7,555 research outputs found

### Some issues about neutrino processes in color superconducting quark matter

Several relevant issues in computing neutrino emissivity in Urca processes in
color superconducting quark matter are addressed. These include: (1) The
constraint on $u$ quark abundance is given from electric neutrality and the
triangle relation among Fermi momenta for participants. (2) The phase space
defined by Fermi momentum reduction of quarks is discussed in QCD and NJL
model. (3) Fermi effective model of weak interaction is reviewed with special
focus on its form in Nambu-Gorkov basis.Comment: Contribution to the proceedings of the Sixth China-Japan Joint
Nuclear Physics Symposium, May 16-20, Shanghai China. aipproc format, 8
pages, 5 figure

### Nonlinear Dirac equations on Riemann surfaces

We develop analytical methods for nonlinear Dirac equations. Examples of such
equations include Dirac-harmonic maps with curvature term and the equations
describing the generalized Weierstrass representation of surfaces in
three-manifolds. We provide the key analytical steps, i.e., small energy
regularity and removable singularity theorems and energy identities for
solutions.Comment: to appear in Annals of Global Analysis and Geometr

### Positive Semi-Definiteness and Sum-of-Squares Property of Fourth Order Four Dimensional Hankel Tensors

A positive semi-definite (PSD) tensor which is not a sum-of-squares (SOS)
tensor is called a PSD non-SOS (PNS) tensor. Is there a fourth order four
dimensional PNS Hankel tensor? Until now, this question is still an open
problem. Its answer has both theoretical and practical meanings. We assume that
the generating vector $v$ of the Hankel tensor $A$ is symmetric. Under this
assumption, we may fix the fifth element $v_4$ of $v$ at $1$. We show that
there are two surfaces $M_0$ and $N_0$ with the elements $v_2, v_6, v_1, v_3,
v_5$ of $v$ as variables, such that $M_0 \ge N_0$, $A$ is SOS if and only if
$v_0 \ge M_0$, and $A$ is PSD if and only if $v_0 \ge N_0$, where $v_0$ is the
first element of $v$. If $M_0 = N_0$ for a point $P = (v_2, v_6, v_1, v_3,
v_5)^\top$, then there are no fourth order four dimensional PNS Hankel tensors
with symmetric generating vectors for such $v_2, v_6, v_1, v_3, v_5$. Then, we
call such a point $P$ PNS-free. We show that a $45$-degree planar closed convex
cone, a segment, a ray and an additional point are PNS-free. Numerical tests
check various grid points, and find that they are also PNS-free

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