Ternary quadratic forms over number fields with small class number

Classifications and representations are two main topics in the theory of quadratic forms. In this paper, we consider these topics of ternary quadratic forms. For a given squarefree integer N N , first we give the classification of positive definite ternary quadratic forms of level 4 N 4N explicitly. Second, we give explicit formulas of the weighted sum of representations over each class in every genus of ternary quadratic forms of level 4 N 4N , which are involved with modified Hurwitz class number. In the proof of the main results, we use the relations among ternary quadratic forms, quaternion algebras, and Jacobi forms. As a corollary, we get the formula for the class number of positive ternary quadratic forms of level 4 N 4N . As applications, we derive an explicit base of Eisenstein series space of modular forms of weight 3 / 2 3/2 and level 4 N 4N , and give new proofs of some interesting identities involving representation number of ternary quadratic forms.

For a positive definite integral ternary quadratic form [Formula: see text], let [Formula: see text] be the number of representations of an integer [Formula: see text] by [Formula: see text]. The famous Minkowski–Siegel formula implies that if the class number of [Formula: see text] is one, then [Formula: see text] can be written as a constant multiple of a product of local densities which are easily computable. In this paper, we consider the case when the spinor genus of [Formula: see text] contains only one class. In this case the above also holds if [Formula: see text] is not contained in a set of finite number of square classes which are easily computable. By using this fact, we prove some extension of the recent results on both the representations of generalized Bell ternary forms and the representations of ternary quadratic forms with some congruence conditions.

In this note, we give an elementary proof of the following classical fact. Any positive definite ternary quadratic form over the rational numbers fails to represent infinitely many positive integers. For any ternary quadratic form (positive definite or indefinite), our method constructs certain congruence classes whose elements, up to a square factor, are the only elements not represented over the rational numbers by that form. In the case of a positive definite ternary form, we show that these classes are non-empty. This shows that the minimum number of variables in a positive definite quadratic form representing all positive integers is four. Our proof is very elementary and only uses quadratic reciprocity of Gauss.

We establish a series of relations between the numbers of solutions of some positive definite ternary quadratic forms and the class numbers of the corresponding imaginary quadratic fields. In particular, we obtain an explicit formula for the ternary quadratic form , where p and q are odd primes. Let h(d) be the class number of , and for a ternary quadratic form Q, we use for the number of integral solutions (x, y, z) of the equation . We give two examples to illustrate our results: (i) For , if m is a square-free positive integer with , then(ii) for , if m is a square-free positive integer with , then

Refining the notion of regularity introduced by Dickson, an integral quadratic form is said to be spinor regular if it represents all integers represented by its spinor genus. Examples of positive definite primitive integral ternary quadratic forms which have this property are presented, and it is proved that there exist only finitely many equivalence classes containing such forms.

Few-shot learning (FSL) is designed to equip models with the capability to quickly adapt to new, unseen domains in open-world scenarios. However, there is a notable discrepancy between the multitude of new concepts encountered in the open world and the limited scale of existing FSL studies, which focus predominantly on a small number of novel classes. This limitation hinders the practical implementation of FSL in real-world situations. To address this issue, we introduce a novel problem called Few-Shot Learning with Many Novel Classes (FSL-MNC), which expands the number of novel classes more than 500 times compared to traditional FSL settings. This new challenge presents two main difficulties: increased computational load during meta-training and reduced classification accuracy due to the larger number of classes during meta-testing. To tackle these problems, we introduce the Simple Hierarchy Pipeline (SHA-Pipeline). In response to the inefficiency of traditional Episode Meta-Learning (EML) protocols, we redesign a more efficient meta-training strategy to manage the increased number of novel classes. Moreover, to distinguish distinct semantic features across a broad array of novel classes, we effectively reconstruct and utilize class hierarchy information during meta-testing. Our experiments demonstrate that the SHA-Pipeline substantially outperforms both the ProtoNet baseline and current leading alternatives across various numbers of novel classes.

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.

We describe the design and use of linear hyperplanes to partition the feature space of a classifier. The objective of the partitioning is to minimize the average entropy of the class distribution in the final partitions. The hyperplanes are characterized by a vector /spl nu//sub n/ and scalar h/sub n/, which are computed with the objective of maximizing the mutual information associated with the partitioning. We show that the problem of designing the /spl nu//sub n/ can be simplified to an approximately equivalent problem, the solution of which, interestingly enough, turns out to be the linear discriminant of the data. We also describe a decision-tree based technique that partitions the feature space in a hierarchical fashion where each node in the decision tree represents a linear hyperplane that further partitions the feature space into two regions. The end result of the tree-growing process is that the entire feature space is partitioned into nonoverlapping regions where each region is bounded by a number of hyperplanes. As the criterion for the design of the hyperplanes is the minimization of the average class entropy of the regions, each region is characterized by the occurrence of only a small number of classes. The partitioning information provided by the decision-tree can then be used to eliminate a number of classes from being considered, and hence simplifies the job of the classifier. We show the application of the decision-tree as a preprocessor to a classifier in the speech recognition problem. Here the classes are modeled with mixtures of tens to hundreds of thousands of Gaussians, and the use of the partitioning information reduces the computation associated with the classification process by factors larger than 20 with negligible degradation in the word error rate (note, however, that the overall decoding process comprises other steps in addition to the classification-the computational speedup mentioned above does not affect the computation of these other steps). We also compare the performance of this quantization scheme with standard Gaussian clustering schemes and show that the decision-tree based quantization provides better performance in most cases.

In this article, we establish the general asymptotic behaviour of the Fourier coefficients of the Hecke eigenforms supported at integers represented by a primitive integral positive-definite binary quadratic form of fixed discriminant $D < 0$ under the assumption that the class number $h(D)=1$. As a consequence, we also obtain a quantitative result for the number of sign changes of the sequence of normalised Fourier coefficients $\lambda_{f}(n)$ of Hecke eigenforms $f$ over the indices, represented by a primitive integral positive definite binary quadratic form of fixed discriminant $D < 0$ when class number $h(D)=1$; in the interval $(X,2X]$ for sufficiently large $X$. Moreover, under the assumption of the Lindel{\"o}f hypothesis, the above-said sequence has at least $X^{\frac{1}{2}-\epsilon}$ many sign changes.

We prove that almost real quadratic fields of a given type have a large ideal class number. For example, the number of ideal classes of the fields , where is the field of rational numbers, grows unbounded with , as ranges through all natural numbers, except for a very sparse sequence. An analogous fact is established for the fields of Ankeny-Brauer-Chowla, Amer. J. Math. 78 (1965), 51-61.

We determine all the simplest cubic fields whose ideal class groups have exponent dividing 3, thus generalizing the determination by G. Lettl of all the simplest cubic fields with class number 1 and the determination by D. Byeon of all all the simplest cubic fields with class number 3. We prove that there are 23 simplest cubic fields with ideal class groups of exponent 3 (and 8 simplest cubic fields with ideal class groups of exponent 1, i.e. with class number one).

We study an asymptotic formula for counting integral points over an equation defined by a non-degenerate indefinite integral ternary quadratic form f representing a non-zero integer a such that $-a\cdot det(\,f)$ is square over a number field. In particular, we prove that the leading coefficient of this asymptotic formula is given by the product of local densities normalized by $1-p^{-1}$ over all finite primes p.

The following is an expanded version of the notes for a survey lecture on the theta correspondence given at Trichy in January, 1996. The aim of this lecture was to give an idea of this theory and some of the directions in which the theory has been developing. Perhaps some of the material contained in sections 2,3 and 9 is new. The theory of theta correspondence gives one of the few general methods of constructing automorphic forms of groups over number fields, or admissible representations of groups over local fields. The method has its origin in the classical construction of theta functions which are modular forms–perhaps of half integral weight on the upper half plane. Here is a special case of this: Let q be a positive definite quadratic form on a lattice L ⊆ IR, q : L→ Z . Let p be a homogeneous polynomial on IR which is harmonic with respect to q, i.e. ∆qp = 0 where ∆q denotes the Laplacian with respect to q (∆q is the unique homogeneous differential operator of order 2 invariant under the orthogonal group of q). Then,

Let {Km}m≥4 be the family of non-normal totally real cubic number fields as- sociated with the Q-irreducible cubic polynomials Pm(x) = x 3 − mx 2 − (m + 1)x − 1, m ≥ 4. We determine all these Km's with class numbers hm ≤ 3: there are 14 such Km's. Assuming the Generalized Riemann hypothesis for all the real quadratic number fields, we also prove that the exponents em of the ideal class groups of these Km go to infinity with m and we determine all these Km's with ideal class groups of exponents em ≤ 3: there are 6 such Km with ideal class groups of exponent 2, and 6 such Km with ideal class groups of exponent 3.

Minimal subschemes of the group association schemes of Mathieu groupsM n (n = 11, 12, 22, 23, 24) are determined. It is proved that for eachM n (n = 11, 12, 22, 23, 24), there is a unique minimal subscheme of . The class numbers of these minimal subschemes are 7, 11, 9, 11 and 20 respectively. A general computer program to determine subschemes of group association schemes of relatively small class numbers is discussd.