Strictly Real Fundamental Theorem of Algebra Using Polynomial Interlacing

Basu, Soham

doi:10.1017/s0004972720001434

Cited by 2 publications

(1 citation statement)

References 5 publications

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“…Notably, we also assumed that Q(•, •) is factorisable. Since it can be approximated by Chebyshev polynomials, and, according to the strictly real fundamental theorem of algebra [21], it is possible to factorise polynomial function to two factors. Alternatively, one can use deep sets [20] as arguments to approximate Q(•, •).…”

Section: Aggregation Of Neighbour-interactionsmentioning

confidence: 99%

Universality of neural dynamics on complex networks

Vasiliauskaite¹,

Antulov-Fantulin²

2023

Preprint

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This paper discusses the capacity of graph neural networks to learn the functional form of ordinary differential equations that govern dynamics on complex networks. We propose necessary elements for such a problem, namely, inductive biases, a neural network architecture and a learning task. Statistical learning theory suggests that generalisation power of neural networks relies on independence and identical distribution (i.i.d.) of training and testing data. Although this assumption together with an appropriate neural architecture and a learning mechanism is sufficient for accurate out-ofsample predictions of dynamics such as, e.g. mass-action kinetics, by studying the out-of-distribution generalisation in the case of diffusion dynamics, we find that the neural network model: (i) has a generalisation capacity that depends on the first moment of the initial value data distribution; (ii) learns the non-dissipative nature of dynamics implicitly; and (iii) the model's accuracy resolution limit is of order O(1/ √ n) for a system of size n.

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Section: Aggregation Of Neighbour-interactionsmentioning

confidence: 99%

Universality of neural dynamics on complex networks

Vasiliauskaite¹,

Antulov-Fantulin²

2023

Preprint

View full text Add to dashboard Cite

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An algebraic approach to the Fundamental Theorem of Algebra

Mandal,

Sheikh

2024

Rend. Circ. Mat. Palermo, II. Ser

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In this paper, we investigate the algebraic counterpart of the Fundamental Theorem of Algebra. We explore the concept of real-closed fields and quadratic forms. We show, by means of Galois theory, that $$\mathcal {F}(\sqrt{-1})$$ F ( - 1 ) is algebraically closed if $$\mathcal {F}$$ F is real-closed. Lastly, we explain the algebraic closure of $$\mathbb {R}(\sqrt{-1})=\mathbb {C}$$ R ( - 1 ) = C by demonstrating the real-closeness of $$\mathbb {R}$$ R .

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Strictly Real Fundamental Theorem of Algebra Using Polynomial Interlacing

Cited by 2 publications

References 5 publications

Universality of neural dynamics on complex networks

Universality of neural dynamics on complex networks

An algebraic approach to the Fundamental Theorem of Algebra

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