scholarly journals Reducing number field defining polynomials: an application to class group computations

2016 ◽  
Vol 19 (A) ◽  
pp. 315-331
Author(s):  
Alexandre Gélin ◽  
Antoine Joux
Keyword(s):  
Number Field ◽  
Class Group ◽  
Main Idea ◽  
Class Groups ◽  
Algebraic Integers ◽  
Small Height ◽  
Subexponential Algorithm ◽  
One To One ◽  
The One

In this paper we describe how to compute smallest monic polynomials that define a given number field $\mathbb{K}$. We make use of the one-to-one correspondence between monic defining polynomials of $\mathbb{K}$ and algebraic integers that generate $\mathbb{K}$. Thus, a smallest polynomial corresponds to a vector in the lattice of integers of $\mathbb{K}$ and this vector is short in some sense. The main idea is to consider weighted coordinates for the vectors of the lattice of integers of $\mathbb{K}$. This allows us to find the desired polynomial by enumerating short vectors in these weighted lattices. In the context of the subexponential algorithm of Biasse and Fieker for computing class groups, this algorithm can be used as a precomputation step that speeds up the rest of the computation. It also widens the applicability of their faster conditional method, which requires a defining polynomial of small height, to a much larger set of number field descriptions.

The Kernel of and the 4-Rank of K2(O)

1992 ◽  
Vol 35 (3) ◽  
pp. 295-302 ◽  
Author(s):  
Ruth I. Berger
Keyword(s):  
Number Field ◽  
Upper Bound ◽  
Class Group ◽  
Number Fields ◽  
Ideal Class ◽  
Ideal Class Group ◽  
Class Groups ◽  
Ring Of Integers

AbstractAn upper bound is given for the order of the kernel of the map on Sideal class groups that is induced by For some special types of number fields F the connection between the size of the above kernel for and the units and norms in are examined. Let K2(O) denote the Milnor K-group of the ring of integers of a number field. In some cases a formula by Conner, Hurrelbrink and Kolster is extended to show how closely the 4-rank of is related to the 4-rank of the S-ideal class group of

scholarly journals 2-Class groups of cyclotomic towers of imaginary biquadratic fields and applications

2021 ◽  
pp. 1-13
Author(s):  
Mohamed Mahmoud Chems-Eddin ◽  
Katharina Müller
Keyword(s):  
Number Field ◽  
Class Group ◽  
Class Groups ◽  
Biquadratic Fields

Let [Formula: see text] be an odd positive square-free integer. In this paper, we shall investigate the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of the imaginary biquadratic number field [Formula: see text] if [Formula: see text] is of specifiic form. Furthermore, we deduce the structure of the [Formula: see text]-class group of the cyclotomic [Formula: see text]-extension of [Formula: see text].

Restricted Determinantal Homomorphisms and Locally Free Class Groups

1990 ◽  
Vol 42 (4) ◽  
pp. 646-658 ◽  
Author(s):  
Victor Snaith
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Number Field ◽  
Class Group ◽  
Class Groups ◽  
A Finite Group

Let K be a number field and let OK denote the integers of K. The locally free class groups, Cl(OK[G]), furnish a fundamental collection of invariants of a finite group, G. In this paper I will construct some new, non-trivial homomorphisms, called restricted determinants, which map the NGH-invariant idèlic units of Ok([Hab] to Cl(OK[G]). These homomorphisms are constructed by means of the Horn-description of Cl(OK[G]), which describes the locally free class group in terms of the representation theory of G, and the technique of Explicit Brauer Induction, which was introduced in [5].

Influence of Rim Run-Out on the Nonuniformity of Tire-Wheel Assemblies

1994 ◽  
Vol 22 (2) ◽  
pp. 99-120 ◽  
Author(s):  
T. B. Rhyne ◽  
R. Gall ◽  
L. Y. Chang
Keyword(s):  
Finite Element ◽  
Radial Force ◽  
Membrane Model ◽  
Model Experiments ◽  
Force Variation ◽  
One To One ◽  
The One

Abstract An analytical membrane model is used to study how wheel imperfections are converted into radial force variation of the tire-wheel assembly. This model indicates that the radial run-out of the rim generates run-out of the tire-wheel assembly at slightly less than the one to one ratio that was expected. Lateral run-out of the rim is found to generate radial run-out of the tire-wheel assembly at a ratio that is dependent on the tire design and the wheel width. Finite element studies of a production tire validate and quantify the results of the membrane model. Experiments using a specially constructed precision wheel demonstrate the behavior predicted by the models. Finally, a population of production tires and wheels show that the lateral run-out of the rims contribute a significant portion to the assembly radial force variation. These findings might be used to improve match-mounting results by taking lateral rim run-out into account.

Near-Perfect Reconstruction Oversampled Nonuniform Cosine-Modulated Filter Banks Based on Frequency Warping and Subband Merging

2012 ◽  
Vol 58 (2) ◽  
pp. 177-192 ◽  
Author(s):  
Marek Parfieniuk ◽  
Alexander Petrovsky
Keyword(s):  
Group Delay ◽  
Filter Banks ◽  
Main Idea ◽  
Perceptual Processing ◽  
Perfect Reconstruction ◽  
Frequency Warping ◽  
Small Loss ◽  
Novel Method ◽  
The One ◽  
Near Perfect Reconstruction

Near-Perfect Reconstruction Oversampled Nonuniform Cosine-Modulated Filter Banks Based on Frequency Warping and Subband MergingA novel method for designing near-perfect reconstruction oversampled nonuniform cosine-modulated filter banks is proposed, which combines frequency warping and subband merging, and thus offers more flexibility than known techniques. On the one hand, desirable frequency partitionings can be better approximated. On the other hand, at the price of only a small loss in partitioning accuracy, both warping strength and number of channels before merging can be adjusted so as to minimize the computational complexity of a system. In particular, the coefficient of the function behind warping can be constrained to be a negative integer power of two, so that multiplications related to allpass filtering can be replaced with more efficient binary shifts. The main idea is accompanied by some contributions to the theory of warped filter banks. Namely, group delay equalization is thoroughly investigated, and it is shown how to avoid significant aliasing by channel oversampling. Our research revolves around filter banks for perceptual processing of sound, which are required to approximate the psychoacoustic scales well and need not guarantee perfect reconstruction.

Tree Rewriting Systems, Combinatory Weak Reduction Systems and Type Free Languages1

1982 ◽  
Vol 5 (3-4) ◽  
pp. 279-299
Author(s):  
Alberto Pettorossi
Keyword(s):  
Combinatory Logic ◽  
Rewriting Systems ◽  
Tree Transducers ◽  
Tree Languages ◽  
One To One ◽  
The One ◽  
Weak Reduction

In this paper we consider combinators as tree transducers: this approach is based on the one-to-one correspondence between terms of Combinatory Logic and trees, and on the fact that combinators may be considered as transformers of terms. Since combinators are terms themselves, we will deal with trees as objects to be transformed and tree transformers as well. Methods for defining and studying tree rewriting systems inside Combinatory Weak Reduction Systems and Weak Combinatory Logic are also analyzed and particular attention is devoted to the problem of finiteness and infinity of the generated tree languages (here defined). This implies the study of the termination of the rewriting process (i.e. reduction) for combinators.

In search of the perfect orthography

2005 ◽  
Vol 7 (2) ◽  
pp. 139-163 ◽  
Author(s):  
Richard L. Venezky
Keyword(s):  
Writing System ◽  
Writing Systems ◽  
International Phonetic Alphabet ◽  
Experienced Reader ◽  
Cultural Considerations ◽  
Advanced Stages ◽  
One To One ◽  
The One ◽  
Psychological Processing ◽  
The Ideal

Philologists, linguists, and educators have insisted for several centuries that the ideal orthography has a one-to-one correspondence between grapheme and phoneme. Others, however, have suggested deviations for such functions as distinguishing homophones, displaying popular alternative spellings, and retaining morpheme identity. If, indeed, the one-to-one ideal were accepted, the International Phonetic Alphabet should become the orthographic standard for all enlightened nations, yet the failure of even a single country to adopt it for practical writing suggests that other factors besides phonology are considered important for a writing system. Whatever the ideal orthography might be, the practical writing systems adopted upon this earth reflect linguistic, psychological, and cultural considerations. Knowingly or unknowingly, countries have adopted orthographies that favour either the early stages of learning to read or the advanced stages, that is, the experienced reader. The more a system tends towards a one-to-one relationship between graphemes and phonemes, the more it assists the new reader and the non-speaker of the language while the more it marks etymology and morphology, the more it favours the experienced reader. The study of psychological processing in reading demonstrates that human capacities for processing print are so powerful that complex patterns and irregularities pose only a small challenge. Orthographic regularity is extracted from lexical input and used to recognise words during reading. To understand how such a system develops, researchers should draw on the general mechanisms of perceptual learning.

A FAST CHINESE CHARACTERS ACCESSING TECHNIQUE USING MANDARIN PHONETIC TRANSCRIPTIONS

1988 ◽  
Vol 02 (01) ◽  
pp. 105-137 ◽  
Author(s):  
C.C. CHANG ◽  
H.C. WU
Keyword(s):  
Main Idea ◽  
Chinese Characters ◽  
One To One ◽  
Hashing Function ◽  
Perfect Hashing ◽  
The Way

In this paper, we consider the problem of how to design a minimal perfect hashing function which is suitable for the Mandarin Phonetic Symbols system. Our main idea is inspired by Chang’s letter-oriented minimal perfect hashing scheme. By using our hashing function, 1303 Mandarin phonetic symbol transcriptions will be hashed to 1303 locations in the way of one-to-one correspondence.

Development of a Single-Code/Default Coding Strategy in Pigeons

2000 ◽  
Vol 11 (3) ◽  
pp. 261-264 ◽  
Author(s):  
Tricia S. Clement ◽  
Thomas R. Zentall
Keyword(s):  
Conditional Discrimination ◽  
The Other ◽  
Matching Accuracy ◽  
Efficient Coding ◽  
Coding Strategy ◽  
One To One ◽  
Retention Intervals ◽  
The Many ◽  
The One ◽  
Better Than

We tested the hypothesis that pigeons could use a cognitively efficient coding strategy by training them on a conditional discrimination (delayed symbolic matching) in which one alternative was correct following the presentation of one sample (one-to-one), whereas the other alternative was correct following the presentation of any one of four other samples (many-to-one). When retention intervals of different durations were inserted between the offset of the sample and the onset of the choice stimuli, divergent retention functions were found. With increasing retention interval, matching accuracy on trials involving any of the many-to-one samples was increasingly better than matching accuracy on trials involving the one-to-one sample. Furthermore, following this test, pigeons treated a novel sample as if it had been one of the many-to-one samples. The data suggest that rather than learning each of the five sample-comparison associations independently, the pigeons developed a cognitively efficient single-code/default coding strategy.

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