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 A = {as + nsZ}s=1 (n1 6 · · · 6 nk) be a system of arithmetic sequences where a1, · · · , ak ∈ Z and n1, · · · , nk ∈ Z+. For m ∈ Z+ system A will be called an (exact) m-cover of Z if every integer is covered by A at least (exactly) m times. In this paper we reveal further connections between the common differences in an (exact) m-cover of Z and Egyptian fractions. Here are some typical results for those m-covers A of Z: (a) For any m1, · · · ,mk ∈ Z+ there are at least m positive integers in the form Σs∈Ims/ns where I ⊆ {1, · · · , k}. (b) When nk−l < nk−l+1 = · · · = nk (0 < l < k), either l > nk/nk−l or Σk−l s=11/ns > m, and for each positive integer λ < nk/nk−l the binomial coefficient ( l λ ) can be written as the sum of some denominators > 1 of the rationals Σs∈I1/ns − λ/nk, I ⊆ {1, · · · , k} if A forms an exact m-cover of Z. (c) If {as + nsZ}s=1 s6=t is not an m-cover of Z, then Σs∈I1/ns, I ⊆ {1, · · · , k} \\ {t} have at least nt distinct fractional parts and for each r = 0, 1, · · · , nt − 1 there exist I1, I2 ⊆ {1, · · · , k} \\ {t} such that r/nt ≡ Σs∈I11/ns − Σs∈I21/ns (mod 1). If A forms an exact m-cover of Z with m = 1 or n1 < · · · < nk−l < nk−l+1 = · · · = nk (l > 0) then for every t = 1, · · · , k and r = 0, 1, · · · , nt − 1 there is an I ⊆ {1, · · · , k} such that Σs∈I1/ns ≡ r/nt (mod 1). 1. Background and Introduction Let N = {0, 1, 2, · · · } be the set of natural numbers and Z = {1, 2, 3, · · · } the set of positve integers. For a ∈ Z and n ∈ Z, we call the infinite set a+ nZ = {a+ jn}+∞ j=−∞ = {· · · , a− 2n, a− n, a, a+ n, a+ 2n, · · · } an arithmetic sequence with common difference n (abbreviated to AS(n)) and its finite subsequence {a+ jn}l+m−1 j=m = {a+mn, a+ (m+ 1)n, · · · , a+ (l +m− 1)n} an arithmetic progression of length l with common difference n (in short, APn(l)). Let m be a positive integer. A finte system A = {as + nsZ}s=1 (a1, · · · , ak ∈ Z and n1, · · · , nk ∈ Z) (1) of arithmetic sequences is said to be an (exact) m-cover of S ⊆ Z if |{1 6 s 6 k : x ∈ as + nsZ}| > m (resp. |{1 6 s 6 k : x ∈ as + nsZ}| = m) for all x ∈ S. Instead of the term ‘1-cover’ we simply use the word ‘cover’. Received by the editors June 7, 1994 and, in revised form, November 10, 1995. 1991 Mathematics Subject Classification. Primary 11B25; Secondary 11A07, 11B75, 11D68. This research is supported by the National Natural Science Foundation of P. R. China. c ©1996 American Mathematical Society 4279

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Covering the integers by arithmetic sequences. II

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let p > 3 be a prime. We prove that

On sums involving products of three binomial coefficients

Congruences for Franel numbers

In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+d}}{64^k}\equiv 0\pmod{p^2}$$ for all $d\in\{0,\ldots,p-1\}$ with $d\equiv (p+1)/2\pmod2$. If $p\equiv 1\pmod4$ and $p=x^2+y^2$ with $x\equiv 1\pmod4$ and $y\equiv 0\pmod2$, then we show $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{2k}{k+1}}{(-8)^k}\equiv 2p-2x^2\pmod{p^2}\ \ \mbox{and}\ \ \sum_{k=0}^{p-1}\frac{\binom{2k}k\binom{2k}{k+1}^2}{(-8)^k}\equiv-2p\pmod{p^2}$$ by means of determining $x$ mod $p^2$ via $$(-1)^{(p-1)/4}\,x\equiv\sum_{k=0}^{(p-1)/2}\frac{k+1}{8^k}\binom{2k}k^2\equiv\sum_{k=0}^{(p-1)/2}\frac{2k+1}{(-16)^k}\binom{2k}k^2\pmod{p^2}.$$ We also solve the remaining open cases of Rodriguez-Villegas' conjectural congruences on $$\sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{3k}k}{108^k},\ \ \sum_{k=0}^{p-1}\frac{\binom{2k}k^2\binom{4k}{2k}}{256^k}, \ \ \sum_{k=0}^{p-1}\frac{\binom{2k}{k}\binom{3k}k\binom{6k}{3k}}{12^{3k}}$$ modulo $p^2$.

Let p be a prime A famous theorem of Lucas states that (np+t) (m) (t) (modp) if m, n, s, t are nonnegative integers with s, t < p In this paper we aim to prove a similar result for generalized binomial coefficients defined in terms of second order recurrent sequences with initial values 0 and 1

/pdf/an-extension-of-lucas-theorem-4yk7m5gc6p.pdf

Zhi-Wei Sun

Papers

Covering the integers by arithmetic sequences. II

On sums involving products of three binomial coefficients

Congruences for Franel numbers

On sums involving products of three binomial coefficients

An extension of Lucas’ theorem