Unique Eigenvalue Research Articles

Eigenvalues of the Ricci Operator on Four-Dimensional Locally Homogeneous (Pseudo)Riemannian Manifolds with a Four-Dimensional Isotropy Subgroup

The problem of restoring a (pseudo)Riemannian manifold from a given Ricci operator was studied in the papers of many mathematicians. This problem was solved by O. Kowalski and S. Nikcevic for the case of three-dimensional locally homogeneous Riemannian manifolds. The work of G. Calvaruso and O. Kowalski contains the answer to the question above for the case of three –dimensional locally homogeneous Lorentzian manifolds.
 For the four-dimensional case, similar studies were carried out only in the case of Lie groups with a left-invariant Riemannian metric. The works of A.G. Kremlyov and Yu.G. Nikonorov presented the possible signatures of the eigenvalues of the Ricci operator. However, the question of recovering a four-dimensional Lie group with a left-invariant Riemannian metric from a given Ricci operator remains open.
 This paper is devoted to the study of the eigenvalues of the Ricci operator on four-dimensional locally homogeneous (pseudo)Riemannian manifolds with a four-dimensional isotropy subgroup. An algorithm for calculating the eigenvalues of the Ricci operator is presented. A theorem on the restoration of such manifolds from a given Ricci operator is proved. It is established that such possibility can happen only in the case when the prescribed operator is diagonalizable and has a unique eigenvalue of multiplicity four.

On similarity of an arbitrary matrix to a block diagonal matrix

Let an n x n -matrix A have m < n (m ? 2) different eigenvalues ?j of the algebraic multiplicity ?j (j = 1,..., m). It is proved that there are ?j x ?j-matrices Aj, each of which has a unique eigenvalue ?j, such that A is similar to the block-diagonal matrix ?D = diag (A1,A2,..., Am). I.e. there is an invertible matrix T, such that T-1AT = ?D. Besides, a sharp bound for the number kT := ||T||||T-1|| is derived. As applications of these results we obtain norm estimates for matrix functions non-regular on the convex hull of the spectra. These estimates generalize and refine the previously published results. In addition, a new bound for the spectral variation of matrices is derived. In the appropriate situations it refines the well known bounds.

Eulerian analysis of the electron flow emitted from an infinitely sharp electrified cone

The axisymmetric steady-state electron flow emitted from an electrified semi-infinite perfectly-conducting filament in vacuum is theoretically analyzed using an Eulerian model. The high symmetry allows for a solution in terms of self-similar functions, with radial dependence described by power-laws, leading to an eigenvalue problem for the angular dependence, which is numerically solved.Our results show a unique eigenvalue compatible with the boundary conditions, leading to a r−1.6308 dependence for the electron number density, r0.3692 for the electric potential, r0.1846 for the velocity field. Hence, for an infinitely thin filament, space charge limited emission is not compatible with our boundary conditions.

Open-Loop Orientation Control Using Dynamic Magnetic Fields

Remote magnetic control of soft magnetic objects has been limited to 2D orientation and 3D position. In this letter, we extend the five degree-of-freedom (5-DoF) control approach to full 6-DoF. We prove that 6-DoF control is possible for objects that have an apparent magnetic susceptibility tensor with unique eigenvalues. We further show that the object’s orientation can be specified with a dynamic magnetic field and can be controlled without orientation feedback.The theory is demonstrated by rotating a soft magnetic object about each of its principle axes using a metronome like dynamic field.

Neumann eigenvalue problems on the exterior domains

For p∈(1,∞), we consider the following weighted Neumann eigenvalue problem on B1c, the exterior of the closed unit ball in RN: (0.1)−Δpϕ=λg|ϕ|p−2ϕinB1c,∂ϕ∂ν=0on∂B1,where Δp is the p-Laplace operator and g∈Lloc1(B1c) is an indefinite weight function. Depending on the values of p and the dimension N, we take g in certain Lorentz spaces or weighted Lebesgue spaces and show that (0.1) admits an unbounded sequence of positive eigenvalues that includes a unique principal eigenvalue. For this purpose, we establish the compact embeddings of W1,p(B1c) into Lp(B1c,|g|) for g in certain weighted Lebesgue spaces. For N>p, we also provide an alternate proof for the embedding of W1,p(B1c) into the Lorentz space Lp∗,p(B1c). Further, we show that the set of all eigenvalues of (0.1) is closed.

Optimal quantum-walk search on Kronecker graphs with dominant or fixed regular initiators

In network science, graphs obtained by taking the Kronecker or tensor power of the adjacency matrix of an initiator graph are used to construct complex networks. In this paper, we analytically prove sufficient conditions under which such Kronecker graphs can be searched by a continuous-time quantum walk in optimal $\Theta(\sqrt{N})$ time. First, if the initiator is regular and its adjacency matrix has a dominant principal eigenvalue, meaning its unique largest eigenvalue asymptotically dominates the other eigenvalues in magnitude, then the Kronecker graphs generated by this initiator can be quantum searched with probability 1 in $\pi\sqrt{N}/2$ time, asymptotically, and we give the critical jumping rate of the walk that enables this. Second, for any fixed initiator that is regular, non-bipartite, and connected, the Kronecker graphs generated by it are quantum searched in $\Theta(\sqrt{N})$ time. This greatly extends the number of Kronecker graphs on which quantum walks are known to optimally search. If the fixed, regular, connected initiator is bipartite, however, then search on its Kronecker powers is not optimal, but is still better than classical computer's $O(N)$ runtime if the initiator has more than two vertices.

Using principal eigenvectors of adjacency matrices with added diagonal weights to compose centrality measures and identify bowtie structures for a digraph

ABSTRACTPrincipal eigenvectors of adjacency matrices are often adopted as measures of centrality for a graph or digraph. However, previous principal-eigenvector-like measures for a digraph usually consider only the strongly connected component whose adjacency submatrix has the largest eigenvalue. In this paper, for each and every strongly connected component in a digraph, we add weights to diagonal elements of its member nodes in the adjacency matrix such that the modified matrix will have the new unique largest eigenvalue and corresponding principal eigenvectors. Consequently, we use the new principal eigenvectors of the modified matrices, based on different strongly connected components, not only to compose centrality measures but also to identify bowtie structures for a digraph.

Perron–Frobenius theorem for hypermatrices in the max algebra

Recently, there have been many intriguing new developments in the study of hypermatrices and their associated eigenvalue problems. In particular, results coming from the matrix setting when studying the max algebra have shown especially attractive combinatorial features. We now extend this max algebra setting into the realm of hypermatrices. Considering that the max algebra has shown particular significance in optimization problems for the matrix setting, we look to examine and extend these results in the higher order conditions. Furthermore, we establish some algebraic properties for hypermatrices and then proceed to extend the Perron–Frobenius Theorem for this setting and prove the existence of a unique eigenvalue. We continue by stating a result from Nussbaum, that the Min–Max theorem holds, and provide a proof for completeness. For strongly increasing hypermatrices, an iterative algorithm which converges to our unique eigenvalue is given. Finally, we conclude with an analysis of our results in the hypergraph setting.

Conformally Flat Algebraic Ricci Solitons on Lie Groups

The paper is devoted to the study of conformally flat Lie groups with left-invariant (pseudo) Riemannianmetric of an algebraic Ricci soliton. Previously conformally flat algebraic Ricci solitons on Lie groups have been studied in the case of small dimension and under an additional diagonalizability condition on the Ricci operator. The present paper continues these studies without the additional requirement that the Ricci operator be diagonalizable. It is proved that any nontrivial conformally flat algebraic Ricci soliton on a Lie group must be steady and have Ricci operator of Segre type {(1... 1 2)} with a unique eigenvalue (equal to 0).

Optimal Shrinkage of Eigenvalues in the Spiked Covariance Model.

We show that in a common high-dimensional covariance model, the choice of loss function has a profound effect on optimal estimation. In an asymptotic framework based on the Spiked Covariance model and use of orthogonally invariant estimators, we show that optimal estimation of the population covariance matrix boils down to design of an optimal shrinker η that acts elementwise on the sample eigenvalues. Indeed, to each loss function there corresponds a unique admissible eigenvalue shrinker η* dominating all other shrinkers. The shape of the optimal shrinker is determined by the choice of loss function and, crucially, by inconsistency of both eigenvalues and eigenvectors of the sample covariance matrix. Details of these phenomena and closed form formulas for the optimal eigenvalue shrinkers are worked out for a menagerie of 26 loss functions for covariance estimation found in the literature, including the Stein, Entropy, Divergence, Fréchet, Bhattacharya/Matusita, Frobenius Norm, Operator Norm, Nuclear Norm and Condition Number losses.

SIMILARITY OF OPERATORS ON TENSOR PRODUCTS OF SPACES AND MATRIX DIFFERENTIAL OPERATORS

Let ${\mathcal{H}}=\mathbb{C}^{n}\otimes {\mathcal{E}}$ be the tensor product of a Euclidean space $\mathbb{C}^{n}$ and a separable Hilbert space ${\mathcal{E}}$. Our main object is the operator $G=I_{n}\otimes S+A\otimes I_{{\mathcal{E}}}$, where $S$ is a normal operator in ${\mathcal{E}}$, $A$ is an $n\times n$ matrix, and $I_{n},I_{{\mathcal{E}}}$ are the unit operators in $\mathbb{C}^{n}$ and ${\mathcal{E}}$, respectively. Numerous differential operators with constant matrix coefficients are examples of operator $G$. In the present paper we show that $G$ is similar to an operator $M=I_{n}\otimes S+\hat{D}\times I_{{\mathcal{E}}}$ where $\hat{D}$ is a block matrix, each block of which has a unique eigenvalue. We also obtain a bound for the condition number. That bound enables us to establish norm estimates for functions of $G$, nonregular on the closed convex hull $\operatorname{co}(G)$ of the spectrum of $G$. The functions $G^{-\unicode[STIX]{x1D6FC}}\;(\unicode[STIX]{x1D6FC}>0)$ and $(\ln G)^{-1}$ are examples of such functions. In addition, in the appropriate situations we improve the previously published estimates for the resolvent and functions of $G$ regular on $\operatorname{co}(G)$. Since differential operators with variable coefficients often can be considered as perturbations of operators with constant coefficients, the results mentioned above give us estimates for functions and bounds for the spectra of differential operators with variable coefficients.

Asymptotics of the bound state induced by δ-interaction supported on a weakly deformed plane

In this paper, we consider the three-dimensional Schrödinger operator with a δ-interaction of strength α > 0 supported on an unbounded surface parametrized by the mapping R2∋x↦(x,βf(x)), where β∈0,∞ and f:R2→R, f ≢ 0, is a C2-smooth, compactly supported function. The surface supporting the interaction can be viewed as a local deformation of the plane. It is known that the essential spectrum of this Schrödinger operator coincides with −14α2,+∞. We prove that for all sufficiently small β > 0, its discrete spectrum is non-empty and consists of a unique simple eigenvalue. Moreover, we obtain an asymptotic expansion of this eigenvalue in the limit β → 0+. In particular, this eigenvalue tends to −14α2 exponentially fast as β → 0+.

Spectral properties of a sequence of matrices connected to each other via Schur complement and arising in a compartmental model

Abstract We consider a sequence of real matrices An which is characterized by the rule that An−1 is the Schur complement in An of the (1,1) entry of An, namely −en, where en is a positive real number. This sequence is closely related to linear compartmental ordinary differential equations. We study the spectrum of An. In particular,we show that An has a unique positive eigenvalue λn and {λn} is a decreasing convergent sequence. We also study the stability of An for small n using the Routh-Hurwitz criterion.

On Exponential Decay and the Markus–Yamabe Conjecture in Infinite Dimensions with Applications to the Cima System

In this work we extend to infinite dimensions a previous result of Lyascenko given in finite dimensions for a class of Lipschitz functions. In particular, this extension substitutes the class of Lipschitz functions by a class of Holder functions. Then we consider applications of these results to analyze its relation with the Markus–Yamabe Conjecture in infinite dimensions. We discuss in detail a celebrated example of Cima et al. ( Adv Math 131(2): 453–457, 1997) of a nonlinear system in dimension three whose Jacobian has a unique eigenvalue \(-1\) of multiplicity 3, and yet an explicit unbounded solution as \(t \rightarrow \infty \) exists. We also present explicit solutions of the same equation that tends to 0 as \(t \rightarrow \infty \). Then we look at this conjecture by considering not the properties of the whole system, but instead the properties of some solutions. Finally we present an application in an infinite dimensional Hilbert space, where we use different techniques to study the local and global asymptotic stabilities.

Experimental Study on Wetland Hydraulic Characteristics of Vegetated Drainage Ditches

Small vegetated drainage ditches play an important role in water and nutrient removal, but may cause water blocking problems. The aim of this study was to investigate vegetated drainage ditches’ hydraulic and wetland hydraulic characters. Field experiment were carried out, small drainage ditches with Juncus, Zizania latifolia, and Acorus calamus were selected under different water flows, and a tracer experiment was also conducted. Research findings suggested that with increased water flow, vegetation roughness coefficient declined, vegetation resistance rose, and drag coefficient declined. Small drainage ditches with emergent vegetation showed a high roughness coefficient value, which was most significant in Juncus, followed by Zizania latifolia, and Acorus calamus. Plants each took on a unique eigenvalue k that was a relative coefficient between the drag coefficient and stem Reynolds number. As small ditches for drainage showed longer residence time and smaller surface hydraulic loading, they featured excellent wetland hydraulic characteristics that could be weakened rapidly as rainfall or flood intensified. This study indicates that the small vegetated drainage ditch shows favorable wetland hydraulic characteristics with good discharge capacity and can be extensively used in irrigated districts.

Complementary eigenvalues of graphs

In this paper, we study the Eigenvalue Complementarity Problem (EiCP) when its matrix A belongs to the class S(G)={A=[aij]:aij=aji≠0 iff ij∈E}, where G=(V,E) is a connected graph. It is shown that if all nondiagonal elements of A∈S(G) are nonpositive, then A has a unique complementary eigenvalue, which is the smallest eigenvalue of A. In particular, zero is the unique complementary eigenvalue of the Laplacian and the normalized Laplacian matrices of a connected graph. The number c(G) of complementary eigenvalues of the adjacency matrix of a connected graph G is shown to be bounded above by the number b(G) of induced nonisomorphic connected subgraphs of G. Furthermore, c(G)=b(G) if the Perron roots of the adjacency matrices of these subgraphs are all distinct. Finally, the maximum number of complementary eigenvalues for the adjacency matrices of graphs is shown to grow faster than any polynomial on the number of vertices.

Potential energy surfaces in atomic structure: The role of Coulomb correlation in the ground state of helium

For the $S$ states of two-electron atoms, we introduce an exact and unique factorization of the internal eigenfunction in terms of a marginal amplitude, which depends functionally on the electron-nucleus distances $r_1$ and $r_2$, and a conditional amplitude, which depends functionally on the interelectronic distance $r_{12}$ and parametrically on $r_1$ and $r_2$. Applying the variational principle, we derive pseudoeigenvalue equations for these two amplitudes, which cast the internal Schr\"odinger equation in a form akin to the Born-Oppenheimer separation of nuclear and electronic degrees of freedom in molecules. The marginal equation involves an effective radial Hamiltonian, which contains a nonadiabatic potential energy surface that takes into account all interparticle correlations in an averaged way, and whose unique eigenvalue is the internal energy. At each point $(r_1,r_2)$, such surface is, in turn, the unique eigenvalue in the conditional equation. Employing the ground state of He as prototype, we show that the nonadiabatic potential energy surface affords a molecularlike interpretation of the structure of the atom, and aids in the analysis of energetic and spatial aspects of the Coulomb correlation, in particular correlation-induced symmetry breaking and quantum phase transition.

The spectra of rectangular lattices of quantum waveguides

We obtain asymptotic formulae for the spectral segments of a thin () rectangular lattice of quantum waveguides which is described by a Dirichlet problem for the Laplacian. We establish that the structure of the spectrum of the lattice is incorrectly described by the commonly accepted quantum graph model with the traditional Kirchhoff conditions at the vertices. It turns out that the lengths of the spectral segments are infinitesimals of order , , and as , and gaps of width and arise between them in the low- frequency and middle- frequency spectral ranges respectively. The first spectral segment is generated by the (unique) eigenvalue in the discrete spectrum of an infinite cross-shaped waveguide . The absence of bounded solutions of the problem in at the threshold frequency means that the correct model of the lattice is a graph with Dirichlet conditions at the vertices which splits into two infinite subsets of identical edges- intervals. By using perturbations of finitely many joints, we construct any given number of discrete spectrum points of the lattice below the essential spectrum as well as inside the gaps.

Long-term treatment effects in chronic myeloid leukemia.

We propose and analyze a simplified version of a partial differential equation (PDE) model for chronic myeloid leukemia (CML) derived from an agent-based model proposed by Roeder et al. This model describes the proliferation and differentiation of leukemic stem cells in the bone marrow and the effect of the drug Imatinib on these cells. We first simplify the PDE model by noting that most of the dynamics occurs in a subspace of the original 2D state space. Then we determine the dominant eigenvalue of the corresponding linearized system that controls the long-term behavior of solutions. We mathematically show a non-monotonous dependence of the dominant eigenvalue with respect to treatment dose, with the existence of a unique minimal negative eigenvalue. In terms of CML treatment, this shows that there is a unique dose that maximizes the decay rate of the CML tumor load over long time scales. Moreover this unique dose is lower than the dose that maximizes the initial tumor load decay. Numerical simulations of the full model confirm that this phenomenon is not an artifact of the simplification. Therefore, while optimal asymptotic dosage might not be the best one at short time scales, our results raise interesting perspectives in terms of strategies for achieving and improving long-term deep response.

Using eigenvectors of perturbed and collapsed adjacency matrices to explore bowtie structures in directed networks

The bowtie structure can illustrate not only the accessibility of the World Wide Web, but also the reachability of other directed networks. In this paper, we use the principal eigenvectors of the adjacency matrix with the unique largest eigenvalue to identify the strongly connected component of a directed network and fit the network into the bowtie structure. To ensure that the largest eigenvalue is unique, we add a little perturbation to the matrix before the eigen analysis. After the revelation of the bowtie structure centered on the strongly connected component with the largest unique eigenvalue, a directed network may have other bowtie structures centered on strongly connected components with smaller eigenvalues. To reveal other bowtie structures, we collapse the perturbed matrix by aggregating nodes of the strongly connected component with the largest eigenvalue into a supernode. Hence, the principal eigenvectors of the perturbed and collapsed matrix can be used to reveal the bowtie structure centered on the strongly connected component with the second largest eigenvalue. Furthermore, repeating the process of collapsing a strongly connected component and finding principal eigenvectors of the perturbed and collapsed matrix, we can reveal all the bowtie structures of a directed network.