Number Of Fields Research Articles

Tamagawa Products for Elliptic Curves Over Number Fields

Tamagawa Products for Elliptic Curves Over Number Fields

Twisted Thue equations with multiple exponents in fixed number fields

Twisted Thue equations with multiple exponents in fixed number fields

On the quantum security of high-dimensional RSA protocol

Abstract The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

Integral points on a del Pezzo surface over imaginary quadratic fields

Chambert-Loir and Tschinkel constructed a framework for a geometric interpretation of the density of integral points on certain varieties, which was refined by Wilsch. By using harmonic analysis and the circle method, it was proved for some partial equivariant compactifications of vector groups over arbitrary number fields and high-dimensional complete intersections over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. Further, there are some examples of using the torsor method for singular del Pezzo surfaces over Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document}. In this paper, we generalise the torsor method for integral points from Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb {Q}}$$\\end{document} to imaginary quadratic number fields. As a first representative example, we characterise integral points of bounded log-anticanonical height on a quartic del Pezzo surface of singularity type A3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\ extbf{A}}_3$$\\end{document} over imaginary quadratic fields with respect to its singularity and its lines. Furthermore, we count these integral points of bounded height by using universal torsors and interpret the count geometrically to prove an analogue of Manin’s conjecture for the set of integral points with respect to the singularity and to a line. Our results coincide with the predicted framework.

Hilbert Poincaré series and kernels for products of L-functions

We study Hilbert Poincaré series associated to general seed functions and construct Cohen’s kernels and double Eisenstein series as series of Hilbert Poincaré series. Then we calculate the Rankin–Cohen brackets of Hilbert Poincaré series and Hilbert modular forms and extend Zagier’s kernel formula to totally real number fields. Finally, we show that the Rankin–Cohen brackets of two different types of Eisenstein series are special values of double Eisenstein series up to a constant.

On monogenity of certain number fields defined by x 2·3 s ·q r − m

Let K = Q ( α ) , where α is a root of the monic irreducible polynomial x 2 · 3 s · q r − m with m ≠ ± 1 is a square-free integer, r and s are two positive integers, and q is a prime of the form 3 k + 2 . In this article, we study the monogenity of the number field K and establish conditions on m for K to be monogenic. Further, we provide sufficient conditions on r, s and m for K to be not monogenic. Finally, we illustrate our results with examples.

Accumulation points of normalized approximations

Building on classical aspects of the theory of Diophantine approximation, we consider the collection of all accumulation points of normalized integer vector translates of points qα with α∈Rd and q∈Z. In the first part of the paper we derive measure theoretic and Hausdorff dimension results about the set of α whose accumulation points are all of Rd. In the second part we focus primarily on the case when the coordinates of α together with 1 form a basis for an algebraic number field K. Here we show that, under the correct normalization, the set of accumulation points displays an ordered geometric structure which reflects algebraic properties of the underlying number field. For example, when d=2, this collection of accumulation points can be described as a countable union of dilates (by norms of elements of an order in K) of a single ellipse, or of a pair of hyperbolas, depending on whether or not K has a non-trivial embedding into C.

Universal quadratic forms and Northcott property of infinite number fields

AbstractWe show that if a universal quadratic form exists over an infinite degree, totally real extension of the field of rationals , then the set of totally positive integers in the extension does not have the Northcott property. In particular, this implies that no universal form exists over the compositum of all totally real Galois fields of a fixed prime degree over . Further, by considering the existence of infinitely many square classes of totally positive units, we show that no classical universal form exists over the compositum of all such fields of degree (for each fixed odd integer ).

Numerical Assessment of MHD Tri-Hybrid Nanoparticles by Tangent Hyperbolic Model with Chemical Reaction: Thermodynamic Analysis

This paper investigates the influence of magneto-tangent hyperbolic nanofluid on the flow of a tri-hybrid nanoliquid consisting of [Formula: see text], and [Formula: see text] particles suspended in [Formula: see text]. The entropy production is encountered in this analysis. The fluid flows over a stretch sheet is considered. In addition, the energy equation also assumes the existence of a uniform heat source or sink and thermal radiation. Furthermore, the concentration equation emphasizes the chemical reaction. The current proposed model yields a set of nonlinear governing equations. The modeled formulation is transformed into a dimensionless system through the application of a suitable alteration. The complex nonlinear equation system was solved using the bvp4c through numerical methods. The main motive of this exploration is to emphasize the rate of heat and mass transfer in a flow of [Formula: see text], and [Formula: see text]-based hybrid nanofluid across a stretch sheet. The graphical study illustrates that Weissenberg number and magnetic field enhancement result in decreasing the velocity. But thermal layer, entropy production, and Bejan number are enhanced with larger values of Weissenberg number and magnetic field. This study focuses on different profiles with various flow parameters. Furthermore, we have compared the tri-hybrid nanofluid with the hybrid and mono nanofluid in all the figures and tabular format. Additionally, we have compared tri-hybrid, hybrid, and mono nanofluid using graphs for velocity, temperature, concentration, entropy production, and Bejan number.

Explicit Formula in an Imaginary Quadratic Number Field

Explicit Formula in an Imaginary Quadratic Number Field

Fermat's Lemma on the Hyperbolic Plane

Generalizing classical mathematical problems to other spaces is an important research direction in mathematics. This paper studies Fermat's Lemma on the hyperbolic plane, where the set of hyperbolic numbers is a commutative ring with zero divisors generated by two real numbers, similar to the complex numbers but more complicated. Compared to the classical Fermat’s Lemma, the Fermat’s Lemma in this paper has a higher dimension and is built on a hyperbolic ring whose algebraic structure is more complex than the real number field. On the basis of a generalization of the classical Fermat’s Lemma, the paper overcomes the difficulty that hyperbolic numbers contain zero divisors, and obtains the proof of the first and second Fermat’s Lemma on the hyperbolic plane through the decomposition of hyperbolic numbers, which will give impetus to theoretical research in hyperbolic analysis.

Entropy generation in radiative motion of tangent hyperbolic nanofluid in the presence of gyrotactic microorganisms and activation energy

In this work, entropy generation is optimized through the application of the second law of thermodynamics. The slip mechanisms, Brownian diffusions, and thermophoresis are elaborated using the tangent hyperbolic nanomaterial model. Magnetohydrodynamic (MHD) fluid is taken into consideration. To characterize the impact of activation energy, a unique model involving the binary chemical reaction is deployed. The effects of mixed convection that is nonlinear in nature, bioconvection, and Joule effect are all taken into consideration. The key partial differential equations (PDEs) are reduced into ordinary differential equations (ODEs) by utilizing appropriate similarity transformations and then solved numerically with the help of a built-in ‘bvp4c’ technique of MATLAB software. Varied flow parameters’ impacts on the nanoparticle volume concentration, entropy number, microorganism concentration, temperature, and velocity fields are analyzed using graphs. Various flow variables are taken into consideration to calculate the total rate of entropy generation. The obtained results show that concentration irreversibility, Joule effect irreversibility, viscous dissipation, and heat irreversibility all influence the entropy. The numerical outcomes were observed by fixing the physical parameters as 0.1<α<4.0, 0.1<M<1.2, 0.1<Nr<2.2, 0.1<Le<2.2, 0.1<Nb<0.4, 0.1<Nt<1.0, 2.0<⁡Pr⁡<5.0, and 0.1<Lb<2.0, as well as their impact on the momentum, thermal, concentration, and microorganism density profiles. From results, an increasing estimate of the variable representing chemical reaction indicates a decline in the concentration. The higher the chemical reaction variable, Hartmann number, and Weissenberg number, the higher the entropy number, while the Bejan number has a contrary behavior. Subsequently, all the outcomes are plotted in graphs and discussed in detail, when subjected to the involving physical quantities.

Cluster consensus of finite-field networks with stochastic communication topology

Most of the multi-agent systems (MASs) in real life are with limited storage and communication capabilities; studies on such systems, including consensus problem, however have been mainly based on real number field models. In this article, the cluster consensus problem of MASs with stochastic communication topology over finite-field known as the cluster consensus of finite-field networks (FFNs), is considered. Using the semi-tensor product of matrices, the dynamics of FFNs with stochastic communication topology are equivalently converted into a logical algebraic form, which facilitates further studies. Then a permutation system is introduced. The relationship between cluster consensus of FFNs with stochastic communication topology and the set stability with probability one (SSPO) of the permutation system is revealed. In such an approach, some verifiable criteria can be derived for the cluster consensus of FFNs with stochastic communication topology, without requesting the connectivity of the whole network. Finally, an example is presented to validate the main results.

An Infinite Family of Number Fields with Given Indices

An Infinite Family of Number Fields with Given Indices

ON THE LOWEST ZERO OF THE DEDEKIND ZETA FUNCTION

Abstract Let $\zeta _K(s)$ denote the Dedekind zeta-function associated to a number field K. We give an effective upper bound for the height of the first nontrivial zero other than $1/2$ of $\zeta _K(s)$ under the generalised Riemann hypothesis. This is a refinement of the earlier bound obtained by Sami [‘Majoration du premier zéro de la fonction zêta de Dedekind’, Acta Arith.99(1) (2000), 61–65].

Bioconvective magnetohydrodynamic flow of tangent hyperbolic nanofluid across a stretched surface, with Arrhenius activation and convective heat and slip conditions

The current study investigates mixed convective flow of an unsteady MHD tangent hyperbolic nanofluid due to a stretching surface with motile microorganisms via convective heat transfer and slip conditions. The flow analysis's governing equations were converted into a nondimensional relation by using the proper alteration. The pertinent similarity transformations are utilized in the main equations, and the solutions are obtained using the integrated bvp4c software. A good agreement with previously published data indicates the efficacy of the numerical technique. The following are the main findings: increasing the magnitude of the radiation and mixed convection parameters improves the velocity profile, whereas the Weissenberg number and magnetic field demonstrate the opposite influence; improvements in heat transfer occur due to the thermal radiation parameter, Brownian movement, and thermophoretic effects; activation energy improves concentration profile, whereas Lewis number shows opposite effect; bioconvection Lewis number and Peclet number declines the Motile microorganism density. Additionally, a tabular analysis of the skin friction, motile microorganism density, Nusselt number, and Sherwood number is also provided.

On Weil–Stark elements, I: Construction and general properties

AbstractWe construct a canonical family of elements in the reduced exterior powers of unit groups of global fields and investigate their detailed arithmetic properties. We then show that these elements specialise to recover the classical theory of cyclotomic elements in real abelian fields and also have connections to the theory of non‐commutative Euler systems for over general number fields.

On the monogenity and Galois group of certain classes of polynomials

Abstract We say a monic polynomial g(x) ∈ ℤ[x] of degree n is monogenic if g(x) is irreducible over ℚ and {1, θ, …, θ n−1} is a basis for the ring ℤ K of integers of number field K = ℚ(θ), where θ is a root of g(x). Let f ( x ) = x n + c ∑ i = 1 n ( a x ) n − i ∈ Z [ x ] and F ( x ) = x n + c ∑ i = 1 n a i − 1 x n − i ∈ Z [ x ] $\begin{array}{} \displaystyle f(x)=x^n+c\sum_{i=1}^{n}(ax)^{n-i} \in \mathbb{Z}[x] \,\text{and}\, F(x)=x^n+c\sum_{i=1}^{n}a^{i-1}x^{n-i} \in \mathbb{Z}[x] \end{array}$ be irreducible polynomials having degree n ≥ 3. In this paper, we provide necessary and sufficient conditions involving only a, c, n for the polynomials f(x) and F(x) to be monogenic. As an application, we also provide a class of polynomials having a non square-free discriminant and Galois group Sn , the symmetric group on n letters.

Experimental and numerical study on the impact of inlet temperature inhomogeneity on the aerodynamic performance of a three-stage turbine

Gas turbines play an important role in power generation and propulsion. The simplification of the real-engine condition during turbine design results in substandard gas turbine performance. Further improvement in understanding of the corresponding effects is essential. At present, the mechanism of the deterioration in turbine aerodynamic performance caused by the full-annulus scale turbine inlet temperature inhomogeneity in real engine are still unclear. In this paper, the corresponding effects are investigated through both gas turbine experiments and full-annulus URANS simulation. The experimental results show that the impact of cold streak on multistage turbine efficiency is about −0.3 %, which is in good quantitative agreement with the simulation results. The simulation results also show a complex reduction of blade work extraction occurs in the cold streak. A strong correlation between angle of attack and Mach number and the deterioration of turbine efficiency is found. It is also found that the “squeezing” effect induced by the cold streak azimuthal migration capability causes differences in Mach number, through-flow capability, and flow field radial distribution on both sides of the cold streak. This paper can help to the understanding of turbine performance characteristics and can also further contribute to improve gas turbine energy efficiency.

Linking invariants for valuations and orderings on fields

The mod-2 arithmetic Milnor invariants, introduced by Morishita, provide a decomposition law for primes in canonical Galois extensions of Q\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}$$\\end{document} with unitriangular Galois groups and contain the Legendre and Rédei symbols as special cases. Morishita further proposed a notion of mod-q arithmetic Milnor invariants, where q is a prime power, for number fields containing the qth roots of unity and satisfying certain class field theory assumptions. We extend this theory from the number field context to general fields, by introducing a notion of a linking invariant for discrete valuations and orderings. We further express it as a Magnus homomorphism coefficient and relate it to Massey product elements in Galois cohomology.