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. A p-adic congruence is called a super congruence if it happens to hold modulo some higher power of p. The topic of super congruences is related to many fields including Gauss and Jacobi sums and hypergeometric series. We prove that 

$$\begin{gathered} \sum\limits_{k = 0}^{p - 1} {\frac{{\left( {_k^{2k} } \right)}} {{2^k }}} \equiv \left( { - 1} \right)^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} - p^2 E_{p - 3} \left( {\bmod p^3 } \right), \hfill \\ \sum\limits_{k = 1}^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} {\frac{{\left( {_k^{2k} } \right)}} {k}} \equiv \left( { - 1} \right)^{{{\left( {p + 1} \right)} \mathord{\left/ {\vphantom {{\left( {p + 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} \frac{8} {3}pE_{p - 3} \left( {\bmod p^2 } \right), \hfill \\ \sum\limits_{k = 0}^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} {\frac{{\left( {_k^{2k} } \right)^2 }} {{16^k }}} \equiv \left( { - 1} \right)^{{{\left( {p - 1} \right)} \mathord{\left/ {\vphantom {{\left( {p - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} + p^2 E_{p - 3} \left( {\bmod p^3 } \right), \hfill \\ \end{gathered}$$

where E0,E1,E2, ... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for π2, π−2 and the constant $K: = \sum\nolimits_{k = 1}^\infty {{{\left( {\tfrac{k} {3}} \right)} \mathord{\left/ {\vphantom {{\left( {\tfrac{k} {3}} \right)} {k^2 }}} \right. \kern-\nulldelimiterspace} {k^2 }}}$ (with (−) the Jacobi symbol), two of which are 

$$\sum\limits_{k = 1}^\infty {\frac{{\left( {10k - 3} \right)8^k }} {{k^3 \left( {_k^{2k} } \right)^2 \left( {_k^{3k} } \right)}} = \frac{{\pi ^2 }} {2}} and \sum\limits_{k = 1}^\infty {\frac{{\left( {15k - 4} \right)\left( { - 27} \right)^{k - 1} }} {{k^3 \left( {_k^{2k} } \right)^2 \left( {_k^{3k} } \right)}} = K.}$$

Super congruences and Euler numbers

Congruences concerning Bernoulli numbers and Bernoulli polynomials

Congruences involving Bernoulli and Euler numbers

numbers. As applications we obtain a new formula for the Fibonacci quotient Fp−( 5 p )/p and a criterion for the relation p |F(p−1)/4 (if p ≡ 1 (mod 4)), where p 6= 5 is an odd prime. We also prove that the affirmative answer to Wall’s question implies the first case of FLT (Fermat’s last theorem); from this it follows that the first case of FLT holds for those exponents which are (odd) Fibonacci primes or Lucas primes.

/pdf/fibonacci-numbers-and-fermat-s-last-theorem-19w3ry0b1w.pdf

Fibonacci numbers and Fermat's last theorem

Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\binom{2k}{k}/2^k=(-1)^{(p-1)/2}-p^2E_{p-3} (mod p^3),$$ $$\sum_{k=1}^{(p-1)/2}\binom{2k}{k}/k=(-1)^{(p+1)/2}8/3*pE_{p-3} (mod p^2),$$ $$\sum_{k=0}^{(p-1)/2}\binom{2k}{k}^2/16^k=(-1)^{(p-1)/2}+p^2E_{p-3} (mod p^3)$$, where E_0,E_1,E_2,... are Euler numbers. Our new approach is of combinatorial nature. We also formulate many conjectures concerning super congruences and relate most of them to Euler numbers or Bernoulli numbers. Motivated by our investigation of super congruences, we also raise a conjecture on 7 new series for $\pi^2$, $\pi^{-2}$ and the constant $K:=\sum_{k>0}(k/3)/k^2$ (with (-) the Jacobi symbol), two of which are $$\sum_{k=1}^\infty(10k-3)8^k/(k^3\binom{2k}{k}^2\binom{3k}{k})=\pi^2/2$$ and $$\sum_{k>0}(15k-4)(-27)^{k-1}/(k^3\binom{2k}{k}^2\binom{3k}k)=K.$$

Zhi-Wei Sun

Papers

Super congruences and Euler numbers

Congruences concerning Bernoulli numbers and Bernoulli polynomials

Congruences involving Bernoulli and Euler numbers

Fibonacci numbers and Fermat's last theorem

Super congruences and Euler numbers