75,279 results on '"eigenvalues"'
Search Results
2. Linear and angular momentum conservation in surface hopping methods.
- Author
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Wu, Yanze, Rawlinson, Jonathan, Littlejohn, Robert G., and Subotnik, Joseph E.
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LINEAR momentum , *ANGULAR momentum (Mechanics) , *DEGREES of freedom , *SPIN-orbit interactions , *ODD numbers , *EIGENVALUES , *CHIRALITY of nuclear particles - Abstract
We demonstrate that, for systems with spin–orbit coupling and an odd number of electrons, the standard fewest switches surface hopping algorithm does not conserve the total linear or angular momentum. This lack of conservation arises not so much from the hopping direction (which is easily adjusted) but more generally from propagating adiabatic dynamics along surfaces that are not time reversible. We show that one solution to this problem is to run along eigenvalues of phase-space electronic Hamiltonians H(R, P) (i.e., electronic Hamiltonians that depend on both nuclear position and momentum) with an electronic–nuclear coupling Γ · P [see Eq. (25)], and we delineate the conditions that must be satisfied by the operator Γ. The present results should be extremely useful as far as developing new semiclassical approaches that can treat systems where the nuclear, electronic orbital, and electronic spin degrees of freedom altogether are all coupled together, hopefully including systems displaying the chiral-induced spin selectivity effect. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
3. Use of FLOSIC for understanding anion-solvent interactions.
- Author
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Pederson, Mark R., Withanage, Kushantha P. K., Hooshmand, Zahra, Johnson, Alex I., Baruah, Tunna, Yamamoto, Yoh, Zope, Rajendra R., Kao, Der-You, Shukla, Priyanka B., Johnson, J. Karl, Peralta, Juan E., and Jackson, Koblar A.
- Subjects
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ELECTRON donor-acceptor complexes , *CROWDSOURCING , *COULOMB potential , *CHROMIUM isotopes , *IONIZATION energy , *EIGENVALUES , *ATMOSPHERE - Abstract
An Achille's heel of lower-rung density-functional approximations is that the highest-occupied-molecular-orbital energy levels of anions, known to be stable or metastable in nature, are often found to be positive in the worst case or above the lowest-unoccupied-molecular-orbital levels on neighboring complexes that are not expected to accept charge. A trianionic example, [ Cr ( C 2 O 4 ) 3 ] 3 − , is of interest for constraining models linking Cr isotope ratios in rock samples to oxygen levels in Earth's atmosphere over geological timescales. Here we describe how crowd sourcing can be used to carry out self-consistent Fermi–Löwdin–Orbital-Self-Interaction corrected calculations (FLOSIC) on this trianion in solution. The calculations give a physically correct description of the electronic structure of the trianion and water. In contrast, uncorrected local density approximation (LDA) calculations result in approximately half of the anion charge being transferred to the water bath due to the effects of self-interaction error. Use of group-theory and the intrinsic sparsity of the theory enables calculations roughly 125 times faster than our initial implementation in the large N limit reached here. By integrating charge density densities and Coulomb potentials over regions of space and analyzing core-level shifts of the Cr and O atoms as a function of position and functional, we unambiguously show that FLOSIC, relative to LDA, reverses incorrect solute-solvent charge transfer in the trianion-water complex. In comparison to other functionals investigated herein, including Hartree–Fock and the local density approximation, the FLOSIC Cr 1s eigenvalues provide the best agreement with experimental core ionization energies. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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4. On the accuracy of the chemically significant eigenvalue method.
- Author
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Holtorf, Flemming and Green, William H.
- Subjects
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EQUATIONS , *EIGENVALUES - Abstract
We study the accuracy and convergence properties of the chemically significant eigenvalues method as proposed by Georgievskii et al. [J. Phys. Chem. A 117, 12146-12154 (2013)] and its close relative, dominant subspace truncation, for reduction of the energy-grained master equation. We formally derive the connection between both reduction techniques and provide hard error bounds for the accuracy of the latter which confirm the empirically excellent accuracy and convergence properties but also unveil practically relevant cases in which both methods are bound to fall short. We propose the use of balanced truncation as an effective alternative in these cases. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
5. Spectrum of the Tudung Saji Graph of Kapal Layar pattern.
- Author
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Zulkfeli, Nabilah and Zamri, Siti Norziahidayu Amzee
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GRAPH theory , *GEOMETRIC connections , *SPECTRAL theory , *EIGENVALUES , *MULTIPLICITY (Mathematics) - Abstract
The study of ethnomathematics has becoming a trend due to the beautiful and uniqueness of its culture. Ethnomathematics explores various cultures and their connection with mathematics. Previously, the ethnomathematics study, particularly on food cover, also known as Tudung Sa ji has been done where Tudung Sa ji Graph has been introduced. In addition, the energy of the Tudung Sa ji Graph of certain patterns has also been determined based on the eigenvalues of the adjacency matrix. In this paper, the exploration of the Tudung Sa ji Graph is extended by focusing on the spectral graph theory. The spectrum of the Tudung Sa ji Graph is a collection of the eigenvalues of the adjacency matrix and multiplicities. Therefore, the spectrum of the Tudung Sa ji Graph of certain Tudung Sa ji patterns will be determined by using reduced row echelon form (RREF) method. The spectrum of the graph is found be S p e c (Γ) = ( λ 1 λ 2 ... λ n m 1 m 2 ... m m ) , where their spectrum is determined by the multiset of its adjacency eigenvalues. This paper also provides other results of The Tudung Sa ji Graph in the context of spectral graph theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
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6. Efficient approximation of molecular kinetics using random Fourier features.
- Author
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Nüske, Feliks and Klus, Stefan
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MOLECULAR kinetics , *STOCHASTIC approximation , *MARKOV processes , *EIGENVALUES , *ALANINE - Abstract
Slow kinetic processes in molecular systems can be analyzed by computing the dominant eigenpairs of the Koopman operator or its generator. In this context, the Variational Approach to Markov Processes (VAMP) provides a rigorous way of discerning the quality of different approximate models. Kernel methods have been shown to provide accurate and robust estimates for slow kinetic processes, but they are sensitive to hyper-parameter selection and require the solution of large-scale generalized eigenvalue problems, which can easily become computationally demanding for large data sizes. In this contribution, we employ a stochastic approximation of the kernel based on random Fourier features (RFFs) to derive a small-scale dual eigenvalue problem that can be easily solved. We provide an interpretation of this procedure in terms of a finite, randomly generated basis set. By combining the RFF approach and model selection by means of the VAMP score, we show that kernel parameters can be efficiently tuned and accurate estimates of slow molecular kinetics can be obtained for several benchmarking systems, such as deca alanine and the NTL9 protein. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
7. Machine learning classification can significantly reduce the cost of calculating the Hamiltonian matrix in CI calculations.
- Author
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Qu, Chen, Houston, Paul L., Yu, Qi, Conte, Riccardo, Pandey, Priyanka, Nandi, Apurba, and Bowman, Joel M.
- Subjects
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MACHINE learning , *SURFACE potential , *EIGENVALUES , *NUCLEAR matrix , *GLYCINE - Abstract
Hamiltonian matrices in electronic and nuclear contexts are highly computation intensive to calculate, mainly due to the cost for the potential matrix. Typically, these matrices contain many off-diagonal elements that are orders of magnitude smaller than diagonal elements. We illustrate that here for vibrational H-matrices of H2O, C2H3 (vinyl), and C2H5NO2 (glycine) using full-dimensional ab initio-based potential surfaces. We then show that many of these small elements can be replaced by zero with small errors of the resulting full set of eigenvalues, depending on the threshold value for this replacement. As a result of this empirical evidence, we investigate three machine learning approaches to predict the zero elements. This is shown to be successful for these H-matrices after training on a small set of calculated elements. For H-matrices of vinyl and glycine, of order 15 552 and 8828, respectively, training on a percent or so of elements is sufficient to obtain all eigenvalues with a mean absolute error of roughly 2 cm−1. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
8. Time-dependent equation-of-motion coupled-cluster simulations with a defective Hamiltonian.
- Author
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Yuwono, Stephen H., Cooper, Brandon C., Zhang, Tianyuan, Li, Xiaosong, and DePrince III, A. Eugene
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ELECTRONIC excitation , *MAGNESIUM fluoride , *OSCILLATOR strengths , *SYSTEM dynamics , *MOLECULAR dynamics , *EQUATIONS of motion , *HAMILTONIAN systems , *EIGENVALUES - Abstract
Simulations of laser-induced electron dynamics in a molecular system are performed using time-dependent (TD) equation-of-motion (EOM) coupled-cluster (CC) theory. The target system has been chosen to highlight potential shortcomings of truncated TD-EOM-CC methods [represented in this work by TD-EOM-CC with single and double excitations (TD-EOM-CCSD)], where unphysical spectroscopic features can emerge. Specifically, we explore driven resonant electronic excitations in magnesium fluoride in the proximity of an avoided crossing. Near the avoided crossing, the CCSD similarity-transformed Hamiltonian is defective, meaning that it has complex eigenvalues, and oscillator strengths may take on negative values. When an external field is applied to drive transitions to states exhibiting these traits, unphysical dynamics are observed. For example, the stationary states that make up the time-dependent state acquire populations that can be negative, exceed one, or even complex-valued. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
9. Subspace recursive Fermi-operator expansion strategies for large-scale DFT eigenvalue problems on HPC architectures.
- Author
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Khadatkar, Sameer and Motamarri, Phani
- Subjects
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GRAPHICS processing units , *CENTRAL processing units , *EIGENVALUES , *POLYNOMIAL chaos , *DENSITY functional theory , *PARALLEL processing , *HIGH performance computing , *ORTHOGRAPHIC projection - Abstract
Quantum mechanical calculations for material modeling using Kohn–Sham density functional theory (DFT) involve the solution of a nonlinear eigenvalue problem for N smallest eigenvector-eigenvalue pairs, with N proportional to the number of electrons in the material system. These calculations are computationally demanding and have asymptotic cubic scaling complexity with the number of electrons. Large-scale matrix eigenvalue problems arising from the discretization of the Kohn–Sham DFT equations employing a systematically convergent basis traditionally rely on iterative orthogonal projection methods, which are shown to be computationally efficient and scalable on massively parallel computing architectures. However, as the size of the material system increases, these methods are known to incur dominant computational costs through the Rayleigh–Ritz projection step of the discretized Kohn–Sham Hamiltonian matrix and the subsequent subspace diagonalization of the projected matrix. This work explores the potential of polynomial expansion approaches based on recursive Fermi-operator expansion as an alternative to the subspace diagonalization of the projected Hamiltonian matrix to reduce the computational cost. Subsequently, we perform a detailed comparison of various recursive polynomial expansion approaches to the traditional approach of explicit diagonalization on both multi-node central processing unit and graphics processing unit architectures and assess their relative performance in terms of accuracy, computational efficiency, scaling behavior, and energy efficiency. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
10. Noncommutative Vieta theorem in Clifford geometric algebras.
- Author
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Shirokov, Dmitry
- Subjects
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COMPUTER vision , *COMPUTER science , *ALGEBRA , *POLYNOMIALS , *EIGENVALUES - Abstract
In this paper, we discuss a generalization of Vieta theorem (Vieta's formulas) to the case of Clifford geometric algebras. We compare the generalized Vieta formulas with the ordinary Vieta formulas for characteristic polynomial containing eigenvalues. We discuss Gelfand–Retakh noncommutative Vieta theorem and use it for the case of geometric algebras of small dimensions. We introduce the notion of a simple basis‐free formula for a determinant in geometric algebra and prove that a formula of this type exists in the case of arbitrary dimension. Using this notion, we present and prove generalized Vieta theorem in geometric algebra of arbitrary dimension. The results can be used in symbolic computation and various applications of geometric algebras in computer science, computer graphics, computer vision, physics, and engineering. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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11. Remarks on the global monopole topological effects on spherical symmetric potentials.
- Author
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Bakke, K.
- Subjects
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WKB approximation , *ENERGY levels (Quantum mechanics) , *EIGENVALUES , *SPACETIME , *EQUATIONS - Abstract
In this paper, we study the topological effects of the global monopole spacetime on the energy eigenvalues of spherical symmetric potentials in the nonrelativistic regime. We deal with the radial equation by using the Wentzel, Kramers and Brillouim (WKB) approximation. In the cases where the energy levels of the ℓ -waves can be achieved, the WKB approximation is used based on the Langer transformation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
12. Stability of smooth periodic traveling waves in the Degasperis–Procesi equation.
- Author
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Geyer, Anna and Pelinovsky, Dmitry E.
- Subjects
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WAVE equation , *STABILITY criterion , *EIGENVALUES - Abstract
We derive a precise energy stability criterion for smooth periodic waves in the Degasperis–Procesi (DP) equation. Compared to the Camassa-Holm (CH) equation, the number of negative eigenvalues of an associated Hessian operator changes in the existence region of smooth periodic waves. We utilize properties of the period function with respect to two parameters in order to obtain a smooth existence curve for the family of smooth periodic waves with a fixed period. The energy stability condition is derived on parts of this existence curve, which correspond to either one or two negative eigenvalues of the Hessian operator. We show numerically that the energy stability condition is satisfied on either part of the curve and prove analytically that it holds in a neighborhood of the boundary of the existence region of smooth periodic waves. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
13. Point-wise behavior of the explosive positive solutions to a degenerate elliptic BVP with an indefinite weight function.
- Author
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López-Gómez, J., Ramos, V.K., Santos, C.A., and Suárez, A.
- Subjects
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BOUNDARY value problems , *EIGENFUNCTIONS , *DEGENERATE differential equations , *EIGENVALUES - Abstract
In this paper we ascertain the singular point-wise behavior of the positive solutions of a semilinear elliptic boundary value problem (1) at the critical value of the parameter, λ , where it begins its metasolution regime. As the weight function m (x) changes sign in Ω, our result is a substantial extension of a previous, very recent, result of Li et al. [8] , where it was imposed the (very strong) condition that m ≥ 0 on a neighborhood of b − 1 ({ 0 }). In this paper, we are simply assuming that m (x 0) > 0 for some x 0 ∈ b − 1 ({ 0 }). • Theorem 1.1 proves that the behavior of the solutions proved by Li et al. [8] also occurs with much weaker hypotheses. • Theorem 3.1 is a substantial extension of Theorem 2.1 of López-Gómez and Sabina de Lis [12]. • Lemma 2.1 provides a useful estimate of eigenfunctions associated to an eigenvalue problem with sign changing weight. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
14. Improved eigenvalue inequalities via two major subclasses of superquadratic functions.
- Author
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Kian, Mohsen
- Subjects
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EIGENVALUES , *CONVEX functions , *CHARACTERISTIC functions , *CONCAVE functions - Abstract
There exist two major subclasses in the class of superquadratic functions, one comprises concave and decreasing functions, while the other consists of convex and monotone increasing functions. Leveraging this distinction, we introduce eigenvalue inequalities for each case. The characteristics of these functions allow us to advance our findings in two ways: firstly, by refining existing results related to eigenvalues for convex functions, and secondly, by deriving complementary inequalities for other function types. To bolster our claims, we will provide illustrative examples. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
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15. Eigenvalue superposition for Toeplitz matrix-sequences with matrix order dependent symbols.
- Author
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Bogoya, M., Grudsky, S.M., and Serra-Capizzano, S.
- Subjects
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TOEPLITZ matrices , *EIGENVALUES , *FINITE difference method , *FRACTIONAL differential equations , *DIFFERENTIAL operators - Abstract
The eigenvalues of Toeplitz matrices T n (f) with a real-valued generating function f , satisfying some conditions and tracing out a simple loop over the interval [ − π , π ] , are known to admit an asymptotic expansion with the form λ j (T n (f)) = f (σ j , n) + c 1 (σ j , n) h + c 2 (σ j , n) h 2 + O (h 3) , where h = 1 / (n + 1) , σ j , n = π j h , and c k are some bounded coefficients depending only on f. The numerical results presented in the literature suggest that the effective conditions for the expansion to hold are weaker and reduce to a fixed smoothness and to having only two intervals of monotonicity over [ − π , π ]. In this article we investigate the superposition caused over this expansion, when considering the following linear combination λ j (T n (f 0) + β n , 1 T n (f 1) + β n , 2 T n (f 2)) , where β n , 1 , β n , 2 are certain constants depending on n and the generating functions f 0 , f 1 , f 2 are either simple loop or satisfy the weaker conditions mentioned before. We formally obtain an asymptotic expansion in this setting under simple-loop related assumptions, and we show numerically that there is much more to investigate, opening the door to linear in time algorithms for the computation of eigenvalues of large matrices of this type including a multilevel setting. The problem is of concrete interest, considering spectral features of matrices stemming from the numerical approximation of standard differential operators and distributed order fractional differential equations, via local methods such as Finite Differences, Finite Elements, and Isogeometric Analysis. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
16. Eigenvalues of laplacian matrices of the cycles with one negative-weighted edge.
- Author
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Grudsky, Sergei M., Maximenko, Egor A., and Soto-González, Alejandro
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LAPLACIAN matrices , *EIGENVALUES , *NEWTON-Raphson method , *TOEPLITZ matrices , *ASYMPTOTIC expansions , *WEIGHTED graphs - Abstract
We study the individual behavior of the eigenvalues of the laplacian matrices of the cyclic graph of order n , where one edge has weight α ∈ C , with Re (α) < 0 , and all the others have weights 1. This paper is a sequel of a previous one where we considered Re (α) ∈ [ 0 , 1 ] (Grudsky et al., 2022 [12]). We prove that for Re (α) < 0 and n > Re (α − 1) / Re (α) , one eigenvalue is negative while the others belong to [ 0 , 4 ] and are distributed as the function x ↦ 4 sin 2 (x / 2). Additionally, we prove that as n tends to ∞, the outlier eigenvalue converges exponentially to 4 Re (α) 2 / (2 Re (α) − 1). We give exact formulas for half of the inner eigenvalues, while for the others we justify the convergence of Newton's method and fixed-point iteration method. We find asymptotic expansions, as n tends to ∞, both for the eigenvalues belonging to [ 0 , 4 ] and the outlier. We also compute the eigenvectors and their norms. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
17. Signal and image reconstruction with a double parameter Hager–Zhang‐type conjugate gradient method for system of nonlinear equations.
- Author
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Ahmed, Kabiru, Waziri, Mohammed Yusuf, Halilu, Abubakar Sani, Murtala, Salisu, and Abdullahi, Habibu
- Subjects
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NONLINEAR equations , *SIGNAL reconstruction , *COMPRESSED sensing , *IMAGE reconstruction , *SIGNAL processing , *CONJUGATE gradient methods , *EIGENVALUES - Abstract
The one parameter conjugate gradient method by Hager and Zhang (Pac J Optim,
2 (1):35–58, 2006) represents a family of descent iterative methods for solving large‐scale minimization problems. The nonnegative parameter of the scheme determines the weight of conjugacy and descent, and by extension, the numerical performance of the method. The scheme, however, does not converge globally for general nonlinear functions, and when the parameter approaches 0, the scheme reduces to the conjugate gradient method by Hestenes and Stiefel (J Res Nat Bur Stand,49 :409–436, 1952), which in practical sense does not perform well due to the jamming phenomenon. By carrying out eigenvalue analysis of an adaptive two parameter Hager–Zhang type method, a new scheme is presented for system of monotone nonlinear equations with its application in compressed sensing. The proposed scheme was inspired by nice attributes of the Hager–Zhang method and the various schemes designed with double parameters. The scheme is also applicable to nonsmooth nonlinear problems. Using fundamental assumptions, analysis of the global convergence of the scheme is conducted and preliminary report of numerical experiments carried out with the scheme and some recent methods indicate that the scheme is promising. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
18. The regularity with respect to domains of the additive eigenvalues of superquadratic Hamilton–Jacobi equation.
- Author
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Bozorgnia, Farid, Kwon, Dohyun, and Tu, Son N.T.
- Subjects
- *
HAMILTON-Jacobi equations , *EIGENVALUES , *OPTIMAL control theory , *CONTINUOUS functions , *VISCOSITY solutions - Abstract
We study the additive eigenvalues on changing domains, along with the associated vanishing discount problems. We consider the convergence of the vanishing discount problem on changing domains for a general scaling type Ω λ = (1 + r (λ)) Ω with a continuous function r and a positive constant λ. We characterize all solutions to the ergodic problem on Ω in terms of r. In addition, we demonstrate that the additive eigenvalue λ ↦ c Ω λ on a rescaled domain Ω λ = (1 + λ) Ω possesses one-sided derivatives everywhere. Additionally, the limiting solution can be parameterized by a real function, and we establish a connection between the regularity of this real function and the regularity of λ ↦ c Ω λ . We provide examples where higher regularity is achieved. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
19. Three-spectra inverse problem for the perturbed Bessel operators.
- Author
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Xu, Xin-Jian and Yang, Chuan-Fu
- Subjects
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EIGENVALUES , *INVERSE problems - Abstract
We study the inverse three spectra problem for the perturbed Bessel operators − D 2 + l (l + 1) x 2 + q (x) in L 2 (0 , 1). We show that eigenvalues of the operators defined on the three intervals (0 , a ] , [ a , 1 ] and (0 , 1 ] , respectively, uniquely determine the potential on (0 , 1 ] , where a ∈ (0 , 1) is fixed. If eigenvalues of the operators defined on the subintervals (0 , a ] and [ a , 1 ] have some common eigenvalues, then the norming constants corresponding to these common eigenvalues are added as the prior data to obtain the uniqueness of the inverse problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
20. On a nonlinear general eigenvalue problem in Musielak–Orlicz spaces.
- Author
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Kassimi, Soufiane, Sabiki, Hajar, and Moussa, Hicham
- Subjects
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NONLINEAR equations , *EIGENVALUES - Abstract
In this paper, we concern the existence result of the following general eigenvalue problem: { 풜 ( u ) = λ ℬ ( u ) in Ω , D α ( u ) = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle{}\mathcal{A}(u)&\displaystyle={\lambda}% \mathcal{B}(u)&&\displaystyle\phantom{}\text{in }{\Omega},\\ \displaystyle D^{\alpha}(u)&\displaystyle=0&&\displaystyle\phantom{}\text{on }% {\partial\Omega},\end{aligned}\right. in an arbitrary Musielak–Orlicz spaces, where 풜 {\mathcal{A}} and ℬ {\mathcal{B}} are quasilinear operators in divergence form of order 2 n {2n} and 2 ( n - 1 ) {2(n-1)} , respectively. The main assumptions in this case are that 풜 {\mathcal{A}} and ℬ {\mathcal{B}} are potential operators with 풜 {\mathcal{A}} being elliptic and monotone. In this study, we intentionally avoid imposing constraints on the growth of a generalized
N -function, including the Δ 2 {\Delta_{2}} -condition for both the generalizedN -function and its conjugate. Consequently, this necessitates the formulation of the approximation theorem and the extensive utilization of modular convergence concepts. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
21. On a class of third-order dissipative differential operator with distributional potentials.
- Author
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Ao, Ji-jun and Li, Fei-fan
- Subjects
- *
BOUNDARY value problems , *DIFFERENTIAL operators , *COMPLETENESS theorem , *DIFFERENTIAL equations , *EIGENVALUES - Abstract
AbstractThis paper aims to investigate a class of boundary value problems generated by a third-order differential equation with distributional potentials and some dissipative boundary conditions. We first construct an operator related to the problem, then prove that the operator is dissipative and show some eigenvalue properties of this operator. Further using Krein’s theorem we prove the completeness theorems of the operator. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
22. Nijenhuis operators with a unity and F$F$‐manifolds.
- Author
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Antonov, Evgenii I. and Konyaev, Andrey Yu.
- Subjects
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VECTOR fields , *CONCORD , *EIGENVALUES - Abstract
The core object of this paper is a pair (L,e)$(L, e)$, where L$L$ is a Nijenhuis operator and e$e$ is a vector field satisfying a specific Lie derivative condition, that is, LeL=Id$\mathcal {L}_{e}L=\operatorname{Id}$. Our research unfolds in two parts. In the first part, we establish a splitting theorem for Nijenhuis operators with a unity, offering an effective reduction of their study to cases where L$L$ has either one real or two complex conjugate eigenvalues at a given point. We further provide the normal forms for gl$\mathrm{gl}$‐regular Nijenhuis operators with a unity around algebraically generic points, along with seminormal forms for dimensions 2 and 3. In the second part, we establish the relationship between Nijenhuis operators with a unity and F$F$‐manifolds. Specifically, we prove that the class of regular F$F$‐manifolds coincides with the class of Nijenhuis manifolds with a cyclic unity. Extending our results from dimension 3, we reveal seminormal forms for corresponding F$F$‐manifolds around singularities. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
23. Asymptotic behavior of Laplacian eigenvalues of subspace inclusion graphs.
- Author
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Lew, Alan
- Subjects
- *
EIGENVALUES , *LOGICAL prediction - Abstract
Let Fln,q$\text{Fl}_{n,q}$ be the simplicial complex whose vertices are the nontrivial subspaces of Fqn$\mathbb {F}_q^n$ and whose simplices correspond to families of subspaces forming a flag. Let Δk+(Fln,q)$\Delta ^{+}_k(\text{Fl}_{n,q})$ be the k$k$‐dimensional weighted upper Laplacian on Fln,q$ \text{Fl}_{n,q}$. The spectrum of Δk+(Fln,q)$\Delta ^{+}_k(\text{Fl}_{n,q})$ was first studied by Garland, who obtained a lower bound on its nonzero eigenvalues. Here, we focus on the k=0$k=0$ case. We determine the asymptotic behavior of the eigenvalues of Δ0+(Fln,q)$\Delta _{0}^{+}(\text{Fl}_{n,q})$ as q$q$ tends to infinity. In particular, we show that for large enough q$q$, Δ0+(Fln,q)$\Delta _{0}^{+}(\text{Fl}_{n,q})$ has exactly n2/4+2$\left\lfloor n^2/4\right\rfloor +2$ distinct eigenvalues, and that every eigenvalue λ≠0,n−1$\lambda \ne 0,n-1$ of Δ0+(Fln,q)$\Delta _{0}^{+}(\text{Fl}_{n,q})$ tends to n−2$n-2$ as q$q$ goes to infinity. This solves the zero‐dimensional case of a conjecture of Papikian. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
24. Nonlinear elliptic eigenvalue problems in cylindrical domains becoming unbounded in one direction.
- Author
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Rawat, Rama, Roy, Haripada, and Roy, Prosenjit
- Subjects
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TOPOLOGICAL degree , *EIGENFUNCTIONS , *EIGENVALUES - Abstract
The aim of this work is to characterize the asymptotic behaviour of the first eigenfunction of the generalised p-Laplace operator with mixed (Dirichlet and Neumann) boundary conditions in cylindrical domains when the length of the cylindrical domains tends to infinity. This generalises an earlier work of Chipot et al. (Asymptot. Anal.85(3–4) (2013) 199–227) where the linear case p = 2 is studied. Asymptotic behavior of all the higher eigenvalues of the linear case and the second eigenvalues of general case (using topological degree) for such problems is also studied. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
25. Exact solutions of Pauli–Schrödinger equation for a particle with position dependent mass and magnetic momentum in a generalized Morse potential and magnetic field.
- Author
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Madouri, Fethi, Alanzi, Abdullah Bnyah, and Merdaci, Abdeldjalil
- Subjects
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MAGNETIC fields , *EIGENFUNCTIONS , *EIGENVALUES , *EQUATIONS - Abstract
The Pauli–Schrodinger equation for a non-relativistic position dependent mass with spin 1/2 and magnetic momentum μ0 in a generalized Morse potential and permanent magnetic field is solved using the Nikiforov–Uvarov method. The energy eigenvalues and the corresponding eigenfunctions are obtained analytically. It is also shown that the results established in a previous work appear to be a special case. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
26. Reduced-Order Model Parameterization for Uncertain LTI SISO Systems.
- Author
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Ansari, Roghaiyeh, Leonessa, Alexander, and Abaid, Nicole
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MATHEMATICAL proofs , *PARAMETERIZATION , *EIGENVALUES - Abstract
The primary goal of this paper is to develop a formal foundation to design an adaptive output feedback predictor for a class of unknown systems where parameters and order are unknown or high-dimensional. We present a reduced-order adaptive output-predictor scheme based on modal reduction and Lyapunov's method. Moreover, the credibility of the proposed reduced-order adaptive output-predictor scheme is validated by mathematical proof and numerical studies. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
27. Remeshing and eigenvalue stabilization in the finite cell method for structures undergoing large elastoplastic deformations.
- Author
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Sartorti, Roman, Garhuom, Wadhah, and Düster, Alexander
- Subjects
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RADIAL basis functions , *CONFIGURATIONS (Geometry) , *ELASTOPLASTICITY , *INVERSE functions , *EIGENVALUES - Abstract
Large strain analysis is a challenging task, especially in fictitious or immersed boundary domain methods, since badly broken elements/cells can lead to an ill-conditioned global tangent stiffness matrix, resulting in convergence problems of the incremental/iterative solution approach. In this work, the finite cell method is employed as a fictitious domain approach, in conjunction with an eigenvalue stabilization technique, to ensure the stability of the solution procedure. Additionally, a remeshing strategy is applied to accommodate highly deformed configurations of the geometry. Radial basis functions and inverse distance weighting interpolation schemes are utilized to map the displacement gradient and internal variables between the old and new meshes during the remeshing process. For the first time, we demonstrate the effectiveness of the remeshing approach using various numerical examples in the context of finite strain elastoplasticity. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
28. Revisiting Kassman's path deletion procedure for the eigenvector coefficients of molecular graphs.
- Author
-
Nag, Madhusree and Mandal, Bholanath
- Subjects
- *
MOLECULAR graphs , *MOLECULAR orbitals , *EIGENVALUES , *FACTORIZATION , *POLYNOMIALS - Abstract
Kassman's path deletion procedure for the determination of eigenvector polynomials (EPs) and hence the eigenvector or molecular orbital (MO) coefficients of a molecular graph is revisited with required proofs and illustrations. As EPs vanish for n-fold degenerate eigenvalues, the calculation of eigenvector (MO) coefficients for such an eigenvalue requires (n-1)-th derivative of each EP. The eigenvector coefficients for other degenerate eigenvalues are then obtained by exploiting the inherent symmetry of the molecular graph. The method of symmetry-factorisation followed by the path deletion procedure makes such calculations straightforward and simple as symmetry-factorisation distributes the degenerate eigenvalues (if there are any) in the fragmented graphs and the problem of degeneracy is thus avoided in calculating the MO coefficients for the individual fragments. The procedure is illustrated with the molecular graphs having non-degenerate and degenerate eigenvalues. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
29. Lower and upper bounds for stokes eigenvalues.
- Author
-
Yue, Yifan, Chen, Hongtao, and Zhang, Shuo
- Subjects
- *
EIGENVALUES , *STOKES equations , *FINITE element method - Abstract
In this paper, we study the lower and upper bounds for Stokes eigenvalues by finite element schemes. For the schemes studied here, roughly speaking, the loss of the local approximation property of the discrete velocity and pressure spaces may lead to different computed bounds of the eigenvalues. Formally theoretical analysis is constructed based on certain mathematical hypotheses, and numerical experiments are given to illustrate the validity of the theory. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
30. Lowest-degree robust finite element schemes for inhomogeneous bi-Laplace problems.
- Author
-
Dai, Bin, Zeng, Huilan, Zhang, Chen-Song, and Zhang, Shuo
- Subjects
- *
SINGULAR perturbations , *CONVEX domains , *DIFFERENTIAL operators , *EIGENVALUES , *INTERPOLATION , *LAPLACE transformation - Abstract
In this paper, we study the numerical method for the bi-Laplace problems with inhomogeneous coefficients; particularly, we propose finite element schemes on rectangular grids respectively for an inhomogeneous fourth-order elliptic singular perturbation problem and for the Helmholtz transmission eigenvalue problem. The new methods use the reduced rectangle Morley (RRM for short) element space with piecewise quadratic polynomials, which are of the lowest degree possible. For the finite element space, a discrete analogue of an equality by Grisvard is proved for the stability issue and a locally-averaged interpolation operator is constructed for the approximation issue. Optimal convergence rates of the schemes are proved, and numerical experiments are given to verify the theoretical analysis. • A discrete analogue of an equality (1.3) by Grisvard [1] on H 2 functions is proved for the reduced rectangular Morley (RRM for short in the sequel) element functions. This discrete equality makes the RRM space usable for bi-Laplacian problems with inhomogeneous coefficients. • Based on piecewise quadratic polynomials, the RRM scheme is the lowest-degree finite element scheme for the inhomogeneous bi-Laplace problems. Compared to other kinds of methods, it does not need tuning parameter or using indirect differential operators. • As revealed by [3] , the RRM element space does not admit a locally-defined projective interpolator. In this paper, however, a locally-defined stable interpolator (not projective) is carefully constructed for the RRM element space, and an optimal approximation is proved rigorously on both convex and nonconvex domains. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
31. Partial Threshold Graphs.
- Author
-
BHAT, K. ARATHI and SHETTY, SHASHWATH S.
- Subjects
- *
SPECTRAL theory , *GRAPH theory , *RESEARCH personnel , *EIGENVALUES , *INTEGERS , *BIPARTITE graphs - Abstract
Chain graphs and threshold graphs play a very important role in Spectral Graph Theory, since the maximizers for the largest eigenvalue of the adjacency matrix (for graphs of fixed order and size, either connected or disconnected) belong to these classes (threshold graphs in the general case, and chain graphs in the bipartite case). Nesting in the neighborhood of vertices in these graphs has gained the attention of various researchers. Motivated by this structure, we generalize and define a new class of graphs named it as 'partial threshold graphs' and study the properties. In this article, we give bounds and expressions for the Wiener index and Hyper-Wiener index of a partial threshold graph. We extend the study further and give a set of integers, except which every other integer is the Wiener index of some partial threshold graph. The highlight of the article is an algorithm for the inverse Wiener index problem of partial threshold graphs. [ABSTRACT FROM AUTHOR]
- Published
- 2024
32. The uniqueness solution for a fractional φ-Laplacian Dirichlet problem and its spectrum.
- Author
-
Lakhdari, Abderrahmane, Tahri, Kamel, and Rhouma, Nedra Belhaj
- Subjects
- *
CRITICAL point theory , *DIRICHLET problem , *EIGENVALUES - Abstract
In this paper, we demonstrate the existence of a unique weak solution for a nonlocal Kirchhoff problem under Dirichlet boundary conditions, involving the fractional φ-Laplacian operator. Our major outcome is acquired by applying variational approaches and critical points theory. In addition, we analyse the spectrum and the eigenvalues associated to this problem. At the end and under some assumptions, we give an application to the previous problems. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
33. A Steklov version of the torsional rigidity.
- Author
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Brasco, L., González, M., and Ispizua, M.
- Subjects
- *
EIGENVALUES - Abstract
Motivated by the connection between the first eigenvalue of the Dirichlet–Laplacian and the torsional rigidity, the aim of this paper is to find a physically coherent and mathematically interesting new concept for boundary torsional rigidity, closely related to the Steklov eigenvalue. From a variational point of view, such a new object corresponds to the sharp constant for the trace embedding of W 1 , 2 (Ω) into L 1 (∂ Ω). We obtain various equivalent variational formulations, present some properties of the state function and obtain some sharp geometric estimates, both for planar simply connected sets and for convex sets in any dimension. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
34. Behavior of the Poincaré constant along the Polchinski renormalization flow.
- Author
-
Serres, Jordan
- Subjects
- *
RENORMALIZATION (Physics) , *SPECTRAL theory , *RENORMALIZATION group , *EIGENVALUES - Abstract
We control the behavior of the Poincaré constant along the Polchinski renormalization flow using a dynamic version of Γ -calculus. We also treat the case of higher order eigenvalues. Our method generalizes a method introduced by Klartag and Putterman to analyze the evolution of log-concave distributions along the heat flow. Furthermore, we apply it to general φ 4 -measures and discuss the interpretation in terms of transport maps. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
35. An Augmented Space Smoothing Method based on the Signal Space in Coherent Scenarios.
- Author
-
Zhao, Jun, Gui, Renzhou, Dong, Xudong, and Sun, Meng
- Subjects
- *
ANGULAR distance , *CROSS correlation , *SIGNAL-to-noise ratio , *SIGNAL processing , *EIGENVALUES - Abstract
In the field of coherent direction of arrival (DOA) estimation, traditional subspace-based algorithms encounter difficulties due to the loss of rank in the signal covariance matrix. To mitigate this concern, we introduce a novel technique called augmented space smoothing (ASS). The proposed method exploits the maximum eigenvectors and their corresponding eigenvalues of the signal subspace, which benefits the utilize both auto-correlation and inter-correlation information. By assigning distinct weights to the auto-correlation and cross-correlation information, our method enables accurate direction-finding estimation of two fully coherent signals located at varying angular intervals. Furthermore, we provide a rigorous proof that the proposed ASS matrix efficiently recovers the matrix rank matching the source quantity. By utilizing these fundamental properties, our approach demonstrates a de-coherence ability to address the rank deficiency issue in traditional subspace-based algorithms used for coherent signal processing tasks. Compared to existing spatial smoothing methods, such as spatial smoothing pre-processing (SSP), modified spatial smoothing pre-processing (MSSP), subarrays cross-correlation (SCC), improved spatial smoothing (ISS), enhanced spatial smoothing (ESS) and enhanced spatial smoothing pre-processing based on signal space (ESS-SS), our proposed algorithm demonstrates a superior estimation performance. Finally, we have verified the effectiveness of our algorithm through the simulation results of signal-to-noise ratio (SNR), the number of snapshots, and angular separation. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
36. Resurgent Asymptotics of Jackiw–Teitelboim Gravity and the Nonperturbative Topological Recursion.
- Author
-
Eynard, Bertrand, Garcia-Failde, Elba, Gregori, Paolo, Lewański, Danilo, and Schiappa, Ricardo
- Subjects
- *
GRAVITY , *STATISTICAL correlation , *EIGENVALUES , *DILATON , *CALCULUS , *INSTANTONS - Abstract
Jackiw–Teitelboim dilaton quantum gravity localizes on a double-scaled random-matrix model, whose perturbative free energy is an asymptotic series. Understanding the resurgent properties of this asymptotic series, including its completion into a full transseries, requires understanding the nonperturbative instanton sectors of the matrix model for Jackiw–Teitelboim gravity. The present work addresses this question by setting-up instanton calculus associated with eigenvalue tunneling (or ZZ-brane contributions), directly in the matrix model. In order to systematize such calculations, a nonperturbative extension of the topological recursion formalism is required—which is herein both constructed and applied to the present problem. Large-order tests of the perturbative genus expansion validate the resurgent nature of Jackiw–Teitelboim gravity, both for its free energy and for its (multi-resolvent) correlation functions. Both ZZ and FZZT nonperturbative effects are required by resurgence, and they further display resonance upon the Borel plane. Finally, the resurgence properties of the multi-resolvent correlation functions yield new and improved resurgence formulae for the large-genus growth of Weil–Petersson volumes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
37. New classes of C1-robustly transitive maps with persistent critical points.
- Author
-
Lizana, C. and Ranter, W.
- Subjects
- *
ENDOMORPHISMS , *EIGENVALUES - Abstract
Recently, the authors proved in [C. Lizana and W. Ranter, Topological obstructions for robustly transitive endomorphisms on surfaces, Adv. Math. 390 (2021), pp. 107901] that every $ C^1 $ C 1 -robustly transitive toral endomorphism displaying critical points must be homotopic to a linear endomorphism having at least one eigenvalue with modulus greater than one. Here, we exhibit some examples of $ C^1 $ C 1 -robustly transitive surface endomorphisms displaying critical points in certain homotopy classes. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
38. An adaptive mesh refinement method based on a characteristic‐compression embedded shock wave indicator for high‐speed flows.
- Author
-
Feng, Yiwei, Lv, Lili, Liu, Tiegang, Xu, Liang, and Yuan, Weixiong
- Subjects
- *
SHOCK waves , *GALERKIN methods , *FLOW simulations , *EIGENVALUES - Abstract
Numerical simulation of high‐speed flows often needs a fine grid for capturing detailed structures of shock or contact wave, which makes high‐order discontinuous Galerkin methods (DGMs) extremely costly. In this work, a characteristic‐compression based adaptive mesh refinement (AMR, h‐adaptive) method is proposed for efficiently improving resolution of the high‐speed flows. In order to allocate computational resources to needed regions, a characteristic‐compression embedded shock wave indicator is developed on incompatible grids and employed as the criterion for AMR. This indicator applies the admissible jumps of eigenvalues to measure the local compression of homogeneous characteristic curves, and theoretically can capture regions of characteristic‐compression which contain structures of shock, contact waves and vortices. Numerical results show that the proposed h‐adaptive DGM is robust, efficient and high‐resolution, it can capture dissipative shock, contact waves of different strengths and vortices with low noise on a rather coarse grid, and can significantly improve resolution of these structures through mild increase of computational resources as compared with the residual‐based h‐adaptive method. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
39. Two Matrix Theorems Arising from Nilpotent Groups.
- Author
-
Zhao, Jing and Liu, Heguo
- Subjects
- *
CHINESE remainder theorem , *NILPOTENT groups , *GROUP theory , *COMPLEX matrices , *EIGENVALUES - Abstract
For a nilpotent group G without π -torsion, and x , y ∈ G , if x n = y n for a π -number n , then x = y ; if x m y n = y n x m for π -numbers m , n , then x y = y x. This is a well-known result in group theory. In this paper, we prove two analogous theorems on matrices, which have independence significance. Specifically, let m be a given positive integer and A a complex square matrix satisfying that (i) all eigenvalues of A are nonnegative, and (ii) rank A 2 = rank A ; then A has a unique m -th root X with rank X 2 = rank X , all eigenvalues of X are nonnegative, and moreover there is a polynomial f (λ) with X = f (A). In addition, let A and B be complex n × n matrices with all eigenvalues nonnegative, and rank A 2 = rank A , rank B 2 = rank B ; then (i) A = B when A r = B r for some positive integer r , and (ii) A B = B A when A s B t = B t A s for two positive integers s and t. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
40. Triangle-free Graphs with Three Positive Eigenvalues.
- Author
-
Duan, Fang
- Subjects
- *
EIGENVALUES - Abstract
A graph G is called triangle-free if G does not contain any triangle as its induced subgraph. Let G n be the set of triangle-free graphs of order n each of which has three positive eigenvalues. In this paper, we find 20 specific graphs in G n , each of which has nullity no more than 2, and we show that in terms of three graph transformations all the other graphs of G n can be constructed from these 20 specific graphs. Hence, we completely characterize the triangle-free graphs with exactly three positive eigenvalues. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
41. Observer-based positive edge consensus of multi-agent systems with directed graphs.
- Author
-
Zhang, Pengyu, Zhang, Wei, Hu, Liduo, and Hu, Zhi
- Subjects
- *
SPANNING trees , *POSITIVE systems , *EIGENVALUES , *COMPUTER simulation , *ALGORITHMS , *DIRECTED graphs - Abstract
This paper concentrates on the positive edge consensus in directed networks with spanning trees using output feedback protocols. First, a novel positive system observer is established, which introduces a parameter to enhance the design freedom of the observer. Furthermore, sufficient conditions are proposed, which are based solely on the network's edge count. Specifically, improved consensus and non-negative conditions are obtained by optimizing the constraints on the eigenvalue information. Subsequently, based on positive edge-consensus design conditions, a programming algorithm is developed. Finally, the effectiveness of the proposed control protocol is verified utilizing numerical simulations. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
42. On the stability of constant higher order mean curvature hypersurfaces in a Riemannian manifold.
- Author
-
Elbert, Maria Fernanda and Nelli, Barbara
- Subjects
- *
ELLIPTIC operators , *RIEMANNIAN manifolds , *STABILITY constants , *HYPERSURFACES , *CURVATURE - Abstract
We propose a notion of stability for constant k$k$‐mean curvature hypersurfaces in a general Riemannian manifold and we give some applications. When the ambient manifold is a Space Form, our notion coincides with the known one, given by means of the variational problem. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
43. Bifurcation for indefinite‐weighted p$p$‐Laplacian problems with slightly subcritical nonlinearity.
- Author
-
Cuesta, Mabel and Pardo, Rosa
- Subjects
- *
ORLICZ spaces , *BOUNDARY value problems , *EIGENVALUES , *LAPLACIAN operator - Abstract
We study a superlinear elliptic boundary value problem involving the p$p$‐Laplacian operator, with changing sign weights. The problem has positive solutions bifurcating from the trivial solution set at the two principal eigenvalues of the corresponding linear weighted boundary value problem.Drabek's bifurcation result applies when the nonlinearity is of power growth. We extend Drabek's bifurcation result to
slightly subcritical nonlinearities. Compactness in this setting is a delicate issue obtained via Orlicz spaces. [ABSTRACT FROM AUTHOR]- Published
- 2024
- Full Text
- View/download PDF
44. Improved spectral cluster bounds for orthonormal systems.
- Author
-
Ren, Tianyi and Zhang, An
- Subjects
- *
RIEMANNIAN manifolds , *INTEGRAL operators , *TORUS , *MATHEMATICS , *EIGENVALUES - Abstract
We improve the work [R. L. Frank and J. Sabin, Spectral cluster bounds for orthonormal systems and oscillatory integral operators in Schatten spaces, Adv. Math. 317 2017, 157–192] concerning the spectral cluster bounds for orthonormal systems at p = ∞ , on the flat torus and spaces of nonpositive sectional curvature, by shrinking the spectral band from [ λ 2 , (λ + 1) 2) to [ λ 2 , (λ + ϵ (λ) ) 2) , where ϵ (λ) is a function of λ that goes to 0 as λ goes to ∞ . In achieving this, we invoke the method developed in [J. Bourgain, P. Shao, C. D. Sogge and X. Yao, On L p -resolvent estimates and the density of eigenvalues for compact Riemannian manifolds, Comm. Math. Phys. 333 2015, 3, 1483–1527]. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
45. Isotopisms of nilpotent Leibniz algebras and Lie racks.
- Author
-
La Rosa, Gianmarco, Mancini, Manuel, and Nagy, Gábor P.
- Subjects
- *
NILPOTENT Lie groups , *LIE algebras , *ALGEBRA , *EIGENVALUES - Abstract
In this paper we study the isotopism classes of two-step nilpotent algebras. We show that every nilpotent Leibniz algebra g with dim [ g , g ] = 1 is isotopic to the Heisenberg Lie algebra or to the Heisenberg algebra l 2 n + 1 J 1 , where J1 is the n × n Jordan block of eigenvalue 1. We also prove that two such algebras are isotopic if and only if the Lie racks integrating them are isotopic. This gives the classification of Lie racks whose tangent space at the unit element is a nilpotent Leibniz algebra with one-dimensional commutator ideal. Eventually, we introduce new isotopism invariants for Leibniz algebras and Lie racks. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
46. Eigenvalues for the Clamped Plate Problem of Lν2 Operator on Complete Riemannian Manifolds.
- Author
-
Zeng, Ling Zhong
- Subjects
- *
DIFFERENTIAL operators , *PROJECTIVE spaces , *EIGENVALUES , *SOLITONS , *CURVATURE - Abstract
L ν operator is an important extrinsic differential operator of divergence type and has profound geometric settings. In this paper, we consider the clamped plate problem of L ν 2 operator on a bounded domain of the complete Riemannian manifolds. A general formula of eigenvalues of L ν 2 operator is established. Applying this general formula, we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds. As several fascinating applications, we discuss this eigenvalue problem on the complete translating solitons, minimal submanifolds on the Euclidean space, submanifolds on the unit sphere and projective spaces. In particular, we get a universal inequality with respect to the L I I operator on the translating solitons. Usually, it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds. Therefore, this work can be viewed as a new contribution to universal estimate. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
47. Randić Matrix and Randić Energy of Uniform Hypergraphs.
- Author
-
Shirdel, Gholam Hassan, Mortezaee, Ameneh, and Alaameri, Laith
- Subjects
HYPERGRAPHS ,EIGENVALUES ,MATHEMATICAL bounds ,MATHEMATICAL models ,MATHEMATICAL analysis - Abstract
The Randić matrix R = [rij] of a graph G = (V;E) was defined as...j if vertices vi and vj are adjacent and rij = 0 otherwise, where di is the degree of the vertex vi 2 V. In this paper, we define the Randić matrix of a uniform hypergraph and study some its spectral properties. We also define the Randić energy of a uniform hypergraph and determine some upper and lower bound for it. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
48. On the Neumann (p, q)-eigenvalue problem in Hölder singular domains.
- Author
-
Garain, Prashanta, Pchelintsev, Valerii, and Ukhlov, Alexander
- Subjects
EIGENFUNCTIONS ,EIGENVALUES - Abstract
In the article we study the Neumann (p, q)-eigenvalue problems in bounded Hölder γ -singular domains Ω γ ⊂ R n . In the case 1 < p < ∞ and 1 < q < p γ ∗ we prove solvability of this eigenvalue problem and existence of the minimizer of the associated variational problem. In addition, we establish some regularity results of the eigenfunctions and some estimates of (p, q)-eigenvalues. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
49. Higher Robin eigenvalues for the p-Laplacian operator as p approaches 1.
- Author
-
Sabina de Lis, José C. and Segura de León, Sergio
- Subjects
EIGENFUNCTIONS ,EIGENVALUES - Abstract
This work addresses several aspects of the dependence on p of the higher eigenvalues λ n to the Robin problem, - Δ p u = λ | u | p - 2 u x ∈ Ω , | ∇ u | p - 2 ∂ u ∂ ν + b | u | p - 2 u = 0 x ∈ ∂ Ω. Here, Ω ⊂ R N is a C 1 bounded domain, ν is the outer unit normal, Δ p u = div (| ∇ u | p - 2 ∇ u) stands for the p-Laplacian operator and b ∈ L ∞ (∂ Ω) . Main results concern: (a) the existence of the limits of λ n as p → 1 , (b) the 'limit problems' satisfied by the 'limit eigenpairs', (c) the continuous dependence of λ n on p when 1 < p < ∞ and (d) the limit profile of the eigenfunctions as p → 1 . The latter study is performed in the one dimensional and radially symmetric cases. Corresponding properties on the Dirichlet and Neumann eigenvalues are also studied in these two special scenarios. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
50. Optimizing Aspect Term Extraction and Sentiment Classification through Attention Mechanism and Sparse Attention Techniques.
- Author
-
Marutho, Dhendra, Muljono, Rustad, Supriadi, and Purwanto
- Subjects
COLOR space ,SENTIMENT analysis ,FEATURE extraction ,FORGERY ,EIGENVALUES - Abstract
Image forgery detection is essential for forensic and security applications. In local locations where key points (KPs) are few, traditional approaches that rely on them suffer from restrictions. To tackle this problem, this paper suggests an innovative strategy that combines pre-processing, feature extraction, and classification methods. We specifically present EFP-AEHO, or Enhanced Flower Pollination-Adaptive Elephant Herd Optimization, for modifying the Deep Belief Network's (DBN) weights. Our method provides better performance by first converting RGB images into the Lab color space, then applying Eigenvalue Asymmetry (EAS) to extract texture features. Its efficacy is highlighted by experimental findings on the F2000 dataset, which show testing accuracy of 99.4%, TPR of 98%, and TNR of 99.12%. [ABSTRACT FROM AUTHOR]
- Published
- 2024
- Full Text
- View/download PDF
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