Classroom note: Using the binomial series to prove the arithmetic mean–geometric mean inequality

In 1968, Leon Gerber compared (1 + x) ^a to its kth partial sum as a binomial series. His result is stated and, as an application of this result, a proof of the arithmetic mean–geometric mean inequality is presented.     

On a family of highly regular graphs by Brouwer,Ivanov, and Klin

Highly regular graphs for which not all regularities are explainable by symmetries are fascinating creatures. Some of them like, e.g., the line graph of W. Kantor’s non-classical $GQ(5^2,5)$, are stumbling stones for existing implementations of graph isomorphism tests. They appear to be extremely rare and even once constructed it is difficult to prove their high regularity. Yet some of them, like the McLaughlin graph on 275 vertices and Ivanov’s graph on 256 vertices are of profound beauty. This alone makes it an attractive goal to strive for their complete classification or, failing this, at least to get a deep understanding of them. Recently, one of the authors discovered new methods for proving high regularity of graphs. Using these techniques, in this paper we study a classical family of strongly regular graphs, originally discovered by A.E. Brouwer, A.V. Ivanov, and M.H. Klin in the late 80s. We analyse their symmetries and show that they are $(3,5)$-regular but not 2-homogeneous. Thus we promote these graphs to the distinguished club of highly regular graphs with few symmetries.     

Modular class primes in the Sundaram sieve

The purpose of this paper is to consider analogues of the twin-prime conjecture in various classes within modular rings.     

Isotactic poly(propylene‐co‐1‐pentene‐co‐1‐hexene) terpolymers: Synthesis,molecular characterization,and evidence of the trigonal polymorph

Terpolymers based on propylene with 1‐pentene and 1‐hexene as comonomeric units are satisfactorily synthesized using a metallocene catalyst. Thus, several terpolymers are prepared with distinct overall compositions in comonomers as well as three different 1‐pentene/1‐hexene ratios at a given composition to evaluate the influence on polymerization activity, intrinsic viscosity, and microstructural details. The new trigonal δ polymorph is observed in those quenched terpolymers with a global content in comonomers of about 14 mol %, independently of the ratio between both comonomers. However, preliminary results indicate a profound influence of that ratio on the crystallization rate. © 2013 Wiley Periodicals, Inc. J. Polym. Sci., Part A: Polym. Chem. 2013, 51, 3251–3259     

一类极限为1的未定式

对于一类00型或∞0型未定式，给出其极限为1的一个充分条件，并通过实例展示其应用。     

关于全国大学生数学竞赛决赛试题的解析与思考

本文对二道全国大学生数学竞赛决赛试题(数学类)给予了诠释与解析.     

Side-lobeless far-field spectrum of a short pulse from a circular aperture with Gaussian form of transmittance

The spectrum change of a Gaussian pulse in the far field is studied with the Fresnel diffraction integral when it is incident on an aperture with Gaussian form transmittance. It is found that the side-lobes of the diffracted spectral intensity, which exist in a normal circular aperture with unit transmittance, can be eliminated completely under such a condition. Also, the red shift and blue shift of the spectral intensity maximum of the incident pulse are presented.     

Associating divisors with quadratic differentials on Klein surfaces

We extend the notion and basic properties of quadratic differential divisors to a Klein surface, using the corresponding form’s divisors from the double cover of the Klein surface.