Proof Technique Research Articles

More on the Fascinating Characterizations of Mulatu’s Numbers

Background The discoveries of Mulatu’s numbers, better known as Mulatu’s sequence, represent revolutionary contributions to the mathematical world. His best-known work is Mulatu’s sequence, in which each new number is the sum of the two preceding numbers. When various operations and manipulations are performed on the numbers in this sequence, beautiful and incredible patterns begin to emerge. This study aimed to identify novel characterizations of Mulatu’s numbers. Methods This study employed a multi-faceted approach to investigate characterizations of Mulatu’s numbers. Mathematical proof techniques such as principle of mathematical induction, proof by contradiction and direct proof were utilized to substantiate findings. GNU Octave (version 9, (1)) software was applied to verify the existence, classify, and conduct computational investigations of findings prior to formal proofs. Results In this study, we provided several characterizations of Mulatu’s numbers. We also investigated the properties and patterns of these fascinating numbers. Moreover, we have also shown that, similar to Fibonacci’s numbers, Mulatu’s numbers also give the so-called golden ratio, which is most applicable in numerical optimization. Furthermore, we formulated a relation among Mulatu’s numbers, Fibonacci numbers, and Lucas numbers. Finally, we provided a generating function for the Mulatu numbers. Conclusions In this study, we uncovered novel characterizations of Mulatu’s numbers and introduced a generating function for them. We investigated relationship between Mulatu’s numbers and the golden ratio. The results discussed offer valuable insights and enhance our understanding of their properties. Furthermore, these findings play a vital role in the boarder context of mathematical sequences, contributing significantly to the field.

Generalized result on the global existence of positive solutions for a parabolic reaction-diffusion model with an m × m diffusion matrix

Abstract The aim of this work is to study the global existence in time of solutions for the tridiagonal system of reaction-diffusion by order m m . Our techniques of proof are based on compact semigroup methods and some L 1 {L}^{1} -estimates. We show that global solutions exist. Our investigation can be applied for a wide class of nonlinear terms of reaction.

Postselection Technique for Optical Quantum Key Distribution with Improved de Finetti Reductions

The postselection technique is an important proof technique for proving the security of quantum key distribution protocols against coherent attacks via the uplift of any security proof against independent identically distributed collective attacks. In this work, we go through multiple steps to rigorously apply the postselection technique to optical quantum key distribution protocols. First, we place the postselection technique on a rigorous mathematical foundation by fixing a technical flaw in the original postselection paper. Second, we extend the applicability of the postselection technique to prepare-and-measure protocols by using a de Finetti reduction with a fixed marginal. Third, we show how the postselection technique can be used for decoy-state protocols by tagging the source. Finally, we extend the applicability of the postselection technique to realistic optical setups by developing a new variant of the flag-state squasher. We also improve existing de Finetti reductions, which reduce the effect of using the postselection technique on the key rate. These improvements can be more generally applied to other quantum information processing tasks. As an example to demonstrate the applicability of our work, we apply our results to the time-bin-encoded three-state protocol. We observe that the postselection technique performs better than all other known proof techniques against coherent attacks. Published by the American Physical Society 2024

Degrees of relations on canonically ordered natural numbers and integers

AbstractWe investigate the degree spectra of computable relations on canonically ordered natural numbers $$(\omega ,<)$$ ( ω , < ) and integers $$(\zeta ,<)$$ ( ζ , < ) . As for $$(\omega ,<)$$ ( ω , < ) , we provide several criteria that fix the degree spectrum of a computable relation to all c.e. or to all $$\Delta _2$$ Δ 2 degrees; this includes the complete characterization of the degree spectra of so-called computable block functions that have only finitely many types of blocks. Compared to Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022), we obtain a more general solution to the problem regarding possible degree spectra on $$(\omega ,<)$$ ( ω , < ) , answering the question whether there are infinitely many such spectra. As for $$(\zeta ,<)$$ ( ζ , < ) , we prove the following dichotomy result: given an arbitrary computable relation R on $$(\zeta ,<)$$ ( ζ , < ) , its degree spectrum is either trivial or it contains all c.e. degrees. This result, and the proof techniques required to solve it, extend the analogous theorem for $$(\omega ,<)$$ ( ω , < ) obtained by Wright (Computability 7:349–365, 2018), and provide initial insight to Wright’s question whether such a dichotomy holds on computable ill-founded linear orders. This article is an extended version of Bazhenov et al. (in: LIPIcs, vol 219, pp 8:1–8:20, 2022).

Public Authentic-Replica Sampling Mechanism in Distributed Storage Environments

With the rapid development of wireless communication and big data analysis technologies, the storage of massive amounts of data relies on third-party trusted storage, such as cloud storage. However, once data are stored on third-party servers, data owners lose physical control over their data, making it challenging to ensure data integrity and security. To address this issue, researchers have proposed integrity auditing mechanisms that allow for the auditing of data integrity on cloud servers without retrieving all the data. To further enhance the availability of data stored on cloud servers, multiple replicas of the original data are stored on the server. However, in existing multi-replica auditing schemes, there is a problem of server fraud, where the server does not actually store the corresponding data replicas. To tackle this issue, this paper presents a formal definition of authentic replicas along with a security model for the authentic-replica sampling mechanism. Based on time-lock puzzles, identity-based encryption (IBE) mechanisms, and succinct proof techniques, we design an authentic replica auditing mechanism. This mechanism ensures the authenticity of replicas and can resist outsourcing attacks and generation attacks. Additionally, our schemes replace the combination of random numbers and replica correspondence tables with Linear Feedback Shift Registers (LFSRs), optimizing the original client-side generation and uploading of replica parameters from being linearly related to the number of replicas to a constant level. Furthermore, our schemes allow for the public recovery of replica parameters, enabling any third party to verify the replicas through these parameters. As a result, the schemes achieve public verifiability and meet the efficiency requirements for authentic-replica sampling in multi-cloud environments. This makes our scheme more suitable for distributed storage environments. The experiments show that our scheme makes the time for generating copy parameters negligible while also greatly optimizing the time required for replica generation. As the amount of replica data increases, the time spent does not grow linearly. Due to the multi-party aggregation design, the verification time is also optimal. Compared to the latest schemes, the verification time is reduced by approximately 30%.

Uncloneable Quantum Advice

The famous no-cloning principle has been shown recently to enable a number of uncloneable cryptographic primitives, including the copy-protection of certain functionalities. Here we address for the first time unkeyed quantum uncloneablity, via the study of a complexity-theoretic tool that enables a computation, but that is natively unkeyed: quantum advice. Remarkably, this is an application of the no-cloning principle in a context where the quantum states of interest are not chosen by a random process. We establish unconditional constructions for promise problems admitting uncloneable quantum advice and, assuming the feasibility of quantum copy-protecting certain functions, for languages with uncloneable advice. Along the way, we note that state complexity classes, introduced by Rosenthal and Yuen (ITCS 2022) — which concern the computational difficulty of synthesizing sequences of quantum states — can be naturally generalized to obtain state cloning complexity classes. We make initial observations on these classes, notably obtaining a result analogous to the existence of undecidable problems. Our proof technique defines and constructs ingenerable sequences of finite bit strings, essentially meaning that they cannot be generated by any uniform circuit family with non-negligible probability. We then prove a generic result showing that the difficulty of accomplishing a computational task on uniformly random inputs implies its difficulty on any fixed, ingenerable sequence. We use this result to derandomize quantum cryptographic games that relate to cloning, and then incorporate a result of Kundu and Tan (arXiv 2022) to obtain uncloneable advice. Applying this two-step process to a monogamy-of-entanglement game yields a promise problem with uncloneable advice, and applying it to the quantum copy-protection of pseudorandom functions with super-logarithmic output lengths yields a language with uncloneable advice.

Geodesics cross any pattern in first-passage percolation without any moment assumption and with possibly infinite passage times

In first-passage percolation, one places nonnegative i.i.d. random variables (T(e)) on the edges of Zd. A geodesic is an optimal path for the passage times T(e). Consider a local property of the time environment. We call it a pattern. We investigate the number of times a geodesic crosses a translate of this pattern. When we assume that the common distribution of the passage times satisfies a suitable moment assumption, it is shown in [Antonin Jacquet. Geodesics in first-passage percolation cross any pattern, arXiv:2204.02021, 2023] that, apart from an event with exponentially small probability, this number is linear in the distance between the extremities of the geodesic. This paper completes this study by showing that this result remains true when we consider distributions with an unbounded support without any moment assumption or distributions with possibly infinite passage times. The techniques of proof differ from the preceding article and rely on a notion of penalized geodesic.

CNN MODEL FOR TRAFFIC SIGN RECOGNITION

The traffic sign recognition framework (TSRS) is an important component of intelligent transportation systems (ITS). Accurately identifying traffic signs can improve driving safety. This research provides a traffic sign recognition approach based on profound learning. It primarily focuses on the location and order of roundabout signs. First, a photograph undergoes pre-processing to highlight important facts. Hough Transform is used to distinguish and find regions. Finally, the unique street traffic signs are analyzed for further understanding. This paper proposes a photo handling-based traffic sign discovery and distinguishing proof technique that is combined with convolutional brain organization (CNN) to sort traffic signs. CNN is useful for recognizing many PC vision tasks due to its high recognition rate. TensorFlow is used to implement CNN. We have a recognition accuracy of over 98.2% for the roundabout image in the German informative collections. KEYWORDS——traffic sign recognition, traffic sign detection, deep learning, convolutional neural network

Douglas-Rudin approximation theorem for operator-valued functions on the unit ball of [formula omitted]

Douglas and Rudin proved that any unimodular function on the unit circle T can be uniformly approximated by quotients of inner functions. We extend this result to the operator-valued unimodular functions defined on the boundary of the open unit ball of Cd. Our proof technique combines the spectral theorem for unitary operators with the Douglas-Rudin theorem in the scalar case to bootstrap the result to the operator-valued case. This yields a new proof and a significant generalization of Barclay's result (2009) [4] on the approximation of matrix-valued unimodular functions on T.

Enhancing Unmanned Aerial Vehicle Security: A Zero-Knowledge Proof Approach with Zero-Knowledge Succinct Non-Interactive Arguments of Knowledge for Authentication and Location Proof.

UAVs are increasingly being used in various domains, from personal and commercial applications to military operations. Ensuring the security and trustworthiness of UAV communications is crucial, and blockchain technology has been explored as a solution. However, privacy remains a challenge, especially in public blockchains. In this work, we propose a novel approach utilizing zero-knowledge proof techniques, specifically zk-SNARKs, which are non-interactive cryptographic proofs. This approach allows UAVs to prove their authenticity or location without disclosing sensitive information. We generated zk-SNARK proofs using the Zokrates tool on a Raspberry Pi, simulating a drone environment, and analyzed power consumption and CPU utilization. The results are promising, especially in the case of larger drones with higher battery capacities. Ethereum was chosen as the public blockchain platform, with smart contracts developed in Solidity and tested on the Sepolia testnet using Remix IDE. This novel proposed approach paves the way for a new path of research in the UAV area.

Absence of Local Conserved Quantity in the Heisenberg Model with Next-Nearest-Neighbor Interaction

We rigorously prove that the S=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S=1/2$$\\end{document} anisotropic Heisenberg chain (XYZ chain) with next-nearest-neighbor interaction, which is anticipated to be non-integrable, is indeed non-integrable in the sense that this system has no nontrivial local conserved quantity. Our result covers some important models including the Majumdar–Ghosh model, the Shastry–Sutherland model, and many other zigzag spin chains as special cases. These models are shown to be non-integrable while they have some solvable energy eigenstates. In addition to this result, we provide a pedagogical review of the proof of non-integrability of the S=1/2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$S=1/2$$\\end{document} XYZ chain with Z magnetic field, whose proof technique is employed in our result.

VERICONDOR: End-to-End Verifiable Condorcet Voting with support for Strict Preference and Indifference

Condorcet voting is widely regarded as one of the most important voting systems in social choice theory. However, it has seen little adoption in practice, due to complex tallying and the need to break ties when there is a Condorcet cycle. Several online Condorcet voting systems have been developed to perform digital tallying and tie-breaking procedures, but they require voters to completely trust the server. Additionally, many end-to-end (E2E) verifiable e-voting systems require trustworthy authorities to perform complex decryption and tallying operations. We propose VERICONDOR, the first E2E verifiable Condorcet e-voting system without tallying authorities. VERICONDOR allows a voter to fully verify the tallying integrity by themselves while providing strong protection of ballot secrecy. We present novel zero-knowledge proof techniques to prove the well-formedness of an encrypted ballot with exceptional efficiency. VERICONDOR supports ranking candidates with strict preference, as well as indifference. The computational cost is exceptionally efficient for strict preferences at \(\mathcal{O}(n^{2})\) per ballot for \(n\) candidates, whilst remaining practical for indifferences at \(\mathcal{O}(n^{3})\) . In the case of ties, we show how to apply known Condorcet methods to break them in a publicly verifiable manner. Finally, we present a proof of concept implementation and evaluate its performance.

A linear algorithm for radio k-coloring of powers of paths having small diameters

The radio k-chromatic number rck(G) of a graph G is the minimum integer λ such that there exists a function ϕ:V(G)→{0,1,⋯,λ} satisfying |ϕ(u)−ϕ(v)|≥k+1−d(u,v), where d(u,v) denotes the distance between u and v. A considerable amount of attention has been given to find the exact values or providing polynomial time algorithms to determine rck(G) for several basic graph families such as paths, cycles, trees, and powers of paths, usually for some specific values of k. In this article, we find the exact values of rck(G) where G is a power of a path with diameter strictly less than k. Our proof readily provides a linear time algorithm for assigning a radio k-coloring of G. Furthermore, our proof technique is a potential tool for solving the same problem for other classes of graphs having “small” diameters.

On time‐fractional partial differential equations of time‐dependent piecewise constant order

This contribution considers the time‐fractional subdiffusion with a time‐dependent variable‐order fractional operator of order . It is assumed that is a piecewise constant function with a finite number of jumps. A proof technique based on the Fourier method and results from constant‐order fractional subdiffusion equations has been designed. This novel approach results in the well‐posedness of the problem.

Empirical optimal transport under estimated costs: Distributional limits and statistical applications

Optimal transport (OT) based data analysis is often faced with the issue that the underlying cost function is (partially) unknown. This is addressed in this paper with the derivation of distributional limits for the empirical OT value when the cost function and the measures are estimated from data. For statistical inference purposes, but also from the viewpoint of a stability analysis, understanding the fluctuation of such quantities is paramount. Our results find direct application in the problem of goodness-of-fit testing for group families, in machine learning applications where invariant transport costs arise, in the problem of estimating the distance between mixtures of distributions, and for the analysis of empirical sliced OT quantities.The established distributional limits assume either weak convergence of the cost process in uniform norm or that the cost is determined by an optimization problem of the OT value over a fixed parameter space. For the first setting we rely on careful lower and upper bounds for the OT value in terms of the measures and the cost in conjunction with a Skorokhod representation. The second setting is based on a functional delta method for the OT value process over the parameter space. The proof techniques might be of independent interest.

Intrinsic universality in automata networks II: Glueing and gadgets

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. This paper is the second of a series about intrinsic universality, i.e. the ability for a family of AN to simulate arbitrary AN. We develop a proof technique to establish intrinsic simulation and universality results which is suitable to deal with families of symmetric networks where connections are non-oriented. It is based on an operation of glueing of networks, which allows to produce complex orbits in large networks from compatible pseudo-orbits in small networks. As an illustration, we give a short proof that the family of networks where each node obeys the rule of the ‘game of life’ cellular automaton is strongly universal. In the third paper of the series, we heavily rely on this proof technique to show intrinsic universality results of various families with particular update schedules.

A verified durable transactional mutex lock for persistent x86-TSO

AbstractThe advent of non-volatile memory technologies has spurred intensive research interest in correctness and programmability. This paper addresses both by developing and verifying a durable (aka persistent) transactional memory (TM) algorithm, $$\text {dTML}_{\text {Px86}}$$ dTML Px86 . Correctness of $$\text {dTML}_{\text {Px86}}$$ dTML Px86 is judged in terms of durable opacity, which ensures both failure atomicity (ensuring memory consistency after a crash) and opacity (ensuring thread safety). We assume a realistic execution model, Px86, which represents Intel’s persistent memory model and extends the Total Store Order memory model with instructions that control persistency. Our TM algorithm, $$\text {dTML}_{\text {Px86}}$$ dTML Px86 , is an adaptation of an existing software transactional mutex lock, but with additional synchronisation mechanisms to cope with Px86. Our correctness proof is operational and comprises two distinct types of proofs: (1) proofs of invariants of $$\text {dTML}_{\text {Px86}}$$ dTML Px86 and (2) a proof of refinement against an operational specification that guarantees durable opacity. To achieve (1), we build on recent Owicki–Gries logics for Px86, and for (2) we use a simulation-based proof technique, which, as far as we are aware, is the first application of simulation-based proofs for Px86 programs. Our entire development has been mechanised in the Isabelle/HOL proof assistant.

Exploring Students’ Misconceptions in Learning Mathematical Proof Techniques at Debark University in Ethiopia

This article focused on “Exploring students’ misconceptions in learning mathematical proof techniques (MPT) at Debark University in Ethiopia." This research aimed to identify students’ misconceptions in learning MPT and rank MPT based on the degree of students’ misconceptions in learning MPT. Data were collected through assignments, follow-up, and structured interviews. Purposive sampling was used to select samples for follow-up interviews, whereas simple random sampling was used to select samples for structured interviews and assignments. The study collected data on the basis of mixed, case study, and pragmatic research approaches, designs, and paradigms from students. The results showed that the identified misconceptions of students in learning MPT were starting the proof with an inappropriate statement, using an ineffective MPT for their proof, providing incorrect symbolic representation, providing unacceptable reasons for each proof’s step, reaching the conclusion without showing necessary steps clearly and neatly, showing non-sequential steps, incorrectly using technical aspects of mathematics, using the premise and conclusion parts interchangeably, incorrect perception regarding the technical concepts of proof, and misusing the pattern in the proof of a certain statement for the proof of another statement.

Local certification of graph decompositions and applications to minor-free classes

Local certification consists in assigning labels to the vertices of a network to certify that some given property is satisfied, in such a way that the labels can be checked locally. In the last few years, certification of graph classes received considerable attention. The goal is to certify that a graph G belongs to a given graph class G. Such certifications with labels of size O(log⁡n) (where n is the size of the network) exist for trees, planar graphs and graphs embedded on surfaces. Feuilloley et al. ask if this can be extended to any class of graphs defined by a finite set of forbidden minors.In this work, we develop new decomposition tools for graph certification, and apply them to show that for every small enough minor H, H-minor-free graphs can indeed be certified with labels of size O(log⁡n). We also show matching lower bounds using a new proof technique.

Spatial invasion of cooperative parasites

In this paper we study invasion probabilities and invasion times of cooperative parasites spreading in spatially structured host populations. The spatial structure of the host population is given by a random geometric graph on [0,1]n, n∈N, with a Poisson(N)-distributed number of vertices and in which vertices are connected over an edge when they have a distance of at most rN with rN of order N(β−1)/n for some 0<β<1. At a host infection many parasites are generated and parasites move along edges to neighbouring hosts. We assume that parasites have to cooperate to infect hosts, in the sense that at least two parasites need to attack a host simultaneously. We find lower and upper bounds on the invasion probability of the parasites in terms of survival probabilities of branching processes with cooperation. Furthermore, we characterize the asymptotic invasion time.An important ingredient of the proofs is a comparison with infection dynamics of cooperative parasites in host populations structured according to a complete graph, i.e. in well-mixed host populations. For these infection processes we can show that invasion probabilities are asymptotically equal to survival probabilities of branching processes with cooperation. Furthermore, we build on proof techniques developed in Brouard and Pokalyuk (2022), where an analogous invasion process has been studied for host populations structured according to a configuration model.We substantiate our results with simulations.