Number Of Fields Research Articles

Explicit Formula in an Imaginary Quadratic Number Field

Explicit Formula in an Imaginary Quadratic Number Field

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.

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 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.

On affine spaces of rectangular matrices with constant rank

Let F be a field, and n≥p≥r>0 be integers. In a recent article, Rubei has determined, when F is the field of real numbers, the greatest possible dimension for an affine subspace of n–by–p matrices with entries in F in which all the elements have rank r. In this note, we generalize her result to an arbitrary field with more than r+1 elements, and we classify the spaces that reach the maximal dimension as a function of the classification of the affine subspaces of invertible matrices of Ms(F) with dimension (s2). The latter is known to be connected to the classification of nonisotropic quadratic forms over F up to congruence.

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits

Pointwise ergodic theorems for nonconventional bilinear polynomial averages along prime orbits Following our proof of the first Goldbach result in number field setting after 60 years (Mitsui 1960s), We give new information on density ternary Goldbach problem after 10 years (Shao 2013), and also design a new strategy to prove density version of representing a number as sum of “certain” primes. We also apply Rosser Sieve arguments to prove represtation of integers into sum of primes coming from a zero density subset. From Ergodic theory side, we give the first pointwise bilinear theorem along the primes, which is the natural pointwise convergence along prime result after Wierdl (1990) and a natural follow up after recent pointwise bilinear break-through by Krause Mirek-Tao (2021). In another project, the sharpest pointwise convergence average along primes for “near” integrable function is given, which is sharp up to Generalized Riemann Hypothesis.

On the torsion part in the K-theory of imaginary quadratic fields

AbstractWe obtain upper bounds for the torsion in the K-groups of the ring of integers of imaginary quadratic number fields, in terms of their discriminants.

NONMONOGENITY OF NUMBER FIELDS DEFINED BY TRUNCATED EXPONENTIAL POLYNOMIALS

Abstract Let p be a prime number. Let $n\geq 2$ be an integer given by $n = p^{m_1} + p^{m_2} + \cdots + p^{m_r}$ , where $0\leq m_1 < m_2 < \cdots < m_r$ are integers. Let $a_0, a_1, \ldots , a_{n-1}$ be integers not divisible by p. Let $K = \mathbb Q(\theta )$ be an algebraic number field with $\theta \in {\mathbb C}$ a root of an irreducible polynomial $f(x) = \sum _{i=0}^{n-1}a_i{x^i}/{i!} + {x^n}/{n!}$ over the field $\mathbb Q$ of rationals. We prove that p divides the common index divisor of K if and only if $r>p$ . In particular, if $r>p$ , then K is always nonmonogenic. As an application, we show that if $n \geq 3$ is an odd integer such that $n-1\neq 2^s$ for $s\in {\mathbb Z}$ and K is a number field generated by a root of a truncated exponential Taylor polynomial of degree n, then K is always nonmonogenic.

Discrete Logarithm Factory

The Number Field Sieve and its variants are the best algorithms to solve the discrete logarithm problem in finite fields (except for the weak small characteristic case). The Factory variant accelerates the computation when several prime fields are targeted. This article adapts the Factory variant to non-prime finite fields of medium and large characteristic. A precomputation, solely dependent on an approximate finite field size and an extension degree, allows to efficiently compute discrete logarithms in a constant proportion of the finite fields of the given approximate size and extension degree. We combine this idea with two other variants of NFS, namely the tower and special variant. This combination improves the asymptotic complexity. We also notice that combining our approach with the MNFS variant would be an unnecessary complication as all the potential gain of MNFS is subsumed by our Factory variant anyway. Furthermore, we demonstrate how Chebotarev's density theorem allows to compute the density of finite fields that can be solved with a given precomputation. Finally, we provide experimental data in order to assess the practical reach of our approach.

Improved computation of polynomial roots over number fields when using complex embeddings

We explore a fairly generic method to compute roots of polynomials over number fields through complex embeddings. Our main contribution is to show how to use a structure of a relative extension to decode in a subfield. Additionally we describe several heuristic options to improve practical efficiency. We provide experimental data from our implementation and compare our methods to the state of the art algorithm implemented in Pari/Gp.

An [formula omitted]-adic algorithm for bivariate Gröbner bases

Let A be a domain, with m⊆A a maximal ideal, and let F⊆A[x,y] be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field K of A). We present a randomized m-adic algorithm to recover the lexicographic Gröbner basis G of 〈F〉⊆K[x,y], or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set F of cardinality greater than two. We use this result to bound the size of the coefficients of G, and to control the probability of choosing a good maximal ideal m⊆A. We give a complete cost analysis over number fields (K=Q(α)) and function fields (▪), and we obtain a complexity that is less than cubic in terms of the dimension of K/〈G〉 and softly linear in the size of its coefficients.

𝐾𝑈-local zeta-functions of finite CW-complexes

Begin with the Hasse-Weil zeta-function of a smooth projective variety over Q \mathbb {Q} . Replace the variety with a finite CW-complex, replace étale cohomology with complex K K -theory K U ∗ KU^* , and replace the p p -Frobenius operator with the p p th Adams operation on K K -theory. This simple idea yields a kind of “ K U KU -local zeta-function” of a finite CW-complex. For a wide range of finite CW-complexes X X with torsion-free K K -theory, we show that this zeta-function admits analytic continuation to a meromorphic function on the complex plane, with a nice functional equation, and whose special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 . We then consider a more general and sophisticated version of the K U KU -local zeta-function, one which is suited to finite CW-complexes X X with nontrivial torsion in their K K -theory. This more sophisticated K U KU -local zeta-function involves a product of L L -functions of complex representations of the torsion subgroup of K U 0 ( X ) KU^0(X) , similar to how the Dedekind zeta-function of a number field factors as a product of Artin L L -functions of complex representations of the Galois group. For a wide range of such finite CW-complexes X X , we prove analytic continuation of the zeta-function, and we show that the special values in the left half-plane recover the K U KU -local stable homotopy groups of X X away from 2 2 if and only if the skeletal filtration on the torsion subgroup of K U 0 ( X ) KU^0(X) splits completely.

There is no 290‐Theorem for higher degree forms

AbstractWe study the universality of forms of degrees greater than 2 over rings of integers of totally real number fields. We show that such universal forms always exist, but cannot be characterized by any variant of the 290‐Theorem of Bhargava–Hanke.

Contribution to didactic research on the completeness/incompleteness of ordered fields of numbers

Contribution to didactic research on the completeness/incompleteness of ordered fields of numbers

On the Selmer group and rank of a family of elliptic curves and curves of genus one violating the Hasse principle

We study an infinite family of j-invariant zero elliptic curves ED:y2=x3+16D and their λ-isogenous curves ED′:y2=x3−27⋅16D, where D and D′=−3D are fundamental discriminants of a specific form, and λ is an isogeny of degree 3. A result of Honda guarantees that for our discriminants D, the quadratic number field KD=Q(D) always has non-trivial 3-class group. We prove a series of results related to the set of rational points ED′(Q)∖λ(ED(Q)), and the SL(2,Z)-equivalence classes of irreducible integral binary cubic forms of discriminant D. By assuming finiteness of the Tate-Shafarevich group, we derive a parity result between the rank of ED and the rank of its 3-Selmer group, and we establish lower and upper bounds for the rank of our elliptic curves. Finally, we give explicit classes of genus-1 curves that correspond to irreducible integral binary cubic forms of discriminant D=48035713, and we show that every curve in these classes violates the Hasse Principle.

The Failure of Galois Descent for p-Selmer Groups of Elliptic Curves

Abstract We show that if F is $\mathbb{Q}$ or a multiquadratic number field, $p\in\left\{{2,3,5}\right\}$ , and $K/F$ is a Galois extension of degree a power of p, then for elliptic curves $E/\mathbb{Q}$ ordered by height, the average dimension of the p-Selmer groups of $E/K$ is bounded. In particular, this provides a bound for the average K-rank of elliptic curves $E/\mathbb{Q}$ for such K. Additionally, we give bounds for certain representation–theoretic invariants of Mordell–Weil groups over Galois extensions of such F. The central result is that: for each finite Galois extension $K/F$ of number fields and prime number p, as $E/\mathbb{Q}$ varies, the difference in dimension between the Galois fixed space in the p-Selmer group of $E/K$ and the p-Selmer group of $E/F$ has bounded average.