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3d0530ba69122cd3894b2c8b2cf0f18818a5e5c7 | subsection | 37 | 38 | Conclusion | We have demonstrated how to achieve zero-cost program and proof reuse
between lists and vectors, which scales to the nested datatype
setting, through the use of identity coercions, which erase to
the identity function. Our technique works for datatypes like lists
and vectors, where vectors are the length-indexed versio... | {
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} | 1802.00787 | Zero-Cost Coercions for Program and Proof Reuse | [
"Larry Diehl",
"Aaron Stump"
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32d5d122539a48e89a9b18a637c5f382a0f7972e | abstract | 0 | 76 | Abstract | We deal with the electromagnetic waves propagation in the harmonic regime. We
derive the Foldy-Lax approximation of the scattered fields generated by a
cluster of small conductive inhomogeneities arbitrarily distributed in a
bounded domain $\Omega$ of $\mathbb{R}^3$.
This approximation is valid under a sufficient but ... | {
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Small Conductive Bodies of Arbitrary Shapes | [
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893635b09c10d176976f9244478b5bc6a20672c5 | abstract | 1 | 76 | Abstract | As this linear algebraic
system comes from the boundary conditions, such a reduction is not
straightforward. | {
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66d23f4e51b859f50f9998a35926a9da9fbd4480 | subsection | 2 | 76 | Introduction and main results | Let (B_i)_{i=1}^m be m open, bounded and simply connected sets containing the origin, with Lipschitz boundaries.
To these sets, we correspond the small bodies (D_i)_{i=1}^m which are defined as the translations and contractions of the m bodies ({B}_i)_{i=1}^m, that isD_i=\epsilon {B}_i+z_i , i=1,...,mwhere z_i, i=1, ..... | {
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795cff34bb8179d22f6674db142fcf63bdcf5182 | subsection | 3 | 76 | Introduction and main results | We set&\epsilon :=\max _{i\in \lbrace 1,...,m\rbrace }{\epsilon _i},
&\delta :=\min _{i\ne j\in \lbrace 1,...,m\rbrace } {\delta _{i,j}}.We suppose in addition that \cup _{i=1}^m\overline{D_i}\subset \Omega , where \Omega is a bounded Lipschitz domain such thatd(\partial \Omega ,\cup _{i=1}^m\overline{D_i})\ge \delta .... | {
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Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
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8443b0da8749eb44f9f4cea9d91e843beabac3ae | subsection | 4 | 76 | Introduction and main results | But when \Im k=0, i.e. in the absence of attenuation, we have the following
behavior (as spherical-waves) of the scattered electric fields far away from the sources D_i'sE^\text{sca}(x)=\frac{e^{ik\vert x\vert }}{\vert x\vert } \lbrace E^{\infty }(\tau ) +O(\vert x\vert ^{-1}) \rbrace , ~~~ \vert x\vert \longmapsto \in... | {
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} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
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5c961ce183d93a11de1ff2e2386feae852328ffb | subsection | 5 | 76 | Introduction and main results | For i=1, ..., m, we recall the single layer operator [S_{ii,_D}^k]: L^2(\partial D_i)\rightarrow H^1(\partial D_i), defined as[S_{ii,_D}^k](\psi )(x):= \int _{\partial D_i}\Phi _k(x,y)\psi (y)~ds(y),\,~ x\in \partial D_i,and the double layer operator [K^k_{ii,_D}]: L^2(\partial D_i)\rightarrow L^2(\partial D_i),[K^k_{i... | {
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a3c803072349980455e2362d7f21141dad0d420e | subsection | 6 | 76 | Introduction and main results | Further, we have the following scales:[\mathcal {P}_{\partial D_i}]=\epsilon ^3[\mathcal {P}_{\partial B_i}],~\text{and }~[\mathcal {T}_{\partial D_i}]=\epsilon ^3[\mathcal {T}_{\partial B_i}].Indeed, Recall that \int _{\partial D_i}[-\frac{1}{2}I+(K^0_{ii,_D})^*]^{-1}(\nu )(y)~ z_i^Tds(y)=\int _{\partial D_i}[-\frac{1... | {
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9c27212259f2cd8547a3e2f8d9d2f4396c8d7cd2 | subsection | 7 | 76 | Introduction and main results | We define\begin{aligned}\mu ^+=\max _{i\in \lbrace 1,...,m\rbrace }((\mu _i^{\mathcal {T}})^{+}, (\mu _i^{\mathcal {P}})^{+}),\\
\mu ^-=\min _{i\in \lbrace 1,...,m\rbrace }((\mu _i^{\mathcal {T}})^{-}, (\mu _i^{\mathcal {P}})^{-}).
\end{aligned}Hence for every vector \mathcal {C}, we get\begin{aligned}\mu ^- *{\mathcal... | {
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} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
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3fb34031df976740a24841398e1cc9b0c1f464a6 | subsection | 8 | 76 | Introduction and main results | \end{aligned}The dyadic Green's function is given by\Pi (x,y):=k^2\Phi _k(x,y)I+\nabla _x\nabla _x \Phi _k(x,y),\\
=k^2\Phi _k(x,y)I-\nabla _x\nabla _y \Phi _k(x,y).We introduce generic functions {{\epsilon }}(\delta ^{s},*{k}^{l}) and {{\epsilon }}_{k,\delta ,m} which express error functions as follows\begin{aligned}&... | {
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} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
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"Ali Bouzekri",
"Mourad Sini"
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40ebeb85b5d1252360b660e774cba8f4a9599468 | subsection | 9 | 76 | Introduction and main results | Now, we are ready to state the main result of this work.Theorem 1.1
Let (\mathcal {A}_i)_{i=1}^m and (\mathcal {B}_i)_{i=1}^m be the solutions of the following linear system\begin{aligned}\mathcal {A}_i&=-\bigl [\mathcal {P}_{\partial D_i} \bigr ]\sum _{(j\ne i)\ge 1}^m \left(\Pi _k(z_i,z_j){\mathcal {A}}_j-k^2\nabla ... | {
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37a74c62ed55aa2a8d107e814e051814728b3dbf | subsection | 10 | 76 | Introduction and main results | Since the pioneering works of Rayleigh till Foldy, the first and original goal of such approximations was to reduce the computation of the fields generated by a cluster of small bodies to inverting an algebraic system
(called the Foldy linear algebraic system), see for more information. With our approximations above, a... | {
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Small Conductive Bodies of Arbitrary Shapes | [
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"Mourad Sini"
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bfd282bc02186d86c170d7e5c72394105e911404 | subsection | 11 | 76 | Introduction and main results | The first key observation here is to derive it in the \mathnormal {L}^{2,Div\,}_t spaces instead of the usual \mathnormal {L}^2 spaces. As a second observation, to derive such estimates,
we used a particular decomposition of the densities, see Key-Decomposition or Theorem REF , which allows to obtain the needed qualita... | {
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"Ali Bouzekri",
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52c540d741f100dc92dad66fda1061c846193fb5 | subsection | 12 | 76 | Preliminaries | Let us recall few properties of the surface divergence which will be important in our later analysis, see (Section 4 in and Chapter 2 in ) for more details.
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Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
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c01bdd5bf6900e5067ffc629510b6a0a0539a5b3 | subsection | 13 | 76 | Preliminaries | \psi ~a\nu =\psi ~(a\nu )=0), the following identity holds, Div\,(\psi a)=\nabla _t\psi a+ \psi Div\,a, and
hence\int _{\partial D_i}\psi ~a~ds
=\int _{\partial D_i}(x-z_i)(\nabla \psi (x)a(x)+\psi (x) Div\,~a(x))~ds(x).Indeed, \int _{\partial D_i}\psi ~a~ds=\int _{\partial D_i}\nabla (x-z_i)~\bigl (\psi ~a\bigr )(x)~d... | {
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} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
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46b7c0cac73ae0f6e1b7dec9a6e0744c17b321de | subsection | 14 | 76 | Existence and uniqueness of the solution | The solution to the problem (REF ) under the boundary condition (REF ) and the radiating conditions (REF ) can be expressed
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0eb8a2be8747d262de93b558ccde9c7efd095d58 | subsection | 15 | 76 | Existence and uniqueness of the solution | Consequently, to solve the scattering problem we need to solve the integral equation[\frac{1}{2}I+M_{\partial D}^k](a)(x)=-\nu \times E^\text{inc},\; ~~ \mbox{ on } \cup ^m_{i=1} \partial D_ior,[\frac{1}{2}I+M_{{\partial D_i}}^k](a)(x_i)+[\sum _{(j\ne i)\ge 1}^m M_{ij,{_D}}^k](a)=
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1873e65c1ec620930ae84794ee531fbc343129e3 | subsection | 16 | 76 | Existence and uniqueness of the solution | Similarly, E^I=(E^\text{inc}_1,......,E^\text{inc}_m) with E^\text{inc}_i=E^\text{inc}/\partial D_i
and \mathcal {M}_D is the diagonal matrix operator given by\mathcal {M}_D:=\left\lbrace \begin{aligned}&M_{ij,_{D}}^k \,\,\text{if }~i=j\\
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] | 2,018 | en | Mathematics | [
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0.013206233270466328,
-0.02038535848259926,
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0.015021994709968567,
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-0.0... | |
ce4a7c438d6ef242751353d2c418747b7990057f | subsection | 17 | 76 | Existence and uniqueness of the solution | As \pm \frac{1}{2}I+\mathcal {M}_D is an isomorphism and \mathcal {M}_N is compact (since the kernel of each component is of class \mathcal {C}^{\infty }),
the operators\pm \frac{1}{2}I+\mathcal {M}_D+\mathcal {M}_N:
\prod _{i=1}^{m}\mathcal {E}(\partial D_i)\longrightarrow \prod _{i=1}^{m}\mathcal {E}(\partial D_i),wh... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-1-4614-4942-3",
"end": 2007,
"openalex_id": "https://openalex.org/W1523113578",
"raw": "D. Colton and R. Kress. Inverse acoustic and electromagnetic scatteringtheory, 3rd edn. Springer, New York., 2013.",
"source_ref_id": ... | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.06622467935085297,
0.009369113482534885,
-0.017258089035749435,
-0.012283609248697758,
-0.018310969695448875,
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0.02511654794216156,
0.03033517487347126,
0.03564535453915596,
-0.06414943188428879,
-0.0048028151504695415,
0.00445948401466012,
0.00... | |
f6e15da32d55c3e757758ccc55be585726a6f878 | subsection | 18 | 76 | A priori estimates of the densities | In order to derive suitable estimates of the densities a_i, i=1, ..., m, we need to use the Helmholtz decomposition based on the following operators, which are isomorphism (see Theorem 5.1 and Theorem 5.3 in ),\begin{aligned}\nu \operatorname{curl}{S^{0}_{{i}{i},_D}}&:\mathnormal {L}^{2,0}_\text{t}(\partial D_i)\longri... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.0012413858203217387,
0.019633345305919647,
-0.027886368334293365,
-0.006990660913288593,
-0.04106679558753967,
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0.03166963905096054,
0.018229873850941658,
-0.05900682136416435,
-0.022409778088331223,
0.006986847147345543,... | |
f103ba569f20d470a530486e7e726920f43fd903 | subsection | 19 | 76 | A priori estimates of the densities | \end{aligned}The following decomposition holds,Proposition 2.1
Each element V of \mathnormal {L}^{p,Div}_\text{t}(\partial D_i) can be decomposed asV=\mathfrak {V} + \nu \times \nabla \mathfrak {v}where\begin{aligned}\mathfrak {V}~&\in \mathnormal {L}^{p,Div}_\text{t}(\partial D_i)\setminus \mathnormal {L}^{p,0}_\text... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.00790342502295971,
0.0198806244879961,
-0.030286291614174843,
0.017805593088269234,
-0.015181289985775948,
0.006888796109706163,
0.05163164436817169,
-0.006247978191822767,
0.03527551889419556,
0.03225452080368996,
-0.038266003131866455,
-0.014098001644015312,
-0.018400639295578003,
0.0... | |
334c935ed5a4930d21fe7814f81ab8d67cc023ed | subsection | 20 | 76 | A priori estimates of the densities | It remains to take the norm to conclude.Concerning (REF ) we have,[\nu \times \nabla {S^{0}_{{i}{i},_D}}](u)(x)
=
\nu \times \nabla \frac{1}{4\pi }\int _{\partial D_i} \frac{1}{*{x-y}}u(y)~ds(y)
=
\nu \times \frac{1}{4\pi }\int _{\partial D_i} \frac{1}{*{x-y}^3} (x-y)u(y)~ds(y),hence, for x_i=\epsilon s_i+z_i and y_i=\... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.005450901575386524,
0.051114123314619064,
-0.05785813555121422,
0.01406782679259777,
-0.0019158259965479374,
0.008719916455447674,
0.015669912099838257,
0.005248733796179295,
0.029051130637526512,
0.028410296887159348,
-0.012404710985720158,
0.013282042928040028,
-0.01268698275089264,
-0... | |
ac036968cb73a329de637ee98312371a42f759ae | subsection | 21 | 76 | A priori estimates of the densities | Finally inverting the left-hand side operator, we have the scales\widehat{[\nu _{x_i}\times \nabla {S^{0}_{{i}{i},_D}}]^{-1}(u)}
=
[\nu _{x_i}\times \nabla {S^{0}_{{i}{i},_B}}]^{-1}\widehat{u}.As{[\nu \times \nabla {S^{0}_{{i}{i},_D}}]^{-1}}_{{\mathcal {L}({\mathnormal {L}_{t}^{2,0}}({\partial D_i}),{\mathnormal {L}_{0... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.007113265804946423,
0.03450982645153999,
-0.06029302999377251,
-0.004378567915409803,
0.004058184567838907,
0.011198149062693119,
0.02242681011557579,
0.04393824189901352,
0.005557119380682707,
0.011503275483846664,
-0.03127548471093178,
0.02888024039566517,
-0.014676593244075775,
0.004... | |
c23b11cc6562ec4053c2b53e327657a133a4ea73 | subsection | 22 | 76 | A priori estimates of the densities | (Of Proposition REF )
It suffices to seek for the solution of the following equation\nu \times \nabla {S^{0}_{{i}{i},_D}}(v)+\nu \times {S^{0}_{{i}{i},_D}}(w)=V.Taking the surface divergence we have, \nu \operatorname{curl}{S^{0}_{{i}{i},_D}}(w)=Div\,V and then using (REF ),
w=[\nu \operatorname{curl}{S^{0}_{{i}{i},_D}... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.031087353825569153,
0.04804270714521408,
-0.025944281369447708,
0.0005083934520371258,
-0.026554735377430916,
0.0564974881708622,
0.03485690429806709,
0.008637920022010803,
0.012567715719342232,
0.0022090792190283537,
0.010568480007350445,
-0.012102244421839714,
-0.012949248775839806,
0... | |
c1259af6728b1091dbfed912f5b6e8feb01858b9 | subsection | 23 | 76 | A priori estimates of the densities | Using (REF )
we get the estimate{w}_{\mathnormal {L}_{t}^{2,0}{(\partial D_i)}}\le {[\nu \operatorname{curl}S_{\partial B_i}^0]^{-1}}_{{\mathcal {L}({\mathnormal {L}_{0}^{2}}({\partial B_i}),{\mathnormal {L}_{t}^{2,0}}({\partial B_i}))}}
{(Div\,V)}_{\mathnormal {L}_{0}^2{(\partial D_i)}}.Put \mathfrak {V}=\nu \times {S... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.059330061078071594,
0.0479157380759716,
-0.020921170711517334,
-0.043795596808195114,
-0.035311151295900345,
0.02064649574458599,
0.04123195260763168,
0.012329909950494766,
0.042269617319107056,
0.03482284024357796,
-0.010269838385283947,
0.023469556123018265,
0.0018655093153938651,
0.0... | |
32868224cbc1c1e36b32e37648200d07b4476071 | subsection | 24 | 76 | A priori estimates of the densities | The inequality () is an immediate consequence. Now, (REF ) becomes \nu \times \nabla {S^{0}_{{i}{i},_D}}(v)=V-\mathfrak {V}, with Div\,(V-\mathfrak {V})=0, then we get successively\begin{aligned}{v}_{\mathnormal {L}_{0}^{2}{(\partial D_i)}}
&\le {[\nu \times \nabla {S^{0}_{{i}{i},_D}}]^{-1}}_{{\mathcal {L}({\mathnormal... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03838538005948067,
0.032526880502700806,
-0.03142841160297394,
-0.018048452213406563,
-0.03133687376976013,
0.071278415620327,
0.04345054179430008,
0.0291857048869133,
0.029887503013014793,
0.013654576614499092,
-0.016507543623447418,
-0.0022579634096473455,
-0.04653235524892807,
0.0235... | |
695e0bbf25b8378a0fca7bf47b7c0c6b5b8e4fd8 | subsection | 25 | 76 | A priori estimates of the densities | Then with (REF ) in mind, we derive the estimate (){\nu \times \nabla \mathfrak {v}}_{\mathnormal {L}^{2,0}_{t}(\partial D_i)}
&={V-\mathfrak {V}}_{\mathnormal {L}^{2,0}_{t}(\partial D_i)}\le {V}_{\mathnormal {L}^{2}_{t}(\partial D_i)}+{\mathfrak {V}}_{\mathnormal {L}^{2}_{t}(\partial D_i)},\\
&\le {V}_{\mathnormal {L}... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.028623506426811218,
0.03381113335490227,
-0.05028948560357094,
-0.034787628799676895,
0.0006727706058882177,
0.0139531958848238,
0.01850762963294983,
-0.0021971133537590504,
0.030286598950624466,
0.007949278689920902,
-0.03878515586256981,
-0.00043341497075743973,
-0.018690722063183784,
... | |
e77d1e14134766b618154022195f3c9b1bb7629e | subsection | 26 | 76 | A priori estimates of the densities | \end{aligned}We prove that under the condition (REF ), we have
(REF ). The properties (REF )-() are immediate conclusions of Key-Decomposition and do not rely on (REF ).We set diam(B):=\max _{i\in \lbrace 1,..,m\rbrace }diam({B}_i) and we suppose that diam(B)\le 1.For every i\in \lbrace 1,...,m\rbrace and x_i=\epsilon ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1138,
"openalex_id": "",
"raw": "D. Mitrea, M. Mitrea, and J. Pipher. Vector potential theory on nonsmooth domains in r 3 and applications to electromagnetic scattering. Journal of Fourier Analysis and Applications, 3(2):131–192, ... | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.018340162932872772,
0.021407030522823334,
-0.026381149888038635,
0.018065519630908966,
-0.024504411965608597,
0.017607778310775757,
0.039762452244758606,
0.010009273886680603,
0.01570052281022072,
0.017363648861646652,
-0.02903604879975319,
-0.01632610335946083,
0.004970306530594826,
0.0... | |
42eb49b54aa49ccb65386f6bec64c812f9a22289 | subsection | 27 | 76 | A priori estimates of the densities | As\left(I+[\frac{1}{2}I+M_{ii,_D}^0]^{-1}[M_{ii,_D}^k-M_{ii,_D}^0]\right)^{-1}=\sum _{n\ge 0}\left(-[\frac{1}{2}I+M_{ii,_D}^0]^{-1}[M_{ii,_D}^k-M_{ii,_D}^0]\right)^nwe have finally{[\frac{1}{2}I+M_{ii,_D}^k]^{-1}}\le \frac{{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}}}{1-{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}}{[M_{ii,_D}^k-M_{ii,_D}^0]}... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03833930939435959,
0.029517604038119316,
-0.04313172027468681,
-0.026114074513316154,
0.01337754912674427,
0.03391319513320923,
-0.0030105209443718195,
0.04609264060854912,
0.018665993586182594,
0.009943494573235512,
-0.04807676002383232,
-0.013087562285363674,
0.019886989146471024,
0.0... | |
8bcded1b2a8adf00fe39b1d5e333f18c8b50f07e | subsection | 28 | 76 | A priori estimates of the densities | In some places of the next computations, we use the notationC_0:=2^6.To justify (REF ) and (REF ), we need the following lemma.Lemma 2.4
For x_i\in \partial D_i, s_i\in \partial B_i, with x_i=\epsilon s_i+z
the following scaling estimation{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}}_{\mathcal {L}(\mathnormal {L}_t^{2}(\partial D... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.045075129717588425,
0.04742501676082611,
-0.04791330546140671,
0.005512319039553404,
-0.015045388601720333,
-0.0027122898027300835,
0.01145189069211483,
-0.006420230492949486,
0.031860820949077606,
0.015274273231625557,
-0.028397025540471077,
0.0163424052298069,
-0.003892956068739295,
0... | |
2367f41e75a3e5bb673c30d07a7ed7b663dae89c | subsection | 29 | 76 | A priori estimates of the densities | With this in mind, considering (\ref {L2div-scaling}) we get
\begin{align*}
{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}}_{\mathcal {L}(\mathnormal {L}_t^{2}(\partial D_i))}&
=\sup _{{(b\ne 0) \in \mathnormal {L}^{2}_{t}(\partial D_i)}}
\frac{{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}(b)}_{\mathnormal {L}^{2}_{t}(\partial D_i)}}{{b}_{\mathn... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.054987888783216476,
0.02088746428489685,
-0.036007609218358994,
0.0031945309601724148,
0.008338202722370625,
0.02166559547185898,
-0.00032255228143185377,
0.01612713746726513,
-0.003350920043885708,
0.0125263761729002,
-0.04589444398880005,
0.0292180385440588,
-0.026868389919400215,
0.0... | |
117ffbcab5e79539b4e05f4d5bf796756e97a4d7 | subsection | 30 | 76 | A priori estimates of the densities | \end{align}
Taking the gradient gives \begin{align}
\nabla _x(\Phi _k(x,y)-\Phi _0(x,y))= \frac{(ik)^2}{4\pi }\int _{0}^{1}\frac{le^{ikl*{x-y}}}{*{x-y}}(x-y)dl,
\end{align} thus, being \Im k=0, *{\nabla _x(\Phi _k(x,y)-\Phi _0(x,y))\times b(y)}\le \frac{*{k}^2}{4\pi }*{ b(y)}\le \frac{*{k}^2}{4\pi }*{ b(y)}, and
\begin... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.014046791940927505,
0.004177413880825043,
-0.03314463049173355,
-0.028826061636209488,
-0.02089088410139084,
0.01043018139898777,
0.06671653687953949,
-0.02433963492512703,
0.030779337510466576,
0.02903970144689083,
-0.045322079211473465,
0.021608101204037666,
-0.01606873609125614,
0.01... | |
530fbb66636cbd5ed4d45ebcfacf726f757fa763 | subsection | 31 | 76 | A priori estimates of the densities | We obtain{[({K^{k}_{{i}{i},_D}}-{K^{0}_{{i}{i},_D}})^*]}_{\mathnormal {L}^2_0(\partial D_i)}^2
\le \left(\frac{*{k}^2(*{\partial {B}_j}*{\partial B_i})^\frac{1}{2}\epsilon ^2}{4\pi }
{Div\,b}_{\mathnormal {L}^2_0(\partial D_i)}\right).Using (), we deduce that*{[k^2\nu {S^{k}_{{i}{i},_D}}](b)}
&=
*{[k^2\nu ({S^{k}_{{i}{... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.021609468385577202,
0.03253629058599472,
-0.04859078302979469,
-0.04038040339946747,
-0.03397081792354584,
0.014879400841891766,
0.02249460108578205,
0.011369388550519943,
0.011010756716132164,
0.02997245453298092,
-0.046484775841236115,
0.024844784289598465,
-0.0027393358759582043,
0.0... | |
4541198eb829ec37f684c18dac462ab7edf3f55a | subsection | 32 | 76 | A priori estimates of the densities | Hence{[M_{ii,_D}^k-M_{ii,_D}^0](b)}_{\mathnormal {L}^{2,Div\,}_\text{t}(\partial D_i)}\le 2C_{_B} *{\partial B} *{k}^2\epsilon {b}_{\mathnormal {L}^2(\partial D_i)}.To prove (REF ), let us recall that we have{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}}&=
\sup _{b\ne 0}\left(\frac{{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}b}_{\mathnormal {L... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03136415779590607,
0.040273044258356094,
-0.023584138602018356,
-0.03984590619802475,
-0.014690508134663105,
0.027306344360113144,
0.010258947499096394,
-0.0051561701111495495,
-0.009275004267692566,
-0.007947823964059353,
-0.0595247782766819,
-0.01911444030702114,
-0.0022176869679242373,... | |
78eeb486e83d5003d21b780ae756531582d23f35 | subsection | 33 | 76 | A priori estimates of the densities | Div\,[\frac{1}{2}I+M_{ii,_D}^0]^{-1}= [\frac{1}{2}I-(K_{ii,_D}^0)^*]^{-1}Div\,, then (REF ) gives{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}}=&
\sup _{b\ne 0}\left(\frac{{[\frac{1}{2}I+M_{ii,_D}^0]^{-1}b}_{\mathnormal {L}^2{(\partial D_i)}}^2+
{[\frac{1}{2}I-(K_{ii,_D}^0)^*]^{-1}Div\,b}_{\mathnormal {L}^2{(\partial D_i})}^2 }{{b}... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.05898728594183922,
0.0017184281023219228,
-0.040128443390131,
-0.028059395030140877,
-0.02142218127846718,
0.039670705795288086,
0.003103088354691863,
0.03314029797911644,
0.00134270079433918,
0.015837766230106354,
-0.05224326625466347,
0.006259579211473465,
-0.04238662123680115,
0.0489... | |
8dade9a6cf56231ee66050157e4eb1ee6f7643c2 | subsection | 34 | 76 | A priori estimates of the densities | \end{aligned}Being -M_{ij,_{D}}^k(b)= -\nu \times \nabla _x\times \int _{\partial D_j} \Phi _k(x_i,y)\times b(y)~ds(y), it comes from (REF )\begin{aligned}-M_{ij,_{D}}^k(b)=&\nu \times \nabla _x\times \int _{\partial D_j}(y-z_j) \Phi _k(x_i,y)Div\,b(y)~ds(y)~\\
&+\nu \times \nabla _x\times \int _{\partial D_j}(y-z_j) \... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.006847555749118328,
0.02288876660168171,
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0.022... | |
00377d92b7f4abf884a34d45b887e9db488a2416 | subsection | 35 | 76 | A priori estimates of the densities | \end{aligned}As Notice that \left((x-y)(x-y)^T\right)b=\left(b(x-y)\right)(x-y) and (x-y)\times (x-y)=0.-\nabla _x\nabla _y\Phi _k(x,y)=(4\pi )^4\Phi _k(x,y)&\Phi _0^2\left\lbrace (\Phi _0-ik)^2+(\Phi _0-ik)\Phi _0+\Phi _0^2\right\rbrace (x,y)\left((x-y)(x-y)^T\right)\\
&+(4\pi )^2\Phi _k\Phi _0(\Phi _0-ik)(x,y) ~I,whe... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.043982911854982376,
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0.... | |
7a3c14115179edf7af623c8d7571c8bd5646bcb4 | subsection | 36 | 76 | A priori estimates of the densities | Draw l spheres (\mathcal {S}_{l{\delta }}(z_i))_{\lbrace l=1,2,...,n\rbrace } centered at z_i with radius l\delta , where n will be determined later, let
R_l=\mathcal {S}_{l+1}-\mathcal {S}_l, and R_0=\mathcal {S}_1 the volume of each R_l is given byVol(R_l)&=\frac{4\pi \left((l+1)\delta \right)^3}{3}-\frac{4\pi (l\del... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
c6b02dd788e6a3bc3ca706fdc89816a82f218aea | subsection | 37 | 76 | A priori estimates of the densities | Now, considering Inter-Lemma, we have{M_{ij,_{D}}^k}_{{\mathcal {L}({\mathnormal {L}_{t}^{2,Div}}({\partial B}),{\mathnormal {L}_{t}^{2,Div}}({\partial B}))}}
\le \frac{4(*{\partial B_i}*{\partial {B}_j})^\frac{1}{2}}{\pi \delta _{i,j}} \bigl (\frac{1}{\delta _{i,j}}+*{k}\bigr )^2 \epsilon ^3
\le \frac{C_{i,j}}{\delta ... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04152575135231018,
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... | |
09346cce499909a234268d23d53ca5585b0cd6ab | subsection | 38 | 76 | A priori estimates of the densities | \end{aligned}Considering (REF ) and (REF ), the condition (REF ) is acquired if\frac{*{\partial B_i}e^{C_{_B}}}{4\pi \, diam(B)^2}(k\epsilon )^2
{[\frac{1}{2}I+M_{ii,_B}^0]^{-1}}_{{\mathcal {L}({\mathnormal {L}_{t}^{2}}({\partial B_i}),{\mathnormal {L}_{t}^{2}}({\partial B_i}))}}
<1,If we set C_{B_i}=\frac{*{\partial B... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0... | |
9463c29936e625ce92ff21e184adadc99c9bd618 | subsection | 39 | 76 | Fields approximation and the linear algebraic systems | Based on the representation (REF ), the expression of the far field pattern is given byE^\infty (\tau )&=\frac{ik}{4\pi }\tau \times \int _{\partial D} a(y)e^{-ik\tau .y}ds(y),where \tau =(x/*{x}) \in \mathbb {S}^2. We put\mathcal {A}_i:=\int _{\partial D_i}a_i^{[1]}ds,~\,~\, \mathcal {B}_i:=\int _{\partial D_i}\nu u_i... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.01... | |
48ba3e789b3d9c41d51f2be441767926ac1338ba | subsection | 40 | 76 | Fields approximation and the linear algebraic systems | \end{aligned}The elements (\mathcal {A}_i)_{i=1}^m and (\mathcal {B}_i)_{i=1}^m are solutions of the following linear algebraic system\begin{aligned}\mathcal {A}_i
=-\bigl [\mathcal {P}_{\partial D_i} \bigr ]&\sum _{(j\ne i)\ge 1}^m \left(\Pi _k(z_i,z_j)\mathcal {A}_j-k^2\nabla \Phi _k(z_i,z_j)\times \mathcal {B}_j\rig... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
adfcdab4796c5cd946a0b1f4346ab904f81cdd4e | subsection | 41 | 76 | Justification of ( | Lemma 3.2
For \Im k=0, the far field pattern can be approximated byE^\infty (\tau )=\frac{ik}{4\pi }
\sum _{i=1}^m e^{-ik\tau .z_i}&\tau \times \left\lbrace \int _{\partial D_i}a_i~ds-\int _{\partial D_i}\left(ik\tau .(y-z_i)\right) a_i(y)~ds(y)\right\rbracewith an error estimate given by O\left( e^{*{k}\epsilon }~ m~... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.042202334851026535,
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... | |
03a75a54ab44f8ce05e2eff5ad0adcf1288f1b10 | subsection | 42 | 76 | Justification of ( | \end{aligned}To prove (REF ), we writeE^\infty (\tau )=\frac{ik}{4\pi }\sum _{i=1}^m e^{-ik\tau .z_i}&\tau \times \int _{\partial D_i}a_m(y)~ds(y)+\frac{ik}{4\pi } \sum _{i=1}^m \tau \times \int _{\partial D_i}\left(e^{-ik\tau .y}-e^{-ik\tau .z_i}\right)a_m(y)~ds(y)for every i\in \lbrace 1,...,m\rbrace and evaluate the... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.015163709409534931,
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0... | |
efa0b3e3eec5f69187c06335696fc07a604ae1cb | subsection | 43 | 76 | Justification of ( | Multiplying by e^{-ik\tau .z_i} and taking the sum over i, we obtain\sum _{i=1}^m~e^{-ik\tau .z_i}\int _{\partial D_i}\hspace{-5.69046pt}ik\tau .(y-z_i) a_i~ds(y)
&=\sum _{i=1}^m~e^{-ik\tau .z_i}\left(\int _{\partial D_i}ik\tau .(y-z_i)\nu \times \nabla u_i~ds(y)+O(*{k}\epsilon ^4)\right),With this last approximation, ... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.0326894074678421,
0.03565007075667381,
-0.015535861253738403,
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-0.0374508872628212,
0.0... | |
cf0f51b9c4f5ca9cbda36ffa968a8c6bb2211d4e | subsection | 44 | 76 | Justification of ( | \end{aligned}For x \in \mathcal {R}^3 \setminus \cup _{i=1}^m \overline{D_i}, using Taylor formula with integral reminder,
we get from the representation (REF )E^\text{sca}(x)&=\sum _{i=1}^m \int _{\partial D_i} \left(\nabla _x \Phi _k(x,z_i)+ \bigl (\nabla _y\nabla _x \Phi _k(x,z_i)(y-z_i)\bigr )\right)\times a_i(y)~d... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04073009639978409,
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0.017619198188185692,
-0.02921278402209282,
... | |
c5f361236c7572de5eefaed1f9a13410754b9e18 | subsection | 45 | 76 | Justification of ( | \end{aligned}Indeed,\int _{\partial D_i} \nabla _x \bigl (\nabla _x \Phi _k(x,z_i)(y-z_i)\bigr )\times a_i(y)~ds(y)= \int _{\partial D_i}\nabla _x\times ~ \bigl [\bigl (\nabla _x \Phi _k(x,z_i)(y-z_i)\bigr ) a_i(y)\bigr ]~ds(y).As we did for the far field approximation, we get in view of decomposition (REF )\int _{\par... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.012416919693350792,
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0.001287073246203363,
... | |
ccd716571942b90b6f8d8df131f04b69f30a0804 | subsection | 46 | 76 | Justification of ( | Multiplying by (REF ) and integrating over \partial D_i, we get\begin{aligned}\int _{\partial D_i}& \psi [\frac{1}{2}I+M_{ii,_D}^k]a~ds+\sum _{(j\ne i)\ge 1}^m\int _{\partial D_i} \psi [M_{ij,_D}^k](a_j)~ds
=-\int _{\partial D_i} \psi \nu _i\times E^\text{inc}~ds. | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.003089560428634286,
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0... | |
7d93beb452de8266589f03b11472b02f04a7a6c9 | subsection | 47 | 76 | Justification of ( | \end{aligned}Recalling the scaling (REF ) and the estimate (REF )With {\mathnormal {L}^2(\partial D_i)} norms., we have\biggl |\int _{\partial D_i}~ \psi [\frac{1}{2}I+{M^{k}_{{i}{i},_D}}]a_i^{[1]}~ds\Biggr |&\le {\psi }\left({[\frac{1}{2}I+{M^{0}_{{i}{i},_D}}]a_i^{[1]}}+
{[{M^{k}_{{i}{i},_D}}-{M^{0}_{{i}{i},_D}}]a_i^{... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03496578335762024,
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... | |
acf8219a1fdb79d3a410d86ef72181aeabd3d01d | subsection | 48 | 76 | Justification of ( | \end{aligned}Now, we show how we choose appropriate candidates \psi to derive the estimates (REF ) and (REF ).Lemma 3.4
There are functions (\psi _l)_{l=1,2,3.} such that
\nu \times \psi _l \in \mathnormal {L}^{2,Div\,}_\text{t}(\partial D_i) and satisfying, for constants C_{_{({M^{0}_{{i}{i},_B}})}}, C_{_{({M^{0}_{{i... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04181348532438278,
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0.0051503838039934635,
0.031... | |
8487fa694326f2c5290d71d1aa04a7c252bb1bff | subsection | 49 | 76 | Justification of ( | Solving (REF ) is amount to solve the following problem (it suffices to take the surface divergence in the identity (REF ))[\frac{1}{2}I+({K^{0}_{{i}{i},_D}})^*](\nu \operatorname{curl}\psi _l)=-\nu _i^l.Further, as it was done in (REF ) and (REF ), the following estimates hold{\nu \times \psi _l}_{\mathnormal {L}^2(\p... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1254,
"openalex_id": "",
"raw": "D. Mitrea, M. Mitrea, and J. Pipher. Vector potential theory on nonsmooth domains in r 3 and applications to electromagnetic scattering. Journal of Fourier Analysis and Applications, 3(2):131–192, ... | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04430079832673073,
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0.... | |
98dc7ae59699d249be69e96e3d5649b9215fa7db | subsection | 50 | 76 | Justification of ( | \end{aligned}Using (REF ) with \psi _l as in (REF ), we getO( (\epsilon ^2+*{k\epsilon }^2\epsilon ) \epsilon ^2 )+\int _{\partial D_i} [\frac{1}{2}I+({K^{0}_{{i}{i},_D}})^*](\nu \operatorname{curl}\psi _l)u_i~ds
=O(\epsilon ^4)+O(*{k}^2\epsilon \epsilon ^4)-\int _{\partial D_i} \nu _i^l u_i~ds,to conclude that\int _{\... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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0.0... | |
dfc9d7b2a0039b3a45e19867f4239e98781c2b50 | subsection | 51 | 76 | Justification of ( | \end{aligned}For the first integral of the right hand side, we get, using Holder's inequality then the Mean-value-theorem, with \mathnormal {L}^2(\partial D_i) norm,\Biggl |\int _{\partial D_i} \psi _l(\nu _{x_i} \times \int _{\partial D_j}&\left(\nabla _x\Phi _k(x_i,y)-\nabla _x\Phi _k(z_i,y)\right)
\times a_j^{[1]}(y... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.008996136486530304,
0.049505457282066345,
-0.05206924304366112,
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0.010735847055912018,
-0.0019428678788244724,
... | |
fcc616eb7c1532cdaed13710516c1c4b5fb5b53c | subsection | 52 | 76 | Justification of ( | \end{aligned}In addition, considering the fact that, for any vectors a, b, c of \mathcal {R}^3 we have a(b\times c)=-c(b\times a ), we write
Recall the definition of \mathcal {A}_j=\int _{\partial D_i} (a_j^{[1]}+a_j^{[2]})~ds=\int _{\partial D_i} a_j^{[1]}~ds.\int _{\partial D_i}\hspace{-5.69046pt} \psi _l(\nu _{x_i} ... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.005939905531704426,
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0.004291839431971312,
-0.002014... | |
73b8994eb6c03f18d7275edabb2c46eabf11614d | subsection | 53 | 76 | Justification of ( | For (REF ),
being a_j^{[2]}=\nu \times \nabla u_j we have (see Lemma 5.11 )[M_{ij,_D}^k]\nu \times \nabla u_j=\nu \times \nabla [K_{i,j}^k]u_j -k^2~\nu \times [S_{ij,_D}^k](\nu _y u_j),then\begin{aligned}\int _{\partial D_i} \psi _l[M_{ij,_D}^k](a_j^{[2]})~ds
= \psi _l(\nu \times \nabla [K_{i,j}^k]u_j -k^2~\nu \times [... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02130311168730259,
0.040195126086473465,
-0.0026018484495580196,
0.0031187846325337887,
-0.035708654671907425,
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0.05188437178730965,
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-0.02072322741150856,
0.021928776055574417,
0.... | |
1473ec3f53a9c4206b5a66c045cf46bb80b3df1a | subsection | 54 | 76 | Justification of ( | \end{aligned}By Taylor formula at the first order, Actually *{\int _{0}^{1}D^3\Phi _k(tx+(1-t)z_i,y)dt\circ (x-z_i)~(x-z_i)}\le \frac{C}{\delta _{i,j}}\bigl (\frac{1}{\delta _{i,j}}+*{k}\bigr )^3\epsilon ^2.\begin{aligned}\nabla _y\left(\Phi _k(x_i,y)-\Phi _k(z_i,y)\right)=\nabla _x\nabla _y \Phi _k(z_i,y)(x-z_i) +
O\B... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.026808807626366615,
0.05737115070223808,
-0.023619825020432472,
0.02133108302950859,
-0.003196610836312175,
0.016570497304201126,
-0.006622095592319965,
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0.038511913269758224,
-0.008094520308077335,
-0.004390571732074022,
0.006313115358352661,... | |
2e0d2cf8f3211efd9b2cfef41aa8574609b217c5 | subsection | 55 | 76 | Justification of ( | \end{aligned}Now, consider the second term of (REF ), With the following product rule u(v\times w)=-w(v\times u ).-k^2 \int _{\partial D_i} \psi _l(\nu \times [S_{ij,_D}^k](\nu _y u_j(y)))~ds
=
k^2\int _{\partial D_i} (\nu _{x_i}\times \psi _l(x))\int _{\partial D_j} \Phi _k(x_i,y)(\nu _y u_j(y))~ds(y)~ds(x),we have\in... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.05599670484662056,
0.0532807894051075,
-0.02799835242331028,
-0.005077085457742214,
-0.02482469752430916,
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0.03814489394426346,
-0.025038309395313263,
-0.0018013926455751061,
-0.012267397716641426,
-0.... | |
b2ca256e452d93ae169f9f487e7aa941c437ab44 | subsection | 56 | 76 | Justification of ( | \end{aligned}It remain to put together (REF ), (REF ) and to sum over j to get the conclusion.Concerning (REF ), doing as in (REF )\int _{\partial D_i}\hspace{-5.69046pt}\psi _l(x)\nu _{x_i} \times E^\text{inc}(x)~ds(x)&=-\int _{\partial D_i}\hspace{-5.69046pt}\nu _{x_i}\times \psi _lE^\text{inc}(z_i)~ds(x) -
\int _{\p... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.005310308188199997,
0.049837544560432434,
-0.034852709621191025,
-0.009850011207163334,
-0.03543257340788841,
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0.03918641060590744,
-0.028993060812354088,
0.014191340655088425,
-0.012428867630660534,
... | |
b9df9fae773dfeff398492343bbe71672f97c731 | subsection | 57 | 76 | Justification of ( | Finally, the approximation for the \mathcal {B}_i's, with \bigl [\mathcal {T}_{\partial D_i} \bigr ] as defined in (REF ),\mathcal {B}_i=\bigl [\mathcal {T}_{\partial D_i} \bigr ]\sum _{(j\ne i)\ge 1}^m&\left( -\nabla _x\Phi _k(z_i,z_j)\times \mathcal {A}_j+\Pi _k(z_i,z_j)\mathcal {B}_j\right)-\bigl [\mathcal {T}_{\par... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03992019593715668,
0.04581056162714958,
-0.04718396067619324,
-0.031862907111644745,
-0.03128302842378616,
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0.03131354600191116,
-0.024400761350989342,
0.016740234568715096,
-0.0012208010302856565,
0.0... | |
d75ab5d94fe08cefb7a1f99778c8a05957005949 | subsection | 58 | 76 | Justification of ( | \end{aligned}Let now \phi be the solution to the following integral equation[-\frac{1}{2}I+{K^{0}_{{i}{i},_D}}](\phi )(x)=(x-z_i),then, as result of (REF ), \phi satisfies the following estimate{\phi }_{\mathnormal {L}^2(\partial D_i)}\le C_{{K^{0}_{{i}{i},_B}}} \epsilon ^2.The tensor \bigl [\mathcal {P}_{\partial D_i}... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04916730895638466,
0.03985878825187683,
-0.026842115446925163,
-0.01602286659181118,
-0.03305288404226303,
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0.021409600973129272,
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0.034609392285346985,
-0.024461574852466583,
0.011124446988105774,
-0.010048625990748405,
0... | |
fb9f69658a716e8e9ec0b82463b066bfb61960d2 | subsection | 59 | 76 | Justification of ( | The justification of (REF ) is a direct consequence of the following expansions.Lemma 3.6
With the previous notation we have the following three approximations\int _{\partial D_i}\phi \biggl ([\frac{1}{2}I-(K_{\partial D_i}^k)^*]Div\,a-&k^2\nu _{x_i}[{S^{k}_{{i}{i},_D}}]a\biggr )~ds=\mathcal {A}_i+O((\epsilon +1)*{k}^... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04971912503242493,
0.04166151210665703,
-0.04306549206376076,
-0.014711244963109493,
-0.027438608929514885,
0.02051028236746788,
0.0008679481688886881,
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0.019854076206684113,
0.03754114359617233,
-0.030673861503601074,
-0.0021498408168554306,
-0.03357338160276413,
... | |
62f6344da6c3d980b63d19e4804b7ada92b94c56 | subsection | 60 | 76 | Justification of ( | \end{aligned}Concerning the second term of the first member of (REF ), we have in view of (REF ) Decomposition-density-estimate\begin{aligned}\int _{\partial D_i}\phi ~k^2\nu _i[S_{ij,_D}^k](a_j)~ds=\int _{\partial D_i}\phi ~k^2\nu _i[S_{ij,_D}^k]\Bigl (a_j^{[1]}+a_j^{[2]}\Bigr )~ds,\\
\end{aligned}and then\int _{\part... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.027464445680379868,
0.04000654071569443,
-0.041898537427186966,
-0.003160318359732628,
-0.019545530900359154,
-0.0025061306077986956,
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0.02906653843820095,
0.04415672644972801,
-0.020476270467042923,
-0.0108866011723876,
-0.01547926664352417,
0... | |
77ffe79c6fa1af4e7ca9e716b1ea759ea99a6753 | subsection | 61 | 76 | Invertibility of the linear system | For \mu ^+ and \mu ^- defined as in (REF ) and {\mathcal {E}}=({\mathcal {E}})_{i=1}^{2m} defined as{\mathcal {E}}_i= \left\lbrace \begin{aligned}& E^\text{inc}(z_i),~i\in \lbrace 1,...,m\rbrace ,\\
&\operatorname{curl}E^\text{inc}(z_{i-m}),~i\in \lbrace m+1,...,2m\rbrace ,
\end{aligned}\right.we have the following pro... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04559999704360962,
0.015703612938523293,
-0.026645781472325325,
-0.004254015628248453,
0.0016481926431879401,
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-0.004070883151143789,
0.06861364841461182,
0.007920481264591217,
0.020053010433912277,
-0.012773493304848671,
0.0018628010293468833,
-0.013025300577282906... | |
7fec63658f3fd06740f8bff5e265dd61e64639fb | subsection | 62 | 76 | Invertibility of the linear system | \end{aligned}Further, if the condition (REF ) is satisfied, then the system could be inverted using Neumann series with the following estimate\begin{aligned}*{\mathcal {A}_i}\le \frac{1}{C_{L^2_i}{\mu ^-}} \epsilon ^3 *{\mathcal {E}_i},\,~
*{\mathcal {B}_i}\le \frac{1}{C_{L^2_i}{\mu ^-}} \epsilon ^3 *{\mathcal {E}_{i+m... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04036666825413704,
0.01983245089650154,
-0.015774426981806755,
0.009321251884102821,
0.0011422729585319757,
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0.02271578460931778,
0.051686421036720276,
0.015049779787659645,
0.008383024483919144,
-0.023875219747424126,
0.020625749602913857,
0.0036079806741327047,
0.... | |
e1d3973e1cc2ea4439f8d2872b178097edc909fa | subsection | 63 | 76 | Invertibility of the linear system | Let (\widehat{\mathcal {C}}_i)_{i\in \lbrace 1,...,2m\rbrace } be defined as\widehat{\mathcal {C}}_i= \left\lbrace \begin{aligned}&\bigl [\mathcal {T}_{\partial D_i} \bigr ]^{-1}\widehat{\mathcal {B}}_i,~i\in \lbrace 1,...,m\rbrace ,\\
&-\bigl [\mathcal {P}_{\partial D_{i-m}}\bigr ]^{-1}\widehat{\mathcal {A}}_{i-m},~i\... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
0.028370484709739685,
0.028675707057118416,
-0.0781981498003006,
0.0012847983743995428,
-0.0009290226735174656,
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0.01149164792150259,
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0.01846599578857422,
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0.0035692022647708654,
-0.014047891832888126,
... | |
ec4b9d6aa94c1082bda729fd6d51883a9d58ec05 | subsection | 64 | 76 | Invertibility of the linear system | \end{aligned}\right.With these notations, solving the system (REF ) is equivalent to solve the equation\widehat{\mathcal {C}}+\Sigma ^k\mathcal {Q}\widehat{\mathcal {C}}+^k\mathcal {Q}\widehat{\mathcal {C}}=\mathcal {E}.If we multiply both sides of the last system by \mathcal {Q}\widehat{\mathcal {C}} we get\begin{alig... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.027553418651223183,
0.03862971067428589,
-0.05031626299023628,
-0.03185578063130379,
0.004348130896687508,
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0.024425815790891647,
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0.02026076428592205,
-0.013807222247123718,
0.03... | |
d85f885376940a743be27a1f8de5fdd96eda43e7 | subsection | 65 | 76 | Invertibility of the linear system | \end{aligned}Hence, we get*{\Pi _k(z_i,z_j)-\Pi _0(z_i,z_j)}
&= *{k^2\Phi _k(z_i,z_j)I-\nabla _x\nabla _y\Phi _k(z_i,z_j)+\nabla _x\nabla _y\Phi _0(z_i,z_j)},\\
&\le *{k^2\Phi _k(z_i,z_j)}+*{\nabla _x\nabla _y\Phi _k(z_i,z_j)-\nabla _x\nabla _y\Phi _0(z_i,z_j)}\le \frac{6*{k}^2}{4\pi \delta _{i,j}}.Now, as\bigl <\Bigl ... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04387780278921127,
0.04247419908642769,
-0.04152829572558403,
-0.03267950192093849,
-0.008314812555909157,
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0.016446547582745552,
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0.004863021429628134,
-0.0459832027554512,
0.02708035334944725,
-0.010130341164767742,
0.... | |
bb79d70bee3a7e524af8381cb6a795f534c098f8 | subsection | 66 | 76 | Invertibility of the linear system | \end{aligned}The inner sum gives, as we did it in (REF ),\sum _{1\le j\ne i}^{m}(\frac{6*{k}^2}{4\pi \delta _{i,j}})^2
\le \sum _{l= 2}^{m^\frac{1}{3}} 7l^2 ~\frac{3^2*{k}^4}{4^2\pi ^2 l^2\delta ^2}
= 7 m^\frac{1}{3}\frac{3^2*{k}^4}{4^2\pi ^2\delta ^2},and then\bigl (\sum _{i=1}^{m} \sum _{1\le j\ne i}^{m}(\frac{3*{k}^... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.027580128982663155,
0.05806021764874458,
-0.025412792339920998,
-0.04484251141548157,
-0.030159566551446915,
0.046460382640361786,
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0.007627653423696756,
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-0.015720825642347336,
-0.04465935751795769,
0.0463382825255394,
-0.012149298563599586,
... | |
4cdc6ccba918bbbd33e3a1ccd600f8a89ac55b5e | subsection | 67 | 76 | Invertibility of the linear system | For the last assertion (REF ), we proceed as follows:\begin{aligned}\bigl <^k\mathcal {Q}\widehat{\mathcal {C}},\mathcal {Q}\widehat{\mathcal {C}}\bigr >_{\mathbb {C}^{3\times 2m}}
=&
\sum _{i=1}^m\bigl (\sum _{j\ne i}^m \nabla \Phi _k(z_i,z_j)\times Q_{j+m}\widehat{\mathcal {C}}_{j+m}\bigr )\overline{Q_i\widehat{\math... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04938479885458946,
0.041388001292943954,
-0.030155859887599945,
-0.06550047546625137,
-0.015840983018279076,
0.011987564153969288,
0.02975907176733017,
-0.0023521112743765116,
-0.0033631566911935806,
0.0033555261325091124,
-0.016054637730121613,
0.02196066826581955,
0.0019820304587483406,... | |
526e7f5322ed0b08dd252a212008124e8162fa5c | subsection | 68 | 76 | Invertibility of the linear system | \end{aligned}Further, the following scaling inequality holds\bigl <S_{{\partial B_{\delta /4}^{z_{i}}}}^0~\mathcal {U},\mathcal {U}\bigr >_{\mathnormal {L}^2({\partial B_{\delta /4}^{z_{i}}})}
&=
\int _{B_{\delta /4}^{z_{i}}}\int _{B_{\delta /4}^{z_{i}}}
(\Phi _0(x,y)\nu _x{Q_i\widehat{\mathcal {C}}_i}~ds(y))\nu _y \ov... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.038847196847200394,
0.03475801646709442,
-0.04766639322042465,
-0.025038588792085648,
0.018904821947216988,
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0.014907191507518291,
-0.00529838539659977,
0.04525560885667801,
0.025450559332966805,
0.0112... | |
4ef067cd5c19f6ff55415d08155b7b01ccd38caf | subsection | 69 | 76 | Invertibility of the linear system | Further, if the previous condition is satisfied, then from(REF ), considering (REF ), we get\left(1- C_{Ls}\frac{\mu ^+\epsilon ^3}{\delta ^3} \right)\mu ^-\epsilon ^3\bigl <\widehat{\mathcal {C}},\widehat{\mathcal {C}}\bigr >_{\mathbb {C}^{3\times 2m}}^\frac{1}{2}\le \mu ^+\epsilon ^3\bigl <\mathcal {E},\mathcal {E}\b... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.04654736444354057,
0.009614701382815838,
-0.03983233496546745,
-0.04966069385409355,
-0.0005575191462412477,
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0.0015461737057194114,
0.00539110042154789,
-0.012415173463523388,
0.022770054638385773,
-0.020618192851543427,
... | |
6c2efa4db1da8ee061a83b1e2481f5090c911e25 | subsection | 70 | 76 | End of the proof of Theorem | With the notations of the previous section, the linear system ((REF ),(REF )) becomes\mathcal {C}+\Sigma ^k\mathcal {Q} \mathcal {C}+^k\mathcal {Q} {\mathcal {C}}=\mathcal {E}+{{\epsilon }}(\epsilon ,\delta ,&*{k},m)\epsilon ^3,with ({\mathcal {C}}_i)_{i\in \lbrace 1,...,2m\rbrace } defined as{\mathcal {C}}_i:= \left\l... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.02504652738571167,
0.01438534539192915,
-0.035898495465517044,
-0.04487311840057373,
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0.010760391131043434,
0.06764546036720276,
0.020757634192705154,
0.005307696294039488,
0.01204247958958149,
0.009020412340760231,
-0.0077230604365468025,
0... | |
73b873c963cda29709c2f143b4bb69bcf609d68e | subsection | 71 | 76 | End of the proof of Theorem | \end{aligned}\right.The difference between (REF ) and (REF ) implies(\mathcal {C}-\widehat{\mathcal {C}})+
\Sigma ^k\mathcal {Q} (\mathcal {C}-\widehat{\mathcal {C}})+^k\mathcal {Q} (\mathcal {C}-\widehat{\mathcal {C}})
={{\epsilon }}(\epsilon ,\delta ,&*{k},m),which gives, with the estimates (REF )\sum _{i=1}^m
\Bigl ... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.017164433375000954,
0.004134721122682095,
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0.015547161921858788,
-0.01901056431233883,
... | |
cd7bf50689fffdc68f456a0d881e6d4cdd313111 | subsection | 72 | 76 | End of the proof of Theorem | \end{aligned}\right.We setO^{{\epsilon }}(\frac{\epsilon ^4}{\delta ^4}):=O\Bigl (\frac{\epsilon ^4}{\delta ^4}
+(1+*{k}){{\epsilon }}_{k,\delta ,m} \epsilon ^4+\max (*{k}^2,1+*{k})\epsilon \Bigr ).Lemma 5.1 We have the following asymptotic approximation for the far field,\begin{aligned}E^\infty (\tau )=&\frac{ik}{4\pi... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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2ba594c6da85b0b909a53292ea4af0dcb180dc5b | subsection | 73 | 76 | End of the proof of Theorem | \end{aligned}For the first term of the right hand side, we have{\frac{ik}{4\pi }
\sum _{i=1}^m e^{-ik\tau .z_i}\tau \times \bigl ((\mathcal {A}_i-\widehat{\mathcal {A}}_i)-ik\tau \times (\mathcal {B}_i-\widehat{\mathcal {B}_i})\bigr )}&\\
&\hspace{-85.35826pt}\le 2~ \frac{*{k}}{4\pi }\max (1,*{k})~m^\frac{1}{2}
\Bigl (... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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d93718a6caee350fc93f8e7ba6e58f0db1255a0a | subsection | 74 | 76 | End of the proof of Theorem | \end{aligned}Let i_0 be as in (REF ),
from the representation of the linear system we have Notice that -_k(x,y)=\nabla _y\times \nabla _x\times \Phi _k(x,y)\; I.\sum _{(i\ne i_0)\ge 1}^m \biggl ( \nabla \Phi _k(z_{i_0},z_i)\times \bigl (\mathcal {A}_i-\widehat{\mathcal {A}}_i\bigr )+&
\operatorname{curl}\operatorname{c... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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... | |
7e05bc649cd7996db171b314c538b04963019a63 | subsection | 75 | 76 | End of the proof of Theorem | \end{aligned}For x\in \partial \Omega , \frac{1}{d_{x,i_0}}=\frac{1}{\delta } and then the first term of the right hand side of (REF )
is smaller then\biggl (\frac{1}{\delta }\bigl (\frac{1}{\delta }+*{k}\bigr ){\mathcal {A}_{i_0}-\widehat{\mathcal {A}}_{i_0}}
+\Bigl (\frac{*{k}^2}{\delta }+\frac{1}{\delta }\bigl (\fra... | {
"cite_spans": []
} | 1802.03082 | The Foldy-Lax Approximation for the Full Electromagnetic Scattering by
Small Conductive Bodies of Arbitrary Shapes | [
"Ali Bouzekri",
"Mourad Sini"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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8acae988b9c19c68523ab14a68541d79f1fd1dc1 | abstract | 0 | 26 | Abstract | To what extent, hiring incentives targeting a specific group of vulnerable
unemployed (i.e. long term unemployed) are more effective, with respect to
generalised incentives (without a definite target), to increase hirings of the
targeted group? Are generalized incentives able to influence hirings of the
vulnerable grou... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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01b0af4172de4e994ff20c0ee908a20e0906ffc9 | subsection | 1 | 26 | Introduction | In spite of the huge amount of international literature on hiring and wage subsidies, targeting a vulnerable category of unemployed (i.e. long-term-unemployed), few is known about the difference between the last and generalised subsidies without a definite target. Moreover, few is known about the mechanism behind them ... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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84b5e2b7d3daacded388635ba5adc1c9f404c8c3 | subsection | 2 | 26 | Introduction | About single study analysis, most of the studies concluded hiring subsidies have a positive and significant effect (, , , , , , ), few of them concluded they have a null effect (, , ) and only two of them concluded they have a negative effect due to targeted group stigmatization (, ).The international literature study ... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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44d49ef75fda13f269171048c6e565e17a370224 | subsection | 3 | 26 | Introduction | The second work, instead, is based on a theoretical model, relying on strong assumptions. Finally, Boo2012 tested for the possibility that firms post-poned some hirings in order to get the subsidy and concluded that wasn't the case.The italian literature on hiring subsidies targeting disadvantaged categories of individ... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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c0ee361a2cdd713a9385c939ace531e3f689a84b | subsection | 4 | 26 | Introduction | Under the first estimation assumptions, the effect estimated on the control group, can be a good representation of the impact, on the treated, of the generalised incentives in absence of Law 407/90 ending.From the analysis, it emerged Law 407/90 had a strong, positive and significant effect on LTU hirings. The estimate... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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67ba7d4242cc436d47d5ef11d25149e91f20f319 | subsection | 5 | 26 | Long-Term Unemployed Stigma | Long-term unemployment status is characterised by a strong duration dependence. Many authors (, , , , ) estimated the probability to exit from unemployment status for different unemployment durations. Indipendently from the considered Country and period, they found that, once taken into account of heterogeneity, such p... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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7cdc0dbd2b4897cb7589eafe1775bf8c622616b7 | subsection | 6 | 26 | Law 407/90 and Generalised Incentives | Law 407 was promulgated on December 29th 1990. According to it, any firm had access to tax credits for a period of 36 months, at the condition of hiring, with a permanent contract, individuals who had been either in unemployment status, or suspended from their job, or in Cassa Integrazione (temporary layoff), for at le... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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... | |
8487f3cac2704e6427e20136a0430bef930ea49a | subsection | 7 | 26 | Law 407/90 and Generalised Incentives | This suggests Law 190 incentives were effective. It is important to notice the peak in hirings in December 2015. This peak can be attributed to the fact that the policy was implemented for a limited and short period of time and its duration was communicated in advance. To compare the two types of incentives in a more g... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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344ee45d471216f896201cbfe9e29d59db4dec75 | subsection | 8 | 26 | Two Types of Hiring Subsidies | The main difference between targeted and untargeted incentives is that the first lower the labour cost in relative terms, while the second lower them only in absolute terms. Hence, if targeted incentives are implemented the LTU become economically more convenient than STU. This is not the case with untargeted incentive... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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79fc5e0ad902d6e2510570b286fd41f5a3871fa5 | subsection | 9 | 26 | Data and Sample Definition | Thanks to an agreement between Dipartimento di Scienze Sociali ed Economiche of Sapienza University of Rome and INAPP (Istituto Nazionale per l'Analisi delle Politiche Pubbliche) we had access to the used data. We used a micro-databases, available, through INAPP, to the participants to the agreement. It is an administr... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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b9284f42daf8cfa72e80c1431fa1e3c5928f2c19 | subsection | 10 | 26 | Data and Sample Definition | Indeed, it is from 2010 that we start having units far enough from the last recorded contract to be possibly considered LTU. The variables recorded in the new database for each unit ij are the share of individuals, in the group, that are hired in day j, the total number of individuals in the group and the share of indi... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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d8010e58095b18177c028381a5cbf988ef9c29fa | subsection | 11 | 26 | Law 407/90 Intention-to-Treatment Effect | To estimate the impact of Law 407/90 we applied a regression discontinuity design with daily fixed effects. It has been demonstrated that, when units at the threshold are considered, the regression discontinuity design is, at the threshold, as reliable as the golden standard of policy evaluation: randomized treatment a... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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216d763eb000d390aed1e86e7c1c8b2b8ad9a8f0 | subsection | 12 | 26 | Law 407/90 Intention-to-Treatment Effect | Hence, we included all units becoming eligibles in two weeks or less and all units who became eligibles since two weeks or less.The results of this estimation are reported in table REF (for obvious reason we excluded the daily parameters).
[Table: Intention to treatment effect of Law 407/90 is given by Treat.]As common... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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52e89e656be09e9e188c0a2eebf398d1dd5814c5 | subsection | 13 | 26 | Bandwidth Choice in a time-varying forcing variable framework | The approach to use in bandwidth selection has been led by the nature of our forcing variable. The last has the particularity to change over time. Consequently we can't widen the bandwidth and use the linear polynomial approach. To explain the differences between a time fixed and a time varying forcing variable we have... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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b883e03722acd7aff9d7f3628a748dfd9629fd83 | subsection | 14 | 26 | Bandwidth Choice in a time-varying forcing variable framework | Indeed, it allows to choose a bandwidth such that, the group of units having an observed forcing variable under the threshold doesn't differ from the group having it over the threshold in terms of selection-into-treatment-determinant characteristics. On the contrary, the approach based on local polynomial, which use a ... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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ed606e9c74d73e6b7d3dfc347d63fd6654275cfd | subsection | 15 | 26 | Are the Assumptions to use Regression Discontinuity Design Plausible in this Context? | Three main assumptions are required in order to use the Regression Discontinuity Design. The first one is the assumption of randomness in the distribution of units around the threshold. It is likely to be satisfied. Indeed, being, at a determined day, unemployed from 23 months and 28 days or 24 months, is not under con... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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7fa345e01eb070b39bcdcec24b1f094826c235a1 | subsection | 16 | 26 | Indirect Effects | Taking into account of the indirect effects of a policy is crucial for two reasons. First of all, it gives a deeper overview on policy effects. Secondly, it allows to check the correctness of the estimated impact. Indeed, if the indirect effects affect the control group the estimation is biased. In their studies, Bro20... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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c2524f7656e102fa5e9ae2540a4b27c43d3728e6 | subsection | 17 | 26 | Indirect Effects | This allows us to attribute the absence of regularity in the difference to the displacement and post-poned hirings effects.In the following image, is it possible to see the Kernel-weighted local polynomial smoothing of the difference between the average outcome before 2015 and after 2014 with respect to the forcing var... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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... | |
45ee85805047ee6636ea952c5b4b644b56a0dc43 | subsection | 18 | 26 | Generalised Incentives Intention-To-Treatment Effect | To estimate the impact of generalised hiring incentives from a LTU perspective, we used another regression discontinuity design. We used time as a forcing variable and exploited the January 1st 2015 threshold defining eligibility. We couldn't estimate the impact of the policy on the exact same group targeted by Law 407... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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a11f0e5d176ef16b0590b25cabf319480136628c | subsection | 19 | 26 | Generalised Incentives Intention-To-Treatment Effect | The results of this estimation are reported in table REF .
[Table: Law 190 incentives ITT on vulnerable group is given by \gamma _3.]The policy had a positive and significant impact. The last was considerably lower than Law 407/90 impact. This suggests a significant component of the impact of Law 190 incentives is prob... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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7bc7f997f4847e398a4a2a9d715b741fe44ab033 | subsection | 20 | 26 | Are the Assumptions to use RDD with time forcing variable Plausible in this Context? | The use of models based on time discontinuity requires some conditions to be satisfied. First of all there can't be an anticipation effect. If policy effect is anticipated with respect to policy implementation starting time, the last can't be used as threshold in the discontinuity analysis. Luckily, the policy was anno... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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870dd7902e2ad8f966ff0f7a14a7f5ec4dc7990a | subsection | 21 | 26 | Conclusions | The previous analysis allows to make several considerations. Nevertheless, making them, it is important to remember that the results of this analysis are only local, given the method we used. The following conclusions should be attributed only to the groups included in the analysis.Law 407/90 had a significant and stro... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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572e23a8774905d8e6548b27cc5e1e6c28d55886 | subsection | 22 | 26 | Estimation for the Mezzogiorno Area | In the following are presented the results of the analysis reduced to the group of individuals having the last working experience in a region of the Mezzogiorno Area. In table REF the results of the estimations of treatment effect of Law 407/90 is reported together with its value under different robustness checks. As m... | {
"cite_spans": []
} | 1802.03343 | Long-Term Unemployed hirings: Should targeted or untargeted policies be
preferred? | [
"Alessandra Pasquini",
"Marco Centra",
"Guido Pellegrini"
] | [
"econ.EM",
"stat.AP"
] | 2,018 | en | Economics | [
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... |
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