A positive proportion of some quadratic number fields with infinite Hilbert 2-class field tower

We show that a positive proportion of real and imaginary quadratic number fields with 2-class rank equal to 2 have 4-rank equal to 1 or 2 and infinite Hilbert 2-class field tower.     

Imaginary quadratic fields with Cl2(k)?(2,2,2)

We characterize those imaginary quadratic number fields, k, with 2-class group of type (2,2,2) and with the 2-rank of the class group of its Hilbert 2-class field equal to 2. We then compute the length of the 2-class field tower of k.     

On 2-ciass field towers of some imaginary quadratic number fields

We construct an infinite family of imaginary quadratic number fields with 2-class groups of type (2, 2, 2) whose Hilbert 2-class fields are finite.     

A remark on the capitulation problem

We determine explicitly an infinite family of imaginary cyclic number fields k, such that the 2-class group of k is elementary with arbitrary large 2-rank and capitulates in an unramified quadratic extension K. The infinitely many number fields k and K have the same Hilbert 2-class field and an infinite Hilbert 2-class field tower.     

On the Hilbert 2-class field tower of some abelian 2-extensions over the field of rational numbers

It is well known by results of Golod and Shafarevich that the Hilbert 2-class field tower of any real quadratic number field, in which the discriminant is not a sum of two squares and divisible by eight primes, is infinite. The aim of this article is to extend this result to any real abelian 2-extension over the field of rational numbers. So using genus theory, units of biquadratic number fields and norm residue symbol, we prove that for every real abelian 2-extension over ? in which eight primes ramify and one of theses primes ≡ ?1 (mod 4), the Hilbert 2-class field tower is infinite.     

Reduction of the number of associate classes of hypercubic association schemes

The reduction of the number of associate classes of some hypercubic association schemes by clubbing certain associate classes has been studied in the paper. It has been found that the reduction of anm-class hypercubic association scheme forv=2 ^m treatments into a 2-class association scheme is always possible. Further it is proved herein that them-class hypercubic association scheme forv=s ^m treatments is reducible (i) to a 3-class association scheme, whens=3 and (ii) to a 2-class association scheme, whens=4, which really hasp ₁₁ ¹ =p ₁₁ ² and hence leads to a series of balanced incomplete block designs.     

On the 2-class field tower of some imaginary biquadratic number fields

Let be an imaginary biquadratic number field with Cl_k,2, the 2-class group of k, isomorphic to Z/2Z × Z/2^mZ, m > 1, with q a prime congruent to 3 mod 4 and d a square-free positive integer relatively prime to q. For a number of fields k of the above type we determine if the 2-class field tower of k has length greater than or equal to 2. To establish these results we utilize capitulation of ideal classes in the three unramified quadratic extensions of k, ambiguous class number formulas, results concerning the fundamental units of real biquadratic number fields, and criteria for imaginary quadratic number fields to have 2-class field tower length 1. 2000 Mathematics Subject Classification Primary—11R29     

Densities for some real quadratic fields with infinite Hilbert 2-class field towers

Let K be a real quadratic field with 2-class rank equal to 4 or 5 and 4-class rank equal to 3. This paper computes density information for such fields to have infinite Hilbert 2-class field towers.     

On a Problem of M. Kambites Regarding Abundant Semigroups

A semigroup is regular if it contains at least one idempotent in each ?-class and in each ?-class. A regular semigroup is inverse if it satisfies either of the following equivalent conditions: (i) there is a unique idempotent in each ?-class and in each ?-class, or (ii) the idempotents commute. Analogously, a semigroup is abundant if it contains at least one idempotent in each ?*-class and in each ?*-class. An abundant semigroup is adequate if its idempotents commute. In adequate semigroups, there is a unique idempotent in each ?* and ?*-class. M. Kambites raised the question of the converse: in a finite abundant semigroup such that there is a unique idempotent in each ?* and ?*-class, must the idempotents commute? In this note, we provide a negative answer to this question.     

A density result for some imaginary quadratic fields with infinite Hilbert 2-class field towers

This paper shows that a positive proportion of the imaginary quadratic fields with 2-class rank equal to 3 have 4-class rank equal to zero and infinite Hilbert 2-class field towers. Received: 14 January 2003     

On the rank of the 2-class group of an extension of degree 8 over $$\mathbb {Q}$$ Q

Let K be an imaginary cyclic quartic number field whose 2-class group is nontrivial, it is known that there exists at least one unramified quadratic extension F of K. In this paper, we compute the rank of the 2-class group of the field F.

     

Four-class skew-symmetric association schemes

An association scheme is called skew-symmetric if it has no symmetric adjacency relations other than the diagonal one. In this paper, we investigate 4-class skew-symmetric association schemes. In recent work by the first author it was discovered that their character tables fall into three types. We now determine their intersection matrices. We then determine the character tables for 4-class skew-symmetric pseudocyclic association schemes, the only known examples of which are cyclotomic schemes. As a result, we answer a question raised by S.Y. Song in 1996. We characterize and classify 4-class imprimitive skew-symmetric association schemes. We also prove that none of 2-class Johnson schemes admits a 4-class skew-symmetric fission scheme. Based on three types of character tables above, a short list of feasible parameters is generated.     

On an estimator in T3-class of linear estimators in sampling with varying probabilities from a finite population

Summary Horvitz and Thompson [4] introduced three classes of linear estimators for estimation of population characteristics on the basis of a sample drawn with varying probabilities and without replacement. TheirT ₃-class of estimators does not admit a best unbiased estimator. In this paper, the variance and an unbiased estimate of variance for an estimator in T₃-class, which is proved to have several good properties by Godambe [2], [3], are derived for sampling with varying probabilities with or without replacement.     

On a 2-class polling model with reneging and $$k_i$$ -limited service

Annals of Operations Research - This paper analyzes a 2-class, single-server polling model operating under a $$k_i$$ -limited service discipline with class-dependent switchover times. Arrivals to...     

几个结合方案族的构造

本文给出了从两个类的结合方案构造三个类结合方案的两种方法,另外得到了一个结合方案族。     

The monoid of strong endomorphisms of a graph

We explicitly find all the idempotents in eachL(R)-class and all the inverses of each element of the strong endomorphism monoid of a graph. The number of these idempotents and inverses is also obtained. The author is deeply indebted to Professor Dr. T. E. Hall for his stimulating questions about this theme and much improvement made to an earlier version of this paper. The author would like to thank Professor Dr. U. Knauer and Dr. E. Wilkeit for helpful comments.     

An Eigenvalue Approach to Analyzing a Finite Source Priority Queueing Model

In this paper, we present a novel approach to determining the steady-state distribution for the number of jobs present in a 2-class, single server preemptive priority queueing model where the low priority source population is finite. Arrivals are assumed to be Poisson with exponential service times. The system investigated is a quasi birth and death process, and the joint distribution is derived via the method of generalized eigenvalues. Using this approach, we are able to obtain all eigenvalues and corresponding eigenvectors explicitly. Furthermore, we link this method to the matrix analytic approach by obtaining an explicit solution for the rate matrix R. Two numerical examples are given to illustrate the procedure and highlight some important computational features.     

On the parity of the class number of multiquadratic number fields

This paper investigates the 2-class group of real multiquadratic number fields. Let p₁,p₂,…,pn be distinct primes and . We draw a list of all fields K whose 2-class group is trivial.     

Association schemes arising from bent functions

We give a construction of 3-class and 4-class association schemes from s-nonlinear and differentially 2 ^s -uniform functions, and a construction of p-class association schemes from weakly regular p-ary bent functions, where p is an odd prime.     

One approach to solving a discrete production planning problem with interval data

In this paper, we develop an approach to solving integer programming problems with interval data based on using the possibilities of varying the relaxation set of the problem. This is illustrated by means of an L-class enumeration algorithm for solving a dicrete production planning problem. We describe the algorithm and a number of its modifications and present results of a computational experiment for families of problems from the OR Library and with randomly generated initial data. This approach is also applied to obtain approximate solutions of the mentioned problem in its conventional setting.