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Products of hyperbolic spaces

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arXiv 2024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 of the Romanian Academy Calea Grivit ¸ei 21 Bucharest010702 Romania Institute for Logic and Data Science Popa Tatu 18 Bucharest010805 Romania

The class of uniformly smooth hyperbolic spaces was recently introduced by Pinto as a common generalization of both CAT(0) spaces and uniformly smooth Banach spaces, in a way that Reich’s theorem on resolvent convergence could still be proven. We define products of such spaces, showing that they are reasonably well-behaved. In this manner, we provide the first example of a space for which Reich’s theorem holds and which is neither a CAT(0) space, nor a convex subset of a normed *** Codes 51F99, 52A55, 53C23 Copyright © 2024, The Authors. All rights reserved.

关键词： Banach spaces

"Vorbes, ti Românes, te?" A Recipe to Train Powerful Romanian LLMs with English Instructions

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arXiv 2024年

作者： Masala, Mihai Ilie-Ablachim, Denis C. Dima, Alexandru Corlatescu, Dragos Zavelca, Miruna Olaru, Ovio Terian, Simina Terian, Andrei Leordeanu, Marius Velicu, Horia Popescu, Marius Dascalu, Mihai Rebedea, Traian Institute for Logic and Data Science Bucharest Romania National University of Science and Technology POLITEHNICA Bucharest Romania Simion Stoilow Institute of Mathematics of The Romanian Academy Bucharest Romania University of Bucharest Romania BRD Groupe Societe Generale Romania Lucian Blaga University Sibiu Romania NVIDIA United States

In recent years, Large Language Models (LLMs) have achieved almost human-like performance on various tasks. While some LLMs have been trained on multilingual data, most of the training data is in English;hence, their performance in English greatly exceeds other languages. To our knowledge, we are the first to collect and translate a large collection of texts, instructions, and benchmarks and train, evaluate, and release open-source LLMs tailored for Romanian. We evaluate our methods on four different categories, including academic benchmarks, MT-Bench (manually translated), and a professionally built historical, cultural, and social benchmark adapted to Romanian. We argue for the usefulness and high performance of RoLLMs by obtaining state-of-the-art results across the board. We publicly release all resources (i.e., data, training and evaluation code, models) to support and encourage research on Romanian LLMs while concurrently creating a generalizable recipe, adequate for other low or less-resourced languages. © 2024, CC BY.

关键词： Open systems

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

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

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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When accurate prediction models yield harmful self-fulfilling prophecies

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arXiv 2023年

作者： van Amsterdam, Wouter A.C. van Geloven, Nan Krijthe, Jesse H. Ranganath, Rajesh Cinà, Giovanni Department of Data Science and Biostatistics Julius Center of Health Sciences Primary Care University Medical Center Utrecht Heidelberglaan 100 Utrecht3584 CX Netherlands University of Utrecht Heidelberglaan 100 Utrecht3584 CX Netherlands Department of Biomedical Data Sciences Leiden University Medical Center Leiden Netherlands Pattern Recognition & Bioinformatics Delft University of Technology Delft Netherlands Courant Institute of Mathematical Science Department of Computer Science Center for Data Science New York University New York City United States Department of Medical Informatics Amsterdam University Medical Center Amsterdam Netherlands Institute for Logic Language and Computation University of Amsterdam Amsterdam Netherlands Pacmed Amsterdam Netherlands

Prediction models are popular in medical research and practice. By predicting an outcome of interest for specific patients, these models may help inform difficult treatment decisions, and are often hailed as the poster children for personalized, data-driven healthcare. We show however, that using prediction models for decision making can lead to harmful decisions, even when the predictions exhibit good discrimination after deployment. These models are harmful self-fulfilling prophecies: their deployment harms a group of patients but the worse outcome of these patients does not invalidate the predictive power of the model. Our main result is a formal characterization of a set of such prediction models. Next we show that models that are well calibrated before and after deployment are useless for decision making as they made no change in the data distribution. These results point to the need to revise standard practices for validation, deployment and evaluation of prediction models that are used in medical decisions. © 2023, CC BY.

关键词： Prediction models

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

Extending a conjecture of Graham and Lovász on the distance characteristic polynomial

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arXiv 2022年

作者： Abiad, Aida Brimkov, Boris Hayat, Sakander Khramova, Antonina P. Koolen, Jack H. Department of Mathematics and Computer Science Eindhoven University of Technology Netherlands Department of Mathematics: Analysis Logic and Discrete Mathematics Ghent University Belgium Department of Mathematics and Data Science Vrije Universiteit Brussel Belgium Department of Mathematics and Statistics Slippery Rock University United States Faculty of Engineering Sciences GIK Institute of Engineering Sciences and Technology Topi Swabi Pakistan School of Mathematical Sciences University of Science and Technology of China China

Graham and Lovász conjectured in 1978 that the sequence of normalized coefficients of the distance characteristic polynomial of a tree of order n is unimodal with the maximum value occurring at ⌊ n2 ⌋. In this paper we investigate this problem for block graphs. In particular, we prove the unimodality part and we establish the peak for several extremal cases of uniform block graphs with small diameter. © 2022, CC BY.

关键词： Polynomials

Mathematical Capabilities of ChatGPT

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arXiv 2023年

作者： Frieder, Simon Pinchetti, Luca Chevalier, Alexis Griffiths, Ryan-Rhys Salvatori, Tommaso Lukasiewicz, Thomas Petersen, Philipp Christian Berner, Julius Department of Computer Science University of Oxford Oxford United Kingdom Institute of Logic and Computation Vienna University of Technology Vienna Austria School of Mathematics Institute for Advanced Study Princeton United States Department of Physics University of Cambridge Cambridge United Kingdom Faculty of Mathematics University of Vienna Vienna Austria Research Network Data Science University of Vienna Vienna Austria VERSES Research Lab Los AngelesCA90016 United States

We investigate the mathematical capabilities of two iterations 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 methodology. In contrast to formal mathematics, where large databases of formal proofs are available (e.g., 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 also test 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 can be used most successfully as a mathematical assistant for querying facts, acting as a mathematical search engine and knowledge base interface. 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 your goal is to use ChatGPT to pass a graduate-level math exam, you would be better off copying from your average peer! Copyright © 2023, The Authors. All rights reserved.

关键词： Benchmarking