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09858f4bc1027dcbea4e33142dc3bfaf7d581832 | subsection | 11 | 36 | Body | In the same way as we
argued before, the enumeration of all structures is equivalent to the
enumeration of \text{Sp}(2,\mathbb {R}) invariants built out of \eta _i
and \tilde{\eta }_i, where now i=1,\ldots ,4. The claim is that these are
in one-to-one correspondence with a three-point function where the third
operator ... | {
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"Andrea Manenti",
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1d20d9be06a26342642eeb0d9b0322585c5c349f | subsection | 12 | 36 | Body | It is sufficient to replace
\rho _1\otimes \rho _2 by \mathrm {S}^2\hspace{1.0pt}\rho _1 if j is even and by
\wedge ^2 \hspace{1.0pt}\rho _1 if j is odd, where \mathrm {S}^2 and \wedge ^2 denote respectively the symmetrized square and the exterior square of representations.Assuming, now, j-{\bar{\jmath }}\hspace{0.9pt}... | {
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} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
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ce556b5be163f1d793708ec1d32c828839cab17e | subsection | 13 | 36 | Body | Finally the (anti)symmetrized products are given by
S2 = k =
2 mod 22 k, 2 = k =
2+1 mod 22 k,[]
where \text{S}^2 and \wedge ^2 inside the parenthesis (\frac{1}{2} j,\frac{1}{2} {\bar{\jmath }}\hspace{0.9pt}), stand for the direct sum of all possible pairs of the resulting irreps.Collecting all the above results,... | {
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theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
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17df3a185050c99f7722cf2d3df8c6922b20aa03 | subsection | 14 | 36 | Body | Therefore,
taking j-{\bar{\jmath }}\hspace{0.9pt} oddThe even case can be treated similarly and it trivially gives zero. one finds
N(j,)(j even) = Res(S2(12,12) (12j,12)(0,12))+ Res(2(12,12) (12j,12)(12,0)) ,N(j,)(j odd) = Res(2(12,12)(12j,12)(0,12))+ Res(S2(12,12) (12j,12)(12,0)) .
[nStructuresOther]
Again notice t... | {
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} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
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fc2012621d77029447dc4fc4044ed75547bbda03 | subsection | 15 | 36 | Body | For simplicity we will omit the superspace coordinate dependence. The constraints we need to impose are
DJJOj,= 0 ,
[DJJO]
DJJOj,= 0 .
[DbJJO][DDbJJO]
These conditions are not independent; in fact there are linear relations
between them. First we can observe that taking the derivative {\hspace{1.94443pt}\overline{\hsp... | {
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"Andrea Manenti",
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4f0ec1b33bf2e9b10b316f7eb0edc05eb15ea20f | subsection | 16 | 36 | Body | To keep this into account one must subtract from the numbers
N2(DJ,DJ,O) ,
[DJDJO]
N2(DJ,DJ,O) .
[DbJDbJO][DDbJDDbJO]
Similarly, given the relation {\hspace{1.94443pt}\overline{\hspace{-1.94443pt}D\hspace{-0.83328pt}}\hspace{0.83328pt}}{}\sim D , we should naîvely subtract from the number
N1(DJ,DJ,O) + N(DJ,DJ,O) + ... | {
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} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
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"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
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f6e026aba6744db956caa9cff0fa6734c264fd7f | subsection | 17 | 36 | Body | Thus for \ell even (odd) we must take the \mathrm {S}^2 (\wedge ^2) product in N(D\mathcal {J},D\mathcal {J},\mathcal {O};-2) and N({\hspace{1.94443pt}\overline{\hspace{-1.94443pt}D\hspace{-0.83328pt}}\hspace{0.83328pt}}{}\mathcal {J},{\hspace{1.94443pt}\overline{\hspace{-1.94443pt}D\hspace{-0.83328pt}}\hspace{0.83328p... | {
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} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
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"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
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7abb8eeab2031b54d6c28322c16e8128461ccea9 | subsection | 18 | 36 | Body | The unitarity bound simply becomes
+2 ,[UniBoundA]
which agrees with the usual non-supersymmetric unitarity
bounds, Eq. .Even \ell
When all Grassmann variables are set to zero, then there are four possible
parity-even structures in t_{\text{A}} for \ell
even . The Ward identity that follows from
will relate these str... | {
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theories | [
"Andrea Manenti",
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b2234716b793936c85df20e6ba8754a0e4301a6a | subsection | 19 | 36 | Body | Examples include the identities in
. In this work we circumvent the need to
impose such idetities by substituting numerical values for the various
quantities that appear. This is equivalent to working in the superconformal
frame. We may express all coefficients in terms of {A_1} and {A_2}
to findThe above expressions a... | {
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theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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4fe148d2d13d0fc19245d7c8f1347041dc80a36c | subsection | 20 | 36 | Body | In the same Appendix we also show the relation between the coefficients defined here and the anomaly coefficients a and c.Case B: (\frac{1}{2}(\ell +2),\frac{1}{2}\ell ) operators
Here we start with
J(1,1,z1)J(2,2,z2)O+2,(3,3,z3)=
1x1311x3112x2322x322
(x312x132x322x232)2
tB(1,2,3,1,2,3,U,) ,
[ThreePFB]
where \mathca... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
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1a5196468855d653579103b59f8559497c30c385 | subsection | 21 | 36 | Body | Similarly to the
even-\ell case, in the lowest component of the three-point function
for general odd \ell there are two structures, which are related to the \lambda ^{(i)} in by
(1)=(2)=A1-A2 , (3)=-(4)=-A2 .[BOddZero]We now need to impose the conservation at the first two points. For
generic \Delta and \ell this... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
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0.0... |
f3862cb77b79ec72d65c0be66b183df7ad293af4 | subsection | 22 | 36 | Body | At the lowest order in the Grassaman variables there is only one structure, and the associated parameter is related to the coefficient \lambda in by
= -A.
[CEvenZero]
Similarly to we need to require
1D tC( even)(i,i,U,)=0 , 1D tC( even)(i,i,U,)=0
.[WardtCEven]
This leads, for generic \Delta and \ell , to three l... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
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2ad42aa71f2d967f22ddead4395ce2b64452d1f5 | subsection | 23 | 36 | Body | Similarly
the order \theta _3{\bar{\theta }}_3 contains contributions from operators of the
form (Q{\hspace{1.94443pt}\overline{\hspace{-1.94443pt}Q\hspace{-0.83328pt}}\hspace{0.83328pt}}{}\mathcal {O})_{j\pm 1,{\bar{\jmath }}\hspace{0.9pt}\pm 1;p}(x), and the order
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"Andrea Manenti",
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13d7da56878496029c479b553a8aa99fedb313b2 | subsection | 24 | 36 | Body | Schematically, we indicate with \lambda ^{(i)}
the OPE coefficients associated to the superconformal primary,
\lambda ^{(i)}_{\pm \pm } or \lambda ^{(i)}_{\pm \mp } those associated to
the superdescendants at order \theta _3,{\bar{\theta }}_3, and finally \hat{\lambda }^{(i)} the coefficients of the order \theta _3^2{\... | {
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"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
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a1230243538e07bbd0d0c133d7a5ea96bca6e3b9 | subsection | 25 | 36 | Body | The \pm \pm subscript refers
to the addition of unity to the \ell labels of Q{\hspace{1.94443pt}\overline{\hspace{-1.94443pt}Q\hspace{-0.83328pt}}\hspace{0.83328pt}}{}\mathcal {O} in the left-hand
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"Andrea Manenti",
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"Alessandro Vichi"
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18b5710c5cc2b145d96e3d8ea932e8dd6b5f4d83 | subsection | 26 | 36 | Body | The first three-point function takes the
form
J(1,1,x1)J(2,2,x2)(QQO)1,1;p(3,3,x3)=K+1,1,0i=15(i)S(i)(i,i,xi) ,[]
and we also have
J(1,1,x1)J(2,2,x2)(QQO)+1,-1;p(3,3,x3)=K+1,-1,2 j=14 +-(j)T(j)(i,i,xi) ,J(1,1,x1)J(2,2,x2)(QQO)-1,+1;p(3,3,x3)=K+1,-1,2 j=14 -+(j)T(j)(i,i,xi) .
[]
The coefficients of these three-point f... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
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9f4cb67a2b916c778fa7d000c80f28b6fff9e8e5 | subsection | 27 | 36 | Body | More specifically, we haveJ(1,1,x1)J(2,2,x2)(QQO)+3,+1;p(3,3,x3)=K+1,+1,2j=14++(j)T(j)(i,i,xi)
,J(1,1,x1)J(2,2,x2)(QQO)+1,-1;p(3,3,x3)=K+1,-1,2j=14–(j)T(j)(i,i,xi)
,[ppBEven]
as well as
J(1,1,x1)J(2,2,x2)(QQO)+3,-1;p(3,3,x3)=0 ,[pmBEven]
and
J(1,1,x1)J(2,2,x2)(QQO)+1,+1;p(3,3,x3)=K+1,+1,0 -+(-)S(-)(i,i,xi) .[mpBEven]... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
-0.03037555329501629,
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-0.0026698752772063017,
0.0004696119867730886... |
d040a6a2cace607227eaa846385c92f4f4df47d8 | subsection | 28 | 36 | Body | We have
J(1,1,x1)J(2,2,x2)(QQO)+3,+1;p(3,3,x3)=K+1,+1,2j=14++(j)T(j)(i,i,xi)
,J(1,1,x1)J(2,2,x2)(QQO)+1,-1;p(3,3,x3)=K+1,-1,2j=14–(j)T(j)(i,i,xi)
,[ppBOdd]
and
J(1,1,x1)J(2,2,x2)(QQO)+3,-1;p(3,3,x3)=+-K+1,-1,4
R(i,i,xi) ,[pmBOdd]
while the symmetric traceless one is
J(1,1,x1)J(2,2,x2)(QQO)+1,+1;p(3,3,x3)=K+1,+1,0 i=1... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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-0.004687011707574... |
36d6bc59a77fa0fd83b88e3f4b8273c7ac41ce93 | subsection | 29 | 36 | Body | More specifically, we have
J(1,1,x1)J(2,2,x2)(Q2Q2O)+2,;p(3,3,x3)=K+2,,2j=14(j)T(j)(i,i,xi) .[threePFBOddQsqQbsq]
After using (REF ) and () we obtainwhere we defined the denominators
9=3 (-2) (--3)(++1) (5(-2)2-(+4)) ,10= (+-2)9 .
[]
Consistently with multiplet shortening \hat{\lambda }^{(j) }=
0 at the unitarity boun... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
692a8e2a6a4849aadb1da241072f4bd156045e4e | subsection | 30 | 36 | Body | We started from the most general
parametrization of such a correlator in superspace and we imposed the
shortening conditions D^\alpha \mathcal {J}_{\alpha {\dot{\alpha }}}=\hspace{1.94443pt}\overline{\hspace{-1.94443pt}D\hspace{-0.83328pt}}\hspace{0.83328pt}^{\dot{\alpha }}\mathcal {J}_{\alpha {\dot{\alpha }}}=0.
Simil... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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-0.... |
bcdceb5d9ab8095f62a4f6d6ee487038026ae5b3 | subsection | 31 | 36 | Body | Introducing a proper basis of tensor structures \mathbb {T}^{(i)}_4 we can schematically write
JJJJO s=1n4 WO(i) T(i)4 ,
[]
where we have omitted a kinematic prefactor, and the partial waves
W_\mathcal {O}^{(i)} represent the contribution of an entire
superconformal multiplet to the four-point function,
WO(i) = O'=O, Q... | {
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{
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"doi": "10.1007/jhep02(2016)183",
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"raw": "A. Castedo Echeverri, E. Elkhidir, D. Karateev & M. Serone, “Seed Conformal Blocks in 4D CFT”, JHEP 1602, 183 (2016), arXiv:1601.05325 [hep-th]... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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d53f7dd6d9f90f10d1f6b3e0975ff7cf49357cbe | subsection | 32 | 36 | Body | In we have the following tensor structures:If \ell = 0 only the structures \mathcal {P}^{(1)}_{\text{A}},\,\mathcal {P}^{(3)}_{\text{A}} and \mathcal {P}^{\hspace{0.5pt}\prime \hspace{0.5pt}(1)}_{\text{A}} are present.
The structures for the odd-spin case appearing in readIf \ell = 1 the structures \tilde{\mathcal {P}}... | {
"cite_spans": []
} | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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-0... |
e10e82573e7bc223cbbdf5aa3a1f23c2d20c6903 | subsection | 33 | 36 | Body | In we have the following tensor structures:
P(1)C= [13][23][31][32]
,P(1)C= [12] [31][32] [3][3] ,P(2)C= [13][23][1
2][3][3] .
[CEvenStructures]
The structures for the odd-\ell case appearing in (REF ) are
P(1)C= [13][23](
[32][3] [
1]
+
[31][3] [
2]
)[33] ,P(2)C= [31][32](
[13][2] [3]
+[23][1] [3]
)[33] ,P(3)C= [31... | {
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{
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"doi": "",
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"raw": "H. Osborn, “\\mathcal {N}=1 superconformal symmetry in four-dimensional quantum field theory”, Annals Phys. 272, 243 (1999), hep-th/9808041",
"source_ref_id": "10c90332bed85ab4b004e12b228b8f9... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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0.004001131281256676,
... |
f2b198e7ea4d269f9fde7020de7d49f3a998a43e | subsection | 34 | 36 | Body | ( odd)(X,X)= 1(XX)3+12-q(i=12 P(i);(123(Ai+Bi 2)+i=36 P(i);(123(CiV+Di )+i=710P(i);(123Ei)U4U) ,2 = V2XX , = UVXXU ,
[tAOddSpin]
where \mathcal {A}_i,\,\mathcal {B}_i,\,\mathcal {C}_i,\,\mathcal {D}_i,\,\mathcal {E}_i are real constants and the tensor structures \mathcal {P}^{(i){}} are given by[]
When all Grassma... | {
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{
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"doi": "",
"end": 2082,
"openalex_id": "",
"raw": "H. Osborn, “\\mathcal {N}=1 superconformal symmetry in four-dimensional quantum field theory”, Annals Phys. 272, 243 (1999), hep-th/9808041",
"source_ref_id": "10c90332bed85ab4b004e12b228b8f9... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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... |
865f90e1b1add96d76e4de8c31f8a19cc96829fd | subsection | 35 | 36 | Body | It is interesting, thus, to relate the coefficients defined
here, {A_1} and {D_2}, with the anomaly coefficients c
(proportional to the central charge C_T) and a (Euler anomaly).
Using together with , or,
equivalently, following and using the
relation between C_J and C_T stemming from supersymmetry one obtains
A1= ... | {
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{
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"doi": "",
"end": 357,
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"source_ref_id": "10c90332bed85ab4b004e12b228b8f97... | 10.1007/JHEP12(2018)108 | 1804.09717 | R-current three-point functions in 4d $\mathcal{N}=1$ superconformal
theories | [
"Andrea Manenti",
"Andreas Stergiou",
"Alessandro Vichi"
] | [
"hep-th"
] | 2,018 | en | Physics | [
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5f11f4b894dd0a6fb862c694b10fe8c81ae01807 | abstract | 0 | 13 | Abstract | $\mu$Dose is a novel compact analytical instrument for assessing low level
$^{238}$U, $^{235}$U, $^{232}$Th decay chains and $^{40}$K radioactivity. The
system is equipped with a dual $\alpha$/$\beta$ scintillator allowing to
discriminate between $\alpha$ and $\beta$ particles. The unique build-in pulse
analyzer measur... | {
"cite_spans": []
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
] | 2,018 | en | Physics | [
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0.01158... |
b9104929afd94177c4de9a6b7465b319031f48fb | subsection | 1 | 13 | Introduction | In trapped charge dating (optically stimulated luminescence, thermoluminescence, electron spin resonance) age is determined from the equivalent total absorbed radiation dose and radiation dose rate. In the natural environment radiation dose rate arises from ^{238}U, ^{235}U, ^{232}Th decay chains and ^{40}K. Commonly t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1259/0007-1285-31-368-397",
"end": 846,
"openalex_id": "https://openalex.org/W2106531776",
"raw": "Turner, R., Radley, J., Mayneord, W., 1958. The alpha-ray activity of human tissues. British Journal of Radiology 31, 397–402.",
"so... | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
] | 2,018 | en | Physics | [
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0.0452... |
aec990d3b9c70d4cb1c1aa4a4275cf1b926d54f3 | subsection | 2 | 13 | Introduction | The second \beta /\alpha pair arises in the ^{238}U series from subsequent decays of ^{214}Bi and ^{214}Po where ^{214}Po has a half-life of 164 \mu s. Therefore, four decay pairs can be used to assess the specific activity of thorium and uranium decay chains as well as the potassium activity.The \mu Dose system is des... | {
"cite_spans": []
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
] | 2,018 | en | Physics | [
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c2c9efaab4fc56c481cedb2d2d3963cbce3a668b | subsection | 3 | 13 | System construction | \mu Dose is a very compact system (shown in Fig. REF ) as it takes just over 20 cm \times 20 cm of desk space and 35 cm height. The entire electronics, including a stable high voltage power supply, a photomultiplier and a pulse analyzer are built into the system and no additional components except a PC (which can contr... | {
"cite_spans": []
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
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ae2af20abf0ed5b4ece0c7cb4ee6723a35f1bb1f | subsection | 4 | 13 | Electronics | The pulse analyzer has been described in detail in , therefore here only a brief description is given. The \alpha and \beta particles produce scintillations in two different scintillator layers. The generated pulse shapes are different for each of the two scintillators, permitting the identification of the source parti... | {
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"raw": "Miłosz, S., Tudyka, K., Walencik-Łata, A., Barwinek, S., Bluszcz, A., Adamiec, G., 2017. Pulse Height, Pulse Shape, and Time Interval Analyzer for Delayed \\alpha /\\beta Coincidence Counting. IEEE ... | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
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"physics.ins-det",
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913510129ce24556633a67351ac500cb87400f05 | subsection | 5 | 13 | Body | Fig. REF shows a typical 2D pulse height vs. pulse shape histogram where the colour indicates relative frequency of the recorded pulses obtained from sample 1 with artificial ^{238}U, ^{235}U, ^{232}Th decay chains and ^{40}K concentrations (see paragraph ”Samples and sample preparation” for detailed description).
[Fig... | {
"cite_spans": []
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
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d600c6c635b6eda284466e9d53a12249dbbedf51 | subsection | 6 | 13 | Body | REF a, b, c, f is virtually absent.The four decay pairs count rates can be used to directly determine the activity per unit of mass\begin{aligned}& r_{Bi-212/Po-212}=k_{Bi-212/Po-212}a_{Bi-212/Po-212},\\
& r_{Bi-214/Po-214}=k_{Bi-214/Po-214}a_{Bi-214/Po-214},\\
& r_{Rn-220/Po-216}=k_{Rn-220/Po-216}a_{Rn-220/Po-216},\\
... | {
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rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
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b98e51859a08f263e294d5d8654723bac845cf63 | subsection | 7 | 13 | Body | (REF ) and (REF ) can be further restricted using a known ^{238}U/^{235}U isotopic ratio , \frac{^{238}U}{^{235}U}=\frac{a_{U-238}/\lambda _{U-238}}{a_{U-235}/\lambda _{U-235}}=137.88.where \lambda is decay constant of radioisotope indicated in subscript. Eq. (REF ) removes one degree of freedom and allows more precise... | {
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{
"arxiv_id": "",
"doi": "10.1016/j.gca.2014.09.034",
"end": 256,
"openalex_id": "https://openalex.org/W1987515641",
"raw": "Uvarova, Y.A., Kyser, T.K., Geagea, M.L., Chipley, D., 2014. Variations in the uranium isotopic compositions of uranium ores from different t... | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
] | 2,018 | en | Physics | [
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7d410536b4e1d406bbd63a85bc4063ae8d2e8063 | subsection | 8 | 13 | System calibration | The \mu Dose system needs to be calibrated with reference materials of known radioactivities, as well as a background sample. In the current work we use IAEA-RGU-1, IAEA-RGTh-1, and IAEA-RGK-1 standards from the International Atomic Energy Agency . The IAEA-RGU-1 and IAEA-RGTh-1 are produced using uranium and thorium o... | {
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{
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"start": 126
}
]
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
] | 2,018 | en | Physics | [
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5ec107ec8bcaab110f40cbc043e242174e71788a | subsection | 9 | 13 | Samples and sample preparation | To test the performance of \mu Dose, activities of five samples were assessed using two additional systems, namely, a high-purity germanium (HPGe) \gamma spectrometer and a conventional TSAC system.Sample 1 was an artificial sample composed from IAEA-RGU-1, IAEA-RGTh-1, and IAEA-RGK-1 mixed in equal weight proportions ... | {
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"raw": "Zöller, L., Pernicka, E., 1989. A note on overcounting in alpha-counters and its elimination. Ancient TL , 11–14.",
"source_ref_id": "18... | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
"nucl-ex"
] | 2,018 | en | Physics | [
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ddfe14f6fa6882f653546bc09aa0d13a3416120a | subsection | 10 | 13 | Measurements | The system set-up and data were evaluated according to .
\mu Dose specific uranium, thorium and potassium radioactivities were obtained using Eqs. (REF -REF ).The results and counting times are summarised in Table REF .
[Table: Specific radioactivity measurements using \mu Dose, a HPGe and a traditional TSAC system. Gi... | {
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{
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"doi": "10.1016/1359-0189(86)90064-6",
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"raw": "Aitken, M., 1985. Thermoluminescence Dating. Academic Press, London.",
"source_ref_id": "5157c28d00c53eec50ab6ad59882f666d02fb403",
... | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
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80fec72db30eac40a6c5f85e0c85a48132f318fe | subsection | 11 | 13 | System performance | As seen in Table REF , there is a very good agreement between the values obtained using the \mu Dose, gamma spectrometry and reference value for sample 1 (a mix of the IAEA standards - see previous subsection). In this case, the TSAC result significantly deviates from the known activities. This might be caused by a dif... | {
"cite_spans": []
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
"physics.ins-det",
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085afd44cfe4c1cac4798a8e2a9ee9301b532db0 | subsection | 12 | 13 | Conclusion | The \mu Dose system allows to detect \alpha and \beta radiation with four different decay pairs arising in the ^{238}U decay chain (^{214}Bi/^{214}Po), ^{232}Th decay chain (^{220}Rn/^{216}Po and ^{212}Bi/^{212}Po) and ^{235}U decay chain (^{219}Rn/^{215}Po). If the sample is close enough to secular equilibrium, the ob... | {
"cite_spans": []
} | 10.1016/J.RADMEAS.2018.07.016 | 1804.09714 | {\mu}Dose: a compact system for environmental radioactivity and dose
rate measurement | [
"Konrad Tudyka",
"Sebastian Miłosz",
"Grzegorz Adamiec",
"Andrzej Bluszcz",
"Grzegorz Poręba",
"Łukasz Paszkowski",
"Aleksander Kolarczyk"
] | [
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da98a8d2a918d4481d4174aba1ffff4933024e3c | abstract | 0 | 26 | Abstract | Let $ \lambda ^2 \in \mathbb N $, and in dimensions $ d\geq 5$, let $
A_{\lambda } f (x)$ denote the average of $ f \;:\; \mathbb Z ^{d} \to \mathbb
R $ over the lattice points on the sphere of radius $\lambda$ centered at $x$.
We prove $ \ell ^{p}$ improving properties of $ A_{\lambda }$.
\begin{equation*} \lVert A_{\... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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1afcdcfc0d5af5e32b95116c231895cb7102d280 | subsection | 1 | 26 | Introduction | The subject of this paper is in discrete harmonic analysis, in which continuous objects are studied in the setting of the integer lattice. Relevant norm properties are much more intricate, with novel
distinctions with the continuous case arising.In the continuous setting, L ^{p}-improving properties of averages over lo... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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e0a7fd12865fab39fda1070502c959a2a832b42b | subsection | 2 | 26 | Introduction | See Ionescu for an endpoint result, and the work of several others which further explore this topic MR1925339,MR2346547,160904313,MR3819049,MR3960006.Theorem B [Magyar, Stein, Wainger, ]
For d \ge 5, there holds\bigl \Vert \sup _{\lambda } \vert A _{\lambda } f \vert \bigr \Vert _{p} \lesssim \Vert f\Vert _{p}, \qquad ... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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33350cb591ab71e5908e5e0d9b7ba9c5dff09e5c | subsection | 3 | 26 | Introduction | For a cube Q \subset \mathbb {R} ^{d} of volume at least one, we set localized and normalized norms to be\langle f \rangle _{Q, p} := \Bigl [
\vert Q\vert ^{-1}
\sum _{n \in Q \cap \mathbb {Z} ^{d}} \vert f (n)\vert ^{p}
\Bigr ] ^{1/p}, \qquad 0< p \le \infty .An equivalent way to phrase our theorem above is the follow... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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9cd276eefa5b9a4916106cd1b119ff73e325db64 | subsection | 4 | 26 | Introduction | Fan Yang and the referee suggested several improvements of the paper. | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
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72476f07c34e77c88ec1a9e8b14fa7f76cd2a805 | subsection | 5 | 26 | Decomposition | Throughout e (x)= e ^{2 \pi i x}.
The Fourier transform on \mathbb {Z} ^{d} is given by\widehat{f} (\xi ) = \sum _{x\in \mathbb {Z} ^{d}} e (-\xi \cdot x) f (x), \qquad \xi \in \mathbb {T} ^{d} \equiv [0,1] ^{d}.We will write { for the inverse Fourier transform.
The Fourier transform on \mathbb {R} ^{d} is
\begin{equat... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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0fa1975099a3f1a1041a19cdaaca1b813ee8fb12 | subsection | 6 | 26 | Decomposition | We havec _{\lambda } (\xi ) &= \sum _{1\le q \le \lambda } c _{\lambda ,q} (\xi ) ,
\\
c _{\lambda , q } (\xi )&= \sum _{\ell \in \mathbb {Z}_q^d}
K (\lambda , q, \ell ) \Phi _{q} (\xi - \ell /q) \widetilde{d \sigma _{\lambda }} ( \xi - \ell /q),
\\
K (\lambda , q, \ell ) & =
q ^{-d} \sum _{ a \in \mathbb {Z}^\times ... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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7681151bef1c53408679ea05a446559266e4fddc | subsection | 7 | 26 | Decomposition | The implied constant only depends upon \eta >0.Concerning the terms \rho (q, \lambda ) , we need this Proposition.Proposition 2.8 We have for N< \lambda and a>1, and all \eta >0\sum _{ q \;:\; N \le q } q ^{-a} \rho (\lambda ,q) &\lesssim N ^{1-a} \sigma _{-1/2} (\lambda ^2 )
,
\\
\sum _{ 1\le q \le N} q ^{\eta } \rho... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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6ed431c4d31d8f927be8d629d7e6812a1b195f39 | subsection | 8 | 26 | Decomposition | The sum over r \ge 0 is just a geometric series, therefore we have to bound\sum _{t \;:\; t\,|\, \lambda ^2 } \sum _{ {s=1\\ st > N} } ^{\infty } \frac{1}{s ^{a} t ^{a- 1/2}}
& \lesssim \sum _{t \;:\; t\,|\, \lambda ^2 } \Bigl ( \frac{t}{N} \Bigr ) ^{a-1} \frac{1}{ t ^{a - 1/2}}
\\
& \lesssim N ^{1-a} \sum _{t \;:\; t... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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630449e6a6e058bfad69a9b169ae156288035725 | subsection | 9 | 26 | Decomposition | (The reader should note that the normalizations
here and in are different.)Lemma 2.11 * Lemma 1, page 71
We have, for all \epsilon >0, uniformly in \lambda \in \Lambda _d ,
\Vert R _{\lambda }\Vert _ {2\rightarrow 2} \lesssim _{\epsilon } \lambda ^{\frac{1-d}{2}+ \epsilon }.For a multiplier m on \mathbb {T} ^{d}, defi... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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bf15d7091bb658f765876b6983d6f94ee376d08c | subsection | 10 | 26 | Decomposition | Using the version formula on that group we have\sum _{\ell \in \mathbb {Z}^d_q } G (a/q, \ell ) e_q ( y \cdot \ell ) = e_q (\vert y\vert ^2 a), \qquad y \in \mathbb {Z} ^{d}_q .Definem ^{a/q} (\xi ) = e _q (-\lambda ^2 a) \sum _{ \ell \in \mathbb {Z}^d_q} G (a/q, \ell ) m (\xi - \ell /q), \qquad a \in \mathbb {Z} _q ^{... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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b2c5b6b6c871ec3e8665b0c748489ded99c8bbcf | subsection | 11 | 26 | Proof | It suffices to show this. For f = \mathbf {1}_{F} \subset E = [0, \lambda ] ^{d}\cap \mathbb {Z} ^{d},
choices of 0< \epsilon <1, and integers N we can writeA _{\lambda } f \le M_1 + M_2 ,
\\
\textup {where} \quad \langle M_1 \rangle _{E, \infty } \lesssim N ^{2} \langle f \rangle _{E}
\\
\textup {and} \quad \langle ... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
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-0.016919607296586037,
... | |
940e1492120f49e2932b6b33673da0db62d0e886 | subsection | 12 | 26 | Proof | We have\sigma _{- 1/2 } (n) \le \prod _{j=1} ^{\omega (n )} (1 - \tfrac{1}{ \sqrt{p_j} }) ^{-1} \lesssim e ^{c \frac{\sqrt{\omega (n)}}{\log \omega (n)}},where 2= p_1 < p_2 < \cdots is the increasing ordering of the primes. This is at most a constant depending upon \omega (n),
the number of distinct prime factors of n.... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.024707898497581482,
0.038091979920864105,
-0.04196832701563835,
-0.0110796382650733,
0.004620331339538097,
0.006535612978041172,
0.008614952675998211,
0.00396791473031044,
0.01693994179368019,
0.02305968850851059,
-0.0381835475564003,
0.018084533512592316,
-0.0381835475564003,
-0.010392... | |
d16452c61c93717f745049e3e610f893c2e61bb0 | subsection | 13 | 26 | Proof | Using the stationary decay estimate (REF ) and the Kloosterman refinement (REF ) to see that\langle M _{2,3} \rangle _{E,2} & \lesssim _{\epsilon } \langle f \rangle _E ^{1/2} \sum _{q\le N}
(q/N) ^{ \frac{d-1}{2}} q ^{\epsilon +\frac{1-d}{2}} \rho (\lambda ^2 , q)
\\
& \lesssim _{\epsilon } \langle f \rangle _E ^{1/2}... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.003340085968375206,
0.023729680106043816,
-0.04037860780954361,
-0.04550604894757271,
0.01748823933303356,
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-0.019914619624614716,
-0.014481971971690655,
0.00521518848836422,
0.023424474522471428,
-0.04217931628227234,
0.023073488846421242,
0.009194296784698963,
0.03... | |
97c1b1e85095413e47c93691bb98475852c09f95 | subsection | 14 | 26 | Proof | Indeed, for g = \mathbf {1}_{G} with G \subset E, we have for any integer N,\vert E\vert ^{-1} \langle A _{\lambda } f, g \rangle \lesssim _{\epsilon } N ^2 \langle f \rangle _E \langle g \rangle _E +
N ^{\epsilon +\frac{3-d}{2} } \sigma _ {- 1/2 } (\lambda ^2 ) \bigl [ \langle f \rangle _{E} \langle g \rangle _E \bigr... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.02184964343905449,
0.019362566992640495,
-0.009124361909925938,
-0.00817072857171297,
0.013686543330550194,
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0.005016110371798277,
-0.006206244695931673,
0.02014073170721531,
0.01563958451151848,
-0.05788934603333473,
-0.000286566762952134,
-0.03545989468693733,
0.0... | |
a407110e3e236cf13ffe73329b1e87f482ee32d7 | subsection | 15 | 26 | Proof | The second contribution to M_2 is the `large q' termM _{2,2} = \sum _{N\le q \le \lambda } C _{\lambda ,q} f .By the Weil estimates for Kloosterman sums (REF ), and Plancherel, we have\langle M _{2,2} \rangle _{E,2}& \lesssim _{\epsilon } \langle f \rangle _{E} ^{1/2} \sum _{N\le q \le \lambda } q ^{ \frac{1-d}{2} + \e... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.0378352627158165,
0.04585999995470047,
-0.05089452862739563,
-0.04805688560009003,
-0.00982496328651905,
-0.012868565507233143,
0.013631373643875122,
-0.00013945078535471112,
0.005835478659719229,
0.011312438175082207,
-0.015820631757378578,
0.022380776703357697,
0.014188222587108612,
0... | |
b82b7e2821619634c086e529d91caec8c9cfed62 | subsection | 16 | 26 | Proof | The definition of M _{1,2} is of the form to which
(REF ) applies.M _{1,2}(n) & \le \sum _{q \le N} q \cdot {_{ \lambda q/N} \ast d \sigma _{\lambda } \ast f (n)
\\ &\lesssim N \langle f \rangle _{E} \sum _{q\le N} 1 \lesssim N ^2 \langle f \rangle _E.
}
Observe that \Phi _{ \lambda q/N} \ast d \sigma _{\lambda } \ast ... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.035009339451789856,
0.005268950946629047,
-0.035314563661813736,
-0.03201812878251076,
0.019030794501304626,
-0.021182633936405182,
-0.003328863065689802,
0.017291011288762093,
0.020556921139359474,
0.004051865544170141,
-0.03656598553061485,
0.004536410328000784,
-0.016054848209023476,
... | |
ee92fb3299b3bdf11d907713be71498dab7940ba | subsection | 17 | 26 | Complements to the Main Theorems | Concerning sharpness of the \ell ^{p} improving estimates in Theorem REF ,
the best counterexample we have been able to find shows that if one has the inequality below,\Vert A _{\lambda } f\Vert _{p^{\prime }} \lesssim \lambda ^{ d (1- \frac{2}{p})} \Vert f\Vert _{p},valid for all \lambda , then necessarily p \ge \frac... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.05600474402308464,
0.011803724803030491,
-0.0012189042754471302,
-0.041446562856435776,
0.008797475136816502,
-0.03378596901893616,
0.04916819557547569,
0.03726528212428093,
0.03567822650074959,
0.00922475941479206,
-0.014886274002492428,
0.01876998133957386,
-0.043613504618406296,
0.02... | |
fb7309b3fa52027867819c1518ee3526421be0ef | subsection | 18 | 26 | Complements to the Main Theorems | We could not find this estimate in the literature.MR3819049article
author=Anderson, Theresa,
author=Cook, Brian,
author=Hughes, Kevin,
author=Kumchev, Angel,
title=Improved \ell ^p-boundedness for integral k-spherical maximal
functions,
journal=Discrete Anal.,
date=2018,
pages=Paper No. 10, 18,
issn=2397-3129,
review=3... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.046967122703790665,
0.0402531735599041,
-0.025406191125512123,
-0.027572965249419212,
-0.006404954008758068,
0.027206750586628914,
0.04391532763838768,
-0.052918121218681335,
0.02989232912659645,
0.03649946674704552,
0.009391898289322853,
0.03387492150068283,
-0.018554912880063057,
0.00... | |
c85195a03097617e042a32f4952baeeaaec22002 | subsection | 19 | 26 | Complements to the Main Theorems | Soc.,
volume=132,
number=5,
pages=14111417,
url=https://doi-org.prx.library.gatech.edu/10.1090/S0002-9939-03-07207-1,
review=2053347,181002240article
author=Kesler, R.,
author=Lacey, M. T.,
author=Mena Arias, D.,
title=Sparse Bound for the Discrete Spherical Maximal
Functions,
date=2018-10,
journal=Pure Appl. Analy., t... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.022932441905140877,
0.01870603859424591,
-0.05416511744260788,
-0.020308105275034904,
-0.01722603291273117,
0.031644634902477264,
0.011199209839105606,
-0.05312758684158325,
0.00529063493013382,
0.05685048550367355,
-0.02523636817932129,
0.02964586578309536,
-0.01573077030479908,
0.0060... | |
58b5c93d7f3a6b8dcd2afe8da17d3371783fd51f | subsection | 20 | 26 | Complements to the Main Theorems | Number Theory,
volume=122,
number=1,
pages=6983,
url=https://doi-org.prx.library.gatech.edu/10.1016/j.jnt.2006.03.006,
review=2287111,MR2346547article
author=Magyar, Akos,
author=Stein, Elias M.,
author=Wainger, Stephen,
title=Maximal operators associated to discrete subgroups of nilpotent
Lie groups,
date=2007,
ISSN=0... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.044738952070474625,
0.0010290187783539295,
-0.025726424530148506,
-0.003997819032520056,
-0.006202723365277052,
0.021438686177134514,
0.06817655265331268,
-0.04806538298726082,
0.0037021788302809,
0.07074003666639328,
-0.02268991246819496,
0.022079559043049812,
-0.032928600907325745,
-0... | |
9e861cca976b83028d60b5c4bdad15469a712aca | subsection | 21 | 26 | Complements to the Main Theorems | Soc.,
volume=16,
number=3,
pages=605638,
url=http://dx.doi.org.prx.library.gatech.edu/10.1090/S0894-0347-03-00420-X,
review=1969206,MR0027006article
author=Weil, André,
title=On some exponential sums,
date=1948,
ISSN=0027-8424,
journal=Proc. Nat. Acad. Sci. U. S. A.,
volume=34,
pages=204207,
review=0027006,Concerning s... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.042609598487615585,
0.024006053805351257,
-0.0026917150244116783,
-0.03726813942193985,
0.01854250207543373,
-0.031102566048502922,
0.04468514025211334,
0.015841247513890266,
0.025501662865281105,
0.010545570403337479,
-0.02631051279604435,
0.02600528486073017,
-0.04154130816459656,
0.0... | |
f80f0ceed649e1d85c66ba66373028f680e58efb | subsection | 22 | 26 | Complements to the Main Theorems | Notice that this estimate concerns the set of solutions n to a pair of quadratic equations
below in which x = (x_1 ,\cdots , x_d) is fixed.n_1 ^2 + \cdots + n _d ^2 &= \lambda ^2 ,
\\
(n_1 -x_1) ^2 + \cdots + (n _d -x_d) ^2 &= \lambda ^2 ,Moreover, we require of x that the set of possible solutions n should be of about... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.0665968582034111,
0.05432741343975067,
-0.042637862265110016,
-0.0257750004529953,
0.007233481388539076,
0.031009353697299957,
0.057043783366680145,
-0.054083243012428284,
0.022020915523171425,
0.019518190994858742,
0.009797247126698494,
0.01514605525881052,
-0.020525384694337845,
0.019... | |
ea3c55994ae3eb9e12facefac8f272577e954fc0 | subsection | 23 | 26 | Complements to the Main Theorems | Soc.,
volume=132,
number=5,
pages=14111417,
url=https://doi-org.prx.library.gatech.edu/10.1090/S0002-9939-03-07207-1,
review=2053347,Iarticle
author=Ionescu, Alexandru D.,
title=An endpoint estimate for the discrete spherical maximal
function,
date=2004,
ISSN=0002-9939,
journal=Proc. Amer. Math. Soc.,
volume=132,
numbe... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.01979040913283825,
0.02009558118879795,
-0.037108924239873886,
-0.025252988561987877,
-0.00745764235034585,
0.032653409987688065,
0.017501618713140488,
-0.04412788152694702,
0.02564971335232258,
0.06143113598227501,
-0.004844606388360262,
0.034759100526571274,
-0.019210582599043846,
0.0... | |
7aad7766d39529e189aabae46d3490ed958914ee | subsection | 24 | 26 | Complements to the Main Theorems | Number Theory,
volume=122,
number=1,
pages=6983,
url=https://doi-org.prx.library.gatech.edu/10.1016/j.jnt.2006.03.006,
review=2287111,MR2346547article
author=Magyar, Akos,
author=Stein, Elias M.,
author=Wainger, Stephen,
title=Maximal operators associated to discrete subgroups of nilpotent
Lie groups,
date=2007,
ISSN=0... | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.044738952070474625,
0.0010290187783539295,
-0.025726424530148506,
-0.003997819032520056,
-0.006202723365277052,
0.021438686177134514,
0.06817655265331268,
-0.04806538298726082,
0.0037021788302809,
0.07074003666639328,
-0.02268991246819496,
0.022079559043049812,
-0.032928600907325745,
-0... | |
82e5799d30a06cd48ece83110c67d30f2e672d6f | subsection | 25 | 26 | Complements to the Main Theorems | Soc.,
volume=16,
number=3,
pages=605638,
url=http://dx.doi.org.prx.library.gatech.edu/10.1090/S0894-0347-03-00420-X,
review=1969206,MR0027006article
author=Weil, André,
title=On some exponential sums,
date=1948,
ISSN=0027-8424,
journal=Proc. Nat. Acad. Sci. U. S. A.,
volume=34,
pages=204207,
review=0027006, | {
"cite_spans": []
} | 1804.09845 | $\ell^p$-improving inequalities for Discrete Spherical Averages | [
"Robert Kesler",
"Michael T. Lacey"
] | [
"math.CA"
] | 2,018 | en | Mathematics | [
-0.035099469125270844,
0.03940296918153763,
-0.02038821391761303,
-0.012422160245478153,
-0.007756219711154699,
-0.007706622593104839,
0.05918075889348984,
0.012315335683524609,
-0.021410677582025528,
0.011590455658733845,
-0.027072373777627945,
0.0109876599162817,
-0.028110098093748093,
-... | |
e3f2c699903b3dd1931baea6583f5f38e69b95bf | abstract | 0 | 94 | Abstract | Linear matrix inequalities (LMIs) are ubiquitous in real algebraic geometry,
semidefinite programming, control theory and signal processing. LMIs with
(dimension free) matrix unknowns are central to the theories of completely
positive maps and operator algebras, operator systems and spaces, and serve as
the paradigm fo... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.0384402722120285,
0.0020917151123285294,
-0.031392887234687805,
0.011326150968670845,
0.03246067464351654,
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0.035938601940870285,
0.04191819950938225,
0.050582513213157654,
0.012546476908028126,
-0.05061302334070206,
0.004137667827308178,
0.025031939148902893,
0.0... |
bf2ca1e357aebeb1ae1a0ad8c8dbc0ef29ed00a9 | subsection | 1 | 94 | Introduction | Fix a positive integer g.
For positive integers n, let M_n(\mathbb {C})^g denote the set of g-tuples X=(X_1,\dots ,X_g) of n\times n matrices with entries from \mathbb {C}.
Given a tuple E=(E_1,\dots ,E_g) of d\times e matrices, the sequence
\mathcal {B}_E= (\mathcal {B}_E(n))_n defined by\mathcal {B}_E(n) =\lbrace X\i... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.027740541845560074,
-0.01051332987844944,
-0.04049691930413246,
-0.009132405743002892,
0.05624402314424515,
-0.06335463374853134,
0.037506189197301865,
0.028122011572122574,
-0.00492478534579277,
0.038238611072301865,
-0.04495249316096306,
-0.009010335430502892,
0.0015735662309452891,
-... |
4a9bb9f27a56b523fd0bc034724fde6d3202ebf3 | subsection | 2 | 94 | Introduction | Thus \mathcal {B}_E(n) is the set of
tuples X\in M_n(\mathbb {C})^g such that \Vert X_j\Vert \le 1 for each j.For A\in M_d(\mathbb {C})^g, let L_A(x,y) denote the monic pencilL_A(x,y) = I+\sum A_jx_j +\sum A_j^* y_j, \index {L_A(x,y)}and letL^{\rm re}_A(x)=L_A(x,x^*)= I+\sum A_jx_j +\sum A_j^* x_j^* \index {L^{\rm re}_... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.023870831355452538,
0.01449191477149725,
-0.022024046629667282,
-0.027304934337735176,
0.036050450056791306,
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0.022771919146180153,
0.03183795139193535,
0.02098618447780609,
0.017750496044754982,
-0.0280222799628973,
0.020467253401875496,
0.015186366625130177,
-0.019... |
b24fcdadbb95e7cc3e7147bc540a4fdc6b2ebc60 | subsection | 3 | 94 | Introduction | Free spectrahedra arise naturally in applications such as systems engineering and in the theories of matrix convex
sets, operator algebras and operator spaces and completely positive maps , , , . They also provide tractable useful
relaxations for spectrahedral inclusion problems that arise in semidefinite programming a... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.021470647305250168,
0.004192651249468327,
-0.043032851070165634,
0.01567189395427704,
0.03201522305607796,
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0.0475497767329216,
0.04269713535904884,
0.08453971892595291,
0.03537239506840706,
-0.014306134544312954,
-0.008080104365944862,
0.02446158230304718,
-0.02900... |
c54e8aee63029f8d5f0651570349f55eef35e1ac | subsection | 4 | 94 | Introduction | In the special case that \mathcal {D}_A=\mathcal {B}_C is also
a spectraball, given b\in \operatorname{int}(\mathcal {B}_C) and
a g\times g matrix M, Corollary REF gives explicit necessary
and sufficient algebraic relations
between E and C for the existence of
a free bianalytic mapping \varphi :\operatorname{int}(\math... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.000307065638480708,
0.006236674729734659,
-0.029676653444767,
-0.004375208634883165,
0.03857202082872391,
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0.018538372591137886,
0.0170736126601696,
0.05016804113984108,
0.014121204614639282,
-0.013877077959477901,
0.01695154793560505,
0.022612236440181732,
0.021162... |
f0d92474a06575e56cea4e8c861e87d79253f20d | subsection | 5 | 94 | Convexotonic maps | A g-tuple of g\times g matrices (\Xi _1,\dots ,\Xi _g)\in M_g(\mathbb {C})^g satisfying\Xi _k \Xi _j = \sum _{s=1}^g (\Xi _j)_{k,s} \Xi _s,for each 1\le j,k\le g, is a convexotonic tuple.
The expressions
p=\begin{pmatrix} p^1 & \cdots & p^g\end{pmatrix}
and q=\begin{pmatrix} q^1 & \cdots & q^g\end{pmatrix}
whose entrie... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.007959195412695408,
0.003952889237552881,
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0.022710034623742104,
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-0.030... |
261b7e3499e8b777a42c518405a921bc5d07a7de | subsection | 6 | 94 | Free bianalytic maps from a spectraball to a free spectrahedron | A tuple E\in M_{d\times e}(\mathbb {C})^g is ball-minimal (for \mathcal {B}_E)
if there does not exist E^{\prime } of size d^{\prime }\times e^{\prime } with
d^{\prime }+e^{\prime }<d+e such that \mathcal {B}_E=\mathcal {B}_{E^{\prime }}.
In fact, if E is ball-minimal and \mathcal {B}_{E^\prime } =\mathcal {B}_E, then ... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.004919573664665222,
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0.015940943732857704,
-0.0... |
9a07cf53dbe7e46cb876b8fe72c70993677c41f9 | subsection | 7 | 94 | Free bianalytic maps from a spectraball to a free spectrahedron | If f:\operatorname{int}(\mathcal {B}_E)\rightarrow \operatorname{int}(\mathcal {D}_A) is a free bianalytic mapping
with f(0)=0 and f^\prime (0)=I_g, then f is convexotonic.If, in addition, A is minimal for \mathcal {D}_A, then
there is convexotonic tuple \Xi \in M_g(\mathbb {C})^g such that
equation (REF ) holds, and f... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
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0.003865067381411791,
-0.002012658631429076,
-0... |
b262d1ff39842ec765f871e1920b0683108a3fd9 | subsection | 8 | 94 | Free bianalytic maps from a spectraball to a free spectrahedron | In particular, f is, up to affine linear equivalence, convexotonic.
Further, with a bit of bookkeeping the algebraic conditions
of equations (REF ) and (REF ) can be
expressed intrinsically in terms of E and A. In the case
\mathcal {D}_A is a spectraball, these conditions are spelled
out in Corollary REF below.
In the... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.013079204596579075,
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0.027928756549954414,
-0.018359722569584846,
-0... |
5972b18034fe158ba97aff39efe5797880201910 | subsection | 9 | 94 | Free bianalytic maps from a spectraball to a free spectrahedron | There exists a free bianalytic mapping
\varphi :\operatorname{int}(\mathcal {B}_E)\rightarrow \operatorname{int}(\mathcal {B}_C) such
that \varphi (0)=b and M=\varphi ^\prime (0) if and only if
E and C have the same size (that is, k=d and \ell = e)
and there exist d\times d and e\times e unitary
matrices W and {V} resp... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.013929431326687336,
-0.009909272193908691,
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0.031245864927768707,
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0.03380900248885155,
0.029369283467531204,
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-0.009443940594792366,
0.04006427899003029,
... |
2dd8d47067170e30dc6aeadad15ee09b7f188545 | subsection | 10 | 94 | Free bianalytic maps from a spectraball to a free spectrahedron | In particular, given a ball-minimal tuple C\in M_{d\times e}(\mathbb {C})^g
and b\in \operatorname{int}(\mathcal {B}_C), if equation (REF ) holds
then, for any choice of M,W and {V} and solving
equation (REF ) for E,
there is a free bianalytic map \varphi :\operatorname{int}(\mathcal {B}_E)\rightarrow \operatorname{int... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.019672496244311333,
-0.0074439565651118755,
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0.050058797001838684,
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0.006982286460697651,
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0.004223710857331753,
0.03476644307374954,
0.0... |
1b801413a97007999e57de271afebf4bb2640ee5 | subsection | 11 | 94 | Main result on maps between free spectrahedra | The article
characterizes the triples (p,A,B) such that p:\mathcal {D}_A\rightarrow \mathcal {D}_B
is bianalytic under
unconventional geometric hypotheses (sketched in Subsection REF below), cf. .
Here we obtain Theorem REF by converting those geometric hypotheses to algebraic irreducibility hypotheses
that we now des... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.06117338687181473,
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0.024817347526550293,
-0.037088654935359955,
0.03641708940267563,
0.00788324698805809,
0.003... |
cd6297488f93231f744e020ec7afe84303d1dc29 | subsection | 12 | 94 | Main result on maps between free spectrahedra | Thus elements of \mathbb {C}\!\mathop {<}\!x\!\mathop {>} are finite \mathbb {C}-linear combinations of words
in the letters \lbrace x_1,\dots ,x_g\rbrace . For each positive integer n,
an element p of \mathbb {C}\!\mathop {<}\!x\!\mathop {>} naturally
induces a function, also denoted p, mapping M_n(\mathbb {C})^g\righ... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.03747306019067764,
-0.020628493279218674,
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0.00014435272896662354,
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0.035398006439208984,
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-0.04607843980193138,
-0.030713872984051704,
... |
d32170289ac239683febffea2781acf017967e97 | subsection | 13 | 94 | Main result on maps between free spectrahedra | As a consequence of Lemma REF (REF )
below, we will see that
if Q_E is an atom, \ker (E)=\lbrace 0\rbrace and \ker (E^*)=\lbrace 0\rbrace ,
then E is ball-minimal.Theorem 1.5
Suppose A\in M_d(\mathbb {C})^g, B\in M_e(\mathbb {C})^g and\mathcal {D}_A is bounded;
Q_A and Q_B are atoms, \ker (B)=\lbrace 0\rbrace
and A... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.01232178695499897,
0.013725630939006805,
-0.015404140576720238,
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0.04071148857474327,
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0.046082720160484314,
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0.02853466384112835,
-0.013527262024581432,
-... |
b577b1a5ce59b2f9fd21dfd3ee9c44d8ada9f1a1 | subsection | 14 | 94 | Geometry of the boundary vs irreducibility | At the core of the proofs of our main theorems in this paper
is a richness of the
geometry of the boundary, \partial \mathcal {B}_E, of a
spectraball, \mathcal {B}_E.
We shall show that a (rather ungainly) key geometric property of the
boundary of \mathcal {B}_E is
equivalent to the defining polynomial Q_E of \mathcal... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
0.011560752987861633,
0.017062602564692497,
-0.015948496758937836,
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0.015597477555274963,
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0.052408747375011444,
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0.028798865154385567,
0.004570885095745325,
0.0068... |
7dd55e16dc8d8aad34075af4a8db89b24f2d7321 | subsection | 15 | 94 | A Nullstellensatz | Theorem REF uses the following Nullstellensatz
whose proof depends upon Theorem REF .Proposition 1.7
Suppose E=(E_1,\dots ,E_g)\in M_{d\times e}(\mathbb {C})^g is ball-minimal and V\in \mathbb {C}\!\mathop {<}\!x\!\mathop {>}^{\ell \times e}
is a (rectangular) matrix polynomial.
If V vanishes on \widehat{\partial \mat... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.015572072006762028,
0.004734184592962265,
-0.0235297754406929,
-0.00011408678983571008,
0.024185923859477043,
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0.05697806552052498,
0.03625599667429924,
0.005760369822382927,
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0.0056039621122181416,
-0.01459547970443964,
... |
321887418873eb05d0a411ac8f4b8d18319abad2 | subsection | 16 | 94 | An overview of the proof of Theorem | We are now in a position to convey, in broad strokes, an outline of the
proof of Theorem REF . The conversely direction
is an immediate consequence of Proposition REF
(see Corollary REF ) of Section
. Its proof reflects the fact that convexotonic maps are bianalytic
between certain special spectrahedral pairs. Proposi... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.055499929934740067,
0.004580498673021793,
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0.001135589904151857,
0.026331191882491112,
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0.029275527223944664,
0.018916962668299675,
0.0523572713136673,
0.018215205520391464,
-0.025827758014202118,
0.015217476524412632,
-0.019756022840738297,
0.... |
2fca3847c1888f3a84b9daf68f8832a01c04fdbf | subsection | 17 | 94 | Free rational maps and convexotonic maps | In this section we review the notions of a free set and free rational function
and provide further background on free functions and mappings. In particular,
convexotonic maps are seen to be free rational mappings. In Subsection REF we show
how algebras of matrices give rise to convexotonic bianalytic maps between free ... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.023151425644755363,
-0.0033613061532378197,
-0.018618812784552574,
0.015123970806598663,
0.02902703545987606,
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0.03171302750706673,
0.04682173579931259,
0.0715450793504715,
0.00905759446322918,
-0.0004893161240033805,
-0.005944285076111555,
-0.008729476481676102,
-0... |
a9139d579db310bac7c35dd2df6e743be1e7fc8a | subsection | 18 | 94 | Free sets, free analytic functions and mappings | Let M(\mathbb {C})^g denote the sequence (M_n(\mathbb {C})^g)_n.
A subset \Gamma of M(\mathbb {C})^g is a sequence (\Gamma _n)_n where
\Gamma _n \subseteq M_n(\mathbb {C})^g. (Sometimes we write \Gamma (n) in place of \Gamma _n.)
The subset \Gamma is a free set if it is closed under direct sums and simultaneous unitary... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.025498542934656143,
-0.030549421906471252,
-0.07898291945457458,
-0.01565619744360447,
0.04818934574723244,
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0.02896243892610073,
0.05963393673300743,
0.02951177954673767,
0.010971545241773129,
-0.031007204204797745,
0.0022393243853002787,
0.016449688002467155,
0.00... |
dcf172e6f8172b2e29f2dffbb71ea80d81bf254d | subsection | 19 | 94 | Free rational functions and mappings | Based on the results of
and a free rational function regular at 0
can, for the purposes of this article,
be defined with minimal overhead as an expression of the formr(x)= c^* \big (I-\Lambda _S(x)\big )^{-1} b,where, for some positive integer s, we have
S\in M_s(\mathbb {C})^g and b,c\in \mathbb {C}^s.
The expression... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.018966378644108772,
-0.005165019538253546,
-0.013664032332599163,
0.025649623945355415,
0.040984466671943665,
-0.019424134865403175,
0.06011869013309479,
0.030593395233154297,
0.051665451377630234,
0.02012602798640728,
-0.029922017827630043,
0.0000883923057699576,
-0.003343529999256134,
... |
2048bfd8047444c2f800e6006b36df70705ebbcc | subsection | 20 | 94 | Algebras and convexotonic maps | Theorem REF below is an expanded version of .
To begin we discuss a sufficient condition
for a tuple X\in M_n(\mathbb {C})^g to lie in \operatorname{dom}(p), the domain of a convexotonic mappingp=\begin{pmatrix} p^1 & \cdots & p^g\end{pmatrix}
= x(I-\Lambda _\Xi (x))^{-1}.Sincep^j = \sum _{k=1}^g x_k \left[ e_k^* (I-\L... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.015239613130688667,
0.004069236107170582,
-0.013645479455590248,
-0.026055315509438515,
0.013683617115020752,
-0.01732953079044819,
0.012142875231802464,
-0.005118008237332106,
0.013454793952405453,
0.019678780809044838,
-0.03819819167256355,
-0.022577205672860146,
-0.041615281254053116,
... |
3074d8a544baf9f6f0f5935063f87d743d896295 | subsection | 21 | 94 | Algebras and convexotonic maps | If there exists
a tuple \Xi \in M_g(\mathbb {C})^g such that\mathfrak {A}_\ell (U-I)\mathfrak {A}_j = \sum _{s=1}^g (\Xi _j)_{\ell ,s} \mathfrak {A}_s,then \Xi is convexotonic and the convexotonic maps
p and q associated to \Xi are
bianalytic maps between \mathcal {D}_\mathfrak {A} and \mathcal {D}_{\mathfrak {B}} in t... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.038339484483003616,
0.0004266355244908482,
-0.03150187060236931,
0.013224986381828785,
0.007684686221182346,
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0.02902933955192566,
0.031623970717191696,
0.02754887379705906,
0.03956048935651779,
-0.025045819580554962,
-0.004376531578600407,
-0.01707877591252327,
-0... |
e15224da3e1e93c5f60b7bc0677aceeb96c6176c | subsection | 22 | 94 | Algebras and convexotonic maps | If X\in \operatorname{dom}(p), but X\notin \operatorname{int}(\mathcal {B}_J), then p(X)\notin \operatorname{int}(\mathcal {D}_J).
If \mathcal {D}_J is bounded, then the domain of p contains \mathcal {B}_J and
p(\partial \mathcal {B}_J)\subseteq \partial \mathcal {D}_J.
If Y\in \operatorname{dom}(q), but Y\notin \m... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04572167247533798,
0.012758116237819195,
-0.022326704114675522,
0.021777311339974403,
0.013269356451928616,
-0.021075310185551643,
0.03183424845337868,
0.03824382647871971,
-0.0011464754352346063,
0.03482538461685181,
-0.045172277837991714,
0.011041264981031418,
-0.01832834631204605,
0.... |
6d3393c341bd8f0db3206aaf4d910313fbcf0987 | subsection | 23 | 94 | Algebras and convexotonic maps | Indeed, it is immediate that I-T is invertible andI+(I-T)^{-1}T + \left( (I-T)^{-1}T\right)^*
= (I-T)^{-1}\, \left(I-TT^*\right) \, (I-T)^{-*} \succ 0.The proof of item (REF ) is similar.[Proof of Proposition REF ]
Compute\begin{split}
\Lambda _{J}(q(x))\, \Lambda _J(x)
& = \sum _{s,k=1}^g q^s(x) x_k J_s J_k
= \sum _{j... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03897876664996147,
0.036903154104948044,
-0.007997210137546062,
0.006959403865039349,
0.012522349134087563,
-0.007943793199956417,
0.03183622285723686,
0.0175206046551466,
0.03678106144070625,
0.022190731018781662,
-0.0064214239828288555,
-0.0015834170626476407,
-0.0325687900185585,
0.0... |
cdf18652b99a6d1776e611e29fc7dd21338169ef | subsection | 24 | 94 | Algebras and convexotonic maps | See , and also , for full details.)
By Lemma REF , {D} contains \mathcal {D}_J,
(as X\in \mathcal {D}_J implies I+\Lambda _{J}(X) is invertible). Hence
the domain of the free rational mapping q contains \mathcal {D}_J.
By Lemma REF and equation (REF ), q maps the interior of \mathcal {D}_{J} into the
interior of \mathc... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03724246099591255,
0.0221165269613266,
-0.016835425049066544,
0.013271442614495754,
0.017171217128634453,
-0.010508785024285316,
0.020254405215382576,
0.00790639128535986,
0.0548868402838707,
0.022223370149731636,
-0.013141704723238945,
0.020025454461574554,
-0.00007572030153824016,
0.0... |
34d84d2797741af01df84786be9176779a9076ca | subsection | 25 | 94 | Algebras and convexotonic maps | The proof of (REF ) is similar.The converse portion of Theorem REF is an immediate consequence
of Proposition REF , stated below as Corollary REF .Corollary 2.5
Suppose E\in M_{d\times e}(\mathbb {C})^g is linearly independent, r\ge \max \lbrace d,e\rbrace ,
the r\times r matrix U is unitary andA= U\begin{pmatrix} 0 &... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.02040008455514908,
0.020857827737927437,
-0.011939465999603271,
-0.003162242006510496,
0.03137066215276718,
-0.027144165709614754,
-0.001382765593007207,
0.024092545732855797,
0.02589300088584423,
0.02586248517036438,
-0.024657094851136208,
0.0107340756803751,
-0.002124690916389227,
0.0... |
591a29cd76fee97ae19146324d1688bd031f52a4 | subsection | 26 | 94 | Algebras and convexotonic maps | Let \Xi \in M_h(\mathbb {C})^h denote the convexotonic tuple associated to J, let
p:\operatorname{int}(\mathcal {B}_J)\rightarrow \operatorname{int}(\mathcal {D}_J) denote the corresponding convexotonic map,
let q denote the inverse of p, and let
\iota :\operatorname{int}(\mathcal {D}_A)\rightarrow \operatorname{int}(\... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.019085202366113663,
0.027247140184044838,
-0.027842123061418533,
-0.017269743606448174,
-0.0004886673996224999,
-0.03240365535020828,
0.010267259553074837,
0.016995135694742203,
0.031518809497356415,
0.05150411278009415,
-0.004595857113599777,
0.0037968263495713472,
-0.01546191330999136,
... |
c7ffac24bc062b20cb090f245c9d25a3b62c84af | subsection | 27 | 94 | Algebras and convexotonic maps | Hence (X^{n_j})_j converges to X
and we conclude that K_* is compact. Thus \iota
is proper. Since q is also proper, f=q\circ \iota is too.
Letting z=(z_1,\dots ,z_{h}) denote an h tuple of freely noncommuting
indeterminates,q(z) = z (I+\Lambda _{\Xi }(z))^{-1}and thus f takes the form of equation (REF ). | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04076888784766197,
0.05181555077433586,
-0.011466249823570251,
0.011580683290958405,
-0.026350250467658043,
-0.006518902722746134,
0.0381140299141407,
0.010932226665318012,
0.015372251160442829,
0.048519860953092575,
-0.010512636043131351,
0.005870445165783167,
-0.022276414558291435,
0.... |
8183fb74505a0af8a2be42d4e5c8e2d525e91a6c | subsection | 28 | 94 | Proof of Theorem | Lemma 2.7
Suppose G\in M_{d\times e}(\mathbb {C})^g
is linearly independent,
C\in M_{e\times d}(\mathbb {C}) and \Psi \in M_g(\mathbb {C})^g.
IfG_\ell C G_j =\sum _{s=1}^g (\Psi _j)_{\ell ,s} G_s,then the tuple \Psi is convexotonic. | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04353610798716545,
0.03642755001783371,
0.002637107390910387,
-0.0055526080541312695,
0.023537566885352135,
-0.10098424553871155,
-0.0024273591116070747,
0.016123918816447258,
0.022027378901839256,
0.038319095969200134,
-0.025673184543848038,
-0.0053199781104922295,
-0.015574759803712368,... |
5a7c6c6bcfeb6d7c4823da917b5d2c17cf2b43a6 | subsection | 29 | 94 | Proof of Theorem | Moreover, letting
T=CG\in M_e(\mathbb {C})^g,G_\ell T^\alpha = \sum _{s=1}^g (\Psi ^\alpha )_{\ell ,s} G_s.In particular,
if A\in M_d(\mathbb {C})^g is linearly independent and spans an algebra,
then the tuple \Psi uniquely determined by equation
(REF ) is convexotonic.Note that the hypothesis implies T spans an algebr... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.017032351344823837,
0.03314898535609245,
-0.02780729904770851,
-0.003712472738698125,
0.01861959509551525,
-0.0651380643248558,
-0.0008441773825325072,
0.015628250315785408,
0.01111834030598402,
0.04154306650161743,
-0.03308793902397156,
0.008455128408968449,
-0.025548527017235756,
-0.0... |
138b811de82d732af9c3ed55f7f5e5ef48320266 | subsection | 30 | 94 | Proof of Theorem | From Lemma REF ,
for words \alpha and 1\le j\le g,A_j R^\alpha = \sum _{s=1}^g (\Xi ^\alpha )_{j,s} A_s.HenceB_j R^\alpha = \sum _{s=1}^g (\Xi ^\alpha )_{j,s} B_s,from which it follows that,
letting \lbrace e_1,\dots , e_g\rbrace denote the standard basis for \mathbb {C}^g,\begin{split}
\Lambda _B(p(x)) & = \sum _s B_s... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.004115563351660967,
0.008620179258286953,
-0.056694839149713516,
-0.03869163617491722,
0.012762442231178284,
-0.04497750103473663,
0.041773539036512375,
0.012876869179308414,
0.007014384493231773,
-0.0382034108042717,
-0.03991219028830528,
0.011335916817188263,
-0.021024081856012344,
0.... |
b821fda839ba787686bcd3c9381aa876a9a1c20d | subsection | 31 | 94 | Proof of Theorem | Finally, using
\Lambda _B(p(x))Q(x)=\Lambda _B(x) as well as R=B-A and B=UA,\begin{split}
Q(Z)^* L^{\rm re}_B(p(Z)) Q(Z) &= Q^*(Z)Q(Z)+ Q(Z)^*\Lambda _B(Z) +\Lambda _B(X)^*Q(Z) \\
& = I+\Lambda _A(Z) +\Lambda _A(Z) +\Lambda _B(Z)^*\Lambda _B(Z)-\Lambda _A(Z)^*\Lambda _A(Z) = L^{\rm re}_A(Z).
\end{split}a routine calcul... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.030939435586333275,
0.02767462283372879,
-0.038170840591192245,
-0.04878911003470421,
0.01684276945888996,
0.018963372334837914,
0.05440336838364601,
-0.0009849739726632833,
0.004599723964929581,
-0.00618636142462492,
-0.030344447121024132,
0.010252120904624462,
0.030497007071971893,
0.... |
3944c3572d5b3e19d8f70e47052866ddc2354b64 | subsection | 32 | 94 | Proof of Theorem | \end{split}Thus, in the notation of equation (REF ), {I}_R\subseteq \operatorname{dom}(p);
that is, if Q(Z)=I-\Lambda _R(Z) is invertible,
then Z\in \operatorname{dom}(p), proving
item (REF ).[Proof of Theorem REF ]
That \Xi is convexotonic follows from Lemma REF .
Let p denote the resulting convexotonic map.
Let R=\ma... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.024417895823717117,
0.02144196443259716,
-0.005234586540609598,
0.0004494895983953029,
0.01265915296971798,
-0.026966514065861702,
0.024402635172009468,
-0.015352752059698105,
0.010568371042609215,
0.027882184833288193,
-0.032964158803224564,
0.024478940293192863,
-0.01412422675639391,
... |
fac4f6b28d76678db628578fe3d48d4b497c43f6 | subsection | 33 | 94 | Proof of Theorem | Hence,
for t real and positive,\begin{split}
\langle &(F_X(te^{i \theta }) + F_X(te^{i\theta })^*)\gamma ,\gamma \rangle \\
& = t^{-m} \langle [e^{-im \theta } \Psi (te^{i\theta })
+e^{im\theta }\Psi (te^{i\theta })^*]\gamma ,\gamma \rangle \\
& = t^{-m} \left[ 2\langle e^{-im\theta } \Psi (0)\gamma ,\gamma \rangle +
\... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.04156980291008949,
0.044011496007442474,
-0.024615304544568062,
0.009888852015137672,
0.0012837958056479692,
-0.03369534760713577,
0.03339013457298279,
-0.00740900868549943,
-0.002596205100417137,
0.02983442135155201,
-0.0227840356528759,
0.01591677777469158,
-0.02426431141793728,
-0.01... |
80ae2336cbd646e32157d358fe4f790f3b2331e0 | subsection | 34 | 94 | Proof of Theorem | Consequently, p:\operatorname{int}(\mathcal {D}_\mathfrak {A})\rightarrow \operatorname{int}(\mathcal {D}_\mathfrak {B})
is bianalytic with inverse q:\operatorname{int}(\mathcal {D}_{\mathfrak {B}})\rightarrow \operatorname{int}(\mathcal {D}_{\mathfrak {A}}),
proving item (REF ).If X\in \operatorname{ext}(\mathcal {D}_... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03991019353270531,
0.02984110824763775,
-0.020794186741113663,
-0.0013921273639425635,
0.006079591810703278,
0.004916307050734758,
0.048392634838819504,
0.0248981025069952,
-0.02502015233039856,
0.033777203410863876,
-0.04479217529296875,
0.02473028376698494,
0.017254751175642014,
-0.01... |
08379ac5b83d26c32640e49295add7e2d3d45424 | subsection | 35 | 94 | Proof of Theorem | Further p(tX)\in \operatorname{int}(\mathcal {D}_A)
for t<1 and p(tX)\in \operatorname{ext}(\mathcal {D}_{\mathfrak {B}}) for t>1.
By continuity, p(X)\in \partial \mathcal {D}_{\mathfrak {B}}, proving item (REF ).We use Proposition REF .
In the terminology of ,
assumptions (REF )
and (REF )
imply that A is eig-generic ... | {
"cite_spans": []
} | 10.1016/j.jfa.2020.108472 | 1804.09743 | Bianalytic free maps between spectrahedra and spectraballs | [
"J. William Helton",
"Igor Klep",
"Scott McCullough",
"Jurij Volčič"
] | [
"math.FA"
] | 2,018 | en | Mathematics | [
-0.03451415151357651,
0.02506929636001587,
-0.03201179951429367,
0.02502352185547352,
-0.002078935969620943,
-0.00026415835600346327,
0.05013859272003174,
0.022551685571670532,
0.006248251534998417,
0.023238306865096092,
-0.028899116441607475,
0.0028418481815606356,
-0.012389695271849632,
... |
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