Nonnegative Eigenvalues Research Articles

Existence and regularity results for a degenerate elliptic nonlinear equation on weighted Sobolev spaces

This paper deals with the existence of many solutions a degenerate weighted elliptic equation which its the number of solutions depend on degeneracy term a i.e., the number of subdomains of Ω\\a −1(0) whose boundary is made by submanifolds with 1-codimension. Moreover, by the Ljusternik-Schnirelman principle, we would study the corresponding eigenvalue problem and obtained the nonnegative eigenvalues sequence of main problem. Finally, we will discuss of the regularity of the solutions on C 1,α (Ω).

Partial Least Squares - Diffusion Tensor Imaging (PLS-DTI): A novel approach for biomarker imaging in breast cancer

Currently, magnetic resonance imaging is the most sensitive imaging technique for detecting cancer processes in early stages. Regarding breast cancer, due to the characteristics of the tissue as it is formed by ducts (tubular structure), anisotropic diffusion should be considered instead of general isotropic Diffusion Weighted Imaging (DWI). Anisotropic diffusion is studied by applying a technique called Diffusion Tensor Imaging (DTI), where the diffusion gradient is applied with several different directions, calculated by Ordinary Least Squares (OLS) in clinical practice. In this paper, we propose a new DTI calculation method based on Partial Least Squares (PLS), which has some advantages over the traditional OLS calculation: i) the PLS model provides valid biomarkers (non-negative eigenvalues) in a larger percentage of pixels, improving the traditional OLS calculation and reducing the effect of noisier images; ii) OLS tensors are calculated pixel-by-pixel, whereas the PLS method calculates only one model taking advantage of the correlation structure between pixels with similar characteristics, obtaining more reliable estimations; iii) PLS performance is quite reliable when lowering the number of directions of the magnetic field, while this is not the case of OLS. PLS keeps providing a good solution even with low functional resolution equipment, reducing costs and acquisition times, which is an important advantage for its widespread use in value-based medicine-oriented clinical practice.

The infimum eigenvalue for degenerate p(x)-biharmonic operator with the Hardy potentiel

The aim of this article is to study the existence of at least one unbounded nondecreasing sequence of nonnegative eigenvalues (λk)k≥1 for a class of elliptic Navier boundary value problems involving the degenerate p(·)-biharmonic operator with q(x)-Hardy inequality by using the variational technique based on the Ljusternik-Schnirelmann theory on C1-manifolds and the theory of the variable exponent Lebesgue spaces. Also, we obtain the positivity of the infimum eigenvalue for the problem.

Signed Complete Graphs with Exactly m Non-negative Eigenvalues

Signed Complete Graphs with Exactly m Non-negative Eigenvalues

On the stability of tracer simulations with opposite-signed diffusivities

Many recent studies have diagnosed opposite-signed diffusion eigenvalues to be a prevalent feature of the transfer tensor for diffusive tracer transport by oceanic mesoscale eddies. This diagnosed tensor, which we refer to as the diffusion tensor, therefore accounts for tracer filamentation effects. The preferential orientation of this filamentation is quantified by the principal axis of the diffusion tensor, namely the diffusion axis. Parameterisations of eddy diffusion commonly invoke a diffusion tensor, typically one with non-negative eigenvalues to avoid numerical issues. Motivated by the need to parameterise tracer filamentation, in this study we examine diffusion of a Gaussian tracer patch with imposed opposite-signed diffusion eigenvalues, and in particular we focus on the time scale for the onset of instability. For a fixed diffusion axis, numerical instability is an inevitable consequence of persistent up-gradient fluxes associated with the negative eigenvalue. For typical oceanic scales and diffusion magnitudes, this time scale is of the order of $100$ days, but is shorter for larger negative eigenvalues or for finer grid resolutions. We show that imposing a time-dependent diffusion axis can lead to simulations with no onset of instability after 100 000 days of tracer evolution. Although motivated by oceanographic fluid dynamics, our results have much broader applications since diffusive processes are present in a wide range of fluid flows.

Eigenvalues for the p-Biharmonic Dirichlet-Steklov problem with weights

This note is concerned with a weighted Dirichlet-Steklov problem driven by the p-biharmonic operator. Our approach is based on variational method and Ljusternick-Schnirelmann principle, we establish that the above problem admits a non-decreasing sequence of non-negative eigenvalues.

Exact algorithms for semidefinite programs with degenerate feasible set

Given symmetric matrices A0,A1,…,An of size m with rational entries, the set of real vectors x=(x1,…,xn) such that the matrix A0+x1A1+⋯+xnAn has non-negative eigenvalues is called a spectrahedron. Minimization of linear functions over spectrahedra is called semidefinite programming. Such problems appear frequently in control theory and real algebra, especially in the context of nonnegativity certificates for multivariate polynomials based on sums of squares.Numerical software for semidefinite programming are mostly based on interior point methods, assuming non-degeneracy properties such as the existence of an interior point in the spectrahedron. In this paper, we design an exact algorithm based on symbolic homotopy for solving semidefinite programs without assumptions on the feasible set, and we analyze its complexity. Because of the exactness of the output, it cannot compete with numerical routines in practice. However, we prove that solving such problems can be done in polynomial time if either n or m is fixed.

Symmetric Positive Semi-Definite FDTD Subgridding Algorithms in Both Space and Time for Accurate Analysis of Inhomogeneous Problems

In this article, we develop a systematic approach to derive symmetric positive semi-definite (SPD) finite-difference time-domain (FDTD) subgridding operators in both space and time for analyzing general inhomogeneous problems in an accurate fashion. The operators are SPD by construction and independent of the grid ratio. The resultant explicit time marching is guaranteed to be stable because such subgridding operators have only nonnegative real eigenvalues. Furthermore, the use of a time step local to the base grid and the subgrid is permitted without sacrificing stability and accuracy. Moreover, the algorithm takes the subgrid information into account to accurately analyze inhomogeneous problems. In addition, we provide an interpretation of the proposed subgridding operators and show how to implement them in the original difference equation-based FDTD framework. Extensive numerical experiments involving both 2- and 3-D subgrids with various grid ratios and inhomogeneities have demonstrated the stability, accuracy, and efficiency of the proposed new SPD subgridding algorithms.

Symmetric Positive Semidefinite FDTD Subgridding Algorithms for Arbitrary Grid Ratios Without Compromising Accuracy

Instability has been a major problem in finite-difference time-domain (FDTD) subgridding methods. Reciprocity has been proposed to overcome the problem but with limited success in producing a symmetric positive semidefinite (SPD) system without compromising accuracy. In this paper, we algebraically derive both 2- and 3-D FDTD subgridding operators, which are SPD by construction, and independent of the grid ratio. Such operators have only nonnegative real eigenvalues, and hence the stability of the resulting explicit time marching is guaranteed. The 3-D operator, the algorithm of which is also applicable to 2-D analysis, further permits the use of a local time step, thus achieving a natural subgridding in both space and time. In addition, the interpretation of the proposed operators in the original FDTD formulation is also provided. Interestingly, not only interface unknowns but also subgrid unknowns are generated in a different way, as compared to the original FDTD, to simultaneously obtain an SPD system and to ensure accuracy. Extensive numerical simulations have demonstrated the accuracy, stability, and efficiency of the proposed new subgridding algorithms.

Small graphs with exactly two non-negative eigenvalues

Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. In this paper we find all simple graphs $G$ such that $G$ has at most twelve vertices and $G$ has exactly two non-negative eigenvalues. In other words we find all graphs $G$ on $n$ vertices such that $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G) 0$, $lambda_2(G)>0$ and $lambda_3(G)<0$. We find that among these $1575$ graphs there are just two integral graphs.

An Optimal Nonnegative Eigenvalue Decomposition for the Freeman and Durden Three-Component Scattering Model

Model-based decomposition allows the physical interpretation of polarimetric radar scattering in terms of various scattering mechanisms. A three-component decomposition proposed by Freeman and Durden has been popular, though significant shortcomings have been identified. In particular, it can result in negative eigenvalues for the component terms and the remainder matrix, hence violating fundamental requirements for physically meaningful decompositions. In addition, since the algorithm solves for the canopy term first, the contribution of the canopy is often over-estimated. In this paper, we show how to determine the parameters for the Freeman–Durden model in a way that minimizes the total power in the remainder matrix without favoring any individual component in the model, while simultaneously satisfying the constraints of nonnegative eigenvalues. We illustrate our analytical solution by comparison with the Freeman–Durden algorithm, as well as the nonnegative eigenvalue decomposition (NNED) proposed by van Zyl et al. The results show that this optimum algorithm generally assigns less power to the volume scattering than either the original Freeman–Durden or the NNED algorithms.

Characterization of graphs with exactly two non-negative eigenvalues

In this paper we characterize all graphs with exactly two non-negative eigenvalues. As a consequence we obtain all graphs G such that λ 3 ( G ) < 0 , where λ 3 ( G ) is the third largest eigenvalue of G .

The critical exponent for generalized doubly nonnegative matrices

It is known that the critical exponent (CE) for conventional, continuous powers of n-by-n doubly nonnegative (DN) matrices is . Here, we consider the larger class of diagonalizable, entrywise nonnegative n-by-n matrices with nonnegative eigenvalues (generalized doubly nonnegative (GDN)). We show that, again, a CE exists and is able to bind with a low-coefficient quadratic. However, the CE is larger than in the DN case; in particular, 2 for . There seems to be a connection with the index of primitivity, and a number of other observations are made and questions raised. It is shown that there is no CE for continuous Hadamard powers of GDN matrices, despite it also being for DN matrices.

A FIXED POINT THEOREM FOR VOLUME PRESERVING LINEAR TRANSFORMATIONS

In this note we derive consequences of the fact that if g ∈ SL(n), where n ≥ 2, and σi(g) are the coefficients of its characteristic polynomial, then g has 1 as an eigenvalue if and only if ∑ n−1 i=1 (−1) σi(g) = 0 or 2 according to the parity of n. These are: Corollary 2. Let D be a real or complex n × n matrix of tr(D) = 0. If D is singular, then one of the following equations holds (according to the parity of n). n−1 ∑ i=1 (−1)σi(ExpD) = 0 n−1 ∑ i=1 (−1)σi(ExpD) = 2. Conversely, if D has no eigenvalues in 2πiZ and the appropriate one of these equations holds, D must be singular. Corollary 3. Let A ∈ Mn(R) = M which acts on M by adA(X) = [A,X]. If A has only non-negative eigenvalues and one of the following equations holds (according to the parity of n) n 2 −1 ∑ i=1 (−1)σi(AdExp(A)) = 0 n 2 −1 ∑ i=1 (−1)σi(AdExp(A)) = 2, then [A,X] = 0 for some X ∈ M, not a linear combination of A and I . Corollary 4. Let A ∈ M and have all non-negative eigenvalues. Then the action of Exp(A) on M by conjugation has a fixed point which is not a linear combination of A and I . AMS Subject Classification: 15A04, 15A16, 15A24, 15A27, 17B40, 20G20, 22E15

Quantum tomography protocols with positivity are compressed sensing protocols

AbstractCharacterising complex quantum systems is a vital task in quantum information science. Quantum tomography, the standard tool used for this purpose, uses a well-designed measurement record to reconstruct quantum states and processes. It is, however, notoriously inefficient. Recently, the classical signal reconstruction technique known as ‘compressed sensing’ has been ported to quantum information science to overcome this challenge: accurate tomography can be achieved with substantially fewer measurement settings, thereby greatly enhancing the efficiency of quantum tomography. Here we show that compressed sensing tomography of quantum systems is essentially guaranteed by a special property of quantum mechanics itself—that the mathematical objects that describe the system in quantum mechanics are matrices with non-negative eigenvalues. This result has an impact on the way quantum tomography is understood and implemented. In particular, it implies that the information obtained about a quantum system through compressed sensing methods exhibits a new sense of ‘informational completeness.’ This has important consequences on the efficiency of the data taking for quantum tomography, and enables us to construct informationally complete measurements that are robust to noise and modelling errors. Moreover, our result shows that one can expand the numerical tool-box used in quantum tomography and employ highly efficient algorithms developed to handle large dimensional matrices on a large dimensional Hilbert space. Although we mainly present our results in the context of quantum tomography, they apply to the general case of positive semidefinite matrix recovery.

Adaptive polarimetric decomposition using incoherent ground scattering models without reflection symmetry assumption

Most of the current incoherent polarimetric decompositions employ coherent models to describe ground scattering; however, this cannot truly reflect the fact especially in natural ground surfaces. This paper proposes a highly adaptive decomposition with incoherent ground scattering models (ADIGSM). In ADIGSM, Neumann’s adaptive model is employed to describe volume scattering, and to explain cross-polarized power in remainder matrix, so that we can obtain orientation angle randomness for both volume scattering and the dominant ground scattering. The computation of volume scattering parameters is strictly constrained for non-negative eigenvalues, while the volume scattering parameters that explain the most cross-polarized power are selected. When applying ADIGSM to NASA’s UAVSAR data, the negative component powers were obtained in quite a few forest pixels. Compared with several newest decompositions, the volume scattering power is obviously lowered, especially in areas dominated by surface scattering or double bounce scattering. The orientation angle randomness of each component is reasonable as well. ADIGSM has potential to be applied in the fields such as PolSAR image classification, land cover mapping, speckle filtering, soil moisture and roughness estimation, etc.

Smoothing Windows for the Synthesis of Gaussian Stationary Random Fields Using Circulant Matrix Embedding

When generating Gaussian stationary random fields, a standard method based on circulant matrix embedding usually fails because some of the associated eigenvalues are negative. The eigenvalues can be shown to be nonnegative in the limit of increasing sample size. Computationally feasible large sample sizes, however, rarely lead to nonnegative eigenvalues. Another solution is to extend suitably the covariance function of interest so that the eigenvalues of the embedded circulant matrix become nonnegative in theory. Though such extensions have been found for a number of examples of stationary fields, the method depends on nontrivial constructions in specific cases.In this work, the embedded circulant matrix is smoothed at the boundary by using a cutoff window or overlapping windows over a transition region. The windows are not specific to particular examples of stationary fields. The resulting method modifies the standard circulant embedding, and is easy to use. It is shown that this straightforward approach works for many examples of interest, with the overlapping windows performing consistently better. The method even outperforms in the cases where extending the covariance leads to nonnegative eigenvalues in theory, in the sense that the transition region is considerably smaller. The Matlab code implementing the method is included in the online supplementary materials and also publicly available at www.hermir.org.

Volume Scattering Power Constraint Based on the Principal Minors of the Coherency Matrix

This letter proposes a constraint for a volume scattering power employing the principal minors, which can be used for polarimetric synthetic aperture radar (POLSAR) model-based decomposition. This constraint effectively allows for avoiding unreasonable results which yield negative eigenvalues. The proposed constraint is derived so that all the principal minors of the coherency matrix after volume scattering subtraction are nonnegative. Mathematically, the constraint is exactly the same as that based on the nonnegative eigenvalues. Thus, it is guaranteed that the result is physically reasonable and that the volume scattering power is not overestimated. A significant advantage of the proposed method compared to the constraint based on the nonnegative eigenvalues is the high computation efficiency, since the maximal volume scattering power can be derived analytically, while the nonnegative eigenvalue constraint requires a numerical calculation. In our experiment, the computation of the maximal power is six times faster using the approach based on the principal minors than that based on the nonnegative eigenvalues.

An Algebraic Analysis of the Graph Modularity

One of the most relevant tasks in network analysis is the detection of community structures, or clustering. Most popular techniques for community detection are based on the maximization of a quality function called modularity, which in turn is based upon particular quadratic forms associated to a real symmetric modularity matrix $M$, defined in terms of the adjacency matrix and a rank-one null model matrix. That matrix could be posed inside the set of relevant matrices involved in graph theory, alongside adjacency and Laplacian matrices. In this paper we analyze certain spectral properties of modularity matrices, which are related to the community detection problem. In particular, we propose a nodal domain theorem for the eigenvectors of $M$; we point out several relations occurring between the graph's communities and nonnegative eigenvalues of $M$; and we derive a Cheeger-type inequality for the graph modularity.

Regular bipartite graphs with three distinct non-negative eigenvalues

We derive some structural and spectral properties of regular bipartite graphs with three distinct non-negative eigenvalues. Next, we consider the relations between these graphs and two-class partially balanced incomplete block designs, and we present a number of situations when the graphs we consider are in fact the incidence graphs of those designs. As a consequence, we give a several constructions of connected regular bipartite graphs with six distinct eigenvalues, and we also determine all such graphs with degree 3, and all such graphs on at most 20 vertices.