Related papers: A Diagrammatic Calculus for Algebraic Effects
Shapley effects are attracting increasing attention as sensitivity measures. When the value function is the conditional variance, they account for the individual and higher order effects of a model input. They are also well defined under…
We introduce a new family of graphical models that consists of graphs with possibly directed, undirected and bidirected edges but without directed cycles. We show that these models are suitable for representing causal models with additive…
Statistics and Optimization are foundational to modern Machine Learning. Here, we propose an alternative foundation based on Abstract Algebra, with mathematics that facilitates the analysis of learning. In this approach, the goal of the…
We present different methods for symbolic computer algebra computations in higher dimensional (\ge9) Clifford algebras using the \Clifford\ and \Bigebra\ packages for \Maple(R). This is achieved using graded tensor decompositions,…
Simplification of fractional powers of positive rational numbers and of sums, products and powers of such numbers is taught in beginning algebra. Such numbers can often be expressed in many ways, as this article discusses in some detail.…
In this paper we use the viewpoint of the formal calculus underlying vertex operator algebra theory to study certain aspects of the classical umbral calculus and we introduce and study certain operators generalizing the classical umbral…
The aim of this paper is to show that there can be either only one or uncountably many contexts in any spectral effect algebra, answering a question posed in [S. Gudder, Convex and Sequential Effect Algebras, (2018), arXiv:1802.01265]. We…
Directed mixed graphs permit directed and bidirected edges between any two vertices. They were first considered in the path analysis developed by Sewall Wright and play an essential role in statistical modeling. We introduce a matrix…
The relationship between algebraic geometry and the inferential framework of the Bayesian Networks with hidden variables has now been fruitfully explored and exploited by a number of authors. More recently the algebraic formulation of…
Can the cross product be generalized? Why are the trace and determinant so important in matrix theory? What do all the coefficients of the characteristic polynomial represent? This paper describes a technique for `doodling' equations from…
There are several ways to define program equivalence for functional programs with algebraic effects. We consider two complementing ways to specify behavioural equivalence. One way is to specify a set of axiomatic equations, and allow proof…
In this work we present a theoretical model for differentiable programming. We construct an algebraic language that encapsulates formal semantics of differentiable programs by way of Operational Calculus. The algebraic nature of Operational…
The syntactic nature of logic and computation separates them from other fields of mathematics. Nevertheless, syntax has been the only way to adequately capture the dynamics of proofs and programs such as cut-elimination, and the finiteness…
We assume that every element of a matrix has a small, individual error, and model it by an external number, which is the sum of a nonstandard real number and a neutrix, the latter being a convex (external) set having the group property. The…
Domain theory has been developed as a mathematical theory of computation and to give a denotational semantics to programming languages. It helps us to fix the meaning of language concepts, to understand how programs behave and to reason…
We consider a class of generalized binomials emerging in fractional calculus. After establishing some general properties, we focus on a particular yet relevant case, for which we provide several ready-for-use combinatorial identities,…
We introduce a class of monotone $\sigma$-complete effect algebras, called representable, which are $\sigma$-homomorphic images of a class of monotone $\sigma$-complete effect algebras of functions taking values in the interval $[0,1]$ and…
Computational effects are commonly modelled by monads, but often a monad can be presented by an algebraic theory of operations and equations. This talk is about monads and algebraic theories for languages for inference, and their…
We present a new formula for umbral operators that yields three main insights. First, it makes explicit a connection between umbral calculus and iteration theory. Second, it leads naturally to a definition of fractional exponents of umbral…
Effect systems are lightweight extensions to type systems that can verify a wide range of important properties with modest developer burden. But our general understanding of effect systems is limited primarily to systems where the order of…