The Infinitude of Q(√ −p) With Class Number Divisible by 16

Abstract. The density of primes p such that the class number h of Q( √ −p) is divisible by 2k is conjectured to be 2−k for all positive integers k. The conjecture is true for 1 ≤ k ≤ 3 but still open for k ≥ 4. For primes p of the form p = a 2 + c 4 with c even, we describe the 8-Hilbert class field of Q( √ −p) in terms of a and c. We then adapt a theorem of Friedlander and Iwaniec to show that there are infinitely many primes p for which h is divisible by 16, and also infinitely many primes p for which h is divisible by 8 but not by 16

A density of ramified primes

Let $K$ be a cyclic totally real number field of odd degree over $\mathbb{Q}$ with odd class number, such that every totally positive unit is the square of a unit, and such that $2$ is inert in $K/\mathbb{Q}$. We define a family of number fields $\{K(p)\}_p$, depending on $K$ and indexed by the rational primes $p$ that split completely in $K/\mathbb{Q}$, such that $p$ is always ramified in $K(p)$ of degree $2$. Conditional on a standard conjecture on short character sums, the density of such rational primes $p$ that exhibit one of two possible ramified factorizations in $K(p)/\mathbb{Q}$ is strictly between $0$ and $1$ and is given explicitly as a formula in terms of $[K:\mathbb{Q}]$. Our results are unconditional in the cubic case. Our proof relies on a detailed study of the joint distribution of spins of prime ideals

ON THE 8-RANK OF NARROW CLASS GROUPS OF Q( √ −4pq), Q( √ −8pq), AND Q( √ 8pq)

Let d∈{−4,−8,8}. We study the 8-part of the narrow class group in thin families of quadratic number fields of the form Q(dpq−−−√), where p≡q≡1mod4 are prime numbers, and we prove new lower bounds for the proportion of narrow class groups in these families that have an element of order 8. In the course of our proof, we prove a general double-oscillation estimate for the quadratic residue symbol in quadratic number fields

On the 16-rank of class groups of quadratic number fields

We prove two new density results about 16-ranks of class groups of quadratic number fields. They can be stated informally as follows. Let C(D) denote the class groups of the quadratic number field of discriminant D. Theorem A. The class group C(-4p) has an element of order 16 for one-fourth of prime numbers p of the form a^2+16c^4. Theorem B. The class group C(-8p) has an element of order 16 for one-eighth of prime numbers p = -1 mod 4. These are the first non-trivial density results about the 16-rank of class groups in a family of quadratic number fields. They prove an instance of the Cohen-Lenstra conjectures. The proofs of these theorems involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. In case of Theorem B, we prove a power-saving error term for a prime-counting function related to the 16-rank of C(-8p), thereby giving strong evidence against a conjecture of Cohn and Lagarias that the 16-rank is governed by a Chebotarev-type criterion.ALGANTNumber theory, Algebra and Geometr

On the 16-rank of class groups of of Q( √−8p) for p ≡ −1 mod 4

We use a variant of Vinogradov’s method to show that the density of the set of prime numbers p ≡ −1 mod 4 for which the class group of the imaginary quadratic number field Q( √−8p) has an element of order 16 is equal to 1/16, as predicted by the Cohen–Lenstra heuristics

On the 16-rank of class groups of Q( √ −2p) for primes p ≡ 1 mod 4

We use Vinogradov’s method to prove equidistribution of a spin symbol governing the 16-rank of class groups of quadratic number fields Q( √ −2p), where p ≡ 1 mod 4 is a prime

Calculated and Observed Settlements of Multistory Building Founded on Loess

ln the Paper are presented the results of laboratory and field tests which were carried out on loess soil in Belgrade In order to determine the type of the foundation for the 13-storey building a preliminary investigation was made. In this phase of investigation exploratory borings and sampling were performed in a standard way. On the basis of the available laboratory and field test results it was concluded that the soil was made of macro porous land loess 14-20 m in thickness. lt was found that the loess on this location had the dry density varying between the limits γd = 15.5 - 15.8 kN/m3. Considering that the subsoil has high values of dry density, the designer adopted the shallow foundations. At the end of the period of three years one part of the building settled considerably and the differential settlements reached very high values. Due to the significant values of the angular distortion the building was seriously damaged. By additional investigation the undisturbed loess samples were cut from blocks and the laboratory results have shown much lower values of the deformation parameters than those obtained in the preliminary investigations. Using the deformation parameters and the coefficients of subsidence for the undisturbed samples cut from blocks, a very good agreement between the calculated and observed settlement was obtained

Modeling of laser-induced breakdown spectroscopic data analysis by an automatic classifier

Laser-induced breakdown spectroscopy (LIBS) is a multi-elemental and real-time analytical technique with simultaneous detection of all the elements in any type of sample matrix including solid, liquid, gas, and aerosol. LIBS produces vast amount of data which contains information on elemental composition of the material among others. Classification and discrimination of spectra produced during the LIBS process are crucial to analyze the elements for both qualitative and quantitative analysis. This work reports the design and modeling of optimal classifier for LIBS data classification and discrimination using the apparatus of statistical theory of detection. We analyzed the noise sources associated during the LIBS process and created a linear model of an echelle spectrograph system. We validated our model based on assumptions through statistical analysis of “dark signal” and laser-induced breakdown spectra from the database of National Institute of Science and Technology. The results obtained from our model suggested that the quadratic classifier provides optimal performance if the spectroscopy signal and noise can be considered Gaussian

A study of optical solitons with Kerr and power law nonlinearities by He's variational principle

This paper studies optical solitons, in presence of perturbation terms, by the aid of He's variational principle. The inter-modal dispersion, self-steepening, nonlinear dispersion and Raman scattering are all treated as perturbation terms. Both Kerr law as well as power law nonlinearities are considered in this paper