q-Inverting pairs of shape 1, 2, 1

Let V denote a vector space with finite positive dimension over an algebraically closed field 𝔽. Let K, K* be a q-inverting pair on V, an ordered pair of invertible linear transformations on V satisfying certain conditions. In this article we study the q-inverting pairs with shape 1, 2, 1. We define a pair of scalars called the parameter pair for K, K*. We give six bases for V and give the action of K, K ^?1, K* and (K*)^?1 on each of these bases, respectively. We classify the q-inverting pairs of shape 1, 2, 1 in terms of the parameter pairs. We conclude with some trace formulae involving the parameter pair.     

Separation and probability of correct classification among two or more distributions

A function on theK-fold product of a set in normed vector space will be called a separation measurement if, for any collection ofK points, the function is bounded below and above, respectively, by maximum and total distance between pairs of points in the collection. Separation measurements are relavent toK-sample hypothesis testing and also to discrimination amongK classes, and several examples are given. In particular, ordinaryL ₁ distance between integrable functions can be generalized to a non-pairwise separation measurement for densitiesf ₁,f ₂,...,f _K inL ₁[μ]; and this separation is a linear transform of the optimal discriminant's probability of correct classification. This research was supported by grant A8044 from the National Research Council of Canada.     

Stable intersections of affine cantor sets

We construct open sets of pairs of affine Cantor sets (K, K′) with the simplest possible combinatorics (i. e., defined by two affine increasing maps) which have stable intersection, while the product of lateral thicknesses τ_R(K) • τ_L (K′) is smaller than one. Thus, in a strong form, the converse to the following classic result due to Newhouse is not true: if the product of the thicknesses of two Cantor sets is bigger than one then there are translations of them which have stable intersection. We also describe the topological structure of K—K′ for typical pairs (K, K′) in this open set.     

Chromatic uniqueness and equivalence of K4 homeomorphs

We derive a new formula for the chromatic polynomial of any K₄ homeomorph. We obtain a large family of chromatically unique K₄ homeomorphs. We obtain seven infinite pairs of chromatically equivalent nonisomorphic K₄ homeomorphs.     

On a problem of Rolewicz about Banach spaces that admit support sets

We construct an example of a nonseparable Banach space which does not admit a support set.² It is a consistent (and necessarily independent from the axioms of ZFC) example of a space C(K) of continuous functions on a compact Hausdorff K with the supremum norm. The construction depends on a construction of a Boolean algebra with some combinatorial properties. The space is also hereditarily Lindelöf in the weak topology but it doesn't have any nonseparable subspace nor any nonseparable quotient which is a C(K) space for K dispersed.     

Remarks on a theorem of Taskinen on spaces of continuous functions

We clarify and prove in a simpler way a result of Taskinen about symmetric operators on C(Kⁿ), K an uncountable metrizable compact space. To do this we prove that, for any compact space K and any n ∈ ?, the symmetric injective n–tensor product of C(K), , is complemented in C(B_C(K)*), a result of independent interest. The techniques we develop allow us to extend the result in several directions. We also show that the hypothesis of metrizability and uncountability cannot be removed.     

Order-two continuous Hausdorff images of compact ordinals

Theorem. LetK be a Hausdorff space. The following conditions are equivalent: (a)K is homeomorphic to a compact scattered ordered space; (b)K is an order-two image of a compact ordinal.     

A Basis of Bideterminants for the Coordinate Ring of the Orthogonal Group

We give a basis of bideterminants for the coordinate ring K[O(n)] of the orthogonal group O(n,K), where K is an infinite field of characteristic not 2. The bideterminants are indexed by pairs of Young tableaux which are O(n)-standard in the sense of King–Welsh. We also give an explicit filtration of K[O(n)] as an O(n,K)-bimodule, whose factors are isomorphic to the tensor product of orthogonal analogs of left and right Schur modules.     

Nonexpansive selections of metric projections in spaces of continuous functions

A subset A of a metric space X is said to be a nonexpansive proximinal retract (NPR) of X if the metric projection from X to A admits a nonexpansive selection. We study the structure of NPR's in the space C(K) of continuous functions on a compact Hausdorff space K. The main results are a characterization of finite-codimensional and of finite-dimensional NPR subspaces of C(K) and a complete characterization of all NPR subsets of .     

Classical theorems of probability on Gelfand pairs—Khinchin’s theorems and Cramér’s theorem

We prove Khinchin’s Theorems for Gelfand pairs (G, K) satisfying a condition (*): (a)G is connected; (b)G is almost connected and Ad (G/M) is almost algebraic for some compact normal subgroupM; (c)G admits a compact open normal subgroup; (d) (G,K) is symmetric andG is 2-root compact; (e)G is a Zariski-connectedp-adic algebraic group; (f) compact extension of unipotent algebraic groups; (g) compact extension of connected nilpotent groups. In fact, condition (*) turns out to be necessary and sufficient forK-biinvariant measures on aforementioned Gelfand pairs to be Hungarian. We also prove that Cramér’s theorem does not hold for a class of Gaussians on compact Gelfand pairs. This author was supported by the European Commission (TMR 1998–2001 Network Harmonic Analysis).     

Computing the radius of pointedness of a convex cone

Let Ξ(H) denote the set of all nonzero closed convex cones in a finite dimensional Hilbert space H. Consider this set equipped with the bounded Pompeiu-Hausdorff metric δ. The collection of all pointed cones forms an open set in the metric space (Ξ(H),δ). One possible way of measuring the degree of pointedness of a cone K is by evaluating the distance from K to the set of all nonpointed cones. The number ρ(K) obtained in this way is called the radius of pointedness of the cone K. The evaluation of this number is, in general, a very cumbersome task. In this note, we derive a simple formula for computing ρ(K), and we propose also a method for constructing a nonpointed cone at minimal distance from K. Our results apply to any cone K whose maximal angle does not exceed 120°. Dedicated to Clovis Gonzaga on the occassion of his 60th birthday.     

Point simpliciality in Choquet theory on nonmetrizable compact spaces

Let H be a function space on a compact space K. The set of simpliciality of H is the set of all points of K for which there exists a unique maximal representing measure. Properties of this set were studied by M. Ba?ák in the paper Point simpliciality in Choquet representation theory, Illinois J. Math. 53 (2009) 289–302, mainly for K metrizable. We study properties of the set of simpliciality for K nonmetrizable.     

Diameters of iterated clique graphs of chordal graphs

The clique graph K(G) of a graph is the intersection graph of maximal cliques of G. The iterated clique graph Kⁿ(G) is inductively defined as K(K^n?1(G)) and K¹(G) = K(G). Let the diameter diam(G) be the greatest distance between all pairs of vertices of G. We show that diam(Kⁿ(G)) = diam(G) — n if G is a connected chordal graph and n ≤ diam(G). This generalizes a similar result for time graphs by Bruce Hedman.     

The Connection Between the Metric and Generalized Projection Operators in Banach Spaces

In this paper we study the connection between the metric projection operator PK ： B →K, where B is a reflexive Banach space with dual space B^＊ and K is a non-empty closed convex subset of B, and the generalized projection operators ∏K ： B → K and πK ： B^＊ → K. We also present some results in non-reflexive Banach spaces.     

A hereditarily indecomposable $ {\mathcal{L}_{\infty}} $-space that solves the scalar-plus-compact problem

We construct a hereditarily indecomposable Banach space with dual space isomorphic to ℓ ₁. Every bounded linear operator on this space is expressible as λI + K, with λ a scalar and K compact.     

Bishop’s theorem and differentiability of a subspace of <Emphasis Type="Italic">C</Emphasis><Subscript><Emphasis Type="Italic">b</Emphasis></Subscript>(<Emphasis Type="Italic">K</Emphasis>)

Let K be a Hausdorff space and C _b (K) be the Banach algebra of all complex bounded continuous functions on K. We study the Gateaux and Fréchet differentiability of subspaces of C _b (K). Using this, we show that the set of all strong peak functions in a nontrivial separating separable subspace H of C _b (K) is a dense G _δ subset of H, if K is compact. This gives a generalized Bishop’s theorem, which says that the closure of the set of all strong peak points for H is the smallest closed norming subset of H. The classical Bishop’s theorem was proved for a separating subalgebra H and a metrizable compact space K.