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

Combinatorial Mathematics: THE PRINCIPLE OF INCLUSION AND EXCLUSION

Arithmetic circuits: The chasm at depth four gets wider

In their paper on the "chasm at depth four", Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2^o(m) also admit arithmetic circuits of depth four and size 2^o(m). This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense the general case. In this paper we show that smaller depth four circuits can be obtained if we start from polynomial size arithmetic circuits. For instance, we show that if the permanent of n*n matrices has circuits of size polynomial in n, then it also has depth 4 circuits of size n^O(sqrt(n)*log(n)). Our depth four circuits use integer constants of polynomial size. These results have potential applications to lower bounds and deterministic identity testing, in particular for sums of products of sparse univariate polynomials. We also give an application to boolean circuit complexity, and a simple (but suboptimal) reduction to polylogarithmic depth for arithmetic circuits of polynomial size and polynomially bounded degree.

Arithmetic circuits: the chasm at depth four gets wider

FELIX KLEIN'S famous enunciation—“Given a manifold and a group of transformations of the same, to develop the theory of invariants relating to that group”—not only describes the spirit of geometry but also reveals that the concept of a group dominated at least one important branch of mathematics. In the fundamental structure of other branches, however, groups are to be found, and the general concepts developed during the latter half of the last century both by Klein and his student friend, Sophus Lie, were responsible for the far-reaching influence which the relevant theory has had on modern mathematical thought. It is not always easy to discover the first important application of a newly developed theory but it is probable that the formulation of the mechanics of the atom made by Heisenberg in 1926 by the application of the theory of matrices—the invention of Cayley in 1858—is a fair illustration of the excursion of that theory into the practical domain. The Theory of Group Representations By Prof. Francis D. Murnaghan. Pp. xi + 369. (Baltimore, Md.: Johns Hopkins Press; London: Oxford University Press, 1938.) 22s. 6d. net.

The Theory of Group Representations

It is shown that any weakly-skew circuit can be converted into a skew circuit with constant factor overhead, while preserving either syntactic or semantic multilinearity. This leads to considering syntactically multilinear algebraic branching programs (ABPs), which are defined by a natural read-once property. A 2n/4size lower bound is proven for ordered syntactically multilinear ABPs computing an explicitly constructed multilinear polynomial in 2nvariables. Without the ordering restriction a lower bound of level i¾?(n3/2/logn) is observed, by considering a generalization of a hypercube covering problem by Galvin [1].

Lower Bounds for Syntactically Multilinear Algebraic Branching Programs

We study structural properties of restricted width arithmetical circuits. It is shown that syntactically multilinear arithmetical circuits of constant width can be efficiently simulated by syntactically multilinear algebraic branching programs of constant width, i.e. that sm-VSC 0 *** sm-VBWBP. Also, we obtain a direct characterization of poly -size arithmetical formulas in terms of multiplicatively disjoint constant width circuits, i.e. MD-VSC0 = VNC1.

For log -width weakly-skew circuits a width efficient multilinearity preserving simulation by algebraic branching programs is given, i.e. weaklyskew-sm-VSC1 *** sm-VBP[width=log2 n ].

Finally, coefficient functions are considered, and closure properties are observed for sm-VSC i , and in general for a variety of other syntactic multilinear restrictions of algebraic complexity classes.

Simulation of Arithmetical Circuits by Branching Programs with Preservation of Constant Width and Syntactic Multilinearity

http://www.cse.buffalo.edu/~regan/papers/pdf/JaRe06-2.pdf

A nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle

We expand the lower bound frontier in arithmetical circuit complexity for several recently considered restricted computational models. The main results are the following. 
First, Shpilka and Wigderson's partial derivative method for sigma-pi-sigma formulas is refined to include additive complexity in the lower bound. Consequently, we strengthen some lower bounds on total sigma-pi-sigma formula size. A supplementary method is derived, that is shown to yield quadratic lower bounds for polynomials for which the partial derivatives method fails. 
Second, we continue the recent study of bilinear circuits with bounded coefficients. Generic lower bounds are proved for bilinear maps that are in the orbit of the circular convolution mapping. This leads to a study of universal-existential quantified predicates about minors of the discrete Fourier matrix. We prove a quantative lower bound on the expected value of the determinant of certain random Vandermonde matrices with nodes on the unit circle in the complex plane. This result is used to establish circuit lower bounds. 
Finally, bounded depth circuits are investigated. We prove a nonlinear lower bound for circular convolution in case the input length is prime. The proof uses a superconcentrator lemma by Raz and Shpilka and the discrete uncertainty principle for cyclic groups of prime order by Tao.

Lower bound frontiers in arithmetical circuit complexity

The determinantal complexity of a polynomial f(x 1,x 2,…,x n ) is the minimum m such that f=det m (L(x 1,x 2,…,x n )), where L(x 1,x 2,…,x n ) is a matrix whose entries are affine forms in the x i s over some field $\mbox {$\mathbb {F}$}$.

Asymptotically tight lower bounds are proven for the determinantal complexity of the elementary symmetric polynomial $S^{d}_{n}$ of degree d in n variables, 2d-fold iterated matrix multiplication of the form 〈u|X 1 X 2…X 2d |v〉, and the symmetric power sum polynomial $\sum_{i=1}^{n} x_{i}^{d}$, for any fixed d>1.

A restriction of determinantal computation is considered in which the underlying affine map λx.L(x) must satisfy a rank lowerability property: L mapping to m×m matrices is said to be r-lowerable, if there exists an $a\in \mbox {$\mathbb {F}$}^{n}$ such that rank (L(a))≤m−r. In this model strongly nonlinear and exponential lower bounds are proved for several polynomial families. For example, for $S^{2d}_{n}$ it is proved that the determinantal complexity using r-lowerable maps is Ω(n d/(2d−r)), for constants d and r with 2≤d+1≤r<2d. For r=2d−1 and d=⌊n 1/5−e ⌋, a lower bound is given for $S^{2d}_{n}$ of magnitude $n^{\Omega(\epsilon n^{1/5-\epsilon})}$, for any e∈(0,1/5).

Maurice J. Jansen

Papers

Lower Bounds for Syntactically Multilinear Algebraic Branching Programs

Simulation of Arithmetical Circuits by Branching Programs with Preservation of Constant Width and Syntactic Multilinearity

A nonlinear lower bound for constant depth arithmetical circuits via the discrete uncertainty principle

Lower bound frontiers in arithmetical circuit complexity

Lower Bounds for the Determinantal Complexity of Explicit Low Degree Polynomials