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In a previous work the authors presented the necessary and sufficient conditions for the invertibility of some circulant matrices that depend on three real parameters and, moreover, we obtained a closed formula for their inverse. The techniques we used allowed us to reduce the computational cost of evaluating the inverse. In this work we complete the theoretical analysis of the mentioned circulant matrices by considering complex parameters and also the singular case. In addition, we explicitely compute the group inverse of a specific kind of this class of matrices using the same techniques that the authors developed for the non singular case; that is, solving first order linear difference equations.

We extend to the singular case the results in [El Bahja H. Regularity for anisotropic quasi-linear parabolic equations with variable growth. Electron J Differ Equ. 2019;2019(104):1–21] concerning t...

We present the singular Hardy-Trudinger-Moser inequality and the existence of their extremal functions on the unit disc \begin{document}$ B $\end{document} in \begin{document}$ \mathbb{R}^2 $\end{document} . As our first main result, we show that for any \begin{document}$ 0 and \begin{document}$ u \in C_0^\infty({B}) $\end{document} satisfying \begin{document}$ \int_{{B}}|\nabla u|^2 dx- \int_{{B}}\frac{u^2}{(1-|x|^2)^2}dx\leq1, $\end{document} there exists a constant \begin{document}$ C_{0}>0 $\end{document} such that the following inequality holds \begin{document}$ \int_{{B}}\frac{e^{4\pi(1-t/2)u^2}}{|x|^t} dx\leq C_{0}. $\end{document} Furthermore, by the method of blow-up analysis, we establish the existence of extremal functions in a suitable function space. Our results extend those in Wang and Ye [ 36 ] from the non-singular case \begin{document}$ t = 0 $\end{document} to the singular case for \begin{document}$ 0 .

In this paper, the modified SSOR (MSSOR) method is further investigated for solving augmented linear systems, including both the nonsingular and singular cases. The selection of quasi-optimal iteration parameters are discussed to minimize the radius of the MSSOR iteration matrix for the nonsingular case. The obtained optimality result is unified, which includes those for some existing methods as special cases. Besides, semi-convergence analysis of the MSSOR method for singular case is given using a singular preconditioning technique, which results in much better convergence behavior than some existing nonsingular preconditioning techniques. Numerical experiments are presented to confirm the theoretical results.

In this paper, we study the global regularity estimates in Lorentz spaces for gradients of solutions to quasilinear elliptic equations with measure data of the form \begin{eqnarray*} \left\{ \begin{array}{rcl} -{\rm div}(\mathcal{A}(x, \nabla u))&=& \mu \quad \text{in} ~\Omega, u&=&0 \quad \text{on}~ \partial \Omega, \end{array}\right. \end{eqnarray*} where $\mu$ is a finite signed Radon measure in $\Omega$, $\Omega \subset \mathbb{R}^n$ is a bounded domain such that its complement $\mathbb{R}^n\backslash\Omega$ is uniformly $p$-thick and $\mathcal{A}$ is a Carath\'eodory vector valued function satisfying growth and monotonicity conditions for the strongly singular case $1

The paper is devoted to the implicit function theorem involving singular mappings.We also discuss the form of the tangent cone to the solution set of the generalized equations in singular case and give some examples of applications to nonlinear programming and complementarity problems., Comment: 11 pages

Stable inversion was an effective method to achieve precise trajectory tracking, for which the tracking problem of periodically time-varying systems was solved in the existing literature. However, iteration techniques were required to approach stable inversion, whose explicit formulas for this tracking problem cannot be obtained. To overcome this drawback, a special kind of stable inversion, named periodic inversion, for the periodic systems is proposed in this study. By means of the lifted technique, the tracking problem of the periodic systems is reformulated to be equivalent to that of lifted systems. In order to obtain precise tracking, the periodic inversion of the periodic systems or the lifted systems is proposed, and is analysed for the non-singular case and the singular case, each of which is further analysed for three cases: the full row rank case, the full column rank case and the rank-deficient case. Thus the periodic inversion of the periodic systems is computed. Accompanied with the optimal state transition method for the time-varying systems, precise trajectory tracking is achieved. The methodology is validated through simulations.

In a previous work (Adimurthi and Yang, 2010 [2] ), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space R N ( N ≥ 2 ) . Precisely, if 0 ≤ β 1 and 0 γ ≤ 1 − β , then there holds for any τ > 0 , sup u ∈ W 1 , N ( R N ) , ∫ R N ( | ∇ u | N + τ | u | N ) d x ≤ 1 ⁡ ∫ R N 1 | x | N β ( e α N γ | u | N N − 1 − ∑ k = 0 N − 2 α N k γ k | u | k N N − 1 k ! ) d x ∞ , where α N = N ω N − 1 1 / ( N − 1 ) and ω N − 1 is the area of the unit sphere in R N . The above inequality is sharp in the sense that if γ > 1 − β , all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case β = 0 has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11] . We shall investigate the singular case 0 β 1 and prove that for all τ > 0 , 0 β 1 and 0 γ ≤ 1 − β , extremal functions for the above inequalities exist. The proof is based on blow-up analysis.

Recently, identifiability results for VAR systems in the context of mixed frequency data have been shown in a number of papers. These results have been extended to VARMA systems, where the MA order is smaller than or equal to the AR order. Here, it is shown that in the VMA case and in the VARMA case, where the MA order exceeds the AR order, results are completely different. Then, for the case, where the innovation covariance matrix is non-singular, “typically” non-identifiability occurs – not even local identifiability. This is due to the fact that, e.g., in the VMA case, as opposed to the VAR case, the not directly observed autocovariances of the output can vary “freely”. In the singular case, i.e., when the innovation covariance matrix is singular, things may be different.

The present paper deals with a linear integral equation of the third kind with fixed singularities in its kernel. We propose and substantiate special generalized methods for its approximate solving in a space of generalized funtions.

In this paper we consider two linear Hamiltonian differential systems on an open unbounded interval. We assume that the systems are completely controllable and satisfy the Sturmian majorant condition and the Legendre condition. We derive a singular Sturmian comparison theorem for the case when the minorant system has a solution, which is principal at both endpoints of the considered interval. The main result is new even for the second order differential equations and it generalizes the singular comparison theorem obtained by Aharonov and Elias (2010). We illustrate our new theory by several examples.

In a recent paper a new iterative method for the solution of the nonsingular symmetric saddle point problem was proposed (Njeru and Guo, 2016). The ASOR method belongs to the family of the SOR-like methods and uses two parameters α and ω . Convergence intervals for the parameters involved were found. In the present work we analyze and study an extension of the above problem to the singular case, and determine optimal values for the two parameters as well as for the semi-convergence factor of the ASOR method. Numerical results are presented to show the efficiency of the optimal singular ASOR method.

We study regularity results for nonlinear parabolic systems of p-Laplacian type with inhomogeneous boundary and initial data, with $$p\in (\frac{2n}{n+2},\infty )$$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calderon–Zygmund theory for such systems. This extends the global result of Bogelein (Calc Var Partial Differ Equ 51(3–4):555–596, 2014) to very rough domains and more general boundary values. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integrability results. Technically interesting is the fact that our parabolic estimates have no scaling deficit with respect to the leading order term. Moreover, in the singular case, $$p\in (\frac{2n}{n+2},2]$$, any scaling deficit can be omitted.

For a Riemannian foliation F on a compact manifold M , J. A. \'Alvarez L\'opez proved that the geometrical tautness of F , that is, the existence of a Riemannian metric making all the leaves minimal submanifolds of M, can be characterized by the vanishing of a basic cohomology class (the \'Alvarez class). In this work we generalize this result to the case of a singular Riemannian foliation K on a compact manifold X. In the singular case, no bundle-like metric on X can make all the leaves of K minimal. In this work, we prove that the \'Alvarez classes of the strata can be glued in a unique global \'Alvarez class. As a corollary, if X is simply connected, then the restriction of K to each stratum is geometrically taut, thus generalizing a celebrated result of E. Ghys for the regular case., Comment: 24 pages, 2 figures. Accepted for publication in RACSAM

Let s be arbitrary integer, we introduce the notion of the matrix of type s, whose nonzero entries are the classical Horadam numbers . In this paper we consider singular case , then the Moore–Penrose inverse of the matrix is given. In the case , we obtain the Pseudoinverse of the generalized Fibonacci matrix . In addition, correlations between the matrix and the generalized Pascal matrices are discussed, and some combinatorial identities involving the Horadam numbers are derived.

This paper presents global existence and boundedness of classical solutions to the fully parabolic chemotaxis system u t = Δ u − ∇ ⋅ ( u χ ( v ) ∇ v ) , v t = Δ v − v + u with the strongly singular sensitivity function χ ( v ) such that 0 χ ( v ) ≤ χ 0 v k ( χ 0 > 0 , k > 1 ) . As to the regular case 0 χ ( v ) ≤ χ 0 ( 1 + α v ) k ( α > 0 , χ 0 > 0 , k > 1 ) , it has been shown, by Winkler (2010), that the system has a unique global classical solution which is bounded in time, whereas this method cannot be directly applied to the singular case. In the present work, a uniform-in-time lower bound for v is established and builds a bridge between the regular case as in Winkler (2010) and the singular one.

Recently, Bai and Wang (2008) [9] presented an efficient accelerated parameterized inexact Uzawa (APIU) method for solving both symmetric and nonsymmetric nonsingular saddle point problems. In this work, we propose a modified APIU (MAPIU) method for solving saddle point problems, which is an extension of the APIU iteration method and covers a series of existing iterative methods. The MAPIU method can be applied not only to the nonsingular case but also to the singular case. The convergence properties for the nonsingular saddle point problems and the semi-convergence properties for the singular one are carefully discussed under suitable restrictions. We prove that the MAPIU method has a wider convergence (or semi-convergence) region than the APIU method.

General (i.e. including non-continuous and non-Archimedean) absolute planes have been classified in different ways, e.g. by using Lambert–Saccheri quadrangles (cf. Greenberg, J Geom 12/1:45-64, 1979; Hartshorne, Geometry; Euclid and beyond, Springer, Berlin, 2000; Karzel and Marchi, Le Matematiche LXI:27–36, 2006; Rostamzadeh and Taherian, Results Math 63:171–182, 2013) or coordinate systems (cf. Pejas, Math Ann 143:212–235, 1961 and, for planes over Euclidean fields, Greenberg, J Geom 12/1:45-64, 1979). Here we consider the notion of quasi-end, a pencil determined by two lines which neither intersect nor have a common perpendicular (an ideal point of Greenberg, J Geom 12/1:45-64, 1979). The cardinality ω of the quasi-ends which are incident with a line is the same for all lines hence it is an invariant \({\omega_\mathcal{A}}\) of the plane \({\mathcal{A}}\) and can be used to classify absolute planes. We consider the case \({\omega_\mathcal{A}=0}\) and, for \({\omega_\mathcal{A} \geq 2}\) (it cannot be 1) we prove that in the singular case \({\omega_\mathcal{A}}\) must be infinite. Finally we prove that for hyperbolic planes, ends and quasi-ends are the same, so \({\omega_\mathcal{A}=2}\) .

This paper is concerned with the parabolic Keller–Segel system { u t = ∇ ⋅ ( ∇ u − u m ∇ v ) in Ω × ( 0 , T ) , Γ v t = Δ v − λ v + u in Ω × ( 0 , T ) , in a domain Ω of R N with N ⩾ 1 , where m , Γ > 0 , λ ⩾ 0 are constants and T > 0 . When Ω ≠ R N , we impose the Neumann boundary conditions on the boundary. Under suitable assumptions, we prove the local nondegeneracy of blow-up points. This seems new even for the classical Keller–Segel system ( m = 1 ). Lower global blow-up estimates are also obtained. In the singular case 0 m 1 , as a prerequisite, local existence and regularity properties are established.

This work concerns a new methodology to solve the ℋ2-optimal disturbance rejection problem by measurement feedback in the singular case: namely, when the plant has no feedthrough terms from the control input and the disturbance input to the controlled output and the measured output, respectively. A necessary and sufficient condition for problem solvability is expressed as the inclusion of two subspaces-a controlled-invariant subspace and a conditioned-invariant subspace. Such subspaces are directly derived from the Hamiltonian systems associated to the ℋ2-optimal control problem and, respectively, to the ℋ2-optimal filtering problem. The proof of sufficiency, which is constructive, provides the computational tools for the synthesis of the feedback regulator. A numerical example is worked out in order to illustrate how to implement the devised procedure.

The Dirichlet problem for the degenerate and singular parabolic p(x)-Laplace equation with one spatial variable is considered. We prove the existence of the unique weak solution such that the derivatives ut and ux of a solution u belong to \({L_{\infty}}\). Moreover for the singular case we prove the existence of the strong solution i.e. such that ut, ux and uxx belong to \({L_{\infty}}\).

We study the perturbation by a critical term and a ▫$(p-1)$▫-superlinear subcritical nonlinearity of a quasilinear elliptic equation containing a singular potential. By means of variational arguments and a version of the concentration-compactness principle in the singular case, we prove the existence of solutions for positive values of the parameter under the principal eigenvalue of the associated singular eigenvalue problem. Obravnavamo perturbacijo kritičnega člena in ▫$(p-1)$▫-superlinearno subkritično nelinearnost kvazilinearne eliptične enačbe s singularnim potencialnom. Z variacijskimi argumenti in verzijo principa koncentracijske kompaktnosti v singularnem primeru dokažemo obstoj rešitev za pozitivne vrednosti parametra pod glavno lastno vrednostjo prirejenega singularnega problema lastne vrednosti.

Of concern is the nonlinear hyperbolic problem with nonlinear dynamic boundary conditions $$\left\{ \begin{array}{lll} u_{tt} ={\rm div} (\mathcal{A} \nabla u)-\gamma (x,u_t), && \quad {\rm in} \; (0, \infty) \times \Omega,\\ u(0, \cdot)=f, \, u_t(0,\cdot)=g, && \quad {\rm in}\; \Omega, \\ u_{tt} + \beta \partial^ \mathcal{A}_\nu u+c(x)u+ \delta (x,u_t)-q \beta \Lambda_{\rm LB} u=0,&& \quad {\rm on} \;(0, \infty ) \times \partial \Omega . \end{array}\right. $$ for t ≥ 0 and \({x \in \Omega \subset \mathbb{R}^N}\) ; the last equation holds on the boundary ∂Ω. Here \({\mathcal{A}= \{a_{ij}(x)\}_{ij}}\) is a real, hermitian, uniformly positive definite N × N matrix; \({\beta \in C(\partial \Omega)}\) , with β > 0; \({\gamma:\Omega \times \mathbb{R} \to \mathbb{R}; \delta:\partial \Omega \times \mathbb{R} \to \mathbb{R}; \,c:\partial \Omega \to \mathbb{R}; \, q \ge 0, \Lambda_{\rm LB}}\) is the Laplace–Beltrami operator on ∂Ω, and \({\partial^\mathcal{A}_\nu u}\) is the conormal derivative of u with respect to \({\mathcal{A}}\) ; everything is sufficiently regular. We prove explicit stability estimates of the solution u with respect to the coefficients \({\mathcal{A},\,\beta,\,\gamma,\,\delta,\,c,\,q}\) , and the initial conditions f, g. Our arguments cover the singular case of a problem with q = 0 which is approximated by problems with positive q.

A unifying theory of plasticity is developed that allows deducing models with either hypo-plastic, elasto-plastic or hyper-plastic constitutive structures. Elasto-plasticity is shown to be a singular case of hypo-plasticity. It is further demonstrated that certain conditions can be imposed to generate a new hierarchy of thermodynamically consistent hypo-plastic models, with a hyper-plastic structure as a singular case. The unifying theory is powerfully bridging between the tools created specifically for advancing models under either of those previous formulations.

In this paper, a simple parasite-host model proposed by Ebert et al.(2000) is reconsidered. The basic epidemiological reproduction number of parasite infection ($R_0$) and the basic demographic reproduction number of infected hosts ($R_1$) are given. The global dynamics of the model is completely investigated, and the existence of heteroclinic and homoclinic orbits is theoretically proved, which implies that the outbreak of parasite infection may happen. The thresholds determining the host extinction in the presence of parasite infection and variation in the equilibrium level of the infected hosts with $R_0$ are found. The effects of $R_0$ and $R_1$ on dynamics of the model are considered and we show that the equilibrium level of the infected host may not be monotone with respect to $R_0$. In particular, it is found that full loss of fecundity of infected hosts may lead to appearance of the singular case.

We study the existence of positive bound states of non-autonomous systems of nonlinear Schrodinger equations. Both the singular case and the regular case are discussed. The proof is based on a nonlinear alternative principle of Leray–Schauder.

In this paper, we study the second-order nonlinear singular Sturm-Liouville boundary value problems with Riemann-Stieltjes integral boundary conditions \begin{equation*}\begin{cases} -(p(t)u'(t))'+q(t)u(t)=f(t,u(t)),\; 0

We consider the system of Fredholm integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{T}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$ and also the system of Volterra integral equations $$\begin{array}{l}\displaystyle u_{i}(t)=\int_{0}^{t}g_{i}(t,s)[h_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))+k_{i}(s,u_{1}(s),u_{2}(s),\ldots,u_{n}(s))]ds,\\[8pt]\quad t\in[0,T],~1\leq i\leq n\end{array}$$ where T>0 is fixed and the nonlinearities h i (t,u 1,u 2,…,u n ) can be singular at t=0 and u j =0 where j∈{1,2,…,n}. Criteria are offered for the existence of constant-sign solutions, i.e., θ i u i (t)≥0 for t∈[0,1] and 1≤i≤n, where θ i ∈{1,−1} is fixed. We also include examples to illustrate the usefulness of the results obtained.

In the present paper we consider the Dirichlet problem for one-dimensional p-Laplacian with nonlinear source. We obtain new a priori estimates of a solution and of the gradient of a solution and formulate conditions guaranteeing the global solvability of this problem. Our consideration includes singular case as well.

Biological feedbacks play a crucial role in determining effects of toxicants, radiation, and other environmental stressors on organisms. Focusing on reactive oxygen species (ROS) that are increasingly recognized as a crucial mediator of many stressor effects, we investigate how feedback strength affects the ability of organisms to control negative effects of exposure. We do this by developing a general theoretical framework for describing effects of a wide range of stressors and species. The framework accounts for positive and negative feedbacks representing cellular processes: (i) production of ROS due to metabolism and the stressor, (ii) ROS reactions with cellular compounds that cause damage, and (iii) cellular control of both ROS and damage. We suggest functional forms that capture generic properties of cellular control mechanisms constituting the feedbacks, assess stability of equilibrium states in the resulting models, and investigate tipping points at which cellular control breaks down causing unregulated increase of ROS and damage. Depending on the chosen functional forms, the models can have zero, one, or two positive steady states ; except in one singular case, the steady state with lowest values of ROS and damage is locally stable. We found two types of tipping points: those induced by changes in the parameters (including the stressor intensity), and those induced by the history of exposure, i.e. ROS and damage levels. The relatively simple models effectively describe several patterns of cellular responses to stress, such as the covariation of ROS and damage, the break-down of cellular control leading to explosion of ROS and/or damage, increase in damage even when ROS is (near)-constant, and the effects of exposure history on the ability of the cell to handle additional stress. The models quantify dynamics of cellular control, and could therefore be used to estimate the metabolic costs of stress and help integrate them into models that use energetic considerations to model organism's response to the environment. Although developed with unicellular organisms in mind, our models can be applied to all multicellular organisms that utilize similar feedbacks when dealing with stress.

We consider the following system of difference equations where μ>0, T≥N>m≥0, the function f may take negative values and f( \cdot ,u_{1},u_{2}, \dddot ,u_{n}) may be singular at u_{j} = 0, j \in \{ 1,2, \dddot ,n \} . Our aim is to establish criteria such that the above system has a constant-sign solution. To illustrate the generality of the results obtained, application to a well known boundary value problem is included. We also extend the above problem to that on {\rm \shade N = \{ 0,1, \dddot \} }

The problem of stabilizability of 2-D singular systems is considered, a necessary and sufficient condition in terms of the existence of a polynomial solution to a Bezout identity has been generalized from 2-D regular systems to the singular case. The BIBO stability condition has been extended to 2-D singular systems. The detectability of 2-D singular system is further discussed, an equivalent condition is obtained. At last it is shown that 2-D singular system is internally stable if and only if it is BIBO stable, detectable and stabilizable.

Here we show that the surjectivity of a natural map such as a Gaussian map for curves or the Zak map in all dimensions forces a projective variety to have only trivial extensions also in the singular case.

In this article we analyze the effect of mass-lumping in the linear triangular finite element approximation of second-order elliptic eigenvalue problems. We prove that the eigenvalue obtained by using mass-lumping is always below the one obtained with exact integration. For singular eigenfunctions, as those arising in non convex polygons, we prove that the eigenvalue obtained with mass-lumping is above the exact eigenvalue when the mesh size is small enough. So, we conclude that the use of mass-lumping is convenient in the singular case. When the eigenfunction is smooth several numerical experiments suggest that the eigenvalue computed with mass-lumping is below the exact one if the mesh is not too coarse. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 19: 653–664, 2003

A new wave of applications of the dynamic Leontief model brought into the forefront the singularity problem of the capital matrix. In these applications the singularity of the capital matrix is a common occurrence which complicates the solution of the model. In the singular case the model cannot be transformed in a direct forward recursive form. The method presented in this paper determines the length of a backward system (τ). Several applications stop at observing singularity while referring to the theoretical possibility of the solution. In particular, the singularity of the capital matrix played a prominent role in Bródy’s extensive contributions to the input-output literature but he never ventured into the details of its various solutions. We demonstrate that a number of papers dealing with the Leontief model with singular capital matrix based their solutions on similar regularity assumptions. Our formulation in this paper offers a brief overview of the approaches that can be followed in a wide range of applications confronting with the singularity problem.

By use of the Cerami-Palais-Smale condition, we generalize the classical Weierstrass minimizing theorem to the singular case by allowing functions which attain infinity at some values. As an application, we study certain singular second-order Hamiltonian systems with strong force potential at the origin and show the existence of new periodic solutions with fixed periods.

In this paper, we study the classical problem of the exponentially small splitting of separatrices of the rapidly forced pendulum. Firstly, we give an asymptotic formula for the distance between the perturbed invariant manifolds in the so-called singular case and we compare it with the prediction of Melnikov theory. Secondly, we give exponentially small upper bounds in some cases in which the perturbation is bigger than in the singular case and we give some heuristic ideas how to obtain an asymptotic formula for these cases. Finally, we study how the splitting of separatrices behaves when the parameters are close to a codimension-2 bifurcation point.

Beauwens' procedure for obtaining lower eigenvalue bounds for (regular) pencils of matrices A − γB is simplified and extended to the singular case. The theory is then compared, through a particular perturbed modified incomplete factorization, with Notay's generalization of another approach, initiated by Gustafsson, and developed by Axelsson and Barker and by Wilmet.

Positive solutions are established for the Volterra integral equation y(t) = ∫ t 0 k(t, s) f(s,y(s)) ds, t ∈[0,T]. Our nonlinearity may be singular at y = 0.

In this paper the robust finite-horizon H 2 estimation problem of a linear time-varying system is addressed in a very general way in order to face the finite-horizon filtering problem and, consequently, the fixed-lag forward-backward non-causal estimation problem. The quantity to be estimated is a generic linear combination of the state, of the input and of a suitable signal representing the uncertainty. In such a way the deconvolution problem and some type of fault detection problems are treated as particular cases. Furthermore, the so-called singular case is treated as well.

A method is proposed for the determination of singular points in the problem of plane-parallel motion of a wheeled vehicle (automobile, aircraft on the runway, etc) which is valid for any arbitrary dependence of lateral reactions on the slip angle. A combination of this method with the Poincare theory of indices enables us to determine the nature of behavior of the zero solution of the equation of perturbed motion. It is shown that in the case of arbitrary law of lateral slip the loss of coordinate origin stability corresponds either to the creation of a multiple singular point or to the merging of singular points there. The singular case of cubic approximation of lateral reactions is considered.

Sturmian comparison theorems are obtained for solutions of a class of normal and singular ultrahyperbolic partial differential equations. In the singular case, solutions are considered which satisfy non-standard boundary conditions.

Wir befassen uns mit der numerischen Losung von parameterabhangigen singularen Randwertaufgaben fur Systeme gewohnlicher Differentialgleichungen 2. Ordnung. Wir zeigen, das die bekannten Prinzipien der Kurvenverfolgung auch auf die singularen Probleme anwendbar sind; wir schreiben die singulare Randwertaufgabe als eine nichtlineare Operatorgleichung und zeigen die Fredholm-Eigenschaft ihrer Linearisierung. Wir geben zwei Algorithmen zur Kurvenverfolgung, basierend auf Kollokationsverfahren und Differenzenverfahren, an und verweisen auf die Ergebnisse der Rechnungen fur Probleme, die das Beulverhalten von dunnwandigen Kugelschalen beschreiben. We discuss the numerical solution of parameterdependent singular boundary value problems for systems of ordinary differential equations of second order. Applicability of the well-known pathfollowing principles is shown for the singular case; we rewrite the singular boundary value problem as a nonlinear operator equation and show the Fredholm-property of its linearization. We present pathfollowing procedures based on collocation and finite differences and refer to the computational results for problems from the buckling theory of thinwalled spherical shells.

Continuous and discontinuous one-dimensional waves of stationary profile are studied in non-linearly hereditary rods, where the heredity kernel is not assumed to be regular (i.e., the existence of a singularity is allowed for in the kernel, an integrable singularity as t → +0). Appropriate near-front asymptotic forms are found. As is well-known /1/, strong discontinuities cannot propagate in hereditary media with singular kernels in a linear situation. Consequently, the very possibility of the existence of solutions with strong discontinuities in the non-linear singular case is not totally trivial.

SYSTEMS of integral equations in convolutions are considered in certain cases when the sets of values of the corresponding operators are not closed, and hence the systems represent ill-posed problems. Regularizing operators [1] are constructed for the systems. The construction is based on the solution of systems of equations of normal type, close to the given systems.

A critical case of m zero roots, n pairs of pure imaginary roots and q roots with negative real parts in which m groups of solutions correspond to the m-tuple zero root and no integral relations exist between the pure imaginary roots is considered, and a case in which asymptotic stability is impossible is singled out. The Kamenkov theorem on stability is extended to an essentially singular case. A theorem on stability for the nonessentially singular cases is established for the critical case of a single zero root, arbitrary number of pairs of pure imaginary roots and roots with negative real parts.

Recent works have shown that, whenA is a Stieltjes matrix, its so-called modified incomplete factorizations provide effective preconditioning matrices for solvingAx=b by polynomially accelerated iterative methods. We extend here these results to the singular case with the conclusion that the latter techniques are able to solve singular systems at the same speed as regular systems.