High-throughput mRNA sequencing (RNA-Seq) promises simultaneous transcript discovery and abundance estimation. However, this would require algorithms that are not restricted by prior gene annotations and that account for alternative transcription and splicing. Here we introduce such algorithms in an open-source software program called Cufflinks. To test Cufflinks, we sequenced and analyzed >430 million paired 75-bp RNA-Seq reads from a mouse myoblast cell line over a differentiation time series. We detected 13,692 known transcripts and 3,724 previously unannotated ones, 62% of which are supported by independent expression data or by homologous genes in other species. Over the time series, 330 genes showed complete switches in the dominant transcription start site (TSS) or splice isoform, and we observed more subtle shifts in 1,304 other genes. These results suggest that Cufflinks can illuminate the substantial regulatory flexibility and complexity in even this well-studied model of muscle development and that it can improve transcriptome-based genome annotation.

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Transcript assembly and quantification by RNA-Seq reveals unannotated transcripts and isoform switching during cell differentiation

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A Course in Universal Algebra

Lattice theory today reflects the general Status of current mathematics: there is a rich production of theoretical concepts, results, and developments, many of which are reached by elaborate mental gymnastics; on the other hand, the connections of the theory to its surroundings are getting weaker and weaker, with the result that the theory and even many of its parts become more isolated. Restructuring lattice theory is an attempt to reinvigorate connections with our general culture by interpreting the theory as concretely as possible, and in this way to promote better communication between lattice theorists and potential users of lattice theory.

Restructuring lattice theory: an approach based on hierarchies of concepts

The theory of directed graphs has developed enormously over recent decades, yet this book (first published in 2000) remains the only book to cover more than a small fraction of the results. New research in the field has made a second edition a necessity. Substantially revised, reorganised and updated, the book now comprises eighteen chapters, carefully arranged in a straightforward and logical manner, with many new results and open problems. As well as covering the theoretical aspects of the subject, with detailed proofs of many important results, the authors present a number of algorithms, and whole chapters are devoted to topics such as branchings, feedback arc and vertex sets, connectivity augmentations, sparse subdigraphs with prescribed connectivity, and also packing, covering and decompositions of digraphs. Throughout the book, there is a strong focus on applications which include quantum mechanics, bioinformatics, embedded computing, and the travelling salesman problem. Detailed indices and topic-oriented chapters ease navigation, and more than 650 exercises, 170 figures and 150 open problems are included to help immerse the reader in all aspects of the subject. Digraphs is an essential, comprehensive reference for undergraduate and graduate students, and researchers in mathematics, operations research and computer science. It will also prove invaluable to specialists in related areas, such as meteorology, physics and computational biology.

Digraphs Theory Algorithms And Applications

On the foundations of combinatorial theory I. Theory of Möbius Functions

Let P be a partially ordered set. Two elements a and b of P are camparable if either a ≧ b or b ≧ a. Otherwise a and b are non-comparable. A subset S of P is independent if every two distinct elements of S are non-comparable. S is dependent if it contains two distinct elements which are comparable. A subset C of P is a chain if every two of its elements are comparable.

A decomposition theorem for partially ordered sets

Algebraic theory of lattices

Consider a lattice S in which the ascending chain condition holds. Then each element of S has at least one reduced1 representation as a cross-cut of irreducibles. Now it is well known that the requirement that this representation be unique considerably restricts the structure of the lattice. For example, Garrett Birkhoff [1] has proved that a modular lattice in which every element is uniquely expressible as a reduced cross-cut of irreducibles is distributive. Furthermore, Morgan Ward has shown that unicity of the irreducible decompositions implies that the lattice is a Birkhoff lattice.2 These results suggest the interesting problem of characterizing a lattice in which every element has a unique reduced representation as a cross-cut of irreducibles in terms of the structure of the lattice. We give here a complete solution of this problem. We show, namely, that such lattices are simply those Birkhoff lattices in which every modular sublattice is distributive.

Lattices with Unique Irreducible Decompositions

In the initial development of lattice theory considerable attention was devoted to the structure of modular lattices. Two of the principal structure theorems which came out of this early work are the following: 
Every complemented modular lattice of finite dimensions is a direct union of a finite number of simple1 complemented modular lattices (Birkhoff [1], Menger [4]).

The Structure of Relatively Complemented Lattices

The classical structure theorems of algebraic systems usually assume some type of finiteness condition. The most common finiteness restriction is a chain condition. Thus the proofs of the fundamental structure and decomposition theorems for lattices have customarily required the ascending chain condition. Moreover, these theorems generally fail to hold in arbitrary lattices. Nevertheless, there are important examples of decomposition theorems for lattices associated with abelian groups and rings in which the ascending chain condition does not hold. These lattices have in common another distinctive property—they are compactly generated. Namely, the lattice is generated by a collection of elements which are finitely dependent in the sense that any such element is contained in the union of a set of lattice elements if and only if it is contained in the union of a finite subset. The compact elements of the lattice of ideals of a ring are the finitely generated ideals. Likewise the compact elements of a lattice of congruence relations are the congruence relations generated by collapsing a finite collection of element pairs. More generally, the lattice of congruence relations and the lattice of subsystems of a universal algebra are compactly generated. Since structure theorems for an algebraic system correspond to decomposition theorems in the lattice of congruence relations, this strongly suggests that compactly generated lattices are the appropriate domain in which to study decomposition theory. Furthermore, since every lattice satisfying the ascending chain condition is trivially compactly generated, it follows that the classical case will be subsumed in the more general theory.

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R. P. Dilworth

Papers

A decomposition theorem for partially ordered sets

Algebraic theory of lattices

Lattices with Unique Irreducible Decompositions

The Structure of Relatively Complemented Lattices

Decomposition Theory for Lattices without Chain Conditions