The small-mass limit and white-noise limit of an infinite dimensional Generalized Langevin Equation

We study asymptotic properties of the Generalized Langevin Equation (GLE) in the presence of a wide class of external potential wells with a power-law decay memory kernel. When the memory can be expressed as a sum of exponentials, a class of Markovian systems in infinite-dimensional spaces is used to represent the GLE. The solutions are shown to converge in probability in the small-mass limit and the white-noise limit to appropriate systems under minimal assumptions, of which no Lipschitz condition is required on the potentials. With further assumptions about space regularity and potentials, we obtain $L^1$ convergence in the white-noise limit

Global estimates for quasilinear parabolic equations on Reifenberg flat domains and its applications to Riccati type parabolic equations with distributional data

In this paper, we prove global weighted Lorentz and Lorentz-Morrey estimates for gradients of solutions to the quasilinear parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=\operatorname{div}(F),$$ in a bounded domain $\Omega\times (0,T)\subset\mathbb{R}^{N+1}$, under minimal regularity assumptions on the boundary of domain and on nonlinearity $A$. Then results yields existence of a solution to the Riccati type parabolic equations: $$u_t-\operatorname{div}(A(x,t,\nabla u))=|\nabla u|^q+\operatorname{div}(F)+\mu,$$ where $q>1$ and $\mu$ is a bounded Radon measure.Comment: to appear Calculus of Variations and Partial Differential Equation

Irreducible restrictions of Brauer characters of the Chevalley group G_2(q) to its proper subgroups

Let $G_2(q)$ be the Chevalley group of type $G_2$ defined over a finite field with q=p^n elements, where p is a prime number and $n$ is a positive integer. In this paper, we determine when the restriction of an absolutely irreducible representation of $G$ in characteristic other than p to a maximal subgroup of $G_2(q)$ is still irreducible. Similar results are obtained for $^2B_2(q)$ and $^2G_2(q)$.Comment: 30 page

Gradient estimates for singular quasilinear elliptic equations with measure data

In this paper, we prove $L^q$-estimates for gradients of solutions to singular quasilinear elliptic equations with measure data $$-\operatorname{div}(A(x,\nabla u))=\mu,$$ in a bounded domain $\Omega\subset\mathbb{R}^{N}$, where $A(x,\nabla u)\nabla u \asymp |\nabla u|^p$, $p\in (1,2-\frac{1}{n}]$ and $\mu$ is a Radon measure in $\Omega$Comment: 20 pages. arXiv admin note: text overlap with arXiv:1511.0621

Low-dimensional complex characters of the symplectic and orthogonal groups

We classify the irreducible complex characters of the symplectic groups $Sp_{2n}(q)$ and the orthogonal groups $Spin_{2n}^\pm(q)$, $Spin_{2n+1}(q)$ of degrees up to the bound D, where $D=(q^n-1)q^{4n-10}/2$ for symplectic groups, $D=q^{4n-8}$ for orthogonal groups in odd dimension, and $D=q^{4n-10}$ for orthogonal groups in even dimension.Comment: 44 pages. Comm. Algebra, to appea

Characters of p'-degree and Thompson's character degree theorem

A classical theorem of John Thompson on character degrees asserts that if the degree of every ordinary irreducible character of a finite group $G$ is 1 or divisible by a prime $p$, then $G$ has a normal $p$-complement. We obtain a significant improvement of this result by considering the average of $p'$-degrees of irreducible characters. We also consider fields of character values and prove several improvements of earlier related results.Comment: 23 page

On the importance of using exact pairing in the study of pygmy dipole resonance

The strength functions of giant dipole resonance (GDR) in oxygen $^{18 - 24}$O, calcium $^{50 - 60}$Ca, and tin $^{120 - 130}$Sn isotopes are calculated within the phonon damping model under three approximations: without superfluid pairing, including BCS pairing, and exact pairing gaps. The analysis of the numerical results shows that exact pairing decreases the two-neutron separation energy in light nuclei, but increases it in heavy nuclei as compared to that obtained within the BCS theory. In neutron-rich medium and heavy nuclei, exact pairing significantly enhances the strength located at the low-energy tail of the GDR, which is usually associated with the pygmy dipole resonance. The line shape of the GDR changes significantly with increasing the neutron number within an isotopic chain if the model parameter is kept fixed at the value determined for the stable isotope.Comment: 26 pages, 19 figures, to appear in Journal of Physics

Self-consistent quasiparticle RPA for multi-level pairing model

Particle-number projection within the Lipkin-Nogami (LN) method is applied to the self-consistent quasiparticle random-phase approximation (SCQRPA), which is tested in an exactly solvable multi-level pairing model. The SCQRPA equations are numerically solved to find the energies of the ground and excited states at various numbers $\Omega$ of doubly degenerate equidistant levels. The use of the LN method allows one to avoid the collapse of the BCS (QRPA) to obtain the energies of the ground and excited states as smooth functions of the interaction parameter $G$. The comparison between results given by different approximations such as the SCRPA, QRPA, LNQRPA, SCQRPA and LNSCQRPA is carried out. While the use of the LN method significantly improves the agreement with the exact results in the intermediate coupling region, we found that in the strong coupling region the SCQRPA results are closest to the exact ones.Comment: Accepted by Phys. Rev.

Comment on arXiv:0709.3700 "Orientation dependence of the optical spectra in graphene at high frequencies"

Zhang et al. reported in [Phys. Rev. B 77, 241402(R) (2008)] a theoretical study of the optical spectra of monolayer graphene employing the Kubo formula within a tight-binding model. Their calculations predicted that at high frequencies the optical conductivity of graphene becomes strongly anisotropic. In particular, at frequencies comparable to the energy separation of the upper and lower bands at the $\Gamma$-point, the optical conductivity is strongly suppressed if the field polarization is along the zigzag direction while it is significantly high for the armchair one. We find that, unfortunately, this result is just a consequence of the incorrect determination of the current operator in k-space. Here, we present the standard scheme to obtain this operator correctly. As a result, we show that the optical conductivity of monolayer graphene is indeed isotropic, which is consistent with the results of other (both theoretical and experimental) studies in the literature.Comment: submitte

Giant dipole resonance in 201^{201}Tl at low temperature

The thermal pairing gap obtained by embedding the exact solutions of the pairing problem into the canonical ensemble is employed to calculate the width and strength function of the giant dipole resonance (GDR) within the phonon damping model. The results of calculations describe reasonably well the data for the GDR width as well as the GDR linearized strength function, recently obtained for $^{201}$Tl in the temperature region between 0.8 and 1.2 MeV, which other approaches that neglect the effect of non-vanishing thermal pairing fail to describe.Comment: 18 page, 4 figures, accepted for publication in Phys. Rev.