Since Eilenberg and Moore [EM], the relative homological algebra, especially the Gorenstein homological algebra ([EJ2]), has been developed to an advanced level. The analogues for the basic notion, such as projective, injective, flat, and free modules, are respectively the Gorenstein projective, the Gorenstein injective, the Gorenstein flat, and the strongly Gorenstein projective modules. One c...

The problem of stack sorting was introduced by Knuth in the 1960s. He described the stack sorting operation as the movement of railway cars across a railroad switching network. The problem is also similar to the childhood game of Hanoi Towers, in which the player is supposed to move concentric discs of varying sizes from one side across to another without placing a bigger disc above a smaller o...

I provide a basic introduction to modern helicity amplitude methods, including color organization, the spinor helicity formalism, and factorization properties. I also describe the BCFW (on-shell) recursion relation at tree level, and explain how similar ideas — unitarity and on-shell methods — work at the loop level. These notes are based on lectures delivered at the 2012 CERN Summer School and...

Modal logic extends classical logic with the ability to express not only ‘P is true’, but also statements like ‘P is known’ or ‘P is necessarily true’. We will define several varieties of modal logic, providing both their semantics and their axiomatic proof systems, and prove their standard soundness and completeness theorems.

derstand some of the basic concepts and terminology of molecular biology. This article is a brief introduction to the extraordinarily complex phenomenon of life and to its molecular basis. We begin with the amazing diversity of life forms and the equally amazing unity in the molecules underlying life’s processes. The challenge of accounting for both the variety and the commonalities among organ...

Vibration phenomena that might be modelled well using linear vibration theory include small amplitude vibrations of long, slender objects like long bridges, aeroplane wings, and helicopter blades; small rocking motions of ships in calm waters; the simplest whirling motions of flexible shafts, and so on. However, interactions between bridges and foundations, between wings/blades and air, between...

Figure 1: Factor Graph In particular, consider a setup where (U1, U2) are two equiprobable bits that are encoded into (X1, X2) = (U1 ⊕ U2, U2). Then, (X1, X2) are mapped to (Y1, Y2) by two independent BMS channels with transition probabilities P(Y1 = y |X1 = x) = P(Y2 = y |X2 = x) = W (y|x). The factor graph for this setup is shown in Figure 1. Since the mapping from (U1, U2) to (X1, X2) is inv...

Supervised learning techniques construct predictive models by learning from a large number of training examples, where each training example has a label indicating its ground-truth output. Though current techniques have achieved great success, it is noteworthy that in many tasks it is difficult to get strong supervision information like fully ground-truth labels due to the high cost of data lab...