Detailed Proof Research Articles

Collective attack free controlled quantum key agreement without quantum memory

Abstract Here we present a new protocol for controlled quantum key agreement and another protocol for key agreement with a specific focus on the security analysis. Specifically, detailed security proof is provided against impersonated fraudulent attack and collective attacks and it is established that the proposed protocols are not only secure, but they also satisfy other desired properties of such schemes (i.e., fairness and correctness). Further, the proposed schemes are critically compared with a set of schemes for quantum key agreement and an existing scheme for controlled quantum key agreement (Tang et al.'s protocol) in terms of efficiency and the required quantum resources. Especially, it is observed that in contrast to the existing schemes, the present scheme does not require quantum memory. In addition, the protocol for controlled quantum key agreement proposed here is found to require quantum resources (Bell state and single photon state) that are easier to produce and maintain compared to the quantum resources (GHZ states) required by the only known existing protocol for the same purpose, i.e., Tang et al.'s protocol.

Hamiltonian structure of single-helicity, incompressible magnetohydrodynamics, and application to magnetorotational instability

A four-field reduced model of single helicity, incompressible magnetohydrodynamic is derived in cylindrical geometry. An appropriate set of noncanonical variables is found, and the Hamiltonian, the Lie–Poisson bracket, and the Casimir invariants are clarified. Detailed proofs of properties of the Lie–Poisson bracket, (i) antisymmetry, (ii) Leibniz rule, and (iii) Jacobi identity are given. Two applications are presented: the first is that the local dispersion relation of axisymmetric magnetohydrodynamics (MRI) is properly reproduced and the second is that linear stability analyses of MRI including negative-energy modes were successfully performed.

Unveiling New Dimensions in Soft Disemigraph Theory: Lexicographic and Restricted Lexicographic Products

Soft set theory provides a systematic approach for handling imprecision and uncertainty by categorizing elements of a set based on specific parameters. In semigraph theory, soft semigraphs utilize this approach, offering a parameterized perspective that has significantly advanced the field through effective parameter management. Building on this foundation, disemigraphs extend semigraphs by incorporating directional relationships among vertices, making them ideal for modeling scenarios where the sequence and direction of connections are crucial. In this paper, we introduce and explore the lexicographic product, and restricted lexicographic product of soft disemigraphs. We provide formal definitions, illustrative examples, and detailed proofs of several theorems to investigate the properties of these product operations.

Exploring the 2-Part of Class Groups in Quadratic Fields: Perspectives on the Cohen–Lenstra Conjectures

Cohen and Lenstra introduced conjectures concerning the distribution of class numbers in quadratic fields, though many of these conjectures remain unproven. This paper investigates the 2-part of class groups in imaginary quadratic fields and examines their alignment with the Cohen–Lenstra heuristics. We provide detailed proofs of key theorems related to ideal decompositions and modular homomorphisms, and we explore the distribution of class groups of imaginary quadratic fields. Our analysis includes constructing imaginary quadratic fields with prescribed 2-class groups and discussing the implications of these findings on the Cohen–Lenstra conjecture.

Enhancing programevaluation and review technique (PERT) for construction project scheduling with Bayesian updating and appropriate probability distributions

The Program Evaluation and Review Technique (PERT) is a popular scheduling technique that takes advantage of the Beta distribution to present uncertainty in activity durations. This study presents an advanced PERT method with an improved Bayesian updating and improved assumed prior distributions, which better represent real-world projects. The method is backed with detailed mathematical proofs and derivations for a solid theoretical foundation. A numerical case study involving a 30-floor building construction project is used to compare the performance of traditional PERT, the Beta-improved Bayesian PERT, and the Log-Normal Bayesian PERT methods. In the example, the activities considered are Formwork, Rebar and Construction, Masonry, Mechanical-Electrical-Plumbing (MEP), and Finishing, which are the main activities in a construction project. The results show that the Beta-improved and the Log-Normal distributions are constructed successfully in the models with converging variance – an observation that delineates the uncertainty reduced along a real project’s course. With enhanced functions, the PERT method can be utilized to support project decision-makers in scheduling and managing complex projects in reality.

Proof of chiral symmetry breaking from anomaly matching in QCD-like theories

We critically reconsider the argument based on ’t Hooft anomaly matching that aims at proving chiral symmetry breaking in confining four-dimensional quantum chromodynamics-like theories with Nc>2 colors and Nf flavors. The main line of reasoning relies on a property of the solutions of the anomaly matching and persistent mass equations called Nf-independence. In previous works, the validity of Nf-independence was assumed based on qualitative arguments, but it was never proven rigorously. We provide a detailed proof and clarify under which (dynamical) conditions it holds. Our results are valid for a generic spectrum of massless composite fermions including baryons and exotics. Published by the American Physical Society 2024

Set-Theoretical Solutions for the Yang–Baxter Equation in GE-Algebras: Applications to Quantum Spin Systems

This manuscript presents set-theoretical solutions to the Yang–Baxter equation within the framework of GE-algebras by constructing mappings that satisfy the braid condition and exploring the algebraic properties of GE-algebras. Detailed proofs and the use of left and right translation operators are provided to analyze these algebraic interactions, while an algorithm is introduced to automate the verification process, facilitating broader applications in quantum mechanics and mathematical physics. Additionally, the Yang–Baxter equation is applied to spin transformations in quantum mechanical spin-12 systems, with transformations like rotations and reflections modeled using GE-algebras. A Cayley table is used to represent the algebraic structure of these transformations, and the proposed algorithm ensures that these solutions are consistent with the Yang–Baxter equation, offering new insights into the role of GE-algebras in quantum spin systems.

Novel predefined-time stability theory and its application in sliding mode control of synchronizing chaotic systems.

Aiming at predefined-time synchronization for chaotic systems, a new predefined-time sliding mode control method is proposed. First, based on the definition of predefined-time stability, a novel predefined-time inequality is proposed, along with a detailed mathematical proof. This inequality differs from existing Lyapunov inequalities and offers greater flexibility. Second, a new sliding mode surface and sliding mode controller are proposed based on this inequality. Since the sliding mode controller introduced in this paper is tunable, the actual convergence time can be adjusted freely within the predefined time. Finally, two sets of numerical simulations demonstrate that the proposed method offers advantages in terms of short synchronization time and high regulatory performance compared to traditional predefined-time sliding mode control, finite-time sliding mode control, and fixed-time sliding mode control.

Main Problems in Constructing Quantum Theory Based on Finite Mathematics

As shown in our publications, quantum theory based on a finite ring of characteristic p (FQT) is more general than standard quantum theory (SQT) because the latter is a degenerate case of the former in the formal limit p→∞. One of the main differences between SQT and FQT is the following. In SQT, elementary objects are described by irreducible representations (IRs) of a symmetry algebra in which energies are either only positive or only negative and there are no IRs where there are states with different signs of energy. In the first case, objects are called particles, and in the second antiparticles. As a consequence, in SQT it is possible to introduce conserved quantum numbers (electric charge, baryon number, etc.) so that particles and antiparticles differ in the signs of these numbers. However, in FQT, all IRs necessarily contain states with both signs of energy. The symmetry in FQT is higher than the symmetry in SQT because one IR in FQT splits into two IRs in SQT with positive and negative energies at p→∞. Consequently, most fundamental quantum theory will not contain the concepts of particle–antiparticle and additive quantum numbers. These concepts are only good approximations at present since at this stage of the universe the value p is very large but it was not so large at earlier stages. The above properties of IRs in SQT and FQT have been discussed in our publications with detailed technical proofs. The purpose of this paper is to consider models where these properties can be derived in a much simpler way.

Based proteomics analyses reveal response mechanisms of Apis mellifera (Hymenoptera: Apidae) against the heat stress.

Heat stress can significantly affect the survival, metabolism, and reproduction of honeybees. It is important to understand the proteomic changes of honeybees under heat stress to understand the molecular mechanism behind heat resistance. However, the proteomic changes of honeybees under heat stress are poorly understood. We analyzed the proteomic changes of Apis mellifera Ligustica (Hymenoptera: Apidae) under heat stress using mass spectrometry-based proteomics with TMT (Tandem mass tags) stable isotope labeling. A total of 3,799 proteins were identified, 85 of which differentially abundance between experimental groups. The most significant categories affected by heat stress were associated with transcription and translation processes, metabolism, and stress-resistant pathways. We found that heat stress altered the protein profiles in A. mellifera, with momentous resist proteins being upregulated in heat groups. These results show a proof of molecular details that A. mellifera can respond to heat stress by increasing resist proteins. Our findings add research basis for studying the molecular mechanisms of honeybees' resistance to heat stress. The differentially expressed proteins identified in this study can be used as biomarkers of heat stress in bees, and provide a foundation for future research on honeybees under heat stress. Our in-depth proteomic analysis provides new insights into how bees cope with heat stress.

On Lie Homomorphisms of Complex IntuitionisticFuzzy Lie Algebras

In this paper, we investigate the properties of complex intuitionistic fuzzy Lie subalgebras and ideals under Lie algebra homomorphisms, focusing on both images and inverse images. We provide detailed proofs for several new results concerning the preservation of complex intuitionistic fuzzy structures through homomorphisms, and we introduce additional homomorphism-related properties for these structures. This work extends known results to the context of complex intuitionistic fuzzy Lie algebras, contributing new insights into their behavior under algebraic mappings.

Integrated cooperative guidance and control of strapdown missiles

This paper develops an innovative integrated cooperative guidance and control (ICGC) law for multiple strapdown missiles, and explicitly considers both the attitude dynamics and field-of-view (FOV) limitations of the seekers. First, a model is rigorously established using strict state feedback to incorporate attitude dynamics, which employs the body-line-of-sight (BLOS) angle to characterize the FOV constraints of strapdown seekers. We propose a nonlinear observer to precisely estimate aerodynamic disturbances, target maneuvers, and unmodeled dynamics. Using the BLOS angle as a critical variable for multi-missile cooperation, we derive a novel ICGC law based on the barrier lyapunov function and the dynamic surface control method. This law enables strategic cooperative geometries among multiple missiles attacking a single maneuvering target, with detailed proof of stability provided. Extensive numerical simulations demonstrate that the proposed ICGC law significantly enhances cooperative engagement capabilities, ensuring effective target attacks while adhering to the FOV constraints. This research substantially advances missile guidance and control systems, offering a robust framework for cooperatively engaging maneuvering targets.

A Unified Framework for Covariate Adjustment Under Stratified Randomisation

ABSTRACTRandomisation, as a key technique in clinical trials, can eliminate sources of bias and produce comparable treatment groups. In randomised experiments, the treatment effect is a parameter of general interest. Researchers have explored the validity of using linear models to estimate the treatment effect and perform covariate adjustment and thus improve the estimation efficiency. However, the relationship between covariates and outcomes is not necessarily linear, and is often intricate. Advances in statistical theory and related computer technology allow us to use nonparametric and machine learning methods to better estimate the relationship between covariates and outcomes and thus obtain further efficiency gains. However, theoretical studies on how to draw valid inferences when using nonparametric and machine learning methods under stratified randomisation are yet to be conducted. In this paper, we discuss a unified framework for covariate adjustment and corresponding statistical inference under stratified randomisation and present a detailed proof of the validity of using local linear kernel‐weighted least squares regression for covariate adjustment in treatment effect estimators as a special case. In the case of high‐dimensional data, we additionally propose an algorithm for statistical inference using machine learning methods under stratified randomisation, which makes use of sample splitting to alleviate the requirements on the asymptotic properties of machine learning methods. Finally, we compare the performances of treatment effect estimators using different machine learning methods by considering various data generation scenarios, to guide practical research.

(q, γ)-Bernstein Basis Functions on the Interval [a ; b

In this paper, we present a novel set of (q, γ)-Bernstein basis functions that are parameterized by q and and defined over an interval. We give detailed proofs for several key properties of these functions, including the partition of unity, recurrence relations, degree elevation, and the Marsden identity. Additionally, we introduce and validate the (q, γ)-De Casteljau algorithm, providing comprehensive examples to illustrate its implementation. These results are analyzed to highlight the theoretical and practical implications of these (q, γ)-Bernstein basis functions in various fields, such as polynomial approximation, numerical methods, and Computer Aided Geometric Design (CAGD). Furthermore, we discuss potential extensions and applications of these functions, considering their impact on future research and developments in the domain. By exploring these aspects, we aim to offer a robust framework for understanding and utilizing (q, γ)-Bernstein basis functions.

Combinatorics of Double Grothendieck Polynomials

We give a thorough combinatorial treatment of double Grothendieck polynomials that is accessible to anyone with a basic knowledge of algebra and combinatorics. The text includes many new combinatorial models for these and related polynomials and provides detailed combinatorial proofs. We generalize the Giambelli formula for double Schubert polynomials to a $k$-theoretic version and our proof specializes to a new combinatorial proof of the former.

On long-time asymptotic behavior and Painlevé asymptotic to the matrix Hirota equation

Abstract The nonlinear descent method is extended to study the long-time asymptotic behavior of the matrix Hirota equation with $4\times 4$ Lax pair in Schwartz space. The implementation of spectral analysis successfully transforms the Cauchy problem of the matrix Hirota equation into the corresponding high-order Riemann–Hilbert (RH) with $4\times 4$ jump matrix, and further analyses the established oscillation RH problem to study the asymptotic behavior of the solution in the space-time plane. Interestingly, the space-time plane $\{(x,t)|-\infty <x<+\infty , t>0\}$ can be divided into three different asymptotic regions based on the phase function and $\xi =x/t$. The first one is the oscillatory region $\xi <\frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated applying the Weber equation with an error of $\mathcal{O}(t^{-1}\log t)$. The second region is the Painlevé region $\xi \approx \frac{\alpha ^{2}}{3\beta }$, whose leading-term can be approximated by the coupled Painlevé II equation, which is related to a $4\times 4$ matrix RH problem with an error of $\mathcal{O}(t^{-\frac{2}{3}})$. The last region is the fast decay region $\xi>\frac{\alpha ^{2}}{3\beta }$, which solution is rapidly decreasing as $t\rightarrow \infty $. Our results provide a detailed proof for the asymptotic analysis for the solution of the matrix Hirota equation on the complete space-time plane.

Reverse engineering in medical application: literature review, proof of concept and future perspectives

Reverse engineering, a process of extracting information or knowledge from existing objects or systems, has gained significant attention in various fields, including medicine. This article presents a comprehensive literature review and a proof of concept on the application of reverse engineering in the medical field. The review particularly focuses on the reverse engineering process, available technologies, and their specific relevance to the medical domain. Various imaging techniques, such as computed tomography and magnetic resonance imaging, are discussed in respect of their integration with reverse engineering methodologies. Furthermore, the article explores the wide range of medical applications facilitated by reverse engineering, including prosthetics, implants, tissue engineering, and surgical planning. The potential of reverse engineering to enhance personalized medicine and patient-specific treatments is highlighted. A detailed proof of concept focusing on femur reconstruction is a significant component of the article. The proof of concept showcases the practical implementation of reverse engineering techniques to assist in designing and manufacturing precise custom-made implants and bone reconstruction. It emphasizes the integration of patient-specific anatomical data obtained through imaging technologies and the subsequent utilization of reverse engineering processes for anatomical reconstruction (solid modeling). Overall, this article provides an extensive overview of reverse engineering in medical applications, incorporating a literature review and a case study. The findings highlight reverse engineering’s potential to advance medical practices, improve patient outcomes, and foster personalized treatments. The review emphasizes the reverse engineering process, available technologies, and their specific relevance to the medical field, as well as their potential and effectiveness in advancing medical practices.

Fast algebraic multigrid for block-structured dense systems arising from nonlocal diffusion problems

Algebraic multigrid (AMG) is one of the most efficient iterative methods for solving large structured systems of equations. However, how to build/check restriction and prolongation operators in practical AMG methods for nonsymmetric structured systems is still an interesting open question in its full generality. The present paper deals with the block-structured dense and Toeplitz-like-plus-cross systems, including nonsymmetric indefinite and symmetric positive definite (SPD) ones, arising from nonlocal diffusion problems. The simple (traditional) restriction operator and prolongation operator are employed in order to handle such block-structured dense and Toeplitz-like-plus-cross systems, which are convenient and efficient when employing a fast AMG. We provide a detailed proof of the two-grid convergence of the method for the considered SPD structures. The numerical experiments are performed in order to verify the convergence with a computational cost of only O(NlogN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathscr {O}(N \ ext{ log } N)$$\\end{document} arithmetic operations, by exploiting the fast Fourier transform, where N is the number of the grid points. To the best of our knowledge, this is the first contribution regarding Toeplitz-like-plus-cross linear systems solved by means of a fast AMG.

The atmospheric Ekman spiral for piecewise-uniform eddy viscosity

We investigate the boundary-value problem of atmospheric Ekman flows with piecewise-uniform eddy viscosity. In addition we present a method for finding more general solutions by considering eddy viscosity as an arbitrary step-function. We discuss the existence and uniqueness of the solutions obtained through this method, providing detailed proofs for cases with one and two ‘jumps’ in eddy viscosity. For scenarios with more ‘jumps,’ we establish results inductively. Furthermore, we examine the angle between the bottom surface of the Ekman layer and geostrophic winds by extremizing variables such as the eddy viscosity and its point of change. These calculations reveal how the angle can differ from 45°, demonstrating that the extreme values of 0° and 90° are achievable, indicating the potential range of the deflection angle.

Quick Introduction into the General Framework of Portfolio Theory

This survey offers a succinct overview of the General Framework of Portfolio Theory (GFPT), consolidating Markowitz portfolio theory, the growth optimal portfolio theory, and the theory of risk measures. Central to this framework is the use of convex analysis and duality, reflecting the concavity of reward functions and the convexity of risk measures due to diversification effects. Furthermore, practical considerations, such as managing multiple risks in bank balance sheets, have expanded the theory to encompass vector risk analysis. The goal of this survey is to provide readers with a concise tour of the GFPT’s key concepts and practical applications without delving into excessive technicalities. Instead, it directs interested readers to the comprehensive monograph of Maier-Paape, Júdice, Platen, and Zhu (2023) for detailed proofs and further exploration.