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

Molecular self-assembly is the spontaneous association of molecules under equilibrium conditions into stable, structurally well-defined aggregates joined by noncovalent bonds. Molecular self-assembly is ubiquitous in biological systems and underlies the formation of a wide variety of complex biological structures. Understanding self-assembly and the associated noncovalent interactions that connect complementary interacting molecular surfaces in biological aggregates is a central concern in structural biochemistry. Self-assembly is also emerging as a new strategy in chemical synthesis, with the potential of generating nonbiological structures with dimensions of 1 to 10(2) nanometers (with molecular weights of 10(4) to 10(10) daltons). Structures in the upper part of this range of sizes are presently inaccessible through chemical synthesis, and the ability to prepare them would open a route to structures comparable in size (and perhaps complementary in function) to those that can be prepared by microlithography and other techniques of microfabrication.

Molecular self-assembly and nanochemistry: A chemical strategy for the synthesis of nanostructures

Computational geometry

In this paper we propose a three participants variation of the Diffie--Hellman protocol. This variation is based on the Weil and Tate pairings on elliptic curves, which were first used in cryptography as cryptanalytic tools for reducing the discrete logarithm problem on some elliptic curves to the discrete logarithm problem in a finite field.

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A One Round Protocol for Tripartite Diffie–Hellman

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 Rothemund and Winfree [6] have considered the program size complexity of constructing squares by self-assembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model defined in [6]. In the generalized model, the Rothemund-Winfree construction of n \times n squares requires time T(n log n) and program size T(log n). We present a new construction for assembling n \times n squares which uses optimal time T(n) and program size T(\frac{log n}{log log n}). This program size is also optimal since it matches the bound dictated by Kolmogorov complexity. Our improved time is achieved by demonstrating a set of tiles for parallel self-assembly of binary counters. Our improved program size is achieved by demonstrating that self-assembling systems can compute changes in the base representation of numbers. Self-assembly is emerging as a useful paradigm for computation. In addition the development of a computational theory of self-assembly promises to provide a new conduit by which results and methods of theoretical computer science might be applied to problems of interest in biology and the physical sciences.

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Running time and program size for self-assembled squares

In this paper, we study the complexity of self-assembly under models that are natural generalizations of the tile self-assembly model. In particular, we extend Rothemund and Winfree's study of the tile complexity of tile self-assembly [Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, Portland, OR, 2000, pp. 459--468]. They provided a lower bound of $\Omega(\frac{\log N}{\log\log N})$ on the tile complexity of assembling an $N\times N$ square for almost all N. Adleman et al. [Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, Heraklion, Greece, 2001, pp. 740--748] gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size $O(\sqrt{\log N})$ which assembles an $N\times N$ square in a model which allows flexible glue strength between nonequal glues. This result is matched for almost all N by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the $\Omega(\frac{\log N}{\log\log N})$ lower bound applies to $N\times N$ squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of $\Omega(\frac{N^{1/k}}{k})$ for the standard model, yet we also give a construction which achieves $O(\frac{\log N}{\log\log N})$ complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape; we show that this problem is NP-hard for three of the generalized models.

Complexities for Generalized Models of Self-Assembly

Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.Self-assembly is the ubiquitous process by which simple objects autonomously assemble into intricate complexes. It has been suggested that intricate self-assembly processes will ultimately be used in circuit fabrication, nano-robotics, DNA computation, and amorphous computing. In this paper, we study two combinatorial optimization problems related to efficient self-assembly of shapes in the Tile Assembly Model of self-assembly proposed by Rothemund and Winfree [18]. The first is the Minimum Tile Set Problem, where the goal is to find the smallest tile system that uniquely produces a given shape. The second is the Tile Concentrations Problem, where the goal is to decide on the relative concentrations of different types of tiles so that a tile system assembles as quickly as possible. The first problem is akin to finding optimum program size, and the second to finding optimum running time for a "program" to assemble the shape.We prove that the first problem is NP-complete in general, and polynomial time solvable on trees and squares. In order to prove that the problem is in NP, we present a polynomial time algorithm to verify whether a given tile system uniquely produces a given shape. This algorithm is analogous to a program verifier for traditional computational systems, and may well be of independent interest. For the second problem, we present a polynomial time $O(\log n)$-approximation algorithm that works for a large class of tile systems that we call partial order systems.

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Combinatorial optimization problems in self-assembly

Self-assembly is a ubiquitous process by which objects autonomously assemble into complexes. In the context of computation, self-assembly is important to both DNA computing and amorphous computing. Thus a well developed mathematical theory of self-assembly will be useful in these and other domains. As a first problem in self-assembly, we will explore the creation of linear polymers. Polymers are chains of molecular units. For examples, a molecule of DNA is a polymer made from the bases adenine, guanine, cytosine and thymine; and proteins are polymers formed from the twenty amino acids. In our model the dynamics of a linear polymerization system is determined by difference and differential equations with initial conditions, and we investigate the equilibrium behavior of such a system. A good characterization of the equilibria of these systems will allow us to predict which computations can be carried out with adequate yields and what quantities of initial substrates these computations require. The rates at which such systems approach equilibria 1Research supported by grants from NASA/JPL, NSF, ONR, and DARPA. 2Research supported by NSF Grant CCR-9820778. 3Research supported by NSF Grant CCR-9820778. 52 L. Adleman, Q. Cheng, A. Goel, M.-D. Huang and H. Wasserman are equally important since convergence rates are an obvious measure of the time complexity of computation through self-assembly.

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Linear Self-Assemblies: Equilibria, Entropy and Convergence Rates

For generalized Reed-Solomon codes, it has been proved [7] that the problem of determining if a received word is a deep hole is co-NP-complete. The reduction relies on the fact that the evaluation set of the code can be exponential in the length of the code - a property that practical codes do not usually possess. In this paper, we first present a much simpler proof of the same result. We then consider the problem for standard Reed-Solomon codes, i.e. the evaluation set consists of all the nonzero elements in the field. We reduce the problem of identifying deep holes to deciding whether an absolutely irreducible hypersurface over a finite field contains a rational point whose coordinates are pairwise distinct and nonzero. By applying Cafure-Matera estimation of rational points on algebraic varieties, we prove that the received vector (f(α))α∈Fpfor the Reed-Solomon [p - 1, k]p, k < p1/4-Ɛ, cannot be a deep hole, whenever f(x) is a polynomial of degree k + d for 1 ≤ d ≤ p3/13-Ɛ.

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Qi Cheng

Papers

Running time and program size for self-assembled squares

Complexities for Generalized Models of Self-Assembly

Combinatorial optimization problems in self-assembly

Linear Self-Assemblies: Equilibria, Entropy and Convergence Rates

On deciding deep holes of Reed-Solomon codes