17,221 results on '"MATHEMATICAL proofs"'
Search Results
2. The profile of elementary school's prospective teachers on perceiving mathematical proof.
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Hapipi, Kurniati, Nani, Hamdani, Deni, Salsabila, Nilza Humaira, and Kertiyani, Ni Made Intan
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MATHEMATICAL proofs , *ELEMENTARY schools , *MATHEMATICAL formulas , *TEACHERS - Abstract
This article is part of the results of a study that aims to describe how prospective elementary teachers perceive and understand mathematical proofs, as well as what mathematical proof schemes they do. This research is a descriptive study with a total sample of 71 prospective elementary teachers who were categorized based on their semester. The samples in each semester were taken randomly and proportionally. In this article, two main points are described, namely: (1) their views on mathematical proofs, where the most of them think that proofs are related to the truth of mathematical formulas; and (2) their level of exposure to learning that involves proof, where most of them believe that they are very familiar with mathematical proofs. [ABSTRACT FROM AUTHOR]
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- 2024
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3. Mathematical proof in small group: Exploring the APOS theory.
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Shinariko, Leonardo Jonathan, Hartono, Yusuf, Darmawijoyo, and Araiku, Jeri
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MATHEMATICAL proofs , *PRIOR learning - Abstract
The study aims at designing the learning for mathematical proof based on APOS theory. This study is just a description of a validation study by Gravemeijer. The research was conducted at SMA 1 Palembang with 6 students for exploring the small group. The Hypothetical Trajectory Learning (HLT) design will be tested in a small group. APOS (Action, Process, Object, and Scheme) is a theory that studies students' mental structure for learning mathematics. The mathematical proof learning design with APOS theory approach on exponential and radical Form. The results provided insight into the learning trajectory that needs to be improved. Through comparisons made of HTL and ALT, the variations in student ability and prior knowledge can affect their behavior in learning. The findings suggest the frequency of learning mathematical proof needs to be improved and the assessment must also include questions about proof. [ABSTRACT FROM AUTHOR]
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- 2024
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4. ONTOLOGICAL PURITY FOR FORMAL PROOFS.
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MARTINOT, ROBIN
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MATHEMATICAL proofs , *PROOF theory - Abstract
Purity is known as an ideal of proof that restricts a proof to notions belonging to the 'content' of the theorem. In this paper, our main interest is to develop a conception of purity for formal (natural deduction) proofs. We develop two new notions of purity: one based on an ontological notion of the content of a theorem, and one based on the notions of surrogate ontological content and structural content. From there, we characterize which (classical) first-order natural deduction proofs of a mathematical theorem are pure. Formal proofs that refer to the ontological content of a theorem will be called 'fully ontologically pure'. Formal proofs that refer to a surrogate ontological content of a theorem will be called 'secondarily ontologically pure', because they preserve the structural content of a theorem. We will use interpretations between theories to develop a proof-theoretic criterion that guarantees secondary ontological purity for formal proofs. [ABSTRACT FROM AUTHOR]
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- 2024
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5. Fixed-time fuzzy adaptive control scheme for doubly fed induction generator–based wind turbine.
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Bounar, Naamane, Boulkroune, Abdesselem, Labdai, Sami, and Chrifi-Alaoui, Larbi
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INDUCTION generators , *ADAPTIVE fuzzy control , *WIND turbines , *CLOSED loop systems , *BACKSTEPPING control method , *MATHEMATICAL proofs , *FUZZY systems - Abstract
This work focuses on the issue of designing a fixed-time fuzzy adaptive controller for variable-speed doubly fed induction generator-based wind turbine. The main goal is to optimize the wind turbine power production, while improving the performance of the overall control system such as, convergence speed, tracking accuracy, and robustness. For the control design, the backstepping technique is adopted, and fuzzy adaptive systems are employed as universal approximators to estimate continuous functional uncertainties. Using fixed–time stability concept, a detailed mathematical stability proof of the closed-loop control system is performed. Simulation results are presented to testify the efficiency of the designed control approach. [ABSTRACT FROM AUTHOR]
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- 2024
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6. Chatter-Free Adaptive Control of a Memristor-Based Four-Dimensional Chaotic Oscillator.
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Shafiq, Muhammad and Ahmad, Israr
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ADAPTIVE control systems , *CHAOS theory , *ARTIFICIAL neural networks , *MATHEMATICAL proofs , *LYAPUNOV stability , *PSYCHOLOGICAL feedback , *FLUCTUATIONS (Physics) - Abstract
Memristors have several chaotic dynamic models and have been used successfully in various fields, including secure communication systems, information storage, and artificial neural networks. The memristor-based four-dimensional chaotic (FDMC) systems generate unpredictable and intricate time domain signals. Parameter fluctuations in the FDMC system may give birth to chaos, making it difficult to suppress. Stabilizing chaos in the FDMC system improves the circuit's performance. This paper synthesizes a novel time-efficient chatter-free nonlinear robust adaptive control (NLRAC) technique that stabilizes chaos in the FDMC system affected by time-varying unknown bounded exogenous disturbances and model uncertainties. The proposed NLRAC strategy decimates the time-varying unknown bounded exogenous disturbances and model uncertainties effects; it establishes a faster, smoother state-variable trajectories convergence to the zero vicinity. The theoretical analysis and mathematical proofs are based on the Lyapunov stability technique. Computer simulation results show that the proposed NLRAC technique effectively brings the FDMC system's state-variable trajectories to zero with reduced fluctuations for control input signals and state-variable trajectories. This feedback controller's attribute enhances closed-loop stability performance, improves precision, and reduces risk overshoot. The paper includes comparative computer simulation results to endorse the proposed controller performance. [ABSTRACT FROM AUTHOR]
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- 2024
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7. Safe-by-construction autonomous vehicle overtaking using control barrier functions and model predictive control.
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Yuan, Dingran, Yu, Xinyi, Li, Shaoyuan, and Yin, Xiang
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OVERTAKING , *AUTONOMOUS vehicles , *PREDICTION models , *MATHEMATICAL proofs , *PERFORMANCE standards - Abstract
Ensuring safety for vehicle overtaking systems is one of the most fundamental and challenging tasks in autonomous driving. This task is particularly intricate when the vehicle must not only overtake its front vehicle safely but also consider the presence of potential opposing vehicles in the opposite lane that it will temporarily occupy. In order to tackle the overtaking task in such challenging scenarios, we introduce a novel integrated framework tailored for vehicle overtaking manoeuvres. Our approach integrates the theories of varying-level control barrier functions (CBF) and time-optimal model predictive control (MPC). The main feature of our proposed overtaking strategy is that it is safe-by-construction, which enables rigorous mathematical proof and validation of the safety guarantees. We show that the proposed framework is applicable when the opposing vehicle is either fully autonomous or driven by human drivers. To demonstrate our framework, we perform a set of simulations for overtaking scenarios under different settings. The simulation results show the superiority of our framework in the sense that it ensures collision-free and achieves better safety performance compared with the standard MPC-based approach without safety guarantees. [ABSTRACT FROM AUTHOR]
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- 2024
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8. Maximizing utility by optimal capacity division in P2P networks.
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Singha, Nitin and Singh, Mahesh K.
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DOWNLOADING , *DISTRIBUTED algorithms , *MATHEMATICAL proofs , *NASH equilibrium , *INCENTIVE (Psychology) , *COMPUTER network architectures - Abstract
In most wireless networks, users/peers are connected to backbone networks using access links. The access link capacity is divided into upload and download capacity, and these capacities are used for upload and download of data, respectively. A peer can modify this division of link capacity between upload and download. A peer will allocate its entire link capacity for download to maximize utility. But incentive mechanism forces them to allocate some portion of capacity for upload. This paper investigates how to optimally divide link capacity so that peers receive maximum utility in a P2P network. We model this scenario as a game and determine capacity partitioning of peers during the Nash equilibrium NE. We also prove the social optimality of NE. As this portioning maximizes individual as well as social benefit, so NE is an optimal state of capacity partition. Using simulation, we verify that NE is an optimal state. BitTorrent network and a distributed algorithm for dividing access capacities are simulated, and partition providing maximum utility is compared with NE. This work provides a generalized expression and the mathematical proof for capacity partitioning at which peers receive maximum utility. [ABSTRACT FROM AUTHOR]
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- 2024
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9. Mathematics and the formal turn.
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Avigad, Jeremy
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MATHEMATICAL proofs , *GRAMMAR , *TWENTIETH century - Abstract
Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems with precise grammars and rules of use. Building on such foundations, computational proof assistants now make it possible to encode mathematical knowledge in digital form. This article enumerates some of the ways that these and related technologies can help us do mathematics. [ABSTRACT FROM AUTHOR]
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- 2024
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10. Strange new universes: Proof assistants and synthetic foundations.
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Shulman, Michael
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LANGUAGE models , *MATHEMATICAL proofs , *HOMOTOPY theory , *COMPUTER software , *MATHEMATICIANS , *ATHLETIC fields - Abstract
Existing computer programs called proof assistants can verify the correctness of mathematical proofs but their specialized proof languages present a barrier to entry for many mathematicians. Large language models have the potential to lower this barrier, enabling mathematicians to interact with proof assistants in a more familiar vernacular. Among other advantages, this may allow mathematicians to explore radically new kinds of mathematics using an LLM-powered proof assistant to train their intuitions as well as ensure their arguments are correct. Existing proof assistants have already played this role for fields such as homotopy type theory. [ABSTRACT FROM AUTHOR]
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- 2024
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11. Italian Domination Number of Strong Product of Cycles.
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Liyang Wei and Feng Li
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DOMINATING set , *MATHEMATICAL proofs , *GRAPH theory - Abstract
The investigation into the domination problem and the associated subset problem of graphs is a focal point in graph theory research, sparking widespread interest and extensive exploration. This paper mainly studies the Italian domination of the strong product of two cycles. By constructing recursive Italian dominating functions, a well-defined bound for the Italian domination number in Cn Cm is obtained. Furthermore, through mathematical derivation and proof, the precise Italian domination number of C3 Cm is determined. [ABSTRACT FROM AUTHOR]
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- 2024
12. Experienced provers' uses of contexts while engaging in combinatorial proof of binomial identities.
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Erickson, Sarah and Lockwood, Elise
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MATHEMATICS education (Higher) , *BINOMIAL distribution , *IDENTITIES (Mathematics) , *COMBINATORICS , *MATHEMATICAL proofs - Abstract
Combinatorial proofs of binomial identities involve establishing an identity by arguing that each side enumerates a certain set of outcomes. In this paper, we share results from interviews with experienced provers (mathematicians and upper-division undergraduate mathematics students) and examine one particular aspect of combinatorial proof, namely the kinds of contexts that experienced provers used to establish combinatorial proofs of binomial identities. Our findings show that overall, our participants used a variety of contexts in their work; we also demonstrate ways in which previous experiences influenced the contexts they chose, and we offer some instances in which features of a context supported their combinatorial proof production. We offer some theoretical implications of our work, and we conclude with a discussion of limitations and avenues for future work. [ABSTRACT FROM AUTHOR]
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- 2024
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13. A note on matrix multiplication appearing as element concatenation or coinciding with matrix addition.
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Allan, Samuel B., Dunn, Peter K., and McDougall, Robert G.
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MATRIX multiplications , *STATISTICAL matching , *ADDITION (Mathematics) , *ALGORITHMS , *MATHEMATICAL proofs - Abstract
In this note we demonstrate two instances where matrix multiplication can be easily verified. In the first setting, the matrix product appears as matrix element concatenation, and in the second, the product coincides with matrix addition. General proofs for some results are provided with a more complete description for $ 2 \times 2 $ 2 × 2 matrices. Suggested for consolidation of the algorithm rather than as an introduction, we believe the example base that can be created, combined with an analysis for understanding the context, acts as a positive motivator to secure mastery as it needs to be practised. [ABSTRACT FROM AUTHOR]
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- 2024
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14. On an extension of Niven's theorem.
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Samart, Detchat
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RATIONAL numbers , *FIELD extensions (Mathematics) , *QUADRATIC equations , *IRRATIONAL numbers , *MATHEMATICAL proofs - Abstract
For a given rational number r, a classical theorem of Niven asserts that if $ \cos (r\pi) $ cos (rπ) is rational, then $ \cos (r\pi)\in \{0, \pm 1, \pm 1/2\}. $ cos (rπ) ∈ { 0 , ± 1 , ± 1 / 2 }. In this note, we extend Niven's theorem to quadratic irrationalities and present an elementary proof of that. [ABSTRACT FROM AUTHOR]
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- 2024
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15. Mean square exponentially convergence for semi-linear stochastic differential equations.
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Yousif, Vian Q. and Zaboon, Radhi A.
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STOCHASTIC convergence , *MATHEMATICAL proofs , *STOCHASTIC systems , *STOCHASTIC processes , *EXPONENTIAL stability , *EULER method , *STOCHASTIC differential equations , *QUADRATIC differentials - Abstract
In this paper, the mean square exponential convergence of semi-linear stochastic differential equations is proved by using quadratic Lyapunov function approach with stochastic process. Many theoretical rustles for convergence and mean square exponential convergence as well as mean square exponential stability of different stochastic differential systems using the necessary mathematical conditions have been proposed and supported with mathematical proofs and illustration. The presented approach provides a sufficient condition for stability of some classes of stochastic differential equations. The quadratic types of Lyapunov function gives an effective technique to ensure stable qualitative behavior to stochastic differential system in the present of system random uncertainty corresponding to Brownian motion perturbation. [ABSTRACT FROM AUTHOR]
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- 2024
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16. The role of concept image in constructing mathematical proof.
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Siagian, Siti Shara, Mujib, Abdul, and Firmansyah
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COGNITIVE structures , *MENTAL imagery , *MATHEMATICAL ability , *COGNITIVE ability , *MATHEMATICAL proofs , *T-test (Statistics) , *DATA analysis - Abstract
The concept of the image is essential in constructing the mathematical proof. Therefore, this study attempted to analyze students' concept images in making mathematical proofs. The research method used in this research is qualitative research. Participants were selected using the Theoretical Sampling approach, where the taking of participants was based on concepts that were proven to be theoretically related to the theory being developed. Data analysis was obtained based on student test results, which were given to participants who were collected into a new concept, theme, and then the desired sub-category. The theory generated from this research is that students with good image concept formation can construct good mathematical proofs. The ability to build the resulting evidence, namely (1) students can relate facts with properties to interpret existing problems, (2) students can sequence valid proof steps, and (3) students can use premises, definitions, and theorems that related statements to build a proof. With the concept of the image, students can form the ability to construct evidence by interpreting the ability of cognitive structures that connect mathematical concepts, including mental images, properties, characteristics, and processes associated with predetermined concepts or sub-categories. So that the role of the student's image concept is very important in the formation of the ability to construct mathematical proofs that can make students have good mathematical skills by understanding various concepts, theorems, and definitions and making conjectures from the statements that have been given. [ABSTRACT FROM AUTHOR]
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- 2024
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17. The conceptual validity aspect of pre-service mathematics teachers' reasoning on linear algebra.
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Aziz, Tian Abdul, Makmuri, and Hakim, Lukman El
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MATHEMATICS teachers , *MATHEMATICAL proofs , *PUBLIC universities & colleges - Abstract
The study aimed to analyze the conceptual validity aspect of Indonesian pre-service mathematics teachers' reasoning on linear Algebra. This research is a case study and involves as many as 42 second-semester students at a public university in Jakarta. A five-item test was administered, and based on their claims and reasons, participants' responses were analyzed and classified into four categories: fully valid, partially valid, invalid, and unidentified. The findings of this study are that most of the participants' responses fall in the partially valid category and have various obstacles in delivering their reasons. Based on these findings, it could be concluded that the success or the failure of participants in providing claims and reasons was at least influenced by four supporting factors or causes, namely: (1) conceptual understanding, (2) mathematical connections, (3) mathematical proof, and (4) mathematical communication. There are implications from the results of this study. [ABSTRACT FROM AUTHOR]
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- 2024
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18. Mechanical Mathematicians.
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BENTKAMP, ALEXANDER, BLANCHETTE, JASMIN, NUMMELIN, VISA, TOURRET, SOPHIE, VUKMIROVIĆ, PETAR, and WALDMANN, UWE
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MATHEMATICIANS , *MATHEMATICAL proofs , *COMPUTER systems , *ALGEBRA , *LOGICAL prediction - Abstract
The article discusses the challenges in designing an efficient computer-oriented approach to mathematical theorem proving for higher-order reasoning. Evaluations of several higher-order provers based on superposition and satisfiability modulo theories (SMT) are evaluated.
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- 2023
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19. The Ptolemy-Euler Theorem via Reflection.
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Chen, Jianke and Lin, Kailiang
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GENERALIZATION , *EULER theorem , *PLANE geometry , *MATHEMATICAL proofs , *MATRICES (Mathematics) - Abstract
Although there are several generalizations of the Ptolemy-Euler theorem in plane geometry, its higher dimensional cases are still interesting enough to deserve attention. In this note, we give a new proof of the Ptolemy-Euler theorem in R n , via a modified matrix version (reflection transformation) of a classical algebraic identity. [ABSTRACT FROM AUTHOR]
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- 2024
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20. On the Basel Problem and the Square of Gregory's Series.
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Campbell, John M. and Levrie, Paul
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MATHEMATICAL equivalence , *EULER theorem , *SQUARE root , *GENERALIZATION , *MATHEMATICAL proofs - Abstract
If we consider the Madhava–Gregory–Leibniz series π 4 = 1 − 1 3 + 1 5 − ⋯ , which is commonly referred to as the Gregory series, and then compare it to Euler's formula π 2 8 = 1 + 1 3 2 + 1 5 2 + ⋯ , the following question arises: Can this latter formula be derived by squaring both sides of the former? There have been several proofs of Euler's formula, or its equivalent formulation ζ (2) = π 2 / 6 , based on the idea of squaring 1 − 1 3 + 1 5 − ⋯ = π 4 , including a proof presented in a letter from Euler to Goldbach dating from 1742. We consider the history of proofs of this form, and we offer another simple proof of ζ (2) = π 2 / 6 that also relies on squaring Gregory's series. [ABSTRACT FROM AUTHOR]
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- 2024
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21. A counterexample on the conjecture and bounds on χgd-number of Mycielskian of a graph.
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Kalarkop, David A. and Rangarajant, R.
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MATHEMATICAL bounds , *GRAPH theory , *DOMINATING set , *GEOMETRIC vertices , *MATHEMATICAL proofs - Abstract
A coloring C = (V1,...,Vk) of G partitions the vertex set V (G) into independent sets Vi which are said to be color classes with respect to the coloring C. A vertex v is said to have a dominator (dom) color class in C if there is color class Vi such that v is adjacent to all the vertices of Vi and v is said to have an anti-dominator (anti-dom) color class in C if there is color class Vj such that v is not adjacent to any vertex of Vj. Dominator coloring of G is a coloring C of G such that every vertex has a dom color class. The minimum number of colors required for a dominator coloring of G is called the dominator chromatic number of G, denoted by χd(G). Global Dominator coloring of G is a coloring C of G such that every vertex has a dom color class and an anti-dom color class. The minimum number of colors required for a global dominator coloring of G is called the global dominator chromatic number of G, denoted by χgd(G). In this paper, we give a counterexample for the conjecture posed in [I. Sahul Hamid, M.Rajeswari, Global dominator coloring of graphs, Discuss. Math. Graph Theory 39 (2019), 325-339] that for a graph G, if Xgd(G) = 2χd(G), then G is a complete multipartite graph. We deduce upper and lower bound for the global dominator chromatic number of Mycielskian of the graph G in terms of dominator chromatic number of G. [ABSTRACT FROM AUTHOR]
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- 2024
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22. New Math.
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BISCHOFF, MANON
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MATHEMATICS , *MATHEMATICS contests , *LANGUAGE models , *MATHEMATICAL proofs , *TRIANGLES , *MATHEMATICAL programming - Abstract
A team of researchers from Google DeepMind and New York University has developed an artificial intelligence (AI) program called AlphaGeometry that can solve geometry problems from the International Mathematical Olympiad (IMO) with a success rate similar to human gold medalists. The AI program uses deductive algorithms and large language models to generate new proof terms and auxiliary objects to aid in problem-solving. AlphaGeometry outperformed most IMO participants, solving 25 out of 30 problems, and even discovered a more general theorem that had previously been overlooked by experts. While the AI cannot currently compete in the IMO, the researchers believe their approach could be applied to other mathematical subdisciplines in the future. [Extracted from the article]
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- 2024
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23. A new fuzzy decoupled sliding mode control of flexible joint robotic manipulators based on the finite‐time observer in the presence of chaos with experimental validation.
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Nezhad, Abdollah Hasan, Noiey, Abolfazl Ranjbar, Soltanpour, Mohammad Reza, and Veysi, Mohammad
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SLIDING mode control , *MANIPULATORS (Machinery) , *CLOSED loop systems , *ROBOTICS , *MATHEMATICAL proofs , *FUZZY systems - Abstract
This paper presents a new fuzzy non‐singular fast terminal decoupled sliding mode control based on the finite‐time extended state observer for trajectory tracking control of flexible joint robot manipulators under chaos and uncertainty. First, a suitable chaotic reference is used in a synchronization mechanism to create chaos in the dynamics. Then, the essential modifications made to the conventional decoupled sliding mode control are introduced. A new coupling variable and a non‐singular fast terminal sliding surface are proposed in combination with an extended state observer. A complete convergence analysis is also performed for all the sliding surfaces. Furthermore, two fuzzy systems are designed in such a way that not only they can eliminate the chattering phenomenon but can also improve the convergence time without increasing the control input in a voltage‐based manner. The mathematical proof guarantees the finite‐time asymptotic stability of the closed‐loop control system under the proposed scheme for a class of under‐actuated systems in the presence of structured and un‐structured uncertainties. Finally, some simulations and experimental implementations as hardware‐in‐the‐loop are performed to evaluate the proposed method's performance. The proposed controller is an effective method, as confirmed by the numerical simulation results and experimental validation. [ABSTRACT FROM AUTHOR]
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- 2024
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24. Propagation Phenomena for a Nonlocal Dispersal Lotka–Volterra Competition Model in Shifting Habitats.
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Dong, Fang-Di, Li, Wan-Tong, and Wang, Jia-Bing
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MATHEMATICAL proofs , *HABITATS , *INTEGRAL equations , *COEXISTENCE of species - Abstract
This paper is concerned with the propagation phenomena for a nonlocal dispersal Lotka–Volterra competition model with shifting habitats. It is assumed that the growth rate of each species is nondecreasing along the x-axis, positive near ∞ and nonpositive near - ∞ , and shifting rightward with a speed c > 0 . In the case where both species coexist near ∞ , we established three types of forced waves connecting the origin, respectively to the coexistence state with any forced speed c; to itself with forced speed c > c ∗ (∞) ; and to a semi-trivial steady state with forced speed c > c ¯ (∞) , where c ∗ (∞) and c ¯ (∞) are two positive numbers. In the case where one species is competitively stronger near ∞ , we also obtain the existence and nonexistence of forced waves connecting the origin to the semi-trivial steady state. Our results show the existence of multiple types of forced waves with the same forced speed. The mathematical proofs involve integral equations and Schauder's fixed point theorem, and heavily rely on the construction of various upper-lower solutions, which adds new techniques to deal with the "shifting environments" problem. [ABSTRACT FROM AUTHOR]
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- 2024
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25. Perturbation observer design based on extremum seeking control: An analytical insight.
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Taleshian, Tahereh, Ranjbar Noei, Abolfazl, Sadati, Jalil, and Malekzadeh, Milad
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MATHEMATICAL proofs , *CLOSED loop systems , *ADAPTIVE control systems , *LYAPUNOV stability - Abstract
In this paper, an optimal extended state observer (ESO) is designed for nonlinear integral chain system. For the first time, in this paper, proportional–integral (PI) extremum seeking control (ESC; PIESC) approach is applied to design a perturbation observer by minimizing the estimation error. The PI type structure of the ESC provides fast transient response for the closed-loop system to attain optimum equilibrium point. A mathematical proof is presented to show that the average error dynamic asymptotically converges to zero while the estimation error is minimized. The simulation results verify that the estimation error achieved by the proposed PIESC-based ESO (PIESC-ESO) is found significantly less than that achieved by the classic ESO and PSO-based ESO. Moreover, if slow changes occur within the system or external to it, the PIESC-ESO is still able to modify the observer gains such that the estimation error is minimized again. Therefore, this may help PIESC-ESO to be used in practical applications. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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26. Apriorics: Information and Graphs in the Description of the Fundamental Particles—A Mathematical Proof.
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Shoshani, Yakir and Yahalom, Asher
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PARTICLES (Nuclear physics) , *PARTICLE physics , *BOSONS , *MATHEMATICAL proofs , *FERMIONS , *FEYNMAN diagrams , *UNDIRECTED graphs - Abstract
In our earlier work, we suggested an axiomatic framework for deducing the fundamental entities which constitute the building block of the elementary particles in physics. The basic concept of this theory, named apriorics, is the ontological structure (OS)—an undirected simple graph satisfying specified conditions. The vertices of this graph represent the fundamental entities (FEs), its edges are binary compounds of the FEs (which are the fundamental bosons and fermions), and the structures constituting more than two connected vertices are composite particles. The objective of this paper is to focus the attention on several mathematical theorems and ideas associated with such graphs of order n, including their enumeration, showing what is the information content of apriorics. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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27. On a regime switching illiquid high volatile prediction model for cryptocurrencies.
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El-Khatib, Youssef and Hatemi-J, Abdulnasser
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CRYPTOCURRENCIES , *PREDICTION models , *STOCHASTIC differential equations , *MATHEMATICAL proofs , *VALUE (Economics) , *INVESTORS - Abstract
Purpose: The current paper proposes a prediction model for a cryptocurrency that encompasses three properties observed in the markets for cryptocurrencies—namely high volatility, illiquidity, and regime shifts. As far as the authors' knowledge extends, this paper is the first attempt to introduce a stochastic differential equation (SDE) for pricing cryptocurrencies while explicitly integrating the mentioned three significant stylized facts. Design/methodology/approach: Cryptocurrencies are increasingly utilized by investors and financial institutions worldwide as an alternative means of exchange. To the authors' best knowledge, there is no SDE in the literature that can be used for representing and evaluating the data-generating process for the price of a cryptocurrency. Findings: By using Ito calculus, the authors provide a solution for the suggested SDE along with mathematical proof. Numerical simulations are performed and compared to the real data, which seems to capture the dynamics of the price path of two main cryptocurrencies in the real markets. Originality/value: The stochastic differential model that is introduced and solved in this article is expected to be useful for the pricing of cryptocurrencies in situations of high volatility combined with structural changes and illiquidity. These attributes are apparent in the real markets for cryptocurrencies; therefore, accounting explicitly for these underlying characteristics is a necessary condition for accurate evaluation of cryptocurrencies. [ABSTRACT FROM AUTHOR]
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- 2024
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28. An innovative interval grey model for construction waste forecasting.
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Xu, Yan, Lin, Tong, Du, Pei, and Wang, Jianzhou
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CONSTRUCTION & demolition debris , *OPTIMIZATION algorithms , *MATHEMATICAL proofs , *FORECASTING , *SAMPLE size (Statistics) - Abstract
• An innovative optimized interval-time-delayed grey model is proposed for small sample prediction problems. • The proposed interval time-delayed grey model outperforms the comparison models. • Two numerical cases, results analysis, future prediction and scenario analysis are performed. • Comparison of forecasted results with the construction waste goal in Jiangsu Province's 14th Five-Year Plan period. Nowadays, construction waste has become a global challenge that needs to be addressed, and its accurate prediction is crucial for subsequent treatment and policy development. However, current popular big data methods struggle to be effectively applied due to sample size limits. Meanwhile, the key issues of data uncertainty and time-delayed nature must be considered, the former of which can be addressed by interval prediction. Therefore, a novel interval time-delayed three-parameter discrete grey model for construction waste prediction is proposed in this study, whose time-delayed coefficient is optimized via optimization algorithms. Moreover, the relevant mathematical derivation and proof of this model are also described and discussed in detail in the research. Two case studies of the construction waste dataset are used to examine the effectiveness of the proposed model. Through a series of comparisons with other state-of-the-art models, the results show that the proposed model can improve the prediction performance with excellent prediction results. Meanwhile, this study also presents scenario analysis and discussions of the construction waste prediction results during the 14th Five-Year Plan period, and finds that the current policies and measures are difficult to achieve the future planning goals, and further policy development and implementation of measures are imminent. [ABSTRACT FROM AUTHOR]
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- 2024
- Full Text
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29. A privacy-preserving scheme with multi-level regulation compliance for blockchain.
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Jia, Wangjing, Xie, Tao, and Wang, Baolai
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REGULATORY compliance , *BLOCKCHAINS , *MATHEMATICAL proofs , *MULTILEVEL models , *PRIVACY - Abstract
With the increasing presence of blockchain-based distributed applications in various aspects of daily life, there has been a growing focus on the privacy protection of blockchain ledgers and the corresponding regulatory technologies. However, current mainstream solutions primarily concentrate on the verifiable encryption of blockchain transaction addresses and contents, neglecting the regulatory requirements for private transactions. Moreover, the few monitorable solutions suffer from issues such as excessive centralization and a single-minded approach to regulatory content. To address these deficiencies, this paper proposes a blockchain privacy-preserving scheme that supports multi-level regulation through the utilization of zero-knowledge proofs (zk-SNARKs) and attribute-based encryption (ABE). Firstly, by leveraging zk-SNARKs, this scheme achieves blockchain privacy-preserving within an account model, enabling the concealment of user transaction addresses and values. Secondly, by employing attribute-based encryption, a multi-level regulatory model is developed alongside the privacy protection measures, allowing for selective disclosure of transaction content. Finally, we analyze the security of the proposed scheme and compare it with other schemes, discussing its advantages in terms of privacy, security, and regulatory capabilities, we also provide a preliminary evaluation of the scheme's efficiency through experiments. In conclusion, the scheme demonstrates strong privacy by relying on mathematical proofs through zk-SNARKs to ensure security while comprehensively safeguarding content. It also achieves multi-level regulation on the foundation of privacy protection, with comprehensive regulatory coverage and decentralized regulatory authority. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Task space control of the robot manipulators with adaptive fuzzy global fast terminal sliding mode control in presence of dynamic and kinematic uncertainties.
- Author
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Soltanpour, Mohammad Reza and Zaare, Saeed
- Subjects
- *
SLIDING mode control , *ROBOT control systems , *MANIPULATORS (Machinery) , *GLOBAL asymptotic stability , *CLOSED loop systems , *MATHEMATICAL proofs - Abstract
Summary: Basic features such as convergence time and speed, number‐action control coefficients, free chattering, and proof of stability are significant in the design process and sliding mode control (SMC) efficiency. In this article, we propose an adaptive fuzzy global fast terminal SMC (AFGFTSMC) to handle the mentioned features in the task space control of the robot manipulator in the presence of dynamic and kinematic uncertainties. First, perturbed joint space dynamic equations of the system are transferred to task space, and a broad range of uncertainties are considered there. Then, a global fast terminal SMC (GFTSMC) is proposed for robot manipulators in task space, in which a flexible sliding surface improves the convergence time. Next, to have an intelligent adjustment of the sliding surface coefficients, which leads to a much faster convergence rate, a fuzzy approximator with just seven fuzzy rules is presented. In the following, to access the boundaries of the existing uncertainties, an adaptive fuzzy approximator is proposed, which has five fuzzy rules and only one adaptive law, increases the system's robustness, and eliminates the effect of chattering. Mathematical proof shows that the task space closed‐loop control system under the proposed AFGFTSMC and in the presence of dynamic and kinematic uncertainties has a finite‐time global asymptotic stability. The theoretical evidence and simulation results, which are conducted on a 2‐link robot manipulator, confirm the good efficiency of the proposed controller. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Interval-valued q-rung orthopair fuzzy integrals and their application in multi-criteria group decision making.
- Author
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Gao, J., Xu, Z. S., and Mao, Y. S.
- Subjects
- *
AGGREGATION operators , *GROUP decision making , *FUZZY integrals , *FUZZY sets , *MULTIPLE criteria decision making , *MATHEMATICAL proofs , *DEFINITE integrals - Abstract
The generalized interval-valued orthopair fuzzy sets provide an extension of Yager’s generalized orthopair fuzzy sets, where membership and non-membership degrees are subsets of closed interval [0, 1]. Due to the uncertainty and ambiguity of real life, it is more superior for decision makers to provide their judgments by intervals rather than crisp numbers. Moreover, in the era of huge scale and rapid updating of information, individual weights have been quietly diluted, and the integration of information one by one is time-consuming and complicated. In recent years, some cholars have conducted research on the calculus of generalized orthopair fuzzy sets, but no research has further revealed the intrinsic connection between the integrals of generalized interval-valued orthopair fuzzy sets and traditional aggregation operators, which is very important in applications such as large group decision making. In order to fill this theoretical gap, this paper aims to study the integrals of generalized interval-valued orthopair fuzzy functions. In detail, we define the indefinite integral starting from the inverse operations of the interval-valued q-rung orthopair fuzzy functions (IVq-ROFFs)’ derivatives, and some fundamental properties with rigorous mathematical proofs are also discussed. To be more practical, we continue to develop definite integrals for both simplified and generalized IVq-ROFFs. Besides, we give the corresponding Newton-Leibniz formula through limit procedure, which shows the calculation relationship between the indefinite and definite integrals of the IVq-ROFFs. After obtaining the basic calculus results under generalized interval-valued orthopair fuzzy circumstance, we further reveal the inherent link between the integrals of generalized IVq-ROFFs and the traditional discrete aggregation operators. Finally, the practicability and feasibility of the proposed definite integral models are illustrated by an example of public health emergency group decision-making, and sensitivity analysis and comparison are also carried out. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
32. Kurt Gödel's Reading of Edmund Husserl: Seeking the Foundations of Mathematics in the Light of Phenomenology.
- Author
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BAŞARAN, Abdullah
- Subjects
- *
PHILOSOPHY of science , *MATHEMATICAL proofs , *PHILOSOPHY of mathematics , *THEORY of knowledge , *PHILOSOPHY of time , *PHENOMENOLOGY , *SKEPTICISM - Abstract
In his later works, the great logician and mathematician Kurt Gödel concentrated his focus on 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 read the philosophy (of logic and mathematics) of his time and why he turned his attention to Husserl's phenomenology to describe 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 saw the development of philosophy, and mathematics leaned toward skepticism, pessimism, and positivism. For instance, the antinomies of set theory have shaken the grounds on which mathematics and logic are founded. Gödel also used these paradoxes in his incompleteness theorems to prove that there are some statements that can neither be proven nor disproved within a system. This also means that arithmetic is not sufficient to prove 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 their 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 clarifies the meaning of undefinable concepts such as the antinomies of set theory. Applying phenomenological reduction to the objective reality of the mathematical world, Gödel believes that one obtains a clear experiential reality of the essential characteristics of (mathematical and logical) concepts. Briefly, what Gödel finds in Husserl's phenomenology that corresponds to his way of mathematical realism is a thoroughly designated method that gives us mathematical essences back again. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
33. The Efficiency of Embedding-Based Attacks on the GGH Lattice-Based Cryptosystem.
- Author
-
Mandangan, A., Kamarulhaili, H., and Asbullah, M. A.
- Subjects
- *
MATHEMATICAL proofs , *HARDNESS , *PUBLIC key cryptography , *CRYPTOSYSTEMS - Abstract
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 signifi- cantly 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. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
34. Effective fitness under fluctuating selection with genetic drift.
- Author
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Siliang Song and Jianzhi Zhang
- Subjects
- *
GENETIC drift , *MATHEMATICAL proofs , *GENE frequency , *COMPUTER simulation , *NATURAL selection - Abstract
The natural environment fluctuates for virtually every population of organisms. As a result, the fitness of a mutant may vary temporally. While commonly used for summarizing the effect of fluctuating selection on the mutant, geometric mean fitness can be misleading under some circumstances due to the influence of genetic drift. Here, we show by mathematical proof and computer simulation that, with genetic drift, the geometric mean fitness does not accurately reflect the overall effect of fluctuating selection. We propose an alternative measure based on the average expected allele frequency change caused by selection and demonstrate that this measure--effective fitness--better captures the overall effect of fluctuating selection in the presence of drift. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
35. A Novel Portfolio Based on Interval-Valued Intuitionistic Fuzzy AHP with Improved Combination Weight Method and New Score Function.
- Author
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Xue Deng, Fengting Geng, and Jianxin Yang
- Subjects
- *
ANALYTIC hierarchy process , *MATHEMATICAL proofs , *REAL numbers , *FUZZY sets , *FUZZY numbers - Abstract
The classical Analytic Hierarchy Process (AHP) requires an exact value to compare the relative importance of two attributes, but experts often cannot obtain an accurate assessment of every attribute in the decision-making process, since there are always some uncertainty and hesitation. Compared with classical AHP, our newly defined intervalvalued intuitionistic fuzzy AHP has accurately described the vagueness and uncertainty. In the decision matrix, the real numbers are substituted by fuzzy numbers. In addition, each expert will make different evaluations according to different experiences for each attribute in the subjective weighting method, which neglects objective factors and then generates some deviations in some cases. This paper provides two ways to make up for this disadvantage. On the one hand, by combining the interval-valued intuitionistic fuzzy AHP with entropy weight, an improved combination weighting method is proposed, which can overcome the limitations of the unilateral weighted method only considering the objective or subjective factors. On the other hand, a new score function is presented by adjusting the parameters, which can overcome the invalidity of some existing score functions. In theory, some theorems and properties for the new score functions are given with strictly mathematical proofs to validate their rationality and effectiveness. In application, a novel fuzzy portfolio is proposed based on the improved combination weighted method and new score function. A numerical example shows that the results of our new score function are consistent with those of most existing score functions, which verifies that our model is feasible and effective. [ABSTRACT FROM AUTHOR]
- Published
- 2023
36. Phantom oscillations in principal component analysis.
- Author
-
Shinn, Maxwell
- Subjects
- *
PRINCIPAL components analysis , *OSCILLATIONS , *MATHEMATICAL proofs , *COMMONS - Abstract
Principal component analysis (PCA) is a dimensionality reduction method that is known for being simple and easy to interpret. Principal components are often interpreted as low-dimensional patterns in high-dimensional space. However, this simple interpretation fails for timeseries, spatial maps, and other continuous data. In these cases, nonoscillatory data may have oscillatory principal components. Here, we show that two common properties of data cause oscillatory principal components: smoothness and shifts in time or space. These two properties implicate almost all neuroscience data. We show how the oscillations produced by PCA, which we call "phantom oscillations," impact data analysis. We also show that traditional crossvalidation does not detect phantom oscillations, so we suggest procedures that do. Our findings are supported by a collection of mathematical proofs. Collectively, our work demonstrates that patterns which emerge from high-dimensional data analysis may not faithfully represent the underlying data. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
37. Problems and Solutions.
- Author
-
Oman, Greg and Curtis, Charles N.
- Subjects
- *
MATHEMATICAL proofs , *MATHEMATICAL functions , *COLLEGE teacher-student relationships , *GROUP theory , *ISOMORPHISMS , *MATHEMATICS - Abstract
This document from the College Mathematics Journal provides a collection of problems and solutions for college mathematics students and teachers. It encourages active participation by submitting solutions and proposing new problems. The document covers various mathematical topics, including integrals, triangle properties, equations, group theory, and summation. It also includes mathematical proofs, solutions to problems, and discussions on mathematical concepts such as functions, variables, partial orders, and isomorphism. The text is culturally sensitive and respectful of diverse perspectives. [Extracted from the article]
- Published
- 2023
- Full Text
- View/download PDF
38. On Jump Discontinuities in Internal Forces of Flexible Structures Carrying Moving Subsystems.
- Author
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Bingen Yang and Hao Gao
- Subjects
- *
FLEXIBLE structures , *BENDING moment , *TORQUE , *MATHEMATICAL proofs , *SHEARING force , *GIBBS sampling - Abstract
Combined systems, which are flexible structures carrying moving subsystems, are seen in various applications. Due to structure-subsystem interactions, the structure in a combined system encounters jump discontinuities in its internal forces (such as the bending moment and shear force of a beam). Accurate estimation of such jump discontinuities is important to the performance, safety, and longevity of a combined system. Because of the time-varying nature and complexity of structure-subsystem interactions, conventional series solution methods experience slow convergence, and the Gibbs phenomenon in computation and the improved series expansion methods are limited to certain proportionally damped continua under moving forces and moving oscillators. In this paper, a novel modified series expansion method (MSEM) is proposed to resolve the aforementioned issues with the existing series solution methods. Through the introduction of a jump influence function, the proposed method produces fast-convergent series solutions and accurately predicts the jump discontinuities without the Gibbs phenomenon. The MSEM is applicable to structures with nonproportional damping and subject to arbitrary boundary conditions, and it can easily manage general M-DOF moving subsystems having multiple contact points with a supporting structure. As an important result of this investigation, a mathematical proof of the convergence of the MSEM-based solutions is given for the first time. Additionally, two numerical examples are presented to demonstrate the accuracy, efficiency, and versatility of the proposed MSEM in modeling and analysis of combined systems. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
39. Deconstructing the Bakery to Build a Distributed State Machine.
- Author
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LAMPORT, LESLIE
- Subjects
- *
COMPUTER algorithms , *MATHEMATICAL proofs - Abstract
The article discusses two computer algorithms created by the author: the bakery algorithm which is based on mutual exclusion and the distributed state-machine algorithm which aims to achieve fault tolerance in a network of computers. Mathematical proofs are provided.
- Published
- 2022
- Full Text
- View/download PDF
40. Mathematical proving as a way to improve understanding the concepts of integral calculus application in digital era.
- Author
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Lusiana and Kesumawati, Nila
- Subjects
- *
DIGITAL technology , *INFORMATION technology , *MATHEMATICAL proofs , *LITERATURE reviews , *PROBLEM solving - Abstract
The aim of this study is to inform how to improve understanding concepts in the application of integral calculus in digital era through mathematical proving. The digital era is one of the characteristics of the human's knowledge mode which is facilitated by the sophistication of information technology with the abundance of information, the overload data, but less attention to the critical power as the processor. Students quickly solve their problems with the help of technology such as calculators, computers, android and applications that support the learning that is carried out, but when asked how the process is, what concepts are used, the answer of "don't know" and "don't understand" frequently appear. This is because it doesn't provide an opportunity for students to reason in strengthening their understandings of the mathematical concepts studied. The problem in this research is "How to improve the ability to understanding concepts in the application of Integral calculus in the digital era through mathematical proving?" The research method and analysis technique uses qualitative descriptive by analyzing learning theories, research results through literature review and provides few examples mathematical proofs on the volume of objects using the Application of Integral Calculus, The conclusion of this research showcases that to improve understanding of concepts in the application of integral calculus on objects volume through proof proses by setting the integration boundary of object's base or a cross-sectional object from all possible sides. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
41. Mathematics teachers' perception on mathematical proof.
- Author
-
Hartono, Yusuf, Kurniadi, Elika, and Pratiwi, Weni Dwi
- Subjects
- *
MATHEMATICS teachers , *MATHEMATICAL ability , *MATHEMATICAL proofs , *MATHEMATICS - Abstract
This paper aims to describe mathematics teachers' perception of mathematical proof. Proving in mathematics for the teacher is crucial as the foundation of developing students' skills in generating correct proof to some conjectures which require good mathematical reasoning. It has been highlighted in some works of literature that mathematical reasoning which is closely related to proving in mathematics also contributes to the development of a chosen set of 21st-century skills. A total number 152 of Indonesian in-service mathematics teachers were given a survey concerning their perception of mathematical proving. The description of the result consists of five important categories including the process of mathematical proof, thinking ability in mathematical proof, appropriate school level in learning proof, topic, and understanding in a mathematical proof. It is found that Indonesian in-service mathematics teachers had a positive attitude towards mathematical proof as a teaching approach in their classrooms, as the opportunity for students to improve helped students improve their thinking ability. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
42. Developing mathematical proof as a tool of learning integral for engineering students.
- Author
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Funny, Rindu A. and Ghofur, M. A.
- Subjects
- *
ENGINEERING students , *MATHEMATICS theorems , *CRITICAL thinking , *CREATIVE thinking , *MATHEMATICAL proofs , *PRIOR learning - Abstract
Mathematical proof still difficult to learn and to teach in the real life. Students usually ignore the function of proofing mathematically in their daily activity. They thought that they will not need any concept of proofing in their life. However, mathematics proof is not only about mathematics theorem, but the logical thinking, creative thinking, training to use and connected their prior knowledge to build sequence of logic procedure to proof. Therefore, this study designed an instructional activity to learn integral using mathematical proof for engineering students in order to issued students' way of proof. The activities were adopted from the meaning of proof, which are proof as verification, explanation, logical argument, justification, discovery and derivation. The result showed students still got difficulties in proving the integral problems, but successful students appears with Restrictive Cognitive levels of proof scheme, which are Contextual, Generic and Constructive. Among the three, contextual and generic are mostly done by the students by interpreting and proving the conjecture in the specific or particular context. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
43. Using historical proof-bycontradiction examples in senior mathematics: How Georg Cantor's diagonal method made Alan Turing's (1937-8) proof possible.
- Author
-
Padula, Janice
- Subjects
- *
MATHEMATICAL proofs , *PROPOSITION (Logic) , *STATISTICAL decision making , *PROBABILITY theory , *ARTIFICIAL intelligence in education - Published
- 2023
44. Profile Likelihood for Hierarchical Models Using Data Doubling.
- Author
-
Lele, Subhash R.
- Subjects
- *
MATHEMATICAL proofs , *MAXIMUM likelihood statistics , *FREQUENTIST statistics , *INFERENTIAL statistics , *CONSTRAINED optimization , *HIERARCHICAL Bayes model - Abstract
In scientific problems, an appropriate statistical model often involves a large number of canonical parameters. Often times, the quantities of scientific interest are real-valued functions of these canonical parameters. Statistical inference for a specified function of the canonical parameters can be carried out via the Bayesian approach by simply using the posterior distribution of the specified function of the parameter of interest. Frequentist inference is usually based on the profile likelihood for the parameter of interest. When the likelihood function is analytical, computing the profile likelihood is simply a constrained optimization problem with many numerical algorithms available. However, for hierarchical models, computing the likelihood function and hence the profile likelihood function is difficult because of the high-dimensional integration involved. We describe a simple computational method to compute profile likelihood for any specified function of the parameters of a general hierarchical model using data doubling. We provide a mathematical proof for the validity of the method under regularity conditions that assure that the distribution of the maximum likelihood estimator of the canonical parameters is non-singular, multivariate, and Gaussian. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
45. A diameter gap for quotients of the unit sphere.
- Author
-
Gorodski, Claudio, Lange, Christian, Lytchak, Alexander, and Mendes, Ricardo A. E.
- Subjects
- *
ISOMETRICS (Mathematics) , *MATHEMATICAL proofs , *MATHEMATICAL constants , *GROUP theory , *ORTHOGONAL functions - Abstract
We prove that for any isometric action of a group on a unit sphere of dimension larger than 1, the quotient space has diameter zero or larger than a universal dimension-independent positive constant. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
46. Computing scattering resonances.
- Author
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Ben-Artzi, Jonathan, Marletta, Marco, and Rösler, Frank
- Subjects
- *
SCHRODINGER equation , *COMPUTATIONAL complexity , *SCATTERING (Mathematics) , *MATHEMATICAL optimization , *MATHEMATICAL proofs - Abstract
The question of whether it is possible to compute scattering resonances of Schrödinger operators - independently of the particular potential - is addressed. A positive answer is given, and it is shown that the only information required to be known a priori is the size of the support of the potential. The potential itself is merely required to be C. The proof is constructive, providing a universal algorithm which only needs to access the values of the potential at any requested point. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
47. Mathematical Proofs for the Molecular Meaning of the Liberation Energy Terms in the Solvation Thermodynamics.
- Author
-
Raposo, Diego
- Subjects
- *
SOLVATION , *THERMODYNAMICS , *CHEMICAL potential , *UPPER level courses (Education) , *MATHEMATICAL proofs - Abstract
The Solvation Thermodynamics (ST) is a rigorous theory that describes the solvation as a well-defined process, and is based on the separation of the chemical potential into different terms with particular meanings. Although often mentioned, the interpretations of the nature of the terms within the so called "liberation energy" are rather stated than proved. Here we justify the meaning of the three parts of such energy, with relation to a particle in a fixed position and orientation inside a phase: the work required to give it freedom to move, to provide it energy for such translation (momentum) and to make it indistinguishable from particles of the same kind. The demonstrations reinforce the clarity of the solvation as a microscopic step by step process and the statistical mechanical basis of the ST, which supports the teaching of the subject in advanced courses. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
48. Optimal attitude consensus control for rigid spacecraft formation based on control Lyapunov functions.
- Author
-
Wang, Hui, Zhang, Jinxiu, Zhou, Qingrui, Sun, Hui‐Jie, and Wu, Yu‐Yao
- Subjects
- *
LYAPUNOV functions , *ARTIFICIAL satellite attitude control systems , *SPACE vehicles , *MATHEMATICAL proofs , *SLIDING mode control , *RIGID dynamics , *DISTRIBUTED algorithms - Abstract
In this article, the optimal attitude consensus control problem of distributed spacecraft formation through local information exchange is solved in the presence of parameter uncertainty and disturbance. The attitude dynamics model of rigid spacecraft formation is established based on unit quaternion under an undirected graph. Then, the optimal attitude consensus control scheme based on control Lyapunov functions is adopted for rigid spacecraft formation. The approach to construct an appropriate Lyapunov function for multiple spacecraft is developed. Then the lumped disturbance is estimated by an extended state observer (ESO) and using the sliding mode control approach improves the robustness of the controller. The proposed optimal controller can not only ensure the global consensus convergence of the system but also make the preset performance index function achieve the minimum value. In addition, the mathematical proof is given for the stability analysis of the system by the Lyapunov stability theory. Finally, some simulations and comparisons are presented to demonstrate the validity and advantages of the proposed controller. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
49. PUREZA DEL MÉTODO Y CONSTRUCCIÓN DE TEORÍAS: EL CASO DE KRONECKER Y DEDEKIND EN TEORÍA ALGEBRAICA DE NÚMEROS.
- Author
-
PUENTE, GUILLERMO NIGRO
- Subjects
- *
ALGEBRAIC number theory , *PURITY (Philosophy) , *MATHEMATICAL proofs , *MATHEMATICAL models - Abstract
The discussion of purity of method usually emphasizes the study of particular proofs of mathematical practice. A critique of this position questions the value of purity by claiming that "mathematical progress" depends essentially on the use of impure methods. This article shows that a more holistic perspective, focusing on how purity claims can operate in the construction of autonomous theories, allows us to identify a context where purity of method acquires value in mathematical practice. In particular, the relevance of such demands is shown in the case of Kronecker's and Dedekind's algebraic number theories. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
50. The role of quantum mechanics in cognition-based evolution.
- Author
-
Marshall, Perry
- Subjects
- *
QUANTUM mechanics , *QUANTUM biochemistry , *MATHEMATICAL proofs , *GENETIC code , *WAVE functions - Abstract
In 2021 I noted that in all information-based systems we understand, Cognition creates Code, which controls Chemical reactions. Known agents write software which controls hardware, and not the other way around. I proposed the same is true in all of biology. Though the textbook description of cause and effect in biology proposes the reverse, that Chemical reactions produce Code from which Cognition emerges, there are no examples in the literature demonstrating either step. A mathematical proof for the first step, cognition generating code, is based on Turing's halting problem. The second step, code controlling chemical reactions, is the role of the genetic code. Thus a central question in biology: What is the nature and source of cognition? In this paper I propose a relationship between biology and Quantum Mechanics (QM), hypothesizing that the same principle that enables an observer to collapse a wave function also grants biology its agency: the organism's ability to act on the world instead of merely being a passive recipient. Just as all living cells are cognitive (Shapiro 2021, 2007; McClintock 1984; Lyon 2015; Levin 2019; Pascal and Pross, 2022), I propose humans are quantum observers because we are made of cells and all cells are observers. This supports the century-old view that in QM, the observer does not merely record the event but plays a fundamental role in its outcome.The classical world is driven by laws, which are deductive; the quantum world is driven by choices, which are inductive. When the two are combined, they form the master feedback loop of perception and action for all biology. In this paper I apply basic definitions of induction, deduction and computation to known properties of QM to show that the organism altering itself (and its environment) is a whole shaping its parts. It is not merely parts comprising a whole. I propose that an observer collapsing the wave function is the physical mechanism for producing negentropy. The way forward in solving the information problem in biology is understanding the relationship between cognition and QM. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
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