A note on the distribution of integer points on spheres

In this note, by applying the works of Böcherer and Schulze-Pillot, we evaluate asymptotically the variance of the Linnik distribution, and show as Y → ∞, $$\label{eq1} \sum _{n \leq Y,\, n \not \equiv 0 \, (\, {\rm mod} 4)} \frac{\sum _{|x|^{2} = n} P\left( \frac{x}{|x|} \right)}{n^{1/4}} {\overline{\frac{\sum _{|z|^{2} = n} Q \left(\frac{z}{|z|} \right)}{n^{1/4}}}} = \, c < P,\, Q > \, \Lambda (1/2,\, P) \, Y + o(Y) ,$$ for any spherical harmonics P, Q on S ² which are Hecke eigenforms, where | · | is the usual Euclidean norm on R ³, and c is an explicit constant.     

Sums of k-th powers and the Whittaker–Fourier coefficients of automorphic forms

The Ramanujan Journal - In this work, we obtain power-saving bounds for shifted convolution sums involving the Whittaker–Fourier coefficients of automorphic forms and $$r_{s, k}(n)$$ , the...     

BOUNDARY VALUE PROBLEM FOR NONLINEAR IMPULSIVE INTEGRO-DIFFERENTIAL EQUATION IN BANACH SPACE

In this paper, by establishing a new comparison result and using the monotone iterative technique, the existence of maximal and minimal solutions of the boundary value problem for second-order impulsive differential equation which depends on x' in Banach space is obtained.     

聚合物载体-稀土金属配合物的研究聚（苯乙烯－丙烯酰胺）载体－氯化钕配合物对?

研究了聚（苯乙烯－丙烯酰胺）（ＰＳＡＣ）载体－氯化钕配合物催化丁二烯聚合的规律，考察了ＰＳＡＣ载体及ＰＳＡＣ·ＮｄＣｌ３配合物的组成与催化活性间的关系．ＰＳＡＣ载体的收率不影响配合物的活性，而配合物的活性却随ＰＳＡＣ载体中Ｃ（Ｏ）ＮＨ２功能团含量的高低而改变，含量低时活性高．ＰＳＡＣ载体与配合物的组成不影响所得聚丁二烯的微观结构     

Sobolev embeddings involving symmetry

Given a bounded regular domain with cylindrical symmetry, then functions having such symmetry and belonging to the first Sobolev space can be embedded compactly into some weighted Lp spaces, with p superior to the critical Sobolev exponent. A simple application to elliptic boundary value problem is also considered.     

Morphology, structure and Raman scattering of carbon nanotubes produced by using mesoporous materials

Carbon nanotubes were prepared by chemical vapor deposition (CVD) of hydrocarbon gas on various substrates. The effect of substrates on the growth, morphology and structure of carbon nanotubes were investigated. Aligned carbon nanotubes with high density and purity were achieved by CVD on mesoporous silica substrate. The Raman scattering of aligned carbon nanotubes was carried out, and the dependence of the phonon properties on the microstructure of the nanotubes has been discussed.