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

We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

Introduction A fundamental objective of cryptography is to enable two persons to communicate over an insecure channel (a public channel such as the internet) in such a way that any other person is unable to recover their message (called the plaintext ) from what is sent in its place over the channel (the ciphertext ). The transformation of the plaintext into the ciphertext is called encryption , or enciphering. Encryption-decryption is the most ancient cryptographic activity (ciphers already existed four centuries b.c.), but its nature has deeply changed with the invention of computers, because the cryptanalysis (the activity of the third person, the eavesdropper, who aims at recovering the message) can use their power. The encryption algorithm takes as input the plaintext and an encryption key K E , and it outputs the ciphertext. If the encryption key is secret, then we speak of conventional cryptography , of private key cryptography , or of symmetric cryptography . In practice, the principle of conventional cryptography relies on the sharing of a private key between the sender of a message (often called Alice in cryptography) and its receiver (often called Bob). If, on the contrary, the encryption key is public, then we speak of public key cryptography . Public key cryptography appeared in the literature in the late 1970s.

/pdf/boolean-models-and-methods-in-mathematics-computer-science-3vbrnbyptk.pdf

Boolean Models and Methods in Mathematics, Computer Science, and Engineering: Boolean Functions for Cryptography and Error-Correcting Codes

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

Recently, it became apparent that a large number of the most interesting structures and phenomena of the world can be described by networks. To develop a mathematical theory of very large networks is an important challenge. This book describes one recent approach to this theory, the limit theory of graphs, which has emerged over the last decade. The theory has rich connections with other approaches to the study of large networks, such as "property testing" in computer science and regularity partition in graph theory. It has several applications in extremal graph theory, including the exact formulations and partial answers to very general questions, such as which problems in extremal graph theory are decidable. It also has less obvious connections with other parts of mathematics (classical and non-classical, like probability theory, measure theory, tensor algebras, and semidefinite optimization). This book explains many of these connections, first at an informal level to emphasize the need to apply more advanced mathematical methods, and then gives an exact development of the theory of the algebraic theory of graph homomorphisms and of the analytic theory of graph limits. This is an amazing book: readable, deep, and lively. It sets out this emerging area, makes connections between old classical graph theory and graph limits, and charts the course of the future. --Persi Diaconis, Stanford University This book is a comprehensive study of the active topic of graph limits and an updated account of its present status. It is a beautiful volume written by an outstanding mathematician who is also a great expositor. --Noga Alon, Tel Aviv University, Israel Modern combinatorics is by no means an isolated subject in mathematics, but has many rich and interesting connections to almost every area of mathematics and computer science. The research presented in Lovasz's book exemplifies this phenomenon. This book presents a wonderful opportunity for a student in combinatorics to explore other fields of mathematics, or conversely for experts in other areas of mathematics to become acquainted with some aspects of graph theory. --Terence Tao, University of California, Los Angeles, CA Laszlo Lovasz has written an admirable treatise on the exciting new theory of graph limits and graph homomorphisms, an area of great importance in the study of large networks. It is an authoritative, masterful text that reflects Lovasz's position as the main architect of this rapidly developing theory. The book is a must for combinatorialists, network theorists, and theoretical computer scientists alike. --Bela Bollobas, Cambridge University, UK

Large Networks and Graph Limits

In this paper, we prove a local limit theorem for the distribution of the number of triangles in the Erdos-Renyi random graph $G(n,p)$, where $p \in (0,1)$ is a fixed constant. Our proof is based on bounding the characteristic function $\psi(t)$ of the number of triangles, and uses several different conditioning arguments for handling different ranges of $t$.

A local central limit theorem for the number of triangles in a random graph

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space U is δ-far in relative Hamming distance from a linear code V - this is the worst-case assumption - then most elements of U are almost-δ-far from V - this is the average case. However, this result was known to hold only below the "double Johnson" function of the relative distance δ_V of the code V, i.e., only when δ < 1-(1-δ_V)^(1/4).
First, we increase the soundness-bound to the "one-and-a-half Johnson" function of δ_V and show that the average distance of U from V is nearly δ for any worst-case distance δ smaller than 1-(1-δ_V)^(1/3). This bound is tight, which is somewhat surprising because the one-and-a-half Johnson function is unfamiliar in the literature on error correcting codes.
To improve soundness further for Reed Solomon codes we sample outside the box. We suggest a new protocol in which the verifier samples a single point z outside the box D on which codewords are evaluated, and asks the prover for the value at z of the interpolating polynomial of a random element of U. Intuitively, the answer provided by the prover "forces" it to choose one codeword from a list of "pretenders" that are close to U. We call this technique Domain Extending for Eliminating Pretenders (DEEP).
The DEEP method improves the soundness of the worst-case-to-average-case reduction for RS codes up their list decoding radius. This radius is bounded from below by the Johnson bound, implying average distance is approximately δ for all δ < 1-(1-δ_V)^(1/2). Under a plausible conjecture about the list decoding radius of Reed-Solomon codes, average distance from V is approximately δ for all δ. The DEEP technique can be generalized to all linear codes, giving improved reductions for capacity-achieving list-decodable codes.
Finally, we use the DEEP technique to devise two new protocols: 
- An Interactive Oracle Proof of Proximity (IOPP) for RS codes, called DEEP-FRI. The soundness of the protocol improves upon that of the FRI protocol of [Ben-Sasson et al., ICALP 2018] while retaining linear arithmetic proving complexity and logarithmic verifier arithmetic complexity. 
- An Interactive Oracle Proof (IOP) for the Algebraic Linking IOP (ALI) protocol used to construct zero knowledge scalable transparent arguments of knowledge (ZK-STARKs) in [Ben-Sasson et al., eprint 2018]. The new protocol, called DEEP-ALI, improves soundness of this crucial step from a small constant < 1/8 to a constant arbitrarily close to 1.

https://drops.dagstuhl.de/opus/volltexte/2020/11690/pdf/LIPIcs-ITCS-2020-5.pdf/

DEEP-FRI: Sampling outside the box improves soundness

Given a pair of finite groups G and H, the set of homomorphisms from G to H form an error-correcting code where codewords differ in at least 1/2 the coordinates. We show that for every pair of abelian groups G and H, the resulting code is (locally) list-decodable from a fraction of errors arbitrarily close to its distance. At the heart of this result is the following combinatorial result: There is a fixed polynomial p(•) such that for every pair of abelian groups G and H, if the maximum fraction of agreement between two distinct homomorphisms from G to H is Λ, then for every e> 0 and every function f:G -> H, the number of homomorphisms that have agreement Λ + e with f is at most p(1/e). We thus give a broad class of codes whose list-decoding radius exceeds the "Johnson bound". Examples of such codes are rare in the literature, and for the ones that do exist, "combinatorial" techniques to analyze their list-decodability are limited. Our work is an attempt to add to the body of such techniques. We use the fact that abelian groups decompose into simpler ones and thus codes derived from homomorphisms over abelian groups may be viewed as certain "compositions" of simpler codes. We give techniques to lift list-decoding bounds for the component codes to bounds for the composed code. We believe these techniques may be of general interest.

Decodability of Group Homomorphisms beyond the Johnson Bound.

We consider codes for space bounded channels. This is a model for communication under noise that was studied by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in one pass, and modifies at most a p fraction of the bits of the codeword. Guruswami and Smith, and later work by Shaltiel and Silbak (RANDOM 2016), gave constructions of list-decodable codes with rate approaching 1-H(p) against channels with space s=c log n, with encoding/decoding time poly(2^s)= poly(n^c). In this paper we show that for every constant 0 ≤ p < ½, and every sufficiently small constant ε>0, there are codes with rate R ≥ 1-H(p)-ε, list size poly(1/ε), and furthermore: • Our codes can handle channels with space s=n^Ω(1), which is much larger than O(log n) achieved by previous work. • We give encoding and decoding algorithms that run in time n polylog(n). Previous work achieved large and unspecified poly(n) time (even for space s=1 ⋅ log n channels). • We can handle space bounded channels that read the codeword in any order, whereas previous work considered channels that read the codeword in the standard order. Our construction builds on the machinery of Guruswami and Smith (with some key modifications) replacing some nonconstructive codes and pseudorandom objects (that are found in exponential time by brute force) with efficient explicit constructions. For this purpose we exploit recent results of Haramaty, Lee and Viola (SICOMP 2018) on pseudorandom properties of “t-wise independence + low weight noise'' which we quantitatively improve using techniques by Forbes and Kelly (FOCS 2018). To make use of such distributions, we give new explicit constructions of binary linear codes that have dual distance of n^Ω(1), and are also polynomial time list-decodable from relative distance ½ -ε, with list size poly(1/ε). To the best of our knowledge, no such construction was previously known. Somewhat surprisingly, we show that Reed-Solomon codes with dimension k<√n, have this property if interpreted as binary codes (in some specific interpretation) which we term: “Raw Reed-Solomon Codes''. A key idea is viewing Reed-Solomon codes as “bundles'' of certain dual-BCH codewords.

Quasilinear time list-decodable codes for space bounded channels.

Algebra and randomness come together rather nicely in computation A central example of this relationship in action is the Schwartz-Zippel lemma and its application to the fast randomized checking of polynomial identities In this thesis, we further this relationship in two ways: (1) by compiling new algebraic techniques that are of potential computational interest, and (2) demonstrating the relevance of these techniques by making progress on several questions in randomness and pseudorandomness The technical ingredients we introduce include: (1) Multiplicity-enhanced versions of the Schwartz-Zippel lemma and the "polynomial method", extending their applicability to "higher-degree" polynomials (2) Conditions for polynomials to have an unusually small number of roots (3) Conditions for polynomials to have an unusually structured set of roots, eg, containing a large linear space Our applications include: (1) Explicit constructions of randomness extractors with logarithmic seed and vanishing "entropy loss" (2) Limit laws for first-order logic augmented with the parity quantifier on random graphs (extending the classical 0-1 law) (3) Explicit dispersers for affine sources of imperfect randomness with sublinear entropy (Copies available exclusively from MIT Libraries, Rm 14-0551, Cambridge, MA 02139-4307 Ph 617-253-5668; Fax 617-253-1690)

Swastik Kopparty

Papers

A local central limit theorem for the number of triangles in a random graph

DEEP-FRI: Sampling outside the box improves soundness

Decodability of Group Homomorphisms beyond the Johnson Bound.

Quasilinear time list-decodable codes for space bounded channels.

Algebraic methods in randomness and pseudorandomness