Computing singular solutions to nonlinear analytic systems

Summary A method to generate an accurate approximation to a singular solution of a system of complex analytic equations is presented. Since manyreal systems extend naturally tocomplex analytic systems, this porvides a method for generating approximations to singular solutions to real systems. Examples include systems of polynomials and systems made up of trigonometric, exponential, and polynomial terms. The theorem on which the method is based is proven using results from several complex variables. No special conditions on the derivatives of the system, such as restrictions on the rank of the Jacobian matrix at the solution, are required. The numerical method itself is developed from techniques of homotopy continuation and 1-dimensional quadrature. A specific implementation is given, and the results of numerical experiments in solving five test problems are presented.     

The Sample Average Approximation Method Applied to Stochastic Routing Problems: A Computational Study

The sample average approximation (SAA) method is an approach for solving stochastic optimization problems by using Monte Carlo simulation. In this technique the expected objective function of the stochastic problem is approximated by a sample average estimate derived from a random sample. The resulting sample average approximating problem is then solved by deterministic optimization techniques. The process is repeated with different samples to obtain candidate solutions along with statistical estimates of their optimality gaps.We present a detailed computational study of the application of the SAA method to solve three classes of stochastic routing problems. These stochastic problems involve an extremely large number of scenarios and first-stage integer variables. For each of the three problem classes, we use decomposition and branch-and-cut to solve the approximating problem within the SAA scheme. Our computational results indicate that the proposed method is successful in solving problems with up to 2¹⁶⁹⁴ scenarios to within an estimated 1.0% of optimality. Furthermore, a surprising observation is that the number of optimality cuts required to solve the approximating problem to optimality does not significantly increase with the size of the sample. Therefore, the observed computation times needed to find optimal solutions to the approximating problems grow only linearly with the sample size. As a result, we are able to find provably near-optimal solutions to these difficult stochastic programs using only a moderate amount of computation time.     

Essential dimension of quadrics

Let X be an anisotropic projective quadric over a field F of characteristic not 2. The essential dimension dim_es(X) of X, as defined by Oleg Izhboldin, is dim_es(X)=dim(X)-i(X) +1, where i(X) is the first Witt index of X (i.e., the Witt index of X over its function field).Let Y be a complete (possibly singular) algebraic variety over F with all closed points of even degree and such that Y has a closed point of odd degree over F(X). Our main theorem states that dim_es(X)dim(Y) and that in the case dim_es(X)=dim(Y) the quadric X is isotropic over F(Y).Applying the main theorem to a projective quadric Y, we get a proof of Izhboldins conjecture stated as follows: if an anisotropic quadric Y becomes isotropic over F(X), then dim_es(X)dim_es(Y), and the equality holds if and only if X is isotropic over F(Y). We also solve Knebuschs problem by proving that the smallest transcendence degree of a generic splitting field of a quadric X is equal to dim_es(X). To the memory of Oleg Izhboldin     

<Emphasis Type="Italic">G</Emphasis>-covering subgroup systems for the classes of supersoluble and nilpotent groups

LetF be a class of groups andG a group. We call a set Σ of subgroups ofG aG-covering subgroup system for the classF (or directly aF-covering subgroup system ofG) ifG ∈F whenever every subgroup in Σ is inF. In this paper, we provide some nontrivial sets of subgroups of a finite groupG which are simultaneouslyG-covering subgroup systems for the classes of supersoluble and nilpotent groups. Research of the first author is supported by the NNSF of China (Grant No. 10171086) and QLGCF of Jiangsu Province and a Croucher Fellowship of Hong Kong. Research of the second author is partially supported by a UGC (HK) grant #2060176 (2001/2002).     

Correlations of two protons in relativisticdp-collisions

Two-proton correlations in thedpppn collisions of relativistic deuterons with protons are considered. These correlations give more information than inclusive proton spectra described in the framework of different deuteron structure function models because of uncertainty in the calculation of secondary interaction contribution (rescattering processes and final state interactions). We present the predictable calculations which show different behaviour of the correlation function for different deuteron structure function models in the kinematic region where contributions of secondary interactions are negligible.     

J/psi production versus centrality, transverse momentum, and rapidity in Au+Au collisions at square root sNN=200 GeV

The PHENIX experiment at the BNL Relativistic Heavy Ion Collider (RHIC) has measured J/psi production for rapidities -2.2相似文献   

Nonlinear Maxwell Equations in Inhomogeneous Media

We study the basic properties of the Maxwell equations for nonlinear inhomogeneous media. Assuming the classical nonlinear optics representation for the nonlinear polarization as a power series, we show that the solution exists and is unique in an appropriate space if the excitation current is not too large. The solution to the nonlinear Maxwell equations is represented as a power series in terms of the solution of the corresponding linear Maxwell equations. This representation holds at least for the time period inversely proportional to the appropriate norm of the solution to the linear Maxwell equation. We derive recursive formulas for the terms of the power series for the solution including an explicit formula for the first significant term attributed to the nonlinearity.