Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

Zeros of orthogonal polynomials

Concrete mathematics, a foundation for computer science, by Ronald L. Graham, Donald E. Knuth and Oren Patashnik. Pp 625. £24·95. 1989. ISBN 0-201-14236-8 (Addison-Wesley)

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

In this paper I present a history of tractability of continuous problems, which has its beginning in the successful numerical tests for highdimensional integration of finance problems. Tractability results will be illustrated for two multivariate problems, integration and linear tensor products problems, in the worst case setting. My talk at FoCM'08 in Hong Kong and this paper are based on the book Tractability of Multivariate Problems , written jointly with Erich Novak. The first volume of our book has been recently published by the European Mathematical Society. Introduction Many people have recently become interested in studying the tractability of continuous problems. This area of research addresses the computational complexity of multivariate problems defined on spaces of functions of d variables, with d that can be in the hundreds or thousands; in fact, d can even be arbitrarily large. Such problems occur in numerous applications including physics, chemistry, finance, economics, and the computational sciences. As with all problems arising in information-based complexity, we want to solve multivariate problems to within ∈, using algorithms that use finitely many functions values or values of some linear functionals. Let n (∈, d ) be the minimal number of function values or linear functionals that is needed to compute the solution of the d -variate problem to within ∈. For many multivariate problems defined over standard spaces of functions n (∈, d ) is exponentially large in d .

Tractability of Multivariate Problems

On 2-adic orders of some binomial sums

In this paper we initiate the study of products and sums divisible by central binomial coefficients. We show that 2(2n+1)binom(2n,n)| binom(6n,3n)binom(3n,n) for every n=1,2,3,... Also, for any nonnegative integers $k$ and $n$ we have $$\binom {2k}k | \binom{4n+2k+2}{2n+k+1}\binom{2n+k+1}{2k}\binom{2n-k+1}n$$ and $$\binom{2k}k | (2n+1)\binom{2n}nC_{n+k}\binom{n+k+1}{2k},$$ where $C_m$ denotes the Catalan number $\binom{2m}m/(m+1)=\binom{2m}m-\binom{2m}{m+1}$. Applying this result we obtain two sums divisible by central binomial coefficients.

Products and sums divisible by central binomial coefficients

For n=1,2,3,... define S(n) as the smallest integer m>1 such that those 2k(k-1) mod m for k=1,...,n are pairwise distinct; we show that S(n) is the least prime greater than 2n-2 and hence the value set of the function S(n) is exactly the set of all prime numbers. For every n=4,5,... we prove that the least prime p>3n with 3|p-1 is just the least positive integer m such that 18k(3k-1) (k=1,...,n) are pairwise distinct modulo m. For d=4,6,12 and n=3,4,...., we prove that the least prime p>2n-2 with p=-1 (mod d) is the smallest integer m such that those (2k-1)^d for k=1,...,n are pairwise distinct modulo m. 
We also pose several challenging conjectures on primes. For example, we find a surprising recurrence for primes, namely, for every n=10,11,... the (n+1)-th prime p_{n+1} is just the least positive integer m such that 2s_k^2 (k=1,...,n) are pairwise distinct modulo m where s_k = sum_{j=1}^k(-1)^{k-j}p_j. We also conjecture that for any positive integer m there are consecutive primes p_k,...,p_n (k<n) not exceeding 2m+2.2*sqrt(m) such that m = p_n-p_{n-1}+...+(-1)^{n-k}p_k.

On functions taking only prime values

Generalized octagonal numbers are those $p_8(x)=x(3x-2)$ with $x\in\mathbb Z$. In this paper we mainly show that every positive integer can be written as the sum of four generalized octagonal numbers one of which is odd. This result is similar to Lagrange's theorem on sums of four squares. Moreover, for $35$ triples $(b,c,d)$ with $1\le b\le c\le d$ (including $(2,3,4)$ and $(2,4,8)$), we prove that any nonnegative integer can be exprssed as $p_8(w)+bp_8(x)+cp_8(y)+dp_8(z)$ with $w,x,y,z\in\mathbb Z$. We also pose several conjectures for further research.

A result similar to Lagrange's theorem

Let \begin{document}$ a,b,c,d,e,f\in\mathbb N $\end{document} with \begin{document}$ a\geq c\geq e>0 $\end{document} , \begin{document}$ b\leq a $\end{document} and \begin{document}$ b\equiv a\ ({\rm{mod}}\ 2) $\end{document} , \begin{document}$ d\leq c $\end{document} and \begin{document}$ d\equiv c\ ({\rm{mod}}\ 2) $\end{document} , \begin{document}$ f\leq e $\end{document} and \begin{document}$ f\equiv e\ ({\rm{mod}}\ 2) $\end{document} . If any nonnegative integer can be written as \begin{document}$ x(ax+b)/2+y(cy+d)/2+z(ez+f)/2 $\end{document} with \begin{document}$ x,y,z\in\mathbb Z $\end{document} , then the ordered tuple \begin{document}$ (a,b,c,d,e,f) $\end{document} is said to be universal over \begin{document}$ \Bbb Z $\end{document} . Recently, Z.-W. Sun found all candidates for such universal tuples over \begin{document}$ \Bbb Z $\end{document} . In this paper, we use the theory of ternary quadratic forms to show that 44 concrete tuples \begin{document}$ (a,b,c,d,e,f) $\end{document} in Sun's list of candidates are indeed universal over \begin{document}$ \mathbb Z $\end{document} . For example, we prove the universality of \begin{document}$ (16,4,2,0,1,1) $\end{document} over \begin{document}$ \Bbb Z $\end{document} which is related to the form \begin{document}$ x^2+y^2+32z^2 $\end{document} .

Zhi-Wei Sun

Papers

On 2-adic orders of some binomial sums

Products and sums divisible by central binomial coefficients

On functions taking only prime values

A result similar to Lagrange's theorem

Some universal quadratic sums over the integers