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1

Bayer Noise symmetric functions and some combinatorial algebraic structures

Abdulwahid Adnan Hashim, Elamami Elgaddafi

Journal of Mathematics and Applications

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2023

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Vol. 46

115--148

EN

Symmetric functions play a crucial role in classifying representations of symmetric groups, and they are largely involved with combinatorial algebras and graph theory. Bayer filter technique is largely applied in most of the professional digital cameras due to the fact that it is a low-cost, and it allows photosensors not only to capture the intensity of light, but also to record the wavelength of light as well. Using Bayer Pattern, we introduce the Bayer Noise symmetric functions and the Bayer Noise Schur functions, and we study some combinatorial structures on the Bayer Noise modules. We study the connection between Bayer Noise symmetric functions and other bases for the algebra of symmetric functions, and we explicitly calculate special cases over a fixed commutative ring k. We also study the compatibility of such algebraic and coalgebraic structures.

2

Coalgebras for modelling observable behaviour of programs

Steingartner W., Novitzka V.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 2

145--157

EN

A useful tool for modelling behaviour in theoretical computer science is the concept of coalgebras. Coalgebras enable one to model execution of programs step by step using categorical structures and polynomial endofunctors. In our paper, we start with a short introduction of basic notions and we use this method for modelling structural operational semantics of a simple imperative language.

3

From Branching to Linear Time, Coalgebraically

Cîrstea C.

Fundamenta Informaticae

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2017

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Vol. 150, nr 3/4

379--406

EN

We consider state-based systems modelled as coalgebras whose type incorporates branching, and show that suitably adapting the definition of coalgebraic bisimulation yields a general and uniform account of the linear-time behaviour of a state in such a coalgebra. By moving away from a boolean universe of truth values, our approach can measure the extent to which a state in a system with branching is able to exhibit a particular linear-time behaviour. This instantiates to measuring the probability of a specific behaviour occurring in a probabilistic system, or measuring the minimal cost of exhibiting a given behaviour in the case of weighted computations.

4

CoCaml: Functional Programming with Regular Coinductive Types

Jeannin J.-B., Kozen D., Silva A.

Fundamenta Informaticae

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2017

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Vol. 150, nr 3/4

347--377

EN

Functional languages offer a high level of abstraction, which results in programs that are elegant and easy to understand. Central to the development of functional programming are inductive and coinductive types and associated programming constructs, such as pattern-matching. Whereas inductive types have a long tradition and are well supported in most languages, coinductive types are subject of more recent research and are less mainstream. We present CoCaml, a functional programming language extending OCaml, which allows us to define recursive functions on regular coinductive datatypes. These functions are defined like usual recursive functions, but parameterized by an equation solver. We present a full implementation of all the constructs and solvers and show how these can be used in a variety of examples, including operations on infinite lists, infinitary γ-terms, and p-adic numbers.

5

Some properties of coalgebras and their role in computer science

Steingartner W., Radakovic D., Valkosak F., Macko P.

Journal of Applied Mathematics and Computational Mechanics

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2016

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Vol. 15, nr 4

145-156

EN

This paper introduces basic theoretical knowledge of coalgebras in computer science. Coalgebras are, specifically in category theory, structures defined according to an endofunctor. For both algebra and coalgebra, a functor is a convenient and general way of defining a signature. We present practical usage of the coalgebras in an example. We observe a behavior of a simple Sequencer developed in SLGeometry framework. We model its behavior with the simple program written in Python, and we describe its behavior within coalgebra of endofunctor. The computation of the values stored in internal states is performed coinductively. Our approach can be used in the teaching process of formal methods for young software engineers.