These notes are based on the two lectures1 I gave in Stat 501, Probability Theory I, as a substitute for Professor Cheng Ouyang. There is a brief discussion of the so-called “basic grouping lemma” concerning independent σ-algebras, followed by a more detailed discussion of the Borel–Cantelli lemma, some applications, and some elaborations on independence. Independence, and the basic grouping le...

We prove a sharp Schwarz lemma type inequality for the Weierstrass–Enneper parameterization of minimal disks. It states following. If F:D→Σ is conformal harmonic disk Σ⊂R3, where D unit and |Σ|=πR2, then |Fx(z)|(1−|z|2)≤R. some z previous equality, surface an affine image disk, F linear up to Möbius transformation disk.

1. Page 13-14: Feature Ball Lemma 1.1 and its proof : Replace bd (B ∩ Σ) with (bd B) ∩ Σ.

Machine ➢ Formalise how the simulator works (program speci cation). ➢ Prove properties about the simulator. ➢ Give greater con dence in the simulation results. ➢ Improve on existing simulators. Andrew Phillips Microsoft Research 2004 23 Machine Data Structures ➢ Machine syntax νn1 νn2 ...νnN (Σ1 ::Σ2 :: ... ::ΣM :: []) : V,U ::= νn V Restriction p A List A,B ::= [] Empty p Σ::A Summation Andrew...

In this supplementary material, we provide the proofs to Theorems 1 and 2 presented in the maintext. We also present more clarifications on the parameter settings in our experiment. Furthermore, we show more experimental results to further substantiate the effectiveness of our method. 1. Proofs to Theorems 1 and 2 We first give the following Lemma [4]. Lemma 1. (von Neumann’s trace inequality) ...

This paper presents the soundness proofs for the type system of the Dynamic Security Domains (DSD) language. Unless otherwise noted, the identifiers and indices used in the proofs directly refer the respective identifiers and indices used in the operational semantics and the typing rules. Also, σ and (s, h) shall refer to the same program state, equally σ1 = (s1, h1), σ′ = (s′, h′), etc. 1 Expr...

Machine for Stochastic Pi-Calculus (SPiM) ➢ Formalise how the simulator works (program specification). ➢ Prove properties about the simulator (program verification). ➢ Give greater confidence in the simulation results. ➢ Improve on existing simulators (BioSpi). Andrew Phillips BioConcur 2004 19 Machine Data Structures ➢ General Machine Term: νn1 νn2 ...νnN (Σ1 ::Σ2 :: ... ::ΣM :: []) ➢ Syntax D...

We erroneously claim that Lemma 5.1 applies to both non-deterministic and deterministic Benenson automata, while it applies only to deterministic ones. The following corrected lemma holds for both deterministic and non-deterministic Benenson automata. It also exponentially improves on the dependence on D for both for circuit depth and size compared to the paper version. Note that compared with ...