On Preserving the Computational Content of Mathematical Proofs: Toy Examples for a Formalising Strategy

2021 ◽  
pp. 285-296
Author(s):  
Angeliki Koutsoukou-Argyraki
Keyword(s):  
Mathematical Proofs

A Novel Simplified AES Algorithm for Lightweight Real-Time Applications: Testing and Discussion

2020 ◽  
Vol 13 (3) ◽  
pp. 435-445 ◽  
Author(s):  
Malik Qasaimeh ◽  
Raad S. Al-Qassas ◽  
Fida Mohammad ◽  
Shadi Aljawarneh
Keyword(s):  
Processing Time ◽  
Block Size ◽  
Statistical Test ◽  
Mathematical Proofs ◽  
The Past ◽  
Aes Algorithm ◽  
Cryptographic Algorithm ◽  
Cryptographic Algorithms ◽  
Compare And Contrast ◽  
Constrained Devices

Background: Lightweight cryptographic algorithms have been the focus of many researchers in the past few years. This has been inspired by the potential developments of lightweight constrained devices and their applications. These algorithms are intended to overcome the limitations of traditional cryptographic algorithms in terms of exaction time, complex computation and energy requirements. Methods: This paper proposes LAES, a lightweight and simplified cryptographic algorithm for constricted environments. It operates on GF(24), with a block size of 64 bits and a key size of 80-bit. While this simplified AES algorithm is impressive in terms of processing time and randomness levels. The fundamental architecture of LAES is expounded using mathematical proofs to compare and contrast it with a variant lightweight algorithm, PRESENT, in terms of efficiency and randomness level. Results: Three metrics were used for evaluating LAES according to the NIST cryptographic applications statistical test suite. The testing indicated competitive processing time and randomness level of LAES compared to PRESENT. Conclusion: The study demonstrates that LAES achieves comparable results to PRESENT in terms of randomness levels and generally outperform PRESENT in terms of processing time.

scholarly journals Mathematics in the Digital Age: The Case of Simulation-Based Proofs

2021 ◽  
Author(s):  
Moritz Lucius Sümmermann ◽  
Daniel Sommerhoff ◽  
Benjamin Rott
Keyword(s):  
Dynamic Geometry ◽  
User Interaction ◽  
Alternative Form ◽  
Sociomathematical Norms ◽  
Mathematical Proofs ◽  
Mathematical Simulations ◽  
Simulation Based ◽  
Actual Use ◽  
Theory Yield ◽  
Functions Of Proof

AbstractDigital transformation has made possible the implementation of environments in which mathematics can be experienced in interplay with the computer. Examples are dynamic geometry environments or interactive computational environments, for example GeoGebra or Jupyter Notebook, respectively. We argue that a new possibility to construct and experience proofs arises alongside this development, as it enables the construction of environments capable of not only showing predefined animations, but actually allowing user interaction with mathematical objects and in this way supporting the construction of proofs. We precisely define such environments and call them “mathematical simulations.” Following a theoretical dissection of possible user interaction with these mathematical simulations, we categorize them in relation to other environments supporting the construction of mathematical proofs along the dimensions of “interactivity” and “formality.” Furthermore, we give an analysis of the functions of proofs that can be satisfied by simulation-based proofs. Finally, we provide examples of simulation-based proofs in Ariadne, a mathematical simulation for topology. The results of the analysis show that simulation-based proofs can in theory yield most functions of traditional symbolic proofs, showing promise for the consideration of simulation-based proofs as an alternative form of proof, as well as their use in this regard in education as well as in research. While a theoretical analysis can provide arguments for the possible functions of proof, they can fulfil their actual use and, in particular, their acceptance is of course subject to the sociomathematical norms of the respective communities and will be decided in the future.

scholarly journals The Multivariate Theory of Functional Connections: Theory, Proofs, and Application in Partial Differential Equations

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1303 ◽  
Author(s):  
Carl Leake ◽  
Hunter Johnston ◽  
Daniele Mortari
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
State Of The Art ◽  
Linear Constraints ◽  
Prior Work ◽  
Partial Differential ◽  
Seminal Paper ◽  
Functional Connections ◽  
Mathematical Proofs ◽  
Multivariate Theory

This article presents a reformulation of the Theory of Functional Connections: a general methodology for functional interpolation that can embed a set of user-specified linear constraints. The reformulation presented in this paper exploits the underlying functional structure presented in the seminal paper on the Theory of Functional Connections to ease the derivation of these interpolating functionals—called constrained expressions—and provides rigorous terminology that lends itself to straightforward derivations of mathematical proofs regarding the properties of these constrained expressions. Furthermore, the extension of the technique to and proofs in n-dimensions is immediate through a recursive application of the univariate formulation. In all, the results of this reformulation are compared to prior work to highlight the novelty and mathematical convenience of using this approach. Finally, the methodology presented in this paper is applied to two partial differential equations with different boundary conditions, and, when data is available, the results are compared to state-of-the-art methods.

Expert and Novice Approaches to Reading Mathematical Proofs

2012 ◽  
Vol 43 (4) ◽  
pp. 358-390 ◽  
Author(s):  
Matthew Inglis ◽  
Lara Alcock
Keyword(s):  
Eye Movements ◽  
Mathematics Education ◽  
Undergraduate Students ◽  
Logical Structure ◽  
Surface Features ◽  
Mathematical Proofs ◽  
Proof Validation ◽  
Expert And Novice ◽  
University Level

This article presents a comparison of the proof validation behavior of beginning undergraduate students and research-active mathematicians. Participants' eye movements were recorded as they validated purported proofs. The main findings are that (a) contrary to previous suggestions, mathematicians sometimes appear to disagree about the validity of even short purported proofs; (b) compared with mathematicians, undergraduate students spend proportionately more time focusing on “surface features” of arguments, suggesting that they attend less to logical structure; and (c) compared with undergraduates, mathematicians are more inclined to shift their attention back and forth between consecutive lines of purported proofs, suggesting that they devote more effort to inferring implicit warrants. Pedagogical implications of these results are discussed, taking into account students' apparent difficulties with proof validation and the importance of this activity in both schooland university-level mathematics education.

Reconsidering Kant’s Rejection of Indirect Arguments in Transcendental Philosophy

2021 ◽  
pp. 1-19
Author(s):  
Marcel Buß
Keyword(s):  
Immanuel Kant ◽  
Explanatory Power ◽  
Transcendental Philosophy ◽  
Point Of View ◽  
Transcendental Arguments ◽  
Mathematical Proofs ◽  
Law Of Excluded Middle ◽  
The Law ◽  
Excluded Middle ◽  
Insight Into

Abstract Immanuel Kant states that indirect arguments are not suitable for the purposes of transcendental philosophy. If he is correct, this affects contemporary versions of transcendental arguments which are often used as an indirect refutation of scepticism. I discuss two reasons for Kant’s rejection of indirect arguments. Firstly, Kant argues that we are prone to misapply the law of excluded middle in philosophical contexts. Secondly, Kant points out that indirect arguments lack some explanatory power. They can show that something is true but they do not provide insight into why something is true. Using mathematical proofs as examples, I show that this is because indirect arguments are non-constructive. From a Kantian point of view, transcendental arguments need to be constructive in some way. In the last part of the paper, I briefly examine a comment made by P. F. Strawson. In my view, this comment also points toward a connection between transcendental and constructive reasoning.

scholarly journals AI and Machine Learning In Nuclear Fusion

2020 ◽  
Author(s):  
Andrew Kamal
Keyword(s):  
Machine Learning ◽  
Algebraic Topology ◽  
Potential Role ◽  
Nuclear Fusion ◽  
Maximum Point ◽  
Maximum Heat ◽  
Mathematical Proofs ◽  
Maximum Heat Transfer ◽  
Entropy Transfer ◽  
Transfer Mechanisms

With the emergence of regressional mathematics and algebraic topology comes advancements in the field of artificial intelligence and machine learning. Such advancements when looking into problems such as nuclear fusion and entropy, can be utilized to analyze unsolved abnormalities in the area of fusion related research. Proof theory will be utilized throughout this paper. For logical mathematical proofs: n represents an unknown number, e represents point of entropy, and m represents maximum point, f represents fusion. This paper will look into analysis of the topic of nuclear fusion and unsolved problems as hardness problems and attempt to formulate computational proofs in relation to entropy, fusion maximum, heat transfer, and entropy transfer mechanisms. This paper will not only be centered around logical proofs but also around computational mechanisms such as distributed computing and its potential role in analyzing computational hardness in relation to fusion related problems. We will summarize a proposal for experimentation utilizing further logical proof formalities and the decentralized-internet SDK for a computational pipeline in order to solve fusion related hardness problems.

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