Proof Technique Research Articles

Research on Blockchain Transaction Privacy Protection Methods Based on Deep Learning

To address the challenge of balancing privacy protection with regulatory oversight in blockchain transactions, we propose a regulatable privacy protection scheme for blockchain transactions. Our scheme utilizes probabilistic public-key encryption to obscure the true identities of blockchain transaction participants. By integrating commitment schemes and zero-knowledge proof techniques with deep learning graph neural network technology, it provides privacy protection and regulatory analysis of blockchain transaction data. This approach not only prevents the leakage of sensitive transaction information, but also achieves regulatory capabilities at both macro and micro levels, ensuring the verification of the legality of transactions. By adopting an identity-based encryption system, regulatory bodies can conduct personalized supervision of blockchain transactions without storing users’ actual identities and key data, significantly reducing storage computation and key management burdens. Our scheme is independent of any particular consensus mechanism and can be applied to current blockchain technologies. Simulation experiments and complexity analysis demonstrate the practicality of the scheme.

Leveraging AI and Blockchain for Privacy Preservation and Security in Fog Computing

INTRODUCTION: Cloud computing's offshoot, fog computing, moves crucial data storage, processing, and networking capabilities closer to the people who need them. There are certain advantages, such improved efficiency and lower latency, but there are also some major privacy and security concerns. For these reasons, this article presents a new paradigm for fog computing that makes use of blockchain and Artificial Intelligence (AI).
 OBJECTIVES: The main goal of this research is to create and assess a thorough framework for fog computing that incorporates AI and blockchain technology. With an emphasis on protecting the privacy and integrity of data transactions and streamlining the management of massive amounts of data, this project seeks to improve the security and privacy of Industrial Internet of Things (IIoT) systems that are cloud-based.
 METHODS: Social network analysis methods are utilised in this study. The efficiency and accuracy of data processing in fog computing are guaranteed by the application of artificial intelligence, most especially Support Vector Machine (SVM), due to its resilience in classification and regression tasks. The network's security and reliability are enhanced by incorporating blockchain technology, which creates a decentralised system that is tamper resistant. To make users' data more private, zero-knowledge proof techniques are used to confirm ownership of data without actually disclosing it.
 RESULTS: When applied to fog computing data, the suggested approach achieves a remarkable classification accuracy of 99.8 percent. While the consensus decision-making process of the blockchain guarantees trustworthy and secure operations, the support vector machine (SVM) efficiently handles massive data analyses. Even in delicate situations, the zero-knowledge proof techniques manage to keep data private. When these technologies are integrated into the fog computing ecosystem, the chances of data breaches and illegal access are greatly reduced.
 CONCLUSION: Fog computing, which combines AI with blockchain, offers a powerful answer to the privacy and security issues with cloud centric IIoT systems. Combining SVM with AI makes data processing more efficient, while blockchain's decentralised and immutable properties make it a strong security measure. Additional security for user privacy is provided via zero-knowledge proofs. Improving the privacy and security of fog computing networks has never been easier than with this novel method.

Performative Federated Learning: A Solution to Model-Dependent and Heterogeneous Distribution Shifts

We consider a federated learning (FL) system consisting of multiple clients and a server, where the clients aim to collaboratively learn a common decision model from their distributed data. Unlike the conventional FL framework that assumes the client's data is static, we consider scenarios where the clients' data distributions may be reshaped by the deployed decision model. In this work, we leverage the idea of distribution shift mappings in performative prediction to formalize this model-dependent data distribution shift and propose a performative FL framework. We first introduce necessary and sufficient conditions for the existence of a unique performative stable solution and characterize its distance to the performative optimal solution. Then we propose the performative FedAvg algorithm and show that it converges to the performative stable solution at a rate of O(1/T) under both full and partial participation schemes. In particular, we use novel proof techniques and show how the clients' heterogeneity influences the convergence. Numerical results validate our analysis and provide valuable insights into real-world applications.

Intrinsic universality in automata networks I: Families and simulations

An automata network (AN) is a finite graph where each node holds a state from a finite alphabet and is equipped with a local map defining the evolution of the state of the node depending on its neighbors. They are studied both from the dynamical and the computational complexity point of view. Inspired from well-established notions in the context of cellular automata, we develop a theory of intrinsic simulations and universality for families of automata networks. We establish many consequences of intrinsic universality in terms of complexity of orbits (periods of attractors, transients, etc) as well as hardness of the standard well-studied decision problems for automata networks (short/long term prediction, reachability, etc). In the way, we prove orthogonality results for these problems: the hardness of a single one does not imply hardness of the others, while intrinsic universality implies hardness of all of them. We also compare our notions of universality to intrinsic universality of cellular automata. This paper is the first of a series of three: in the second one, we develop a proof technique to establish intrinsic universality based on an operation of glueing; in the third one we study the effect of update schedules on intrinsic universality for concrete symmetric families of automata networks.

A Spectral Extremal Problem on Non-Bipartite Triangle-Free Graphs

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda(G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $\lambda_1^2(G) + \lambda_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $\lambda (G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $\lambda (G) \le \beta (m)$, where $\beta(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

Problem in Geometry, with Four Solutions, One of Them with Dynamic Research

Problem solving and proofs have always played a major role in mathematics. They are, in fact, the heart and soul of the discipline. The using of a number of different proof techniques for one specific problem can display the beauty, and elegance of mathematics. In this paper, we present one specific, interesting geometry problem, and present four different proofs for it, using methods, including using GeoGebra, that is dynamic geometry software (DGS) application, to make the initial conjecture. Encouraging students to derive results in such various ways will enhance their appreciation of mathematics, give them incentive to derive even more elegant solutions on their own, and provide them the opportunity to come up with some original insights on how proofs work. As a bonus, the problem we used, also let them know Ptolemy's theorem and the fourth theorem of the overlap of triangles, which are not included in the curriculum of some countries. The implication of our findings is that mathematics educators should be encouraged to introduce many such authentic multiple-proof problems into the teaching program. In addition, the effect of such exercises on students' mathematical understanding should be studied.

Images of Gaussian and other stochastic processes under closed, densely-defined, unbounded linear operators

Gaussian Processes (GPs) are widely-used tools in spatial statistics and machine learning and the formulae for the mean function and covariance kernel of a GP [Formula: see text] that is the image of another GP [Formula: see text] under a linear transformation [Formula: see text] acting on the sample paths of [Formula: see text] are well known, almost to the point of being folklore. However, these formulae are often used without rigorous attention to technical details, particularly when [Formula: see text] is an unbounded operator such as a differential operator, which is common in many modern applications. This note provides a self-contained proof of the claimed formulae for the case of a closed, densely-defined operator [Formula: see text] acting on the sample paths of a square-integrable (not necessarily Gaussian) stochastic process. Our proof technique relies upon Hille’s theorem for the Bochner integral of a Banach-valued random variable.

Introducing a Supportive Framework to Address Students’ Misconceptions and Difficulties in the Learning Mathematical Proof Techniques: A Case of Debark University in Ethiopia

This research article is about "Introducing a Supportive Framework to Address Students' Misconceptions and Difficulties in Learning Mathematical proof techniques (MPT): A Case of Debark University”. This study aims to develop, introduce, and implement a supportive framework to overcome students’ misconceptions and difficulties in MPT. The framework, named IR2CP2CE, was developed, introduced, and implemented at Debark University in Ethiopia using various data-gathering instruments such as questionnaires, interviews, classroom observations, and document analysis from students and instructors. The study collected data over four months, including the implementation of a supportive framework using mixed, quasi-experimental, and pragmatism research approaches, designs, and paradigms respectively. The internal reliability of the data-gathering instruments was interpreted using Cronbach’s coefficient, Spearman-Brown, Spearman correlations, Kuder-Richardson 20 and 21, and difficulty and discrimination indices. The results showed that the implementation of the supportive framework led to significant improvements in students’ academic performance in MPT, regardless of factors such as gender, academic year category, and preliminary knowledge and proving skills. This study recommends additional imperatives for practice and future research.

Classification aggregation without unanimity

A classification is a surjective mapping from a set of objects to a set of categories. A classification aggregation function aggregates every vector of classifications into a single one. We show that every citizen sovereign and independent classification aggregation function is essentially a dictatorship. This impossibility implies an earlier result of Maniquet and Mongin (2016), who show that every unanimous and independent classification aggregation function is a dictatorship. The relationship between the two impossibilities is reminiscent to the relationship between Wilson’s (1972) and Arrow’s (1951) impossibilities in preference aggregation. Moreover, while the Maniquet-Mongin impossibility rests on the existence of at least three categories, we propose an alternative proof technique that covers the case of two categories, except when the number of objects is also two. We also identify all independent and unanimous classification aggregation functions for the case of two categories and two objects.

On Construction of Bounded Sets Not Admitting a General Type of Riesz Spectrum

We construct a bound set that does not admit a Riesz spectrum containing a nonempty periodic set for which the period is a rational multiple of a fixed constant. As a consequence, we obtain a bounded set V with an arbitrarily small Lebesgue measure such that for any positive integer N, the set of exponentials with frequencies in any union of cosets of NZ cannot be a frame for the space of square integrable functions over V. These results are based on the proof technique of Olevskii and Ulanovskii from 2008.

The infinite-dimensional Pontryagin maximum principle for optimal control problems of fractional evolution equations with endpoint state constraints

<abstract><p>In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $ \mathsf{X} $-valued left Caputo fractional evolution equation with the analytic semigroup, where $ \mathsf{X} $ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $ \mathsf{X} $. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of $ \mathsf{X}^* $, the dual space of $ \mathsf{X} $. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.</p></abstract>

Generalizing Horn's conditions for preemptive scheduling on identical parallel machines via network flow techniques

AbstractWe study the use of flow‐based algorithmic and proof techniques applied to preemptive scheduling of jobs on parallel identical machines. For the classical problem in which the jobs have individual release dates and must be finished by a common deadline, we present and prove unified necessary and sufficient conditions for the existence of a feasible schedule by examining the structure of minimum cuts in a special network. We then study an enhanced model that allows the presence of additional resources, provided that some jobs at any time of their processing require one unit of a particular resource. We extend our argument developed for the classical case to this enhanced model. The generalized necessary and sufficient conditions for the existence of a feasible schedule are presented and proved using the max‐flow/min‐cut reasoning.

Multiplicity results for generalized quasilinear critical Schrödinger equations in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document}

Multiplicity results are proved for solutions both with positive and negative energy, as well as nonexistence results, of a generalized quasilinear Schrödinger potential free equation in the entire RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {R}}^N$$\\end{document} involving a nonlinearity which combines a power-type term at a critical level with a subcritical term, both with weights. The equation has been derived from models of several physical phenomena such as superfluid film in plasma physics as well as the self-channelling of a high-power ultra-short laser in matter. Proof techniques, also in the symmetric setting, are based on variational tools, including concentration compactness principles, to overcome lack of compactness, and the use of a change of variable in order to deal with a well defined functional.

Proximity and Flatness Bounds for Linear Integer Optimization

This paper deals with linear integer optimization. We develop a technique that can be applied to provide improved upper bounds for two important questions in linear integer optimization. Given an optimal vertex solution for the linear relaxation, how far away is the nearest optimal integer solution (if one exists; proximity bounds)? If a polyhedron contains no integer point, what is the smallest number of integer parallel hyperplanes defined by an integral, nonzero, normal vector that intersect the polyhedron (flatness bounds)? This paper presents a link between these two questions by refining a proof technique that has been recently introduced by the authors. A key technical lemma underlying our technique concerns the areas of certain convex polygons in the plane; if a polygon [Formula: see text] satisfies [Formula: see text], where τ denotes [Formula: see text] counterclockwise rotation and [Formula: see text] denotes the polar of K, then the area of [Formula: see text] is at least three. Funding: J. Paat was supported by the Natural Sciences and Engineering Research Council of Canada [Grant RGPIN-2021-02475]. R. Weismantel was supported by the Einstein Stiftung Berlin.

Maximal Dissent: A State-Dependent Way to Agree in Distributed Convex Optimization

Consider a set of agents collaboratively solving a distributed convex optimization problem asynchronously under stringent communication constraints. When an agent becomes active, it is allowed to communicate with only one of its neighbors. We propose new state-dependent gossip algorithms where the agents with maximal dissent average their estimates. We prove the almost sure convergence of max-dissent subgradient methods using a unified framework applicable to other state-dependent distributed optimization algorithms. Furthermore, our proof technique bypasses the need to establish the information flow between any two agents within a time interval of uniform length by intelligently studying the convergence properties of the Lyapunov function used in our analysis.

Proving exact values for the $2$-limited broadcast domination number on grid graphs

We establish exact values for the $2$-limited broadcast domination number of various grid graphs, in particular $C_m\square C_n$ for $3 \leq m \leq 6$ and all $n\geq m$, $P_m \square C_3$ for all $m \geq 3$, and $P_m \square C_n$ for $4\leq m \leq 5$ and all $n \geq m$. We also produce periodically optimal values for $P_m \square C_4$ and $P_m \square C_6$ for $m \geq 3$, $P_4 \square P_n$ for $n \geq 4$, and $P_5 \square P_n$ for $n \geq 5$. Our method completes an exhaustive case analysis and eliminates cases by combining tools from linear programming with various mathematical proof techniques.

Extensible Proof Systems for Infinite-State Systems

This article revisits soundness and completeness of proof systems for proving that sets of states in infinite-state labeled transition systems satisfy formulas in the modal mu-calculus in order to develop proof techniques that permit the seamless inclusion of new features in this logic. Our approach relies on novel results in lattice theory, which give constructive characterizations of both greatest and least fixpoints of monotonic functions over complete lattices. We show how these results may be used to reason about the sound and complete tableau method for this problem due to Bradfield and Stirling. We also show how the flexibility of our lattice-theoretic basis simplifies reasoning about tableau-based proof strategies for alternative classes of systems. In particular, we extend the modal mu-calculus with timed modalities, and prove that the resulting tableau method is sound and complete for timed transition systems.

Convergence rate in [formula omitted] sense of tamed EM scheme for highly nonlinear neutral multiple-delay stochastic McKean–Vlasov equations

This paper focuses on the numerical scheme of highly nonlinear neutral multiple-delay stochastic McKean-Vlasov equation (NMSMVE) by virtue of the stochastic particle method. First, under general assumptions, the results about propagation of chaos in Lp sense are revealed, where the convergence rate loses a little due to the proof technique. Then the tamed Euler–Maruyama scheme to the corresponding particle system is established and the convergence rate in Lp sense is obtained. Furthermore, combining these two results gives the convergence error in Lp sense between the objective NMSMVE and numerical approximation, which is related to the particle number and step size. Finally, two numerical examples are provided to support the finding.

Multi-layer state evolution under random convolutional design *

Signal recovery under generative neural network priors has emerged as a promising direction in statistical inference and computational imaging. Theoretical analysis of reconstruction algorithms under generative priors is, however, challenging. For generative priors with fully connected layers and Gaussian i.i.d. weights, this was achieved by the multi-layer approximate message (ML-AMP) algorithm via a rigorous state evolution. However, practical generative priors are typically convolutional, allowing for computational benefits and inductive biases, and so the Gaussian i.i.d. weight assumption is very limiting. In this paper, we overcome this limitation and establish the state evolution of ML-AMP for random convolutional layers. We prove in particular that random convolutional layers belong to the same universality class as Gaussian matrices. Our proof technique is of an independent interest as it establishes a mapping between convolutional matrices and spatially coupled sensing matrices used in coding theory.

Constructive proofs for localised radial solutions of semilinear elliptic systems on

Ground state solutions of elliptic problems have been analysed extensively in the theory of partial differential equations, as they represent fundamental spatial patterns in many model equations. While the results for scalar equations, as well as certain specific classes of elliptic systems, are comprehensive, much less is known about these localised solutions in generic systems of nonlinear elliptic equations. In this paper we present a general method to prove constructively the existence of localised radially symmetric solutions of elliptic systems on . Such solutions are essentially described by systems of non-autonomous ordinary differential equations. We study these systems using dynamical systems theory and computer-assisted proof techniques, combining a suitably chosen Lyapunov–Perron operator with a Newton–Kantorovich type theorem. We demonstrate the power of this methodology by proving specific localised radial solutions of the cubic Klein–Gordon equation on , the Swift–Hohenberg equation on , and a three-component FitzHugh–Nagumo system on . These results illustrate that ground state solutions in a wide range of elliptic systems are tractable through constructive proofs.