R. F. Harrington, Field computation by moment methods, Krieger Publishing Co., Malabar, FL, 1982. 24  Home/Search   Document Not in Database   Summary   Related Articles   Check

....half length correspondingly) If the charge density is given, then equation (1) determines the corresponding potential u. Conversely, if the potential u is known, then the corresponding # may be found from an integral equation (1) The formalism known as the moment method described below is from [5]. A similar technique called the charge density method [6] is commonly used in electron and ion optics. 0 7803 5573 3 99 10.00 1999 IEEE. 2781 3 SLICE FORMALISM The original charge density method is too slow to be used repeatedly during step by step PIC simulation. We have developed a ....

R.E.Harrington, "Field Computation by Moment Methods", Macmillan, New York (1968).

....the reduced computational complexity. We call this the multilevel fast multipole algorithm (MLFMA) 2. Description of the Algorithm A boundary integral equation for E z incident wave is GammaOE inc (r) Gamma Z C dl g (r; r ) J (r ) r 2 C: 1) It can be discretized so that [9] GammaOE inc (r j ) Gamma 4l g (r j ; r i ) J (r i ) g ji b i ; j = 1; 2; Delta Delta Delta ; N (2) where in the two dimensional case, g ji = ae H 0 (k jr j Gamma r i j) i 6= j; 1 2i ln(0:163805k Deltal) i = j; and b i is proportional to the unknown current. The ....

R. F. Harrington, Field Computation by Moment Methods, reprint ed., Krieger Pub. Co., Malabar, FL, 1983.

....x lim = LuL lim lim = lim v , yv l , m lim yv l , k l k Bases and can be the same or they can be different. Depending on and , 2. 34) can result in different well known discretization schemes, such as Method of Moments [2], Galerkin method [2] Finite Differences [3] and Finite Elements [4] For example, if and are interpolating polynomials on an interval and order of polynomial is 2 more than the order of polynomial, the discretization scheme becomes Finite Differences. For arbitrary and the method is usually ....

.... lim lim = lim v , yv l , m lim yv l , k l k Bases and can be the same or they can be different. Depending on and , 2. 34) can result in different well known discretization schemes, such as Method of Moments [2] Galerkin method [2], Finite Differences [3] and Finite Elements [4] For example, if and are interpolating polynomials on an interval and order of polynomial is 2 more than the order of polynomial, the discretization scheme becomes Finite Differences. For arbitrary and the method is usually referred to as Method of ....

[Article contains additional citation context not shown here]

R. F. Harrington, Field Computation by Moment Methods, New York: MacMillan, 1968.

....[3] and method of line [8] algorithms have been performed. We have developed a new EM TABLE I PERFORMANCE VARIATION SUMMARY OF VARIOUS BOND WIRE INTERCONNECT TEST STRUCTURES model for wire bond and simulated using a commercially available full wave EM simulator by method of moment (MoM) 13] [14]. This model is implemented for two different wire bond lengths; 15 and 25 mils. The approximated geometry of a tight loop wire bond is illustrated in Fig. 6(a) In this figure, and indicate the thickness of the microstrip substrate onto which the wire is bonded and the wire height above the ....

R. F. Harrington, Field Computation by Moment Methods, 1993.

....mod elisation filaire moyennant des hypoth eses sur le courant, et a partir d approximations guid ees par la physique. A partir de l equation etablie par Pocklington, de nombreux travaux ont et e men es pour proposer des m ethodes de r esolution soit de l equation elle meme, soit de variantes [18, 20, 28]. La premi ere analyse math ematique de l equation de Pocklington est due a D. S. Jones, qui a par ailleurs largement contribu e a l etude des antennes filaires, 22, 21, 23] Depuis, d autres math ematiciens se sont pench es sur les mod eles filaires propos es par les physiciens pour les ....

.... de premi ere esp ece dont le noyau est analytique et l inversion de telles equations est tr es d elicate car elle est par nature mal pos ee, 10] N eanmoins, cette formulation a l avantage de s adapter facilement a des g eom etries d antennes complexes et est a la base de nombreuses m ethodes [28, 20, 8]. Elle s av ere tr es efficace tant que la longueur des segments discr etisant l antenne est au moins de l ordre de 4 ou 5 fois le rayon de l antenne. Les instabilit es apparaissent lorsqu on raffine le maillage pour obtenir une meilleure pr ecision. Signalons que Mazari a pr esent e des ....

R. F. Harrington. Field computation by moment methods. Macmillan, New York, 1968. 26

....more reliant on accurate computer simulation tools. Signal integrity problems, like ground plane noise, are particular hard to simulate because so much of the problem geometry must be included to achieve accurate results. Multipole and precorrected FFT accelerated Method of Moments techniques [1, 3, 2, 4, 8] are one of the few techniques that are fast enough to analyze signal integrity problems, so optimizing these techniques seem worthwhile even if the resulting optimizations are somewhat incremental. In this paper we describe two optimizations to accelerated method of moments simulation, the first ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....equations (PDEs) lead to very di#erent computational approaches. With computational methods derived from IEs, a three dimensional boundary value problem reduces to a two dimensional problem over the boundary of the domain of interest (e.g. boundary element methods or the method of moments [59, 98]) However, even with a significant reduction in the number of unknowns, the computational cost of generating the full system matrix and di#culties in solving the linear equations often makes this approach more costly than comparable PDE methods [77] It is also di#cult to formulate the ....

....absorb outgoing waves; for a numerical study of implementations of various artificial boundary conditions, see [104] It is not necessary to assign boundary conditions on the artificial boundary with certain formulations. One alternative is to use boundary integrals (e.g. boundary element methods [59]) to connect the problem within the finite computational domain to the unbounded exterior region. The use of infinite elements also allows a computational framework that incorporates the behav30 ior of the fields at infinity into the solution [47] Both of these techniques have some advantages but ....

R.F. Harrington. Field Computation by Moment Methods. MacMillan, 1968.

....are constructed, respectively: o g.a g 376 5o o 429 fpg . q r L .is o r k r 512 (9) k o g . 38370 1 k o 4272 f q r L . 5u o r r 5045 (10) s and s are the coefficients to be determined by using the generalized Galerkin s method [9], with the power function set 5 ] bW 140 2 ) j (from Eqn. 6) being test functions. For each , we substitute an appropriate number of test functions from 5 b to 4 K into Eqn. 9, then a set of V b b linear equations are obtained, with ] s o.a s o 2W ....

.... 7 399 F2= 10 3 . 3 , s . s .32 s 2 . s 202 , 7 530 F2= 0 (21) and 42 d . u . 22) 2 . are coefficients to be determined by using fitting functions. Using the generalized Galerkin s method [9], we choose 424 ] bW 280 2 as fitting functions . in the first line of Eqn. 21, then it follows: 0. s . 2 # ( 23) 2 7 s . 2 s . 2 2 # ( 24) which results in . A . 2 b A . Doing the same operations to the second line of Eqn. ....

R. F. Harrington, Field Computation by Moment Methods. Macmillan, NY, 1962.

....the RCS of a target. Our goal is to highlight the complicated relationship that exists between target state (e.g. position and orientation) and RCS. The method of moments (MoM) is a numeric technique which has found widespread use in the solution of scattering problems involving complex targets [18]. The method of moments applies to general linear operator equations, such as 435 (3) where is a linear operator, is an unknown response, and is a known excitation. The unknown response is expanded as the sum of basis functions 7698 : 1 . To solve for the unknowns ....

R. F. Harrington, Field Computation by Moment Methods. New York, NY: MacMillan, 1968.

....the RCS of a target. Our goal is to highlight the complicated relationship that exists between target state (e.g. position and orientation) and RCS. The method of moments (MoM) is a numeric technique that has found widespread use in the solution of scattering problems involving complex targets [23]. The method of moments applies to general linear operator equations, such as (2.3) where is a linear operator, is an unknown response, and is a known excitation. The unknown response is expanded as a sum of basis functions, 8 solve for the unknowns 8 ....

R. F. Harrington, Field Computation by Moment Methods. New York, NY: MacMillan, 1968.

....(see Figure 2b) That is, the current is constant on each panel but discontinuous between panels. Because of the Dirichlet condition, the potentialoverapanelmustalsobeconstant thepanels form an equipotential region. Based on this discretization, the collocation method or the Galerkin method [18] can be used to obtain a system of equations with U as the vector of N panel potentials, X as the vector of (unknown) panel currents and G as the elastance matrix describing the potential at panel i due to a unit current in panel j: U = GX (1) The BEM then continues by defining an incidence ....

R.F. Harrington, Field Computation by Moment Methods. New York: The Macmillan Company, 1968.

....= 0 for non contact surfaces, we have 4v0(x) fs (x )da j We now break up the conductor surfaces into N small tiles or panels. It is then assumed that on each panel l, the potential , its normal derivative , and On l the external current density J, are all constants. A collocation scheme [9], in which (3) is enforced at the centtold in each of N panels, is used to generate a system of N equations. The result is a dense N x N linear system pj, 4) where = a e is the dielectric relaxation time. n, jest are N vectors whose elements represent the potential, its normal ....

R. F. Harrington. Field Computation by Moment Methods. Macmillan, New York, 1968.

.... work on techniques for rapid electrostatic analysis for capacitance extraction have been based on random walk methods [3] partitioning heuristics combined with techniques from matrix extension theory [4] 5] finite difference [6] 7] or finiteelement methods [8] 9] or method of moments [10] techniques. Algorithms using method of moments [10] or weighted residuals [11] 12] based discretizations of integral equation formulations, also known as boundary element methods [13] are commonly used to perform electrostatic analyzes, but such approaches generate dense matrix problems which ....

....for capacitance extraction have been based on random walk methods [3] partitioning heuristics combined with techniques from matrix extension theory [4] 5] finite difference [6] 7] or finiteelement methods [8] 9] or method of moments [10] techniques. Algorithms using method of moments [10] or weighted residuals [11] 12] based discretizations of integral equation formulations, also known as boundary element methods [13] are commonly used to perform electrostatic analyzes, but such approaches generate dense matrix problems which are computationally expensive to solve, and this ....

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R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....that for . To numerically solve (6) for at noncontact surfaces and for at contact surfaces, the conductor surfaces are broken into small tiles, or panels. It is then assumed that on each panel , there is a constant potential and a constant external supply current density . A collocation scheme [8], in which (6) is enforced at the centroid of each panel, is used to generate a system of equations. The result is a dense linear system (7) where , represent the discretized panel potentials and external supply current densities. The elements of the dense matrices and are (8) 9) where is the ....

R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968.

....inside filament is a unit vector along the length of the filament and is the weighting function which has a value of zero outside filament , and inside, where is the cross sectional area. By defining the inner product of two vector functions, and ,by (7) and following the method of moments [8], a system of equations can be generated by taking the inner product of each of the weighting functions with the vector integral equation, 2) In matrix form, 2) becomes (8) where C is the vector of filament currents, is the diagonal matrix of filament DC resistances (9) is the dense, ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....show that the accuracy of this new approach is nearly independent of the permittivity ratios and superior to the ECF for realistic interconnect structures. Index Terms Boundary element methods, capacitance, dielectric materials. I. INTRODUCTION T HE boundary element, or method of moments [1], technique for computing capacitances in complicated threedimensional (3 D) geometries has rebounded in popularity in the last decade. This is due, in part, to the development of very fast solution algorithms based on sparsifying the associated dense matrices [2] 5] The combination of the ....

R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968.

....interaction between the interconnect and attached driving circuitry. Since the introduction of PEEC methods [9] the dominate approach to accurate coupled interconnect circuitry analysis was modeling the interconnect with a densely coupled equivalent circuit derived using the method of moments [4]. PEEC generated equivalent circuits were then combined with the nonlinear drivers and receivers and used as input to a circuit simulation program. And since circuit simulators use Gaussian elimination, the computational cost of that en tire approach grews as n 3, where n was the number of ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

.... for performing 3 D capacitance extraction have focussed on three techniques, the floating random walk method [10] improvements to the finite difference and finite element methods [3, 2] and the so called fast methods based on acceleration of the method of moments or boundary element approach [8, 9, 7]. In this paper we present a new multiscale, or wavelet like, approach to accelerating the boundary element method, and demonstrate the method on several examples. We show that this method has two important features: it can accurately represent the entries of the dense boundary element matrix ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....vector along the length of the filament and wi(r) is the weighting function which has a value of zero outside filament i, and 1 ai inside, where ai is the cross sectional area. By defining the inner product of two vector functions, a and b, by b) f b (7) and following ;he me;hod of momen;s [5], a sys;em of b equa;ions can be genera;ed by ;aking ;he inner produc; of each of ;he weigh;ing func;ions wi;h ;he vec;or in;egral equa;ion, 4) This gives ] 1 ai a where li is the length of filament i, ai is the cross section, qbA and qbB are the potentials on the filament end faces, and V and ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....branch currents 65 where I i is the current in mesh i, l i is a unit vector along the length of the mesh and w i (r)istheweighting function whichhasavalue of zero outside mesh i, and 1=a i inside, where a i is the cross sectional area of the filaments in mesh i. Following the method of moments [52], a system of m equations can be generated by taking the inner product of each of the weighting functions with the vector integral equation (4.37) Then multiplying both sides with a i gives, r Phi(r) dL = Vm i (4.40) where oe is the ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....design of high performance integrated circuits and integrated circuit packaging, there are many cases where accurate estimates of the capacitances of complicated three dimensional structures are important for determining final circuit speeds and func tionality. Algorithms using method of moments [1] based discretizations of integral equation formulations are commonly used to compute these capacitances, but such approaches generate dense matrix problems which are computationally expensive to solve, and this limits the complexity of problems which can be analyzed. In [2] a rapid method, based ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968. [5]

....nalog Devices. sumed that on each panel k, a charge, q, is uniformly distributed. Then for each panel, an equation is writ ten that relates the potential at the center of that k panel, denoted p, to the sum of the contributions to that potential from the n charge distributions on all n panels[3]. The result is a dense linear system, 2) where P R Xn is the matrix of potential coefficients; q, are the vectors of panel charges and given panel potentials respectively, and lfr 1 Pc = a ; I1 da , 3) where x is the center of the k ta panel and at is the area of the Ita panel. The ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....the basis for the boundary element method (BEM) used in [19] 7] 24] To numerically solve (1) the domain discretized into disjoint, rectangular panels ( An example of panel discretization for a three contact layout is given in Figure 2. In the Galerkin scheme [10], the current density on each panel ( is assumed to be uniform. Then linear equations are constructed by evaluating the average potential i over each panel result is a discretized version of (1) 132 Permission to make digital or hard copies of all or part of this work for ....

R. F. Harrington. Field Computation by Moment Methods. Macmillan, New York, 1968.

....By convolving with the magnetic field of a small current element, the magnetic field can be easily found once the current distribution on the receiver is known. A Galerkin method of moments was used to solve for the current distribution on the loopless receiver (i.e. a dipole antenna) [6]. This method also yields the input resistance of the antenna. Piecewise sinusoids were used as the testing basis function, as described previously [7] To model the effect of insulation, the volume equivalence theorem was used introducing polarization currents to account for the effect of ....

P F. Harrington, Field Computation by Moment Methods. New York: IEEE Press, 1993.

....cell is more than a wavelength long) 12 3 (a)r= 1. Sd, k= 0 rid o (c)r 6d, kd i 10 s (b)r = 6d, k = 0 r d o (d) r = 6d, kd = 5 10 4 2 4 6 rid 8 o 4 rid o Figure 3: Error in grid approximation of potential of 100 charges of random strength ( G [0, 1] located at random positions inside a cube of side length 2d centered at the origin. Collocation sphere radius is rc = 1.5d (left figure) rc = 6d (right figure) Solid line: p = 3, order 7 quadrature rule. Dash line: p = 4, order ll quadrature rule. Dash dotted line: p = 5, order 14 quadrature ....

R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

....to less accurate system performance predictions. This problem can be avoided if information about the mutual coupling is somehow included in the system analysis. The mutual coupling effects information may be extracted from an antenna array if it could be represented as an N port network [2] 10] [8] [13] 16] Thus, the circuit parameters associated with such a representation could be used to form a mutual impedance matrix, Z] Therefore, by using [Z] in the derivation of the system capacity perfor mance we are able to include the effects of mutual coupling and thus provide more accurate ....

....the induced EMF method is shown in Figure 4. A comparison of beampatterns using mutual coupling information generated by various methods is per formed in Section 5. 2. 2 Method of moments The method of moments is another technique of obtain ing the mutual impedance matrix [Z] as described in [8] [13] Using basic equations from electromagnetics and assuming that the current flows in one direction, the current and charge densities are approximated by filaments of current and charge on the wire axis, and viewing the antenna array as a system of N small increments which constitute an N port ....

R. F. Harrington. Field Computation by Moment Method. MacMillan Co., 1968.

....the traditional structure becomes impractical. Unfortunately, exact synthesis of tapped line filters is not straightforward. Available techniques are not directly applicable to microstrip configurations. Final validation using rigorous models is considered necessary. The method of moments (MoM) [6], 7] and the finiteelement method (FEM) 8] have been used successfully Manuscript received January 28, 1997; revised January 28, 1997. This work was supported in part by Optimization Systems Associates, Inc. in part by the Natural Sciences and Engineering Research Council of Canada under Grant ....

R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968.

....or stripline. III. Structure Geometry and Modeling The EM modeling of the SSS structure in Fig. 4 begins with the development of the Green s functions following the approach in [21] and using the MPIE and MoM techniques presented in [22] and [23] First, using the equivalence principle [24], the center conductor at z = 0 in Fig. 4 is removed and replaced by an equivalent electric surface current density, J s . Then the slot, at the z = h plane, is removed and replaced by perfect electric conductors and the equivalent magnetic surface current density flowing at z = h is M int s ....

R. F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968.

....and dielectrics are divided into a 3 D grid of filaments. Fig. 3 shows an example of the 3D volume discretization of a dielectric parallelepiped. Each filament carries a constant current. Other basis functions choices are possible for the interior of the conductors [18] A Galerkin method [19] can be used to transform the Mixed Potentials Integral Equations (1) 4) into an algebraic form R sL 1 s 0 0 0 Pol 0 0 0 0 0 0 0 0 P I c I d q c q d V c V d f c f d (11) where I c , I d , q c and q d are vectors of basis ....

....V c , V d , f C and f d are the vectors generated by inner products of the basis functions with the potential gradient and with the potential itself. The resistance matrix R, the inductance matrix L and the coefficients of potential matrix P are all derived directly from the Galerkin condition [19], R R c 0 0 0 (12) L L cc L cd L dc L dd (13) P P cc P cd P dc P dd (14) L and P are frequency dependent when using a full wave kernel as in (5) and frequency independent when using a quasi static kernel as in (6) Matrix Pol in (11) is a diagonal matrix carrying ....

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R. F. Harrington. Field Computation by Moment Methods. MacMillan, 1968.

....(nonmagnetic medium) and for TM and in plane incidence is assumed [16] Also, in the above equation (7) where is the propagation constant in medium . The above formulation gives two integral equations in two unknowns ( and or and ) on the surface profile. Applying a point matching MOM technique [17] results in a matrix equation in terms of the unknown pulse basis function expansion coefficients of these fields which can be written as (8) where is a vector containing the expansion coefficients, and contains the incident field evaluated at points on the surface profile. Elements of the ....

R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968.

....at arbitrary observation points. A numerical technique which is particularly well suited to the analysis of HF and VHF wire antennas is solution of the electric field integral equation by the method of moments [32] This technique was 28 pioneered in the mid 1960 s by Richmond [33] Harrington [34], and others. In this technique, wires and plates are broken down into straight segments and flat patches, each of which are small compared to wavelength (so that an assumption of a constant value of current across the segment patch is valid) Once the geometry of the structure has been defined, a ....

Harrington, R. F.: Field Computation by Moment Methods. MacMillian Co., New York, 1968.

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R. F. Harrington, Field computation by moment methods, Krieger Publishing Co., Malabar, FL, 1982. 24

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R. F. Harrington, Field computation by moment methods, Krieger Publishing Co., Malabar, FL, 1982.

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HARRINGTON, R.F.: "Field computation by moment methods", Macmillan, New York, 1968.

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R. Harrington, Field computation by moment methods. New York: Macmillan, 1968.

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R.F. Harrington, Field Computation by Moment Methods, Malabar, FL, Krieger, 1968

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R. F. Harrington. Field Computation by Moment Methods. IEEE Press, Piscataway, NJ, 1993.

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R.F. Harrington, Field computation by moment methods, Macmillan, New York, 1968.

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R.F. Harrington, "Field Computation by Moment Methods", IEEE Press Series on EM Waves, 1993.

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Roger F. Harrington, Field Computation by Moment Methods, IEEE Press, 1993.

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R. F. Harrington, Field Computation by Moment Methods, Krieger Publishing Company, Inc., Malabar, FL, 1982.

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Harrington, R. F., Field computation by moment method, Krieger, Mar-

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R. F. Harrington, Field Computations by Moment Methods, Macmillan, 1968.

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R. F. Harrington, Field Computation by Moment Methods, 1968, Robert E. Krieger Publishing Company, Inc., p. 229.

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R. F. Harrington. Field Computations by Moment Methods. The Macmillan Co., New York, 1968.

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R.F. Harrington, Field Computation by Moment Methods, Macmillan, New York, 1968.

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R. F. Harrington, Field Computation by Moment Methods. New York: Macmillan, 1968.

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R. F. HARRINGTON, Field Computation by Moment Methods, Macmillan, New York, 1968.

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R. F. Harrington, Field Computation by Moment Methods. New York: MacMillan, 1968.

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R. F. Harrington, Field Computation by Moment Methods (Krieger, Malarbar, Fla., 1982).

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