Explicit Matrix Research Articles

Linking habitat use to mortality and population viability to disarm an ecological trap

Animal excursions out of protected areas are a source of human-wildlife conflict and can lead animals into ecological traps. These arise when animals prefer areas of their habitat conferring lower fitness than other available areas. Ecological traps should become increasingly common as humans continue to alter habitats, yet their impact on population viability has rarely been documented and there is limited knowledge on how to disarm them. Moreover, although spatial factors such as the proportion of trap habitat in the landscape are crucial in determining the probability of extinction, few studies have attempted to link animal use of space to demography to obtain insights into how to release trapped populations. Here we tackle these gaps using a stochastic, spatially explicit matrix model parametrized with empirical data. We show that a free-ranging population of plains bison (Bison bison bison) caught in an ecological trap caused by legal but unregulated hunting has a 66% probability of extinction over the next 50 years under current conditions. By linking the time bison spent in fields with hunting permission to survival and population persistence, we show that bison use of such fields must decrease by 70% to ensure population viability. Our approach narrowed down the ecological trap to <1% of the population's range during the hunting period. Targeting this limited portion of the landscape would hence be a cost-effective strategy to disarm the trap. We show that a spatially explicit approach to demography can refine conservation strategies to avoid potentially onerous and ineffective interventions.

Explicit resistance matrix for a Hall disk with multiple peripheral contacts: Application to a van der Pauw type method for extended contacts

A direct analytical method for determining the Resistance Matrix of a Hall disk with an arbitrary number of extended peripheral contacts has been developed. The method does not require the use of any conformal mappings. It works also in the case of large magnetic fields. The resulting explicit formulas involve the angular coordinates of the asymmetrical contacts ends, the sheet resistance, and the Hall angle θH as inputs. The formulas are obtained through the calculation of some definite integrals of analytical functions with integrable singularities at the end of the peripheral contacts. The method can be used for determining the sheet resistance and the Hall mobility of a circular plate with extended contacts on its boundary by utilizing two measurements similar to those used by van der Pauw's method for pointlike contacts.

Iwasawa theory for Rankin-Selberg products of p-nonordinary eigenforms

Let $f$ and $g$ be two modular forms which are non-ordinary at $p$. The theory of Beilinson-Flach elements gives rise to four rank-one non-integral Euler systems for the Rankin-Selberg convolution $f \otimes g$, one for each choice of $p$-stabilisations of $f$ and $g$. We prove (modulo a hypothesis on non-vanishing of $p$-adic $L$-fuctions) that the $p$-parts of these four objects arise as the images under appropriate projection maps of a single class in the wedge square of Iwasawa cohomology, confirming a conjecture of Lei-Loeffler-Zerbes. Furthermore, we define an explicit logarithmic matrix using the theory of Wach modules, and show that this describes the growth of the Euler systems and $p$-adic $L$-functions associated to $f \otimes g$ in the cyclotomic tower. This allows us to formulate "signed" Iwasawa main conjectures for $f\otimes g$ in the spirit of Kobayashi's $\pm$-Iwasawa theory for supersingular elliptic curves; and we prove one inclusion in these conjectures under our running hypotheses.

An explicit solution to a three-dimensional wedge problem considering two edges effect

The paper presents an explicit matrix algorithm to solve the problem of an elastic wedge with three loaded surfaces. The algorithm makes use of a recently published concept of transformation matrix, by which the original surface loads are converted to equivalent loads in half-space. The three loaded edges are considered simultaneously. The developed algorithm is used to study the effects of two free edges of a steel block and tapered rollers with different contact angles. The two load-free edges can substantially increase deformation if the two edges are close in distance. The results of the tapered roller simulation show that deformation is considerably sensitive to the contact angle of the tapered roller. The largest deformation appears at the big end of the roller. Furthermore, empirical formulae for correction factors for the calculation of block or quarter-space deformation based on half-space solutions are summarized.

Scalable statistical inference for averaged implicit stochastic gradient descent

AbstractStochastic gradient descent (SGD) provides a scalable way to compute parameter estimates in applications involving large‐scale data or streaming data. As an alternative version, averaged implicit SGD (AI‐SGD) has been shown to be more stable and more efficient. Although the asymptotic properties of AI‐SGD have been well established, statistical inferences based on it such as interval estimation remain unexplored. The bootstrap method is not computationally feasible because it requires to repeatedly resample from the entire data set. In addition, the plug‐in method is not applicable when there is no explicit covariance matrix formula. In this paper, we propose a scalable statistical inference procedure, which can be used for conducting inferences based on the AI‐SGD estimator. The proposed procedure updates the AI‐SGD estimate as well as many randomly perturbed AI‐SGD estimates, upon the arrival of each observation. We derive some large‐sample theoretical properties of the proposed procedure and examine its performance via simulation studies.

Explicit Matrix Representation for the Hamiltonian of the One Dimensional Spin 1/2 Ising Model in Mutually Orthogonal External Magnetic Fields

We derive an explicit matrix representation for the Hamiltonian of the Ising model in mutually orthogonal external magnetic fields, using as basis the eigenstates of a system of non-interacting spin 1/2 particles in external magnetic fields. We subsequently apply our results to obtain an analytical expression for the ground state energy per spin, to the fourth order in the exchange integral, for the Ising model in perpendicular external fields.

Recursive SURE for image recovery via total variation minimization

Recently, total variation regularization has become a standard technique, and even a basic tool for image denoising and deconvolution. Generally, the recovery quality strongly depends on the regularization parameter. In this work, we develop a recursive evaluation of Stein’s unbiased risk estimate (SURE) for the parameter selection, based on specific reconstruction algorithms. It enables us to monitor the evolution of mean squared error (MSE) during the iterations. In particular, to deal with large-scale data, we propose a Monte Carlo simulation for the practical computation of SURE, which is free of any explicit matrix operation. Experimental results show that the proposed recursive SURE could lead to highly accurate estimate of regularization parameter and nearly optimal restoration performance in terms of MSE.

On the mixed-type generalized inverses of the products of two operators

Let A, B and be closed range operators. The explicit matrix expressions for various generalized inverses are obtained by using block operator matrix methods. Some subtle relationships between the properties of sub-blocks in operator matrices A, B and their range relations are built. New necessary and sufficient conditions for the equivalent relations, inclusion relations and mixed-type generalized inverses relations are presented. Some recent mixed-type reverse-order laws results are covered and many new mixed-type generalized inverses relations are established by using this block-operator matrix technique.

Matrix Rigidity and the Croot-Lev-Pach Lemma

Matrix rigidity is a notion put forth by Valiant as a means for proving arithmetic circuit lower bounds. A matrix is rigid if it is far, in Hamming distance, from any low rank matrix. Despite decades of efforts, no explicit matrix rigid enough to carry out Valiant's plan has been found. Recently, Alman and Williams showed, contrary to common belief, that the $2^n \times 2^n$ Hadamard matrix could not be used for Valiant's program as it is not sufficiently rigid. In this note we observe a similar `non rigidity' phenomena for any $q^n \times q^n$ matrix $M$ of the form $M(x,y) = f(x+y)$, where $f:F_q^n \to F_q$ is any function and $F_q$ is a fixed finite field of $q$ elements ($n$ goes to infinity). The theorem follows almost immediately from a recent lemma of Croot, Lev and Pach which is also the main ingredient in the recent solution of the cap-set problem.

Estimating range expansion of wildlife in heterogeneous landscapes: A spatially explicit state-space matrix model coupled with an improved numerical integration technique.

Dispersal as well as population growth is a key demographic process that determines population dynamics. However, determining the effects of environmental covariates on dispersal from spatial‐temporal abundance proxy data is challenging owing to the complexity of model specification for directional dispersal permeability and the extremely high computational loads for numerical integration. In this paper, we present a case study estimating how environmental covariates affect the dispersal of Japanese sika deer by developing a spatially explicit state‐space matrix model coupled with an improved numerical integration technique (Markov chain Monte Carlo with particle filters). In particular, we explored the environmental drivers of inhomogeneous range expansion, characteristic of animals with short dispersal. Our model framework successfully reproduced the complex population dynamics of sika deer, including rapid changes in densely populated areas and distribution fronts within a decade. Furthermore, our results revealed that the inhomogeneous range expansion of sika deer seemed to be primarily caused by the dispersal process (i.e., movement barriers in fragmented forests) rather than population growth. Our state‐space matrix model enables the inference of population dynamics for a broad range of organisms, even those with low dispersal ability, in heterogeneous landscapes, and could address many pressing issues in conservation biology and ecosystem management.

Pullbacks and nontriviality of associated noncommutative vector bundles

Our main theorem is that the pullback of an associated noncommutative vector bundle induced by an equivariant map of quantum principal bundles is a noncommutative vector bundle associated via the same finite-dimensional representation of the structural quantum group. At the level of K0-groups, we realize the induced map by the pullback of explicit matrix idempotents. We also show how to extend our result to the case when the quantum-group representation is infinite dimensional, and then apply it to the Ehresmann–Schauenburg quantum groupoid. Finally, using noncommutative Milnor’s join construction, we define quantum quaternionic projective spaces together with noncommutative tautological quaternionic line bundles and their duals. As a key application of the main theorem, we show that these bundles are stably non-trivial as noncommutative complex vector bundles.

3D free vibration analysis of multi-directional FGM parallelepipeds using the quadrature element method

• A novel element for analyzing 3D vibration of multidirectional FGM solid is established. • Stiffness matrix is explicitly given when integration points do not coincide with nodes to achieve higher accuracy. • Comparisons show the accuracy of the proposed multidirectional FGM parallelepiped element. In this paper, the existing quadrature element method (QEM) is extended to analyze the three-dimensional (3D) free vibration of multi-directional functionally graded material (FGM) parallelepipeds under general boundary conditions. The material properties vary continuously in multiple directions according to a power-law form. Formulations of the multi-directional FGM parallelepiped element are given. Since the element nodes do not coincide with the integration points, a different way is used to compute the strains at integration points to derive explicit element stiffness matrix. A number of case studies on multi-directional FGM square and rhombic thick plates with general boundary conditions and different power-law exponents have been conducted. The effect of power-law exponents and boundary conditions on the free vibration behavior of FGM plates is investigated. The natural frequencies are in excellent agreement with either existing results or data obtained by ABAQUS using fine meshes. It is demonstrated that the extended QEM is simple in formulations and capable of capturing the 3D vibration behavior of the multi-directional FGM plates with high precision. New results are presented and may be used as benchmark solutions for future researches.

Nonlinear free vibration analysis of shell-type structures by the variational differential quadrature method in the context of six-parameter shell theory

In the present paper, an attempt is made to adopt the variational differential quadrature (VDQ) technique for the large-amplitude vibration analysis of shell-type structures based on the six-parameter shell theory. The functional of energy in quadratic form is derived based on Hamilton's principle which is then directly discretized by the VDQ method. Although the derived formulation is general, the focus of paper is on the cylindrical and spherical shells. The nonlinear vibration problem is solved by means of the time periodic discretization method. The results reveal that the present numerical method can solve the problem accurately. It is also easy to implement due to its compact and explicit matrix formulation. Comprehensive numerical results are presented to study the effects of geometrical properties and boundary conditions on the frequency-response curves of cylindrical and spherical shells. Moreover, comparison studies are presented between the results of the six-parameter shell theory and the first-order shear deformation shell theory.

A Meshfree Higher Order Mass Matrix Formulation for Structural Vibration Analysis

An accurate meshfree formulation with higher order mass matrix is proposed for the structural vibration analysis with particular reference to the 1D rod and 2D membrane problems. Unlike the finite element analysis with an explicit mass matrix, the mass matrix of Galerkin meshfree formulation usually does not have an explicit expression due to the rational nature of meshfree shape functions. In order to develop a meshfree higher order mass matrix, a frequency error measure is derived by using the entries of general symmetric stiffness and mass matrices. The frequency error is then expressed as a series expansion of the nodal distance, in which the coefficients of each term are related to the meshfree stiffness and mass matrices. It is theoretically proved that the constant coefficient in the frequency error vanishes identically provided with the linear completeness condition, which does not rely on any specific form of the shape functions. Furthermore, a meshfree higher order mass matrix is developed through a linear combination of the consistent and lumped mass matrices, in which the optimal mass combination coefficient is attained via eliminating the lower order error terms. In particular, the proposed higher order mass matrix with Galerkin meshfree formulation achieves a fourth-order accuracy when the moving least squares or reproducing kernel (RK) meshfree approximation with linear basis function is employed; nonetheless, the conventional meshfree method only gives a second-order accuracy for the frequency computation. In the multidimensional formulation, the optimal mass combination coefficient is a function of the wave propagation angle so that the proposed accurate meshfree method is applicable to the computation of frequencies associated with any wave propagation direction. The superconvergence of the proposed meshfree higher order mass matrix formulation is validated via numerical examples.

On duals and parity-checks of convolutional codes over [formula omitted

A convolutional code C over Zpr((D)) is a Zpr((D))-submodule of Zprn((D)) that admits a polynomial set of generators, where Zpr((D)) stands for the ring of (semi-infinity) Laurent series. In this paper we study several structural properties of its dual C⊥. We use these results to provide a constructive algorithm to build an explicit generator matrix of C⊥. Moreover, we show that the transpose of such a matrix is a parity-check matrix (also called syndrome former) of C.

Nonlinear large deformation analysis of shells using the variational differential quadrature method based on the six-parameter shell theory

The present work is concerned with the application of the variational differential quadrature (VDQ) method (Faghih Shojaei and Ansari, 2017), in the area of computational mechanics, to the nonlinear large deformation analysis of shell-type structures. To this end, based on the six-parameter shell model, the functional of energy in quadratic form is derived based on Hamilton’s principle which is then directly discretized by the VDQ technique. The formulation of article is presented in a general form so that it can be readily used for different structures such as beams, annular plates, cylindrical shells and hemispherical shells under various loading conditions. In order to reveal the accuracy of developed solution strategy, it is tested in several popular benchmark problems for the geometric nonlinear analysis of shells. The results show that the present numerical method is capable of yielding highly accurate solution in the nonlinear large deformation analysis of shells. It is also easy to implement due to its compact and explicit matrix formulation.

Parallel Direct Solution of the Covariance-Localized Ensemble Square Root Kalman Filter Equations with Matrix Functions

Abstract Recently, the serial approach to solving the square root ensemble Kalman filter (ESRF) equations in the presence of covariance localization was found to depend on the order of observations. As shown previously, correctly updating the localized posterior covariance in serial requires additional effort and computational expense. A recent work by Steward et al. details an all-at-once direct method to solve the ESRF equations in parallel. This method uses the eigenvectors and eigenvalues of the forward observation covariance matrix to solve the difficult portion of the ESRF equations. The remaining assimilation is easily parallelized, and the analysis does not depend on the order of observations. While this allows for long localization lengths that would render local analysis methods inefficient, in theory, an eigenpair-based method scales as the cube number of observations, making it infeasible for large numbers of observations. In this work, we extend this method to use the theory of matrix functions to avoid eigenpair computations. The Arnoldi process is used to evaluate the covariance-localized ESRF equations on the reduced-order Krylov subspace basis. This method is shown to converge quickly and apparently regains a linear scaling with the number of observations. The method scales similarly to the widely used serial approach of Anderson and Collins in wall time but not in memory usage. To improve the memory usage issue, this method potentially can be used without an explicit matrix. In addition, hybrid ensemble and climatological covariances can be incorporated.

Constitutive modeling of principal stress rotation by considering inherent and induced anisotropy of soils

In order to simulate the soil response during principal stress rotation, anisotropic unified hardening (UH) model is developed within the framework of elastoplastic theory. Without introducing any additional mechanism to display the role of stress rotation specifically, this model achieves the simulation by considering the material anisotropy. The effect of inherent anisotropy is reflected using the anisotropic transformed stress method, but a new formula for the stress mapping is adopted to keep the mean stress unchanged. Analysis indicates that from the view of the transformed stress tensor, the anisotropic soil is subjected to loading during pure rotation of principal stress axes, so that plastic strains can be calculated. To represent the induced anisotropy, a fabric evolution law is proposed based on laboratory and numerical test results. At the critical state, the fabric tensor reaches a stable value determined by the stress state, while the critical state line is unique in the plane of void ratio versus mean stress. The anisotropic UH model has concise formulation and explicit elastoplastic flexibility matrix and can provide reasonable predictions for the deformation of anisotropic soils when principal stresses rotate.

An efficient iterative method for computing deflections of Bernoulli–Euler–von Karman beams on a nonlinear elastic foundation

An efficient iterative method is developed for the static analysis of large deflections of an infinite beam with variable cross-section resting on a nonlinear foundation. A pseudo spring constant is added and explicit matrix operators are introduced to perform differentiation through Green’s function. The nonlinearity of the problem is handled with quasilinearization. To compute the solution of the quasilinear differential equation with prescribed accuracy, a new discretization method for solving quasilinear differential equations involving up to the 4th order derivative is used. The discretization method is based on relating discretizations of up to the fourth order derivative of the solution with a discretization of the solution by using a suitable Green function. Numerical experiments show that the error incurred by the discretization can be made small for the two first derivatives and that the method proposed in the paper converges fast and has good accuracy.

GPU-accelerated simulations of isolated black holes

We present a port of the numerical relativity code SpEC which is capable of running on NVIDIA GPUs. Since this code must be maintained in parallel with SpEC itself, a primary design consideration is to perform as few explicit code changes as possible. We therefore rely on a hierarchy of automated porting strategies. At the highest level we use TLoops, a C++ library of our design, to automatically emit CUDA code equivalent to tensorial expressions written into C++ source using a syntax similar to analytic calculation. Next, we trace out and cache explicit matrix representations of the numerous linear transformations in the SpEC code, which allows these to be performed on the GPU using pre-existing matrix-multiplication libraries. We port the few remaining important modules by hand. In this paper we detail the specifics of our port, and present benchmarks of it simulating isolated black hole spacetimes on several generations of NVIDIA GPU.