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1. Fractal Nature Bridge between Neural Networks and Graph Theory Approach within Material Structure Characterization

   https://doi.org/10.3390/fractalfract6030134  

   Randjelovic, Branislav M.; Mitic, Vojislav V.; Ribar, Srdjan; Milosevic, Dusan M.; Lazovic, Goran; Fecht, Hans J.; Vlahovic, Branislav (March 2022, Fractal and Fractional)

   Many recently published research papers examine the representation of nanostructures and biomimetic materials, especially using mathematical methods. For this purpose, it is important that the mathematical method is simple and powerful. Theory of fractals, artificial neural networks and graph theory are most commonly used in such papers. These methods are useful tools for applying mathematics in nanostructures, especially given the diversity of the methods, as well as their compatibility and complementarity. The purpose of this paper is to provide an overview of existing results in the field of electrochemical and magnetic nanostructures parameter modeling by applying the three methods that are “easy to use”: theory of fractals, artificial neural networks and graph theory. We also give some new conclusions about applicability, advantages and disadvantages in various different circumstances. 

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2. Higher Dimensional Algebraic Geometry: A Volume in Honor of V. V. Shokurov

   https://doi.org/10.1017/9781009396233  

   Cheltsov, Ivan; Sarikyan, Arman; Zhuang, Ziquan (December 2024, Cambridge University Press)

   Hacon, Christopher; Xu, Chenyang (Ed.)

   Arising from the 2022 Japan-US Mathematics Institute, this book covers a range of topics in modern algebraic geometry, including birational geometry, classification of varieties in positive and zero characteristic, K-stability, Fano varieties, foliations, the minimal model program and mathematical physics. The volume includes survey articles providing an accessible introduction to current areas of interest for younger researchers. Research papers, written by leading experts in the field, disseminate recent breakthroughs in areas related to the research of V.V. Shokurov, who has been a source of inspiration for birational geometry over the last forty years. 

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3. Reduction games, provability and compactness

   https://doi.org/10.1142/S021906132250009X  

   Dzhafarov, Damir D.; Hirschfeldt, Denis R.; Reitzes, Sarah (December 2022, Journal of Mathematical Logic)

   Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between [Formula: see text] principles over [Formula: see text]-models of [Formula: see text]. They also introduced a version of this game that similarly captures provability over [Formula: see text]. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication [Formula: see text] between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H. Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between [Formula: see text] principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is “purely proof-theoretic”, as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of [Formula: see text], uncovering new differences between their logical strengths. 

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4. Exercises Integrating High School Mathematics with Robot Motion Planning

   https://doi.org/10.1109/FIE43999.2019.9028394  

   Greenberg, Ronald I.; Thiruvathukal, George K. (October 2019, 2019 IEEE Frontiers in Education Conference)

   This Innovative Practice Work in Progress presents progress in developing exercises for high school students incorporating level-appropriate mathematics into robotics activities. We assume mathematical foundations ranging from algebra to precalculus, whereas most prior work on integrating mathematics into robotics uses only very elementary mathematical reasoning or, at the other extreme, is comprised of technical papers or books using calculus and other advanced mathematics. The exercises suggested are relevant to any differential-drive robot, which is an appropriate model for many different varieties of educational robots. They guide students towards comparing a variety of natural navigational strategies making use of typical movement primitives. The exercises align with Common Core State Standards for Mathematics. 

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5. Meta-Xenakis: Developing Style-Agnostic, Stochastic Algorithmic Music

   Manaris, Bill; Forgette, Anna (January 2023, International Computer Music Conference proceedings)

   Iannis Xenakis was a pioneer in algorithmic composition of music and art. He combined architecture, mathematics, music, and performance art to create avant-garde compositions and performances that are still being analyzed, performed widely, and discussed today. Xenakis’s musical contributions are deeply algorithmic in nature, inspired by his appreciation and understanding of mathematics and controlled randomness, i.e., stochastic processes. This paper argues that Xenakis's algorithmic approach is style-agnostic, as it may be used with different sonification choices to produce pieces with more traditional aesthetics, possibly bringing broader acceptance, appreciation, and application of his techniques and ideas to more traditional compositional spaces. Also, today’s music technology has evolved tremendously, through the integration of artificial intelligence, advanced computing algorithms, and human computer interaction – techniques and technology that were unavailable to Xenakis, but which would have been inline with his pioneering spirit. We explore some of Xenakis’s early works in algorithmic and stochastic music, and reimagine the types of music Xenakis could possibly be making today, having access to the modern technology of smartphones and computing devices. Examples include a retelling of Xenakis's “Concret PH” using smartphones for sound spatialization and audience interaction / participation; “Éolienne PH”, an example piece which utilizes the same statistical distributions as “Concret PH” to produce a completely different sound aesthetic; “on the Fractal Nature of Being…” which combines Xenakis's stochastic / aleatoric techniques with traditional music theory and modern mathematics / fractal geometry; and, finally “Nereides / Νηρηΐδες”, a piano miniature piece, which is made using statistical distributions from a cloudy sky. We close with some general ideas on the future of the Algorithmic Arts. 

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