Iskanje po repozitoriju

Opis: Let ▫$\Gamma$▫ denote a bipartite distance-regular graph with diameter ▫$D \geq 4$▫ and valency ▫$k \geq 3$▫. Let ▫$X$▫ denote the vertex set of ▫$\Gamma$▫, and let ▫$A$▫ denote the adjacency matrix of ▫$\Gamma$▫. For ▫$x \in X$▫ and for ▫$0 \leq i \leq D$▫, let ▫$\operatorname{\Gamma}_i(x)$▫ denote the set of vertices in ▫$X$▫ that are distance ▫$i$▫ from vertex ▫$x$▫. Define a parameter ▫$\operatorname{\Delta}_2$▫ in terms of the intersection numbers by ▫$\operatorname{\Delta}_2 = (k - 2)(c_3 - 1) -(c_2 - 1) p_{22}^2$▫. We first show that ▫$\operatorname{\Delta}_2 = 0$▫ implies that ▫$D \leq 5$▫ or ▫$c_2 \in \{1, 2 \}$▫. For ▫$x \in X$▫ let ▫$T = T(x)$▫ denote the subalgebra of ▫$\text{Mat}_X(\mathbb{C})$▫ generated by ▫$A, E_0^\ast, E_1^\ast, \ldots, E_D^\ast$▫, where for ▫$0 \leq i \leq D$, $E_i^\ast$▫ represents the projection onto the▫ $i$▫th subconstituent of ▫$\Gamma$▫ with respect to ▫$x$▫. We refer to ▫$T$▫ as the Terwilliger algebra of ▫$\Gamma$▫ with respect to ▫$x$▫. By the endpoint of an irreducible ▫$T$▫-module ▫$W$▫ we mean ▫$\min \{i | E_i^\ast W \ne 0 \}$▫. In this paper we assume ▫$\Gamma$▫ has the property that for ▫$2 \leq i \leq D - 1$▫, there exist complex scalars ▫$\alpha_i$▫, ▫$\beta_i$▫ such that for all ▫$x, y, z \in X$▫ with ▫$\partial(x, y) = 2$▫, ▫$\partial(x, z) = i$▫, ▫$\partial(y, z) = i$▫, we have ▫$\alpha_i + \beta_i | \operatorname{\Gamma}_1(x) \cap \operatorname{\Gamma}_1(y) \cap \operatorname{\Gamma}_{i - 1}(z) | = | \operatorname{\Gamma}_{i - 1}(x) \cap \operatorname{\Gamma}_{i - 1}(y) \cap \operatorname{\Gamma}_1(z) |$▫. We additionally assume that▫ $\operatorname{\Delta}_2 = 0$▫ with ▫$c_2 = 1$▫. Under the above assumptions we study the algebra ▫$T$▫. We show that if ▫$\Gamma$▫ is not almost 2-homogeneous, then up to isomorphism there exists exactly one irreducible ▫$T$▫-module with endpoint 2. We give an orthogonal basis for this ▫$T$▫-module, and we give the action of ▫$A$▫ on this basis.
Ključne besede: distance-regular graphs, terwilliger algebra, subconstituent algebra
Objavljeno v RUP: 14.11.2017; Ogledov: 2363; Prenosov: 143
Povezava na celotno besedilo

Opis: Let ▫$H_n$▫ be the set of all ▫$n \times n$▫ hermitian matrices over ▫$\mathbb{C}$▫, ▫$n \ge 3$▫. It is said that ▫$A,B \in H_n$▫ quasi-commute if there exists a nonzero ▫$\xi \in \mathbb{C}$▫ such that ▫$AB = \xi BA$▫ Bijective not necessarily linear maps on hermitian matrices which preserve quasi-commutativity in both directions are classified.
Ključne besede: mathematics, linear algebra, general preserver, hermitian matrices, quasi-commutativity
Objavljeno v RUP: 03.04.2017; Ogledov: 2179; Prenosov: 255
Povezava na celotno besedilo

Opis: For a finite-dimensional linear subspace ▫{$\mathscr{S}} \subseteq {\mathscr{L}} (V,W)$▫ and a positive integer ▫$k$▫, the ▫$k$▫-reflexivity defect of ▫$\mathscr{S}$▫ is defined by ▫${\mathrm{rd}}_k ({\mathscr{S}}) = \dim({\mathrm{Ref}}_k (\mathscr{S})/\mathscr{S})$▫ where ▫${\mathrm{Ref}}_k$▫ is the ▫$k$▫-reflexive closure of ▫$\mathscr{S}$▫. We study this quantity for two-dimensional spaces of operators and for single generated algebras and their commutants.
Ključne besede: mathematics, operator theory, reflexivity defect, reflexivity, two-dimensional space of operators, single generated algebra, commutant
Objavljeno v RUP: 03.04.2017; Ogledov: 2147; Prenosov: 193
Povezava na celotno besedilo

Opis: Let ▫$H$▫ be a separable Hilbert space and▫ ${\mathcal B}_{sa}(H)▫$ the set of all bounded linear self-adjoint operators. We say that ▫$A, B \in {\mathcal B}_{sa}(H)$▫ quasi-commute if there exists a nonzero ▫$\xi \in \mathbb{C}$▫ suchthat ▫$AB=\xi BA$▫. Bijective maps on ▫${\mathcal B}_{sa}(H)$▫ which preserve quasi-commutativity in both directions are classified.
Ključne besede: mathematics, linear algebra, general preserver, self-adjoint operator, quasi-commutativity
Objavljeno v RUP: 03.04.2017; Ogledov: 2165; Prenosov: 76
Povezava na celotno besedilo

Opis: It is shown that matrices over algebraically closed fields that are farthest apart in the commuting graph must be non-derogatory. Rank-one matrices and diagonalizable matrices are also characterized in terms of the commuting graph.
Ključne besede: matematika, linearna algebra, teorija grafov, komutirajoči grafi, matrična algebra, algebraično zaprt obseg, centralizator, razdalja v grafih, mathematics, linear algebra, graph theory, commuting graph, matrix algebra, algebraically closed field, centralizer, distance in graphs
Objavljeno v RUP: 03.04.2017; Ogledov: 2315; Prenosov: 256
Povezava na celotno besedilo