A multiplicative cascade can be thought of as a randomization of a measure on the boundary of a tree, constructed from an iid collection of random variables attached to the tree vertices. Given an initial measure with certain regularity properties, we construct a continuous time, measure-valued process whose value at each time is a cascade of the initial one. We do this by replacing the random ...

In this tutorial, we provide a broad introduction to the topic of interaction between the effects of exposures. We discuss interaction on both additive and multiplicative scales using risks, and we discuss their relation to statistical models (e.g. linear, log-linear, and logistic models). We discuss and evaluate arguments that have been made for using additive or multiplicative scales to asses...

1. Historical survey 2. Convergence of Sequences 2.1 Definition of Convergence 2.2. The Basic Property of Real Numbers. 2.3. Real Line 3. Continuous Functions 3.1 Continuous Functions and Their Limits 3.2 Properties of Continuous Functions. 3.2.1 The Intermediate Value Theorem 3.2.2. Maxima and Minima of Continuous Functions 3.3. The graph of a function 4. Differential Calculus 4.1 Derivative 4...

Abstract The logic of Bunched Implications (BI) freely combines additive and multiplicative connectives, including implications; however, despite its well-studied proof theory, proof-search in BI has always been a difficult problem. focusing principle is restriction the space that can capture various goal-directed procedures. In this paper we show focused complete for by first reformulating tra...

Given a syntax S endowed with some rewrite rules, and given a denotational semantics D for S (i.e. a semantics which gives to any term t of S an interpretation t D that is invariant under the rewrite rules), we say that D is injective with respect to S if, for any two normal terms t and t of S, t D = t D implies t = t. In categorical terms, injectivity corresponds to faithfulness of the " inter...

This paper presents a tableau calculus for finding a model for a set-satisfiable finite set of formulas of a fuzzy logic BL△∼, a fuzzy logic BL with additional Baaz connective △ and the involutive negation ∼, if such a model exists. The calculus is a generalisation of a tableau calculus for BL, which is based on the decomposition theorem for a continuous t-norm. The aforementioned tableau calcu...