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The number and nature of real-valued solutions to the nonlinear power flow equations are not completely understood and there are no proven simple methods for computing all solutions. It has been noted that in practice the equations tend to admit many fewer real-valued solutions than what might be estimated using traditional bounds. In this paper we examine a means to calculate the distribution of number of solutions as a function of multiple parameters. This method relies on a reduction technique that resolves to an equation in one variable for which the calculation of real-valued solutions is straightforward. Importantly we examine the fundamental question of whether a real-valued solution in one variable necessarily implies all variables will be simultaneously realvalued. We present topologies for which this favorable property is retained and topologies for which it fails.

Analyses of power engineering networks rely on an understanding of the power flow equations, a system of quadratic equations relating power injections and voltages at each node. Finding the number of nontrivial, real solutions to these equations is an active area of study. In this paper we propose that in general for a fixed topology, a single, univariate polynomial whose coefficients are polynomials in the network susceptances suffices for determining the number of nontrivial, real solutions to the power flow equations. We then exploit this polynomial to gain geometric insight to the solution regions for a fixed network where a solution region is the region in the space of susceptances wherein the number of real solutions to the system is some fixed constant. This type of analysis allows easier insight to robustness analysis for varying susceptances. We observe that some solution regions resemble thickened hyperplanes and therefore appear as stripes in our figures.

Operation and planning of electric grids involve the analysis of certain power flow equations, which exhibit multiple solutions. One solution generally corresponds to a preferred operating condition, while other less desirable solutions are often useful for assessing dynamic angular stability and static voltage stability margins. The number and nature of solutions to these network equations has been studied for decades, yet without revealing a complete understanding. In light of observations that the number of solutions encountered in practice is smaller than the number predicted by the usual mathematical bounds, the most recent theoretical papers on this topic have sought to produce more accurate bounds on the number of solutions, accounting for network structure. In this paper we further refine the analysis to describe distributions of solutions. We present algebraic results for a certain small system and empirical results for others.

Quasi-Quadratic Residue Codes (QQR Codes) are a family of double circulant codes. They are important in at least two respects: Firstly, they have very good minimum distances when p ≡ 3 (mod 8), providing a critical test case for Goppa's Conjecture. Secondly, they serve as a powerful tool for studying point distributions on hyperelliptic curves. We will use the result of their weight distributions to prove a variant on a result of Larsen [1] on asymptotic normal distribution of numbers of points on hyperelliptic curves.

Consider a generic $r$-dimensional subspace of $\mathbb{R}^d$, $r, Comment: To appear in Proc. of IEEE ISIT, 2015

In network information theory, Shannon-type inequalities are not enough to describe the entropy regions if there are more than three random variables. The Ingleton inequality is one of the most interesting and useful non-Shannon-type inequalities in four random variables. It is satisfied by linear network codes, but not by all network codes. A measure of how much it fails by is given by the Ingleton score. The Four-Atom Conjecture of R. Dougherty, C Freiling, and K. Zeger states that the Ingleton score cannot exceed 0.089373. Using groups to characterize the entropy region, we propose a two-dimensional extension of the Ingleton score, which yields finer information. In particular, we conjecture constraints on this two-dimensional Ingleton score, i.e. a refinement of the Four-Atom Conjecture. Also, we present two families of examples that between them produce all permissible two-dimensional Ingleton scores.

In network information theory, non-Shannon-type inequalities arise in determining capacity/entropy regions where there are more than three random variables. The most famous such inequality is the Ingleton inequality, which is satisfied by linear network codes. To produce better network codes raises the question of finding points in the region that violate the Ingleton inequality and this problem can be translated into group theory. Abelian and small groups do not produce violations, and Mao, Thill, and Hassibi computed that the smallest Ingleton-violating group is the symmetric group on five letters, of order 120. They then generalized this example to other matrix groups. In each of their cases, the Ingleton ratio (which is less than or equal to 1 if and only if the inequality holds) is smaller than 4/3. We go beyond their work to show that examples of Ingleton-violating groups abound, construct explicit examples with arbitrarily large Ingleton ratio, give a systematic approach to finding good examples and a much simplified formula for the Ingleton ratio, and give supporting evidence for the Four-Atom Conjecture of Dougherty, Freiling, and Zeger, which would bound the ratio in terms of the group order.

The Belgian chocolate problem asks for which values of a process parameter δ there exist three stable polynomials satisfying a certain controller design equation. For small n, earlier papers have employed global optimization techniques to obtain solutions involving polynomials of degree ≤ n, with each successive paper having a slightly larger δ. We note that these solutions converge to a critical case and we introduce algebraic methods to identify that case. The previously obtained solutions are simply approximations to this critical case, with the particular δ artificially constrained by the precision to which the authors were working. As a demonstration of the efficacy of algebraic methods, we see that the record value of δ can be increased from 0.973974 to 0.976461. We discover a surprising connection with the abc theorem for polynomials and present ideas for systematically increasing this record.

We present a new algebraic framework for linear trellises which yields a new and simpler proof of the fundamental Factorization Theorem by Koetter and Vardy [4], and which sheds light on several other foundational questions that were not considered before. The techniques used within this framework are new, and comprise algebraic tools that can be used to analyze systematically the structure of linear trellises. In fact our methods and tools produce several subsequent results, the most important of which are: characterization of linear trellises isomorphy, uniqueness of linear structure of linearizable trellises, methods for determining all possible factorizations of trellises. These same algebraic methods have a potential for extending the Factorization Theorem to the case of group trellises. In fact this is very important, since the Factorization Theorem for group trellises as formulated by Koetter and Vardy is false.

Tail-biting trellises and their pseudocodewords are very important for modern decoding techniques like iterative decoding. We introduce a useful matrix representation of trellises, give its fundamental properties, and use it to enumerate and describe the distribution of trellis pseudocodewords. We give several examples, a couple of which lead to important open problems. Next, we prove that the pseudocodeword weight enumerator introduced in [2] always satisfies a recurrence equation, and, for certain trellises including the Golay trellis given in [3], that it is invariant under generalized MacWilliams transformations, allowing invariant theory to be used for computing it. Computation for the Golay trellis shows then that pseudocodewords of period at most 4 must have AWGN pseudoweight at least 8.

This paper addresses an old, yet challenging issue — curvature estimation from discrete sampling points over a curve. We introduce a novel algorithm based on performing line integrals. The proposed method is computationally more efficient than the previous integration-based methods because of the constant computation time. Qualitative tests on synthesized shapes in the presence of noise are performed, which show the robustness of our approach. Also, the discriminative capability of the estimated curvature is evaluated by conducting experiments on the FRGC (Face Recognition Grand Challenge) v2.0 dataset which contains 4007 3D facial images recorded from 466 subjects. The results show that recognition performance is significantly improved by using our curvature estimation method. This novel approach presents potential for a broad class of multimedia applications.

Applications of summation invariant features to multi-region face recognition are explored in this work. Earlier, we have demonstrated the potential benefits of this approach. In this paper, we provide a systematic, thorough comparison of all the summation invariant features derived to-date, and propose a new multi-feature fusion approach to further improve the overall performance. We also identify summation invariant features that yield superior performance for face recognition applications. Special attention is given to the implementation of 3D summation invariants. Extensive experimental results with the FRGC (Face Recognition Grand Challenge) 2.0 data set confirms the advantage of summation invariant features for 3D face recognition.

Recently a novel family of geometrically invariant features, called summation invariant, has been developed and applied to object recognition. The range of this family of features is expanded here beyond the Euclidean and affine transformation groups to planar projective transformations. Whereas other methods require small changes in view, or collinear points, this method removes those limitations and allows recognition of general planar objects over wide ranges of viewpoint. The derivation of these new features requires the innovation of deriving the invariants in the homogeneous coordinate space, yet yields results formulated in terms of Cartesian coordinates. Simulations demonstrate the effectiveness of this new approach to object recognition under projective transformations like those encountered in camera networks.

We present a novel 3D face recognition method that incorporates summation invariant features extracted from multiple sub-regions of a facial range images, and optimal fusion of similarity scores between corresponding sub-regions. The key innovation of this paper is the development of the fusion-based face recognition algorithm that delivers significant performance enhancement while requiring very little computation. Experiments on the FRGC (Face Recognition Grand Challenge) version 2 dataset show that our algorithm improves the recognition performance significantly in the presence of facial expressions.

We propose a handwritten digit recognition algorithm that uses 4×4 2D hidden Markov models to extract basic features from an unclassified image. The novel idea given here is that we use powerful techniques from the emerging mathematical fields of tropical geometry and algebraic statistics to determine parameters for the model. The distance between the unclassified images and prototypes is calculated in stages, where estimates of the distance become finer as obviously distant prototypes are discarded from the pool of possible K-nearest neighbors. Our algorithm achieves a 95.51 percent recognition rate with zero rejection on the MNIST database of handwritten digits.

In this paper, we introduce a novel 3D surface landmark detection method using a 3D integral invariant feature extended from that proposed by Manay et al. for 2D contours. We apply this new feature to detect the nose tips of 2.5D range images of human faces. Using the Face Recognition Grand Challenge 2.0 dataset, our method compares favorably with a recently proposed competing method.

Digital video watermarking is increasingly important. Video signals are very susceptible to attacks like frame averaging, dropping, swapping, collusion. This paper presents a new video watermarking method. The most important information to be watermarked such as a company's name is hidden in a sequence of statistical data, which is then embedded into each frame. Every segment of the statistical data is typically enough for the key watermark extraction. A very detailed example of this method is given, based on the cryptanalysis. The most important information is used as the key to encrypt a plaintext. The plaintext can be some less important information about the product. The ciphertext is then embedded in the video frames. With statistical knowledge of the English language, we can quickly recover the key. Mathematical analysis and simulation results are given in the paper.

A novel summation invariant of curves under transformation group action is proposed. This new invariant is less sensitive to noise than the differential invariant and does not require an analytical expression for the curve as the integral invariant does. We exploit this summation invariant to define a shape descriptor called a semi-local summation invariant and use it as a new feature for shape recognition. Tested on a database of noisy shapes of fish, it was observed that the summation invariant feature exhibited superior discriminating power compared to that of wavelet-based invariant features.

A novel family of geometrically invariant features, called summation invariants are proposed for the recognition of the 3D surface of human faces. In particular, a 2D semi-local summation invariant feature is extracted from each column and each row of a rectangular region surrounding the nose of a 3D facial depth map. Through extensive experimentation, we empirically identify the most efficient 2D summation invariant features. We also investigate the proper pre-processing method for the 2D summation invariant features. Tested with the 3D facial data from the Face Recognition Grand Challenge v1.0 dataset, the proposed new features exhibit significant performance improvement over the baseline algorithm distributed with the dataset.

A novel family of 2D and 3D geometrically invariant features, called summation invariants is proposed for the recognition of the 3D surface of human faces. Focusing on a rectangular region surrounding the nose of a 3D facial depth map, a subset of the so called semi-local summation invariant features is extracted. Then the similarity between a pair of 3D facial depth maps is computed to determine whether they belong to the same person. Out of many possible combinations of these set of features, we select, through careful experimentation, a subset of features that yields best combined performance. Tested with the 3D facial data from the on-going Face Recognition Grand Challenge v1.0 dataset, the proposed new features exhibit significant performance improvement over the baseline algorithm distributed with the datase

In this paper, we developed a family of 2D and 3D invariant features with applications to 3D human faces recognition. The main contributions of this paper are: (a) systematically deriving a family of novel features, called summation invariant that are invariant to Euclidean transformation in both 2D and 3D; (b) developing an effective method to apply summation invariant to the 3D face recognition problem. Tested with the 3D data from the Face Recognition Grand Challenge v1.0 dataset, the proposed new features exhibit achieves a performance that rivals the best 3D face recognition algorithms reported so far.

Transmit beamforming with receive combining is a low-complexity solution that achieves the full diversity afforded by a multi-antenna channel. Building on our recent result which shows that even channel statistics are sufficient to achieve perfect feedback performance (in the limit of antenna dimensions) with beamforming and combining, we propose near-optimal codebook designs for correlated channels with a focus on few antennas at the transmitter and the receiver. In the process, we refine the answer to the question: When are channel statistics sufficient to achieve near perfect feedback performance? We show that the condition number of the transmit and receive covariance matrices hold the key to this question. We partition the transmit and receive covariance spaces into 4 regions based on well and ill-conditioning of the covariance matrices and show that the number of bits required for near perfect feedback performance is dependent on the condition numbers of these matrices.

Transmit beamforming and receive combining are low complexity techniques that help in achieving the full diversity afforded by the multi-antenna channel. Complete channel state information at the transmitter may be possible only under rare instances. Thus quantized beamforming with limited feedback on a reverse link has been a topic that has attracted great attention recently. But almost all of the work to date has focussed on modeling the channel with independent, identically distributed (i.i.d.) Rayleigh fading between antenna pairs. In this paper, we consider the correlated channel case, and show that Grassmannian line packing is an artificial artifact of the i.i.d. assumption. We show that there are dominant peaks in the eigen-domain when correlation is imposed and the codebook construction should be matched to the correlation in the channel, which renders the problem in this case easier than for i.i.d. channels

The IEEE Signal Processing Society (2000) offered a prize of $1000 for proving or disproving Makhoul's conjecture, which says that, given a causal all-pass digital signal x/sub n/ of order p, with nonzero x/sub 0/, the location of the peak of x/sub n/ always lies between n = 0 and n = 2p-1. The case of p = 1 is trivial, and no further progress had been made in 25 years until Lertniphonphun, Rajagopal, and Wenzel gave counter examples for large p. In this paper, Makhoul's conjecture is proven for p = 2. It is also shown that the conjecture fails dramatically in the case of complex coefficients.