Mathematical Explanation Research Articles

The New Way to Solve Physical Problems Described by ODE of the Second Order with the Special Structure

In the last decade, many researchers have studied extensively theoretical and practical problems of natural sciences using ODEs as a means to analyze and understand them. Specifically, second-order ODEs with special complex structures provide the necessary tools to construct mathematical models for several physical - and other- processes such as the Schturm-Liouville, Schrölinger, Population, etc. As a result, it is of great importance to construct special stable methods of a higher order as a means to solve differential equations. One of the most important efficiency methods for solving these problems is the Stёrmer-Verlet method which consists of hybrid methods with constant coefficients. In this paper, we expand on recent studies that prove that the hybrid methods are more precise than the Stёrmer-Verlet method while investigating the convergence variable. This paper aims to prove the existence of a new, stable hybrid method using a special structure of degree(p)=3k+2, where k is the order of the multistep methods. Lastly, we also provide a detailed mathematical explanation of how to construct stable methods on the intersection of multistep and hybrid methods having a degree(p)≤3k+3.

Fractals: The Secret Code of Creation

Fractals: The Secret Code of Creation

Human motor behaviour. An outline of a system-theoretical approach.

The author presents a system-theoretical approach (STA) to the issue of the construction and control of motor operations in humans. It is based on the theory of Nikolai Bernstein, who developed a neurophysiological model called the ‘brain skyscraper’, of a systemic nature. Taking it as a model, a functional equivalent of a skyscraper, a modalities’ ladder, was developed. The common denominator of both these mental structures is the system-theoretical basis for the construction and control of motor operations in humans. The science of human motor behaviour – anthropokinetics – eludes mathematical description and explanation, which works well in physics. STA, however, seems to be a promising method of revealing specific aspects of anthropokinetics. This is one of the possible perspectives that allows us to analyse such a multifaceted and complex problem as human motor behaviour, which is the only observable manifestation of any mental activity. This is probably the greatest intellectual challenge for all modern science. STA is an original method of thinking about human motor behaviour, which probably reveals its specific aspects and thus may contribute to a better understanding of anthropokinetics. It also enables the construction of an original, novel family tree of science as a whole.

Mathematical Explanation in Computer Science

AbstractThis note scouts a broad but underexplored class of explanations found in contemporary computer science. These explanations, which I call limitative explanations, explain why certain problems cannot be solved computationally. Limitative explanations are philosophically rich, but have not received the attention they deserve. The primary goals of this note are to isolate limitative explanations and provide a preliminary account of what makes them explanatory. On the account I favor, limitative explanations are a kind of non-causal mathematical explanation which depend on highly idealized models of computation.

On the Insensitivity of Bit Density to Read Noise in One-Bit Quanta Image Sensors

The one-bit quanta image sensor (QIS) is a photon-counting device that produces binary measurements where each bit represents the presence or absence of a photon. The sensor quantizes the analog voltage into the binary bits using a threshold value <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${q}$ </tex-math></inline-formula> . The average number of ones in the bitstream is known as the bit density and is the sufficient statistics for signal estimation. An intriguing phenomenon is observed when the quanta exposure is at the unity and the threshold is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${q} = {0.5}$ </tex-math></inline-formula> . The bit density demonstrates an insensitivity as long as the read noise level does not exceed a certain limit. In other words, the bit density stays at a constant independent of the amount of read noise. This article provides a mathematical explanation of the phenomenon by deriving conditions under which the phenomenon happens. It was found that the insensitivity holds when some forms of the symmetry of the underlying Poisson–Gaussian distribution hold.

Approximation bounds for random neural networks and reservoir systems

This work studies approximation based on single-hidden-layer feedforward and recurrent neural networks with randomly generated internal weights. These methods, in which only the last layer of weights and a few hyperparameters are optimized, have been successfully applied in a wide range of static and dynamic learning problems. Despite the popularity of this approach in empirical tasks, important theoretical questions regarding the relation between the unknown function, the weight distribution, and the approximation rate have remained open. In this work it is proved that, as long as the unknown function, functional, or dynamical system is sufficiently regular, it is possible to draw the internal weights of the random (recurrent) neural network from a generic distribution (not depending on the unknown object) and quantify the error in terms of the number of neurons and the hyperparameters. In particular, this proves that echo state networks with randomly generated weights are capable of approximating a wide class of dynamical systems arbitrarily well and thus provides the first mathematical explanation for their empirically observed success at learning dynamical systems.

IMPORTANCE OF SUFFICIENT TIME POINTS FOR EFFICIENT PHARMACOKINETIC (PK) COMPARTMENTAL MODELING

Objective: Modeling and simulation are the two widely used terms, usually simultaneously mentioned in most PK discussions. There are several modeling strategies to model pharmacokinetic (PK) profiles. Compartmental modeling divides the body into different compartments based on the observed C-t profile and model comparison functions. Most C-t profiles are efficiently modeled using at max three compartments model (one, two, or three compartments). While there are many important applications of classical compartmental models, it emphasizes the importance of selecting the best model to explain the observed data. Therefore, initial data generation is very important. In many instances, insufficient data collection might not lead to the best model, which can be proved later costly by underpredicting or overpredicting PK parameters. This paper illustrates that adequate data collection can lead to correct model selection. Methods: Data was generated using the three-compartmental model's explicit equation for twenty-five simulated patients with 15% random variability. Generated data were fitted to different compartmental models using sufficient time points (case a) and without enough time points (case b). Results: In the case of a, generated data from three compartmental models was explained best by three compartmental models. In the case of b, the same data was presented better by two compartmental models. Finally, in the case of b, with sufficient time points, data generated from three compartmental models could be explained better by three compartmental models. Conclusion: With sufficient time points, the compartmental PK model can converge to an accurate one. Although almost all pharmacometricians know the importance of time points, there is no paper with a mathematical explanation of this incident. This paper will help the current and future pharmacometricians to help design efficient in vivo works.

Logical Priority in Aristotle’s Metaphysics M.2

In Metaphysics M, Aristotle aims to refute the Platonic view that mathematical objects are substantially prior to sensible things. For Aristotle, mathematical objects are the abstracted attributes of sensible substances required for geometrical analysis and proof. Yet, despite this derivative status of the objects of mathematics, Aristotle insists that they are logically prior to individual substances. This paper examines the distinction between logical and substantial priority, arguing that it underwrites Aristotle’s conception of mathematical necessity and explanation.

Numerical study on oxygen transport pattern in porous transport layer of proton exchange membrane electrolysis cells

A transient, two-dimensional and two-phase numerical model is established to study the nature of oxygen transport in proton exchange membrane electrolysis cells (PEMEC). Randomly generated circle fibers are used to construct a patterned porous transport layer (PTL), and the volume of fluid (VOF) method is employed to capture accurate gas-liquid interfaces. The patterns of oxygen transport and the relationship between multiple transport paths are analyzed in detail. We propose for the first time the concept of ‘isolation belt’, which is composed of a series of narrow throats, hindering the spontaneous combination of different oxygen transport paths. The influence of PTL perforation and spatially-graded fiber diameters on oxygen transport are explored. The numerical results show that perforation can probabilistically destroy the ‘isolation belt’, forming a transport path with high priority, which helps to reduce oxygen saturation. Moreover, when the porosity of PTL remains unchanged, reducing the fiber diameter in the region near the catalyst layer (CL) can significantly reduce the local oxygen saturation. The pressure variation reveals its consistency with the maximum throat width of the oxygen transport path, which indicates the law of capillary pressure. From the perspective of mathematical probability theory, a novel functional expression of the profile of oxygen saturation with respect to the structural properties of PTL is proposed to make a concise explanation of gas bubble transport in the context of PEMEC or similar gas-liquid porous flow. The highlights in this paper include: • Comprehensive spatiotemporal profile for in the process of oxygen penetration. • The concept of isolation belt for spontaneous merging of oxygen pathways. • A mathematical explanation for probability of oxygen saturation distribution. • Influence of PTL perforation and graded fiber diameters on oxygen saturation.

A Problem of Accuracy of Computer Calculations

The paper presented the results of the research related to the analysis of the reliability of computer calculations. Relevant examples of incorrect program operation were demonstrated: both quite simple and much less obvious, such as S. Rump's example. In addition to mathematical explanations, authors focused on purely software capabilities for controlling the accuracy of complex calculations. For this purpose, examples of effective use of the functionality of the decimal and fraction modules in Python 3.x were given.

The Effect of Cognitive Relevance of Directed Actions on Mathematical Reasoning

Theories of grounded and embodied cognition offer a range of accounts of how reasoning and body-based processes are related to each other. To advance theories of grounded and embodied cognition, we explore the cognitive relevance of particular body states to associated math concepts. We test competing models of action-cognition transduction to investigate the cognitive relevance of directed actions to students' mathematical reasoning in the area of geometry. The hypotheses we test include (1) that cognitively relevant directed actions have a direct effect on performance (direct cognitive relevance hypothesis), (2) that cognitively relevant directed actions lead to more frequent production of gestures during explanations, which leads to improved performance (mediated cognitive relevance hypothesis), and (3) that performance effects of directed actions are influenced by the presence or absence of gesture production during mathematical explanations (moderated cognitive relevance hypothesis). We explore these hypotheses in an experiment where high school students (N = 85) evaluated the truth of geometry conjectures after performing cognitively relevant or cognitively irrelevant directed actions while playing a movement-based video game. Contrary to the direct and mediated cognitive relevance hypotheses, we found no overall differences in performance or gesture production between relevant and irrelevant conditions. Consistent with the moderated cognitive relevance hypothesis, cognitive relevance influenced mathematical performance, as measured by the accuracy of students' intuitions, insights, and the validity of their proofs, provided that students produced certain kinds of gestures during mathematical explanations (i.e., with explanatory gestures as the moderator). Implications for theories of grounded and embodied cognition and the design of embodied forms of educational interventions are discussed.

Statistical analysis of locally parameterized shapes

In statistical shape analysis, the establishment of correspondence and defining shape representation are crucial steps for hypothesis testing to detect and explain local dissimilarities between two groups of objects. Most commonly used shape representations are based on object properties that are either extrinsic or noninvariant to rigid transformation. Shape analysis based on noninvariant properties is biased because the act of alignment is necessary, and shape analysis based on extrinsic properties could be misleading. Besides, a mathematical explanation of the type of dissimilarity, e.g., bending, twisting, stretching, etc., is desirable. This work proposes a novel hierarchical shape representation based on invariant and intrinsic properties to detect and explain locational dissimilarities by using local coordinate systems. The proposed shape representation is also superior for shape deformation and simulation. The power of the method is demonstrated on the hypothesis testing of simulated data as well as the left hippocampi of patients with Parkinson’s disease and controls.

A Unifying Theory for Quantum Physics, Part 2:

The quantum world is allegedly strange. But is it? What if there’s a simple mathematical explanation, and a simple solution? The success of quantum mechanics (QM) arises from the accuracy of its probability predictions, which are obtained by squaring amplitudes (the Born rule). Suppose for a moment that nature uses the negative of QM’s equations. When squared they would yield the same probabilities, confirmed by the same experiments and technological triumphs. If that were true, if nature uses the negative of QM’s equations, then the quantum world would become transparent, easy to understand. No more Schrödinger’s-cat. No quantum-eraser. No backwards-in-time cause-and-effect. No paradoxes nor enigmas. But, what’s a negative quantum equation? It could mean that particles follow zero-energy waves backwards, instead of forwards. That’s still an eccentric idea, a residual strangeness. Overall, it’s a bargain. We could swap one odd idea for another, because wave-particle duality is odd. We are accustomed to wave-particle duality. But we’ll show that experiments support the other arrangement: quantum particles follow zero-energy waves backwards. How could that possibly be true? We present substantial experimental evidence, new mathematics, and six dozen colorful illustrations. This is the Theory of Elementary Waves (TEW).

Overcoming the trade-off between the static sensitivity and resonant frequency of an FBG accelerometer

Although the method to overcome the tradeoff between the static sensitivity and resonant frequency of an FBG (fiber Bragg grating) accelerometer was reported, by narrowing the distance between the two fixed ends of the FBG, an explicit mathematical explanation has never been revealed until now. Besides the revelation, a novel mechanical structure is also proposed to effectively shorten the distance by moving the inertial object below the two fixed ends and hanging it on the FBG to apply forces. The crest-to-trough sensitivity 1.79 nm/g at 5 Hz and the resonant frequency 90 Hz are experimentally demonstrated.

Do mathematicians and undergraduates agree about explanation quality?

Offering explanations is a central part of teaching mathematics, and understanding those explanations is a vital activity for learners. Given this, it is natural to ask what makes a good mathematical explanation. This question has received surprisingly little attention in the mathematics education literature, perhaps because the field has no agreed method by which explanation quality can be reliably assessed. In this paper, we explore this issue by asking whether mathematicians and undergraduates agree with each other about explanation quality. A corpus of 10 explanations produced by 10 mathematicians was used. Using a comparative judgement method, we analysed 320 paired comparisons from 16 mathematicians and 320 from 32 undergraduate students. We found that both mathematicians and undergraduates were able to reliably assess the quality of a set of mathematical explanations. Furthermore, the assessments were largely consistent across the two groups. Implications for theories of mathematical explanation are discussed. We conclude by arguing that comparative judgement is a promising technique for exploring explanation quality.

Dynamics of dipolar quantum droplets in an extended Gross–Pitaevskii equation in the presence of time-dependent harmonic trapping potential and a damping term

The purpose of this paper is to study the dynamics of solutions to an extended Gross–Pitaevskii equation that models the formation of droplets in a dipolar Bose–Einstein condensate (BEC). The formation of these droplets has been recently discovered by driving the BEC into the strongly dipolar regime. Surprisingly, instead of collapsing, the system formed stable droplets. So far, no rigorous mathematical explanation has been proved. To the best of our knowledge, only experimental results have been obtained. The goal of this paper is to validate this breakthrough discovery. Many predictions/ conjectures properties of these droplets have been stated by some research groups in physics and engineering. In particular, it has been claimed that the stability of these droplets is a consequence of the presence of the damping term in the extended Gross–Pitaevskii equation under study. This term describes the three-body loss process. To accurately model the dynamics of formation of these droplets, it is necessary to consider a time-dependent harmonic trapping potential as well as other terms with different types of nonlinearity among them that describe the Lee–Huang–Yang (LHY). This presents some challenges that will be solved in this paper.

SYSTEMATIC APPROACH TO VALUE EVALUATION OF THE PATENT PORTFOLIO

The article points out the advantages of conducting agreements on purchase and sale of intellectual property not individual patents, but a portfolio of patents. A systematic analysis of the range of valuation of intellectual property and the patent portfolio was conducted. A mathematical explanation of the "maximum-minimum" values is given, as well as the separation of the actual market price is substantiated. A graphical analysis of the stages of life cycles is given. Based on the proposed mathematical model, the value of the basic invention according to the patent UA85101 and the general portfolio of patents of JSC "Ukrainian Power Machines" (Kharkiv, Ukraine) as of 01.02.2022, which is 1755810 UAH. These calculations assume the best economic effect of using the rights to this portfolio of patents both in-house and obtaining income from the sale of a non-exclusive license. The practical applicability of the study allows JSC "Ukrainian Energy Machines" to assess the possibility of improving the efficiency of intellectual property management, which provides maximum benefit from the commercialization of innovations.

Nonlinear biological population model; computational and numerical investigations

This research paper provides precise solutions for nonlinear fractional population biology (FBP) models by implementing the generalized Khater (GK) technique and utilizing Atangana's conformable fractional (ACF) derivative operator. Because people have natural death and a birth rate, we can get demographic information using a model. Many physical features, such as exponential, hyperbolic, and trigonometric functions, were given a mathematical explanation. To get a better understanding of such methods, one should draw them in various ways. Many novel analytical solutions are obtained in distinct formulas, such as rational, hyperbolic, parabolic, etc.Additionally, we use Hamiltonian system properties to confirm the stability of the solutions we get. Finally, the trigonometric- quantic-B- spline (TQBS) scheme tests the accuracy of the obtained solutions through the novel applied analytical method by revealing potential problems and eliminating absolute errors. The power and effectiveness of the two employed analytical and numerical techniques are verified through their performances.

A metaphysical foundation for mathematical philosophy

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.

Examining technology-supported teacher responding and students’ written mathematical explanations

Examining technology-supported teacher responding and students’ written mathematical explanations