Mathematical Explanation Research Articles

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

Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle

Biological processes are governed by the expression of proteins, and for some proteins, their level of expression can fluctuate periodically over time (i.e., they oscillate). Many oscillatory proteins (e.g., cell cycle proteins and those from the HES family of transcription factors) are connected in complex ways, often within large networks. This complexity can be elucidated by developing intuitive mathematical models that describe the underlying critical aspects of the relationships between these processes. Here, we provide a mathematical explanation of a recently discovered biological phenomenon: the phasic position of the gene Hes1’s oscillatory expression at the beginning of the cell cycle of an individual human breast cancer stem cell can have a predictive value on how long that cell will take to complete a cell cycle. We use a two-component model of coupled oscillators to represent Hes1 and the cell cycle in the same cell with minimal assumptions. Inputting only the initial phase angles, we show that this model is capable of predicting the dynamic mitosis to mitosis behaviour of Hes1 and predicting cell cycle length patterns as found in real-world experimental data. Moreover, we discover that bidirectional coupling between Hes1 and the cell cycle is critical within the system for the data to be reproduced and that nonfixed asymmetry in the interactions between the oscillators is required. The phase dynamics we present here capture the complex interplay between Hes1 and the cell cycle, helping to explain nongenetic cell cycle variability, which has critical implications in cancer treatment contexts.

Cognitive bias and attitude distortion of a priority decision

The resource saving bias is a cognitive bias describing how resource savings from improvements of high-productivity units are overestimated compared to improvements of less productive units. Motivational reasoning describes how attitudes, here towards private/public health care, distort decisions based on numerical facts. Participants made a choice between two productivity increase options with the goal of saving doctor resources. The options described productivity increases in low-/high-productivity private/public emergency rooms. Jointly, the biases produced 78% incorrect decisions. The cognitive bias was stronger than the motivational bias. Verbal justifications of the decisions revealed elaborations of the problem beyond the information provided, biased integration of quantitative information, change of goal of decision, and motivational attitude biases. Most (83%) of the incorrect decisions were based on (incorrect) mathematical justifications illustrating the resource saving bias. Participants who had better scores on a cognitive test made poorer decisions. Women who gave qualitative justifications to a greater extent than men made more correct decision. After a first decision, participants were informed about the correct decision with a mathematical explanation. Only 6.3% of the participants corrected their decisions after information illustrating facts resistance. This could be explained by psychological sunk cost and coherence theories. Those who made the wrong choice remembered the facts of the problem better than those who made a correct choice.

About the Teaching and Learning of Differentiability for Piecewise Functions in Science Degrees’ First-Year Calculus Courses

In this work, we present an issue we have observed over the last several years in the first year of Spanish science degrees’ (such as mathematics, physics or engineering) calculus courses. It is related to the study of differentiability of piecewise one real variable real functions at a point. We have analyzed how students study the differentiability of piecewise functions at a point, explaining the students’ work and reasoning and commenting about the common misunderstandings we have found. Then, we have researched about how to help the students in their learning process related to this calculus concept, and we have also used several activities or ways to work: student meeting groups to talk about their worries and misunderstandings, mathematical definitions, explanations and theory and also working out specific examples.

A Fair and Cooperative MAC Protocol for Heterogeneous Cognitive Radio Enabled Vehicular Ad-Hoc Networks

Advancements in intelligent transportation necessitate a dependable medium access control (MAC) protocol to enable high-priority safety broadcasts. Implementing cognitive radio-based vehicular ad-hoc networks (CR-VANETs) alleviates the significant growth in spectrum demand. The architecture must deploy multiple networks to make CR-VANET viable; however, the MAC design is a substantial challenge. The hardware-limited vehicles have varied characteristics and do not attain the same performance when provided with equal resources, posing a fairness concern. The resource allocation should balance two conflicting objectives: performance should be increased, prioritizing a few users under better conditions, and fairness among users should be emphasized, which reduces the overall performance. This paper presents a Fair and Cooperative MAC protocol for heterogeneous CR-VANETs (FCCR-MAC) to find this balance. In the event of a primary user appearance, cooperation allows pro-active channel switching. In FCCR-MAC, mode selection and retransmission of unsuccessful broadcasts are enabled by including additional control signals. By assisting users in determining the identities of others, cooperation also improves the system’s fairness. Markov Chain and the Karush–Kuhn–Tucker (KKT) conditions are utilized to offer a unified mathematical explanation of the throughput versus fairness trade-off optimization problem. According to the findings, the FCCR-MAC protocol outperforms other protocols with better fairness.

Reallocation of cutaneous and global blood circulation during sauna bathing through a closed-loop model

ObjectiveSauna bathing (SB) is an important strategy in cardiovascular protection, but there is no mathematical explanation for the reallocation of blood circulation during heat-induced superficial vasodilation. We sought to reveal such reallocation via a simulated hemodynamic model. MethodsA closed-loop cardiovascular model with a series of electrical parameters was constructed. The body surface was divided into seven blocks and each block was modeled by a lumped resistance. These resistances were adjusted to increase skin blood flow (SBF), with the aim of reflecting heat-induced vasodilation during SB. Finally, the blood pressure was compared before and after SB, and the blood flow inside the aorta and visceral arteries were also analyzed. ResultsWith increasing SBF in this model, the systolic, diastolic, and mean blood pressure in the arterial trunk decreased by 13–29, 18–36, and 19–37 mmHg, respectively. Despite the increase in the peak and mean blood flow in the arterial trunk, the diastolic blood flow reversal in the thoracic and abdominal aortas increased significantly. Nevertheless, the blood supply to the heart, liver, stomach, spleen, kidney, and intestine decreased by at least 25%. Moreover, the pulmonary blood flow increased significantly. ConclusionSimulated heat-induced cutaneous vasodilation in this model lowers blood pressure, induces visceral ischemia, and promotes pulmonary circulation, suggesting that the present closed-loop model may be able to describe the effect of sauna bathing on blood circulation. However, the increase of retrograde flow in the aortas found in this model deserves further examination.

Benfordness of the Generalized Gamma Distribution

The generalized gamma distribution shows up in many problems related to engineering, hydrology as well as survival analysis. Earlier work has been done that estimated the deviation of the exponential and the Weibull distribution from Benford's Law. We give a mathematical explanation for the Benfordness of the generalized gamma distribution and present a measure for the deviation of the generalized gamma distribution from the Benford distribution.

Eulerian Routing in Practice

The Königsberg bridge problem has played a central role in recent philosophical discussions of mathematical explanation. In this paper I look at it from a novel perspective, which is independent of explanatory concerns. Instead of restricting attention to the solved Königsberg bridge problem, I consider Euler’s construction of a solution method for the problem and discuss two later integrations of Euler’s approach into a more structured methodology, arisen in operations research and genetics respectively. By examining Euler’s work and its later developments, I achieve two main goals. First, I offer an analysis of the role played by mathematics as a problem-solving instrument within scientific enquiry. Second, I shed light on the broader significance of well known contributions to the debate on mathematical explanation. I suggest that these contributions, which are tied to a localised explanatory context, achieve a greater relevance and attain a sharper formulation when they are referred to scientific enquiry at large, as opposed to its possible explanatory outcomes alone.

Not so distinctively mathematical explanations: topology and dynamical systems

So-called ‘distinctively mathematical explanations’ (DMEs) are said to explain physical phenomena, not in terms of contingent causal laws, but rather in terms of mathematical necessities that constrain the physical system in question. Lange argues that the existence of four or more equilibrium positions of any double pendulum has a DME. Here we refute both Lange’s claim itself and a strengthened and extended version of the claim that would pertain to any n-tuple pendulum system on the ground that such explanations are actually causal explanations in disguise and their associated modal conditionals are not general enough to explain the said features of such dynamical systems. We argue and show that if circumscribing the antecedent for a necessarily true conditional in such explanations involves making a causal analysis of the problem, then the resulting explanation is not distinctively mathematical or non-causal. Our argument generalises to other dynamical systems that may have purported DMEs analogous to the one proposed by Lange, and even to some other counterfactual accounts of non-causal explanation given by Reutlinger and Rice.

Research on Mathematics Classroom Teaching Optimization Model Based on GA Neural Network

The teaching characteristic of colleges and universities is determined by the teaching quality of teachers, but it is difficult to calculate with a linear mathematical explanation how to judge the teaching quality of teachers. In most colleges and universities, expert evaluations, supervision groups, peers listening to lectures in the classroom, and students' after-class evaluations are still used to determine the teaching ability of professors. It is undeniable that these judgment methods have certain practicality. Especially for new teachers, experts and peers can find problems in time and help new teachers quickly correct them, and they can also know how to better practice the “student-centered” teaching concept from the feedback of students after class. However, these judgment standards also have their limitations. For example, the classroom quality judgment standards for experts and peers are mostly set by some administrative departments according to social requirements, leadership requirements, and subjective cognition. Various institutions should have different weights for the same criterion, and if they all employ the same criteria, it will easily lead to imprecise conclusions. In addition, the expert group and students are also subjective and incomplete in the stage of teacher evaluation. Based on different nonquantitative factors, the evaluation standards for teachers in colleges and universities also need the scientific theoretical basis. Therefore, many scientists are considering whether to use a more intelligent and reasonable way to judge the quality of teaching. Nowadays, with the popularization of artificial intelligence (AI), the use of neural networks (NNs) is becoming more and more comprehensive. According to the characteristics of NNs, this paper uses the GA NNs algorithm to signify that this algorithm can be effectively used in the evaluation of teachers' teaching quality. In this paper, a large number of experimental results show that the use of GA NNs can be used to evaluate the quality of teaching, and the evaluation results can be obtained quantitatively in this way. For the assessment of teaching quality, quantitative results are also more powerful in teaching evaluation.

A Petrov–Galerkin finite element approach for the unsteady boundary layer upper-convected rotating Maxwell fluid flow and heat transfer analysis

A numerical investigation is carried out in the present study to view heat transfer framing indifferent Cattaneo–Christov hyperbolic heat flux equation. A numerical scheme built on the Petrov–Galerkin finite element method (PG-FEM) with shooting technique is incorporated successfully to examine the flow regime. In addition, mathematical explanation of energy equation in detail is studied which is in a transient state and unmatched in the accessible literature. Pertinent unsteady parameter for boundary layer flow is analyzed for physical implication. Moreover, momentum and energy profiles are inspected with various physical effects adjacent to thermal relaxation time, rotational parameters, Prandtl and Deborah numbers. A refined numerical strategy based on the PG-FEM scheme is implemented and validated for the results. These results found strong agreement in limiting cases to Mustafa [Cattaneo-Christov heat flux model for rotating flow and heat transfer of upper-convected Maxwell fluid. AIP AdvAIP Advances. 2015;5:047109] and with other numerical results available in existing literature as well.

Inference to the best explanation as supporting the expansion of mathematicians’ ontological commitments

This paper argues that in mathematical practice, conjectures are sometimes confirmed by “Inference to the Best Explanation” (IBE) as applied to some mathematical evidence. IBE operates in mathematics in the same way as IBE in science. When applied to empirical evidence, IBE sometimes helps to justify the expansion of scientists’ ontological commitments. Analogously, when applied to mathematical evidence, IBE sometimes helps to justify mathematicians' in expanding the range of their ontological commitments. IBE supplements other forms of non-deductive reasoning in mathematics, avoiding obstacles sometimes faced by enumerative induction or hypothetico-deductive reasoning. Both platonist and non-platonist interpretations of mathematics ought to accommodate explanation in mathematics and ought to recognize IBE in mathematics, though these interpretations disagree on the ontological commitments that mathematicians ought to have. This paper offers an inductive account of why mathematical IBE tends to lead to mathematical truths.

Mathematical Neural Networks

ANNs succeed in several tasks for real scenarios due to their high learning abilities. This paper focuses on theoretical aspects of ANNs to enhance the capacity of implementing those modifications that make ANNs absorb the defining features of each scenario. This work may be also encompassed within the trend devoted to providing mathematical explanations of ANN performance, with special attention to activation functions. The base algorithm has been mathematically decoded to analyse the required features of activation functions regarding their impact on the training process and on the applicability of the Universal Approximation Theorem. Particularly, significant new results to identify those activation functions which undergo some usual failings (gradient preserving) are presented here. This is the first paper—to the best of the author’s knowledge—that stresses the role of injectivity for activation functions, which has received scant attention in literature but has great incidence on the ANN performance. In this line, a characterization of injective activation functions has been provided related to monotonic functions which satisfy the classical contractive condition as a particular case of Lipschitz functions. A summary table on these is also provided, targeted at documenting how to select the best activation function for each situation.

An Investigation of Mathematics Teachers' Specialized Content Knowledge Related to Basic Concepts about Sets

This research was carried out in order to examine the specialized content knowledge of mathematics teachers related to teaching basic concepts about sets within the MKT Model. In this study, a holistic single-case study design is used in qualitative research methods. The study group consists of 18 mathematics teachers at different educational background and professional experience in teaching sets selected through the sampling of criterion, which is working in different high schools in one province in the Mediterranean region in the academic year 2016-2017. Research data were obtained using semi-structured interview, observation and document analysis techniques. In the analysis of obtained data, descriptive analysis, content analysis, continuous comparison techniques and special data analysis methods were used together with each data collection tool. The findings of the research show that teachers gave correct answers to the questions asked about the basic concepts of "Why?" in the sets. However, it is clear that the teachers cannot write sufficient reason why the asked questions are "Why?". As a result of the research, it has been determined that the teachers have superficial knowledge about the concepts of "Set, Universal Set, Infinite Set" and "Equal Set". However, it was determined that teachers made explanations with a general mathematical knowledge based on the characteristics of the concept rather than providing persuasive mathematical explanations to the questions asked.

An analysis of random elections with large numbers of voters

In an election in which each voter ranks all of the candidates, we consider the head-to-head results between each pair of candidates and form a labeled directed graph, called the margin graph, which contains the margin of victory of each candidate over each of the other candidates. A central issue in developing voting methods is that there can be cycles in this graph, where candidate A defeats candidate B, B defeats C, and C defeats A. It is known that such cycles are unlikely to occur. Under the Impartial Culture assumption, in a random election with three candidates and a very large number of voters there is a 91.23% chance of avoiding a cycle. By studying the geometry of the space of random elections, we give a mathematical explanation of why this is the case. Our main result is that margin graphs that are more cyclic in a certain precise sense are less likely to occur. This connects the probabilistic study of voting methods to Zwicker’s analysis of Condorcet’s paradox in terms of cycles and cuts.

¿CUÁNTO TIEMPO TARDA EN DUPLICARSE SU DINERO O DEUDA? EXPLICANDO LA REGLA DEL 72 A LOS ESTUDIANTES

The ‘rule of 72’ provides a useful approximation of when an investment or debt will double. Students can apply it for an estimate, avoiding mistakes later when using technology for a precise answer. On standardized tests, moreover, such devices may be disallowed. In job interviews, too, quickly approximating the doubling answer demonstrates the impressive problem-solving ability. Illustrations abound online and in traditional media showing how to use it. This is not the same as explaining why it works or the limitations. Inquiring students want to know. This paper combines familiar territories in math and application to provide a relatively simple mathematical explanation.

MODELING OF FLEXIBLE FUNCTIONING INDICATORS OF PRODUCTION AND ECONOMIC SYSTEM

The created economic and mathematical models allowed to calculate the quantitative ratio of product costs and investment costs for progressive technological renewal of production, and established interval limits to predict possible changes in production as a result of new technologies and plan the results of the enterprise depending on its investment and innovation activity. During the analysis, the parameters that characterize the quality of the developed models and indicate their adequacy, and feasibility in practice are calculated. Scientific novelty of the research is a mathematical explanation of the laws of interaction of innovation and production activities, estimated quantitative impact of new technologies on the growth of efficiency of industrial enterprises. The article considers the current problems of development of the production and economic system. The authors present a modeling of the prerequisites for step-by-step flexible adjustment of strategies to achieve the expected (predicted) value of the components of the flexible functioning of a particular production and economic system (market conditions). As a result, it was determined that the distance between two points on the plane (t, R) in management understanding and economic perception is a choice at a certain point in time t, belonging to the interval [t0, T] of the appropriate strategy of flexible dynamic transition (with the greatest benefit or with the lowest costs), from one (t0, R (t0)), specified in certain units of measurement of their coordinates, to another (T, R (T)), the parameters of which correspond to the value of their possible achievement, i.e. a certain expected (planned) or predictable and even intuitive forecast indicator. Determining the value of the distance between two indicators creates the preconditions for step-by-step flexible adjustment of strategies to achieve its expected (predicted) value. At any time t1, the value of the final T = T ± △ t can be adjusted, the strategy for achieving it fi (t) (1, 2… 7) can be changed, and the total amount of possible costs can be changed: V_i=∫_(t_i)^T▒〖f_(i ) (t)dt〗. Keywords: production and economic system, flexible dynamic operation, modeling, forecasting, market conditions.

Inhibitive performance of 4-Methoxyphenethylamine on low-carbon steel in 1 M hydrochloric acid: Kinetics, theoretical, and mathematical views

The current study looked at the inhibition performance of 4-Methoxyphenethylamine (MPTA) in 1 M HCl. Besides the experimental investigation, a theoretical and mathematical attempt to correlate some chemical characteristics and inhibitor efficacy was also made. The inhibiting effect was investigated by potentiodynamic polarization, electrochemical impedance spectroscopy, and weight loss techniques. Corrosion rate was evaluated as function of time, MPTA concentration, and temperature. The results show that MPTA suppressed both cathodic and anodic processes of low-carbon steel corrosion in acidic solutionby adsorption on the surface, which followed a Langmuir adsorption isotherm. Maximum corrosion inhibition efficiency was 91% at 30 °C and 200 ppm. Kinetics studies revealed that the corrosion reaction was behaved according to the first order mechanism. Mathematical studies showed a high correlation coefficient was obtained between predicated and experimental corrosion rate. The 6-311G (d, p) base set and Density Functional Theory (DFT) were employed. MPTA's inhibitory capabilities were discovered to be connected to the charge on the nitrogen atom as well as the total of the net charges of the six atoms in the cyclic ring. MPTA was studied in a three states, ground state (1), oxygen protonated state (2), nitrogen protonated state (3). Physical characteristics such as dipole moment, hardness, ionization energy, and softness of MPTA and its protonated states were computed. Theoretical order of inhibitor efficiency was 2 > 3 > 1. It is concerned with an experimental, theoretical, and mathematical explanations between electronic and structural characteristics of MPTA and its inhibitor efficiency.

A Theological Approach to the Simplest Mathematical Explanation of the Dynamics of the Temporal Universe

Guided heuristically by the theism of ancient Egypt where religion underpinned cosmology, to explain the dynamics of the temporal universe (gravitation, rotation, and translation), this paper starts from the existence of individualities and the law of causality to build a cosmological argument, into a natural systematic theology (NST). According to this NST, God (the greatest possible being) includes the creator as one of his celestial manifestations. One deduces from the NST that, manifesting an individuality included in the indivisible and immutable God, the creator expresses the fullness of the Most High in an individual manner. This fullness (the Logos) is thus, like God, a constant power working in the creator. The existence of the temporal universe in God requires the existence of a principle of the mutability of God at the occurrence of creation ex nihilo; in accord to the law of causality this principle must greater than God, which is impossible. Moreover, the existence of creation outside of God implies an entity greater than God including him and creation, this is also impossible as God is the greatest possible being. Therefore, the temporal universe exists in the temporal consciousness of the creator as a mere appearance of, or perspective on, the celestial reality. Thus, the NST dictates the existence of an absolute space-time that includes the Euclidean space-time known in Newtonian physics and corresponds to the temporal consciousness of the creator. The NST proves that to create, the creator had first to leave the eternal plane for the temporal one. Then on, as a constant power, the Logos impels the creator to accelerate back toward the celestial-eternal level. This isotropic acceleration of the creator causes the absolute space-time to accelerate towards its nothingness, its non-existence in the celestial realm. Therefore, this background acceleration of the absolute space-time is the simplest, exhaustive, deterministic, and mathematical explanation of the dynamics of the temporal universe (gravitation, rotation, translation). This explanation offers the advantage of using elementary notions of algebra and analysis and the result applies to the astronomical level as well as to the subatomic one.