Number Of Fields Research Articles

Arthur’s multiplicity formula for even orthogonal and unitary groups

Let F be a number field and G an even orthogonal or unitary group over a number field. Based on the method used by Gan and Ichino (2018), we prove Arthur’s multiplicity formula for the generic part of the automorphic discrete spectrum of G by using the theta lift. Enhancing this method, we also obtain a description of the full automorphic discrete spectrum of G of F -rank \le1 .

On indices of quintic number fields defined by x 5 + ax + b

Abstract The goal of this paper is to calculate explicitly the field index of any quintic number field K generated by a complex root α of a monic irreducible trinomial F(x) = x 5 + ax + b ∈ ℤ[x]. In such a way we provide a complete answer to the Problem 22 of Narkiewicz [36] for this class of number fields. Namely, for every prime integer p, we evaluate the highest power of p dividing i(K). In particular, we give sufficient conditions on a and b, which guarantee the non-monogenity of K.

Generalizing the Classical Remainder Theorem: A Reflection-Based Methodological Strategy

The framework of this paper is the presentation of a case study in which university students are required to extend a particular problem of division of polynomials in one variable over the field of real numbers (as generalizing action) clearly influenced by prior strategies (as reflection generalization). Specifically, the objective of this paper is to present a methodology for generalizing the classical Remainder Theorem to the case in which the divisor is a product of binomials (x−a1)n1(x−a2)n2⋯(x−ak)nk, where a1,a2,⋯,ak∈R and n1,n2,⋯,nk∈N. A first approach to this issue is the Taylor expansion of the dividend P(x) at a point a, which clearly shows the quotient and the remainder of the division of P(x) by (x−a)k, where the degree of P(x), say n, must be greater than or equal to k. The methodology used in this paper is the proof by induction which allows to obtain recurrence relations different from those obtained by other scholars dealing with the generalization of the classical Remainder Theorem.

LOWER BOUNDS FOR THE MAHLER MEASURE AND INERTIA DEGREES OF PRIMES

Abstract We investigate the relationship between lower bounds for the Mahler measure and splitting of primes, and prove various lower bounds for the Mahler measure of algebraic integers in terms of the least common multiples of all inertia degrees of primes. The results generalise work of the second author and Kumar [‘Lehmer’s problem and splitting of rational primes in number fields’, Acta Math. Hungar.169(2) (2023), 349–358].

Open image computations for elliptic curves over number fields

Open image computations for elliptic curves over number fields

Featuring the aspects with temperature dependent viscosity of inclined MHD Williamson fluid along with heat source/sink, Soret and Dufour effects: A predictor-corrector approach

The boundary layer flow analysis of Williamson fluid moving on the stretched cylinder has applications in the polymer processing, drug delivery system, cooling systems, rheology of food products, dyeing, and finishing. The numerical solutions of governing Navier-Stokes equations at different scenarios of thermal and solutal aspects have lots of practical implications, including biomedical engineering, cooling and heating processes, chemical reactors, aerodynamics, environmental science, and thermal storage systems. From these practical applications, the authors were motivated to conduct a study in which the numerical behavior is examined by applying the Adams-Bashforth predictor and corrector approach of numerical analysis for the Williamson fluid model in the presence of varying viscosity, natural convection, and the inclined magnetic field. Furthermore, Joule heating, thermal radiation, and heat source/sink effects are included in the thermal aspects because these are the important concepts in thermal management and engineering and have applications in solar thermal energy systems, combustion engines, heat exchangers, nuclear reactors, thermal imaging, and safety. Apart from thermal aspects, the analysis of mass transfer focuses on the influence of chemical reactions and Soret-Dufour effects. The similarity transformations are adopted to convert partial differential equations into ordinary differential equations. Specifically, the well-known predictor and corrector numerical method named the Adams-Bashforth method in combination with the Runge-Kutta 4th order scheme is used to achieve the numerical solutions. The temperature, concentration, and velocity regions are displayed on the graphical and numerical conducts. A comparison is also made between the results obtained by the shooting method and the numerical findings of skin friction, Nusselt number, and Sherwood number determined by the Adams-Bashforth method. The determining results concluded that the restrictions in the flow of Williamson fluid are caused by variations in the inclination magnetic field, the surface curvature, and the Williamson parameter, whereas the motion of the fluid increases due to convection, which occurs due to thermal phenomena. The determined results indicate that the temperature distribution is enhanced due to the addition of thermal radiation, Eckert number, and magnetic field. The surface curvature, Soret number, and reaction coefficient marks the decrease impact on the concentration region. The temperature-dependent viscosity increases the drag force and friction coefficient. The Nusselt number is amplified by applying thermal radiation to the surface of the cylinder. The chemical reaction is the source of the increment in the Sherwood number.

Vortex structure and control in a lid-driven cavity with magnetic field

Ferrofluids are magnetic field-sensitive colloidal solutions of ultra-fine, single-domain superpara-magnetic nanoparticles of metallic materials. An external magnetic field can be used to regulate the magnetic nanoparticles in the ferrofluid, which permits the fluid to flow and distribute within the domain in a predetermined way. This study examines the ferrofluid flow properties in a lid-driven cavity exposed to an external magnetic field. We investigate the impact of different aspect ratios, Reynolds number, and magnetic field strength on generation of vortices and their behavior. We have simulated aspect ratios of 1, 0.5, and 2 in this investigation, using Reynolds number values of 100, 400, and 1000 and magnetic field strength of 0, 1, and 4 tesla. Our results show that the introduction of magnetic fields greatly modifies the behavior of vortex, affecting the flow stability, and vorticity patterns inside the cavity. These findings have ramifications for actuation mechanisms, drug delivery systems, and microfluidic devices. In particular, we find that the flow patterns in the cavity evolve from basic recirculation to more intricate vortex structures. The commencement of these transitions is largely dependent on the Reynolds numbers. Along the sidewalls, secondary vortices form at higher Reynolds numbers. Practical applications can benefit from the enhanced mixing provided by these sidewall vortices. Additionally, we talk about how ferrofluids might improve fluid control and manipulation in constrained geometries. The interaction of magnetism and fluid dynamics creates opportunities for new types of actuation mechanisms in which the ferrofluid flow can be steered and controlled by magnetic fields. Our research opens up new avenues for creative applications in microfluidics and other fields by providing insightful information on ferrofluid behavior.

Data-Driven Turbulent Prandtl Number Modeling for Hypersonic Shock–Boundary-Layer Interactions

We develop a neural-network-based variable turbulent Prandtl number model for the k–ϵ turbulence model for improved wall heating predictions in hypersonic shock–boundary-layer interactions (SBLIs). The model is developed by performing a finite-dimensional field inference for a spatially varying turbulent Prandtl number on six canonical SBLIs: three compression ramps at Mach 8 and three impinging shocks at Mach 5. The inference results identify a turbulent Prandtl number that reduces wall heating by systematically directing heat transfer away from the wall. An ensemble of Lipschitz-continuous neural networks is then trained on the inferred turbulent Prandtl number fields to develop a predictive model. We evaluate the resulting variable turbulent Prandtl number model on a suite of test cases, including the hollow cylinder flare and HIFiRE ground test experiments. The machine-learning-augmented model systematically increases Prt near the wall to reduce negative turbulent heat flux while decreasing Prt away from the wall to enhance positive turbulent heat flux, collectively reducing overall heat transfer to the surface. Results show that the learned model consistently improves peak heating predictions by 40–70% compared to the baseline k–ϵ model, a k–ϵ model augmented with various high-speed corrections, and the shear stress transport model across a range of conditions.

On Index Divisors and Monogenity of Certain Octic Number Fields Defined by x8 + ax3 + b

On Index Divisors and Monogenity of Certain Octic Number Fields Defined by x8 + ax3 + b

Advancements and Prospects in Large Integer Factorization: A Comprehensive Review of the Number Field Sieve Method

The Number Field Sieve (NFS) stands as one of the most effective algorithms for the factorization of large integers, playing a critical role in cryptographic security, particularly in breaking RSA encryption systems. This review paper provides an in-depth analysis of NFS, highlighting its structure, key processes, and computational challenges. NFS is notably effective for integers larger than 110 digits and is recognized for its sub-exponential computational complexity, making it superior to other classical factorization algorithms. The core processes of NFSpolynomial selection, number pair sieving, matrix solving, and square root extractionare systematically examined to illustrate their roles in factorization. This study also reviews the latest advancements in NFS, including improvements in polynomial selection methods and optimizations in sieving and matrix-solving stages, which have significantly enhanced the algorithms efficiency. Moreover, the paper discusses the future prospects of NFS, emphasizing the need for further optimization to reduce computational resource demands and increase practicality in large-scale applications. The ongoing evolution of NFS continues to push the boundaries of cryptographic analysis, driving future research towards even more efficient factorization methods.

Birational Automorphism Groups of Severi–Brauer Surfaces Over the Field of Rational Numbers

Birational Automorphism Groups of Severi–Brauer Surfaces Over the Field of Rational Numbers

Sparsity of stable primes for dynamical sequences

AbstractWe show that a dynamical sequence of polynomials over a number field whose set of stable primes is of positive density must necessarily have a very restricted, and, in particular, virtually prosolvable dynamical Galois group. Together with existing heuristics, our results suggest, moreover, that a polynomial all of whose iterates are irreducible modulo a positive density subset of the primes must necessarily be a composition of linear functions, monomials, and Dickson polynomials.

First order Stickelberger modules over imaginary quadratic fields

Let K/k be a finite abelian extension of number fields of Galois group G with k imaginary quadratic. Let n≥2 be a rational integer, and for a certain finite set S of places of k, let OK,S be the ring of S-integers of K. We use generalized Stark elements to construct first order Stickelberger modules in odd higher algebraic K-groups of OK,S. We show that the Fitting ideal (resp. index) of these modules inside the corresponding odd K-groups is exactly the Fitting ideal (resp. cardinality) of the even higher algebraic K-group K2n−2(OK,S).

Counting wild quartics with prescribed discriminant and Galois closure group

Given a 2-adic field K, we give a formula for the number of totally ramified quartic field extensions L/K with a given discriminant valuation and Galois closure group. We use these formulae to prove refinements of Serre's mass formula, which will have applications to the arithmetic statistics of number fields.

Efficient computations in central simple algebras using Amitsur cohomology

We introduce a presentation for central simple algebras over a field k using Amitsur cohomology. We provide efficient algorithms for computing a cocycle corresponding to any such algebra given by structure constants. If k is a number field, we use this presentation to prove that the explicit isomorphism problem (i.e., finding an isomorphism between central simple algebras given by structure constants) reduces to S-unit group computation and other related number theoretical computational problems. This also yields, conditionally on the generalised Riemann hypothesis, the first polynomial quantum algorithm for the explicit isomorphism problem over number fields.

Fast and Globally Consistent Normal Orientation based on the Winding Number Normal Consistency

Estimating consistently oriented normals for point clouds enables a number of important applications in computer graphics such as surface reconstruction. While local normal estimation is possible with simple techniques like principal component analysis (PCA), orienting these normals to be globally consistent has been a notoriously difficult problem. Some recent methods exploit various properties of the winding number formula to achieve global consistency with state-of-the-art performance. Despite their exciting progress, these algorithms either have high space/time complexity, or do not produce accurate and consistently oriented normals for imperfect data. In this paper, we propose a novel property from the winding number formula, termed Winding Number Normal Consistency (WNNC ), to tackle this problem. The derived property is based on the simple observation that the normals (negative gradients) sampled from the winding number field should be codirectional to the normals used to compute the winding number field. Since the WNNC property itself does not resolve the inside/outside orientation ambiguity, we further propose to incorporate an objective function from Parametric Gauss Reconstruction (PGR). We propose to iteratively update normals by alternating between WNNC-based normal updates and PGR-based gradient descents, which leads to an embarrassingly simple yet effective iterative algorithm that allows fast and high-quality convergence to a globally consistent normal vector field. Furthermore, our proposed algorithm only involves repeatedly evaluating the winding number formula and its derivatives, which can be accelerated and parallelized using a treecode-based approximation algorithm due to their special structures. Exploiting this fact, we implement a GPU-accelerated treecode-based solver. Our GPU (and even CPU) implementation can be significantly faster than the recent state-of-the-art methods for normal orientation from raw points. Our code is integrated with the popular PyTorch framework to facilitate further research into winding numbers, and is publicly available at https://jsnln.github.io/wnnc/index.html.

Enhancing ultrasonic-assisted drying of low-porosity products through pulsed electric field (PEF) pretreatment: The case of butternut squash

Ultrasonic-assisted drying is an effective technique for accelerating drying processes, particularly for products with high porosity. The structural changes induced by pulsed electric field (PEF) treatment can make low-porosity products more susceptible to the effects of ultrasound during drying. This study aimed to investigate the influence of PEF treatment on the structure of low-porosity products, such as butternut squash, and to evaluate its effect on ultrasonic-assisted drying. PEF pretreatment altered the physicochemical and microstructural properties of butternut squash. Thus, the higher the energy input, the higher the cell disintegration rate, the lower the shearing force and the lower the water holding capacity. For the same energy input applied, no influence was observed from the different combinations of pulse number and electric field intensity used. The microstructural analysis also showed greater effects with increasing intensity of PEF treatments. All these changes affected the subsequent drying, increasing the drying rate of conventional drying. Moreover, PEF pretreatment enhanced the ultrasound effects when applied during drying, reducing drying time by up to 47% when moderate PEF intensity was used. Therefore, PEF pretreatment under the appropriate conditions could make ultrasound-assisted drying of low-porosity products, such as butternut squash, more feasible.

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.