Foundations

In axisymmetric fusion devices like tokamaks, the winding of the magnetic field is characterized by its safety profile $q=q_{\mathbf{B}}$. Similarly, the winding of the current density field is characterized by $q_{\mathbf{J}}$. Currently, the relationship between $q_{\mathbf{B}}$ and $q_{\mathbf{J}}$ profiles and their effect on tokamak plasma confinement properties remains unexplored, as the $q_{\mathbf{J}}$ profile is neither computed nor considered. This study presents a reconstruction of the current density winding profile from experimental data in the quiescent H-mode. The topology analysis derived from $(q_{\mathbf{B}},q_{\mathbf{J}})$ was carried out using Hamada coordinates. It shows a large central plasma region unaffected by current filamentation-driven resonant magnetic perturbations, while the outer region harbors a spectrum of magnetic resonant modes, induced by current filaments located within the core plasma, which degrade peripheral confinement. These results suggest a QH-mode signature pattern needing further validation with additional data. Implementing $(q_{\mathbf{B}},q_{\mathbf{J}})$ real-time monitoring could provide insights into tokamak confinement regimes with significant implications. Full article

Using the results of numerical simulations and astrophysical observations (mainly in the WMAP and Planck frequency bands), it is shown that Galactic foreground emission becomes more sensitive to the mean magnetic field with the frequency, resulting in the appearance of two levels of its randomization due to the chaotic/turbulent dynamics of a magnetized interstellar medium dominated by magnetic helicity. The galactic foreground emission is more randomized at higher frequencies. The Galactic synchrotron and polarized dust emissions have been studied in detail. It is shown that the magnetic field imposes its level of randomization on the synchrotron and dust emission. The main method for the theoretical consideration used in this study is the Kolmogorov–Iroshnikov phenomenology in the frames of distributed chaos notion. Despite the vast differences in the values of physical parameters and spatio-temporal scales between the numerical simulations and the astrophysical observations, there is a quantitative agreement between the results of the astrophysical observations and the numerical simulations in the frames of the distributed chaos notion. Full article

Mean–geometric mean (MGM) linking compares group differences on a latent variable $\theta$ within the two-parameter logistic (2PL) item response theory model. This article investigates three specifications of MGM linking that differ in the weighting of item difficulty differences: unweighted (UW), discrimination-weighted (DW), and precision-weighted (PW). These methods are evaluated under conditions where random DIF effects are present in either item difficulties or item intercepts. The three estimators are analyzed both analytically and through a simulation study. The PW method outperforms the other two only in the absence of random DIF or in small samples when DIF is present. In larger samples, the UW method performs best when random DIF with homogeneous variances affects item difficulties, while the DW method achieves superior performance when such DIF is present in item intercepts. The analytical results and simulation findings consistently show that the PW method introduces bias in the estimated group mean when random DIF is present. Given that the effectiveness of MGM methods depends on the type of random DIF, the distribution of DIF effects was further examined using PISA 2006 reading data. The model comparisons indicate that random DIF with homogeneous variances in item intercepts provides a better fit than random DIF in item difficulties in the PISA 2006 reading dataset. Full article

We show numerical evidence for a bipartite $d\times d$ pure state entanglement witness that is readily calculated from the wavefunction coefficients directly, without the need for the numerical computation of eigenvalues. This is accomplished by using an approximate analytic diagonalization of the bipartite state that captures dominant contributions to the negativity of the partially transposed state. We relate this entanglement witness to the Log Negativity, and show that it exactly agrees with it for the class of pure states whose quantum amplitudes form a positive Hermitian matrix. In this case, the Log Negativity is given by the negative logarithm of the purity of the amplitudes considered a density matrix. In other cases, the witness forms a lower bound to the exact, numerically computed Log Negativity. The formula for the approximate Log Negativity achieves equality with the exact Log Negativity for the case of an arbitrary pure state of two qubits, which we show analytically. We compare these results to a witness of entanglement given by the linear entropy. Finally, we explore an attempt to extend these pure state results to mixed states. We show that the Log Negativity for this approximate formula is exact for the class of pure state decompositions, for which the quantum amplitudes of each pure state form a positive Hermitian matrix. Full article

In this survey, we have included the recent results on Lyapunov-type inequalities for differential equations of fractional order associated with Dirichlet, nonlocal, multi-point, anti-periodic, and discrete boundary conditions. Our results involve a variety of fractional derivatives such as Riemann–Liouville, Caputo, Hilfer–Hadamard, $\psi$-Riemann–Liouville, Atangana–Baleanu, tempered, half-linear, and discrete fractional derivatives. Full article

This paper revisits some aspects connected with the methods for the determination of thermodynamically consistent models. While the concepts apply to the general context of continuum physics, the details are developed for the modelling of deformable dielectrics. The symmetry condition arising from the balance of angular momentum is viewed as a constraint for the constitutive equations and is shown to be satisfied by sets of objective fields that account jointly for deformation and electric field. The second law of thermodynamics is considered in a generalized form where the entropy production is given by a constitutive function possibly independent of the other constitutive functions. Furthermore, a representation formula is applied for solving the Clausius–Duhem inequality with respect to the chosen unknown fields. Full article

I dispute the conventional claim that the second law of thermodynamics is saved from a “Maxwell’s demon” by the entropy cost of information erasure and show that instead it is measurement that incurs the entropy cost. Thus, Brillouin, who identified measurement as savior of the second law, was essentially correct, and putative refutations of his view, such as Bennett’s claim to measure without entropy cost, are seen to fail when the applicable physics is taken into account. I argue that the tradition of attributing the defeat of Maxwell’s demon to erasure rather than to measurement arose from unphysical classical idealizations that do not hold for real gas molecules, as well as a physically ungrounded recasting of physical thermodynamical processes into computational and information-theoretic conceptualizations. I argue that the fundamental principle that saves the second law is the quantum uncertainty principle applying to the need to localize physical states to precise values of observables in order to effect the desired disequilibria aimed at violating the second law. I obtain the specific entropy cost for localizing a molecule in the Szilard engine and show that it coincides with the quantity attributed to Landauer’s principle. I also note that an experiment characterized as upholding an entropy cost of erasure in a “quantum Maxwell’s demon” actually demonstrates an entropy cost of measurement. Full article

Ramsey analysis is applied to the problem of the relativistic and quantum synchronization of clocks. Various protocols of synchronization are addressed. Einstein and Eddington special relativity synchronization procedures are considered, and quantum synchronization is discussed. Clocks are seen as the vertices of the graph. Clocks may be synchronized or unsynchronized. Thus, introducing complete, bi-colored, Ramsey graphs emerging from the lattices of clocks becomes possible. The transitivity of synchronization plays a key role in the coloring of the Ramsey graph. Einstein synchronization is transitive, while general relativity and quantum synchronization procedures are not. This fact influences the value of the Ramsey number established for the synchronization graph arising from the lattice of clocks. Any lattice built of six clocks, synchronized with quantum entanglement, will inevitably contain the mono-chromatic triangle. The transitive synchronization of logical clocks is discussed. Interrelation between the symmetry of the clock lattice and the structure of the synchronization graph is addressed. Ramsey analysis of synchronization is important for the synchronization of computers in networks, LIGO, and Virgo instruments intended for the registration of gravitational waves and GPS tame-based synchronization. Full article

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