Mathematical Proof Research Articles

Stability on the Robust Dissipative Coding in Quantum System for Thermal Noise

In this paper, we deal with the stability of the improved quantum coding method based on Markovian dissipative dynamics that is robust against thermal noise. This coding method was proposed by Nishino et al. and the suppression of the effect of thermal noise was demonstrated by numerical simulations, however the mathematical proof was not given. In this paper, we analyze the impact of the thermal noise in this method and mathematically prove that the quantum states converge to a neighborhood around the target code words. The bound of the neighborhood set is also given strictly. We also demonstrate numerical simulations, which confirm the theoretical results.

Identification of Multievent Stochastic Fuzzy Discrete Event Systems.

We recently introduced a novel category of fuzzy discrete event systems (FDESs) termed stochastic FDESs (SFDESs), wherein multiple fuzzy automata occur randomly with different probabilities. We also developed two techniques for identifying event transition matrices in single-event SFDES employing the max-product fuzzy inference. One of them, named the equation-systems-based technique, focuses on the single-event SFDES identification, where the fuzzy automaton of each FDES has only one event. Expanding on our research, this article delves into multievent SFDES identification, allowing each FDES to encompass a sequence of events. This is a new research direction that has not been mentioned in the literature before. Upon activation of an FDES, all its events occur sequentially. Our mathematical proof first establishes the associativity of the max-product inference operation, leading to the introduction of a pivotal and novel concept called an equivalent overall event transition matrix for a consecutive event sequence. This concept establishes a theoretical framework for utilizing the equation-systems-based technique in a novel three-step method for identifying multievent SFDESs. The technique is employed in the first two steps to: 1) determine the number of fuzzy automata in an SFDES and 2) calculate their occurrence frequencies. In the third step, multievent transition matrices of the SFDES are learned by using the stochastic-gradient-descent-based algorithms that we previously developed for multievent FDESs, provided the numbers of consecutive events for each fuzzy automaton within the SFDES are known. Theoretical analysis reveals, for the first time, the interconnections between the event transition matrices learned by the algorithms, the equivalent overall event transition matrices derived from these matrices, and the target event transition matrices. To illustrate our findings, we present an informative example.

Fast Quasi-Optimal Power Flow of Flexible DC Traction Power Systems

This paper proposes a quasi-optimal power flow (OPF) algorithm for flexible DC traction power systems (TPSs). Near-optimal OPF solutions can be solved with high computational efficiency by quasi-OPF. Unlike conventional OPF algorithms, quasi-OPF does not utilize mathematical optimization algorithms but adopts a new methodology. First, we adopt a new modeling method and successfully reveal the physical meaning of OPF solutions in flexible DC TPSs. Then, by converting the physical meaning of OPF solutions into mathematical expressions, a simple mapping from the power flow solution to the near-optimal OPF solution is obtained, and the quasi-OPF algorithm is designed based on power flow and this mapping. Since calculating power flow is computationally cheap and the mapping is based on simple arithmetic, the quasi-OPF algorithm can solve OPF with much less execution time, achieving subsecond level calculation and a speed-up of 57 times compared to the primal-dual interior point method. The effectiveness is verified by mathematical proofs and a case study with Beijing Metro Line 13. This study provides insight into the physical meaning of OPF solutions and is a powerful tool for flexible DC TPSs to analyze the effects of coordinated control, design real-time coordinated control strategies, and solve operational problems in planning.

A Predictive Out-of-Step Identification Method Based on Wide-Area Velocity-Acceleration Trajectory

Online identification of the transient instability is essential in power system analysis, as it facilitates the grid operator to decide and coordinate system failure correction control actions. The rapid technological development of phasor measurement units (PMUs) provides an opportunity for online transient out-of-step identification in real-time. This paper analyzes the characteristics of angular velocity-acceleration trajectories in detail, and then proposes a predictive transient out-of-step identification method among coherent groups of generators based on the velocity-acceleration trajectory sampled by PMUs. To prove that the method can distinguish the transient out-of-step event following disturbance reliably, a mathematical proof of the criterion extended to multi-machine system is made, and the influences of the exciter and emergency control are taken into consideration to a certain extent. Finally, the effectiveness and predictability of the proposed method are demonstrated on a 39-bus New England test system and two large-scale power system simulation models of China by numerical studies.

Exponential Synchronization of Coupled Inertial Neural Networks With Hybrid Delays and Stochastic Impulses.

The synchronization problem of the coupled delayed inertial neural networks (DINNs) with stochastic delayed impulses is studied. Based on the properties of stochastic impulses and the definition of average impulsive interval (AII), some synchronization criteria of the considered DINNs are obtained in this article. In addition, compared with previous related works, the requirement on the relationship among the impulsive time intervals, system delays, and impulsive delays is removed. Furthermore, the potential effect of impulsive delay is studied by rigorous mathematical proof. It is shown that within a certain range, the larger the impulsive delay, the faster the system converges. Numerical examples are provided to show the correctness of the theoretical results.

Learning Disentangled Priors for Hyperspectral Anomaly Detection: A Coupling Model-Driven and Data-Driven Paradigm.

Accurately distinguishing between background and anomalous objects within hyperspectral images poses a significant challenge. The primary obstacle lies in the inadequate modeling of prior knowledge, leading to a performance bottleneck in hyperspectral anomaly detection (HAD). In response to this challenge, we put forth a groundbreaking coupling paradigm that combines model-driven low-rank representation (LRR) methods with data-driven deep learning techniques by learning disentangled priors (LDP). LDP seeks to capture complete priors for effectively modeling the background, thereby extracting anomalies from hyperspectral images more accurately. LDP follows a model-driven deep unfolding architecture, where the prior knowledge is separated into the explicit low-rank prior formulated by expert knowledge and implicit learnable priors by means of deep networks. The internal relationships between explicit and implicit priors within LDP are elegantly modeled through a skip residual connection. Furthermore, we provide a mathematical proof of the convergence of our proposed model. Our experiments, conducted on multiple widely recognized datasets, demonstrate that LDP surpasses most of the current advanced HAD techniques, exceling in both detection performance and generalization capability.

Specialized Knowledge of Prospective Mathematics Teachers on the Concept of Mathematical Proof

[Objetivo] En este artículo, se presentan los resultados de una investigación cualitativa de carácter descriptivo que tiene como objetivo caracterizar el conocimiento especializado de los profesores de matemáticas en formación inicial de la Universidad Nacional en Costa Rica (UNA), sobre el concepto de la demostración matemática, mediante el modelo Mathematics Teacher’s Specialized Knowledge (MTSK). [Metodología] La investigación se posiciona en el paradigma interpretativo y tiene un enfoque cualitativo. Se aplicó un cuestionario, durante el primer semestre del 2021, a 42 profesores de matemáticas en formación inicial de cuarto y quinto año de la carrera Bachillerato y Licenciatura en Enseñanza de la Matemática de la Universidad Nacional de Costa Rica. Para examinar la información, se empleó el análisis de contenido y se hicieron agrupaciones de las respuestas, con el propósito de generar ideas centrales sobre el concepto de demostración. [Resultados] De los resultados se desprenden cuatro ideas centrales sobre lo que es una demostración matemática para los sujetos de la investigación, cercanas a aspectos formales lógico-sintácticos y matemáticos (ALSM) o aspectos informales semánticos (AIS). Se encontraron evidencias de las cinco funciones de De Villiers (1993) para la demostración y surgieron nuevas funciones relacionadas con esta en las matemáticas y en las matemáticas escolares. [Conclusiones] Los resultados brindan insumos a formadores de profesores de matemáticas e investigadores, en la revisión y análisis de programas de formación docente, y contribuyen en la búsqueda de nuevas áreas de investigación relacionadas con el tema.

Sentiment analysis of prospective mathematics teacher on reasoning and proof questions using Naïve-Bayes classification

Reasoning and proof is the ability to arrange patterns, make conjectures, test conjectures, and carry out logical proof, which can be seen as negative or positive for students when solving proof problems requiring reasoning skills. Prospective mathematics teachers in the Mathematics Education Study Program of FKIP Universitas Sriwijaya became the subject of this study. As the research instrument, the questionnaire was designed to explore students' opinions and responses to problems with significant logical difficulty. Sentiment analysis was used to analyze the opinions of prospective mathematics teachers on reasoning and proof questions by grouping positive and negative opinions. The technique used in this study uses the Naïve Bayes algorithm. The classification results in this study were 58.2% positive and 41.86.7% negative, with a total of 70 data. The final result achieved an accuracy value of 53.33%, signifying the reliability of the Naïve Bayes algorithm in understanding and classifying the complex spectrum of sentiments expressed by students. The implications of these findings go beyond sentiment analysis, providing valuable insights for educators, curriculum developers, and policymakers in designing learning strategies and educational policies that can improve mathematics students' reasoning and proof abilities.

ANALISIS KEMAMPUAN PEMBUKTIAN MATEMATIS PADA MATA KULIAH TEORI RING DITINJAU DARI PENDIDIKAN SEKOLAH MENENGAH

Mathematical proof is a crucial component in studying mathematics, especially at advanced levels. Previous research has reported that many students face difficulties in solving mathematical proof problems, including in rings theory course. This study aims to determine the level of students' mathematical proof abilities in ring theory course and to compare these abilities based on their school backgrounds. This research employs a quantitative descriptive approach followed by comparative analysis. The subjects were 30 students from the mathematics education program taking rings theory course at Tidar University. The research instruments include tests, questionnaire, and interviews guidelines. Data analysis of the test results was conducted using descriptive statistic and one-way ANOVA followed by Tukey HSD post hoc tests. The research findings indicate that students' overall mathematical proof abilities fall into the low category, with an average score of 41.67 (on a scale of 100). From these results, 84% of the respondents are in the low category, 13% in the medium category, and 3% in the high category. Furthermore, when considering the backgrounds of secondary schools MA, SMK, and SMA, the average scores for mathematical proof abilities in all three groups are in the low category. The one-way ANOVA and Tukey HSD post hoc tests reveal a significant difference in mathematical proof abilities between the MA and SMA groups, while there was no significant difference between the MA and SMK groups or the SMK and SMA groups

Subject, functionality and level of proofs preferred by pre-service elementary mathematics teachers

This study examined the subject(s) that elementary mathematics teacher candidates find most suitable for proving in analysis courses, the functional structure of proof they remember most, the level of proof, and the reasons for preferring this proof. For this reason, this research aims to reveal the pre-service teachers' preferences for proof in analysis courses, the proof they keep in mind the most and its functional structure, their level of proof, and their views on proof relationally and holistically. In this study, which was conducted with a qualitative research approach, a form consisting of open-ended questions was applied to teacher candidates. In this form, teacher candidates were asked various questions about the mathematical proofs they made. With descriptive analysis, the answers of the pre-service teachers who participated in the research were systematically defined, and data were tried to be defined through content analysis. Accordingly, while the subject that the pre-service teachers found the most appropriate application of the proof approach to be the subject of trigonometry, it was determined that the proof that remained in their minds the most was also related to the subject of trigonometry. By examining the functional structure of these proofs written by pre-service teachers, it has been seen that they have the function of explanation and systematization. In addition, the reasons for preferring the proof they made were asked of the pre-service teachers, and the answers were gathered on the fact that proof provides the most permanence and causal learning. It was emphasized that theorems that require formula memorization generally become more understandable with the proof method. According to the results of the research, it is stated that the common opinion of the prospective teachers is that teaching how to obtain the proof method of formulas in trigonometry, instead of memorizing them, is beneficial in ensuring both meaningful and permanent learning. In light of the findings of these studies, more sensible suggestions can be made to improve pre-service teachers' knowledge systems and classroom teaching on proof. By determining which topics and theorems students have difficulty in proving in addition to trigonometry, additional learning on these subjects can be recommended.

Kurt Gödel’in Edmund Husserl Okuması: Matematiğin Temellerini Fenomenolojinin Işığında Aramak

In his later works, the great logician and mathematician Kurt Gödel concentrates his focus on the philosophical problems such as the implications of set theory, the grammar and philosophy of language, objectivity and relativity, the ontological proof of God’s existence, and phenomenology as an exact method. This essay explores how Gödel reads the philosophy (of logic and mathematics) of his time and why he turns his attention to Husserl’s phenomenology for describing the foundations of mathematics. To begin with, Gödel employs Husserl’s significant distinction between Weltanschauung (worldview) philosophy and philosophy as rigorous science: According to the Weltanschauung philosophy, the spirit of time constantly changes so that the ideas discussed and goals attempted are meant to be temporal, and not for the sake of eternal truths, but for that of their own perfection; philosophy as rigorous science, on the other hand, is supratemporal so that its aim is to discover absolute and timeless values. As for the worldview of his time, Gödel sees the development of philosophy and mathematics leaned toward skepticism, pessimism, and positivism. The antinomies of set theory, for instance shaked the grounds on which mathematics and logic are founded. Gödel, too, uses these paradoxes in his incompleteness theorems in order to prove that there are some statements which can neither be proved nor disproved within a system. That also means that arithmetic is not eligible to prove its own consistency. From this, however, Gödel does not come to a conclusion for a nihilism in mathematics and logic: These mere antinomies of set theory do not “necessarily” lead us to logical positivism, and neither to such a materialism, nor to any kind of pessimistic theory of knowledge. The incompleteness theorems assert that there are arithmetical propositions that are true but neither provable nor unprovable within its own calculus, so that arithmetic is intrinsically incomplete. However, instead of Alfred Tarski’s pathological view of examining the detections within the faulty system and then reforming the system all together, Gödel holds that we need to change our methods to find new patterns that describe the antinomies pointing to the unrecoverable reality of the mathematical world. Thus, Gödel does not follow any variation of the Weltanschauung philosophy of his time, either attempting to reduce mathematical realities to mathematical proofs in order to get rid of antinomies, or endeavoring to rescue a complete system of truths by a closed formal system, both Weltanschauung philosophies fail to set forth a realistic method. In this context, Gödel finds the task of phenomenology analogous to what he pursues in terms of a systematic framework for the foundations of mathematics. Husserl’s phenomenology, in Gödel’s account, proliferates the intuition of (mathematical) essences and provides a clarification of meaning of undefinable concepts, such as the antinomies of set theory. Applying the phenomenological reduction to the objective reality of the mathematical world, Gödel believes one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly put, what Gödel finds in Husserl’s phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method giving us mathematical essences back again.

Oscillatory Solution of a Convolutional Volterra Integral Equation

Oscillatory solutions play a pivotal role in understanding functional differential and integral equations, offering insights into the behaviour of these equations' solutions, and assisting in understanding their growth, stability, and convergence properties. This study establishes the oscillatory solution of a convolutional Volterra integral equation using mathematical proofs. Theorems for oscillatory solutions are proposed and proven based on well-defined assumptions, along with an illustrated example. The proofs presented herein reveal that the convolutional Volterra integral equation can exhibit oscillatory or non-oscillatory behavior, contingent upon the characteristics of the function within the integral.

Differgram: A convex optimization-based method for extracting optimal frequency band for fault diagnosis of rotating machinery

The extraction of fault resonance bands from a full frequency band has always stood as a classical and effective strategy for fault diagnosis in rotating machinery. Among the existing techniques for capturing fault frequency bands, methods such as fast kurtogram and its variants have been prevalent. However, these methods encounter two main limitations. Firstly, they are subject to the constraints of index performance, potentially leading to the identification of incorrect frequency bands. Secondly, they do not fully harness the potential of healthy signals, thus hindering accurate fault frequency band localization. While approaches like SKRgram and accugram endeavor to leverage healthy signals to highlight differences between healthy and faulty signals, SKRgram falls short in thoroughly exploring the significance of healthy signals. Accugram relies on classifier accuracy to pinpoint fault frequency bands, rendering it less robust and less interpretable. To capitalize more effectively on healthy signals' potential, this study proposed a novel approach called differgram based on the convex optimization model. By inputting both healthy and faulty signals into the differgram, optimal fault frequency bands can be extracted. By solving the constructed convex optimization model, the optimal difference frequency bands between healthy and faulty signals are automatically identified. The method is substantiated through mathematical proofs and physical explanations, enhancing its interpretability. Through simulations and experimental validation, the proposed diffetergram demonstrates its efficiency in the extraction of fault frequency bands. In comparison to fast kurtogram, SKRgram, and accugram, the differgram showcases heightened robustness and noise immunity.

ANALYZING STUDENTS' ABILITIES AS PROSPECTIVE TEACHERS OF MATHEMATICS IN CONSTRUCTING PROOFS: A STUDY OF GEOMETRY AND ALGEBRA

Mathematical proof demands accuracy and precision in formulating correct and logical arguments. Students need to develop their ability to produce accurate and precise proofs. This study aims to analyze the ability of prospective mathematics teachers in constructing geometry and algebraic proofs. The research subjects were 8 prospective mathematics teacher students, four male and four female. This descriptive qualitative research begins with giving tests to research subjects, then we conduct interviews. The results showed that male students were better at compiling geometry proofs than compiling algebraic proofs. While female students are better at compiling algebraic proofs than geometry. This is due to the spatial ability of men better than women. When compiling geometry proofs apply procedural proofs, syntactic proofs, and semantic proofs. When compiling algebraic proofs only apply procedural proofs. Female students when compiling geometry proofs only apply procedural proofs. When compiling algebraic proofs, they apply procedural proofs and syntactic proofs.

Modeling the dynamic semantics of a computer program

In this paper, two approaches are used to construct the dynamic semantics of computer programs: the first is the representation of mathematical models of computational processes generated by a computer program in the form of a discrete transformer; the second is the representation of the mathematical model of a computer program in terms of functional grammars. The study focuses on the computational process generated by a computer program. Application of the discrete transducers concept to describe computational processes allows us to obtain a mathematical model or dynamic semantics of a program, which is a composition of two discrete systems B and C. In this case, component A is a control component (a model of the program block diagram), and component B is a model of the program memory. The decomposition of a program into two components B and B is convenient when studying the process of modelling a computational process. This is due to the fact that the prospects of searching for invariants in critical nodes of a programme for mathematical proof of correctness of a computer programme are opened. The comparative analysis of the two approaches leads to the theorem on the regularity of the representation of a function generated by a programme within the framework of functional grammar.

Conditional spatial transition reduction data encoding technique for VLSI interconnects

In this paper the DSM (Deep sub micron) and lower node technologies device dimensions are reduced. Compared to logic functional blocks interconnects dynamic power consumption plays most precarious factor. The most effective way to save the dynamic power in wires is to reduce the number of transitions by using the data encoder. In this paper we proposed a inverse based algorithm with mathematical proof and simple architecture. it is having the different options lower bit inversion, upper bit inversion, all bit inversions and no bit inversion, the data selection based on the which option saves the more power. It is having the simple and compact architecture and it offers less delay compared to related methods.

A New Class of Simple, General and Efficient Finite Volume Schemes for Overdetermined Thermodynamically Compatible Hyperbolic Systems

In this paper, a new efficient, and at the same time, very simple and general class of thermodynamically compatible finite volume schemes is introduced for the discretization of nonlinear, overdetermined, and thermodynamically compatible first-order hyperbolic systems. By construction, the proposed semi-discrete method satisfies an entropy inequality and is nonlinearly stable in the energy norm. A very peculiar feature of our approach is that entropy is discretized directly, while total energy conservation is achieved as a mere consequence of the thermodynamically compatible discretization. The new schemes can be applied to a very general class of nonlinear systems of hyperbolic PDEs, including both, conservative and non-conservative products, as well as potentially stiff algebraic relaxation source terms, provided that the underlying system is overdetermined and therefore satisfies an additional extra conservation law, such as the conservation of total energy density. The proposed family of finite volume schemes is based on the seminal work of Abgrall [1], where for the first time a completely general methodology for the design of thermodynamically compatible numerical methods for overdetermined hyperbolic PDE was presented. We apply our new approach to three particular thermodynamically compatible systems: the equations of ideal magnetohydrodynamics (MHD) with thermodynamically compatible generalized Lagrangian multiplier (GLM) divergence cleaning, the unified first-order hyperbolic model of continuum mechanics proposed by Godunov, Peshkov, and Romenski (GPR model) and the first-order hyperbolic model for turbulent shallow water flows of Gavrilyuk et al. In addition to formal mathematical proofs of the properties of our new finite volume schemes, we also present a large set of numerical results in order to show their potential, efficiency, and practical applicability.

The Efficiency of Embedding-Based Attacks on the GGH Lattice-Based Cryptosystem

The Goldreich-Goldwasser-Halevi (GGH) cryptosystem is declared broken due to the modified versions of the embedding attacks, known as Nguyen's σ, Nguyen's 2σ and Lee-Hahn's attacks. Despite using the same approach as the original embedding attack, these attacks deployed different strategies and resulted in different performances for breaking the GGH cryptosystem. In this paper, we described those strategies in detail. Moreover, we investigated the mathematical factors behind these attacks' ability and performance discrepancies. Mathematical proof examines and discusses the factors that triggered those variances. As a result, the expected lattice gap and implemented lattice dimensions are mathematically proven as the factors that significantly influenced these attacks' performance. By demonstrating how the attacks manipulated these factors, any lattice-based cryptosystem that relies on the hardness of the CVP could avoid repeating the same slipup as the GGH. Hence, precautionary action could be proactively taken to prevent it from being threatened by embedding-based attacks.

A re-look into the modeling aspects of Eringen’s strain-driven nonlocal Euler-Bernoulli nanobeam bending problems

There has long been debate on the applicability of Eringen’s strain-driven nonlocal model to the study of nanostructures. Previous studies based on Eringen’s differential and integral nonlocal models to simulate the static bending behavior of nanobeams have shown shortcomings in nanobeam elastostatics solutions. The solution methodologies employed to solve the differential and integro-differential equations arising from Eringen’s strain-driven nonlocal constitutive model have shown inconsistencies in ensuring the fulfillment of essential and natural boundary conditions. This study aims to conduct an in-depth investigation of the modeling aspects of Euler-Bernoulli nanobeams under different loading and boundary conditions. Dirac-delta identity and generalized functions-based approach shall be utilized for the same. A rigorous mathematical proof establishing the equivalence between Eringen’s differential and integral nonlocal models is presented here for the first time.

Curved boundary integral method for electromagnetic fields.

The angular spectrum method is a rigorous method to synthesize near and far-field electromagnetic beams from planar field distributions. However, this limitation of planar surfaces has restricted its applicability to beams with simple focal planes. We propose a curved boundary integral method (CBIM) to synthesize electromagnetic beams from arbitrary surfaces to address this limitation and expand the method's scope to synthesize beams from and between shaped objects. This study presents a detailed theoretical framework behind the CBIM and validates its effectiveness and accuracy with a comprehensive set of simulations. Additionally, we present mathematical proof to support our proposal. The proposed method satisfies Maxwell's equations and significantly benefits optical systems and inverse beam design. It allows for analyzing electromagnetic forward/backward propagation between optical elements using a single method. It is also valuable for optical force beam design and analysis.