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

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26. CMB 2011 (vol 54 pp. 249)

Dattori da Silva, Paulo L.
A Note about Analytic Solvability of Complex Planar Vector Fields with Degeneracies
This paper deals with the analytic solvability of a special class of complex vector fields defined on the real plane, where they are tangent to a closed real curve, while off the real curve, they are elliptic.

Keywords:semi-global solvability, analytic solvability, normalization, complex vector fields, condition~($\mathcal P$)
Categories:35A01, 58Jxx

27. CMB 2011 (vol 54 pp. 302)

Kurka, Ondřej
Structure of the Set of Norm-attaining Functionals on Strictly Convex Spaces
Let $X$ be a separable non-reflexive Banach space. We show that there is no Borel class which contains the set of norm-attaining functionals for every strictly convex renorming of $X$.

Keywords:separable non-reflexive space, set of norm-attaining functionals, strictly convex norm, Borel class
Categories:46B20, 54H05, 46B10

28. CMB 2010 (vol 54 pp. 21)

Bouali, S.; Ech-chad, M.
Generalized D-symmetric Operators II
Let $H$ be a separable, infinite-dimensional, complex Hilbert space and let $A, B\in{\mathcal L }(H)$, where ${\mathcal L}(H)$ is the algebra of all bounded linear operators on $H$. Let $\delta_{AB}\colon {\mathcal L}(H)\rightarrow {\mathcal L}(H)$ denote the generalized derivation $\delta_{AB}(X)=AX-XB$. This note will initiate a study on the class of pairs $(A,B)$ such that $\overline{{\mathcal R}(\delta_{AB})}= \overline{{\mathcal R}(\delta_{A^{\ast}B^{\ast}})}$.

Keywords:generalized derivation, adjoint, D-symmetric operator, normal operator
Categories:47B47, 47B10, 47A30

29. CMB 2009 (vol 53 pp. 295)

Guo, Boling; Huo, Zhaohui
The Global Attractor of a Damped, Forced Hirota Equation in $H^1$
The existence of the global attractor of a damped forced Hirota equation in the phase space $H^1(\mathbb R)$ is proved. The main idea is to establish the so-called asymptotic compactness property of the solution operator by energy equation approach.

Keywords:global attractor, Fourier restriction norm, damping system, asymptotic compactness
Categories:35Q53, 35B40, 35B41, 37L30

30. CMB 2009 (vol 52 pp. 424)

Martini, Horst; Spirova, Margarita
Covering Discs in Minkowski Planes
We investigate the following version of the circle covering problem in strictly convex (normed or) Minkowski planes: to cover a circle of largest possible diameter by $k$ unit circles. In particular, we study the cases $k=3$, $k=4$, and $k=7$. For $k=3$ and $k=4$, the diameters under consideration are described in terms of side-lengths and circumradii of certain inscribed regular triangles or quadrangles. This yields also simple explanations of geometric meanings that the corresponding homothety ratios have. It turns out that basic notions from Minkowski geometry play an essential role in our proofs, namely Minkowskian bisectors, $d$-segments, and the monotonicity lemma.

Keywords:affine regular polygon, bisector, circle covering problem, circumradius, $d$-segment, Minkowski plane, (strictly convex) normed plane
Categories:46B20, 52A21, 52C15

31. CMB 2008 (vol 51 pp. 508)

Cavicchioli, Alberto; Spaggiari, Fulvia
A Result in Surgery Theory
We study the topological $4$-dimensional surgery problem for a closed connected orientable topological $4$-manifold $X$ with vanishing second homotopy and $\pi_1(X)\cong A * F(r)$, where $A$ has one end and $F(r)$ is the free group of rank $r\ge 1$. Our result is related to a theorem of Krushkal and Lee, and depends on the validity of the Novikov conjecture for such fundamental groups.

Keywords:four-manifolds, homotopy type, obstruction theory, homology with local coefficients, surgery, normal invariant, assembly map
Categories:57N65, 57R67, 57Q10

32. CMB 2008 (vol 51 pp. 261)

Neeb, Karl-Hermann
On the Classification of Rational Quantum Tori and the Structure of Their Automorphism Groups
An $n$-dimensional quantum torus is a twisted group algebra of the group $\Z^n$. It is called rational if all invertible commutators are roots of unity. In the present note we describe a normal form for rational $n$-dimensional quantum tori over any field. Moreover, we show that for $n = 2$ the natural exact sequence describing the automorphism group of the quantum torus splits over any field.

Keywords:quantum torus, normal form, automorphisms of quantum tori

33. CMB 2008 (vol 51 pp. 81)

Kassel, Christian
Homotopy Formulas for Cyclic Groups Acting on Rings
The positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

Keywords:group cohomology, norm map, cyclic group, homotopy
Categories:20J06, 20K01, 16W22, 18G35

34. CMB 2008 (vol 51 pp. 15)

Aqzzouz, Belmesnaoui; Nouira, Redouane; Zraoula, Larbi
The Duality Problem for the Class of AM-Compact Operators on Banach Lattices
We prove the converse of a theorem of Zaanen about the duality problem of positive AM-compact operators.

Keywords:AM-compact operator, order continuous norm, discrete vector lattice
Categories:46A40, 46B40, 46B42

35. CMB 2007 (vol 50 pp. 610)

Rychtář, Jan; Spurný, Jiří
On Weak$^*$ Kadec--Klee Norms
We present partial positive results supporting a conjecture that admitting an equivalent Lipschitz (or uniformly) weak$^*$ Kadec--Klee norm is a three space property.

Keywords:weak$^*$ Kadec--Klee norms, three-space problem
Categories:46B03, 46B2

36. CMB 2007 (vol 50 pp. 268)

Manuilov, V.; Thomsen, K.
On the Lack of Inverses to $C^*$-Extensions Related to Property T Groups
Using ideas of S. Wassermann on non-exact $C^*$-algebras and property T groups, we show that one of his examples of non-invertible $C^*$-extensions is not semi-invertible. To prove this, we show that a certain element vanishes in the asymptotic tensor product. We also show that a modification of the example gives a $C^*$-extension which is not even invertible up to homotopy.

Keywords:$C^*$-algebra extension, property T group, asymptotic tensor $C^*$-norm, homotopy
Categories:19K33, 46L06, 46L80, 20F99

37. CMB 2006 (vol 49 pp. 185)

Averkov, Gennadiy
On the Inequality for Volume and Minkowskian Thickness
Given a centrally symmetric convex body $B$ in $\E^d,$ we denote by $\M^d(B)$ the Minkowski space ({\em i.e.,} finite dimensional Banach space) with unit ball $B.$ Let $K$ be an arbitrary convex body in $\M^d(B).$ The relationship between volume $V(K)$ and the Minkowskian thickness ($=$ minimal width) $\thns_B(K)$ of $K$ can naturally be given by the sharp geometric inequality $V(K) \ge \alpha(B) \cdot \thns_B(K)^d,$ where $\alpha(B)>0.$ As a simple corollary of the Rogers--Shephard inequality we obtain that $\binom{2d}{d}{}^{-1} \le \alpha(B)/V(B) \le 2^{-d}$ with equality on the left attained if and only if $B$ is the difference body of a simplex and on the right if $B$ is a cross-polytope. The main result of this paper is that for $d=2$ the equality on the right implies that $B$ is a parallelogram. The obtained results yield the sharp upper bound for the modified Banach--Mazur distance to the regular hexagon.

Keywords:convex body, geometric inequality, thickness, Minkowski space, Banach space, normed space, reduced body, Banach-Mazur compactum, (modified) Banach-Mazur distance, volume ratio
Categories:52A40, 46B20

38. CMB 2005 (vol 48 pp. 195)

Daniel, D.; Nikiel, J.; Treybig, L. B.; Tuncali, H. M.; Tymchatyn, E. D.
On Suslinian Continua
A continuum is said to be Suslinian if it does not contain uncountably many mutually exclusive nondegenerate subcontinua. We prove that Suslinian continua are perfectly normal and rim-metrizable. Locally connected Suslinian continua have weight at most $\omega_1$ and under appropriate set-theoretic conditions are metrizable. Non-separable locally connected Suslinian continua are rim-finite on some open set.

Keywords:Suslinian continuum, Souslin line, locally connected, rim-metrizable,, perfectly normal, rim-finite
Categories:54F15, 54D15, 54F50

39. CMB 2005 (vol 48 pp. 121)

Mollin, R. A.
Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$
We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc $ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$.

Keywords:quadratic Diophantine equations, simple continued fractions,, norms of ideals, infrastructure of real quadratic fields
Categories:11A55, 11D09, 11R11

40. CMB 2004 (vol 47 pp. 49)

Lindström, Mikael; Makhmutov, Shamil; Taskinen, Jari
The Essential Norm of a Bloch-to-$Q_p$ Composition Operator
The $Q_p$ spaces coincide with the Bloch space for $p>1$ and are subspaces of $\BMOA$ for $0
Keywords:Bloch space, little Bloch space, $\BMOA$, $\VMOA$, $Q_p$ spaces,, composition operator, compact operator, essential norm
Categories:47B38, 47B10, 46E40, 46E15

41. CMB 2001 (vol 44 pp. 323)

Schuman, Bertrand
Une classe d'hamiltoniens polynomiaux isochrones
Soit $H_0 = \frac{x^2+y^2}{2}$ un hamiltonien isochrone du plan $\Rset^2$. On met en \'evidence une classe d'hamiltoniens isochrones qui sont des perturbations polynomiales de $H_0$. On obtient alors une condition n\'ecessaire d'isochronisme, et un crit\`ere de choix pour les hamiltoniens isochrones. On voit ce r\'esultat comme \'etant une g\'en\'eralisation du caract\`ere isochrone des perturbations hamiltoniennes homog\`enes consid\'er\'ees dans [L], [P], [S]. Let $H_0 = \frac{x^2+y^2}{2}$ be an isochronous Hamiltonian of the plane $\Rset^2$. We obtain a necessary condition for a system to be isochronous. We can think of this result as a generalization of the isochronous behaviour of the homogeneous polynomial perturbation of the Hamiltonian $H_0$ considered in [L], [P], [S].

Keywords:Hamiltonian system, normal forms, resonance, linearization
Categories:34C20, 58F05, 58F22, 58F30

42. CMB 2001 (vol 44 pp. 370)

Weston, Anthony
On Locating Isometric $\ell_{1}^{(n)}$
Motivated by a question of Per Enflo, we develop a hypercube criterion for locating linear isometric copies of $\lone$ in an arbitrary real normed space $X$. The said criterion involves finding $2^{n}$ points in $X$ that satisfy one metric equality. This contrasts nicely to the standard classical criterion wherein one seeks $n$ points that satisfy $2^{n-1}$ metric equalities.

Keywords:normed spaces, hypercubes
Categories:46B04, 05C10, 05B99

43. CMB 2000 (vol 43 pp. 406)

Borwein, David
Weighted Mean Operators on $l_p$
The weighted mean matrix $M_a$ is the triangular matrix $\{a_k/A_n\}$, where $a_n > 0$ and $A_n := a_1 + a_2 + \cdots + a_n$. It is proved that, subject to $n^c a_n$ being eventually monotonic for each constant $c$ and to the existence of $\alpha := \lim \frac{A_n}{na_n}$, $M_a \in B(l_p)$ for $1 < p < \infty$ if and only if $\alpha < p$.

Keywords:weighted means, operators on $l_p$, norm estimates
Categories:47B37, 47A30, 40G05

44. CMB 1999 (vol 42 pp. 118)

Rao, T. S. S. R. K.
Points of Weak$^\ast$-Norm Continuity in the Unit Ball of the Space $\WC(K,X)^\ast$
For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak$^\ast$-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.

Keywords:Points of weak$^\ast$-norm continuity, space of vector valued weakly continuous functions, $M$-ideals
Categories:46B20, 46E40

45. CMB 1999 (vol 42 pp. 87)

Kittaneh, Fuad
Some norm inequalities for operators
Let $A_i$, $B_i$ and $X_i$ $(i=1, 2, \dots, n)$ be operators on a separable Hilbert space. It is shown that if $f$ and $g$ are nonnegative continuous functions on $[0,\infty)$ which satisfy the relation $f(t)g(t) =t$ for all $t$ in $[0,\infty)$, then $$ \Biglvert \,\Bigl|\sum^n_{i=1} A^*_i X_i B_i \Bigr|^r \,\Bigrvert^2 \leq \Biglvert \Bigl( \sum^n_{i=1} A^*_i f (|X^*_i|)^2 A_i \Bigr)^r \Bigrvert \, \Biglvert \Bigl( \sum^n_{i=1} B^*_i g (|X_i|)^2 B_i \Bigr)^r \Bigrvert $$ for every $r>0$ and for every unitarily invariant norm. This result improves some known Cauchy-Schwarz type inequalities. Norm inequalities related to the arithmetic-geometric mean inequality and the classical Heinz inequalities are also obtained.

Keywords:Unitarily invariant norm, positive operator, arithmetic-geometric mean inequality, Cauchy-Schwarz inequality, Heinz inequality
Categories:47A30, 47B10, 47B15, 47B20
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