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1. CMB Online first

Vetrik, Tomas
 The metric dimension of circulant graphs A subset $W$ of the vertex set of a graph $G$ is called a resolving set of $G$ if for every pair of distinct vertices $u, v$ of $G$, there is $w \in W$ such that the~distance of $w$ and $u$ is different from the distance of $w$ and $v$. The~cardinality of a~smallest resolving set is called the metric dimension of $G$, denoted by $dim(G)$. The circulant graph $C_n (1, 2, \dots , t)$ consists of the vertices $v_0, v_1, \dots , v_{n-1}$ and the~edges $v_i v_{i+j}$, where $0 \le i \le n-1$, $1 \le j \le t$ $(2 \le t \le \lfloor \frac{n}{2} \rfloor)$, the indices are taken modulo $n$. Grigorious et al. [On the metric dimension of circulant and Harary graphs, Applied Mathematics and Computation 248 (2014), 47--54] proved that $dim(C_n (1,2, \dots , t)) \ge t+1$ for $t \lt \lfloor \frac{n}{2} \rfloor$, $n \ge 3$, and they presented a~conjecture saying that $dim(C_n (1,2, \dots , t)) = t+p-1$ for $n=2tk+t+p$, where $3 \le p \le t+1$. We disprove both statements. We show that if $t \ge 4$ is even, there exists an infinite set of values of $n$ such that $dim(C_n (1,2, \dots , t)) = t$. We also prove that $dim(C_n (1,2, \dots , t)) \le t + \frac{p}{2}$ for $n=2tk+t+p$, where $t$ and $p$ are even, $t \ge 4$, $2 \le p \le t$ and $k \ge 1$. Keywords:metric dimension, resolving set, circulant graph, Cayley graph, distanceCategories:05C35, 05C12

2. CMB Online first

Karpenko, Nikita A.
 Incompressibility of products of pseudo-homogeneous varieties We show that the conjectural criterion of $p$-incompressibility for products of projective homogeneous varieties in terms of the factors, previously known in a few special cases only, holds in general. Actually, the proof goes through for a wider class of varieties which includes the norm varieties associated to symbols in Galois cohomology of arbitrary degree. Keywords:algebraic groups, projective homogeneous varieties, Chow groups and motives, canonical dimension and incompressibilityCategories:20G15, 14C25

3. CMB Online first

Dolžan, David
 The metric dimension of the total graph of a finite commutative ring We study the total graph of a finite commutative ring. We calculate its metric dimension in the case when the Jacobson radical of the ring is nontrivial and we examine the metric dimension of the total graph of a product of at most two fields, obtaining either exact values in some cases or bounds in other, depending on the number of elements in the respective fields. Keywords:total graph, finite ring, metric dimensionCategories:13M99, 05E40

4. CMB 2016 (vol 59 pp. 403)

Zargar, Majid Rahro; Zakeri, Hossein
 On Flat and Gorenstein Flat Dimensions of Local Cohomology Modules Let $\mathfrak{a}$ be an ideal of a Noetherian local ring $R$ and let $C$ be a semidualizing $R$-module. For an $R$-module $X$, we denote any of the quantities $\mathfrak{d}_R X$, $\operatorname{\mathsf{Gfd}}_R X$ and $\operatorname{\mathsf{G_C-fd}}_RX$ by $\operatorname{\mathsf{T}}(X)$. Let $M$ be an $R$-module such that $\operatorname{H}_{\mathfrak{a}}^i(M)=0$ for all $i\neq n$. It is proved that if $\operatorname{\mathsf{T}}(X)\lt \infty$, then $\operatorname{\mathsf{T}}(\operatorname{H}_{\mathfrak{a}}^n(M))\leq\operatorname{\mathsf{T}}(M)+n$ and the equality holds whenever $M$ is finitely generated. With the aid of these results, among other things, we characterize Cohen-Macaulay modules, dualizing modules and Gorenstein rings. Keywords:flat dimension, Gorenstein injective dimension, Gorenstein flat dimension, local cohomology, relative Cohen-Macaulay module, semidualizing moduleCategories:13D05, 13D45, 18G20

5. CMB 2015 (vol 58 pp. 741)

Gao, Zenghui
 Homological Properties Relative to Injectively Resolving Subcategories Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\operatorname{Ext}_R^1(G,M)=0$ (resp. $\operatorname{Tor}_1^R(N,G)=0$) for any $G\in\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that $\mathcal{E}^+\subseteq\mathcal{F}$ and $\mathcal{F}^+\subseteq\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules. Keywords:injectively resolving subcategory, \mathcal{E}-injective module (dimension), \mathcal{E}-flat module (dimension), cover, preenvelope, \mathcal{E}-(semi)hereditary ringCategories:16E30, 16E10, 16E60

6. CMB 2015 (vol 59 pp. 144)

Laterveer, Robert
 A Brief Note Concerning Hard Lefschetz for Chow Groups We formulate a conjectural hard Lefschetz property for Chow groups, and prove this in some special cases: roughly speaking, for varieties with finite-dimensional motive, and for varieties whose self-product has vanishing middle-dimensional Griffiths group. An appendix includes related statements that follow from results of Vial. Keywords:algebraic cycles, Chow groups, finite-dimensional motivesCategories:14C15, 14C25, 14C30

7. CMB 2015 (vol 58 pp. 787)

 Non-branching RCD$(0,N)$ Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups In this paper, we generalize the finite generation result of Sormani to non-branching $RCD(0,N)$ geodesic spaces (and in particular, Alexandrov spaces) with full support measures. This is a special case of the Milnor's Conjecture for complete non-compact $RCD(0,N)$ spaces. One of the key tools we use is the Abresch-Gromoll type excess estimates for non-smooth spaces obtained by Gigli-Mosconi. Keywords:Milnor conjecture, non negative Ricci curvature, curvature dimension condition, finitely generated, fundamental group, infinitesimally HilbertianCategories:53C23, 30L99

8. CMB 2015 (vol 58 pp. 664)

Vahidi, Alireza
 Betti Numbers and Flat Dimensions of Local Cohomology Modules Assume that $R$ is a commutative Noetherian ring with non-zero identity, $\mathfrak{a}$ is an ideal of $R$ and $X$ is an $R$--module. In this paper, we first study the finiteness of Betti numbers of local cohomology modules $\operatorname{H}_\mathfrak{a}^i(X)$. Then we give some inequalities between the Betti numbers of $X$ and those of its local cohomology modules. Finally, we present many upper bounds for the flat dimension of $X$ in terms of the flat dimensions of its local cohomology modules and an upper bound for the flat dimension of $\operatorname{H}_\mathfrak{a}^i(X)$ in terms of the flat dimensions of the modules $\operatorname{H}_\mathfrak{a}^j(X)$, $j\not= i$, and that of $X$. Keywords:Betti numbers, flat dimensions, local cohomology modulesCategories:13D45, 13D05

9. CMB 2015 (vol 58 pp. 519)

Kang, Su-Jeong
 Refined Motivic Dimension We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration. Keywords:motivic dimension, generalized Hodge conjectureCategories:14C30, 14C25

10. CMB 2015 (vol 58 pp. 402)

Tikuisis, Aaron Peter; Toms, Andrew
 On the Structure of Cuntz Semigroups in (Possibly) Nonunital C*-algebras We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras as measured by their densely defined lower semicontinuous traces. We first prove that a unital simple $\mathrm{C}^*$-algebra whose extreme tracial boundary is nonempty and finite contains positive operators of every possible rank, independent of the property of strict comparison. We then turn to nonunital simple algebras and establish criteria that imply that the Cuntz semigroup is recovered functorially from the Murray-von Neumann semigroup and the space of densely defined lower semicontinuous traces. Finally, we prove that these criteria are satisfied by not-necessarily-unital approximately subhomogeneous algebras of slow dimension growth. Combined with results of the first-named author, this shows that slow dimension growth coincides with $\mathcal Z$-stability, for approximately subhomogeneous algebras. Keywords:nuclear C*-algebras, Cuntz semigroup, dimension functions, stably projectionless C*-algebras, approximately subhomogeneous C*-algebras, slow dimension growthCategories:46L35, 46L05, 46L80, 47L40, 46L85

11. CMB 2014 (vol 57 pp. 814)

Hou, Ruchen
 On Global Dimensions of Tree Type Finite Dimensional Algebras A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k-$algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum of possible global dimensions is $n-1$. Keywords:global dimension, tree type finite dimensional $k-$algebra, quiverCategories:16D40, 16E10, , 16G20

12. CMB 2014 (vol 57 pp. 673)

 Complexifying Lie Group Actions on Homogeneous Manifolds of Non-compact Dimension Two If $X$ is a connected complex manifold with $d_X = 2$ that admits a (connected) Lie group $G$ acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification $\widehat{G}$ of $G$ on $X$ except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of $X$. Keywords:homogeneous complex manifold, non-compact dimension two, complexificationCategory:32M10

13. CMB 2013 (vol 57 pp. 245)

Brodskiy, N.; Dydak, J.; Lang, U.
 Assouad-Nagata Dimension of Wreath Products of Groups Consider the wreath product $H\wr G$, where $H\ne 1$ is finite and $G$ is finitely generated. We show that the Assouad-Nagata dimension $\dim_{AN}(H\wr G)$ of $H\wr G$ depends on the growth of $G$ as follows: \par If the growth of $G$ is not bounded by a linear function, then $\dim_{AN}(H\wr G)=\infty$, otherwise $\dim_{AN}(H\wr G)=\dim_{AN}(G)\leq 1$. Keywords:Assouad-Nagata dimension, asymptotic dimension, wreath product, growth of groupsCategories:54F45, 55M10, 54C65

14. CMB 2013 (vol 57 pp. 335)

Karassev, A.; Todorov, V.; Valov, V.
 Alexandroff Manifolds and Homogeneous Continua ny homogeneous, metric $ANR$-continuum is a $V^n_G$-continuum provided $\dim_GX=n\geq 1$ and $\check{H}^n(X;G)\neq 0$, where $G$ is a principal ideal domain. This implies that any homogeneous $n$-dimensional metric $ANR$-continuum is a $V^n$-continuum in the sense of Alexandroff. We also prove that any finite-dimensional homogeneous metric continuum $X$, satisfying $\check{H}^n(X;G)\neq 0$ for some group $G$ and $n\geq 1$, cannot be separated by a compactum $K$ with $\check{H}^{n-1}(K;G)=0$ and $\dim_G K\leq n-1$. This provides a partial answer to a question of Kallipoliti-Papasoglu whether any two-dimensional homogeneous Peano continuum cannot be separated by arcs. Keywords:Cantor manifold, cohomological dimension, cohomology groups, homogeneous compactum, separator, $V^n$-continuumCategories:54F45, 54F15

15. CMB 2013 (vol 56 pp. 795)

MacDonald, Mark L.
 Upper Bounds for the Essential Dimension of $E_7$ This paper gives a new upper bound for the essential dimension and the essential 2-dimension of the split simply connected group of type $E_7$ over a field of characteristic not 2 or 3. In particular, $\operatorname{ed}(E_7) \leq 29$, and $\operatorname{ed}(E_7;2) \leq 27$. Keywords:$E_7$, essential dimension, stabilizer in general positionCategories:20G15, 20G41

16. CMB 2013 (vol 57 pp. 159)

Oral, Kürşat Hakan; Özkirişci, Neslihan Ayşen; Tekir, Ünsal
 Strongly $0$-dimensional Modules In a multiplication module, prime submodules have the property, if a prime submodule contains a finite intersection of submodules then one of the submodules is contained in the prime submodule. In this paper, we generalize this property to infinite intersection of submodules and call such prime submodules strongly prime submodule. A multiplication module in which every prime submodule is strongly prime will be called strongly 0-dimensional module. It is also an extension of strongly 0-dimensional rings. After this we investigate properties of strongly 0-dimensional modules and give relations of von Neumann regular modules, Q-modules and strongly 0-dimensional modules. Keywords:strongly 0-dimensional rings, Q-module, Von Neumann regular moduleCategories:13C99, 16D10

17. CMB 2013 (vol 56 pp. 745)

Fu, Xiaoye; Gabardo, Jean-Pierre
 Dimension Functions of Self-Affine Scaling Sets In this paper, the dimension function of a self-affine generalized scaling set associated with an $n\times n$ integral expansive dilation $A$ is studied. More specifically, we consider the dimension function of an $A$-dilation generalized scaling set $K$ assuming that $K$ is a self-affine tile satisfying $BK = (K+d_1) \cup (K+d_2)$, where $B=A^t$, $A$ is an $n\times n$ integral expansive matrix with $\lvert \det A\rvert=2$, and $d_1,d_2\in\mathbb{R}^n$. We show that the dimension function of $K$ must be constant if either $n=1$ or $2$ or one of the digits is $0$, and that it is bounded by $2\lvert K\rvert$ for any $n$. Keywords:scaling set, self-affine tile, orthonormal multiwavelet, dimension functionCategory:42C40

18. CMB 2012 (vol 56 pp. 737)

Elliott, George A.; Niu, Zhuang
 On the Radius of Comparison of a Commutative C*-algebra Let $X$ be a compact metric space. A lower bound for the radius of comparison of the C*-algebra $\operatorname{C}(X)$ is given in terms of $\operatorname{dim}_{\mathbb{Q}} X$, where $\operatorname{dim}_{\mathbb{Q}} X$ is the cohomological dimension with rational coefficients. If $\operatorname{dim}_{\mathbb{Q}} X =\operatorname{dim} X=d$, then the radius of comparison of the C*-algebra $\operatorname{C}(X)$ is $\max\{0, (d-1)/2-1\}$ if $d$ is odd, and must be either $d/2-1$ or $d/2-2$ if $d$ is even (the possibility of $d/2-1$ does occur, but we do not know if the possibility of $d/2-2$ also can occur). Keywords:Cuntz semigroup, comparison radius, cohomology dimension, covering dimension

19. CMB 2012 (vol 56 pp. 683)

Nikseresht, A.; Azizi, A.
 Envelope Dimension of Modules and the Simplified Radical Formula We introduce and investigate the notion of envelope dimension of commutative rings and modules over them. In particular, we show that the envelope dimension of a ring, $R$, is equal to that of the $R$-module $R^{(\mathbb{N})}$. Also we prove that the Krull dimension of a ring is no more than its envelope dimension and characterize Noetherian rings for which these two dimensions are equal. Moreover we generalize and study the concept of simplified radical formula for modules, which we defined in an earlier paper. Keywords:envelope dimension, simplified radical formula, prime submoduleCategories:13A99, 13C99, 13C13, 13E05

20. CMB 2012 (vol 56 pp. 551)

Handelman, David
 Real Dimension Groups Dimension groups (not countable) that are also real ordered vector spaces can be obtained as direct limits (over directed sets) of simplicial real vector spaces (finite dimensional vector spaces with the coordinatewise ordering), but the directed set is not as interesting as one would like, i.e., it is not true that a countable-dimensional real vector space that has interpolation can be represented as such a direct limit over the a countable directed set. It turns out this is the case when the group is additionally simple, and it is shown that the latter have an ordered tensor product decomposition. In the Appendix, we provide a huge class of polynomial rings that, with a pointwise ordering, are shown to satisfy interpolation, extending a result outlined by Fuchs. Keywords:dimension group, simplicial vector space, direct limit, Riesz interpolationCategories:46A40, 06F20, 13J25, 19K14

21. CMB 2012 (vol 56 pp. 491)

Bahmanpour, Kamal
 A Note on Homological Dimensions of Artinian Local Cohomology Modules Let $(R,{\frak m})$ be a non-zero commutative Noetherian local ring (with identity), $M$ be a non-zero finitely generated $R$-module. In this paper for any ${\frak p}\in {\rm Spec}(R)$ we show that $\operatorname{{\rm injdim_{_{R_{\frak p}}}}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ and ${\rm fd}_{R_{\p}} H^{i-\dim(R/{\frak p})}_{{\frak p}R_{\frak p}}(M_{\frak p})$ are bounded from above by $\operatorname{{\rm injdim_{_{R}}}} H^i_{\frak m}(M)$ and ${\rm fd}_R H^i_{\frak m}(M)$ respectively, for all integers $i\geq \dim(R/{\frak p})$. Keywords:cofinite modules, flat dimension, injective dimension, Krull dimension, local cohomologyCategory:13D45

22. CMB 2012 (vol 56 pp. 630)

Sundar, S.
 Inverse Semigroups and Sheu's Groupoid for the Odd Dimensional Quantum Spheres In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid $C^{*}$-algebras. We show that the $C^{*}$-algebra $C(S_{q}^{2\ell+1})$ is generated by an inverse semigroup $T$ of partial isometries. We show that the groupoid $\mathcal{G}_{tight}$ associated with the inverse semigroup $T$ by Exel is exactly the same as the groupoid considered by Sheu. Keywords:inverse semigroups, groupoids, odd dimensional quantum spheresCategories:46L99, 20M18

23. CMB 2011 (vol 56 pp. 354)

Hare, Kathryn E.; Mendivil, Franklin; Zuberman, Leandro
 The Sizes of Rearrangements of Cantor Sets A linear Cantor set $C$ with zero Lebesgue measure is associated with the countable collection of the bounded complementary open intervals. A rearrangment of $C$ has the same lengths of its complementary intervals, but with different locations. We study the Hausdorff and packing $h$-measures and dimensional properties of the set of all rearrangments of some given $C$ for general dimension functions $h$. For each set of complementary lengths, we construct a Cantor set rearrangement which has the maximal Hausdorff and the minimal packing $h$-premeasure, up to a constant. We also show that if the packing measure of this Cantor set is positive, then there is a rearrangement which has infinite packing measure. Keywords:Hausdorff dimension, packing dimension, dimension functions, Cantor sets, cut-out setCategories:28A78, 28A80

24. CMB 2011 (vol 56 pp. 292)

Dai, Mei-Feng
 Quasisymmetrically Minimal Moran Sets M. Hu and S. Wen considered quasisymmetrically minimal uniform Cantor sets of Hausdorff dimension $1$, where at the $k$-th set one removes from each interval $I$ a certain number $n_{k}$ of open subintervals of length $c_{k}|I|$, leaving $(n_{k}+1)$ closed subintervals of equal length. Quasisymmetrically Moran sets of Hausdorff dimension $1$ considered in the paper are more general than uniform Cantor sets in that neither the open subintervals nor the closed subintervals are required to be of equal length. Keywords:quasisymmetric, Moran set, Hausdorff dimensionCategories:28A80, 54C30

25. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
 From Matrix to Operator Inequalities We generalize LÃ¶wner's method for proving that matrix monotone functions are operator monotone. The relation $x\leq y$ on bounded operators is our model for a definition of $C^{*}$-relations being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators, and "positive" noncommutative $*$-polynomials. Keywords:$C*$-algebras, matrices, bounded operators, relations, operator norm, order, commutator, exponential, residually finite dimensionalCategories:46L05, 47B99
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