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Artículos de revistas sobre el tema "E-eigenvalues"

Autor: Grafiati
Publicado: 10 de marzo de 2023
Última modificación: 11 de marzo de 2023

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1

Grinevich, Petr y Roman Novikov. "TRANSMISSION EIGENVALUES FOR MULTIPOINT SCATTERERS". Eurasian Journal of Mathematical and Computer Applications 9, n.º 4 (diciembre de 2021): 17–25. http://dx.doi.org/10.32523/2306-6172-2021-9-4-17-25.

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We study the transmission eigenvalues for the multipoint scatterers of the Bethe- Peierls-Fermi-Zeldovich-Beresin-Faddeev type in dimensions d = 2 and d = 3. We show that for these scatterers: 1) each positive energy E is a transmission eigenvalue (in the strong sense) of infinite multiplicity; 2) each complex E is an interior transmission eigenvalue of infinite multiplicity. The case of dimension d = 1 is also discussed.
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2

Ahmad, Sk y Rafikul Alam. "On Wilkinson's problem for matrix pencils". Electronic Journal of Linear Algebra 30 (8 de febrero de 2015): 632–48. http://dx.doi.org/10.13001/1081-3810.3145.

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Suppose that an n-by-n regular matrix pencil A -\lambda B has n distinct eigenvalues. Then determining a defective pencil E−\lambda F which is nearest to A−\lambda B is widely known as Wilkinson’s problem. It is shown that the pencil E −\lambda F can be constructed from eigenvalues and eigenvectors of A −\lambda B when A − \lambda B is unitarily equivalent to a diagonal pencil. Further, in such a case, it is proved that the distance from A −\lambda B to E − \lambdaF is the minimum “gap” between the eigenvalues of A − \lambdaB. As a consequence, lower and upper bounds for the “Wilkinson distance” d(L) from a regular pencil L(\lambda) with distinct eigenvalues to the nearest non-diagonalizable pencil are derived.Furthermore, it is shown that d(L) is almost inversely proportional to the condition number of the most ill-conditioned eigenvalue of L(\lambda).
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3

VAIDYA, SAMIR K. y KALPESH POPAT. "On Equienergetic, Hyperenergetic and Hypoenergetic Graphs". Kragujevac Journal of Mathematics 44, n.º 4 (diciembre de 2020): 523–32. http://dx.doi.org/10.46793/kgjmat2004.523v.

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The eigenvalue of a graph G is the eigenvalue of its adjacency matrix and the energy E(G) is the sum of absolute values of eigenvalues of graph G. Two non-isomorphic graphs G1 and G2 of the same order are said to be equienergetic if E(G1) = E(G2). The graphs whose energy is greater than that of complete graph are called hyperenergetic and the graphs whose energy is less than that of its order are called hypoenergetic graphs. The natural question arises: Are there any pairs of equienergetic graphs which are also hyperenergetic (hypoenergetic)? We have found an affirmative answer of this question and contribute some new results.
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4

G, Sridhara y Rajesh Kanna. "Bounds on Energy and Laplacian Energy of Graphs". Journal of the Indonesian Mathematical Society 23, n.º 2 (24 de diciembre de 2017): 21–31. http://dx.doi.org/10.22342/jims.23.2.316.21-31.

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Let G be simple graph with n vertices and m edges. The energy E(G) of G, denotedby E(G), is dened to be the sum of the absolute values of the eigenvalues of G. Inthis paper, we present two new upper bounds for energy of a graph, one in terms ofm,n and another in terms of largest absolute eigenvalue and the smallest absoluteeigenvalue. The paper also contains upper bounds for Laplacian energy of graph.
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5

Hall, Richard L. "A simple eigenvalue formula for the quartic anharmonic oscillator". Canadian Journal of Physics 63, n.º 3 (1 de marzo de 1985): 311–13. http://dx.doi.org/10.1139/p85-048.

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The eigenvalues Enl(λ) of the Hamiltonian H = −Δ + r2 + λr4 are analysed with the help of "potential envelopes" and "kinetic potentials." The result is the following simple approximate eigenvalue formiula:[Formula: see text]where E ≥ P = (4n + 2l − 1) and Q = 3(An + Bl − C)4/322/3. E is a lower bound to Enl if (A, B, C) = (1, 1/2, 1/4) and a good approximation if (A, B, C) = (1.125, 0.509, 0.218).
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6

Korek, M. y K. Fakhreddine. "A canonical approach for computing the eigenvalues of the Schrödinger equation for double-well potentials". Canadian Journal of Physics 78, n.º 11 (1 de noviembre de 2000): 969–76. http://dx.doi.org/10.1139/p00-072.

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The problem of obtaining the eigenvalues of the Schrödinger equation for a double-well potential function is considered. By replacing the differential Schrödinger equation by a Volterra integral equation the wave function will be given by [Formula: see text] where the coefficients ai are obtained from the boundary conditions and the fi are two well-defined canonical functions. Using these canonical functions, we define an eigenvalue function F(E) = 0; its roots E1, E2, ... are the eigenvalues of the corresponding double-well potential. The numerical application to analytical potentials (either symmetric or asymmetric) and to a numerical potential of the (2)1 [Formula: see text] state of the molecule Na2 shows the validity and the high accuracy of the present formulation. PACS No.: 03.65Ge
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7

Tan, Shenyang, Tiren Huang y Wenbin Zhang. "Estimates for Eigenvalues of the Elliptic Operator in Divergence Form on Riemannian Manifolds". Advances in Mathematical Physics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/387953.

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We investigate the Dirichlet weighted eigenvalue problem of the elliptic operator in divergence form on compact Riemannian manifolds(M,g,e-ϕdv). We establish a Yang-type inequality of this problem. We also get universal inequalities for eigenvalues of elliptic operators in divergence form on compact domains of complete submanifolds admitting special functions which include the Hadamard manifolds with Ricci curvature bounded below and any complete manifolds admitting eigenmaps to a sphere.
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8

Kobeissi, Hafez, Majida Kobeissi y Chafia H. Trad. "On nonintegral E corrections in perturbation theory: application to the perturbed Morse oscillator". Canadian Journal of Physics 72, n.º 1-2 (1 de enero de 1994): 80–85. http://dx.doi.org/10.1139/p94-013.

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A new formulation of the Rayleigh–Schrödinger perturbation theory is applied to the derivation of the vibrational eigenvalues of the perturbed Morse oscillator (PMO). This formulation avoids the conventional projection of the Ψ corrections on the basis of unperturbed eigenfunctions [Formula: see text], or the projection of the nonhomogeneous Schrödinger equations on [Formula: see text], it gives simple expressions for each E correction [Formula: see text] free of summations and integrals. When the PMO is characterized by the potential U = UM + UP (where UM is the unperturbed Morse potential), the eigenvalue of a vibrational level ν is given by: [Formula: see text]. According to the new formulation the correction £, [Formula: see text] is given by [Formula: see text], where σp(r) is a particular solution of the nonhomogeneous differential equation y″ + f y = sp; here [Formula: see text], sp is well known for each p: for p = 0, [Formula: see text]; for [Formula: see text]. For the numerical application one single routine is used, that of integrating y″ + f y = s, where the coefficients are known as well as the initial values. An example is presented for the Huffaker PMO of the (carbon monoxide) CO-X1Σ+ state. The vibrational eigenvalues Eν are obtained to a good accuracy (with p = 4) even for high levels. This result confirms the validity of this new formulation and gives a semianalytic expression for the PMO eigenvalues.
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9

Abolarinwa, Abimbola. "Eigenvalues of the weighted Laplacian under the extended Ricci flow". Advances in Geometry 19, n.º 1 (28 de enero de 2019): 131–43. http://dx.doi.org/10.1515/advgeom-2018-0022.

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Abstract Let ∆φ = ∆ − ∇φ∇ be a symmetric diffusion operator with an invariant weighted volume measure dμ = e−φ dν on an n-dimensional compact Riemannian manifold (M, g), where g = g(t) solves the extended Ricci flow. We study the evolution and monotonicity of the first nonzero eigenvalue of ∆φ and we obtain several monotone quantities along the extended Ricci flow and its volume preserving version under some technical assumption. We also show that the eigenvalues diverge in a finite time for n ≥ 3. Our results are natural extensions of some known results for Laplace–Beltrami operators under various geometric flows.
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10

Loginov, B., O. Makeeva y E. Foliadova. "Pseudoperturbation method for computation of E. Schmidt eigenvalues". PAMM 6, n.º 1 (diciembre de 2006): 643–44. http://dx.doi.org/10.1002/pamm.200610302.

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11

Tamura, Hideo. "The efimov effect of three-body schrödinger operators: Asymptotics for the number of negative eigenvalues". Nagoya Mathematical Journal 130 (junio de 1993): 55–83. http://dx.doi.org/10.1017/s0027763000004426.

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The Efimov effect is one of the most remarkable results in the spectral theory for three-body Schrödinger operators. Roughly speaking, the effect will be explained as follows: If all three two-body subsystems have no negative eigenvalues and if at least two of these two-body subsystems have resonance states at zero energy, then the three-body system under consideration has an infinite number of negative eigenvalues accumulating at zero. This remarkable spectral property was first discovered by Efimov [1] and the problem has been discussed in several physical journals. For related references, see, for example, the book [3]. The mathematically rigorous proof of the result has been given by the works [4, 8, 9]. The aim of the present work is to study the asymptotic distribution of these negative eigenvalues below zero (bottom of essential spectrum). Denote by N(E), E > 0, the number of negative eigenvalues less than – E. Then the main result obtained here is, somewhat loosely stating, that N(E) behaves like | log E | as E → 0. We first formulate precisely the main theorem and then make a brief comment on the recent related result obtained by Sobolev [7].
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12

Taylor, Dane, Juan G. Restrepo y François G. Meyer. "Ensemble-based estimates of eigenvector error for empirical covariance matrices". Information and Inference: A Journal of the IMA 8, n.º 2 (3 de julio de 2018): 289–312. http://dx.doi.org/10.1093/imaiai/iay010.

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Abstract Covariance matrices are fundamental to the analysis and forecast of economic, physical and biological systems. Although the eigenvalues $\{\lambda _i\}$ and eigenvectors $\{\boldsymbol{u}_i\}$ of a covariance matrix are central to such endeavours, in practice one must inevitably approximate the covariance matrix based on data with finite sample size $n$ to obtain empirical eigenvalues $\{\tilde{\lambda }_i\}$ and eigenvectors $\{\tilde{\boldsymbol{u}}_i\}$, and therefore understanding the error so introduced is of central importance. We analyse eigenvector error $\|\boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2$ while leveraging the assumption that the true covariance matrix having size $p$ is drawn from a matrix ensemble with known spectral properties—particularly, we assume the distribution of population eigenvalues weakly converges as $p\to \infty $ to a spectral density $\rho (\lambda )$ and that the spacing between population eigenvalues is similar to that for the Gaussian orthogonal ensemble. Our approach complements previous analyses of eigenvector error that require the full set of eigenvalues to be known, which can be computationally infeasible when $p$ is large. To provide a scalable approach for uncertainty quantification of eigenvector error, we consider a fixed eigenvalue $\lambda $ and approximate the distribution of the expected square error $r= \mathbb{E}\left [\| \boldsymbol{u}_i - \tilde{\boldsymbol{u}}_i \|^2\right ]$ across the matrix ensemble for all $\boldsymbol{u}_i$ associated with $\lambda _i=\lambda $. We find, for example, that for sufficiently large matrix size $p$ and sample size $n> p$, the probability density of $r$ scales as $1/nr^2$. This power-law scaling implies that the eigenvector error is extremely heterogeneous—even if $r$ is very small for most eigenvectors, it can be large for others with non-negligible probability. We support this and further results with numerical experiments.
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13

Rakhimov, D. G. "Reductional Method in Perturbation Theory of Generalized Spectral E. Schmidt Problem". Contemporary Mathematics. Fundamental Directions 65, n.º 1 (15 de diciembre de 2019): 72–82. http://dx.doi.org/10.22363/2413-3639-2019-65-1-72-82.

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In this a paper perturbations of multiple eigenvalues of E. Schmidt spectral problems is considered. At the usage of the reductional method suggested in the articles [10, 11] the investigation of the multiple E. Schmidt perturbation eigenvalues is reduced to the investigation of perturbation of simple ones. At the end, as application of the obtained results the problem about the boundary perturbation for the system of two Sturm-Liouville problems with E. Schmidt spectral parameter is considered.
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14

Crafter, E. C., R. M. Heise, C. O. Horgan y J. G. Simmonds. "The Eigenvalues for a Self-Equilibrated, Semi-Infinite, Anisotropic Elastic Strip". Journal of Applied Mechanics 60, n.º 2 (1 de junio de 1993): 276–81. http://dx.doi.org/10.1115/1.2900790.

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The linear theory of elasticity is used to study an homogeneous anisotropic seminfinite strip, free of tractions on its long sides and subject to edge loads or displacements that produce stresses that decay in the axial direction. If one seeks solutions for the (dimensionless) Airy stress function of the form φ = e−γxF(y), γ constant, then one is led to a fourth-order eigenvalue problem for F(y) with complex eigenvalues γ. This problem, considered previously by Choi and Horgan (1977), is the anisotropic analog of the eigenvalue problem for the Fadle-Papkovich eigenfunctions arising in the isotropic case. The decay rate for Saint-Venant end effects is given by the eigenvalue with smallest positive real part. For an isotropic strip, where the material is described by two elastic constants (Young’s modulus and Poisson’s ratio), the associated eigencondition is independent of these constants. For transversely isotropic (or specially orthotropic) materials, described by four elastic constants, the eigencondition depends only on one elastic parameter. Here, we treat the fully anisotropic strip described by six elastic constants and show that the eigencondition depends on only two elastic parameters. Tables and graphs for a scaled complex-valued eigenvalue are presented. These data allow one to determine the Saint-Venant decay length for the fully anisotropic strip, as we illustrate by a numerical example for an end-loaded off-axis graphite-epoxy strip.
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15

CORTEZ, MARÍA ISABEL, FABIEN DURAND y SAMUEL PETITE. "Eigenvalues and strong orbit equivalence". Ergodic Theory and Dynamical Systems 36, n.º 8 (21 de julio de 2015): 2419–40. http://dx.doi.org/10.1017/etds.2015.26.

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We give conditions on the subgroups of the circle to be realized as the subgroups of eigenvalues of minimal Cantor systems belonging to a determined strong orbit equivalence class. Actually, the additive group of continuous eigenvalues $E(X,T)$ of the minimal Cantor system $(X,T)$ is a subgroup of the intersection $I(X,T)$ of all the images of the dimension group by its traces. We show, whenever the infinitesimal subgroup of the dimension group associated with $(X,T)$ is trivial, the quotient group $I(X,T)/E(X,T)$ is torsion free. We give examples with non-trivial infinitesimal subgroups where this property fails. We also provide some realization results.
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16

SEKE, J., A. V. SOLDATOV y N. N. BOGOLUBOV. "THE COMPLETE TREATMENT OF THE TIME EVOLUTION IN THE CASE OF A DISCRETIZED ATOM-FIELD INTERACTION MODEL". Modern Physics Letters B 15, n.º 21 (10 de septiembre de 2001): 883–94. http://dx.doi.org/10.1142/s021798490100266x.

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The dynamics of a discretized atom-field interaction model with a physically relevant form factor is analyzed. It is shown that after some short time interval only a small fraction of eigenvalues and eigenstates (belonging to the close vicinity of the excited atomic state energy E = ω0/2) contributes to the nondecay probability amplitudes in the long-time regime, whereas the contribution of all other eigenstates and eigenvalues is negligible. Nevertheless, to describe correctly the non-Markovian dynamics in the short-time regime the contribution of all eigenstates and eigenvalues must be taken into account.
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17

Drmač, Zlatko y Ivana Šain Glibić. "An Algorithm for the Complete Solution of the Quartic Eigenvalue Problem". ACM Transactions on Mathematical Software 48, n.º 1 (31 de marzo de 2022): 1–34. http://dx.doi.org/10.1145/3494528.

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The quartic eigenvalue problem (λ 4 A +λ 3 B +λ 2 C +λ D + E ) x = 0 naturally arises in a plethora of applications, such as when solving the Orr–Sommerfeld equation in the stability analysis of the Poiseuille flow, in theoretical analysis and experimental design of locally resonant phononic plates, modeling a robot with electric motors in the joints, calibration of catadioptric vision system, or, for example, computation of the guided and leaky modes of a planar waveguide. This article proposes a new numerical method for the full solution (all eigenvalues and all left and right eigenvectors) that, starting with a suitable linearization, uses an initial, structure-preserving reduction designed to reveal and deflate a certain number of zero and infinite eigenvalues before the final linearization is forwarded to the QZ algorithm. The backward error in the reduction phase is bounded column wise in each coefficient matrix, which is advantageous if the coefficient matrices are graded. Numerical examples show that the proposed algorithm is capable of computing the eigenpairs with small residuals, and that it is competitive with the available state-of-the-art methods.
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18

Li, Jianxi, Chee Shiu y An Chang. "On the laplacian estrada index of a graph". Applicable Analysis and Discrete Mathematics 3, n.º 1 (2009): 147–56. http://dx.doi.org/10.2298/aadm0901147l.

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Let G be a graph of order n. Let ?1 , ?2 , . . . , ?n be the eigenvalues of the adjacency matrix of G, and let ?1 , ?2 , . . . , ?n be the eigenvalues of the Laplacian matrix of G. Much studied Estrada index of the graph G is defined n as EE = EE(G)= ?n/i=1 e?i . We define and investigate the Laplacian Estrada index of the graph G, LEE=LEE(G)= ?n/i=1 e(?i - 2m/n). Bounds for LEE are obtained, as well as some relations between LEE and graph Laplacian energy.
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19

Redžepović, Izudin y Ivan Gutman. "Comparing Energy and Sombor Energy - An Empirical Study". Match Communications in Mathematical and in Computer Chemistry 88, n.º 1 (2022): 133–40. http://dx.doi.org/10.46793/match.88-1.133r.

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The Sombor index is a recently invented vertex-degree-based topological index, to which a matrix – called Sombor matrix – is associated in a natural manner. The graph energy E(G) is the sum of absolute values of the eigenvalues of the adjacency matrix of the graph G. Analogously, the Sombor energy ESO(G) is the sum of absolute values of the eigenvalues of the Sombor matrix. In this paper, we present computational results on the relations between ESO(G) and E(G) for various classes of (molecular) graphs, and establish the respective regularities. The correlation between ESO(G) and E(G) if found to be much more perplexed than earlier reported.
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20

PEARCE, PAUL A. "ROW TRANSFER MATRIX FUNCTIONAL EQUATIONS FOR A–D–E LATTICE MODELS". International Journal of Modern Physics A 07, supp01b (abril de 1992): 791–804. http://dx.doi.org/10.1142/s0217751x9200404x.

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Determinantal functional equations satisfied by the row transfer matrix eigenvalues of critical A–D–E lattice spin models are presented. These are obtained for models associated with the Lie algebras [Formula: see text], [Formula: see text], AL, DL and E6,7,8 by exploiting connections with functional equations satisfied by the row transfer matrix eigenvalues of the six-vertex model at rational values of the crossing parameter λ=sπ/h where h is the Coxeter number. In addition, fusion is used to derive special functional equations, called inversion identity hierarchies, which provide the key to the direct calculation of finite-size corrections, central charges and conformal weights for the critical A–D–E lattice models.
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21

Chamberland, Marc. "Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues". Canadian Mathematical Bulletin 46, n.º 3 (1 de septiembre de 2003): 323–31. http://dx.doi.org/10.4153/cmb-2003-034-4.

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AbstractRecent papers have shown that C1 maps whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or F is a polynomial. Specifically, F = (u, v) must take the formfor some constants a, b, c, d, e, f , α, β and a C1 function ϕ in one variable. If, in addition, the function ϕ is not affine, thenThis paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are ±1/2 and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge—Ampère equation.
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22

Shulemovich, Alexander. "Advanced theory of vibration of uniform beams". International Journal of Engineering & Technology 7, n.º 1 (19 de enero de 2018): 70. http://dx.doi.org/10.14419/ijet.v7i1.8748.

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In classical theory the equation of a dynamic Euler – Lagrange beam is solved by using the composition of the displacements into the sum of harmonic vibrations to obtain the ordinary differential equation. The solution of this equation with prescribed set of boundary conditions is a typical Sturm – Liouville problem with the infinite, discrete eigenvalues and modes of vibration. The purpose of this paper is to reveal that an elastic beam is a limited continuum with limited domain of physically existing, continuous eigenvalues and modes of vibration. In contrast to the classical theory, the advanced theory of free vibration of beams without damping in present investigation is based on the analysis with transversal and angular stiffness, initiated by external transient excitations and inherited by beams in compliance with energy conservation law. The output of this investigation demonstrates the fundamental distinction between the dynamic characteristics of uniformed beams established by classical theory with infinite, discrete eigenvalues and derived characteristics of beams with continuous eigenvalues and modes of vibration in limited domains. The theoretical investigation shows that only few, natural, discrete eigenvalues and normal modes of vibration physically exist in limited domains.Nomenclature: k4 = ω2a−2, a2 = E I g (A γ) −1, E − modulus of elasticity, g − gravitational acceleration, A − area of the beam’s cross - section, γ − specific gravity of the beam’s material, ω = 2πf, f − frequency of vibration per second, angular frequency p = a (kl)2/l2, I – length of beam. I − moment of inertia of area.
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23

Saad Naji Abood y Narjis Zamil Abdulzahra. "Numerical and perturbation solutions for the gauss potential". Global Journal of Engineering and Technology Advances 10, n.º 2 (28 de febrero de 2022): 043–59. http://dx.doi.org/10.30574/gjeta.2022.10.2.0033.

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We will focus on the Schrodinger eigenvalue problem for a Gauss potential in this study. When high and relatively large values of the coupling constant g2 are involved, we will compare eigenvalues E determined numerically with those obtained using the asymptotic series. However, we were interested in the mathematical elements of this comparison throughout the course of this work and explored it for considerably larger, albeit no longer physically plausible, values of g2. Even for power potentials where the Gaussian is a common example, Muller's perturbation method shows some fascinating mathematical characteristics of the Schrodinger equation. The solution's overall analytic features are very similar to well-known periodic differential equations like the Mathieu equation.
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24

Aràndiga, F. y V. Caselles. "Approximations of positive operators and continuity of the spectral radius III". Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 57, n.º 3 (diciembre de 1994): 330–40. http://dx.doi.org/10.1017/s1446788700037733.

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AbstractWe prove estimates on the speed of convergence of the ‘peripheral eigenvalues’ (and principal eigenvectors) of a sequence Tn of positive operators on a Banach lattice E to the peripheral eigenvalues of its limit operator T on E which is positive, irreducible and such that the spectral radius r(T) of T is a Riesz point of the spectrum of T (that is, a pole of the resolvent of T with a residuum of finite rank) under some conditions on the kind of approximation of Tn to T. These results sharpen results of convergence obtained by the authors in previous papers.
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25

Mufti, Zeeshan Saleem, Rukhshanda Anjum, Qin Xin, Fairouz Tchier, Iram Anwar-ul-Haq y Yaé Ulrich Gaba. "Computing the Energy and Estrada Index of Different Molecular Structures". Journal of Chemistry 2022 (28 de enero de 2022): 1–7. http://dx.doi.org/10.1155/2022/6227093.

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Graph energy is an invariant that is derived from the spectrum of the adjacency matrix of a graph. Graph energy is actually the absolute sum of all the eigenvalues of the adjacency matrix of a graph i.e. E = ∑ i = 1 n λ i , and the Estrada index of a graph G is elaborated as EE G = ∑ i = 1 n e λ i , where, λ 1 , λ 2 , … , λ n are the eigenvalues of the adjacency matrix of a graph. In this paper, energy E G and Estrada index EE G of different molecular structures are obtained and also established inequalities among the exact and estimated values of energies and Estrada index of TUC 4 C 8 nanosheet and naphthalene.
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26

Das, Kinkar y Seyed Ahmad Mojalal. "On Energy and Laplacian Energy of Graphs". Electronic Journal of Linear Algebra 31 (5 de febrero de 2016): 167–86. http://dx.doi.org/10.13001/1081-3810.3272.

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Let $G=(V,E)$ be a simple graph of order $n$ with $m$ edges. The energy of a graph $G$, denoted by $\mathcal{E}(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. The Laplacian energy of the graph $G$ is defined as \[ LE = LE(G)=\sum^n_{i=1}\left|\mu_i-\frac{2m}{n}\right| \] where $\mu_1,\,\mu_2,\,\ldots,\,\mu_{n-1},\,\mu_n=0$ are the Laplacian eigenvalues of graph $G$. In this paper, some lower and upper bounds for $\mathcal{E}(G)$ are presented in terms of number of vertices, number of edges, maximum degree and the first Zagreb index, etc. Moreover, a relation between energy and Laplacian energy of graphs is given.
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27

Taghvaee, Fatemeh y Gholam Hossein Fath-Tabar. "Trees with Four and Five Distinct Signless Laplacian Eigenvalues". Journal of the Indonesian Mathematical Society 25, n.º 3 (31 de octubre de 2019): 302–13. http://dx.doi.org/10.22342/jims.25.3.557.302-313.

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‎‎Let $G$ be a simple graph with vertex set $V(G)=\{v_1‎, ‎v_2‎, ‎\cdots‎, ‎v_n\}$ ‎and‎‎edge set $E(G)$‎.‎The signless Laplacian matrix of $G$ is the matrix $‎Q‎‎=‎D‎+‎A‎‎$‎, ‎such that $D$ is a diagonal ‎matrix‎%‎‎, ‎indexed by the vertex set of $G$ where‎‎%‎$D_{ii}$ is the degree of the vertex $v_i$ ‎‎‎ and $A$ is the adjacency matrix of $G$‎.‎%‎ where $A_{ij} = 1$ when there‎‎%‎‎is an edge from $i$ to $j$ in $G$ and $A_{ij} = 0$ otherwise‎.‎The eigenvalues of $Q$ is called the signless Laplacian eigenvalues of $G$ and denoted by $q_1$‎, ‎$q_2$‎, ‎$\cdots$‎, ‎$q_n$ in a graph with $n$ vertices‎.‎In this paper we characterize all trees with four and five distinct signless Laplacian ‎eigenvalues.‎‎‎
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28

Ita, Benedict Iserom, P. Ekuri, Idongesit O. Isaac y Abosede O. James. "BOUND STATE SOLUTIONS OF SCHRÖDINGER EQUATION FOR A MORE GENERAL EXPONENTIAL SCREENED COULOMB POTENTIAL VIA NIKIFOROVUVAROV METHOD". Eclética Química Journal 35, n.º 3 (17 de enero de 2018): 103. http://dx.doi.org/10.26850/1678-4618eqj.v35.3.2010.p103-107.

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The arbitrary angular momentum solutions of the Schrödinger equation for a diatomic molecule with the general exponential screened coulomb potential of the form V(r) = (− a / r){1+ (1+ br )e−2br } has been presented. The energy eigenvalues and the corresponding eigenfunctions are calculated analytically by the use of Nikiforov-Uvarov (NU) method which is related to the solutions in terms of Jacobi polynomials. The bounded state eigenvalues are calculated numerically for the 1s state of N2 CO and NO
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29

Klaus, M. "Nondassical eigenvalue distribution of one-dimensional Schrödinger operators". Proceedings of the Royal Society of Edinburgh: Section A Mathematics 101, n.º 1-2 (1985): 149–58. http://dx.doi.org/10.1017/s030821050002624x.

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SynopsisWe consider differential operators of the form H = −d2/dx2 + q(x) acting on u ∈ L2(0,∞) with boundary condition u(0) = 0. The potential q(x) is such that H has essential spectrum [0,∞) and an infinite sequence of negative eigenvalues converging to zero. Let n(E) denote the number of eigenvalues of H which are less than E. Under certain conditions on q(x), the well-known formula n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E}, E↑0, holds. We shall study the validity of this formula for potentials which show oscillatory behaviour as x →∞, like e.g. q(x) = −(1 + x)−α(a + b sin x) with 0<α <2, a≧0, b≠0. We shall obtain the leading-order behaviour of both n(E) and vol n(E)∼(2φ)−1 vol {x, p | p2 + q(x)<E} as E↑0 for a certain class of q's, and we shall see that the classical formula fails in most cases, but there are some noteworthy exceptions.
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30

Chen, Jiao-Kai. "Numerical solutions of the Maung-Norbury-Kahana equation with the coulomb potential in momentum space". Revista Mexicana de Física 64, n.º 1 (30 de octubre de 2017): 8. http://dx.doi.org/10.31349/revmexfis.64.8.

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In this paper, the numerical solutions of the Maung-Norbury-Kahana equation which has the complicated form of the eigenvalues are presented. Taken as examples, the bound states $e^+e^-$, $\mu^+\mu^-$ and $\mu^+e^-$ are discussed by employing the Maung-Norbury-Kahana equation with the Coulomb potential.
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31

Pirzada, S., H. A. Ganie y A. M. Alghamdi. "On the sum of signless Laplacian spectra of graphs". Carpathian Mathematical Publications 11, n.º 2 (31 de diciembre de 2019): 407–17. http://dx.doi.org/10.15330/cmp.11.2.407-417.

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For a simple graph $G(V,E)$ with $n$ vertices, $m$ edges, vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$ and edge set $E(G)=\{e_1, e_2,\dots, e_m\}$, the adjacency matrix $A=(a_{ij})$ of $G$ is a $(0, 1)$-square matrix of order $n$ whose $(i,j)$-entry is equal to 1 if $v_i$ is adjacent to $v_j$ and equal to 0, otherwise. Let $D(G)={diag}(d_1, d_2, \dots, d_n)$ be the diagonal matrix associated to $G$, where $d_i=\deg(v_i),$ for all $i\in \{1,2,\dots,n\}$. The matrices $L(G)=D(G)-A(G)$ and $Q(G)=D(G)+A(G)$ are respectively called the Laplacian and the signless Laplacian matrices and their spectra (eigenvalues) are respectively called the Laplacian spectrum ($L$-spectrum) and the signless Laplacian spectrum ($Q$-spectrum) of the graph $G$. If $0=\mu_n\leq\mu_{n-1}\leq\cdots\leq\mu_1$ are the Laplacian eigenvalues of $G$, Brouwer conjectured that the sum of $k$ largest Laplacian eigenvalues $S_{k}(G)$ satisfies $S_{k}(G)=\sum\limits_{i=1}^{k}\mu_i\leq m+{k+1 \choose 2}$ and this conjecture is still open. If $q_1,q_2, \dots, q_n$ are the signless Laplacian eigenvalues of $G$, for $1\leq k\leq n$, let $S^{+}_{k}(G)=\sum_{i=1}^{k}q_i$ be the sum of $k$ largest signless Laplacian eigenvalues of $G$. Analogous to Brouwer's conjecture, Ashraf et al. conjectured that $S^{+}_{k}(G)\leq m+{k+1 \choose 2}$, for all $1\leq k\leq n$. This conjecture has been verified in affirmative for some classes of graphs. We obtain the upper bounds for $S^{+}_{k}(G)$ in terms of the clique number $\omega$, the vertex covering number $\tau$ and the diameter of the graph $G$. Finally, we show that the conjecture holds for large families of graphs.
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32

Joshi, Prajakta Bharat y Mayamma Joseph. "𝒫-energy of graphs". Acta Universitatis Sapientiae, Informatica 12, n.º 1 (1 de julio de 2020): 137–57. http://dx.doi.org/10.2478/ausi-2020-0009.

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AbstractGiven a graph G = (V, E), with respect to a vertex partition 𝒫 we associate a matrix called 𝒫-matrix and define the 𝒫-energy, E𝒫 (G) as the sum of 𝒫-eigenvalues of 𝒫-matrix of G. Apart from studying some properties of 𝒫-matrix, its eigenvalues and obtaining bounds of 𝒫-energy, we explore the robust(shear) 𝒫-energy which is the maximum(minimum) value of 𝒫-energy for some families of graphs. Further, we derive explicit formulas for E𝒫 (G) of few classes of graphs with different vertex partitions.
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33

Sasaki, Ryu. "Exactly solvable piecewise analytic double well potential VD(x) = min[(x + d)2, (x − d)2] and its dual single well potential VS(x) = max[(x + d)2, (x − d)2]". Journal of Mathematical Physics 64, n.º 2 (1 de febrero de 2023): 022102. http://dx.doi.org/10.1063/5.0127371.

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By putting two harmonic oscillator potentials x2 side by side with a separation 2 d, two exactly solvable piecewise analytic quantum systems with a free parameter d > 0 are obtained. Due to the mirror symmetry, their eigenvalues { E} for the even and odd parity sectors are determined exactly as the zeros of certain combinations of the confluent hypergeometric function [Formula: see text] of d and E, which are common to VD and VS but in two different branches. The eigenfunctions are the piecewise square integrable combinations of [Formula: see text], the so-called U functions. By comparing the eigenvalues and eigenfunctions for various values of the separation d, vivid pictures unfold showing the tunneling effects between the two wells.
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34

Davey, A. y H. Salwen. "On the stability of flow in an elliptic pipe which is nearly circular". Journal of Fluid Mechanics 281 (25 de diciembre de 1994): 357–69. http://dx.doi.org/10.1017/s0022112094003149.

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In an earlier paper (Davey 1978) the first author investigated the linear stability of flow in a straight pipe whose cross-section was an ellipse, of small ellipticity e, by regarding the flow as a perturbation of Poiseuille flow in a circular pipe. That paper contained some serious errors which we correct herein. We show analytically that for the most important mode n = 1, for which the circular problem has a double eigenvalue c0 as the ‘swirl’ can be in either direction, the ellipticity splits the double eigenvalue into two separate eigenvalues c0 ± e2c12, to leading order, when the cross-sectional area of the pipe is kept fixed. The imaginary part of c12 is non-zero and so the ellipticity always makes the flow less stable. This specific problem is generic to a much wider class of fluid dynamical problems which are made less stable when the symmetry group of the dynamical system is reduced from S1 to Z2.In the Appendix, P. G. Drazin describes simply the qualitative structure of this problem, and other problems with the same symmetries, without technical detail.
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35

Zorin, Alexander V., Mikhail D. Malykh y Leonid A. Sevastianov. "Complex eigenvalues in Kuryshkin-Wodkiewicz quantum mechanics". Discrete and Continuous Models and Applied Computational Science 30, n.º 2 (3 de mayo de 2022): 139–48. http://dx.doi.org/10.22363/2658-4670-2022-30-2-139-148.

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One of the possible versions of quantum mechanics, known as Kuryshkin-Wodkiewicz quantum mechanics, is considered. In this version, the quantum distribution function is positive, but, as a retribution for this, the von Neumann quantization rule is replaced by a more complicated rule, in which an observed value AA is associated with a pseudodifferential operator O^(A){\hat{O}(A)}. This version is an example of a dissipative quantum system and, therefore, it was expected that the eigenvalues of the Hamiltonian should have imaginary parts. However, the discrete spectrum of the Hamiltonian of a hydrogen-like atom in this theory turned out to be real-valued. In this paper, we propose the following explanation for this paradox. It is traditionally assumed that in some state ψ{\psi} the quantity AA is equal to λ{\lambda} if ψ{\psi} is an eigenfunction of the operator O^(A){\hat{O}(A)}. In this case, the variance O^((A-λ)2)ψ{\hat{O}((A-\lambda)2)\psi} is zero in the standard version of quantum mechanics, but nonzero in Kuryshkins mechanics. Therefore, it is possible to consider such a range of values and states corresponding to them for which the variance O^((A-λ)2){\hat{O}((A-\lambda)2)} is zero. The spectrum of the quadratic pencil O^(A2)-2O^(A)λ+λ2E^{\hat{O}(A2)-2\hat{O}(A)\lambda + \lambda 2 \hat{E}} is studied by the methods of perturbation theory under the assumption of small variance D^(A)=O^(A2)-O^(A)2{\hat{D}(A) = \hat{O}(A2) - \hat{O}(A) 2} of the observable AA. It is shown that in the neighborhood of the real eigenvalue λ{\lambda} of the operator O^(A){\hat{O}(A)}, there are two eigenvalues of the operator pencil, which differ in the first order of perturbation theory by ±i⟨D^⟩{\pm i \sqrt{\langle \hat{D} \rangle}}.
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36

Farkhondeh, Masoumeh, Mohammad Habibi, Doost Ali Mojdeh y Yongsheng Rao. "Some Bicyclic Graphs Having 2 as Their Laplacian Eigenvalues". Mathematics 7, n.º 12 (12 de diciembre de 2019): 1233. http://dx.doi.org/10.3390/math7121233.

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If G is a graph, its Laplacian is the difference between the diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs G 1 and G 2 is a graph G = G 1 ⊙ u v G 2 with V ( G ) = V ( G 1 ) ∪ V ( G 2 ) and E ( G ) = E ( G 1 ) ∪ E ( G 2 ) ∪ { e = u v } where u ∈ V ( G 1 ) and v ∈ V ( G 2 ) . In this paper, we study some structural conditions ensuring the presence of 2 in the Laplacian spectrum of bicyclic graphs of type G 1 ⊙ u v G 2 . We also provide a condition under which a bicyclic graph with a perfect matching has a Laplacian eigenvalue 2. Moreover, we characterize the broken sun graphs and the one-edge connection of two broken sun graphs by their Laplacian eigenvalue 2.
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37

Abdesselam, B., A. Chakrabarti y R. Chakrabarti. "On ${\mathcal U}_h({\rm sl}(2))$, ${\mathcal U}_h(e(3))$ and Their Representations". International Journal of Modern Physics A 12, n.º 13 (20 de mayo de 1997): 2301–19. http://dx.doi.org/10.1142/s0217751x97001341.

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By solving a set of recursion relations for the matrix elements of the [Formula: see text] generators, the finite dimensional highest weight representations of the algebra were obtained as factor representations. Taking a nonlinear combination of the generators of the two copies of the [Formula: see text] algebra, we obtained [Formula: see text] algebra. The latter, on contraction, yields [Formula: see text] algebra. A nonlinear map of [Formula: see text] algebra on its classical analog e(3) was obtained. The inverse mapping was found to be singular. It signifies a physically interesting situation, where in the momentum basis, a restricted domain of the eigenvalues of the classical operators is mapped on the whole real domain of the eigenvalues of the deformed operators.
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38

Filipovski, Slobodan y Robert Jajcay. "Bounds for the Energy of Graphs". Mathematics 9, n.º 14 (18 de julio de 2021): 1687. http://dx.doi.org/10.3390/math9141687.

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Let G be a graph on n vertices and m edges, with maximum degree Δ(G) and minimum degree δ(G). Let A be the adjacency matrix of G, and let λ1≥λ2≥…≥λn be the eigenvalues of G. The energy of G, denoted by E(G), is defined as the sum of the absolute values of the eigenvalues of G, that is E(G)=|λ1|+…+|λn|. The energy of G is known to be at least twice the minimum degree of G, E(G)≥2δ(G). Akbari and Hosseinzadeh conjectured that the energy of a graph G whose adjacency matrix is nonsingular is in fact greater than or equal to the sum of the maximum and the minimum degrees of G, i.e., E(G)≥Δ(G)+δ(G). In this paper, we present a proof of this conjecture for hyperenergetic graphs, and we prove an inequality that appears to support the conjectured inequality. Additionally, we derive various lower and upper bounds for E(G). The results rely on elementary inequalities and their application.
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39

Ghasemian, E. y G. H. Fath-Tabar. "On signed graphs with two distinct eigenvalues". Filomat 31, n.º 20 (2017): 6393–400. http://dx.doi.org/10.2298/fil1720393g.

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Let G? be a signed graph with the underlying graph G and with sign function ? : E(G) ? {?}. In this paper, we characterize the signed graphs with two distinct eigenvalues whose underlying graphs are triangle-free. Also, we classify all 3-regular and 4-regular signed graphs whose underlying graphs are triangle-free and give their adjacency matrices as well.
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40

Ramane, Harishchandra S., B. Parvathalu, K. Ashoka y Daneshwari Patil. "On A-energy and S-energy of certain class of graphs". Acta Universitatis Sapientiae, Informatica 13, n.º 2 (1 de diciembre de 2021): 195–219. http://dx.doi.org/10.2478/ausi-2021-0009.

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Abstract Let A and S be the adjacency and the Seidel matrix of a graph G respectively. A-energy is the ordinary energy E(G) of a graph G defined as the sum of the absolute values of eigenvalues of A. Analogously, S-energy is the Seidel energy ES(G) of a graph G defined to be the sum of the absolute values of eigenvalues of the Seidel matrix S. In this article, certain class of A-equienergetic and S-equienergetic graphs are presented. Also some linear relations on A-energies and S-energies are given.
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41

Du, Zhibin. "The sum of the first two largest signless Laplacian eigenvalues of trees and unicyclic graphs". Electronic Journal of Linear Algebra 35 (1 de febrero de 2019): 449–67. http://dx.doi.org/10.13001/1081-3810.3405.

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Let $G$ be a graph on $n$ vertices with $e(G)$ edges. The sum of eigenvalues of graphs has been receiving a lot of attention these years. Let $S_2 (G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and define $f(G) = e (G) +3 - S_2 (G)$. Oliveira et al. (2015) conjectured that $f(G) \geqslant f(U_{n})$ with equality if and only if $G \cong U_n$, where $U_n$ is the $n$-vertex unicyclic graph obtained by attaching $n-3$ pendent vertices to a vertex of a triangle. In this paper, it is proved that $S_2(G) < e(G) + 3 -\frac{2}{n}$ when $G$ is a tree, or a unicyclic graph whose unique cycle is not a triangle. As a consequence, it is deduced that the conjecture proposed by Oliveira et al. is true for trees and unicyclic graphs whose unique cycle is not a triangle.
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42

RIEUTORD, M. y L. VALDETTARO. "Inertial waves in a rotating spherical shell". Journal of Fluid Mechanics 341 (25 de junio de 1997): 77–99. http://dx.doi.org/10.1017/s0022112097005491.

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The structure and spectrum of inertial waves of an incompressible viscous fluid inside a spherical shell are investigated numerically. These modes appear to be strongly featured by a web of rays which reflect on the boundaries. Kinetic energy and dissipation are indeed concentrated on thin conical sheets, the meridional cross-section of which forms the web of rays. The thickness of the rays is in general independent of the Ekman number E but a few cases show a scaling with E1/4 and statistical properties of eigenvalues indicate that high-wavenumber modes have rays of width O(E1/3). Such scalings are typical of Stewartson shear layers. It is also shown that the web of rays depends on the Ekman number and shows bifurcations as this number is decreased.This behaviour also implies that eigenvalues do not evolve smoothly with viscosity. We infer that only the statistical distribution of eigenvalues may follow some simple rules in the asymptotic limit of zero viscosity.
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43

Gheorghiu, Călin-Ioan. "Accurate Spectral Collocation Computation of High Order Eigenvalues for Singular Schrödinger Equations". Computation 9, n.º 1 (29 de diciembre de 2020): 2. http://dx.doi.org/10.3390/computation9010002.

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We are concerned with the study of some classical spectral collocation methods, mainly Chebyshev and sinc as well as with the new software system Chebfun in computing high order eigenpairs of singular and regular Schrödinger eigenproblems. We want to highlight both the qualities as well as the shortcomings of these methods and evaluate them in conjunction with the usual ones. In order to resolve a boundary singularity, we use Chebfun with domain truncation. Although it is applicable with spectral collocation, a special technique to introduce boundary conditions as well as a coordinate transform, which maps an unbounded domain to a finite one, are the special ingredients. A challenging set of “hard”benchmark problems, for which usual numerical methods (f. d., f. e. m., shooting, etc.) fail, were analyzed. In order to separate “good”and “bad”eigenvalues, we have estimated the drift of the set of eigenvalues of interest with respect to the order of approximation and/or scaling of domain parameter. It automatically provides us with a measure of the error within which the eigenvalues are computed and a hint on numerical stability. We pay a particular attention to problems with almost multiple eigenvalues as well as to problems with a mixed spectrum.
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44

Koolen, Jack, Vincent Moulton, Ivan Gutman y Dusica Vidovic. "More hyperenergetic molecular graphs". Journal of the Serbian Chemical Society 65, n.º 8 (2000): 571–75. http://dx.doi.org/10.2298/jsc0008571k.

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If G is a molecular graph and ?1,?2,... ?n are its eigenvalues, then the energy of G is equal to E(G) = |?1|+|?2|+ ... +|?n|. This energy cannot exceed the value n?n-1 ? n3/2. The graph G is said to be hyperenergetic if E(G)>2n-2. We describe the construc?tion of hyperenergetic graphs G for which E(G)?1/2 n3/2.
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45

Merajuddin, M., S. Bhatnagar y S. Pirzada. "On spectral radius and Nordhaus-Gaddum type inequalities of the generalized distance matrix of graphs". Carpathian Mathematical Publications 14, n.º 1 (23 de junio de 2022): 185–93. http://dx.doi.org/10.15330/cmp.14.1.185-193.

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If $Tr(G)$ and $D(G)$ are respectively the diagonal matrix of vertex transmission degrees and distance matrix of a connected graph $G$, the generalized distance matrix $D_{\alpha}(G)$ is defined as $D_{\alpha}(G)=\alpha ~Tr(G)+(1-\alpha)~D(G)$, where $0\leq \alpha \leq 1$. If $\rho_1 \geq \rho_2 \geq \dots \geq \rho_n$ are the eigenvalues of $D_{\alpha}(G)$, the largest eigenvalue $\rho_1$ (or $\rho_{\alpha}(G)$) is called the spectral radius of the generalized distance matrix $D_{\alpha}(G)$. The generalized distance energy is defined as $E^{D_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i -\frac{2\alpha W(G)}{n}\right|$, where $W(G)$ is the Wiener index of $G$. In this paper, we obtain the bounds for the spectral radius $\rho_{\alpha}(G)$ and the generalized distance energy of $G$ involving Wiener index. We derive the Nordhaus-Gaddum type inequalities for the spectral radius and the generalized distance energy of $G$.
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46

G, Sangeetha y J. Kavitha. "Results on Graph Energy". Journal of Physics: Conference Series 2332, n.º 1 (1 de septiembre de 2022): 012008. http://dx.doi.org/10.1088/1742-6596/2332/1/012008.

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Abstract Adjacency matrix A(G)=[aij] yields the graph energy, which is equal to the addition of absolute values of the eigenvalues G. This research investigates the energy graph class in terms of another graph class after removing a vertex. After deleting a vertex, a relationship between the energy of a complete graph E[kn ] and energy of a splitting graph E(S’ [kn ]) is discovered.
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47

Gutman, I., Y. Hou, H. B. Walikar, H. S. Ramane y P. R. Hampiholi. "No Hückel graph is hyperenergetic". Journal of the Serbian Chemical Society 65, n.º 11 (2000): 799–801. http://dx.doi.org/10.2298/jsc0011799g.

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If G is a molecular graph with n vertices and if ?1, ?2, ..., ?n are its eigenvalues, then the energy of G is equal to E(G) = |?1| + |?2|+ ... + |?n|. If E(G) > 2n - 2, then G is said to be hyperenergetic. We show that no H?ckel graph (= the graph representation of a conjugated hydrocarbon within the H?ckel molecular orbital model) is hyperenergetic.
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48

Kuo, Wentang y M. Ram Murty. "On a Conjecture of Birch and Swinnerton-Dyer". Canadian Journal of Mathematics 57, n.º 2 (1 de abril de 2005): 328–37. http://dx.doi.org/10.4153/cjm-2005-014-0.

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AbstractLet E/Q be an elliptic curve defined by the equation For a prime p, defineAs a precursor to their celebrated conjecture, Birch and Swinnerton-Dyer originally conjectured that for some constant c,Let αp and βp be the eigenvalues of the Frobenius at p. Define
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49

Algazin, S. D. "About calculation with the high accuracy of eigenvalues of the operator Laplace in the ellipse (with the regional condition of Neumann)". Доклады Академии наук 486, n.º 2 (27 de mayo de 2019): 143–46. http://dx.doi.org/10.31857/s0869-56524862143-146.

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The method of the computing experiment investigates the task about fluctuations of the elliptic membrane with the regional condition of Neumann. It is shown that on the grid 40×61 it is possible to define up to 50 eigenvalues with several right significant figures for eccentricity e = 0.99999.
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50

Miao, Qing. "Eigenvalues for a Neumann Boundary Problem Involving thep(x)-Laplacian". Advances in Mathematical Physics 2015 (2015): 1–5. http://dx.doi.org/10.1155/2015/632745.

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We study the existence of weak solutions to the following Neumann problem involving thep(x)-Laplacian operator: -Δp(x)u+e(x)|u|p(x)-2u=λa(x)f(u),in Ω,∂u/∂ν=0,on ∂Ω. Under some appropriate conditions on the functionsp, e, a, and f, we prove that there existsλ¯>0such that anyλ∈(0,λ¯)is an eigenvalue of the above problem. Our analysis mainly relies on variational arguments based on Ekeland’s variational principle.
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