Search results for: non-uniform field

Authors: Ming Su, Ziqiang Mu

Abstract:

This paper proposes a synthesis method for nonuniform linear antenna arrays that combine Gaussian process regression (GPR) and genetic algorithm (GA). In this method, the GPR model can be used to calculate the array radiation pattern in the presence of mutual coupling effects, and then the GA is used to optimize the excitations and locations of the elements so as to generate the desired radiation pattern. In this paper, taking a 9-element nonuniform linear array as an example and the desired radiation pattern corresponding to a Chebyshev distribution as the optimization objective, optimize the excitations and locations of the elements. Finally, the optimization results are verified by electromagnetic simulation software CST, which shows that the method is effective.

Keywords: nonuniform linear antenna arrays, GPR, GA, mutual coupling effects, active element pattern

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Authors: Nileshkumar Vishnav, Aditya Tatu

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A recent approach of representing graph signals and graph filters as polynomials is useful for graph signal processing. In this approach, the adjacency matrix plays pivotal role; instead of the more common approach involving graph-Laplacian. In this work, we follow the adjacency matrix based approach and corresponding algebraic signal model. We further expand the theory and introduce the concept of similarity of two graphs. The similarity of graphs is useful in that key properties (such as filter-response, algebra related to graph) get transferred from one graph to another. We demonstrate potential applications of the relation between two similar graphs, such as nonuniform filter design, DTMF detection and signal reconstruction.

Keywords: graph signal processing, algebraic signal processing, graph similarity, isospectral graphs, nonuniform signal processing

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Authors: Sen Yung Lee, Chih Cheng Huang, Te Wen Tu

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A solution methodology without using integral transformation is proposed to develop analytical solutions for transient heat conduction in nonuniform hollow cylinders with time-dependent boundary condition at the outer surface. It is shown that if the thermal conductivity and the specific heat of the medium are in arbitrary polynomial function forms, the closed solutions of the system can be developed. The influence of physical properties on the temperature distribution of the system is studied. A numerical example is given to illustrate the efficiency and the accuracy of the solution methodology.

Keywords: analytical solution, nonuniform hollow cylinder, time-dependent boundary condition, transient heat conduction

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Authors: Abdulatif Abdusalam, Mohamed Shaban

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In this paper, we consider the nonlinear pulse propagation through a nonuniform birefringent fiber Bragg grating (FBG) whose index modulation depth varies along the propagation direction. Here, the pulse propagation is governed by the nonlinear birefringent coupled mode (NLBCM) equations. To form the Bragg soliton outside the photonic bandgap (PBG), the NLBCM equations are reduced to the well known NLS type equation by multiple scale analysis. As we consider the pulse propagation in a nonuniform FBG, the pulse propagation outside the PBG is governed by inhomogeneous NLS (INLS) rather than NLS. We, then, discuss the formation of soliton in the FBG known as Bragg soliton whose central frequency lies outside but close to the PBG of the grating structure. Further, we discuss Bragg soliton compression due to a delicate balance between the SPM and the varying grating induced dispersion. In addition, Bragg soliton collision, Bragg soliton switching and possible logic gates have also been discussed.

Keywords: Bragg grating, non uniform fiber, non linear pulse

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Authors: Mounir Belattar

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In this paper the Analysis of voltage waves that propagate along a lossless exponential nonuniform line is presented. For this analysis the parameters of this line are assumed to be varying function of the distance x along the line from the source end. The approach is based on the tow-port networks cascading presentation to derive the ABDC parameters of transmission using Picard-Carson Method which is a powerful method in getting a power series solution for distributed network because it is easy to calculate poles and zeros and solves differential equations such as telegrapher equations by an iterative sequence. So the impedance, admittance voltage and current along the line are expanded as a Taylor series in x/l where l is the total length of the line to obtain at the end, the main transmission line parameters such as voltage response and transmission and reflexion coefficients represented by scattering parameters in frequency domain.

Keywords: ABCD parameters, characteristic impedance exponential nonuniform transmission line, Picard-Carson's method, S parameters, Taylor's series

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Authors: T. H. Young, Y. J. Tsai

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A method for determining the stress distribution of a rectangular plate subjected to two pairs of arbitrarily distributed in-plane edge shear loads is proposed, and the free vibration and buckling of such a rectangular plate are investigated in this work.  The method utilizes two stress functions to synthesize the stress-resultant field of the plate with each of the stress functions satisfying the biharmonic compatibility equation. The sum of stress-resultant fields due to these two stress functions satisfies the boundary conditions at the edges of the plate, from which these two stress functions are determined. Then, the free vibration and buckling of the rectangular plate are investigated by the Galerkin method. Numerical results obtained by this work are compared with those appeared in the literature, and good agreements are observed.

Keywords: stress analysis, free vibration, plate buckling, nonuniform in-plane edge shear

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Authors: T. H. Young, S. J. Huang, Y. S. Chiu

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This paper investigates the parametric stability of an axially moving web subjected to nonuniform in-plane edge excitations on two opposite, simply-supported edges. The web is modeled as a viscoelastic plate whose constitutive relation obeys the Kelvin-Voigt model, and the in-plane edge excitations are expressed as the sum of a static tension and a periodical perturbation. Due to the in-plane edge excitations, the moving plate may bring about parametric instability under certain situations. First, the in-plane stresses of the plate due to the nonuniform edge excitations are determined by solving the in-plane forced vibration problem. Then, the dependence on the spatial coordinates in the equation of transverse motion is eliminated by the generalized Galerkin method, which results in a set of discretized system equations in time. Finally, the method of multiple scales is utilized to solve the set of system equations analytically if the periodical perturbation of the in-plane edge excitations is much smaller as compared with the static tension of the plate, from which the stability boundaries of the moving plate are obtained. Numerical results reveal that only combination resonances of the summed-type appear under the in-plane edge excitations considered in this work.

Keywords: axially moving viscoelastic plate, in-plane periodic excitation, nonuniformly distributed edge tension, dynamic stability

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Authors: Renu Tomar, Hitendra K. Malik, Raj P. Dahiya

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Similar to the sound waves in air, plasmas support the propagation of ion waves, which evolve into the solitary structures when the effect of non linearity and dispersion are balanced. The ion acoustic solitary waves have been investigated in details in homogeneous plasmas, inhomogeneous plasmas, and magnetized plasmas. The ion acoustic solitary waves are also found to reflect from a density gradient or boundary present in the plasma after propagating. Another interesting feature of the solitary waves is their collision. In the present work, we carry out analytical calculations for the head-on collision of solitary waves in a magnetized plasma which has dust grains in addition to the ions and electrons. For this, we employ Poincar´e-Lighthill-Kuo (PLK) method. To lowest nonlinear order, the problem of colliding solitary waves leads to KdV (modified KdV) equations and also yields the phase shifts that occur in the interaction. These calculations are accomplished for the uniform and nonuniform plasmas, and the results on the soliton properties are discussed in detail.

Keywords: inhomogeneous magnetized plasma, dust charging, soliton collisions, magnetized plasma

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Authors: Semih Erdogan

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In this paper, the normal and shear stress distributions in Vlasov beams are determined by two-dimensional triangular finite element formulations. The proposed formulations take into account the warping effects along the beam axis. The shape of the considered beam sections may be arbitrary and varied throughout its length. The stiffness matrices and force vectors are derived for transversal forces, uniform torsion, and nonuniform torsion. The proposed finite element algorithm is validated by comparing the analytical solutions, structural engineering books, and related articles. The numerical examples include beams with different cross-section types such as solid, thick-walled, closed-thin-walled, and open-thin-walled sections. Materials defined in the examples are homogeneous, isotropic, and linearly elastic. Through these examples, the study demonstrates the capability of the proposed method to address a wide range of practical engineering scenarios.

Keywords: Vlasov beams, warping function, nonuniform torsion, finite element method, normal and shear stresses, cross-section properties

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Authors: Jeronimo Cox, Tomonari Furukawa

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Magnetometers have become more popular in inertial measurement units (IMU) for their ability to correct estimations using the earth's magnetic field. Accelerometer and gyroscope-based packages fail with dead-reckoning errors accumulated over time. Localization in robotic applications with magnetometer-inclusive IMUs has become popular as a way to track the odometry of slower-speed robots. With high-speed motions, the accumulated error increases over smaller periods of time, making them difficult to track with IMU. Tracking a high-speed motion is especially difficult with limited observability. Visual obstruction of motion leaves motion-tracking cameras unusable. When motions are too dynamic for estimation techniques reliant on the observability of the gravity vector, the use of magnetometers is further justified. As available magnetometer calibration methods are limited with the assumption that background magnetic fields are uniform, estimation in nonuniform magnetic fields is problematic. Hard iron distortion is a distortion of the magnetic field by other objects that produce magnetic fields. This kind of distortion is often observed as the offset from the origin of the center of data points when a magnetometer is rotated. The magnitude of hard iron distortion is dependent on proximity to distortion sources. Soft iron distortion is more related to the scaling of the axes of magnetometer sensors. Hard iron distortion is more of a contributor to the error of attitude estimation with magnetometers. Indoor environments or spaces inside ferrite-based structures, such as building reinforcements or a vehicle, often cause distortions with proximity. As positions correlate to areas of distortion, methods of magnetometer localization include the production of spatial mapping of magnetic field and collection of distortion signatures to better aid location tracking. The goal of this paper is to compare magnetometer methods that don't need pre-productions of magnetic field maps. Mapping the magnetic field in some spaces can be costly and inefficient. Dynamic measurement fusion is used to track the motion of a multi-link system with us. Conventional calibration by data collection of rotation at a static point, real-time estimation of calibration parameters each time step, and using two magnetometers for determining local hard iron distortion are compared to confirm the robustness and accuracy of each technique. With opposite-facing magnetometers, hard iron distortion can be accounted for regardless of position, Rather than assuming that hard iron distortion is constant regardless of positional change. The motion measured is a repeatable planar motion of a two-link system connected by revolute joints. The links are translated on a moving base to impulse rotation of the links. Equipping the joints with absolute encoders and recording the motion with cameras to enable ground truth comparison to each of the magnetometer methods. While the two-magnetometer method accounts for local hard iron distortion, the method fails where the magnetic field direction in space is inconsistent.

Keywords: motion tracking, sensor fusion, magnetometer, state estimation

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Authors: M. Abouelatta, A. Shaker, M. El-Banna, G. T. Sayah, C. Gontrand, A. Zekry

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In this paper, a methodology for physically modeling the intrinsic MOS part and the drift region of the n-channel Laterally Double-diffused MOSFET (LDMOS) is presented. The basic physical effects like velocity saturation, mobility reduction, and nonuniform impurity concentration in the channel are taken into consideration. The analytical model is implemented using MATLAB. A comparison of the simulations from technology computer aided design (TCAD) and that from the proposed analytical model, at room temperature, shows a satisfactory accuracy which is less than 5% for the whole voltage domain.

Keywords: LDMOS, MATLAB, RESURF, modeling, TCAD

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Authors: Rishabh Beri, Sahil Shah

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We have applied a range of filters and processing in order to extract out the various features of the football game, like the field lines of a football field. Another important aspect was the detection of the players in the field and tagging them according to their teams distinguished by their jersey colours. This extracted information combined about the players and field helped us to create a virtual field that consists of the playing field and the players mapped to their locations in it.

Keywords: Detect, Football, Players, Virtual

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Authors: Tracy B.E. Omorogiuwa

Abstract:

Understanding the role of social work in improving societal well-being cannot be separated from the place of field education, which is an integral aspect of social work education. Field learning provides students with knowledge and opportunities to experience solving issues in the field and giving them a clue of the practice situation. Despite being a crucial component in social work curriculum, field education occupies a large space in learning outcome, given the issues and challenges pertaining to its purpose and significance in the society. The drive of this paper is to provide insight on the specific ways in which field education has been conceived, realized and valued in the society. Emphasis is on the significance of field instruction; the link with classroom learning; and the structure of field experience in social work education. Given documented analysis and experience, this study intends to contribute to the development of social work curriculum, by analyzing the pattern, issues and challenges fronting the social work field education in the University of Benin, Nigeria.

Keywords: challenges, curriculum, field education, social work education

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Authors: Sirapat Lookrak, Anol Paisal

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Space debris has numerous manifestations, including ferro-metalize and non-ferrous. The electric field will induce negative charges to split from positive charges inside the space debris. In this research, we focus only on conducting materials. The assumption is that the electric charge density of a conducting surface is proportional to the electric field on that surface due to Gauss's Law. We are trying to find the induced charge density from an external electric field perpendicular to a conducting spherical surface. An object is a sphere on which the external electric field is not uniform. The electric field is, therefore, considered locally. The localised spherical surface is a tangent plane, so the Gaussian surface is a very small cylinder, and every point on a spherical surface has its own cylinder. The electric field from a circular electrode has been calculated in near-field and far-field approximation and shown Explanation Touchless maneuvering space debris orbit properties. The electric charge density calculation from a near-field and far-field approximation is done.

Keywords: near-field approximation, far-field approximation, localized Gauss’s law, electric charge density

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Authors: Mohammed T. Shukri

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The failure of bridges due to excessive local scour during floods poses a challenging problem to hydraulic engineers. The failure of bridges piers is due to many reasons such as localized scour combined with general riverbed degradation. In this paper, we try to estimate the temporal variation of scour depth at nonuniform cylindrical bridge pier, by experimental work conducted in hydraulic laboratories of Gaziantep University Civil Engineering Department on a flume having dimensions of 8.3 m length, 0.8 m width and 0.9 m depth. The experiments will be carried on 20 cm depth of sediment layer having d₅₀=0.4 mm. Three bridge pier shapes having different scaled models will be constructed in a 1.5m of test section in the channel.

Keywords: scour, local scour, bridge piers, scour depth

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Authors: Fatemeh Shahi, Mehdi Sharifian, Laia Shahrassai, Elham Eskandari A.

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A new mechanism is reported here for magnetic field generation in laser-plasma interaction by means of nonlinear ponderomotive force. The plasma considered here is unmagnetized inhomogeneous plasma with an exponentially decreasing profile. A damped periodic magnetic field with a relatively lower frequency is obtained using the ponderomotive force exerted on plasma electrons. Finally, with an electric field and by using Faraday’s law, the magnetic field profile in the plasma has been obtained. Because of the negative exponential density profile, the generated magnetic field is relatively slowly oscillating and damped through the plasma.

Keywords: magnetic field generation, laser-plasma interaction, ponderomotive force, inhomogeneous plasma

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Authors: M. Sandeep Kumar, Vasu Velagapudi, A. Venugopal

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When machining is carried out on a workpiece, residual stresses are induced in the workpiece due to nonuniform thermal and mechanical loads. These stresses play a vital role in the surface integrity of the final product or the output. Inconel 718 is commonly used in critical structural components of aircraft engines due to its properties at high temperatures. Therefore it is important to keep down the stresses induced due to machining. This can be achieved through proper lubricating conditions. In this work, experiments were carried out to check the influence of the developed nanofluid as cutting fluids on residual stresses developed during the course of machining. The results of MQL/Nanofluids were compared with MQL/Vegetable oil and dry machining lubricating condition. Results indicate the reduction in residual stress with the use of MQL/Nanofluid.

Keywords: nanofluids, MQL, residual stress, Inconel-718

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Authors: Tun Myat Aung, Ni Ni Hla

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This paper begins by describing basic properties of finite field and elliptic curve cryptography over prime field and binary field. Then we discuss the discrete logarithm problem for elliptic curves and its properties. We study the general common attacks on elliptic curve discrete logarithm problem such as the Baby Step, Giant Step method, Pollard’s rho method and Pohlig-Hellman method, and describe in detail experiments of these attacks over prime field and binary field. The paper finishes by describing expected running time of the attacks and suggesting strong elliptic curves that are not susceptible to these attacks.c

Keywords: discrete logarithm problem, general attacks, elliptic curve, prime field, binary field

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Authors: Amal Machtalay

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In this survey, we aim to review the rapidly growing literature on numerical methods to solve different forms of mean field problems, namely mean field games (MFG), mean field controls (MFC), potential MFGs, and master equations, as well as their corresponding recent applications. Here, we distinguish two families of numerical methods: iterative methods based on mesh generation and those called mesh-free, normally related to neural networking and learning frameworks.

Keywords: mean-field games, numerical schemes, partial differential equations, complex systems, machine learning

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Authors: Yanchun Zuo, Yingao Liu, Wei Liu, Le Yu, Run Huang, Lixin Guo

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Electric field detection is important in many application scenarios. The traditional strategy is measuring the electric field with a man walking around in the area under test. This strategy cannot provide a satisfactory measurement accuracy. To solve the mentioned problem, an autopilot measurement system is divided. A mini-car is produced, which can travel in the area under test according to respect to the program within the CPU. The electric field measurement platform (EFMP) carries a central computer, two horn antennas, and a vector network analyzer. The mini-car stop at the sampling points according to the preset. When the car stops, the EFMP probes the electric field and stores data on the hard disk. After all the sampling points are traversed, an electric field map can be plotted. The proposed system can give an accurate field distribution description of the chamber.

Keywords: autopilot mini-car measurement system, electric field detection, field map, static zone measurement

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Authors: D. Ravinder

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Nickel ferrites and Ni doped Lithium nano ferrites [Li0.5Fe0.5]1-xNixFe2O4 with x= 0.8 and 1.0 synthesized by citrate-gel auto combustion method. The broad peaks in the X-ray diffraction pattern (XRD) indicate a crystalline behavior of the prepared samples. Low temperature magnetization studies i,e Field Cooled (FC) and Zero Field Cooled (ZFC) magnetic studies of the investigated samples are measured by using vibrating sample magnetometer (VSM). The magnetization of the prepared samples as a function of an applied magnetic field 10 T was measured at two different temperatures 5 K and 310 K. Field Cooled (FC) and Zero Field Cooled (ZFC) magnetization measurements under an applied field of 100 Oe and 1000 Oe in the temperature range of 5–375 K were carried out.

Keywords: ferro-spinels, field cooled (FC), Zero Field Cooled (ZFC) and blocking temperature, superpara magnetism, drug delivery applications

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Authors: P. W. Chen, C. T. Chang, Y. Peng, J. Y. Wu, D. J. Jan, Md. Manirul Ali

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In the last decade, the nature of cathode spot splitting and the current per spot depended on an oblique magnetic field was investigated. This model for cathode current splitting is developed that we have investigated with relationship the magnetic pressures produced by kinetic pressure, self-magnetic pressure, and changed with an external magnetic field. We propose a theoretical model that has been established to an external magnetic field with components normal and tangential to the cathode surface influenced on magnetic pressure strength. We mainly focus on developed to understand the current per spot influenced with the tangential magnetic field strength and normal magnetic field strength.

Keywords: cathode spot, vacuum arc discharge, oblique magnetic field, tangential magnetic field

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Authors: Tatyana Rybitskaya, Alexandr Starostenko, Kseniya Ryabchenko

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Collector ring (CR) of FAIR project is a large acceptance storage ring and field quality plays a major role in the magnet design. The CR will use normal conducting dipole magnets. There will be 24 H-type sector magnets with a maximum field value of 1.6 T. The integrated over the length of the magnet field quality as a function of radius is ∆B.l/B.l = ±1x10⁻⁴. Below 1.6 T the value ∆B.l/B.l can be higher with a linear approximation up to ±2.5x10⁻⁴ at the field level of 0.8 T. An iron-dominated magnet with required field quality is produced with standard technology as the quality is dominated by the yoke geometry.

Keywords: conventional magnet, iron yoke dipole, harmonic terms, particle accelerators

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Authors: Alexander Aurell

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One of the latest trends in the modeling of human crowds is the mean-field game approach. In the mean-field game approach, the motion of a human crowd is described by a nonstandard stochastic optimal control problem. It is nonstandard since congestion is considered, introduced through a dependence in the performance functional on the distribution of the crowd. This study extends the class of mean-field pedestrian crowd models to allow for non-local congestion and arbitrary, but finitely, many interacting crowds. The new congestion feature grants pedestrians a 'personal space' where crowding is undesirable. The model is treated as a mean-field type game which is derived from a particle picture. This, in contrast to a mean-field game, better describes a situation where the crowd can be controlled by a central planner. The latter is suitable for decentralized situations. Solutions to the mean-field type game are characterized via a Pontryagin-type Maximum Principle.

Keywords: congestion, crowd dynamics, interacting populations, mean-field approximation, optimal control

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Authors: Bing Gu

Abstract:

Abstract—Most of Conventional facial recognition systems are based on image features, such as LBP, SIFT. Recently some DBN-based 2D facial recognition systems have been proposed. However, we find there are few DBN-based 3D facial recognition system and relative researches. 3D facial images include all the individual biometric information. We can use these information to build more accurate features, So we present our DBN-based face recognition system using Light Field. We can see Light Field as another presentation of 3D image, and Light Field Camera show us a way to receive a Light Field. We use the commercially available Light Field Camera to act as the collector of our face recognition system, and the system receive a state-of-art performance as convenient as conventional 2D face recognition system.

Keywords: DBN, face recognition, light field, Lytro

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Authors: N. Q. Bau, N. V. Nghia

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The acoustomagnetoelectric (AME) field in a rectangular quantum wire with an infinite potential (RQWIP) is calculated in the presence of an external magnetic field (EMF) by using the quantum kinetic equation for the distribution function of electrons system interacting with external phonons and electrons scattering with internal acoustic phonon in a RQWIP. We obtained ananalytic expression for the AME field in the RQWIP in the presence of the EMF. The dependence of AME field on the frequency of external acoustic wave, the temperature T of system, the cyclotron frequency of the EMF and the intensity of the EMF is obtained. Theoretical results for the AME field are numerically evaluated, plotted and discussed for a specific RQWIP GaAs/GaAsAl. This result has shown that the dependence of the AME field on intensity of the EMF is nonlinearly and it is many distinct maxima in the quantized magnetic region. We also compared received fields with those for normal bulk semiconductors, quantum well and quantum wire to show the difference. The influence of an EMF on AME field in a RQWIP is newly developed.

Keywords: rectangular quantum wire, acoustomagnetoelectric field, electron-phonon interaction, kinetic equation method

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Authors: Rahisham Abd Rahman

Abstract:

In this paper, a nonlinear pollution model has been proposed to compute electric field distribution over the polymeric insulator surface under wet contaminated conditions. A 2D axial-symmetric insulator geometry, energized with 11kV was developed and analysed using Finite Element Method (FEM). A field-dependent conductivity with simplified assumptions was established to characterize the electrical properties of the pollution layer. Comparative field studies showed that simulation of dynamic pollution model results in a more realistic field profile, offering better understanding on how the electric field behaves under wet polluted conditions.

Keywords: electric field distributions, pollution layer, dynamic model, polymeric outdoor insulators, finite element method (FEM)

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Authors: Ashkan Javadzadegan, Aitak Javadzadegan, Babak Fakhim

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One of possible ways to retard movement of fluid is through applying an external magnetic field. In this regard, this study is focused on the effect of a uniform transverse magnetic field on fluid flow behavior inside a channel with a local symmetric constriction. Also, Ellis Non-Newtonian model is implemented to address the effects of shear-dependent viscosity. According to the results, the flow separation downstream of the constriction can be controlled by applying an external magnetic field and/or manipulating the shear-thinning degree of fluid. It is also demonstrated that pressure drop increases by an increase in the strength of the magnetic field.

Keywords: magnetic field, non-Newtonian, separation, shear thinning

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Authors: Olfat A. Fadali, Mohamed S. Mahmoud, Omnia H. Abdelraheem, Shimaa G. Mohammed

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The effect of electromagnetic field (EMF) on the removal of edible oil from oil-in-water emulsion by means of electrocoagulation was investigated in rectangular batch electrochemical cell with DC current. Iron (Fe) plate anodes and stainless steel cathodes were employed as electrodes. The effect of different magnetic field intensities (1.9, 3.9 and 5.2 tesla), three different positions of EMF (below, perpendicular and parallel to the electrocoagulation cell), as well as operating time; had been investigated. The application of electromagnetic field (5.2 tesla) raises percentage of oil removal from 72.4% for traditional electrocoagulation to 90.8% after 20 min.

Keywords: electrocoagulation, electromagnetic field, Oil-water emulsion, edible oil

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Authors: Shouji Yujiro, Memei Dukovic, Mist Yakubu

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Fields are important objects of study in algebra since they provide a useful generalization of many number systems, such as the rational numbers, real numbers, and complex numbers. In particular, the usual rules of associativity, commutativity and distributivity hold. Fields also appear in many other areas of mathematics; see the examples below. When abstract algebra was first being developed, the definition of a field usually did not include commutativity of multiplication, and what we today call a field would have been called either a commutative field or a rational domain. In contemporary usage, a field is always commutative. A structure which satisfies all the properties of a field except possibly for commutativity, is today called a division ring ordivision algebra or sometimes a skew field. Also non-commutative field is still widely used. In French, fields are called corps (literally, body), generally regardless of their commutativity. When necessary, a (commutative) field is called corps commutative and a skew field-corps gauche. The German word for body is Körper and this word is used to denote fields; hence the use of the blackboard bold to denote a field. The concept of fields was first (implicitly) used to prove that there is no general formula expressing in terms of radicals the roots of a polynomial with rational coefficients of degree 5 or higher. An extension of a field k is just a field K containing k as a subfield. One distinguishes between extensions having various qualities. For example, an extension K of a field k is called algebraic, if every element of K is a root of some polynomial with coefficients in k. Otherwise, the extension is called transcendental. The aim of Galois Theory is the study of algebraic extensions of a field. Given a field k, various kinds of closures of k may be introduced. For example, the algebraic closure, the separable closure, the cyclic closure et cetera. The idea is always the same: If P is a property of fields, then a P-closure of k is a field K containing k, having property, and which is minimal in the sense that no proper subfield of K that contains k has property P. For example if we take P (K) to be the property ‘every non-constant polynomial f in K[t] has a root in K’, then a P-closure of k is just an algebraic closure of k. In general, if P-closures exist for some property P and field k, they are all isomorphic. However, there is in general no preferable isomorphism between two closures.

Keywords: field theory, mechanic maths, supertech, rolltech

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