In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement an...

Let Σ be compact orientable surface, and write Γ = π1(Σ). Let H be a hyperbolic space (in the sense of Gromov [Gr]) admitting a discrete action by Γ. and write M = H/Γ. Let X(Σ) be the set of homotopy classes of essential non-peripheral closed curves in Σ. Let G(Σ) be the curve graph with vertex set V (G) = X(Σ) (i.e. the 1-skeleton of the curve complex [H]). To each γ ∈ X(Σ) we can associate a...

and Applied Analysis 3 such that a ∈⇓ d. Thus d ∈⇑ a ⊆ U. Therefore D ∩ U ̸ = 0, hence U ∈ σ(P). This proves τ(P) ⊆ σ(P). (2) Now assume that P is strongly continuous. Then P is continuous. In a strongly continuous lattice, the relation ≪ and ⇐ are the same by the Theorem 2.5 of [5]. Also by [2] in a continuous lattice, every Scott open set A satisfies the condition A = ↑≪A, it follows that ever...

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement an...

Motivated by the Beck-Fiala conjecture, we study discrepancy bounds for random sparse set systems. Concretely, these are set systems (X,Σ), where each element x ∈ X lies in t randomly selected sets of Σ, where t is an integer parameter. We provide new bounds in two regimes of parameters. We show that when |Σ| ≥ |X| the hereditary discrepancy of (X,Σ) is with high probability O( √ t log t); and ...

We will present the proofs of Lemma 4.1, Lemma 4.3 and Theorem 4.2 here. For completeness, first recall Lyapunov’s theorem in probability theory. Let Yn = ∑kn i=1 Yni, for n ∈ IN , be a triangular array of random variables such that for each n, the random variables Yni, for 1 ≤ i ≤ kn are independent with E(Yni) = 0 and E(Y 2 ni) = σ ni, where E(X) stands for the expected value of the random va...

and Applied Analysis 3 ΔA t ,ΔB t ,ΔD t KF t Ea, Eb, Ed , 2.2 where K,Ea, Eb, Ed are the known constant matrices with appropriate dimensions; F t is unknown continuous time-varying matrix function satisfying F t F t ≤ I, for all t ≥ 0. The nominal Σ0 of Σ can be defined as Σ0 : ⎧ ⎪⎨ ⎪⎩ ẋ t − Cẋ t − τ2 Ax t Bx t − τ1 D ∫ t t−h x s ds, t > 0, x t φ t , t ∈ −τ, 0 . 2.3 For further discussion, we f...

Given many independent and identically-distributed (i.i.d.) copies of a quantum system described either by the state ρ or σ (called null and alternative hypotheses, respectively), what is the optimal measurement to learn the identity of the true state? In asymmetric hypothesis testing one is interested in minimizing the probability of mistakenly identifying ρ instead of σ, while requiring that ...

A sequence over an alphabet Σ is called disjunctive [13] if it contains all possible finite strings over Σ as its substrings. Disjunctive sequences have been recently studied in various contexts, e.g. [12, 9]. They abound in both category and measure senses [5]. In this paper we measure the complexity of a sequence x by the complexity of the language P (x) consisting of all prefixes of x. The l...

APPENDIX A USEFUL KNOWN RESULTS Lemma A.1 (Minimal Spanning Tree, [1]): Let Xi, 1 ≤ i < ∞, denote independent random variables with values in R, d ≥ 2, and let Mn denote the cost of a minimal spanning tree of a complete graph with vertex set {Xi}i=1, where the cost of an edge (Xi, Xj) is given by Ψ(|Xi − Xj |). Here, |Xi−Xj | denotes the Euclidean distance between Xi and Xj and Ψ is a monotone ...