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We examine which $p$-groups of order $\le p^6$ are Beauville. We completely classify them for groups of order $\le p^4$. We also show that the proportion of 2-generated groups of order $p^5$ which are Beauville tends to 1 as $p$ tends to infinity; this is not true, however, for groups of order $p^6$. For each prime $p$ we determine the smallest non-abelian Beauville $p$-group., Comment: Some typos corrected

In 1990 Hendrik W. Lenstra, Jr. asked the following question: if G is a transitive permutation group of degree n and A is the set of elements of G that move every letter, then can one find a lower bound (in terms of n) for f(G) = |A|/|G|? Shortly thereafter, Arjeh Cohen showed that 1 n is such a bound. Lenstra’s problem arose from his work on the number field sieve [2]. A simple example of how f(G) arises in number theory is the following: if h is an irreducible polynomial over the integers, consider the proportion

In [10], Mazur showed that the p-adic lifts of a given absolutely irreducible representation are parametrized by a universal deformation ξ:Gℚ, S → GL2() where has the form . (Here Gℚ, S is the Galois group over ℚ of a maximal algebraic extension unramified outside a finite set S of rational primes.) In [1, 3, 10], situations were investigated where the universal deformation ring turned out to be ℚp[[T1T2, T3]] (i.e. r = 3, I = (0)). In [2], the tame relation of algebraic number theory led to more complicated universal deformation rings, ones whose prime spectra consist essentially of four-dimensional sheets.

An efficient method based on projective joint invariant signatures is presented for distributed matching of curves in a camera network. The fundamental projective joint invariants for curves in the real projective space are the volume cross-ratios. A curve in m-dimensional projective space is represented by a signature manifold comprising n-point projective joint invariants, where n is at least m + 2. The signature manifold can be used to establish equivalence of two curves in projective space. However, without correspondence between the two curves, matching signature manifolds is a computational challenge. In this paper we overcome this challenge by finding discriminative sections of signature manifolds consistently across varying viewpoints and scoring the similarity between these sections. This motivates a simple yet powerful method for distributed curve matching in a camera network. Experimental results with real data demonstrate the classification performance of the proposed algorithm with respect to the size of the sections of the invariant signature in various noisy conditions.

The task of curvature estimation from discrete sampling points along a curve is investigated. A novel curvature estimation algorithm based on performing line integrals over an adaptive data window is proposed. The use of line integrals makes the proposed approach inherently robust to noise. Furthermore, the accuracy of curvature estimation is significantly improved by using wild bootstrapping to adaptively adjusting the data window for line integral. Compared to existing approaches, this new method promises enhanced performance, in terms of both robustness and accuracy, as well as low computation cost. A number of numerical examples using synthetic noisy and noiseless data clearly demonstrated the advantages of this proposed method over state-of-the-art curvature estimation algorithms.

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.

For each n, we consider whether there exists an (n + 1)-dimensional space of n by n matrices over GF(2) in which each nonzero matrix has rank n−1. Examples are given for n =3 ,4, and 5, together with evidence for the conjecture that none exist for n>8.

We prove more special cases of the Fontaine–Mazur conjecture regarding p -adic Galois representations unramified at p , and we present evidence for and consequences of a generalization of it.

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.

In the preceding paper, Calegari and Dunfield exhibit a sequence of hyperbolic 3-manifolds which have increasing injectivity radius, and which, subject to some conjectures in number theory, are rational homology spheres. We prove unconditionally that these manifolds are rational homology spheres, and give a sufficient condition for a tower of hyperbolic 3-manifolds to have first Betti number 0 at each level. The methods involved are purely pro-p group theoretical., Comment: This is the version published by Geometry & Topology on 2 April 2006

We produce infinitely many finite 2-groups that do not embed with index 2 in any group generated by involutions. This disproves a conjecture of Lemmermeyer and restricts the possible Galois groups of unramified 2-extensions, Galois over Q, of quadratic number fields.

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.

This paper presents a novel 3-D multiregion face recognition algorithm that consists of new geometric summation invariant features and an optimal linear feature fusion method. A summation invariant, which captures local characteristics of a facial surface, is extracted from multiple subregions of a 3-D range image as the discriminative features. Similarity scores between two range images are calculated from the selected subregions. A novel fusion method that is based on a linear discriminant analysis is developed to maximize the verification rate by a weighted combination of these similarity scores. Experiments on the Face Recognition Grand Challenge V2.0 dataset show that this new algorithm improves the recognition performance significantly in the presence of facial expressions.

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.

Galois groups of infinite p-extensions of number fields unramified at p are a complete mystery. We find by computer a family of pro-p groups that satisfy everything that such a Galois group must, and give evidence for the conjecture that these are the only such groups. This suggests that these mysterious Galois groups indeed have a specific form of presentation. There are surprising connections with knot theory and quantum field theory. Finally, the Fontaine-Mazur conjecture reduces to a purely group-theoretic conjecture, and evidence for this conjecture and an extension of it is given.

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.

Applications of computer algebra range over the entire spectrum of research, development, production, and education. Computer algebra problems arise in industry, commerce, software engineering, and also in banking and insurance applications, although sometimes hidden. We compiled several interesting applications which are exemplary for the most important areas.

Hyperelliptic curve cryptography with genus larger than one has not been seriously considered for cryptographic purposes because many existing implementations are significantly slower than elliptic curve versions with the same level of security. In this paper, the first ever complete hardware implementation of a hyperelliptic curve coprocessor is described. This coprocessor is designed for genus two curves over F2113. Additionally, a modification to the Extended Euclidean Algorithm is presented for the GCD calculation required by Cantor's algorithm. On average, this new method computes the GCD in one-fourth the time required by the Extended Euclidean Algorithm.

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.

We study the intimate interactions between the theory of p-adic Galois representations and the structure of pro-p Galois groups. In particular, information passes in both directions. Algebraic geometry, for instance in the guise of elliptic curves and modular forms, yields naturally occurring Galois representations, whereas on the other side, co-homological techniques and variants on class field theory tell us about the generators and relations of the pro-p Galois groups. In the case of pro-p extensions ramified at (primes above) p, this combination works together rather well to elucidate the structure of the set of Galois representations. In the case of pro-p extensions unramified at p,both sides are poorly understood, but there is the fundamental conjecture of Fontaine—Mazur claiming that such representations should have finite image (since algebraic geometry can produce no others). This has very interesting consequences for the corresponding pro-p Galois groups, possibly producing a new family of just-infinite pro-p groups.

There is an old problem that asks for the probability that two integers chosen at random are relatively prime. The informal solution goes as follows.

In [6] Mazur showed how there is a “versal deformation” parametrising the collection of p-adic representations of a profinite group G lifting a given representation \(\bar p\,:\,G\, \to \,G{L_2}\,\left( {{\mathbb{F}_p}} \right).\) Of particular interest are the \(\bar p\,\) associated to modular forms and elliptic curves in which G is the Galois group of a maximal algebraic extension of Q unramified outside a finite set S of rational primes containing p.

In this paper a problem in group theory is solved to produce an interesting class of groups. The motivation, however, comes from algebraic number theory and in particular the Artin representation. To describe this representation we follow [5, Chap. IV and VI]. Let K and L be local fields, i.e., fields complete with respect to a discrete valuation and with perfect residue field. Let L have valuation ring A, and valuation v, extending the valuation on K. Suppose that L/K is a finite Galois extension with Galois group H. Then H has a normal series consisting of the ramification groups Hj = {O E H: V~(UX X) > j + 1 V x E A,}, j = -1, 0, I,2 ,..., which produces a character of H, due to Artin, as described in Theorem 1 below. The notation adopted in this paper is as follows. If G is a finite group, then Irr(G) denotes its set of irreducible characters. l,, rG, and U, = rc 1, are its principal, regular, and augmentation characters, respectively, whilst unless otherwise stated xc and xG will mean the restriction of x to G and the character of G induced from x.