A case study in meta-automation: automatic generation of congruence automata for combinatorial sequences

In this paper, which may be considered a sequel to a recent article by Eric Rowland and Reem Yassawi, we present yet another approach for the automatic generation of automata (and an extension that we call congruence linear schemes) for the fast (log-time) determination of congruence properties, modulo small (and not so small!) prime powers, for a wide class of combinatorial sequences. Even more interesting than the new results that could be obtained is the illustrated methodology, that of designing ‘meta-algorithms’ that enable the computer to develop algorithms, that it (or another computer) can then proceed to use to actually prove (potentially!) infinitely many new results. This paper is accompanied by a Maple package, AutoSquared, and numerous sample input and output files, that readers can use as templates for generating their own, thereby proving many new ‘theorems’ about congruence properties of many famous (and, of course, obscure) combinatorial sequences.     

一类含临界耦合非线性项的奇异椭圆方程组的正解

本文研究了一类含临界指数与耦合非线性项的奇异椭圆方程组. 利用变分方法与极大值原理, 通过证明对应的能量泛函满足局部的 (PS)c 条件, 得到了这类方程组正解的存在性, 推广了单个方程与方程组中的相应结果.     

Positive periodic solutions of delayed Nicholson’s blowflies models with a nonlinear density-dependent mortality term

This paper is concerned with the periodic solutions for a new fishery model with a nonlinear density-dependent mortality term, which is a generalization of the classical Nicholson’s blowflies model. By using coincidence degree theory, some criteria are obtained to guarantee that the existence of positive periodic solutions of the model. Moreover, an example and its numerical simulation are given to illustrate our main results.     

Existence and Uniqueness of Periodic Solutions for a Kind of Second Order Neutral Functional Differential Equations with Constant Delays

In this paper, we use the coincidence degree theory to establish new results on the existence and uniqueness of T-periodic solutions for the second order neutral functional differential equation with constant delays of the form （x（t）＋Bx（t-δ））＂＋Cx＇（t）＋g（x（t-τ））=p（t）.     

The growth irregularity of slowly growing entire functions

We show that entire transcendental functions f satisfying $\log M(r,f) = o(\log ^2 r),r \to \infty (M(r,f): = \mathop {\max }\limits_{|z| = r} |f(z)|)$ necessarily have growth irregularity, which increases as the growth diminishes. In particular, if 1 < p < 2, then the asymptotics $\log M(r,f) = \log ^p r + o(\log ^{2 - p} r),r \to \infty ,$ , is impossible. It becomes possible if “o” is replaced by “O.”     

On a problem of D.H. Lehmer over short intervals

Let p be an odd prime and a be an integer coprime to p. Denote by N(a,p) the number of pairs of integers b,c with bc≡a (mod p), and with b,c having different parity. The main purpose of this paper is to study the sum , and obtain a sharp asymptotic formula.     

Necessary and sufficient conditions for oscillation of first order nonlinear neutral differential equations

In this paper, we prove that every solution of the first order nonlinear neutral differential equation

     

Neutral loss of water from the b ions with histidine at the C-terminus and formation of the c ions involving lysine side chains

Neutral loss of water from the amide bond induced by the His side chain has been reported. The proposed fragmentation pathway is a retro-Ritter reaction catalyzed by the imidazole nitrogen. In our MS/MS study of the neuropeptide GAHKNYLRFamide, we observed that the neutral loss of water from the b(3) ion is abundant. The b(3) ion has a His residue at the C-terminus. As reported previously, in the b ions with His at the C-terminus, the imidazole residue is connected to the carbonyl carbon to form a five-membered ring. Therefore, it is unlikely that the neutral loss of water from the b(3) ion is catalyzed by the imidazole nitrogen. Through MS2 and MS3 studies of a synthetic peptide standard AGHKLL and its chemically labeled and isotope-encoded forms, we discovered that the water loss from the b(3) ion involves the carbonyl group of His, the hydrogen connected to the alpha-carbon of Gly, and the amide hydrogen of His. We also discovered the formation of an unusual c(x) ion in peptides with a Lys or Arg residue at the (x + 1) position of the peptide.     

A differentially interpolated direct forcing immersed boundary method for predicting incompressible Navier–Stokes equations in time-varying complex geometries

A dispersion-relation-preserving dual-compact scheme developed in Cartesian grids is applied together with the immersed boundary method to solve the flow equations in irregular and time-varying domains. The artificial momentum forcing term applied at certain points in cells containing fluid and solid allows an imposition of velocity condition to account for the motion of solid body. We develop in this study a differential-based interpolation scheme which can be easily extended to three-dimensional simulation. The results simulated from the proposed immersed boundary method agree well with other numerical and experimental results for the chosen benchmark problems. The accuracy and fidelity of the IB flow solver developed to predict flows with irregular boundaries are therefore demonstrated.     

具有分布时滞的二阶非线性中立型时标动力方程的振动定理

利用广义Riccati变换技术, 研究时标上具有分布时滞的二阶非线性中立型动力方程 \[\big(r(t)\big(\big(y(t)+p(t)y(\tau(t))\big)^{\it \Delta}\big)^\beta\big)^{\it \Delta}+\int_c^dF(t,\xi,y(\delta(t,\xi))){\it \Delta}\xi=0 \]的振动性, 其中$\beta>0$是两个正奇数之比, 获得了方程所有解振动的几个充分条件,推广和改进了一些已知的结果, 并给出了几个应用实例.