Factorization formulas of KK-kk-Schur functions II

Subsequently to the author's preceding paper, we give full proofs of some explicit formulas about factorizations of $K$-$k$-Schur functions associated with any multiple $k$-rectangles.Comment: 28 page

Factorization formulas of KK-kk-Schur functions I

We give some new formulas about factorizations of $K$-$k$-Schur functions $g^{(k)}_{\lambda}$, analogous to the $k$-rectangle factorization formula $s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}$ of $k$-Schur functions, where $\lambda$ is any $k$-bounded partition and $R_t$ denotes the partition $(t^{k+1-t})$ called \textit{$k$-rectangle}. Although a formula of the same form does not hold for $K$-$k$-Schur functions, we can prove that $g^{(k)}_{R_t}$ divides $g^{(k)}_{R_t\cup\lambda}$, and in fact more generally that $g^{(k)}_{P}$ divides $g^{(k)}_{P\cup\lambda}$ for any multiple $k$-rectangles $P=R_{t_1}^{a_1}\cup\dots\cup R_{t_m}^{a_m}$ and any $k$-bounded partition $\lambda$. We give the factorization formula of such $g^{(k)}_{P}$ and the explicit formulas of $g^{(k)}_{P\cup\lambda}/g^{(k)}_{P}$ in some cases, including the case where $\lambda$ is a partition with a single part as the easiest example.Comment: 36 page

On the Pieri rules of stable and dual stable Grothendieck polynomials

We give an explanation for the Pieri coefficients for the stable and dual stable Grothendieck polynomials; their non-leading terms are obtained by taking an alternating sum of meets (or joins) of their leading terms.Comment: 8 pages, revised thoroughly, results added and title change

A Pieri-type formula and a factorization formula for sums of KK-kk-Schur functions

We give a Pieri-type formula for the sum of $K$-$k$-Schur functions $\sum_{\mu\le\lambda} g^{(k)}_{\mu}$ over a principal order ideal of the poset of $k$-bounded partitions under the strong Bruhat order, which sum we denote by $\widetilde{g}^{(k)}_{\lambda}$. As an application of this, we also give a $k$-rectangle factorization formula $\widetilde{g}^{(k)}_{R_t\cup\lambda}=\widetilde{g}^{(k)}_{R_t} \widetilde{g}^{(k)}_{\lambda}$ where $R_t=(t^{k+1-t})$, analogous to that of $k$-Schur functions $s^{(k)}_{R_t\cup\lambda}=s^{(k)}_{R_t}s^{(k)}_{\lambda}$.Comment: 33 pages, revised thoroughly, results unchange

Mini radio lobes in AGNs core illumination and their hadronic gamma-ray afterlight

Recent radio observations reveal the existence of mini radio lobes in active galaxies with their scales of $\sim 10 {\rm pc}$. The lobes are expected to be filled with shock accelerated electrons and protons. In this work, we examine the photon spectra from the mini lobes, properly taking the hadronic processes into account. We find that the resultant broadband spectra contain the two distinct hadronic bumps in $\gamma$-ray bands, i.e., the proton synchrotron bump at $\sim$ MeV and the synchrotron bump at $\sim$ GeV due to the secondary electrons/positrons produced via photo-pion cascade. Especially when the duration of particle injection is shorter than the lobe age, radio-dark $\gamma$-ray lobes are predicted. The existence of the $\gamma$-ray lobes could be testable with the future TeV-$\gamma$ telescope {\it CTA}.Comment: 5 pages, 3 figures. Accepted for publication in MNRAS Letter

On the origin of Fanaroff-Riley classification of radio galaxies: Deceleration of supersonic radio lobes

We argue that the origin of "FRI/FRI{-.1em}I dichotomy" -- the division between Fanaroff-Riley class I (FRI) with subsonic lobes and class I{-.1em}I (FRI{-.1em}I) radio sources with supersonic lobes is sharp in the radio-optical luminosity plane (Owen-White diagram) -- can be explained by the deceleration of advancing radio lobes. The deceleration is caused by the growth of the effective cross-sectional area of radio lobes. We derive the condition in which an initially supersonic lobe turns into a subsonic lobe, combining the ram-pressure equilibrium between the hot spots and the ambient medium with the relation between "the hot spot radius" and "the linear size of radio sources" obtained from the radio observations. We find that the dividing line between the supersonic lobes and subsonic ones is determined by the ratio of the jet power $L_{\rm j}$ to the number density of the ambient matter at the core radius of the host galaxy $\bar{n}_{\rm a}$. It is also found that there exists the maximal ratio of $(L_{\rm j}/\bar{n}_{\rm a})$ and its value resides in $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}\approx 10^{44-47} {\rm erg} {\rm s}^{-1} {\rm cm}^{3}$, taking account of considerable uncertainties. This suggests that the maximal value $(L_{\rm j}/\bar{n}_{\rm a})_{\rm max}$ separates between FRIs and FRI{-.1em}Is.Comment: 5 pages, 1 figure, accepted for publication in ApJ Letter

Evidence for a significant mixture of electron/positron pairs in FRII jets constrained by cocoon dynamics

We examine the plasma composition of relativistic jets in four FRII radio galaxies by analyzing the total cocoon pressure in terms of partial pressures of thermal and non-thermal electrons/positrons and protons. The total cocoon pressure is determined by cocoon dynamics via comparison of theoretical model with the observed cocoon shape. By inserting the observed number density of non-thermal electrons/positrons and the upper limit of thermal electron/positron number density into the equation of state, the number density of protons is constrained. We apply this method to four FRII radio galaxies (Cygnus A, 3C 219, 3C 223 and 3C 284), for which the total cocoon pressures have been already evaluated. We find that the positron-free plasma comprising of protons and electrons is ruled out, when we consider plausible particle distribution functions. In other words, the mixture of positrons is required for all four FRII radio galaxies; the number density ratio of electrons/positrons to protons is larger than two. Thus, we find that the plasma composition is independent of the jet power and the size of cocoons. We also investigate the additional contribution of thermal electrons/positrons and protons on the cocoon dynamics. When thermal electrons/positrons are absent, the cocoon is supported by the electron/ proton plasma pressure, while both electron/positron pressure supported and electron/proton plasma pressure supported cocoons are allowed if the number density of thermal electrons/positrons is about 10 times larger than that of non-thermal ones.Comment: 14 pagese, 9 figures, Accepted for publication in MNRA

Low temperature limit of lattice QCD

We study the low temperature limit of lattice QCD by using a reduction formula for a fermion determinant. The reduction formula, which is useful in finite density lattice QCD simulations, contains a reduced matrix defined as the product of $N_t$ block-matrices. It is shown that eigenvalues of the reduced matrix follows a scaling law with regard to the temporal lattice size $N_t$. The $N_t$ scaling law leads to two types of expressions of the fermion determinant in the low temperature limit; one is for small quark chemical potentials, and the other is for larger quark chemical potentials.Comment: 7 pages, 4figures. Proceedings of The 30 International Symposium on Lattice Field Theory, June 24-29, 2012, Cairns, Australi

A fast solver for multi-particle scattering in a layered medium

In this paper, we consider acoustic or electromagnetic scattering in two dimensions from an infinite three-layer medium with thousands of wavelength-size dielectric particles embedded in the middle layer. Such geometries are typical of microstructured composite materials, and the evaluation of the scattered field requires a suitable fast solver for either a single configuration or for a sequence of configurations as part of a design or optimization process. We have developed an algorithm for problems of this type by combining the Sommerfeld integral representation, high order integral equation discretization, the fast multipole method and classical multiple scattering theory. The efficiency of the solver is illustrated with several numerical experiments

A fast and robust solver for the scattering from a layered periodic structure containing multi-particle inclusions

We present a solver for plane wave scattering from a periodic dielectric grating with a large number $M$ of inclusions lying in each period of its middle layer.Such composite material geometries have a growing role in modern photonic devices and solar cells. The high-order scheme is based on boundary integral equations, and achieves many digits of accuracy with ease. The usual way to periodize the integral equation---via the quasi-periodic Green's function---fails at Wood's anomalies. We instead use the free-space Green's kernel for the near field, add auxiliary basis functions for the far field, and enforce periodicity in an expanded linear system; this is robust for all parameters. Inverting the periodic and layer unknowns, we are left with a square linear system involving only the inclusion scattering coefficients. Preconditioning by the single-inclusion scattering matrix, this is solved iteratively in $O(M)$ time using a fast matrix-vector product. Numerical experiments show that a diffraction grating containing $M=1000$ inclusions per period can be solved to 9-digit accuracy in under 5 minutes on a laptop