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

Let p > 3 be a prime. We show that p 1 X k=0 � −1/(p + 1) k �p+1 � 0 (mod p 5 ) and p 1 X k=0 � 1/(p − 1) k �p 1 � 0 (mod p 4 ). For any positive integer m 㘑 0 (mod p), we prove that p 1 X k=0 (−1) km � p/m − 1 k �m � 0 (mod p 4 ),

Some new super-congruences modulo prime powers

We prove that for any nonnegative integers $n$ and $r$ the binomial sum $$ \sum_{k=-n}^n\binom{2n}{n-k}k^{2r} $$ is divisible by $2^{2n-\min\{\alpha(n),\alpha(r)\}}$, where $\alpha(n)$ denotes the number of 1's in the binary expansion of $n$. This confirms a recent conjecture of Guo and Zeng.

On 2-adic orders of some binomial sums

In this paper we establish some explicit congruences for Euler polynomials modulo a general positive integer. As a consequence, if a,m ∈ ℤ and 2 ∤ m then 

$$ \frac{{m^{k + 1} }} {2}E_k \left( {\frac{{x + a}} {m}} \right) - \frac{{\left( { - 1} \right)^a }} {2}E_k \left( x \right) \in \mathbb{Z}\left[ x \right] for every k = 0,1,2,..., $$

which may be regarded as a refinement of a multiplication formula.

Explicit congruences for euler polynomials

Let m be a positive integer, and define ζm(s) =∑n=1∞(−e2πi/m)ω(n) ns andζm∗(s) =∑ n=1∞(−e2πi/m)Ω(n) ns , for ℛ(s) > 1, where ω(n) denotes the number of distinct prime factors of n, and Ω(n) represents the total number of prime factors of n (counted with multiplicity). In this paper, we study these two zeta functions and related arithmetical functions. We show that ∑ n=1n is squarefree∞(−e2πi/m)ω(n) n = 0if m > 4, which is similar to the known identity ∑n=1∞μ(n)/n = 0 equivalent to the Prime Number Theorem. For m > 4, we prove that ζm(1) :=∑n=1∞(−e2πi/m)ω(n) n = 0andζm∗(1) :=∑ n=1∞(−e2πi/m)Ω(n) n = 0. We also raise a hypothesis on the parities of Ω(n) − n which implies the Riemann Hypothesis.

On a pair of zeta functions

Let F be a uniform function of two complex variables to an additive abelian group, i.e., F satisfies the functional equation n 1 X r=0

Zhi-Wei Sun

Papers

Some new super-congruences modulo prime powers

On 2-adic orders of some binomial sums

Explicit congruences for euler polynomials

On a pair of zeta functions

A reciprocity law for uniform functions