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Register a new userSome mathematics for quasi-symmetry
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Quasi-symmetry of a steady magnetic field means integrability of first-order guiding-centre motion. Here we derive many restrictions on the possibilities for a quasi-symmetry. We also derive an analogue of the Grad-Shafranov equation for the flux function in a quasi-symmetric magnetohydrostatic field.
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We consider the alternating sign matrices of the odd order that have some kind of central symmetry. Namely, we deal with matrices invariant under the half-turn, quarter-turn and flips in both diagonals. In all these cases, there are two natural structures in the centre of the matrix. For example, for the matrices invariant under the half-turn the central element is equal $pm 1$. It was recently found that $A^+_{HT}(2m+1)/A^-_{HT}(2m+1)$=(m+1)/m. We conjecture that similar very simple relations are valid in the two remaining cases.
Motivated by fractional quantum Hall effects, we introduce a universal space of statistics interpolating Bose-Einstein statistics and Fermi-Dirac statistics. We connect the interpolating statistics to umbral calculus and use it as a bridge to study the interpolation statistics by the principle maximum entropy by deformed entropy functions. On the one hand this connection makes it possible to relate fractional quantum Hall effects to many different mathematical objects, including formal group laws, complex bordism theory, complex genera, operads, counting trees, spectral curves in Eynard-Orantin topological recursions, etc. On the other hand, this also suggests to reexamine umbral calculus from the point of view of quantum mechanics and statistical mechanics.
We reconsider the Gram-Hadamard bound as it is used in constructive quantum field theory and many body physics to prove convergence of Fermionic perturbative expansions. Our approach uses a recursion for the amplitudes of the expansion, discovered originally by Djokic arXiv:1312.1185. It explains the standard way to bound the expansion from a new point of view, and for some of the amplitudes provides new bounds, which avoid the use of Fourier transform, and are therefore superior to the standard bounds for models like the cold interacting Fermi gas.
Consider a function $F(X,Y)$ of pairs of positive matrices with values in the positive matrices such that whenever $X$ and $Y$ commute $F(X,Y)= X^pY^q.$ Our first main result gives conditions on $F$ such that ${rm Tr}[ X log (F(Z,Y))] leq {rm Tr}[X(plog X + q log Y)]$ for all $X,Y,Z$ such that ${rm Tr} Z = {rm Tr} X$. (Note that $Z$ is absent from the right side of the inequality.) We give several examples of functions $F$ to which the theorem applies. Our theorem allows us to give simple proofs of the well known logarithmic inequalities of Hiai and Petz and several new generalizations of them which involve three variables $X,Y,Z$ instead of just $X,Y$ alone. The investigation of these logarithmic inequalities is closely connected with three quantum relative entropy functionals: The standard Umegaki quantum relative entropy $D(X||Y) = {rm Tr} [X(log X-log Y])$, and two others, the Donald relative entropy $D_D(X||Y)$, and the Belavkin-Stasewski relative entropy $D_{BS}(X||Y)$. They are known to satisfy $D_D(X||Y) leq D(X||Y)leq D_{BS}(X||Y)$. We prove that the Donald relative entropy provides the sharp upper bound, independent of $Z$, on ${rm Tr}[ X log (F(Z,Y))]$ in a number of cases in which $(Z,Y)$ is homogeneous of degree $1$ in $Z$ and $-1$ in $Y$. We also investigate the Legendre transforms in $X$ of $D_D(X||Y)$ and $D_{BS}(X||Y)$, and show how our results for these lead to new refinements of the Golden-Thompson inequality.
We introduce an application of the Quasi-Gasdynamic method for a solution of ideal magnetohydrodynamic equations in the modeling of compressible conductive gas flows. A time-averaging procedure is applied for all physical parameters in order to obtain the quasi-gas-dynamic system of equations for magnetohydrodynamics. Evolution of all physical variables is presented in an unsplit divergence form. Divergence-free evolution of the magnetic field is provided by using a constrained transport method based on Stokes theorem. Accuracy and convergence of this method are verified on a large set of standard 1D and 2D test cases.
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