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

In this paper we derive a general combinatorial identity in terms of polynomials with dual sequences of coefficients. Moreover, combinatorial identities involving Bernoulli and Euler polynomials are deduced. Also, various other known identities are obtained as particular cases.

Combinatorial identities in dual sequences

For integers b and c the generalized central trinomial coefficient Tn(b, c) denotes the coefficient of xn in the expansion of (x2 + bx + c)n. Those Tn = Tn(1, 1) (n = 0, 1, 2, …) are the usual central trinomial coefficients, and Tn(3, 2) coincides with the Delannoy number $D_n = \sum\nolimits_{k = 0}^n {\left( {\begin{array}{*{20}c} n \\ k \\ \end{array} } \right)} \left( {\begin{array}{*{20}c} {n + k} \\ k \\ \end{array} } \right)$ in combinatorics. We investigate congruences involving generalized central trinomial coefficients systematically. Here are some typical results: For each n = 1, 2, 3, …, we have 

$$\sum\limits_{k = 0}^{n - 1} {\left( {2k + 1} \right)T_k \left( {b,c} \right)^2 \left( {b^2 - 4c} \right)^{n - 1 - k} \equiv 0 \left( {\bmod n^2 } \right)}$$

and in particular $\left. {n^2 } \right|\sum\nolimits_{k = 0}^{n - 1} {\left( {2k + 1} \right)D_k^2 }$ is an odd prime then 

$$\sum\limits_{k = 0}^{p - 1} {T_k^2 \equiv \left( {\frac{{ - 1}} {p}} \right)\left( {\bmod p} \right) and } \sum\limits_{k = 0}^{p - 1} {D_k^2 \equiv \left( {\frac{2} {p}} \right)\left( {\bmod p} \right), }$$

where (−) denotes the Legendre symbol. We also raise several conjectures some of which involve parameters in the representations of primes by certain binary quadratic forms.

Congruences involving generalized central trinomial coefficients

New identities involving Bernoulli and Euler polynomials

Supercongruences involving products of two binomial coefficients

By a very simple argument, we prove that if $l,m,n$ are nonnegative integers then $$\sum_{k=0}^l(-1)^{m-k}\binom{l}{k}\binom{m-k}{n}\binom{2k}{k-2l+m} =\sum_{k=0}^l\binom{l}{k}\binom{2k}{n}\binom{n-l}{m+n-3k-l}. 
On the basis of this identity, for $d,r=0,1,2,...$ we construct explicit $F(d,r)$ and $G(d,r)$ such that for any prime $p>\max\{d,r\}$ we have 
\sum_{k=1}^{p-1}k^r C_{k+d}\equiv \cases F(d,r)(mod p)& if 3|p-1, \\G(d,r)\ (mod p)& if 3|p-2, 
where $C_n$ denotes the Catalan number $(n+1)^{-1}\binom{2n}{n}$. For example, when $p\geq 5$ is a prime, we have 
\sum_{k=1}^{p-1}k^2C_k\equiv\cases-2/3 (mod p)& if 3|p-1, \1/3 (mod p)& if 3|p-2; 
and 
\sum_{0<k<p-4}\frac{C_{k+4}}k \equiv\cases 503/30 (mod p)& if 3|p-1, -100/3 (mod p)& if 3|p-2. 
This paper also contains some new recurrence relations for Catalan numbers.

Zhi-Wei Sun

Papers

Combinatorial identities in dual sequences

Congruences involving generalized central trinomial coefficients

New identities involving Bernoulli and Euler polynomials

Supercongruences involving products of two binomial coefficients

A combinatorial identity with application to Catalan numbers