Science China-mathematics 

About: Science China-mathematics is an academic journal published by Springer Nature. The journal publishes majorly in the area(s): Bounded function & Estimator. It has an ISSN identifier of 1869-1862. Over the lifetime, 4299 publications have been published receiving 39993 citations. The journal is also known as: Zhongguo kexue..

Existence of periodic and subharmonic solutions for second-order superlinear difference equations

Abstract: By critical point theory, a new approach is provided to study the existence and multiplicity results of periodic and subharmonic solutions for difference equations. For secord-order difference equations $$\Delta ^2 x_{n - 1} + f(n, x_n ) = 0,$$ some new results are obtained for the above problems when f(t, z) has superlinear growth at zero and at infinity in z.

Survey on normal distributions, central limit theorem, Brownian motion and the related stochastic calculus under sublinear expectations

Abstract: This is a survey on normal distributions and the related central limit theorem under sublinear expectation. We also present Brownian motion under sublinear expectations and the related stochastic calculus of Ito’s type. The results provide new and robust tools for the problem of probability model uncertainty arising in financial risk, statistics and other industrial problems.

The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

Abstract: We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation $$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$ , where Ω is a bounded domain of R N with Lipschitz boundary, λ is a real parameter, N ≥ 3, $$2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)$$ is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

Super congruences and Euler numbers

Abstract: Let p > 3 be a prime. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that $$\begin{gathered} \sum\limits_{k = 0}^{p - 1} {\frac{{\left( {_k^{2k} } \right)}} {{2^k }}} \equiv \left( { - 1} \right)^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern- ulldelimiterspace} 2}} - p^2 E_{p - 3} \left( {\bmod p^3 } \right), \hfill \\ \sum\limits_{k = 1}^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern- ulldelimiterspace} 2}} {\frac{{\left( {_k^{2k} } \right)}} {k}} \equiv \left( { - 1} \right)^{{{\left( {p + 1} \right)} \mathord{\left/ {\vphantom {{\left( {p + 1} \right)} 2}} \right. \kern- ulldelimiterspace} 2}} \frac{8} {3}pE_{p - 3} \left( {\bmod p^2 } \right), \hfill \\ \sum\limits_{k = 0}^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern- ulldelimiterspace} 2}} {\frac{{\left( {_k^{2k} } \right)^2 }} {{16^k }}} \equiv \left( { - 1} \right)^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern- ulldelimiterspace} 2}} + p^2 E_{p - 3} \left( {\bmod p^3 } \right), \hfill \\ \end{gathered}$$ where E0,E1,E2, ... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π−2 and the constant \(K: = \sum olimits_{k = 1}^\infty {{{\left( {\tfrac{k} {3}} \right)} \mathord{\left/ {\vphantom {{\left( {\tfrac{k} {3}} \right)} {k^2 }}} \right. \kern- ulldelimiterspace} {k^2 }}}\) (with (−) the Jacobi symbol), two of which are $$\sum\limits_{k = 1}^\infty {\frac{{\left( {10k - 3} \right)8^k }} {{k^3 \left( {_k^{2k} } \right)^2 \left( {_k^{3k} } \right)}} = \frac{{\pi ^2 }} {2}} and \sum\limits_{k = 1}^\infty {\frac{{\left( {15k - 4} \right)\left( { - 27} \right)^{k - 1} }} {{k^3 \left( {_k^{2k} } \right)^2 \left( {_k^{3k} } \right)}} = K.}$$

Some dimensional results for homogeneous Moran sets

Abstract: Let M ({ nk } k ≥1,{ ck }k≥1) be the collection of homogeneous Moran sets determined by { n k}k≥1and { ck }k≥1, where { nk }k≥1 is a sequence of positive integers and { ck }k≥1 a sequence of positive numbers. Then the maximal and minimal values of Hausdorff dimensions for elements in M are determined. The result is proved that for any value s between the maximal and minimal values, there exists an element in M { nk } k ≥1, { ck } k ≥1) such that its Hausdorff dimension is equal to s. The same results hold for packing dimension. In the meantime, some other properties of homogeneous Moran sets are discussed.