Recent progress in random metric theory and its applications to conditional risk measures

The purpose of this paper is to give a selective survey on recent progress in random metric theory and its applications to conditional risk measures. This paper includes eight sections. Section 1 is a longer introduction, which gives a brief introduction to random metric theory, risk measures and conditional risk measures. Section 2 gives the central framework in random metric theory, topological structures, important examples, the notions of a random conjugate space and the Hahn-Banach theorems for random linear functionals. Section 3 gives several important representation theorems for random conjugate spaces. Section 4 gives characterizations for a complete random normed module to be random reflexive. Section 5 gives hyperplane separation theorems currently available in random locally convex modules. Section 6 gives the theory of random duality with respect to the locally $L^{0}-$convex topology and in particular a characterization for a locally $L^{0}-$convex module to be $L^{0}-$pre$-$barreled. Section 7 gives some basic results on $L^{0}-$convex analysis together with some applications to conditional risk measures. Finally, Section 8 is devoted to extensions of conditional convex risk measures, which shows that every representable $L^{\infty}-$type of conditional convex risk measure and every continuous $L^{p}-$type of convex conditional risk measure ($1\leq p<+\infty$) can be extended to an $L^{\infty}_{\cal F}({\cal E})-$type of $\sigma_{\epsilon,\lambda}(L^{\infty}_{\cal F}({\cal E}), L^{1}_{\cal F}({\cal E}))-$lower semicontinuous conditional convex risk measure and an $L^{p}_{\cal F}({\cal E})-$type of ${\cal T}_{\epsilon,\lambda}-$continuous conditional convex risk measure ($1\leq p<+\infty$), respectively.Comment: 37 page

The Schrodinger-like Equation for a Nonrelativistic Electron in a Photon Field of Arbitrary Intensity

The ordinary Schrodinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schrodinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The "eigenvalue" in the new equation is an operator in a Cartan subalgebra. An approximation consistent with the nonrelativistic energy level derived from its relativistic value replaces the "eigenvalue" operator by an ordinary number, recovering the ordinary Schrodinger eigenvalue equation used in the formal scattering formalism. The Schrodinger-like equation for the multimode case is also presented.Comment: Tex file, 13 pages, no figur

Two novel nonlinear companding schemes with iterative receiver to reduce PAPR in multi-carrier modulation systems

Companding transform is an efficient and simple method to reduce the Peak-to-Average Power Ratio (PAPR) for Multi-Carrier Modulation (MCM) systems. But if the MCM signal is only simply operated by inverse companding transform at the receiver, the resultant spectrum may exhibit severe in-band and out-of-band radiation of the distortion components, and considerable peak regrowth by excessive channel noises etc. In order to prevent these problems from occurring, in this paper, two novel nonlinear companding schemes with a iterative receiver are proposed to reduce the PAPR. By transforming the amplitude or power of the original MCM signals into uniform distributed signals, the novel schemes can effectively reduce PAPR for different modulation formats and sub-carrier sizes. Despite moderate complexity increasing at the receiver, but it is especially suitable to be combined with iterative channel estimation. Computer simulation results show that the proposed schemes can offer good system performances without any bandwidth expansion

Spin singlet pairing in the superconducting state of NaxCoO2\cdot1.3H2O: evidence from a ^{59}Co Knight shift in a single crystal

We report a ^{59}Co Knight shift measurement in a single crystal of the cobalt oxide superconductor Na_{x}CoO_2\cdot1.3H_2O (T_c=4.25 K). We find that the shift due to the spin susceptibility, K^s, is substantially large and anisotropic, with the spin shift along the a-axis K^s_a being two times that along the c-axis K^s_c. The shift decreases with decreasing temperature (T) down to T\sim100 K, then becomes a constant until superconductivity sets in. Both K^s_a and K^s_c decrease below T_c. Our results indicate unambiguously that the electron pairing in the superconducting state is in the spin singlet form.Comment: 4 pages, 5 figure

Modeling Magnetic Field Structure of a Solar Active Region Corona using Nonlinear Force-Free Fields in Spherical Geometry

We test a nonlinear force-free field (NLFFF) optimization code in spherical geometry using an analytical solution from Low and Lou. Several tests are run, ranging from idealized cases where exact vector field data are provided on all boundaries, to cases where noisy vector data are provided on only the lower boundary (approximating the solar problem). Analytical tests also show that the NLFFF code in the spherical geometry performs better than that in the Cartesian one when the field of view of the bottom boundary is large, say, $20^\circ \times 20^\circ$. Additionally, We apply the NLFFF model to an active region observed by the Helioseismic and Magnetic Imager (HMI) on board the Solar Dynamics Observatory (SDO) both before and after an M8.7 flare. For each observation time, we initialize the models using potential field source surface (PFSS) extrapolations based on either a synoptic chart or a flux-dispersal model, and compare the resulting NLFFF models. The results show that NLFFF extrapolations using the flux-dispersal model as the boundary condition have slightly lower, therefore better, force-free and divergence-free metrics, and contain larger free magnetic energy. By comparing the extrapolated magnetic field lines with the extreme ultraviolet (EUV) observations by the Atmospheric Imaging Assembly (AIA) on board SDO, we find that the NLFFF performs better than the PFSS not only for the core field of the flare productive region, but also for large EUV loops higher than 50 Mm.Comment: 34 pages, 8 figures, accepted for publication in Ap

Co-Clustering Network-Constrained Trajectory Data

Recently, clustering moving object trajectories kept gaining interest from both the data mining and machine learning communities. This problem, however, was studied mainly and extensively in the setting where moving objects can move freely on the euclidean space. In this paper, we study the problem of clustering trajectories of vehicles whose movement is restricted by the underlying road network. We model relations between these trajectories and road segments as a bipartite graph and we try to cluster its vertices. We demonstrate our approaches on synthetic data and show how it could be useful in inferring knowledge about the flow dynamics and the behavior of the drivers using the road network

Almost Perfect Privacy for Additive Gaussian Privacy Filters

We study the maximal mutual information about a random variable $Y$ (representing non-private information) displayed through an additive Gaussian channel when guaranteeing that only $\epsilon$ bits of information is leaked about a random variable $X$ (representing private information) that is correlated with $Y$. Denoting this quantity by $g_\epsilon(X,Y)$, we show that for perfect privacy, i.e., $\epsilon=0$, one has $g_0(X,Y)=0$ for any pair of absolutely continuous random variables $(X,Y)$ and then derive a second-order approximation for $g_\epsilon(X,Y)$ for small $\epsilon$. This approximation is shown to be related to the strong data processing inequality for mutual information under suitable conditions on the joint distribution $P_{XY}$. Next, motivated by an operational interpretation of data privacy, we formulate the privacy-utility tradeoff in the same setup using estimation-theoretic quantities and obtain explicit bounds for this tradeoff when $\epsilon$ is sufficiently small using the approximation formula derived for $g_\epsilon(X,Y)$.Comment: 20 pages. To appear in Springer-Verla

Vacuum State of Lattice Gauge Theory with Fermions in 2+1 Dimensions

We investigate the vacuum state of the lattice gauge theory with fermions in 2+1 dimensions. The vacuum in the Hermite form for the fermion part is obtained; the vacuum in the unitary form has been proposed by Luo and Chen. It is shown that the Hermite vacuum has a lower energy than the unitary one through the variational method.Comment: 16 pages, 5 embedded PS figures, LaTeX with special styl