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

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1. CMB 2011 (vol 55 pp. 418)

Vinh, Le Anh
Maximal Sets of Pairwise Orthogonal Vectors in Finite Fields
Given a positive integer $n$, a finite field $\mathbb{F}_q$ of $q$ elements ($q$ odd), and a non-degenerate symmetric bilinear form $B$ on $\mathbb{F}_q^n$, we determine the largest possible cardinality of pairwise $B$-orthogonal subsets $\mathcal{E} \subseteq \mathbb{F}_q^n$, that is, for any two vectors $\mathbf{x}, \mathbf{y} \in \mathcal{E}$, one has $B (\mathbf{x}, \mathbf{y}) = 0$.

Keywords:orthogonal sets, zero-distance sets
Category:05B25

2. CMB 2011 (vol 55 pp. 821)

Perez-Garcia, C.; Schikhof, W. H.
New Examples of Non-Archimedean Banach Spaces and Applications
The study carried out in this paper about some new examples of Banach spaces, consisting of certain valued fields extensions, is a typical non-archimedean feature. We determine whether these extensions are of countable type, have $t$-orthogonal bases, or are reflexive. As an application we construct, for a class of base fields, a norm $\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm, without non-zero vectors that are $\|\cdot\|$-orthogonal and such that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$ into a valued field.

Keywords:non-archimedean Banach spaces, valued field extensions, spaces of countable type, orthogonal bases
Categories:46S10, 12J25

3. CMB 2011 (vol 55 pp. 767)

Martini, Horst; Wu, Senlin
On Zindler Curves in Normed Planes
We extend the notion of Zindler curve from the Euclidean plane to normed planes. A characterization of Zindler curves for general normed planes is given, and the relation between Zindler curves and curves of constant area-halving distances in such planes is discussed.

Keywords:rc length, area-halving distance, Birkhoff orthogonality, convex curve, halving pair, halving distance, isosceles orthogonality, midpoint curve, Minkowski plane, normed plane, Zindler curve
Categories:52A21, 52A10, 46C15

4. CMB 2011 (vol 55 pp. 597)

Osękowski, Adam
Sharp Inequalities for Differentially Subordinate Harmonic Functions and Martingales
We determine the best constants $C_{p,\infty}$ and $C_{1,p}$, $1 < p < \infty$, for which the following holds. If $u$, $v$ are orthogonal harmonic functions on a Euclidean domain such that $v$ is differentially subordinate to $u$, then $$ \|v\|_p \leq C_{p,\infty} \|u\|_\infty,\quad \|v\|_1 \leq C_{1,p} \|u\|_p. $$ In particular, the inequalities are still sharp for the conjugate harmonic functions on the unit disc of $\mathbb R^2$. Sharp probabilistic versions of these estimates are also studied. As an application, we establish a sharp version of the classical logarithmic inequality of Zygmund.

Keywords: harmonic function, conjugate harmonic functions, orthogonal harmonic functions, martingale, orthogonal martingales, norm inequality, optimal stopping problem
Categories:31B05, 60G44, 60G40

5. CMB 2011 (vol 55 pp. 462)

Campbell, Peter S.; Stokke, Anna
Hook-content Formulae for Symplectic and Orthogonal Tableaux
By considering the specialisation $s_{\lambda}(1,q,q^2,\dots,q^{n-1})$ of the Schur function, Stanley was able to describe a formula for the number of semistandard Young tableaux of shape $\lambda$ in terms of the contents and hook lengths of the boxes in the Young diagram. Using specialisations of symplectic and orthogonal Schur functions, we derive corresponding formulae, first given by El Samra and King, for the number of semistandard symplectic and orthogonal $\lambda$-tableaux.

Keywords:symplectic tableaux, orthogonal tableaux, Schur function
Categories:05E05, 05E10

6. CMB 2009 (vol 52 pp. 95)

Miranian, L.
Matrix Valued Orthogonal Polynomials on the Unit Circle: Some Extensions of the Classical Theory
In the work presented below the classical subject of orthogonal polynomials on the unit circle is discussed in the matrix setting. An explicit matrix representation of the matrix valued orthogonal polynomials in terms of the moments of the measure is presented. Classical recurrence relations are revisited using the matrix representation of the polynomials. The matrix expressions for the kernel polynomials and the Christoffel--Darboux formulas are presented for the first time.

Keywords:Matrix valued orthogonal polynomials, unit circle, Schur complements, recurrence relations, kernel polynomials, Christoffel-Darboux
Category:42C99

7. CMB 2001 (vol 44 pp. 337)

Vinet, Luc; Zhedanov, Alexei
Spectral Transformations of the Laurent Biorthogonal Polynomials, II. Pastro Polynomials
We continue to study the simplest closure conditions for chains of spectral transformations of the Laurent biorthogonal polynomials ($\LBP$). It is shown that the 1-1-periodic $q$-closure condition leads to the $\LBP$ introduced by Pastro. We introduce classes of semi-classical and Laguerre-Hahn $\LBP$ associated to generic closure conditions of the chain of spectral transformations.

Keywords:Laurent orthogonal polynomials, Pastro polynomials, spectral transformations
Category:33D45

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