# Parton distribution amplitudes of light vector mesons

A rainbow-ladder truncation of QCD’s Dyson-Schwinger equations is used to calculate rho- and phi-meson valence-quark (twist-two parton) distribution amplitudes (PDAs) via a light-front projection of their Bethe-Salpeter wave functions, which possess S- and D-wave components of comparable size in the meson rest frame. All computed PDAs are broad concave functions, whose dilation with respect to the asymptotic distribution is an expression of dynamical chiral symmetry breaking. The PDAs can be used to define an ordering of valence-quark light-front spatial-extent within mesons: this size is smallest within the pion and increases through the perp-polarisation to the parallel-polarisation of the vector mesons; effects associated with the breaking of SU(3)-flavour symmetry are significantly smaller than those associated with altering the polarisation of vector mesons. Notably, the predicted pointwise behaviour of the rho-meson PDAs is in quantitative agreement with that inferred recently via an analysis of diffractive vector-meson photoproduction experiments.

# Symmetric Relative Equilibria in the Four-Vortex Problem with Three Equal Vorticities

>Ernesto Perez-Chavela, Manuele Santoprete, Claudia Tamayo

We examine in detail the relative equilibria of the 4-vortex problem when three vortices have equal strength, that is, Γ1=Γ2=Γ3=1, and Γ4 is a real parameter. We give the exact number of relative equilibria and bifurcation values. We also study the relative equilibria in the vortex rhombus problem.

# Adiabatic theorem for a class of quantum stochastic equations

We derive an adiabatic theory for a stochastic differential equation, εdX(s)=L1(s)X(s)ds+ε√L2(s)X(s)dBs, under a condition that instantaneous stationary states of L1(s) are also stationary states of L2(s). We use our results to derive the full statistics of tunneling for a driven stochastic Schr\”{o}dinger equation describing a dephasing process.

# Building a Foolproof Navigation System: Fuzzy Logic Emulating the Brain

Subhajit Ganguly

## Description

Our aim is to help build a machine that can reduce the possibility of mishaps in navigation to zero. For that devise a new system of numbers, in which the real numbers are represented on the y-axis and complex numbers on the x-axis. Inside such a system, we incorporate the equivalent Ideal Fuzzy Logic that can be used by the machine to predict and avoid mishaps.

Natural processes are vastly emergent phenomena and each new result is always the source of new emergence. To cope with nonlinear control problems, binary logic is no longer sufficient. What we need is an ideal Fuzzy Logic that not only can process complex numbers with utmost efficiency, but also ‘thinks’ in terms of complex numbers. Such a system also needs to be as simple as possible for us and for machines to work with.

A perfectly efficient navigation system will be able to receive inputs that have all sets of possible values. These values may be real or imaginary. Such a system will be particularly efficient in dealing with imaginary numbers and will be able to reduce the probability of a mishap to zero.

**share**.

http://dx.doi.org/10.6084/m9.figshare.1093898

# Theory of orthogonality of eigenfunctions of the characteristic equations as a method of solution boundary problems for model kinetic equations

We consider two classes of linear kinetic equations: with constant collision frequency and constant mean free path of gas molecules (i.e., frequency of molecular collisions, proportional to the modulus molecular velocity). Based homogeneous Riemann boundary value problem with a coefficient equal to the ratio of the boundary values dispersion function, develops the theory of the half-space orthogonality of generalized singular eigenfunctions corresponding characteristic equations, which leads separation of variables. And in this two boundary value problems of the kinetic theory (diffusion light component of a binary gas and Kramers problem about isothermal slip) shows the application of the theory orthogonality eigenfunctions for analytical solutions these tasks.

# Abstraction and Structures in Energy

Subhajit Ganguly

Zero postulation and the principles of the Theory of Abstraction are used to study structures of energy inside a black hole, which is incredibly heavy and incredibly small. We chase the questions, how matter (with various structures) is formed from energy and the energy making up matter has to be in what orientation to form the matter that we see. We arrive at the fundamental model and the equations describing the formation of structure in energy.

**share**.

http://dx.doi.org/10.6084/m9.figshare.1035785