Mathematical Rigor Research Articles

Ructions over fluxions: Maclaurin’s draft, The Analyst Controversy and Berkeley’s anti-mathematical philosophy

Following the publication of Berkeley's caustic critique of calculus, The Analyst (1734), numerous figures in the scientific world responded to the text's allegations of rigour violations and a culture of mathematics-laced heresy. Maclaurin's Treatise on Fluxions (1742) shaped the reception of Berkeley's mathematical views. The published version of the Treatise focused on securing the foundations of Newton's fluxions against Berkeley's claims that they were unrigorous. However, an earlier draft has a different, ideological focus, and engaging with this text sheds new light on both Maclaurin and Berkeley. I will argue that Maclaurin was concerned with much more than simply mathematical foundations and rigour and that his initial response helps us to understand Berkeley's hostilities to mathematics in their proper context. The ultimate aim of this paper is to demonstrate that what was at stake were issues of mathematical rhetoric and reputation, and questions about the morality of mathematicians in an emerging culture of mathematical authority.

Unveiling the Physics of Partial Differential Equations with Heuristics

Heuristic arguments and order of magnitude estimates for partial differential equations highlight essential features of the physics they describe. We present order of magnitude estimates, and their limitations, for the three classic second order PDEs of mathematical physics (wave, heat, and Laplace equations), for first order transport equations, and for two nonlinear wave equations. It is beneficial to expose the beginning student to these considerations before jumping into more rigorous mathematics. Yet these simple arguments are missing from physics textbooks.

Spreading Equilibria Under Mildly Singular Potentials: Pancakes Versus Droplets

We study global minimizers of a functional modeling the free energy of thin liquid layers over a solid substrate under the combined effect of surface, gravitational, and intermolecular potentials. When the latter ones have a mild repulsive singularity at short ranges, global minimizers are compactly supported and display a microscopic contact angle of pi /2. Depending on the form of the potential, the macroscopic shape can either be droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient for purely repulsive potentials. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed.

Closure to “Discussion: Selective-Compliance-Based Lagrange Model and Multilevel Noncollocated Feedback Control of a Humanoid Robot” (Spyrakos-Papastavridis, E., Dai, J. S., Childs, P. R. N., and Tsagarakis, N. G., 2018, ASME J. Mech. Rob., 10(3), p. 031009)

Abstract This closure exposes the technical inconsistencies, erroneous comments, and misconceptions contained in the Discussion Article (Medrano-Cerda, 2020, “Discussion: ‘Selective Compliance-Based Lagrange Model and Multilevel Noncollocated Feedback Control of a Humanoid Robot’,” ASME J. Mech. Rob., 10.1115/1.4049005 [1]). The majority of incorrect comments relating to stability are not based on mathematical rigor and equations, but on the reproduction of more generic statements made in some textbooks published in the early 1990s. The comments in the Discussion Article fail to demonstrate how any of these generic statements could be applied to the specific cases presented in the Paper (Spyrakos-Papastavridis et al., 2018, “Selective-Compliance-Based Lagrange Model and Multilevel Noncollocated Feedback Control of a Humanoid Robot,” ASME J. Mech. Rob., 10(3). 10.1115/1.4039394 [2]) criticized by the Discussion Article. Further, the only equation that appears in the Discussion Article is incorrect, as will be demonstrated in this closure. Apart from being expressed incorrectly, the text accompanying this equation reveals the fundamentally incorrect mathematical assumption that a diagonal matrix is not equal to its transpose. Moreover, a significant proportion of the comments in the Discussion Article relies upon misconceptions about flexible-joint robot control and humanoid robotics, in terms of their theory and practice, standing in contrast to results reported in these research fields over the last decades.

The Role of Imagination and Anticipation in the Acceptance of Computability Proofs: A Challenge to the Standard Account of Rigor

ABSTRACT In a 2022 paper, Hamami claimed that the orthodox view in mathematics is that a proof is rigorous if it can be translated into a derivation. Hamami then developed a descriptive account that explains how mathematicians check proofs for rigor in this sense and how they develop the capacity to do so. By exploring introductory texts in computability theory, we demonstrate that Hamami’s descriptive account does not accord with actual mathematical practice with respect to computability theory. We argue instead for an alternative account in which imagination, anticipation, and interpretations of natural language play roles in establishing mathematical rigor.

M-Theory and F-Theory over Theoretical Analysis on Cosmic Strings and Calabi-Yau Manifolds Subject to Conifold Singularity with Randall-Sundrum Model

String theory always comes with heavy mathematical rigor as it questions the most significant and impossible attempt to make a scale-invariant phenomenology between general relativity and quantum theory. Thus, steps have been taken to simplify the theory a bit thereby making it accessible to general yet enthusiastic readers of physics. However, as there is numerous mathematics involved in the modeling of this theory, thus, any chance to make a purely non-mathematical approach towards strings would prove vacuous and intimidating making the pathway of this marvelous theory chocked with unnecessary assumptions resulting in false analogies (or hopes) relating to this theory. Thus, where it’s almost impossible to proceed without any equations, we have given a few just to wipe out some logical confusion arising to the first readers of strings. Few necessary diagrams are included along with intense theory and least mathematics for making this significant approach of theoretical physicists accessible to general learners or readers.
 Topics: Bosonic string theory, supersymmetric string theory, M – theory, F – theory, dualities and interconnectedness, viability, Randall – Sundrum model for tackling the hierarchy problem of particle physics, conifold singularities, Branes, Bulks, Extremal black holes, Ekpyrotic cosmology, topological aspects of Calabi – Yau (CY) manifolds, A and B models, Mirror Symmetries; AdS/CFT, cosmic strings, all in a way accessible to every reader.
 Methods: Theoretical analogies, deductions, principles behind the origin, development along with the probable conclusion of this theory, the roots of its origin, the necessary difficulty for detecting those strings, and approaches done by theorists to work out the pathway of achieving Einstein’s dream of unification irrespective of several hindrances.
 Results: String theory itself is not a complete theory. Rather it’s in the process of further development through the increment of time resulting in more applications of mathematics by developing or incorporating them in due needs. Thus, without stating any concrete results, the theory has been tackled in this paper with a viable hypothesis based on the current understanding, and previous attempts are stated to have been made for its success.

Today’s Civil Rights Fight: What’s Math Got to Do With It?

Research shows student success in advanced-level science, technology, engineering, and mathematics (STEM) programs is connected to future educational and economic opportunities. Yet, Black students have been excluded from participating in rigorous mathematics courses that prepare them to engage in STEM majors and careers. This article reviews research on Black student mathematics participation at the pre-college, post-secondary, and professional levels. Three systemic barriers to Black student participation are limited access, limiting mindsets and beliefs, and lack of support. Eliminating these barriers can result in an increase in Black students’ enrollment, persistence, and achievement in advanced mathematics courses. The article provides strategies proven successful in urban contexts to ensure equitable learning environments that maximize the full potential of Black students. Educational leaders must address systemic inequities rooted in racism, purposefully engage Black students in rigorous and extended learning opportunities, and provide them with peer and faculty support.

Characterizing the propagation pathway of neuropathological events of Alzheimer's disease using harmonic wavelet analysis.

Empirical imaging biomarkers such as the level of the regional pathological burden are widely used to measure the risk of developing neurodegenerative diseases such as Alzheimer's disease (AD). However, ample evidence shows that the brain network (wirings of white matter fibers) plays a vital role in the progression of AD, where neuropathological burdens often propagate across the brain network in a prion-like manner. In this context, characterizing the spreading pathway of AD-related neuropathological events sheds new light on understanding the heterogeneity of pathophysiological mechanisms in AD. In this work, we propose a manifold-based harmonic network analysis approach to explore a novel imaging biomarker in the form of the AD propagation pattern, which eventually allows us to identify the AD-related spreading pathways of neuropathological events throughout the brain. The backbone of this new imaging biomarker is a set of region-adaptive harmonic wavelets that represent the common network topology across individuals. We conceptualize that the individual's brain network and its associated pathology pattern form a unique system, which vibrates as do all natural objects in the universe. Thus, we can computationally excite such a brain system using selected harmonic wavelets that match the system's resonance frequency, where the resulting oscillatory wave manifests the system-level propagation pattern of neuropathological events across the brain network. We evaluate the statistical power of our harmonic network analysis approach on large-scale neuroimaging data from ADNI. Compared with the other empirical biomarkers, our harmonic wavelets not only yield a new imaging biomarker to potentially predict the cognitive decline in the early stage but also offer a new window to capture the in-vivo spreading pathways of neuropathological burden with a rigorous mathematics insight.

Spectral invariants of ergodic symbolic systems for pattern recognition and anomaly detection.

Despite tangible advances in machine learning (ML) over the last few decades, many of the ML techniques still suffer from fundamental issues like overfitting and lack of explainability. These issues mandate requirements for mathematical rigor to ensure robust learning from observed data. In this context, topological invariants in data manifolds provide a rich representation of the underlying dynamical system, which can be utilized for developing a mathematically rigorous ML tool to characterize the dynamical behaviour and operational phases of the underlying process. This paper aims to investigate spectral invariants of symbolic systems for detecting changes in topological characteristics of data manifolds. A novel ML approach is proposed, where commutator norms are used on sequences of endomorphisms to symbolically describe dynamical systems on probability spaces with ergodic measures. The objective here is to detect topological invariants of data manifolds that can be used for signal processing, pattern recognition, and anomaly detection. The proposed ML approach is validated on models of selected chaotic dynamical systems for prompt detection of phase transitions. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

On the Inertial Range Bounds of K-41-like Magnetohydrodynamics Turbulence.

The spectral slope of magnetohydrodynamic (MHD) turbulence varies depending on the spectral theory considered; is the spectral slope in Kraichnan–Iroshnikov–Dobrowolny (KID) theory, in Marsch–Matthaeus–Zhou and Goldreich–Sridhar theories, also called Kolmogorov-like (K-41-like) MHD theory, the combination of the and scales in Biskamp, and so on. A rigorous mathematical proof to any of these spectral theories is of great scientific interest. Motivated by the 2012 work of A. Biryuk and W. Craig (Physica D 241(2012) 426–438), we establish inertial range bounds for K-41-like phenomenon in MHD turbulent flow through a mathematical rigor; a range of wave numbers in which the spectral slope of MHD turbulence is proportional to is established and the upper and lower bounds of this range are explicitly formulated. We also have shown that the Leray weak solution of the standard MHD model is bonded in the Fourier space, the spectral energy of the system is bounded and its average over time decreases in time.

Fractal dimension of Posidonia oceanica meadows for the assessment of their ecological condition

Ecological analyses are aimed at characterizing the complexity of the structure of natural objects, yet their heterogeneity is hardly described by the Euclidean concepts. For such purpose, the fractal geometry can be best suited due to its ability in describing, with mathematical rigor, the inherent irregularity of nature. Fractal dimension provides indeed a measurement of the complexity of the analyzed object in terms of space occupation. In this study, we applied the fractal geometry to Posidonia oceanica in order to characterize the structural complexity of its meadows, which are widely recognized as one of the most important coastal ecosystems in the Mediterranean basin. For achieving our aim, we developed an ad hoc implementation of the Box-Counting algorithm based on the Moore neighborhood analysis. Our approach allowed to render the structural complexity of P. oceanica meadows spatially explicit, thus expressing an intrinsic ecological property. The fractal analysis suggested that the complexity of meadows structure is intimately connected with the ecological conditions of P. oceanica. In fact, meadows in living and mixed conditions showed a higher fractal dimension, suggesting a largely uniform and smooth structure. While the fractal dimension associated to the regressed ecological condition of P. oceanica meadows exhibited lower values, highlighting a more jagged and rough structure. Therefore, the fractal theory may prove useful to both fundamental and applied ecological research focusing on P. oceanica and its interactions with Mediterranean coastal ecosystems. In fact, the fractal analysis we performed could result in an effective and straightforward approach for assessing the condition of P. oceanica at large spatial scale, enhancing an integrated maritime spatial planning over the whole Mediterranean basin.

Testing the NCTM 2020 Standards Using Rigorous Mathematics and Multiple Solutions to a Single Geometric Problem

Testing the NCTM 2020 Standards Using Rigorous Mathematics and Multiple Solutions to a Single Geometric Problem

Examining Pre-service Mathematics and Science Teachers’ Plans to Implement Culturally Relevant Pedagogy

ABSTRACT There is a need for greater guidance in preparing pre-service teachers to enact culturally relevant pedagogy while maintaining the content and rigor of mathematics and science in middle and secondary grades. This study investigated a group of pre-service teachers’ plans to implement culturally relevant pedagogy in middle/secondary mathematics and science classrooms. Through this action research study, we sought to apply an existing culturally relevant pedagogy framework to coursework showcasing instructional planning in middle/secondary mathematics/science methods courses. This analysis would serve to help teacher educators systematically define evidence of successful culturally relevant pedagogy-aligned instructional planning, as well as identifying areas for improvement. In the coursework reviewed, there were non-traditional and traditional approaches to structuring learning tasks and activities in order to integrate elements of culturally relevant instruction. Among our results, we concluded that methods instruction could emphasize instructional planning that begins with culturally-conscious frameworks that aim to disrupt historical power dynamics, rather than conclude by making culturally significant connections, thereby supporting more pre-service teachers in possibly developing radically transformed mathematics and science instructional plans.

Ab initio method of calculating invariant measure for turbulent flow

We present a method of calculating the invariant measure (IM) of hydrodynamical systems describing incompressible viscous fluid flow in bounded 3D volume V. Using a basis { w I } of zero-divergence vector functions on V satisfying boundary conditions we transform the Navier–Stokes equations (NSE) for the velocity field v ( x , t) = ∑ I v I (t) w I ( x ) into a simple system of ordinary differential equations for v I (t). The fact that fluid consists of a finite number of molecules in thermal motion implies that the number N of variables v I is finite too and that the system of equations for v I (t) contains a white-noise term. We prove that all solutions of this system are global, the IM exists and is unique. Its density ψ(v) defined on the phase space satisfies an elliptic partial differential equation (PDE). Expanding into Hermite functions He μ (v) φ ε (v) (where φ ε is a Gaussian function on , He μ are Hermite polynomials orthonormal with the weight φ ε and are IM mean values of He μ ) we transform this PDE into an infinite system of linear algebraic equations for and suggest some methods for solving it numerically. From known first- and second-order we could easily calculate mean turbulent velocity and Reynolds stress at any point x . Cylindrical pipe flow is used to illustrate the method everywhere in the paper. All conclusions are derived from the mathematical model by rigorous mathematics, with no further assumptions. All approximations can be arbitrarily improved.

SpillwayPro: Integrated Water Surface Profile, Cavitation, and Aerated Flow Analysis for Smooth and Stepped Chutes

Spillways of large and small dams experience flow conditions that exceed the capabilities of common water surface profile computer programs. Unique channel geometries, steep slopes, and streamline curvature demand rigorous mathematics, while phenomena associated with aerated flow and potential cavitation require special analyses, especially for stepped chutes. SpillwayPro is a generalized, energy-based, water surface profile calculation tool for both smooth and stepped chutes that integrates cavitation analysis and aerated flow effects, with separate modules supporting aerator design and the development of optimized structure profiles that meet cavitation parameter objectives. Simultaneous calculation of smooth and stepped-chute flow profiles enables the rapid assessment of the energy dissipation benefits of steps, as well as their unique aerated flow and cavitation issues. SpillwayPro’s technical basis is presented, and its results are compared to the data from prototype case studies and empirical methods developed from large-scale laboratory studies. SpillwayPro’s fundamental energy-based methods are useful for cases that differ from idealized empirical approaches and would benefit from a more rapid analysis than can be accomplished with physical or computational fluid dynamics (CFD) modeling.

Secure Text Transfer Using Doubly Encrypted Chat Application Based on Cloud

One of the most important ways to preserve information security is to use cryptographic techniques. It provides digital signature, authentication, secret sub-storage, system security, and other capabilities in addition to keeping the information confidential. As a result, the encryption and decryption solution can secure information secrecy, as well as information integrity and certainty, to avoid tampering, forgery, and counterfeiting. The security of encryption and decryption algorithms is determined by the algorithm's internal structure and mathematical rigour, as well as the key secrecy. The key in the encryption algorithm plays a crucial role; if the key is released, anybody may use the encryption system to encrypt and decrypt data, rendering the encryption process ineffective. As a result, the encryption and decryption solution can secure information secrecy, as well as information integrity and certainty, to avoid tampering, forgery, and counterfeiting. The security of encryption and decryption algorithms is determined by the algorithm's internal structure and mathematical rigour, as well as the key secrecy. The key in an encryption algorithm plays a critical role; if the key is released, anybody may use the encryption system to encrypt and decrypt data, rendering the encryption technique ineffective. As a result, throughout the encryption and decryption process, the type of data you pick to be a key, how you disseminate the private key, and how you preserve both data transmission keys are all critical considerations.

How-to Compute EPRL Spin Foam Amplitudes

Spin foam theory is a concrete framework for quantum gravity where numerical calculations of transition amplitudes are possible. Recently, the field became very active, but the entry barrier is steep, mainly because of its unusual language and notions scattered around the literature. This paper is a pedagogical guide to spin foam transition amplitude calculations. We show how to write an EPRL-FK transition amplitude, from the definition of the 2-complex to its numerical implementation using sl2cfoam-next. We guide the reader using an explicit example balancing mathematical rigor with a practical approach. We discuss the advantages and disadvantages of our strategy and provide a novel look at a recently proposed approximation scheme.

Data-adaptive global full-waveform inversion

SUMMARYWe present a novel approach to global-scale full-waveform inversion (FWI) that can reduce computational cost by over an order of magnitude, compared to previously published methods, without sacrificing physical and mathematical rigour. This is based on data-adaptation, and thereby application-oriented specialization, on two complementary levels. On the simulation level, we exploit the approximate azimuthal symmetry of seismic wavefields by implementing wavefield-adapted meshes and discrete adjoints, thereby lowering numerical simulation cost. On the measurement level, we use a quasi-stochastic approach where variable mini-batches of data are used during an iterative misfit minimization in order to promote a parsimonious exploitation of data. In addition to the methodological developments, we present an inversion of long-period (100–200 s) seismic waveforms from 1179 earthquakes for 3-D whole-mantle structure. The computational cost of the 72 iterations in the inversion approximately equals one third of a single iteration using an FWI approach with widely used cubed-sphere-based meshes and non-stochastic gradient optimization. The resulting LOng-Wavelength earth model (LOWE) constitutes the first global FWI constructed entirely from a spherically symmetric initial mantle structure. While mostly serving as a showcase for the method, LOWE contains a wealth of regional-scale structures that compare well to earlier tomographic images. Being conservatively smooth and based on minimal assumptions, it may therefore serve as starting model for future inversions at shorter period or smaller scales.

Derivation of Liouville-like equations for the n-state probability density of an open system with thermalized particle reservoirs and its link to molecular simulation

A physico-mathematical model of open systems proposed in a previous paper (Delle Site and Klein 2020 J. Math. Phys. 61 083102) can represent a guiding reference in designing an accurate simulation scheme for an open molecular system embedded in a reservoir of energy and particles. The derived equations and the corresponding boundary conditions are obtained without assuming the action of an external source of heat that assures thermodynamic consistency of the open system with respect to a state of reference. However, in numerical schemes the temperature in the reservoir must be controlled by an external heat bath otherwise thermodynamic consistency cannot be achieved. In this perspective, the question to address is whether the explicit addition of an external heat bath in the theoretical model modifies the equations of the open system and its boundary conditions. In this work we consider this aspect and explicitly describe the evolution of the reservoir employing the Bergmann–Lebowitz statistical model of thermostat. It is shown that the resulting equations for the open system itself are not affected by this change and an example of numerical application is reviewed where the current result shows its conceptual relevance. Finally, a list of pending mathematical and modelling problems is discussed the solution of which would strengthen the mathematical rigour of the model and offer new perspectives for the further development of a new multiscale simulation scheme.

An electromagnetic simulation assisted small signal modeling method for InP double-heterojunction bipolar transistors

We present a convenient and practical electromagnetic (EM) assisted small-signal model extraction method for InP double-heterojunction bipolar transistors (DHBTs). Parasitic parameters of pad and electrode fingers are extracted by means of 3D EM simulation. The simulations with a new excitation scheme are closer to the actual on-wafer measurement conditions. Appropriate simulation settings are calibrated by comparing measurement and simulation of OPEN and SHORT structures. A simpler π-type topology is proposed for the intrinsic model, in which the base-collector resistance Rμ , output resistance R ce are deleted, and a capacitance C ce is introduced to characterize the capacitive parasitic caused by the collector finger and emitter ground bar. The intrinsic parameters are all extracted by exact equations that are derived from rigorous mathematics. The method is characterized by its ease of implementation and the explicit physical meaning of extraction procedure. Experimental validations are performed at four biases for three InGaAs/InP HBT devices with 0.8 × 7 μm, 0.8 × 10 μm and 0.8 × 15 μm emitter, and quite good fitting results are obtained in the range of 0.1–50 GHz.