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

. For x>0 let π(x) be the number of primes not exceeding x. Theasymptotic behaviors of the prime-counting function π(x) and the n-th primep n have been studied intensively in analytic number theory. Surprisingly, weﬁnd that π(x) and p n have many combinatorial properties which should not beignored. In this paper we pose 60 open problems on combinatorial propertiesof primes for further research. For example, we conjecture that for any integern>1 one of the nnumbers π(n),π(2n),...,π(n 2 ) is prime; we also conjecturethat for each n= 1,2,3,... there is a number k∈ {1,...,n} such that thenumber of twin prime pairs not exceeding knis a square. 1. IntroductionPrime numbers play important roles in number theory. For x >0 let π(x)denote the number of primes not exceeding x. The celebrated Prime NumberTheorem states thatπ(x) ∼ Li(x) as x → +∞,where Li(x) =R x2dtlogt ∼ xlogx . This has the following equivalent version:p n ∼ nlogn as n → +∞,where p n denotes the n-th prime. To get sharp estimation for π(x) is a mainresearch topic in analytic number theory. It is known (cf. [Sc]) that Riemann’sHypothesis implies thatπ(x) = Li(x)+O(√xlogx) and p

Properties of primes

For each $n=0,1,2,\ldots$ the central trinomial coefficient $T_n$ is the coefficient of $x^n$ in the expansion of $(x^2+x+1)^n$. In 2016 the second author conjectured that for any prime $p>3$ and positive integer $n$ the quotient $(T_{pn}-T_n)/(pn)^2$ is a $p$-adic integer. In this paper we confirm this conjecture and prove further that $$\frac{T_{pn}-T_n}{(pn)^2}\equiv\frac{T_{n-1}}6\left(\frac p3\right)B_{p-2}\left(\frac13\right)\pmod p,$$ where $(\frac p3)$ is the Legendre symbol and $B_{p-2}(x)$ is the Bernoulli polynomial of degree $p-2$.

Supercongruences for central trinomial coefficients.

Let A and B be two finite subsets of a field F. In this paper we provide a nontrivial lower bound for |{a+b: a in A, b in B, and P(a,b) not=0}| where $P(x,y)\in F[x,y]$.

A lower bound for |{a+b: a\in A, b\in B, P(a,b)\not=0}|

This paper deals with combinatorial aspects of finite covers of groups by cosets or subgroups. Let $a_1G_1,...,a_kG_k$ be left cosets in a group $G$ such that ${a_iG_i}_{i=1}^k$ covers each element of $G$ at least $m$ times but none of its proper subsystems does. We show that if $G$ is cyclic, or $G$ is finite and $G_1,...,G_k$ are normal Hall subgroups of $G$, then $k\geq m+f([G:\bigcap_{i=1}^kG_i])$, where $f(\prod_{t=1}^r p_t^{\alpha_t})=\sum_{t=1}^r\alpha_t(p_t-1)$ if $p_1,...,p_r$ are distinct primes and $\alpha_1,...,\alpha_r$ are nonnegative integers. When all the $a_i$ are the identity element of $G$ and all the $G_i$ are subnormal in $G$, we prove that there is a composition series from $\bigcap_{i=1}^kG_i$ to $G$ whose factors are of prime orders. The paper also includes some other results and two challenging conjectures.

Finite covers of groups by cosets or subgroups

Let E be a real quadratic field with discriminant d and let p be an odd prime not dividing d. For \rho=1 or -1, we determine $\prod_{0<c<d, (d/c)=\rho} binomial coeff.{p-1}{\lfloor pc/d\rfloor}$ modulo p^2 in terms of Lucas numbers, the fundamental unit and the class number of E, where (d/c) is the Kronecker symbol.

Zhi-Wei Sun

Papers

Properties of primes

Supercongruences for central trinomial coefficients.

A lower bound for |{a+b: a\in A, b\in B, P(a,b)\not=0}|

Finite covers of groups by cosets or subgroups

Binomial Coefficients and Quadratic Fields