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 Bn(t) be the nth Bernoulli polynomial. We show that Bp-ι(a/q) — Bp-ι = q(Up — l)/2p (mod p), where Un is a certain linear recurrence of order [q/2] which depends only on α, q and the least positive residue of p (mod q). This can be re-written as a sum of linear recurrence sequences of order < φ(q)/2, and so we can recover the classical results where φ(q) < 2 (for instance, Bp-ϊ(l/6)-Bp-1 = (3 -3)/2p+ (2-2)/p (modp)). Our results provide the first advance on the question of evaluating these polynomials when φ(q) > 2, a problem posed by Emma Lehmer in 1938.

/pdf/values-of-bernoulli-polynomials-eldkz3czn9.pdf

Values of Bernoulli polynomials

Algebraic Approaches to Periodic Arithmetical Maps

Let p be any prime, and let a and n be nonnegative integers. Let $r\in Z$ and $f(x)\in Z[x]$. We establish the congruence 
$$p^{\deg f}\sum_{k=r(mod p^a)}\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\sum_{i=a}^{\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then 
$$\frac{1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)} \binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\frac {1}{[n/p^{a-1}]!} \sum_{k=r(mod p^a)}\binom{n}{k}\binom{s}{t}(-1)^k((k-r)/p^{a-1})^l (mod p).$$ 
We also present an application of the first congruence to Bernoulli polynomials, and apply the second congruence to show that a p-adic order bound given by the authors in a previous paper can be attained when p=2.

Combinatorial congruences modulo prime powers

Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3\equiv (-1)^{(p-1)/2}p\pmod{p^3}.$$ In this paper we show further that \begin{align*}\sum_{k=0}^{p-1}(4k+1)\binom{-1/2}k^3\equiv &\sum_{k=0}^{(p-1)/2}(4k+1)\binom{-1/2}k^3 \\\equiv & (-1)^{(p-1)/2}p+p^3E_{p-3} \pmod{p^4},\end{align*}where $E_0,E_1,E_2,\ldots$ are Euler numbers. We also prove that if $p>3$ then $$\sum_{k=0}^{(p-1)/2}\frac{20k+3}{(-2^{10})^k}\binom{4k}{k,k,k,k}\equiv(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2)\pmod{p^4}.$$

A refinement of a congruence result by van Hamme and Mortenson

For n=1,2,3,... let S_n be the sum of the first n primes. We show that the sequence a_n=(S_n/n)^{1/n} (n=1,2,3,...) is strictly decreasing, and conjecture that the sequence 
a_{n+1}/a_n (n=5,6,...) is strictly increasing.

 10.1017/S0004972712000986

/pdf/on-a-sequence-involving-sums-of-primes-3h2qxne27t.pdf

Zhi-Wei Sun

Papers

Values of Bernoulli polynomials

Algebraic Approaches to Periodic Arithmetical Maps

Combinatorial congruences modulo prime powers

A refinement of a congruence result by van Hamme and Mortenson

On a sequence involving sums of primes