We study the problem of matching interior and exterior solutions of Einstein’s equations along a particular hypersuface. We present the main aspects of the C3 matching approach that involves third-order derivatives of the corresponding metric tensors in contrast to the standard matching procedures known in general relativity, which impose conditions on the second-order derivatives only. The C3 alternative approach does not depend on coordinates and allows us to determine the matching surface by using the invariant properties of the eigenvalues of the curvature tensor.

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, that 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.

In this paper, we demonstrate a useful interaction between the theory of clique partitions, edge clique covers of a graph, and the spectra of graphs. Using a clique partition and an edge clique cover of a graph we introduce the notion of a vertex-clique incidence matrix for a graph and produce new lower bounds for the negative eigenvalues and negative inertia of a graph. Moreover, utilizing these vertex-clique incidence matrices, we generalize several notions such as the signless Laplacian matrix, and develop bounds on the incidence energy and the signless Laplacian energy of the graph. %The tight upper bounds for the energies of a graph and its line graph are given. More generally, we also consider the set $S(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent. An important parameter in this setting is $q(G)$, and is defined to be the minimum number of distinct eigenvalues over all matrices in $S(G)$. For a given graph $G$ the concept of a vertex-clique incidence matrix associated with an edge clique cover is applied to establish several classes of graphs with $q(G)=2$.

In this work we discuss the numerical challenges involved in the computation of the complex eigenvalues of damped multi-flexible-body problems. Aiming at the highest generality, the candidate method must be able to deal with arbitrary rigid body modes (free-free mechanisms), arbitrary algebraic constraints, and must be able to exploit the sparsity pattern of Jacobians of large systems. We propose a custom implementation of the Krylov-Schur method, proving its robustness and its accuracy in a variety of different complex test cases.

We overview the main equations of the Rayleigh-Ritz variational method and discuss their connection with the problem of simultaneous diagonalization of two Hermitian matrices.

This paper discusses about the quantum energy levels of muonic hydrogen considering only Coulomb interaction. The computational analysis in this paper will help the reader get a good understanding of how different a muonic hydrogen is from its electronic counterparts. Energy eigenvalues are calculated by numerical method and the probability density distribution of muon around the nucleus is plotted for different quantum states using special functions. The code is written in python. A comparative study of emission spectra between the muonic hydrogen and hydrogen has also been discussed.

In this study, we consider delta fractional Sturm--Liouville (DFSL) initial value problems in the sense of delta Caputo and delta Riemann-Liouville (R--L) operators. One of the properties of delta fractional difference operators which makes it different from nabla counterpart is to shift its domain. This feature makes it more complex than the nabla fractional operator. We obtain sum representation of solutions for DFSL initial value problems with the help of $\mathcal{Z}-$ transformation. Moreover, we get analytical solutions of homogeneous DFSL problem within Riemann-Liouville (R--L) and Caputo sense, discrete Sturm--Liouville (DSL) problem, continuous fractional Sturm--Liouville (FSL) problem in the sense of R--L and Caputo operators, and continuous Sturm--Liouville (SL) differential problem. From this point of view, we compare all the solutions with each other. Consequently, we show that all results for these four eigenvalue problems are compatible with each other and approach to each other while the orders tends to one, i.e. $\Delta^{\mu }\left( \Delta x\left( t-\mu \right) \right)\cong D_{0^{+}}^{\mu }\left( x^{\prime }\left( t\right) \right)\cong \Delta^2x(n-1) \cong x^{\prime \prime }\left( t\right) =\lambda x\left( t\right),\ \mu\rightarrow1 $ . We support our results comparatively by tables and simulations in detail.

In this paper, we study a differential operator of parabolic type with a variable and unbounded coefficient, defined on an infinite strip. Sufficient conditions for the existence and compactness of the resolvent are established, and an estimate for the maximum regularity of solutions of the equation Lu=f∈L_2(Ω) is obtained. Two-sided estimates for the distribution function of approximation numbers are obtained. As is known, estimates of approximation numbers show the rate of best approximation of the resolvent of an operator by finite-dimensional operators. The paper proves the assertion about the existence of positive eigenvalues among the eigenvalues of the given operator and finds two-sided estimates for them.

In the modern power grid, characterized by the increased penetration of power electronics and extensive utilization of renewable energy, inverters-based power plants play a pivotal role as the principal interface of renewable energy sources (RESs) and the grid. Considering the stability characteristics of grid-following (GFL) inverters when the grid is relatively weak, the application of grid-forming (GFM) controls becomes imperative to enhance the stability of the entire power plant. Thus, there is an urgent need for suitable and effective models to study the interaction and stability of the paralleled inverters employing GFL and GFM controls. Thus, the small-signal modeling with full-order state space model and eigenvalues analysis is presented in this paper. First, the small-signal state-space model of the individual GFL and GFM inverters is obtained considering the control loop, interaction, reference frame, transmissions, and time delays. Then, the models of the individual inverter are extended to the hybrid inverters to study the effects of the GFM inverters on the small- signal stability of the entire system. And, the impacts of the inertia and damping are analyzed by the eigenvalues of the state-transition matrix. Case comprising three parallel GFL inverters and two GFL inverters with one GFM inverter, respectively, are studied to examine the effectiveness and accuracy of the model. Finally, the stability margin obtained from the eigenvalue analysis of the entire system is verified by time-domain simulations.

This paper explores innovative approaches to determining eigenvalues and eigenvectors through the combination of neural networks and traditional numerical methods. Traditional methods like the Newton-Raphson and Durand-Kerner algorithms are proven effective but often require good initial approximations and can be sensitive to the choice of starting points. In this study, we propose a hybrid approach that uses neural networks to generate better initial approximations, which are then iteratively refined using traditional methods. By combining the pattern recognition power of neural networks with the adaptability of traditional numerical methods, we demonstrate faster convergence and higher accuracy in determining eigenvalues and eigenvectors. Experimental results on various polynomials of different degrees and characteristics show that our hybrid approach outperforms individual methods, providing a more robust and efficient solution.

The article considers an internal boundary value problem of the distribution of an electrostatic field in a lens formed by two identical semi-infinite circular cylinders coaxially located inside an infinite external cylinder. The problem is reduced to solving a system of singular Wiener-Hopf integral equations, which is further solved by the Wiener-Hopf method using factorized Bessel functions. Solutions to the problem for each region inside the infinite outer cylinder are presented as exponentially converging series in terms of eigenfunctions and eigenvalues.