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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
edd7adbf3105f4cffe6dd2387256c14f77fd86e3 | subsection | 36 | 94 | Minimality and indecomposability | A monic pencil L_A=L_A(x,y) of size e is indecomposable
if its coefficients \lbrace A_1,\dots ,A_g,A_1^*,\dots ,A_g^*\rbrace generate
M_{e}(\mathbb {C}) as a \mathbb {C}-algebra.Previously, in
such pencils were called irreducible.
A collection of sets \lbrace S_1,\dots ,S_k\rbrace is irredundant if
\bigcap _{j\ne \ell... | {
"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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0.0434885174036026,
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0.00242810882627964,
0.003091881051659584,
0.001... |
b7e09d8a99646059261b6c5bd8caf09ff2207437 | subsection | 37 | 94 | Minimality and indecomposability | If A\in M_N(\mathbb {C})^g and A_mA_j=0 for all 1\le j,m\le g
then, \dim \operatorname{rg}A + \dim \operatorname{rg}A^* \le N and for any s\ge \dim \operatorname{rg}A
and t\ge \dim \operatorname{rg}A^* with s+t= N,
there exists a tuple F \in M_{s\times t}(\mathbb {C})^g such that A is unitarily equivalent to
\begin{pm... | {
"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... |
92ce616a678e2498e434b4362b34f4e412196f00 | subsection | 38 | 94 | Minimality and indecomposability | Hence
\mathcal {B}_E=\mathcal {B}_F if and only if \Phi is completely isometric.(REF ) Straightforward.(REF ) By (REF ), Q_E and \mathbb {L}_{E} are stably associated, cf.
. Hence \mathbb {L}_{E} does not factor in \mathbb {C}\!\mathop {<}\!x,y\!\mathop {>}^{(d+e)\times (d+e)}
if and only if Q_E does not factor in \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 | [
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... |
65b51865d1942800b8991710c4ed4307fe3e77fc | subsection | 39 | 94 | Minimality and indecomposability | Now any choice of s\ge \dim {R} and t\ge \dim {R_*} with s+t=N applied to (REF ) gives the desired decomposition.(REF )
Since L_A is minimal,
by Lemma REF , L_A is
unitarily equivalent to L_{A^1}\oplus \cdots \oplus L_{A^k} for some indecomposable irredundant monic pencils L_{A^1},\dots ,L_{A^k}. Let N_j denote the siz... | {
"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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0.02307193912565708,
0.026291634887456894,
0.01479... |
f08d7de3fe9438caddb0987ed17cf7c560bbf65d | subsection | 40 | 94 | Minimality and indecomposability | Thus \mathbb {L}^{\rm re}_E is minimal
and hence E is ball-minimal by item (REF ).(REF )
LetA= \begin{pmatrix} 0&E\\0 &0\end{pmatrix} \in M_{d+e}(\mathbb {C})^g.By item (REF ), L^{\rm re}_A=\mathbb {L}^{\rm re}_E is minimal.
Since \mathbb {L}^{\rm re}_F defines
\mathcal {B}_E, there is a reducing subspace {M} forB= \be... | {
"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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0.005662159062922001,
0.011034341529011726,
... |
14ab718e1c49fc0a1a8060fea9c69108c63992e0 | subsection | 41 | 94 | Minimality and indecomposability | Similarly, X_{22}^* is isometric on \operatorname{rg}F^*
and hence X_{22}^* extends to a unitary V on all of \mathbb {C}^\ell such that
V F^*= X_{21}^* F^*. Finally, UG=X_{11}G=FX_{22}=FV^*. Hence
equation (REF ) holds, which implies \mathcal {B}_E=\mathcal {B}_F= \mathcal {B}_E\cap \mathcal {B}_R.
Thus \mathcal {B}_E\... | {
"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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0.011665191501379013,
0.... |
4d70e38ad8bd3bcab9486ebf5a5deda92cc29a0f | subsection | 42 | 94 | Minimality and indecomposability | Because F=\mathfrak {H}A \mathfrak {H} it is now evident that {N}
is reducing for A.Now consider the special case b=0. A subspace {M}
reduces A if and only if it reduces M{\cdot }A. Combining
these two special cases proves item (REF ).Finally we prove item (REF ).
By Lemma REF , L_A
is unitarily equivalent to
\bigoplus... | {
"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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0.005248387809842825,
0.0294... |
471f99d55b2b8c0bb225ff1bf6927ea7a62e0d01 | subsection | 43 | 94 | Characterizing bianalytic maps between spectrahedra | In this section we prove Theorem REF and
Proposition REF , stated as
Propositions REF and REF below.
A major accomplishment, exposited in Subsection REF , is the
reduction of the eig-generic type hypotheses of
to various natural and cleaner algebraic
conditions on the corresponding pencils defining spectrahedra.Lemma ... | {
"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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0.0001300542789977044,
0.017211509868502617,
-0.017730295658111572,
0.... |
0ec33835cc5cbf630bd670e3a24e8cfe4e6e61ff | subsection | 44 | 94 | The detailed boundary | Let \rho be a hermitian d \times d free matrix polynomial with \rho (0) =I_d.
Thus \rho \in \mathbb {C}\!\mathop {<}\!x,y\!\mathop {>}^{d\times d} and \rho (X,X^*)^*=\rho (X,X^*) for all X\in M(\mathbb {C})^g.
The detailed boundary of \mathcal {D}_{\rho } is the sequence of sets\widehat{\partial \mathcal {D}_{\rho } }(... | {
"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 | [
0.0377766489982605,
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-0.0071517787873744965,
-0.00... |
49959bbe9a35c30e966465bc5e34aabbd3259657 | subsection | 45 | 94 | Boundary hair spans | In this subsection we connect the notion of boundary hair to ball-minimal ity.
Given a tuple E\in M_{d\times e}(\mathbb {C})^g, a subset {S}\subseteq \widehat{ \partial ^1 \mathcal {B}_E} is
closed under unitary similarity if for each n, each (X,v)\in \widehat{ \partial ^1 \mathcal {B}_E}(n)
and each n\times n 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 | [
-0.03744633495807648,
0.028153419494628906,
-0.04785318300127983,
0.04013197496533394,
0.004211568273603916,
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0.03668336942791939,
0.006843798793852329,
0.034791216254234314,
-0.03857552632689476,
0.020447470247745514,
0.03173935413360596,
-0.... |
eed793a1e6d336772f7c71d48d82a56173e39838 | subsection | 46 | 94 | Boundary hair spans | Moreover, if \pi (\operatorname{hair}\mathcal {B}_E) spans \mathbb {C}^e,
then there exists a positive integer rWhile it is not needed
here, r can be chosen at most e. and pairs
(\alpha ^a,\gamma ^a) \in \widehat{\partial ^1 \mathcal {B}_E(r)} for
1\le a\le e such that, writing
\gamma ^a = \sum _{t=1}^r \delta ^a_t\oti... | {
"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 | [
-0.04268873482942581,
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0.022534402087330818,
-0.002740515861660242,
-... |
0fec104b646e2593904c1a7570873606d044b44e | subsection | 47 | 94 | Boundary hair spans | Since are convex sets containing 0 in their interiors, and their boundaries are
contained in \mathcal {Z}_{\mathbb {L}_E} and \mathcal {Z}_{\mathbb {L}_{EW}} respectively,
the inclusion \mathcal {Z}_{\mathbb {L}_E}\subseteq \mathcal {Z}_{\mathbb {L}_{EW}} implies \mathcal {B}_{EW}\subseteq \mathcal {B}_E.
Indeed, if X\... | {
"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 | [
-0.040010519325733185,
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0.019242282956838608,
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0.04458838328719139,
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0.04507668688893318,
-0.01271119900047779,
-0.... |
66069336657e6ff4d3c24d28ee6533eedcafef78 | subsection | 48 | 94 | Boundary hair spans | Note that e^\prime \ne 0 since \ker (F^*)=\lbrace 0\rbrace and furtherQ_F = V^* \begin{pmatrix} Q_E &0 \\ 0 & Q_R \end{pmatrix} V = V^* (Q_E\oplus Q_R) V.Without loss of generality, we may assume V=I.Suppose X\in \partial ^1 \mathcal {B}_F(n) and 0\ne v \in \mathbb {C}^\ell \otimes \mathbb {C}^n is in the kernel
of Q^{... | {
"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 | [
-0.02203889936208725,
0.030570579692721367,
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0.02029898762702942,
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-0.003224181244149804,
-0.021886276081204414,
0.00... |
6f8c1366e1063906cb2d816cbeb786e9adcd7dd0 | subsection | 49 | 94 | From basis to hyperbasis | Call an e+1-element subset \mathcal {U}= \lbrace u^1,\dots ,u^{e+1}\rbrace of \mathbb {C}^e a hyperbasis if each
e-element subset of \mathcal {U} is a basis.
This notion critically enters the genericity
conditions considered in .Lemma 4.3
Given E\in M_{d\times e}(\mathbb {C})^g and n\in \mathbb {N}, if \mathcal {Z}_{Q... | {
"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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0.03244130313396454,
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0.0022545638494193554,
-0.01907414384186268,
... |
4976177242572bc814e5e5f865da2678c59a9853 | subsection | 50 | 94 | From basis to hyperbasis | Moreover,
for every k=1,\dots ,e there exists 1\le i_k\le er such that
\ker (Q^{\rm re}_E(X^k)) = \operatorname{span}(\operatorname{adj}Q^{\rm re}_E(X^k))_{(i_k)},
and hence (I\otimes e_1^*) \operatorname{adj}(Q^{\rm re}_E(X^k))_{(i_k)} = \mu _k \delta ^k_1
for some \mu _k\ne 0.
Now considerv(t,X,Y):=\sum _{k=1}^e t_k\... | {
"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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0.011734627187252045,
0.007133859675377607,
-0.0011950168991461396,
-0.024476386606693268... |
f49512f7182e5389d3178e120a8b93e3d7155b6a | subsection | 51 | 94 | From basis to hyperbasis | In particular, if U(X,X^*)\ne 0, then
u(X,X^*)\in \pi (\operatorname{hair}\mathcal {B}_E).0\ne u(X^k,X^{k*}) =\nu _k \delta ^1_k,for each k and hence u(X^1,X^{1*}),\dots ,u(X^e,X^{e*}) form a basis of \mathbb {C}^e.
Therefore,u(X,Y)=\sum _{k=1}^e r_k(X,Y) u(X^k,X^{k*})for (X,Y)\in \mathcal {Z}_{Q_E}(n), where r_k are p... | {
"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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0.00199891603551805,
-0.012596223503351212,
-0.01... |
48ebfb2885e10fa3dddf4208ab886694110466f6 | subsection | 52 | 94 | From basis to hyperbasis | It follow that Q_E is an
atom if and only if Q_{\widehat{E}} is an atom; \ker (E)=\lbrace 0\rbrace
if and only if \ker (\widehat{E})=\lbrace 0\rbrace ; and \pi (\operatorname{hair}\mathcal {B}_E)
contains a hyperbasis of \mathbb {C}^e if and only if \pi (\operatorname{hair}\mathcal {B}_{\widehat{E}})
does. Thus, by 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 | [
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... |
8f707e40daf6295d8f9b3e36808bb00ad1deac6f | subsection | 53 | 94 | From basis to hyperbasis | Thus \mathbb {L}^{\rm re}_E decomposes non-trivially as \mathbb {L}^{\rm re}_{E^1}\oplus \mathbb {L}^{\rm re}_{E^2} by Lemma REF (REF ). Hence Q_E decomposes as Q_{E^1}\oplus Q_{E^2}. Letting e_i\ge 1 denote the size of Q_{E^i},\pi (\operatorname{hair}\mathcal {B}_E)\subseteq \left(\mathbb {C}^{e_1}\oplus \lbrace 0\rbr... | {
"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 | [
-0.0008687313529662788,
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0.010138695128262043,
-0.00046058971202000976... |
efab0675ae9d6ab344315a286a96dc7769de3d1b | subsection | 54 | 94 | The eig-generic conditions | In this subsection we connect the various genericity assumptions on tuples in M_d(\mathbb {C})^g used in
to clean, purely algebraic conditions of the
corresponding hermitian monic pencils, see Proposition REF .
We begin by recalling these assumptions precisely.Definition 4.6 ()
A tuple A\in M_d(\mathbb {C})^g is weak... | {
"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 | [
-0.05140006169676781,
0.03476524353027344,
-0.020236486569046974,
-0.03513151779770851,
0.03623032942414284,
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0.0010101073421537876,
0.03916050121188164,
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0.0018609249964356422,
-0.028141839429736137,
0.0161769799888134,
-0.0014631772646680474,
0.0... |
70f2a5337950ce348ee9765951e715dd0e4fdb36 | subsection | 55 | 94 | The eig-generic conditions | Then A is weakly *-generic and \ker (A)=\lbrace 0\rbrace if and only if {A^*\iota } is ball-minimal.It is immediate from the definitions that if \pi (\operatorname{hair}\mathcal {B}_A) contains
a hyperbasis, then A is eig-generic. On the other hand,
if (\alpha ,u) \in \widehat{\partial ^1 \mathcal {B}_E}.
then u is an ... | {
"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 | [
-0.0192984938621521,
0.011533329263329506,
-0.025995757430791855,
0.01820008084177971,
0.0247295331209898,
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0.040244605392217636,
-0.04552308842539787,
0.05855147913098335,
0.014996379613876343,
0.00148... |
788db6d8a11b36ff47e9fe3e9775dccd65619c28 | subsection | 56 | 94 | Bianalytic maps between spectraballs and free spectrahedra | In this section we prove the rest of our
main results, Proposition REF ,
and then Theorem REF and its Corollary 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 | [
-0.004174029920250177,
0.04047359153628349,
0.0023159380070865154,
-0.0269366092979908,
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0.017810212448239326,
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0.0006271537276916206,
-0.025105224922299385,
0... |
cb11544cfd4f21dc29268e261f65eee327fa6e5c | subsection | 57 | 94 | The proof of Proposition | Throughout this subsection, we
fix a tuple E\in M_{d\times e}(\mathbb {C})^g, a positive integer M
and an F\in \mathbb {C}\!\mathop {<}\!x\!\mathop {>}^{1\times e} of degree degree at most M.
Write F=\begin{pmatrix} F^1 & \cdots & F^e\end{pmatrix} andF^s = \sum _{|w|\le M} F^s_w w,where |w| denotes the length of the wo... | {
"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 | [
-0.020504266023635864,
-0.01106833666563034,
0.005763010587543249,
-0.0336245559155941,
0.009542722254991531,
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0.008055247366428375,
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0.016858045011758804,
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-0.007094109896570444,
-0.0028929475229233503,
... |
029d55cfc07364985fc89af5a8c2a9e6f7499041 | subsection | 58 | 94 | The proof of Proposition | For N=1 and w=x_t,(\beta {\cdot }S)^w= \sum _{k=1}^g \beta _{t,k} \otimes S_k =\sum _{k} \widehat{\beta }_{x_k,x_t} S_k
= \sum _{|u|=1} \widehat{\beta }_{u,x_t} S^uNow suppose the result holds for N. Let v be a word of length N
and consider the word w=vx_t of length N+1. Using the induction
hypothesis and 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 | [
0.005375721957534552,
0.03137682005763054,
-0.014955876395106316,
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0.010316502302885056,
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-0.0077946060337126255,
-0.026935838162899017,... |
4a27ff760467996cb2b89317dc7821f2b5efe6dc | subsection | 59 | 94 | The proof of Proposition | In particular, B(\beta ,1) = \begin{pmatrix} \beta _{k,j} \end{pmatrix}_{j,k=1}^g.Lemma 5.2
For each positive integer N the set of \beta \in M_g(M_r(\mathbb {C})) such that
B(\beta ,N) is invertible is open and dense.For the second statement, observe that B(I,N) is the identity
matrix since, with \beta _{j,k}=_{j,k}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 | [
0.009806155227124691,
0.0006472253007814288,
-0.028556134551763535,
0.0164529737085104,
0.019078122451901436,
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0.012469461187720299,
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0.04178871959447861,
0.03376064449548721,
-0.026282021775841713,
-0.00453677773475647,
0.03659946843981743,
-0.... |
3368dec5409184a905bce064bd4987761d000a60 | subsection | 60 | 94 | The proof of Proposition | If\sum _{s=1}^e F^s(\beta {\cdot }S)[\gamma _s \otimes 1] =0,then, for 1\le N\le M and each word u of length N,\sum _{|w|=N} \widehat{\beta }_{u,w} [\sum _{s=1}^e F^s_w \gamma _s] = 0.Moreover, if B(\beta ,N) is invertible, then\sum _{s=1}^e F^s_w \gamma _s =0for each word |w|=N.Since F^s_w\in \mathbb {C}, by Lemma 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 | [
0.016797086223959923,
0.03048190474510193,
-0.015050249174237251,
-0.025172743946313858,
0.01847526803612709,
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0.01081665139645338,
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0.025142231956124306,
-0.03533338010311127,
0.008688410744071007,
-0.004256482236087322,
-0... |
e236c65d2b2f3796713d3ce98cb5181cc1b0a367 | subsection | 61 | 94 | The proof of Proposition | If there exist a positive integer r and
(\beta ^a, \gamma ^a)\in M_g(M_r(\mathbb {C}))\times [\mathbb {C}^e\otimes \mathbb {C}^r]
for 1\le a\le e such that,writing
\gamma ^a =\sum _{t=1}^r \delta ^a_t\otimes \varrho _t
the vectors \lbrace \delta ^a_1: 1\le a\le e\rbrace span \mathbb {C}^e;
B(\beta ^a,N) is invertib... | {
"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 | [
-0.03374899923801422,
0.013716251589357853,
0.01032151747494936,
-0.013700994662940502,
0.01994120329618454,
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0.024060655385255814,
0.006358451675623655,
0.03083486668765545,
-0.023633452132344246,
-0.013967996463179588,
-0.007159456145018339,
-0.... |
9f3aebe8c4017d7ee75bdfe65df76a36ebb3b71c | subsection | 62 | 94 | The proof of Proposition | Thus F^s_w=0 for 1\le s\le e
and |w|=N.Given \beta =(\beta _{j,k}) \in M_g(M_r(\mathbb {C})), let(E{\cdot }\beta )_k = \sum _{j=1}^g E_j\otimes \beta _{j,k}Lemma 5.5
For \beta \in M_g(M_r(\mathbb {C})),\begin{split}
\Lambda _E(\beta {\cdot }S) & = \sum _k (E{\cdot }\beta )_k \otimes S_k\\
Q^{\rm re}_E(\beta {\cdot }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 | [
-0.03375294804573059,
0.01785305142402649,
-0.020447084680199623,
-0.02742045558989048,
0.018219267949461937,
-0.0239871758967638,
-0.0033493544906377792,
0.02592507191002369,
0.008010984398424625,
0.03363087773323059,
-0.032867927104234695,
0.01544212643057108,
0.0019130994332954288,
-0.0... |
62bb2b6b1521fdd20d75100820cd6f2f9f40636b | subsection | 63 | 94 | The hair spanning condition | A subset \lbrace (\alpha ^a,\gamma ^a):1\le a\le e\rbrace \subseteq M_r(\mathbb {C})^g \times [\mathbb {C}^e\otimes \mathbb {C}^r] is a boundary spanning set for \mathcal {B}_E if
each (\alpha ^a,\gamma ^a)\in \widehat{\partial \mathcal {B}_E} and,
writing \gamma ^a=\sum _{t=1}^r \delta ^a_t \otimes \varrho _t,
the set... | {
"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 | [
-0.022408349439501762,
0.0067919110879302025,
-0.05592172220349312,
-0.02008971758186817,
0.03718962147831917,
-0.042498067021369934,
0.030386270955204964,
0.036609966307878494,
0.008412664756178856,
0.017160920426249504,
-0.04509127512574196,
0.021691404283046722,
-0.00488132843747735,
-0... |
f739f2a21f8eceb12b7bd5596e9a55004ed5d097 | subsection | 64 | 94 | The hair spanning condition | There is an \epsilon >0 such that, if
\tau ^a=\sum _{t=1}^g \tau ^a_t\otimes \rho _t and
\Vert \zeta ^a-\tau ^a\Vert <\epsilon for each 1\le a\le e,
then the set \lbrace \tau ^a_1: 1\le a\le e\rbrace spans \mathbb {C}^e.Fix 1\le a\le e and let, for 1\le j,k\le g,\widetilde{\beta }^a_{j,k} = {\left\lbrace \begin{array}{... | {
"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 | [
-0.03937748447060585,
0.0340050533413887,
-0.02753371372818947,
-0.020726598799228668,
0.029105760157108307,
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0.025992192327976227,
0.03373032435774803,
0.017826706171035767,
0.02451172098517418,
-0.017445141449570656,
0.01906297542154789,
-0.015995195135474205,
-0.013... |
2498106bb872985775b7bce3695502532cb93b1c | subsection | 65 | 94 | Proof of Proposition | Suppose E is ball-minimal It is enough to assume that
PE is ball-minimal, where P is the projection of \mathbb {C}^d onto
\operatorname{rg}(E). and F\in \mathbb {C}\!\mathop {<}\!x\!\mathop {>}^{1\times e} vanishes on
\widehat{\partial \mathcal {B}_E} and has degree at most M.Fix 1\le N\le M.
By Proposition REF , there... | {
"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 | [
-0.024522028863430023,
0.015450251288712025,
-0.01418371219187975,
0.00843850802630186,
0.046633053570985794,
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0.03241119533777237,
0.052675820887088776,
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0.039339009672403336,
-0.03299105539917946,
0.01702961139380932,
-0.012726428918540478,
0.001... |
9d888e9e522692ec8743126f346416ec16775f87 | subsection | 66 | 94 | Theorem | In this subsection we prove the first part Theorem REF .
(The conversely portion was already proved as Corollary REF .)A free analytic mapping f into M(\mathbb {C})^h defined in a neighborhood of 0
of M(\mathbb {C})^g has a power series expansion (, , ),f(x) = \sum _{j=0}^\infty G_j(x) =\sum _{j=0}^\infty \sum _{|\alph... | {
"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 | [
-0.03479830548167229,
0.02609873004257679,
-0.02058899775147438,
-0.015506613999605179,
0.047649260610342026,
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0.00983662623912096,
0.032875243574380875,
0.010378442704677582,
-0.02588505670428276,
-0.005608937703073025,
-0.019825877621769905,
-0.... |
bdee08063bde0dbcd8662f27e1e236ca545d18d4 | subsection | 67 | 94 | Theorem | Thus L^{\rm re}_B(p(X))\succeq 0.
Arguing by contradiction, suppose
L^{\rm re}_B(p(X))\succ 0; that is p(X)\in \operatorname{int}(\mathcal {D}_B(n)). Hence
there is an \eta such that\overline{B}_\eta (p(X)):=\lbrace Y\in M_n(\mathbb {C})^g: \Vert Y-p(X)\Vert \le \eta \rbrace
\subseteq \operatorname{int}(\mathcal {D}_B... | {
"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 | [
-0.0093685956671834,
0.02856353670358658,
0.0018033402739092708,
-0.00207894342020154,
0.022872190922498703,
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0.01893555000424385,
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0.027007190510630608,
-0.026656249538064003,
0.012008282355964184,
-0.011062267236411572,
-0.0... |
ad92a01a96e870d3c1b3ccfd1019111ee06026b3 | subsection | 68 | 94 | Theorem | Since \lbrace B_1\mathfrak {V},\dots ,B_g\mathfrak {V}\rbrace is
linearly independent, the subspace {M} of {B}
spanned by \lbrace B_1,\dots ,B_g\rbrace has dimension g and satisfies
{M}\cap {N}=\lbrace 0\rbrace . Thus there is a g\le t\le h
such that h-t is the dimension
of {N}. Choose a basis \lbrace J_{t+1},\dots ,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 | [
-0.04094722867012024,
0.016659602522850037,
-0.018292000517249107,
-0.007872119545936584,
0.03810960426926613,
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0.011411522515118122,
-0.04509687423706055,
-0.0134787168353796,
-0.0013358575524762273,
0.... |
3cb4d20cff174b4ae3e9b8ac6ce59dce709621ab | subsection | 69 | 94 | Theorem | Since
f:\operatorname{int}(\mathcal {B}_E)\rightarrow \operatorname{int}(\mathcal {D}_A)
is proper with f(0)=0 and f^\prime (0)=I
(and S_X is nilpotent), f(S_X)=S_X
(see Remark REF ), and S_X\in \partial \mathcal {D}_{A}.
Thus I+\Lambda _{A}(S_X)+\Lambda _{A}(S_X)^* is positive semidefinite
and has a (non-trivial) kern... | {
"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 | [
-0.03048243187367916,
0.02567664161324501,
0.00838343147188425,
-0.01434871181845665,
0.014691982418298721,
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0.03923964500427246,
-0.01690417155623436,
0.0016295819077640772,
0.0045769414864480495,
-0.02686... |
add1b5cb8afec6b70534cf45646265eb1f8ccd1d | subsection | 70 | 94 | Proof of Theorem | We assume, without loss of generality, that E is ball-minimal.
We will now show f is convexotonic.Lemma REF applied to the
proper free analytic mapping f:\operatorname{int}(\mathcal {B}_E)\rightarrow \operatorname{int}(\mathcal {D}_A)
gives \mathcal {B}_E=\mathcal {B}_A.
Applying Lemma REF (REF ) there exist
r\times r ... | {
"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 | [
-0.01630067452788353,
0.02819039858877659,
-0.004769626073539257,
-0.0008084516157396138,
0.03177715837955475,
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0.05360296741127968,
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0.025382041931152344,
0.0524735189974308,
-0.028419340029358864,
0.029869306832551956,
-0.0007407229277305305,
-0... |
12e160ac12a51f1b138fd3627f17c55d180b3111 | subsection | 71 | 94 | Proof of Theorem | For now observeA_j \mathfrak {V}= U \begin{pmatrix} E_j \\ 0 \end{pmatrix}.Thus, since \lbrace E_1,\dots ,E_g\rbrace is linearly independent,
the set \lbrace A_1\mathfrak {V},\dots , A_g\mathfrak {V}\rbrace is linearly independent.We now apply Lemma REF to A
in place of B and obtain
a basis \lbrace J_1,\dots ,J_h\rbrac... | {
"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 | [
-0.02010766789317131,
0.0313362292945385,
-0.016613999381661415,
-0.009558008052408695,
0.037011533975601196,
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0.020351767539978027,
0.048240095376968384,
0.050803136080503464,
0.00991652812808752,
-0.03691999614238739,
0.008581611327826977,
-0.020489072427153587,
-0.0... |
acee18a36bc883ef565cd967cdd51c820fa5197e | subsection | 72 | 94 | Proof of Theorem | Thus,H_j = \begin{pmatrix} H_j^{1} & \dots & H_j^{h} \end{pmatrix}and H_1(x) = \begin{pmatrix} x & 0 \end{pmatrix}.
Likewise,{F}_{x_j}(x) = \begin{pmatrix} 0 & \dots &0 & x_j & 0 &\dots & 0\end{pmatrix}for 1\le j\le g and {F}_{x_j}=0 for j>g.The next objective is to show H_m^s=0 for m\ge 2 and s\le t.
Given a positive ... | {
"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 | [
-0.015045053325593472,
0.043639808893203735,
-0.0000027417929686635034,
-0.0006833025254309177,
0.006874032784253359,
-0.015182381495833397,
0.010269087739288807,
0.007503453176468611,
0.036681853234767914,
0.04171721637248993,
-0.0320432148873806,
-0.007293646223843098,
-0.00105284852907061... |
fc290248a08e244143df2cfc767b510c23d97980 | subsection | 73 | 94 | Proof of Theorem | Thus,I-\Lambda _A(Y)^* \Lambda _A(Y) -\Lambda _J(H_m(Y))^* \Lambda _J(H_m(Y)) \succeq 0.Multiplying on the right by \mathfrak {V}\otimes I and on the left by \mathfrak {V}^*\otimes I,I - \Lambda _{A\mathfrak {V}}(Y)^* \Lambda _{A\mathfrak {V}}(Y)
- \Lambda _{J\mathfrak {V}}(H_m(Y))^* \Lambda _{J\mathfrak {V}}(H_m(Y)) \... | {
"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 | [
-0.03749460354447365,
0.04204032942652702,
-0.0002919732069130987,
-0.021340500563383102,
0.01674901321530342,
0.0035580212716013193,
0.01527699176222086,
0.03078279457986355,
0.024025224149227142,
0.01144821010529995,
-0.04179626330733299,
-0.004602927714586258,
-0.0010029576951637864,
0.... |
84dc2c86baef681ac92cb78b1df14b13e1bed34f | subsection | 74 | 94 | Proof of Theorem | Hence,{F}(x) =
\begin{pmatrix} x & 0 & \Psi (x)\end{pmatrix}where the 0 has length t-g and \Psi has length h-t and moreover,
\Psi (0)=0 and \Psi ^\prime (0)=0.Let \psi denote the inverse of \varphi ,\psi (x) = x(I+\Lambda _{\Xi }(x))^{-1}.Thus, \psi \circ {F}= \iota \circ f = \begin{pmatrix} f(x) &0&0\end{pmatrix}
and ... | {
"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 | [
-0.00798884779214859,
-0.004776522051542997,
-0.029147466644644737,
0.0039181215688586235,
-0.01974702626466751,
-0.006851943675428629,
0.062079526484012604,
0.03044460527598858,
0.0028823181055486202,
0.04797886684536934,
-0.0189077015966177,
0.014306675642728806,
-0.0038189284969121218,
... |
79674c5939152f7af83e3edc8970ef86805e4d8a | subsection | 75 | 94 | Proof of Theorem | Hence, the right hand side of equation (REF ), for g<k\le t (so that
I_{\ell ,k}=0 for \ell \le g) is,\sum _{\ell =1}^g f^\ell (x)\,
\big (I+\Lambda _{\Xi }( \begin{pmatrix} x & 0 & \Psi (x)\end{pmatrix})\big )_{\ell ,k}
= \sum _{j,\ell =1}^g (\Xi _j)_{\ell ,k} f^\ell (x)\, x_jand similarly, for 1\le k\le g,\sum _{\ell... | {
"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 | [
-0.04473577067255974,
-0.008735071867704391,
-0.016798801720142365,
-0.047909386456012726,
0.0074648624286055565,
0.00606496213003993,
0.030958009883761406,
0.030454503372311592,
0.02857779711484909,
0.02201695740222931,
-0.02714356780052185,
0.006904139649122953,
-0.03680173307657242,
-0.... |
e83593754ca02b6a3c69eda1c0582dc9eb435a3d | subsection | 76 | 94 | Proof of Theorem | \end{split}Multiplying equation (REF ) on the left by U^*
and using equation (REF ) gives\begin{pmatrix} E_\ell & 0\\0 & R_\ell \end{pmatrix}\,
(-U) \, \begin{pmatrix} E_j \\ 0 \end{pmatrix}
= \begin{pmatrix} \sum _{s=1}^g (\Xi _j)_{\ell ,s} E_s \\ 0 \end{pmatrix}.Using equation (REF ), it follows thatE_\ell (-U_{11}) ... | {
"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 | [
-0.027595799416303635,
0.020578626543283463,
-0.02218037284910679,
-0.0070248013362288475,
0.0007584459381178021,
-0.02899923548102379,
0.03825885429978371,
0.04936429485678673,
-0.013065673410892487,
0.04289628937840462,
-0.042408138513565063,
0.04347597062587738,
-0.031973905861377716,
-... |
a26dfedf19360af613fe26aec01342cb3023eae8 | subsection | 77 | 94 | Proof of Theorem | \end{split}Thus B spans an algebra and, by Proposition REF ,
the convexotonic map f of equation (REF ) is a bianalytic map
f:\operatorname{int}(\mathcal {B}_B)\rightarrow \operatorname{int}(\mathcal {D}_B). On the other hand, \mathcal {B}_B=\mathcal {B}_E=\mathcal {B}_A. Thus,
as f:\operatorname{int}(\mathcal {B}_E)\ri... | {
"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 | [
-0.028860071673989296,
0.018787281587719917,
-0.015376267954707146,
0.016360655426979065,
0.012239967472851276,
-0.0071654170751571655,
0.035621050745248795,
0.021519144997000694,
0.006749533116817474,
0.04258043318986893,
-0.03546843305230141,
0.04459499195218086,
-0.014513975940644741,
-... |
65e82d9e4eec2a948f5adc80b4f0058b8ac602b3 | subsection | 78 | 94 | Corollary | This subsection begins by illustrating Corollary REF
in the case of free automorphism of free matrix balls and
free polydiscs
before turning to the proof of the corollary. | {
"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 | [
-0.023732144385576248,
0.009561535902321339,
0.0048876008950173855,
0.010004129260778427,
-0.00021318867220543325,
0.020832397043704987,
0.01422402448952198,
0.01689484529197216,
0.02942480705678463,
-0.003172552678734064,
0.008836599066853523,
-0.0069555784575641155,
0.014399535953998566,
... |
d53d5657b356f26be449ffdd2c0ef50af8a67e79 | subsection | 79 | 94 | Automorphisms of free polydiscs | Let \lbrace e_1,\dots ,e_{g}\rbrace denote the usual orthonormal basis for \mathbb {C}^{g}
and let E_j = e_j e_j^*. The spectraball \mathcal {B}_E is then the free polydisc
with\operatorname{int}(\mathcal {B}_E(n)) = \lbrace X\in M_n(\mathbb {C})^g: \Vert X_j\Vert <1\rbrace .Let b\in \operatorname{int}(\mathcal {B}_E(1... | {
"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 | [
-0.011774800717830658,
-0.00856973510235548,
-0.014384640380740166,
-0.012385290116071701,
0.037575580179691315,
-0.057935379445552826,
0.015918493270874023,
0.01552930660545826,
0.027777235954999924,
0.03473680838942528,
-0.0334242582321167,
-0.005650836043059826,
0.03485890477895737,
-0.... |
8d15940ab7caf5d8d23ccbddf61f9c231bfcab6e | subsection | 80 | 94 | Automorphisms of free polydiscs | Hence, the automorphisms
of the free polydisc are given by\varphi (x) =
\left(\rho _{\pi (1)} (x_{\pi (1)}+c_{\pi (1)})(1+c_{\pi (1)}^*x_{\pi (1)})^{-1},\ldots , \rho _{\pi (g)} (x_{\pi (g)}+c_{\pi (g)})(1+c_{\pi (g)}^* x_{\pi (g)})^{-1},
\right)for c=(c_1,\dots ,c_g)\in \mathbb {D}^g, unimodular \rho _j and a permutat... | {
"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 | [
-0.03396867588162422,
-0.01039203442633152,
-0.04120189696550369,
0.007114957552403212,
-0.0017816006438806653,
-0.034762192517519,
0.057469017803668976,
0.03058096393942833,
0.028154630213975906,
0.002075354801490903,
0.000029327740776352584,
-0.017075898125767708,
0.00772535614669323,
0.... |
1fcee2062cfec81641831cecddf0317da3c94d22 | subsection | 81 | 94 | Automorphisms of free matrix balls | Let (E_{ij})_{i,j=1}^{d,e} denote the matrix units in M_{d\times e}(\mathbb {C})
and view E\in M_{d\times e}(\mathbb {C})^{de}. We consider automorphisms of
\mathcal {B}_E, the free d\times e matrix ball.Before
proceeding further, note, since \lbrace E_{ij}: 1\le i\le d,\, 1\le j\le e\rbrace
spans all of M_{d\times e}... | {
"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 | [
-0.00993991270661354,
-0.004268135875463486,
-0.019559428095817566,
0.0046495599672198296,
0.017789620906114578,
-0.02836269699037075,
-0.005400965455919504,
0.034968964755535126,
0.01044339220970869,
0.01704202964901924,
-0.041651513427495956,
0.005831975024193525,
0.019162748008966446,
-... |
b6af363d8a178c62d93216b90049d55fcdc53130 | subsection | 82 | 94 | Automorphisms of free matrix balls | \end{split}Hence,M_{(i,j),(u,v)} = [e_u^* D_{\Lambda _E(b)^*} We_i]\,
[e_j^* {V}^* D_{\Lambda _E(b)} e_v].Next observe that,- E_{ij} {V}^* \Lambda _E(b)^* WE_{st} = - e_ie_j^* {V}^* b^* We_se_t^*
= -(e_j^* {V}^* b^* We_s) E_{it}.Hence, letting \beta _{js} =-(e_j^* {V}^* b^* We_s) for 1\le j\le e
and 1\le s\le d, the tu... | {
"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 | [
-0.012189796194434166,
-0.005332082509994507,
-0.04418228939175606,
-0.000019606748537626117,
0.01128967385739088,
-0.027461368590593338,
0.018414372578263283,
0.033411331474781036,
0.0358523428440094,
0.03063468262553215,
-0.03408261016011238,
-0.011816016398370266,
0.012807676568627357,
... |
1f0ed3d934e476f1b3fab2d591bb668922ee7766 | subsection | 83 | 94 | Automorphisms of free matrix balls | For a tuple y= (y_{s,t})_{s,t=1}^{d,e} of indeterminates,\begin{split}
(y{\cdot }M)_{u,v} &= \sum _{i,j} M_{(i,j),(u,v)} y_{i,j} \\
&= \sum _{i,j} [e_u^* D_{\Lambda _E(b)^*} W]\, y_{i,j} e_i e_j^* \,[{V}^* D_{\Lambda _E(b)} e_v]\\
&= e_u^* \, [D_{\Lambda _E(b)^*} W]\, \operatorname{mat}(y) \,[{V}^* D_{\Lambda _E(b)}] \... | {
"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 | [
-0.0006983843632042408,
0.04442201182246208,
-0.05128667876124382,
0.005274351220577955,
0.010792779736220837,
-0.06760932505130768,
0.029197709634900093,
0.031165581196546555,
0.022333044558763504,
0.038106519728899,
-0.031730007380247116,
0.030311310663819313,
-0.01700911484658718,
-0.00... |
f4fe65d1fb816ea1e51c045a5c3f3cd661ac6552 | subsection | 84 | 94 | Automorphisms of free matrix balls | Thus,\begin{split}
\operatorname{row}(x) & (I-\Lambda _{\Xi }(x))^{-1} \\
& = {\small \begin{pmatrix} (\operatorname{mat}(x)[I-\beta \operatorname{mat}(x)]^{-1})_{11} &
(\operatorname{mat}(x)[I-\beta \operatorname{mat}(x)]^{-1})_{12} &
\dots & (\operatorname{mat}(x)[I-\beta \operatorname{mat}(x)]^{-1})_{de} \end{pmatri... | {
"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 | [
0.016296952962875366,
0.01614435948431492,
-0.0629294365644455,
0.01602228544652462,
-0.02501002512872219,
-0.028183963149785995,
0.04760907590389252,
0.027405736967921257,
0.016571620479226112,
0.013412941247224808,
-0.030366046354174614,
0.003971237689256668,
0.02172926999628544,
-0.0020... |
bd6916325113707861a6ef9d6f3b66341691442a | subsection | 85 | 94 | Automorphisms of free matrix balls | Consequently, using, in order,
equations (REF ), (REF ), and (REF )
together with the definition of c in the first
three equalities followed by some algebra,\begin{split}
\operatorname{mat}(\varphi (x)) &= \operatorname{mat}(\psi (x){\cdot }M) + b \\
&= D_{\Lambda _E(b)^*} W\operatorname{mat}(\psi ) {V}^* D_{\Lambda _E... | {
"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 | [
-0.0008652781834825873,
0.026224365457892418,
-0.03551502153277397,
0.0015942095778882504,
-0.03780336305499077,
-0.009122844785451889,
0.04707876220345497,
0.008169370703399181,
0.016918454319238663,
0.020823884755373,
-0.03212828189134598,
0.021464619785547256,
0.030389143154025078,
0.02... |
7459d0e2acfe9b3bd164ec456ec53053e53b8db1 | subsection | 86 | 94 | Proof of Corollary | Suppose E=(E_1,\dots ,E_g)\in M_{d\times e}(\mathbb {C})^g and
C=(C_1,\dots ,C_g)\in M_{k\times \ell }(\mathbb {C})^g are linearly independent and
ball-minimal and \varphi :\operatorname{int}(\mathcal {B}_E)\rightarrow \operatorname{int}(\mathcal {B}_C) is bianalytic.Let
\widehat{C} denote the tuple\widehat{C}_j = \beg... | {
"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 | [
-0.0314270444214344,
0.01231146790087223,
-0.04485218971967697,
0.009008577093482018,
0.04372325539588928,
-0.028421642258763313,
0.0012814911315217614,
0.05760607495903969,
0.005945966113358736,
0.028925085440278053,
-0.03954315558075905,
-0.02372284047305584,
0.04683544859290123,
-0.0176... |
dd8e8f7fcea8e521bd1a92550771a68f8857912c | subsection | 87 | 94 | Proof of Corollary | An application of Theorem REF now implies that
there is a convexotonic tuple \Xi such that equation (REF ) holds,
f is the corresponding convexotonic map and there are unitaries
V and W of size r such thatA= W \begin{pmatrix} 0_{d,r-e} & E \\ 0_{r-d,r-e} & 0_{r-d,e} \end{pmatrix} V^*.In particular, \varphi (x)=f(x){\cd... | {
"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 | [
-0.0076348367147147655,
0.04326153174042702,
-0.04332255199551582,
-0.003689662553369999,
-0.01336287148296833,
-0.005552608519792557,
0.02687828615307808,
0.014384916983544827,
0.018823953345417976,
0.034352950751781464,
-0.03810553997755051,
0.025917258113622665,
-0.02443758025765419,
-0... |
1824600774b415b7ec252a5290ebd25eb8bd3e99 | subsection | 88 | 94 | Proof of Corollary | Thus E and C have the same size d\times e.Since E and C are both d\times e and r=d+e, the matrices V and W decompose asV=
\begin{pmatrix} V_{11} & V_{12}\\V_{21} & V_{22} \end{pmatrix},
\quad \quad W=
\begin{pmatrix} W_{11} & W_{12}\\W_{21} & W_{22} \end{pmatrix}with respect to the decomposition \mathbb {C}^r = \mathbb... | {
"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 | [
0.011090482585132122,
0.022257240489125252,
-0.03584956377744675,
-0.0020060501992702484,
-0.005472779273986816,
-0.010480277240276337,
0.042714376002550125,
0.017711210995912552,
0.024926887825131416,
0.022486066445708275,
-0.014950031414628029,
-0.018229885026812553,
0.02770332247018814,
... |
7ef7e9858dc6a8d8df25d55018e3737d69fa46a4 | subsection | 89 | 94 | Proof of Corollary | It follows thatW_{21} \sum E_j E_j^* = -D_\Lambda ^{-1} \Lambda ^* W_{11} \sum E_j E_j^*.Thus, again using that E is ball-minimal (so that \ker (E^*)=\lbrace 0\rbrace ),W_{21} = -D_\Lambda ^{-1}\Lambda ^* W_{11}.Hence,I = W_{11}^* W_{11} + W_{21}^* W_{21} = W_{11}^*[I+\Lambda D_{\Lambda }^{-2} \Lambda ^*] W_{11}
= W_{1... | {
"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 | [
-0.011289330199360847,
0.048513609915971756,
-0.0029405655805021524,
-0.02137344889342785,
-0.012570821680128574,
-0.01126644667237997,
0.022822754457592964,
0.03255598992109299,
-0.005171733908355236,
0.027079137042164803,
-0.03273905813694,
0.01986311934888363,
0.011464772745966911,
-0.0... |
fa91149ad8bdc34cd8aa67f86d540ea8f8909435 | subsection | 90 | 94 | Proof of Corollary | Since
the pair (A,\Xi ) satisfies equation (REF ),\begin{pmatrix} E_k & 0 \\ 0 & 0 \end{pmatrix} \, U \,
\begin{pmatrix} E_j & 0 \\ 0 & 0 \end{pmatrix} =
\sum _s (\Xi _j)_{k,s} \begin{pmatrix} E_s & 0 \\ 0 & 0 \end{pmatrix},item (REF ) holds.To prove the converse,
suppose E,C\in M_{d\times e}(\mathbb {C})^g and b\in \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 | [
-0.009117779321968555,
0.02000570483505726,
-0.022798264399170876,
-0.01895277388393879,
0.002294704783707857,
-0.03238147124648094,
0.009033850394189358,
0.016053395345807076,
0.023195020854473114,
-0.0005946585442870855,
-0.0337548591196537,
-0.004356696270406246,
0.006145917810499668,
-... |
45bfa3a8101632daee9f75c36484279a74254e36 | subsection | 91 | 94 | Proof of Corollary | \end{split}Thus, using item (REF ),\begin{split}
A_j A_k &= \begin{pmatrix} 0 & W_{11} E_j {V}^* W_{21} E_k {V}^* \\
0 & W_{21} E_j {V}^* W_{21} E_k {V}^* \end{pmatrix}
= \sum _s (\Xi _k)_{j,s} \begin{pmatrix} 0 & W_{11} E_s {V}^* \\
0 & W_{21} E_s {V}^* \end{pmatrix}
= \sum _s (\Xi _k)_{j,s} A_s.
\end{split}Thus A spa... | {
"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 | [
-0.0035436172038316727,
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0.045193515717983246,
-0.02581614814698696,
-0.... |
b5800ea28ebd49ca93880dae7f60663f143484cb | subsection | 92 | 94 | Convex sets defined by rational functions | In this section we employ a variant of the main result of
to extend Theorem REF to cover
birational maps from a matrix convex set to a spectraball.
A free set is matrix convex
if it is closed with respect to isometric conjugation.
We refer the reader to , , , ,
for the theory of matrix convex sets.
For expository con... | {
"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 | [
-0.018434662371873856,
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0.013909929431974888,
-0.018068410456180573,
0.00863743294030428,
-0.0003304848214611411,
-0.0... |
e5b02baf89659c7146fe05c7e3f408aab5d6febe | subsection | 93 | 94 | Convex sets defined by rational functions | Hence
Corollary REF follows from Theorem REF .Corollary 6.3
Suppose p:M(\mathbb {C})^g\rightarrow M(\mathbb {C})^g is a free polynomial mapping, E\in M_{d\times e}(\mathbb {C})^g
is linearly independent and let{C} := \lbrace X: \Vert \Lambda _E(p(X))\Vert <1\rbrace .If {C} is bounded, convex and contains 0,
then there... | {
"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 | [
-0.0333205908536911,
0.013219914399087429,
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-0... |
c661de3e0b82f7436ebd92faf75ecd0f8d9f42c1 | abstract | 0 | 45 | Abstract | The aim of this paper is to prove stability of traveling waves for
integro-differential equations connected with branching Markov processes. In
other words, the limiting law of the left-most particle of a (time-continuous)
branching Markov process with a L\'{e}vy non-branching part is demonstrated.
The key idea is to a... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.022233668714761734,
-0.0014401526423171163,
0.0002443967678118497,
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0.00849213171750307,
-0.002449213294312358... | |
5bcd5ca844ed27e94cf2652266da2a768cf8ca6f | subsection | 1 | 45 | Introduction | The Fisher-KPP equation and its analogues have been attracting growing attention over the last decade (see e.g. , , , , , , and references therein).
Common results in the PDE literature are existence and uniqueness of traveling waves for speed c\ge c_*, where c_* is called the minimal speed of propagation (see referenc... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.038508106023073196,
-0.030574582517147064,
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0.007727557793259621,
0... | |
a9e783a08f945672c1b9296924d944e00fe8f980 | subsection | 2 | 45 | Introduction | (t\widehat{f}\,) \right|_{\mathbb {R}} (x),
solves
\begin{equation}
u(x,t) = T_t^0f(x) + (\mathbf {K}\widehat{u})(x,t),
\end{equation}
where, f is bounded Borel, f(z) = j=1n f(zj), u(z,t) = j=1n u(zj,t), zRsymn; for X0 = (X0t,Px0),
\begin{gather}
T_t^0f(x) = \mathbb {E}_x^0[f(X^0_t), t<\tau ], \\
K(x;dt\,dy) = \mathbb... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.029871361330151558,
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0.03194618597626686,
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0.023707913234829903,
0.016751151531934738,
... | |
bc908fa26214511620c574b6411328433aaffa22 | subsection | 3 | 45 | Introduction | Finally, taking k\rightarrow \infty we will obtain the statement for continuous time.Together with the S-equation let us consider the following auxiliary linear equations,v_\lambda (x,t) &= (T_t^0 e_\lambda )(x) + (\mathbf {K}{_\lambda )(x,t), \\
w_{\lambda ,\mu }(x,t) &= (T_t^0 e_{\lambda +\mu })(x) + \mathbf {K}({_{\... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.034240901470184326,
0.03884907811880112,
-0.03646869584918022,
0.01372535154223442,
0.005805999506264925,
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0.026870867237448692,
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-0.015884481370449066,
0.002164851874113083,
0.0... | |
9617215c9208acdfe9bfc2817e119afa94f99227 | subsection | 4 | 45 | Introduction | \end{equation}
With the following assumption we ensure that (\mathbf {X}_n) survives with a positive probability and its left-most particle propagates linearly (asymptotically equivalent to - n c_*) as n\rightarrow \infty on the set of non-extinction,
{\equation
\psi (0)\in (0,\infty )\text{ and }\psi (\lambda )<\inft... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.043280087411403656,
0.000024977902285172604,
-0.025546850636601448,
-0.004696561023592949,
0.00225671473890543,
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0.001763594918884337,
0.040380507707595825,
0.021212738007307053,
0.005333706270903349,
-0.02127378061413765,
-0.0004289761127438396,
-0.00149080529808998... | |
89c6800410fd82ccb25fce70c13b15a02db6ee62 | subsection | 5 | 45 | Introduction | Then the following statements hold true:
\begin{}
\item The left-most particle of \mathbf {X}, M_t:= \min \lbrace y\in \mathbb {R}: y\in \mathbf {X}_t\rbrace ,
satisfies,
\begin{equation}
\lim \limits _{t\rightarrow \infty } ¶[\lbrace 0\rbrace ][M_t + c_*t - \frac{3}{2\lambda _*} \ln t + C \ge -x] = \phi (x), \quad x\i... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02497771382331848,
0.017440151423215866,
-0.04006810113787651,
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-0.006698360666632652,
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0.03411739319562912,
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0.04336387664079666,
0.03878640756011009,
0.0033186639193445444,
-0.00674413563683629,
0.013167847879230976,
0.03... | |
b677bbab9f10ef18779496580c661fd303b715cc | subsection | 6 | 45 | Introduction | (t\widehat{g}) \right|_{\widehat{\mathbb {R}}}(x) to (\ref {eq:S}) satisfies,
\phi (x) \le \liminf _{t\rightarrow \infty } u_g(x+\theta (t),t) \le \limsup _{t\rightarrow \infty } u_g(x+\theta (t),t) \le \phi (x+h),
with \phi given by (\ref {eq:traveling_wave}) and \theta (t) = c_*t-\frac{3}{2\lambda _*}\ln t +C.
\end... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.052834928035736084,
0.0044486946426332,
-0.06046562269330025,
0.03357505425810814,
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0.026554817333817482,
0.021411728113889694,
0... | |
b658f92fd64376de3a782925eb67acb593570f35 | subsection | 7 | 45 | Introduction | In the case of Example~\ref {exmp:X2+P3}, it was shown in \cite {FKT2018ii} that the traveling wave (\ref {eq:traveling_wave}) with the minimal speed can have different asymptotic behaviour depending on the choice of X^0.
Namely, additionally to the expected asymptotic behaviour \phi (x)\sim x e^{-\lambda _*x}, it is a... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.046843040734529495,
0.0012693015160039067,
-0.046812526881694794,
0.035002585500478745,
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0.04382189363241196,
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0.02589336782693863,
0.0010795261478051543,
... | |
55e4fe926d0c36c77383a41111a613ae777365fa | subsection | 8 | 45 | Related PDEs | In order to pass from the S-equation () to a partial differential equation (PDE) one need to differentiate both sides of () with respect to the time variable.
This is equivalent to the definition of a generator of the underlying (X^0,\pi )-branching Markov process \mathbf {X}, that requires additional regularity assump... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.0025870029348880053,
0.015552543103694916,
-0.0333639420568943,
-0.0004180509131401777,
-0.0034932170528918505,
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0.035683851689100266,
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0.04478033632040024,
0.026083705946803093,
-0.035103876143693924,
0.010531162843108177,
-0.0037488648667931557,... | |
272f7aba0142f00668c2b0047cb24d0088fdebb1 | subsection | 9 | 45 | Related PDEs | Thus, for f\in \text{B}(\mathbb {R}), the generator T^0 of the process X^0 has the following form, \begin{align*}
T_t^0f(x) &= \mathbb {E}_x^0[f(X^0_t), \tau >t] = e^{-t} f(x).
\end{align*}
\item
\textit {Pure-jump process.} Let the non-branching part X^0 of a branching Markov process \mathbf {X} be the pure-jump Mark... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.005818959791213274,
0.004182019736617804,
-0.02997622825205326,
0.023001108318567276,
0.0016054605366662145,
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0.030312011018395424,
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-0.0023810039274394512,
-0.007181168999522924,
... | |
172a6059b03968b71be8fffd974368a6dd6a2c87 | subsection | 10 | 45 | Related PDEs | Then the branching law \pi has the following form,
\begin{align*}
\pi (y,d\mathbf {z}) &= {1}_{\mathbb {R}_{sym}^2}(\mathbf {z}) \delta _{y}(dz_1)\delta _{y}(dz_2).
\end{align*}
\end{}\item
We consider the following generalization of the branching law \ref {item:P1}. We assume that a particle gives birth to n children... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.016437705606222153,
0.0027491566725075245,
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0.011958163231611252,
-0.0019087656401097775,
... | |
dc26182c6fe68a7d283b0fb7401c1e17c8b1dffc | subsection | 11 | 45 | Related PDEs | The Laplace transform v_\lambda (x,t) of \mathbf {X}, defined by () satisfies (REF ), which reads now as follows,\partial _t v_\lambda (x,t) = (b*v_\lambda )(x,t),\quad v_\lambda (x,0)=e^{ -\lambda x}, \quad x\in \mathbb {R},\ t>0.Therefore, the log-Laplace transform of \mathbf {X}_1 equals\psi (\lambda ) = \ln v_\lamb... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.031088244169950485,
0.028020625934004784,
-0.04706732556223869,
0.018344659358263016,
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-0.005040202755481005,
0.0197945274412632,
0.021... | |
a101d143b95bb9043f70f31864e955ff546fca65 | subsection | 12 | 45 | Related PDEs | The idea to consider the logistic reaction term u-u^2 is usually referred to Fisher .The Laplace transform v_\lambda (x,t) of \mathbf {X}, defined by () satisfies (REF ), which reads now as follows,\partial _t v_\lambda (x,t) = (a*v_\lambda )(x,t),\quad v_\lambda (x,0)=e^{ -\lambda x}, \quad x\in \mathbb {R},\ t>0.Ther... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.04746561497449875,
-0.014415200799703598,
-0.031089214608073235,
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0.0080050528049469,
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0.020115653052926064,
-0.0033424340654164553,
... | |
474cf9b99d62748156955bba5fe3fbaaec9ccac7 | subsection | 13 | 45 | Related PDEs | We refer again to , .The Laplace transform v_\lambda (x,t) of \mathbf {X} satisfies,\partial _t v_\lambda (x,t) = (a*v_\lambda )(x,t) - 2v_\lambda (x,t) + (\sum _{n\ge 1} np_n) v_\lambda (x,t),\quad v_\lambda (x,0)=e^{ -\lambda x}.Therefore, the log-Laplace transform of \mathbf {X}_1 equals\psi (\lambda ) = \ln v_\lamb... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.007104660850018263,
0.0033462876453995705,
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0.017887568101286888,
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0.025137221440672874,
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-0.001933559076860547,
0.02487776055932045,
0.0... | |
65704723e4c6d9e7a87e3801421ad2cab1d82533 | subsection | 14 | 45 | Related PDEs | Then the (X_0,\tau )-branching Markov process \mathbf {X} given by and satisfies conditions of Theorem .We omit the proof, since it repeats the one of Proposition REF .Example 2.9 (X3+P2)
The (X_0,\pi )-branching Markov process \mathbf {X} defined by and correspond to the following S-equation\partial _t u(x,t) = \frac... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.025710532441735268,
0.012733198702335358,
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0.024093134328722954,
... | |
3b5ef6abb641d38efb6d88a213b6964e8cd16568 | subsection | 15 | 45 | A relation between branching Markov processes and evolution equations | The purpose of this section is to formulate results of Ikeda, Nagasawa and Watanabe on connection between branching Markov processes and evolution equations. We will follow the notations of , , .We will denote \mathbb {N}_0=\mathbb {N}\cup \lbrace 0\rbrace =\lbrace 0,1,2,\dots \rbrace ,
\widehat{\mathbb {R}}=\mathbb {R... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.01408997643738985,
-0.0065224310383200645,
-0.015623319894075394,
-0.012846519239246845,
-0.008589774370193481,
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0.018781550228595734,
-0.02140577882528305,
-0.002687165280804038,
-0.028500353917479515... | |
5ada9ed1f4d3116f61cd53570a097001ad7773de | subsection | 16 | 45 | A relation between branching Markov processes and evolution equations | The norm of f\in \text{B}(\widehat{\mathbb {R}}) and g\in \text{B}(\widehat{\mathbf {R}}) will be denoted correspondingly\Vert f\Vert = \operatornamewithlimits{ess\,sup}_{x\in \widehat{\mathbb {R}}} |f(x)|, \qquad \Vert g\Vert = \operatornamewithlimits{ess\,sup}_{x\in \widehat{\mathbf {R}}} |g(x)|.The bold symbols {x},... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.002601390006020665,
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0.028287256136536598,
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0.024396613240242004,
-0.03155234456062317,
0.012717058882117271,
0.02914167195558548,
0.03... | |
c678df14d1f6a6a06b24055ead536535c73fdaec | subsection | 17 | 45 | A relation between branching Markov processes and evolution equations | \ (t\widehat{f}\,) \right|_{\widehat{\mathbb {R}}}}{}({x}), \quad {x}\in \widehat{\mathbf {R}},\ t\ge 0.for every f\in \text{B}(\widehat{\mathbb {R}}), \Vert f\Vert <1.Let \lbrace \tau _j\rbrace _{j\ge 1} be splitting (or branching) times of the process \mathbf {X}. We denote\tau _* = \lim \limits _{n\rightarrow \infty... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.011137627996504307,
-0.025402946397662163,
-0.03200923651456833,
0.008070966228842735,
-0.011923365294933319,
0.0120530491694808,
0.0072699724696576595,
0.03875284269452095,
0.026440424844622612,
0.022900793701410294,
0.011900478973984718,
-0.023526331409811974,
0.009245757013559341,
0.0... | |
68d3a6bee307e4c9497fd4f27482cb48ff079ae1 | subsection | 18 | 45 | A relation between branching Markov processes and evolution equations | There exists a stochastic kernel \pi (x,E)\text{ on }\widehat{\mathbb {R}}\times \widehat{\mathbf {R}} such that for each \lambda >0,\ x\in \widehat{\mathbb {R}}, and E - Borel in \widehat{\mathbf {R}}, we have a.s. on \lbrace \tau <\infty \rbrace ,
[\lbrace x\rbrace ][e^{-\lambda \tau },\, \mathbf {X}_{\tau }{\in } E\... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
0.012585342861711979,
-0.0013157404027879238,
-0.020197568461298943,
0.01707030087709427,
0.0130353644490242,
-0.007894442416727543,
0.02780979312956333,
0.033438876271247864,
0.019465330988168716,
0.008207169361412525,
0.015895670279860497,
-0.004595557227730751,
0.0013424365315586329,
0.... | |
8434fa0832d49d89d43147430e06b240609b4843 | subsection | 19 | 45 | A relation between branching Markov processes and evolution equations | Namely, for {x}\in \widehat{\mathbf {R}}\backslash \lbrace \varnothing ,\mathit {\Delta }\rbrace , t\ge 0,¶[{x}][\tau _\mathit {\Delta }> t] = t\widehat{f}({x}), \quad f\equiv 1.The following theorem states that the semigroup of the (X^0,\pi )-branching Markov process \mathbf {X} started from a one-point configuration ... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0078027620911598206,
0.01908358745276928,
-0.038502778857946396,
0.005236164201050997,
0.010907086543738842,
0.009282464161515236,
0.049882758408784866,
0.06123223155736923,
0.03819768503308296,
0.016063163056969643,
0.002715330570936203,
-0.01764964871108532,
0.005983642768114805,
0.02... | |
db607d683c21e000f12e0bbf5c8f949ae74aadd4 | subsection | 20 | 45 | A relation between branching Markov processes and evolution equations | Then v and w are the minimal solutions in the class of non-negative functions to the following equations correspondingly
\begin{align}
v(x,t) &= (T_t^0 f)(x) + (\mathbf {K}{)(x,t), \\
w(x,t) &= (T_t^0 fg)(x) + \mathbf {K}({ +{_f{_g- {}{{{*[{v_f v_g}]{\hspace{-0.6pt}\bigvee \hspace{-0.6pt}}{\rule [-/2]{1ex}{}}}{}}{0.5ex... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.01755794696509838,
0.013477364555001259,
-0.010556125082075596,
0.00346849556080997,
0.02439196966588497,
-0.003384595736861229,
0.03777018189430237,
-0.014537553302943707,
0.004702204372733831,
0.017008785158395767,
0.005095008295029402,
-0.018961362540721893,
0.005697561427950859,
0.0... | |
d5773e69e98de70074bc2eaef67a013e7ad0c397 | subsection | 21 | 45 | A relation between branching Markov processes and evolution equations | Then for f\ge 0, g\ge 0, f,g\in \text{B}(\mathbb {R}), \gamma \in (0,1], k,l\in \mathbb {N}_0, {x}\in \mathbb {R}_{sym}^n, x\in \mathbb {R}, t\ge 0,
\begin{align}
t\big (\widehat{\gamma }\,({)^k({)^l\big )({x}) = \sum _{{k_1+\dots +k_n=k\\k_1,\dots ,k_n\in \mathbb {N}_0}} &\sum _{{l_1+\dots +l_n=l\\l_1,\dots ,l_n\in \m... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03277064487338066,
0.03670678287744522,
-0.02895655669271946,
-0.034967560321092606,
0.031000906601548195,
0.001879391842521727,
0.027797073125839233,
0.010214127600193024,
-0.012830591760575771,
0.037225499749183655,
-0.0031771352514624596,
-0.013318795710802078,
-0.031168727204203606,
... | |
f4c3d9ccff33c29febb529838841bf19c3014327 | subsection | 22 | 45 | A relation between branching Markov processes and evolution equations | We have {}{{{*[{e^{\lambda f+\mu g}}]{\hspace{-0.6pt}\bigwedge \hspace{-0.6pt}}{\rule [-/2]{1ex}{}}}{}}{0.5ex}}[1pt]{e^{\lambda f+\mu g}}{} = e^{\lambda {+ \mu {}, and for {y}\in \mathbf {R}\backslash \lbrace \varnothing \rbrace ,\ |\lambda |<\lambda _0,\ |\mu |<\mu _0,
\begin{equation}
{}{{{*[{e^{\lambda f+\mu g}}]{\h... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.036435529589653015,
-0.002683458384126425,
-0.011977341026067734,
-0.0358252190053463,
0.019240034744143486,
-0.007941664196550846,
0.008056096732616425,
0.03152253106236458,
0.038998834788799286,
0.025297366082668304,
-0.050075966864824295,
0.011260226368904114,
-0.04919101670384407,
0... | |
c864cd77302de4844afc5a9905a9c5913e6ae89f | subsection | 23 | 45 | A relation between branching Markov processes and evolution equations | By the monotone convergence theorem, taking \gamma \rightarrow 1_{-} we extend (\ref {eq:branching_for_product}) to \gamma =1.
}Next, by Theorem~2.3,
q(x,t) := t ( {}{{{*[{\gamma e^{\lambda f+\mu g}}]{\hspace{-0.6pt}\bigwedge \hspace{-0.6pt}}{\rule [-/2]{1ex}{}}}{}}{0.5ex}}[1pt]{\gamma e^{\lambda f+\mu g}}{})(\lbrace ... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.053559016436338425,
0.06341631710529327,
-0.02154567837715149,
-0.04660092294216156,
0.007675266359001398,
-0.02163723297417164,
0.02694735676050186,
-0.005603860132396221,
0.033813897520303726,
0.02540620043873787,
-0.0024318997748196125,
-0.010238775052130222,
-0.030289074406027794,
-... | |
380f0bde711cb2209b89262c93777214b5e9ec36 | subsection | 24 | 45 | A relation between branching Markov processes and evolution equations | (t{) _{\widehat{\mathbb {R}}}}{}({x}), \right.\\
t({{)({x}) &= {}{{{*[{\left. (t{{) _{\widehat{\mathbb {R}}}}]{\hspace{-0.6pt}\bigvee \hspace{-0.6pt}}{\rule [-/2]{1ex}{}}}{}\right.{0.5ex}}[1pt]{\left. (t{{) _{\widehat{\mathbb {R}}}}{}({x}) + t{({x})\, t{({x}) - {}{{{*[{ \left. (t{) _{\widehat{\mathbb {R}}} \! \left. (t... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.013486500829458237,
0.03771338239312172,
-0.035791099071502686,
-0.03936105594038963,
-0.0015942752361297607,
0.0027461200952529907,
-0.022853821516036987,
-0.0010021431371569633,
0.012357540428638458,
-0.0004550645244307816,
-0.01972629688680172,
-0.013814509846270084,
-0.009733470156788... | |
030ea8c8f7d917afc3aec21014099dadaa30645c | subsection | 25 | 45 | A relation between branching Markov processes and evolution equations | By Lemma~\ref {lem:prod_of_sum}, for \gamma \in (0,1],
w_\gamma (x,t):=t(\gamma {{)(\lbrace x\rbrace ),
satisfies the following equation
\begin{equation}
w_\gamma (x,t) = (T_t^0 \gamma f g)(x) + (\mathbf {K}\, \widehat{\gamma }({_\gamma +{_f {_g - {))(x,t).
}
Denote inductively, for \gamma \in (0,1],
T_{t}^{\gamma ,... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03550243750214577,
0.0483236089348793,
-0.026176560670137405,
-0.027992893010377884,
0.013050122186541557,
-0.026008663699030876,
0.0221317857503891,
-0.0030297650955617428,
-0.0056741321459412575,
0.03504453971982002,
-0.0014576452085748315,
-0.015782251954078674,
-0.0021273226011544466,... | |
3f3061278c9de0bb0d110f0256b43dc21080f521 | subsection | 26 | 45 | A relation between branching Markov processes and evolution equations | Denote
f_m=\min \lbrace f, m\rbrace ,\quad g_m=\min \lbrace g, m\rbrace ,\quad w_m(x,t) =t({_m{_m)(\lbrace x\rbrace ),\quad m\in \mathbb {N}.
Since
\sum _{j=0}^{n} (T_t^{\gamma ,j} f_m g_m) \le \sum _{j=0}^{n} (T_t^{\gamma ,j} f g) \le \sum _{j=0}^{n} (T_t^{\gamma ,j} f g) + (T_t^{\gamma ,n+1} w),
then
w_m(x,t) = ... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.029784180223941803,
0.010612139478325844,
-0.03619266301393509,
-0.021773578599095345,
-0.006000322289764881,
-0.023421473801136017,
0.027525953948497772,
0.0038527180440723896,
0.02072075568139553,
0.03857295587658882,
-0.009719530120491982,
-0.016066977754235268,
-0.00041125857387669384... | |
bb842ac2873cfda7b3b2ff0a83751832743c52ca | subsection | 27 | 45 | A relation between branching Markov processes and evolution equations | \end{}
}For the shift operator on \mathbb {R} we will write S_y(x) = x+y, x,y\in \mathbb {R}. For {x}\in \mathbf {R}, y\in \mathbb {R}, with abuse of notations we denote
S_y({x})={x}+y:=\widehat{S_y}({x}),
where we put S_y(\varnothing ) := \varnothing .
}\begin{}
We call a Markov process spatially homogeneous if its ... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.016722718253731728,
0.002118948381394148,
-0.04128802567720413,
0.009139514528214931,
0.046658825129270554,
0.007564130704849958,
0.00020824723469559103,
0.02793731354176998,
0.06328999251127243,
0.026396261528134346,
-0.0132362749427557,
-0.01850789785385132,
0.006332051008939743,
0.00... | |
da67ee50f78bc1c2022122003952d2ad76c58872 | subsection | 28 | 45 | A relation between branching Markov processes and evolution equations | \end{align*}
Hence, by (\ref {eq:minimal_sol_to_S_equation}), Definition~\ref {defn:X0_pi_BMP}, and the dominated convergence theorem,
\begin{align*}
\Vert t\widehat{f}-\widehat{f}\Vert &\le [\lbrace 0\rbrace ][\Vert \widehat{f}(\cdot + \mathbf {X}_t)-f(\cdot )\Vert , \tau >t] + [\lbrace 0\rbrace ][\tau \le t] \\
&= \m... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03618746995925903,
0.053823523223400116,
-0.04012354463338852,
-0.001616193214431405,
-0.0029596835374832153,
-0.011564124375581741,
0.014012728817760944,
0.019497297704219818,
0.031976789236068726,
0.043785009533166885,
0.0028567048721015453,
-0.00858918484300375,
-0.015156936831772327,
... | |
99b4800a324065d49c05020455f848b730f772ad | subsection | 29 | 45 | A relation between branching Markov processes and evolution equations | (\ref {eq:explosion_and_first_branching}), (\ref {eq:space_homogeneous}),
\begin{align*}
u(x+\tilde{x},t)-&u(x,t) = [\lbrace 0\rbrace ][S_{\tilde{x}}\widehat{f}(x+\mathbf {X}_t)-\widehat{f}(x+\mathbf {X}_t)]\\
&\le [\lbrace 0\rbrace ][S_{\tilde{x}}\widehat{f}(x+\mathbf {X}_t)-\widehat{f}(x+\mathbf {X}_t),\, \tau _n > T... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0025936951860785484,
0.023389026522636414,
-0.06902280449867249,
0.013464328832924366,
0.012594678439199924,
0.005923542194068432,
0.013937297277152538,
0.013914411887526512,
0.03426729142665863,
0.009909440763294697,
0.0030647560488432646,
-0.00786499958485365,
-0.02270246110856533,
0.... | |
5313a112101e2f8f2f9b98374bf69c93c2db618e | subsection | 30 | 45 | A relation between branching Markov processes and evolution equations | The branching property of (\mathbf {X}_n) follows form the branching property of \mathbf {X}, namely, by (\ref {eq:branching_property}),
\begin{equation}
n\widehat{f}({x}) = \widehat{(n\widehat{f})\big \vert _{\widehat{\mathbb {R}}}}({x}).
\end{equation}
where f\in \text{B}(\widehat{\mathbb {R}}),\ \Vert f\Vert <1.
By ... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.019808385521173477,
0.0032848650589585304,
-0.03387874737381935,
0.02516488917171955,
-0.008774900808930397,
-0.02441711537539959,
-0.006894019898027182,
0.015733778476715088,
0.01423060055822134,
-0.00745485071092844,
-0.026980912312865257,
0.019136914983391762,
-0.010644336231052876,
... | |
efde57388d5d26dc2cae05990ff4ffa50c7c5521 | subsection | 31 | 45 | A relation between branching Markov processes and evolution equations | \end{equation}Now we will formulate results on the position of the left-most particle of a branching random walk.
}Assume,
\begin{align}
&[\lbrace 0\rbrace ][{_{\!\!1}(\mathbf {X}_1)] = 0, & \quad &[\lbrace 0\rbrace ][{_{\!\!2}(\mathbf {X}_1)]<\infty , \\
&[\lbrace 0\rbrace ][{_{\!\!3}(\mathbf {X}_1) (\ln _+ {_{\!\!3}(... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.019078020006418228,
0.0173381045460701,
-0.019230645149946213,
0.00881404522806406,
-0.0053265830501914024,
-0.014148259535431862,
0.006734541151672602,
0.005151065532118082,
0.028800178319215775,
0.02557981014251709,
-0.025045624002814293,
0.015933962538838387,
-0.017185481265187263,
0... | |
ac451e01c73fc93ec74577984745f90882cf6914 | subsection | 32 | 45 | A relation between branching Markov processes and evolution equations | \begin{}
Under (\ref {assum:supercritical}), (\ref {assum:c_*}), (\ref {assum:H1}), (\ref {assum:H2}), if the distribution of \mathbf {X}_1 is non-lattice, then there exists a constant C_*>0, such that for any x\in \mathbb {R},
\lim \limits _{n\rightarrow \infty } [\lbrace 0\rbrace ][M_n + c_* n - \frac{3}{2\lambda _*... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06507289409637451,
0.020724434405565262,
-0.03879345580935478,
0.007401583716273308,
-0.006905601359903812,
-0.04614925757050514,
0.009973062202334404,
0.029331637546420097,
0.0006643303204327822,
0.017870627343654633,
-0.022311579436063766,
-0.020632868632674217,
0.023303544148802757,
... | |
0680addcfdb70603d052667887472d212873a1e1 | subsection | 33 | 45 | A relation between branching Markov processes and evolution equations | \end{equation}
In particular v_{\lambda +\mu }(0,s)<\infty , w_{\lambda ,\mu }(0,s)<\infty , s\in [0,t].
\end{}
\end{}
\begin{}
Denote
e_{\lambda ,n}(y):= \min \lbrace n, e^{- \lambda y}\rbrace .
First, we note that v_{\lambda }(0,t)<\infty implies v_{\lambda }(0,s)<\infty , s\in [0,t].
Indeed, by the Markov property... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.02761039510369301,
0.0024458628613501787,
-0.02574833482503891,
0.012301812879741192,
0.0035734018310904503,
0.01817798800766468,
-0.00454449700191617,
0.004330818075686693,
0.05094720795750618,
0.017735367640852928,
-0.016911176964640617,
0.009936079382896423,
-0.02880089357495308,
-0.... | |
d369726c7a5034b09c4f731e477f35f895772bc1 | subsection | 34 | 45 | A relation between branching Markov processes and evolution equations | \end{equation}
In particular v_0(0,t)<\infty implies v_0(0,s)<\infty , s\in [0,t].
}Let us prove the opposite inequality.
Denote
e_{\lambda ,n,m}(y):= \max \lbrace \min \lbrace n, e_\lambda (y)\rbrace , \frac{1}{m}\rbrace .
Then, for m\le n, n,m\in \mathbb {N}, the inequality holds,
e_{\lambda ,n,m^2}(x+y) \le e_{\l... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.039526719599962234,
0.019076602533459663,
-0.033574819564819336,
0.025730520486831665,
-0.0016787409549579024,
0.002016396727412939,
-0.02786709927022457,
0.017733609303832054,
0.03751223161816597,
0.012994981370866299,
-0.007851929403841496,
0.024387527257204056,
-0.022571435198187828,
... | |
0e880884ea83f2c3c6f5c9efc25d21484d2f8e5c | subsection | 35 | 45 | A relation between branching Markov processes and evolution equations | By (\ref {eq:v2_t_leq_v_2t}) and (\ref {eq:v2_t_geq_v_2t}) the equation (\ref {eq:Laplace_is_time_multiplicative}) holds.
}Similar to the previous consideration, we have by (\ref {eq:nice_formula}) and (\ref {eq:another_nice_formula}),
\begin{align*}
&[\lbrace 0\rbrace ][{(\mathbf {X}_t) {(\mathbf {X}_t)] = [\lbrace 0\... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0621299147605896,
0.04308813437819481,
-0.020887980237603188,
0.014967938885092735,
0.005489007104188204,
-0.036740873008966446,
0.010108317248523235,
0.00027344905538484454,
0.00613365089520812,
-0.009398828260600567,
-0.01400669477880001,
-0.00890294834971428,
-0.0175465140491724,
0.0... | |
7073370535888e25ab45a66a2323a20d0491e087 | subsection | 36 | 45 | A relation between branching Markov processes and evolution equations | Therefore,
[\lbrace 0\rbrace ][{_{\!\!2}(\mathbf {X}_1)] \le C (v_0(0,1) + v_{\delta +\lambda _*}(0,1)) <\infty .
Next, c_* = \frac{\partial }{\partial \lambda } \psi (\lambda _*) implies \frac{\partial }{\partial _\lambda } v_{\lambda _*}(0,1) = c_* v_{\lambda _*}(0,1) which yields [\lbrace 0\rbrace ][{_{\!\!1}(\mat... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06352575123310089,
0.04174026474356651,
-0.025645243003964424,
0.019924264401197433,
0.01780368760228157,
-0.015172037295997143,
0.03234260156750679,
0.03341051936149597,
0.009115426801145077,
0.027109812945127487,
-0.038658563047647476,
-0.020229382440447807,
-0.014401611872017384,
0.0... | |
7f51bd93e24d407d2ed61bcd2e2d07ab2ea32eb9 | subsection | 37 | 45 | A relation between branching Markov processes and evolution equations | Hence,
[\lbrace 0\rbrace ][{_{\!\!4}(\mathbf {X}_1)\ln _+ {_{\!\!4}(\mathbf {X}_1)] \le C w_{0,0}(0,1)<\infty .
As a result, (\ref {assum:H2}) holds and the proof is fulfilled.
}
}\section {Proof of Theorem~\ref {thm:main}}
Let u be the minimal non-negative solution to the S-equation (\ref {eq:S}) with the initial co... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.045311637222766876,
0.06865399330854416,
-0.03832418471574783,
0.021984534338116646,
-0.004649401176720858,
-0.011419447138905525,
0.02520364336669445,
-0.007391746621578932,
0.025539284572005272,
0.027293777093291283,
-0.007082803640514612,
0.00665943743661046,
0.02819390594959259,
0.0... | |
8ca80b907d376352fd67c69de5c65716f9555ed3 | subsection | 38 | 45 | A relation between branching Markov processes and evolution equations | The corresponding derivative martingale (D_n(k)) satisfies
D_n(k)\rightarrow D_\infty (k),\ n\rightarrow \infty ,\ a.s.
Since D_{2^kn}(k) = D_n(1), then a.s. D_\infty (k)=D_\infty (1)=D_\infty , k\in \mathbb {N}.
Denote
M_n(k) := \min \lbrace x\in \mathbb {R}\,\vert \, x\in \mathbf {X}_n(k)\rbrace ,\quad n\in \mathb... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06205559894442558,
0.023278476670384407,
-0.021783528849482536,
-0.04951633885502815,
-0.0455806627869606,
-0.00171423249412328,
0.023461531847715378,
-0.005491646006703377,
0.010273953899741173,
0.051285870373249054,
-0.00012227492698002607,
0.012935876846313477,
0.0020593672525137663,
... | |
b1a88fd571913382d49f276557f6c1eedcb811e7 | subsection | 39 | 45 | A relation between branching Markov processes and evolution equations | Hence, taking k\rightarrow \infty in (\ref {eq:descrete_k_limit}),
\phi (x) \le \liminf _{t\rightarrow \infty } u_h(x+\theta (t),t) \le \limsup _{t\rightarrow \infty } u_h(x+\theta (t),t) \le \phi (x+h).
On the other hand, u_h(x-h, t) \le u(x, t) \le u_h(x, t), implies
\phi (x-h) \le \liminf _{t\rightarrow \infty } ... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.0728980079293251,
-0.0018611809937283397,
-0.06004622206091881,
-0.021338235586881638,
-0.027458859607577324,
-0.04536283016204834,
0.026497265323996544,
0.013759956695139408,
0.021368762478232384,
0.02771833725273609,
-0.028008341789245605,
-0.011554394848644733,
-0.0005060771363787353,
... | |
f8f5f53643b7254f3c033b7c4a4e9ee0c42efb85 | subsection | 40 | 45 | A relation between branching Markov processes and evolution equations | ({t_0} S_{c_*t_0} {t} \widehat{f}\,) \right|_{\mathbb {R}} (x+c_*t- \frac{3}{2\lambda _*} \ln t + C) = \left. ({t_0} S_{c_*t_0} \widehat{\phi }) \right|_{\mathbb {R}}(x).
\end{align*}
}This finishes the proof of Theorem~\ref {thm:main}.
\end{align*}}\section {Proof of Proposition~\ref {prop:nl+logistic}}
\begin{}
By co... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.06720735877752304,
0.07843910902738571,
-0.01148354634642601,
0.008355138823390007,
-0.004143232945352793,
-0.0035175515804439783,
-0.0276368148624897,
0.005551016889512539,
0.004906259477138519,
-0.009835409931838512,
-0.017397001385688782,
0.012567044235765934,
0.001756868208758533,
0... | |
40344949fbaf2b213aee8d0d7d35ed3b5c3bc212 | subsection | 41 | 45 | A relation between branching Markov processes and evolution equations | Therefore, c_* =\frac{\partial }{\partial \lambda } \psi (\lambda _*).
Next, for \lambda ,\mu \ge 0, \lambda +\mu <\lambda _0,
w_{\lambda ,\mu }(x,t) = e^{-(\lambda +\mu )x} \Big ( e^{t(\mathfrak {L}a)(\lambda +\mu )} + \int [0][t] e^{(t-s)(\mathfrak {L}a)(\lambda +\mu )} e^{s(\mathfrak {L}a)(\lambda )} e^{s(\mathfrak... | {
"cite_spans": []
} | 1808.00411 | On stability of traveling wave solutions for integro-differential
equations related to branching Markov processes | [
"Pasha Tkachov"
] | [
"math.PR"
] | 2,018 | en | Mathematics | [
-0.03404947370290756,
0.014271184802055359,
-0.04195163771510124,
0.0073873018845915794,
-0.010495536960661411,
-0.015575499273836613,
-0.007082199212163687,
-0.04915206506848335,
0.025796443223953247,
0.06535302102565765,
0.0031330245546996593,
0.03584958240389824,
0.019892703741788864,
0... |
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