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

A major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists, since expander graphs, the protagonists of our story, come up in numerous and often surprising contexts in both fields But, perhaps, we should start with a few words about graphs in general They are, of course, one of the prime objects of study in Discrete Mathematics However, graphs are among the most ubiquitous models of both natural and human-made structures In the natural and social sciences they model relations among species, societies, companies, etc In computer science, they represent networks of communication, data organization, computational devices as well as the flow of computation, and more In mathematics, Cayley graphs are useful in Group Theory Graphs carry a natural metric and are therefore useful in Geometry, and though they are “just” one-dimensional complexes, they are useful in certain parts of Topology, eg Knot Theory In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such systems The study of these models calls, then, for the comprehension of the significant structural properties of the relevant graphs But are there nontrivial structural properties which are universally important? Expansion of a graph requires that it is simultaneously sparse and highly connected Expander graphs were first defined by Bassalygo and Pinsker, and their existence first proved by Pinsker in the early ’70s The property of being an expander seems significant in many of these mathematical, computational and physical contexts It is not surprising that expanders are useful in the design and analysis of communication networks What is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness In mathematics, we will encounter eg their role in the study of metric embeddings, and in particular in work around the Baum-Connes Conjecture Expansion is closely related to the convergence rates of Markov Chains, and so they play a key role in the study of Monte-Carlo algorithms in statistical mechanics and in a host of practical computational applications The list of such interesting and fruitful connections goes on and on with so many applications we will not even

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Expander graphs and their applications

We develop an automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction. Using this approach we obtain a new improved bound on the matrix multiplication exponent ω

/pdf/multiplying-matrices-faster-than-coppersmith-winograd-1xvjqg5bxt.pdf

Multiplying matrices faster than coppersmith-winograd

This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound $\omega<2.3728639$ on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

Powers of Tensors and Fast Matrix Multiplication

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions.

The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we discuss the recent lower bounds for multilinear circuits and formulas, the advances in the question of deterministically checking polynomial identities, and the results regarding reconstruction of arithmetic circuits. We do, however, also cover part of the classical works on arithmetic circuits. In order to keep this monograph at a reasonable length, we do not give full proofs of most theorems, but rather try to convey the main ideas behind each proof and demonstrate it, where possible, by proving some special cases.

https://www.cs.tau.ac.il/~shpilka/publications/SY10.pdf

Arithmetic Circuits: A Survey of Recent Results and Open Questions

In this work we study two, seemingly unrelated, notions Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions In particular we obtain the following results: (1) We show that if $E: \mathbb{F}^n \mapsto \mathbb{F}^m$ is a linear LDC with two queries, then $m = \exp(\Omega(n))$ Previously this was known only for fields of size $\ll 2^n$ [O Goldreich e, Comput Complexity, 15 (2006), pp 263-296] (2) We show that from every depth 3 arithmetic circuit ($\Sigma\Pi\Sigma$ circuit), ${\cal C}$, with a bounded (constant) top fan-in that computes the zero polynomial, one can construct an LDC More formally, assume that ${\cal C}$ is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates) Denote by $d$ the degree of the polynomial computed by ${\cal C}$ and by $r$ the rank of the linear functions appearing in ${\cal C}$ Then we can construct a linear LDC with two queries that encodes messages of length $r/{\operatorname{polylog}(d)}$ by codewords of length $O(d)$ (3) We prove a structural theorem for $\Sigma\Pi\Sigma$ circuits, with a bounded top fan-in, that compute the zero polynomial In particular we show that if such a circuit is simple, minimal, and of polynomial size, then its rank, $r$, is only polylogarithmic in the number of variables (a priori it could have been linear) (4) We give new PIT algorithms for $\Sigma\Pi\Sigma$ circuits with a bounded top fan-in: (a) a deterministic algorithm that runs in quasipolynomial time, and (b) a randomized algorithm that runs in polynomial time and uses only a polylogarithmic number of random bits Moreover, when the circuit is multilinear, our deterministic algorithm runs in polynomial time Previously deterministic subexponential time algorithms for PIT in bounded depth circuits were known only for depth 2 circuits (in the black box model) [D Grigoriev, M Karpinski, and M F Singer, SIAM J Comput, 19 (1990), pp 1059-1063; M Ben-Or and P Tiwari, Proceedings of the 20th Annual ACM Symposium on Theory of Computing, ACM Press, New York, 1988, pp 301-309; A R Klivans and D Spielman, Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2001, pp 216-223] In particular, for the special case of depth 3 circuits with three multiplication gates our result resolves an open question asked by Klivans and Spielman

Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits

We give a deterministic polynomial time algorithm for polynomial identity testing in the following two cases: 
Non-commutative arithmetic formulas: The algorithm gets as an input an arithmetic formula in the non-commuting variables x1,···,xn and determines whether or not the output of the formula is identically 0 (as a formal expression)
Pure arithmetic circuits: The algorithm gets as an input a pure set-multilinear arithmetic circuit (as defined by Nisan and Wigderson) in the variables x1,···,xn and determines whether or not the output of the circuit is identically 0 (as a formal expression).



One application is a deterministic polynomial time identity testing for set-multilinear arithmetic circuits of depth 3. We also give a deterministic polynomial time identity testing algorithm for non-commutative algebraic branching programs as defined by Nisan. Finally, we obtain an exponential lower bound for the size of pure setmultilinear arithmetic circuits for the permanent and for the determinant. (Only lower bounds for the depth of pure circuits were previously known.)

Deterministic polynomial identity testing in non-commutative models

In this paper we prove quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero. Such bounds are obtained for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant. As corollaries we get the first nontrivial lower bounds for computing polynomials of constant degree, and a gap between the power of depth-3 arithmetic circuits and depth-4 arithmetic circuits. We also give new shorter formulae of constant depth for the elementary symmetric functions.¶The main technical contribution relates the complexity of computing a polynomial in this model to the wealth of partial derivatives it has on every affine subspace of small co-dimension. Lower bounds for related models utilize an algebraic analog of the Neciporuk lower bound on Boolean formulae.

Depth-3 arithmetic circuits over fields of characteristic zero

We consider the problem of approximating the support size of a distribution from a small number of samples, when each element in the distribution appears with probability at least $\frac{1}{n}$. This problem is closely related to the problem of approximating the number of distinct elements in a sequence of length $n$. Charikar, Chaudhuri, Motwani, and Narasayya [in Proceedings of the Nineteenth ACM SIGMOD-SIGACT-SIGART Symposium on Principles of Database Systems, 2000, pp. 268-279] and Bar-Yossef, Kumar, and Sivakumar [in Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM Press, New York, 2001, pp. 266-275] proved that multiplicative approximation for these problems within a factor $\alpha>1$ requires $\Theta(\frac{n}{\alpha^2})$ queries to the input sequence. Their lower bound applies only when the number of distinct elements (or the support size of a distribution) is very small. For both problems, we prove a nearly linear in $n$ lower bound on the query complexity, applicable even when the number of distinct elements is large (up to linear in $n$) and even for approximation with additive error. At the heart of the lower bound is a construction of two positive integer random variables, $\mathsf{X}_1$ and $\mathsf{X}_2$, with very different expectations and the following condition on the first $k$ moments: $\mathsf{E}[\mathsf{X}_1]/\mathsf{E}[\mathsf{X}_2] = \mathsf{E}[\mathsf{X}_1^2]/\mathsf{E}[\mathsf{X}_2^2] = \cdots = \mathsf{E}[\mathsf{X}_1^k]/\E[\mathsf{X}_2^k]$. It is related to a well-studied mathematical question, the truncated Hamburger problem, but differs in the requirement that our random variables have to be supported on integers. Our lower bound method is also applicable to other problems and, in particular, gives a new lower bound for the sample complexity of approximating the entropy of a distribution.

Amir Shpilka

Papers

Arithmetic Circuits: A Survey of Recent Results and Open Questions

Locally Decodable Codes with Two Queries and Polynomial Identity Testing for Depth 3 Circuits

Deterministic polynomial identity testing in non-commutative models

Depth-3 arithmetic circuits over fields of characteristic zero

Strong Lower Bounds for Approximating Distribution Support Size and the Distinct Elements Problem