Gamma Ray Burst Prompt Emission Variability in Synchrotron and Synchrotron Self-Compton Lightcurves

Gamma Ray Burst prompt emission is believed to originate from electrons accelerated in a highly relativistic outflow. "Internal shocks" due to collisions between shells ejected by the central engine is a leading candidate for electron acceleration. While synchrotron radiation is generally invoked to interpret prompt gamma-ray emission within the internal shock model, synchrotron self-Compton (SSC) is also considered as a possible candidate of radiation mechanism. In this case, one would expect a synchrotron emission component at low energies, and the naked-eye GRB 080319B has been considered as such an example. In the view that the gamma-ray lightcurve of GRB 080319B is much more variable than its optical counterpart, in this paper we study the relative variability between the synchrotron and SSC components. We develop a "top-down" formalism by using observed quantities to infer physical parameters, and subsequently to study the temporal structure of synchrotron and SSC components of a GRB. We complement the formalism with a "bottom-up" approach where the synchrotron and SSC lightcurves are calculated through a Monte-Carlo simulations of the internal shock model. Both approaches lead to the same conclusion. Small variations in the synchrotron lightcurve can be only moderately amplified in the SSC lightcurve. The SSC model therefore cannot adequately interpret the gamma-ray emission properties of GRB 080319B.Comment: 13 pages, 4 figures, accepted for publication in MNRA

Mutual Interlacing and Eulerian-like Polynomials for Weyl Groups

We use the method of mutual interlacing to prove two conjectures on the real-rootedness of Eulerian-like polynomials: Brenti's conjecture on $q$-Eulerian polynomials for Weyl groups of type $D$, and Dilks, Petersen, and Stembridge's conjecture on affine Eulerian polynomials for irreducible finite Weyl groups. For the former, we obtain a refinement of Brenti's $q$-Eulerian polynomials of type $D$, and then show that these refined Eulerian polynomials satisfy certain recurrence relation. By using the Routh--Hurwitz theory and the recurrence relation, we prove that these polynomials form a mutually interlacing sequence for any positive $q$, and hence prove Brenti's conjecture. For $q=1$, our result reduces to the real-rootedness of the Eulerian polynomials of type $D$, which were originally conjectured by Brenti and recently proved by Savage and Visontai. For the latter, we introduce a family of polynomials based on Savage and Visontai's refinement of Eulerian polynomials of type $D$. We show that these new polynomials satisfy the same recurrence relation as Savage and Visontai's refined Eulerian polynomials. As a result, we get the real-rootedness of the affine Eulerian polynomials of type $D$. Combining the previous results for other types, we completely prove Dilks, Petersen, and Stembridge's conjecture, which states that, for every irreducible finite Weyl group, the affine descent polynomial has only real zeros.Comment: 28 page

Uncertainty Relation for a Quantum Open System

We derive the uncertainty relation for a quantum open system comprised of a Brownian particle interacting with a bath of quantum oscillators at finite temperature. We examine how the quantum and thermal fluctuations of the environment contribute to the uncertainty in the canonical variables of the system. We show that upon contact with the bath (assumed ohmic in this paper) the system evolves from a quantum-dominated state to a thermal-dominated state in a time which is the same as the decoherence time in similar models in the discussion of quantum to classical transition. This offers some insight into the physical mechanisms involved in the environment-induced decoherence process. We obtain closed analytic expressions for this generalized uncertainty relation under the conditions of high temperature and weak damping separately. We also consider under these conditions an arbitrarily-squeezed initial state and show how the squeeze parameter enters in the generalized uncertainty relation. Using these results we examine the transition of the system from a quantum pure state to a nonequilibrium quantum statistical state and to an equilibrium quantum statistical state. The three stages are marked by the decoherence time and the relaxation time respectively. With these observations we explicate the physical conditions when the two basic postulates of quantum statistical mechanics become valid. We also comment on the inappropriateness in the usage of the word classicality in many decoherence studies of quantum to classical transition.Comment: 36 pages,Tex,umdpp93-162,(submitted to Phys. Rev. A

A Rate-Splitting Based Bound-Approaching Transmission Scheme for the Two-User Symmetric Gaussian Interference Channel with Common Messages

This paper is concerned with a rate-splitting based transmission strategy for the two-user symmetric Gaussian interference channel that contains common messages only. Each transmitter encodes its common message into multiple layers by multiple codebooks that drawn from one separate code book, and transmits the superposition of the messages corresponding to these layers; each receiver decodes the messages from all layers of the two users successively. Two schemes are proposed for decoding order and optimal power allocation among layers respectively. With the proposed decoding order scheme, the sum-rate can be increased by rate-splitting, especially at the optimal number of rate-splitting, using average power allocation in moderate and weak interference regime. With the two proposed schemes at the receiver and the transmitter respectively, the sum-rate achieves the inner bound of HK without time-sharing. Numerical results show that the proposed optimal power allocation scheme with the proposed decoding order can achieve significant improvement of the performance over equal power allocation, and achieve the sum-rate within two bits per channel use (bits/channel use) of the sum capacity

Avoiding vincular patterns on alternating words

A word $w=w_1w_2\cdots w_n$ is alternating if either $w_1w_3\cdots$ (when the word is up-down) or $w_1>w_2w_4<\cdots$ (when the word is down-up). The study of alternating words avoiding classical permutation patterns was initiated by the authors in~\cite{GKZ}, where, in particular, it was shown that 123-avoiding up-down words of even length are counted by the Narayana numbers. However, not much was understood on the structure of 123-avoiding up-down words. In this paper, we fill in this gap by introducing the notion of a cut-pair that allows us to subdivide the set of words in question into equivalence classes. We provide a combinatorial argument to show that the number of equivalence classes is given by the Catalan numbers, which induces an alternative (combinatorial) proof of the corresponding result in~\cite{GKZ}. Further, we extend the enumerative results in~\cite{GKZ} to the case of alternating words avoiding a vincular pattern of length 3. We show that it is sufficient to enumerate up-down words of even length avoiding the consecutive pattern $\underline{132}$ and up-down words of odd length avoiding the consecutive pattern $\underline{312}$ to answer all of our enumerative questions. The former of the two key cases is enumerated by the Stirling numbers of the second kind.Comment: 25 pages; To appear in Discrete Mathematic