On certain modified boundary value problems for ordinary differential equations with irregular singularity

We consider a linear second-order differential equation with irregularly singular point at the beginning of the interval. For the corresponding homogeneous differential equation, we obtain the asymptotics of the solutions and their derivatives near the singular point. Using some modified Green functions and taking into account the asymptotics, we consider three boundary value problems with various boundary conditions (including a weighted one) at the singular point, proving theorems on the existence and uniqueness of the solutions and giving their structure. Lithuanian Mathematical Journal, Vol. 49, No. 1, 2009, pp. 109–121     

Stable approximation of solutions to irregular nonlinear operator equations in a Hilbert space under large noise

The class of regularized Gauss-Newton methods for solving inexactly specified irregular nonlinear equations is examined under the condition that additive perturbations of the operator in the problem are close to zero only in the weak topology. By analogy with the well-understood conventional situation where the perturbed and exact operators are close in norm, a stopping criterion is constructed ensuring that the approximate solution is adequate to the errors in the operator.     

新疆女娄菜属花粉形态研究

新疆女娄菜属Melandrium Roehl.有11种,对该属的花粉形态目前尚无专门论述,为了给女娄菜属植物的种间识别提供依据,作者对新疆所产的该属8个的花粉形态作了研究,并进行了扫描电镜的观察和照相。结果表明:女娄菜属花粉粒近球形或椭球形,极面观时近圆形,大小为21.0×21.0～31.5×33.0(μm~2),属于中型花粉;具12～18个萌发孔,孔径2.5～7.0μm,孔间距4.0～9.0μm;外壁两层,厚2.5～4.0μm,外壁纹饰分为两大类:一类为颗粒状,另一类为网状。     

Irregular wavelet frames on<Emphasis Type="Italic">L</Emphasis><Superscript>2</Superscript>(ℝ<Superscript>n</Superscript>)

In this paper, we present the conditions on dilation parameter {s _j}_j that ensure a discrete irregular wavelet system {s _j ^n/2ψ(s _j ·−bk)} _{j∈ℤ,k∈ℤ} ⁿ to be a frame on L²(ℝⁿ), and for the wavelet frame we consider the perturbations of translation parameter b and frame function ψ respectively.     

Comments on the number of sensitivity centres in silver halide grains

A comment on the number of sensitivity centres in silver halide grains of nuclear emulsions is made and a theory for its evaluation at different temperatures is presented. The results at room temperature agree satisfactorily with assumptions made by various workers.     

First-hit analysis of algorithms for computing quadratic irregularity

The author has previously extended the theory of regular and irregular primes to the setting of arbitrary totally real number fields. It has been conjectured that the Bernoulli numbers, or alternatively the values of the Riemann zeta function at odd negative integers, are uniformly distributed modulo for every . This is the basis of a well-known heuristic, given by Siegel, estimating the frequency of irregular primes. So far, analyses have shown that if is a real quadratic field, then the values of the zeta function at negative odd integers are also distributed as expected modulo for any . We use this heuristic to predict the computational time required to find quadratic analogues of irregular primes with a given order of magnitude. We also discuss alternative ways of collecting large amounts of data to test the heuristic.

     

Comparison of algorithms to calculate quadratic irregularity of prime numbers

In previous work, the author has extended the concept of regular and irregular primes to the setting of arbitrary totally real number fields , using the values of the zeta function at negative integers as our ``higher Bernoulli numbers'. In the case where is a real quadratic field, Siegel presented two formulas for calculating these zeta-values: one using entirely elementary methods and one which is derived from the theory of modular forms. (The author would like to thank Henri Cohen for suggesting an analysis of the second formula.) We briefly discuss several algorithms based on these formulas and compare the running time involved in using them to determine the index of -irregularity (more generally, ``quadratic irregularity') of a prime number.

     

定向凝固过程中的不规则固液界面形貌

固液界面通常为规则界面, 但有时为不规则界面,如倾斜枝晶、退化枝晶和海藻状晶体界面.以琥珀腈为研究对象,用设计的定向凝固实验体系, 研究了不同温度梯度和界面生长速度对固液界面形貌的影响.实验结果表明:界面生长速度一定时,增加温度梯度界面由倾斜枝晶逐渐转变为退化枝晶,最终成为海藻状晶体;温度梯度一定时,降低界面生长速度界面会发生类似的变化.温度梯度和界面生长条件在一定范围内变化时,界面可以从一种生长方式变为另一种方式,如可以从海藻状晶体连续地变为倾斜枝晶.在某些条件下,海藻状晶体和倾斜枝晶可能同时出现在界面上,并竞相生长.退化枝晶界面处于动态变化中,二次枝晶臂不断地改变着生长方向.     

A Cartesian grid method for two‐phase gel dynamics on an irregular domain

We develop a numerical method for simulating models of two‐phase gel dynamics in an irregular domain using a regular Cartesian grid. The models consist of transport equations for the volume fractions of the two phases, polymer network and solvent; coupled momentum equations for the two phases; and a volume‐averaged incompressibility constraint. Multigrid with Vanka‐type box relaxation scheme is used as a preconditioner for the Krylov subspace solver (GMRES) to solve the momentum and incompressibility equations. Ghost points are used to enforce no‐slip boundary conditions for the velocity field of each phase, and no‐flux boundary conditions for the volume fractions. The behavior of the new method, including its rate of convergence, is explored through numerical experiments for a problem in which strong phase separation develops from an initially (almost) homogeneous phase distribution. We also use the method to explore situations, motivated by biology, which show that imposed boundary velocities can cause substantial redistribution of network and solvent. Copyright © 2010 John Wiley & Sons, Ltd.     

Pressure and velocity of air during drying and storage of cereal grains

A common method of drying cereal grains is to ventilate a large static mass of grain with an even flow of air at near ambient temperature. After the grain has been dried it is often stored in the same container and kept cool by aeration with a lower velocity of air than is used in drying. To analyse the airflow through this mass of grain a nonlinear momentum equation for flow through porous media is used where the resistance to flow is a + b ¦ν¦. This equation, together with the assumption that the air is incompressible, defines the problem which is solved numerically, using the finite element method, and the results compared with experimental values. The small parameter ε = bν _r /a, where ν _r is the velocity scale, is used in a perturbation analysis to examine the nonlinear effects of the resistance on the airflow. When ε = 0 the equations reduce to those for potential flow, while for small values of ε there are first-order corrections to the pressure p ₁ and the stream function χ ₁. The nonlinear problem is simplified by changing to curvilinear coordinates (s, t) where s is constant on the potential flow isobars while t is constant on the streamlines. General conclusions are derived for p ₁ and χ ₁, for example that they depend on the curvature of the potential flow solution with a large curvature of the isobars leading to larger values of p ₁ and similarly for the streamlines. The potential flow solution p ₀ and the first order solution p ₀ + εp ₁ are close to the solution of the full nonlinear problem when ε is small. To illustrate this for a typical grain storage problem, the solution p ₀ is shown to be very close to the finite element solution (with a difference of less than 1%) when ε < 0.03 while for the first order solution p ₀ + εp ₁ the difference is less than 1% when ε < 0.1.