Imaginary Number Research Articles

Determination of all imaginary quadratic fields for which their Hilbert 2-class fields have 2-class groups of rank 2

We determine those imaginary quadratic number fields whose 2-class groups have rank 3 and 4-rank ≤2 and such that the 2-class groups of their Hilbert 2-class fields have rank 2. This result, along with previous work, gives a complete determination of all complex quadratic number fields for which the 2-class groups of their 2-class fields have rank at most 2.

Corrections to the hadron resonance gas from lattice QCD and their effect on fluctuation-ratios at finite density

The hadron resonance gas (HRG) model is often believed to correctly describe the confined phase of QCD. This assumption is the basis of many phenomenological works on QCD thermodynamics and of the analysis of hadron yields in relativistic heavy ion collisions. We use first-principle lattice simulations to calculate corrections to the ideal HRG. Namely, we determine the sub-leading fugacity expansion coefficients of the grand canonical free energy, receiving contributions from processes like kaon-kaon or baryon-baryon scattering. We achieve this goal by performing a two dimensional scan on the imaginary baryon number chemical potential ($\mu_B$) - strangeness chemical potential ($\mu_S$) plane, where the fugacity expansion coefficients become Fourier coefficients. We carry out a continuum limit estimation of these coefficients by performing lattice simulations with temporal extents of $N_\tau=8,10,12$ using the 4stout-improved staggered action. We then use the truncated fugacity expansion to extrapolate ratios of baryon number and strangeness fluctuations and correlations to finite chemical potentials. Evaluating the fugacity expansion along the crossover line, we reproduce the trend seen in the experimental data on net-proton fluctuations by the STAR collaboration.

An approach for computing generators of class fields of imaginary quadratic number fields using the Schwarzian derivative

Let N N be one of the 38 38 distinct square-free integers such that the arithmetic group Γ 0 ( N ) + \Gamma _0(N)^+ has genus one. We constructed canonical generators x N x_N and y N y_N for the associated function field (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 25 (2016), pp. 295–319]). In this article we study the Schwarzian derivative of x N x_N , which we express as a polynomial in y N y_N with coefficients that are rational functions in x N x_N . As a corollary, we prove that for any point e e in the upper half-plane which is fixed by an element of Γ 0 ( N ) + \Gamma _0(N)^+ , one can explicitly evaluate x N ( e ) x_N(e) and y N ( e ) y_N(e) . As it turns out, each value x N ( e ) x_N(e) and y N ( e ) y_N(e) is an algebraic integer which we are able to understand in the context of explicit class field theory. When combined with our previous article (see Jorgenson, L. Smajlović, and H. Then [Exp. Math. 29 (2020), pp. 1–27]), we now have a complete investigation of x N ( τ ) x_N(\tau ) and y N ( τ ) y_N(\tau ) at any CM point τ \tau , including elliptic points, for any genus one group Γ 0 ( N ) + \Gamma _0(N)^+ . Furthermore, the present article when combined with the two aforementioned papers leads to a procedure which we expect to yield generators of class fields, and certain subfields, using the Schwarzian derivative and which does not use either modular polynomials or Shimura reciprocity.

Kinetic Analysis on Hydrogen Entry into Steel By Complex Hydrogen Penetration Ratio

Degradation of steels due to the hydrogen embrittlement, which is related to the hydrogen penetration reaction (HPR) into steel, is a serious problem from the standpoint of infrastructure preservation. A Devanatnan-Stachurski permeation technique [1, 2] is innovative method to determine a hydrogen penetration rate in metals. In this technique, two independent electrochemical cells are arranged on two surfaces of metal sheet and anodic and cathodic polarizations can be performed with each cell using the three-electrode system. The hydrogen penetration rate can be evaluated by measuring the oxidation current of hydrogen atoms diffused in the metal sheet at the anode side. The electrochemical impedance spectroscopy (EIS) was combined with the Devanatnan cell to investigate HPR mechanisms because the time constants observed in impedance spectrum can be discriminated [3-5]. In this study, we propose a complex hydrogen penetration ratio (CHPR) as a new transfer function related to hydrogen penetration and diffusion in metal sheet with the Devanatnan-Stachurski permeation technique. Figure 1 shows a scheme of hydrogen entry and detection at two electrode surfaces of metal sheet. The CHPR, ι, is defined from ratio of ΔI c and ΔI en, where the ΔI c is the cathode current on cathode (entry) side, and the ΔI per is the current response related to the hydrogen penetrated through metal sheet on anode (detection) side. The theoretical equation (1) of CHPR, ι, is shown at lower part in Fig. 1. In the equation (1), σab is the frequency-dependent constant including a kinetic parameter of hydrogen entry into metal sheet, h is the thickness of metal sheet, j is imaginary number unit, and D is the hydrogen diffusion coefficient.The CHPRs of carbon steel sheet were measured in this study. The experimental results were compared with the simulated ones according to the equation (1), and rate constant of hydrogen entry and diffusion coefficient were determined.

The Hecke Group H λ 4 Acting on Imaginary Quadratic Number Fields

Let H λ 4 be the Hecke group x , y : x 2 = y 4 = 1 and, for a square-free positive integer n , consider the subset ℚ ∗ − n = a + − n / c | a , b = a 2 + n / c ∈ ℤ , c ∈ 2 ℤ of the quadratic imaginary number field ℚ − n . Following a line of research in the relevant literature, we study the properties of the action of H λ 4 on ℚ ∗ − n . In particular, we calculate the number of orbits arising from this action for every such n . Some illustrative examples are also given.

Nonanalyticity, sign problem, and Polyakov line in a Z3 -symmetric heavy quark model at low temperature: Phenomenological model analyses

The nonanalyticity and the sign problem in the Z3-symmetric heavy quark model at low temperature are studied phenomenologically. For the free heavy quarks, the nonanalyticity is analyzed in the relation to the zeros of the grand canonical partition function. The Z3-symmetric effective Polyakov-line model (EPLM) in strong coupling limit is also considered as an phenomenological model of Z3-symmetric QCD with large quark mass at low temperature. We examine how the Z3-symmetric EPLM approaches to the original one in the zero-temperature limit. The effects of the Z3-symmetry affect the structure of zeros of the microscopic probability density function at the nonanalytic point. The average value of the Polyakov line can detect the structure, while the other thermodynamic quantities are not sensible to the structure in the zero-temperature limit. The effect of the imaginary quark chemical potential is also discussed. The imaginary part of the quark number density is very sensitive to the symmetry structure at the nonanalytical point. For a particular value of the imaginary quark number chemical potential, large quark number may be induced in the vicinity of the nonanalytical point.

Sensitivity analysis of system reliability using the complex-step derivative approximation

Sensitivity analysis of system failure probability is performed to evaluate the dependence of system reliability on design parameters or component events. Sensitivity analysis is integral for the implementation of efficient gradient-based optimization algorithms in system reliability-based design optimization, and for risk-informed decision-making. This paper presents a method for sensitivity analysis of system failure probability using complex-step differentiation. Many derivative approximations use a small step size to minimize subtractive cancellation errors. The complex-step approximation utilizes an imaginary number, such that the subtractive cancellation is not included in the formulation, resulting in calculations without the associated round-off error. The level of accuracy in sensitivity analysis using the finite difference method (FDM) can vary with change in the step size, which is generally selected arbitrarily, as the actual effect of the step size on the result itself is difficult to predict prior to actual calculation. The complex-step approximation, however, is not confined by the step size, as subtraction cancellation is not included. Compared to the FDM, the complex step approximation has only one limit, the numerical precision of evaluating the function. The proposed method integrates complex-step differentiation into a numerical integration scheme, for the assessment of system reliability and sensitivity. System failure probability and sensitivity are obtained by taking the real and imaginary part of the cumulative distribution function, which is numerically evaluated using the proposed method. The computational efficiency for system reliability problems involving high dimensionality is improved with the utilization of a dimension reduction technique. Numerical examples of sensitivity analysis for series, parallel, and general systems are presented to illustrate the performance of the proposed method.

Reliability-Based Design Optimization of Structures Using the Second-Order Reliability Method and Complex-Step Derivative Approximation

This paper proposes a reliability-based design optimization (RBDO) approach that adopts the second-order reliability method (SORM) and complex-step (CS) derivative approximation. The failure probabilities are estimated using the SORM, with Breitung’s formula and the technique established by Hohenbichler and Rackwitz, and their sensitivities are analytically derived. The CS derivative approximation is used to perform the sensitivity analysis based on derivations. Given that an imaginary number is used as a step size to compute the first derivative in the CS derivative method, the calculation stability and accuracy are enhanced with elimination of the subtractive cancellation error, which is commonly encountered when using the traditional finite difference method. The proposed approach unifies the CS approximation and SORM to enhance the estimation of the probability and its sensitivity. The sensitivity analysis facilitates the use of gradient-based optimization algorithms in the RBDO framework. The proposed RBDO/CS–SORM method is tested on structural optimization problems with a range of statistical variations. The results demonstrate that the performance can be enhanced while satisfying precisely probabilistic constraints, thereby increasing the efficiency and efficacy of the optimal design identification. The numerical optimization results obtained using different optimization approaches are compared to validate this enhancement.

Reliability-Based Design Optimization of Structures Using Complex-Step Approximation with Sensitivity Analysis

Structural optimization aims to achieve a structural design that provides the best performance while satisfying the given design constraints. When uncertainties in design and conditions are taken into account, reliability-based design optimization (RBDO) is adopted to identify solutions with acceptable failure probabilities. This paper outlines a method for sensitivity analysis, reliability assessment, and RBDO for structures. Complex-step (CS) approximation and the first-order reliability method (FORM) are unified in the sensitivity analysis of a probabilistic constraint, which streamlines the setup of optimization problems and enhances their implementation in RBDO. Complex-step approximation utilizes an imaginary number as a step size to compute the first derivative without subtractive cancellations in the formula, which have been observed to significantly affect the accuracy of calculations in finite difference methods. Thus, the proposed method can select a very small step size for the first derivative to minimize truncation errors, while achieving accuracy within the machine precision. This approach integrates complex-step approximation into the FORM to compute sensitivity and assess reliability. The proposed method of RBDO is tested on structural optimization problems across a range of statistical variations, demonstrating that performance benefits can be achieved while satisfying precise probabilistic constraints.

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

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].

Finiteness of lattice points on varieties F(y) = F(g(𝕏)) + r(𝕏) over imaginary quadratic fields

We construct affine varieties over \mathbb{Q} and imaginary quadratic number fields \mathbb{K} with a finite number of \alpha-lattice points for a fixed \alpha\in \mathcal{O}_\mathbb{K}, where \mathcal{O}_\mathbb{K} denotes the ring of algebraic integers of \mathbb{K}. These varieties arise from equations of the form F(y) = F(g(x_1,x_2,\ldots, x_k))+r(x_1,x_2\ldots, x_k), where F is a rational function, g and r are polynomials over \mathbb{K}, and the degree of r is relatively small. We also give an example of an affine variety of dimension two, with a finite number of algebraic integral points. This variety is defined over the cyclotomic field \mathbb{Q}(\xi_3)=\mathbb{Q}(\sqrt{-3}).

Diophantine approximation with prime restriction in real quadratic number fields

The distribution of $$\alpha p$$ modulo one, where p runs over the rational primes and $$\alpha $$ is a fixed irrational real, has received a lot of attention. It is natural to ask for which exponents $$\nu >0$$ one can establish the infinitude of primes p satisfying $$||\alpha p||\le p^{-\nu }$$ . The latest record in this regard is Kaisa Matomäki’s landmark result $$\nu =1/3-\varepsilon $$ which presents the limit of currently known technology. Recently, Glyn Harman, and, jointly, Marc Technau and the first-named author, investigated the same problem in the context of imaginary quadratic fields. Glyn Harman obtained an analog for $$\mathbb {Q}(i)$$ of his result in the context of $$\mathbb {Q}$$ , which yields an exponent of $$\nu =7/22$$ . Marc Technau and the first-named author produced an analogue of Bob Vaughan’s result $$\nu =1/4-\varepsilon $$ for all imaginary quadratic number fields of class number 1. In the present article, we establish an analog of the last-mentioned result for real quadratic fields of class number 1 under a certain Diophantine restriction. This setting involves the additional complication of an infinite group of units in the ring of integers. Moreover, although the basic sieve approach remains the same (we use an ideal version of Harman’s sieve), the problem takes a different flavor since it becomes truly 2-dimensional. We reduce it eventually to a counting problem which is, interestingly, related to roots of quadratic congruences. To approximate them, we use an approach by Christopher Hooley based on the theory of binary quadratic forms.

Imaginary multiquadratic number fields with class group of exponent $3$ and $5$

Imaginary multiquadratic number fields with class group of exponent $3$ and $5$

On the rank of the 2-class group of some imaginary triquadratic number fields

Let d be an odd square-free integer and \(\zeta _8\) a primitive 8-th root of unity. The purpose of this paper is to investigate the rank of the 2-class group of the fields \(L_d={\mathbb {Q}}(\zeta _8,\sqrt{d})\).

On the distribution ofαpmodulo one in imaginary quadratic number fields with class number one

We investigate the distribution of $\alpha p$ modulo one in imaginary quadratic number fields $\mathbb{K}\subset\mathbb{C}$ with class number one, where $p$ is restricted to prime elements in the ring of integers $\mathcal{O} = \mathbb{Z}[\omega]$ of $\mathbb{K}$. In analogy to classical work due to R. C. Vaughan, we obtain that the inequality $\lVert\alpha p\rVert_\omega < \mathrm{N}(p)^{-1/8+\epsilon}$ is satisfied for infinitely many $p$, where $\lVert\varrho\rVert_\omega$ measures the distance of $\varrho\in\mathbb{C}$ to $\mathscr{O}$ and $\mathrm{N}(p)$ denotes the norm of $p$. The proof is based on Harman's sieve method and employs number field analogues of classical ideas due to Vinogradov. Moreover, we introduce a smoothing which allows us to make conveniently use of the Poisson summation formula.

Unification Archimedes Constant π, Golden Ratio φ, Euler's Number e and Imaginary Number i

In the first chapter we will present the definitions,history,relations and arithmetics approximations of Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 220 expressions. In the second chapter we will present exact expressions between Archimedes constant π,golden ratio φ,Euler's number e and imaginary number i. A total of 127 expressions. In the third chapter we will present approximations expressions between Archimedes' constant π,the golden ratio φ,the number Euler e and the imaginary number i. A total of 112 expressions.

Quadratic fields with a class group of large 3-rank

We prove that there are $\gg X^{1/30}/\!\log X$ imaginary quadratic number fields with an ideal class group of $3$-rank at least $5$ and discriminant bounded in absolute value by $X$. This improves on an earlier result of Craig, who proved the infinitude

Hyperbolic tessellations and generators of for imaginary quadratic fields

AbstractWe develop methods for constructing explicit generators, modulo torsion, of the$K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic$3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for$ K_3 $of any field, predict the precise power of$2$that should occur in the Lichtenbaum conjecture at$ -1 $and prove that this prediction is valid for all abelian number fields.

On local and global bounds for Iwasawa λ-invariants

It is an open problem whether the Iwasawa λ-invariants of the Zp-extensions of a fixed number field are bounded. Using the class-field theoretic tool of logarithmic class groups, we obtain bounds for the λ-invariants of Zp-extensions of suitable field extensions of imaginary quadratic number fields. We also prove the Gross-Kuz'min Conjecture for certain families of non-cyclotomic Zp-extensions.

A Bombieri-type theorem for convolution with application on number field

Let $$K$$ be an imaginary quadratic number field and $$\mathcal{O}_K$$ be its ring of integers. We show that, if the arithmetic functions $$f, g\colon\mathcal{O}_K\rightarrow \mathbb{C}$$ both have level of distribution $$\vartheta$$ for some $$0<\vartheta\leq 1/2$$ then the Dirichlet convolution $$f*g$$ also has level of distribution $$\vartheta$$ . As an application we also obtain an analogue of the Titchmarsh divisor problem for product of two primes in imaginary quadratic fields.