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41dc408c5146b7d94017c9a060818b0a2d3dc412 | subsection | 74 | 93 | evaluation of | One can show this by
carrying out the z-integral formally,C_{ab,T=0} = \sum _{c=a,b} \lambda _c \int ^{\infty }_{-\infty }
d\tau \!\ G(\tau ) e^{-u_{c}\Lambda |\tau |},andG(\tau ) &\equiv \int ^{\infty }_{-\infty } d\xi F_{a}(\xi ;\tau ) F_{b}(\Lambda -\xi ;\tau ) \!\ d\xi , \\
F_{a}(\xi ;\tau ) & \equiv \int ^{\infty ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0.... |
d7546ea61f0550e04ee15f40ed392141385297fb | subsection | 75 | 93 | evaluation of | When the Luttinger parameters
get much smaller/larger than 1, \lambda _{a}+\lambda _b \rightarrow +\infty ,
the upper bound of C_{ab,T=0} as well as |A_{ab,T=0}| diverge;&C_{\rm u} \rightarrow \frac{\alpha }{\Lambda _{\cal E}} \Gamma \Big (\frac{1}{2}\Big )
\big (\lambda _{a}+\lambda _b\big )^{\frac{1}{2}}, \\
&|A_{ab,... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0.029653962701559067,... |
edba2ce2c2ffdcc2f875ce54ccf9a94648d8fdbb | subsection | 76 | 93 | parameters used in Fig. | To obtain theoretical phase diagram at finite temperature as in Fig. REF ,
we solved numerically the RG equations Eqs. (REF ,,) for
H<H_0 and Eqs.(REF ,,) for H_{0}<H<H_1. Thereby, a
set of parameters in the
RG equations are chosen in the following way.C_{ab} has an engineering dimension of [length]/[energy]. From
Eq. ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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... |
2a7833e2827be35ed4189ab6a3f9ccfb913099e6 | subsection | 77 | 93 | parameters used in Fig. | (REF ) (c=1,2,3,4) is set to the high-energy
cutoff in the energy scale, \Lambda _{\cal E}=40 [meV] .
For the Luttinger parameters K_a in Eq. (REF ),
we use Eq. (). The intra-pocket forward scattering strengths
in Eq. () are set as,g_{4,a=1} = g_{4,a=4} &= \tilde{g}, \\
g_{4,a=2} = g_{4,a=3}&= \tilde{g}, \\
g_{2,a=1} =... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0.012117939069867134,
0... |
79b781fba957b3be29b234557e06c357fcc9ae83 | subsection | 78 | 93 | calculation of optical conductivity | In the main text, we describe how the longitudinal
optical conductivity along the field direction behaves in the SNEI phases
as well as the metal-insulator transition points at H=H_{c,1} and H=H_{c,2}. According to the
linear response theory, the conductivity is given by a retarded correlation function between
an elect... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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4498918870108bff6b5c27989fb9e769a1741263 | subsection | 79 | 93 | calculation of optical conductivity | The Fourier transform is taken with respect to the
spatial coordinate z, imaginary time \tau and
the chain index j (y_{j}\equiv 2\pi l^2 j/L_x);\phi _{a,j}(z,\tau ) \equiv \frac{1}{\beta L_z N} \sum _{K} e^{ik_z z + ik y_j - i\omega _n \tau } \phi _{a,{K}}.In the following, we briefly summarize how to calculate the ret... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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-0.013787982985377... |
e2c613a7638b0ace1dcda8099ac096bc75f103f1 | subsection | 80 | 93 | calculation of optical conductivity | They are given by{A}_{K} &\equiv \left[\begin{array}{cccc}
\frac{u_1}{\pi K_1}k_z^2+2M(0) & 2M(0) & 2M^*(k) & 2M^*(k) \\
2M(0) & \frac{u_4}{\pi K_4}k_z^2+2M(0) & 2M^*(k) & 2M^*(k) \\
2M(k) & 2M(k) & \frac{u_2}{\pi K_2}k_z^2+2M(0) & 2M(0) \\
2M(k) & 2M(k) & 2M(0) & \frac{u_3}{\pi K_3}k_z^2+2M(0) \\
\end{array}\right], \... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0.02689... |
546f00e8cc3cb5363bff630cedbea26e2ae799b4 | subsection | 81 | 93 | calculation of optical conductivity | With the new basis, the gaussian action is given by\mathcal {S}_{\rm MF}=\frac{1}{2\beta L_zN}\sum _{{K}}\begin{pmatrix}
\vec{\Phi }^{\dagger }_{K} & \vec{\Theta }^{\dagger }_{K}
\end{pmatrix}[{M}_{c,{K}}]\begin{pmatrix}
\vec{\Phi }_{K} \\ \vec{\Theta }_{K}
\end{pmatrix},and[{M}_{c,{K}}] \equiv \left[\begin{array}{cc}
... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0... |
7dc9481914a3f756a81f565a6cc51f7ab37232cb | subsection | 82 | 93 | calculation of optical conductivity | The real part of the retarded correlation function
is nothing but the optical conductivity \sigma _{zz}(\omega );\sigma _{zz}(\omega ) &= {\rm Re} \!\ \Big \lbrace \overline{\sigma _{zz}(i\omega _n)}_{|i\omega _n =\omega +i\eta }\Big \rbrace , \\
\overline{\sigma _{zz}(i\omega _n)} &= {\vec{e}_{+}}^{\!\ T} \!\ {U}^{-1}... | {
"cite_spans": [
{
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"doi": "10.5517/cc6wfdw",
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"start": 146... | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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... |
ca3a5bb2ea336c6050faafa3ab2187f4db290e4a | subsection | 83 | 93 | calculation of optical conductivity | (REF ,,REF ,REF ).
[{P}(i\omega _n)] is a 4 by 4 diagonal matrix that represents an effect of the disorder,[{P}(i\omega _n)] \equiv \left[\begin{array}{cccc}
g_y m(i\omega _n) & & & \\
& 0 & & \\
& & 0 & \\
& & & 0 \\
\end{array}\right].m(i\omega _n) is a sum of the (\Phi _{+},\Phi _{+})-component of the inverse of the... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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fc8d5f51cffecc3ce548b3e090d98d148b7d2342 | subsection | 84 | 93 | calculation of optical conductivity | (REF ), we obtain the imaginary-time optical
conductivity as&\overline{\sigma _{zz}(i\omega _n)} = \\
& \ \ \ \ \frac{e^2 \omega _n}{\pi ^2 l^2} {\vec{e}_{+}}^{\!\ T}
\Big [\frac{\pi ^2}{k^2_z} \big ({D}{A} - {D}{T}{P}{T}\big )
+ \omega ^2_n {1}_{4\times 4} \Big ]^{-1}_{|{k}={0}}
{U}^{-1} \!\ \vec{e}_{+}.The {k}={0} li... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
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... |
38aebfd46af58f09980f6b1b261d1f5ca2bc2bd3 | subsection | 85 | 93 | calculation of optical conductivity | To see this, use Taylor expansions of {A} and {D} in
small k;{A}_{K} = 2M(0) {A}_0 + k^2_z {A}_1 + {\cal O}(k), \\
{D}_{K} = 2M(0) {D}_0 + k^2_z {D}_1 + {\cal O}(k),with{A}_0 \equiv \left[\begin{array}{cccc}
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 \\
\end{array}\right], \ {D}_0 \equiv \left[\be... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0... |
dde084fd15693965895f522e7f2b6879e122b10e | subsection | 86 | 93 | calculation of optical conductivity | (REF ) as,&\overline{\sigma _{zz}(i\omega _n)} = \frac{e^2 \omega _n}{\pi ^2 l^2} \!\ {\vec{e}_{+}}^{\!\ T}
\bigg [2\pi ^2 M(0) {D}_0{A}_1
\\
&\hspace{0.0pt} + \pi \Big (2M(0)-\frac{g_y m(i\omega _n)}{4}\Big )
{U}^{-1} {A}_0
+ \omega ^2_n {1}_{4\times 4}\bigg ]^{-1}
{U}^{-1} \!\ \vec{e}_{+} \\
& \ = \frac{e^2 \omega _n... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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0.0... |
4e00e92d2ca5a4f537d6638745b3498cee6a9a2d | subsection | 87 | 93 | calculation of optical conductivity | With use of M(k)=0 for k\gg 1/l , we
obtain the following expression for m(i\omega _n),m(i\omega _n) = \frac{\pi }{2} \Big (\frac{K_1}{\sqrt{\omega ^2_n+\omega ^2_1}}
+ \frac{K_2}{\sqrt{\omega ^2_n+\omega ^2_2}}\Big ),with \omega ^2_1\equiv 4\pi M(0) u_1 K_1<4\pi M(0) u_2 K_2 \equiv \omega ^2_2.
After the analytic cont... | {
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... | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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388bee1f72faf69930137fece30da4966b7b1749 | subsection | 88 | 93 | calculation of optical conductivity | (REF ) is one and only one
solution of g(\omega )=0 within 0<\omega <\omega _{1}. The renormalized
gap \omega _{*} becomes progressively smaller, when the disorder strength increases.
There exists a critical value of the disorder,g_{y,c} \equiv \frac{1}{\pi ^2 uK} \frac{8 \omega ^2_{g} \omega _1 \omega _2}{K_1 \omega _... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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86943d95d6880dda487a8bae0350af67f7c01c52 | subsection | 89 | 93 | magnetism and spin nematicity in SNEI phases | SNEI phases introduced in the main text are characterized by particle-hole pairings between
n=0 LL with \uparrow (\downarrow ) spin and n=-1 LL with \downarrow (\uparrow )
spins. The phases break the U(1) spin rotational symmetry around the field direction. Nonetheless,
neither A-carbon site \pi -orbital electron spin ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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563a003558c5dc9897a7d6fe162d2bf8d82c6a14 | subsection | 90 | 93 | magnetism and spin nematicity in SNEI phases | Such 2nd rank spin tensor has two components,Q^{AB}_{+-}({r}) &\equiv \langle S_{A,+}({r}) S_{B,-}({r}) \rangle , \\
Q^{AB}_{++}({r}) &\equiv \langle S_{A,+}({r}) S_{B,+}({r}) \rangle .In the SNEI-I phase, Q^{AB}_{+-}({r}) vanishes identically, while
Q^{AB}_{++}({r}) exhibits both a ferro-type and a density-wave-type o... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
"cond-mat.dis-nn",
"cond-mat.str-el"
] | 2,018 | en | Physics | [
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4a5bcda33e9d2505d7fa5b39dbd670c722353cf4 | subsection | 91 | 93 | magnetism and spin nematicity in SNEI phases | (REF ,,REF ) and\langle \psi ^{\dagger }_{1,+,j}(z) \psi _{4,-,m}(z) \rangle &= \delta _{jm} i\sigma _{\overline{4}1,m}
e^{i(\phi _{1}+\phi _4)+i(\theta _4-\theta _1)}, \\
\langle \psi ^{\dagger }_{1,-,j}(z) \psi _{4,+,m}(z) \rangle &= \delta _{jm} i\sigma _{4\overline{1},m}
e^{-i(\phi _{1}+\phi _4)+i(\theta _4-\theta ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
"Xiao-Tian Zhang",
"Ryuichi Shindou"
] | [
"cond-mat.mes-hall",
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"cond-mat.str-el"
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6ff3c8a6c5b6478deecbc2af11b40d9ed658430e | subsection | 92 | 93 | magnetism and spin nematicity in SNEI phases | (REF ) and finite expectation values of the following
two quantities in the SNEI phases;\langle \psi ^{\dagger }_{\uparrow }({r},A) \psi _{\downarrow }({r},B) \rangle &= \frac{\sqrt{2} i v}{\pi l^2} e^{-i\Theta _{-}} \cos \big ((k_{F,1}+k_{F,4})z + \Phi _{-}\big ), \\
\langle \psi ^{\dagger }_{\downarrow }({r},A) \psi ... | {
"cite_spans": []
} | 10.1103/PhysRevB.98.205121 | 1802.10253 | Theory of metal-insulator transitions in graphite under high magnetic
field | [
"Zhiming Pan",
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"Ryuichi Shindou"
] | [
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4192b88f1025402ccc3c7f9773a0a655aa39bbdc | abstract | 0 | 13 | Abstract | A matrix is called acyclic if replacing the diagonal entries with $0$, and
the nonzero diagonal entries with $1$, yields the adjacency matrix of a forest.
In this paper we show that null space and the rank of a acyclic matrix with $0$
in the diagonal is obtained from the null space and the rank of the adjacency
matrix ... | {
"cite_spans": []
} | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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474c6be3b6f0ba29ba52ed2982118c8263ad07ee | subsection | 1 | 13 | Introduction | Throughout this article, all graphs are assumed to be finite, undirected and without loops or multiple edges.
The vertices of a graph G are denoted by V(G) and its edges by E(G).
We also assume that \mathbb {F} denotes an arbitrary field.
Following the notation in , we denote by \mathcal {M}_{\mathbb {F}}(G)
the set of... | {
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$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
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] | 2,018 | en | Mathematics | [
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4f9d4604e7795eb73840f1e01809661d9d6e7ce9 | subsection | 2 | 13 | Introduction | It was
implicitly shown
that \operatorname{{Null}}(A(F)) coincides with the intersection of all the maximum independent sets of F.
In an optimal time algorithm for finding a sparsest (i.e., has the fewest nonzeros)
\lbrace -1,0,1\rbrace basis for the null
space of a forest has been found. It is important to notice that... | {
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{
"arxiv_id": "",
"doi": "10.1137/0607059",
"end": 509,
"openalex_id": "https://openalex.org/W1967717294",
"raw": "T. F. Coleman and A. Pothen. The null space problem. I. Complexity. SIAM J. Algebraic Discrete Methods, 7(4):527–537, 1986.",
"source_ref_id": "1... | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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7bf97e6ae26f5ab6d5be86cfadc65c4347c5d011 | subsection | 3 | 13 | On the null space | The null space of a graph is the direct sum of the null spaces of its connected components.
In a similar fashion, the null space of a matrix M\in \mathcal {M}_{\mathbb {F}}(G)
is the direct sum of the null spaces of M over the connected components of G.
Because of this, we study the null space of matrices over trees an... | {
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$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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695eb78fc7195561646452bae8ae664400b364ab | subsection | 4 | 13 | On the null space | The following statements are true.If u_1,u_2\in \operatorname{{Supp}}(M),
D^{(M,v)}_{u_2,u_2}=D^{(M,v)}_{u_1,u_1}*M_{u,u_1}^{-1}*M_{u,u_2},
if u_1\in \operatorname{{Supp}}(M), u_2\notin \operatorname{{Supp}}(M),
D^{(M,v)}_{u_2,u_2}=D^{(M,v)}_{u_1,u_1}*M_{u,u_1}^{-1},
if u_1\notin \operatorname{{Supp}}(M), u_2\in ... | {
"cite_spans": []
} | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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03aea947d2d54abfa55c2c3376690e7f53e307b2 | subsection | 5 | 13 | On the null space | If \overrightarrow{x}_{w_1}\ne 0, then applying Lemma REF we get:(A(T)D\overrightarrow{x})_{u}&=D_{w_1,w_1}\overrightarrow{x}_{w_1}+ \sum _{{w\sim u,\\ w\ne w_1}}D_{w,w}\overrightarrow{x}_{w}
\\
&=D_{w_1,w_1}\overrightarrow{x}_{w_1}+\sum _{{w\sim u,\\ w\ne w_1}}\overrightarrow{x}_{w}D_{w_1,w_1}
\left(M_{u,w_1}^{-1}M_{u... | {
"cite_spans": []
} | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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75e2d2d8eb227dcfa9d9c7c6f79863251bc79dd3 | subsection | 6 | 13 | On the null space | We have the following.Corollary 5 Given a forest F and a matrix M\in \mathcal {M}_{\mathbb {F},0}(F), a vector \overrightarrow{x} is in \operatorname{{Null}}(M) if
and only if D^{(M,U)}\overrightarrow{x} is in \operatorname{{Null}}(A(F)) for every set U supp-transversal of M.Corollary 6
Let F be a forest, and M,N\in \... | {
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"doi": "10.1016/j.disc.2017.11.019",
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$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
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e7c0ae847e3ea8a0ee00564c965185b007ed7184 | subsection | 7 | 13 | On the null space | Given a matrix M\in \mathcal {M}_{\mathbb {F},0}(F), and G an induced subgraph of F, we denote by M[G]
the matrix obtained by deleting the rows and columns of vertices not in G. We do the same for
vectors, \overrightarrow{x}[G] denotes the vector obtained from \overrightarrow{x} by deleting the coordinates
correspoding... | {
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{
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"source_ref_id": "68f1a3c8200d4b503eb2b1... | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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55ef62ae73d927584f39da31bffc5de3802a224d | subsection | 8 | 13 | On the null space | Hence a sparsest basis for the null space of M can be found in optimal time.[H]for finding a sparsest basis of the null space a acyclic matrix with 0 in the diagonal.INPUT: M, a tree-patterned matrix with 0 in the diagonal.
Find F such that M\in \mathcal {M}_{\mathbb {F},0}(F).
Apply the algorithms from to find a spa... | {
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{
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"doi": "",
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"raw": "D. A. Jaume, G. Molina, A. Pastine, and M. Safe. A \\lbrace -1,0,1\\rbrace - and sparsest basis for the null space of a forest in optimal time, 2017. Preprint available as arXiv:1710.01639.",
... | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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cc0e4fe031a8de44c7d967f8cfcdddfe4b802d95 | subsection | 9 | 13 | On the rank | In the previous section we proved that given a forest F and M\in \mathcal {M}_{\mathbb {F},0}(F), \operatorname{{Null}}(M)
is a non-singular diagonal multiplication of \operatorname{{Null}}(F). In this section show that \operatorname{{Rank}}(M)
is a non-singular diagonal
multiplication of \operatorname{{Rank}}(F). In o... | {
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{
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"doi": "10.1145/253168.253189",
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"raw": "D. A. Jaume, G. Molina, and R. Sota. S-trees, 2017. Preprint available as arXiv:1709.03865.",
"source_ref_id": "717ee9622a3c7ce2b5fa63f467a... | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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11b6c9d4589a45de39a5d92cd0650e41dde29a5c | subsection | 10 | 13 | On the rank | We define the v-normalization of M as the non-singular diagonal matrix
withC^{(M,v)}_{w,w}=M_{v,\pi (v,w)}.We define the rank-normalization of M, R^M, as the product of C^{(M,v)} over all vertices v\notin \operatorname{{Supp}}(M).R^M=\prod _{v\notin \operatorname{{Supp}}(M)}C^{(M,v)}Let v\in T, then we say that v is a ... | {
"cite_spans": []
} | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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0.... | |
b4c39df454fb86e574d000ff324540b678036742 | subsection | 11 | 13 | On the rank | ThereforeR^M\overrightarrow{s}_v(T)=&\prod _{{u\notin \operatorname{{Supp}}(M),\\ u\ne v}}M(u,\pi (u,v))\sum _{w\in \operatorname{{Supp}}(M)\cap N(v)}C^{(M,v)}\overrightarrow{e}_{w}\\
=&\prod _{{u\notin \operatorname{{Supp}}(M),\\ u\ne v}}M(u,\pi (u,v))\sum _{w\in \operatorname{{Supp}}(M)\cap N(v)}M(v,w)\overrightarrow... | {
"cite_spans": []
} | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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5c72b414fc2fe348227746cffd8e35c0fa58695b | subsection | 12 | 13 | Conclusion | There is a strong relation between the rank and the null space of a tree-patterned (acyclic)
matrix with
diagonal 0, and its underlying tree (forest).
It would be interesting to study what happens when non-zero diagonal entries are allowed, or when a different graph
is used. We conjecture that there will still be a str... | {
"cite_spans": []
} | 1802.10142 | On the structure of the fundamental subspaces of acyclic matrices with
$0$ in the diagonal | [
"Daniel A. Jaume",
"Adrián Pastine"
] | [
"math.CO"
] | 2,018 | en | Mathematics | [
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b413620d059f2fd3a3508d43472462ba267ac3cd | abstract | 0 | 55 | Abstract | Starting with two supercompact cardinals we produce a generic extension of
the universe in which a principle that we call ${\rm GM}^+(\omega_3,\omega_1)$
holds. This principle implies ${\rm ISP}(\omega_2)$ and ${\rm ISP}(\omega_3)$,
and hence the tree property at $\omega_2$ and $\omega_3$, the Singular Cardinal
Hypothe... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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60475c60c468b9210556f16a28331ce010bc525f | subsection | 1 | 55 | Introduction | In C. Weiß formulated some combinatorial principles that capture the essence of some large cardinal properties, but can hold at small cardinals.
These principles usually have two parameters, a regular uncountable cardinal \kappa and a cardinal \lambda \ge \kappa .
Among them there are, in increasing strength, the prin... | {
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{
"arxiv_id": "",
"doi": "10.1016/j.apal.2011.12.017",
"end": 146,
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"raw": "Christoph Weiß. The combinatorial essence of supercompactness. Ann. Pure Appl. Logic, 163(11):1710–1717, 2012.",
"source_ref_id": "68c... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
f3fbc82f792f6fd52b8cc2549076523d409a81d8 | subsection | 2 | 55 | Introduction | He started with a cardinal \kappa that is \kappa ^+-Mahlo, and built an involved forcing construction
yielding a model in which I[\omega _2]\! \upharpoonright \!S_{\omega _2}^{\omega _1} is the non stationary ideal on S_{\omega _2}^{\omega _1}.
One feature of this construction is that it uses \square _\kappa in the gro... | {
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{
"arxiv_id": "",
"doi": "10.1215/00294527-2420666",
"end": 2037,
"openalex_id": "https://openalex.org/W2088750390",
"raw": "Itay Neeman. Forcing with sequences of models of two types. Notre Dame J. Formal Logic, 55(2):265–298, 2014.",
"source_ref_id": "ad2140... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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726d29d8a86c689f06fb3a832dae9c8adfdde637 | subsection | 3 | 55 | Introduction | Using this type of models as side conditions allows us
not only to generalize Neeman's iteration theory to semiproper forcing, but also to formulate and prove iteration
theorems for large classes of forcing notions preserving two uncountable cardinals, such as \omega _1 and \omega _2.
This theory is presented in and .
... | {
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"doi": "",
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"start":... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.0459502711892128,
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6f9bf2f631f152399eb8c7b7841fed9937472f1c | subsection | 4 | 55 | Preliminaries | Throughout this paper by a model M we mean a set or a class such that
(M,\in ) satisfies a sufficient fragment of {\rm ZFC}.
For a model M, we let \overline{M} denote its transitive collapse and we let \pi _M be the collapsing map.
For a set X and an uncountable regular cardinal \kappa , we let \mathcal {P}_\kappa (X) ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/j.apal.2011.12.015",
"end": 922,
"openalex_id": "https://openalex.org/W2067542931",
"raw": "Matteo Viale. Guessing models and generalized Laver diamond. Ann. Pure Appl. Logic, 163(11):1660–1678, 2012.",
"source_ref_id": "21dec... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03942055627703667,
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... | |
e98e70164a89d5df28e3ffee610a5ffd78b5ee89 | subsection | 5 | 55 | Preliminaries | Let us also mention that in Trang showed the consistency of {\rm GM}(\omega _3,\omega _2)
assuming the existence of a supercompact cardinal. In his model the Continuum Hypothesis holds.We also recall the related notion of the \gamma -approximation property, introduced by Hamkins in .Our plan is to strengthen the princi... | {
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"arxiv_id": "",
"doi": "10.1007/s11856-016-1390-x",
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"openalex_id": "https://openalex.org/W2963024593",
"raw": "Nam Trang. PFA and guessing models. Israel Journal of Mathematics, 215(2):607–667, Sep 2016.",
"source_ref_id": "ac39e331e644d95765c257... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
dcfb8bc26cbfc2b45c007f19ebefe78f7751b29b | subsection | 6 | 55 | Preliminaries | The principle {\rm FS}(\kappa ^+,\gamma ) asserts that,
for every X\in H_{\kappa ^+}, there is a collection \mathcal {G} of \gamma -guessing models of cardinality \kappa all containing X
such that \lbrace M\cap \kappa ^+: M\in \mathcal {G} \rbrace is \kappa -closed and unbounded in \kappa ^+.0Proposition
1.0propositio... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
fcb11c6f0aece21136347dad9352c4d302d1ad62 | subsection | 7 | 55 | Preliminaries | We state it for any pair of uncountable regular cardinals \gamma \le \kappa .For a powerful model R and regular cardinals \gamma and \kappa , we let\mathfrak {G}^+_{\kappa ^{++},\gamma }(R) = \lbrace M\in {\mathcal {P}}_{\kappa ^{++}}(R): M\prec R \mbox{ and } M \mbox{ is strongly $\gamma $-guessing} \rbrace .Clearly {... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
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"raw": "William J. Mitchell. Adding closed unbounded subsets of \\omega _2 with finite forcing. Notre Dame J. Formal Logic, 46(3):357–371, 07 2005.",
"source_ref_id": "4f0e6656181c2c482d16fc3efc50a3f... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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1cc4112de4f84cdb1078d602731131d5dfc4d7d7 | subsection | 8 | 55 | Preliminaries | Let \alpha be an ordinal, \dot{X} a \mathbb {P}-name, and suppose some condition p\in \mathbb {P}
forces that \dot{X}\subseteq \alpha and \dot{X}\cap \check{Z}\in V, for all Z\in V with |Z|^V< \kappa .
Fix a sufficiently large regular cardinal \theta . By the stationarity of \mathcal {S}, we can find M\prec H_{\theta }... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/bf02771565",
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"raw": "M. Magidor. On the role of supercompact and extendible cardinals in logic. Israel Journal of Mathematics, 10(2):147–157, Jun 1971.",
"source_... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
f5c3b3f2c522dccf6b6399d26dc77f0a8d485ff1 | subsection | 9 | 55 | Virtual Models | In this section we review the notion of virtual models introduced in and .
In we used virtual models of two types: countable and internally club (I.C.) models of size \aleph _1.
In the current situation we replace the I.C. models by models that have a much stronger closure property that we
call Magidor models.
We shall... | {
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"start": ... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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4107a99960b5dd90ede9ba56f44b0b0e96814997 | subsection | 10 | 55 | Virtual Models | Since \alpha \le \beta , it follows that y^{\prime }\in A_\beta , a contradiction.The main reason we have defined the \rm Hull operation in this way is that it allows us to define
the Skolem hull of M and X without referring explicitly to the ambient model A.For each \gamma \in A, let {\rm id}_\gamma be the identity fu... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0012184894876554608,
... | |
1172d6eda56acb261526ea387cd7f2d2bcfc1289 | subsection | 11 | 55 | Virtual Models | Being a member of F_\alpha will be expressed by a \Sigma _1-formula with parameter V_\alpha
and similarly for R_\alpha and O_\alpha .
If A is another suitable structure we can interpret these formulas in A and obtain
families F_\alpha ^A, relations R_\alpha ^A and operations O_\alpha ^A. In this section we shall only ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
8da501f44fdcc3d3cea9a01d48ca2f5c941e8d94 | subsection | 12 | 55 | Virtual Models | \alpha to be \pi M0,
i.e. the image of M under the collapsing map of Hull(M,V).Note also that if A\in A_\alpha then V_\alpha ^A \subseteq V_\alpha .
Therefore, if A,B\in A_\alpha , M \in V^{A}, and N \in V^B, we can
still write M \cong _\alpha N if M \upharpoonright \alpha = N \upharpoonright \alpha . This is
of course... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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7e285936246602a3b57dbe7bdb32b1c2c70ac55e | subsection | 13 | 55 | Virtual Models | If \pi is the Mostowski collapse map of {\rm Hull}(N,V_\alpha ), then \pi (M^{\prime })\in N\upharpoonright \alpha .
On the other hand, since | M | < \kappa <| V_\alpha |, we have that {\rm Hull}(M^{\prime },V_\alpha ) \subseteq {\rm Hull}(N,V_\alpha ) and
\pi
M'
Proposition2.0propositiontheoremplain Let \alpha ,\be... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
4454ffcdbd42430c0744526234bcfae5f5e76b80 | subsection | 14 | 55 | Virtual Models | Then witnesses that M and N are -isomorphic.In our forcing we will use two types of virtual models, the countable ones and some nice models of size less than \kappa defined below.First note that since \lambda is of uncountable cofinality E is unbounded and thus club in \lambda .
Suppose M is a countable elementary subm... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... | |
7097c4b27b72eee584641b0ae346f990f79d4d95 | subsection | 15 | 55 | Virtual Models | Then by alphasharpinE \gamma \in E_A.
Since \gamma \in M, we have that E_A\cap (\gamma +1)\in M and therefore
we can compute \alpha in M as the the next element of E_A\cap (\gamma +1) above \beta .
Thus, in this case we have \alpha \in M.It will be convenient to also have the following definition.Note that a(M) is a cl... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.01546553149819374,
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0.0033551885280758142,
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0.025315556675195694,
0.00082592194667086,
... | |
fbbc4d13dec5d6118b9b32afb5bc12af74403807 | subsection | 16 | 55 | Virtual Models | If we let \pi _\gamma be the collapsing map of {\rm Hull}(M,V_\gamma ) to A_\gamma , we then have that, for every \gamma < \delta ,
the following diagram commutes:\begin{}
{\rm Hull}(M,V_\gamma ) {r}{{\rm id}} [swap]{d}{\pi _\gamma } & {\rm Hull}(M,V_\delta ) {d}{\pi _\delta } \\
A_\gamma {r}{\sigma _{\gamma ,\delta }}... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.04986220970749855,
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0.03701521456241608,
0.003682830836623907,
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0.02543460950255394,
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0.0274944007396698,
0.012... | |
4332f0906222069dd9b35b3c6723bce5b35fac4f | subsection | 17 | 55 | Virtual Models | Set\eta = \sup (\sup (\pi ^{-1}NM0ORD)E(+1)).We define the meet ofand M to be NM=-1[NM].To make sense of the above definition, we need to prove the following.0Proposition
2.0propositiontheoremplain
Under the assumptions of the above definition, N\wedge M\in C_\eta .Since \eta (N)\ge \alpha we can form the model A={\rm... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.008203250356018543,
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0.004731176886707544,
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... | |
da218c98edecfa607d6745eedb4aa7d21fca3fd1 | subsection | 18 | 55 | Virtual Models | Now, \overline{N} is also the transitive collapse of N\upharpoonright \alpha . In fact, if \sigma is the \alpha -isomorphism between N and N\upharpoonright \alpha ,
and \pi and \pi ^{\prime } are the collapsing maps of N and N^{\prime } respectively, then \pi = \pi ^{\prime } \circ \sigma . Therefore,
\sigma \upharpoon... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.010897142812609673,
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0.03400396928191185,
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... | |
8fd429e4f269c5f0afa24ade649f2a006c0da517 | subsection | 19 | 55 | Virtual Models | Since \sigma (P^{\prime })\in M, by the transitivity of \cong _\alpha we get that P is \alpha -isomorphic to \sigma (P^{\prime }).
This implies that P\in _\alpha M.Now assume P\in _\alpha N and P\in _\alpha M. By active-Magidor we know that \alpha \in N.
Since P is an \alpha -model, we conclude that P\in N.
If also \al... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.03240864723920822,
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... | |
2d067d0577e5556c718d2d533026128664c73aa0 | subsection | 20 | 55 | Virtual Models | Suppose j > i+1. If M_j is a Magidor model or if there are no Magidor models between M_i and M_j
by intermediate we conclude that M_i\in _\alpha M_j. Otherwise let k < j be the largest such that M_k is a Magidor model.
Then again by intermediate, we conclude that M_i\in _\alpha M_k\in _\alpha M_j.Let \alpha \in E and l... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.029969310387969017,
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0.... | |
3830653ef83e5196e48caba4e5d192ff78753ce5 | subsection | 21 | 55 | Main Forcing | We fix an inaccessible cardinal \kappa and a cardinal \lambda > \kappa with
{\rm cof}(\lambda )\ge \kappa such that (V_\lambda ,\in ,\kappa ) is suitable.
We start by defining the forcing notions \mathbb {M}^\kappa _\alpha , for all \alpha \in E\cup \lbrace \lambda \rbrace .If \kappa < \lambda are supercompact cardinal... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1215/00294527-2420666",
"end": 2232,
"openalex_id": "https://openalex.org/W2088750390",
"raw": "Itay Neeman. Forcing with sequences of models of two types. Notre Dame J. Formal Logic, 55(2):265–298, 2014.",
"source_ref_id": "ad2140... | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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3db2b437d43ae45424e800b9d5b21a29b7a0ada5 | subsection | 22 | 55 | Main Forcing | In a stronger condition this finite set is allowed to increase.
The main point is that d_p(M) controls what models can be added \in -above M in stronger conditions.
In our situation there are some complications. First, we have not one chain, but a \delta -chain, for each \delta \in E.
It is therefore reasonable to have... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0... | |
272cfd36c9257361f1607be48c1ee7ffecc19119 | subsection | 23 | 55 | Main Forcing | The ordering on \mathbb {P}^\kappa _\lambda is clearly transitive. We will say that q is stronger than p
if q forces that p belongs to the generic filter, in order words, any r\le q is compatible with p.
We write p\sim q if each of p and q is stronger than the other. We identify equivalent conditions, often without say... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0eeb3937595ab3295fff7a73f6f4d594a4686a87 | subsection | 24 | 55 | Main Forcing | Note that \mathcal {M}_r is closed under meets.
We define d_r by letting d_r(M)=d_q(M) if M\in {\rm dom}(d_q), and d_r(M)= d_p(M) if M\in {\rm dom}(d_p) with \eta (M) > \alpha .
It is straightforward that r is as required.
[]
Our goal is to prove that our poset \mathbb {P}^\kappa _\lambda is strongly proper for an appr... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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-0.003701364621520042... | |
9b74cd75fe60bac4e2b7a5ec12a4e67289e1e45a | subsection | 25 | 55 | Main Forcing | Indeed, if there is P\in \mathcal {M}_{p^M} such that
N\in _{\bar{\delta }} P and P is not strongly active at \bar{\delta }, then P\in M, and hence \eta (P)\ge \delta .
Moreover, P is active but not strongly active at \delta as well.
Since N\in _{\bar{\delta }}P and N,P\in M it follows that N\in _{\gamma } P, for unbou... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
11661702e7711d1a234a141f511933645198493d | subsection | 26 | 55 | Main Forcing | It is straightforward to check that every N\in {\rm dom}(d_{p\upharpoonright M}) is \mathcal {M}_{p\upharpoonright M}-free,
and (*) from PF holds.
Finally, the fact that p\le p\upharpoonright M follows from the definition.
Let d_r = d_q \cup d_p \! \upharpoonright \!({\rm dom}(d_p)\setminus M).
Let us check that every ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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-0... | |
e32a6c5877ce5aed557b3832fee1efae8982d0a7 | subsection | 27 | 55 | Main Forcing | Then, for any Magidor model N\in (P,M)_p^\delta , we have P\in _\delta N by intermediate.
Then by meetactiveness we have that P\in _\delta N\wedge M.
Conversely, suppose P is in (\emptyset ,M)^\delta _{p}, but not in (\mathcal {M}_p\upharpoonright M)^\delta .
Then, by intermediate again, there must be a Magidor model N... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
a2a410bb49268617e02075938604d11825beeb12 | subsection | 28 | 55 | Main Forcing | \upharpoonright \!M)^\delta is a Magidor
model, and the interval [M,V_\lambda )_p^\delta . Consider one such interval, say [N\wedge M,N)_p^\delta .
If P is the last model of \mathcal {M} before N then P\in _\delta M by the assumption that \mathcal {M}\in _\delta M,
and P\in _\delta N by intermediate. Hence by MPI we ha... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
2c294f37fe6aba19149a3eecfa0385506bced57f | subsection | 29 | 55 | Main Forcing | Now, let \lbrace P_i : i < k\rbrace list all countable models on the chain [N\wedge M,N)_p^\delta below the first Magidor model, if it exists.
Then P_0= N\wedge M and P= P_j, for some j. Note that Q\in _\delta P_i, for all i<k, again by intermediate.
Now let S be the \in _\delta ^*-predecessor of N on the \delta -chain... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
e9f364aa392917f24407d59e36c433b3ac3598da | subsection | 30 | 55 | Main Forcing | We will let V^M_\alpha = V^A_\alpha \cap M, and
(\mathbb {P}^\kappa _\alpha )^M= (\mathbb {P}^ \kappa _\alpha )^A\cap M, if \alpha \in E_A\cap M.
Suppose N\in \mathcal {M}_p and N\in _\delta M, for some \delta \in a(M)\cap a(N),
Then by full-countable, N\in _\alpha M, where \alpha = \alpha (M,N).
Note that if M is a st... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.0... | |
7bac208128d7b18c2287e4d6afa8b4e905522220 | subsection | 31 | 55 | Main Forcing | Note that if P\in \mathcal {M}_{p\upharpoonright M} is a \delta -model that is active at \gamma then a(P) is cofinal in \delta .
Moreover, a(P)\in M and since \bar{\delta }= \sup (M\cap \delta ) we have that \bar{\delta }\in a(P).
This implies that \mathcal {M}_{p\upharpoonright M}^\gamma \! \upharpoonright \!\bar{\del... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.004504027310758829,
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167a507077660089270070442b8f7f614724b4c3 | subsection | 32 | 55 | Main Forcing | Since \eta = \sup (M\cap \eta ^*) and N\in M we must have that N is strongly active at \eta ^* as well.
This also establishes (*) from PF. Indeed, if P^*\in _{\eta ^*} N then P\in _\eta N\! \upharpoonright \!\eta , and
hence d_p(P)\subseteq N, since N\! \upharpoonright \!\eta \in \mathcal {L}(\mathcal {M}_p), and p is ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.021633541211485863,
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... | |
f1eb6e6af578e7767899252a847ac2c168fe4c6b | subsection | 33 | 55 | Main Forcing | By the assumption, P\notin _\eta M, hence by gapmeet,
there must be a Magidor model N\in (\mathcal {M}_p\! \upharpoonright \!M)^\eta such that P is in the interval [N\wedge M\! \upharpoonright \!\eta ,N)_p^\eta .
By intermediate, we have that P\in _\eta N and thus d_q(P)\in N.
Note that P also belongs to the interval [... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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a2108485a2c8f5ba18cd9ed7462b13990b991b58 | subsection | 34 | 55 | Main Forcing | Now, suppose P\in {\rm dom}(d_q). Let \delta (P) be the largest ordinal \gamma \in E\cap (\eta (P)+1) such that M is strongly active at \gamma .
Let
D_q= \lbrace P\! \upharpoonright \!\delta (P) : P\in {\rm dom}(d_q)\rbrace .
Note that, for every P\in {\rm dom}(d_q), we have (P \! \upharpoonright \!\delta (P))\! \uph... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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... | |
55eeee35a12a53109b9ce87fe433d6e3d8041294 | subsection | 35 | 55 | Main Forcing | \upharpoonright \!M)\in R\! \upharpoonright \!M.
Since (R\! \upharpoonright \!M) \cap V_\kappa = R\cap V_\kappa , we get that d_q(P\! \upharpoonright \!M)\in R, and hence d_r(P)\in R.
Moreover, since R\! \upharpoonright \!M is strongly active at \rho , it follows that R is strongly active at \eta .
This shows that all ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.012879256159067154,
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0caab9c471825501ec30551566a458a703cdc1d0 | subsection | 36 | 55 | Main Forcing | By intermediate, we have that
\in _\alpha is transitive on U_\alpha . Note that if P,Q \in U_\alpha then P\cap V_\alpha \subseteq Q\cap V_\alpha .
Now, a standard density argument using the stationarity of U shows that, for every x\in V_\alpha , there is P\in U_\alpha such that x\in P.
It follows that \lbrace P\cap V_\... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.00271497294306755... | |
e0de2448e149b52b2af412738ffbc7ab6c079cb9 | subsection | 37 | 55 | Main Forcing | Case 1. Suppose M is strongly active at \alpha .
Since P is \mathcal {M}_p-free and we may assume that P\in {\rm dom}(d_p), by defining d_p(P)=\emptyset if necessary.
Since \gamma < \kappa _M, we can find \delta \in M such that \gamma \le \delta < \kappa _M.
Define a condition q as follows. Let \mathcal {M}_q=\mathcal ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.005174218211323023,
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... | |
b11edbcb34dacbf43b968834a942f5ecf64dfdbc | subsection | 38 | 55 | Main Forcing | \upharpoonright \!\alpha : Q^*\in \mathcal {M}^*\rbrace . Then \mathcal {M}\in _\alpha M, is an \alpha -chain closed under meets
that are active at \alpha , and (\mathcal {M}_q\! \upharpoonright \!M)^\alpha \subseteq \mathcal {M}.
We now define a condition r. Let \mathcal {M}_r be the closure of \mathcal {M}_p and \mat... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.011351319961249828,
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0.03... | |
92b63500ffe728f513f35182fabcfcd472df7b60 | subsection | 39 | 55 | Main Forcing | Let us fix a V-generic filter G_\alpha over \mathbb {P}^\kappa _\alpha , and let \mathbb {Q}_\alpha denote the quotient forcing.
Recall that \mathbb {Q}_\alpha consists of all p\in \mathbb {P}^\kappa _\lambda such that p\! \upharpoonright \!\alpha \in G_\alpha ,
with the induced ordering. Forcing with this poset over V... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.005829531233757734,
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0.015855714678764343,
... | |
af6c1ce94693cd8fcf45d8243a81e51cb443fcc9 | subsection | 40 | 55 | Main Forcing | Since \dot{F}\in M^*, it follows that p^{M^{\prime }} forces that M is closed under \dot{F}.
It also forces that M^{\prime } belongs to \dot{\mathcal {M}}_{\alpha }, hence it forces that M belongs to \dot{C}^\alpha _{\rm st}.
Work in V[G_\alpha ]. Let p\in \mathbb {Q}_\alpha , and M\in C^\alpha _{\rm st} be such that p... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.024085847660899162,
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0.007669922895729542,
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-0... | |
60cc6c71f41e7b9693811b3f53303388e0ee49da | subsection | 41 | 55 | Main Forcing | It follows that q and r are compatible in \mathbb {Q}_\alpha .
Now, by C-stationary, quotient-alpha-sp, and guessingbystronglyproper, we get the following.
0Corollary
4.0corollarytheoremplain The pair (V
G
Corollary4.0corollarytheoremplain The pair (VG0,V[G]) has the 1-approximation property.
[]
Suppose now N\in U.... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
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0.01600622944533825,
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-0.01940888725221157,
-0.001975983614102006,
0.043029140681028366,
0.024795159697532654,
0.0012311750324442983,
-0.017013294622302055,
0.005138319917023182,
... | |
9d591fe0aba73b55cb089e58c463fc9a0502cdeb | subsection | 42 | 55 | Main Forcing | Now, p^{N\wedge M} is (N\wedge M, \mathbb {P}_N)-strongly generic, hence also (N\cap M, \mathbb {P}_N)-strongly generic,
and therefore it is (M^*,\mathbb {P}_N)-generic. It follows that p^{N\wedge M} forces that M\in {\dot{C}}^N_{\rm st} and
is closed under \dot{F}.
Let \mathbb {Q}_N denotes the quotient forcing (\math... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.00801882054656744,
0.006466174963861704,
-0.025300102308392525,
-0.061586979776620865,
0.009155645966529846,
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0.00891912542283535,
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0.014557477086782455,
-0.013268057256937027,
0.... | |
801cf9bc3c1e6bc08a6d92bcb2b01a187290b6cb | subsection | 43 | 55 | Main Forcing | \upharpoonright \!N \in G_N.
It follows that r and q are compatible in \mathbb {Q}_N.
Suppose G is a V-generic filter over \mathbb {P}^\kappa _\lambda .
As before, for \alpha \in E, let G_\alpha = G\cap \mathbb {P}^\kappa _\alpha .
0Lemma
4.0lemmatheoremplain
Let \alpha \in E. Suppose N\in \mathcal {M}_G is a Magidor ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.027856415137648582,
-0.01105255912989378,
-0.014385870657861233,
-0.0026124923024326563,
0.041616812348365784,
-0.03783346712589264,
-0.012021278962492943,
0.010991537012159824,
0.004164732526987791,
0.04744438827037811,
-0.019313374534249306,
-0.020167678594589233,
-0.002622026950120926,... | |
f010886c56ad6108608153bafa01e8937feb7bff | subsection | 44 | 55 | Main Forcing | 0Theorem
4.0theoremtheoremplain The principle {\rm GM}(\omega _2,\omega _1) holds in V
G
Theorem4.0theoremtheoremplain The principle {\rm GM}(\omega _2,\omega _1) holds in VG0.
[]
0Theorem
4.0theoremtheoremplain The principle {\rm FS}(\omega _2,\omega _1) holds in V
G
Theorem4.0theoremtheoremplain The principle ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03634020686149597,
-0.012891466729342937,
-0.034967150539159775,
0.007502223365008831,
0.016324106603860855,
-0.06377080827951431,
0.003562316531315446,
0.017147939652204514,
0.013822093605995178,
0.009855488315224648,
-0.047416187822818756,
-0.01765139400959015,
-0.001491290982812643,
... | |
4976d14b99aa239871f7f48ae120ba668f509a75 | subsection | 45 | 55 | Main Forcing | Note that {\rm cof}(\bar{\lambda })\ge \kappa , and hence
the transitive collapse \overline{N[G]} of N[G] equals V_{\bar{\gamma }}[G_{\bar{\lambda }}].
On the other hand, by approx-alpha, the pair (V[G_{\bar{\lambda }}],V) has the \omega _1-approximation property.
Moreover, by quotient-alpha-sp, {\mathcal {P}}_{\omega ... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03305891528725624,
-0.0077877710573375225,
-0.020787738263607025,
0.004952732473611832,
0.03192947804927826,
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0.03461570665240288,
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0.011111214756965637,
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-0.01414085179567337,
-0.013492189347743988,
0... | |
a140b4bf27cfae29fbfe3fbfa0cbf43099d954ea | subsection | 46 | 55 | Main Forcing | Let G be V-generic over \mathbb {P}^\kappa _\lambda .
Then in VG0 the principle GM+(3,1) holds.
[] | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03356913477182388,
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0.001819599769078195,
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0.008758492767810822,
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-0.0022163258399814367,
-0.003093701088801026... | |
b9d233b8a6601aeb0c576e5438a0be312a3eaa2f | subsection | 47 | 55 | Guessing Models in | We assume \kappa is supercompact and \lambda is inaccessible and analyze \omega _1-guessing models
in the the generic extension by \mathbb {P}^\kappa _\lambda .
Suppose \alpha \in E. We have already established in alpha-complete-suborder that \mathbb {P}^\kappa _\alpha
is a complete suborder of \mathbb {P}^\kappa _\la... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.021331805735826492,
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0.007320410571992397,
-0.03317263722419739,
-0.02340700291097164,
0.018981950357556343,
... | |
a7289da2c7015c88813cf81694a6599d50807470 | subsection | 48 | 55 | Guessing Models in | Note that \sigma (q)=q, for all q\in M\cap \mathbb {P}^\kappa _\alpha .
Hence, M \cap \mathbb {P}^\kappa _\alpha = M^{\prime } \cap \mathbb {P}^\kappa _\alpha .
Therefore, p^{M^{\prime }} is also (M,\mathbb {P}^\kappa _\alpha )-strongly generic, and thus it is (M^*,\mathbb {P}^\kappa _\alpha )-generic.
Since \dot{F}\in... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.025201771408319473,
-0.017818201333284378,
-0.02591877244412899,
-0.06462149322032928,
0.02250158227980137,
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0.022135455161333084,
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0... | |
6f95f98e23e8aeb0e68952699d148cd778a53970 | subsection | 49 | 55 | Guessing Models in | \upharpoonright \!\alpha \wedge q\! \upharpoonright \!\alpha =(r\wedge q)\! \upharpoonright \!\alpha . Since r\! \upharpoonright \!\alpha , q\! \upharpoonright \!\alpha \in G_\alpha ,
we conclude that r\! \upharpoonright \!\alpha \wedge q\! \upharpoonright \!\alpha \in \mathbb {Q}_\alpha .
It follows that q and r are c... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.005598405376076698,
0.01691725105047226,
-0.04310924559831619,
-0.00950356014072895,
0.032949741929769516,
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0.030264947563409805,
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0.007242078892886639,
0.006677662022411823,
... | |
d21ba00a496331aaf61a685ecc172c05fcdb43c5 | subsection | 50 | 55 | Guessing Models in | Let M=M^*\cap V_\lambda , and note that M \in C_{\rm st}. Since N\in _{\eta (N)}M, the meet N\wedge M is defined.
Let \eta = \eta (N\wedge M) and let \sigma be the \eta -isomorphism between N\cap M and N\wedge M.
Note that \sigma (q)=q, for all q\in \mathbb {P}_N.
Now, p^{N\wedge M} is (N\wedge M, \mathbb {P}_N)-strong... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
0.025539586320519447,
0.012991014868021011,
-0.033869702368974686,
-0.05016377568244934,
0.0015409189509227872,
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0.012502802535891533,
-0.011450095102190971,
... | |
c94004f2e1029fd33ee0a335df508487ade2873b | subsection | 51 | 55 | Guessing Models in | \upharpoonright \!N exists,
and r\! \upharpoonright \!N \wedge q\! \upharpoonright \!N = (r\wedge q)\! \upharpoonright \!N.
Since r\! \upharpoonright \!N, q\! \upharpoonright \!N \in G_N, we have that r\! \upharpoonright \!N \wedge q\! \upharpoonright \!N \in G_N.
It follows that r and q are compatible in \mathbb {Q}_N... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03109552524983883,
-0.005115198437124491,
-0.017836442217230797,
-0.0029447677079588175,
0.04620080813765526,
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-0.006038299296051264,
... | |
d7f9b619a13afd079de39f14a8ba628de39f4b9f | subsection | 52 | 55 | Guessing Models in | Thus, N[G_N] remains an \omega _1-guessing model in V[G].A similar argument shows the following.0Lemma
4.0lemmatheoremplain Suppose \mu > \lambda and N\prec V_\mu is a \kappa -Magidor model
containing all the relevant parameters. Then N
G
Lemma4.0lemmatheoremplain Suppose \mu > \lambda and N\prec V_\mu is a \kappa -... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03784231096506119,
-0.020065579563379288,
-0.028107071295380592,
-0.002937356708571315,
0.023773515596985817,
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0.016540752723813057,
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-0.05032417178153992,
-0.010955959558486938,
... | |
27cbce265f6be1acf57fee4e9728570e65eed041 | subsection | 53 | 55 | Guessing Models in | In fact, we show that for all \mu >\lambda the set of strong \omega _1-guessing models is stationary in {\mathcal {P}}_{\omega _3}(V_\mu [G]).0Lemma
4.0lemmatheoremplain Suppose \mu > \lambda and N\prec V_\mu is a \lambda -Magidor model
containing all the relevant parameters. Then N
G
Lemma4.0lemmatheoremplain Suppo... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.036814361810684204,
-0.020315056666731834,
-0.027595506981015205,
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... | |
02af97272a14c2aabf1f3c00dc8e2433f9d3f6ea | subsection | 54 | 55 | Guessing Models in | Moreover, if M\in \mathcal {M}_G^\delta is a limit of such Magidor models
then by continuity-Magidor,M\cap V_{\bar{\gamma }} = \bigcup \lbrace Q\cap V_\delta : Q\in _\delta M \mbox{ and } Q\in \mathcal {M}_G^\delta \rbrace .Hence if we let \mathcal {G} be the collection of the models (M\cap V_{\bar{\gamma }})[G_{\bar{\... | {
"cite_spans": []
} | 1802.10125 | Guessing models and the approachability ideal | [
"Rahman Mohammadpour",
"Boban Velickovic"
] | [
"math.LO"
] | 2,018 | en | Mathematics | [
-0.03595176711678505,
-0.012589222751557827,
-0.018357375636696815,
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... | |
9b36b2eb27c5bb92674a17ed8926369ed12dc3e9 | abstract | 0 | 62 | Abstract | We study the derivative nonlinear wave equation \( - \partial_{tt} u + \Delta
u = |\nabla u|^2 \) on \( \mathbb{R}^{1+3} \). The deterministic theory is
determined by the Lorentz-critical regularity \( s_L = 2 \), and both local
well-posedness above \( s_L \) as well as ill-posedness below \( s_L \) are
known. In this ... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.06793444603681564,
0.048585642129182816,
-0.022278591990470886,
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23cae391124fe26e692ef812fc6a079a36fe7d18 | subsection | 1 | 62 | Introduction | We consider the Cauchy problem for the nonlinear wave equation{\left\lbrace \begin{array}{ll}
-\partial _{tt} u + \Delta u = |\nabla u|^2 \qquad \qquad \text{for} ~ (t,x)\in \mathbb {R}^{1+d} \\
u|_{t=0}=f_0 , ~ \partial _t u|_{t=0}=f_1
\end{array}\right.}~,with initial data (f_0,f_1)\in H_x^{s}(\mathbb {R}^d)\times H_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.48550/arxiv.1208.4706",
"end": 951,
"openalex_id": "https://openalex.org/W4302317592",
"raw": "Jalal Shatah and Michael Struwe. Geometric wave equations, volume 2 of Courant Lecture Notes in Mathematics. New York University, Courant Inst... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.021745294332504272,
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0.014... | |
9a27f594cf69cf32ab32a79019c75079a9234929 | subsection | 2 | 62 | Introduction | Let (\Omega , {F}, \mathbb {P} ) be a probability space and let
\lbrace g_k(\omega ) \colon k \in \mathbb {Z}^d \rbrace be a family of independent standard complex Gaussians. Then, we define\widehat{f^\omega }(\xi ) = \sum _{k\in \mathbb {Z}^d} g_k(\omega ) \varphi (\xi -k) \widehat{f}(\xi )~.Thus, f^\omega is a rando... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 849,
"openalex_id": "https://openalex.org/W2803371764",
"raw": "Sagun Chanillo, Magdalena Czubak, Dana Mendelson, Andrea Nahmod, and Gigliola Staffilani. Almost sure boundedness of iterates for derivative nonlinear wave equations,... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.050179623067379,
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0.019290463998913765,
0.0075238910503685474,
-0.0... | |
a768c148b04c0ffed1b9b82a71fe2841f33a3b5b | subsection | 3 | 62 | Introduction | In addition, let 0 < T_0 \ll 1 and \sigma =1.1 . Then, there exists a random function u and random times 0 < T(\omega ) \le T_0 such that\begin{aligned}u&\in \big ( L_\omega ^2 C_t^0 H_x^s L_\omega ^2 L_t^2 W_{x}^{\sigma ,\infty } \big )(\Omega \times [0,T_0] \times )~,\\
\partial _t u &\in \big ( L_\omega ^2 C_t^0 H_x... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 849,
"openalex_id": "",
"raw": "J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. I. Schrödinger equations. Geom. Funct. Anal., 3(2):107–156, 1993."... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
-0.03514755144715309,
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0.00013479235349223018... | |
3ebd320977f355b5fbc7ef4aa88436e2c662f51a | subsection | 4 | 62 | Introduction | Heuristically, we have for any \nu >s_d>s that|\nabla |^{\nu } \nabla P_1 F \cdot \nabla P_{\gg 1} F^\omega \simeq \int _0^t \sin ((t-s)|\nabla |)~ \nabla P_1 F \cdot |\nabla |^{\nu -1} \nabla P_{\gg 1} FThus, the linear evolution F_n^\omega (t) is attacked by more than s derivatives. Since the Duhamel integral does no... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0012-9593(00)00109-9",
"end": 507,
"openalex_id": "https://openalex.org/W2058332344",
"raw": "Damiano Foschi and Sergiu Klainerman. Bilinear space-time estimates for homogeneous wave equations. Ann. Sci. École Norm. Sup. (4), 33(2)... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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06304111c82d3a72bc63a93c818bafe42460369e | subsection | 5 | 62 | Introduction | Then, the equation for v_n reads{\left\lbrace \begin{array}{ll}
-\partial _{tt} v_n + \Delta v_n = |\nabla v_n|^2 + 2 \nabla u_{n-1} \cdot \nabla v_n \\
v_n|_{t=0} = Q_{N} f_0^\omega ~,~ ~ \partial _t v_n |_{t=0} = Q_{N} f_1^\omega ~.
\end{array}\right.}To control v_n uniformly in n \ge 0 , it is necessary to decompose... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1811,
"openalex_id": "",
"raw": "A. de Bouard and A. Debussche. A stochastic nonlinear Schrödinger equation with multiplicative noise. Comm. Math. Phys., 205(1):161–181, 1999.",
"source_ref_id": "6c001de4c3fea21ec0082d4f4a8f... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
] | 2,018 | en | Mathematics | [
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eb20f151f8992b0cb7d36c9d005d6c498e34efdd | subsection | 6 | 62 | Introduction | For this, we use a result of Geba and Tataru and the re-centered Besov-type spaces from Section REF . Finally, we control the nonlinear component w_n . To handle the low-high interaction term \nabla P_1 w \cdot \nabla F_n^\omega , we place w_n in a function space that is concentrated at frequencies \sim N . | {
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"doi": "10.1081/pde-200059294",
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"raw": "Daniel Tataru and Dan-Andrei Geba. Dispersive estimates for wave equations. Comm. Partial Differential Equations, 30(4-6):849–880, 2005.",
... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
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fd79620901fdbaf70e807266403c0ec2e5a623e4 | subsection | 7 | 62 | Notation and Preliminaries | In this section, we will provide the necessary notation and preliminaries for the rest of the paper. In Section REF , we construct spaces of frequency-localized functions. In Section REF , we recall the Strichartz estimates for the wave equation. In particular, we describe the refinement of Klainerman and Tataru . | {
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"raw": "Sergiu Klainerman and Daniel Tataru. On the optimal local regularity for Yang-Mills equations in {\\bf R}^{4+1}. J. Amer. Math. Soc., 12(1):93–116, 1999.",
"source_ref_id": "97eec346c2b33cef96... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
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"Bjoern Bringmann"
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7b34d72edbddf0b9e9d45f5e132305a5776838d5 | subsection | 8 | 62 | Function Spaces | For any function f\in L^1(\mathbb {R}^d) , we define its Fourier transform \widehat{f} by\widehat{f}(\xi ) := \frac{1}{(2\pi )^{\frac{d}{2}}} \int _{\mathbb {R}^d} \exp (-i x \cdot \xi ) f(x) ~.Let \varphi \colon \mathbb {R}^d \rightarrow \mathbb {R} be a smooth, compactly supported function s.t. \varphi |_{B(0,1)} \eq... | {
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"doi": "10.48550/arxiv.math/0010068",
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"raw": "Terence Tao. Global regularity of wave maps. I. Small critical Sobolev norm in high dimension. Internat. Math. Res. Notices, (6):299–328, 2... | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
] | [
"math.AP"
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66ce4833a4fc1b4fe0b99e7854a7814fdcdd7e1a | subsection | 9 | 62 | Function Spaces | We define frequency localized versions of the L_t^\infty L_x^2 -norm by\Vert u \Vert _{}&:= \sum _{M\ge 1} c_{N,D}(M) \Vert P_M u \Vert _{L_t^\infty L_x^2([0,T]\times )}~,\\
\Vert u \Vert _{}&:= \sum _{M\ge 1} c_{\le N,D}(M) \Vert P_M u \Vert _{L_t^\infty L_x^2([0,T]\times )}~.Similarly, we define frequency localized v... | {
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"Bjoern Bringmann"
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afca54a8d90484a2babfae967f1134152fd6a58c | subsection | 10 | 62 | Function Spaces | Since \Vert u \Vert _{([0,t])} is a uniform limit of the partial sums in M \ge 1 , the result follows.
Equipped with the functions spaces above, we are now ready to define the function space for the solution w_n of (REF ). For given parameters \nu > 2 , \sigma = \nu -1 - , and \eta ,D > 0 , we set\begin{aligned}([0,T]... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
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"math.AP"
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e871097e9e8fa13d8ec3a7febd9d1477fffda525 | subsection | 11 | 62 | Function Spaces | We define the weight functionc^{\rho ,\gamma }_{k,D}(M) := M^\rho \max \left( 1, \frac{M}{N^\gamma } \right)^{D}~.Using this weight function, we set\Vert f \Vert _{} := \sum _{M\ge 1} c^{\rho ,\gamma }_{k,D}(M) \Vert P_{M;k} f \Vert _{L_x^2()} \quad \text{and} \quad := \lbrace f \in L_x^2()\colon \Vert f \Vert _{} < \i... | {
"cite_spans": []
} | 1809.00220 | Almost sure local well-posedness for a derivative nonlinear wave
equation | [
"Bjoern Bringmann"
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"math.AP"
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f6372387bc064d1136d6bd26beab33c2db5ba4e0 | subsection | 12 | 62 | Strichartz Estimates | First, we state a local Strichartz estimate in the form needed for this paper.[Strichartz Estimate]
Let \nu > 2 and let \sigma = \nu - 1 - \delta , where \delta >0 is small. Let 0 < T \le 1 and let u be a solution of{\left\lbrace \begin{array}{ll}
-\partial _{tt} u + \Delta u = F \qquad \qquad \text{for} ~ (t,x)\in [0,... | {
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"doi": "10.1353/ajm.1998.0039",
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"raw": "Markus Keel and Terence Tao. Endpoint Strichartz estimates. Amer. J. Math., 120(5):955–980, 1998.",
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