On the controversy concerning the definition of quark and gluon angular momentum

A major controversy has arisen in QCD as to how to split the total angular momentum into separate quark and gluon contributions, and as to whether the gluon angular momentum can itself be split, in a gauge invariant way, into a spin and orbital part. Several authors have proposed various answers to these questions and offered a variety of different expressions for the relevant operators. I argue that none of these is acceptable and suggest that the canonical expression for the momentum and angular momentum operators is the correct and physically meaningful one. It is then an inescapable fact that the gluon angular momentum operator cannot, in general, be split in a gauge invariant way into a spin and orbital part. However, the projection of the gluon spin onto its direction of motion i.e. its helicity is gauge invariant and is measured in deep inelastic scattering on nucleons. The Ji sum rule, relating the quark angular momentum to generalized parton distributions, though not based on the canonical operators, is shown to be correct, if interpreted with due care. I also draw attention to several interesting aspects of QED and QCD, which, to the best of my knowledge, are not commented upon in the standard textbooks on Field Theory.Comment: 41 pages; Some incorrect statements have been rectified and a detailed discussion has been added concerning the momentum carried by quarks and the Ji sum rule for the angular momentu

Parton distribution functions of proton in a light-front quark-diquark model

We present the parton distribution functions (PDFs) for un- polarised, longitudinally polarized and transversely polarized quarks in a proton using the light-front quark diquark model. We also present the scale evolution of PDFs and calculate axial charge and tecsor charge for $u$ and $d$ quarks at a scale of experimental findings.Comment: XXII DAE-BRNS High Energy Physics Symposium, December 12-16, 2016, University of Delhi, India; 4 pages, 1 figur

On the non-vanishing of the Collins mechanism for single spin asymmetries

The Collins mechanism provides a non-perturbative explanation for the large single spin asymmetries found in hard semi-inclusive reactions involving a transversely polarized nucleon. However, there are seemingly convincing reasons to suspect that the mechanism vanishes, and indeed it does vanish in the naive parton model where a quark is regarded as an essentially 'free' particle. We give an intuitive analysis which highlights the difference between the naive picture and the realistic one, and shows how the Collins mechanism arises when the quark is described as an off-shell particle by a field in interaction. A typographical error is corrected in this version.Comment: 15 pages, 2 figure

Set Systems Containing Many Maximal Chains

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed $\alpha>0$ there is a family of $\alpha 2^n$ sets containing $(\alpha+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.Comment: 5 page

Spin Structure of the Nucleon - Status and Recent Results

After the initial discovery of the so-called "spin crisis in the parton model" in the 1980's, a large set of polarization data in deep inelastic lepton-nucleon scattering was collected at labs like SLAC, DESY and CERN. More recently, new high precision data at large x and in the resonance region have come from experiments at Jefferson Lab. These data, in combination with the earlier ones, allow us to study in detail the polarized parton densities, the Q^2 dependence of various moments of spin structure functions, the duality between deep inelastic and resonance data, and the nucleon structure in the valence quark region. Together with complementary data from HERMES, RHIC and COMPASS, we can put new limits on the flavor decomposition and the gluon contribution to the nucleon spin. In this report, we provide an overview of our present knowledge of the nucleon spin structure and give an outlook on future experiments. We focus in particular on the spin structure functions g_1 and g_2 of the nucleon and their moments.Comment: 69 pages, 46 figures. Report to be published in "Progress in Particle and Nuclear Physics". v2 with added references and minor edit

Power iterations and the dominant eigenvalue problem

The orbits of an iterative numerical method for the dominant eigenvalue problem are analyzed from a discrete dynamical systems perspective. It is shown that the method can extract more information than the standard power method but at greater computational cost.http://archive.org/details/poweriterationsd00leadApproved for public release; distribution is unlimited

Neural network identification of keystream generators

Applications such as stream ciphers and spread spectra require the generation of binary keystreams to implement, and the simulation of such keystreams to break. Most cryptanalytic attacks are of the known generator type, that is, they assume knowledge of the method used to generate the keystream. We show that a neural network can be used to identify the generator, and in some cases to simulate the keystream.http://archive.org/details/neuralnetworkide00leadApproved for public release; distribution is unlimited

Analytical Solution of the Symmetric Circulant Tridiagonal Linear System

A circulant tridiagonal system is a special type of Toeplitz system that appears in a variety of problems in scientific computation. In this paper we give a formula for the inverse of a symmetric circulant tridiagonal matrix as a product of a circulant matrix and its transpose, and discuss the utility of this approach for solving the associated system

Correlation for permutations

In this note we investigate correlation inequalities for `up-sets' of permutations, in the spirit of the Harris--Kleitman inequality. We focus on two well-studied partial orders on $S_n$, giving rise to differing notions of up-sets. Our first result shows that, under the strong Bruhat order on $S_n$, up-sets are positively correlated (in the Harris--Kleitman sense). Thus, for example, for a (uniformly) random permutation $\pi$, the event that no point is displaced by more than a fixed distance $d$ and the event that $\pi$ is the product of at most $k$ adjacent transpositions are positively correlated. In contrast, under the weak Bruhat order we show that this completely fails: surprisingly, there are two up-sets each of measure $1/2$ whose intersection has arbitrarily small measure. We also prove analogous correlation results for a class of non-uniform measures, which includes the Mallows measures. Some applications and open problems are discussed