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AppliedMath, Volume 4, Issue 3 (September 2024) – 4 articles

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12 pages, 248 KiB  
Article
From Algebro Geometric Solutions of the Toda Equation to Sato Formulas
by Pierre Gaillard
AppliedMath 2024, 4(3), 856-867; https://doi.org/10.3390/appliedmath4030046 - 9 Jul 2024
Viewed by 420
Abstract
We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the [...] Read more.
We know that the degeneracy of solutions to PDEs, given in terms of theta functions on Riemann surfaces, provides important results about particular solutions, as in the case of the NLS equation. Here, we degenerate the so called finite gap solutions of the Toda lattice equation from the general formulation in terms of abelian functions when the gaps tend to points. This degeneracy allows us to recover the Sato formulas without using inverse scattering theory or geometric or representation theoretic methods. Full article
13 pages, 690 KiB  
Article
Long- and Medium-Term Financial Strategies on Equities Using Dynamic Bayesian Networks
by Karl Lewis, Mark Anthony Caruana and David Paul Suda
AppliedMath 2024, 4(3), 843-855; https://doi.org/10.3390/appliedmath4030045 - 3 Jul 2024
Viewed by 410
Abstract
Devising a financial trading strategy that allows for long-term gains is a very common problem in finance. This paper aims to formulate a mathematically rigorous framework for the problem and compare and contrast the results obtained. The main approach considered is based on [...] Read more.
Devising a financial trading strategy that allows for long-term gains is a very common problem in finance. This paper aims to formulate a mathematically rigorous framework for the problem and compare and contrast the results obtained. The main approach considered is based on Dynamic Bayesian Networks (DBNs). Within the DBN setting, a long-term as well as a short-term trading strategy are considered and applied on twelve equities obtained from developed and developing markets. It is concluded that both the long-term and the medium-term strategies proposed in this paper outperform the benchmark buy-and-hold (B&H) trading strategy. Despite the clear advantages of the former trading strategies, the limitations of this model are discussed along with possible improvements. Full article
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15 pages, 1478 KiB  
Article
Network Goodness Calculus Propositions
by Marina Bershadsky, Božidar Ivanković and Marko Pušić
AppliedMath 2024, 4(3), 828-842; https://doi.org/10.3390/appliedmath4030044 - 2 Jul 2024
Viewed by 255
Abstract
We coin the term “network goodness” for a value we define for a network embedded in a given environment as a metric that describes the suitability of that network for meeting a demand. Three formulas are proposed to calculate the metric from three [...] Read more.
We coin the term “network goodness” for a value we define for a network embedded in a given environment as a metric that describes the suitability of that network for meeting a demand. Three formulas are proposed to calculate the metric from three variable values. The first variable considers parts of the environment gravitated by the network. For these parts of the environment, we define a value that measures user costs refusing them the use of the network. Last but not least, the network maintenance costs are considered. The results are obtained after focusing on infrastructure and transport networks, but can be used for other types of networks as well. Full article
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22 pages, 342 KiB  
Article
The Mechanics Underpinning Non-Deterministic Computation in Cortical Neural Networks
by Elizabeth A. Stoll
AppliedMath 2024, 4(3), 806-827; https://doi.org/10.3390/appliedmath4030043 - 26 Jun 2024
Viewed by 635
Abstract
Cortical neurons integrate upstream signals and random electrical noise to gate signaling outcomes, leading to statistically random patterns of activity. Yet classically, the neuron is modeled as a binary computational unit, encoding Shannon entropy. Here, the neuronal membrane potential is modeled as a [...] Read more.
Cortical neurons integrate upstream signals and random electrical noise to gate signaling outcomes, leading to statistically random patterns of activity. Yet classically, the neuron is modeled as a binary computational unit, encoding Shannon entropy. Here, the neuronal membrane potential is modeled as a function of inherently probabilistic ion behavior. In this new model, each neuron computes the probability of transitioning from an off-state to an on-state, thereby encoding von Neumann entropy. Component pure states are integrated into a physical quantity of information, and the derivative of this high-dimensional probability distribution yields eigenvalues across the multi-scale quantum system. In accordance with the Hellman–Feynman theorem, the resolution of the system state is paired with a spontaneous shift in charge distribution, so this defined system state instantly becomes the past as a new probability distribution emerges. This mechanistic model produces testable predictions regarding the wavelength of free energy released upon information compression and the temporal relationship of these events to physiological outcomes. Overall, this model demonstrates how cortical neurons might achieve non-deterministic signaling outcomes through a computational process of noisy coincidence detection. Full article
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