chunk_uid stringlengths 40 40 | chunk_type stringclasses 2
values | chunk_index int64 0 6.71k | total_chunks int64 1 6.71k | section_title stringlengths 1 157 | embed_text stringlengths 1 83.3k | spans dict | paper_doi stringlengths 0 63 | paper_id_arxiv stringlengths 9 16 | title stringlengths 7 245 | authors listlengths 1 768 | categories listlengths 1 7 | year int64 2k 2.02k | language stringclasses 2
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values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
ed6d29662ea453c99c1b589c0648af0bb2ce9e8c | subsection | 93 | 150 | First invariant subspace | To see that it yields zero when tested with vectors (0,0,D_3)^\top , we use the same reasoning as above ({\it cf.} (\ref {psi_id})):
\begin{equation*}
\begin{aligned}\int _Q A &\bigl (\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_5^{(2)}+iX{\mathfrak {u}}_4^{(3)} \bigr ): \overline{i X(0,0,D_3)^\top }
=\int _Q A \bigl ... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.007055350113660097,
0.032052427530288696,
-0.017064101994037628,
0.005929699167609215,
-0.01669778861105442,
-0.006696667987853289,
0.02787034958600998,
0.033792417496442795,
0.004445366561412811,
0.033792417496442795,
-0.022726697847247124,
0.016941998153924942,
0.010882562957704067,
0.... |
77753bb35c7406fe5b23ef8c0bc88ce353afc0d2 | subsection | 94 | 150 | First invariant subspace | \end{equation}
\vspace{5.69046pt}{\it Step 4.} To complete the proof, we define the approximate solution
\begin{align*}
U&=\bigl (-i\chi _1 x_3\bigl (m_3+m_3^{(1)}+m_3^{(2)}\bigr ), -i\chi _2 x_3\bigl (m_3+m_3^{(1)}+m_3^{(2)}\bigr ),m_3+m_3^{(1)}+m_3^{(2)}\bigr )^\top +{\mathfrak {u}}_2+{\mathfrak {u}}_3^{(1)}+{\mathfr... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.005344246048480272,
0.03134070336818695,
-0.04989489167928696,
-0.007827547378838062,
-0.005218364764004946,
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0.022170420736074448,
0.012931473553180695,
-0.014747221022844315,
-0.008979554288089275,
-0.01029177475720644,
-0.0002794190077111125,
0.0006208251579664648,
... |
80b1d1a796882a881e6de6dd240e60b7ba54de03 | subsection | 95 | 150 | First invariant subspace | \end{equation}
It is easy to see that, due to the estimate (\ref {bukal1000}), there exists \rho _1>0 such that for all \chi \in Q^{\prime }_{\rm r}, \vert \chi \vert \le \rho _1, one has
\bigl \Vert A({\rm sym}\nabla +iX)z\bigr \Vert ^2_{L^2(Q, {\mathbb {C}}^{3\times 3})} \ge \frac{1}{2}\bigl (\Vert {\rm sym}\nabla z... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.004763601813465357,
0.029916487634181976,
-0.03240317106246948,
-0.009771294891834259,
-0.019390033558011055,
0.027826452627778053,
0.06181621924042702,
0.005549272056668997,
-0.017574600875377655,
-0.03566789627075195,
-0.02643818035721779,
0.011113799177110195,
0.004340255167335272,
0... |
a42f0ea092f330d6961c99fb47e69dadc838674c | subsection | 96 | 150 | First invariant subspace | \end{equation*}
Finally, as a consequence of (\ref {sinisa1002}), (\ref {julian101}), (\ref {julian100}), (\ref {julian102}), (\ref {julian103}), (\ref {julian104}), (\ref {julian105}), (\ref {julian110}), (\ref {julian111}), (\ref {julian112}), we obtain
\begin{equation}
\begin{aligned}\Vert u_\alpha +i \chi _\alpha m... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.029329538345336914,
0.03699000924825668,
-0.06744877994060516,
-0.02844446338713169,
-0.012261333875358105,
-0.03949263319373131,
0.022920377552509308,
0.03906535729765892,
0.03467050567269325,
-0.000997616327367723,
-0.006042922381311655,
0.03467050567269325,
-0.01623653993010521,
0.03... |
6319825b0a863d92e7f5d6dcdc9a4c64d024dcc7 | subsection | 97 | 150 | First invariant subspace | \begin{} Denote by \widetilde{m} _3 \in \mathbb {C} the solution to the identity
\begin{equation*}
\bigl (\vert \chi \vert ^{-4}A^{\textrm {\rm hom},1}_{\chi }+1\bigr )\widetilde{m} _3\,\overline{d_3} =\int _Q|\chi |^{-1}\bigl ({\!\!_1, {\!\!_2\bigr )^\top \cdot \overline{e_\chi (-i\chi _1 x_3 d_3, -i\chi _2 x_3d_3)^\... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.02294113114476204,
0.016774652525782585,
-0.03306087106466293,
-0.022422170266509056,
0.0042776125483214855,
-0.02487960271537304,
0.06532803922891617,
0.01031053438782692,
0.05623095482587814,
-0.026711229234933853,
-0.02634490467607975,
0.019094713032245636,
-0.003533514216542244,
0.00... |
8aa560f68575dd723a184d1173024deefc4012b1 | subsection | 98 | 150 | First invariant subspace | An approximating problem for (m_1, m_2)^\top in the estimates (\ref {korrre1oo}) then takes the form
\begin{equation}
\begin{aligned}\bigl (A^{\textrm {hom},2}_{\chi } +|\chi |^2\bigr )(m_1,m_2)^\top \cdot \overline{(d_1,d_2)^\top }
=|\chi |^2\int _Q\bigl (\widehat{f}_1, \widehat{f}_2\bigr )^\top \cdot \overline{e_\chi... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.009431579150259495,
0.018115347251296043,
-0.03690220043063164,
-0.01980936899781227,
0.019748322665691376,
-0.009454471990466118,
0.015505638904869556,
0.008981366641819477,
0.021717051044106483,
0.011468983255326748,
-0.03937455639243126,
-0.001197070349007845,
-0.019641492515802383,
... |
e015f8700025ad9cb1f7361ac68064e1a2cf2a91 | subsection | 99 | 150 | First invariant subspace | Define {\mathfrak {u}}_2\in H^1_{\#} (Q, \mathbb {C}^3) that satisfies
\begin{equation}
\begin{aligned}({\rm sym}\nabla )^{*}A\,{\rm sym}\nabla {\mathfrak {u}}_2^{(1)}&= i\bigl \lbrace X^{*}A\,{\rm sym}\nabla {\mathfrak {u}}_1-({\rm sym}\nabla )^{*}A(X{\mathfrak {u}}_1)+X^{*} A \Xi (\chi ,m_1,m_2)\bigr \rbrace \\[0.7em... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.02673269994556904,
0.0014876930508762598,
-0.02419980801641941,
-0.022551901638507843,
-0.005378582514822483,
0.0032042621169239283,
0.03805442526936531,
0.03802391141653061,
0.03613187000155449,
0.027587169781327248,
-0.021636398509144783,
-0.0008797415648587048,
0.01763870008289814,
0... |
563a3e9d3350c577b43dbaf4b8e1f17aa07f3edb | subsection | 100 | 150 | First invariant subspace | \end{aligned}
\end{equation}
The following estimate is a consequence of (\ref {sinisa10revision}), (\ref {revision11111}):
\begin{equation}
\bigl |\bigl (m_1^{(1)}, m_2^{(1)}\bigr )^\top \bigr | \le C|\chi |\Vert f
\Vert _{L^2(Q, \mathbb {C}^3)}.
\end{equation}
Next, we define
\begin{equation*}
({\rm sym}\nabla )^{*}... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.04480065032839775,
0.032623905688524246,
-0.03237976133823395,
-0.028534473851323128,
0.007747795898467302,
-0.003356996923685074,
0.04016188904643059,
0.05777086317539215,
0.04785246402025223,
0.014084127731621265,
-0.018524518236517906,
0.02081337943673134,
-0.008773968555033207,
0.01... |
74d82182c3e0aab843ca26d4a82fc5c89aa9a73a | subsection | 101 | 150 | First invariant subspace | To see that it vanishes when tested with vectors (D_1,D_2,0)^\top , we use the fact that
\begin{align*}
\int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_2^{(2)}&+\Xi \bigl (\chi ,m_1^{(1)},m_2^{(1)}\bigr ) \bigr ): \overline{i X(D_1,D_2,0)^\top }
=A^{\textrm {hom},2}_{\chi }\bigl (m_1^{(1)},m_2^{(1)}\bigr... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.012564071454107761,
0.013456601649522781,
-0.031886957585811615,
-0.030712174251675606,
-0.010634071193635464,
0.001950024743564427,
0.03194798529148102,
0.011190949007868767,
0.016492728143930435,
0.017835335806012154,
-0.020078103989362717,
-0.013838023878633976,
0.005439091008156538,
... |
ddb8b5de3b3db8b2b7a6a244f15c43a7d1d533aa | subsection | 102 | 150 | First invariant subspace | \end{aligned}
\end{equation*}
It follows that the error z:=u
-U satisfies
\begin{equation*} ({\rm sym}\nabla +iX)^{*}A ({\rm sym}\nabla +iX)z+|\chi |^2 z=R_4,
\end{equation*}
and hence, in the same way as before, we obtain (see the argument between (\ref {revision10000}) and (\ref {korrre1}))
\begin{equation*}
\Vert z\... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.01727115362882614,
0.025693129748106003,
-0.060021836310625076,
-0.0049852910451591015,
0.006301224697381258,
0.008414347656071186,
0.02822582609951496,
-0.0017440890660509467,
-0.015226688235998154,
-0.030453376471996307,
-0.0018833109643310308,
-0.00006955138815101236,
-0.002565117320045... |
93eb7da8e24208a4b51c040a0442ae1aaf2a6a7e | subsection | 103 | 150 | First invariant subspace | \end{aligned}
\end{equation}
\begin{}
One can set {\!\!_3=0 when deriving
the first two estimates in (\ref {korrre1oo}),
by virtue of the inequalities (\ref {additional_est}) and the fact that m_1, m_2
do not depend on {\!\!_3, see (\ref {sinisa3*second}).
}
}\end{}}}\end{equation}\end{equation}}}}}\end{equation}\sub... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.010560017079114914,
0.016328349709510803,
-0.02597275748848915,
-0.025347091257572174,
0.017717022448778152,
-0.017884884029626846,
0.0329313799738884,
0.009041632525622845,
0.03641069307923317,
0.006500819697976112,
-0.01831216737627983,
0.02902478538453579,
0.0073592024855315685,
0.00... |
d7e3fd1163f151b682dda0adca164e27c09b12f3 | subsection | 104 | 150 | First invariant subspace | \end{aligned}In the same way as in Remark , it can be shown that (REF ) is equivalent to the identity\bigl (\vert \chi \vert ^{-4}A^{\textrm {hom}}_{\chi }+1\bigr )m\cdot \overline{d} =\int _Q|\chi |^{-1}\bigl (f\,\!\!_1, f\,\!\!_2\bigr )^\top \cdot \overline{e_\chi (d_1-i\chi _1 x_3 d_3,d_2 -i\chi _2 x_3d_3)^\top }+\i... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.0022910679690539837,
-0.0005031383479945362,
-0.028950247913599014,
-0.018298020586371422,
-0.01928999088704586,
0.03699282929301262,
0.0778924971818924,
0.011903633363544941,
0.026142209768295288,
-0.017107658088207245,
-0.04089966416358948,
0.008157040923833847,
0.004704224411398172,
... |
63c77b643a3e521b1d6ecc2dc98a6b5b84d10a83 | subsection | 105 | 150 | First invariant subspace | In order to determine the “corrector” term {\mathfrak {u}}_2, we solve({\rm sym}\nabla )^{*}A ({\rm sym}\nabla ) {\mathfrak {u}}_2=-({\rm sym}\nabla )^{*}A\bigl (\Xi (\chi , m_1,m_2)-ix_3 \Upsilon (\chi ,m_3)\bigr ),\ \ \ \ {\mathfrak {u}}_2 \in H^1_{\#}(Q, \mathbb {C}^3), \quad \int _{Q} {\mathfrak {u}}_2=0,so that, d... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.041172005236148834,
-0.02337861806154251,
-0.036166660487651825,
-0.022493526339530945,
0.015641698613762856,
-0.01459637563675642,
0.0029890905134379864,
0.010376932099461555,
0.005810318980365992,
0.024523131549358368,
-0.036441344767808914,
0.02180681750178337,
0.01603846438229084,
0... |
d7a7e5d086320f82d8b011450ec7bac0b4a4222c | subsection | 106 | 150 | First invariant subspace | We update m \in \mathbb {C}^3 with m^{(1)}=\bigl (m^{(1)}_1, m^{(1)}_2, m^{(1)}_3\bigr )^\top \in \mathbb {C}^3 such that\begin{aligned}&A^{\textrm {hom}}_{\chi } m^{(1)}\cdot \overline{d}+|\chi |^4\bigl (m_1^{(1)}-i\chi _1 x_3 m_3^{(1)},m_2^{(1)}-i\chi _2 x_3 m_3^{(1)}, m_3^{(1)}\bigr )^\top \cdot \overline{(d_1-i\chi... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.02262296713888645,
-0.015010789036750793,
-0.035635367035865784,
-0.03249286487698555,
0.0037908342201262712,
-0.046100206673145294,
0.03365223482251167,
0.019236385822296143,
-0.020761873573064804,
-0.014286182820796967,
-0.03652014955878258,
-0.009122409857809544,
-0.010197877883911133,
... |
739dbd75e57050d23b00eaa6eb48c977dc162cb4 | subsection | 107 | 150 | First invariant subspace | \end{aligned}It is straightforward to see that\bigl |\bigl (m_1^{(1)}, m_2^{(1)}\bigr )^\top \bigr |\le C|\chi |^2 \Vert f\Vert _{L^2(Q,\mathbb {C}^3)}, \quad \bigl |m_3^{(1)}\bigr |\le C|\chi | \Vert f\Vert _{L^2(Q,\mathbb {C}^3)}.Furthermore, we define {\mathfrak {u}}_3^{(2)}\in H^1_{\#}(Q, \mathbb {C}^3) as the solu... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.025613000616431236,
0.018839819356799126,
-0.044940974563360214,
-0.049517448991537094,
0.010144517756998539,
-0.0035753704141825438,
0.02899959124624729,
0.00335989473387599,
0.005465073045343161,
0.01897711306810379,
-0.044391799718141556,
-0.003291247645393014,
0.03947971761226654,
0... |
3b649f01614a5b6567fdf0fe3eb98a29c4e36350 | subsection | 108 | 150 | First invariant subspace | \end{aligned}As before, the right-hand side of (REF ) vanishes when tested with constant vectors,
in view of the identity (). Thus (REF ) has a unique solution, and\bigl \Vert \mathfrak {u}_4^{(1)}\bigr \Vert _{H^1(Q,\mathbb {C}^3)}\le C|\chi |^4 \Vert f\Vert _{L^2(Q, \mathbb {C}^3)}.Step 3. | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.035775016993284225,
0.025686582550406456,
-0.005795888137072325,
-0.047404948621988297,
-0.006406383588910103,
-0.01297302171587944,
0.045451365411281586,
0.022420434281229973,
0.02843381091952324,
0.005368541926145554,
-0.022313598543405533,
0.001037841779179871,
0.042459938675165176,
... |
65741cb9eaa8b9a149200e994c9cb8ba32cda29c | subsection | 109 | 150 | First invariant subspace | We again update m \in \mathbb {C}^3 with m^{(2)} \in \mathbb {C}^3 such that\begin{aligned}&A^{\textrm {hom}}_{\chi } m^{(2)}\cdot \overline{d}+|\chi |^4\bigl (m_1^{(2)}-i\chi _1 x_3 m_3^{(2)},m_2^{(2)}-i\chi _2 x_3 m_3^{(2)}, m_3^{(2)}\bigr )^\top \cdot \overline{(d_1-i\chi _1 x_3 d_3,d_2-i\chi _2 x_3 d_3,d_3)^\top }\... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.019246848300099373,
-0.015606583096086979,
-0.0321442186832428,
-0.02182632125914097,
0.010981845669448376,
-0.04105790704488754,
0.02291000634431839,
0.010165265761315823,
-0.012630267068743706,
0.007704081013798714,
-0.03565474599599838,
0.009280002675950527,
-0.005910659674555063,
0.... |
bfbea40f4835a94450485404f70835e7f8739adf | subsection | 110 | 150 | First invariant subspace | We update m \in \mathbb {C}^3 with m^{(3)} \in \mathbb {C}^3
in the same way as above by defining \mathfrak {u}_5^{(2)}, \mathfrak {u}_6 \in H^1_{\#}(Q, \mathbb {C}^3).Step 5.
In the same way as in Section , it follows that\begin{aligned}\bigl \Vert u_\alpha -(m_{\alpha }-i \chi _\alpha m_3 x_3)e_\chi \bigr \Vert _{H^1... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.0065805851481854916,
-0.011696274392306805,
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-0.029145315289497375,
0.009880414232611656,
-0.03094591572880745,... |
910ca7be8dfce530635747999db1e1dba3346300 | subsection | 111 | 150 | Second scaling | Asymptotic equation. An approximating problem for () has the form\begin{aligned}\vert \chi \vert ^{-2}A^{\textrm {hom}}_{\chi }m\cdot \overline{d} &+\int _Q (m_1-i\chi _1 x_3 m_3, m_2-i\chi _2 x_3m_3,m_3)^\top \cdot \overline{(d_1-i\chi _1 x_3 d_3, d_2-i\chi _2 x_3d_3,d_3)^\top }
\\[0.3em] &=\int _Qf
\cdot \overline{e_... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.013327614404261112,
-0.0026433991733938456,
-0.017332766205072403,
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0.02033853530883789,
-0.00333190360106527... |
2fe30c3629b17851878d89cc9a23bc8c0d5440ed | subsection | 112 | 150 | Second scaling | We define {\mathfrak {u}}_1 \in H^1_{\#} (Q, \mathbb {C}^3) as the solution to({\rm sym}\nabla )^{*}A\,{\rm sym}\nabla {\mathfrak {u}}_1=-({\rm sym}\nabla )^{*}A\bigl ( \Xi (\chi , m_1,m_2)-ix_3\Upsilon (\chi ,m_3) \bigr ),\quad {\mathfrak {u}}_1 \in H^1_{\#}(Q, \mathbb {C}^3), \quad \int _Q {\mathfrak {u}}_1=0,and inf... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.019421815872192383,
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0.042779866605997086,
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0.023907292634248734,
0.03249683231115341,
... |
bc9e7610005e955756c61d23d47d87eeef597265 | subsection | 113 | 150 | Second scaling | We update m \in \mathbb {C}^3 with m^{(1)} \in \mathbb {C}^3, which we define to satisfy\begin{aligned}&A^{\textrm {hom}}_{\chi } m^{(1)}\cdot \overline{d}+|\chi |^2\bigl (m_1^{(1)}-i\chi _1 x_3 m_3^{(1)},m_2^{(1)}-i\chi _2 x_3 m_3^{(1)},m_3^{(1)}\bigr )^\top \cdot \overline{(d_1-i\chi _1 x_3 d_3,d_2-i\chi _2 x_3 d_3,d... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
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-0.00994314718991518,
-0.007405851501971483,
0... |
61c9a4c3d0b4b4019ac94d91bada05b775d363ae | subsection | 114 | 150 | Second scaling | \end{aligned}Recalling (REF ), we obtain the estimates\bigl |m^{(1)}\bigr |\le C |\chi | \Vert f\Vert _{L^2 (Q,\mathbb {C}^3)},\quad \bigl |m_3^{(1)}
\bigr | \le C |\chi |^2\Vert f\Vert _{L^2(Q, \mathbb {C}^3)},where for the second inequality we set
d_1=d_2=0, d_3=m_3^{(1)}
in (REF ).Next, we define {\mathfrak {u}}_2^{... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.03219792619347572,
0.015587764792144299,
-0.02853560447692871,
-0.051150452345609665,
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0.03177065774798393,
-0.03015313111245632,
0.018357396125793457,
0.020142778754234314,
0.011... |
91815bc2ed2e2d85b41a12fa0431d3bf5b7f82bf | subsection | 115 | 150 | Second scaling | \end{aligned}It is easy to see that, as a consequence of (REF ), (REF ) and (REF ), the right-hand side of (REF ) vanishes when tested with constant vectors.
Thus (REF ) has a unique solution, and\Vert {\mathfrak {u}}_3\Vert _{H^1(Q, {\mathbb {C}}^3)} \le C|\chi |^3\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)}.Step 3. Simila... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.047173574566841125,
-0.002103511244058609,
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0.0001741403975756839,
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0.03701264411211014,
0.024654753506183624,
-0.014211572706699371,
0.026699144393205643,
-0.00510335061699152... |
6655df3666a72a4b285c86ff679da1bc56ed314f | subsection | 116 | 150 | Second scaling | An approximating problem for () has the form\begin{aligned}\vert \chi \vert ^{-2}A^{\textrm {hom}}_{\chi }m\cdot \overline{d} &+\int _Q (m_1-i\chi _1 x_3 m_3, m_2-i\chi _2 x_3m_3,m_3)^\top \cdot \overline{(d_1-i\chi _1 x_3 d_3, d_2-i\chi _2 x_3d_3,d_3)^\top }
\\[0.3em] &=\int _Qf
\cdot \overline{e_\chi (d_1-i\chi _1 x_... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.003876962698996067,
0.005225222557783127,
-0.040276169776916504,
-0.028910355642437935,
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0.018902339041233063,
-0.003457419341430068,... |
9e17b7dbe8f4e3ca44387609069af3befd1f7944 | subsection | 117 | 150 | Second scaling | We define {\mathfrak {u}}_1 \in H^1_{\#} (Q, \mathbb {C}^3) as the solution to({\rm sym}\nabla )^{*}A\,{\rm sym}\nabla {\mathfrak {u}}_1=-({\rm sym}\nabla )^{*}A\bigl ( \Xi (\chi , m_1,m_2)-ix_3\Upsilon (\chi ,m_3) \bigr ),\quad {\mathfrak {u}}_1 \in H^1_{\#}(Q, \mathbb {C}^3), \quad \int _Q {\mathfrak {u}}_1=0,and inf... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.019421815872192383,
-0.014364209957420826,
-0.024639615789055824,
-0.024624358862638474,
0.012670712545514107,
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0.042779866605997086,
-0.010496629402041435,
0.023907292634248734,
0.03249683231115341,
... |
84fecaa06b2de086dede9871b3a6b07e18b1c926 | subsection | 118 | 150 | Second scaling | We update m \in \mathbb {C}^3 with m^{(1)} \in \mathbb {C}^3, which we define to satisfy\begin{aligned}&A^{\textrm {hom}}_{\chi } m^{(1)}\cdot \overline{d}+|\chi |^2\bigl (m_1^{(1)}-i\chi _1 x_3 m_3^{(1)},m_2^{(1)}-i\chi _2 x_3 m_3^{(1)},m_3^{(1)}\bigr )^\top \cdot \overline{(d_1-i\chi _1 x_3 d_3,d_2-i\chi _2 x_3 d_3,d... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.026677854359149933,
-0.025243232026696205,
-0.04581631347537041,
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0.009843944571912289,
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-0.03580448776483536,
-0.00994314718991518,
-0.007405851501971483,
0... |
3f6b757608822e5a33965e736c40aad233cf8271 | subsection | 119 | 150 | Second scaling | \end{aligned}Recalling (REF ), we obtain the estimates\bigl |m^{(1)}\bigr |\le C |\chi | \Vert f\Vert _{L^2 (Q,\mathbb {C}^3)},\quad \bigl |m_3^{(1)}
\bigr | \le C |\chi |^2\Vert f\Vert _{L^2(Q, \mathbb {C}^3)},where for the second inequality we set
d_1=d_2=0, d_3=m_3^{(1)}
in (REF ).Next, we define {\mathfrak {u}}_2^{... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.03219792619347572,
0.015587764792144299,
-0.02853560447692871,
-0.051150452345609665,
0.005043324548751116,
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0.03177065774798393,
-0.03015313111245632,
0.018357396125793457,
0.020142778754234314,
0.011... |
867e072199d2cacc93155cea1ee298959dd1dcba | subsection | 120 | 150 | Second scaling | \end{aligned}It is easy to see that, as a consequence of (REF ), (REF ) and (REF ), the right-hand side of (REF ) vanishes when tested with constant vectors.
Thus (REF ) has a unique solution, and\Vert {\mathfrak {u}}_3\Vert _{H^1(Q, {\mathbb {C}}^3)} \le C|\chi |^3\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)}.Step 3. Simila... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.04707743227481842,
-0.0022329743951559067,
-0.016536597162485123,
-0.04515528306365013,
0.0007727693300694227,
0.0017409954452887177,
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0.0366734117269516,
0.025293059647083282,
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0.022028455510735512,
-0.003518221899867058,
... |
faeb059967d98ea2be71740c85db346e7c1a8032 | subsection | 121 | 150 | Norm-resolvent estimates for the infinite plate | Here we interpret the error estimates obtained above in terms the original family of elasticity operators {\mathcal {A}}^\varepsilon , \varepsilon >0, in L^2(\Pi ^h , {\mathbb {R}}^3), \Pi ^h ={\mathbb {R}}^2\times (-h/2, h/2), given by the differential expressions (cf. (REF ), where h=\varepsilon )({\rm sym}\nabla )^{... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.027443138882517815,
0.009297717362642288,
-0.03597049415111542,
0.011891009286046028,
0.014209718443453312,
0.009084152057766914,
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0.02210400626063347,
0.027336355298757553,
0.01334020309150219,
-0.026466840878129005,
0.018122538924217224,
-0.011410487815737724,
0.03... |
d8a282d95aead20b7582be2b45400bea2fbabcf0 | subsection | 122 | 150 | Case of planar-symmetric elasticity tensor | Following a convention similar to that of Section , we attach the overscripts \,\widehat{}\, and { to the force components that are even and odd in x_3, respectively.
}\subsubsection {First invariant subspace: Theorem \ref {main_result_first}}Proof of the L^2\rightarrow L^2 estimate (). For each \chi \in Q^{\prime }_{\... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.0018766240682452917,
0.05651232227683067,
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-0.01731681637465954,
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0.010771517641842365,
-0.012556598521769047,... |
fb1e8e725e9ddf628212a6cdfa034365e77d45b5 | subsection | 123 | 150 | Case of planar-symmetric elasticity tensor | Furthermore, we use
the Riesz integral representation\left(\varepsilon ^{-\gamma -2} \mathcal {A}_{\chi }+I\right)^{-1}=\frac{1}{2\pi i} \oint _{\Gamma }f_{\varepsilon ,\chi } (\zeta ) \left(\zeta I-|\chi |^{-4} \mathcal {A}_{\chi }\right)^{-1}d\zeta .Here we choose \rho _2>0 and a contour \Gamma \subset \lbrace \zeta ... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.010626335628330708,
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0.017819900065660477,
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0.047723155468702316,
0.01623319648206234,
0.02... |
01b1ac6742a52953fefaa7c39d36416d1f5ed911 | subsection | 124 | 150 | Case of planar-symmetric elasticity tensor | To obtain an approximation in the L^2(\Pi ^h , {\mathbb {R}})\rightarrow H^1(\Pi ^h , {\mathbb {R}}) operator norm, we use the third and fourth estimates in () and include the corrector \mathfrak {u}_2 in the first two components.Notice that neglecting the spectral projection onto the eigenspace of an order-one eigenva... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.041188087314367294,
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0.03533022478222847,
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-0.02341618947684765,
0.036764178425073624,
-0.0... |
6558505131de8bd25df642f106562cf3ccf6e700 | subsection | 125 | 150 | Case of planar-symmetric elasticity tensor | As before, the error is maximised when
\varepsilon ^{2-\gamma }|\theta |^4 \sim 1,
and therefore the overall approximation error is of order\max \bigl \lbrace \varepsilon ^{p(\gamma +2)/4},\varepsilon ^{(p+1)(\gamma +2)/4-1}\bigr \rbrace \max \lbrace \varepsilon ^{(\gamma +2)/4-\delta },1\rbrace .As a result, there exi... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.02990676648914814,
0.017989834770560265,
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0.027267036959528923,
0.0019702608697116375,
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0.015540837310254574,
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0.02206386998295784,
-0.0009078839793801308,
... |
845f4a79633f97f3925938274d9342600839f302 | subsection | 126 | 150 | Case of planar-symmetric elasticity tensor | This corrector seems to be unknown in the homogenisation theory, although its derivation is similar to the elliptic argument of \cite {BirmanSuslina}.
We leave the details to the interested reader, noting that the associated error is bounded by
C\varepsilon ^p|\theta |^p\left(\max \left\lbrace \varepsilon ^{2-\gamma }... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.025473512709140778,
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0.026419803500175476,
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0.005506033543497324,
-0.0007135330815799534... |
09043a25ec6fdc11b60d2aa8f6dff6b706065738 | subsection | 127 | 150 | Case of planar-symmetric elasticity tensor | As a result, there exists C>0 such that
\begin{equation*}
\begin{aligned}&\Bigl \Vert P_{\alpha } \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}\bigl (\varepsilon ^{-\delta }{\!\!_1,\varepsilon ^{-\delta }{\!\!_2, \widehat{F}_3\bigr )^\top \\[0.8em]
&\hspace{21.25pt}+\left(\varepsilon x_3\parti... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.024023672565817833,
0.042186424136161804,
-0.03269295394420624,
0.01083659939467907,
0.00403701514005661,
0.0011761907953768969,
0.02358105219900608,
0.025275224819779396,
0.03102930635213852,
0.002770201303064823,
-0.020391123369336128,
0.009256898425519466,
-0.005567112471908331,
-0.0... |
725dff3a767d6ae42440e2de9861d8cbd75b0fcb | subsection | 128 | 150 | Case of planar-symmetric elasticity tensor | In the approximation on each fibre we neglect the projections onto the eigenspaces determined by the latter
eigenvalues, which results in an error of the same order \varepsilon ^{\gamma +2} for the operators (\varepsilon ^{-\gamma }{\mathcal {A}}^\varepsilon +I)^{-1}.
On the other hand, the first two estimates in (\ref... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.017488326877355576,
0.026934465393424034,
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-0.020143622532486916,
-0.008889135904610157,... |
a5bbf63013670ba1d7db71b311d1e04b441451a2 | subsection | 129 | 150 | Case of planar-symmetric elasticity tensor | The resulting error is of order
\max \bigl \lbrace \varepsilon |\theta |, \varepsilon |\theta |^2\bigr \rbrace \left(\max \left\lbrace \varepsilon ^{-\gamma }|\theta |^2,1\right\rbrace \right)^{-1}\!\!,
so that
\begin{equation*}
\begin{aligned}\Bigl \Vert \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\ri... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.000602184038143605,
0.012533819302916527,
-0.032497912645339966,
0.0015095127746462822,
0.007376873400062323,
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0.0101308049634099,
-0.02856154553592205,
0.015402179211378098,
-0.025784729048609734,
0.0... |
aea044caa33d63fb2d97b667ed20e5fc4aec9d91 | subsection | 130 | 150 | Case of planar-symmetric elasticity tensor | In terms of the original operator family {\mathcal {A}}^\varepsilon
this yields the estimate
\begin{equation*}
\begin{aligned}\Bigl \Vert \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}\bigl (\widehat{F}_1,\widehat{F}_2, {\!\!_3\bigr )^\top -\left(\left( \varepsilon ^{-\gamma }\mathcal {A}^{{\r... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.031316835433244705,
0.02051161229610443,
-0.06812785565853119,
0.0043075913563370705,
0.0078444704413414,
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0.0003612710861489177,
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0.008287057280540466,
-0.008355734869837761,
0.... |
dc37f53d921f246bb435c838cd1ca7f517f0cff3 | subsection | 131 | 150 | Case of planar-symmetric elasticity tensor | In the approximation we neglect the projections onto the eigenspaces of the latter eigenvalues, which results in an error of the same order \varepsilon ^{\gamma +2} for the operators (\varepsilon ^{-\gamma }{\mathcal {A}}^\varepsilon +I)^{-1}.
}Under two different scalings for the fibres {\mathcal {A}}_\chi , we obtain... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.006692821159958839,
0.015720879659056664,
-0.057053059339523315,
-0.009890417568385601,
0.036203812807798386,
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0.024848148226737976,
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-0.008905954658985138,
0.0019155052723363042,
... |
94c793d93bf9cd9dc5411d4817fa49fac6c04d17 | subsection | 132 | 150 | Case of planar-symmetric elasticity tensor | This yields the estimates
\begin{equation*}
\begin{aligned}&\Bigl \Vert P_{\alpha }\left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}F
\\[0.5em]
& \hspace{42.5pt}
-\left(\left(P_{\alpha }-\varepsilon \partial _{\alpha }P_3\right)\left( \varepsilon ^{-\gamma }\mathcal {A}^{\rm hom}+I \right)^{-1}+\v... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.005420143250375986,
0.020749880000948906,
-0.04006557911634445,
-0.02639506757259369,
0.005130255129188299,
-0.040157120674848557,
0.0157912690192461,
0.03472553566098213,
0.002998052630573511,
-0.009406103752553463,
-0.03472553566098213,
0.0605713427066803,
-0.046687230467796326,
0.023... |
77bcaf4aedf185a7c0648ec4e3bac4ef883d68a6 | subsection | 133 | 150 | Case of planar-symmetric elasticity tensor | \end{aligned}
\end{equation*}
Finally, using (\ref {marita10000}), (\ref {otto1revision0000}), (\ref {marita10002}), (\ref {marita10001}), (\ref {marita10003}), (\ref {revision1003000pon0}), (\ref {sinisa101000}), (\ref {sinisa3*secondoooo}), we can define a corrector {\mathfrak {K}}_2 so that
\begin{equation*}
\begin{... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.024552039802074432,
0.05053844302892685,
-0.051820214837789536,
-0.011040025390684605,
0.00755711505189538,
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0.03152548894286156,
-0.004383203107863665,
0.018921397626399994,
0.023407600820064545,
-0.03137289732694626,
0.019012954086065292,
0.0026798956096172333,
0... |
ebe4c0a332c1157d8fca3ccb47b77721833d8031 | subsection | 134 | 150 | Case of planar-symmetric elasticity tensor | IV has been supported by the Croatian Science Foundation under Grant agreement No.~9477 (MAMPITCoStruFl) and Grant agreement No. IP-2018-01-8904 (Homdirestroptcm).
}\begin{}{9}
\end{}\bibitem {BLP}
Bensoussan, A., Lions, J.-L., and Papanicolaou, G. C., 1978. {\it Asymptotic Analysis for Periodic Structures,} North-Holl... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.04718819633126259,
0.030812732875347137,
-0.04303710162639618,
0.018298398703336716,
0.013742875307798386,
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0.002647849963977933,
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0.012773778289556503,
0.024143507704138756,
-0.033849749714136124,
0.006432672962546349,
0.015917623415589333,
-0... |
051418f7d456820340c3c69c0ef8ae1b29ad5798 | subsection | 135 | 150 | Case of planar-symmetric elasticity tensor | Velčić, I., 2013. Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity. Math. Models Methods Appl. Sci. 23(14), 2701–2748.Oleinik, O. A., Shamaev, A. S., Yosifian, G. A., 1992. Mathematical Problems in Elasticity and Homogenization, Amstredam: North-Holland.Panasenko, G., 2005. Multi-scale m... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.0154373524710536,
0.026405194774270058,
-0.04387136921286583,
0.002730519976466894,
-0.002551281824707985,
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0.03941711410880089,
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0.008611053228378296,
-0.021035680547356606,
0.004892817232757807,
-0.007924609817564487,
0.... |
af2b22da5d0dd01e55ad424de1e6c3376f6669db | subsection | 136 | 150 | Case of planar-symmetric elasticity tensor | Indeed, for each \varepsilon >0, \chi \in Q^{\prime }_{\rm r}\setminus \lbrace 0\rbrace , we write\varepsilon ^{-\gamma -2} \mathcal {A}_{\chi }+I=\varepsilon ^{-\gamma -2}|\chi |^4\bigl (|\chi |^{-4} \mathcal {A}_{\chi } \bigr )+Iand notice that for all \eta >0 the functionf_{\varepsilon ,\chi }(\zeta ):=\bigl (\varep... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.00931032095104456,
0.017643820494413376,
-0.02266528829932213,
-0.013553384691476822,
0.02091006562113762,
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0.06465330719947815,
0.007284181192517281,
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0.015003353357315063,
-0.0503978356719017,
0.03134983405470848,
-0.0174301415681839,
0.01565... |
0c48e0c8e5fa9bfdb3f233c608cf1631db5afb10 | subsection | 137 | 150 | Case of planar-symmetric elasticity tensor | (Notice that analogous estimates ensure that for a suitable choice of \rho _2 the contour \Gamma possesses the properties 1 and 3 above in relation to the eigenvalue of {\mathcal {A}}_\chi ^{{\rm hom},1} whenever |\chi |<\rho _2.) For quasimomenta \chi such that |\chi |>\rho _2 the required bound holds automatically.By... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.007424086797982454,
-0.003244699677452445,
-0.05038917809724808,
0.0025370081420987844,
-0.007958193309605122,
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0.048924196511507034,
-0.02698003128170967,
-0.0044941287487745285,
0.006863274611532688,
-0.043155841529369354,
0.0020410516299307346,
0.02861287258565426... |
ebb26f979c3b36cefc98a655348ffd3ddb20dbe1 | subsection | 138 | 150 | Case of planar-symmetric elasticity tensor | The error of neglecting the spectral projections of (\varepsilon ^{-\gamma }{\mathcal {A}}^\varepsilon +I)^{-1} onto the eigenspaces corresponding to higher eigenvalues is of order \varepsilon ^{\gamma +1-\delta } and, with the corrector
{\mathfrak {K}}^{\rm b}_1 included,
the H^1 error in each fibreRecall that under t... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.018703995272517204,
0.015431558713316917,
-0.028834054246544838,
0.007399215362966061,
0.01821579970419407,
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0.04781265929341316,
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0.010351273231208324,
-0.04125253111124039,
-0.018703995272517204,
-0.021297534927725792,
... |
78741b59609e4c004eb8ea13faaccd7453ef8aa1 | subsection | 139 | 150 | Case of planar-symmetric elasticity tensor | As before, the error is maximised when
\varepsilon ^{2-\gamma }|\theta |^4 \sim 1,
and therefore the overall approximation error is of order\max \bigl \lbrace \varepsilon ^{p(\gamma +2)/4},\varepsilon ^{(p+1)(\gamma +2)/4-1}\bigr \rbrace \max \lbrace \varepsilon ^{(\gamma +2)/4-\delta },1\rbrace .As a result, there exi... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.02990676648914814,
0.017989834770560265,
-0.052764080464839935,
0.027267036959528923,
0.0019702608697116375,
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0.019729310646653175,
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0.015540837310254574,
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0.02206386998295784,
-0.0009078839793801308,
... |
57012133c3133d88dd3b1bc30b2835556fc20b79 | subsection | 140 | 150 | Case of planar-symmetric elasticity tensor | This corrector seems to be unknown in the homogenisation theory, although its derivation is similar to the elliptic argument of \cite {BirmanSuslina}.
We leave the details to the interested reader, noting that the associated error is bounded by
C\varepsilon ^p|\theta |^p\left(\max \left\lbrace \varepsilon ^{2-\gamma }... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.025473512709140778,
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0.026419803500175476,
0.010989172384142876,
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0.005506033543497324,
-0.0007135330815799534... |
9a8a11b32715c4d85062cc9f251c6f110e87103f | subsection | 141 | 150 | Case of planar-symmetric elasticity tensor | As a result, there exists C>0 such that
\begin{equation*}
\begin{aligned}&\Bigl \Vert P_{\alpha } \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}\bigl (\varepsilon ^{-\delta }{\!\!_1,\varepsilon ^{-\delta }{\!\!_2, \widehat{F}_3\bigr )^\top \\[0.8em]
&\hspace{21.25pt}+\left(\varepsilon x_3\parti... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.024023672565817833,
0.042186424136161804,
-0.03269295394420624,
0.01083659939467907,
0.00403701514005661,
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0.025275224819779396,
0.03102930635213852,
0.002770201303064823,
-0.020391123369336128,
0.009256898425519466,
-0.005567112471908331,
-0.0... |
9aa566afabdcdcee3e78cf9c2e24950bb3cc4a6a | subsection | 142 | 150 | Case of planar-symmetric elasticity tensor | In the approximation on each fibre we neglect the projections onto the eigenspaces determined by the latter
eigenvalues, which results in an error of the same order \varepsilon ^{\gamma +2} for the operators (\varepsilon ^{-\gamma }{\mathcal {A}}^\varepsilon +I)^{-1}.
On the other hand, the first two estimates in (\ref... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.017488326877355576,
0.026934465393424034,
-0.03351166099309921,
-0.009812385775148869,
0.02656821720302105,
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0.031985629349946976,
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0.012482942081987858,
-0.043064624071121216,
-0.020143622532486916,
-0.008889135904610157,... |
efbcc5e07708981c7f878635a7fa97752c9e0b44 | subsection | 143 | 150 | Case of planar-symmetric elasticity tensor | The resulting error is of order
\max \bigl \lbrace \varepsilon |\theta |, \varepsilon |\theta |^2\bigr \rbrace \left(\max \left\lbrace \varepsilon ^{-\gamma }|\theta |^2,1\right\rbrace \right)^{-1}\!\!,
so that
\begin{equation*}
\begin{aligned}\Bigl \Vert \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\ri... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.000602184038143605,
0.012533819302916527,
-0.032497912645339966,
0.0015095127746462822,
0.007376873400062323,
0.012320217676460743,
0.010069775395095348,
0.007979534566402435,
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0.0101308049634099,
-0.02856154553592205,
0.015402179211378098,
-0.025784729048609734,
0.0... |
308952ddbf949e12eacf1f3a593a990d50d8365d | subsection | 144 | 150 | Case of planar-symmetric elasticity tensor | In terms of the original operator family {\mathcal {A}}^\varepsilon
this yields the estimate
\begin{equation*}
\begin{aligned}\Bigl \Vert \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}\bigl (\widehat{F}_1,\widehat{F}_2, {\!\!_3\bigr )^\top -\left(\left( \varepsilon ^{-\gamma }\mathcal {A}^{{\r... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.031316835433244705,
0.02051161229610443,
-0.06812785565853119,
0.0043075913563370705,
0.0078444704413414,
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0.03974124789237976,
0.007016527932137251,
0.036688923835754395,
0.0003612710861489177,
-0.021808849647641182,
0.008287057280540466,
-0.008355734869837761,
0.... |
1f92d335317f041e09d0870f669f7f35df5c68b3 | subsection | 145 | 150 | Case of planar-symmetric elasticity tensor | In the approximation we neglect the projections onto the eigenspaces of the latter eigenvalues, which results in an error of the same order \varepsilon ^{\gamma +2} for the operators (\varepsilon ^{-\gamma }{\mathcal {A}}^\varepsilon +I)^{-1}.
}Under two different scalings for the fibres {\mathcal {A}}_\chi , we obtain... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
0.006692821159958839,
0.015720879659056664,
-0.057053059339523315,
-0.009890417568385601,
0.036203812807798386,
0.009737787768244743,
0.03757748380303383,
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0.024848148226737976,
-0.039805877953767776,
-0.008905954658985138,
0.0019155052723363042,
... |
b779e0f97695d68c2dc1f5edd3e64b1254f94627 | subsection | 146 | 150 | Case of planar-symmetric elasticity tensor | This yields the estimates
\begin{equation*}
\begin{aligned}&\Bigl \Vert P_{\alpha }\left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}F
\\[0.5em]
& \hspace{42.5pt}
-\left(\left(P_{\alpha }-\varepsilon \partial _{\alpha }P_3\right)\left( \varepsilon ^{-\gamma }\mathcal {A}^{\rm hom}+I \right)^{-1}+\v... | {
"cite_spans": []
} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
"math.AP",
"math-ph",
"math.MP",
"math.SP"
] | 2,018 | en | Mathematics | [
-0.005420143250375986,
0.020749880000948906,
-0.04006557911634445,
-0.02639506757259369,
0.005130255129188299,
-0.040157120674848557,
0.0157912690192461,
0.03472553566098213,
0.002998052630573511,
-0.009406103752553463,
-0.03472553566098213,
0.0605713427066803,
-0.046687230467796326,
0.023... |
40e11be25ec906410d83379cfb5ea0bfe6a01b91 | subsection | 147 | 150 | Case of planar-symmetric elasticity tensor | \end{aligned}
\end{equation*}
Finally, using (\ref {marita10000}), (\ref {otto1revision0000}), (\ref {marita10002}), (\ref {marita10001}), (\ref {marita10003}), (\ref {revision1003000pon0}), (\ref {sinisa101000}), (\ref {sinisa3*secondoooo}), we can define a corrector {\mathfrak {K}}_2 so that
\begin{equation*}
\begin{... | {
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} | 10.1112/jlms.12543 | 1802.02639 | Sharp operator-norm asymptotics for thin elastic plates with rapidly
oscillating periodic properties | [
"Kirill Cherednichenko",
"Igor Velčić"
] | [
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f1fd067ce0f3ce5297e6a3aec5f8ce937a607369 | subsection | 148 | 150 | Case of planar-symmetric elasticity tensor | IV has been supported by the Croatian Science Foundation under Grant agreement No.~9477 (MAMPITCoStruFl) and Grant agreement No. IP-2018-01-8904 (Homdirestroptcm).
}\begin{}{9}
\end{}\bibitem {BLP}
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9ad8183e386a66f83b9b07c3518e329cf1ee1fc1 | subsection | 149 | 150 | Case of planar-symmetric elasticity tensor | Velčić, I., 2013. Derivation of a homogenized von-Kármán plate theory from 3D nonlinear elasticity. Math. Models Methods Appl. Sci. 23(14), 2701–2748.Oleinik, O. A., Shamaev, A. S., Yosifian, G. A., 1992. Mathematical Problems in Elasticity and Homogenization, Amstredam: North-Holland.Panasenko, G., 2005. Multi-scale m... | {
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0836c5dbe36e4e45a1ecc3a143299a5b1a380a32 | abstract | 0 | 12 | Abstract | In the present work, we systematically study the $\mathcal{\alpha}$ decay
preformation factors $P_{\alpha}$ within the cluster-formation model and
$\mathcal{\alpha}$ decay half-lives by the proximity potential 1977 formalism
for nuclei around $Z=82$, $N=126$ closed shells. The calculations show that the
realistic $P_{\... | {
"cite_spans": []
} | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
around $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closure within the
cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
"Jie-Cheng Zhao",
"Peng-Cheng Chu",
"Xiao-Hua Li"
] | [
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9e41081629197d4f63feb78d5b6c9f6da7717ce5 | subsection | 1 | 12 | Introduction | In 1928, the phenomenon of \mathcal {\alpha } decay for nuclei was independently explained by Gurney and Condon and Gamow using the quantum tunnel theory. Since then, \mathcal {\alpha } decay has long been perceived as one of the most powerful tools to investigate unstable nuclei, neutron-deficient nuclei and superheav... | {
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... | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
around $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closure within the
cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
"Jie-Cheng Zhao",
"Peng-Cheng Chu",
"Xiao-Hua Li"
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8d1f566f68a6e99093316ce57ccdeba00c9849f7 | subsection | 2 | 12 | Introduction | Very recently, Ahmed et al. and Deng et al. extended CFM to odd-A and doubly-odd nuclei through modifying the formation energy of interior \mathcal {\alpha } cluster for various types of nuclei (i.e. even Z- odd N, odd Z-even N and doubly-odd nuclei) and considering the effects of unpaired nucleon , , , . In 2011, Seif... | {
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"end": ... | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
around $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closure within the
cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
"Jie-Cheng Zhao",
"Peng-Cheng Chu",
"Xiao-Hua Li"
] | [
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e44643546cbbf2a9e82572d3a9aa6cf9d4bba825 | subsection | 3 | 12 | the cluster-formation model | Within the cluster-formation model (CFM) , , , , , the total clsuterization state \Psi of parent nuclei is assumed as a linear combination of all its n possible clusterization states \Psi _i. It can be represented as\ \Psi =\sum _{i=1}^n a_i{\Psi }_i
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cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
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0b20dcf4167bfa711f183928ed8502aaa61163e5 | subsection | 4 | 12 | the cluster-formation model | In the framework of CFM , , , , , the \mathcal {\alpha } cluster formation energy E_{f\alpha } and total energy E of considered system can be expressed as four different cases.Case I for even-even nuclei,E_{f\alpha }=&&3B(A,Z)+B(A-4,Z-2)\\
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... | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
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cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
"Jie-Cheng Zhao",
"Peng-Cheng Chu",
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26d71705f98c23b487001e54a1389ffa04d15ee9 | subsection | 5 | 12 | Body | The \mathcal {\alpha } decay half-life can be calculated by decay width \Gamma or decay constant \mathcal {\lambda } and expressed as\ T_{1/2}=\frac{{\hbar }ln2}{\Gamma }=\frac{ln2}{\lambda }
,where \hbar is the Planck constant. In the framework of the Proximity potential 1977 formalism (Prox.1977) , the \mathcal {\alp... | {
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"Jun-Gang Deng",
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c9ed29f7243b994a687e23bbcf9b7ae5e7309d46 | subsection | 6 | 12 | Body | V(r) and Q_{\alpha } are the total \mathcal {\alpha }-core potential and \mathcal {\alpha } decay energy, respectively. r_{\text{in}} and r_{\text{out}} are the classical turning points, they satisfy the conditions V (r_{\text{in}}) = V (r_{\text{out}}) =Q_{\alpha }.The total interaction potential V(r) between \mathcal... | {
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} | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
around $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closure within the
cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
"Jie-Cheng Zhao",
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"Xiao-Hua Li"
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cacf6948a3cdeb4bb9d395338f71143780d39227 | subsection | 7 | 12 | Body | On the basis of the conservation laws of angular momentum and parity , the minimum angular momentum l_{\text{min}} taken away by the \mathcal {\alpha } particle can be obtained by\ l_{\text{min}}=\left\lbrace \begin{array}{llll}
\end{array}{\Delta }_j,&\text{for even${\Delta }_j$ and ${\pi }_p$= ${\pi }_d$},\\
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f904877e9602f9ab663a2b8836f6c686d778c9f8 | subsection | 8 | 12 | RESULTS AND DISCUSSION | The aims of this work are to study the \mathcal {\alpha } preformation factors and \mathcal {\alpha } decay half-lives of nuclei around Z=82, N=126 shell closures. Many researchers suggested that the smaller valance nucleons (holes) nuclei have, the smaller \mathcal {\alpha } preformation factors be , , . In 2011, Seif... | {
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... | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
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cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
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"Xiao-Hua Li"
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03541fd6a4742d30c87ada25f466fa2906822c6b | subsection | 9 | 12 | RESULTS AND DISCUSSION | In order to have a deeper insight into P_{\alpha }, we plot the relationship between P_{\alpha } and \frac{N_pN_n}{Z_0+N_0} of even-even nuclei, odd-A nuclei (including favored and unfavored \mathcal {\alpha } decay cases) and doubly-odd nuclei (including favored and unfavored \mathcal {\alpha } decay cases) around Z=8... | {
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e467169ce1d1f10d8b2461a1b452a0b16038ce08 | subsection | 10 | 12 | RESULTS AND DISCUSSION | The last three ones are calculated \mathcal {\alpha } decay half-life by Prox.1977 without considering P_{\alpha }, with taking P_{\alpha } by CFM and with fitting P_{\alpha } calculated by Eq. (REF ) and parameters listed in Table REF , which are denoted as {T_{1/2}^{\text{calc1}}}, {T_{1/2}^{\text{calc2}}} and {T_{1/... | {
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} | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
around $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closure within the
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6b0e59d3c8c38f38e6958667b9b26cc4b9bcd84a | subsection | 11 | 12 | Summary | In summary, we preformed the systematically study of \mathcal {\alpha } preformation factors within the cluster-formation model (CFM) and \mathcal {\alpha } decay half-lives within the proximity potential 1977 formalism (Prox.1977) for nuclei around Z=82, N=126 closed shells. Our results indicate that the realistic P_{... | {
"cite_spans": []
} | 10.1103/PhysRevC.97.044322 | 1804.06010 | Systematic study of $\boldsymbol{\mathcal{\alpha}}$ decay of nuclei
around $\boldsymbol{Z=82}$, $\boldsymbol{N=126}$ shell closure within the
cluster-formation model and proximity potential 1977 formalism | [
"Jun-Gang Deng",
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1c51421c601a6f6ed02c78c1b4a0ca727f315c04 | abstract | 0 | 10 | Abstract | This paper presents an open-source enforcement learning toolkit named CytonRL
(https://github.com/arthurxlw/cytonRL). The toolkit implements four recent
advanced deep Q-learning algorithms from scratch using C++ and NVIDIA's
GPU-accelerated libraries. The code is simple and elegant, owing to an
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"Xiaolin Wang"
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3fb89f4945292933afb228e96b158ffb629e576d | subsection | 1 | 10 | Introduction | Reinforcement learning (RL) is self learning what to do under an environment, in other words, how to map situations to actions, so as to maximize a numerical reward signal . RL is an meaningful artificial intelligence task, and will be extremely useful if it works. However, traditional real-world RL systems were usuall... | {
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Implemented in C++ | [
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c4e79809ca8aa6534091954a84a0a4423815a08b | subsection | 2 | 10 | Method | CytonRL has implemented four recent advanced deep Q-learning algorithms proposed by , , , . The following subsections first introduce the background knowledge of reinforcement learning, and then present the details of these four algorithms. | {
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Implemented in C++ | [
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6475f43826b7264dd2d967273b28ffd1d071e5c0 | subsection | 3 | 10 | Background | Suppose an agent interacts with an environment \mathcal {E} in a sequence of actions, observations, and rewards . At each time-step,the agent selects an action a_t from the set of legal game actions, A =\lbrace 1, \ldots , K\rbrace . The action is passed to \mathcal {E} and modifies its internal state. The agent both r... | {
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Implemented in C++ | [
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64437f73cb5d1e52361049d1d50e2d4d09e3833c | subsection | 4 | 10 | Deep Q-Network with Experience Replay | Deep Q-Network (DQN) uses a neural network to approximate the optimal action-value function Q^*(s,a). The network is trained by minimizing a sequence of loss functions at each iteration i, asL_i(\theta _i) = \mathop {\mathbb {E}}_{s,a \sim \rho (\cdot )} \left[\left(y_i - Q(s,a;\theta _i)\right)^2\right]where y_i = \ma... | {
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45f87f6f548813dd3898f59e8d4c919a984bbf77 | subsection | 5 | 10 | Deep Q-Network with Experience Replay | Each step of experience is potentially used in many weight updates, which allows for greater data efficiency.Deep Q-Network with Experience Replay
[1]Initialize the replay memory \mathcal {D} to capacity NInitialize the action-value function \mathcal {Q} with random weightsepisode = 1, MInitialize sequence s_1 = \lbr... | {
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8e011009b9d537ba450e00ed0ae5d61601ba4913 | subsection | 6 | 10 | Double Deep Q-Network | proposed double DQN to reduce the over-estimations caused by the max operation in the equation REF .The standard DQN used a training target as,Y_t^{DQN} = r_t+\gamma \mathop {max}_a Q(s_{t+1}, a ; \theta _t^-),where \theta _t^- is the parameters of a target network which is copied periodically from the online network. ... | {
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Implemented in C++ | [
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977029dbd6e95eada2a106ae83976d96ceb18514 | subsection | 7 | 10 | Prioritized Experience Replay | proposed prioritized experience replay to improve the learning efficiency of DQN, presented by the algorithm REF . The intuition of the method is to replay important transitions more frequently.The probability of sampling a transition i is defined as,P(i) = \frac{p_i ^ \alpha }{\sum _k p_k ^ \alpha }where p_i > 0 is th... | {
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32d667c7fc74df07c56b3105c3339cdc6ba84dad | subsection | 8 | 10 | Dueling DQN | proposed an dueling neural network architecture for DQN, which decomposed the Q function into two separate estimators; one for the state-value function and one for the state-dependent action-advantage function.The Q function of dueling DQN is formulated as,Q(s,a; \theta , \alpha , \beta ) & = & V(s; \theta , \beta ) +
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636deb9ea7b05ed1277be86298cd00e5fcfc6d3d | subsection | 9 | 10 | Implementation | CytonRL is implemented using the C++ language with a dependency on OpenCV https://github.com/opencv/opencv to down-sample the input images, and a dependency on NVIDIA's GPU-accelerated libraries – cuda, cublas and cudnn to use GPUs. CytonLib – a general purpose C++ neural network library – is shipped together with the ... | {
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4ab94e9a3481ac8bc3dec003393814f93ba72f35 | abstract | 0 | 33 | Abstract | Computing uniformization maps for surfaces has been a challenging problem and
has many practical applications. In this paper, we provide a theoretically
rigorous algorithm to compute such maps via combinatorial Calabi flow for
vertex scaling of polyhedral metrics on surfaces, which is an analogue of the
combinatorial Y... | {
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} | 1806.02166 | Combinatorial Calabi flow with surgery on surfaces | [
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418d8a4a26afee7e7d9e39529323544ca2613bd6 | subsection | 1 | 33 | Backgrounds and main results | One of the central topics in modern geometry concerns with
the canonical metrics on a given manifold, which is related to
special geometric structures on manifolds.
The flow method is an important approach for such problems.
To achieve this goal, Hamilton introduced the Ricci flow and
Calabi , introduced the Calabi flo... | {
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991dc74178ab22565463763433dfc01c691b6822 | subsection | 2 | 33 | Backgrounds and main results | The finiteness of surgeries along the combinatorial Yamabe flow was proved by Wu .
The combinatorial Yamabe flow with surgery provides an effective algorithm to compute the uniformization maps on surfaces .
Unlike the circle packing case , , , , , the convergence of the combinatorial Yamabe flow with surgery in ,
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1f19240dcba7b51507615b170caa8b7779fd1857 | subsection | 3 | 33 | Notations and definitions | Here we give some notations and definitions used in the main results.
Suppose S is a closed surface and V is a finite subset of S, (S, V) is called a marked surface.
A piecewise linear metric on (S, V) is a flat cone metric with cone points contained in V.
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c89ad75965de6cc0cb4669fa8b1f3a4544989442 | subsection | 4 | 33 | Notations and definitions | \end{aligned}where u: V\rightarrow \mathbb {R} is the conformal factor and
\Delta ^{\mathbb {E},\mathcal {T}} is the Euclidean discrete Laplace operator of u*d_0
on (S, V, \mathcal {T}) defined as\begin{aligned}(\Delta ^{\mathbb {E}, \mathcal {T}} f)_i=\sum _{j; j\sim i} \omega _{ij}^{\mathbb {E}, \mathcal {T}}(f_j-f_i... | {
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cc918f95617c8747f1ff8342b28480463ae88c04 | subsection | 5 | 33 | Notations and definitions | We replace the triangulation \mathcal {T} by a new triangulation \mathcal {T}^{\prime } at time t=T
by replacing two triangles \triangle ijk and \triangle ijl
adjacent to \lbrace ij\rbrace by two new triangles \triangle ikl and \triangle jkl.
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a93aea33b8370924c02964423dd61b49ff0b0a8b | subsection | 6 | 33 | Organization of the paper | The paper is organized as follows.
In Section , we give some preliminaries on discrete conformal geometry,
including the definitions of polyhedral metrics, vertex scaling, discrete curvature and discrete Laplace operators.
In Section , we study the Euclidean combinatorial Calabi flow on surfaces
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eb12f6ff57df9803a61c070ce6efd08781b04642 | subsection | 7 | 33 | Polyhedral metrics on surfaces | The definition of piecewise linear metric on marked surfaces has been given in Section .
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7f89c7f93dca56fe2aaec90001d122f386a5c638 | subsection | 8 | 33 | Polyhedral metrics on surfaces | Note that a polyhedral metric on a marked surface is independent of the triangulations.Suppose \mathcal {T}=\lbrace V,E,F\rbrace is a geometric triangulation of (S, V) with a PL or PH metric d,
then the metric d determines a map\begin{aligned}d: E&\longrightarrow (0, +\infty )\\
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1a195c13d2c41ed566377eb944e86b4f44027863 | subsection | 9 | 33 | Discrete curvature | On a marked surface, the well-known combinatorial curvature is defined as follows.Definition 2.2 Suppose (S, V) is a marked surface with a polyhedral metric,
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33ce6838604b97e16a2a08804798a33e85bb7dc0 | subsection | 10 | 33 | Vertex scaling of polyhedral metrics | Vertex scaling of PL metrics on a triangulated surface was introduced
by Luo and Rǒcek-Williams independently as an analogy of
the conformal transformation of Riemannian metrics.Definition 2.3 (, )
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8fb40396d62597e85712c53b9efe64f4020aeebf | subsection | 11 | 33 | Vertex scaling of polyhedral metrics | Gu-Luo-Sun-Wu observed
that the notions of vertex scaling are related to the Ptolemy identities for all polyhedral surfaces.
Note that the definition of vertex scaling for polyhedral metrics depends on
the triangulation of the marked surface (S, V). | {
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85859677105926eef136ac00693e357e6a6de7ba | subsection | 12 | 33 | Laplace operators on triangulated surfaces | The discrete Laplace operator of a PL metric on a triangulated surface, known as finite elements Laplacian,
has been extensively studied in geometry and computer graphics and is defined as follows.Definition 2.5
Suppose (S, V, \mathcal {T}) is a triangulated surface with a PL metric d.
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ed2042a186b4079f24ca3f43322e418c0141e97a | subsection | 13 | 33 | Laplace operators on triangulated surfaces | In this case, the Euclidean discrete Laplace operator \Delta ^{\mathbb {E}, \mathcal {T}} is intrinsic
in the sense that it is
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2be97f7dbe6472ab961a49d6d5e0210e37065bf5 | subsection | 14 | 33 | Laplace operators on triangulated surfaces | If the triangulation \mathcal {T} is Delaunay in u*d, then the
Euclidean discrete Laplace operator \Delta ^{\mathbb {E}, \mathcal {T}}, as a matrix-valued map,
is defined on\Omega ^{\mathbb {E}, \mathcal {T}}_{D}(d)\triangleq \lbrace u\in \mathbb {R}^V| \mathcal {T} \text{ is Delaunay in } u*d \rbrace ,which is a subsp... | {
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9c5f51f83dee70cb635ae1dff0c21fa4a69451f3 | subsection | 15 | 33 | Laplace operators on triangulated surfaces | If the triangulation \mathcal {T} is Delaunay in u*d, then the
hyperbolic discrete Laplace operator \Delta ^{\mathbb {E}, \mathcal {T}}, as a matrix-valued map,
is defined on\Omega ^{\mathbb {H}, \mathcal {T}}_{D}(d)\triangleq \lbrace u\in \mathbb {R}^V| \mathcal {T} \text{ is Delaunay in } u*d \rbrace ,which is a subs... | {
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"Xiang Zhu",
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b1cf4efeaa1fc0c1cd62765871dca5288f0b18cc | subsection | 16 | 33 | Euclidean combinatorial Calabi flow on triangulated surfaces | The definition of Euclidean combinatorial Calabi flow on a triangulated marked surface (S, V, \mathcal {T})
is given in Definition REF , which could be written in the following
matrix form \frac{du}{dt}=-LK.Note that this is essential an ODE system, therefor the Euclidean combinatorial Calabi flow
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... | 1806.02166 | Combinatorial Calabi flow with surgery on surfaces | [
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7a873f4c46da263b1756b0aaa1fabdad6ccda431 | subsection | 17 | 33 | Euclidean combinatorial Calabi flow on triangulated surfaces | Furthermore, suppose there is a constant combinatorial curvature PL metric d^*=u^**d_0
on (S, V, \mathcal {T}) with \sum _{i=1}^nu^*_{i}=0,
there exists a constant \delta >0
such that if the initial modified combinatorial Calabi energy ||K(u(0))-K(u^*)||^2<\delta ,
then the Euclidean combinatorial Calabi flow (\ref {Eu... | {
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"doi": "10.1017/9781108120241.010",
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"raw": "L.S. Pontryagin, Ordinary differential equations, Addison-Wesley Publishing Company Inc., Reading, 1962.",
"source_ref_id": "1fab2591f... | 1806.02166 | Combinatorial Calabi flow with surgery on surfaces | [
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ddb878738e0ef7600c2fa362eaaad76a25de243b | subsection | 18 | 33 | Euclidean combinatorial Calabi flow on triangulated surfaces | Note that, for j\sim i, the weight of the Euclidean discrete Laplace operator \Delta ^{\mathbb {E}, \mathcal {T}} is\begin{aligned}\omega ^{\mathbb {E}, \mathcal {T}}_{ij}
=-\frac{\partial K_i}{\partial u_j}=\cot \theta _{k}^{ij}+\cot \theta _l^{ij}
=\frac{\sin (\theta _{k}^{ij}+\theta _{l}^{ij})}{\sin \theta _{k}^{ij}... | {
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c35325ab404ec29b4b09d30033a27a75d476f7c5 | subsection | 19 | 33 | Gu-Luo-Sun-Wu's work on discrete uniformization theorem | One of the main tools used in the proof of Theorem REF for convergence of combinatorial Calabi flow
with surgery is
the discrete conformal theory developed by Gu-Luo-Sun-Wu .
In this subsection, we will briefly recall the theory. For details of the theory,
please refer to .Definition 3.2 ( Definition 1.1)
Two piecewis... | {
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... | 1806.02166 | Combinatorial Calabi flow with surgery on surfaces | [
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faf87295542a2bce4529a22305e21a129e719f58 | subsection | 20 | 33 | Gu-Luo-Sun-Wu's work on discrete uniformization theorem | Furthermore, the metric d^{\prime } can be found using a finite dimensional (convex) variational principle.Denote the Teichimüller space of all PL metrics on (S, V) by T_{PL}(S, V) and decorated
Teichimüller space of all equivalence class of decorated hyperbolic metrics on S-V by T_D(S-V).
In the proof of Theorem REF ,... | {
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