Hasil untuk "Mathematics"

B. Efron, R. Tibshirani

A. Kirsch

M. Henningsen, M. Stein

Johan Commelin, Mateja Jamnik, Rodrigo Ochigame et al.

Artificial intelligence is transforming mathematics at a speed and scale that demand active engagement from the mathematical community. We examine five areas where this transformation is particularly pressing: values, practice, teaching, technology, and ethics. We offer recommendations on safeguarding our intellectual autonomy, rethinking our practice, broadening curricula, building academically oriented infrastructure, and developing shared ethical principles - with the aim of ensuring that the future of mathematics is shaped by the community itself.

Randy Davila, Boris Brimkov, Ryan Pepper

We present four open conjectures in graph theory generated by the automated conjecturing system \texttt{TxGraffiti}. Each conjecture is concise, grounded in natural graph invariants, and empirically validated across hundreds of graphs. Despite extensive effort, these statements remain unresolved--defying both proof and counterexample. They are not only mathematical challenges but creative expressions--born of symbolic pattern recognition and mathematician-defined heuristics, refined through years of human dialogue, and now offered back to the community as collaborative artifacts. These conjectures invite not only formal proof, but also reflection on how machines can evoke wonder, spark curiosity, and contribute to the raw material of discovery. By highlighting these problems, we aim to inspire both human mathematicians and AI systems to engage with them--not only to solve them, but to reflect on what it means when machines participate meaningfully in the creative process of mathematical thought.

Muhammad Zafarullah Baber, Nauman Ahmed, Muhammad Waqas Yasin et al.

Abstract In this study, we consider the coupled nonlinear Schrödinger equation under the influence of the multiplicative time noise. The coupled nonlinear Schrödinger equation, which shows the complex envelope amplitudes of the two modulated weakly resonant waves in two polarisations and is used to describe the pulse propagation in high birefringence fibre, has several uses in optical fibres.query:Journal instruction requires a city for affiliations; however, these are missing in affiliation [6]. Please verify if the provided city are correct and amend if necessary. The underlying model is analyzed numerically and analytically as well. For the computational results, the proposed stochastic backward Euler scheme is developed and its consistency is derived in the mean square sense. For the linear stability analysis, Von-Neumann criteria is used, given proposed stochastic scheme is unconditionally stable. The exact optical soliton solutions are constructed with the help of the $$\phi ^6$$ -model expansion technique, which provided us with the Jacobi elliptic function solutions that will explore optical solitons and solitary waves as well. The initial and boundary conditions are constructed for the numerical result by some optical soliton solutions. The 3D, 2D and corresponding contour plot are drawn for the different values of noise. Mainly, the comparison of results is shown graphically in 3D and line plots for some newly constructed solutions by selecting suitable parameters value.

Nana Liu, Qisheng Wang, Mark M. Wilde et al.

Abstract Matrix geometric means between two positive definite matrices can be defined from distinct perspectives—as solutions to certain nonlinear systems of equations, as points along geodesics in Riemannian geometry, and as solutions to certain optimisation problems. We devise quantum subroutines for the matrix geometric means, and construct solutions to the algebraic Riccati equation—an important class of nonlinear systems of equations appearing in machine learning, optimal control, estimation, and filtering. Using these subroutines, we present a new class of quantum learning algorithms, for both classical and quantum data, called quantum geometric mean metric learning, for weakly supervised learning and anomaly detection. The subroutines are also useful for estimating geometric Rényi relative entropies and the Uhlmann fidelity, in particular achieving optimal dependence on precision for the Uhlmann and Matsumoto fidelities. Finally, we provide a BQP-complete problem based on matrix geometric means that can be solved by our subroutines.

Longjian Piao, Laurens de Vries, Mathijs de Weerdt et al.

Future energy markets for low voltage AC and DC distribution systems will facilitate prosumer participation in the market. To comply with market regulations and grid constraints, a tailored market design reflecting (DC) operational requirements is needed. Our previous work identified a locational energy market design. However, its real-life implementation faces challenges due to uncertainties in system operation, prosumer preferences, and bidding strategies. This article tests the market design under uncertain scenarios. To this end, we develop an agent-based model that simulates typical electric vehicle user preferences and bidding strategies, influenced by varying degrees of range anxiety. The market design is tested in challenging scenarios with a high share of solar panels and electric vehicles, modelled using the high-resolution Pecan Street database. Simulations indicate that the proposed market design maintains both economic efficiency and system reliability under real-life uncertainties. This in turn indicates the practical feasibility of locational energy markets in helping to integrate renewable generation sources and bidirectional power flows.

Fatih Hezenci, Hüseyin Budak

Abstract In this article, trapezoid-type inequalities are proved by means of convex functions according to tempered fractional integral operators. The resulting trapezoid-type inequalities generalize several studies on this subject, including those involving Riemann-Liouville fractional integrals. To obtain these types of inequalities, we utilize the Hölder and power-mean inequalities. Finally, new results are obtained through specific parameter choices.

Anton Freund

A predilator is a particularly uniform transformation of linear orders. We have a dilator when the transformation preserves well-foundedness. Over the theory $\mathsf{ACA}_0$ from reverse mathematics, any $Π^1_2$-formula is equivalent to the statement that some predilator is a dilator. We show how this completeness result breaks down without arithmetical comprehension: over $\mathsf{RCA}_0+\mathsf{PA}$, the statements from a large part of the reverse mathematics zoo are not equivalent to some predilator being a dilator.

Hongzhang Xu, Rowena Ball

We present a scoping review of published literature on ethnomathematics and Indigenous mathematics as a step towards a goal to decolonize the prevailing Eurocentric view of the provenance of mathematics. Mathematical practices were identified globally from 169 included studies. We map three development stages of ethnomathematical research from 1984 to 2023 and identify 20 categories of Indigenous and traditional cultural activities that evidence mathematical design and expression. We address two challenges of investigating non-Western based mathematics: where to look for mathematical knowledge, and how to decode it from cultural practices. These two hurdles are overcome by cluster analysis of the keywords of included studies. Existing research falls into two categories: I. identification of mathematical concepts used in Indigenous societies, and II. systematizing identified mathematical concepts. Both approaches are essential for research on Indigenous mathematics to flourish, in order to empower Indigenous knowledge holders and deconstruct restrictive colonial boundaries of mathematical knowledge and education.

László Csató, Sergey Ilyin

UEFA declares that it is committed to respecting the fundamental values of sports. However, the qualification rules of the post-2024 UEFA Champions League are shown to be unfair: a game with misaligned incentives was narrowly avoided in the 2023/24 German Bundesliga. We develop a mathematical model to reveal how incentives for losing can be reduced or eliminated. Since UEFA repeatedly commits the same theoretical mistake in designing the qualification system of its competitions, governing bodies in sports are called to work more closely together with the scientific community.

Yasena Chantova

This article presents a discussion on polysemy in a crossword by Perec. We apply Culioli’s modelling approach in terms of operation of location and construction of a notion and its notional domain. Polysemy is seen as part of the intangible cultural heritage within languages.

Kohli Mathieu

Aesop's famous fable The Ant and the Grasshopper emphasizes the choice that living beings have to make between survival (the Ant) and reproduction (the Grasshopper). Apis Mellifera colonies face the same dilemma : must they rather collect nectar in order to produce honey that will enable them to endure the cold of winter or should they choose to bring pollen back to the hive so as to raise young bees ? Both, of course, but in which proportion ? And where should they fly to find the best resources ? We adress these question through a mathematical perspective, by considering the geometry of resources around the hive.

Lee-Peng Teo

This is the first volume of a textbook for a two-semester course in mathematical analysis. This first volume is about analysis of functions of a single variable. The topics covered include completeness axiom, Archimedean property, sequentially compact subsets of $\mathbb{R}$, limits of functions, continuous functions, intermediate value theorem, extreme value theorem, differentiation, mean value theorem, l'Hopital's rule, Riemann integrals, improper integrals, elementary transcendental functions, sequences and series of numbers, infinite products, sequences and series of functions, uniform convergence, power series, Taylor series and Taylor polynomials. At the end of the book, we include some classical examples such as the irrationality of the number $e$, the existence of a non-analytic infinitely differentiable function, the existence of a nowhere differentiable continuous function. The book is concluded with the proof of the Weierstrass approximation theorem.

Jouko Väänänen

I have argued elsewhere that second order logic provides a foundation for mathematics much in the same way as set theory does, despite the fact that the former is second order and the latter first order, but second order logic is marred by reliance on ad hoc {\em large domain assumptions}. In this paper I argue that sort logic, a powerful extension of second order logic, provides a foundation for mathematics without any ad hoc large domain assumptions. The large domain assumptions are replaced by ZFC-like axioms. Despite this resemblance to set theory sort logic retains the structuralist approach to mathematics characteristic of second order logic. As a model-theoretic logic sort logic is the strongest logic. In fact, every model class definable in set theory is the class of models of a sentence of sort logic. Because of its strength sort logic can be used to formulate particularly strong reflection principles in set theory.

Amer Hassan Albargi

The aim of this paper is to define generalized rational contractions in the setting of graphical b-metric spaces and obtain some fixed-point theorems. Our results are significant generalizations and extensions of some well-known results in the existing theory. We also supply a nontrivial example to show the validity of the obtained theorems. As applications, we obtain some results on rational expressions in the background of graphical metric spaces.

Xin Zhang, Pingping Wei, Qingling Wang

Anomaly detection of high-dimensional data is a challenge because the sparsity of the data distribution caused by high dimensionality hardly provides rich information distinguishing anomalous instances from normal instances. To address this, this article proposes an anomaly detection method combining an autoencoder and a sparse weighted least squares-support vector machine. First, the autoencoder is used to extract those low-dimensional features of high-dimensional data, thus reducing the dimension and the complexity of the searching space. Then, in the low-dimensional feature space obtained by the autoencoder, the sparse weighted least squares-support vector machine separates anomalous and normal features. Finally, the learned class labels to be used to distinguish normal instances and abnormal instances are outputed, thus achieving anomaly detection of high-dimensional data. The experiment results on real high-dimensional datasets show that the proposed method wins over competing methods in terms of anomaly detection ability. For high-dimensional data, using deep methods can reconstruct the layered feature space, which is beneficial for gaining those advanced anomaly detection results.

Dag Normann, Sam Sanders

The aim of Reverse Mathematics(RM for short)is to find the minimal axioms needed to prove a given theorem of ordinary mathematics. These minimal axioms are almost always equivalent to the theorem, working over the base theory of RM, a weak system of computable mathematics. The Big Five phenomenon of RM is the observation that a large number of theorems from ordinary mathematics are either provable in the base theory or equivalent to one of only four systems; these five systems together are called the 'Big Five'. The aim of this paper is to greatly extend the Big Five phenomenon as follows: there are two supposedly fundamentally different approaches to RM where the main difference is whether the language is restricted to second-order objects or if one allows third-order objects. In this paper, we unite these two strands of RM by establishing numerous equivalences involving the second-order Big Five systems on one hand, and well-known third-order theorems from analysis about (possibly) discontinuous functions on the other hand. We both study relatively tame notions, like cadlag or Baire 1, and potentially wild ones, like quasi-continuity. We also show that slight generalisations and variations of the aforementioned third-order theorems fall far outside of the Big Five.

Xin Liu, Martin Valcke, Kajsa Yang Hansen et al.

This paper attempts to demonstrate the usefulness of the linkage data from two international large-scale assessment studies, Teaching and Learning International Survey 2013 (TALIS) 2013 and Programme for International Student Assessment (PISA) 2012, in examining the effects of schools. Data from seven educational systems are used to link, and four critical issues with five selection criteria are applied to the data selected. The linking dataset facilitates the investigation of mathematics performance while considering individual learner characteristics, mathematics teacher variables in the classroom environment and the school-level variables. We extend the new avenue of research by developing a linked database geared to the specific mathematics teaching and learning domain to reflect the school mathematics educational environment. The case study using Singapore linkage data demonstrated the feasibility and potential of exploring school effectiveness. In Singapore, schools with teachers of a higher level of education and self-efficacy in teaching mathematics related to a higher level of school mathematics performance. The study offers a guideline and inspiration to the research community to exploit the rich information in both TALIS and PISA studies to facilitate school effectiveness studies.