Nonzero Matrix Research Articles

A note on matrices with prescribed off-diagonal submatrix and characteristic polynomial

Let A∈Fq×p be a nonzero matrix over a field F, and let f be any monic polynomial of degree n=p+q with coefficients in F. A completion problem proposed and solved by de Oliveira asks for the existence of a matrix of order n with characteristic polynomial f and such that its q×p submatrix in the left down corner is equal to A. Let C(f)∈Fn×n be the companion matrix of f. We will construct a matrix XA∈Fn×n (which depends exclusively on A) such that XA−1C(f)XA is a closed form solution of this completion problem.

Orientation dependent electro-optic properties of K0.95Li0.05Ta0.41Nb0.59O3 single crystal: Experiment and simulation

Large low frequency linear electro-optic responses of K0.95Li0.05Ta0.41Nb0.59O3 single crystal were determined at the laser wavelength of 632.8 nm: γ33 = 223.7 pm/V, γ13 = −35.2 pm/V, and γ51 = 11060 pm/V by using the autoscanning Mach-Zehnder interferometer technique and modified AC modulating method, respectively. Crystal orientation dependent electro-optic properties have been simulated based on the non-zero matrix of experimental measured electro-optic coefficients. The simulation results demonstrate the maximum electro-optic coefficient occurred at 54.53° away [001] direction is 8548.3 pm/V. In addition to the high ferroelectric phase transition temperature (∼430 K), good piezoelectric properties and high optical crystal quality, the ultrahigh electro-optic performance of K0.95Li0.05Ta0.41Nb0.59O3 single crystal forecasts its outstanding potential in various electro-mechanical-optical coupled applications.

Factorization in large-scale many-body calculations

One approach for solving interacting many-fermion systems is the configuration-interaction method, also sometimes called the interacting shell model, where one finds eigenvalues of the Hamiltonian in a many-body basis of Slater determinants (antisymmetrized products of single-particle wavefunctions). The resulting Hamiltonian matrix is typically very sparse, but for large systems the nonzero matrix elements can nonetheless require terabytes or more of storage. An alternate algorithm, applicable to a broad class of systems with symmetry, in our case rotational invariance, is to exactly factorize both the basis and the interaction using additive/multiplicative quantum numbers; such an algorithm recreates the many-body matrix elements on the fly and can reduce the storage requirements by an order of magnitude or more. We discuss factorization in general and introduce a novel, generalized factorization method, essentially a ‘double-factorization’ which speeds up basis generation and set-up of required arrays. Although we emphasize techniques, we also place factorization in the context of a specific (unpublished) configuration-interaction code, BIGSTICK, which runs both on serial and parallel machines, and discuss the savings in memory due to factorization.

Hybrid textures of Majorana neutrino mass matrix and current experimental tests

Motivated by recent measurements of a relatively large \theta_{13} in the Daya Bay and RENO reactor neutrino experiments, we carry out a systematic analysis of the hybrid textures of Majorana neutrino mass matrix $M_\nu$, which contain one texture zero and two equal nonzero matrix elements. We show that three neutrino masses (m_1, m_2, m_3) and three leptonic CP-violating phases (\delta, \rho, \sigma) can fully be determined from two neutrino mass-squared differences (\delta m^2, \Delta m^2) and three flavor mixing angles (\theta_{12}, \theta_{23}, \theta_{13}). Out of sixty logically possible patterns of $M_\nu$, thirty-nine are found to be compatible with current experimental data at the 3\sigma level. We demonstrate that the texture zero of $M_\nu$ is stable against one-loop quantum corrections, while the equality between two independent elements not. Phenomenological implications of $M_\nu$ for the neutrinoless double-beta decay and leptonic CP violation are discussed, and a realization of the texture zero and equality by means of discrete flavor symmetries is illustrated.

Two-parameter neutrino mass matrices with two texture zeros

We reanalyze Majorana neutrino mass matrices with two texture zeros, by searching for viable hybrid textures in which the non-zero matrix elements of have simple ratios. Referring to the classification scheme of Frampton, Glashow and Marfatia, we find that the mass matrix denoted by A1 allows the ratios and . There are analogous ratios for texture A2. With these two hybrid textures, one obtains, for instance, good agreement with the data if one computes the three mixing angles in terms of the experimentally determined mass-squared differences and . We could not find viable hybrid textures based on mass matrices different from those of cases A1 and A2.

Modeling the scattering properties of mineral aerosols using concave fractal polyhedra

The single-scattering properties of concave fractal polyhedra are investigated, with particle size parameters ranging from the Rayleigh to geometric-optics regimes. Two fractal shape parameters, irregularity and aspect ratio, are used to iteratively construct "generations" of irregular fractal particles. The pseudospectral time-domain (PSTD) method and the improved geometric-optics method (IGOM) are combined to compute the single-scattering properties of fractal particles over the range of size parameters. The effects of fractal generation, irregularity, and aspect ratio on the single-scattering properties of fractals are investigated. The extinction efficiency, absorption efficiency, and asymmetry factor, calculated by the PSTD method for fractal particles, with small-to-moderate size parameters, smoothly bridges the gap between those size parameters and size parameters for which solutions given by the IGOM may be used. Somewhat surprisingly, excellent agreement between values of the phase function of randomly oriented fractal particles calculated by the two numerical methods is found, not only for large particles, but in fact extends as far down in equivalent-projected-area size parameters as 25. The agreement in the case of other nonzero phase matrix elements is not as good at so small a size. Furthermore, the numerical results of ensemble-averaged phase matrix elements of a single fractal realization are compared with dust particle measurements, and good agreement is found by using the fractal particle model to represent data from a study of feldspar aerosols.

Reactive transport modelling of acid mine drainage within discretely fractured porous media: Plume evolution from a surface source zone

A numerical model is developed for investigating the transport behaviour and geochemical evolution of acidic mine drainage (AMD) in discretely fractured porous media. The simulation approach is tested using a conceptual model of a reactive mine waste system in which an active source of AMD overlies a fractured silicate-rich porous host rock with a low but non-zero matrix permeability. Source composition is based on measured data from an existing mine tailings site. The numerical model includes groundwater flow, AMD infiltration, multi-component advective-dispersive transport, equilibrium geochemical speciation and water-rock pH-buffering reactions within a discrete fracture network (DFN). An analytical solution for parallel-fractures is used to verify the model, which is then applied to simulate the evolution of pH, the major aqueous species from the AMD source, as well as selected mineral buffers. As the acidic drainage water infiltrates into the initially uncontaminated fracture networks, high concentration gradients develop within the matrix along fracture interfaces inducing diffusion-limited pH buffering and precipitation of secondary minerals within the rock matrix. A comparison of AMD evolution in three fracture networks shows that even within a densely fractured network, AMD plume evolution can be significantly different from that obtained from assuming an equivalent porous medium (EPM). The paper also addresses issues of time scales and matrix diffusion. The results have implications for predicting environmental impacts of acid mine drainage in complex mining environments and for coupling of hydro-geochemical and geotechnical models. The model can also be applied to other hydrogeological systems including fractured clays and tills, to other contaminants including hydrocarbons or organic solvents, and to simulate geochemical evolution in natural flow systems.

On characterizing proteomics maps by using weighted Voronoi maps

In contrast to the standard construction of Voronoi regions, in which the boundaries between different regions are at equal distance from the given points, we consider the construction of modified Voronoi regions obtained by giving greater weights to spots reported to have higher abundance. Specifically we are interested in applying this approach to 2-D proteomics maps and their numerical characterization. As will be seen, the boundaries of the weighted Voronoi regions are sensitive to the relative abundances of the protein spots and thus the abundances of protein spots, the z component of the (x, y, z) triplet, are automatically incorporated in the numerical analysis of the adjacency matrix, rather than used to augment the adjacency matrix as non-zero diagonal matrix elements. The outlined approach is general and it may be of interest for numerical analyses of other maps that are defined by triplets (x, y, z) as input information.

The Perturbed Puma Model

The puma model on the basis of the Lorentz and CPT violation may bring an economical interpretation to the conventional neutrinos oscillation and part of the anomalous oscillations. We study the effect of the perturbation to the puma model. In the case of the first-order perturbation which keeps the (23) interchange symmetry, the mixing matrix element Ue3 is always zero. The nonzero mixing matrix element Ue3 is obtained in the second-order perturbation that breaks the (23) interchange symmetry.

Specification of Matrix Cleanup Goals in Fractured Porous Media

Semianalytical transient solutions have been developed to evaluate what level of fractured porous media (e.g., bedrock or clay) matrix cleanup must be achieved in order to achieve compliance of fracture pore water concentrations within a specified time at specified locations of interest. The developed mathematical solutions account for forward and backward diffusion in a fractured porous medium where the initial condition comprises a spatially uniform, nonzero matrix concentration throughout the domain. Illustrative simulations incorporating the properties of mudstone fractured bedrock demonstrate that the time required to reach a desired fracture pore water concentration is a function of the distance between the point of compliance and the upgradient face of the domain where clean groundwater is inflowing. Shorter distances correspond to reduced times required to reach compliance, implying that shorter treatment zones will respond more favorably to remediation than longer treatment zones in which back-diffusion dominates the fracture pore water response. For a specified matrix cleanup goal, compliance of fracture pore water concentrations will be reached sooner for decreased fracture spacing, increased fracture aperture, higher matrix fraction organic carbon, lower matrix porosity, shorter aqueous phase decay half-life, and a higher hydraulic gradient. The parameters dominating the response of the system can be measured using standard field and laboratory techniques.

Polar Kerr Effect and Time Reversal Symmetry Breaking in Bilayer Graphene

The unique sensitivity of optical response to different types of symmetry breaking can be used to detect and identify spontaneously ordered many-body states in bilayer graphene. We predict a strong response at optical frequencies, sensitive to electronic phenomena at low energies, which arises because of nonzero interband matrix elements of the electric current operator. In particular, the polar Kerr rotation and reflection anisotropy provide fingerprints of the quantum anomalous Hall state and the nematic state, characterized by spontaneously broken time-reversal symmetry and lattice rotation symmetry, respectively. These optical signatures, which undergo a resonant enhancement in the near-infrared regime, lie well within reach of existing experimental techniques.

A proof of the [formula omitted] conjecture: Centralizers, Jacobians and spectrally arbitrary sign patterns

We prove the so-called Tn conjecture: for every real-monic polynomial p(x) of degree n⩾2 there exists an n by n matrix with sign patternTn=-+0⋯0-0⋱⋮0⋱⋱⋱0⋮⋱0+0⋯0-+,whose characteristic polynomial is p(x). The proof converts the problem of determining the nonsingularity of a certain Jacobi matrix to the problem of proving the non-existence of a nonzero matrix B that commutes with a nilpotent matrix with sign pattern Tn and has zeros in positions (1,1), and (j+1,j) for j=2,…,n-1.

Fully-3D PET Image Reconstruction Using Scanner-Independent, Adaptive Projection Data and Highly Rotation-Symmetric Voxel Assemblies

For iterative, fully 3D positron emission tomography (PET) image reconstruction intrinsic symmetries can be used to significantly reduce the size of the system matrix. The precalculation and beneficial memory-resident storage of all nonzero system matrix elements is possible where sufficient compression exists. Thus, reconstruction times can be minimized independently of the used projector and more elaborate weighting schemes, e.g., volume-of-intersection (VOI), are applicable. A novel organization of scanner-independent, adaptive 3D projection data is presented which can be advantageously combined with highly rotation-symmetric voxel assemblies. In this way, significant system matrix compression is achieved. Applications taking into account all physical lines-of-response (LORs) with individual VOI projectors are presented for the Siemens ECAT HR+ whole-body scanner and the Siemens BrainPET, the PET component of a novel hybrid-MR/PET imaging system. Measured and simulated data were reconstructed using the new method with ordered-subset-expectation-maximization (OSEM). Results are compared to those obtained by the sinogram-based OSEM reconstruction provided by the manufacturer. The higher computational effort due to the more accurate image space sampling provides significantly improved images in terms of resolution and noise.

ON THE EXISTENCE OF NOWHERE-ZERO VECTORS FOR LINEAR TRANSFORMATIONS

AbstractA matrix A over a field F is said to be an AJT matrix if there exists a vector x over F such that both x and Ax have no zero component. The Alon–Jaeger–Tarsi (AJT) conjecture states that if F is a finite field, with |F|≥4, and A is an element of GL n (F) , then A is an AJT matrix. In this paper we prove that every nonzero matrix over a field F, with |F|≥3 , is similar to an AJT matrix. Let AJTn (q) denote the set of n×n, invertible, AJT matrices over a field with q elements. It is shown that the following are equivalent for q≥3 : (i) AJTn (q)=GL n (q) ; (ii) every 2n×n matrix of the form (A∣B)t has a nowhere-zero vector in its image, where A,B are n×n, invertible, upper and lower triangular matrices, respectively; and (iii) AJTn (q) forms a semigroup.

Lowest eigenvalue of the nuclear shell model Hamiltonian

In this paper we investigate regular patterns of matrix elements of the nuclear shell model Hamiltonian $H$, by sorting the diagonal matrix elements from the smaller to larger values. By using simple plots of non-zero matrix elements and lowest eigenvalues of artificially constructed "sub-matrices" $h$ of $H$, we propose a new and simple formula which predicts the lowest eigenvalue with remarkable precisions.

On bar frameworks, stress matrices and semidefinite programming

A bar framework G(p) in r-dimensional Euclidean space is a graph G on the vertices 1, 2, . . . , n, where each vertex i is located at point p i in $${\mathbb{R}^r}$$ . Given a framework G(p) in $${\mathbb{R}^r}$$ , a problem of great interest is that of determining whether or not there exists another framework G(q), not obtained from G(p) by a rigid motion, such that ||q i − q j ||2 = ||p i − p j ||2 for each edge (i, j) of G. This problem is known as either the global rigidity problem or the universal rigidity problem depending on whether such a framework G(q) is restricted to be in the same r-dimensional space or not. The stress matrix S of a bar framework G(p) plays a key role in these and other related problems. In this paper, semidefinite programming (SDP) theory is used to address, in a unified manner, several problems concerning universal rigidity. New results are presented as well as new proofs of previously known theorems. In particular, we use the notion of SDP non-degeneracy to obtain a sufficient condition for universal rigidity, and we show that this condition yields the previously known sufficient condition for generic universal rigidity. We present new results concerning positive semidefinite stress matrices and we use a semidefinite version of Farkas lemma to characterize bar frameworks that admit a nonzero positive semidefinite stress matrix S.

Chemical bonding analysis for solid-state systems using intrinsic oriented quasiatomic minimal-basis-set orbitals

A chemical bonding scheme is presented for the analysis of solid-state systems. The scheme is based on the intrinsic oriented quasiatomic minimal-basis-set orbitals (IO-QUAMBOs) previously developed by Ivanic and Ruedenberg for molecular systems. In the solid-state scheme, IO-QUAMBOs are generated by a unitary transformation of the quasiatomic orbitals located at each site of the system with the criteria of maximizing the sum of the fourth power of interatomic orbital bond order. Possible bonding and antibonding characters are indicated by the single particle matrix elements, and can be further examined by the projected density of states. We demonstrate the method by applications to graphene and (6,0) zigzag carbon nanotube. The oriented-orbital scheme automatically describes the system in terms of $s{p}^{2}$ hybridization. The effect of curvature on the electronic structure of the zigzag carbon nanotube is also manifested in the deformation of the intrinsic oriented orbitals as well as a breaking of symmetry leading to nonzero single particle density matrix elements. In an additional study, the analysis is performed on the ${\text{Al}}_{3}\text{V}$ compound. The main covalent bonding characters are identified in a straightforward way without resorting to the symmetry analysis. Our method provides a general way for chemical bonding analysis of ab initio electronic structure calculations with any type of basis sets.

Efficient methodologies for system matrix modelling in iterative image reconstruction for rotating high-resolution PET

A fully 3D iterative image reconstruction algorithm has been developed for high-resolution PET cameras composed of pixelated scintillator crystal arrays and rotating planar detectors, based on the ordered subsets approach. The associated system matrix is precalculated with Monte Carlo methods that incorporate physical effects not included in analytical models, such as positron range effects and interaction of the incident gammas with the scintillator material. Custom Monte Carlo methodologies have been developed and optimized for modelling of system matrices for fast iterative image reconstruction adapted to specific scanner geometries, without redundant calculations. According to the methodology proposed here, only one-eighth of the voxels within two central transaxial slices need to be modelled in detail. The rest of the system matrix elements can be obtained with the aid of axial symmetries and redundancies, as well as in-plane symmetries within transaxial slices. Sparse matrix techniques for the non-zero system matrix elements are employed, allowing for fast execution of the image reconstruction process. This 3D image reconstruction scheme has been compared in terms of image quality to a 2D fast implementation of the OSEM algorithm combined with Fourier rebinning approaches. This work confirms the superiority of fully 3D OSEM in terms of spatial resolution, contrast recovery and noise reduction as compared to conventional 2D approaches based on rebinning schemes. At the same time it demonstrates that fully 3D methodologies can be efficiently applied to the image reconstruction problem for high-resolution rotational PET cameras by applying accurate pre-calculated system models and taking advantage of the system's symmetries.

Fuzzy symmetric solutions of fuzzy linear systems

In this paper, we propose a simple and practical method to solve a fuzzy linear system AX̃=b̃, where X̃ and b̃ are fuzzy triangular vectors with non-zero spreads and matrix A is nonsingular with real coefficients. The aim of this paper is twofold. First, we obtain the crisp solution of a fuzzy linear system. To this end, we solve the 1-cut of a fuzzy linear system. Second, we allocate some unknown symmetric spreads to any rows of fuzzy linear system in 1-cut position. Thus, a fuzzy linear system in 1-cut will be transformed to a system of interval equations. The symmetric spreads of each element of a fuzzy vector solution are derived by solving such an interval system. Moreover, based on the obtained symmetric spreads we derive three types of solutions. However, one of the mentioned spreads has pessimistic/optimistic attitude that is determined with a decision maker. It seems that such a solution is a connection solution between Tolerable Solution Set (TSS) and Controllable Solution Set (CSS). Also, we derive the maximal solution and the minimal solution of an original fuzzy linear system which are placed in TSS and CSS, respectively. Finally, some numerical examples are given to illustrate the efficiency and ability of the proposed method.

Generating Ultrashort Coherent Soft X-Ray Radiation in Storage Rings Using Angular-Modulated Electron Beams

A technique is proposed to generate ultrashort coherent soft x-ray radiation in storage rings using angular-modulated electron beams. In the scheme a laser operating in the TEM01 mode is first used to modulate the angular distribution of the electron beam in an undulator. After passing through a special beam line with nonzero transfer matrix element R54, the angular modulation is converted to density modulation which contains considerable high harmonic components of the laser. It is found that the harmonic number can be 1 or 2 orders of magnitude higher than the standard coherent harmonic generation method which relies on beam energy modulation. The technique has the potential of generating femtosecond coherent soft x-ray radiation directly from an infrared seed laser and may open new research opportunities for ultrafast sciences in storage rings.