where the second equality holds because of the change of measure with dP̃ dP |Ft = e Xt−rt. Because St has a continuous distribution under both P and P̃ , then f1(k) and f2(k) are in C. Besides, P̃ ( St ≥ e−k ) and P ( St ≥ e−k ) are both increasing in k. Therefore, applying Lemma 3.1 yields that both f1(k) and f2(k) satisfy Assumption 1. Note that for any σ ∈ (−1,−σl − 1), ∫ +∞ −∞ ef2(k)dk =e −rt...

The authors and Fischer recently proved that any hereditary property of two-dimensional matrices (where the row and column order is not ignored) over a finite alphabet is testable with a constant number of queries, by establishing the following (ordered) matrix removal lemma: For any finite alphabet Σ, any hereditary property P of matrices over Σ, and any > 0, there exists fP( ) such that for a...

ion clause, we use A -{x:σ} to denote the set difference, i.e., the type assignment defined by removing x:σ from A. G(x) = { s⊆t }, {x:s} ⊃ x:t G(MN) = let C1, A1 ⊃ M:σ = G(M) C2, A2 ⊃ N:τ = G(N), with type variables renamed to be disjoint from those in G(M) S = UNIFY({α=β | x:α∈A1 and x:β∈A2} ∪ {σ=τ→t}) where t is a fresh type variable in SC1∪SC2∪{St⊆u}, SA1∪SA2 ⊃ MN:u where u is a fresh type ...

and Applied Analysis 3 2. Preliminaries and Lemmas For the convenience of the reader, we present here some definitions from fractional calculus which are to be used in the later sections. Definition 2.1 see 18–20 . The Riemann-Liouville fractional integral of order α > 0 of a function x : 0, ∞ → R is given by Ix t 1 Γ α ∫ t 0 t − s α−1x s ds 2.1 provided that the right-hand side is pointwise de...

Measures And Measurable Functions The Lebesgue integral is much better than the Rieman integral. This has been known for over 100 years. It is much easier to generalize to many dimensions and it is much easier to use in applications. That is why I am going to use it rather than struggle with an inferior integral. It is also this integral which is most important in probability. However, this int...

We consider three theorems in ergodic theory concerning a fixed aperiodic measure preserving transformation σ of a Lebesgue probability space (X,A, μ) and show that these theorems are all equivalent. Two of these results concern the existence of a partition of the space X with special properties. The third theorem asserts that the conjugates of σ are dense in the uniform topology on the space o...

Lemma 1.1. Let V be a finite-dimensional vector space over a field F. Let β, β′ be two bases for V . Let T : V → V be a linear transformation. Define Q := [IV ] ′ β . Then [T ] β β and [T ] ′ β′ satisfy the following relation [T ] ′ β′ = Q[T ] β βQ −1. Theorem 1.2. Let A be an n× n matrix. Then A is invertible if and only if det(A) 6= 0. Exercise 1.3. Let A be an n×n matrix with entries Aij, i,...

The asymptotic form of the number of n-quasigroups of order 4 is 3 n+1 2 2 n +1 (1 + o(1)). An algebraic system that consists of a set Σ of cardinality |Σ| = k and an n-ary operation f : Σ n → Σ uniquely invertible by each of its arguments is called an n-quasigroup of order k. The function f can also be referred to as an n-quasigroup of order k (see [Bel72]). The value table of an n-quasigroup ...

The purpose of this paper is to survey our recent research in computability and definability over continuous data types such as the real numbers, real-valued functions and functionals. We investigate the expressive power and algorithmic properties of the language of Σ-formulas intended to represent computability over the real numbers. In order to adequately represent computability we extend the...

This note presents details of a result of Okada and Terui[2] which shows that the equational theory of residuated lattices is decidable and gives an effective algorithm based on a Gentzen system for propositional intuitionistic linear logic. The variety of residuated lattices is denoted by RL. An algebra (L,∨,∧, ∗, \, /, 1) is a member of this variety if (L,∨,∧) is a lattice, (L, ∗, 1) is a mon...