Decomposition algorithms for locating minimal cuts in a network

This paper provides decomposition algorithms for locating minimal cuts in a large directed network. The main theorem provides several cases for the algorithms. In the worst case, it is shown that the efficiency of one of the proposed algorithms is of the same order as a no-decomposition algorithm. As in linear programming, the obvious advantage of the proposed decomposition procedure is its ability to potentially handle larger problems than a no-decomposition algorithm.     

The inverse problem of stationary covariance generation

The paper considers the problem of passing from a stationary covariance, or spectral matrix, associated with the output of a constant linear finite-dimensional system excited by white noise to the set of all possible systems of minimum possible dimension which will generate this covariance. The problem, originally posed by R. E. Kalman in 1965, is solved by identifying each possible system with the solution of a quadratic matrix inequality; an algorithm for the solution of the inequality is also presented.Work supported by the Australian Research Grants Committee. University of New castle Technical Report EE-6816.     

A Flaw in the Use of Minimal Defining Sets for Secret Sharing Schemes

It is shown that in some cases it is possible to reconstruct a block design uniquely from incomplete knowledge of a minimal defining set for . This surprising result has implications for the use of minimal defining sets in secret sharing schemes.     

On active redundancy allocation for coherent systems—from the viewpoint of minimal cut decomposition

This paper deals with coherent systems with one active redundancy. For systems with the subclass of minimal cuts associated with one component covering that of another, assigning the redundancy to the former component is proved to bring forth a more reliable redundant system. As for symmetric systems with lower tail permutation decreasing component lifetimes, allocating the redundancy to the less reliable component results in a longer system lifetime. Several numerical examples are also presented to illustrate the new findings.     

An Intrinsic Rigidity Theorem for Closed Minimal Hypersurfaces in S5 with Constant Nonnegative Scalar Curvature

Let M⁴ be a closed minimal hypersurface in \(\mathbb{S}^5\) with constant nonnegative scalar curvature. Denote by f₃ the sum of the cubes of all principal curvatures, by g the number of distinct principal curvatures. It is proved that if both f₃ and g are constant, then M⁴ is isoparametric. Moreover, the authors give all possible values for squared length of the second fundamental form of M⁴. This result provides another piece of supporting evidence to the Chern conjecture.     

Moving-centre monotonicity formulae for minimal submanifolds and related equations

Monotonicity formulae play a crucial role for many geometric PDEs, especially for their regularity theories. For minimal submanifolds in a Euclidean ball, the classical monotonicity formula implies that if such a submanifold passes through the centre of the ball, then its area is at least that of the equatorial disk. Recently Brendle and Hung proved a sharp area bound for minimal submanifolds when the prescribed point is not the centre of the ball, which resolved a conjecture of Alexander, Hoffman and Osserman. Their proof involves asymptotic analysis of an ingeniously chosen vector field, and the divergence theorem.In this article we prove a sharp ‘moving-centre’ monotonicity formula for minimal submanifolds, which implies the aforementioned area bound. We also describe similar moving-centre monotonicity formulae for stationary p-harmonic maps, mean curvature flow and the harmonic map heat flow.     

THE PLATEAU-B(E)ZIER PROBLEM WITH WEAK-AREA FUNCTIONAL

In this paper, we present a new method to solve the Plateau-Bézier problem. A new energy functional called weak-area functional is proposed as the objective functional to obtain the approximate minimal Bézier surface from given boundaries. This functional is constructed based on Dirichlet energy and weak isothermal parameterization condition. Experimental comparisons of the weak-area functional method with existing Dirichlet, quasi-harmonic, the strain energy-minimizing, harmonic and biharmonic masks are performed which show that the weak-area functional method are among the best by choosing appropriate parameters.     

Lorentz空间型中正常2-调和超曲面的分类

本文对Lorentz空间型中的正常2-调和超曲面进行了完全分类,它的形状算子的极小多项式的阶数至多是2.     

Filling polygonal holes with minimal energy surfaces on Powell-Sabin type triangulations

In this paper we present two different methods for filling in a hole in an explicit 3D surface, defined by a smooth function f in a part of a polygonal domain D⊂R². We obtain the final reconstructed surface over the whole domain D. We do the filling in two different ways: discontinuous and continuous. In the discontinuous case, we fill the hole with a function in a Powell-Sabin spline space that minimizes a linear combination of the usual seminorms in an adequate Sobolev space, and approximates (in the least squares sense) the values of f and those of its normal derivatives at an adequate set of points. In the continuous case, we will first replace f outside the hole by a smoothing bivariate spline s_f, and then we fill the hole also with a Powell-Sabin spline minimizing a linear combination of given seminorms. In both cases, we obtain existence and uniqueness of solutions and we present some graphical examples, and, in the continuous case, we also give a local convergence result.     

二值命题逻辑中有限理论带误差结论集的结构

二值命题逻辑系统中理论Г的带误差结论集是近似推理研究的基本对象,对其结构进行分析是近似推理研究中需要解决的问题。通过公式是有限理论Г的带误差结论的充要条件,利用集合划分方法,对有限理论Г的带误差结论集分别基于真度相等关系和逻辑等价关系进行分类,得到了基于两类等价关系的包含等价类个数和代表元表示形式的分类定理,进一步体现了二值命题逻辑系统近似推理研究中理论Г的带误差结论集的特征。