arXiv subjects

Feng Tian

Publications and source records attributed to Feng Tian.

161 records · Page 9Linked to original sources

arXiv2014

Frames and Factorization of Graph Laplacians

Using functions from electrical networks (graphs with resistors assigned to edges), we prove existence (with explicit formulas) of a canonical Parseval frame in the energy Hilbert space $\mathscr{H}_{E}$ of a prescribed infinite (or finite) network. Outside degenerate cases, our Parseval frame is not an orthonormal basis. We apply our frame to prove a number of explicit results: With our Parseval frame and related closable operators in $\mathscr{H}_{E}$ we characterize the Friedrichs extension of the $\mathscr{H}_{E}$-graph Laplacian. We consider infinite connected network-graphs $G=\left(V,E\right)$, $V$ for vertices, and \emph{E} for edges. To every conductance function $c$ on the edges $E$ of $G$, there is an associated pair $\left(\mathscr{H}_{E},Δ\right)$ where $\mathscr{H}_{E}$ in an energy Hilbert space, and $Δ\left(=Δ_{c}\right)$ is the $c$-Graph Laplacian; both depending on the choice of conductance function $c$. When a conductance function is given, there is a current-induced orientation on the set of edges and an associated natural Parseval frame in $\mathscr{H}_{E}$ consisting of dipoles. Now $Δ$ is a well-defined semibounded Hermitian operator in both of the Hilbert $l^{2}\left(V\right)$ and $\mathscr{H}_{E}$. It is known to automatically be essentially selfadjoint as an $l^{2}\left(V\right)$-operator, but generally not as an $\mathscr{H}_{E}$ operator. Hence as an $\mathscr{H}_{E}$ operator it has a Friedrichs extension. In this paper we offer two results for the Friedrichs extension: a characterization and a factorization. The latter is via $l^{2}\left(V\right)$.

math.FA↗

arXiv2014

von Neumann indices and classes of positive definite functions

With view to applications, we establish a correspondence between two problems: (i) the problem of finding continuous positive definite extensions of functions $F$ which are defined on open bounded domains $Ω$ in $\mathbb{R}$, on the one hand; and (ii) spectral theory for elliptic differential operators acting on $Ω$, (constant coefficients.) A novelty in our approach is the use of a reproducing kernel Hilbert space $\mathscr{H}_{F}$ computed directly from $\left(Ω,F\right)$, as well as algorithms for computing relevant orthonormal bases in $\mathscr{H}_{F}$.

math.FA↗

arXiv2014

Harmonic analysis of a class of reproducing kernel Hilbert spaces arising from groups

We study two extension problems, and their interconnections: (i) extension of positive definite (p.d.) continuous functions defined on subsets in locally compact groups $G$; and (ii) (in case of Lie groups $G$) representations of the associated Lie algebras $La\left(G\right)$, i.e., representations of $La\left(G\right)$ by unbounded skew-Hermitian operators acting in a reproducing kernel Hilbert space $\mathscr{H}_{F}$ (RKHS). Our analysis is non-trivial even if $G=\mathbb{R}^{n}$, and even if $n=1$. If $G=\mathbb{R}^{n}$, (ii), we are concerned with finding systems of strongly commuting selfadjoint operators $\left\{ T_{i}\right\} $ extending a system of commuting Hermitian operators with common dense domain in $\mathscr{H}_{F}$. Specifically, we consider partially defined positive definite (p.d.) continuous functions $F$ on a fixed group. From $F$ we then build a reproducing kernel Hilbert space $\mathscr{H}_{F}$, and the operator extension problem is concerned with operators acting in $\mathscr{H}_{F}$, and with unitary representations of $G$ acting on $\mathscr{H}_{F}$. Our emphasis is on the interplay between the two problems, and on the harmonic analysis of our RKHSs $\mathscr{H}_{F}$.

math.FA↗

arXiv2013

High Stellar FUV/NUV Ratio and Oxygen Contents in the Atmospheres of Potentially Habitable Planets

Recent observations of several planet-hosting M dwarfs show that most have FUV/NUV flux ratios 1000 times greater than that of the Sun. Here we show that the atmospheric oxygen contents (O2 and O3) of potentially habitable planets in this type of UV environment could be 2~3 orders of magnitude greater than those of their counterparts around Sun-like stars as a result of decreased photolysis of O3, H2O2, and HO2. Thus detectable levels of atmospheric oxygen, in combination with the existence of H2O and CO2, may not be the most promising biosignatures on planets around stars with high FUV/NUV ratios such as the observed M dwarfs.

astro-ph.EP↗

arXiv2013

Momentum Operators in The Unit Square

We investigate the skew-adjoint extensions of a partial derivative operator acting in the direction of one of the sides a unit square. We investigate the unitary equivalence of such extensions and the spectra of such extensions. It follows from our results, that such extensions need not have discete spectrum. We apply our techniques to the problem of finding commuting skew-adjoint extensions of the partial derivative operators acting in the directions of the sides of the unit square. While our results are most easily stated for the unit square, they are established for a larger class of domains, including certain fractal domains.

math.SP↗

arXiv2012

The Ultraviolet Radiation Environment Around M dwarf Exoplanet Host Stars

The spectral and temporal behavior of exoplanet host stars is a critical input to models of the chemistry and evolution of planetary atmospheres. At present, little observational or theoretical basis exists for understanding the ultraviolet spectra of M dwarfs, despite their critical importance to predicting and interpreting the spectra of potentially habitable planets as they are obtained in the coming decades. Using observations from the Hubble Space Telescope, we present a study of the UV radiation fields around nearby M dwarf planet hosts that covers both FUV and NUV wavelengths. The combined FUV+NUV spectra are publically available in machine-readable format. We find that all six exoplanet host stars in our sample (GJ 581, GJ 876, GJ 436, GJ 832, GJ 667C, and GJ 1214) exhibit some level of chromospheric and transition region UV emission. No "UV quiet" M dwarfs are observed. The bright stellar Ly-alpha emission lines are reconstructed, and we find that the Ly-alpha line fluxes comprise ~37-75% of the total 1150-3100A flux from most M dwarfs; > 10^{3} times the solar value. The F(FUV)/F(NUV) flux ratio, a driver for abiotic production of the suggested biomarkers O2 and O3, is shown to be ~0.5-3 for all M dwarfs in our sample, > 10^{3} times the solar ratio. For the four stars with moderate signal-to-noise COS time-resolved spectra, we find UV emission line variability with amplitudes of 50-500% on 10^{2} - 10^{3} s timescales. Finally, we observe relatively bright H2 fluorescent emission from four of the M dwarf exoplanetary systems (GJ 581, GJ 876, GJ 436, and GJ 832). Additional modeling work is needed to differentiate between a stellar photospheric or possible exoplanetary origin for the hot (T(H2) \approx 2000-4000 K) molecular gas observed in these objects.

astro-ph.EP↗

arXiv2012

Design and Implementation of Flight Visual Simulation System

The design requirement for flight visual simulation system is studied and the overall structure and development process are proposed in this paper. Through the construction of 3D scene model library and aircraft model, the rendering and interaction of visual scene are implemented. The changes of aircraft flight attitude in visual system are controlled by real-time calculation of aircraft aerodynamic and dynamic equations and flight simulation effect is enhanced by this kind of control. Several key techniques for optimizing 3D model and relative methods for large terrain modeling are explored for improving loading ability and rendering speed of the system. Experiment shows that, with specific function and performance guaranteed as a premise, the system achieves expected results, that is, precise real-time calculation of flight attitude and smooth realistic screen effect.

cs.CY↗

arXiv2012

Time-resolved Ultraviolet Spectroscopy of the M-dwarf GJ 876 Exoplanetary System

Extrasolar planets orbiting M-stars may represent our best chance to discover habitable worlds in the coming decade. The ultraviolet spectrum incident upon both Earth-like and Jovian planets is critically important for proper modeling of their atmospheric heating and chemistry. In order to provide more realistic inputs for atmospheric models of planets orbiting low-mass stars, we present new near- and far-ultraviolet (NUV and FUV) spectroscopy of the M-dwarf exoplanet host GJ 876 (M4V). Using the COS and STIS spectrographs aboard the Hubble Space Telescope, we have measured the 1150-3140A spectrum of GJ 876. We have reconstructed the stellar HI LyA emission line profile, and find that the integrated LyA flux is roughly equal to the rest of the integrated flux (1150-1210A + 1220-3140A) in the entire ultraviolet bandpass (F(LyA)/F(FUV+NUV) ~0.7). This ratio is ~ 2500x greater than the solar value. We describe the ultraviolet line spectrum and report surprisingly strong fluorescent emission from hot H2 (T(H2) > 2000 K). We show the light-curve of a chromospheric + transition region flare observed in several far-UV emission lines, with flare/quiescent flux ratios >= 10. The strong FUV radiation field of an M-star (and specifically LyA) is important for determining the abundance of O2 -- and the formation of biomarkers -- in the lower atmospheres of Earth-like planets in the habitable zones of low-mass stars.

astro-ph.EP↗

arXiv2012

Restrictions and extensions of semibounded operators

We study restriction and extension theory for semibounded Hermitian operators in the Hardy space of analytic functions on the disk D. Starting with the operator zd/dz, we show that, for every choice of a closed subset F in T=bd(D) of measure zero, there is a densely defined Hermitian restriction of zd/dz corresponding to boundary functions vanishing on F. For every such restriction operator, we classify all its selfadjoint extension, and for each we present a complete spectral picture. We prove that different sets F with the same cardinality can lead to quite different boundary-value problems, inequivalent selfadjoint extension operators, and quite different spectral configurations. As a tool in our analysis, we prove that the von Neumann deficiency spaces, for a fixed set F, have a natural presentation as reproducing kernel Hilbert spaces, with a Hurwitz zeta-function, restricted to FxF, as reproducing kernel.

math.SP↗

arXiv2012

Spectral Theory of Multiple Intervals

We present a model for spectral theory of families of selfadjoint operators, and their corresponding unitary one-parameter groups (acting in Hilbert space.) The models allow for a scale of complexity, indexed by the natural numbers $\mathbb{N}$. For each $n\in\mathbb{N}$, we get families of selfadjoint operators indexed by: (i) the unitary matrix group U(n), and by (ii) a prescribed set of $n$ non-overlapping intervals. Take $Ω$ to be the complement in $\mathbb{R}$ of $n$ fixed closed finite and disjoint intervals, and let $L^{2}(Ω)$ be the corresponding Hilbert space. Moreover, given $B\in U(n)$, then both the lengths of the respective intervals, and the gaps between them, show up as spectral parameters in our corresponding spectral resolutions within $L^{2}(Ω)$. Our models have two advantages: One, they encompass realistic features from quantum theory, from acoustic wave equations and their obstacle scattering; as well as from harmonic analysis. Secondly, each choice of the parameters in our models, $n\in\mathbb{N}$, $B\in U(n)$, and interval configuration, allows for explicit computations, and even for closed-form formulas: Computation of spectral resolutions, of generalized eigenfunctions in $L^{2}(Ω)$ for the continuous part of spectrum, and for scattering coefficients. Our models further allow us to identify embedded point-spectrum (in the continuum), corresponding, for example, to bound-states in scattering, to trapped states, and to barriers in quantum scattering. The possibilities for the discrete atomic part of spectrum includes both periodic and non-periodic distributions.

math.SP↗

arXiv2012

Translation Representations and Scattering By Two Intervals

Studying unitary one-parameter groups in Hilbert space (U(t),H), we show that a model for obstacle scattering can be built, up to unitary equivalence, with the use of translation representations for L2-functions in the complement of two finite and disjoint intervals. The model encompasses a family of systems (U (t), H). For each, we obtain a detailed spectral representation, and we compute the scattering operator, and scattering matrix. We illustrate our results in the Lax-Phillips model where (U (t), H) represents an acoustic wave equation in an exterior domain; and in quantum tunneling for dynamics of quantum states.

math.SP↗

arXiv2011

Momentum Operators in Two Intervals: Spectra and Phase Transition

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs. Compared to earlier studies, it yields a more direct link between spectrum and geometry.

math.SP↗

arXiv2010

Functional Analysis

Notes from a course taught by Palle Jorgensen in the fall semester of 2009. The course covered central themes in functional analysis and operator theory, with an emphasis on topics of special relevance to such applications as representation theory, harmonic analysis, mathematical physics, and stochastic integration.

math.FA↗

arXiv2009

Dynamic spectrum sharing game by lease

We propose and analyze a dynamic implementation of the property-rights model of cognitive radio. A primary link has the possibility to lease the owned spectrum to a MAC network of secondary nodes, in exchange for cooperation in the form of distributed space-time coding (DSTC). The cooperation and competition between the primary and secondary network are cast in the framework of sequential game. On one hand, the primary link attempts to maximize its quality of service in terms of signal-to-interference-plus-noise ratio (SINR); on the other hand, nodes in the secondary network compete for transmission within the leased time-slot following a power control mechanism. We consider both a baseline model with complete information and a more practical version with incomplete information, using the backward induction approach for the former and providing approximate algorithm for the latter. Analysis and numerical results show that our models and algorithms provide a promising framework for fair and effective spectrum sharing, both between primary and secondary networks and among secondary nodes.

cs.GT↗

arXiv2008

A Relaying Incentive Scheme in Multihop Cellular Networks Based on Coalitional Game with Externalities

Cooperative multihop communication can greatly increase network throughput, yet packet forwarding for other nodes involves opportunity and energy cost for relays. Thus one of the pre-requisite problems in the successful implementation of multihop transmission is how to foster cooperation among selfish nodes. Existing researches mainly adopt monetary stimulating. In this manuscript, we propose instead a simple and self-enforcing forwarding incentive scheme free of indirect monetary remunerating for asymmetric (uplink multihop, downlink single-hop) cellar network based on coalitional game theory, which comprises double compensation, namely, Inter- BEA, global stimulating policy allotting resources among relaying coalitions according to group size, and Intra-BEA, local compensating and allocating rule within coalitions. Firstly, given the global allotting policy, we introduce a fair allocation estimating approach which includes remunerating for relaying cost using Myerson value for partition function game, to enlighten the design of local allocating rules. Secondly, given the inter- and intra-BEA relay fostering approach, we check stability of coalition structures in terms of internal and external stability as well as inductive core. Theoretic analysis and numerical simulation show that our measure can provide communication opportunities for outer ring nodes and enlarge system coverage, while at the same time provide enough motivation with respect to resource allocation and energy saving for nodes in inner and middle ring to relay for own profits.

cs.GT↗

arXiv2006

The Number of Complete Maps on Surfaces

A map is a connected topological graph cellularly embedded in a surface and a complete map is a cellularly embedded complete graph in a surface. In this paper, all automorphisms of complete maps of order n are determined by permutations on its vertices. Applying a scheme for enumerating maps on surfaces with a given underlying graph, the numbers of unrooted complete maps on orientable or non-orientable surfaces are obtained.

math.GM↗

arXiv1996

Burau representation and random walk on string links

Using a probabilistic interpretation of the Burau representation of the braid group offered by Vaughan Jones, we generalize the Burau representation to a representation of the semigroup of string links. This representation is determined by a linear system, and is dominated by finite type string link invariants. For positive string links, the representation matrix can be interpreted as the transition matrix of a Markov process. For positive non-separable links, we show that all states are persistent. (This is an extensively revised version of the first author's original paper titled "Burau representation and random walk on knots".)

q-alg↗