Who of us would not be glad to lift the veil behind which the future lies hidden; to cast a glance at the next advances of our science and at the secrets of its development during future centuries? What particular goals will there be toward which the leading mathematical spirits of coming generations will strive? What new methods and new facts in the wide and rich field of mathematical thought will the new centuries disclose? History teaches the continuity of the development of science. We know that every age has its own problems, which the following age either solves or casts aside as profitless and replaces by new ones. If we would obtain an idea of the probable development of mathematical knowledge in the immediate future, we must let the unsettled questions pass before our minds and look over the problems which the science of to-day sets and whose solution we expect from the future. To such a review of problems the present day, lying at the meeting of the centuries, seems to me well adapted. For the close of a great epoch not only invites us to look back into the past but also directs our thoughts to the unknown future. The deep significance of certain problems for the advance of mathematical science in general and the important rôle which they play in the work of the individual investigator are not to be denied. As long as a branch of science offers an abundance of problems, so long is it alive; a lack of problems foreshadows extinction or the cessation of independent development. Just as every human undertaking pursues certain objects, so also mathematical research requires its problems. It is by the solution of problems that the investigator tests the temper of his steel; he finds new methods and new outlooks, and gains a wider and freer horizon. It is difficult and often impossible to judge the value of a problem correctly in advance; for the final award depends upon the grain which science obtains from the problem. Nevertheless we can ask whether there are general criteria which mark a good mathematical problem. An old French mathematician said: “A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street.” This clearness and ease of comprehension, here insisted on for a mathematical theory, I should still more demand for a mathematical problem if it is to be perfect; for what is clear and easily comprehended attracts, the complicated repels us. Moreover a mathematical problem should be difficult in order to entice us, yet not completely inaccessible, lest it mock at our efforts. It should be to us a guide
AI for Mathematics (AI4Math) has emerged as a distinct field that leverages machine learning to navigate mathematical landscapes historically intractable for early symbolic systems. While mid-20th-century symbolic approaches successfully automated formal logic, they faced severe scalability limitations due to the combinatorial explosion of the search space. The recent integration of data-driven approaches has revitalized this pursuit. In this review, we provide a systematic overview of AI4Math, highlighting its primary focus on developing AI models to support mathematical research. Crucially, we emphasize that this is not merely the application of AI to mathematical activities; it also encompasses the development of stronger AI systems where the rigorous nature of mathematics serves as a premier testbed for advancing general reasoning capabilities. We categorize existing research into two complementary directions: problem-specific modeling, involving the design of specialized architectures for distinct mathematical tasks, and general-purpose modeling, focusing on foundation models capable of broader reasoning, retrieval, and exploratory workflows. We conclude by discussing key challenges and prospects, advocating for AI systems that go beyond facilitating formal correctness to enabling the discovery of meaningful results and unified theories, recognizing that the true value of a proof lies in the insights and tools it offers to the broader mathematical landscape.
We study topology, particularly compactness, as an extension of Shulman's work on constructive mathematics via affine logic, while allowing propositional impredicativity. We introduce a notion of compactness in affine logic and prove the fundamental properties of compactness, including the extreme value theorem and the Heine-Borel theorem for 'cuts', which are a version of Dedekind cuts in affine logic. Moreover, from the antithesis translation of the Heine-Borel theorem for cuts to intuitionistic logic, we derive the Heine-Borel theorem for one-sided reals intuitionistically, and have verified the proof with an interactive theorem prover. The code is available at https://github.com/hziwara/CutsHeineBorel.
Niloofar Asefi, Leonard Lupin-Jimenez, Tianning Wu
et al.
Reconstructing ocean dynamics from observational data is fundamentally limited by the sparse, irregular, and Lagrangian nature of spatial sampling, particularly in subsurface and remote regions. This sparsity poses significant challenges for forecasting key phenomena such as eddy shedding and rogue waves. Traditional data assimilation methods and deep learning models often struggle to recover mesoscale turbulence under such constraints. We leverage a deep learning framework that combines neural operators with denoising diffusion probabilistic models to reconstruct high-resolution ocean states from extremely sparse Lagrangian observations. By conditioning the generative model on neural operator outputs, the framework accurately captures small-scale, high-wavenumber dynamics even at 99% sparsity (for synthetic data) and 99.9% sparsity (for real satellite observations). We validate our method on benchmark systems, synthetic float observations, and real satellite data, demonstrating robust performance under severe spatial sampling limitations as compared to other deep learning baselines.
This note presents a brief account of the School of Mathematical Physics of the National Group of Mathematical Physics of the Istituto Nazionale di Alta Matematica (INdAM), written on the occasion of its 50th anniversary.
In this study, we examine osculating-type curves with Frenet-type frame in Myller configuration for Euclidean 3-space E3. We present the necessary characterizations for a curve to be an osculating-type curve. Characterizations originating from the natural structure of Myller configuration are a generalization of osculating curves according to the Frenet frame. Also, we introduce some new results that are not valid for osculating curves. Then, we give an illustrative numerical example supported by a figure.
Francesco Toscano, Costanza Fiorentino, Nicola Capece
et al.
Digital Precision Agriculture (DPA) is a comprehensive approach to agronomic management that utilizes advanced technologies, such as sensor data analysis and automation, to optimize crop productivity, enhance farm income, and minimize environmental impacts. DPA encompasses various agricultural domains, including pest control, pest management, fertilization, irrigation management, sowing, transplanting, crop health monitoring, yield forecasting, harvesting, and post-harvest stages. Among the enabling technologies for DPA, Unmanned Aerial Vehicles (UAVs) have gained significant attention and market growth. The advancements in control systems, robotics, electronics, and artificial intelligence have led to the development of sophisticated agricultural drones. UAVs offer advantages such as versatility, quick and accurate remote sensing capabilities, and high-quality imaging at affordable prices. Furthermore, the miniaturization of sensors and advancements in nanotechnology enable UAVs to perform multiple operations simultaneously without compromising flight autonomy. However, various variables, including aircraft mass, payload capacity, size, battery characteristics, flight autonomy, cost, and environmental conditions, impact the performance and applicability of UAV systems in agriculture. The economic considerations involve the purchase of drones, equipment, and the expertise of trained pilots for flight management and data processing. Payload capacity, flight range, and financial factors influence agriculture’s choice and implementation of UAVs. The research and patent trends show the growing interest in UAVs for agricultural applications. This paper provides a general review of UAV types, construction architectures, and their diverse applications in agriculture until 2022.
Despite the extensive amount of scholarly work done on Indian mathematics in the last 200 years, the conditions under which it originated and evolved is still not clear. Often, one reads the ancient texts with the present concepts and methods in mind. The fact of absence of script over a long stretch of Indian history in ancient times also gets overlooked in such readings. The purpose of this article is to explore the journey of mathematics by examining what the ancient texts tell us about the nature of mathematics in their times. What one finds from the investigation of arithmetic, geometry and algebra is that while it was concrete and context bound, rooted in solving practical problems in ancient times, Indian mathematics transitioned to context free, abstract stage with the advent of algebra supported by writing.
Fullerenes are hollow carbon molecules where each atom is connected to exactly three other atoms, arranged in pentagonal and hexagonal rings. Mathematically, they can be combinatorially modeled as planar, 3-regular graphs with facets composed only of pentagons and hexagons. In this work, we outline a few of the many open questions about fullerenes, beginning with the problem of generating fullerenes randomly. We then introduce an infinite family of fullerenes on which the generalized Stone-Wales operation is inapplicable. Furthermore, we present numerical insights on a graph invariant, called \textit{character} of a fullerene, derived from its adjacency and degree matrices. This descriptor may lead to a new method for linear enumeration of all fullerenes.
Christoph Dlapa, Martin Helmer, Georgios Papathanasiou
et al.
Abstract We provide evidence through two loops, that rational letters of polylogarithmic Feynman integrals are captured by the Landau equations, when the latter are recast as a polynomial of the kinematic variables of the integral, known as the principal A-determinant. Focusing on one loop, we further show that all square-root letters may also be obtained, by re-factorizing the principal A-determinant with the help of Jacobi identities. We verify our findings by explicitly constructing canonical differential equations for the one-loop integrals in both odd and even dimensions of loop momenta, also finding agreement with earlier results in the literature for the latter case. We provide a computer implementation of our results for the principal A-determinants, symbol alphabets and canonical differential equations in an accompanying Mathematica file. Finally, we study the question of when a one-loop integral satisfies the Cohen-Macaulay property and show that for almost all choices of kinematics the Cohen-Macaulay property holds. Throughout, in our approach to Feynman integrals, we make extensive use of the Gel’fand, Graev, Kapranov and Zelevinskiĭ theory on what are now commonly called GKZ-hypergeometric systems whose singularities are described by the principal A-determinant.
Nuclear and particle physics. Atomic energy. Radioactivity
Izabela Babiarz, Victor P. Goncalves, Wolfgang Schäfer
et al.
One of the main goals of future electron-ion colliders is to improve our understanding of the structure of hadrons. In this letter, we study the exclusive ηc production by γ⁎γ interactions in eA collisions and demonstrate that future experimental analysis of this process can be used to improve the description of the ηc transition form factor. The rapidity, transverse momentum and photon virtuality distributions are estimated considering the energy and target configurations expected to be present at the EIC, EicC and LHeC and assuming different predictions for the light-front wave function of the ηc meson. Our results indicate that the electron-ion colliders can be considered an alternative to providing supplementary data to those obtained in e−e+ colliders.
Abstract In this paper, we consider fractional neutral differential equations with multipoint boundary value conditions involving Hadamard derivatives and integrals. We obtain the existence and uniqueness of the solution of the equation by using several fixed point theorems, and we also consider the Ulam–Hyers stability of the solution. In addition, we study the differential inclusion problem with multipoint boundary value conditions and prove the existence of the solution of the boundary value problem when the multivalued map has convex values. We also give several examples to illustrate the feasibility of the results.
We propose the concepts of philomatics and psychomatics as hybrid combinations of philosophy and psychology with mathematics. We explain four motivations for this combination which are fulfilling the desire of analytical philosophy, proposing science of philosophy, justifying mathematical algorithms by philosophy, and abstraction in both philosophy and mathematics. We enumerate various examples for philomatics and psychomatics, some of which are explained in more depth. The first example is the analysis of relation between the context principle, semantic holism, and the usage theory of meaning with the attention mechanism in mathematics. The other example is on the relations of Plato's theory of forms in philosophy with the holographic principle in string theory, object-oriented programming, and machine learning. Finally, the relation between Wittgenstein's family resemblance and clustering in mathematics is explained. This paper opens the door of research for combining philosophy and psychology with mathematics.
Muhammad Sohail, Umar Nazir, Essam R. El-Zahar
et al.
Abstract The mechanism of thermal transport can be enhanced by mixing the nanoparticles in the base liquid. This research discusses the utilization of nanoparticles (tri-hybrid) mixture into Carreau–Yasuda material. The flow is assumed to be produced due to the stretching of vertical heated surface. The phenomena of thermal transport are modeled by considering Joule heating and heat generation or absorption involvement. Additionally, activation energy is engaged to enhance heat transfer rate. The mathematical model composing transport of momentum, heat and mass species is developed in Cartesian coordinate system under boundary layer investigation in the form of coupled nonlinear partial differential equations. The complex partial differential equations are converted into coupled nonlinear ordinary differential equations by using the appropriate similarity transformation. The conversion of PDEs into ODEs make the problem easy to handle and it overcome the difficulties to solve the PDEs. The transformed ordinary differential equations are solved with the help of help of finite element scheme. The obtained solution is plotted against numerous involved parameters and comparative study is established for the reliability of method and accuracy of obtained results. An enhancement in fluid temperature is recorded against magnetic parameter and Eckert number. Also, decline in velocity is recorded for Weissenberg number and concentration is controlled against higher values of Schmidt number. Furthermore, it is recommended that the finite element scheme can be implemented to handle complex coupled nonlinear differential equation arising in modeling of several phenomena occurs in mathematical physics.
In this article we describe special type of mathematical problems that may help develop teaching methods that motivate students to explore patterns, formulate conjectures and find solutions without only memorizing formulas and procedures. These are problems that either their solutions do not make sense in a real life context and/or provide a contradiction during the solution process. In this article we call these problems incorrect problems. We show several examples that can be applied in undergraduate mathematics courses and provide possible ways these examples can be used to motivate critical mathematical thinking. We also discuss the results of exposing a group of 168 undergraduate engineering students to an incorrect problem in a Differential Equations course. This experience provided us with important insight on how well prepared our students are to "out of the box" thinking and on the importance of previous mathematical skills in order to master further mathematical analysis to solve such a problem.
This monograph contains revised and enlarged materials from previous lecture notes of undergraduate and graduate courses and seminars delivered by both authors over the last years on a subject that is central both in abstract operator theory and in applications to quantum mechanics: to decide whether a given densely defined and symmetric operator on Hilbert space admits a unique self-adjoint realisation, namely its operator closure, or whether instead it admits an infinite multiplicity of distinct self-adjoint extensions, and in the latter case to classify them and characterise their main features (operator and quadratic form domains, spectrum, etc.) This is at the same time a very classical, well established field, corresponding to the first part of the monograph, and a territory of novel, modern applications, a selection of which, obviously subjective to some extent, but also driven by a pedagogical criterion, is presented in depth in the second part. A number of models are discussed, which are receiving today new or renewed interest in mathematical physics, in particular from the point of view of realising certain operators of interests self-adjointly, classifying their self-adjoint extensions as actual quantum Hamiltonians, studying their spectral and scattering properties, and the like, but also from the point of view of intermediate technical questions that have theoretical interest per se, such as characterising the corresponding operator closures and adjoints.
The Korea Society Of Educational Studies In Mathematics, Tiong Seah Wee, Hee-jeong Kim
et al.
The 21st Century is characterised by technological advances which is the Fourth Industrial Revolution, climate change, and the COVID19 pandemic, for examples. The role of mathematics in each of these phenomena has been central and crucial. As such, it is an opportune time now to take stock of events that are (re-)shaping the world, so that we can better facilitate mathematics education in schools. Three themes are identified and discussed in this article, namely the convergence of mathematics pedagogical approaches, mathematics proficiencies, and students’ mathematical wellbeing.
We give a sufficient condition on coefficients $a_k$ of an algebraic polynomial $P(z)=\sum\limits_{k=0}^{n}a_kz^k$, $a_n\not=0,$ such that the pointwise Bernstein inequality $|P'(z)|\le n|P(z)|$ is true for all $z,\ |z|\le 1$.
Ayo Stephen Adebowale, Adeniyi Francis Fagbamigbe, Joshua Odunayo Akinyemi
et al.
Background: The coronavirus disease (COVID-19) remains a global public health issue due to its high transmission and case fatality rate. There is apprehension on how to curb the spread and mitigate the socio-economic impacts of the pandemic, but timely and reliable daily confirmed cases' estimates are pertinent to the pandemic's containment. This study therefore conducted a situation assessment and applied simple predictive models to explore COVID-19 progression in Nigeria as at 31 May 2020. Methods: Data used for this study were extracted from the websites of the European Centre for Disease Control (World Bank data) and Nigeria Centre for Disease Control. Besides descriptive statistics, four predictive models were fitted to investigate the pandemic natural dynamics. Results: The case fatality rate of COVID-19 was 2.8%. A higher number of confirmed cases of COVID-19 was reported daily after the relaxation of lockdown than before and during lockdown. Of the 36 states in Nigeria, including the Federal Capital Territory, 35 have been affected with COVID-19. Most active cases were in Lagos (n = 4064; 59.2%), followed by Kano (n = 669; 9.2%). The percentage of COVID-19 recovery in Nigeria (29.5%) was lower compared to South Africa (50.3%), but higher compared to Kenya (24.1%). The cubic polynomial model had the best fit. The projected value for COVID-19 cumulative cases for 30 June 2020 in Nigeria was 27,993 (95% C.I: 27,001–28,986). Conclusion: The daily confirmed cases of COVID-19 are increasing in Nigeria. Increasing testing capacity for the disease may further reveal more confirmed cases. As observed in this study, the cubic polynomial model currently offers a better prediction of the future COVID-19 cases in Nigeria.