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Energy-Efficient Trajectory Planning with B-Splines and the Schoenberg Quasi-Interpolant

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Energy-Efficient Trajectory Planning with B-Splines and the ...

IEEE Conference on Decision and Control

作者： Vincent Marguet Florin Stoican Ionela Prodan Univ. Grenoble Alpes Grenoble INP Institute of Engineering and Management Univ. Grenoble Alpes. LCIS Valence France Faculty of Automatic Control and Computers Politehnica University of Bucharest ACSE Bucharest Romania Institute for Logic and Data Science Bucharest Romania

ISBN: (数字)9798350316339

ISBN: (纸本)9798350316346

Few studies address the challenge of minimizing energy consumption during trajectory generation, particularly in the context of multicopter dynamics, characterized by strong non-linearity. This paper introduces a novel approach by formulating energy consumption as a function of $\mathbf{B}$-splines control points, enabling comprehensive optimization of total energy expenditure throughout the trajectory. To mitigate the complexity of representation, we leverage the Schoenberg quasi-interpolant, which not only facilitates the solution of the optimization problem but also effectively reduces computational overhead. The approach is validated through two distinct scenarios: a conventional trajectory used in precision agriculture, and a scenario involving more aggressive maneuvers.

关键词： Precision agriculture Energy consumption Trajectory planning Approximation error Energy efficiency Trajectory Complexity theory Splines (mathematics) Optimization Drones

Normal forms and representable functions in Moisil logic

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arXiv 2023年

作者： Sipoş, Andrei Department of Computer Science Faculty of Mathematics and Computer Science University of Bucharest Academiei 14 Bucharest010014 Romania Simion Stoilow Institute of Mathematics The Romanian Academy Calea Grivitęi 21 Bucharest010702 Romania Institute for Logic and Data Science Popa Tatu 18 Bucharest010805 Romania

In this note, we determine, by a disjunctive normal form theorem, which functions on the standard n-nuanced Lukasiewicz-Moisilalgebra are representable by formulas and we show how this result may help in establishing ... 详细信息

关键词： Many valued logics

The computational content of super strongly nonexpansive mappings and uniformly monotone operators

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arXiv 2023年

作者： Sipoş, Andrei Department of Computer Science Faculty of Mathematics and Computer Science University of Bucharest Academiei 14 Bucharest010014 Romania Simion Stoilow Institute of Mathematics The Romanian Academy Calea Grivit ¸ei 21 Bucharest010702 Romania Institute for Logic and Data Science Popa Tatu 18 Bucharest010805 Romania

Recently, Liu, Moursi and Vanderwerff have introduced the class of super strongly nonexpansive mappings as a counterpart to operators which are maximally monotone and uniformly monotone. We give a quantitative study of these notions in the style of proof mining, providing a modulus of super strong nonexpansiveness, giving concrete examples of it and connecting it to moduli associated to uniform monotonicity. For the supercoercive case, we analyze the situation further, yielding a quantitative inconsistent feasibility result for this class (obtaining effective uniform rates of asymptotic regularity), a result which is also qualitatively *** Codes 47H05, 47H09, 47J25, 03F10 Copyright © 2023, The Authors. All rights reserved.

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Empowering LLMs with logical Reasoning: A Comprehensive Survey

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arXiv 2025年

作者： Cheng, Fengxiang Li, Haoxuan Liu, Fenrong van Rooij, Robert Zhang, Kun Lin, Zhouchen Institute for Logic Language and Computation University of Amsterdam Netherlands Center for Data Science Peking University China Machine Learning Department MBZUAI Department of Philosophy Tsinghua University China Department of Philosophy CMU United States Institute for Artificial Intelligence Peking University China Peng Cheng Laboratory China National Key Lab of General AI School of Intelligence Science and Technology Peking University China

Large language models (LLMs) have achieved remarkable successes on various natural language tasks. However, recent studies have found that there are still significant challenges to the logical reasoning abilities of LLMs. This paper summarizes and categorizes the main challenges into two aspects: (1) logical question answering, LLMs often fail to generate the correct answer within complex logical problem which requires sophisticated deductive, inductive or abductive reasoning given a collection of premises and constrains. (2) logical consistency, LLMs are prone to producing responses contradicting themselves across different questions. For example, a state-of-the-art Macaw question-answering LLM answers Yes to both questions Is a magpie a bird? and Does a bird have wings? but answers No to Does a magpie have wings?. To facilitate this research direction, we comprehensively investigate the most cutting-edge methods and propose detailed taxonomies of these methods. Specifically, to accurately answer complex logic questions, previous methods can be categorized based on reliance on external solvers, prompts, pretraining, and fine-tuning. To avoid logical contradictions, we discuss concepts and solutions of various logical consistencies, including implication, negation, transitivity, factuality consistency, and their composites. In addition, we review commonly used benchmark datasets and evaluation metrics, and discuss promising research directions, such as extensions to modal logic to account for uncertainty, and efficient algorithms satisfying multiple logical consistencies simultaneously. © 2025, CC BY.

关键词： Birds

Large Language Models for Mathematicians

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arXiv 2023年

作者： Frieder, Simon Berner, Julius Petersen, Philipp Lukasiewicz, Thomas Department of Computer Science University of Oxford Oxford United Kingdom Department of Computing and Mathematical Sciences Caltech Pasadena United States Faculty of Mathematics and Research Network Data Science University of Vienna Vienna Austria Institute of Logic and Computation Vienna University of Technology Vienna Austria

Large language models (LLMs) such as ChatGPT have received immense interest for their general-purpose language understanding and, in particular, their ability to generate high-quality text or computer code. For many professions, LLMs represent an invaluable tool that can speed up and improve the quality of work. In this note, we discuss to what extent they can aid professional mathematicians. We first provide a mathematical description of the transformer model used in all modern language models. Based on recent studies, we then outline best practices and potential issues and report on the mathematical abilities of language models. Finally, we shed light on the potential of LLMs to change how mathematicians work. Copyright © 2023, The Authors. All rights reserved.

关键词： Computational linguistics

ON MODIFIED HALPERN AND TIKHONOV-MANN ITERATIONS

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arXiv 2022年

作者： Cheval, Horaţiu Kohlenbach, Ulrich Leuştean, Laurenţiu LOS Faculty of Mathematics and Computer Science University of Bucharest Romania Department of Mathematics Technische Universität Darmstadt Germany Simion Stoilow Institute of Mathematics of the Romanian Academy Romania Institute for Logic and Data Science Bucharest Romania

We show that the asymptotic regularity and the strong convergence of the modified Halpern iteration due to T.-H. Kim and H.-K. Xu and studied further by A. Cuntavenapit and B. Panyanak and the Tikhonov-Mann iteration introduced by H. Cheval and L. Leuştean as a generalization of an iteration due to Y. Yao et al. that has recently been studied by Bot ¸ et al. can be reduced to each other in general geodesic settings. This, in particular, gives a new proof of the convergence result in Bot ¸ et al. together with a generalization from Hilbert to CAT(0) spaces. Moreover, quantitative rates of asymptotic regularity and metastability due to K. Schade and U. Kohlenbach can be adapted and transformed into rates for the Tikhonov-Mann iteration corresponding to recent quantitative results on the latter of H. Cheval, L. Leuştean and B. Dinis, P. Pinto respectively. A transformation in the converse direction is also possible. We also obtain rates of asymptotic regularity of order O(1/n) for both the modified Halpern (and so in particular for the Halpern iteration) and the Tikhonov-Mann iteration in a general geodesic setting for a special choice of *** Codes 47J25, 47H09, 03F10 © 2022, CC BY.

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Rates of asymptotic regularity for the alternating Halpern-Mann iteration

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arXiv 2022年

作者： Leuştean, Laurenţiu Pinto, Pedro LOS Faculty of Mathematics and Computer Science University of Bucharest Romania Department of Mathematics Technische Universität Darmstadt Germany Simion Stoilow Institute of Mathematics The Romanian Academy Romania Institute for Logic and Data Science Bucharest Romania

In this paper we extend to UCW-hyperbolic spaces the quantitative asymptotic regularity results for the alternating Halpern-Mann iteration obtained by Dinis and the second author for CAT(0) spaces. These results are new even for uniformly convex normed spaces. Furthermore, for a particular choice of the parameter sequences, we compute linear rates of asymptotic regularity in W-hyperbolic spaces and quadratic rates of T- and U-asymptotic regularity in CAT(0) *** Codes 47J25, 47H09, 03F10 © 2022, CC BY.

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Mathematical capabilities of ChatGPT 23

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Mathematical capabilities of ChatGPT

Proceedings of the 37th International Conference on Neural Information Processing Systems

作者： Simon Frieder Luca Pinchetti Alexis Chevalier Ryan-Rhys Griffiths Tommaso Salvatori Thomas Lukasiewicz Philipp Petersen Julius Berner Department of Computer Science University of Oxford Oxford UK School of Mathematics Institute for Advanced Study Princeton Department of Physics University of Cambridge Cambridge UK VERSES AI Research Lab Los Angeles and Institute of Logic and Computation Vienna University of Technology Vienna Austria Institute of Logic and Computation Vienna University of Technology Vienna Austria and Department of Computer Science University of Oxford Oxford UK Faculty of Mathematics University of Vienna Vienna Austria and Research Network Data Science University of Vienna Vienna Austria Department of Computing and Mathematical Sciences Caltech Pasadena

We investigate the mathematical capabilities of two versions of ChatGPT (released 9-January-2023 and 30-January-2023) and of GPT-4 by testing them on publicly available datasets, as well as hand-crafted ones, using a novel evaluation scheme. In contrast to formal mathematics, where large databases of formal proofs are available (e.g., mathlib, the Lean Mathematical Library), current datasets of natural-language mathematics used to benchmark language models either cover only elementary mathematics or are very small. We address this by publicly releasing two new datasets: GHOSTS and miniGHOSTS. These are the first natural-language datasets curated by working researchers in mathematics that (1) aim to cover graduate-level mathematics, (2) provide a holistic overview of the mathematical capabilities of language models, and (3) distinguish multiple dimensions of mathematical reasoning. These datasets test, by using 1636 human expert evaluations, whether ChatGPT and GPT-4 can be helpful assistants to professional mathematicians by emulating use cases that arise in the daily professional activities of mathematicians. We benchmark the models on a range of fine-grained performance metrics. For advanced mathematics, this is the most detailed evaluation effort to date. We find that ChatGPT and GPT-4 can be used most successfully as mathematical assistants for querying facts, acting as mathematical search engines and knowledge base interfaces. GPT-4 can additionally be used for undergraduate-level mathematics but fails on graduate-level difficulty. Contrary to many positive reports in the media about GPT-4 and ChatGPT's exam-solving abilities (a potential case of selection bias), their overall mathematical performance is well below the level of a graduate student. Hence, if you aim to use ChatGPT to pass a graduate-level math exam, you would be better off copying from your average peer!

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Arithmetic-term representations for the greatest common divisor

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arXiv 2024年

作者： Prunescu, Mihai Shunia, Joseph M. Faculty of Mathematics and Computer Science University of Bucharest Academiei 14 BucharestRO-010014 Romania Institute of Logic and Data Science Bucharest Romania Simion Stoilow Institute of Mathematics of the Romanian Academy Research unit 5 P. O. Box 1-764 BucharestRO-014700 Romania Department of Mathematics The Johns Hopkins University BaltimoreMD United States Veeam Software ColumbusOH United States

We construct a new arithmetic-term representation for the function gcd(a,b). As a byproduct, we also deduce a representation gcd(a,b) by a modular term in integer *** Codes 11A05, 05A15, 11B37 © 2024, CC BY.

关键词： Integer linear programming