434,371 research outputs found
Differential inclusions, non-absolutely convergent integrals and the first theorem of complex analysis
In the theory of complex valued functions of a complex variable arguably the
first striking theorem is that pointwise differentiability implies
regularity. As mentioned in Ahlfors's standard textbook there have been a
number of studies proving this theorem without use of complex integration but
at the cost of considerably more complexity. In this note we will use the
theory of non-absolutely convergent integrals to firstly give a very short
proof of this result without complex integration and secondly (in combination
with some elements of the theory of elliptic regularity) provide a far reaching
generalization
Extending Equational Monadic Reasoning with Monad Transformers
There is a recent interest for the verification of monadic programs using proof assistants. This line of research raises the question of the integration of monad transformers, a standard technique to combine monads. In this paper, we extend Monae, a Coq library for monadic equational reasoning, with monad transformers and we explain the benefits of this extension. Our starting point is the existing theory of modular monad transformers, which provides a uniform treatment of operations. Using this theory, we simplify the formalization of models in Monae and we propose an approach to support monadic equational reasoning in the presence of monad transformers. We also use Monae to revisit the lifting theorems of modular monad transformers by providing equational proofs and explaining how to patch a known bug using a non-standard use of Coq that combines impredicative polymorphism and parametricity
Holonomic Gradient Descent and its Application to Fisher-Bingham Integral
We give a new algorithm to find local maximum and minimum of a holonomic
function and apply it for the Fisher-Bingham integral on the sphere ,
which is used in the directional statistics. The method utilizes the theory and
algorithms of holonomic systems.Comment: 23 pages, 1 figur
On theories of random variables
We study theories of spaces of random variables: first, we consider random
variables with values in the interval , then with values in an arbitrary
metric structure, generalising Keisler's randomisation of classical structures.
We prove preservation and non-preservation results for model theoretic
properties under this construction: i) The randomisation of a stable structure
is stable. ii) The randomisation of a simple unstable structure is not simple.
We also prove that in the randomised structure, every type is a Lascar type
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