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Field Theory Handbook
Abstract
Let us first state exactly what this book is and what it is not. It is a compendium of equations for the physicist and the engineer working with electrostatics, magne tostatics, electric currents, electromagnetic fields, heat flow, gravitation, diffusion, optics, or acoustics. It tabulates the properties of 40 coordinate systems, states the Laplace and Helmholtz equations in each coordinate system, and gives the separation equations and their solutions. But it is not a textbook and it does not cover relativistic and quantum phenomena. The history of classical physics may be regarded as an interplay between two ideas, the concept of action-at-a-distance and the concept of a field. Newton's equation of universal gravitation, for instance, implies action-at-a-distance. The same form of equation was employed by COULOMB to express the force between charged particles. AMPERE and GAUSS extended this idea to the phenomenological action between currents. In 1867, LUDVIG LORENZ formulated electrodynamics as retarded action-at-a-distance. At almost the same time, MAXWELL presented the alternative formulation in terms of fields. In most cases, the field approach has shown itself to be the more powerful.
... In the case where h = ε, from Eq. (14) we see that the 2 × 2 block at the upper left corner of the matrix (a j i ) represents an ordinary rotation about the origin on the plane R 3 = 0, hence, under these transformations (R 1 ) 2 + (R 2 ) 2 is invariant and one finds that ...
... and l, m are separation constants. This equation appears in the solution by separation of variables of the Laplace equation in the three-dimensional Euclidean space in prolate spheroidal equations (with l being an integer) (see, e.g., Ref. [13], Eq. (8.6.7) or Ref. [14], Table 1.06) and in toroidal coordinates (where l is a half-integer) (see, e.g, Ref. [13], Eq. (8.10.11) or Ref. [14], Sec. IV). ...
... and l, m are separation constants. This equation appears in the solution by separation of variables of the Laplace equation in the three-dimensional Euclidean space in prolate spheroidal equations (with l being an integer) (see, e.g., Ref. [13], Eq. (8.6.7) or Ref. [14], Table 1.06) and in toroidal coordinates (where l is a half-integer) (see, e.g, Ref. [13], Eq. (8.10.11) or Ref. [14], Sec. IV). ...
We explicitly show that the groups of $2 \times 2$ unitary matrices with determinant equal to 1 whose entries are double or dual numbers are homomorphic to ${\rm SO}(2,1)$ or to the group of rigid motions of the Euclidean plane, respectively, and we introduce the corresponding two-component spinors. We show that with the aid of the double numbers we can find generating functions for separable solutions of the Laplace equation in the $(2 + 1)$ Minkowski space, which contain special functions that also appear in the solution of the Laplace equation in the three-dimensional Euclidean space, in spheroidal and toroidal coordinates.
... Namely, the theoretical analysis considers a two-region problem, which corresponds to the dielectric barrier tube of the DBD and the external grounded electrode, wherein our goal is to calculate the temperature distribution in the axisymmetric configuration of the setup. In order to construct the related initial and boundary value problem and proceed to the solution, our analytical methodology is primarily based on the selection of a suitably located cylindrical coordinate system [21] for modeling purposes with respect to the current investigation. Therein, the domain of field activity is divided into the two subsectors of the dielectric and the electrode-metal, in which classical heat diffusion partial differential equations [22] are considered. ...
... Substituting (46) and (47) into (8) and (9) (taking into account the azimuthal independence), while attaching (21) and (22) Thus, the initial conditions (7) or equivalently (20), by virtue of (48) and (49) , they also depend upon the physical and geometrical characteristics of the system under consideration. ...
We claim an analytical solution for the thermal boundary value problem that arises in DBD-based plasma jet systems as a preliminary and consistent approach to a simplified geometry. This approach involves the outline of a coaxial plasma jet reactor and the consideration of the heat transfer to the reactor solids, namely, the dielectric barrier and the grounded electrode. The non-homogeneous initial and boundary value thermal problem is solved analytically, while a simple cut-off technique is applied to deal with the appearance of infinite series relationships, being the outcome of merging dual expressions. The results are also implemented numerically, supporting the analytical solution, while a Finite Integration Technique (FIT) is used for the validation. Both the analytical and numerical data reveal the temperature pattern at the cross-section of the solids in perfect agreement. This analytical approach could be of importance for the optimization of plasma jet systems employed in tailored applications where temperature-sensitive materials are involved, like in plasma biomedicine.
... Toroidal coordinates (η, θ, ϕ) for a point x = (x 0 , x 1 , x 2 ) in three-dimensional Euclidean space are defined [17,30] by the relations x 0 = sin θ cosh η − cos θ , x 1 = sinh η cos ϕ cosh η − cos θ , x 2 = sinh η sin ϕ cosh η − cos θ (1) in the range η ∈ (0, ∞), θ ∈ [−π, π], ϕ ∈ (−π, π]. For any fixed η 0 > 0, these coordinates define the interior domain η 0 = {x : η 0 < η ≤ ∞} ∪ S 1 where the limiting value η = ∞ corresponds respectively to the singular subset S 1 = {x ∈ R 3 : x 0 = 0, x 2 1 + x 2 2 = 1}. ...
... Proof Take a +,+ 0,0 = ( √ 2/π )q 0,0 , which gives f +,+ 0 = 1 identically. Then apply (30) to evaluate α(η 0 ) ...
The Fourier method approach to the Neumann problem for the Laplacian operator in the case of a solid torus contrasts in many respects with the much more straight forward situation of a ball in 3-space. Although the Dirichlet-to-Neumann map can be readily expressed in terms of series expansions with toroidal harmonics, we show that the resulting equations contain undetermined parameters which cannot be calculated algebraically. A method for rapidly computing numerical solutions of the Neumann problem is presented with numerical illustrations. The results for interior and exterior domains combine to provide a solution for the Neumann problem for the case of a shell between two tori.
... Finally, by virtue of the representation theory, it is obvious that spherical geometry approximates sufficiently well most basic problems in linear isotropic elasticity. Nevertheless, the extension to spheroidal, ellipsoidal or even more complicated geometries [21,41] provides a challenging area for future investigation. ...
... comprise a complete set of eigenfunctions for harmonic functions and belong to the kernel space of the Laplace's operator from (10), i.e. and mn , while they are obtained once the classical method of separation of variables[21,22] is applied.Adopting the above mathematical analysis, the harmonic function A in differential representation (3) admits series expansion in terms of functions (13), i.e. ...
Linear isotropic elasticity is an interesting branch of continuum mechanics, described by the fundamental laws of Hooke and Newton, which are combined in order to construct the governing generalized Navier equation of the displacement within any material. Implying time-independence and in the absence of external body forces, the latter is reduced to the corresponding form of a homogeneous second-order partial differential equation, whose solution is given via the Papkovich differential representation, which expresses the displacement field in terms of harmonic functions. On the other hand, spherical geometry provides the most widely used framework in real-life applications, concerning interior and exterior problems in elasticity. The present work aims to provide a little progress, by producing ready-to-use basic functions for linear isotropic elasticity in spherical coordinates. Hence, we calculate the Papkovich eigensolutions, generated by the spherical harmonic eigenfunctions, obtaining connections between Navier and spherical harmonic kernels. A set of useful results are provided at the end of the paper in the form of examples, regarding the evaluation of displacement field inside and outside a sphere.
... The spherical coordinate system is a way of representing a vector position that lies in a 3-dimensional space with three values, see Fig. 2a, one for the radius r of a sphere centered at the origin, one for the polar angle φ and the other for the azimuthal angle θ, in order to determine a position on such sphere's surface [15]. This coordinate system is used for representing vectors in problems that mainly deal with rotation operations, and therefore is appropriate for representing the W vectors in a version space bounded by a unit radius hypersphere, see Fig. 2b. ...
... Equation (17) is related to the property of complementary slackness and is satisfied because multiplying Eq. (18) by W T we obtain 2 · λ · ||W || 2 ≈ m , which implies that ||W || 2 = 1 because λ > 0. Consequently, Eqs. (15) and (16) are fulfilled. Therefore, considering λ = m/2, the KKT conditions can be simplified to the vector equation: ...
Classification is an essential task in the field of Machine Learning, where developing a classifier that minimizes errors on unknown data is one of its central problems. It is known that the analytic center is a good approximation of the center of mass of the version space that is consistent with the Bayes-optimal decision surface. Therefore, in this work, we propose an evolutionary algorithm, relying on the convexity properties of the version space, that evolves a population of perceptron classifiers in order to find a solution that approximates its analytic center. Hyperspherical coordinates are used to guarantee feasibility when generating new individuals and enabling exploration to be uniformly distributed through the search space. To evaluate the individuals we consider using a potential function that employs a logarithmic barrier penalty. Experiments were performed on real datasets, and the obtained results indicate concrete possibilities for applying the proposed algorithm for solving practical problems.
... When ε dd = 0.3, the trajectory of the vortex is almost perfectly circular and is characterised by only a small ellipticity with a semi-major axis alongŷ. A larger degree of ellipticity is seen for ε dd = 0.6, though, and for ε dd = 0.9 the vortex trajectories are clearly no longer elliptical but are instead reminiscent of Cassini ovals [65]. ...
The static and dynamic properties of vortices in dipolar Bose-Einstein condensates (dBECs) can be considerably modified relative to their nondipolar counterparts by the anisotropic and long-ranged nature of the dipole-dipole interaction. Working in a uniform dBEC, we analyze the structure of single vortices and the dynamics of vortex pairs, investigating the deviations from the nondipolar paradigm. For a straight vortex line, we find that the induced dipolar interaction potential is axially anisotropic when the dipole moments have a nonzero projection orthogonal to the vortex line. This results in a corresponding elongation of the vortex core along this projection as well as an anisotropic superfluid phase and enhanced compressibility in the vicinity of the vortex core. Consequently, the trajectories of like-signed vortex pairs are described by a family of elliptical and oval-like curves rather than the familiar circular orbits. Similarly for opposite-signed vortex pairs their translation speeds along the binormal are found to be dipole interaction-dependent. We expect that these findings will shed light on the underlying mechanisms of many-vortex phenomena in dBECs such as quantum turbulence, vortex reconnections, and vortex lattices.
... The types of potential problems defined by (19) to (21) have been discussed in connection with problems in electrostatic potential theory, fluid mechanics and elasticity and authoritative treatises covering these areas are given by [48][49][50][51][52][53][54][55][56] and others. It can be shown that the harmonic function that satisfies the boundary condition (20) and the regularity condition (21) takes the form ...
The permeability of intact geologic media features prominently in many geo-environmental endeavours. The laboratory estimation of permeability is an important adjunct to the field estimation of bulk permeability values, which involves a great deal of supplementary in situ investigations to correctly interpret field data. Laboratory permeability estimation is also a viable method if core samples are recovered from in situ geological mapping of the region under study. The basic methodologies for permeability estimation rely on either steady-state or transient tests of the geologic material depending on the anticipated permeability value. This paper presents a steady flow test conducted on a partially drilled cavity located on the axis of a cylindrical specimen. Certain compact theoretical relationships are proposed for the estimation of steady flow from a cavity of finite dimensions located along the axis of a cylindrical specimen. The relationships are used to estimate the permeability of a cylinder of Lac du Bonnet granite obtained from the western flank of the Canadian Shield. The results from the cavity flow permeability experiments are compared with other estimates for the permeability of granitic rocks reported in the literature.
... Let n = (γ 2 , −γ 1 ) be the unit exterior normal to Ω and κ = det γ , γ : T L → R be the signed curvature. We define the open tubular neighborhood, see Moon & Spencer (1988), by ...
It is well known that classical optical cavities can exhibit localized phenomena associated with scattering resonances, leading to numerical instabilities in approximating the solution. This result can be established via the ‘quasimodes to resonances’ argument from the black box scattering framework. Those localized phenomena concentrate at the inner boundary of the cavity and are called whispering gallery modes. In this paper we investigate scattering resonances for unbounded transmission problems with sign-changing coefficient (corresponding to optical cavities with negative optical properties, e.g. made of metamaterials). Due to the change of sign of optical properties, previous results cannot be applied directly, and interface phenomena at the metamaterial-dielectric interface (such as the so-called surface plasmons) emerge. We establish the existence of scattering resonances for arbitrary two-dimensional smooth metamaterial cavities. The proof relies on an asymptotic characterization of the resonances, and shows that problems with sign-changing coefficient naturally fit the black box scattering framework. Our asymptotic analysis reveals that, depending on the metamaterial’s properties, scattering resonances situated close to the real axis are associated with surface plasmons. Examples for several metamaterial cavities are provided.
In this paper, we present new axisymmetric and reflection symmetric vacuum solutions to the Einstein field equations. They are obtained using the Hankel integral transform method and all three solutions exhibit naked singularities. Our results further reinforce the importance and special character of axisymmetric solutions in general relativity and highlight the role of integral transforms methods in solving complex problems in this field. We compare our results to already existing solutions which exhibit the same type of singularities. In this context we notice that most known axial-symmetric solutions possess naked singularities. A discussion of characteristic features of the newly found metrics, e.g., blueshift and the geometry of the singularities, is given.
Applications of superconducting magnetic shielding are discussed for magnetic measurements. Analytical and numerical modeling techniques are reviewed for superconductors in the presence of dipole source and external noise fields. Further applications include eddy current nondestructive testing, fault current limiters, and the shielding of space radiation.
In recent network architectures, multi-MEC cooperative caching has been introduced to reduce the transmission latency of VR videos, in which MEC servers’ computing and caching capability are utilized to optimize the transmission process. However, many solutions that use the computing capability of MEC servers ignore the additional arithmetic power consumed by the codec process, thus making them infeasible. Besides, the minimum cache unit is usually the entire VR video, which makes caching inefficient.To address these challenges, we split VR videos into tile files for caching based on the current popular network architecture and provide a reliable transmission mechanism and an effective caching strategy. Since the number of different tile files N is too large, the current cooperative caching algorithms do not cope with such large-scale input data. We further analyze the problem and propose an optimized k-shortest paths (OKSP) algorithm with an upper bound time complexity of \(O((K \cdot M + N) \cdot M \cdot \log N))\), and suitable for shortest paths with restricted number of edges, where K is the total number of tiles that all M MEC servers can cache in the collaboration domain. And we prove the OKSP algorithm can compute the caching scheme with the lowest average latency in any case, which means the solution given is the exact solution. The simulation results show that the OKSP algorithm has excellent speed for solving large-scale data and consistently outperforms other caching algorithms in the experiments.Keywords360\(^\circ \) virtual reality videoVirtual realityMobile edge computingK-shortest paths
Neuroscientists employ many different techniques to observe the activity of the brain, from single-channel recording to functional imaging (fMRI). Many practical books explain how to use these techniques, but in order to extract meaningful information from the results it is necessary to understand the physical and mathematical principles underlying each measurement. This book covers an exhaustive range of techniques, with each chapter focusing on one in particular. Each author, a leading expert, explains exactly which quantity is being measured, the underlying principles at work, and most importantly the precise relationship between the signals measured and neural activity. The book is an important reference for neuroscientists who use these techniques in their own experimental protocols and need to interpret their results precisely; for computational neuroscientists who use such experimental results in their models; and for scientists who want to develop new measurement techniques or enhance existing ones.
Eccentric compound drops, which are ubiquitous in many naturally inspired and engineering systems, can migrate under the sole presence of a uniform electric field, unlike the case of isolated single drops. Here, we report the migration of eccentric compound drops under a uniform electric field, imposed parallel to the line of centres of the constituting drops, by developing an approximate analytical model that applies to low Reynolds number limits under negligible droplet deformation, following axisymmetric considerations. In contrast to the sole influence of the electrohydrodynamic forces that has thus far been established to be emphatic for the eccentric configuration, here we report the additional effects induced by the dielectrophoretic forces to result in decisive manipulation in the drop migration. We show that the relative velocity between the inner and outer drops, which is a function of the eccentricity itself, dictates the dynamical evolution of the eccentricity variation under the competing electrohydrodynamic and dielectrophoretic interactions. This brings out four distinct regimes of the migration characteristics of the two drops based on their relative electro-physical properties. Our results reveal that an increase in eccentricity and the size ratio of the inner and outer droplets may induce monotonic or non-monotonic variation in the drop velocities, depending on the operating regime. We show how the interplay of various properties holds the control of selectively increasing or suppressing the eccentricity with time. These findings open up various avenues of electrically manipulative motion of encapsulated fluidic phases in various applications encompassing engineering and biology.
Magnetic hopfions are localized magnetic solitons with a nonzero 3D topological charge (Hopf index). Herein, an analytical calculation of the magnetic hopfion gyrovector is presented and it is shown that it does not vanish even in an infinite sample. The calculation method is based on the concept of the emergent magnetic field. The particular case of the simplest nontrivial toroidal hopfion with the Hopf index in the cylindrical magnetic dot is considered and dependencies of the gyrovector components on the dot sizes are calculated. Nonzero hopfion gyrovector is important in any description of the hopfion dynamics within the collective coordinate Thiele's approach.
In this work, a 3D fully coupled hygro-elastic model is proposed. The moisture content profile is a primary variable of the model’s displacements. This generic fully coupled 3D exact shell model allows the investigations into the consequences arising from moisture content and elastic fields in terms of stresses and deformations on different plate and shell configurations embedded in composite and laminated layers. Cylinders, plates, cylindrical and spherical shells are analyzed in the orthogonal mixed curvilinear reference system. The 3D equilibrium equations and the 3D Fick diffusion equation for spherical shells are fully coupled in a dedicated system. The main advantage of the orthogonal mixed curvilinear coordinates is related to the degeneration of the equations for spherical shells to simpler geometries thanks to basic considerations of the radii of curvature. The exponential matrix method is used to solve this fully coupled model based on partial differential equations in the thickness direction. The closed-form solution is related to simply supported sides and harmonic forms for displacements and the moisture content. The moisture content amplitudes are directly applied at the top and bottom outer faces through steady-state hypotheses. The final system is based on a set of coupled homogeneous second-order differential equations. A first-order differential equation system is obtained by redoubling the number of variables. The moisture field implications are evaluated for the static analysis of the plates and shells in terms of displacement and stress components. After preliminary validations, new benchmarks are proposed for several thickness ratios, geometrical and material data, lamination sequences and moisture values imposed at the external surfaces. In the proposed results, there is clearly accordance between the uncoupled hygro-elastic model (where the 3D Fick diffusion law is separately solved) and this new fully coupled hygro-elastic model: the differences between the investigated variables (displacements, moisture contents, stresses and strains) are always less than 0.3%. The main advantages of the 3D coupled hygro-elastic model are a more compact mathematical formulation and lower computational costs. Both effects connected with the thickness layer and the embedded materials are included in the conducted hygro-elastic analyses.
Using two different approaches, we study imaging in the strong-lensing regime taking into account the effects of plasmatic environments on light propagation. First, we extend the use of a perturbative approach that allows us to quickly and analytically calculate the position and shape of the images of a circular source lensed by a galaxy. Such an approach will be compared with that obtained from the numerical solution of the lens equations. Secondly, we introduce a three-dimensional spheroidal model to describe the spacetime associated with the dark matter halo around the lens galaxy and an associated optical metric to incorporate the presence of the plasma medium. The (chromatic) deformation on caustic and critical curves and associated multiplicity of images is also analyzed for particular configurations.
Axisymmetric, steady-state, Darcian flows in homogeneous and isotropic aquifers towards a toroid or disk intake are analytically studied. Both unbounded (infinite) and bounded (by a an equipotential soil surface or by an impermeable horizontal caprock-bedrock) aquifers are considered. The Gauss closed-form solution from astronomy for a gravitating circle having a uniform mass distribution and the Weber solution from electrostatics for an equipotential disk are utilized. The scalar/vector fields of piezometric head (potential)/specific discharge allow for reconstruction of stream lines, isobars, isochrones, and isotachs. An air-filled toroid drains much more water than equipotential, or – inversely - at a given flow rate, the size of an empty toroid is much smaller than that of a water-filled one. The hydraulic gradients in the vicinity of modeled wells/tunnels are very high, triggering colmation and suffusion. The functionals of dissipation and drawdown over a specified zone in the far field are evaluated.
In this work, a coupled 3D thermo-elastic shell model is presented. The primary variables are the scalar sovra-temperature and the displacement vector. This model allows for the thermal stress analysis of one-layered and sandwich plates and shells embedding Functionally Graded Material (FGM) layers. The 3D equilibrium equations and the 3D Fourier heat conduction equation for spherical shells are put together into a set of four coupled equations. They automatically degenerate in those for simpler geometries thanks to proper considerations about the radii of curvature and the use of orthogonal mixed curvilinear coordinates α, β, and z. The obtained partial differential governing the equations along the thickness direction are solved using the exponential matrix method. The closed form solution is possible assuming simply supported boundary conditions and proper harmonic forms for all the unknowns. The sovra-temperature amplitudes are directly imposed at the outer surfaces for each geometry in steady-state conditions. The effects of the thermal environment are related to the sovra-temperature profiles through the thickness. The static responses are evaluated in terms of displacements and stresses. After a proper and global preliminary validation, new cases are presented for different thickness ratios, geometries, and temperature values at the external surfaces. The considered FGM is metallic at the bottom and ceramic at the top. This FGM layer can be embedded in a sandwich configuration or in a one-layered configuration. This new fully coupled thermo-elastic model provides results that are coincident with the results proposed by the uncoupled thermo-elastic model that separately solves the 3D Fourier heat conduction equation. The differences are always less than 0.5% for each investigated displacement, temperature, and stress component. The differences between the present 3D full coupled model and the the advantages of this new model are clearly shown. Both the thickness layer and material layer effects are directly included in all the conducted coupled thermal stress analyses.
An analytical–numerical method for determining the two-dimensional (2D) thermal field in a layer-inhomogeneous elliptic cylinder (elliptical roller) was developed in the article. A mathematical model was formulated in the form of a boundary problem for Poisson equations with an external boundary condition of the third kind (Hankel’s). The conditions of continuity of temperature and heat flux increment were assumed at the inner boundaries of material layers. The eigenfunctions of the boundary problem were determined analytically. Hankel’s condition was subjected to appropriate mathematical transformations. As a result, a system of algebraic equations with respect to the unknown coefficients of the eigenfunctions was obtained. The above-mentioned system of equations was solved numerically (iteratively). As an example of an application of the aforementioned method, an analysis of the thermal field in an elliptical electric wire was presented. The system consists of an aluminum core and two layers of insulation (PVC and rubber). In addition to the field distribution, the steady-state current rating was also determined. The thermal conductivities of PVC and rubber are very similar to each other. For this reason, apart from the real model, a test system was also considered. Significantly different values of thermal conductivity were assumed in individual layers of the test model. The temperature distributions were presented graphically. The graphs showed that the temperature drop is almost linear in the insulation of an electrical conductor. On the other hand, in the analogous area of the test model, a broken line was observed. It was also found that the elliptical layer boundaries are not isothermal. The results obtained by the method presented in this paper were verified numerically.
Written by a recognized expert in the field, Electrostatics of Conducting Cylinders and Spheres presents the theory in a simple and physical way, beginning with basic principles and moving onto the details of electrostatics of pairs of spheres and of pairs of cylinders. The difficult topic of a finite charged cylinder is also covered. The book presents exact solutions, in closed form where possible, with recent and new results included. Applications to engineering, physical, chemical, and biological problems are also presented.
Key topics include:Fundamentals of the polarizability tensor and capacitance coefficients of an assembly of conductors.Rigorous and thorough theoretical investigation of the electrostatics of spheres and cylinders.Uniform formalism used in the solution for a rich array of problems.
Theoretical and experimental physicists, chemists, and engineers will find this an invaluable reference. It is also suitable for advanced undergraduate and graduate students in electrodynamics.
Accurate calculation of the tunneling currents in a scanning tunneling microscope (STM) is needed for developing image processing algorithms that convert raw data of the STM into surface topographic images. In this paper, an accurate calculation of the tunneling current for several tip–sample distances, bias voltages, and tips of a hyperboloidal shape with several radii of curvature is carried out. The main features of this calculation are the following. Non-WKB exact solutions to the trapezoidal (linear) potential in the barrier region are used to calculate the tunneling probabilities. Pauli blocking effects on both forward and reverse current densities are introduced. Finite temperature (viz. [Formula: see text]) calculation in which electrons belonging to a narrow band of energy about the Fermi level contribute to tunneling is carried out. Integration over a field line method is used to obtain tunneling currents for the nonplanar hyperboloidal shaped tips, using the expressions obtained in the paper, for planar model current densities. An estimate of the lateral resolution is introduced. Earlier works do not consider all these aspects together in a single calculation. Tunneling currents are found to increase rapidly with increasing bias voltage and decrease exponentially with increasing tip–sample distances. Airy function determined currents are a more accurate function of a tip–sample distance than the WKB determined currents. Pauli effects are found to not always reduce currents from their non-Pauli values. The lateral resolution is found to be degraded for blunter tips, larger bias voltages, and larger tip–sample distances.
Nonlocal generalization of classical (Newtonian) gravity field theory is proposed by using the general fractional calculus in the Luchko form. Nonlocal analogs of the standard Gauss’s law and the “local” potentiality of gravitational field are suggested in integral and differential forms. Nonlocality is described by the pairs of Sonin kernels that belong to the Luchko set. The general fractional vector calculus, which can be considered as nonlocal vector calculus, is used as a mathematical tool for formulation of nonlocal field theory. General fractional (GF) vector operators, such as GF flux, GF circulation, GF divergence, GF curl operators and GF gradient are defined to describe nonlocalities in space. Examples of using the general nonlocal Gauss’s law to calculate gravity fields are proposed for the case of spherically symmetric nonlocality and mass distribution. The nonlocal effects caused by nonlocality in space are discussed. Among such effects, the following effects are described: effects of mass shielding by nonlocality, violation of local potentiality by nonlocality, and violation of local solenoidality by nonlocality (massiveness of nonlocality). A possibility of using the nonlocal theory of gravity to explain some nonstandard properties of dark matter and dark energy through the properties of nonlocality is discussed.
Realization of “FT-IR on a chip” holds the potential of a disruptive technology for downhole chemical analysis within the oil & gas industry. One of the critical obstacles to downhole integration though has been the cooling requirements of conventional technologies. Here we report the design and numerical analysis of an uncooled miniaturized FT-MIR spectrometer compatible with downhole thermal environments, enabled by a broadband mid-infrared metasurface detector/source combination derived from a geometric inversion of a set of conformal mapping contours. The metasurface is numerically found to exhibit an NZIM behavior with absorption characterized by surface plasmon resonances confined to the ultrathin (λ/300) metasurface plane, making the absorption properties of the microbolometer design much less sensitive to the remaining support structure than in typical designs. This feature allows the metasurface to be integrated on a single VO
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substrate operated at elevated downhole temperatures that coincide with the metal-insulator-transition region. Within this transition region the VO
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material exhibits enhanced thermometric properties, enabling an uncooled microbolometer design with predicted maximum detectivity D* = 1.5×10
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cm√Hz/W and noise equivalent difference temperature NEDT of 1 mK at a modulation frequency of 500 Hz. These parameters approach entry-level lab FT-MIR spectrometers and could represent a significant step in deploying mid-infrared spectroscopy into oilfield downhole logging applications.
This study addresses the first feasible, and comprehensive approach to demonstrate a compact resistance-inductance-capacitance-conductance (
RLCG
) model for a multi-walled carbon nanotube bundle (MWB) and multilayered graphene nanoribbon (MLGNR) based tapered through silicon via (
T
-TSV) along with the different shaped bumps. The physical structures of bumps accurately considered the effect of the high frequency resistive impact and the inter-metal dielectric (IMD) layer. A mathematical framework has been designed for the parasitics of the cylindrical, barrel, hourglass and the tapered bump structures. The bump and via parasitics have been computed by utilizing the current continuity expression, partial inductance method, splitting infinitesimally thin slices of bump and triangular arrangement of tube assemblage. In order to validate the proposed model, the EM simulation is performed and compared against the analytical results. A remarkable consistency of the analytical and EM simulation-based results supports the proposed model accuracy. Furthermore, when compared to the MWB based structures, the MLGNR -based tapered TSV shows a substantial improvement in power loss and crosstalk. Furthermore, regardless of via height, the TSV with tapered bump structure reduces the overall crosstalk induced delay by 33.22%, 28.90%, and 21.61%, respectively, when compared to the barrel, cylindrical and the hourglass structure.
The composition of the multiphase flow rates from hydrocarbon reservoirs has direct impact on the programs to optimize hydrocarbon production and for identification of problems with production. However, accurate measurement of the flow rates of oil, brine, and gas in three-phase regimes remains an intractable problem for current state-of-the-art sensor technologies. Here we describe a piezoelectric acoustic leaf-cell sensor array providing an in situ real-time measurement of the distribution in three-phase flow fluid density over the wellbore cross-section. The sensor array measurements are evaluated from experimental three-phase flow loop test data under a range of flow regimes comprised of 0%-95% GVF and water/liquid volume ratio between 0- 100%. The experimental data includes 3-phase flow transitions well into the nonlinear gas-subsonic velocity region predicted by the Wood equation and show an average absolute error in 3-phase fluid density prediction less than 0.05 g/cc over the entire test regimen.
Analytical models are useful in the design of electromagnetic devices. In this paper, a model is developed for a rotary actuator whose stator curvature is elliptically shaped to have a reluctance torque that restores the rotor to the maximum torque per ampere position. The total torque is decomposed into the coil torque as well as a reluctance torque. The rotor is a permanent magnet represented by equivalent Amperian currents. The stator geometry is simplified to an ellipse having surface current densities at the interpolar regions which are equivalent to the stator currents. Then, the field solution within the ellipse is obtained using Laplace’s equation in elliptical coordinates, so that the coil torque can be obtained by the Lorentz force. The reluctance torque is derived by the energy method and an approach referred to here as differential flux tubes, which is similar to the use of conventional flux tubes in magnetic equivalent circuits. A rotating reference frame on the rotor is also adopted to simplify the mathematics. The finite element method is also used in the field analysis and development of the proposed model. Finally, the actuator is prototyped and experimental results are employed to evaluate the results obtained from the analytical model and finite element method.
A new coordinate system is proposed for the purposes of analysis of far-field radiation from a localized source at the origin. The proposed stereographic coordinates exhibit no singularities over the far-field hemisphere, making them a good choice for automated radiation pattern analysis. Finally, it is shown that Ludwig polarization basis, commonly used to represent polarization of the far-field radiation, arises naturally, as a coordinate basis when one uses stereographic coordinates.
This paper proposes an analytical approach to model the evaporation of multicomponent drops of general shape, which is based on the solution to Stefan–Maxwell equations. The model predicts the quasi-steady molar fractions and temperature distributions in the gas phase as well as the heat rate and the species evaporation rates. The model unifies previous approaches to this problem, namely, for spherical and spheroidal drops, under a unique model and proposes solutions for other shapes and geometries, such as sessile drops and drop pairs. To assess the model, a comparison with a numerical solution to the conservation equations is also reported for both different drop configurations and different compositions.
We present a versatile perturbative calculation scheme to determine the leading-order correction to the mobility matrix for particles in a low-Reynolds-number fluid with spatially variant viscosity. To this end, we exploit the Lorentz reciprocal theorem and the reflection method in the far field approximation. To demonstrate how to apply the framework to a particular choice of a viscosity field, we first study particles in a finite-size, interface-like, linear viscosity gradient. The extent of the latter should be significantly larger than the particle separation. Both situations of symmetrically and asymmetrically placed particles within such an odd symmetric viscosity gradient are considered. As a result, long-range flow fields are identified that decay by one order slower than their constant-viscosity counterparts. Self-mobilities for particle rotations and translations are affected, while for asymmetric placement, additional correction appears for the latter. The mobility terms associated with hydrodynamic interactions between the particles also need to be corrected, in a placement-specific manner. While the results are derived for the system of two particles, they apply also to many-particle systems. Furthermore, we treat the viscosity gradients induced by two particles with temperatures different from that of the surrounding fluid. Assuming a linear relation between fluid temperature and viscosity, we find that both the self-mobilities of the particles as well as the mobility terms for hydrodynamic interactions increase for hot particles and decrease for cold particles.
We discuss integrable discretizations of 3-dimensional cyclic systems, that is, orthogonal coordinate systems with one family of circular coordinate lines. In particular, the underlying circle congruences are investigated in detail and characterized by the existence of a certain flat connection. Within the developed framework, discrete cyclic systems with a family of discrete flat fronts in hyperbolic space and discrete cyclic systems, where all coordinate surfaces are discrete Dupin cyclides, are investigated.
Low emittance machines require lattices with many magnets of short length. Furthermore, successful lattices require strong gradients with strong dipole fields. These lattices are popular today, as MAX IV has demonstrated that a diffraction limited synchrotron light source can be based on such a lattice. This paper gives an overview of the various magnets used by the various diffraction limited light sources – available, under construction or designed. It compares the different magnet design and technology used and presents the magnet parameters in a consistent fashion.
A systematic approach, with the help of electric field theory, develops a vivid understanding of the behavior of dielectrics under various field conditions. The electric field, produced due to potential on a body, stresses the dielectric (electric insulation) with “electric stress.” The parameter that determines the magnitude of electric stress on the dielectrics is known as the “electric field intensity.” This chapter classifies the field configurations, and then describes different analytical and numerical methods of field estimation. Methods of stress control and numerical optimization techniques of electric stress are also explained. The increase in electric stress on insulation beyond a certain magnitude may lead to partial or total breakdown of the insulating material. Such critical magnitudes of electric stress are briefly explained. The source of “electric field intensity” is the electric charge. The electric field theory is intimately associated with the properties of electric charges.
The present paper concerns the formulation and the evolution of the non symmetrical growth of an avascular cancerous cell colony in an analytical mathematical fashion. Although most of the existing research considers spherical tumours, here we work in the frame of a more general case of the prolate spheroidal geometry. The tumour lies inside a host spheroidal shell which provides vital nutrients, receives the debris of the dead cells and also transmittes to the tumour the pressure imposed by the surrounding on its exterior boundary. Under the aim of studying the evolution of the exterior tumour boundary, we focus on the exterior conditions under which such a geometrical structure can be sustained. To that purpose, the corresponding nutrient concentration, the inhibitor concentration and the pressure field are calculated analytically providing the necessary data for the evolution equation to be solvable. It turns out that an avascular tumour can exhibit a prolate spheroidal growth only if the external conditions for the nutrient supply and the transversally isotropic pressure field have a specific form, which is consistent with the tumour evolution. Additionally, our model exhibits a geometrical reduction to special cases and, mainly, to the spherical geometry in order to recover the existing results for the sphere.
The forward electroencephalography (EEG) problem is studied in the framework of a multilayered structure, which models the scalp, skull, cerebrospinal fluid, and brain. Both the exterior and all inner boundaries are perturbed spheres so that special localized defects in head‐brain imaging are considered in analytic fashion. Linear perturbation analysis is implemented, providing exact expression for the first significant term of the forward EEG solution of the perturbed problem. Comparison with the solution of the corresponding unperturbed spherical problem is included, together with numerical demonstration of the produced errors in special deformation cases. The results suggest that significant errors are caused when large inaccuracies in the head‐brain structure in the vicinity of EEG source are not taken into account. Moreover, the suggested procedure provides a mathematical tool for evaluating quantitatively the impact of special deformations in the head representation on the EEG forward analytic solution.
Lamé introduced a triaxial ellipsoidal system, and via some ingenious arguments, he managed to spectrally decompose the Laplace operator and define ellipsoidal harmonic functions. Since then, many authors modified the Lamé system and proposed some related ellipsoidal coordinate variables. Perhaps the most important of them is the system introduced by Jacobi, in connection with the geodesic curves on an ellipsoid. All other attempts to introduce ellipsoidal coordinates are basically modifications of the Jacobi or the Lamé system. However, an important question in the theory of ellipsoidal harmonics is the way in which this theory reduces to the spheroidal and spherical systems. This is by no means a straightforward procedure, since the relative limiting cases lead to generic underdetermined forms. The basic difficulty is due to the different dimensionality of the singularity regions corresponding to each system. Note that this region has zero dimensions in the case of the sphere, one dimension in the case of the prolate spheroid, and two dimensions in the case of the oblate spheroid and the ellipsoid. The present work provides a systematic way to obtain these reductions and to establish a correspondence between ellipsoidal, spheroidal, and spherical harmonics. We have also demonstrated how our approach applies to the proposed modified Jacobi ellipsoidal system when we connect the Lamé with the Jacobi system through certain non‐degenerable relations. The importance of these geometrical degeneracies lies in the fact that the solution of any boundary value problem in ellipsoidal geometry provides immediately the solutions in all special geometries of prolate and oblate spheroids, discs, needles, or spheres.
We present unbiased, finite-variance estimators of energy derivatives for real-space diffusion Monte Carlo calculations within the fixed-node approximation. The derivative dλE is fully consistent with the dependence E(λ) of the energy computed with the same time step. We address the issue of the divergent variance of derivatives related to variations of the nodes of the wave function both by using a regularization for wave function parameter gradients recently proposed in variational Monte Carlo and by introducing a regularization based on a coordinate transformation. The essence of the divergent variance problem is distilled into a particle-in-a-box toy model, where we demonstrate the algorithm.
The boundary-value problem for the perturbation of an electric potential by a homogeneous anisotropic dielectric sphere in vacuum was formulated. The total potential in the exterior region was expanded in series of radial polynomials and tesseral harmonics, as is standard for the Laplace equation. A bijective transformation of space was carried out to formulate a series representation of the potential in the interior region. Boundary conditions on the spherical surface were enforced to derive a transition matrix that relates the expansion coefficients of the perturbation potential in the exterior region to those of the source potential. Far from the sphere, the perturbation potential decays as the inverse of the distance squared from the center of the sphere, as confirmed numerically when the source potential is due to either a point charge or a point dipole.
The study of the response of divergence-free electric fields near corners and edges, resembling singularities that accumulate charges, is significant in modern engineering technology. A sharp point can mathematically be modelled with respect to the tip of the one sheet of a double cone. Here, we investigate the behaviour of the generated harmonic potential function close to the apex of a single-sheeted two-hyperboloid with elliptic cross-section, whose asymptote is the corresponding elliptic double cone with one sheet present. Hence, the electrostatic potential problem, involving a single sheet of a two-hyperboloid, is developed using the theory of ellipsoidal-hyperboloidal harmonics, wherein the particular consideration enforces as solution in terms of generalised Lamé functions of non-integer order. A numerical method to determine these functions is outlined and tested. We demonstrate our technique to the solution of a classical boundary value problem in electrostatics, referring to a metallic and charged single-sheeted elliptic two-hyperboloid and its double-cone limit. Semi-analytical expressions for the related fields are derived, all cases being accompanied by the necessary numerical implementation.
We study the planar two-dimensional relative induced-charge electrophoretic (ICEP) motion of a pair of identical, ideally polarizable circular cylindrical microparticles carrying no net charge and freely suspended in an unbounded electrolyte solution under a uniform steady (DC) external electric field acting in an arbitrary direction relative to the instantaneous orientation of their line-of-centers (LOC). Within the framework of the thin electric-double-layer (EDL) limit and sufficiently weak fields description of particle paths is obtained via integration of the quasisteady kinematic equations of motion based on the instantaneous geometric configuration. Owing to the inherent nonlinearity of the ICEP mechanism, interaction of the effects of external-field components parallel and perpendicular, respectively, to the LOC results in its rotation which, in turn, determines that particles undergo transient pairing eventually moving apart in the general direction perpendicular to the external field. Dielectrophoresis is demonstrated to only have a secondary effect on the resulting motion.
We introduce the Tapered-Impedance Half Impulse Radiating Antenna (TI-HIRA). This innovative version of the HIRA allows the direct connection of COTS, 50 Ω generators to the antenna, without requiring external transformers and without using the large, blocking, plates that would be required in the traditional constant-impedance case. The idea relies on the use of feeders having a characteristic impedance that increases progressively, as a function of the radial distance between the focal point and the parabolic reflector. The additional effort required for manufacturing the new profile is offset by the reduced number of electrical and mechanical parts required for operating the radiating system. Various impedance profiles for the feeders are presented and compared. The design is discussed and validated, both analytically and by full-wave simulations demonstrating that the performances of the original HIRA remains.
The analytical modelling of heat and mass transfer phenomena relies on the analytical solutions to partial differential equations, which are used to describe the conservation of mass, chemical species, momentum and energy and their transfer mechanisms, as it will be shown in Part II of this book. Analytical methods for this kind of problems are widely used (see [1]) and among the several available techniques to solve Partial Differential Equations (PDE), separation of variable is generally the most valuable one since it may yield solutions in a form that is easily implementable for routine calculations. Separability of a PDE depends on the chosen coordinate system and this chapter is devoted to analyse conditions and methods for PDE separation.
Orthogonal curvilinear coordinates occupy a special place among general coordinate systems, due to their special properties. There exists a number of such coordinate systems where the Laplace or Helmholtz equations may be separable, thus yielding a powerful tool to solve them. Operations like gradients, divergence, Laplacian take on much simpler forms in orthogonal coordinates. In this chapter the summation convention will not be used.
In Chap. 10 we have seen that when an evaporating drop has the shape of a sphere, a spheroid or an ellipsoid, and the boundary conditions are uniform over the drop surface, the whole problem simplifies when proper coordinate systems are used and one-dimensional solutions of the conservation equations can be found. When the drop assumes different shapes, or the boundary conditions are not uniform, two- or three-dimensional solutions appear, even using proper coordinate systems. In this chapter we will explore some cases of practical interest when 2-D or even 3-D solutions can be found analytically.
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