Let Ai (i = 1,2, · · · ,k) be bounded linear operators on a Hilbert space. This paper aims to show a characterization of operator order Ak Ak−1 · · · A2 A1 > 0 in terms of operator inequalities. Afterwards, an application of the characterization is given to operator equalities due to Douglas’s majorization and factorization theorem. Mathematics subject classification (2010): 47A63.

and Applied Analysis 3 in K0,∞ . In particular, the inequality λ K ≥ K holds for all K ∈ 1,∞ with the best possible constant c a. 2 g K logηK t − log t / K − 1 is convex in 1,∞ for fixed t ∈ 0,∞ . 3 If t ≥ 1 and r √ t/ 1 t , then h K logηK t − log t / K − 1/K is strictly increasing from 1,∞ onto 2K r K′ r /π,πK r /K′ r . Theorem 1.2. Let t ∈ 0,∞ , r √ t/ 1 t , a 4/π K 1/√2 , b a/2,A r π2/ 2μ r ...

We introduce the Tier-based k-Strictly Local Inference Algorithm (kTSLIA), an algorithm that learns the class of Tier-based Strictly k-Local (TSLk) formal languages in polynomial time on a sample of positive data whose size is bounded by a constant. The TSLk languages are useful in modeling the cognition of sound patterns in natural language [6, 11], and it is known that they can be efficiently...

It is shown that in metric spaces each (α, φ)-meagre set A is uniformly very porous and its index of uniform v-porosity is not smaller than k−α 3k+α , provided that φ is a strictly k-monotone family of Lipschitz functions and α < k. The paper contains also conditions implying that a k-monotone family of Lipschitz functions is strictly k-monotone.

We consider billiard dynamics inside a smooth strictly convex curve. For each pair of integers (k,n), we focus our attention on the billiard trajectory that traces a closed polygon with n sides and makes k turns inside the billiard table, called a (k,n)-orbit. Birkhoff proved that a strictly convex billiard always has at least two (k,n)-orbits for any relatively prime integers k and n such that...

Suppose that f x is strictly increasing, strictly concave, and twice continuously differentiable on a nonempty interval I ⊆ R, and f ′ x is strictly convex on I. Suppose that xk ∈ a, b ⊆ I, where 0 < a < b, and pk ≥ 0 for k 1, · · · , n, and suppose that ∑n k 1pk 1. Let x ∑n k 1pkxk, and σ 2 ∑n k 1pk xk − x . We show ∑n k 1pkf xk ≤ f x−θ1σ 2 , ∑n k 1pkf xk ≥ f x−θ 2σ 2 , for suitably chosen θ1 ...

Strictly Chordality-k graphs (SCk) are graphs which are either cycle-free or every induced cycle is of length exactly k, k ≥ 3. Strictly chordality-3 and strictly chordality-4 graphs are well known chordal and chordal bipartite graphs, respectively. For k ≥ 5, the study has been recently initiated in [1] and various structural and algorithmic results are reported. In this paper, we show that ma...