In this note we discuss a lemma from [3]. Before stating the lemma, we need to establish some notation. This is done in Section 1. We then state Sapozhenko's result in Section 2; that section also previews the rest of the note. 1 Setup Throughout, Σ is a d-regular bipartite graph with bipartition V = V (Σ) = X ∪ Y. For u, v ∈ V and A, B ⊆ V we write u ∼ v if there is an edge in Σ joining u and ...

ion from Y yields T∆+AI(QX) (r̂)X ←−−−− AI ( Q(ΣX)). Now we need the coproduct of monads: With T∆ = |∆ | and Lemma 6.2.4.1 we can write the rule as |∆ + ( AI(QX) )? | (r̂)X ←−−−− AI ( Q(ΣX)). Abstracting from X gives using from X gives us | | · (∆ + ) · ( ) · AI · Q r̂ ←−− AI · Q · Σ. We have the adjunctions Q a U and AI a ΛI as in Subsection 6.4.3 and ( ) ? a | | from Corollary 4.4.4.6. Now we us...

We use G-stable pieces to construct some equidimensional varieties and as a consequence, obtain Lusztig’s dimension estimates [L2, section 4]. This is a generalization of [HL]. In Lemma 1.1 and Proposition 1.2, we assume that G is arbitrary connected algebraic group and G̃ is an algebraic group with identity component G. Lemma 1.1. Let g ∈ G̃. Define i : G̃ → G̃ by i(h) = ghgıhı. For any closed sub...

1 Proofs of Theorems and Lemmas in the Paper We provide detailed proofs of the theorems and lemmas. As stated in the paper, we define f0 Q ∑ i=1 π (i) f (i) max σ 2 0 ∥K∥ 2 2 f0 (1) Let L, C > 0 be constants which only depend on the VC characteristics of the Gaussian kernel K. For all λ ≥ C and σ > 0, we define E σ 2 log (1 + λ/4L) λLσ 2 (2) Lemma 1. For any P XY ∈ P XY , there exists a n 0 whi...

Given a multi-variable polynomial, there is an associated divided symmetrization (in particular turning it into a symmetric function). Postinkov has found the volume of a permutohedron as a divided symmetrization (DS) of the power of a certain linear form. The main task in this paper is to exhibit and prove closed form DS-formulas for a variety of polynomials. We hope the results to be valuable...

In [1] Akama, Berardi, Hayashi and Kohlenbach used a monotone modified realizability interpretation to establish the relative independence of Σ n+1-DNE from Π 0 n+1-LEM over HA, and hence the independence of Σ n+1-LEM from Π 0 n+1-LEM over HA, for all n ≥ 0. We show that the same relative independence results hold for these arithmetical principles over Kleene and Vesley’s system FIM of intuitio...

iff px ∈ [0, 1) for all x. When the events are not independent, an answer is given by the Lovász local lemma (and its variations). Definition 14.1. We say that G is the dependency graph of (Ax)x∈X if for all x ∈ X, Ax is independent of the σ-algebra generated by the collection {Ay | y / ∈ Γ∗(x)} (where Γ(x) is the set of neighbours of x in G and Γ∗(x) := Γ(x) ∪ {x}). Theorem 14.2 (Lovász local ...

and Applied Analysis 3 Definition 2.1. Let E be a real Banach space. A nonempty closed convex set P ⊂ E is called a cone of E if it satisfies the following two conditions: 1 x ∈ P, σ > 0 implies σx ∈ P ; 2 x ∈ P,−x ∈ P implies x θ. Definition 2.2. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets. Let E be a real Banach space, E∗ the dual...

2 of Lemma 4 and 4 of Proposition 2 in our paper [1] entitled “*Continuous Idempotent Semirings and Their Ideal Completion” are wrong. This note shows counter examples of them. The counter examples are given in linguistic models of *-Continuous Idempotent Semirings. 1 Monodic Tree Languages A ranked set is a couple (Σ, r) of a set Σ of (function) symbols and a mapping r from Σ to the set N of n...