arXiv subjects

Po-Ning Chen

Publications and source records attributed to Po-Ning Chen.

45 records · Page 3Linked to original sources

arXiv2014

Conserved quantities on asymptotically hyperbolic initial data sets

In this article, we consider the limit of quasi-local conserved quantities [31,9] at the infinity of an asymptotically hyperbolic initial data set in general relativity. These give notions of total energy-momentum, angular momentum, and center of mass. Our assumption on the asymptotics is less stringent than any previous ones to validate a Bondi-type mass loss formula. The Lorentz group acts on the asymptotic infinity through the exchange of foliations by coordinate spheres. For foliations aligning with the total energy-momentum vector, we prove that the limits of quasi-local center of mass and angular momentum are finite, and evaluate the limits in terms of the expansion coefficients of the metric and the second fundamental form.

math.DG↗

arXiv2014

Rigidity of time-flat surfaces in the Minkowski spacetime

A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to constant torsion condition for curves in three dimensional space and has been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this article, we address the question in the title and prove several local and global rigidity theorems for such surfaces in the Minkowski spacetime.

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arXiv2014

Maximum-likelihood Soft-decision Decoding for Binary Linear Block Codes Based on Their Supercodes

Based on the notion of supercodes, we propose a two-phase maximum-likelihood soft-decision decoding (tpMLSD) algorithm for binary linear block codes in this work. The first phase applies the Viterbi algorithm backwardly to a trellis derived from the parity-check matrix of the supercode of the linear block code. Using the information retained from the first phase, the second phase employs the priority-first search algorithm to the trellis corresponding to the linear block code itself, which guarantees finding the ML decision. Simulations on Reed-Muller codes show that the proposed two-phase scheme is an order of magnitude more efficient in average decoding complexity than the recursive maximum-likelihood decoding (RMLD) [1] when the signal-to-noise ratio per information bit is 4.5 dB.

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arXiv2014

Quasilocal angular momentum and center of mass in general relativity

For a spacelike 2-surface in spacetime, we propose a new definition of quasi-local angular momentum and quasi-local center of mass, as an element in the dual space of the Lie algebra of the Lorentz group. Together with previous defined quasi-local energy-momentum, this completes the definition of conserved quantities in general relativity at the quasi-local level. We justify this definition by showing the consistency with the theory of special relativity and expectations on an axially symmetric spacetime. The limits at spatial infinity provide new definitions for total conserved quantities of an isolated system, which do not depend on any asymptotically flat coordinate system or asymptotic Killing field. The new proposal is free of ambiguities found in existing definitions and presents the first definition that precisely describes the dynamics of the Einstein equation.

math.DG↗

arXiv2014

Conserved quantities in general relativity: from the quasi-local level to spatial infinity

We define quasi-local conserved quantities in general relativity by using the optimal isometric embedding in [26] to transplant Killing fields in the Minkowski spacetime back to the 2-surface of interest in a physical spacetime. To each optimal isometric embedding, a dual element of the Lie algebra of the Lorentz group is assigned. Quasi-local angular momentum and quasi-local center of mass correspond to pairing this element with rotation Killing fields and boost Killing fields, respectively. They obey classical transformation laws under the action of the Poincaré group. We further justify these definitions by considering their limits as the total angular momentum and the total center of mass of an isolated system. These expressions were derived from the Hamilton-Jacobi analysis of gravitation action and thus satisfy conservation laws. As a result, we obtained an invariant total angular momentum theorem in the Kerr spacetime. For a vacuum asymptotically flat initial data set of order $1$, it is shown that the limits are always finite without any extra assumptions. We also study these total conserved quantities on a family of asymptotically flat initial data sets evolving by the vacuum Einstein evolution equation. It is shown that the total angular momentum is conserved under the evolution. For the total center of mass, the classical dynamical formula relating the center of mass, energy, and linear momentum is recovered, in the nonlinear context of initial data sets evolving by the vacuum Einstein evolution equation. The definition of quasi-local angular momentum provides an answer to the second problem in classical general relativity on Penrose's list [18].

math.DG↗

arXiv2011

A Generalized Poor-Verdu Error Bound for Multihypothesis Testing and the Channel Reliability Function

A lower bound on the minimum error probability for multihypothesis testing is established. The bound, which is expressed in terms of the cumulative distribution function of the tilted posterior hypothesis distribution given the observation with tilting parameter theta larger than or equal to 1, generalizes an earlier bound due the Poor and Verdu (1995). A sufficient condition is established under which the new bound (minus a multiplicative factor) provides the exact error probability in the limit of theta going to infinity. Examples illustrating the new bound are also provided. The application of this generalized Poor-Verdu bound to the channel reliability function is next carried out, resulting in two information-spectrum upper bounds. It is observed that, for a class of channels including the finite-input memoryless Gaussian channel, one of the bounds is tight and gives a multi-letter asymptotic expression for the reliability function, albeit its determination or calculation in single-letter form remains an open challenging problem. Numerical examples regarding the other bound are finally presented.

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arXiv2007

Maximum-Likelihood Priority-First Search Decodable Codes for Combined Channel Estimation and Error Protection

The code that combines channel estimation and error protection has received general attention recently, and has been considered a promising methodology to compensate multi-path fading effect. It has been shown by simulations that such code design can considerably improve the system performance over the conventional design with separate channel estimation and error protection modules under the same code rate. Nevertheless, the major obstacle that prevents from the practice of the codes is that the existing codes are mostly searched by computers, and hence exhibit no good structure for efficient decoding. Hence, the time-consuming exhaustive search becomes the only decoding choice, and the decoding complexity increases dramatically with the codeword length. In this paper, by optimizing the signal-tonoise ratio, we found a systematic construction for the codes for combined channel estimation and error protection, and confirmed its equivalence in performance to the computer-searched codes by simulations. Moreover, the structural codes that we construct by rules can now be maximum-likelihoodly decodable in terms of a newly derived recursive metric for use of the priority-first search decoding algorithm. Thus,the decoding complexity reduces significantly when compared with that of the exhaustive decoder. The extension code design for fast-fading channels is also presented. Simulations conclude that our constructed extension code is robust in performance even if the coherent period is shorter than the codeword length.

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arXiv2007

Analysis of Sequential Decoding Complexity Using the Berry-Esseen Inequality

his study presents a novel technique to estimate the computational complexity of sequential decoding using the Berry-Esseen theorem. Unlike the theoretical bounds determined by the conventional central limit theorem argument, which often holds only for sufficiently large codeword length, the new bound obtained from the Berry-Esseen theorem is valid for any blocklength. The accuracy of the new bound is then examined for two sequential decoding algorithms, an ordering-free variant of the generalized Dijkstra's algorithm (GDA)(or simplified GDA) and the maximum-likelihood sequential decoding algorithm (MLSDA). Empirically investigating codes of small blocklength reveals that the theoretical upper bound for the simplified GDA almost matches the simulation results as the signal-to-noise ratio (SNR) per information bit ($γ_b$) is greater than or equal to 8 dB. However, the theoretical bound may become markedly higher than the simulated average complexity when $γ_b$ is small. For the MLSDA, the theoretical upper bound is quite close to the simulation results for both high SNR ($γ_b\geq 6$ dB) and low SNR ($γ_b\leq 2$ dB). Even for moderate SNR, the simulation results and the theoretical bound differ by at most \makeblue{0.8} on a $\log_{10}$ scale.

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arXiv2007

Analysis of the Sufficient Path Elimination Window for the Maximum-Likelihood Sequential-Search Decoding Algorithm for Binary Convolutional Codes

A common problem on sequential-type decoding is that at the signal-to-noise ratio (SNR) below the one corresponding to the cutoff rate, the average decoding complexity per information bit and the required stack size grow rapidly with the information length. In order to alleviate the problem in the maximum-likelihood sequential decoding algorithm (MLSDA), we propose to directly eliminate the top path whose end node is $Δ$-trellis-level prior to the farthest one among all nodes that have been expanded thus far by the sequential search. Following random coding argument, we analyze the early-elimination window $Δ$ that results in negligible performance degradation for the MLSDA. Our analytical results indicate that the required early elimination window for negligible performance degradation is just twice of the constraint length for rate one-half convolutional codes. For rate one-third convolutional codes, the required early-elimination window even reduces to the constraint length. The suggestive theoretical level thresholds almost coincide with the simulation results. As a consequence of the small early-elimination window required for near maximum-likelihood performance, the MLSDA with early-elimination modification rules out considerable computational burdens, as well as memory requirement, by directly eliminating a big number of the top paths, which makes the MLSDA with early elimination very suitable for applications that dictate a low-complexity software implementation with near maximum-likelihood performance.

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