Higher order global solution and normalized flux for singularly perturbed reaction-diffusion problems

In this paper a computational technique is proposed for obtaining a higher order global solution and global normalized flux of singularly perturbed reaction-diffusion two-point boundary-value problems. The HOC (higher order compact) finite difference scheme developed in Gracia et al. (2001) [4] and which is constructed on an appropriate piecewise uniform Shishkin mesh, has been considered to find an almost fourth order convergent solution at mesh points. Using these values, piecewise cubic interpolants based approximations for solution and normalized flux in whole domain have been defined. It has been shown that the global solution and the global normalized flux are also uniformly convergent. Moreover, for the global solution, the order of uniform convergence in the whole domain is optimal, i.e., it is the same as this one obtained at mesh points, whereas, for the global normalized flux, the uniform convergence is almost third order, except at midpoints of the mesh, where it is also almost fourth order. Theoretical error bounds have been provided along with some numerical examples, which corroborate the efficiency of the proposed technique to find good approximations to the global solution and the global normalized flux.     

A Nonlinear Parabolic Equation with Noise

In this paper we study a class of parabolic equations with a nonlinear gradient term. The system is disturbed by white noise in time. We show that the unique solution of this problem can be represented as the Wick product between a normalized random variable of exponential form and the solution of a nonlinear parabolic equation. We allow random initial data which might be anticipating. A relation between the Wick product with a normalized exponential and translation is proved in order to establish our results.     

Ricci Flow on Surfaces with Conical Singularities

This paper studies the normalized Ricci flow on surfaces with conical singularities. It’s proved that the normalized Ricci flow has a solution for a short time for initial metrics with conical singularities. Moreover, the solution makes good geometric sense. For some simple surfaces of this kind, for example, the tear drop and the football, it’s shown that they admit a Ricci soliton metric.     

An estimate of the curvature of level curves in a class of univalent functions

In this note a class of normalized functions which are univalent in the unit disk is considered. An exact solution of the problem of an upper estimate of curvature of level curves for functions of this class is obtained. In particular, an exact solution of this problem for univalent functions with real coefficients at the points of the interval (0, 1) is obtained.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 381–388, March, 1976.The author is thankful to the referee for taking interest in this note.     

Starshaped Compact Hypersurfaces with Prescribed M-th Mean Curvature in Elliptic Space

We consider the problem of finding a compact starshaped hypresurface in a space form for which the normalized m-th elementary symmetric function of principal curvatures is a prescribed function. In this paper the conditions for the existence of at least one solution to a nonlinear second order elliptic equation of that problem are established in case of a space form with positive sectional curvature.     

Computer extension and analytic continuation of problems of porous thrust bearings

Summary In this paper we propose new type of series solution, the semi-analytical, semi-numerical technique for the steady flow of a viscous fluid between two parallel disks in which the fluid is injected through the lower porous disk. We develop a double series expansion of the solution function and sufficiently large number of terms (30 terms-universal coefficients) in the expansion are obtained by delegating routine complex algebra to computer. The expression obtained in the form of power series for the normalized lift is analysed by means of Padé approximants. The change of roles of dependent and independent variables and the use of bilinear Euler transformation increases the region of validity of the corresponding power series. The results obtained from double series expansion method for small as well as moderately large values of cross-flow Reynolds number,R are more accurate and the computing time required in this method is negligible compared with pure numerical methods [1]. Besides this, we find that the method proposed by Phan-Thien and Bush [2] has to be implemented for each value ofR separately, whereas the one proposed by us has advantage of yielding, at a stretch, the results for larger range ofR which agree with exact values and at the same time requires less computer time. It is of interest to note that the pure numerical results in respect of normalized lift coefficient agree closely with the analytic continuation of our findings. In addition various Padé approximants are found to bracket the numerical results.     

厚壁圆柱壳开孔应力集中问题的复变函数解法

本文基于考虑横向剪切变形影响的厚壳理论建立了求解圆柱壳开孔应力集中问题的复变函数方法，得到了此种问题的一般解和满足任意形开孔边界条件的表达式·该应力集中问题可以简化为求解无穷代数方程组的问题·用复变函数方法可以规范地求解应力集中问题·文中给出了圆柱壳开小圆孔和椭圆孔时应力集中系数的数值结果·     

The logarithmic least squares and the generalized pseudoinverse in estimating ratios

This paper concerns the pairwise-comparison method used in Analytic Hierarchy Process (AHP). The logarithmic least square method is one of the methods used to rank a finite number of stimuli based on their pairwise-comparison. In the case of one decision-maker the problem can be solved using the geometric mean method. It is then assumed that the solution is geometrically normalized. In the case of multiple decision makers a set of linear equations is obtained and if we have a different number of judgments for each pair of the compared objects the geometric normalization assumption can not be used directly. The aim of this paper is to show that applying the generalized pseudoinverse we obtain the solution that is geometrically normalized and consistent with the case of one decision maker. To define the pseudoinverse the spectral decomposition is used. The structure of the general solution is presented and the existence of the general solution is discussed.     

一类矩阵方程的非奇异解

利用分块矩阵的等价标准形讨论了矩阵方程Am×nXn×nBn×l=Cm×l有解的充分必要条件,给出了一般解的表达式.在此基础上,进一步讨论了这类矩阵方程有非奇异解的条件,并且给出非奇异解的一般表达形式.     

Improved approximations of the solution and derivatives to a singularly perturbed reaction-diffusion equation based on the solution decomposition method

In the case of the Dirichlet problem for a singularly perturbed ordinary differential reaction-diffusion equation, a new approach is used to the construction of finite difference schemes such that their solutions and their normalized first- and second-order derivatives converge in the maximum norm uniformly with respect to a perturbation parameter ɛ ∈(0, 1]; the normalized derivatives are ɛ-uniformly bounded. The key idea of this approach to the construction of ɛ-uniformly convergent finite difference schemes is the use of uniform grids for solving grid subproblems for the regular and singular components of the grid solution. Based on the asymptotic construction technique, a scheme of the solution decomposition method is constructed such that its solution and its normalized first- and second-order derivatives converge ɛ-uniformly at the rate of O(N ⁻²ln² N), where N + 1 is the number of points in the uniform grids. Using the Richardson technique, an improved scheme of the solution decomposition method is constructed such that its solution and its normalized first and second derivatives converge ɛ-uniformly in the maximum norm at the same rate of O(N ⁻⁴ln⁴ N).     

Optimization reformulations of the generalized Nash equilibrium problem using regularized indicator Nikaidô–Isoda function

In this paper, we extend the literature by adapting the Nikaidô–Isoda function as an indicator function termed as regularized indicator Nikaidô–Isoda function, and this is demonstrated to guarantee existence of a solution. Using this function, we present two constrained optimization reformulations of the generalized Nash equilibrium problem (GNEP for short). The first reformulation characterizes all the solutions of GNEP as global minima of the optimization problem. Later this approach is modified to obtain the second optimization reformulation whose global minima characterize the normalized Nash equilibria. Some numerical results are also included to illustrate the behaviour of the optimization reformulations.     

Two-parameter asymptotic approximations in the analysis of a thin solid fixed on a small part of its boundary

Planar elasticity problems are considered for thin domains fixedalong a small part of the end region boundary. The analysisinvolves two small parameters: the normalized thickness of thebody and the normalized length of the fixed part of the boundary.The aim of the paper is to derive an asymptotic approximationof the solution to a boundary-value problem in such a domainand, in particular, analyze the ‘effective boundary conditions’,which occur for the leading-order terms of the asymptotics.We include applications for problems of both anti-plane shearand plane strain elasticity.     

r-regular shape reconstruction from unorganized points

In this paper, the problem of reconstructing a surface, given a set of scattered data points is addressed. First, a precise formulation of the reconstruction problem is proposed. The solution is mathematically defined as a particular mesh of the surface called the normalized mesh. This solution has the property to be included inside the Delaunay graph. A criterion to detect faces of the normalized mesh inside the Delaunay graph is proposed. This criterion is proved to provide the exact solution in 2D for points sampling a r-regular shapes with a sampling path < sin(π/8)r. In 3D, this result cannot be extended and the criterion cannot retrieve every face. A heuristic is proposed in order to complete the surface.     

On a new edge function on complete weighted graphs and its application for locating Hamiltonian cycles of small weight

In this paper, we define a new combinatorial function on the edges of complete weighted graphs. This function assigns to each edge of the graph the sum of the weights of all Hamiltonian cycles that contain the edge. Since this function involves the factorial function, whose exact calculation is intractable due to its superexponential asymptotic rate of increase, we also introduce a normalized version of the function that is efficiently computable. From this version, we derive an upper bound to the weight of the minimum weight Hamiltonian cycle of the graph based on the weights of the graph edges. Then we investigate the possible algorithmic applications of this normalized function using the Nearest Neighbor Heuristic and a smallest edge first heuristic. As evidence for its applicability, we show that the use of this function as a criterion for the selection of the next edge, improves the performance of both heuristics for approximating the minimum weight Hamiltonian cycles in Euclidean plane graphs. Moreover, our experimental results show that the use of the function is more suitable with the structure of the smallest edge first heuristic since it provides a solution closer to the best known solution of known hard TSP instances but in \(O(n^3)\) time.     

Analytical estimation of liquid film thickness in two-phase annular flow using electrical resistance measurement

This paper presents an analytical solution to estimate the liquid film thickness in two-phase annular flow through a circular pipe using electrical resistance tomography. Gas–liquid flow with circular gas core surrounded by a liquid film is considered. Conformal mapping is employed to obtain the analytic solution for annular flow with an eccentric circular gas core. The liquid film thickness for an arbitrary annular flow is estimated by comparing the resistance values for concentric and eccentric annular flows. The film thickness estimation has a good performance when the normalized distance between the gas core center and the flow center is less than 0.2 and the void fraction is greater than 0.4, the estimated error of the normalized thickness is less than 0.04.     

The Path-Connectivity of s-Elementary Frame Wavelets with Frame MRA

MRA wavelets have been widely studied in recent years due to their applications in signal processing. In order to understand the properties of the various MRA wavelets, it makes sense to study the topological structure of the set of all MRA wavelets. In fact, it has been shown that the set of all MRA wavelets (in any given dimension with a fixed expansive dilation matrix) is path-connected. The current paper concerns a class of functions more general than the MRA wavelets, namely normalized tight frame wavelets with a frame MRA structure. More specifically, it focuses on the parallel question on the topology of the set of all such functions (in the given dimension with a fixed dilation matrix): is this set path-connected? While we are unable to settle this general path-connectivity problem for the set of all frame MRA normalized tight frame wavelets, we show that this holds for a subset of it. An s-elementary frame MRA normalized tight frame wavelets (associated with a given expansive matrix A as its dilation matrix) is a normalized tight frame wavelet whose Fourier transform is of the form $\frac{1}{\sqrt{2\pi}}\chi_{E}$ for some measurable set E?? ^d . In this paper, we show that for any given d×d expansive matrix A, the set of all (A-dilation) s-elementary normalized tight frame wavelets with a frame MRA structure is also path-connected.     

The Ricci flow in a class of solvmanifolds

In this paper, we study the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension one, by using the bracket flow. We prove that solutions to the Ricci flow are immortal, the ω-limit of bracket flow solutions is a single point, and that for any sequence of times there exists a subsequence in which the Ricci flow converges, in the pointed topology, to a manifold which is locally isometric to a flat manifold. We give a functional which is non-increasing along a normalized bracket flow that will allow us to prove that given a sequence of times, one can extract a subsequence converging to an algebraic soliton, and to determine which of these limits are flat. Finally, we use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative in finite time.     

Stability of Runge-Kutta Methods for Trajectory Problems

A solution of a system of m autonomous differential equationsdefines a trajectory in m-dimensional space and, in particular,may give a closed orbital path. Typical trajectories are describedby a model nonlinear problem introduced in this article. Forthis problem, a trajectory lies on a surface characterized bya real symmetric matrix. It is shown that some Runge-Kutta methodspossess a property which ensures that, for this model problem,the numerical solution lies on the same surface as the trajectory.When m = 2, the numerical solution lies on the trajectory. Thisproperty is related to algebraic stability. A weaker propertysuffices for normalized differential systems.     

Weak Convergence of a Sequence of Semimartingales to a Diffusion with Discontinuous Drift and Diffusion Coefficients

This paper presents a set of sufficient conditions for a sequence of semimartingales to converge weakly to a solution of a stochastic differential equation (SDE) with discontinuous drift and diffusion coefficients. This result is closely related to a well-known weak-convergence theorem due to Liptser and Shiryayev (see [27]) which proves the weak convergence to a solution of a SDE with continuous drift and diffusion coefficients in the Skorokhod–Lindvall J ₁-topology.The goal of this paper is to obtain a stronger result in order to solve outstanding problems in the area of large-scale queueing networks – in which the weak convergence of normalized queueing length is a solution of a SDE with discontinuous coefficients. To do this we need to make the stronger assumptions: (1) replacing the convergence in probability of the triplets of a sequence of semimartingales in the original Liptser and Shiryayev's theorem by stronger convergence in L ₂, (2) assuming the diffusion coefficient is coercive, and (3) assuming the discontinuity sets of the coefficients of the limit diffusion processs are of Lebesgue measure zero.     

Multisummability of formal power series solutions of partial differential equations with constant coefficients

In an earlier paper of the author's, partial differential equations with constant coefficients have been studied. Under a certain (restrictive) assumption upon the equation, those initial conditions were characterized for which the normalized formal solution of a corresponding Cauchy problem is k-summable. Here we treat the general situation and prove an analogous result, using multisummability instead of k-summability. The appropriate multisummability type is shown to depend upon the given PDE only, and can be determined from a corresponding Newton polygon.