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 q>1 and m>0 be relatively prime integers. We find an explicit period $\nu_m(q)$ such that for any integers n>0 and r we have $[n+\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\sum_{k=r(mod m)}\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.

Congruences for sums of binomial coefficients

Let $G$ be any additive abelian group with cyclic torsion subgroup, and let $A$, $B$ and $C$ be finite subsets of $G$ with cardinality $n>0$. We show that there is a numbering $\{a_i\}_{i=1}^n$ of the elements of $A$, a numbering $\{b_i\}_{i=1}^n$ of the elements of $B$ and a numbering $\{c_i\}_{i=1}^n$ of the elements of $C$, such that all the sums $a_i+b_i+c_i\ (1\ls i\ls n)$ are (pairwise) distinct. Consequently, each subcube of the Latin cube formed by the Cayley addition table of $\Z/N\Z$ contains a Latin transversal. This additive theorem is an essential result which can be further extended via restricted sumsets in a field.

https://www.intlpress.com/site/pub/files/_fulltext/journals/mrl/2008/0015/0006/MRL-2008-0015-0006-a015.pdf

An additive theorem and restricted sumsets

Lagrange's four-square theorem asserts that any $n\in\mathbb N=\{0,1,2,\ldots\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \begin{gather*} x,\ 2x,\ x-y,\ 2x-2y,\ a(x^2-y^2)\ (a=1,2,3),\ x^2-3y^2,\ 3x^2-2y^2, \\x^2+ky^2\ (k=2,3,5,6,8,12),\ (x+4y+4z)^2+(9x+3y+3z)^2, \\x^2y^2+y^2z^2+z^2x^2,\ x^4+8y^3z+8yz^3, x^4+16y^3z+64yz^3. \end{gather*} We also pose some conjectures for further research; for example, our 1-3-5-Conjecture states that any $n\in\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\in\mathbb N$ such that $x+3y+5z$ is a square.

Refining Lagrange's four-square theorem

Let p be any odd prime. We mainly show that $$\sum_{k=1}^{p-1}binomial(3k,k)*2^k/k=0 (mod p)$$ and $$\sum_{k=1}^{p-1}2^{k-1}C_k^{(2)}=(-1)^{(p-1)/2}-1 (mod p),$$ where $C_k^{(2)}=binomial(3k,k)/(2k+1)$ is the $k$th Catalan number of order 2.

Some congruences for the second-order Catalan numbers

It is known that $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)4^k)=\pi/2$ and $\sum_{k=0}^\infty\binom{2k}{k}/((2k+1)16^k)=\pi/3$. In this paper we obtain their p-adic analogues such as $$\sum_{p/2 3 is a prime and E_0,E_1,E_2,... are Euler numbers. Besides these, we also deduce some other congruences related to central binomial coefficients. In addition, we pose some conjectures one of which states that for any odd prime p we have $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=4x^2-2p (mod p^2)$$ if (p/7)=1 and p=x^2+7y^2 with x,y integers, and $$\sum_{k=0}^{p-1}\binom{2k}{k}^3=0 (mod p^2)$$ if (p/7)=-1, i.e., p=3,5,6 (mod 7).

Zhi-Wei Sun

Papers

Congruences for sums of binomial coefficients

An additive theorem and restricted sumsets

Refining Lagrange's four-square theorem

Some congruences for the second-order Catalan numbers

On congruences related to central binomial coefficients