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
values | discipline stringclasses 8
values | dense_vector listlengths 1.02k 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
d53dc1ec261f1defffb4c966fbf3216398f88de0 | subsection | 88 | 95 | Implementation for systems with rate | We present the implementation of (4,2,1), (10,5,1) and (20,10,1) systems on Amazon EC2 clusters. We use M4.large EC2 instances from Amazon web services (AWS) for our implementation. We assign the Master's job to an instance located in Virginia and the workers job to instances located in Ohio. We plot in Figures REF , R... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 182,
"openalex_id": "",
"raw": "https://aws.amazon.com/ec2.",
"source_ref_id": "fc5845feaaef934648a171a86a6d14f331b496fe",
"start": 97
}
]
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02311181090772152,
-0.014629852958023548,
-0.058336347341537476,
-0.008329405449330807,
-0.004179957788437605,
-0.005606330465525389,
0.012997533194720745,
0.013706905767321587,
-0.015789257362484932,
0.03609408810734749,
-0.011067735031247139,
0.023401662707328796,
-0.01908440701663494,
... |
3b8d6f2a0e2fd23c67134e3a71fcdf23bd5f747f | subsection | 89 | 95 | Implementation on | We present the trace of a (4,2,1) system implemented at different dates and times on Amazon EC2 clusters. We follow the same setting as before except that A is a 42000 \times 250 matrix generated using the LFW dataset of public facesTo obtain the data matrix A, we convert the first 56 faces to 3 matrices each. Each mat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-319-25958-1_8",
"end": 446,
"openalex_id": "https://openalex.org/W2474608001",
"raw": "E. Learned-Miller, G. B. Huang, A. Roy Chowdhury, H. Li, and G. Hua, “Labeled faces in the wild: A survey,” in Advances in face detection a... | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02195301651954651,
-0.02511095628142357,
-0.024851607158780098,
-0.009061912074685097,
-0.02039693109691143,
0.008886471390724182,
-0.02134278602898121,
0.04781150445342064,
-0.012151200324296951,
0.028619777411222458,
0.00916870217770338,
0.020534232258796692,
0.027872245758771896,
0.0... |
c8e93b6b97df256a526937060e521413878a12bb | subsection | 90 | 95 | Conclusion and open problems | We consider the problem of secure coded computing. We propose the use of a new family of secret sharing codes called Staircase codes that reduces the delays caused by stragglers. We show that Staircase codes always lead to smaller waiting time compared to classical secret sharing codes, e.g., Shamir secret sharing code... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02491517923772335,
-0.032467544078826904,
-0.06298215687274933,
0.015005559660494328,
0.009932505898177624,
0.013670545071363449,
0.004989138804376125,
0.009634988382458687,
-0.022458752617239952,
0.04006568342447281,
-0.01963615231215954,
0.006675071083009243,
-0.05007447302341461,
0.0... |
42e6826f435f8b5e2c1927b5a1af6ec1f9cbc4a8 | subsection | 91 | 95 | Proof of Theorem | For the clarity of presentation, we restate Theorem REF .Theorem 2 (Exact expression of \mathbb {E}{[T_\text{SC}]} for systems with up to 2 stragglers)
The mean waiting time of the Master for (k+1,k,z) and (k+2,k,z) systems is given in (REF ) and (), respectively.\mathbb {E}\left[T_{\text{SC}}(k+1,k,z)\right] &= \fra... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.0561777725815773,
-0.01512008998543024,
-0.05913770943880081,
0.0041195000521838665,
0.008383944630622864,
0.005835958290845156,
-0.007182424422353506,
-0.0069230482913553715,
-0.03664829209446907,
0.0002469792671035975,
0.011259966529905796,
0.03506151959300041,
-0.02509843371808529,
0... |
24f501d0ab554dc30a69b0e03848aff470c947d6 | subsection | 92 | 95 | Proof of Theorem | Applying Theorem REF for the case of n=k+1, we get\bar{F}_{T_{\text{SC}}(k+1,k,z)}(t) &= 1- F_{T^{\prime }}(t_{k+1})^{k+1}-F_{T^{\prime }}(t_k)^k\bar{F}_{T^{\prime }}(t_{k+1})(k+1), \quad \text{for } t>0.Recall that t_k and t_{k+1} are defined as
t_k = \max \left\lbrace t - \frac{c}{k-z}, 0 \right\rbrace and t_{k+1} = ... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02157747931778431,
0.02055506780743599,
-0.04941151291131973,
-0.010781110264360905,
0.023973282426595688,
0.02493465505540371,
-0.0003471624222584069,
-0.00959083903580904,
0.01991415210068226,
0.016968993470072746,
0.003446826944127679,
0.03769192099571228,
-0.0572856143116951,
0.0326... |
284eee979e922282fd8345ad0f390cc771862f38 | subsection | 93 | 95 | Proof of Theorem | Since F_{T^{\prime }}(0) = 0, we can compute the Master's mean waiting time \mathbb {E}\left[T_{\text{SC}}(k+2,k,z)\right] as\mathbb {E}\left[T_{\text{SC}}(k+2,k,z)\right]&= \int _{0}^{\infty }(1 - F_{T^{\prime }}(t_{k+2})^{k+2}) dt - \int _{0}^{\infty }(k+2)\bar{F}_{T^{\prime }}(t_{k+2})F_{T^{\prime }}(t_{k+1})^{k+1}d... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.049836598336696625,
-0.010132119059562683,
-0.03646952658891678,
0.012962703593075275,
0.02410956099629402,
0.00037742717540822923,
0.00037695030914619565,
-0.008003458380699158,
0.0015745603013783693,
0.02169860526919365,
-0.016769878566265106,
0.02760392054915428,
-0.03952137008309364,
... |
2cd60ec4b9597480892c3840af2aa8f4e8f63de3 | subsection | 94 | 95 | Hiding the attribute vectors | Throughout the paper we assumed privacy over one iteration, i.e., the Master needs to hide only A. In the following we describe how our scheme can be generalized to achieve privacy over the whole algorithm, i.e., the Master needs to hide A and the attribute vectors \mathbf {x}^1,\mathbf {x}^2,\dots . Since the algorith... | {
"cite_spans": []
} | 10.1109/TCOMM.2020.2988506 | 1802.02640 | Minimizing Latency for Secure Coded Computing Using Secret Sharing via
Staircase Codes | [
"Rawad Bitar",
"Parimal Parag",
"Salim El Rouayheb"
] | [
"cs.IT",
"math.IT"
] | 2,018 | en | Computer Science | [
-0.02505466900765896,
-0.008560091257095337,
-0.040282782167196274,
0.0007080957293510437,
0.017104923725128174,
-0.00942983292043209,
0.015312034636735916,
-0.024215444922447205,
-0.007850565016269684,
0.012420523911714554,
-0.021484149619936943,
0.0055693998001515865,
0.004650067072361708,... |
07df7e34ab8ad441b86d9568f451111ff5f490a7 | abstract | 0 | 150 | Abstract | We analyse a system of partial differential equations describing the
behaviour of an elastic plate with periodic moduli in the two planar
directions, in the asymptotic regime when the period and the plate thickness
are of the same order of smallness. Assuming that the displacement gradients of
the points of the plate a... | {
"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 | [
-0.027020854875445366,
0.000465350691229105,
-0.03780783712863922,
0.003472965443506837,
-0.026959825307130814,
-0.04864059016108513,
0.02630375698208809,
-0.023008158430457115,
0.019224323332309723,
0.04031004756689072,
-0.01077172439545393,
0.026181697845458984,
-0.013525684364140034,
0.... |
6531e61402af6655625e74b542d0844e7a4a9262 | subsection | 1 | 150 | Introduction | This work is a contribution to the analysis of the asymptotic behaviour of solutions u_\varepsilon to partial differential equations (PDE) with periodic coefficients, of the form{\mathcal {D}}^*\bigl (A(\cdot /\varepsilon ){\mathcal {D}}U^\varepsilon \bigr )+U^\varepsilon =f,\qquad u_\varepsilon \in X, f\in X^*,\qquad ... | {
"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 | [
-0.03589550405740738,
-0.020542241632938385,
-0.05713978409767151,
-0.02762366645038128,
0.022343121469020844,
-0.012949547730386257,
0.06220666691660881,
0.008546548895537853,
0.013193734921514988,
0.005028728395700455,
-0.021442681550979614,
0.016574200242757797,
-0.024128738790750504,
0... |
9b5ff655d79f789f75d48668f906877dd79334a8 | subsection | 2 | 150 | Introduction | In this approach, the control of the resolvent in the sense of the operator norm is obtained by means of a careful analysis of the remainder estimates for the power series, taking advantage of the related Poincaré-type inequalities (or Korn-type inequalities for vector problems) that bound the L^2-norm of the solution ... | {
"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 | [
-0.0661020278930664,
0.032654277980327606,
-0.04715033993124962,
-0.001503966050222516,
-0.0035877744667232037,
-0.010872043669223785,
-0.005119397770613432,
-0.02035551890730858,
0.02194245532155037,
0.0498664416372776,
-0.03973446041345596,
0.0031605223193764687,
0.0334782674908638,
0.01... |
16e285e4f1fffee2fcf2ae69a251c2375333f320 | subsection | 3 | 150 | Introduction | However, their estimates (due to the scaling of the spectrum in one of the two invariant subspaces) imply that the error of the approximation explodes on any compact frequency interval as h\rightarrow 0. In Section we obtain an operator-norm resolvent approximation for the infinite plate,
which allows us to provide mor... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/1-4020-2982-9",
"end": 938,
"openalex_id": "https://openalex.org/W626314958",
"raw": "Panasenko, G., 2005. Multi-scale modelling for structures and composites. Springer, Dordrecht.",
"source_ref_id": "e0ccaaa8eddf5f7c48b9ac0db... | 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 | [
-0.04407062008976936,
-0.00162327301222831,
-0.04144591838121414,
0.038363415747880936,
-0.0003125897201243788,
-0.04120175912976265,
0.016679082065820694,
-0.031801652163267136,
0.039034850895404816,
0.040652401745319366,
-0.04519985616207123,
0.004093470983207226,
0.052219413220882416,
-... |
56d64eed160468f6bc49e9511a61faaeee25aec7 | subsection | 4 | 150 | Problem formulation and main results | Denote Q_{\rm r}:=[0,1)^2\subset {\mathbb {R}}^2 and
consider a function on Q_{\rm r} with values in the space of fourth-order tensors, i.e.A(y)=\bigl \lbrace A_{ijkl}(y)\bigr \rbrace _{i, j,k,l=1}^3,\quad y\in Q_{\rm r}.We assume that A is measurable, symmetric, bounded, and uniformly positive definite: for a.e. y\in ... | {
"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 | [
-0.047736503183841705,
0.05753408372402191,
-0.02737828902900219,
-0.021502790972590446,
0.003431824268773198,
0.001734988414682448,
-0.0019085826352238655,
-0.02342568151652813,
0.025684313848614693,
0.012552197091281414,
-0.011399989947676659,
0.002626804867759347,
-0.000024381881303270347... |
1afb305dd3523615b458d63096ead275edb3e8cc | subsection | 5 | 150 | Problem formulation and main results | For all h>0 we denote \Pi ^h:={\mathbb {R}}^2\times (-h/2, h/2), and for each \varepsilon >0, h>0 suppose that the tensor of elastic moduli at any point
(x_1/\varepsilon , x_2/\varepsilon , x_3)\in \Pi ^h is given by A(x_1/\varepsilon , x_2/\varepsilon ).For given “body-force densities” F^h\in L^2(\Pi ^h, {\mathbb {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 | [
-0.017728354781866074,
-0.02657727524638176,
-0.02073393575847149,
-0.00773517694324255,
-0.021588314324617386,
-0.04137632995843887,
0.03092545084655285,
0.014677002094686031,
0.006457423325628042,
0.016538327559828758,
-0.06859438866376877,
0.015600036829710007,
0.0027042606379836798,
-0... |
7ad6502170f4c19862056b834209dd29589b7000 | subsection | 6 | 150 | Problem formulation and main results | (REF ))\int _{\Pi ^h } A\biggl (\frac{x_1}{\varepsilon }, \frac{x_2}{\varepsilon }\biggr ) \,{\rm sym} \nabla U:{\rm sym}\nabla \Phi ,\qquad U,\Phi \in H^1(\Pi ^h , {\mathbb {R}}^3).Next, denote by \mathcal {L} the following symmetric tensor of order 4:\begin{aligned}\mathcal {L}(M_1,M_2) : (M_1,M_2):= \inf _{\psi \in ... | {
"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 | [
-0.033694297075271606,
0.030306555330753326,
-0.005978143308311701,
-0.018693620339035988,
0.005684386473149061,
-0.00286508328281343,
0.019059862941503525,
0.013192354701459408,
0.037387240678071976,
0.03726515918970108,
-0.06061311066150665,
0.00806496199220419,
0.03830284625291824,
-0.0... |
1b969f0ee7b35a8273f019e304c8576fd2a65db7 | subsection | 7 | 150 | Problem formulation and main results | (REF ) below)The transform (REF ) is a bounded extension of the mapping defined on C_0^\infty (\Pi , {\mathbb {C}}) by the same formula, see Section REF , where the transform defined by (REF ) is extended to L^2(\Pi , {\mathbb {C}}^3).u\in L^2(\Pi , {\mathbb {C}})\ni u\mapsto \frac{\varepsilon ^2}{2\pi }\sum _{n\in \ma... | {
"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 | [
-0.0391228124499321,
0.013999680057168007,
-0.030410204082727432,
-0.011329441331326962,
0.0054320283234119415,
-0.03127994015812874,
0.03512508422136307,
0.0263819582760334,
0.0099180294200778,
0.02104148082435131,
-0.0654895156621933,
0.006740445736795664,
0.004672918003052473,
0.0145108... |
642c1c7626f176285280015ad387363c1966e04e | subsection | 8 | 150 | Problem formulation and main results | Then for each \gamma >0 there exists C=C(A)>0, independent of F^h, such that for all \varepsilon >0 the following estimate holds:
\begin{equation}
\Bigl \Vert \left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}F^h
-\left(\varepsilon ^{-\gamma }\mathcal {A}^{{\rm hom}, 2} +I \right)^{-1}\bigl (S \wid... | {
"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 | [
-0.04049462825059891,
-0.0001734401157591492,
-0.018507907167077065,
0.02548079565167427,
-0.0013817991130053997,
-0.04607904329895973,
0.04827618971467018,
-0.01821800507605076,
0.042234037071466446,
0.00611844239756465,
-0.04595698043704033,
0.023344680666923523,
0.0356120802462101,
-0.0... |
0f08f7e1316e2b8ead25bc274b78960c907e9a06 | subsection | 9 | 150 | Problem formulation and main results | For each \gamma >0 there exists C=C(A)>0, independent of F^h, such that for all \varepsilon >0 the following estimates hold:
\begin{equation}
\begin{aligned}&\Bigl \Vert P_{\alpha }\left(\varepsilon ^{-\gamma } \mathcal {A}^{\varepsilon }+I\right)^{-1}F^h-\left(P_{\alpha }-\varepsilon x_3\partial _{\alpha }P_3\right)\l... | {
"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 | [
-0.021586213260889053,
0.010976169258356094,
-0.030113911256194115,
-0.003842422040179372,
-0.0011355644091963768,
-0.07517798990011215,
0.038626354187726974,
0.00031440166640095413,
0.019267410039901733,
0.0017371942522004247,
-0.03542274609208107,
0.031578417867422104,
0.002568606752902269... |
2768f40fe78a4ef18a0424b952474c2b4cb6e4b8 | subsection | 10 | 150 | Problem formulation and main results | \end{aligned}\section {Auxiliary results}
\end{equation}In what follows we set h=\varepsilon for simplicity.
}We rewrite (\ref {original_identity}) on the scaled domainwith the solution UH1(, R3), as follows:
\begin{align}
\int _\Pi A\biggl (\dfrac{x_1}{\varepsilon }, \dfrac{x_2}{\varepsilon }\biggr ) \,{\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 | [
-0.03707466274499893,
-0.003490053117275238,
-0.008955895900726318,
0.0022713951766490936,
0.017484594136476517,
-0.0009578574681654572,
0.056420620530843735,
-0.024075645953416824,
0.05175195634365082,
0.008650754578411579,
-0.0728677362203598,
0.038447797298431396,
0.0003864328027702868,
... |
e75e08bf909c0156492cc0705795611d12e9f128 | subsection | 11 | 150 | Floquet transform and an equivalent family of problems | We naturally embed L^2(\Pi , {\mathbb {R}}^3) into L^2(\Pi , {\mathbb {C}}^3) and apply a unitary transform to functions in L^2(\Pi , {\mathbb {C}}^3),
so that for each \varepsilon the problem ()
is replaced by an equivalent family of problems on L^2(Q, {\mathbb {C}}^3), parametrised by an auxiliary variable \chi \in Q... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1007/978-3-0348-8573-7",
"end": 1477,
"openalex_id": "https://openalex.org/W2092987417",
"raw": "Kuchment, P., 1993. Floquet Theory for Partial Differential Equations, Birkhäuser.",
"source_ref_id": "23f1a52516db98c10c4143963fd223c... | 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 | [
-0.02310228906571865,
0.02684077061712742,
-0.04837137460708618,
0.00550472317263484,
-0.005729794967919588,
-0.042115140706300735,
0.030472438782453537,
-0.011947881430387497,
-0.0036602786276489496,
0.014442745596170425,
-0.0289007518440485,
0.030899694189429283,
0.029846815392374992,
0.... |
b9d5f8b73b91af0e84879639fcc9b498f5d57b6e | subsection | 12 | 150 | Korn inequalities | For \chi \in Q^{\prime }_{\rm r}, we define the space H^1_\chi (Q, {\mathbb {C}}^3) as the closure in H^1(Q, {\mathbb {C}}^3) of the set of smooth functions u on \overline{\Pi } that are \chi -quasiperiodic with respect to y_1, y_2, i.e.u(y+e_\alpha , x_3)=u(y, x_3)\exp (i\chi _\alpha ),\ \ \ \ \ e_\alpha :=(\delta _{\... | {
"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 | [
-0.0380532369017601,
0.001252102549187839,
-0.057674914598464966,
-0.04394279420375824,
-0.005813265219330788,
-0.01988106220960617,
-0.004218813497573137,
0.04131843149662018,
0.01456367876380682,
0.017897533252835274,
-0.009391246363520622,
0.02624361217021942,
-0.0017088863532990217,
0.... |
981778cc22dd0ea8584ec5e127bab7215a58dde3 | subsection | 13 | 150 | Korn inequalities | We prove the following Korn-type inequalities, which will inform us about the structure of solutions to (REF ) in Section .Lemma 2.2 There exists a constant C>0 such that for all \chi \in Q^{\prime }_{\rm r} and u=(u_1, u_2, u_3)^\top \in H^1_\chi (Q, {\mathbb {C}}^3),
there are a_1, a_2, c_1, c_2, c_3 \in \mathbb {C} ... | {
"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 | [
-0.029272152110934258,
0.020237160846590996,
-0.02214488573372364,
-0.03571263700723648,
0.0028062653727829456,
0.006654149387031794,
0.021595461294054985,
0.03891761600971222,
-0.026982881128787994,
-0.0039642550982534885,
0.0023121642880141735,
0.04123741015791893,
-0.008729754947125912,
... |
c3a20d7f652e39002eb3100d494867f55ec3fe15 | subsection | 14 | 150 | Korn inequalities | The standard “second” Korn inequality, see e.g. , gives\Vert u_1- a_1x_3-dy_2-c_1 \Vert _{H^1(Q, {\mathbb {C}})} &\le & C\bigl \Vert \textrm {sym}\nabla u\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})}, \\[0.5em]
\Vert u_2- a_2x_3+dy_1-c_2 \Vert _{H^1(Q, {\mathbb {C}})} &\le & C\bigl \Vert \textrm {sym}\nabla u\bigr \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "10.1016/s0168-2024(08)x7009-2",
"end": 705,
"openalex_id": "https://openalex.org/W650857255",
"raw": "Oleinik, O. A., Shamaev, A. S., Yosifian, G. A., 1992. Mathematical Problems in Elasticity and Homogenization, Amstredam: North-Holland.",... | 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 | [
-0.03211427479982376,
0.032572176307439804,
-0.03626592829823494,
-0.018712975084781647,
-0.00802856869995594,
0.011218626983463764,
-0.016637148335576057,
0.02718418277800083,
-0.027077339589595795,
0.010707302019000053,
0.020300373435020447,
-0.02225409261882305,
0.0008013304905034602,
0... |
3d1967c7d9fb75d6fb983fc3826f81254073e947 | subsection | 15 | 150 | Korn inequalities | Noting thatd= \int _{Q_{\rm r}}\left\lbrace \partial _2 \int _I u_1d\,x_3-\partial _1 \int _I u_2\,dx_3 \right\rbrace dy_1dy_2,and applying (REF ) to the vector\overline{u}:=\left(\int _I u_1\,dx_3,\int _I u_2\,dx_3 \right)^\top : Q_{\rm r} \rightarrow \mathbb {R}^2,we infer that|d| \le C\bigl \Vert {\rm sym} \nabla \o... | {
"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 | [
-0.0162673182785511,
0.0073096114210784435,
-0.028490697965025902,
-0.00214405101723969,
-0.011620603501796722,
0.03183266893029213,
0.039859507232904434,
0.04709281772375107,
-0.021348947659134865,
0.019502470269799232,
-0.01225389912724495,
0.01509991753846407,
-0.007748340722173452,
0.0... |
0d84beb8d56c5eb56f73385bca1d0054659a5fe7 | subsection | 16 | 150 | Korn inequalities | Notice first that from (REF )–(), using the trace inequality and the fact that u is \chi -quasiperodic, one has&\int \limits _{\lbrace (y_1, y_2, x_3)\in Q: y_\beta =1\rbrace }\Bigl |\bigl (\exp {(i\chi _\beta )} -1\bigr ) (a_\alpha x_3+c_\alpha )\Bigr |^2dy_2dy_3\le C\bigl \Vert {\rm sym} \nabla u\bigr \Vert ^2_{L^2(Q... | {
"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 | [
-0.030980553478002548,
0.027729883790016174,
-0.05655858665704727,
0.00165681098587811,
0.008004581555724144,
-0.00892789289355278,
-0.002067913766950369,
0.043189637362957,
0.006562382914125919,
0.01533003244549036,
-0.01930561475455761,
0.022937817499041557,
0.010431136935949326,
0.01973... |
b7ebb7d457ab394f84b40e36b8aec8c12ff32557 | subsection | 17 | 150 | Structure of the leading-order field | Taking into account
(), we infer that ()
are equivalent to the estimates|i\chi _\alpha c_3+a_\alpha | \le C\bigl \Vert {\rm sym} \nabla u\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3})},
\qquad \alpha =1,2,In particular, from (REF )–(), (REF )
we obtain\Vert u_\alpha -(c_\alpha -i \chi _\alpha c_3 x_3)e_\chi \Vert _{H^1(Q, {\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 | [
-0.0022884563077241182,
0.026881733909249306,
-0.04808809608221054,
-0.022548923268914223,
-0.0007847498054616153,
0.0007404109928756952,
0.019848544150590897,
0.01900944486260414,
0.011831318959593773,
-0.005709698423743248,
-0.022915076464414597,
0.02434917539358139,
-0.0014493557391688228... |
f538e92e222f4425f70f05122e1f79505bc5a4d6 | subsection | 18 | 150 | Structure of the leading-order field | (REF )){{\mathfrak {b}}_\chi }(u,\varphi ):=\int _QA\,\,{\rm sym}\nabla u:\overline{{\rm sym}\nabla \varphi },\qquad u, \varphi \in H^1_{\chi } (Q,\mathbb {C}^3).It follows from the estimates (REF )–(REF ), by examining appropriate Rayleigh quotients, that the smallest eigenvalue of
\mathcal {A}_{\chi } is of order |\c... | {
"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 | [
-0.002760542556643486,
0.03284415975213051,
-0.007989739067852497,
-0.028891263529658318,
-0.00031025102362036705,
0.022542204707860947,
0.032233674079179764,
0.02272535115480423,
0.011393199674785137,
0.011171898804605007,
-0.008920731022953987,
0.004670219961553812,
0.024129468947649002,
... |
9793bb5c77b83b0c23c4cd28db332d09f007e88d | subsection | 19 | 150 | Structure of the leading-order field | Indeed, as a consequence of (), (), (REF )–(REF ) as well as coercivity and boundedness of the tensor A, the following proposition holds.Proposition 2.3 There exists C>0
such that for all \chi \in Q^{\prime }_{\rm r} the following bounds hold:& {\mathfrak {b}}_{\chi } (u,u)\ge C\nu |\chi |^4 \Vert u\Vert ^2_{L^2(Q,\mat... | {
"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 | [
-0.002440001582726836,
0.04047464579343796,
-0.032385822385549545,
-0.03894845396280289,
-0.0014994848752394319,
-0.008630623109638691,
-0.01416307408362627,
0.0009138082386925817,
0.012552940286695957,
0.029073981568217278,
-0.02211454138159752,
0.015132206492125988,
0.01916898787021637,
... |
4dc90ad4ca06385ec0bbbe1a44333bb8adfc97f2 | subsection | 20 | 150 | Structure of the leading-order field | Next, using () and the fact that every two-dimensional subspace
of H^1_{\chi } (Q, \mathbb {C}^3) contains a vector orthogonal to (-i\chi _1 x_3,-i\chi _2 x_3,1)^{\top }e_{\chi }, we infer that \lambda _\chi ^{(2)} \ge C\nu |\chi |^2. Furthermore, combining () and (REF ), where k=3 and {\mathfrak {U}} is taken to be th... | {
"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 | [
-0.04163037985563278,
-0.013780144043266773,
-0.010407594963908195,
-0.00046711141476407647,
0.002737334929406643,
0.007298287935554981,
0.006092716008424759,
-0.02818596549332142,
0.01407772209495306,
0.04535391926765442,
-0.0159929022192955,
0.025500133633613586,
0.031680598855018616,
0.... |
7f6f1cae354156c2e9bf1cd578092ea0af76bf77 | subsection | 21 | 150 | A priori estimates for solutions of ( | We consider separately the case when the elasticity tensor A is planar-symmetric and the general, not necessarily symmetric, case: in the former, one is able to separate the study of (REF ) into two different problems, due to the fact that one can identify two invariant subspaces for the operator {\mathcal {A}}_\chi an... | {
"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 | [
-0.020843615755438805,
-0.0009813359938561916,
-0.03598041459918022,
0.006137880031019449,
0.0002040873805526644,
-0.02758803591132164,
-0.0008263631025329232,
0.012062136083841324,
0.034301936626434326,
0.02278149127960205,
-0.025772230699658394,
0.039520472288131714,
0.011665405705571175,
... |
a8bd9cb9c11c1c7675f5837522b2b946e1232e8a | subsection | 22 | 150 | The case of planar-symmetric elasticity tensor | In this section we work under the assumption (REF ) on the elasticity tensor A in (REF ). We derive estimates that provide an informed guess about the asymptotic behaviour of solutions to (REF )
as \varepsilon \rightarrow 0 for each of two
subspaces L^2(Q, {\mathbb {C}}^3) invariant with respect to the operator {\mathc... | {
"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 | [
-0.0225108340382576,
0.014963981695473194,
-0.011797203682363033,
-0.0038535497151315212,
-0.004784506279975176,
-0.029058054089546204,
0.04071790352463722,
0.012560282833874226,
0.0171998031437397,
-0.004368627909570932,
-0.0172150656580925,
0.036444660276174545,
0.00069917127257213,
0.00... |
d71ab37b76cd3d643ed5a18c7f95b7e16abb64bd | subsection | 23 | 150 | First invariant subspace | In the first subspace we scale with |\chi |^{-4} the operator {\mathcal {A}}_\chi (equivalently, the form {{\mathfrak {b}}_\chi } with \chi -dependent domain)
and with |\chi |^{-1} the horizontal components of the force density on its right-hand side, so the equation (REF ) is replaced with\frac{1}{|\chi |^4}{{\mathfra... | {
"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 | [
-0.004237077664583921,
0.005724440328776836,
-0.03761884197592735,
-0.02967098355293274,
0.012196375988423824,
0.017817845568060875,
0.06172174960374832,
0.040273211896419525,
0.04823632538318634,
-0.009061472490429878,
-0.03673405200242996,
-0.014332076534628868,
0.0019316649995744228,
-0... |
16ccca074237b9f882927f772364dffa611bda1b | subsection | 24 | 150 | First invariant subspace | \end{aligned}
\end{equation}
Setting \varphi =u in (\ref {jdba10_new}) and applying (\ref {revision1})
to the right-hand side of the resulting equation, we obtain
\begin{equation}
\bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} \le C|\chi |^2
\bigl \Vert ({\!\!_{1},{\!\!_{2}, \wideha... | {
"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 | [
0.01881267875432968,
0.037381235510110855,
-0.05538525804877281,
-0.0389375165104866,
-0.01280116569250822,
0.010787156410515308,
0.03683196008205414,
0.04140925407409668,
0.010924475267529488,
-0.006248006597161293,
-0.01748526282608509,
0.021818434819579124,
0.00891809444874525,
-0.00242... |
cc7c2eb81fefd355ff385778b74c82a9e0fdd620 | subsection | 25 | 150 | First invariant subspace | \end{equation*}
In combination with (\ref {janprva})--(\ref {jantreca}), (\ref {sym_nabla_est}) this implies the existence of c_3 \in \mathbb {C} such that
\begin{align*}
\Vert u_\alpha +i \chi _\alpha c_3 x_3e_\chi \Vert _{H^1(Q, {\mathbb {C}})}&\le C|\chi |^2 \bigl \Vert ({\!\!_{1},{\!\!_{2}, \widehat{f}_3)^\top \big... | {
"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 | [
-0.0020927335135638714,
0.01471208781003952,
-0.06025240942835808,
-0.011217203922569752,
0.02096930332481861,
-0.017749127000570297,
0.008027549833059311,
0.03284275159239769,
0.03909996896982193,
-0.019702598452568054,
-0.013323291204869747,
-0.00281002395786345,
-0.006314446218311787,
0... |
422b3a5bbf83f1162c7ac3e6735cd1b7812bf35b | subsection | 26 | 150 | First invariant subspace | Setting \varphi =u in (\ref {jdba10_new}),
we obtain
\begin{equation}
\Vert u\Vert _{L^2(Q, {\mathbb {C}}^3)}\le C\bigl \Vert (\widehat{f}_{1}, \widehat{f}_2, {\!\!_{3})^\top \bigr \Vert _{L^2(Q, {\mathbb {C}}^3)},\qquad \bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} \le C|\chi |
\bi... | {
"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 | [
-0.00036555511178448796,
0.027806032449007034,
-0.053994372487068176,
-0.01727575808763504,
-0.006978215649724007,
-0.03143821656703949,
0.01671108976006508,
0.028279133141040802,
0.03244546055793762,
-0.000034934000723296776,
-0.00603964738547802,
-0.0006877109408378601,
0.00951540190726518... |
53bdc135005fe16e26e865e9056f52be1ffd4381 | subsection | 27 | 150 | First invariant subspace | We demonstrate the latter in Section \ref {asymptotic_proc}, see the first estimate in (\ref {korrre1oo}).
\end{equation}\end{equation}Notice also that
setting \widehat{f}_{\alpha }=0, \alpha =1,2, in
(\ref {jdba10_newsecond}), we obtain
\begin{equation}
\bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {... | {
"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 | [
-0.0008384166867472231,
0.018893467262387276,
-0.024250177666544914,
-0.020114369690418243,
0.02740926295518875,
-0.04230427369475365,
0.010057184845209122,
0.011270456947386265,
0.030171554535627365,
0.005326187703758478,
-0.027165083214640617,
0.003906888421624899,
0.01311707217246294,
0... |
ebbc1e391e6036fba8ab865f493800db806b9072 | subsection | 28 | 150 | First invariant subspace | \end{equation}
In the same way as in Section
\ref {first_subspace}, we obtain the estimates ({\it cf.} (\ref {sym_nabla_est}), (\ref {revision11}))
\begin{align*}
\bigl \Vert \textrm {sym}\nabla u\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} &\le C|\chi |^2\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)}, \\[0.6em]
\bigl \Ve... | {
"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 | [
-0.01137434970587492,
0.04986099898815155,
-0.03039262816309929,
-0.011191261000931263,
-0.016035467386245728,
0.023694653064012527,
0.03527497872710228,
0.015974437817931175,
0.019865060225129128,
0.010031702928245068,
-0.0009540607570670545,
0.029568731784820557,
0.015196314081549644,
0.... |
4381fa81dc23bf859b9d6f870286a622befdb452 | subsection | 29 | 150 | First invariant subspace | Namely, first setting \varphi =u in (\ref {scalingdva}) we obtain ({\it cf.} (\ref {init_est}))
\begin{equation}
\Vert u\Vert _{L^2(Q, {\mathbb {C}}^3)} \le C\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)},\qquad \bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} \le C|\chi |
\bigl \Vert f\Vert ... | {
"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 | [
-0.0186309777200222,
0.0360717698931694,
-0.047759998589754105,
-0.0165862999856472,
-0.014404293149709702,
-0.006565092597156763,
0.013854976743459702,
0.0325012132525444,
0.0330505296587944,
-0.024322504177689552,
0.00005692242120858282,
-0.003614424727857113,
0.0092239361256361,
-0.0166... |
84f18d68b9e9db6e61019b3083fb21f1e89b3917 | subsection | 30 | 150 | First invariant subspace | Therefore ({\it cf.} (\ref {revision1second2}))
\Vert u_3\Vert _{H^1(Q, {\mathbb {C}})} \le C\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)}
and ({\it cf.} (\ref {approx21}))
\bigl \Vert u_\alpha -c_{\alpha }e_\chi \bigr \Vert _{H^1(Q, {\mathbb {C}})}\le C|\chi |\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)},\qquad \alpha =1,2.
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 | [
0.01708975061774254,
0.007873492315411568,
-0.05404633283615112,
-0.007110556587576866,
-0.014961160719394684,
-0.01840199902653694,
0.0070495218969881535,
-0.028945764526724815,
0.0006136860465630889,
0.037200722843408585,
-0.023879874497652054,
-0.010536136105656624,
0.009063671343028545,
... |
140e749a1cec10d770291e31332f9f90840d80ab | subsection | 31 | 150 | First invariant subspace | In addition, we identify the correctors required for L^2\rightarrow H^1 estimates as well as for higher-precision L^2\rightarrow L^2 estimates, see Section \ref {normresolvent} for details. By adopting particular scalings for the operator of (\ref {original_identity}) and the force density, we thus also recover a versi... | {
"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 | [
-0.02335863560438156,
0.012503202073276043,
-0.04052288457751274,
-0.01034432090818882,
-0.007720097899436951,
-0.031978901475667953,
0.04989074915647507,
-0.008757581003010273,
0.017682990059256554,
-0.008437181822955608,
-0.040156714618206024,
0.015897909179329872,
0.02221435308456421,
0... |
eae87d9f519511887eb1b0717713d7b9a26760f1 | subsection | 32 | 150 | First invariant subspace | \end{equation}
We also define, for m_1,m_2,m_3 \in \mathbb {C},
\begin{align*}
\Xi (\chi , m_1,m_2)&:=i{\mathfrak {I}}\left(\begin{array}{ccc} \chi _1 m_1 & \dfrac{1}{2}(\chi _1 m_2+ \chi _2 m_1)\\[0.8em]
\dfrac{1}{2}(\chi _1 m_2 +\chi _2 m_1) & \chi _2 m_2\end{array} \right),\quad \Upsilon (\chi , m_3):=im_3{\mathfrak... | {
"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 | [
0.023068351671099663,
-0.011442635208368301,
-0.04094937816262245,
-0.015439928509294987,
-0.0022408494260162115,
-0.03933215141296387,
0.024197358638048172,
0.003360320581123233,
-0.001016487367451191,
-0.0005130114732310176,
-0.04134605452418327,
0.007391942199319601,
0.011831684969365597,... |
dfe5c720f063618046bd3fc1024945288a05f036 | subsection | 33 | 150 | First invariant subspace | \end{align*}
Notice that
\begin{equation}
\begin{aligned}\max \bigl \lbrace |m_1|, |m_2|\bigr \rbrace +\max \bigl \lbrace |\chi _1|,|\chi _2|\bigr \rbrace |m_3|&
\le C\vert \chi \vert ^{-1}
\Bigl (\bigl |\Xi (\chi ,m_1,m_2)\bigr |+\bigl |\Upsilon (\chi , m_3)\bigr |\Bigr ),\\[0.5em]
|m_3| &\le C\vert \chi \vert ^{-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 | [
-0.012822909280657768,
-0.0058164591901004314,
-0.047752175480127335,
-0.01042004395276308,
-0.014195974916219711,
0.005366398952901363,
0.011899903416633606,
0.030970262363553047,
0.014043412171304226,
0.000778547371737659,
-0.017544729635119438,
0.026683244854211807,
-0.010732797905802727,... |
7d64de59f3317db68c029fd62a8981b921caf948 | subsection | 34 | 150 | First invariant subspace | Therefore,
\begin{equation*}
A^{\textrm {hom}}_{\chi } m\cdot \overline{d}
=A^{\textrm {hom},1}_{\chi }m_3\,\overline{d_3}+A^{\textrm {hom},2}_{\chi }(m_1,m_2)^\top \cdot \overline{(d_1,d_2)^\top },
\end{equation*}
where, for \chi \in Q^{\prime }_{\rm r}, m_3, d_3\in {\mathbb {C}}, (m_1, m_2)^\top , (d_1, d_2)^\top \in... | {
"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 | [
0.05772370845079422,
-0.022390449419617653,
-0.026206135749816895,
-0.02812924236059189,
-0.012782550416886806,
-0.01888001710176468,
-0.004250674974173307,
0.02203940600156784,
-0.012110989540815353,
-0.019551578909158707,
-0.04175887256860733,
0.011691263876855373,
0.015545107424259186,
... |
d4fbb50ee5bf1b2f50b7bec724ad04eefced8087 | subsection | 35 | 150 | First invariant subspace | \end{align*}
}\begin{}
Notice that the following coercivity estimate holds:
\begin{equation*}
\bigl \Vert \Theta \varphi \bigr \Vert _{L^2(Q, {\mathbb {C}}^3)} \ge \frac{1}{4}|\theta | \Vert \varphi \Vert _{L^2(Q, {\mathbb {C}}^3)},
\end{equation*}
which is obtained by combining
\begin{eqnarray*}
\bigl \Vert \Theta \va... | {
"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 | [
-0.01670956052839756,
0.04916731268167496,
-0.03311392664909363,
-0.030138250440359116,
0.019349519163370132,
0.008141142316162586,
0.0073972237296402454,
0.0059284609742462635,
0.023958001285791397,
0.03445679321885109,
-0.0340295173227787,
0.007820685394108295,
0.015450621955096722,
-0.0... |
f9946b49432da7dcfb18ca2d0259eed9f50e0c31 | subsection | 36 | 150 | First invariant subspace | After that we have the equation
\begin{align*}
\frac{1}{\varepsilon ^4|\theta |^4}\int _Q A \ {\rm sym} \nabla ( u_1, u_2, u_3):\overline{ {\rm sym} \nabla (\varphi _1,\varphi _2, \varphi _3)}+ \int _Q u_\alpha \overline{\varphi _\alpha }+\int _Q u_3\overline{\varphi _3}\\[0.4em]
=\frac{1}{\varepsilon |\theta |}\int _Q... | {
"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 | [
0.049899425357580185,
0.04407019540667534,
-0.03071790374815464,
-0.023011723533272743,
0.010689464397728443,
-0.04410071671009064,
0.05426371842622757,
0.014206839725375175,
0.04632864147424698,
-0.0014945031143724918,
-0.025132829323410988,
0.00038602433050982654,
-0.001207428751513362,
... |
56d9cbf49c29242b1b3e558a2e8d9e5697596ab8 | subsection | 37 | 150 | First invariant subspace | For clarity of the argument, we keep the original form of the second term on the left-hand side of (\ref {sinisa1000revision}).
}
\begin{equation}
\begin{aligned}\vert \chi \vert ^{-4}A^{\textrm {hom},1}_{\chi }m_3\,\overline{d_3} &+\int _Q (-i\chi _1 x_3 m_3, -i\chi _2 x_3m_3,m_3)^\top \cdot \overline{(-i\chi _1 x_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 | [
0.006733328569680452,
0.013115685433149338,
-0.016953492537140846,
-0.020264839753508568,
-0.0012961181346327066,
-0.01113192830234766,
0.04321538284420967,
0.009460994973778725,
0.002142075914889574,
0.011254005134105682,
-0.02266060747206211,
0.02137879654765129,
0.009972193278372288,
0.... |
c2fb31fbdeffb51cd826b64fc3229c283d428e74 | subsection | 38 | 150 | First invariant subspace | \end{equation}
In addition, the following symmetry properties\footnote {Similar symmetry properties hold for all terms in the asymptotics series, see {\it e.g.} (\ref {sinisa1005}).} hold:
\begin{equation}
({\mathfrak {u}}_{2})_\alpha (\cdot ,-x_3)=-({\mathfrak {u}}_{2})_{\alpha }(\cdot ,x_3), \quad \alpha =1,2,
\qqua... | {
"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 | [
-0.027485521510243416,
-0.002575336955487728,
-0.04077807441353798,
-0.02740921638906002,
0.012651581317186356,
-0.037084851413965225,
0.06183860823512077,
0.01991593837738037,
0.02084687538444996,
0.018069326877593994,
-0.013162833638489246,
0.03072090819478035,
0.004860709886997938,
0.00... |
4371ca6a47f28d017295d0ebe1930f824888ed01 | subsection | 39 | 150 | First invariant subspace | To verify that it also vanishes when tested with vectors (0,0,D_3)^\top , we use the fact that
\begin{equation}
\begin{aligned}\int _Q A\bigl (-ix_3 \Upsilon (\chi ,m_3)+\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_2\bigr ):\overline{iX(0,0,D_3)^\top }&=
\int _Q A\bigl ( -ix_3 \Upsilon (\chi ,m_3)+\mathop {\mathrm {s... | {
"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 | [
-0.0059352139942348,
0.023130550980567932,
-0.01957552507519722,
-0.01701224595308304,
-0.010779508389532566,
-0.02494620718061924,
0.05971832200884819,
0.027005987241864204,
0.00882653333246708,
0.0005473670898936689,
-0.015593287535011768,
0.008277258835732937,
0.019712844863533974,
-0.0... |
8db04286a1b6c23195f36805e1af333e73e6ba1b | subsection | 40 | 150 | First invariant subspace | \end{equation}
\end{aligned}Next, we seek {\mathfrak {u}}_4^{(1)} \in H^1_{\#}(Q,\mathbb {C}^3) such that
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_4^{(1)} &= i \bigl \lbrace X^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_3^{(1)}-(\mathop {... | {
"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 | [
0.014745084568858147,
0.03215999901294708,
-0.03484508395195007,
-0.03933039680123329,
0.009352030232548714,
-0.007212352938950062,
0.044761594384908676,
0.01071745716035366,
-0.00211297906935215,
0.02128235250711441,
-0.0018841365817934275,
-0.0204432625323534,
0.03356356546282768,
-0.036... |
776faa0bf77a30fd2d304e7faae0f7e83aa2e885 | subsection | 41 | 150 | First invariant subspace | To ensure it also vanishes when tested with vectors (0,0,D_3)^\top , we observe that ({\it cf.} (\ref {psi_id}))
\begin{equation*}
\begin{aligned}& \int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_3^{(1)}+ iX{\mathfrak {u}}_2 \bigr ): \overline{iX(0,0,D_3)^\top }
=\int _Q A \bigl (\mathop {\mathrm {sym}}\... | {
"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 | [
-0.0005110370693728328,
0.011006365530192852,
-0.03621498495340347,
-0.02449926920235157,
-0.005198085680603981,
-0.000968682172242552,
0.044940751045942307,
0.02610102668404579,
0.01066313125193119,
0.04051684960722923,
-0.009229176677763462,
0.0035772593691945076,
-0.00026052401517517865,
... |
93279a13335ad13b4ba3cfac28f86f4714eed75f | subsection | 42 | 150 | First invariant subspace | To this end, we define a ``correction^{\prime \prime } m_3^{(1)} to the value m_3 as the solution to
\begin{equation}
\begin{aligned}
A^{\textrm {hom},1}_{\chi }m_3^{(1)}\,\overline{d_3}
&+|\chi |^4\int _Q\bigl (-i\chi _1 x_3 m_3^{(1)}, -i\chi _2 x_3m_3^{(1)},m_3^{(1)}\bigr )^\top \cdot \overline{(-i\chi _1 x_3 d_3, -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 | [
0.011068008840084076,
-0.014401378110051155,
-0.026590680703520775,
-0.03774259611964226,
-0.0037967616226524115,
-0.02378363162279129,
0.04958101734519005,
0.01386742852628231,
0.021052861586213112,
0.00968736782670021,
-0.04741470888257027,
-0.012753763236105442,
0.012318975292146206,
0.... |
f354b1f23414592d872ba1bb4b3b7c7a22b02b36 | subsection | 43 | 150 | First invariant subspace | \end{equation}
Second, consider {\mathfrak {u}}_4^{(2)} \in H^1_{\#} (Q, \mathbb {C}^3) satisfying ({\it cf.} (\ref {sinisa1004}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_4^{(2)}&= i \bigl \lbrace X^{*} \cdot A \mathop {\mathrm {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 | [
0.009341063909232616,
0.04988006129860878,
-0.026008453220129013,
-0.01804107427597046,
-0.0021387527231127024,
-0.02967161498963833,
0.04374426603317261,
-0.009142642840743065,
0.030434774234890938,
0.030770564451813698,
0.0061663235537707806,
-0.005780928302556276,
0.03379267454147339,
0... |
51f23eb410b873a30130ccd5cc548499904ad40f | subsection | 44 | 150 | First invariant subspace | \end{equation}
\end{equation*}Finally, we define {\mathfrak {u}}_5^{(1)} \in H^1_{\#} (Q, \mathbb {C}^3) as the solution to ({\it cf.} (\ref {sinisa1010}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_5^{(1)} &= i \Bigl (X^{*}A \mathop {\mathrm {sy... | {
"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 | [
-0.003299608128145337,
0.01224478892982006,
-0.036925096064805984,
-0.012969558127224445,
-0.000723815755918622,
-0.020781809464097023,
0.02960110828280449,
0.04171620309352875,
0.04162465035915375,
0.03277483582496643,
-0.015563469380140305,
0.007121813017874956,
0.025557657703757286,
0.0... |
20bcab4d9fd5fff9fe225bae5f35003c327f0054 | subsection | 45 | 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{align*}
\int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_4^{(2)}+iX{\mathfrak {u}}_3^{(2)}\bigr )&: \overline{i X(0,0,D_3)^\top }
=\int _Q A \bigl (\mathop {\mathrm {... | {
"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 | [
-0.005458633415400982,
0.01859978772699833,
-0.007873249240219593,
-0.011069849133491516,
-0.01670776680111885,
-0.01192430965602398,
0.032957784831523895,
0.022856835275888443,
-0.008689564652740955,
0.0331103689968586,
-0.026823975145816803,
0.02201763167977333,
0.013282292522490025,
0.0... |
49d46ef3310af0a09172e8cbbac6e99755477530 | subsection | 46 | 150 | First invariant subspace | Defining m_3^{(2)} so that
\begin{align}
A^{\textrm {hom},1}_{\chi }m_3^{(2)}\,\overline{d_3}
&+|\chi |^4\int _Q\bigl (-i\chi _1 x_3 m_3^{(2)}, -i\chi _2 x_3m_3^{(2)},m_3^{(2)}\bigr )^\top \cdot \overline{(-i\chi _1 x_3 d_3, -i\chi _2 x_3d_3,d_3)^\top }
\\[0.3em] &
=-\int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\ma... | {
"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 | [
0.020778939127922058,
0.00502691650763154,
-0.03246518597006798,
-0.019909335300326347,
0.018475253134965897,
-0.009725824929773808,
0.030848028138279915,
0.014218774624168873,
-0.00487816845998168,
0.010961576364934444,
-0.03920842334628105,
-0.0077234492637217045,
0.005412135273218155,
0... |
71208b259201f06bc4cd95b11e9ee1e52f0cec13 | subsection | 47 | 150 | First invariant subspace | \end{equation}
Second, consider {\mathfrak {u}}_5^{(2)} \in H^1_{\#} (Q, \mathbb {C}^3) that satisfies ({\it cf.} (\ref {sinisa1104oo}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_5^{(2)}&= i \bigl \lbrace X^{*}A \mathop {\mathrm {sym}}\nabla {\ma... | {
"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 | [
0.004358930047601461,
0.0270307045429945,
-0.04353591054677963,
-0.007878870703279972,
-0.005750889424234629,
-0.03085954673588276,
0.0287086833268404,
0.025932392105460167,
0.013202638365328312,
0.02944089286029339,
-0.00873311422765255,
-0.029593436047434807,
0.02105100080370903,
0.01225... |
e9c6406d5dff380e589d42febf04bc0a098f002e | subsection | 48 | 150 | First invariant subspace | \end{equation}
Finally, we define {\mathfrak {u}}_6 \in H^1_{\#} (Q, \mathbb {C}^3) as the solution to ({\it cf.} (\ref {sinisa1022oo}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}&A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_6= i \Bigl (X^{*}A \mathop {\mathrm {sym}}\nabla \bigl ({\mathfra... | {
"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 | [
0.006128324195742607,
0.012737151235342026,
-0.01159309595823288,
-0.028174273669719696,
-0.01691676862537861,
-0.017969299107789993,
0.02523023821413517,
0.02631327696144581,
0.04060634598135948,
0.0067423004657030106,
-0.009388882666826248,
0.0033635233994573355,
-0.0008208598592318594,
... |
083bf47f4a5c3e6bad1a1155d3cce174fdef8070 | subsection | 49 | 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.... |
80f3567d8975a24fc9885e3eae6f219acf396891 | subsection | 50 | 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,
0.02061406709253788,
0.022170420736074448,
0.012931473553180695,
-0.014747221022844315,
-0.008979554288089275,
-0.01029177475720644,
-0.0002794190077111125,
0.0006208251579664648,
... |
46b20cd57ad90151430940040a5d9ef8c21bac44 | subsection | 51 | 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... |
de874e990b40c169b0bafb9ba4bbdb693996b5b8 | subsection | 52 | 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... |
6fdca292055aa782136d98cee31c696f18a687f2 | subsection | 53 | 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... |
c6f91d6d9ff4ab8222a3a3483bf03b711180cabc | subsection | 54 | 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,
... |
966d7199e9b4531f300800752b86d1a67da1b44b | subsection | 55 | 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... |
c79f3e1ab072006a46a5a276c6063078b9e9b2c2 | subsection | 56 | 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... |
55e8560b1362c4305d3e9ae30e24e6304dff53ab | subsection | 57 | 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,
... |
8fb4009a13a3c07a62d7a4c8e56d1f62d77faa2c | subsection | 58 | 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... |
d607e193856190b52671933b147de8eeefc7eb22 | subsection | 59 | 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... |
954c8b2142794a2646b37c1f03377447508cf02f | subsection | 60 | 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,
... |
1ec02920c3c98e4256e5921bae43e8c8a0cb9fb1 | subsection | 61 | 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... |
fda849aa3e73465b4fec48cce0279061bf699afb | subsection | 62 | 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,
... |
3064a58320a37b0a2ff0c81ca1061b53f293a4fe | subsection | 63 | 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... |
8e68afe64012057f8f3b56e7e1e6d94365d36e5e | subsection | 64 | 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,
... |
c8725bccfc324752c6125cc838f4a9f56a2e3e96 | subsection | 65 | 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.... |
e540224fd2593d325dd103cd658cb8b663731228 | subsection | 66 | 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 | [
0.014386679977178574,
0.0010679401457309723,
-0.04036813974380493,
-0.04772166907787323,
0.011999070644378662,
0.005290118046104908,
0.025600051507353783,
0.024577880278229713,
0.04186325520277023,
-0.006392384879291058,
-0.04463990032672882,
0.005751620512455702,
-0.009519923478364944,
0.... |
66adb9909189c45189864fc8228837c733a4e808 | subsection | 67 | 150 | First invariant subspace | Here
we assume that the components f_\alpha ={\!\!_{\alpha }, \alpha =1,2, are odd in the x_3 variable, while f_3=\widehat{f}_3 is even. The estimates (\ref {cetvrta})--(\ref {peta}), (\ref {janprva})--(\ref {jantreca}) imply that
\begin{equation}
\begin{aligned}\bigl \Vert (u_1, u_2)^\top \bigr \Vert _{H^1(Q, {\mathbb... | {
"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 | [
0.0029958407394587994,
0.03113691136240959,
-0.04219730570912361,
-0.019878195598721504,
0.020335866138339043,
-0.0071129766292870045,
0.05662921071052551,
0.03902411460876465,
0.0012995961587876081,
-0.00907714944332838,
-0.013440283946692944,
-0.0018592900596559048,
-0.023829424753785133,
... |
2874a951e3d9b87a3d91803cd1f07969ea3a86f0 | subsection | 68 | 150 | First invariant subspace | \end{aligned}
\end{equation}
Setting \varphi =u in (\ref {jdba10_new}) and applying (\ref {revision1})
to the right-hand side of the resulting equation, we obtain
\begin{equation}
\bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} \le C|\chi |^2
\bigl \Vert ({\!\!_{1},{\!\!_{2}, \wideha... | {
"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 | [
0.01881267875432968,
0.037381235510110855,
-0.05538525804877281,
-0.0389375165104866,
-0.01280116569250822,
0.010787156410515308,
0.03683196008205414,
0.04140925407409668,
0.010924475267529488,
-0.006248006597161293,
-0.01748526282608509,
0.021818434819579124,
0.00891809444874525,
-0.00242... |
d213bc9f7f0b71336be83b4849461076ed2c2c19 | subsection | 69 | 150 | First invariant subspace | \end{equation*}
In combination with (\ref {janprva})--(\ref {jantreca}), (\ref {sym_nabla_est}) this implies the existence of c_3 \in \mathbb {C} such that
\begin{align*}
\Vert u_\alpha +i \chi _\alpha c_3 x_3e_\chi \Vert _{H^1(Q, {\mathbb {C}})}&\le C|\chi |^2 \bigl \Vert ({\!\!_{1},{\!\!_{2}, \widehat{f}_3)^\top \big... | {
"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 | [
-0.0020927335135638714,
0.01471208781003952,
-0.06025240942835808,
-0.011217203922569752,
0.02096930332481861,
-0.017749127000570297,
0.008027549833059311,
0.03284275159239769,
0.03909996896982193,
-0.019702598452568054,
-0.013323291204869747,
-0.00281002395786345,
-0.006314446218311787,
0... |
daca4b7f1e0e970350b4317f7e0b0930b13267bf | subsection | 70 | 150 | First invariant subspace | Setting \varphi =u in (\ref {jdba10_new}),
we obtain
\begin{equation}
\Vert u\Vert _{L^2(Q, {\mathbb {C}}^3)}\le C\bigl \Vert (\widehat{f}_{1}, \widehat{f}_2, {\!\!_{3})^\top \bigr \Vert _{L^2(Q, {\mathbb {C}}^3)},\qquad \bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} \le C|\chi |
\bi... | {
"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 | [
-0.00036555511178448796,
0.027806032449007034,
-0.053994372487068176,
-0.01727575808763504,
-0.006978215649724007,
-0.03143821656703949,
0.01671108976006508,
0.028279133141040802,
0.03244546055793762,
-0.000034934000723296776,
-0.00603964738547802,
-0.0006877109408378601,
0.00951540190726518... |
2fc044e6b72ce4fb27d3c9f11cb401a0d2a3618a | subsection | 71 | 150 | First invariant subspace | We demonstrate the latter in Section \ref {asymptotic_proc}, see the first estimate in (\ref {korrre1oo}).
\end{equation}\end{equation}Notice also that
setting \widehat{f}_{\alpha }=0, \alpha =1,2, in
(\ref {jdba10_newsecond}), we obtain
\begin{equation}
\bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {... | {
"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 | [
-0.0008384166867472231,
0.018893467262387276,
-0.024250177666544914,
-0.020114369690418243,
0.02740926295518875,
-0.04230427369475365,
0.010057184845209122,
0.011270456947386265,
0.030171554535627365,
0.005326187703758478,
-0.027165083214640617,
0.003906888421624899,
0.01311707217246294,
0... |
1e88379e76c8e18985ddcd336fe089903631d80e | subsection | 72 | 150 | First invariant subspace | \end{equation}
In the same way as in Section
\ref {first_subspace}, we obtain the estimates ({\it cf.} (\ref {sym_nabla_est}), (\ref {revision11}))
\begin{align*}
\bigl \Vert \textrm {sym}\nabla u\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} &\le C|\chi |^2\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)}, \\[0.6em]
\bigl \Ve... | {
"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 | [
-0.01137434970587492,
0.04986099898815155,
-0.03039262816309929,
-0.011191261000931263,
-0.016035467386245728,
0.023694653064012527,
0.03527497872710228,
0.015974437817931175,
0.019865060225129128,
0.010031702928245068,
-0.0009540607570670545,
0.029568731784820557,
0.015196314081549644,
0.... |
ac5b622d0f18a55330cc466bb9f435b87e044742 | subsection | 73 | 150 | First invariant subspace | Namely, first setting \varphi =u in (\ref {scalingdva}) we obtain ({\it cf.} (\ref {init_est}))
\begin{equation}
\Vert u\Vert _{L^2(Q, {\mathbb {C}}^3)} \le C\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)},\qquad \bigl \Vert \textrm {sym}\nabla u
\bigr \Vert _{L^2(Q, {\mathbb {C}}^{3\times 3})} \le C|\chi |
\bigl \Vert f\Vert ... | {
"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 | [
-0.0186309777200222,
0.0360717698931694,
-0.047759998589754105,
-0.0165862999856472,
-0.014404293149709702,
-0.006565092597156763,
0.013854976743459702,
0.0325012132525444,
0.0330505296587944,
-0.024322504177689552,
0.00005692242120858282,
-0.003614424727857113,
0.0092239361256361,
-0.0166... |
7c395a90f671886f1eae775424f1891ccc3096dd | subsection | 74 | 150 | First invariant subspace | Therefore ({\it cf.} (\ref {revision1second2}))
\Vert u_3\Vert _{H^1(Q, {\mathbb {C}})} \le C\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)}
and ({\it cf.} (\ref {approx21}))
\bigl \Vert u_\alpha -c_{\alpha }e_\chi \bigr \Vert _{H^1(Q, {\mathbb {C}})}\le C|\chi |\Vert f\Vert _{L^2(Q, {\mathbb {C}}^3)},\qquad \alpha =1,2.
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 | [
0.01708975061774254,
0.007873492315411568,
-0.05404633283615112,
-0.007110556587576866,
-0.014961160719394684,
-0.01840199902653694,
0.0070495218969881535,
-0.028945764526724815,
0.0006136860465630889,
0.037200722843408585,
-0.023879874497652054,
-0.010536136105656624,
0.009063671343028545,
... |
f0b19aab718a2edf27e46287495d67a6734e06a5 | subsection | 75 | 150 | First invariant subspace | In addition, we identify the correctors required for L^2\rightarrow H^1 estimates as well as for higher-precision L^2\rightarrow L^2 estimates, see Section \ref {normresolvent} for details. By adopting particular scalings for the operator of (\ref {original_identity}) and the force density, we thus also recover a versi... | {
"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 | [
-0.02335863560438156,
0.012503202073276043,
-0.04052288457751274,
-0.01034432090818882,
-0.007720097899436951,
-0.031978901475667953,
0.04989074915647507,
-0.008757581003010273,
0.017682990059256554,
-0.008437181822955608,
-0.040156714618206024,
0.015897909179329872,
0.02221435308456421,
0... |
f61132720abe655cf87a509f2e7779eb4ff725b2 | subsection | 76 | 150 | First invariant subspace | \end{equation}
We also define, for m_1,m_2,m_3 \in \mathbb {C},
\begin{align*}
\Xi (\chi , m_1,m_2)&:=i{\mathfrak {I}}\left(\begin{array}{ccc} \chi _1 m_1 & \dfrac{1}{2}(\chi _1 m_2+ \chi _2 m_1)\\[0.8em]
\dfrac{1}{2}(\chi _1 m_2 +\chi _2 m_1) & \chi _2 m_2\end{array} \right),\quad \Upsilon (\chi , m_3):=im_3{\mathfrak... | {
"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 | [
0.023068351671099663,
-0.011442635208368301,
-0.04094937816262245,
-0.015439928509294987,
-0.0022408494260162115,
-0.03933215141296387,
0.024197358638048172,
0.003360320581123233,
-0.001016487367451191,
-0.0005130114732310176,
-0.04134605452418327,
0.007391942199319601,
0.011831684969365597,... |
e6f313b9f91c9edd5ac13e4716875a18f6af2d43 | subsection | 77 | 150 | First invariant subspace | \end{align*}
Notice that
\begin{equation}
\begin{aligned}\max \bigl \lbrace |m_1|, |m_2|\bigr \rbrace +\max \bigl \lbrace |\chi _1|,|\chi _2|\bigr \rbrace |m_3|&
\le C\vert \chi \vert ^{-1}
\Bigl (\bigl |\Xi (\chi ,m_1,m_2)\bigr |+\bigl |\Upsilon (\chi , m_3)\bigr |\Bigr ),\\[0.5em]
|m_3| &\le C\vert \chi \vert ^{-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 | [
-0.012822909280657768,
-0.0058164591901004314,
-0.047752175480127335,
-0.01042004395276308,
-0.014195974916219711,
0.005366398952901363,
0.011899903416633606,
0.030970262363553047,
0.014043412171304226,
0.000778547371737659,
-0.017544729635119438,
0.026683244854211807,
-0.010732797905802727,... |
6a79dd637878ff5a00e9d467fa53110cbe243912 | subsection | 78 | 150 | First invariant subspace | Therefore,
\begin{equation*}
A^{\textrm {hom}}_{\chi } m\cdot \overline{d}
=A^{\textrm {hom},1}_{\chi }m_3\,\overline{d_3}+A^{\textrm {hom},2}_{\chi }(m_1,m_2)^\top \cdot \overline{(d_1,d_2)^\top },
\end{equation*}
where, for \chi \in Q^{\prime }_{\rm r}, m_3, d_3\in {\mathbb {C}}, (m_1, m_2)^\top , (d_1, d_2)^\top \in... | {
"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 | [
0.05772370845079422,
-0.022390449419617653,
-0.026206135749816895,
-0.02812924236059189,
-0.012782550416886806,
-0.01888001710176468,
-0.004250674974173307,
0.02203940600156784,
-0.012110989540815353,
-0.019551578909158707,
-0.04175887256860733,
0.011691263876855373,
0.015545107424259186,
... |
460889d8e6dab1d16a8b043922ed98188a72a9b7 | subsection | 79 | 150 | First invariant subspace | \end{align*}
}\begin{}
Notice that the following coercivity estimate holds:
\begin{equation*}
\bigl \Vert \Theta \varphi \bigr \Vert _{L^2(Q, {\mathbb {C}}^3)} \ge \frac{1}{4}|\theta | \Vert \varphi \Vert _{L^2(Q, {\mathbb {C}}^3)},
\end{equation*}
which is obtained by combining
\begin{eqnarray*}
\bigl \Vert \Theta \va... | {
"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 | [
-0.01670956052839756,
0.04916731268167496,
-0.03311392664909363,
-0.030138250440359116,
0.019349519163370132,
0.008141142316162586,
0.0073972237296402454,
0.0059284609742462635,
0.023958001285791397,
0.03445679321885109,
-0.0340295173227787,
0.007820685394108295,
0.015450621955096722,
-0.0... |
20b17da6b4eb9456cb60ff6d814214a8b4641cab | subsection | 80 | 150 | First invariant subspace | After that we have the equation
\begin{align*}
\frac{1}{\varepsilon ^4|\theta |^4}\int _Q A \ {\rm sym} \nabla ( u_1, u_2, u_3):\overline{ {\rm sym} \nabla (\varphi _1,\varphi _2, \varphi _3)}+ \int _Q u_\alpha \overline{\varphi _\alpha }+\int _Q u_3\overline{\varphi _3}\\[0.4em]
=\frac{1}{\varepsilon |\theta |}\int _Q... | {
"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 | [
0.049899425357580185,
0.04407019540667534,
-0.03071790374815464,
-0.023011723533272743,
0.010689464397728443,
-0.04410071671009064,
0.05426371842622757,
0.014206839725375175,
0.04632864147424698,
-0.0014945031143724918,
-0.025132829323410988,
0.00038602433050982654,
-0.001207428751513362,
... |
1507e2e0ce0e1e34f3a6a62595eb1aef0df28cd8 | subsection | 81 | 150 | First invariant subspace | For clarity of the argument, we keep the original form of the second term on the left-hand side of (\ref {sinisa1000revision}).
}
\begin{equation}
\begin{aligned}\vert \chi \vert ^{-4}A^{\textrm {hom},1}_{\chi }m_3\,\overline{d_3} &+\int _Q (-i\chi _1 x_3 m_3, -i\chi _2 x_3m_3,m_3)^\top \cdot \overline{(-i\chi _1 x_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 | [
0.006733328569680452,
0.013115685433149338,
-0.016953492537140846,
-0.020264839753508568,
-0.0012961181346327066,
-0.01113192830234766,
0.04321538284420967,
0.009460994973778725,
0.002142075914889574,
0.011254005134105682,
-0.02266060747206211,
0.02137879654765129,
0.009972193278372288,
0.... |
5243eda65dd3aaef880c186ee7eb1c3e6355b537 | subsection | 82 | 150 | First invariant subspace | \end{equation}
In addition, the following symmetry properties\footnote {Similar symmetry properties hold for all terms in the asymptotics series, see {\it e.g.} (\ref {sinisa1005}).} hold:
\begin{equation}
({\mathfrak {u}}_{2})_\alpha (\cdot ,-x_3)=-({\mathfrak {u}}_{2})_{\alpha }(\cdot ,x_3), \quad \alpha =1,2,
\qqua... | {
"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 | [
-0.027485521510243416,
-0.002575336955487728,
-0.04077807441353798,
-0.02740921638906002,
0.012651581317186356,
-0.037084851413965225,
0.06183860823512077,
0.01991593837738037,
0.02084687538444996,
0.018069326877593994,
-0.013162833638489246,
0.03072090819478035,
0.004860709886997938,
0.00... |
8332e42f058c21317e4d2fff4ea59fa2cf7da6a3 | subsection | 83 | 150 | First invariant subspace | To verify that it also vanishes when tested with vectors (0,0,D_3)^\top , we use the fact that
\begin{equation}
\begin{aligned}\int _Q A\bigl (-ix_3 \Upsilon (\chi ,m_3)+\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_2\bigr ):\overline{iX(0,0,D_3)^\top }&=
\int _Q A\bigl ( -ix_3 \Upsilon (\chi ,m_3)+\mathop {\mathrm {s... | {
"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 | [
-0.0059352139942348,
0.023130550980567932,
-0.01957552507519722,
-0.01701224595308304,
-0.010779508389532566,
-0.02494620718061924,
0.05971832200884819,
0.027005987241864204,
0.00882653333246708,
0.0005473670898936689,
-0.015593287535011768,
0.008277258835732937,
0.019712844863533974,
-0.0... |
eb89d88ffb10bc2c434581c59bc23766826cdd6c | subsection | 84 | 150 | First invariant subspace | \end{equation}
\end{aligned}Next, we seek {\mathfrak {u}}_4^{(1)} \in H^1_{\#}(Q,\mathbb {C}^3) such that
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_4^{(1)} &= i \bigl \lbrace X^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_3^{(1)}-(\mathop {... | {
"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 | [
0.014745084568858147,
0.03215999901294708,
-0.03484508395195007,
-0.03933039680123329,
0.009352030232548714,
-0.007212352938950062,
0.044761594384908676,
0.01071745716035366,
-0.00211297906935215,
0.02128235250711441,
-0.0018841365817934275,
-0.0204432625323534,
0.03356356546282768,
-0.036... |
1e153574866605d4bc30f86585f3459887c8f775 | subsection | 85 | 150 | First invariant subspace | To ensure it also vanishes when tested with vectors (0,0,D_3)^\top , we observe that ({\it cf.} (\ref {psi_id}))
\begin{equation*}
\begin{aligned}& \int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_3^{(1)}+ iX{\mathfrak {u}}_2 \bigr ): \overline{iX(0,0,D_3)^\top }
=\int _Q A \bigl (\mathop {\mathrm {sym}}\... | {
"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 | [
-0.0005110370693728328,
0.011006365530192852,
-0.03621498495340347,
-0.02449926920235157,
-0.005198085680603981,
-0.000968682172242552,
0.044940751045942307,
0.02610102668404579,
0.01066313125193119,
0.04051684960722923,
-0.009229176677763462,
0.0035772593691945076,
-0.00026052401517517865,
... |
fa7f2fdf782c89d9b2e0ac2df48fd90a316154ca | subsection | 86 | 150 | First invariant subspace | To this end, we define a ``correction^{\prime \prime } m_3^{(1)} to the value m_3 as the solution to
\begin{equation}
\begin{aligned}
A^{\textrm {hom},1}_{\chi }m_3^{(1)}\,\overline{d_3}
&+|\chi |^4\int _Q\bigl (-i\chi _1 x_3 m_3^{(1)}, -i\chi _2 x_3m_3^{(1)},m_3^{(1)}\bigr )^\top \cdot \overline{(-i\chi _1 x_3 d_3, -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 | [
0.011068008840084076,
-0.014401378110051155,
-0.026590680703520775,
-0.03774259611964226,
-0.0037967616226524115,
-0.02378363162279129,
0.04958101734519005,
0.01386742852628231,
0.021052861586213112,
0.00968736782670021,
-0.04741470888257027,
-0.012753763236105442,
0.012318975292146206,
0.... |
ebc02e8aeaa35e93be862f49df6fcb4267b1ec02 | subsection | 87 | 150 | First invariant subspace | \end{equation}
Second, consider {\mathfrak {u}}_4^{(2)} \in H^1_{\#} (Q, \mathbb {C}^3) satisfying ({\it cf.} (\ref {sinisa1004}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_4^{(2)}&= i \bigl \lbrace X^{*} \cdot A \mathop {\mathrm {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 | [
0.009341063909232616,
0.04988006129860878,
-0.026008453220129013,
-0.01804107427597046,
-0.0021387527231127024,
-0.02967161498963833,
0.04374426603317261,
-0.009142642840743065,
0.030434774234890938,
0.030770564451813698,
0.0061663235537707806,
-0.005780928302556276,
0.03379267454147339,
0... |
358a1224836e224f1202bc47552cbed21da70e7c | subsection | 88 | 150 | First invariant subspace | \end{equation}
\end{equation*}Finally, we define {\mathfrak {u}}_5^{(1)} \in H^1_{\#} (Q, \mathbb {C}^3) as the solution to ({\it cf.} (\ref {sinisa1010}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_5^{(1)} &= i \Bigl (X^{*}A \mathop {\mathrm {sy... | {
"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 | [
-0.003299608128145337,
0.01224478892982006,
-0.036925096064805984,
-0.012969558127224445,
-0.000723815755918622,
-0.020781809464097023,
0.02960110828280449,
0.04171620309352875,
0.04162465035915375,
0.03277483582496643,
-0.015563469380140305,
0.007121813017874956,
0.025557657703757286,
0.0... |
8deb47a79b0f092af3477fbbd734e62be1c9bf3e | subsection | 89 | 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{align*}
\int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_4^{(2)}+iX{\mathfrak {u}}_3^{(2)}\bigr )&: \overline{i X(0,0,D_3)^\top }
=\int _Q A \bigl (\mathop {\mathrm {... | {
"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 | [
-0.005458633415400982,
0.01859978772699833,
-0.007873249240219593,
-0.011069849133491516,
-0.01670776680111885,
-0.01192430965602398,
0.032957784831523895,
0.022856835275888443,
-0.008689564652740955,
0.0331103689968586,
-0.026823975145816803,
0.02201763167977333,
0.013282292522490025,
0.0... |
a2e7eff3ef8cfbdbc4dec4d8bd58c5da5518e939 | subsection | 90 | 150 | First invariant subspace | Defining m_3^{(2)} so that
\begin{align}
A^{\textrm {hom},1}_{\chi }m_3^{(2)}\,\overline{d_3}
&+|\chi |^4\int _Q\bigl (-i\chi _1 x_3 m_3^{(2)}, -i\chi _2 x_3m_3^{(2)},m_3^{(2)}\bigr )^\top \cdot \overline{(-i\chi _1 x_3 d_3, -i\chi _2 x_3d_3,d_3)^\top }
\\[0.3em] &
=-\int _Q A \bigl (\mathop {\mathrm {sym}}\nabla {\ma... | {
"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 | [
0.020778939127922058,
0.00502691650763154,
-0.03246518597006798,
-0.019909335300326347,
0.018475253134965897,
-0.009725824929773808,
0.030848028138279915,
0.014218774624168873,
-0.00487816845998168,
0.010961576364934444,
-0.03920842334628105,
-0.0077234492637217045,
0.005412135273218155,
0... |
000ac683c70fe80ca3d047ea590e2ca2795a30f3 | subsection | 91 | 150 | First invariant subspace | \end{equation}
Second, consider {\mathfrak {u}}_5^{(2)} \in H^1_{\#} (Q, \mathbb {C}^3) that satisfies ({\it cf.} (\ref {sinisa1104oo}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_5^{(2)}&= i \bigl \lbrace X^{*}A \mathop {\mathrm {sym}}\nabla {\ma... | {
"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 | [
0.004358930047601461,
0.0270307045429945,
-0.04353591054677963,
-0.007878870703279972,
-0.005750889424234629,
-0.03085954673588276,
0.0287086833268404,
0.025932392105460167,
0.013202638365328312,
0.02944089286029339,
-0.00873311422765255,
-0.029593436047434807,
0.02105100080370903,
0.01225... |
259bd729a6583b7595f27428f0a705429938ec94 | subsection | 92 | 150 | First invariant subspace | \end{equation}
Finally, we define {\mathfrak {u}}_6 \in H^1_{\#} (Q, \mathbb {C}^3) as the solution to ({\it cf.} (\ref {sinisa1022oo}))
\begin{equation}
\begin{aligned}(\mathop {\mathrm {sym}}\nabla )^{*}&A \mathop {\mathrm {sym}}\nabla {\mathfrak {u}}_6= i \Bigl (X^{*}A \mathop {\mathrm {sym}}\nabla \bigl ({\mathfra... | {
"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 | [
0.006128324195742607,
0.012737151235342026,
-0.01159309595823288,
-0.028174273669719696,
-0.01691676862537861,
-0.017969299107789993,
0.02523023821413517,
0.02631327696144581,
0.04060634598135948,
0.0067423004657030106,
-0.009388882666826248,
0.0033635233994573355,
-0.0008208598592318594,
... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.