相关论文: A Monadic, Functional Implementation of Real Numbe…

We define a counting function that is related to the binomial coefficients. An explicit formula for this function is proved. In some particular cases, simpler explicit formuls are derived. We also derive a formula for the number of…

Finite metric spaces arise in many different contexts. Enormous bodies of data, scientific, commercial and others can often be viewed as large metric spaces. It turns out that the metric of graphs reveals a lot of interesting information.…

We propose a realizability interpretation of a system for quantifier free arithmetic which is equivalent to the fragment of classical arithmetic without "nested" quantifiers, called here EM1-arithmetic. We interpret classical proofs as…

Implementing a complex concept as an executable model in a strongly typed, purely functional language hits a sweet spot between mere simulation and formal specification. For research and education it is often desirable to enrich the…

Motivated by the developing mathematics of deep learning, we build universal functions approximators of continuous maps between arbitrary Polish metric spaces $\mathcal{X}$ and $\mathcal{Y}$ using elementary functions between Euclidean…

Linear algebra is a major field of numerical computation and is widely applied. Most linear algebra libraries (in most programming languages) do not statically guarantee consistency of the dimensions of vectors and matrices, causing runtime…

We revisit and generalize the geometric procedure of regularizing a sequence of real numbers with respect to a so-called regularizing function. This approach was studied by S. Mandelbrojt and becomes useful and necessary when working with…

Shallow embeddings that use monads to represent effects are popular in proof-oriented languages because they are convenient for formal verification. Once shallowly embedded programs are verified, they are often extracted to mainstream…

The algebraic properties of the combination of probabilistic choice and nondeterministic choice have long been a research topic in program semantics. This paper explains a formalization in the Coq proof assistant of a monad equipped with…

The calculation and manipulation of large multi-variable rational functions is a key bottleneck in multi-loop calculations. In these conference proceedings, based on my article [Chawdhry (2023) arXiv:2312.03672], I present a technique to…

The execution of sequential programs allows them to be represented using mathematical functions formed by the composition of statements following one after the other. Each such statement is in itself a partial function, which allows only…

In this article we describe the construction of logarithmic models in both real and complex cases. A logarithmic model is a germ of closed meromorphic 1-form with simple poles - and the analytic foliation defined by it - produced upon some…

We consider a generic basic semi-algebraic subset $\mathcal{S}$ of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an…

The subject of this paper is regularity-preserving aggregation of regular norms on finite-dimensional linear spaces. Regular norms were introduced in [5] and are closely related to ``type 2'' spaces [9, Chapter 9] playing important role in…

This paper studies the design of programming languages with handlers of higher-order effectful operations -- effectful operations that may take in computations as arguments or return computations as output. We present and analyse a core…

Taking inspiration from the monadicity of complete atomic Boolean algebras, we prove that profinite modal algebras are monadic over Set. While analyzing the monadic functor, we recover the universal model construction - a construction…

Within Bishop-style constructive mathematics we study the classical McShane-Whitney theorem on the extendability of real-valued Lipschitz functions defined on a subset of a metric space. Using a formulation similar to the formulation of…

We propose a general framework to allow: (a) specifying the operational semantics of a programming language; and (b) stating and proving properties about program correctness. Our framework is based on a many-sorted system of hybrid modal…

In this work, a functional variant of the polynomial analogue of the classical Gandy's fixed point theorem is obtained. Sufficient conditions have been found to ensure that the complexity of the recursive function does not go beyond the…

A new and extensive formalism is developed for monads and galaxies in non-standard enlargements. It is shown that monads and galaxies can be manipulated using order-preserving and order-reversing set-to-set maps, and that set properties…