On the Electro-magnetic Wave-Surface

1897 ◽  
Vol 21 ◽  
pp. 165-166
Author(s):  
Tait
Keyword(s):  
Linear Function ◽  
Plane Waves ◽  
Normal Velocity ◽  
Wave Surface ◽  
Magnetic Wave ◽  
Electro Magnetic ◽  
The Moment ◽  
Image Position

We may write the electro-magnetic equations of Clerk-Maxwell asFor plane waves, running with normal velocity vα= −μ−1, we havewhence at once[For the moment, we assume that φ and ψ are self-conjugate, so that a linear function of them is also self-conjugate. And we employ the method sketched in Tait's Quaternions, §§ 438–9.]

scholarly journals AVERAGING OF AN INCREASING NUMBER OF MOMENT CONDITION ESTIMATORS

2014 ◽  
Vol 32 (1) ◽  
pp. 30-70 ◽  
Author(s):  
Xiaohong Chen ◽  
David T. Jacho-Chávez ◽  
Oliver Linton
Keyword(s):  
Moment Condition ◽  
Moment Conditions ◽  
Parameter Heterogeneity ◽  
Class Of Estimators ◽  
Optimal Weighting ◽  
The Mean ◽  
The Moment ◽  
Image Position ◽  
Nonlinear Estimators ◽  
Available Information

We establish the consistency and asymptotic normality for a class of estimators that are linear combinations of a set of$\sqrt n$-consistent nonlinear estimators whose cardinality increases with sample size. The method can be compared with the usual approaches of combining the moment conditions (GMM) and combining the instruments (IV), and achieves similar objectives of aggregating the available information. One advantage of aggregating the estimators rather than the moment conditions is that it yields robustness to certain types of parameter heterogeneity in the sense that it delivers consistent estimates of the mean effect in that case. We discuss the question of optimal weighting of the estimators.

scholarly journals Applied Optimization and Approximation

2014 ◽  
Vol 3 (4) ◽  
pp. 37-48
Author(s):  
Octav Olteanu
Keyword(s):  
Optimal Control ◽  
Cost Function ◽  
Linear Function ◽  
Polynomial Approximation ◽  
Moment Problem ◽  
Quadratic Forms ◽  
Two Dimensional ◽  
Several Variables ◽  
Applied Optimization ◽  
The Moment

The present work deals with optimization in kinematics, generalizing previous results of the author. A second theme is maximizing the constrained gain linear function and minimizing the constrained cost function. Elementary notions of optimal control are considered as well. Finally, polynomial approximation results on unbounded subsets in several variables are applied to the moment problem. The existence of the solution of a two dimensional moment problem is characterized in terms of quadratic forms.

scholarly journals The Product of Two Legendre Polynomials

1953 ◽  
Vol 1 (3) ◽  
pp. 121-125 ◽  
Author(s):  
John Dougall
Keyword(s):  
Linear Function ◽  
Spherical Harmonics ◽  
Legendre Polynomials ◽  
Image Position ◽  
Two Sides

1. It is known that any polynomial in μ. can be expanded as a linear function of Legendre polynomials [1]. In particular, we haveThe earlier coefficients, say A0, A2, A4 may easily be found by equating the coefficients of μp+q, μp+q-2, μp+q-4 on the two sides of (1). The general coefficient A2k might then be surmised, and the value verified by induction. This may have been the method followed by Ferrers, who stated the result as an exercise in his Spherical Harmonics (1877). A proof was published by J. C. Adams [2]. The proof now to be given follows different lines from his.

Three-phase boundary motions under constant velocities. I: The vanishing surface tension limit

1996 ◽  
Vol 126 (4) ◽  
pp. 837-865 ◽  
Author(s):  
Fernando Reitich ◽  
H. Mete Soner
Keyword(s):  
Surface Tension ◽  
Formal Analysis ◽  
Local Equilibrium ◽  
Convergence Result ◽  
Normal Velocity ◽  
Infinitely Many Solutions ◽  
Weak Formulation ◽  
Material Interfaces ◽  
Internal Surfaces ◽  
Image Position

In this paper, we deal with the dynamics of material interfaces such as solid–liquid, grain or antiphase boundaries. We concentrate on the situation in which these internal surfaces separate three regions in the material with different physical attributes (e.g. grain boundaries in a polycrystal). The basic two-dimensional model proposes that the motion of an interface Гij between regions i and j (i, j = 1, 2, 3, i ≠ j) is governed by the equationHere Vij, kij, μij and fij denote, respectively, the normal velocity, the curvature, the mobility and the surface tension of the interface and the numbers Fij stand for the (constant) difference in bulk energies. At the point where the three phases coexist, local equilibrium requires thatIn case the material constants fij are small, and ε ≪ 1, previous analyses based on the parabolic nature of the equations (0.1) do not provide good qualitative information on the behaviour of solutions. In this case, it is more appropriate to consider the singular case with fij = 0. It turns out that this problem, (0.1) with fij = 0, admits infinitely many solutions. Here, we present results that strongly suggest that, in all cases, a unique solution—‘the vanishing surface tension (VST) solution’—is selected by letting ε→0. Indeed, a formal analysis of this limiting process motivates us to introduce the concept of weak viscosity solution for the problem with ε = 0. As we show, this weak solution is unique and is therefore expected to coincide with the VST solution. To support this statement, we present a perturbation analysis and a construction of self-similar solutions; a rigorous convergence result is established in the case of symmetric configurations. Finally, we use the weak formulation to write down a catalogue of solutions showing that, in several cases of physical relevance, the VST solution differs from results proposed previously.

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

On the linear exponential family

1968 ◽  
Vol 64 (2) ◽  
pp. 481-483 ◽  
Author(s):  
J. K. Wani
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Scalar Product ◽  
Exponential Family ◽  
Characterization Theorem ◽  
Random Variable ◽  
Moment Generating Function ◽  
The Moment ◽  
Image Position

In this paper we give a characterization theorem for a subclass of the exponential family whose probability density function is given bywhere a(x) ≥ 0, f(ω) = ∫a(x) exp (ωx) dx and ωx is to be interpreted as a scalar product. The random variable X may be an s-vector. In that case ω will also be an s-vector. For obvious reasons we will call (1) as the linear exponential family. It is easy to verify that the moment generating function (m.g.f.) of (1) is given by

The triode oscillation generator and amplifier: limitations on sinoidal performance

1930 ◽  
Vol 26 (4) ◽  
pp. 507-527
Author(s):  
L. B. Turner ◽  
L. A. Meacham
Keyword(s):  
Linear Function ◽  
Fundamental Frequency ◽  
Alternating Current ◽  
Anode Potential ◽  
Anode Current ◽  
Simple Analysis ◽  
Oscillation Generator ◽  
Image Position ◽  
Characteristic Relationship ◽  
Grid Potential

The characteristic relationship subsisting between anode current ia, anode potential ea, and grid potential eg in a well-evacuated thermionic triode, viz.leads to very simple analysis of the circuit conditions when μ, is a constant and φ is a linear function. In all the usual applications of triodes, μ is very nearly a constant; in some applications, e.g. in good acoustic amplifiers, over the working range φ is very nearly linear; in others, e.g. in rectifiers, non-linearity is essential to the performance. In oscillators, and in power amplifiers used with them, both conditions are met. Where a sinoidal oscillation is desired, i.e. an alternating current as devoid of harmonics of the fundamental frequency as can be contrived, the working range must be sensibly linear. A feature of such a régime is that the major portion of the power supplied to the triode must be dissipated in heating the anode. Where higher efficiency is required, the cycle must be made to extend beyond the linear range, and harmonics are necessarily introduced.

A multi-objective topology optimisation for 2D electro-magnetic wave problems with the level set method and BEM

2016 ◽  
Vol 25 (1-2) ◽  
pp. 165-193 ◽  
Author(s):  
Hiroshi Isakari ◽  
Kenta Nakamoto ◽  
Tatsuya Kitabayashi ◽  
Toru Takahashi ◽  
Toshiro Matsumoto
Keyword(s):  
Level Set Method ◽  
Level Set ◽  
Topology Optimisation ◽  
Magnetic Wave ◽  
Multi Objective ◽  
Electro Magnetic ◽  
Wave Problems
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