Search

In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian. [ABSTRACT FROM AUTHOR]

The region in the complex plane containing the eigenvalues of all n × n stochastic matrices was described by Karpelevič in 1951, and it is since then known as the Karpelevič region. The boundary of the Karpelevič region is the union of arcs called the Karpelevič arcs. We provide a complete characterization of the Karpelevič arcs that are powers of some other Karpelevič arc. Furthermore, we find the necessary and sufficient conditions for a sparsest stochastic matrix associated with the Karpelevič arc of order n to be a power of another stochastic matrix. [ABSTRACT FROM AUTHOR]

Estimation of genetic diversity is vital in sunflower breeding. Principal component analysis (PCA) is one of the best statistical approaches to study diversity among genotypes by using eigenvalue and PC-1/PC-2 based-biplots. An experiment commenced at the College of Agriculture, University of Sargodha, Pakistan to find out the diversity in 28 sunflower hybrids and select stable hybrids for normal and terminal heat-stress conditions. Growing sunflower hybrids in randomized complete block design had three replications. Under normal condition, sunflower hybrid sowing ensued in February 2019, while to check the effect of terminal stress, growing these hybrids transpired at the end of March 2019. The collected data of seven morpho-phenological traits underwent the principal component analysis, found greater than one for the first three and four PCs under normal conditions. In terminal stress environments, these respectively exposed the presence of ample genetic variation among sunflower hybrids in both environments. Bi-plot analysis signified that SF-18100, SF-18045, and SF-19025 were stable hybrids for most studied traits under normal sowing environment. Meanwhile, the performance of SF-18035 and SF-19010 proved better under the heat-stress environment for studied traits. Hence, these hybrids showed better adaptability under the current scenario of climate change. [ABSTRACT FROM AUTHOR]

Let λ (t) be the first eigenvalue of the operator − ∆ + a R b on locally three-dimensional homogeneous manifolds along the backward Ricci flow, where a , b are real constants and R is the scalar curvature. In this paper, we study the properties of λ (t) on Bianchi classes. We begin by deriving an evolution equation for the quantity λ (t) on three-dimensional homogeneous manifolds in the context of the backward Ricci flow. Utilizing this equation, we subsequently establish a monotonic quantity that is contingent upon λ (t) . Additionally, we present both upper and lower bounds for λ (t) within the framework of Bianchi classes. [ABSTRACT FROM AUTHOR]

The authors use a temporal stability analysis to examine the hydrodynamics performance of flow response quantities to investigate the impacts of pertained parameters on Casson nanofluid over a porous shrinking wedge. Thermal analysis is performed in the current flow with thermal radiation and the viscous dissipation effect. Boungiorno's model is used to develop flow equations for Casson nanofluid over a shrinking wedge. An efficient similarity variable is used to change flow equations (PDEs) into dimensionless ordinary differential equations (ODEs) and numerical results are evaluated using MATLAB built-in routine bvp4c. The consequence of this analysis reveals that the impact of active parameters on momentum, thermal and concentration boundary layer distributions are calculated. The dual nature of flow response output i.e. Cƒx is computed for various values of βT = 2.5, 3.5, 4.5, and the critical value is found to be -1.544996, -1.591, and -1.66396. It is perceived that the first (upper branch) solution rises for the temperature profile when the value of thermal radiation is increased and it has the opposite impact on the concentration profile. Thermal radiation has the same critical value for Nux and Shx. The perturbation scheme is applied to the boundary layer problem to obtain the eigenvalues problem. The unsteady solution ƒ (η, τ) converges to steady solution ƒo (η) for τ → ∞ when γ ≥ 0. However, an unsteady solution ƒ (η, τ) diverges to a steady solution ƒo (η) for τ → ∞ when γ < 0. It is found that the boundary layer thickness for the second (lower branch) solution is higher than the first (upper branch) solution. This investigation is the evidence that the first (upper branch) solution is stable and reliable. [ABSTRACT FROM AUTHOR]

Let G be a connected graph. The toughness of G is defined as t (G) = min | S | c (G - S) , in which the minimum is taken over all proper subsets S ⊂ V (G) such that c (G - S) ≥ 2 where c (G - S) denotes the number of components of G - S . Confirming a conjecture of Brouwer, Gu (SIAM J Discrete Math 35:948–952, 2021) proved a tight lower bound on toughness of regular graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu (Eur J Combin 110:103701, 2023) then studied the toughness of simple graphs from the spectral radius perspective. While the toughness is an important concept in graph theory, it is also very interesting to study |S| for which c (G - S) ≥ l for a given integer l ≥ 2 . This leads to the concept of the l-connectivity, which is defined to be the minimum number of vertices of G whose removal produces a disconnected graph with at least l components or a graph with fewer than l vertices. Gu (Eur J Combin 92:103255, 2021) discovered a lower bound on the l-connectivity of regular graphs via the second largest absolute eigenvalue. As a counterpart, we discover the connection between the l-connectivity of simple graphs and the spectral radius. We also study similar problems for digraphs and an edge version. [ABSTRACT FROM AUTHOR]

The current engineering theories on bending vibrations and the stability of beam structures are based on solving eigenvalue problems through similarly formulated differential equations. Solving the eigenvalue problem for engineering calculations is particularly laborious, especially for non-classical supports, where factors like the stiffness of supports, axial forces, or temperature must be considered. In this case, the solution can be obtained only by numerical methods using specially created programs, which makes it difficult to select supports for a given planar beam structure in engineering practice. This work utilizes established solutions from eigenvalue problems in the theory of vibrations and stability of beams, incorporating factors such as axial forces, temperature, and support stiffness. This combined solution is applicable to beam structures of any type and cross-section, as it is determined solely by the selected support conditions (stiffness) and loading (axial force, temperature). Approximation of eigenvalue problem solutions through continuous functions allows the readers to use them for the analytical solution of the design problem of choosing a support system to ensure the frequency of vibrations and stability of the planar beam structure. At the same time, the known solutions given in the reference books on bending vibrations and stability become their particular solutions. This approach is applicable to solving problems of vibrations and loss of stability of various types (torsional, longitudinal, etc.), and is also applicable in other disciplines where solving problems for eigenvalues is required. [ABSTRACT FROM AUTHOR]

This paper improves the ill-condition of bone-conducted (BC) speech signal by reducing the eigenvalue expansion. BC speech commonly contains a large spectral dynamic range that causes ill-condition for the classical linear prediction (LP) methods. In the field of numerical analysis, we often face the situation where an ill-conditioned case occurs in finding the solution. Principally, eigenvalue expansion causes ill-condition in numerical analysis. To mitigate this problem, the regularized least squares (RLS) technique is commonly used. Motivated by the RLS concept, we derive the regularized modified covariance (RMC) method for BC speech analysis in this study. The RMC method reduces eigenvalue expansion by compressing the spectral dynamic range of the speech signal. Thus, the RMC method resolves the ill-conditioned problem of LP. In experiments, we show that the RMC method provides compressed eigenvalue expansion than the conventional methods for BC speech where synthetic and real BC speeches are considered. The performance of the RMC method is affected by the setting of the regularization parameter. In this paper, the regularization parameter in practice is iteratively and rule-based derived. The RMC method with such a setting provides the best performance for BC speech analysis. [ABSTRACT FROM AUTHOR]

Let k and n be two coprime positive integers. In this paper, we show that any $ n\times n $ n × n k-circulant operator matrix $ A_{k,n} $ A k , n with the first row $ A_{1},\dots,A_{n}\in B(H) $ A 1 , ... , A n ∈ B (H) is unitarily equivalent to a generalized permutation operator matrix, an operator matrix that has at most one nonzero entry in each row and in each column. Among other applications in the subject of invertibility, eigenvalues and eigenvectors, using this unitary equivalence in the particular cases of $ k^{2}\pm 1=n $ k 2 ± 1 = n $ ({\rm mod} n) $ (mod n) , we give the following new formulas for computing the numerical radius of these operators: \[ w(A_{k,n})= \begin{cases} \frac{1}{2}\displaystyle \max_{1\leq i\leq n} \sup_{\theta \in \mathbb{R}} \left\| \begin{bmatrix} {\rm e}^{\mathrm{i}\theta}B_{i+1} & B^{*}_{ki+1}\\ B^{*}_{-ki+1} & {\rm e}^{\mathrm{i}\theta}B_{-i+1} \end{bmatrix} \right\|, & {\rm if} \ k^{2}+1\\ & \quad =n \ ({\rm mod}\,n)\\ \displaystyle\max_{1\leq i \leq n}w \left(\begin{bmatrix} 0 & B_{k(i-1)+1}\\ B_{i} & 0 \end{bmatrix} \right), & {\rm if} \ k^{2}-1\\ & \quad =n \ ({\rm mod}\,n), \end{cases} \] w (A k , n) = { 1 2 max 1 ≤ i ≤ n sup θ ∈ R ‖ [ e i θ B i + 1 B ki + 1 ∗ B − ki + 1 ∗ e i θ B − i + 1 ] ‖ , if k 2 + 1 = n (mod n) max 1 ≤ i ≤ n w ([ 0 B k (i − 1) + 1 B i 0 ]) , if k 2 − 1 = n (mod n) , where $ B_{j}=\sum _{s=1}^{n}\omega ^{(s-1)(j-1)}A_{s} $ B j = ∑ s = 1 n ω (s − 1) (j − 1) A s and $ \omega ={\rm e}^{\frac {2\pi \mathrm {i}}{n}} $ ω = e 2 π i n , the nth root of unity. [ABSTRACT FROM AUTHOR]

Summary: Multiple‐input multiple‐output (MIMO) radar, employing multiple transmitters and receivers, enhances radar capabilities. It detects and tracks objects like aircraft and ships using radio waves. Compared with traditional phased‐array radar, MIMO systems offer greater flexibility, improving angular resolution and target detection. Researchers focus on direction of arrival (DoA) evaluation for closely spaced targets. Effective beamforming and accurate DoA estimation are crucial for MIMO radar performance. This study explores two methods: Capon beamforming with Dolph–Chebyshev windowing and the MUSIC algorithm with Eigenvalue thresholding. Tested under low signal‐to‐noise ratio (SNR) and fewer snapshots, these techniques notably reduce side lobes and enhance angular resolution, validated by experiments. Additionally, the suppression of side lobes significantly improves the clarity and accuracy of target detection, minimizing potential interference and false targets. This enhancement in side lobe suppression facilitates a more precise spatial differentiation between multiple targets, thus contributing to the overall effectiveness and reliability of MIMO radar systems. [ABSTRACT FROM AUTHOR]

We present an approach yielding closed analytical formulas for the Kirchhoff indices of circulant graphs with even and odd vertex valency respectively and the prism-like graphs based on circulant graphs. Inspecting the asymptotics of the Kirchhoff index we show that in each of the above-mentioned cases the index can be expressed as the sum of a cubic polynomial and an exponentially vanishing remainder term. [ABSTRACT FROM AUTHOR]

We consider the Schrödinger operator in a bounded domain $ \mathbf{R}^n (n=2,3) $ R n (n = 2 , 3) with Neumann boundary condition. We suppose that this domain contains small deformable inclusions, i.e. regions where the potentials do not have the same values as the exterior medium. Our goal is to construct an asymptotic formula for the case of multiple and simple eigenvalue problems. We find an expansion that highlights the relation between the deformation parameters and the eigenvalues. The problem is devoted to a domain containing a finite number of inhomogeneities. Also, we focus on the interface of a small inclusion. [ABSTRACT FROM AUTHOR]

Consider the product of m independent n-by-n Ginibre matrices and their inverses, where m = p + q , p is the number of Ginibre matrices, and q is the number of inverses of Ginibre matrices. The maximum absolute value of the eigenvalues of the product matrices is known as the spectral radius. In this paper, we explore the limiting spectral radii of the product matrices as n tends to infinity and m varies with n. Specifically, when q ≥ 1 is a fixed integer, we demonstrate that the limiting spectral radii display a transition phenomenon when the limit of p/n changes from zero to infinity. When q = 0 , the limiting spectral radii for Ginibre matrices have been obtained by Jiang and Qi [J Theor Probab 30: 326–364, 2017]. When q diverges to infinity as n approaches infinity, we prove that the logarithmic spectral radii exhibit a normal limit, which reduces to the limiting distribution for spectral radii for the spherical ensemble obtained by Chang et al. [J Math Anal Appl 461: 1165–1176, 2018] when p = q . [ABSTRACT FROM AUTHOR]

We are concerned with the linear problem - Δ u + κ | x | 2 x · ∇ u = λ K (| x |) u , x ∈ R N , u (x) > 0 , x ∈ R N , [ 2 e x ] u (x) → 0 , | x | → ∞ , where λ is a positive parameter, κ ∈ [ 0 , N - 2) , N > 2 and K : R N → (0 , ∞) is continuous and satisfies certain decay assumptions. We obtain the existence of the principal eigenvalue λ 1 rad and the corresponding positive eigenfunction φ 1 satisfies lim | x | → ∞ φ 1 (| x |) = c | x | N - 2 - κ for some c > 0 . As applications, we also study the existence of connected component of positive solutions for nonlinear infinite semipositone elliptic problems by bifurcation techniques. [ABSTRACT FROM AUTHOR]

Copyright of Systems Engineering & Electronics is the property of Journal of Systems Engineering & Electronics Editorial Department and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)

The article investigates a Dirichlet-type problem with conjugation conditions for a degenerate equation of high even order with variable coefficients, including the Riemann–Liouville fractional derivative, in a rectangular region consisting of two subdomains ( y > 0 and y < 0 ), in each of which the equation has various kind.The solution is constructed as a series in terms of eigenfunctions of the one-dimensional problem, the existence of eigenfunctions is proved by the method of the theory of integral equations with symmetric kernels. The theorem of expansion in terms of the system of obtained eigenfunctions is proved. Sufficent conditions are found for boundary functions under which the solution in the form of a series converges uniformly. When justifying the convergence of a series, the problem of "small denominators" arises.This problem has been successfully solved in the article. The uniqueness of the solution is proved by the spectral method. [ABSTRACT FROM AUTHOR]

Let β > 0 . Motivated by the notion of jumbled graphs introduced by Thomason, the expander mixing lemma and Haemers's vertex separation inequality, we say that a graph G with n vertices is a weakly (n , β) -graph if | X | | Y | (n - | X |) (n - | Y |) ≤ β 2 holds for every pair of disjoint proper subsets X, Y of V(G) with no edge between X and Y. It is an (n , β) -graph if in addition X and Y are not necessarily disjoint. Using graph eigenvalues, we show that every graph can be an (n , β) -graph and/or a weakly (n , β) -graph for some specific value β . For instances, the expander mixing lemma implies that a d-regular graph on n vertices with the second largest absolute eigenvalue at most λ is an (n , λ / d) -graph, and Haemers's vertex separation inequality implies that every graph is a weakly (n , β) -graph with β ≥ μ n - μ 2 μ n + μ 2 , where μ i denotes the i-th smallest Laplacian eigenvalue. This motivates us to study (n , β) -graph and weakly (n , β) -graph in general. Our main results include the following. (i) For any weakly (n , β) -graph G, the matching number α ′ (G) ≥ min 1 - β 1 + β , 1 2 · (n - 1) . If in addition G is a (U, W)-bipartite graph with | W | ≥ t | U | where t ≥ 1 , then α ′ (G) ≥ min { t (1 - 2 β 2) , 1 } · | U | . (ii) For any (n , β) -graph G, α ′ (G) ≥ min 2 - β 2 (1 + β) , 1 2 · (n - 1). If in addition G is a (U, W)-bipartite graph with | W | ≥ | U | and no isolated vertices, then α ′ (G) ≥ min { 1 / β 2 , 1 } · | U | . (iii) If G is a weakly (n , β) -graph for 0 < β ≤ 1 / 3 or an (n , β) -graph for 0 < β ≤ 1 / 2 , then G has a fractional perfect matching. In addition, G has a perfect matching when n is even and G is factor-critical when n is odd. (iv) For any connected (n , β) -graph G, the toughness t (G) ≥ 1 - β β . For any connected weakly (n , β) -graph G, t (G) > 5 (1 - β) 11 β and if n is large enough, then t (G) > 1 2 - ε 1 - β β for any ε > 0 . The results imply many old and new results in spectral graph theory, including several new lower bounds on matching number, fractional matching number and toughness from eigenvalues. In particular, we obtain a new lower bound on toughness via normalized Laplacian eigenvalues that extends a theorem originally conjectured by Brouwer from regular graphs to general graphs. [ABSTRACT FROM AUTHOR]

The authors use a temporal stability analysis to examine the hydrodynamics performance of flow response quantities to investigate the impacts of pertained parameters on Casson nanofluid over a porous shrinking wedge. Thermal analysis is performed in the current flow with thermal radiation and the viscous dissipation effect. Boungiorno's model is used to develop flow equations for Casson nanofluid over a shrinking wedge. An efficient similarity variable is used to change flow equations (PDEs) into dimensionless ordinary differential equations (ODEs) and numerical results are evaluated using MATLAB built-in routine bvp4c. The consequence of this analysis reveals that the impact of active parameters on momentum, thermal and concentration boundary layer distributions are calculated. The dual nature of flow response output i.e. Cfx is computed for various values of βT=2.5,3.5,4.5, and the critical value is found to be −1.544996, −1.591, and −1.66396. It is perceived that the first (upper branch) solution rises for the temperature profile when the value of thermal radiation is increased and it has the opposite impact on the concentration profile. Thermal radiation has the same critical value for Nux and Shx. The perturbation scheme is applied to the boundary layer problem to obtain the eigenvalues problem. The unsteady solution f(η,τ) converges to steady solution fo(η) for τ→∞ when γ≥0. However, an unsteady solution f(η,τ) diverges to a steady solution fo(η) for τ→∞ when γ

The current engineering theories on bending vibrations and the stability of beam structures are based on solving eigenvalue problems through similarly formulated differential equations. Solving the eigenvalue problem for engineering calculations is particularly laborious, especially for non-classical supports, where factors like the stiffness of supports, axial forces, or temperature must be considered. In this case, the solution can be obtained only by numerical methods using specially created programs, which makes it difficult to select supports for a given planar beam structure in engineering practice. This work utilizes established solutions from eigenvalue problems in the theory of vibrations and stability of beams, incorporating factors such as axial forces, temperature, and support stiffness. This combined solution is applicable to beam structures of any type and cross-section, as it is determined solely by the selected support conditions (stiffness) and loading (axial force, temperature). Approximation of eigenvalue problem solutions through continuous functions allows the readers to use them for the analytical solution of the design problem of choosing a support system to ensure the frequency of vibrations and stability of the planar beam structure. At the same time, the known solutions given in the reference books on bending vibrations and stability become their particular solutions. This approach is applicable to solving problems of vibrations and loss of stability of various types (torsional, longitudinal, etc.), and is also applicable in other disciplines where solving problems for eigenvalues is required.

We consider a Riemannian compact manifold $ M $, the associated Laplacian $ \Delta $ and the corresponding Brownian motion $ X_t $, $ t\geq 0. $ Given a Lipschitz function $ V:M\to{\mathbb R} $ we consider the operator $ \frac{1}{2}\Delta+V $, which acts on differentiable functions $ f: M\to{\mathbb R} $ via the expression$ \frac{1}{2} \Delta f(x)+\,V(x)f(x) \,, $for all $ x\in M $.Denote by $ P_t^V $, $ t \geq 0, $ the semigroup acting on functions $ f: M\to{\mathbb R} $ given by$ P_t^V(f)(x): = \mathbb{E}_x\left[e^{\int_0^t V\left(X_r\right) d r} f\left(X_t\right)\right]. $We will derive results that show that this semigroup is a continuous-time version of the discrete-time Ruelle operator.Consider the positive differentiable eigenfunction $ F: M \to \mathbb{R} $ associated with the main eigenvalue $ \lambda $, for the semigroup $ P_t^V $, $ t \geq 0 $. From the function $ F $, in a procedure similar to the one used in discrete-time Thermodynamic Formalism, we can associate by way of a coboundary procedure, a certain stationary Markov semigroup. We show that the probability on the Skorohod space obtained from this new stationary Markov semigroup meets the requirements to be called stationary Gibbs state associated with the potential $ V $. We define entropy, pressure, and the continuous-time Ruelle operator. Also, we present a variational principle of pressure for such a setting. [ABSTRACT FROM AUTHOR]

By using functional analysis we study well-posedness, asymptotic behavior of the time-dependent solution and asymptotic behavior of some time-dependent performance measures of the M/G/1 retrial queueing model with general retrial times, negative customers, feedback and repairs. This queueing model is described by infinitely many partial differential equations with integral boundary conditions. Under some conditions, by using C 0 - semigroup theory and spectral theory of linear operators, we prove that this model has a unique positive time-dependent solution which satisfies the probability condition and its time-dependent solution converges strongly to its steady-state solution. Moreover, we give asymptotic behavior of some time-dependent indices of the queueing system. Finally, we conclude with some further research problems. [ABSTRACT FROM AUTHOR]

We consider the magnetic Dirichlet Laplacian with constant magnetic field on domains of finite measure. First, in the case of a disk, we prove that the eigenvalue branches with respect to the field strength behave asymptotically linear with an exponentially small remainder term as the field strength goes to infinity. We compute the asymptotic expression for this remainder term. Second, we show that for sufficiently large magnetic field strengths, the spectral bound corresponding to the Pólya conjecture for the non-magnetic Dirichlet Laplacian is violated up to a sharp excess factor which is independent of the domain. [ABSTRACT FROM AUTHOR]

In this study, a skewness estimation method (SEM) and kurtosis estimation method (KEM) are introduced to determine the number of inactive cycles in Monte Carlo eigenvalue calculations. The SEM and KEM can determine the number of inactive cycles on the basis that fully converged fission source distributions may follow normal distributions without asymmetry or outliers. Two convergence criteria values and a minimum cycle length for the SEM and KEM were determined from skewness and kurtosis analyses of the AGN-201K benchmark and 1D slab problems. The SEM and KEM were then applied to two OECD/NEA slow convergence benchmark problems to evaluate the performance and reliability of the developed methods. Results confirmed that the SEM and KEM provide appropriate and effective convergence cycles when compared to other methods and fission source density fraction trends. Also, the determined criterion value of 0.5 for both ε1 and ε2 was concluded to be reasonable. The SEM and KEM can be utilized as a new approach for determining the number of inactive cycles and judging whether Monte Carlo tally values are fully converged. In the near future, the methods will be applied to various practical problems to further examine their performance and reliability, and optimization will be performed for the convergence criteria and other parameters as well as for improvement of the methodology for practical usage.

In this paper, we consider an eigenvalue problem for fourth-order ordinary differential equations with a spectral parameter contained in all boundary conditions. We characterize the location of eigenvalues on the real axis, find their multiplicities, and study the oscillatory properties of eigenfunctions. Moreover, we obtain refined asymptotic formulas for the eigenvalues, as well as for the values at the endpoints of the interval of eigenfunctions and their derivatives. Then, using these results and the operator-theoretic formulation, we establish sufficient conditions for the system of eigenfunctions of this problem to form a basis in the space L p , 1 < p < ∞ , after removing four functions. [ABSTRACT FROM AUTHOR]

A complex unit gain graph, or T -gain graph, is a triple Φ = (G , T , φ) comprised of a simple graph G as the underlying graph of Φ, the set of unit complex numbers T = { z ∈ C : | z | = 1 } , and a gain function φ : E → → T with the property that φ (e i j) = φ (e j i) − 1. A cactus graph is a connected graph in which any two cycles have at most one vertex in common. In this paper, we firstly show that there does not exist a complex unit gain graph with nullity n (G) − 2 m (G) + 2 c (G) − 1 , where n (G) , m (G) and c (G) are the order, matching number, and cyclomatic number of G. Next, we provide a lower bound on the nullity for connected complex unit gain graphs and an upper bound on the nullity for complex unit gain bipartite graphs. Finally, we characterize all non-singular complex unit gain bipartite cactus graphs, which generalizes a result in Wong et al. (2022) [30]. [ABSTRACT FROM AUTHOR]

Let G be a simple connected graph and A (G) be the adjacency matrix of G. A diagonal matrix with diagonal entries ±1 is called a signature matrix. If A (G) is nonsingular and X = S A (G) − 1 S − 1 is entrywise nonnegative for some signature matrix S , then X can be viewed as the adjacency matrix of a unique weighted graph. It is called the inverse of G , denoted by G +. A graph G is said to have the reciprocal eigenvalue property (property(R)) if A (G) is nonsingular, and 1 λ is an eigenvalue of A (G) whenever λ is an eigenvalue of A (G). Further, if λ and 1 λ have the same multiplicity for each eigenvalue λ , then G is said to have the strong reciprocal eigenvalue property (property (SR)). It is known that for a tree T , the following conditions are equivalent: a) T + is isomorphic to T , b) T has property (R), c) T has property (SR) and d) T is a corona tree (it is a tree which is obtained from another tree by adding a new pendant at each vertex). Studies on the inverses, property (R) and property (SR) of bipartite graphs are available in the literature. However, their studies for the non-bipartite graphs are rarely done. In this article, we study the inverse and property (SR) for non-bipartite graphs. We first introduce an operation, which helps us to study the inverses of non-bipartite graphs. As a consequence, we supply a class of non-bipartite graphs for which the inverse graph G + exists and G + is isomorphic to G. It follows that each graph G in this class has property (SR). [ABSTRACT FROM AUTHOR]

We say that a simple graph G is Seidel integral if its Seidel spectrum consists entirely of integers. In this work we establish a characterization of Seidel integral graphs which belong to the class αKa ∪ βKb,b, where mG denotes the m-fold union of the graph G. [ABSTRACT FROM AUTHOR]

The traditional multi-period soliton method takes too long to solve and the solution results are not satisfactory. To address the above problems, this study proposes a multi-period soliton method for the discrete Riccati equation. In this method, the multi-period soliton solution index of the discrete Riccati equation is firstly selected, and the multi-period soliton solution of the discrete Riccati equation is determined according to the index. The correlation matrix is solved and a positive semi-definite matrix is established. Next, the lower bound and eigenvalues of the matrix are solved using the matrix inequality property inequality and the perturbation parameter. The boundary estimation and eigenvalue properties of the reference matrix and the solution matrix are obtained. Another positive semi-definite matrix is defined. Finally, the soliton solution is constructed to complete the multi-period soliton solution of the discrete Riccati equation. The simulation results showed that the traditional method took more than 5 minutes, while the multi-period soliton solution of the discrete Riccati equation designed in the research only took 2.5 minutes. The solution time of the proposed method was shorter than the traditional method. The optimal solution could be obtained. It showed that the multi-period soliton discrete Riccati equation solution could deepen the understanding for multi-period soliton and nonlinear fluctuation phenomenon, providing important mathematical models and tools for practical application and basic research. These instructions provide guidelines for preparing papers. The multi-periodic soliton method for the discrete Riccati equation proposed in the study can effectively improve the solution time and efficiency, and has certain applications in controller optimization and other aspects. [ABSTRACT FROM AUTHOR]

The inverse Sturm–Liouville problem with smooth potentials is considered. The high-order estimate of the entire function associated with two Sturm–Liouville problems is established. Applying this estimate expression to inverse Sturm–Liouville problems, we proved that the conclusion in [L. Amour, J. Faupin and T. Raoux, Inverse spectral results for Schrödinger operators on the unit interval with partial information given on the potentials, J. Math. Phys. 50 2009, 3, Article ID 033505] remains true for more general case. [ABSTRACT FROM AUTHOR]

In response to the challenges posed by complex boundary conditions and singularities in molecular systems and quantum chemistry, accurately determining energy levels (eigenvalues) and corresponding wavefunctions (eigenfunctions) is crucial for understanding molecular behavior and interactions. Mathematically, eigenvalues and normalized eigenfunctions play crucial role in proving the existence and uniqueness of solutions for nonlinear boundary value problems (BVPs). In this paper, we present an iterative procedure for computing the eigenvalues (μ ) and normalized eigenfunctions of novel fractional singular eigenvalue problems, D 2 α y (t) + k t α D α y (t) + μ y (t) = 0 , 0 < t < 1 , 0 < α ≤ 1 , with boundary condition, y ′ (0) = 0 , y (1) = 0 , where D α , D 2 α represents the Caputo fractional derivative, k ≥ 1 . We propose a novel method for computing Lagrange multipliers, which enhances the variational iteration method to yield convergent solutions. Numerical findings suggest that this strategy is simple yet powerful and effective. [ABSTRACT FROM AUTHOR]

In this note, we give simple proofs of two recent extensions of the Hua determinant inequality. We also answer a question of Poon affirmatively on a Young type inequality, which appears in his investigation of inequalities on eigenvalues arising from the Hua determinant inequality. [ABSTRACT FROM AUTHOR]

This paper focuses on computing the eigenvalues of the generalized collocation matrix of the rational Said–Ball basis, also called as the quasi-rational Said–Ball–Vandermonde (q-RSBV) matrix, with high relative accuracy. To achieve this, we propose explicit expressions for the minors of the q-RSBV matrix and develop a high-precision algorithm to compute these parameters. Additionally, we present perturbation theory and error analysis to further analyze the accuracy of our approach. Finally, we provide some numerical examples to demonstrate the high relative accuracy of our algorithms. [ABSTRACT FROM AUTHOR]

Consider (M, g) as an n-dimensional compact Riemannian man- ifold. Our main aim in this paper is to study the first eigenvalue of (p, q)- Laplacian system on C-totally real submanifold in Sasakian space of form M2m+1 (K). Also in the case of p, q > n we show that for λ1,p,q arbitrary large there exists a Riemannian metric of volume one conformal to the standard metric of Sn. [ABSTRACT FROM AUTHOR]

Background: Diffusion imaging holds great potential for the non‐invasive assessment of the glymphatic system in humans. One technique, diffusion tensor imaging along the perivascular space (DTI‐ALPS), has introduced the ALPS‐index, a novel metric for evaluating diffusivity within the perivascular space. However, it still needs to be established whether the observed reduction in the ALPS‐index reflects axonal changes, a common occurrence in neurodegenerative diseases. Purpose: To determine whether axonal alterations can influence change in the ALPS‐index. Study Type: Retrospective. Population: 100 participants (78 cognitively normal and 22 with mild cognitive impairments) aged 50–90 years old. Field Strength/Sequence: 3T; diffusion‐weighted single‐shot spin‐echo echo‐planar imaging sequence, T1‐weighted images (MP‐RAGE). Assessment: The ratio of two radial diffusivities of the diffusion tensor (i.e., λ2/λ3) across major white matter tracts with distinct venous/perivenous anatomy that fulfill (ALPS‐tracts) and do not fulfill (control tracts) ALPS‐index anatomical assumptions were analyzed. Statistical Tests: To investigate the correlation between λ2/λ3 and age/cognitive function (RAVLT) while accounting for the effect of age, linear regression was implemented to remove the age effect from each variable. Pearson correlation analysis was conducted on the residuals obtained from the linear regression. Statistical significance was set at p < 0.05. Results: λ2 was ~50% higher than λ3 and demonstrated a consistent pattern across both ALPS and control tracts. Additionally, in both ALPS and control tracts a reduction in the λ2/λ3 ratio was observed with advancing age (r = −0.39, r = −0.29, association and forceps tract, respectively) and decreased memory function (r = 0.24, r = 0.27, association and forceps tract, respectively). Data Conclusions: The results unveil a widespread radial asymmetry of white matter tracts that changes with aging and neurodegeration. These findings highlight that the ALPS‐index may not solely reflect changes in the diffusivity of the perivascular space but may also incorporate axonal contributions. Level of Evidence: 3 Technical Efficacy: Stage 2 [ABSTRACT FROM AUTHOR]

The free vibrations of a rod are governed by a differential equation of the form (a(x)y')' + λa(x)y(x) = 0, where a(x) is the cross sectional area and λ is an eigenvalue parameter. Using the finite element method (FEM) we transform this equation to a generalized matrix eigenvalue problem of the form (K, M)u = 0 and, for given a(x), we correct the eigenvalues Λ of the matrix pair (K - ΛM)u to approximate the eigenvalues of the rod equation. The results show that with step size h the correction technique reduces the error from O(h²i4) to O(h²i²) for the i-th eigenvalue. We then solve the inverse spectral problem by imposing numerical algorithms that approximate the unknown coefficient a(x) from the given spectral data. The cross section is obtained by solving a nonlinear system using Newton's method along with a regularization technique. Finally, we give numerical examples to illustrate the efficiency of the proposed algorithms. [ABSTRACT FROM AUTHOR]

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]

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]

In this manuscript, Conformal Mapping Technique (CMT) is applied to predict the resonant frequency of an Equilateral Triangular Microstrip Antenna (ETMA). The ETMA is mapped into a simple circular geometry with the aid of CMT. The wave equation is solved in the transformed domain using Galerkin technique to obtain the approximate solution of eigenfunction and eigenvalue. Twenty-three measured data are collected from the open literature to validate our theory. Our theoretical results are also compared with several CAD models to show the effectiveness of the present theory. It is found that our proposed theory can predict the resonant frequency of the ETMA with an error of less than 1.5%, whereas those CAD models can produce a large error (sometimes, 10-19% error). Internal field patterns, input impedance, far-field radiation patterns, quality factor, bandwidth gain, and directivity of the ETMA for the dominant mode are also investigated using CMT. [ABSTRACT FROM AUTHOR]

In this paper, we study dual quaternion, dual complex unit gain graphs, and their spectral properties in a unified frame of dual unit gain graphs. Unit dual quaternions represent rigid movements in the 3D space, and have wide applications in robotics and computer graphics. Dual complex numbers have found application in brain science recently. We establish the interlacing theorem for dual unit gain graphs, and show that the spectral radius of a dual unit gain graph is always not greater than the spectral radius of the underlying graph, and these two radii are equal if, and only if, the dual gain graph is balanced. By using dual cosine functions, we establish the closed form of the eigenvalues of adjacency and Laplacian matrices of dual complex and quaternion unit gain cycles. We then show the coefficient theorem holds for dual unit gain graphs. Similar results hold for the spectral radius of the Laplacian matrix of the dual unit gain graph too. [ABSTRACT FROM AUTHOR]

We consider two spectral problems on an equilateral rooted tree with the standard (continuity and Kirchhoff's type) conditions at the interior vertices (except of the root if it is interior) and Dirichlet conditions at the pendant vertices (except of the root if it is pendant). For the first (Neumann) problem we impose the standard conditions (if the root is an interior vertex) or Neumann condition (if the root is a pendant vertex) at the root, while for the second (Dirichlet) problem we impose the Dirichlet condition at the root. We show that for caterpillar trees the spectra of the Neumann problem and of the Dirichlet problem uniquely determine the shape of the tree. Also, we present an example of co-spectral snowflake graphs. [ABSTRACT FROM AUTHOR]

In this paper, we give some new bounds for the eigenvalues of non-monic operator polynomials by applying several numerical radius inequalities to the Frobenius companion matrices of these polynomials. Our bounds refine the existing bounds for monic matrix polynomials. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new class of sets namely analytically imaged sets in the space of m x n matrices. A sufficient condition is obtained for an analytically imaged subset of the set of all n x n matrices to have a dense subset in terms of algebraic multiplicities of the eigenvalues. Also, the counterparts of this result have been studied for singular values of rectangular matrices and it has been shown that all the results hold for convex subsets of matrices. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Chen and Moore determinants of quaternion Hermitian matrices to dual quaternion Hermitian matrices. We show the Chen determinant of dual quaternion Hermitian matrices is invariant under addition, switching, multiplication, and unitary operations at the both hand sides. We then show the Chen and Moore determinants of dual quaternion Hermitian matrices are equal to each other, and they are also equal to the products of eigenvalues. The characteristic polynomial of a dual quaternion Hermitian matrix is also studied. [ABSTRACT FROM AUTHOR]

In this paper, we obtain a new estimate for the (product) γ -diagonally dominant degree of the Schur complement of matrices. As applications we discuss the localization of eigenvalues of the Schur complement and present several upper and lower bounds for the determinant of strictly γ -diagonally dominant matrices, which generalizes the corresponding results of Liu and Zhang (SIAM J. Matrix Anal. Appl. 27 (2005) 665-674). [ABSTRACT FROM AUTHOR]

In this paper we consider an initial-boundary value problem (direct problem) for a fourth order equation with the fractional Caputo derivative. Two inverse problems of determining the right-hand side of the equation by a given solution to the direct problem at some point are studied. The unknown of the first problem is a one-dimensional function depending on a spatial variable, while in the second problem a function depending on a time variable is found. Using eigenvalues and eigenfunctions, a solution to the direct problem is found in the form of Fourier series. Sufficient conditions are established for the given functions, under which the solution to this problem is classical. Using the results obtained for the direct problem and applying the method of integral equations, we study the inverse problems. Thus, the uniqueness and existence theorems of the direct and inverse problems are proved. [ABSTRACT FROM AUTHOR]

In this paper, we define extensions of the classical eigenvalues of the matrix A ∈ Mm(C). These extensions are eigenvalues matrices for the block matrix A ∈ Mm(Mn), where Mm(Mn) is the set of all m×m block complex matrices with each block in Mn(C), they are.....Among other equalities and inequalities, we prove equalities which relate our new definitions with tr1(A) and tr2(A) as follows,... These relations are extensions of the classical relation tr(A) = Σmh=1λh(A), where A ∈Mm(C). Several new relations and properties of our new definitions are also given. [ABSTRACT FROM AUTHOR]