Foundations

Parameter identification problems in partial differential equations (PDEs) consist in determining one or more functional coefficient in a PDE. In this article, the Bayesian nonparametric approach to such problems is considered. Focusing on the representative example of inferring the diffusivity function in an elliptic PDE from noisy observations of the PDE solution, the performance of Bayesian procedures based on Gaussian process priors is investigated. Building on recent developments in the literature, we derive novel asymptotic theoretical guarantees that establish posterior consistency and convergence rates for methodologically attractive Gaussian series priors based on the Dirichlet–Laplacian eigenbasis. An implementation of the associated posterior-based inference is provided and illustrated via a numerical simulation study, where excellent agreement with the theory is obtained. Full article

In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every other vertex based on their distances from these chosen vertices. Building on existing work, we explore fault tolerance by considering the minimal number of vertices needed to ensure that all other vertices remain uniquely identifiable even if a specified number of these vertices fails. We compute the fault-tolerant metric dimension of various chemical graphs, namely fullerenes, benzene, and polyphenyl graphs. Full article

Vectors are almost always introduced as objects having magnitude and direction. Following that idea, textbooks and courses introduce the concept of a vector norm and the angle between two vectors. While this is correct and useful for vectors in two- or three-dimensional Euclidean space, these concepts make no sense for more general vectors, that are defined in abstract, non-metric vector spaces. This is even the case when an inner product exists. Here, we analyze how several textbooks are imprecise in presenting the restricted validity of the expressions for the norm and the angle. We also study one concrete example, the so-called ‘vector-based sustainability analytics’, in which scientists have gone astray by mistaking an abstract vector for a Euclidean vector. We recommend that future textbook authors introduce the distinction between vectors that have and that do not have magnitude and direction, even in cases where an inner product exists. Full article

Artificial neural networks are widely used in applications from various scientific fields and in a multitude of practical applications. In recent years, a multitude of scientific publications have been presented on the effective training of their parameters, but in many cases overfitting problems appear, where the artificial neural network shows poor results when used on data that were not present during training. This text proposes the incorporation of a three-stage evolutionary technique, which has roots in the differential evolution technique, for the effective training of the parameters of artificial neural networks and the avoidance of the problem of overfitting. The new method effectively constructs the parameter value range of the artificial neural network with one processing level and sigmoid outputs, both achieving a reduction in training error and preventing the network from experiencing overfitting phenomena. This new technique was successfully applied to a wide range of problems from the relevant literature and the results were extremely promising. From the conducted experiments, it appears that the proposed method reduced the average classification error by 30%, compared to the genetic algorithm, and the average regression error by 45%, as compared to the genetic algorithm. Full article

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunction seeming to be the primary lens. Our topic is a theory showing “where adjoints come from”. The theory is based on object-to-object “chimera morphisms”, “heteromorphisms”, or “hets” between the objects of different categories (e.g., the insertion of generators as a set-to-group map). After showing that heteromorphisms can be treated rigorously using the machinery of category theory (bifunctors), we show that all adjunctions between two categories arise (up to an isomorphism) as the representations (i.e., universal models) within each category of the heteromorphisms between the two categories. The conventional treatment of adjunctions eschews the whole concept of a heteromorphism, so our purpose is to shine a new light on this notion by showing its origin as a het between categories being universally represented within each of the two categories. This heteromorphic treatment of adjunctions shows how they can be split into two separable universal constructions. Such universals can also occur without being part of an adjunction. We conclude with the idea that it is the universal constructions (adjunctions being an important special case) that are really the foundational concepts to pick out what is important in mathematics and perhaps in other sciences, not to mention in philosophy. Full article

In glass physics, order parameters have long been used in the thermodynamic description of glasses, but their nature is not yet clear. The difficulty is how to find order in disordered systems. This study provides a coherent understanding of the nature of order parameters for glasses and crystals, starting from the fundament of the definition of state variables in thermodynamics. The state variable is defined as the time-averaged value of a dynamical variable under the constraints, when equilibrium is established. It gives the same value at any time it is measured as long as the equilibrium is maintained. From this definition, it is deduced that the state variables of a solid are the time-averaged positions of all atoms constituting the solid, and the order parameters are essentially the same as state variables. Therefore, the order parameters of a glass are equilibrium atom positions. Full article

In spite of its unbroken PT symmetry, the popular imaginary cubic oscillator Hamiltonian $H^{\left(IC\right)}=p^2+\mathrm{i}{x}^3$ does not satisfy all of the necessary postulates of quantum mechanics. This failure is due to the “intrinsic exceptional point” (IEP) features of $H^{\left(IC\right)}$ and, in particular, to the phenomenon of a high-energy asymptotic parallelization of its bound-state-mimicking eigenvectors. In this paper, it is argued that the operator $H^{\left(IC\right)}$ (and the like) can only be interpreted as a manifestly unphysical, singular IEP limit of a hypothetical one-parametric family of certain standard quantum Hamiltonians. For explanation, ample use is made of perturbation theory and of multiple analogies between IEPs and conventional Kato’s exceptional points. Full article

In this paper, we consider the application of brute force computational techniques (BFCTs) for solving computational problems in mathematical analysis and matrix algebra in a floating-point computing environment. These techniques include, among others, simple matrix computations and the analysis of graphs of functions. Since BFCTs are based on matrix calculations, the program system MATLAB^® is suitable for their computer realization. The computations in this paper are completed in double precision floating-point arithmetic, obeying the 2019 IEEE Standard for binary floating-point calculations. One of the aims of this paper is to analyze cases where popular algorithms and software fail to produce correct answers, failing to alert the user. In real-time control applications, this may have catastrophic consequences with heavy material damage and human casualties. It is known, or suspected, that a number of man-made catastrophes such as the Dharhan accident (1991), Ariane 5 launch failure (1996), Boeing 737 Max tragedies (2018, 2019) and others are due to errors in the computer software and hardware. Another application of BFCTs is finding good initial guesses for known computational algorithms. Sometimes, simple and relatively fast BFCTs are useful tools in solving computational problems correctly and in real time. Among particular problems considered are the genuine addition of machine numbers, numerically stable computations, finding minimums of arrays, the minimization of functions, solving finite equations, integration and differentiation, computing condensed and canonical forms of matrices and clarifying the concepts of the least squares method in the light of the conflict remainders vs. errors. Usually, BFCTs are applied under the user’s supervision, which is not possible in the automatic implementation of computational methods. To implement BFCTs automatically is a challenging problem in the area of artificial intelligence and of mathematical artificial intelligence in particular. BFCTs allow to reveal the underlying arithmetic in the performance of computational algorithms. Last but not least, this paper has tutorial value, as computational algorithms and mathematical software are often taught without considering the properties of computational algorithms and machine arithmetic. Full article

Low-grade gliomas are a group of brain tumors that mostly affect people in early adulthood. A glioma is a tumor that originates in glial cells. They are classified into four levels according to their level of proliferation, with grades 1 and 2 called low-grade gliomas. In this research, we conduct an analysis focused on a mathematical model that emulates the behavior of low-grade gliomas with chemotherapy treatment based on a system of nonlinear differential equations. An analysis of the model is carried out in the absence of treatment, resulting in a predictive solution for the behavior of glioma if it left untreated for some reason. In turn, a stability analysis is carried out on the system of equations to find the critical points for treating glioma. In addition, the numerical results of the model are obtained, presenting the state variables at each instant. Finally, some simulations are presented, varying the moments of treatment initiation and the applied doses of Temozolomide. Full article

This study employs advanced structural characterization techniques, including anomalous small-angle X-ray scattering (ASAXS), small-angle X-ray scattering (SAXS), and X-ray photoelectron spectroscopy (XPS), to investigate erbium (Er³⁺)-doped silica-based glass-ceramic thin films synthesized via the sol–gel method. This research examines the SiO₂-TiO₂ and SiO₂-TiO₂-PO_2.5 systems, focusing on the formation, dispersion, and structural integration of Er³⁺-containing nanocrystals within the amorphous matrix under different thermal treatments. Synchrotron radiation tuned to the L_III absorption edge of erbium enabled ASAXS measurements, providing element-specific details about the localization of Er³⁺ ions. The findings confirm their migration into crystalline phases, such as erbium phosphate (EPO) and erbium titanate (ETO). SAXS and Guinier analysis quantified nanocrystal sizes, revealing trends influenced by their composition and heat treatment. Complementary XPS analysis of the Er 5p core-level states provided detailed information on the chemical and electronic environment of the Er³⁺ ions, confirming their stabilization within the crystalline structure. Transmission electron microscopy (TEM) highlighted the nanoscale morphology, verifying the aggregation of Er³⁺ ions into well-defined nanocrystals. The results offer a deeper understanding of their size, distribution, and interaction with the surrounding matrix. Full article

In fixed item parameter calibration (FIPC), an item response theory (IRT) model is estimated with item parameters fixed at reference values to estimate the distribution parameters within a specific group. The presence of random differential item functioning (DIF) within this group introduces additional variability in the distribution parameter estimates, which is captured by the linking error (LE). Conventional LE estimates, based on item jackknife methods, are subject to positive bias due to sampling errors. To address this, this article introduces a bias-corrected LE estimate. Moreover, the use of statistical inference is examined using the newly proposed bias-corrected total error, which includes both the sampling error and LE. The proposed error estimates were evaluated through a simulation study, and their application is illustrated using PISA 2006 data for the reading domain. Full article

In this study, we explore the order–disorder transition in the dynamics of a straightforward master equation that describes the evolution of a probability distribution between two states, $p_1$ and $p_2$ (with $p_1+p_2=1$). We focus on (1) the behavior of entropy S, (2) the distance D from the uniform distribution ($p_1=p_2=1/2$), and (3) the free energy F. To facilitate understanding, we introduce two price-ratios: ${\eta }_S={\displaystyle \frac{dS/dt}{dF/dt}}$ and ${\eta }_D={\displaystyle \frac{dD/dt}{dF/dt}}$. They respectively define the energetic costs of modifying (1) S and (2) D. Our findings indicate that both energy costs diverge to plus and minus infinity as the system approaches the uniform distribution, marking a critical transition point where the master equation temporarily loses its physical meaning. Following this divergence, the system stabilizes itself into a new well-behaved regime, reaching finite values that signify a new steady state. This two-regime behavior showcases the intricate dynamics of simple probabilistic systems and offers valuable insights into the relationships between entropy, distance in probability space, and free energy within the framework of statistical mechanics, making it a useful case study that highlights the underlying principles of the system’s evolution and equilibrium. Our discussion revolves about the order–disorder contrast that is important in various scientific disciplines, including physics, chemistry, and material science, and even in broader contexts like philosophy and social sciences. Full article

Stocking–Lord (SL) linking is a widely used linking method based on item response theory (IRT). This article examines the variability in SL linking parameter estimates within the two-parameter logistic (2PL) model. The uncertainty in SL linking arises from the sampling variability (standard error) and item selection (linking error), which can induce variability due to random differential item functioning (DIF). Three linking error estimation approaches are compared in this paper: the conventional jackknife linking error method, a newly developed approximate jackknife linking error method, and a Taylor approximation-based estimate. Simulation studies showed that the approximate jackknife method closely aligns with the traditional jackknife linking error method and outperforms the linking error estimation approach based on Taylor approximation. The adequacy of coverage rates for SL linking parameter estimates was also assessed using estimates of the total error. Results from a simulation study demonstrate that the bias-corrected total error provides superior coverage rates compared to both the conventional total error and the standard error, which does not account for item-related uncertainty due to random DIF. Full article

In this work, we present a new theoretical approach to interpreting and reproducing quantum mechanics using trajectory-guided wavelets. Inspired by the 1925 work of Louis de Broglie, we demonstrate that pulses composed of a difference between a delayed wave and an advanced wave (known as antisymmetric waves) are capable of following quantum trajectories predicted by the de Broglie–Bohm theory (also known as Bohmian mechanics). Our theory reproduces the main results of orthodox quantum mechanics and unlike Bohmian theory, is local in the Bell sense. We show that this is linked to the superdeterminism and past–future (anti)symmetry of our theory. Full article

This work explores the hyperbolic cosine and hyperbolic secant functions within the framework of the maximum entropy principle, deriving these probability distribution functions from first principles. The resulting maximum entropy solutions are applied to various physical systems, including the repulsive oscillator and solitary wave solutions of the advection equation, using the method of moments. Additionally, a different moment analysis using experimental and theoretical inputs is employed to address non-linear systems described by the non-linear Schrödinger equation, non-linear diffusion equation, and Korteweg–de Vries equation, demonstrating the versatility of this approach. These findings demonstrate the broad applicability of maximum entropy methods in solving different differential equations, with potential implications for future research in non-linear dynamics and transport physics. Full article

The rising demand for both water and energy has intensified the urgency of addressing the water–energy nexus. Energy is required for water treatment and distribution, and energy production processes require water. The increasing demand for energy requires substantial amounts of water, primarily for cooling. The emergence of new persistent contaminants has necessitated the use of advanced, energy-intensive water treatment methods. Coupled with the energy demands of water distribution, this has significantly strained the already limited energy resources. Regrettably, no straightforward, universal model exists for estimating water usage and energy consumption in power and water treatment plants, respectively. Current approaches rely on data from direct surveys of plant operators, which are often unreliable and incomplete. This has significantly undermined the efficiency of the plants as these surveys often miss out on complex interactions, lack robust predictive power and fail to account for dynamic temporal changes. The study thus aims to evaluate the potential of mathematical modeling and simulation in the water–energy nexus. It formulates a mathematical framework and subsequent simulation in Java programming to estimate the water use in hydroelectric power and geothermal energy, the energy consumption of the advanced water treatment processes focusing on advanced oxidation processes and membrane separation processes and energy demands of water distribution. The importance of mathematical modeling and simulation in the water–energy nexus has been extensively discussed. The paper then addresses the challenges and prospects and provides a way forward. The findings of this study strongly demonstrate the effectiveness of mathematical modeling and simulation in navigating the complexities of the water–energy nexus. Full article

The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generalization). Specifically, the objective of this paper is to present a methodology for generalizing the classical Remainder Theorem to the case in which the divisor is a product of binomials ${(x-a_1)}^{n_1}{(x-a_2)}^{n_2}\cdots {(x-a_k)}^{n_k}$, where $a_1,a_2,\cdots ,a_k\in \mathbb{R}$ and $n_1,n_2,\cdots ,n_k\in \mathbb{N}$. A first approach to this issue is the Taylor expansion of the dividend $P\left(x\right)$ at a point a, which clearly shows the quotient and the remainder of the division of $P\left(x\right)$ by ${(x-a)}^k$, where the degree of $P\left(x\right)$, say n, must be greater than or equal to k. The methodology used in this paper is the proof by induction which allows to obtain recurrence relations different from those obtained by other scholars dealing with the generalization of the classical Remainder Theorem. Full article

The empirical wavelet transform is a fully adaptive time-scale representation that has been widely used in the last decade. Inspired by the empirical mode decomposition, it consists of filter banks based on harmonic mode supports. Recently, it has been generalized to build the filter banks from any generating function using mappings. In practice, the harmonic mode supports can have a low-constrained shape in 2D, leading to numerical difficulties to estimate mappings adapted to the construction of empirical wavelet filters. This work aims to propose an efficient numerical scheme to compute empirical wavelet coefficients using the demons registration algorithm. Results show that the proposed approach is robust, accurate, and continuous wavelet filters permitting reconstruction with a low signal-to-noise ratio. An application for texture segmentation of scanning tunneling microscope images is also presented. Full article

The Wilson–Cowan model has been widely applied for the simulation of electroencephalography (EEG) waves associated with neural activities in the brain. The Runge–Kutta (RK) method is commonly used to numerically solve the Wilson–Cowan equations. In this paper, we focus on enhancing the accuracy of the numerical method by proposing a strategy to construct a class of fourth-order RK methods using a generalized iterated Crank–Nicolson procedure, where the RK coefficients depend on a free parameter $c_2$. When $c_2$ is set to 0.5, our method becomes a special case of the classical fourth-order RK method. We apply the proposed methods to solve the Wilson–Cowan equations with two and three neuron populations, modeling EEG epileptic dynamics. Our simulations demonstrate that when $c_2$ is set to 0.4, the proposed RK4-04 method yields smaller errors compared to those obtained using the classical fourth-order RK method. This is particularly visible when the spectral radius of the connection matrix or the excitation-inhibition coupling coefficient is relatively large. Full article