In light of the nonsmooth and complex dynamic characteristics of the impact capsule system, this paper proposes a novel finite-time tracking control strategy. Compared with conventional asymptotic control methods, the proposed approach not only effectively suppresses basin hopping induced by noise and significantly enhances system robustness but also enables the capsule robot to achieve fast and accurate trajectory tracking. Furthermore, through rigorous dynamic analysis and mathematical proof based on Lyapunov stability theory, this study establishes finite-time stability criteria for the closed-loop system, providing a theoretical basis for system performance evaluation. Numerical simulation results demonstrate that the designed controller excels in multistability regulation and chaos suppression. Regardless of whether the initial state lies in a multistable or chaotic region, the system can rapidly and accurately track the desired motion trajectory. The research outcomes also offer valuable insights for the control design of other nonsmooth dynamical systems.

One of the key claims in the paper “Robust Contract Designs: Linear Contracts and Moral Hazard” by Yu and Kong (2020) is proposition 4. This proposition states that when designing the best robust contract—assuming the agent’s utility function is piecewise linear and concave—the optimal solution involves only progressive fixed payments and linear rewards with progressive commission rates. In this note, we provide a clear counterexample backed by a detailed proof to show that this claim is actually incorrect. We further use numerical examples to demonstrate that the contract design suggested by Yu and Kong can perform significantly worse than expected. Lastly, we identify and explain the specific mistakes in their mathematical proofs that led them to this incorrect conclusion.

This paper reviews the fracture mechanics parameters associated with the variability in the crack growth curves associated with forty-two different tests that range from additively manufactured (AM) steels to cold spray additively manufactured (CSAM) 316L steel. As a result of this review, it is found that, to a first approximation, the effects of different building processes and R-ratios on the relationship between ΔK and the crack growth rate (da/dN) can be captured by allowing for changes in the fatigue threshold and the apparent cyclic toughness in the Schwalbe crack driving force (Δκ). Whilst this observation, when taken in conjunction with similar findings for AM Ti-6Al-4V, Inconel 718, Inconel 625, and Boeing Space Intelligence and Weapon Systems (BSI&WS) laser powder bed (LPBF)-built Scalmalloy®, as well as for a range of CSAM pure metals, go a long way in making a point; it is NOT a mathematical proof. It is merely empirical evidence. As a result, this review highlights that for AM and CSAM materials, it is advisable to plot the crack growth rate (da/dN) against both ΔK and Δκ. The observation that, for the AM and CSAM steels examined in this study, the da/dN versus Δκ curves are similar, when coupled with similar observation for a range of other AM materials, supports a prior study that suggested using fracture toughness measurements in conjunction with the flight load spectrum and the operational life requirement to guide the choice of the building process for AM Ti-6Al-4V parts. The observations outlined in this study, when taken together with related findings given in the open literature for AM Ti-6Al-4V, AM Inconel 718, AM Inconel 625, and BSI&WS LPFB-built Scalmalloy®, as well as for a range of CSAM-built pure metals, have implications for the implementation and certification of limited-life AM parts.

A mode sorter separates a set of M orthogonal spatial modes in a shared input channel into M different output channels. Here we present an analytic derivation and experimental validation of a single plane device for sorting spatial modes from a diverse variety of mode families, including Hermite-Gaussian (HG), Laguerre-Gaussian (LG), and Bessel-Gaussian (BG), with almost no cross-talk. This sorting capability is required for a wide range of applications that employ classical or quantum light. We also show that applying this design in order to sort a set of Orbital Angular Momentum (OAM) modes with zero radial index reproduces the well-known Fork grating configuration. Furthermore, by taking the limit of M → ∞, we present an analytical expression for sorting all the modes of a given family. By operating this device in reverse, it can be used to generate arbitrary modes by illuminating it with a Gaussian beam. The power transmission coefficient for this sorter goes as O ( M −1 ), and we provide a mathematical proof that this is optimal for any typical arrangement of the detector positions. We further study the sorter's sensitivity to wavelength and random phase noise.

This paper presents a mathematical explanation of one of the Multi-Attribute Decision-Making (MADM) methods—the Ranking based on Distance and Range (RADAR) method—along with its modified variant, RADAR II. Through mathematical proofs, the influence of each step of the method on the final ranking of alternatives is analyzed. The methods are tested on three numerical examples with varying criterion weights. The robustness of the methods, as well as their fundamental characteristics, is demonstrated. A comparative analysis reveals that although both methods prioritize alternatives based on their stability across all criteria—particularly the most important ones—the RADAR II method is somewhat more rigorous and stringent, whereas the original RADAR method is more flexible and yields more objective results.

Abstract This tutorial reviews the mathematical foundations of single-antenna radio polarimetry with the aim of fostering a conceptual understanding of the relationships between a physical description of signal propagation (gain, delay, reflection, down-conversion, etc.), the corresponding transformations of the electric field vector, and the equivalent operations on the Stokes parameters. The adopted framework is based on the work of Britton and Hamaker and applied to analyze the signal path described by Hamaker et al. with additional corrections for phase convention and reflection. Some objective criteria for selecting a model of the instrumental response are introduced and discussed, along with some practical guidelines that facilitate polarimetric calibration. Further relevant background material and lengthier mathematical proofs are included in the appendix, which introduces the vector, matrix, and tensor notation and concepts of linear algebra used in this work. The appendix also reviews some of the basics of analog and digital signal processing that are relevant to radio astronomy, and discusses some numerical instabilities that arise when modeling observations.

This paper presents mathematical proofs of N.A. Kozyrev's postulates, confirming his theory of time. To visualize the proofs, vector diagrams constructed based on the dual equations derived by the author are used. In addition to Kozyrev's time flow, the author uses another time flow, which he obtained theoretically. Furthermore, the paper utilizes the author's previous research results. Their application made the model of the cause-effect interaction process simple and understandable. An example implementing the cause-effect model is provided at the end of the article.

This paper outlines a plan for designing a corpus-linguistic project from which an analysis of frames specific to mathematical proof texts shall be derived. Previous work has already developed instances of frames for mathematical texts. We have argued that these frames are well suited to model how background knowledge enriches explicitly given information. To this end, a collection of mathematical texts needs to be annotated.We describe the idea behind the corpus annotation, frame semantics and how we adapted frames for mathematical texts in particular by distinguishing between structural proof frames and ontological ones. Ontological frames for instance correspond to proof techniques, while structural frames model domain knowledge like exact definitions of mathematical structures. We describe annotation principles, explain qualifications annotators should have and how such annotations can be evaluated.We explain potential linguistic research questions: The corpus will allow us to study the linguistic means by which frames are introduced and signaled. We plan to generalize beyond previous case-based analyses and to provide a foundation for broader empirical research on mathematical language and practice. We argue that these can be used in deeper semantic parsing and the development of interactive theorem-proving software. We furthermore assume that this perspective on mathematical text will also have implications on the philosophy of mathematics and its practice.

The expansion rate of the Universe has been a debatable discussion among all minds and with the establishment of the homogeneity model and retrieval of CMB data, several equations and metrics have been derived, which compute the expansion rate of the Universe with an assumption of homogeneity and isotropy throughout. However, it is also notable that the scalability of the homogeneity cannot be validated overall on a small scale. Hence, concepts have been used to establish metrics that help in computing the expansion rate of the Universe by assuming heterogeneous frames and large scaling. A thorough review of isotropy and its deduction to homogeneity is done in this further to check on the scalability of the expansion rate of the Universe in the homogeneity scale. Besides, an assumption that if the Copernican assumption is denied for a case and the Universe is considered to be heterogeneous, what would be the expansion rate of the Universe then and if it is valid then what changes do we need to make in the current cosmological model CDM are noted and mentioned. We currently have a semi-proven cosmological model which has proven its existence observationally to a huge extent but needs enough mathematical proof to confirm the observations. We take note of the assumptions and based on them we proceed further in concluding all the research areas that require more work to make this a mathematically stable model completely. The main focus of this paper will stick to the assumptions needed for the FLRW metric and further works on the same will come up.

This research proposes and details a quantum authentication methodology as a promising and essential solution for enhancing data security in credit card transactions, specifically focusing on data leakage protection. Unlike conventional approaches such as blockchain, quantum authentication offers unbroken spoofing protection and does not require extensive data archiving. The practical implementation involves the establishment of secure communication channels using Quantum Key Distribution (QKD) between the Point-of-Sale (POS) terminal and the bank, the dynamic generation of a quantum-enhanced CVV, and the encryption of transaction data with post-quantum algorithms, including Lattice-based Cryptography, designed to resist future quantum attacks. Data Leakage Protection (DLP) systems, incorporating tokenization and encryption, are integral to secure data storage and processing throughout the entire transaction lifecycle. The methodology is underpinned by rigorous mathematical proofs and integrates various quantum authentication methods, such as Key-Controlled Maximally Mixed Quantum State Encryption, Three-Factor Quantum Biometric Authentication, and a Triple Security Mechanism, to provide robust security against unauthorized access and fraudulent activities. While offering significant advancements in security, challenges remain in its widespread adoption, including complexities in implementation efficiency, potential for message collisions, seamless integration with existing systems, and the requirement for specialized hardware. Furthermore, tokenization, though crucial, faces limitations such as technical complexity, interoperability issues, ongoing maintenance, and susceptibility to Electromagnetic Interference (EMI). This research highlights the critical need for continued development to fully leverage quantum technologies in securing financial data.

In communication environments with limited computing resources, securely and efficiently transmitting image data has become a challenging problem. However, most existing image data protection schemes are based on high-dimensional chaotic systems as key generators, which suffer from issues such as high algorithmic complexity and large computational overhead. To address this, this paper presents new designs for a 1D Sine Fractional Chaotic Map (1D-SFCM) as a random sequence generator and provides mathematical proofs related to the boundedness and fixed points of this model. Furthermore, this paper improves the traditional 2D compressive sensing (2DCS) algorithm by using the newly designed 1D-SFCM map to generate a chaotic measurement matrix, which can effectively enhance the quality of image recovery and reconstruction. Moreover, referring to the principle of gene mutation in biogenetics, this paper designs an image encryption algorithm based on DNA base substitution. Finally, the security of the proposed encryption scheme and the quality of image compression and reconstruction are verified through indicators such as key space, information entropy, and Number of Pixel Change Rate (NPCR).

Quantum thermalization describes how closed quantum systems can effectively reach thermal equilibrium, resolving the apparent incongruity between the reversibility of Schrödinger's equation and the second law of thermodynamics. Despite its ubiquity and conceptual significance, the precise conditions that give rise to quantum thermalization are still not well understood. After nearly a century of efforts, we have yet to find a complete mathematical proof that an effective statistical description naturally emerges the underlying quantum dynamics in generic settings. Here, we prove that quantum thermalization must occur in any qubit system with local interactions under three conditions: (i) high effective temperature, (ii) translation invariance, and (iii) no perfect resonances in the energy spectrum. Specifically, we show that a typical, low-complexity pure state drawn from any ensemble with large entropy and well-defined effective temperature becomes locally indistinguishable from a Gibbs state upon unitary evolution. In this setting, our rigorous results prove the widely anticipated notion that statistical physics should be understood as an emergent phenomenon, explicitly derived from the first principles of quantum mechanics.

This study aims to improve students' mathematical reasoning abilities in geometry learning that integrates a realistic approach with an Artificial Intelligence (AI) platform. This research was conducted at the Mathematics Education Study Program, Department of Mathematics, Faculty of Mathematics and Natural Sciences, Medan State University in the odd semester of the 2025/2026 academic year with a design research model by Freudenthal. The subjects of the study were 26 students from the PSPM class of 2024 and 25 students from the MESP class of 2024 taken by purposive sampling. The learning process was carried out in 12 meetings with 4 tests. The results showed that the mathematical reasoning ability of PSPM class students reached an average of 75.2 with a classical completeness of 92%. Meanwhile, the mathematical reasoning ability of MESP class students reached an average of 87.0 with a classical completeness of 100%. Thus, geometry learning based on a realistic approach assisted by AI can improve students' mathematical reasoning abilities and can strengthen the mathematical proof and argumentation abilities of students in the Mathematics Education study program, Medan State University.

We present an exposition of the Chain Bounding Lemma, which is a common generalization of both Zorn’s lemma and the Bourbaki-Witt fixed point theorem. The proofs of these results through the use of Chain Bounding are amongst the simplest ones that we are aware of. As a by-product, we show that for every poset P and function f from the powerset of P into P, there exists a maximal well-ordered chain whose family of initial segments is appropriately closed under f. We also provide an introduction to the process of “computer formalization” of mathematical proofs by using proof assistants. As an illustration, we verify our main results with the Lean proof assistant.

Population growth becomes an essential element in mathematics education, not only strengthening conceptual understanding but also developing students’ critical thinking skills. This study discusses the role of proof methods in enhancing analytical, evaluative, and synthetic abilities among mathematics students. Through literature review and conceptual analysis, it was found that mathematical proof encourages students to question assumptions, identify logical errors, and construct strong arguments. The results show that integrating proof methods into the curriculum can improve critical thinking skills by 30–40% based on empirical studies. Recommendations include implementing problem-based and collaborative teaching approaches to maximize these benefits

Diagrams as joint epistemic actions: A dialogical account of diagrams in mathematical proofs

We introduce a novel class of G 2 continuous splines constructed using an innovative blending method, which guarantees precise interpolation of given control points. These splines are designed to achieve local curvature maxima specifically at these control points and possess compact local support, thereby eliminating the need for global optimization processes. The formulation ensures the splines are free from cusps and self-intersections and, notably, prevents adjacent segments from intersecting—a significant improvement over prior blending-based curve techniques. This framework utilizes quadratic Bézier splines in conjunction with quartic Bézier blending functions. A constructive algorithm is presented that generates these curvature-controlled curves without relying on global optimization. Through parametric adjustments of curvatures, the curve's geometry near control points can be tuned to create features ranging from smooth to sharp, thus broadening the design possibilities. Rigorous mathematical proofs and visual demonstrations validate all claimed properties of the framework.

Abstract This study aims to analyze the global convergence and uniqueness properties of the successive approximations method when applied to Caputo tempered fractional differential equations. We provide rigorous mathematical proofs to establish the convergence of the method, ensuring that the iterative process converges to a unique solution. Furthermore, we investigate the impact of the nonlocal conditions on the uniqueness of the solution obtained through the successive approximations method.

A novel neural network-based nonlinear controller for shipboard rotary cranes against random wave interference.

Non-compliance in randomised controlled trials poses a significant challenge to accurately estimate treatment effects. Traditional methods of analysis, such as intention-to-treat (ITT) and per-protocol (PP) analyses, are known to have limitations. The ITT analysis tends to underestimate the effect of treatment, while PP is prone to selection bias. The Complier Average Causal Effect (CACE) framework has been proposed to address the issue of non-compliance. In this paper, we propose a novel method that assumes the CACE framework based on logistic regression using multiple imputation technique to estimate the unknown compliance status, with a lemma that provides mathematical proof of identification. Its performance is compared with ITT, PP, instrumental variable method and latent class regression method. The performance of these five methods was evaluated via intensive simulations assuming varying compliance rates, sample sizes and effect sizes. Scenarios that account for selection bias are also considered. The simulation results demonstrate that our proposed method has smaller bias and mean squared error, with wider coverage and larger power, even in the presence of significant selection bias. The methods were also compared using data from the JOBS II randomised clinical trial on depression, where the non-compliance rate was 55%. The performance aligned with simulation results and demonstrated the good potential of LMI in estimating treatment effect in RCTs subject to non-compliance.Supplementary InformationThe online version contains supplementary material available at 10.1186/s13063-025-09203-z.