Data-driven approaches to study the spectral properties of chemical structures.

The molecular energy, which is the sum of all eigenvalues, is crucial in determining the total π-electron energy of conjugated hydrocarbon molecules. We used machine learning techniques to calculate the energy, inertia, nullity, signature, and Estrada index of molecular graphs for bismuth tri-iodide and benzene rings embedded in P-type surfaces within 2D networks. We applied MATLAB to extract the actual eigenvalues from the data and developed general equations for these molecular properties. We then used these equations to estimate the values and compared them to the actual values through graphical analysis. Our results demonstrate the potential of data-driven techniques in predicting molecular properties and enhancing our understanding of spectral theory.

Approximate resolutions of the Schrodinger theory applying the WKB approximation for certain diatomic molecular interactions.

CONTEXT: The Analyzing of energetic bond spectra of diatomic compounds is crucial to understanding their qualities because it allows one to evaluate their attributes. Diatoms compounds' spectral properties and bound energies are presented in this study. These energies are found by solving the Schrodinger equation while making consideration of the employing of the Kratzer Feus potential. METHOD: This study focuses on the calculation of bound states for diatomic molecules using the WKB approximation. The final energy spectrum equation is utilized to compute the bound states of specific diatomic molecules for varying quantum numbers n and l through the utilization of the Mathematica software. The method produced the desired and anticipated results, as shown by a comparison of the eigenvalue results with earlier studies.

Derivation of Bose's Entropy Spectral Density from the Multiplicity of Energy Eigenvalues.

The modern textbook analysis of the thermal state of photons inside a three-dimensional reflective cavity is based on the three quantum numbers that characterize photon's energy eigenvalues coming out when the boundary conditions are imposed. The crucial passage from the quantum numbers to the continuous frequency is operated by introducing a three-dimensional continuous version of the three discrete quantum numbers, which leads to the energy spectral density and to the entropy spectral density. This standard analysis obscures the role of the multiplicity of energy eigenvalues associated to the same eigenfrequency. In this paper we review the past derivations of Bose's entropy spectral density and present a new analysis of energy spectral density and entropy spectral density based on the multiplicity of energy eigenvalues. Our analysis explicitly defines the eigenfrequency distribution of energy and entropy and uses it as a starting point for the passage from the discrete eigenfrequencies to the continuous frequency.

Bernstein collocation technique for a class of Sturm-Liouville problems.

Sturm-Liouville problems have yielded the biggest achievement in the spectral theory of ordinary differential operators. Sturm-Liouville boundary value issues appear in many key applications in natural sciences. All the eigenvalues for the standard Sturm-Liouville problem are guaranteed to be real and simple, and the related eigenfunctions form a basis in a suitable Hilbert space. This article uses the weighted residual collocation technique to numerically compute the eigenpairs of both regular and singular Strum Liouville problems. Bernstein polynomials over [0,1] has been used to develop a weighted residual collocation approach to achieve an improved accuracy. The properties of Bernstein polynomials and the differentiation formula based on the Bernstein operational matrix are used to simplify the given singular boundary value problems into a matrix-based linear algebraic system. Keeping this fact in mind such a polynomial with space defined collocation scheme has been studied for Strum Liouville problems. The main reasons to use the collocation technique are its affordability, ease of use, well-conditioned matrices, and flexibility. The weighted residual collocation method is found to be more appealing because Bernstein polynomials vanish at the two interval ends, providing better versatility. A multitude of test problems are offered along with computation errors to demonstrate how the suggested method behaves. The numerical algorithm and its applicability to particular situations are described in detail, along with the convergence behavior and precision of the current technique.

Integration of statistical and simulation analyses for ternary hybrid nanofluid over a moving surface with melting heat transfer.

In industrial and engineering fields including lamination, melt-spinning, continuous casting, and fiber spinning, the flow caused by a continually moving surface is significant. Therefore, the problem of ternary hybrid nanofluid flow over a moving surface is studied. This study explores the stability and statistical analyses of the magnetohydrodynamics (MHD) forced flow of the ternary hybrid nanofluid with melting heat transfer phenomena. The impacts of viscous dissipation, Joule heating, and thermal radiation are also included in the flow. Different fluids including ternary hybrid nanofluid, hybrid nanofluids, and nanofluids with base fluid ethylene glycol (EG) are examined and compared, where magnetite (Fe3O4) and silica (SiO2) are taken as the magnetic nanomaterials while silver (Ag) is chosen as the nonmagnetic nanomaterial. The skin friction coefficient and the local Nusselt number are estimated through regression analysis. By employing similarity transformations, the governing partial differential equations are converted into non-linear ordinary differential equations. Then, the least square method is applied to solve the equations analytically. Dual solutions are established in a particular range of moving parameterλ. Due to this, a stability test is implemented to find the stable solution by using the bvp4c function in MATLAB software. It is found that the first solution is the stable one while the second is unstable. The use of ternary hybrid nanomaterials improves the heat transport rate. The increasing values of the Eckert number enlarge the heat passage. The fluid velocity and temperature profiles for nonmagnetic nanomaterials are higher than that of magnetic nanomaterials. The uniqueness and originality of this study stems from the fact that, to the best of the authors' knowledge, it is the first to use this combination technique.

The unreasonable effectiveness of the total quasi-steady state approximation, and its limitations.

In this note, we discuss the range of parameters for which the total quasi-steady-state approximation of the Michaelis-Menten reaction mechanism holds validity. We challenge the prevalent notion that total quasi-steady-state approximation is "roughly valid" across all parameters, showing that its validity cannot be assumed, even roughly, across the entire parameter space - when the initial substrate concentration is high. On the positive side, we show that the linearized one-dimensional equation for total substrate is a valid approximation when the initial reduced substrate concentration s0/KM is small. Moreover, we obtain a precise picture of the long-term time course of both substrate and complex.

How collectively integrated are ecological communities?

Beyond abiotic conditions, do population dynamics mostly depend on a species' direct predators, preys and conspecifics? Or can indirect feedback that ripples across the whole community be equally important? Determining where ecological communities sit on the spectrum between these two characterizations requires a metric able to capture the difference between them. Here we show that the spectral radius of a community's interaction matrix provides such a metric, thus a measure of ecological collectivity, which is accessible from imperfect knowledge of biotic interactions and related to observable signatures. This measure of collectivity integrates existing approaches to complexity, interaction structure and indirect interactions. Our work thus provides an original perspective on the question of to what degree communities are more than loose collections of species or simple interaction motifs and explains when pragmatic reductionist approaches ought to suffice or fail when applied to ecological communities.

On the connection between uniqueness from samples and stability in Gabor phase retrieval.

Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of R2). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in L2(R). Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

Energy spectrum of selected diatomic molecules (H2, CO, I2, NO) by the resolution of Schrodinger equation for combined potentials via NUFA method.

CONTEXT: To determine the properties of diatomic molecules, studying their chemical bond energy spectrum is essential since it enables the assessment of their characteristics. This research presents diatomic molecules spectroscopic characteristics and rovibrational energy (H2, CO, I2, NO). The Schrodinger equation is solved to determine these energies, considering the presence of a combination of two distinct potentials. the inverse quadratic Yukawa potential in combination with the screened modified Kratzer. METHOD: This work used the Greene-Aldrich assumption and the Nikiforov-Uvarov functional analysis approach as analytical tools to solve the Schrodinger equation and determine the energy spectrum of diatomic molecules (H2, CO, I2, NO). The use of Mathematica software allows for the calculation of the eigenvalues of energy of the previously mentioned diatomic molecules (H2, CO, I2, NO) based on their rovibrational energies in the final equation. By comparing the eigenvalue findings with previous research, it was seen that the technique yielded the expected and desirable outcomes.

Adaptation in a heterogeneous environment II: to be three or not to be.

We propose a model to describe the adaptation of a phenotypically structured population in a H-patch environment connected by migration, with each patch associated with a different phenotypic optimum, and we perform a rigorous mathematical analysis of this model. We show that the large-time behaviour of the solution (persistence or extinction) depends on the sign of a principal eigenvalue, [Formula: see text], and we study the dependency of [Formula: see text] with respect to H. This analysis sheds new light on the effect of increasing the number of patches on the persistence of a population, which has implications in agroecology and for understanding zoonoses; in such cases we consider a pathogenic population and the patches correspond to different host species. The occurrence of a springboard effect, where the addition of a patch contributes to persistence, or on the contrary the emergence of a detrimental effect by increasing the number of patches on the persistence, depends in a rather complex way on the respective positions in the phenotypic space of the optimal phenotypes associated with each patch. From a mathematical point of view, an important part of the difficulty in dealing with [Formula: see text], compared to [Formula: see text] or [Formula: see text], comes from the lack of symmetry. Our results, which are based on a fixed point theorem, comparison principles, integral estimates, variational arguments, rearrangement techniques, and numerical simulations, provide a better understanding of these dependencies. In particular, we propose a precise characterisation of the situations where the addition of a third patch increases or decreases the chances of persistence, compared to a situation with only two patches.

Albertson (Alb) spectral radii and Albertson (Alb) energies of graph operation.

The sum of the absolute eigenvalues of the adjacency matrix make up graph energy. The greatest absolute eigenvalue of the adjacency matrix is represented by the spectral radius of the graph. Both molecular computing and computer science have uses for graph energies and spectral radii. The Albertson (Alb) energies and spectral radii of generalized splitting and shadow graphs constructed on any regular graph is the main focus of this study. The only thing that may be disputed is the comparison of the (Alb) energies and (Alb) spectral radii of the newly formed graphs to those of the base graph. By concentrating on splitting and shadow graph, we compute new correlations between the Alb energies and spectral radius of the new graph and the prior graph.

Fitting parameters and therapies of ODE tumor models with senescence and immune system.

This work is devoted to introduce and study two quasispecies nonlinear ODE systems that model the behavior of tumor cell populations organized in different states. In the first model, replicative, senescent, extended lifespan, immortal and tumor cells are considered, while the second also includes immune cells. We fit the parameters regulating the transmission between states in order to approximate the outcomes of the models to a real progressive tumor invasion. After that, we study the identifiability of the fitted parameters, by using two sensitivity analysis methods. Then, we show that an adequate reduced fitting process (only accounting to the most identifiable parameters) gives similar results but saving computational cost. Three different therapies are introduced in the models to shrink (progressively in time) the tumor, while the replicative and senescent cells invade. Each therapy is identified to a dimensionless parameter. Then, we make a fitting process of the therapies' parameters, in various scenarios depending on the initial tumor according to the time when the therapies started. We conclude that, although the optimal combination of therapies depends on the size of initial tumor, the most efficient therapy is the reinforcement of the immune system. Finally, an identifiability analysis allows us to detect a limitation in the therapy outcomes. In fact, perturbing the optimal combination of therapies under an appropriate therapeutic vector produces virtually the same results.

Matrix stability and bifurcation analysis by a network-based approach.

In this paper, we develop a network-based methodology to investigate the problems related to matrix stability and bifurcations in nonlinear dynamical systems. By matching a matrix with a network, i.e., interaction graph, we propose a new network-based matrix analysis method by proving a theorem about matrix determinant under which matrix stability can be considered in terms of feedback loops. Especially, the approach can tell us how a node, a path, or a feedback loop in the interaction graph affects matrix stability. In addition, the roles played by a node, a path, or a feedback loop in determining bifurcations in nonlinear dynamical systems can also be revealed. Therefore, the approach can help us to screen optimal node or node combinations. By perturbing them, unstable matrices can be stabilized more efficiently or bifurcations can be induced more easily to realize desired state transitions. To illustrate feasibility and efficiency of the approach, some simple matrices are used to show how single or combinatorial perturbations affect matrix stability and induce bifurcations. In addition, the main idea is also illustrated through a biological problem related to T cell development with three nodes: TCF-1, GATA3, and PU.1, which can be considered to be a three-variable nonlinear dynamical system. The approach is especially helpful in understanding crucial roles of single or molecule combinations in biomolecular networks. The approach presented here can be expected to analyze other biological networks related to cell fate transitions and systematic perturbation strategy selection.

On the Stability and Null-Controllability of an Infinite System of Linear Differential Equations.

In this work, the null controllability problem for a linear system in ℓ2 is considered, where the matrix of a linear operator describing the system is an infinite matrix with λ∈ℝ on the main diagonal and 1s above it. We show that the system is asymptotically stable if and only if λ ≤- 1, which shows the fine difference between the finite and the infinite-dimensional systems. When λ ≤- 1 we also show that the system is null controllable in large. Further we show a dependence of the stability on the norm, i.e. the same system considered ℓ∞ is not asymptotically stable if λ = - 1.

Missing value imputation in a data matrix using the regularised singular value decomposition.

Some statistical analysis techniques may require complete data matrices, but a frequent problem in the construction of databases is the incomplete collection of information for different reasons. One option to tackle the problem is to estimate and impute the missing data. This paper describes a form of imputation that mixes regression with lower rank approximations. To improve the quality of the imputations, a generalisation is proposed that replaces the singular value decomposition (SVD) of the matrix with a regularised SVD in which the regularisation parameter is estimated by cross-validation. To evaluate the performance of the proposal, ten sets of real data from multienvironment trials were used. Missing values were created in each set at four percentages of missing not at random, and three criteria were then considered to investigate the effectiveness of the proposal. The results show that the regularised method proves very competitive when compared to the original method, beating it in several of the considered scenarios. As it is a very general system, its application can be extended to all multivariate data matrices. •The imputation method is modified through the inclusion of a stable and efficient computational algorithm that replaces the classical SVD least squares criterion by a penalised criterion. This penalty produces smoothed eigenvectors and eigenvalues that avoid overfitting problems, improving the performance of the method when the penalty is necessary. The size of the penalty can be determined by minimising one of the following criteria: the prediction errors, the Procrustes similarity statistic or the critical angles between subspaces of principal components.

Sombor spectra of chain graphs.

We study the Sombor index and the Sombor spectral properties of chain graphs. In particular, an explicit formula for the Sombor index is given, the Sombor eigenvalues are discussed, bounds on the largest and the smallest Sombor eigenvalues are presented, chain graphs with the simple Sombor eigenvalue are characterized, formulae for the Frobenius norm and the determinant of the Sombor quotient matrix of chain graphs are given, the Sombor spread bound and the Sombor energy bounds are presented along with the characterization of graphs attaining them.

Some stable and closed-shell structures of anticancer drugs by graph theoretical parameters.

The eigenvalues are significant in mathematics, but they are also relevant in other domains like as chemistry, economics, and a variety of others. In terms of our research, eigenvalues are used in chemistry to represent not only the form of energy but also the various physicochemical aspects of a chemical substance. We must comprehend the connection between mathematics and chemistry. The antibonding level is related to positive eigenvalues, the bonding level is associated to negative eigenvalues, and the nonbonding level is linked to zero eigenvalues. In this work, we studied some anticancer drug structures in terms of nullity, matching number, eigenvalues of adjacency matrix, and characteristics polynomials. As a result, Carmustine, Caulibugulone-E, Aspidostomide-E anticancer drug structures are stable, closed-shell molecules since their nullity is equal to zero.

Computing tensor Z-eigenpairs via an alternating direction method.

Tensor eigenproblems have wide applications in blind source separation, magnetic resonance imaging, and molecular conformation. In this study, we explore an alternating direction method for computing the largest or smallest Z-eigenvalue and corresponding eigenvector of an even-order symmetric tensor. The method decomposes a tensor Z-eigenproblem into a series of matrix eigenproblems that can be readily solved using off-the-shelf matrix eigenvalue algorithms. Our numerical results show that, in most cases, the proposed method converges over two times faster and could determine extreme Z-eigenvalues with 20-50% higher probability than a classical power method-based approach.

Computation of Eigenvalues and Eigenfunctions in the Solution of Eddy Current Problems.

The solution of the eigenvalue problem in bounded domains with planar and cylindrical stratification is a necessary preliminary task for the construction of modal solutions to canonical problems with discontinuities. The computation of the complex eigenvalue spectrum must be very accurate since losing or misplacing one of the thereto linked modes will have an important impact on the field solution. The approach followed in a number of previous works is to construct the corresponding transcendental equation and locate its roots in the complex plane using the Newton-Raphson method or Cauchy-integral-based techniques. Nevertheless, this approach is cumbersome, and its numerical stability decreases dramatically with the number of layers. An alternative, approach consists in the numerical evaluation of the matrix eigenvalues for the weak formulation for the respective 1D Sturm-Liouville problem using linear algebra tools. An arbitrary number of layers can thus be easily and robustly treated, with continuous material gradients being a limiting case. Although this approach is often used in high frequency studies involving wave propagation, this is the first time that has been used for the induction problem arising in an eddy current inspection situation. The developed method is implemented in Matlab and is used to deal with the following problems: magnetic material with a hole, a magnetic cylinder, and a magnetic ring. In all the conducted tests, the results are obtained in a very short time, without missing a single eigenvalue.

Moisture Detection in Tree Trunks in Semiarid Lands Using Low-Cost Non-Invasive Capacitive Sensors with Statistical Based Anomaly Detection Approach.

This paper focuses on building a non-invasive, low-cost sensor that can be fitted over tree trunks growing in a semiarid land environment. It also proposes a new definition that characterizes tree trunks' water retention capabilities mathematically. The designed sensor measures the variations in capacitance across its probes. It uses amplification and filter stages to smooth the readings, requires little power, and is operational over a 100 kHz frequency. The sensor sends data via a Long Range (LoRa) transceiver through a gateway to a processing unit. Field experiments showed that the system provides accurate readings of the moisture content. As the sensors are non-invasive, they can be fitted to branches and trunks of various sizes without altering the structure of the wood tissue. Results show that the moisture content in tree trunks increases exponentially with respect to the measured capacitance and reflects the distinct differences between different tree types. Data of known healthy trees and unhealthy trees and defective sensor readings have been collected and analysed statistically to show how anomalies in sensor reading baseds on eigenvectors and eigenvalues of the fitted curve coefficient matrix can be detected.