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The concept of tricomplex valued S-metric space is introduced in this article, and some properties are derived. Also, some fixed point results of contraction maps satisfying various types of rational inequalities are proved for tricomplex valued S-metric spaces. Moreover, an example is given to illustrate our main result. Furthermore, an existence theorem for the unique solution to the linear system of equations is obtained by using our main result. [ABSTRACT FROM AUTHOR]

Let R be an associative ring and M be a left R-module. We introduce the concept of the incidence module I(X, M) of a locally finite partially ordered set X over M. We study the properties of I(X, M) and give the necessary and sufficient conditions for the incidence module to be an IN-module, EIN-module, nil injective module and nonsingular module, respectively. Furthermore, we show that the class of EIN-modules is closed under direct product and upper triangular matrix modules. [ABSTRACT FROM AUTHOR]

This paper presents a nonsingular sliding mode fault-tolerant control method with fixed-time convergence for a class of robot manipulators with uncertainties, external disturbances, and actuator failures. We estimate self-friction and external disturbances by designing a disturbance observer. Furthermore, based on the disturbance observer, we propose a sliding mode control method for the considered uncertain robot manipulator. Finally, the effectiveness of the proposed method is demonstrated by a numerical example. [ABSTRACT FROM AUTHOR]

This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result.

In this paper, we propose a generalized shift-HSS (denoted by SFHSS) iteration method for solving nonsingular saddle point systems with nonsymmetric positive definite (1, 1)-block sub-matrix, and theoretically verify its convergence property. In addition, we discuss the algebraic properties of the resulted SFHSS preconditioner and estimate the sharp eigenvalue bounds of the related preconditioned matrix. Finally, numerical experiments are given to support our theoretical results and reveal that the new method is feasible and effective.

In this article, for robotic systems with uncertain dynamics and time-varying asymmetric output constraints, we present a composite learning finite-time control scheme with salient features benefited from two design steps. In the first step, unlike existing composite adaptive/learning control algorithms, which result in either exponential convergence or finite-time convergence but exhibit a potential singularity issue, a modified nonsingular terminal sliding mode-based composite learning controller is adopted such that both tracking error and parameter estimation error converge to zero in finite time without singularity. The unknown parameter learning law is constructed by using online historical data and regressor extension, which gives a benefit of relaxing the typically required stringent persistent excitation with a much weaker excitation condition termed interval excitation. In the second step, a universal time-varying asymmetric barrier function (UTABF) is adopted to directly constrain the system output rather than the most commonly used barrier Lyapunov function, which indirectly converts the output constraints into the conservative tracking error constraints. Moreover, the UTABF can handle both constrained and unconstrained cases uniformly without the need for changing the control structure. Both theoretical analysis and experiments results on an industrial manipulator confirm the benefits and effectiveness of the proposed control scheme. [ABSTRACT FROM AUTHOR]

This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result. [ABSTRACT FROM AUTHOR]

A nonsingular distributed cooperative terminal guidance law is proposed for the problem of multiple missiles attacking ground stationary target simultaneously. By selecting the estimated attack time as consistency variable, the consistency algorithm is combined with nonsingular guidance law, and the limitation caused by the assumption of small angle and specified attack time can be avoided. The closed-loop stabilities of the consensus error and the lead angle are proved based on Lyapunov stability theory, which can verify the effectiveness of the proposed distributed cooperative guidance law. Numerical contrast simulations further verify that the miss distance of the designed guidance law under large initial lead angle is one order of magnitude smaller than that of the compared distributed guidance law, demonstrating the effectiveness and superiority of the proposed guidance law.

This study proposes a nonsingular barrier-function-based terminal sliding mode control technique for $n$ th-order nonlinear dynamic systems. Its main objective is to guarantee the finite-time tracking performance in the presence of unmodeled dynamics, external disturbances and parameter variations. The proposed approach is synthesized using a novel barrier function-based terminal sliding surface to ensure the effective estimation of the system perturbations using the barrier adaptation laws, and thereby achieve the desired tracking performance. The dynamics of chaotic models are strongly dependent upon the initial conditions, parameters of the system, parametric uncertainty and external disturbances which are required to be controlled/synchronized by a robust nonlinear control technique. By designing the terminal sliding mode control approach combined with the adaptive control law, the tracking problem of the nth-order nonlinear dynamical system with unmodeled dynamics, parametric variations and external disturbances is investigated. Moreover, the application of the proposed method is studied using the color-image scrambling system. The scrambling keys are created by transmitter chaotic systems, where using the chaotic keys and scrambling techniques, the original color image is encrypted. The performance and applicability of the proposed design is assessed using two practical applications: a chaotic hyper-jerk system and a color image encryption system. The simulation and analytical results obtained with both systems confirmed the ability of the proposed control method to guarantee the finite time convergence of sliding surface, ensure chattering-free dynamics and avoid the singularity problems.

In this study, a new fast variable structure adaptive fuzzy controller is presented for nonlinear statedelay systems which are subjected to external disturbances and uncertainties. The undesirable chattering and singularity of the variable structure scheme are eliminated by using a novel fast robust high-precision continuous nonsingular control law which is able to accelerate the finite-time convergence both in reaching and sliding phases of the motion. A fuzzy logic system with a neural network adaptive law is used to approximate the dynamics of the nonlinear system containing the current state and the delayed state. The superiority of the proposed fuzzy neural network in online adjusting the weights of the network is the fast convergence rate of the approximation error to the optimum value in a very short time. The stability of the closed-loop system is proved by using an extended finite-time Lyapunov criterion such that the convergence of the position tracking error, velocity tracking error, and the estimation error to the bounded region is guaranteed in a very short time. Two second-order uncertain nonlinear simulation examples with external disturbances are given to evaluate the efficacy of the proposed control technique. The simulation results show that faster and high-precision tracking performance is obtained compared with the existing recent works focused on robust control of nonlinear state-delay systems with uncertainties. [ABSTRACT FROM AUTHOR]

Let (G , σ) be a signed graph with order n (G) and size e (G). The rank r (G , σ) of (G , σ) is the rank of A (G , σ) , where A (G , σ) is the adjacency matrix of (G , σ). Let m (G) be the matching number of G , m ⁎ (G) be the fractional matching number of G , and c (G) = e (G) − n (G) + θ (G) be the cyclomatic number of G with θ (G) the number of connected components. In this paper, we prove that 2 m (G) − 2 κ (G) ≤ r (G , σ) ≤ 2 m ⁎ (G) for any signed graph, κ (G) is the number of even cycles in G. This improves the main result in He and Hao (2019) [10] saying that 2 m (G) − 2 c (G) ≤ r (G , σ) ≤ 2 m (G) + c (G). Moreover, we characterize all signed graphs with r (G , σ) = 2 m (G) − 2 κ (G) when κ (G) = c (G) o r c (G) − 1. Furthermore, all nonsingular bipartite cycle-disjoint signed graphs are achieved in this paper. [ABSTRACT FROM AUTHOR]

This paper proposes a fault-tolerant nonsingular fixed-time control scheme based on neural networks (NNs) for spacecraft maneuver mission subject to actuator faults, unknown external disturbance and parametric uncertainty. The pose dynamic model of the rigid spacecraft is derived with unknown external disturbance and parametric uncertainty. A novel adaptive neural control law is proposed to estimate and compensate for the lumped system faults or uncertainties. Based on the adaptive neural control law, a nonsingular fixed-time terminal sliding-mode (NFTSM) controller is developed to ensure the fixed-time stability of the closed-loop system via the Lyapunov analysis. The singularity in the controller design can be directly avoided with no prior knowledge of lumped disturbance' upper bound. The controller can speed up the convergence rate with improved control accuracy. Moreover, the settling time for the system states is independent of the initial conditions. Finally, comparative simulations are carried out to prove that the proposed control scheme has strong robustness and fault tolerance. [ABSTRACT FROM AUTHOR]

We derive the necessary conditions and a sufficient condition for the nonsingularity of a class of block three-by-three saddle-point problems. When it is singular, the sufficient conditions for the solvability are also discussed. [ABSTRACT FROM AUTHOR]

Designing a controller to stabilize maneuvering hovercrafts is an important challenge in amphibious vehicles. Hovercrafts are implemented in several applications, such as military missions, transportation, and scientific tasks. Thus, to improve their performance, it is crucial to control the system and compensate uncertainties and disruptions. In this paper, both classic and intelligent approaches are combined to design an observer-based controller. The system is assumed to be both controllable and observable. An adaptive neural network observer with guaranteed stability is derived for the nonlinear dynamics of a hovercraft, which is controlled via a nonsingular super-twisting terminal sliding-mode method. The main merits of the proposed method are as follows: (1) the Lyapunov stability of the overall closed-loop system, (2) the convergence of the tracking and observer errors to zero, (3) the robustness against uncertainties and disturbances, and (4) the reduction of the chattering phenomena. The simulation results validate the excellent performance of the derived method. [ABSTRACT FROM AUTHOR]

This investigation attempts to study a possible controller in improving the dynamic stability of capacitive microstructures through mitigating the effects of disturbances and uncertainties in their resultant dynamic behavior. Consequently, a nonsingular terminal sliding mode control strategy is suggested in this regard. The main features of this particular control strategy are its high response speed and its non-reliance on powerful controller forces. The stability of the controller was investigated using Lyapunov theory. For this purpose, a suitable Lyapunov function was introduced to prove the stability of a controller, and the singularity conditions and methods to overcome these conditions are presented. The achieved results proved the high capability of the applied technique in stabilizing of the microstructure as well as mitigating the effects of disturbances and uncertainties.

We present singularity-free solution for cracks within polar media in which material points possess both position and orientation. The plane strain problem is addressed in this study for which the generalized continua including micropolar, nonlocal micropolar, and gradient micropolar elasticity theories are employed. For the first time, the variationally consistent boundary conditions are derived for gradient micropolar elasticity. Moreover, having reviewed the solution to line defects including glide edge dislocation and wedge disclination from the literature, the fields of a climb edge dislocation within micropolar, nonlocal micropolar and gradient micropolar elasticity are derived. This completes the collection of line defects needed for an inplane strain analysis. Afterward, as the main contribution, using both types of line defects (i.e., dislocation being displacement discontinuity and disclination being rotational discontinuity), the well-established dislocation-based fracture mechanics is systematically generalized to the dislocation- and disclination-based fracture mechanics of polar media for which we have three translations together with three rotations as degrees of freedom. Due to the application of the line defects, incompatible elasticity is employed throughout the paper. Cracks under three possible loadings including stress and couple stress components are analyzed, and the corresponding line defect densities and stress and couple stress fields are reported. The singular fields are obtained once using the micropolar elasticity, while nonlocal micropolar, and gradient micropolar elasticity theories give rise to singularity-free fracture mechanics. [ABSTRACT FROM AUTHOR]

This article proposes a combination of linear and nonlinear sliding surfaces to design a new structure for terminal sliding mode control, capable of accepting a definite final time as an input data. The structures of both single-input-single-output and multi-input-multi-output systems are expressed. The controller operates in two modes: first, reaching the states to linear sliding surface, defining control parameters and rise time; second, switching to nonlinear sliding surface and defining a convergence time. Sum of rise time and convergence time, both of which as inputs, sets the final time. The control gains are adaptively tuned and parameter uncertainty in dynamics is considered in the design. The proposed method is implemented theoretically and experimentally on Scout robot in point-to-point motion and trajectory tracking. The results are compared to conventional terminal sliding mode control and finite-time state-dependent Riccati equation to assess the improvement. [ABSTRACT FROM AUTHOR]

In an entry pattern matrix A , all entries are indeterminates and the same indeterminate may appear in multiple positions. For a field F , an F -completion of A results from assigning a value from F to each indeterminate entry. We say that a square entry pattern matrix is almost-nonsingular over a field F if all of its F -completions are nonsingular, except for those in which all indeterminates are assigned the same value. This work investigates bounds for the maximum number of indeterminates of almost-nonsingular entry pattern matrices over some fields, including the real field, the rational field and finite fields. [ABSTRACT FROM AUTHOR]

Aiming at the non-Hermitian saddle point problem， an improved Uzawa-PSS iteration method is constructed based on the existing Uzawa-PSS method. The main idea of the new method is to solve two linear subsystems in each iteration step of Uzawa-PSS method， whose coefficient matrices are αI+P and αI+S， respectively. The first subsystem can be solved by CG method， but the second subsystem is very difficult to solve. The improved algorithm uses the single-step PSS iteration method to approximate the problem. Then the new method is used to solve the non-singular and singular saddle point problems respectively， and the corresponding convergence analysis is given. The numerical simulation also proves that the improved Uzawa-PSS iteration method has obvious advantages in iteration steps， CPU time and relative residuals. [ABSTRACT FROM AUTHOR]

In this paper, we present nonsingular permutation polynomials from nonsingular feedback shift registers and examine nonlinearity and algebraic degree of nonsingular polynomials of certain forms. The upper bound on the nonlinearity of nonsingular Boolean functions is investigated. We also present n -variable nonsingular Boolean functions with algebraic degree n − 1 and highest possible nonlinearity for odd n. [ABSTRACT FROM AUTHOR]

In this study, a nonsingular adaptive-gain super-twisting (AGSTW) guidance law is proposed to intercept a target with an impact angle constraint. The stability and convergence characteristic of the AGSTW guidance law are analyzed. The control chattering in system could be mitigated in the presence of uncertainty with unknown boundary. To validate the superiority of the proposed strategy, simulation comparisons with a conventional super-twisting and a sliding mode guidance law are carried out. The results indicate that the proposed AGSTW guidance law could make a missile intercept the target with more favorable miss distance and angle tracking error. The requirement of intercepting different targets could be met in a more satisfactory manner. [ABSTRACT FROM AUTHOR]

A backstepping nonsingular fast terminal sliding mode control with the extended state observer (ESO) is proposed for the uncertain factors of nonlinear spinning missile. Based on Lyapunov stability theory, the virtual control as sliding mode is taken in the backstepping design and then the tracking differentiator (TD) is employed to eliminate the "explosion of terms". In the last step of the backstepping design, nonsingular fast terminal sliding mode control is utilized to drive the angular velocity tracking error to converge to the origin in a finite period of time. To estimate the chattering phenomenon caused by disturbance influence variable in the system, an ESO is applied to estimate and compensate the impact from uncertainties and disturbances. The stability of the closed-loop system is proved. The simulation results show the effectiveness of the proposed control method and the stability of the controller. [ABSTRACT FROM AUTHOR]

In this paper, a class of mismatched uncertain systems is investigated. And a novel sliding mode control design is presented. During the design process of the proposed control, a transformation scheme is proposed, which can easily transform the mismatched uncertain systems into matched systems. Based on the matched systems, a novel full order sliding mode surface is designed, which can avoid the singular issue of terminal sliding mode control. And a chattering alleviating scheme is also proposed, which can ensure that the sliding mode control is chattering free regardless of whether the conventional sliding mode control or the terminal sliding mode control is used. Compared with the recurrent controller, the complexity of the proposed controller is reduced. It is much simpler and easy to implement. Also, the need of system knowledge is also reduced. The control performance is validated by simulation. [ABSTRACT FROM AUTHOR]

In a spatially-flat Friedmann-Lemaître-Robertson-Walker universe, the incorporation of bulk viscous process in general relativity leads to an appearance of a nonsingular background of the universe that both at early and late times depicts an accelerated universe. These early and late scenarios of the universe can be analytically calculated and mimicked, in the context of general relativity, by a single scalar field whose potential could also be obtained analytically where the early inflationary phase is described by a one-dimensional Higgs potential and the current acceleration is realized by an exponential potential. We show that the early inflationary universe leads to a power spectrum of the cosmological perturbations which match with current observational data, and after leaving the inflationary phase, the universe suffers a phase transition needed to explain the reheating of the universe via gravitational particle production. Furthermore, we find that at late times, the universe enters into the de Sitter phase that can explain the current cosmic acceleration. Finally, we also find that such bulk viscous-dominated universe attains the thermodynamical equilibrium, but in an asymptotic manner. [ABSTRACT FROM AUTHOR]

It is shown that a ring R is semiprime right Goldie if and only if R is right non- singular and every nonsingular right R-module M has a direct decomposition M = I⊕N, where I is injective and N is a reduced module such that N does not contain any extending submodule of infinite Goldie dimension. [ABSTRACT FROM AUTHOR]

When the computational point is approaching the poles, the variance and covariance formulae of the disturbing gravity gradient tensors tend to be infinite, and this is a singular problem. In order to solve the problem, the authors deduced the practical non-singular computational formulae of the first-and second-order derivatives of the Legendre functions and two kinds of spherical harmonic functions, and then constructed the nonsingular formulae of variance and covariance function of disturbing gravity gradient tensors.

We prove the Hasse principle and weak approximation for varieties defined by the smooth complete intersection of three quadratics in at least 19 variables, over arbitrary number fields. [ABSTRACT FROM AUTHOR]

This paper is the sequel of a companion Part I paper devoted to dislocation-based antiplane fracture mechanics within nonlocal and gradient elasticity of bi-Helmholtz type. In the present paper, the inplane analysis is carried out to study cracks of Modes I and II. Generalized continua including nonlocal elasticity of bi-Helmholtz type and gradient elasticity of bi-Helmholtz type (second strain gradient elasticity) offer nonsingular frameworks for the discrete dislocations. Consequently, the dislocation-based fracture mechanics within these frameworks is expected to result in a regularized fracture theory. By distributing the (climb and glide) edge dislocations, (Modes I and II) cracks are modeled. Distinctive features are captured for crack solutions within second-grade theories (nonlocal and gradient elasticity of bi-Helmholtz type) comparing with solutions within first-grade theories (nonlocal and gradient elasticity of Helmholtz type) as well as classical elasticity. Other than the total stress tensor, all of the field quantities are regularized within second-grade theories, while first-grade theories give singular double stress and dislocation density and classical elasticity leads to singularity in the stress field and dislocation density. Similar to gradient elasticity of Helmholtz type (first strain gradient elasticity), crack tip plasticity is captured in gradient elasticity of bi-Helmholtz type without any assumption of the cohesive zone. [ABSTRACT FROM AUTHOR]

The H-matrices are an important class in the matrix theory, and have many applications. Recently, this concept has been extended to higher order ℋ-tensors. In this paper, we establish important properties of diagonally dominant tensors and ℋ-tensors. Distributions of eigenvalues of nonsingular symmetric ℋ-tensors are given. An ℋ-tensor is semi-positive, which enlarges the area of semi-positive tensor from M-tensor to ℋ-tensor. The spectral radius of Jacobi tensor of a nonsingular (resp. singular) ℋ-tensor is less than (resp. equal to) one. In particular, we show that a quasi-diagonally dominant tensor is a nonsingular ℋ-tensor if and only if all of its principal sub-tensors are nonsingular ℋ-tensors. An irreducible tensor A is an ℋ-tensor if and only if it is quasi-diagonally dominant. [ABSTRACT FROM AUTHOR]

In the present paper, the dislocation-based antiplane fracture mechanics is employed for the analysis of Mode III crack within nonlocal and (strain) gradient elasticity of bi-Helmholtz type. These frameworks are appropriate candidates of generalized continua for regularization of classical singularities of defects such as dislocations. Within nonlocal elasticity of bi-Helmholtz type, nonlocal stress is regularized, while the strain field remain singular. Interestingly, gradient elasticity of bi-Helmholtz type (second strain gradient elasticity) eliminates all physical singularities of discrete dislocation including stress and strain fields and dislocation density while the so-called total stress tensor still contains singularity at the dislocation core. Based on the distribution of dislocations, a fracture theory with nonsingular stress field is formulated in these nonlocal and gradient theories. Strain and displacement fields within nonlocal fracture theory are identical to the classical ones. In contrast, gradient elasticity of bi-Helmholtz type leads to a full nonsingular fracture theory in which stress, strain and dislocation density are regularized. However, the singular total stress of a discrete dislocation results in singular total stress of the plane weakened by a crack. Within classical fracture mechanics, Barenblatt’s cohesive fracture theory assumes that cohesive forces is distributed ahead of the crack tip to model crack tip plasticity and remove the stress singularity. Here, considering the dislocations as the carriers of plasticity, the crack tip plasticity is captured without any assumption. Once the crack is modeled by distributing the dislocations along its surface, due to the gradient theory, the distribution function gives rise to a non-zero plastic distortion ahead of the crack. Consequently, regularized solutions of crack are developed incorporating crack tip plasticity. [ABSTRACT FROM AUTHOR]

We first show that the eigenvector of a tensor is well defined. The differences between the eigenvectors of a tensor and its E-eigenvectors are the eigenvectors on the nonsingular projective variety. We show that a generic tensor has no eigenvectors on. Actually, we show that a generic tensor has no eigenvectors on a proper nonsingular projective variety in. By these facts, we show that the coefficients of the E-characteristic polynomial are algebraically dependent. Actually, a certain power of the determinant of the tensor can be expressed through the coefficients besides the constant term. Hence, a nonsingular tensor always has an E-eigenvector. When a tensoris nonsingular and symmetric, its E-eigenvectors are exactly the singular points of a class of hypersurfaces defined byand a parameter. We give explicit factorization of the discriminant of this class of hypersurfaces which completes Cartwright and Strumfels’ formula. We show that the factorization contains the determinant and the E-characteristic polynomial of the tensoras irreducible factors. [ABSTRACT FROM PUBLISHER]

Abstract: This paper deals with a new type of matrices. The structure of this class of matrices is similar to circulant matrix, we call it H-circulant matrix. Furthermore, the sum, the difference, the product, the inverse and the adjoint matrix of these matrices are also H-circulant matrixes. We not only give six discriminance for H-circulant matrices but also discuss the diagonalization and nonsingularity of these matrices. Finally, author gives a numerical example. [Copyright &y& Elsevier]

The distributive property can be studied through bilinear maps and various morphisms between these maps. The adjoint-morphisms between bilinear maps establish a complete abelian category with projectives and admits a duality. Thus the adjoint category is not a module category but nevertheless it is suitably familiar. The universal properties have geometric perspectives. For example, products are orthogonal sums. The nonsingular bilinear maps are the simple bimaps with respect to nondegenerate adjoint-morphisms. That formalizes the understanding that the atoms of linear geometries are algebraic objects with no zero-divisors. Adjoint-isomorphism coincides with principal isotopism; hence, semifields can be studied within this framework. [ABSTRACT FROM PUBLISHER]

We investigate nonsingular hypercubes, and prove several results, stating a condition for a hypercube to be the product of hypercubes of smaller dimensions. There is a shortage of higher dimensional nonsingular hypercubes in the literature. However, we show that the product of two nonsingular hypercubes is always nonsingular. Then we show how to construct four-dimensional nonsingular hypercubes that are not the products of two three-dimensional hypercubes. It is noted that higher dimensional nonsingular hypercubes, that are not products of smaller ones, correspond to many semifields. [ABSTRACT FROM AUTHOR]

A ring is clean (almost clean) if each of its elements is the sum of a unit (regular element) and an idempotent. A module is clean (almost clean) if its endomorphism ring is clean (almost clean). We show that every quasi-continuous and nonsingular module is almost clean and that every right CS (i.e. right extending) and right nonsingular ring is almost clean. As a corollary, all right strongly semihereditary rings, including finite AW-algebras and noetherian Leavitt path algebras in particular, are almost clean. We say that a ring R is special clean (special almost clean) if each element a can be decomposed as the sum of a unit (regular element) u and an idempotent e with aR ∩ eR = 0. The Camillo-Khurana Theorem characterizes unit-regular rings as special clean rings. We prove an analogous theorem for abelian Rickart rings: an abelian ring is Rickart if and only if it is special almost clean. As a corollary, we show that a right quasi-continuous and right nonsingular ring is left and right Rickart. If a special (almost) clean decomposition is unique, we say that the ring is uniquely special (almost) clean. We show that (1) an abelian ring is unit-regular (equiv. special clean) if and only if it is uniquely special clean, and that (2) an abelian and right quasi-continuous ring is Rickart (equiv. special almost clean) if and only if it is uniquely special almost clean. Finally, we adapt some of our results to rings with involution: a *-ring is *-clean (almost *-clean) if each of its elements is the sum of a unit (regular element) and a projection (self-adjoint idempotent). A special (almost) *-clean ring is similarly defined by replacing 'idempotent' with 'projection' in the appropriate definition. We show that an abelian *-ring is a Rickart *-ring if and only if it is special almost *-clean, and that an abelian *-ring is *-regular if and only if it is special *-clean. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the structure of the endomorphism rings of nonsingular semiperfect progenerator uniform CS modules with uniform submodule. It is shown that such rings are direct sums of indecomposable right CS rings and a ring with no uniform right ideal. In particular, right nonsingular semiperfect right uniform CS rings with uniform right ideal are direct sums of indecomposable right CS rings and a ring with no uniform right ideal. Also it is shown that a progenerator R-module is uniform CS if and only if its endomorphism ring is right uniform CS. [ABSTRACT FROM AUTHOR]

There is growing notion that black holes may not contain curvature singularities (and that indeed nature in general may abhor such spacetime defects). This notion could have implications on our understanding of the evolution of primordial Black holes (PBHs) and possibly on their contribution to cosmic energy. This paper discusses the evolution of a nonsingular black hole (NSBH) based on a recent model [M. R. Mbonye and D. Kazanas, Phys. Rev. D. 72 (2005) 024016]. The model is used to discuss the time evolution of a primordial black hole (PBH), through the early radiation era of the universe to present, under the assumption that PBHs are nonsingular. In particular, we track the evolution of two benchmark PBHs, namely the one radiating up to the end of the cosmic radiation domination era, and the one stopping to radiate currently, and in each case determine some useful features including the initial mass mf and the corresponding time of formation tf. It is found that along the evolutionary history of the universe the distribution of PBH remnant masses (PBH-RM) PBH-RMs follows a power law. We believe such a result can be a useful step in a study to establish current abundance of PBH-MRs. [ABSTRACT FROM AUTHOR]

Abstract: An affine column independent matrix is a matrix whose entries are polynomials of degree at most 1 in a number of indeterminates where no indeterminate appears with a nonzero coefficient in two different columns. A completion is a matrix obtained by giving values to each of the indeterminates. Affine column independent matrices are more general than partial matrices where each entry is either a constant or a distinct indeterminate. We determine when the rank of all completions of an affine column independent matrix is bounded by a given number, generalizing known results for partial matrices. We also characterize the square partial matrices over a field all of whose completions are nonsingular. The maximum number of free entries in such matrices of a given order is determined as well as the partial matrices with this maximum number of free entries. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, a new efficient computational algorithm is presented for solving nearly penta-diagonal linear systems based on the use of any penta-diagonal linear solver. The implementation of the algorithm using computer algebra systems (CAS) such as MAPLE and MATLAB is straightforward. Numerical examples are given to illustrate the effectiveness of our method. [Copyright &y& Elsevier]

In this article we show, among others, that if R is a prime ring which is not a domain, then R is right nonsingular, right max-min CS with uniform right ideal if and only if R is left nonsingular, left max-min CS with uniform left ideal. The above result gives, in particular, Huynh et al. (2000) Theorem for prime rings of finite uniform dimension. [ABSTRACT FROM AUTHOR]

We investigate when modules which satisfy the descending (respectively, ascending) chain condition on non-essential submodules are uniform or Artinian (respectively, Noetherian). [ABSTRACT FROM AUTHOR]

In this paper, some properties of centrosymmetric matrices, which often appear in the construction of orthonormal wavelet basis in wavelet analysis, are investigated. [Copyright &y& Elsevier]

We consider the functional difference system ( A ) Δ x i ( n )= f i ( n ; X ), 1≤ i ≤ k , where X =( x 1 ,…, x k ) and f 1 (·; X ),…, f k (·; X ) are real-valued functionals of X , which may depend quite arbitrarily on values of X ( l ) for multiple values of l ∈ Z . We give sufficient conditions for ( A ) to have solutions that approach specified constant vectors as n →∞. Some of the results guarantee only that the solutions are defined for n sufficiently large, while others are global. The proof of the main theorem is based on the Schauder-Tychonoff theorem. Applications to specific quasi-linear systems are included. [ABSTRACT FROM AUTHOR]