Quartic fields with large class numbers

This paper examines the possibilities of using Landsat Thematic Mapper images for updating land-use information on topographic maps. Information obtained from satellite images can be kept up-todate and a large number of classes can be distinguished. However, the quality of the information still leaves much to be desired, with a reliability of no more than 40 - 90 %. By using digitized landuse information from a topographic map as a source of contest information the reliability could be improved considerably. The method permitted the Identification of a large number of additional classes and provided quantitative insight into the dynamics of landscap change. Verification using 1:18,000 scale aerial photographs and field data showed the results to have a reliability of over 95 %. It can be expected that whereever topographic maps permit the extraction of land-use information in a similar way as described in this paper, remote sensing information will meet operational accuracy requirements for many fields of application e.g. physical planning, map udating, land-use planning etc. Paper presented at the IGARSS'88 conference Remote Sensing - moving towards the 21st century in Edinburgh, Scotland, 12-16 September 1988

The label tree-based classification is one of the most popular approaches for reducing the testing complexity to sublinear with the large number of classes. One of the popular approaches to generate the label tree is to apply recursively a spectral clustering algorithm to an affinity matrix for partition set of class labels into subsets, each subset corresponds to a child node of the tree. To obtain the affinity matrix from confusion matrix, a set of N binary one-versus-all classifiers is trained and applied on validation set, where N is number of classes. However, these approaches are not reliable when there are a large number of classes because it is too costly to train these classifiers. Furthermore, the affinity matrix could not reflect the real similarity among classes due to the classification accuracy can be low. In addition, the resulting label tree may not be balanced due to the objective function of spectral clustering penalizes unbalanced partitions. In this paper, to achieve better similarity measurement between classes and without using one-versus-all classifiers, we adopt the sum match kernel to get similarity matrix. Moreover, we propose a heuristic for learning balanced tree structure by adjusting the number of class labels in clusters after the spectral clustering is done. The experimental results on benchmark datasets SUN-397 and Caltech-256 show that the performance of the proposed approach outperforms significantly the other approaches.

A learning-based classification problem with a large number of classes is considered. The error-correcting-output-codes (ЕСОС) scheme is optimized. An initial binary matrix is formed at random so that the number of its rows is equal to the number of classes and each column corresponds to the union of several classes in two macroclasses. In the ЕСОС approach, a binary classification problem is solved for every object to be recognized and for every union. The object is assigned to the class with the nearest code row. A generalization of the ЕСОС approach is presented in which a discrete optimization problem is solved to find optimal unions, probabilities of correct classification are used in dichotomy problems, and the degree of dichotomy informativeness is taken into account. If the solution algorithms for the dichotomy problems are correct, the recognition algorithm for the original problem is correct as well.

Classifier combination holds the potential of improving performance by combining the results of multiple classifers. For domains with very large numbers of classes, such as biometrics, we present an axiomatic framework of desirable mathematical properties for combination functions of rank-based classifiers. This framework represents a continuum of combination rules, including the Borda Count, Logistic Regression, and Highest Rank combination methods as extreme cases [11], [23], [4], [13]. Intuitively, this framework captures how the two complementary concepts of general preference for specific classifiers and the confidence it has in any specific result (as indicated by ranks) can be balanced while maintaining consistent rank interpretation. Mixed Group Ranks (MGR) is a new combination function that balances preference and confidence by generalizing these other functions. We demonstrate that MGR is an effective combination approach by performing multiple experiments on data sets with large numbers of classes and classifiers from the FERET face recognition study.

In this paper, we study the Mordell-Weil group of an elliptic curve as a Galois module. We consider an elliptic curve E defined over a number field K whose Mordell-Weil rank over a Galois extension F is 1, 2 or 3. We show that E acquires a point (points) of infinite order over a field whose Galois group is one of Cn×Cm (n = 1, 2, 3, 4, 6, m = 1, 2), Dn×Cm (n = 2, 3, 4, 6, m = 1, 2), A4×Cm (m = 1, 2), S4 × Cm (m = 1, 2). Next, we consider the case where E has complex multiplication by the ring of integers of an imaginary quadratic field contained in K. Suppose that the -rank over a Galois extension F is 1 or 2. If ≠ and and h (class number of ) is odd, we show that E acquires positive -rank over a cyclic extension of K or over a field whose Galois group is one of SL2(/3), an extension of SL2(/3) by /2, or a central extension by the dihedral group. Finally, we discuss the relation of the above results to the vanishing of L-functions.

We now apply the cohomology theory of the previous chapter to the study of the Brauer group. However, we shall have to use a slightly modified construction which takes into account the fact that the absolute Galois group of a field is determined by its finite quotients. This is the cohomology theory of profinite groups, which we develop first. As a fruit of our labours, we identify the Brauer group of a field with a second, this time commutative, cohomology group of the absolute Galois group. As applications, we obtain easy proofs of some basic facts concerning the Brauer group, and give an important characterization of the index of a central simple algebra. Last but not least, one of the main objects of study in this book makes its appearance: the Galois symbol. The cohomology theory of profinite groups was introduced in the late 1950s by John Tate, motivated by sheaf-theoretic considerations of Alexander Grothendieck. His original aim was to find the appropriate formalism for developing class field theory. Tate himself never published his work, which thus became accessible to the larger mathematical community through the famous account of Serre [4], which also contains many original contributions. It was Brauer himself who described the Brauer group as a second cohomology group, using his language of factor systems. We owe to Serre the insight that descent theory can be used to give a more conceptual proof. The Galois symbol was defined by Tate in connection with the algebraic theory of power residue symbols, a topic extensively studied in the 1960s by Bass, Milnor, Moore, Serre and others. Profinite groups and Galois groups It can be no surprise that the main application of the cohomological techniques of the previous chapter will be in the case when G is the Galois group of a finite Galois extension. However, it will be convenient to consider the case of infinite Galois extensions as well, and first and foremost that of the extension k s |k , where k s is a separable closure of k .

We construct infinite families of elliptic curves with given torsion group structures over quartic number fields. In a 2006 paper, the first two authors and Park determined all of the group structures which occur infinitely often as the torsion of elliptic curves over quartic number fields. Our result presents explicit examples of their theoretical result. This paper also presents an efficient way of finding such families of elliptic curves with prescribed torsion group structures over quadratic or quartic number fields.

Conventional application of convolutional neural networks (CNNs) for image classification and recognition is based on the assumption that all target classes are equal (i.e. no hierarchy) and exclusive of one another (i.e. no overlap). CNN-based image classifiers built on this assumption, therefore, cannot take into account an innate hierarchy among target classes (e.g. cats and dogs in animal image classification) or additional information that can be easily derived from the data (e.g. numbers larger than five in the recognition of handwritten digits), thereby resulting in scalability issues when the number of target classes is large. Combining two related but slightly different ideas of hierarchical classification and logical learning by auxiliary inputs, we propose a new learning framework called hierarchical auxiliary learning, which not only address the scalability issues with a large number of classes but also could further reduce the classification/recognition errors with a reasonable number of classes. In the hierarchical auxiliary learning, target classes are semantically or non-semantically grouped into superclasses, which turns the original problem of mapping between an image and its target class into a new problem of mapping between a pair of an image and its superclass and the target class. To take the advantage of a superclass as a hint during the learning phase, we introduce an auxiliary block into a neural network, which generates auxiliary scores used as additional information for final classification/recognition; in this paper, we add the auxiliary block between the last residual block and the fully-connected output layer of the ResNet. Experimental results show that the proposed hierarchical auxiliary learning reduces classification errors up to 0.56, 1.6 and 3.56 percent with MNIST, SVHN and CIFAR-10 datasets, respectively.

Let $G$ be a finite group and $\pi$ be a set of primes. We study finite groups with a large number of conjugacy classes of $\pi$-elements. In particular, we obtain precise lower bounds for this number in terms of the $\pi$-part of the order of $G$ to ensure the existence of a nilpotent or abelian Hall $\pi$-subgroup in $G$.

This paper proposes a growable switch architecture with multiple modules comprising a CORE module to support highspeed switching and EDGE modules with large buffers, providing a large number of traffic classes. For open-loop traffic such as CBR and VBR, the proposed switch uses a weighted fair queueing scheduler at each EDGE buffer module based on the equivalent bandwidth allocated by the call admission control (CAC). On the other hand, for best effort class services (e.g., ABR and UBR), closed-loop control using adaptive rate control at the actual source and at the virtual source queue of the input EDGE module is proposed to avoid internal congestion and to achieve fair throughput performance among the flows competing on the same link. Simulation results show that the proposed architecture achieves ATM switches having hundreds of Gbps capacity, while providing a large number of QoS classes with high throughput and fair sharing of bandwidth by all flows.

Visual thalamic neurons were stained in a Golgilike fashion following cortical injections of horseradish peroxidase (HRP). Such staining allows neurons to be classified on the basis of their dendritic morphology, and it enables the direct relation of morphologically identified neurons with their respective cortical projection target to be visualized. All classes of neurons categorized on the basis of Golgi material (Guillery, '66) were identified in the dorsal part of the lateral geniculate body (LGBd): class 1 and class 2 neurons in the laminae A, A1, and C; class 3 neurons in laminae A and A1; class 4 cells in the C‐laminae. Three subtypes of class 4 neurons were differentiated: round cells, spindle cells, and fusiform cells. The round subtype is similar in dendritic morphology to the class 3 neurons, but its soma is larger (mean soma diameter 17 μm compared with 13 μm). Other types of neurons were identified in the laminae A, A1, and C, having morphological characteristics intermediate between class 1 and class 2, and between class 2 and class 3.Class 1 neurons, class 2 neurons, and flat neurons were identified in the medial interlaminar nucleus (MIN). The flat type is similar in morphology to the fusiform type in the C‐laminae; however, its soma is larger (mean soma diameter 23 μm compared with 19 μm), and its dendrites are free of appendages.In the extrageniculate nuclei three types of neurons were found: class 2 neurons in the pulvinar, medium‐sized cells—probably class 5—in the pulvinar and in the nucleus lateralis posterior, and neurons with an isodendritic branching pattern in the intralaminar nuclei.The cortical projections of the various thalamic cell classes were analyzed by injection of HRP into acallosal parts, or callosal parts of area 17 or area 18. Area 17 receives input from a large number of class 2 neurons and from a moderate number of class 1 neurons in the LGBd. In contrast, area 18 receives a dominant input from class 1 neurons and only weak input from class 2 neurons (up to 10% of the population of projection cells in the individual cases).Class 3 neurons in the A‐laminae and class 4 neurons in the C‐laminae project to either cortical area. Class 2 neurons in MIN project only to callosal parts of area 18, whereas class 2 neurons in pulvinar project in addition to acallosal parts of area 18. The significant change in the relative number of class 1 and class 2 neurons in the LGBd occurs gradually across the callosal zone from area 17 to area 18; it is not strictly related to the 17/18 border as defined by cytoarchitectonic or physiological criteria. The class 1/class 2 relation in the projection to area 18 is independent of eccentricity in the A‐laminae. An increase in the frequency of occurrence with eccentricity is noted, however, for class 4 neurons in the C‐laminae and for flat neurons in MIN. The correlation between neuronal function and morphology is discussed. It is concluded that, in addition to strong input with y‐characteristics, area 18 receives significant inputs with non‐y properties. latter inputs arise mainly from the laminae of the LGBd and from extra‐geniculate nuclei, but also may be seen to originate from MIN and the A‐laminae of the LGBd.

Question classification is an important phase in question answering systems. In this paper, we propose to apply i) hierarchical classifiers, ii) hierarchical classifiers in combination with semi-supervised learning and iii) hierarchy expansion for question classification for improving the precision. When the number of classes is large, the performance of classification algorithms may be affected. In order to improve the performance by reducing the number of classes for each classifier, we propose to use hierarchical classifiers according to the question taxonomy, in which each internal node is attached a classifier. We try to use semi-supervised learning to consume unlabeled questions with expectation to improve the performance of classifiers in the hierarchy. We explored different applications of learning methods in for each classifier of the hierarchy: a) supervised learning for all classifiers at all levels; b) semi-supervised learning for the first-level classifier and supervised learning for other classifiers; c) semi-supervised learning for all classifiers. The experiments show that the first method (a) has better results than those of flat classification; the second method (b) produces better results than those of the first method while the effort to increase the performance of fine classifiers in the last method (c) is not so successful. As another effort, we propose to automatically group question classes by clustering in order to expand a node which has a large number of classes in the question taxonomy. The experiment also shows that the overall precision is improved.

Evaluating the effects of similar-class combination on class integration test order generation

Let $X$ be a Berkovich space over a valued field. We prove that every finite group is a Galois group over $\Ms(B)(T)$, where $\Ms(B)$ is the field of meromorphic functions over a part $B$ of $X$ satisfying some conditions. This gives a new geometric proof that every finite group is a Galois group over $K(T)$, where $K$ is a complete valued field with non-trivial valuation. Then we switch to Berkovich spaces over ${\bf Z}$ and use a similar strategy to give a new proof of the following theorem by D. Harbater: every finite group is a Galois group over a field of convergent arithmetic power series. We believe our proof to be more geometric and elementary that the original one. We have included the necessary background on Berkovich spaces over ${\bf Z}$.

This paper presents a constructive method for deriving an updated discriminant eigenspace for classification when bursts of data that contains new classes is being added to an initial discriminant eigenspace in the form of random chunks. Basically, we propose an incremental linear discriminant analysis (ILDA) in its two forms: a sequential ILDA and a Chunk ILDA. In experiments, we have tested ILDA using datasets with a small number of classes and small-dimensional features, as well as datasets with a large number of classes and large-dimensional features. We have compared the proposed ILDA against the traditional batch LDA in terms of discriminability, execution time and memory usage with the increasing volume of data addition. The results show that the proposed ILDA can effectively evolve a discriminant eigenspace over a fast and large data stream, and extract features with superior discriminability in classification, when compared with other methods.