Response to "Reply to comment on 'Divergent and Ultrahigh Thermal Conductivity in Millimeter-Long Nanotubes'"

More than one year ago, Prof. Chih-Wei Chang and the co-authors published "Divergent and Ultrahigh Thermal Conductivity in Millimeter-Long Nanotubes" in PRL and we submitted a comment. After some while we received Prof. Chang et al.'s reply, which is almost the same as their arXiv preprint, and responded to the reply promptly. On the request of some readers, I personally post here the detailed response to "Reply to comment on 'Divergent and Ultrahigh Thermal Conductivity in Millimeter-Long Nanotubes'"

Constraining the Compressed Top Squark and Chargino along the W Corridor

Studying superpartner production together with a hard initial state radiation (ISR) jet has been a useful strategy for searches of supersymmetry with a compressed spectrum at the Large Hadron Collider (LHC). In the case of the top squark (stop), the ratio of the missing transverse momentum from the lightest neutralinos and the ISR momentum, defined as $\bar{R}_M$, turns out to be an effective variable to distinguish the signal from the backgrounds. It has helped to exclude the stop mass below 590 GeV along the top corridor where $m_{\tilde{t}} - m_{\tilde{\chi}_1^0} \approx m_t$. On the other hand, the current experimental limit is still rather weak in the $W$ corridor where $m_{\tilde{t}} - m_{\tilde{\chi}_1^0} \approx m_W +m_b$. In this work we extend this strategy to the parameter region around the $W$ corridor by considering the one lepton final state. In this case the kinematic constraints are insufficient to completely determine the neutrino momentum which is required to calculate $\bar{R}_M$. However, the minimum value of $\bar{R}_M$ consistent with the kinematic constraints still provides a useful discriminating variable, allowing the exclusion reach of the stop mass to be extended to $\sim 550$ GeV based on the current 36 fb$^{-1}$ LHC data. The same method can also be applied to the chargino search with $m_{\tilde{\chi}_1^\pm} -m_{\tilde{\chi}_1^0} \approx m_W$ because the analysis does not rely on $b$ jets. If no excess is present in the current data, a chargino mass of 300 GeV along the $W$ corridor can be excluded, beyond the limit obtained from the multilepton search.Comment: 29 pages,8 figure

On the Number of Zeros and Poles of Dirichlet Series

This paper investigates lower bounds on the number of zeros and poles of a general Dirichlet series in a disk of radius $r$ and gives, as a consequence, an affirmative answer to an open problem of Bombieri and Perelli on the bound. Applications will also be given to Picard type theorems, global estimates on the symmetric difference of zeros, and uniqueness problems for Dirichlet series.Comment: 24 page

A characterization of rational functions

We give an elementary characterization of rational functions among meromorphic functions in the complex plane

Transport Protocols in Cognitive Radio Networks: A Survey

Cognitive radio networks (CRNs) have emerged as a promising solution to enhance spectrum utilization by using unused or less used spectrum in radio environments. The basic idea of CRNs is to allow secondary users (SUs) access to licensed spectrum, under the condition that the interference perceived by the primary users (PUs) is minimal. In CRNs, the channel availability is uncertainty due to the existence of PUs, resulting in intermittent communication. Transmission control protocol (TCP) performance may significantly degrade in such conditions. To address the challenges, some transport protocols have been proposed for reliable transmission in CRNs. In this paper we survey the state-of-the-art transport protocols for CRNs. We firstly highlight the unique aspects of CRNs, and describe the challenges of transport protocols in terms of PU behavior, spectrum sensing, spectrum changing and TCP mechanism itself over CRNs. Then, we provide a summary and comparison of existing transport protocols for CRNs. Finally, we discuss several open issues and research challenges. To the best of our knowledge, our work is the first survey on transport protocols for CRNs.Comment: to appear in KSII Transactions on Internet and Information System

Ensemble Kalman Inversion: mean-field limit and convergence analysis

Ensemble Kalman Inversion (EKI) has been a very popular algorithm used in Bayesian inverse problems. It samples particles from a prior distribution, and introduces a motion to move the particles around in pseudo-time. As the pseudo-time goes to infinity, the method finds the minimizer of the objective function, and when the pseudo-time stops at $1$, the ensemble distribution of the particles resembles, in some sense, the posterior distribution in the linear setting. The ideas trace back further to Ensemble Kalman Filter and the associated analysis, but to today, when viewed as a sampling method, why EKI works, and in what sense with what rate the method converges is still largely unknown. In this paper, we analyze the continuous version of EKI, a coupled SDE system, and prove the mean field limit of this SDE system. In particular, we will show that 1. as the number of particles goes to infinity, the empirical measure of particles following SDE converges to the solution to a Fokker-Planck equation in Wasserstein 2-distance with an optimal rate, for both linear and weakly nonlinear case; 2. the solution to the Fokker-Planck equation reconstructs the target distribution in finite time in the linear case

Stability of Stationary Inverse Transport Equation in Diffusion Scaling

We consider the inverse problem of reconstructing the optical parameters for stationary radiative transfer equation (RTE) from velocity-averaged measurement. The RTE often contains multiple scales characterized by the magnitude of a dimensionless parameter---the Knudsen number ($K_n$). In the diffusive scaling ($K_n \ll 1$), the stationary RTE is well approximated by an elliptic equation in the forward setting. However, the inverse problem for the elliptic equation is acknowledged to be severely ill-posed as compared to the well-posedness of inverse transport equation, which raises the question of how uniqueness being lost as $K_n \rightarrow 0$. We tackle this problem by examining the stability of inverse problem with varying $K_n$. We show that, the discrepancy in two measurements is amplified in the reconstructed parameters at the order of $K_n^p~ (p = 1\text{ or} ~2)$, and as a result lead to ill-posedness in the zero limit of $K_n$. Our results apply to both continuous and discrete settings. Some numerical tests are performed in the end to validate these theoretical findings

Stability of inverse transport equation in diffusion scaling and Fokker-Planck limit

We consider the inverse problem of reconstructing the scattering and absorption coefficients using boundary measurements for a time dependent radiative transfer equation (RTE). As the measurement is mostly polluted by errors, both experimental and computational, an important question is to quantify how the error is amplified in the process of reconstruction. In the forward setting, the solution to the RTE behaves differently in different regimes, and the stability of the inverse problem vary accordingly. In particular, we consider two scalings in this paper. The first one concerns with a diffusive scaling whose macroscopic limit is a diffusion equation. In this case, we showed, following the similar approach as in [Chen, Li and Wang, arXiv:1703.00097], that the stability degrades when the limit is taken. The second one considers a highly forward peaked scattering, wherein the scattering operator is approximated by a Fokker-Planck operator as a limit. In this case, we showed that a fully recover of the scattering coefficient is less possible in the limit, whereas obtaining a rescaled version of the scattering coefficient becomes more practice friendly

Parameter Reconstruction for general transport equation

We consider the inverse problem for the general transport equation with external field, source term and absorption coefficient. We show that the source and the absorption coefficients can be uniquely reconstructed from the boundary measurement, in a Lipschitz stable manner. Specifically, the uniqueness and stability are obtained by using the Carleman estimate in which a special weight function is designed to pick up information on the desired parameter.Comment: 23 pages, 3 figure

Batalin-Vilkovisky quantization and the algebraic index

Into a geometric setting, we import the physical interpretation of index theorems via semi-classical analysis in topological quantum field theory. We develop a direct relationship between Fedosov's deformation quantization of a symplectic manifold X and the BV quantization of a one-dimensional sigma model with target X. This model is a quantum field theory of AKSZ type and is quantized rigorously using Costello's homotopic theory of effective renormalization. We show that Fedosov's Abelian connections on the Weyl bundle produce solutions to the effective quantum master equation. Moreover, BV integration produces a natural trace map on the deformation quantized algebra. This formulation allows us to exploit a (rigorous) localization argument in quantum field theory to deduce the algebraic index theorem via semi-classical analysis, i.e., one-loop Feynman diagram computations.Comment: V2: Significant re-write, 51 pages, 9 figure