dimension for Non-Invertible Maps We study the case of an Axiom A holomorphic non-degenerate (hence non-invertible) map $f\from\mathbb P^2 \mathbb C \to \mathbb P^2 \mathbb C$, where $\mathbb P^2 \mathbb C$ stands for the complex projective space of dimension 2. Let $\Lambda$ denote a basic set for $f$ of unstable index 1, and $x$ an arbitrary point of $\Lambda$; we denote by $\delta^s(x)$ the Hausdorff dimension of $W^s_r(x) \cap \Lambda$, where $r$ is some fixed positive number and $W^s_r(x)$ is the local stable manifold at $x$ of size $r$; $\delta^s(x)$ is called \emph{the stable dimension at} $x$. Mihailescu and Urba\'nski introduced a notion of inverse topological pressure, denoted by $P^-$, which takes into consideration preimages of points. Manning and McCluskey study the case of hyperbolic diffeomorphisms on real surfaces and give formulas for Hausdorff dimension. Our non-invertible situation is different here since the local unstable manifolds are not uniquely determined by their base point, instead they depend in general on whole prehistories of the base points. Hence our methods are different and are based on using a sequence of inverse pressures for the iterates of $f$, in order to give upper and lower estimates of the stable dimension. We obtain an estimate of the oscillation of the stable dimension on $\Lambda$. When each point $x$ from $\Lambda$ has the same number $d'$ of preimages in $\Lambda$, then we show that $\delta^s(x)$ is independent of $x$; in fact $\delta^s(x)$ is shown to be equal in this case with the unique zero of the map $t \to P(t\phi^s - \log d')$. We also prove the Lipschitz continuity of the stable vector spaces over $\Lambda$; this proof is again different than the one for diffeomorphisms (however, the unstable distribution is not always Lipschitz for conformal non-invertible maps). In the end we include the corresponding results for a real conformal setting. Keywords:Hausdorff dimension, stable manifolds, basic sets, inverse topological pressureCategories:37D20, 37A35, 37F35

18. CJM 2007 (vol 59 pp. 332)

Leuschke, Graham J.
 Endomorphism Rings of Finite Global Dimension For a commutative local ring $R$, consider (noncommutative) $R$-algebras $\Lambda$ of the form $\Lambda = \operatorname{End}_R(M)$ where $M$ is a reflexive $R$-module with nonzero free direct summand. Such algebras $\Lambda$ of finite global dimension can be viewed as potential substitutes for, or analogues of, a resolution of singularities of $\operatorname{Spec} R$. For example, Van den Bergh has shown that a three-dimensional Gorenstein normal $\mathbb{C}$-algebra with isolated terminal singularities has a crepant resolution of singularities if and only if it has such an algebra $\Lambda$ with finite global dimension and which is maximal Cohen--Macaulay over $R$ (a `noncommutative crepant resolution of singularities''). We produce algebras $\Lambda=\operatorname{End}_R(M)$ having finite global dimension in two contexts: when $R$ is a reduced one-dimensional complete local ring, or when $R$ is a Cohen--Macaulay local ring of finite Cohen--Macaulay type. If in the latter case $R$ is Gorenstein, then the construction gives a noncommutative crepant resolution of singularities in the sense of Van den Bergh. Keywords:representation dimension, noncommutative crepant resolution, maximal Cohen--Macaulay modulesCategories:16G50, 16G60, 16E99

19. CJM 2006 (vol 58 pp. 449)

Agarwal, Ravi P.; Cao, Daomin; Lü, Haishen; O'Regan, Donal
 Existence and Multiplicity of Positive Solutions for Singular Semipositone $p$-Laplacian Equations Positive solutions are obtained for the boundary value problem $\begin{cases} -( | u'| ^{p-2}u')' =\lambda f( t,u),\;t\in ( 0,1) ,p>1\\ u( 0) =u(1) =0. \end{cases}$ Here $f(t,u) \geq -M,$ ($M$ is a positive constant) for $(t,u) \in [0\mathinner{,}1] \times (0,\infty )$. We will show the existence of two positive solutions by using degree theory together with the upper-lower solution method. Keywords:one dimensional $p$-Laplacian, positive solution, degree theory, upper and lower solutionCategory:34B15

20. CJM 2002 (vol 54 pp. 1280)

Skrzypczak, Leszek
 Besov Spaces and Hausdorff Dimension For Some Carnot-CarathÃ©odory Metric Spaces We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$ satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian $\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic decomposition of the spaces is given. In consequence we get the distributional characterization of the Hausdorff dimension of Borel subsets with the Haar measure zero. Keywords:Besov spaces, sub-elliptic operators, Carnot-CarathÃ©odory metric, Hausdorff dimensionCategories:46E35, 43A15, 28A78

21. CJM 1999 (vol 51 pp. 673)

Barlow, Martin T.; Bass, Richard F.
 Brownian Motion and Harmonic Analysis on Sierpinski Carpets We consider a class of fractal subsets of $\R^d$ formed in a manner analogous to the construction of the Sierpinski carpet. We prove a uniform Harnack inequality for positive harmonic functions; study the heat equation, and obtain upper and lower bounds on the heat kernel which are, up to constants, the best possible; construct a locally isotropic diffusion $X$ and determine its basic properties; and extend some classical Sobolev and Poincar\'e inequalities to this setting. Keywords:Sierpinski carpet, fractal, Hausdorff dimension, spectral dimension, Brownian motion, heat equation, harmonic functions, potentials, reflecting Brownian motion, coupling, Harnack inequality, transition densities, fundamental solutionsCategories:60J60, 60B05, 60J35

22. 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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