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Search: All articles in the CMB digital archive with keyword dimension

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1. 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 groups
Categories:54F45, 55M10, 54C65

2. 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, $V^n$-continuum
Categories:54F45, 54F15

3. 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 position
Categories:20G15, 20G41

4. 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 module
Categories:13C99, 16D10

5. 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 function
Category:42C40

6. 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

7. 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 submodule
Categories:13A99, 13C99, 13C13, 13E05

8. 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 interpolation
Categories:46A40, 06F20, 13J25, 19K14

9. 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 cohomology
Category:13D45

10. 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 spheres
Categories:46L99, 20M18

11. 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 set
Categories:28A78, 28A80

12. 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 dimension
Categories:28A80, 54C30

13. CMB 2011 (vol 55 pp. 339)

Loring, Terry A.
From Matrix to Operator Inequalities
We generalize Löwner'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 dimensional
Categories:46L05, 47B99

14. CMB 2011 (vol 54 pp. 619)

Dibaei, Mohammad T.; Vahidi, Alireza
Artinian and Non-Artinian Local Cohomology Modules
Let $M$ be a finite module over a commutative noetherian ring $R$. For ideals $\mathfrak{a}$ and $\mathfrak{b}$ of $R$, the relations between cohomological dimensions of $M$ with respect to $\mathfrak{a}, \mathfrak{b}$, $\mathfrak{a}\cap\mathfrak{b}$ and $\mathfrak{a}+ \mathfrak{b}$ are studied. When $R$ is local, it is shown that $M$ is generalized Cohen-Macaulay if there exists an ideal $\mathfrak{a}$ such that all local cohomology modules of $M$ with respect to $\mathfrak{a}$ have finite lengths. Also, when $r$ is an integer such that $0\leq r< \dim_R(M)$, any maximal element $\mathfrak{q}$ of the non-empty set of ideals $\{\mathfrak{a} : \textrm{H}_\mathfrak{a}^i(M) $ is not artinian for some $ i, i\geq r \}$ is a prime ideal, and all Bass numbers of $\textrm{H}_\mathfrak{q}^i(M)$ are finite for all $i\geq r$.

Keywords:local cohomology modules, cohomological dimensions, Bass numbers
Categories:13D45, 13E10

15. CMB 2010 (vol 53 pp. 629)

Chinen, Naotsugu; Hosaka, Tetsuya
Asymptotic Dimension of Proper CAT(0) Spaces that are Homeomorphic to the Plane
In this paper, we investigate a proper CAT(0) space $(X,d)$ that is homeomorphic to $\mathbb R^2$ and we show that the asymptotic dimension $\operatorname{asdim} (X,d)$ is equal to $2$.

Keywords:asymptotic dimension, CAT(0) space, plane
Categories:20F69, 54F45, 20F65

16. CMB 2010 (vol 53 pp. 564)

Watanabe, Yoshiyuki; Suh, Young Jin
On $6$-Dimensional Nearly Kähler Manifolds
In this paper we give a sufficient condition for a complete, simply connected, and strict nearly Kähler manifold of dimension 6 to be a homogeneous nearly Kähler manifold. This result was announced in a previous paper by the first author.

Keywords:Nearly Kähler manifold, 6-dimension, Homogeneous, The 1st Chern Class, Einstein manifolds
Categories:53C40, 53C15

17. CMB 2010 (vol 53 pp. 503)

Kurenok, V. P.
The Time Change Method and SDEs with Nonnegative Drift
Using the time change method we show how to construct a solution to the stochastic equation $dX_t=b(X_{t-})dZ_t+a(X_t)dt$ with a nonnegative drift $a$ provided there exists a solution to the auxililary equation $dL_t=[a^{-1/\alpha}b](L_{t-})d\bar Z_t+dt$ where $Z, \bar Z$ are two symmetric stable processes of the same index $\alpha\in(0,2]$. This approach allows us to prove the existence of solutions for both stochastic equations for the values $0<\alpha<1$ and only measurable coefficients $a$ and $b$ satisfying some conditions of boundedness. The existence proof for the auxililary equation uses the method of integral estimates in the sense of Krylov.

Keywords:One-dimensional SDEs, symmetric stable processes, nonnegative drift, time change, integral estimates, weak convergence
Categories:60H10, 60J60, 60J65, 60G44

18. CMB 2010 (vol 53 pp. 438)

Chigogidze, A.; Nagórko, A.
Near-Homeomorphisms of Nöbeling Manifolds
We characterize maps between $n$-dimensional Nöbeling manifolds that can be approximated by homeomorphisms.

Keywords:n-dimensional Nöbeling manifold, Z-set unknotting, near-homeomorphism
Categories:55M10, 54F45

19. CMB 2010 (vol 53 pp. 327)

Luor, Dah-Chin
Multidimensional Exponential Inequalities with Weights
We establish sufficient conditions on the weight functions $u$ and $v$ for the validity of the multidimensional weighted inequality $$ \Bigl(\int_E \Phi(T_k f(x))^q u(x)\,dx\Bigr)^{1/q} \le C \Bigl (\int_E \Phi(f(x))^p v(x)\,dx\Bigr )^{1/p}, $$ where 0<$p$, $q$<$\infty$, $\Phi$ is a logarithmically convex function, and $T_k$ is an integral operator over star-shaped regions. The condition is also necessary for the exponential integral inequality. Moreover, the estimation of $C$ is given and we apply the obtained results to generalize some multidimensional Levin--Cochran-Lee type inequalities.

Keywords:multidimensional inequalities, geometric mean operators, exponential inequalities, star-shaped regions
Categories:26D15, 26D10

20. CMB 2008 (vol 51 pp. 236)

21. CMB 2007 (vol 50 pp. 481)

Blanlœil, Vincent; Saeki, Osamu
Concordance des nœuds de dimension $4$
We prove that for a simply connected closed $4$-dimensional manifold, its embeddings into the sphere of dimension $6$ are all concordant to each other.

Keywords:concordance, cobordisme, n{\oe}ud de dimension $4$, chirurgie plongée
Categories:57Q45, 57Q60, 57R40, 57R65, 57N13

22. CMB 2007 (vol 50 pp. 588)

Labute, John; Lemire, Nicole; Mináč, Ján; Swallow, John
Cohomological Dimension and Schreier's Formula in Galois Cohomology
Let $p$ be a prime and $F$ a field containing a primitive $p$-th root of unity. Then for $n\in \N$, the cohomological dimension of the maximal pro-$p$-quotient $G$ of the absolute Galois group of $F$ is at most $n$ if and only if the corestriction maps $H^n(H,\Fp) \to H^n(G,\Fp)$ are surjective for all open subgroups $H$ of index $p$. Using this result, we generalize Schreier's formula for $\dim_{\Fp} H^1(H,\Fp)$ to $\dim_{\Fp} H^n(H,\Fp)$.

Keywords:cohomological dimension, Schreier's formula, Galois theory, $p$-extensions, pro-$p$-groups
Categories:12G05, 12G10

23. CMB 2006 (vol 49 pp. 247)

Myjak, Józef; Szarek, Tomasz; Ślȩczka, Maciej
A Szpilrajn--Marczewski Type Theorem for Concentration Dimension on Polish Spaces
Let $X$ be a Polish space. We will prove that $$ \dim_T X=\inf \{\dim_L X': X'\text{ is homeomorphic to } X\}, $$ where $\dim_L X$ and $\dim_T X$ stand for the concentration dimension and the topological dimension of $X$, respectively.

Keywords:Hausdorff dimension, topological dimension, Lévy concentration function, concentration dimension
Categories:11K55, 28A78

24. CMB 2005 (vol 48 pp. 614)

Tuncali, H. Murat; Valov, Vesko
On Finite-to-One Maps
Let $f\colon X\to Y$ be a $\sigma$-perfect $k$-dimensional surjective map of metrizable spaces such that $\dim Y\leq m$. It is shown that for every positive integer $p$ with $ p\leq m+k+1$ there exists a dense $G_{\delta}$-subset ${\mathcal H}(k,m,p)$ of $C(X,\uin^{k+p})$ with the source limitation topology such that each fiber of $f\triangle g$, $g\in{\mathcal H}(k,m,p)$, contains at most $\max\{k+m-p+2,1\}$ points. This result provides a proof the following conjectures of S. Bogatyi, V. Fedorchuk and J. van Mill. Let $f\colon X\to Y$ be a $k$-dimensional map between compact metric spaces with $\dim Y\leq m$. Then: \begin{inparaenum}[\rm(1)] \item there exists a map $h\colon X\to\uin^{m+2k}$ such that $f\triangle h\colon X\to Y\times\uin^{m+2k}$ is 2-to-one provided $k\geq 1$; \item there exists a map $h\colon X\to\uin^{m+k+1}$ such that $f\triangle h\colon X\to Y\times\uin^{m+k+1}$ is $(k+1)$-to-one. \end{inparaenum}

Keywords:finite-to-one maps, dimension, set-valued maps
Categories:54F45, 55M10, 54C65

25. CMB 2005 (vol 48 pp. 340)

Andruchow, Esteban
Short Geodesics of Unitaries in the $L^2$ Metric
Let $\M$ be a type II$_1$ von Neumann algebra, $\tau$ a trace in $\M$, and $\l2$ the GNS Hilbert space of $\tau$. We regard the unitary group $U_\M$ as a subset of $\l2$ and characterize the shortest smooth curves joining two fixed unitaries in the $L^2$ metric. As a consequence of this we obtain that $U_\M$, though a complete (metric) topological group, is not an embedded riemannian submanifold of $\l2$

Keywords:unitary group, short geodesics, infinite dimensional riemannian manifolds.
Categories:46L51, 58B10, 58B25
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