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 | sparse_indices listlengths 1 1.02k | sparse_values listlengths 1 1.02k |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2a58249fb2940aba0083966a6d2e96dc9473be80 | subsection | 80 | 83 | Morphism for corollas | As shown in , the subspace of \mathsf {PL} spanned by trees
that are not corollas is a two-sided pre-Lie ideal.The quotient pre-Lie algebra is isomorphic to the following pre-Lie
algebra. Let us identify the image of the corolla \mathtt {Crl}_{n+1} with
n leaves to x^n for all n\ge 0. In particular, the tree \includegr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 111,
"openalex_id": "",
"raw": "F. Chapoton. Rooted trees and an exponential-like series. arXiv.org:math/0209104, 2002.",
"source_ref_id": "3b3b2ae3dcac41b1b3b6ce4f863c245e2e7f3fd1",
"start": 0
},
{
"arxi... | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
127887,
1614,
65421,
125458,
420,
21130,
27734,
14534,
1360,
90,
450,
621,
959,
8231,
12116,
7,
83,
6626,
8752,
479,
92314,
6397,
41502,
18750,
144,
429,
2844,
13882,
178851,
1771,
47,
25632,
135812,
29569,
3062,
54123,
21748,
653,
31358,
... | [
0.024322509765625,
0.16552734375,
0.224853515625,
0.03729248046875,
0.1142578125,
0.1436767578125,
0.1087646484375,
0.020477294921875,
0.17236328125,
0.0631103515625,
0.006805419921875,
0.0513916015625,
0.105712890625,
0.1201171875,
0.2293701171875,
0.0170135498046875,
0.057373046875... | |
1ab84d682397449535c109ee23f2bfa48b361bf2 | subsection | 81 | 83 | Morphism to a pre-Lie algebra of vector fields | There exists an interesting morphism from \mathsf {PL} to a pre-Lie algebra
of vector fields. We describe it here only as a side remark, as the
image of \Omega _q seems to have no special property.Consider the the vector space V=\mathbb {Q}[x]_+, endowed with the following
pre-Lie product:(f \curvearrowleft g)=x f \, \... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
32316,
49041,
178851,
8780,
1295,
41872,
125458,
420,
21130,
47,
479,
92314,
144,
429,
2844,
173,
18770,
44457,
98363,
5609,
10015,
29569,
670,
87849,
101,
864,
37202,
765,
110,
5361,
57266,
30542,
32628,
310,
5125,
2737,
425,
1328,
246,
... | [
0.1224365234375,
0.1669921875,
0.257080078125,
0.1824951171875,
0.0750732421875,
0.02984619140625,
0.047393798828125,
0.138916015625,
0.181396484375,
0.111083984375,
0.15234375,
0.2362060546875,
0.05267333984375,
0.1279296875,
0.151611328125,
0.15771484375,
0.188232421875,
0.205566... | |
d542c9ef2c9828d54cb58e907846ac75a636281e | subsection | 82 | 83 | First terms of some expansions | \Omega =\includegraphics [height=5mm]{a0.eps}-\frac{1}{2}\includegraphics [height=5mm]{a10.eps}+\frac{1}{3}
\includegraphics [height=5mm]{a110.eps}+\frac{1}{12}\includegraphics [height=5mm]{a200.eps} -\frac{1}{4}
\includegraphics [height=5mm]{a1110.eps}-\frac{1}{12}\includegraphics [height=5mm]{a1200.eps}-\frac{1}{12}\... | {
"cite_spans": []
} | 0807.1830 | A rooted-trees q-series lifting a one-parameter family of Lie
idempotents | [
"Frédéric Chapoton"
] | [
"math.QA"
] | 2,008 | en | Mathematics | [
6,
41872,
670,
87849,
2203,
217028,
48461,
7,
378,
1106,
22553,
114997,
2276,
268,
24854,
11,
2389,
5,
4517,
8152,
9,
132076,
418,
304,
963,
1328,
363,
36053,
1530,
5955,
20,
617,
1662,
96740,
3559,
997,
13273,
758,
2839,
39425,
36605,
... | [
0.0872802734375,
0.1029052734375,
0.1243896484375,
0.29150390625,
0.1173095703125,
0.25830078125,
0.319580078125,
0.03631591796875,
0.03497314453125,
0.1883544921875,
0.2313232421875,
0.15087890625,
0.1761474609375,
0.035247802734375,
0.0350341796875,
0.037567138671875,
0.10803222656... | |
ce0717b60c78a06a55d75a2d5d6372c61d2207e5 | abstract | 0 | 7 | Abstract | Chern-Simons modified gravity models in 4-dimensions are shown to be special
cases of low energy effective string models to first order in the string
constant. | {
"cite_spans": []
} | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
70535,
19,
133620,
5245,
73197,
64002,
939,
115774,
23,
7306,
91403,
127887,
47,
186,
5361,
50218,
27226,
48302,
60266,
79315,
5117,
12989,
53697
] | [
0.1414794921875,
0.080078125,
0.122314453125,
0.08880615234375,
0.1480712890625,
0.182861328125,
0.05462646484375,
0.21142578125,
0.00762939453125,
0.162353515625,
0.1748046875,
0.089599609375,
0.049041748046875,
0.005950927734375,
0.06439208984375,
0.07000732421875,
0.1361083984375,... |
179483c05684ee372b5179ac3a9d15c0310ff360 | subsection | 1 | 7 | Introduction | A Chern-Simons modified gravity in D=4 dimensions
proposed by Jackiw and Pi , has attracted
a lot of attention recently . The Jackiw-Pi
model is derived from an action that consists of the usual
Einstein-Hilbert term plus a topological term with a cosmic scalar
field \theta appearing as a Lagrange multiplier. It was sh... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 122,
"openalex_id": "",
"raw": "R Jackiw, S Y Pi, Phys. Rev. D 68 (2003) 104012",
"source_ref_id": "bdaa802c27e6c8fd9afb4d8b5a48538473ddfb56",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end":... | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
70535,
19,
9,
133620,
5245,
73197,
297,
64002,
939,
391,
107947,
158208,
26171,
71,
21763,
14,
434,
136,
3065,
6,
110281,
10,
5915,
111,
35743,
78684,
29348,
3299,
16406,
1295,
22631,
450,
58055,
7,
70,
115723,
119225,
379,
16466,
13579,
... | [
0.1727294921875,
0.140380859375,
0.05303955078125,
0.1705322265625,
0.15625,
0.197021484375,
0.04254150390625,
0.2318115234375,
0.1422119140625,
0.12890625,
0.209228515625,
0.196044921875,
0.1297607421875,
0.005615234375,
0.1676025390625,
0.139892578125,
0.2003173828125,
0.04953002... |
15f7d2a7c305115a2d7ae3c4b9882faf706da82a | subsection | 2 | 7 | Effective String Theory Field Equations | We will start by examining the bosonic field equations
arising from the action I = \int _M L where M is a 4-dimensional
manifold and the 4-form L is given in terms of a dilaton 0-form
\phi , and a 3-form field H. We take an actionL_0 = e^{-\phi } \left( R_{ab} \wedge *e^{ab} -\alpha d\phi \wedge *d\phi + \beta H \wedge... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1866,
"openalex_id": "",
"raw": "T Dereli, R W Tucker, Class. Quant. Grav. 4 (1987) 791",
"source_ref_id": "0f8843c376217e56dcf0534f72511ab02027c45e",
"start": 1713
}
]
} | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
1401,
1221,
4034,
42276,
337,
1681,
1771,
44457,
28,
13722,
5256,
187,
72219,
1295,
22631,
87,
2203,
6,
4288,
594,
339,
276,
7306,
157955,
17174,
42822,
5037,
83,
34475,
23,
69407,
111,
45,
143,
1507,
757,
9,
19379,
4,
5691,
572,
5646... | [
0.010284423828125,
0.04302978515625,
0.06781005859375,
0.09490966796875,
0.1395263671875,
0.1810302734375,
0.125,
0.2225341796875,
0.04290771484375,
0.1986083984375,
0.0433349609375,
0.0289764404296875,
0.037353515625,
0.012451171875,
0.2086181640625,
0.0799560546875,
0.0187835693359... |
fbb2a5a498d71fca7507e0623fd7fd05d3d2d491 | subsection | 3 | 7 | Effective String Theory Field Equations | We keep it in the action but it
doesn't give any contribution to the variational field equations.In order to derive the effective string theory field
equations, the action I=\int _M (L_0 + L_1 + L_C) is going to be
varied as a functional of the variables \lbrace e^a , \phi , H,
\omega ^a{}_b \rbrace in a fixed local co... | {
"cite_spans": []
} | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
1401,
13695,
442,
23,
22631,
1284,
22027,
18,
8337,
2499,
127752,
143834,
289,
44457,
28,
13722,
5256,
12989,
122,
5844,
60266,
79315,
154453,
87,
4288,
594,
866,
2389,
997,
339,
115187,
454,
441,
7730,
10861,
297,
123309,
77336,
48543,
9... | [
0.0804443359375,
0.180908203125,
0.044677734375,
0.09002685546875,
0.223388671875,
0.05206298828125,
0.022705078125,
0.06884765625,
0.033355712890625,
0.033294677734375,
0.1124267578125,
0.2113037109375,
0.08349609375,
0.241455078125,
0.0941162109375,
0.2442626953125,
0.1045532226562... |
a20d412c4676c40556b63eeb19dd1c4fb0807cc7 | subsection | 4 | 7 | Effective String Theory Field Equations | Since d^2\mu =0, the H-field equation may be
replaced by\beta d(e^{-\phi } *H) = 0 .We also vary the connection 1-forms \omega ^a{}_b as independent
variables and obtain- e^{-\phi } d\phi \wedge *e_{ab} + 4\epsilon \beta e^{-\phi } R_{ab}
\wedge *H = \frac{1}{2}(e_a \wedge \lambda _b - e_b \wedge \lambda _a).These are ... | {
"cite_spans": []
} | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
66016,
104,
8353,
304,
561,
6,
145407,
572,
9,
28394,
28,
5490,
2320,
1543,
91995,
71,
390,
59865,
13,
19379,
51912,
841,
16,
2203,
757,
285,
53,
94878,
4317,
5037,
306,
2765,
11,
8152,
275,
41371,
77336,
7,
113054,
24243,
429,
454,
... | [
0.057464599609375,
0.158203125,
0.07843017578125,
0.18798828125,
0.1917724609375,
0.0115966796875,
0.1719970703125,
0.158203125,
0.0357666015625,
0.2130126953125,
0.1256103515625,
0.235595703125,
0.08013916015625,
0.067138671875,
0.1795654296875,
0.0345458984375,
0.013427734375,
0.... |
9ec159c0a19e416902cc776205e6eb97ae46ef90 | subsection | 5 | 7 | Effective String Theory Field Equations | The reduced field equations becomeG_a - \beta \tau _a[H] +
\lambda *e_a + 8\epsilon \beta D(\imath ^b
(R_{ba} \wedge *H)) & & \\
- 2\epsilon \beta e_a \wedge D( \imath ^b
\imath ^c (R_{bc} \wedge *H)) &=& 0 , \\
2 \beta H \wedge *H - \lambda *1 &=& 0 , \\
dH - \epsilon R_{ab} \wedge R^{ab} &=& 0 ,\\
\beta (d*H ) &=& 0 ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 878,
"openalex_id": "",
"raw": "T L Smith, A L Erickcek, R R Caldwell, M Kamionkowski, Phys. Rev. D 77 (2008) 024015",
"source_ref_id": "abe66fbd12d4fdcb40639713752d2f1250a67858",
"start": 786
},
{
"arxiv... | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
34390,
71,
44457,
28,
13722,
5256,
24209,
724,
454,
11,
20,
59865,
50104,
841,
268,
997,
6492,
85,
382,
15759,
391,
132,
1352,
927,
275,
1052,
402,
8152,
24243,
429,
16,
116,
6,
41872,
24854,
65037,
1369,
757,
572,
4,
627,
2055,
113... | [
0.218505859375,
0.09716796875,
0.25390625,
0.1060791015625,
0.224853515625,
0.1287841796875,
0.1123046875,
0.06854248046875,
0.06231689453125,
0.10638427734375,
0.0107421875,
0.239501953125,
0.22021484375,
0.1466064453125,
0.000152587890625,
0.13720703125,
0.151123046875,
0.0792846... |
f2b126e44490045ef5c642aa3d398cc385e42797 | subsection | 6 | 7 | Concluding Comments | We first comment on variational field equations with
dynamical torsion. In this case, independent
\omega ^a{}_b-variations of the action \int _M (L_0 + L_1) yield
the field equationse^{-\phi } \left( T^a \wedge *e_{abc} - \frac{1}{2} d\phi \wedge e^a \wedge *e_{abc} + 4 \epsilon \beta R_{ab} \wedge *H \right) = 0 .Thes... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 989,
"openalex_id": "",
"raw": "T Dereli, R W Tucker, Class. Quant. Grav. 4 (1987) 791",
"source_ref_id": "0f8843c376217e56dcf0534f72511ab02027c45e",
"start": 956
},
{
"arxiv_id": "",
"doi": "",
... | 10.1140/epjc/s10052-012-1979-0 | 0807.1832 | String-Inspired Chern-Simons Modified Gravity In 4-Dimensions | [
"M. Adak",
"T. Dereli"
] | [
"gr-qc"
] | 2,008 | en | Physics | [
5117,
6868,
143834,
289,
44457,
28,
13722,
5256,
678,
84079,
9983,
6889,
41371,
306,
2765,
275,
21690,
21094,
22631,
4288,
594,
866,
2389,
997,
339,
115187,
11180,
19388,
19379,
384,
8353,
24243,
2055,
132076,
304,
201,
59865,
841,
757,
9... | [
0.03485107421875,
0.082275390625,
0.1915283203125,
0.057586669921875,
0.1727294921875,
0.04388427734375,
0.177734375,
0.02874755859375,
0.0892333984375,
0.1419677734375,
0.21044921875,
0.138916015625,
0.11279296875,
0.0565185546875,
0.1490478515625,
0.08416748046875,
0.2044677734375,... |
ccb9edaad5a9c3b15c32b1f630bce99012fbab87 | abstract | 0 | 19 | Abstract | Micro-optomechanical systems are central to a number of recent proposals for
realizing quantum mechanical effects in relatively massive systems. Here we
focus on a particular class of experiments which aim to demonstrate massive
quantum superpositions, although the obtained results should be generalizable
to similar ex... | {
"cite_spans": []
} | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
37992,
2146,
188,
282,
17032,
21533,
76519,
9879,
14012,
17309,
152132,
14823,
110436,
135969,
289,
93425,
35845,
106750,
11853,
32153,
17311,
18507,
28007,
464,
106804,
1601,
40322,
50339,
5608,
4537,
21373,
7968,
22443,
94418,
13,
52768,
2060... | [
0.1304931640625,
0.1734619140625,
0.10162353515625,
0.00030517578125,
0.168701171875,
0.0445556640625,
0.1766357421875,
0.1396484375,
0.060272216796875,
0.0714111328125,
0.1298828125,
0.15087890625,
0.1990966796875,
0.2099609375,
0.02484130859375,
0.2030029296875,
0.05120849609375,
... |
415fff16f9e620d679808c2b663231f73743a54e | subsection | 1 | 19 | Introduction | Micro-optomechanical systems have recently attracted significant interest as a potential architecture for observing quantum mechanical effects on scales many orders of magnitude more massive than previous experiments.
Proposals include entangling states of mechanical resonators to each other , , or cavity fields , , th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 604,
"openalex_id": "",
"raw": "S. Mancini, D. Vitali, V. Giovannetti, and P. Tombesi. Stationary entanglement between macroscopic mechanical oscillators. Eur. Phys. J. D, 22:417–422, March 2003.",
"source_ref_id": "d1aaba8b... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
37992,
2146,
188,
282,
17032,
21533,
76519,
78684,
110281,
297,
88551,
33946,
10,
38516,
6,
159958,
66339,
214,
110436,
135969,
289,
93425,
98,
105994,
7,
12989,
111,
101668,
13,
1286,
106750,
3501,
96362,
28007,
43796,
232,
16765,
22,
1452... | [
0.145263671875,
0.185546875,
0.1119384765625,
0.0241546630859375,
0.1885986328125,
0.0170745849609375,
0.184814453125,
0.05841064453125,
0.0780029296875,
0.011566162109375,
0.06597900390625,
0.08966064453125,
0.011566162109375,
0.1519775390625,
0.011505126953125,
0.1949462890625,
0.1... |
de46cdc7da09552dcc818806c92127ca53304644 | subsection | 2 | 19 | Quantum Mechanical Description | A more detailed analysis of the system begins with the Hamiltonian, given by Law :H = \hbar \omega _a \left[ {a^{\dagger }}a+ {b}^{\dagger }b\right] + \hbar \omega _c \left[{c}^{\dagger }c- \kappa {a^{\dagger }}a\left(c+ {c}^{\dagger }\right)\right],where \omega _a is the frequency of the optical field, {a^{\dagger }}... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 620,
"openalex_id": "",
"raw": "C. K. Law. Interaction between a moving mirror and radiation pressure: A Hamiltonian formulation. Phys. Rev. A, 51:2537–2541, March 1995.",
"source_ref_id": "f6ac58f32a2b544bdb264354feca585f25... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
62,
1286,
185688,
114137,
111,
70,
5426,
9842,
678,
94674,
3378,
34475,
390,
36293,
841,
2203,
6,
1299,
306,
2765,
11,
41872,
133,
1065,
8353,
24854,
85,
21407,
47391,
1328,
10666,
275,
8152,
51912,
268,
101,
238,
2480,
9,
161,
7495,
... | [
0.003570556640625,
0.0777587890625,
0.170166015625,
0.2271728515625,
0.051055908203125,
0.0650634765625,
0.2379150390625,
0.096923828125,
0.052581787109375,
0.2705078125,
0.2484130859375,
0.0911865234375,
0.017669677734375,
0.216064453125,
0.141357421875,
0.012420654296875,
0.0273895... |
40bfe484f890341830cef29e048f357a4ecc3d66 | subsection | 3 | 19 | Coherent State | If we consider a cantilever initially in a coherent state with complex amplitude \beta , the total initial state is given by
\vert \, \Psi (0)\rangle = \frac{1}{\sqrt{2}} \left( \vert \, 0,1\rangle _{n_a, n_b} + \vert \, 1,0\rangle _{n_a, n_b} \right) \otimes \vert \, \beta \rangle _c.
Under the action of the unitary o... | {
"cite_spans": []
} | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
4263,
16916,
10,
831,
118,
27571,
61475,
538,
23,
241463,
11341,
678,
27140,
217269,
13,
59865,
3622,
34475,
11549,
683,
172,
6649,
5445,
133,
2203,
132076,
418,
864,
3198,
304,
72966,
81830,
70141,
22631,
25072,
6635,
39933,
5,
5980,
773... | [
0.0079345703125,
0.1021728515625,
0.042572021484375,
0.19921875,
0.19091796875,
0.3408203125,
0.1885986328125,
0.0115966796875,
0.0207366943359375,
0.2464599609375,
0.221923828125,
0.000396728515625,
0.1785888671875,
0.18115234375,
0.0143585205078125,
0.2349853515625,
0.194091796875,... |
3b137d76ac7df3344c1d327b4da7532d46a24333 | subsection | 4 | 19 | Coherent State | This is equivalent to stipulating that a measurement of the cantilever state alone is sufficient to determine which path a photon took with a reasonable fidelity.
As will be discussed in section-experiment, obtaining this large a value of \kappa poses the most significant barrier to experimental realization.In practice... | {
"cite_spans": []
} | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
183234,
197551,
72350,
674,
831,
118,
27571,
11341,
75447,
129980,
83324,
3129,
60875,
16186,
19,
34739,
678,
169022,
46735,
2481,
186,
297,
23,
40059,
230648,
113054,
21334,
34292,
6,
41872,
161,
7495,
159931,
70,
2684,
88551,
125861,
195935... | [
0.108154296875,
0.0728759765625,
0.224853515625,
0.06085205078125,
0.1729736328125,
0.1539306640625,
0.2783203125,
0.1734619140625,
0.1751708984375,
0.156005859375,
0.1556396484375,
0.036346435546875,
0.194091796875,
0.2059326171875,
0.0885009765625,
0.1265869140625,
0.00912475585937... |
4bb856fff3e38246042a928ee406cc11394acfe8 | subsection | 5 | 19 | The cantilever at finite temperatures | At finite temperatures the exact wavefunction of the cantilever is unknown, so the state is instead described by a density matrix:\rho _c(0) = \frac{\sum _n e^{-E_n/k_B T } \vert \, n\rangle \langle n \, | }{ \sum _n e^{-E_n/k_B T } } = \frac{1}{\pi \bar{n}} \int d^2\beta e^{-|\beta |^2 / \bar{n} } \vert \, \beta \rang... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1478,
"openalex_id": "",
"raw": "J. Zsolt Bernád, L. Diósi, and T. Geszti. Quest for quantum superpositions of a mirror: high and moderately low temperatures. Physical Review Letters, 97(25):250404, December 2006.",
"source_... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
1913,
94418,
13,
52768,
7,
24763,
259,
272,
137175,
831,
118,
27571,
51,
69723,
19,
11341,
64457,
151552,
390,
168,
2481,
50944,
425,
497,
238,
177609,
2203,
6,
41872,
132076,
24854,
11832,
28,
8353,
9,
647,
454,
64,
92,
571,
384,
519... | [
0.03900146484375,
0.199951171875,
0.07489013671875,
0.2412109375,
0.092041015625,
0.10382080078125,
0.1083984375,
0.0732421875,
0.1632080078125,
0.175048828125,
0.1700439453125,
0.323974609375,
0.0306243896484375,
0.14453125,
0.095458984375,
0.1982421875,
0.04010009765625,
0.165649... |
60e9ee70c552a8f18444cc18e5e43645ae1e6934 | subsection | 6 | 19 | The cantilever at finite temperatures | Provided that the opto-mechanical coupling strength \kappa is relatively well known (e.g., by independently measuring m, \omega _c, L, etc.) and the instantaneous quantum state of the cantilever is regarded as some random coherent state (as should be the case for the weakly mechanically damped systems discussed here) i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2194,
"openalex_id": "",
"raw": "M. A. Nielsen, I. L. Chuang, and D. F. V. James. Quantum Computation and Quantum Information. Physics Today, 54:60–62, November 2001.",
"source_ref_id": "1490452bea3905cb8f6a2e7f6c6857ccd980a... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
233,
188,
17032,
21533,
14974,
2069,
90254,
161,
7495,
51529,
41371,
163,
162,
347,
306,
2765,
34648,
6130,
110436,
11341,
831,
118,
27571,
28601,
96759,
241463,
642,
344,
135969,
22539,
20051,
76519,
72546,
83324,
3229,
1601,
40322,
5608,
... | [
0.13525390625,
0.07769775390625,
0.120361328125,
0.1365966796875,
0.130615234375,
0.0460205078125,
0.1549072265625,
0.060211181640625,
0.1915283203125,
0.0914306640625,
0.052520751953125,
0.05908203125,
0.08642578125,
0.030853271484375,
0.0241851806640625,
0.088623046875,
0.09765625,... |
e5198d27a5930ed2057e9778a76a278afebd99cc | subsection | 7 | 19 | The Wigner Function and the Classical Limit | To study transitions between the quantum and the classical regimes, it is often convenient to refer to quasi-probability distributions, with which quantum mechanics can be formulated in the common classical phase space.
One such distribution was proposed in 1932 by Wigner and can be obtained from the density matrix \rh... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 445,
"openalex_id": "",
"raw": "E. Wigner. On the Quantum Correction For Thermodynamic Equilibrium. Physical Review, 40:749–759, June 1932.",
"source_ref_id": "3cad6ff12c4b24db9d45aaac9b8b16f96f136320",
"start": 220
... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
35187,
149307,
17721,
110436,
136,
54704,
289,
63647,
7,
4,
27983,
142267,
15005,
12404,
79122,
41159,
113068,
135969,
831,
186,
26168,
3674,
70,
39210,
93402,
32628,
26171,
71,
23,
60775,
390,
5140,
51086,
113054,
297,
1295,
168,
2481,
509... | [
0.1312255859375,
0.1824951171875,
0.06793212890625,
0.1805419921875,
0.09722900390625,
0.221923828125,
0.1280517578125,
0.2281494140625,
0.068359375,
0.013397216796875,
0.07659912109375,
0.15966796875,
0.07000732421875,
0.2205810546875,
0.228271484375,
0.1756591796875,
0.256103515625... |
c50cda3bb3bd014a1a4eecd7be0fa982c146a875 | subsection | 8 | 19 | The Wigner Function and the Classical Limit | This projection is equivalent to detecting a single photon at one output of the interferometer, where the phase term in the projection accounts for path length differences in the arms.
Generally speaking, varying \theta shifts the interference peaks but does not modify the Wigner function in a significant way; hereafte... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1356,
"openalex_id": "",
"raw": "M. Paternostro, D. Vitali, S. Gigan, M. S. Kim, C. Brukner, J. Eisert, and M. Aspelmeyer. Creating and Probing Multipartite Macroscopic Entanglement with Light. Physical Review Letters, 99(25):2504... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
3293,
13452,
1830,
183234,
96391,
11001,
16186,
19,
1632,
140992,
1940,
2875,
102220,
93402,
13579,
15426,
60875,
140909,
60212,
121641,
9082,
285,
38543,
2347,
102,
122925,
193943,
3956,
37185,
2811,
5140,
51086,
32354,
46327,
5423,
47,
757,
... | [
0.020172119140625,
0.1971435546875,
0.07159423828125,
0.1373291015625,
0.1719970703125,
0.1326904296875,
0.1141357421875,
0.069580078125,
0.0877685546875,
0.09185791015625,
0.1077880859375,
0.1458740234375,
0.13818359375,
0.1624755859375,
0.1318359375,
0.04833984375,
0.10693359375,
... |
cdf36fc581d779d5ba3aba9cab2397e94debcdcd | subsection | 9 | 19 | Decoherence | In addition to “classical” phase scrambling caused by the initial thermal motion of the cantilever as discussed above, there are other effects which cause “quantum” decoherence of the cantilever.
The signature of this type of decoherence is a reduction of the visibility's revival peak – this is caused by information lo... | {
"cite_spans": []
} | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
67413,
93402,
9023,
2198,
79298,
143434,
61475,
70,
42,
2749,
78112,
831,
118,
27571,
36917,
3789,
93425,
3129,
22304,
44764,
316,
8,
587,
3334,
6620,
138256,
111,
903,
10644,
83,
456,
77391,
108625,
939,
15226,
1405,
280,
344,
390,
4677,... | [
0.0877685546875,
0.1649169921875,
0.03143310546875,
0.1732177734375,
0.167236328125,
0.13330078125,
0.06646728515625,
0.0269927978515625,
0.1058349609375,
0.0478515625,
0.108642578125,
0.1605224609375,
0.151123046875,
0.31201171875,
0.001129150390625,
0.06256103515625,
0.206787109375... |
42d9a4a8ea867031cf42e1ab63e972306eb0760e | subsection | 10 | 19 | Environmentally Induced Decoherence | Environmentally induced decoherence is due to the coupling of the system to a finite temperature bath, and results in a finite lifetime for the quantum superposition of the cantilever.
Decoherence happens when the thermal bath measures the state of the cantilever while the photon is in the cavity, introducing a phase s... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1449,
"openalex_id": "",
"raw": "R. P. Feynman and F. L. Vernon, Jr. The theory of a general quantum system interacting with a linear dissipative system. Annals of Physics, 24:118–173, October 1963.",
"source_ref_id": "15140... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
231527,
538,
135989,
297,
8,
587,
3334,
6620,
83,
4743,
47,
70,
14974,
2069,
111,
5426,
94418,
13,
52768,
101086,
4,
50339,
23,
10,
6897,
6032,
110436,
1601,
40322,
831,
118,
27571,
5,
262,
96276,
3229,
42,
2749,
72350,
7,
11341,
1618... | [
0.2313232421875,
0.0714111328125,
0.229248046875,
0.1253662109375,
0.2362060546875,
0.2158203125,
0.3212890625,
0.1724853515625,
0.05657958984375,
0.0919189453125,
0.01092529296875,
0.052398681640625,
0.1336669921875,
0.008056640625,
0.010986328125,
0.1734619140625,
0.15673828125,
... |
aa12f2649e505d025454251bbcb81818be4827f1 | subsection | 11 | 19 | Environmentally Induced Decoherence | The equation is valid in the Markovian regime when memory effects in the bath can be neglected; this is satisfied when the coupling to the bath is weak (Q \gg 1) and the thermal energy is much higher than the phonon energy (k_B T_b \gg \hbar \omega _c).
Both conditions are easily satisfied for realistic devices.
[Figur... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 646,
"openalex_id": "",
"raw": "W. H. Zurek. Decoherence, einselection, and the quantum origins of the classical. Reviews of Modern Physics, 75:715–775, May 2003.",
"source_ref_id": "5719721ba818b3da7a5447ae179b9f41ca824043"... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
28,
5490,
2320,
35604,
23,
7880,
3677,
66,
63647,
3229,
98323,
93425,
101086,
831,
124789,
89829,
83,
214521,
70,
14974,
2069,
642,
344,
2737,
6,
9815,
42,
48302,
5045,
77546,
53073,
6431,
15,
92,
571,
384,
41872,
1299,
306,
2765,
27289... | [
0.056915283203125,
0.2347412109375,
0.0850830078125,
0.17529296875,
0.00634765625,
0.1484375,
0.17431640625,
0.0712890625,
0.2115478515625,
0.009613037109375,
0.17138671875,
0.143798828125,
0.2147216796875,
0.036285400390625,
0.114013671875,
0.041015625,
0.000762939453125,
0.152465... |
80633a39df25bda8bf296a35df810cc74294e62c | subsection | 12 | 19 | Gravitationally Induced Quantum Collapse | To explain the apparent classicality of the macroscopic world, it has been suggested that there may be a quantum state collapse mechanism for large objects, possibly induced by mass.
Several proposals have been made which lead to such a collapse, among them reformulations of quantum mechanics , and the use of the intri... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 396,
"openalex_id": "",
"raw": "G. C. Ghirardi, A. Rimini, and T. Weber. Unified dynamics for microscopic and macroscopic systems. Physical Review D, 34:470–491, July 1986.",
"source_ref_id": "d20121ec41dec0c94ceeb5f4bccc0da... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
717,
73342,
173676,
54704,
134393,
111789,
12064,
18695,
8999,
442,
1556,
2809,
42459,
297,
2685,
1543,
110436,
11341,
3365,
127966,
191619,
100,
21334,
36746,
7,
4,
144681,
135989,
390,
46889,
5,
48752,
289,
152132,
765,
7228,
3129,
37105,
... | [
0.026336669921875,
0.1634521484375,
0.1451416015625,
0.156982421875,
0.115966796875,
0.10028076171875,
0.1331787109375,
0.07318115234375,
0.1231689453125,
0.046356201171875,
0.0184783935546875,
0.045318603515625,
0.136474609375,
0.0462646484375,
0.004180908203125,
0.08807373046875,
0... |
7673fcebbebdf531a95168d46637254f4d64b6de | subsection | 13 | 19 | Gravitationally Induced Quantum Collapse | If the atomic spacing is much larger than the effective mass radius, the energy due to the interaction between different atomic sites is negligible and the gravitational self-energy is given by:\Delta E = 2 G m m_1 \left( \frac{6}{5 a} - \frac{1}{\Delta x} \right) \quad \textrm {(given: } \Delta x \ge 2 a \textrm {)}.I... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1707,
"openalex_id": "",
"raw": "M. Arndt, O. Nairz, J. Vos-Andreae, C. Keller, G. van der Zouw, and A. Zeilinger. Wave-particle duality of C_{60} molecules. Nature, 401:680–682, October 1999.",
"source_ref_id": "1cb976aae1f... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
4263,
34627,
1771,
5623,
21896,
5045,
150679,
3501,
60266,
46889,
4567,
223,
48302,
4743,
70,
182809,
17721,
12921,
15271,
83,
22024,
735,
2661,
137352,
43315,
15970,
39060,
3432,
34475,
390,
58598,
102,
241,
2203,
116,
527,
347,
115187,
6,... | [
0.046478271484375,
0.1806640625,
0.08062744140625,
0.201904296875,
0.126953125,
0.08984375,
0.127685546875,
0.0295867919921875,
0.15087890625,
0.1468505859375,
0.1549072265625,
0.056640625,
0.2080078125,
0.0079345703125,
0.0160675048828125,
0.1807861328125,
0.050384521484375,
0.106... |
71fcebff39f26fc8189af05bbb7b6d8f1dc74fe6 | subsection | 14 | 19 | Optomechanical Devices | In practice, the experimental realization of a macroscopic quantum superposition is severely technically demanding.
Perhaps the most challenging aspect is achieving sufficient optical quality, which is required to put the cantilever into a distinguishable state via interaction with a single photon, i.e. \kappa \gtrsim ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1100,
"openalex_id": "",
"raw": "D. Kleckner, W. Marshall, M. J. A. de Dood, K. N. Dinyari, B.-J. Pors, W. T. M. Irvine, and D. Bouwmeester. High Finesse Opto-Mechanical Cavity with a Movable Thirty-Micron-Size Mirror. Phys. Rev. ... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
360,
41361,
195935,
25558,
111789,
12064,
18695,
110436,
1601,
40322,
83,
141591,
538,
121392,
35968,
214,
181799,
70,
2684,
223920,
43585,
10,
55391,
129980,
233,
70760,
31897,
4,
56065,
3884,
831,
118,
27571,
3934,
157167,
1495,
2886,
11341... | [
0.009368896484375,
0.1846923828125,
0.2008056640625,
0.17333984375,
0.106201171875,
0.112548828125,
0.04510498046875,
0.19189453125,
0.2025146484375,
0.2030029296875,
0.017547607421875,
0.07861328125,
0.0093994140625,
0.1510009765625,
0.1290283203125,
0.03594970703125,
0.018051147460... |
f96bf9d8b18defcbd5615d448849373ed07847b3 | subsection | 15 | 19 | Optomechanical Devices | Although the magnitude of this effect should be smaller for high finesse optomechanical systems due to lower absorption and incident light levels, at temperatures of order 1 mK absorption of even single photons should produce non-negligible heating .
[Figure: A comparison of opto-mechanical devices, showing the finesse... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 250,
"openalex_id": "",
"raw": "D. Kleckner and D. Bouwmeester. Sub-kelvin optical cooling of a micromechanical resonator. Nature, 444:75–78, November 2006.",
"source_ref_id": "f12c67ade912ca7ef8ada017b8f8fabc54b9b280",
... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
106073,
101668,
903,
21543,
5608,
164917,
11192,
72124,
184,
43922,
13450,
17032,
21533,
76519,
92319,
1563,
4970,
254,
1363,
45559,
22729,
90926,
52768,
7,
111,
12989,
106,
347,
605,
3853,
11001,
16186,
1779,
27489,
351,
86,
177,
7883,
266... | [
0.034515380859375,
0.0986328125,
0.047271728515625,
0.21533203125,
0.05889892578125,
0.1302490234375,
0.06964111328125,
0.1966552734375,
0.09161376953125,
0.18359375,
0.000885009765625,
0.2210693359375,
0.0653076171875,
0.1580810546875,
0.1151123046875,
0.1116943359375,
0.17614746093... |
9d817f8311c65688948fa09551c3bc6dd639d03e | subsection | 16 | 19 | Optical Cooling | As stated above, unambiguous observation of a macroscopic quantum superposition is possible only when the cantilever's fundamental mode is in a low phonon quantum number state.
Given that this requires temperatures of less than 1 \mu K for kHz resonators, the only way to practically obtain this is optical feedback cool... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1466,
"openalex_id": "",
"raw": "D. Kleckner and D. Bouwmeester. Sub-kelvin optical cooling of a micromechanical resonator. Nature, 444:75–78, November 2006.",
"source_ref_id": "f12c67ade912ca7ef8ada017b8f8fabc54b9b280",
... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
36917,
6332,
150556,
111789,
12064,
18695,
110436,
1601,
40322,
7722,
4734,
831,
118,
27571,
20531,
13736,
27226,
53073,
6431,
14012,
11341,
144570,
52768,
40715,
561,
341,
472,
93423,
92526,
76,
22230,
3917,
138155,
113054,
233,
70760,
58269,
... | [
0.000732421875,
0.0169677734375,
0.2020263671875,
0.114501953125,
0.120849609375,
0.0771484375,
0.1759033203125,
0.205810546875,
0.1904296875,
0.116943359375,
0.034698486328125,
0.1732177734375,
0.1619873046875,
0.29296875,
0.1888427734375,
0.1678466796875,
0.12353515625,
0.1125488... |
4a32fd16ece13740a2cf99acd2752fdc849794b1 | subsection | 17 | 19 | Optical Cooling | In the limit of low pumping power and minimal cooling, this ratio remains constant, but begins to rapidly decrease when the ground state is approached.
When the ratio is less than half the low power value, the mean phonon number, \bar{n}, is less than one, providing a clear indication of ground state cooling.
Because t... | {
"cite_spans": []
} | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
17475,
27226,
42874,
214,
14537,
136,
20187,
21185,
903,
70460,
47143,
53697,
9842,
25545,
227204,
3229,
61585,
11341,
51515,
14847,
40715,
3501,
23552,
70,
34292,
29459,
53073,
6431,
14012,
41872,
1299,
19,
1632,
101904,
34735,
14585,
10644,
... | [
0.180908203125,
0.172119140625,
0.1964111328125,
0.150634765625,
0.18017578125,
0.00030517578125,
0.1387939453125,
0.251220703125,
0.0273284912109375,
0.265625,
0.0718994140625,
0.162109375,
0.013031005859375,
0.052093505859375,
0.1605224609375,
0.0028076171875,
0.2044677734375,
0.... |
02b6971f384b2082a2baa1ebe18ff65b23feee2a | subsection | 18 | 19 | Conclusion | A detailed analysis of the effects of finite temperature on the proposed massive superposition experiments show that a fully unambiguous demonstration requires low fundamental mode temperatures, \bar{n} \lesssim 1.
Despite this, observation of a revival of the interference visibility can be used to strongly imply the e... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 388,
"openalex_id": "",
"raw": "W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester. Towards Quantum Superpositions of a Mirror. Phys. Rev. Lett., 91(13):130401, September 2003.",
"source_ref_id": "0b2d1952cdf5c818a901fbeb... | 10.1088/1367-2630/10/9/095020 | 0807.1834 | Creating and Verifying a Quantum Superposition in a Micro-optomechanical
System | [
"D. Kleckner",
"I. Pikovski",
"E. Jeffrey",
"L. Ament",
"E. Eliel",
"J. van den Brink",
"D. Bouwmeester"
] | [
"quant-ph"
] | 2,008 | en | Physics | [
185688,
114137,
93425,
94418,
13,
52768,
26171,
106750,
1601,
40322,
28007,
7639,
89554,
220,
6332,
1234,
32837,
144570,
27226,
20531,
13736,
7,
1299,
9393,
5072,
615,
150556,
15226,
1405,
193943,
3956,
108625,
939,
831,
11814,
37515,
21980,
... | [
0.05010986328125,
0.1114501953125,
0.1868896484375,
0.2030029296875,
0.050445556640625,
0.2359619140625,
0.1241455078125,
0.124267578125,
0.19287109375,
0.1788330078125,
0.209228515625,
0.0271148681640625,
0.0570068359375,
0.0142669677734375,
0.083984375,
0.077392578125,
0.1944580078... |
9cfaf6a67d64e47c0a032a33d36e0b5a32b8d9c1 | abstract | 0 | 18 | Abstract | In this paper biharmonic maps between doubly warped product manifolds are
studied. We show that the inclusion maps of Riemannian manifolds $B$ and $F$
into the doubly warped product $_{f}B\times_{b}F$ can not be proper biharmonic
maps. Also we analyze the conditions for the biharmonicity of projections
$_{f}B\times_{b}... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
903,
15122,
333,
88975,
238,
22288,
7,
17721,
54,
34,
38526,
1631,
20051,
12996,
17174,
42822,
22282,
7639,
190440,
41419,
127613,
66,
571,
136,
919,
3934,
420,
70141,
275,
831,
959,
186,
27798,
7968,
27289,
60089,
13452,
17514,
335,
188,... | [
0.0208740234375,
0.05096435546875,
0.1563720703125,
0.26416015625,
0.079833984375,
0.2030029296875,
0.0419921875,
0.1383056640625,
0.052276611328125,
0.103515625,
0.131103515625,
0.1689453125,
0.1424560546875,
0.2109375,
0.07135009765625,
0.1627197265625,
0.07470703125,
0.025878906... | |
c9226905085c573f31e1e22bfee853ee0ac60f71 | subsection | 1 | 18 | Introduction | The study of biharmonic maps between Riemannian manifolds, as a
generalization of harmonic maps, was suggested by J. Eells and J. H. Sampson
in . The energy of a smooth map \varphi :(B,g_{B})\rightarrow (F,g_{F}) between two Riemannian manifolds is defined by E(\varphi )=\frac{1}{2}\int _{D}|d\varphi |^{2}v_{g_{B}} and... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 145,
"openalex_id": "",
"raw": "J. Eells, J.H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964) 109-160.",
"source_ref_id": "6c10598c378ff715736fb543f89ac3604b0b55ac",
"start": 0
},
... | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
35187,
333,
88975,
238,
22288,
7,
17721,
41419,
127613,
66,
17174,
42822,
4,
10,
4537,
47691,
111,
22313,
42459,
297,
390,
821,
241,
88510,
5,
3362,
254,
1681,
6,
48302,
156100,
1961,
19379,
571,
177,
454,
24854,
8152,
16,
41872,
54969,... | [
0.1236572265625,
0.19970703125,
0.343017578125,
0.1728515625,
0.256591796875,
0.1090087890625,
0.1220703125,
0.174072265625,
0.2257080078125,
0.0889892578125,
0.105712890625,
0.1947021484375,
0.0382080078125,
0.03826904296875,
0.135986328125,
0.042938232421875,
0.038238525390625,
0... | |
2aac1603828d1184a072a7a0b040a85a676c01d4 | subsection | 2 | 18 | Introduction | A doubly
warped product manifold is a product manifold B\times F of two Riemannian
manifolds (B,g_{B})\ and (F,g_{F}) endowed with the metric g=f^{2}g_{B}\oplus b^{2}g_{F} where b:B\rightarrow (0,\infty ) and f:F\rightarrow (0,\infty ) are smooth functions. The canonical leaves \lbrace x_{0}\rbrace \times F and B\times... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 494,
"openalex_id": "",
"raw": "R. Ponge, H. Reckziegel, Twisted Products in Pseudo-Riemannian Geometry, Geom. Dedicata 48 (1993) 15-25.",
"source_ref_id": "36876886dc8b9d8d615701d421afbd64c8152734",
"start": 260
}... | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
62,
54,
34,
38526,
1631,
20051,
12996,
17174,
42822,
83,
10,
335,
41872,
70141,
563,
111,
6626,
41419,
127613,
66,
7,
15,
571,
177,
454,
24854,
8152,
16,
136,
919,
22,
246,
186518,
706,
420,
8353,
304,
31,
32108,
876,
12,
54969,
118... | [
0.0931396484375,
0.14892578125,
0.1907958984375,
0.1956787109375,
0.233642578125,
0.2291259765625,
0.275390625,
0.1436767578125,
0.2039794921875,
0.114990234375,
0.013458251953125,
0.1109619140625,
0.015838623046875,
0.195068359375,
0.1533203125,
0.022003173828125,
0.143310546875,
... | |
1c4256e816a21a93aa05b9c12ed1a2195bed4296 | subsection | 3 | 18 | Biharmonic maps between Riemannian manifolds | Let (B,g_{B}) and (F,g_{F}) be Riemann manifolds and \varphi :(B,g_{B})\rightarrow (F,g_{F}) be a smooth map. The tension field of \varphi is given by\tau (\varphi )=trace\nabla d\varphi ,where \nabla d\varphi is the second fundamental form of \varphi .Biharmonic maps \varphi :(B,g_{B})\rightarrow (F,g_{F}) between Rie... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
10842,
571,
177,
8152,
16,
136,
919,
186,
41419,
5761,
17174,
42822,
7,
6,
1961,
19379,
24854,
41872,
54969,
118201,
156100,
22288,
63672,
44457,
34475,
50104,
1388,
39989,
13,
76,
7119,
104,
70,
17932,
20531,
3173,
16882,
88975,
238,
454... | [
0.05767822265625,
0.09234619140625,
0.0421142578125,
0.0144195556640625,
0.014190673828125,
0.011474609375,
0.092041015625,
0.015625,
0.1541748046875,
0.20166015625,
0.060821533203125,
0.1685791015625,
0.03082275390625,
0.014251708984375,
0.1832275390625,
0.283203125,
0.0142822265625... | |
5f463a489389d9d7520f9307d86423f30f480870 | subsection | 4 | 18 | Biharmonic maps between Riemannian manifolds | Here \Delta is
the rough Laplacian on sections of the pull-back bundle \varphi ^{-1}(TF)
defined by, for an orthonormal frame field \lbrace B_{j}\rbrace _{j=1}^{m} on B,\Delta v &=&-trace_{g_{b}}(\nabla ^{\varphi })^{2}v \\
&=&-\sum _{j=1}^{m}\lbrace \nabla _{B_{j}}^{\varphi }\nabla _{B_{j}}^{\varphi }v-\nabla _{\nabla... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
11853,
41872,
58598,
102,
83,
70,
166904,
239,
7290,
69438,
40059,
50065,
12620,
57134,
133,
1961,
19379,
5759,
30992,
61924,
707,
24948,
33176,
123789,
44457,
99407,
335,
170,
33000,
39,
81,
39989,
177,
76,
7119,
304,
334,
11832,
571,
73... | [
0.06658935546875,
0.0302276611328125,
0.2200927734375,
0.2210693359375,
0.108154296875,
0.0350341796875,
0.230224609375,
0.0872802734375,
0.1571044921875,
0.1859130859375,
0.08447265625,
0.1651611328125,
0.2052001953125,
0.1429443359375,
0.1063232421875,
0.072265625,
0.2064208984375,... | |
c501eb00d938398f052cbad434b75e43f3f6a1a7 | subsection | 5 | 18 | Doubly warped product manifolds | Let (B,g_{B}) and (F,g_{F}) be Riemannian manifolds of dimensions m
and n, respectively and let b:B\rightarrow (0,\infty ) and f:F\rightarrow (0,\infty ) be smooth functions. As a generalization of the
warped product of two Riemannian manifolds, a doubly warped product of
Riemannian manifolds (B,g_{B}) and (F,g_{F}) wi... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
10842,
571,
177,
136,
919,
454,
186,
41419,
127613,
66,
17174,
42822,
158208,
347,
653,
107013,
2633,
876,
22085,
46632,
939,
1238,
156100,
32354,
4537,
47691,
1631,
20051,
12996,
6626,
10,
54,
34,
38526,
678,
10366,
83,
335,
70141,
563,
... | [
0.0870361328125,
0.1253662109375,
0.0584716796875,
0.0845947265625,
0.130615234375,
0.004364013671875,
0.0262298583984375,
0.156005859375,
0.2254638671875,
0.085693359375,
0.09954833984375,
0.1885986328125,
0.1270751953125,
0.1165771484375,
0.05413818359375,
0.013763427734375,
0.0753... | |
f2234e4eda0771a75c66aedbcdcc74499ed86d0e | subsection | 6 | 18 | Doubly warped product manifolds | The Levi-Civita connection of doubly warped product manifold _{f}B\times _{b}F is defined by\overline{\nabla }_{X}Y &=&\nabla _{X}Y+\frac{1}{2b^{2}}X_{1}(b^{2})(0,Y_{2})+\frac{1}{2b^{2}}Y_{1}(b^{2})(0,X_{2}) \\
&&+\frac{1}{2f^{2}}X_{2}(f^{2})(Y_{1},0)+\frac{1}{2f^{2}}Y_{2}(f^{2})(X_{1},0) \\
&&-\frac{1}{2}g_{B}(X_{1},Y... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
581,
56890,
9,
40658,
16741,
94878,
111,
54,
34,
38526,
1631,
20051,
12996,
17174,
42822,
101,
420,
571,
70141,
275,
919,
83,
61924,
71,
390,
5465,
2256,
76,
7119,
51912,
454,
24854,
1542,
8152,
1723,
619,
1369,
1230,
1328,
41872,
13207... | [
0.034332275390625,
0.3291015625,
0.104736328125,
0.1474609375,
0.282958984375,
0.253173828125,
0.002410888671875,
0.09027099609375,
0.1556396484375,
0.1295166015625,
0.1949462890625,
0.1585693359375,
0.24560546875,
0.117919921875,
0.192626953125,
0.022369384765625,
0.1357421875,
0.... | |
465443f09e577bd4e4ccfc960a01aa72e176198d | subsection | 7 | 18 | Biharmonicity of the inclusion maps | Let (_{f}B\times _{b}F,g) be a doubly warped product manifold. For y_{0}\in F, let us consider the inclusion map of Bi_{y_{0}}:(B,g_{B}) &\rightarrow &(_{f}B\times _{b}F,g) \\
x &\rightarrow &(x,y_{0})at the point y_{0} level in _{f}B\times _{b}F and for x_{0}\in B leti_{x_{0}} :(F,g_{F})&\rightarrow & (_{f}B\times _{b... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
10842,
420,
571,
70141,
275,
919,
177,
186,
54,
34,
38526,
1631,
20051,
12996,
17174,
42822,
113,
454,
2389,
8152,
73,
563,
2633,
16916,
190440,
191,
22288,
1843,
53,
24854,
16,
41872,
54969,
118201,
132,
101,
4,
18991,
1022,
619,
425,
... | [
0.10003662109375,
0.1060791015625,
0.1351318359375,
0.165283203125,
0.056365966796875,
0.1390380859375,
0.10650634765625,
0.06756591796875,
0.053375244140625,
0.104248046875,
0.111572265625,
0.157958984375,
0.1796875,
0.2071533203125,
0.08038330078125,
0.1710205078125,
0.1181640625,
... | |
ae2f83c41921dc89787e14dc10b5fbc789453f5f | subsection | 8 | 18 | Biharmonicity of the inclusion maps | We have\nabla _{B_{j}}\tau (i_{y_{0}}) &=&-\frac{m}{2}\nabla _{B_{j}}(0,{grad}\ f^{2})|_{i_{y_{0}}} \\
&=&-\frac{m}{2}(\overline{\nabla }_{(B_{j},0)}(0,{grad}\ f^{2}))|_{i_{y_{0}}}
\\
&=&-m(\frac{1}{4f^{2}}|{grad}\ f^{2}|^{2}(B_{j},0)+\frac{1}{4b^{2}}(B_{j},0)(b^{2})(0,{grad}\ f^{2}))|_{i_{y_{0}}}.Then\nabla _{B_{j}}\n... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
1401,
765,
41872,
76,
7119,
101,
571,
454,
170,
47391,
50104,
15,
14,
53,
24854,
2389,
16,
619,
1369,
1230,
9,
132076,
39,
8152,
304,
132,
4,
8961,
1238,
8353,
58745,
18991,
5465,
2256,
51912,
77495,
418,
617,
420,
1328,
275,
51421,
... | [
0.0711669921875,
0.119873046875,
0.06890869140625,
0.20703125,
0.2452392578125,
0.05889892578125,
0.1617431640625,
0.09808349609375,
0.248779296875,
0.04315185546875,
0.348388671875,
0.0426025390625,
0.1368408203125,
0.1439208984375,
0.042724609375,
0.1895751953125,
0.04248046875,
... | |
a9f0faf92603df778ae6aea7c14812758a2c742b | subsection | 9 | 18 | Biharmonicity of the inclusion maps | Therefore the equation (REF ) has the form\tau _{2}(i_{y_{0}}) &=&\lbrace -\frac{m^{2}}{8b^{2}}|{grad}\ f^{2}|^{2}({grad}\ b^{2},0) \\
&&+\frac{m}{2}\Delta (\ln b)(0,{grad}\ f^{2})+\frac{m^{2}}{8}(0,{grad}(|{grad}\ f^{2}|^{2})\rbrace |_{i_{y_{0}}}.This completes the proof.Corollary 4.1 The inclusion map i_{y_{0}}:(B,g_... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
228072,
70,
28,
5490,
2320,
15,
11766,
919,
1388,
1556,
3173,
50104,
101,
304,
8152,
14,
24854,
53,
454,
2389,
47391,
16,
619,
1369,
1230,
41872,
99407,
20,
132076,
39,
8353,
1019,
275,
8961,
1238,
58745,
132,
876,
4,
18991,
1328,
585... | [
0.08984375,
0.044464111328125,
0.0887451171875,
0.239013671875,
0.1251220703125,
0.009368896484375,
0.1602783203125,
0.228759765625,
0.002227783203125,
0.13037109375,
0.21337890625,
0.2626953125,
0.016693115234375,
0.1815185546875,
0.0091552734375,
0.0643310546875,
0.0091552734375,
... | |
47a0e25d73fe862174759da4f0bca16c2b501be8 | subsection | 10 | 18 | Biharmonicity of the inclusion maps | We have,Theorem 4.2 The bitension field of the inclusion map i_{x_{0}}:(F,g_{F})\rightarrow (_{f}B\times _{b}F,g) is given by\tau _{2}(i_{x_{0}}) &=&\lbrace \frac{n^{2}}{8}({grad}(|{grad}\ b^{2}|^{2}),0)+\frac{n}{2}\Delta (\ln b)({grad}\ b^{2},0) \\
&&-\frac{n^{2}}{8f^{2}}|{grad}b^{2}|^{2}(0,{grad}f^{2})\rbrace |_{i_{x... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
1401,
765,
3957,
58391,
63708,
581,
333,
128872,
44457,
190440,
191,
22288,
17,
454,
425,
2389,
919,
177,
118201,
420,
571,
70141,
275,
83,
34475,
390,
50104,
304,
14,
99407,
132076,
19,
1019,
8961,
876,
1328,
58598,
563,
42,
186,
707,
... | [
0.07440185546875,
0.1231689453125,
0.046844482421875,
0.1790771484375,
0.20166015625,
0.00518798828125,
0.211181640625,
0.314453125,
0.2469482421875,
0.208984375,
0.0902099609375,
0.189697265625,
0.1348876953125,
0.037109375,
0.09600830078125,
0.15966796875,
0.09197998046875,
0.073... | |
c0a3ebf4610569084fd2a89efbaacbe7d4dc7b1d | subsection | 11 | 18 | Biharmonicity of the inclusion maps | Since\nabla _{F_{r}}\tau (i_{x_{0}}) &=&-\frac{n}{2}\nabla _{F_{r}}({grad}\ b^{2},0)|_{i_{x_{0}}} \\
&=&-\frac{n}{2}(\overline{\nabla }_{(0,F_{r})}({grad}\ b^{2},0))|_{i_{x_{0}}}
\\
&=&-\frac{n}{4}\lbrace \frac{1}{f^{2}}(0,F_{r})(f^{2})({grad}\ b^{2},0)+\frac{1}{b^{2}}|{grad}\ b^{2}|^{2}(0,F_{r})\rbrace |_{i_{x_{0}}}th... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
66016,
41872,
76,
7119,
101,
919,
454,
24854,
42,
47391,
50104,
14,
425,
2389,
16,
619,
1369,
1230,
9,
132076,
19,
8152,
304,
132,
8961,
876,
8353,
58745,
18991,
5465,
2256,
51912,
4,
77495,
617,
48543,
99407,
420,
51421,
1328,
418,
2... | [
0.1209716796875,
0.062255859375,
0.1976318359375,
0.2413330078125,
0.0548095703125,
0.1458740234375,
0.076171875,
0.06927490234375,
0.095703125,
0.03759765625,
0.33544921875,
0.137939453125,
0.1318359375,
0.1888427734375,
0.0374755859375,
0.050140380859375,
0.037139892578125,
0.061... | |
884537c7e728ef91206c32028af18481ce86578c | subsection | 12 | 18 | Biharmonicity of the inclusion maps | So we haveCorollary 4.6 Let (_{f}B\times _{b}F,g) be a doubly warped product manifold with
non-constant warping functions b and f. Then the inclusion map of the
manifold (F,g_{F}) into the doubly warped product manifold (_{f}B\times _{b}F,g) is never a proper biharmonic map.Remark 4.2 If f=1, _{f}B\times _{b}F becomes ... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
1061,
642,
765,
50886,
12116,
1294,
115459,
10842,
420,
571,
70141,
919,
177,
186,
54,
34,
38526,
1631,
20051,
12996,
17174,
42822,
351,
42539,
10366,
32354,
876,
136,
1238,
190440,
22288,
8152,
16,
3934,
83,
8306,
10,
27798,
333,
88975,
... | [
0.0091552734375,
0.035430908203125,
0.06292724609375,
0.04180908203125,
0.1590576171875,
0.09521484375,
0.1671142578125,
0.06219482421875,
0.0770263671875,
0.07684326171875,
0.1185302734375,
0.10443115234375,
0.052642822265625,
0.0261688232421875,
0.00653076171875,
0.045257568359375,
... | |
5e636a1445efcc14e083a15853dd0e3f315221bb | subsection | 13 | 18 | Product maps | Let I_{B}:B\rightarrow B be the identity map on B and \varphi :F\rightarrow F be a harmonic map. Obviuosly \Psi =I_{B}\times \varphi :B\times F\rightarrow B\times F is a harmonic map. Now suppose the product
manifold B\times F (either as domain or codomain) with the doubly warped
product metric tensor g=f^{2}g_{B}\oplu... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
10842,
87,
571,
118201,
335,
186,
182324,
22288,
98,
6,
1961,
19379,
54969,
563,
10,
22313,
238,
34,
232,
538,
683,
172,
2203,
568,
24854,
70141,
83,
139124,
12996,
17174,
42822,
15,
1399,
9319,
237,
77758,
552,
56206,
73,
16,
678,
70... | [
0.06390380859375,
0.033355712890625,
0.1298828125,
0.05426025390625,
0.1734619140625,
0.03338623046875,
0.218994140625,
0.2322998046875,
0.011322021484375,
0.011962890625,
0.0955810546875,
0.2000732421875,
0.0108642578125,
0.1925048828125,
0.0545654296875,
0.28076171875,
0.1645507812... | |
fe10f103ae1cbb838030e9291bfceef643de701e | subsection | 14 | 18 | Product maps | Since \varphi is harmonic,\tau (\overline{\Psi }) &=&trace_{g}\nabla d\overline{\Psi } \\
&=&\frac{1}{f^{2}}\sum _{j=1}^{m}\nabla d\overline{\Psi }((B_{j},0),(B_{j},0))+\frac{1}{b^{2}}\sum _{r=1}^{n}\nabla d\overline{\Psi }((0,F_{r}),(0,F_{r})) \\
&=&\frac{1}{f^{2}}\sum _{j=1}^{m}\lbrace \nabla _{(B_{j},0)}^{\overline{... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
66016,
41872,
1961,
19379,
83,
22313,
238,
50104,
5465,
2256,
24854,
683,
172,
51912,
16,
1369,
39989,
13,
454,
177,
8152,
76,
7119,
104,
18991,
619,
1230,
132076,
418,
420,
8353,
304,
47391,
11832,
170,
33000,
39,
132,
571,
4,
77495,
... | [
0.0653076171875,
0.024169921875,
0.1656494140625,
0.28955078125,
0.08740234375,
0.2445068359375,
0.158447265625,
0.300537109375,
0.1361083984375,
0.177001953125,
0.0197296142578125,
0.09765625,
0.1685791015625,
0.019500732421875,
0.0197601318359375,
0.0196533203125,
0.1619873046875,
... | |
e07f4f7fa01d0a0dddd4f5637ecf4278d16a62a3 | subsection | 15 | 18 | Product maps | By a straightforward
calculation, we get \tau (\bar{\pi })=n({grad}\ln b)\circ \bar{\pi }, and
the bitension field of \bar{\pi } is\tau _{2}(\bar{\pi }) &=&\frac{n}{f^{2}}trace_{g_{b}}\nabla ^{2}{grad}\ln b
\\
&&+\frac{n}{f^{2}}Ricc^{B}({grad}\ln b)+\frac{n^{2}}{2}{grad}(|{grad}\ln b|^{2}).By using the bitension field ... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
3311,
80560,
2472,
19364,
74481,
1363,
642,
2046,
41872,
50104,
1299,
24854,
1434,
51912,
16,
19,
132,
8961,
8152,
876,
82063,
6,
4,
70,
333,
128872,
44457,
83,
304,
1369,
132076,
420,
8353,
47391,
39989,
13,
454,
177,
275,
76,
7119,
... | [
0.0137939453125,
0.06207275390625,
0.0908203125,
0.0227813720703125,
0.153076171875,
0.015960693359375,
0.005157470703125,
0.05548095703125,
0.0234222412109375,
0.314697265625,
0.17236328125,
0.0232086181640625,
0.232177734375,
0.0230712890625,
0.023162841796875,
0.071044921875,
0.02... | |
9b2d0c0708d3d39fd9eb11021deaf31aa994de20 | subsection | 16 | 18 | Product maps | Then \widetilde{\Psi }=\widetilde{\varphi \times I_{F}}:\,_{f}B\times _{b}F\rightarrow B\times F is a proper
biharmonic map if and only if b is a non-constant solution of0=-\frac{1}{f^{2}}J_{\varphi }(d\varphi ({grad}\ln b))+\frac{n}{2}{grad}(|d\varphi ({grad}\ln b)|^{2})and f is a non-constant solution of0=\frac{1}{b^... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
113458,
3675,
112,
683,
172,
51912,
1961,
19379,
6,
70141,
87,
24854,
919,
47391,
12,
41872,
4,
454,
420,
8152,
275,
54969,
118201,
335,
563,
83,
10,
27798,
333,
88975,
238,
22288,
136,
4734,
2174,
876,
351,
9,
2271,
42539,
29806,
111... | [
0.1519775390625,
0.2421875,
0.221923828125,
0.0689697265625,
0.1541748046875,
0.002838134765625,
0.08056640625,
0.20703125,
0.0028076171875,
0.20654296875,
0.0225982666015625,
0.003173828125,
0.1021728515625,
0.002960205078125,
0.0029296875,
0.002899169921875,
0.0028076171875,
0.00... | |
237ab387730937d4bd4dbfd810152678434fba53 | subsection | 17 | 18 | Product maps | We will see
that the energy density of the harmonic map \varphi :F\rightarrow F has an
important role for the biharmonicity of the product map \widehat{\Psi }=\widehat{I_{B}\times \varphi }. We haveTheorem 5.3 Let (B,g_{B}) and (F,g_{F}) be Riemannian manifolds of dimensions m
and n, respectively and let b:B\rightarrow... | {
"cite_spans": []
} | 0807.1836 | Biharmonic maps between doubly warped product manifolds | [
"Selcen Yüksel Perktaş",
"Erol Kılıç"
] | [
"math.DG"
] | 2,008 | en | Mathematics | [
1221,
1957,
48302,
168,
7,
2481,
22313,
238,
22288,
1961,
19379,
919,
118201,
563,
1556,
142,
5526,
31486,
100,
333,
88975,
60089,
12996,
113458,
2943,
683,
172,
568,
571,
70141,
1401,
765,
3957,
58391,
129274,
10842,
177,
136,
186,
41419... | [
0.04534912109375,
0.0662841796875,
0.205078125,
0.2061767578125,
0.1240234375,
0.095703125,
0.265380859375,
0.1064453125,
0.2122802734375,
0.1065673828125,
0.224365234375,
0.1802978515625,
0.04443359375,
0.1888427734375,
0.035552978515625,
0.000030517578125,
0.114501953125,
0.14074... | |
0f96e2fb9f259f4f7e3a90c27fd97afa7eebb72c | abstract | 0 | 6 | Abstract | We find evidence for decaying magnons at strong magnetic field in the square
lattice spin-1/2 Heisenberg antiferromagnet. The results are obtained using
Quantum Monte Carlo simulations combined with a Bayesian inference technique to
obtain dynamics and are consistent with predictions from spin wave theory. | {
"cite_spans": []
} | 10.1103/PhysRevB.78.180413 | 0807.1837 | Numerical evidence for unstable magnons at high fields in the Heisenberg
antiferromagnet on the square lattice | [
"Olav F. Syljuasen"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
7413,
77950,
8,
408,
38543,
1697,
6431,
7,
99,
37515,
214706,
44457,
108047,
10495,
24494,
25927,
118551,
19614,
48467,
2874,
2875,
516,
155116,
50339,
113054,
17368,
75344,
316,
17629,
83551,
40226,
5256,
70163,
9631,
90,
3378,
53498,
6620,
... | [
0.0721435546875,
0.1959228515625,
0.1588134765625,
0.1739501953125,
0.136962890625,
0.0804443359375,
0.248779296875,
0.06549072265625,
0.0887451171875,
0.1270751953125,
0.156494140625,
0.15283203125,
0.1319580078125,
0.09039306640625,
0.1549072265625,
0.169921875,
0.175537109375,
0... |
21487a890055c1763829f9c066648cc233baba8b | subsection | 1 | 6 | Body | Numerical evidence for unstable magnons at high fields in the Heisenberg antiferromagnet on the square lattice
Olav F. Syljuåsen
Department of Physics, University of Oslo, P. O. Box 1048 Blindern, N-0316 Oslo, Norway75.10.Jm,05.10.Ln,75.40.Gb,75.50.EeWe find evidence for decaying magnons at strong magnetic field in the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1323,
"openalex_id": "",
"raw": "M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Lett. 82, 4536 (1999).",
"source_ref_id": "4f9db308b38f93c8e42099db197b784c3d088422",
"start": 1175
},
{
"arxiv_id": "",... | 10.1103/PhysRevB.78.180413 | 0807.1837 | Numerical evidence for unstable magnons at high fields in the Heisenberg
antiferromagnet on the square lattice | [
"Olav F. Syljuasen"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
93511,
77950,
2234,
22819,
1697,
6431,
7,
99,
11192,
44457,
19614,
48467,
2874,
2875,
516,
155116,
70,
6,
108047,
10495,
24494,
140437,
563,
4843,
12356,
57670,
33,
63557,
111,
165712,
4,
12535,
7397,
5,
180,
27271,
9285,
141524,
9022,
97... | [
0.09619140625,
0.190673828125,
0.11328125,
0.2305908203125,
0.1138916015625,
0.2354736328125,
0.051727294921875,
0.06378173828125,
0.14599609375,
0.1820068359375,
0.1275634765625,
0.25439453125,
0.143310546875,
0.14892578125,
0.07391357421875,
0.1739501953125,
0.0721435546875,
0.02... |
24038f44c99e6961e2f0b96338672444239cad35 | subsection | 2 | 6 | Body | Thus the spinon picture works well for explaining the (\pi ,0) anomaly qualitatively, however being essentially a projected mean field calculation it is not of the level of precision needed in order to compare quantitative details with exact results and conventional spin wave theory.In a magnetic field the distinction ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 559,
"openalex_id": "",
"raw": "O. F. Syljuåsen and P. A. Lee, Phys. Rev. Lett. 88, 207207 (2002).",
"source_ref_id": "fc124401c9fbe14c2503ece81e768e1454dbbb4a",
"start": 376
},
{
"arxiv_id": "",
"d... | 10.1103/PhysRevB.78.180413 | 0807.1837 | Numerical evidence for unstable magnons at high fields in the Heisenberg
antiferromagnet on the square lattice | [
"Olav F. Syljuasen"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
25927,
191,
49726,
43240,
5299,
73342,
214,
41872,
1434,
6,
77495,
10,
31079,
538,
150234,
21286,
4,
49903,
30378,
71,
29459,
44457,
74481,
1363,
83,
959,
111,
70,
22619,
1830,
23,
47,
69101,
88779,
5844,
41653,
678,
24763,
50339,
136,
... | [
0.2080078125,
0.1756591796875,
0.24365234375,
0.0771484375,
0.0804443359375,
0.1334228515625,
0.002685546875,
0.002777099609375,
0.1663818359375,
0.002166748046875,
0.142822265625,
0.021484375,
0.137451171875,
0.002197265625,
0.201171875,
0.0018310546875,
0.00274658203125,
0.048370... |
5bee322ffbb2771c503f07d104793431e23a9343 | subsection | 3 | 6 | Body | In contrast the antiferromagnetic signal at (\pi ,\pi ) remains even at high magnetic fields for the transverse spin components.
For high fields note that D^{\alpha }(\vec{q},0) becomes nearly independent of \vec{q} except close to \vec{q}=0 and \vec{q}=(\pi ,\pi ) for the longitudinal and transverse polarization respe... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1448,
"openalex_id": "",
"raw": "A. W. Sandvik and R. R. P. Singh, Phys. Rev. Lett. 86, 528 (2001).",
"source_ref_id": "18cb75e79e38b8ef75dd5e44a9bdfbf4e5b19d61",
"start": 1331
}
]
} | 10.1103/PhysRevB.78.180413 | 0807.1837 | Numerical evidence for unstable magnons at high fields in the Heisenberg
antiferromagnet on the square lattice | [
"Olav F. Syljuasen"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
69822,
2874,
2875,
516,
155116,
1771,
26073,
99,
1434,
6,
1388,
47143,
3853,
11192,
214706,
44457,
7,
3900,
37676,
25927,
82761,
391,
289,
14612,
35259,
864,
8152,
77495,
24209,
110518,
41371,
111,
20903,
47,
24854,
145407,
136,
41872,
1369... | [
0.1209716796875,
0.119384765625,
0.1199951171875,
0.067138671875,
0.1544189453125,
0.03558349609375,
0.21435546875,
0.0831298828125,
0.1756591796875,
0.012847900390625,
0.012847900390625,
0.0908203125,
0.021881103515625,
0.109619140625,
0.2152099609375,
0.2103271484375,
0.01287841796... |
6c0c09ffc2dd3cff34bfa5dc32f8c9b63c38eb61 | subsection | 4 | 6 | Body | (REF ) in a canted coordinate system, characterized by the canting angle \theta , and expressing the spin operators by boson operators according to the Holstein-Primakoff transformation one gets terms with all orders of bosonic creation and annihilation operators, including linear and cubic terms. Minimizing the energy... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 951,
"openalex_id": "",
"raw": "M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Lett. 82, 4536 (1999).",
"source_ref_id": "4f9db308b38f93c8e42099db197b784c3d088422",
"start": 678
},
{
"arxiv_id": "",
... | 10.1103/PhysRevB.78.180413 | 0807.1837 | Numerical evidence for unstable magnons at high fields in the Heisenberg
antiferromagnet on the square lattice | [
"Olav F. Syljuasen"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
11766,
919,
23,
831,
3674,
176866,
13,
5426,
4,
62816,
29367,
390,
70,
1916,
55291,
6,
2347,
102,
36510,
214,
25927,
39933,
7,
337,
1681,
47,
186872,
145111,
344,
16713,
167201,
1632,
62163,
69407,
756,
12989,
111,
1771,
166635,
4483,
1... | [
0.1585693359375,
0.2386474609375,
0.036346435546875,
0.16357421875,
0.156982421875,
0.1759033203125,
0.0192108154296875,
0.1561279296875,
0.0193023681640625,
0.07794189453125,
0.01910400390625,
0.019317626953125,
0.019378662109375,
0.1015625,
0.135986328125,
0.0192108154296875,
0.138... |
7c242dd649ae53c77f061f6d76972785c27f8bd6 | subsection | 5 | 6 | Body | Here the system size is L=16 and the inverse temperature is \beta J=200.]For magnetic fields above H \approx 3J the self-energy \Sigma acquires an imaginary part due to decay of magnons thru the three-boson vertex in Eq. (REF ). This implies that the perturbation expression Eq. (REF ) will not work well for finding the... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 377,
"openalex_id": "",
"raw": "M. E. Zhitomirsky and A. L. Chernyshev, Phys. Rev. Lett. 82, 4536 (1999).",
"source_ref_id": "4f9db308b38f93c8e42099db197b784c3d088422",
"start": 230
},
{
"arxiv_id": "",
... | 10.1103/PhysRevB.78.180413 | 0807.1837 | Numerical evidence for unstable magnons at high fields in the Heisenberg
antiferromagnet on the square lattice | [
"Olav F. Syljuasen"
] | [
"cond-mat.str-el"
] | 2,008 | en | Physics | [
11853,
5426,
13267,
339,
2485,
70,
23,
37676,
52768,
6,
59865,
821,
5955,
214706,
44457,
7,
36917,
572,
2631,
64101,
138,
1375,
15970,
39060,
3432,
294,
872,
192,
163629,
142,
114135,
53,
2831,
4743,
47,
8,
408,
111,
1697,
6431,
5675,
... | [
0.021728515625,
0.135986328125,
0.09747314453125,
0.015045166015625,
0.1112060546875,
0.0167388916015625,
0.02166748046875,
0.176025390625,
0.1868896484375,
0.01641845703125,
0.10113525390625,
0.0654296875,
0.0880126953125,
0.1944580078125,
0.1614990234375,
0.0166778564453125,
0.1070... |
53f76556f39ed10f10245d88d330e40921a4f831 | abstract | 0 | 22 | Abstract | There is presented an algorithm for computing the topological degree for a
large class of polynomial mappings. As an application there is given an
effective algebraic formula for the intersection number of a polynomial
immersion M --> R^2m, where M is an m-dimensional algebraic manifold. | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
8622,
8121,
71,
142,
234873,
100,
242122,
70,
2663,
109622,
79385,
21334,
18507,
111,
35874,
1687,
15403,
291,
26783,
38415,
34475,
60266,
144,
429,
2844,
26168,
1940,
7,
58994,
14012,
3807,
6889,
276,
129908,
627,
8353,
304,
39,
347,
157... | [
0.00811767578125,
0.1192626953125,
0.00555419921875,
0.011871337890625,
0.264404296875,
0.0177459716796875,
0.19140625,
0.0631103515625,
0.1807861328125,
0.17578125,
0.202880859375,
0.0760498046875,
0.1318359375,
0.056060791015625,
0.1580810546875,
0.147216796875,
0.06365966796875,
... | |
2eda50bef775bff12d8063ad351f0d686863d248 | subsection | 1 | 22 | Quotients of ideals in | Lemma 1.1
Let f_1,\ldots ,f_r be polynomials with real coefficients. Let
S_{\mathbb {R}} (resp. S_{) denote the ideal in \mathbb {R}[x] (resp.
x]) generated by f_1,\ldots ,f_r. Then
\begin{}
\item [(i)] S_{\mathbb {R}}=S_{\cap \mathbb {R}[x], \item [(ii)]S_{\mathbb {R}}=\mathbb {R}[x]\Leftrightarrow S_{=x].
}
}
{\em P... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
636,
18023,
38245,
10842,
1238,
115187,
30591,
420,
454,
42,
186,
35874,
1687,
15403,
7,
678,
2773,
552,
4240,
11044,
35066,
159,
125458,
5125,
1052,
24854,
8,
48345,
70,
6397,
23,
425,
1022,
139392,
6820,
217,
15644,
1573,
294,
118201,
... | [
0.0865478515625,
0.230224609375,
0.1978759765625,
0.1278076171875,
0.214111328125,
0.2049560546875,
0.138916015625,
0.1077880859375,
0.1173095703125,
0.096923828125,
0.086669921875,
0.19580078125,
0.1871337890625,
0.12744140625,
0.01416015625,
0.0849609375,
0.220458984375,
0.112792... | |
0f1f4b489ae8a2c3370c195dc1b74114353919aa | subsection | 2 | 22 | Quotients of ideals in | Of course
J_{\mathbb {R}}:I_{\mathbb {R}}=\lbrace h\in \mathbb {R}[x]: hg_i \in J_{\mathbb {R}}, \quad \textrm {for each} \quad 1 \le i \le s\rbrace
J_{:I_{=\lbrace h\in x]: hg_i \in J_{, \quad \textrm {for
each} \quad 1 \le i \le s\rbrace .
}Take h\in (J_{\mathbb {R}}:I_{\mathbb {R}})_{, then there exist w_1, \ldots ... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
6619,
15411,
821,
454,
24854,
125458,
5125,
1052,
47391,
12,
568,
41872,
1369,
48543,
99407,
1096,
73,
6,
8152,
425,
268,
177,
14,
4,
91526,
22829,
42,
39,
2472,
12638,
106,
133,
17,
1022,
10666,
91,
5,
51912,
51654,
1375,
16,
7068,
... | [
0.1007080078125,
0.1845703125,
0.301025390625,
0.1375732421875,
0.06390380859375,
0.1553955078125,
0.218017578125,
0.252197265625,
0.120361328125,
0.136962890625,
0.2066650390625,
0.07586669921875,
0.186279296875,
0.057403564453125,
0.207275390625,
0.25341796875,
0.1728515625,
0.03... | |
f3b18f39d3d9706b2ce99b0e2ac225a1f1c6bf21 | subsection | 3 | 22 | Quotients of ideals in | \hfill \Box
}If S_{\mathbb {K}} is an ideal in \mathbb {K}[x], where \mathbb {K} is either or
\mathbb {R}, denote
V(S_{\mathbb {K}})=\lbrace p\in \mathbb {K}^n\ :\ f(p)=0\ \mbox{for all}\ f\in S_{\mathbb {K}}\rbrace .
}Consider ideals J_{\mathbb {R}}\subset I_{\mathbb {R}} \subset \mathbb {R}[x], such that
\dim _{\mat... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
41872,
127,
116115,
72295,
51912,
91306,
159,
454,
125458,
5125,
605,
47391,
83,
142,
6397,
23,
425,
268,
4,
6,
8152,
40101,
707,
1052,
8,
48345,
310,
294,
16,
48543,
99407,
915,
10666,
8353,
19,
152,
1238,
132,
254,
145407,
11728,
24... | [
0.0155792236328125,
0.0633544921875,
0.15087890625,
0.1944580078125,
0.011444091796875,
0.046417236328125,
0.162841796875,
0.08056640625,
0.1229248046875,
0.2255859375,
0.1910400390625,
0.044769287109375,
0.08306884765625,
0.0909423828125,
0.28125,
0.077392578125,
0.1187744140625,
... | |
4980e07e50fb1b95a6e537c34b47974508b12aa0 | subsection | 4 | 22 | Quotients of ideals in | For p\in V(J_{)\setminus V(I_{) , we define an -algebra
\mathcal {A}_{p}:=\frac{x]}{J_{+m_{p}^k}, \quad \textrm {where}
\quad m_{p}=\lbrace f\in x]:f(p)=0\rbrace . Of course, f\in J_{+m_{p}^k if and only if \overline{f}\in J_{+m_{\overline{p}}^k. In particular
\begin{equation} \mathbb {R}[x]\cap (J_{+m_{p}^k)=\mathbb {... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
1326,
915,
73,
310,
1375,
454,
24854,
3509,
34731,
568,
642,
61924,
142,
20,
289,
429,
2844,
125458,
6827,
284,
254,
132076,
425,
1328,
39,
8353,
92,
91526,
22829,
347,
99407,
1238,
1022,
420,
145407,
821,
2174,
5465,
2256,
6820,
5490,
... | [
0.009735107421875,
0.1826171875,
0.108154296875,
0.212646484375,
0.1685791015625,
0.053070068359375,
0.051300048828125,
0.1112060546875,
0.20947265625,
0.07086181640625,
0.012420654296875,
0.1702880859375,
0.04144287109375,
0.0777587890625,
0.096435546875,
0.1671142578125,
0.19970703... | |
368b50f1b9d95e978cf2ec71dc586017867160d7 | subsection | 5 | 22 | Quotients of ideals in | Using Lemma \ref {postac j:i 2} and
(\ref {wazne1}) we get
\begin{align*}
\ker \pi & = \mathbb {R}[x] \cap \bigcap _{i=1}^m (J_{\mathbb {R}}+m_{\mathbb {R},p_i}^k) \cap \bigcap _{j=1}^r (J_{+m_{q_j}^k)=\\
& =\mathbb {R}[x] \cap \bigcap _{i=1}^m
(J_{\mathbb {R}}+m_{\mathbb {R},p_i}^k)_{ \cap \bigcap _{j=1}^r
(J_{+m_{q_j... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
345,
6953,
636,
18023,
29087,
16063,
238,
1647,
12,
14,
116,
8152,
136,
73079,
86,
418,
16,
642,
2046,
372,
6820,
143420,
1639,
6,
1728,
1434,
619,
2203,
41872,
125458,
5125,
1052,
425,
268,
15644,
32976,
24854,
33000,
8353,
39,
15,
1... | [
0.0736083984375,
0.050262451171875,
0.04541015625,
0.24951171875,
0.285888671875,
0.135986328125,
0.1063232421875,
0.14208984375,
0.059356689453125,
0.0672607421875,
0.148193359375,
0.009429931640625,
0.0518798828125,
0.1314697265625,
0.164794921875,
0.064697265625,
0.009124755859375... | |
97d0f6e3d5448ed338e3d3d6c25bfabd778945d1 | subsection | 6 | 22 | Quotients of ideals in | For p\in \mathbb {K}^n, let {\cal O}_{\mathbb {K},p} denote the ring of germs at p of analytic functions
\mathbb {K}^n\longrightarrow \mathbb {K}. There is a natural homomorphism
\eta :\mathbb {K}[x]=\mathbb {K}[x_1, \ldots ,x_n]\longrightarrow {\cal O}_{\mathbb {K},p}. Let
m_{\mathbb {K},p}=\lbrace f\in \mathbb {K}[x]... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
1326,
915,
41872,
73,
6,
125458,
5125,
605,
8353,
19,
2633,
6827,
180,
8152,
254,
8,
48345,
15789,
111,
117979,
7,
99,
140815,
32354,
10666,
8622,
83,
10,
6083,
12840,
178851,
8780,
4241,
1065,
425,
268,
1369,
115187,
4,
10617,
54969,
... | [
0.0745849609375,
0.2120361328125,
0.03717041015625,
0.11376953125,
0.03759765625,
0.1129150390625,
0.1712646484375,
0.186767578125,
0.0654296875,
0.1495361328125,
0.07861328125,
0.1866455078125,
0.1922607421875,
0.022979736328125,
0.19189453125,
0.1221923828125,
0.1636962890625,
0.... | |
e37a15ca21a0f544c8569103cc2a564388ee7881 | subsection | 7 | 22 | Quotients of ideals in | If that is the case then p
is isolated in V(S_{\mathbb {R}}), and \eta (m^k_{\mathbb {K},p}) \subset S_{\mathbb {K},p} for all k large enough.
\begin{}
If p\in V(S_{\mathbb {K}}) is isolated in H_{^{-1}(0) and
S_{\mathbb {K},p}=J_{\mathbb {K},p}, then \eta induces an isomorphism of
\mathbb {K}--algebras \eta : \mathbb ... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
4263,
450,
7225,
7068,
915,
83,
54015,
3674,
23,
310,
294,
454,
41872,
125458,
5125,
1052,
47391,
247,
136,
6,
4241,
8353,
92,
24854,
605,
8152,
4,
254,
16,
22144,
3509,
159,
756,
472,
21334,
20174,
372,
6820,
73,
10666,
572,
5759,
... | [
0.060333251953125,
0.03033447265625,
0.0887451171875,
0.01611328125,
0.237060546875,
0.058563232421875,
0.246337890625,
0.1630859375,
0.1197509765625,
0.2327880859375,
0.0933837890625,
0.035400390625,
0.004791259765625,
0.06317138671875,
0.144775390625,
0.0850830078125,
0.01768493652... | |
0d8d16199b3088b8e2a23dbb8a0fa1517e5a595b | subsection | 8 | 22 | Quotients of ideals in | Obviously, {\mathcal {B}} is a finite dimensional
\mathbb {R}--algebra.
}}Let u\in \mathbb {R}[x], and let \varphi :{\mathcal {B}} \longrightarrow \mathbb {R}
be a linear functional. Then there are bilinear symmetric forms
\Phi ,\,\Psi :{\mathcal {B}} \times {\mathcal {B}}\longrightarrow \mathbb {R} given by
\Phi (a,b)... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
3545,
686,
79850,
125458,
6827,
571,
47391,
83,
10,
94418,
13,
157955,
5125,
1052,
429,
2844,
124480,
75,
73,
425,
2633,
1961,
19379,
10617,
54969,
186,
192617,
123309,
2685,
621,
43286,
28575,
230612,
238,
3173,
45689,
14,
683,
172,
7014... | [
0.03387451171875,
0.0665283203125,
0.06219482421875,
0.06201171875,
0.1685791015625,
0.2080078125,
0.07867431640625,
0.1656494140625,
0.08367919921875,
0.207275390625,
0.0860595703125,
0.2410888671875,
0.07427978515625,
0.132568359375,
0.0894775390625,
0.124267578125,
0.0630493164062... | |
3b13e413b4df899210c38007dce96b6279e35943 | subsection | 9 | 22 | Quotients of ideals in | There is the natural projection \mathbb {R}[x,x^{\prime }]\longrightarrow {\mathcal {B}}\otimes {\mathcal {B}} given by
x_1^{\alpha _1}\cdots x_n^{\alpha _n}(x_1^{\prime })^{\beta _1}\cdots (x_n^{\prime })^{\beta _n}
\mapsto x_1^{\alpha _1}\cdots x_n^{\alpha _n}\otimes (x_1^{\prime })^{\beta _1}\cdots (x_n^{\prime })^{... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
8622,
83,
70,
6083,
13452,
1830,
125458,
5125,
1052,
425,
114654,
54969,
118201,
571,
31,
70141,
34475,
1022,
115187,
14612,
418,
15464,
19,
59865,
62346,
384,
8,
48345,
29569,
3667,
13786,
104,
5771,
66596,
28,
30591,
13,
71,
3173,
18231... | [
0.1217041015625,
0.0791015625,
0.0155792236328125,
0.2293701171875,
0.2275390625,
0.1485595703125,
0.022735595703125,
0.1107177734375,
0.12548828125,
0.05419921875,
0.1561279296875,
0.037506103515625,
0.09710693359375,
0.13232421875,
0.063232421875,
0.1961669921875,
0.093017578125,
... | |
298505f4e36a23cdd4cde8ca34362092c467b26a | subsection | 10 | 22 | Quotients of ideals in | \end{}
\hfill \Box
\end{equation}}}\section {Topological degree}
}Let h_1,\ldots ,h_n\in \mathbb {R}[x_1,\ldots ,x_n], and let
H_{\mathbb {K}}=(h_1,\ldots ,h_n):\mathbb {K}^n\longrightarrow \mathbb {K}^n. Denote by
J_{\mathbb {K}} the ideal in \mathbb {K}[x] generated by h_1,\ldots ,h_n, so
that H_{^{-1}(0)=V(J_{).
}A... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
127,
116115,
72295,
3611,
13,
5490,
2320,
58994,
77888,
109622,
79385,
124480,
1096,
115187,
30591,
454,
19,
1052,
2633,
572,
605,
125458,
262,
48345,
821,
6397,
425,
139392,
5759,
177609,
1369,
856,
132,
1375,
24854,
66596,
2685,
142,
87,
... | [
0.06622314453125,
0.1312255859375,
0.1697998046875,
0.007171630859375,
0.00164794921875,
0.1697998046875,
0.047210693359375,
0.070556640625,
0.134033203125,
0.10595703125,
0.194580078125,
0.0782470703125,
0.091064453125,
0.145263671875,
0.0987548828125,
0.0144500732421875,
0.07983398... | |
95a0ac3b62fc2a85e76c00a88999f16f1162adcd | subsection | 11 | 22 | Quotients of ideals in | Hence each point p\in V(S_{)=H_{^{-1}(0)\setminus V(I_{) satisfies the assumptions of Lemma \ref {pi indukuje}.
}Put H_{\mathbb {R}}^{-1}(0)\setminus V(I_{\mathbb {R}})=\lbrace p_1,\ldots ,p_m\rbrace and
(H_{^{-1}(0)\setminus V(I_{))\setminus \mathbb {R}^n=
\lbrace q_1,\overline{q_1},\ldots ,q_r,\overline{q_r}\rbrace \... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
6620,
12638,
6275,
915,
73,
310,
294,
454,
1369,
841,
5759,
177609,
3509,
34731,
40407,
3387,
237259,
7,
636,
18023,
29087,
1434,
116401,
1411,
27559,
572,
5125,
1052,
99407,
115187,
30591,
8096,
5465,
75358,
581,
58391,
73079,
299,
445,
... | [
0.04913330078125,
0.1473388671875,
0.1854248046875,
0.1275634765625,
0.0894775390625,
0.212890625,
0.054718017578125,
0.04827880859375,
0.00592041015625,
0.0865478515625,
0.075927734375,
0.1260986328125,
0.0897216796875,
0.178466796875,
0.1756591796875,
0.0660400390625,
0.19836425781... | |
1f689b4a67f3f2ebaa2f1b83e0864daa33c8610d | subsection | 12 | 22 | Quotients of ideals in | \end{}
\hfill \Box
}Let \Psi _T be the bilinear form on \mathcal {A} given by
\Psi _T(a,b)=\varphi _T(uab). Using the same arguments as in
\cite [Theorem 1.5, p. 304]{Szafraniec} one may prove
\begin{} \Psi _T is non-degenerate if and only if u(p)\ne 0 for each p\in H_{^{-1}(0)\setminus V(I_{). If that is
the case the... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
116115,
72295,
124480,
683,
172,
101,
618,
186,
43286,
28575,
3173,
98,
125458,
6827,
284,
34475,
275,
1961,
19379,
34,
2055,
5701,
10750,
47959,
58391,
22410,
112986,
6000,
133110,
1543,
23534,
351,
112,
48281,
2174,
75,
254,
757,
12638,
... | [
0.0574951171875,
0.1007080078125,
0.07318115234375,
0.114501953125,
0.2034912109375,
0.0653076171875,
0.2156982421875,
0.0618896484375,
0.177978515625,
0.259765625,
0.200439453125,
0.021697998046875,
0.03802490234375,
0.1207275390625,
0.0634765625,
0.00244140625,
0.040069580078125,
... | |
e0e51579145115e2317e15658689726193b044bf | subsection | 13 | 22 | Quotients of ideals in | According to the Lashof and
Smale \cite [Theorem 3.1]{LashofSmale},
d(G)=2I(g).
}Now assume that f=(f_1,\ldots \,f_n):\mathbb {R}^{n+m} \longrightarrow \mathbb {R}^n is a C^1 mapping, such that M=f^{-1}(0) and M is a
complete intersection, i.e. for each p\in M the rank of Df(p)
equals n.
}We shall say that vectors v_1,... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
129551,
70,
5599,
21676,
136,
159,
49100,
47959,
3957,
58391,
45151,
2729,
7,
104,
724,
55257,
568,
177,
41591,
1238,
420,
115187,
30591,
19,
125458,
5125,
1052,
1328,
39,
54969,
118201,
83,
10,
313,
8353,
418,
291,
26783,
276,
5759,
17... | [
0.045166015625,
0.033172607421875,
0.158203125,
0.304443359375,
0.1239013671875,
0.1490478515625,
0.251708984375,
0.017852783203125,
0.0570068359375,
0.1278076171875,
0.10638427734375,
0.0887451171875,
0.14404296875,
0.1982421875,
0.1739501953125,
0.1292724609375,
0.1099853515625,
... | |
b856734985cb096bc239ee3819c9a8f513e0599d | subsection | 14 | 22 | Quotients of ideals in | As \operatorname{rank}\left[
\begin{array}{c}
D\overline{g}\\
Df\\
\end{array} \right]=\operatorname{rank}\left[
\begin{array}{ccc}
1 & 0 & 0\\
0 & 1 & 0\\
z & 0 & x\\
0 & z & y\\
2x & 2y & 2z\\
\end{array} \right]
has a non-zero (3\times 3)--minor at each point p\in \mathbb {R}^3\setminus \lbrace 0\rbrace , then g=\ov... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
1301,
206469,
11627,
36467,
8152,
2480,
6,
372,
6820,
19305,
24854,
238,
391,
5465,
2256,
177,
420,
13273,
41872,
3611,
53,
54969,
268,
1369,
133,
10060,
106,
619,
757,
97,
1022,
113,
116,
425,
169,
1556,
10,
351,
80510,
6896,
70141,
... | [
0.052947998046875,
0.1341552734375,
0.0703125,
0.2406005859375,
0.0052490234375,
0.007659912109375,
0.0035400390625,
0.00274658203125,
0.06036376953125,
0.1358642578125,
0.003387451171875,
0.03851318359375,
0.19873046875,
0.0716552734375,
0.14111328125,
0.17822265625,
0.19140625,
0... | |
43c6ce1ac87aeb5903f53bec20e93f0b02aca94b | subsection | 15 | 22 | Quotients of ideals in | \end{}
\hfill \Box
}A homotopy h_t\colon M\longrightarrow \mathbb {R}^{2m} is called \emph {a regular
homotopy}, if at each stage it is an immersion and the induced
homotopy of the tangent bundle is continuous.
}As in \cite {whitneySelfInter} we say that an immersion g\colon M\longrightarrow \mathbb {R}^{2m} has \emph... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
116115,
72295,
284,
12840,
13784,
53,
1096,
18,
22796,
276,
1052,
304,
39,
35839,
195,
11727,
20324,
2174,
12638,
36541,
3807,
6889,
135989,
25269,
2517,
57134,
133,
62005,
47959,
11400,
10186,
104475,
44851,
5154,
706,
1556,
15970,
10433,
... | [
0.045654296875,
0.10595703125,
0.044708251953125,
0.2335205078125,
0.275146484375,
0.1859130859375,
0.04730224609375,
0.06402587890625,
0.0180511474609375,
0.1229248046875,
0.10296630859375,
0.1273193359375,
0.1903076171875,
0.1536865234375,
0.110595703125,
0.2467041015625,
0.2670898... | |
2bb4882ee3059636b25288acdb7087ea0d0239f5 | subsection | 16 | 22 | Quotients of ideals in | Then each
self--intersection is represented by a single point in H^{-1}(0)\cap \lbrace u>0\rbrace \setminus \Delta .
}\begin{}
Suppose that m>1 is odd, and
\begin{}
\item [(a)] M=f^{-1}(0) is an oriented, compact m--dimensional
complete intersection, \item [(b)] g=\overline{g}|M\colon M\longrightarrow \mathbb {R}^{2m} ... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
47009,
12638,
15970,
9,
10433,
7,
58994,
83,
33636,
297,
390,
11001,
6275,
23,
572,
8353,
5759,
177609,
15644,
99407,
75,
2389,
3509,
34731,
58598,
102,
121691,
8364,
347,
2740,
418,
70270,
6820,
217,
276,
420,
142,
23509,
94928,
157955,
... | [
0.058197021484375,
0.1663818359375,
0.2452392578125,
0.0285186767578125,
0.1944580078125,
0.20068359375,
0.208740234375,
0.1075439453125,
0.251220703125,
0.10009765625,
0.0455322265625,
0.1629638671875,
0.1727294921875,
0.036102294921875,
0.127197265625,
0.035888671875,
0.14440917968... | |
deee7b7322f716792d2f8cdb39774d963f2646b1 | subsection | 17 | 22 | Quotients of ideals in | \hfill \Box
}}If f_1,\ldots ,f_n,g_1,\ldots ,g_{2m} are polynomials then
H=(h_1,\ldots ,h_{2n+2m}) is a polynomial mapping. Let J_{\mathbb {R}}
denote the ideal in \mathbb {R}[x,y]=\mathbb {R}[x_1,\ldots ,x_{n+m}, y_1,\ldots ,y_{n+m}] generated by h_1,\ldots ,h_{2n+2m}, and I_{\mathbb {R}} the
one generated by f_1(x),... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
127,
116115,
72295,
47391,
91306,
1238,
115187,
30591,
420,
454,
19,
177,
304,
39,
621,
35874,
1687,
15403,
7,
7068,
572,
1369,
54651,
83,
10,
291,
26783,
10842,
821,
125458,
5125,
1052,
8,
48345,
70,
6397,
53,
1328,
113,
7344,
139392,
... | [
0.1387939453125,
0.105712890625,
0.1121826171875,
0.0203857421875,
0.0841064453125,
0.1256103515625,
0.19189453125,
0.1044921875,
0.08856201171875,
0.04229736328125,
0.1026611328125,
0.076416015625,
0.1031494140625,
0.157470703125,
0.076416015625,
0.238037109375,
0.2236328125,
0.15... | |
0695419203a1e7627ffb28009e938b1c07826346 | subsection | 18 | 22 | Quotients of ideals in | As a consequence of Theorem \ref {teczka}
and Proposition \ref {odd} we get
\begin{}
If m>1 is odd and \det [\Psi ]\ne 0, then I(g)\equiv \dim _{\mathbb {R}}\mathcal {A}+1+(\operatorname{sgn}\det [\Phi ]+\operatorname{sgn}\det [\Psi ])/2\mod {2}.
\end{}
\hfill \Box
}}}\begin{}
Let us consider the mapping g=(g_1,g_2,g_... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
179804,
6620,
581,
58391,
29087,
18,
89451,
8152,
136,
1250,
40322,
13606,
2046,
6,
6820,
4263,
347,
2740,
418,
83,
70270,
3667,
683,
172,
10114,
757,
87,
177,
16,
3181,
334,
5771,
24854,
41872,
5125,
1052,
47391,
125458,
6827,
284,
217... | [
0.1226806640625,
0.001190185546875,
0.040802001953125,
0.19091796875,
0.249755859375,
0.0204925537109375,
0.2218017578125,
0.000823974609375,
0.053436279296875,
0.11328125,
0.1485595703125,
0.214111328125,
0.05279541015625,
0.0003662109375,
0.025909423828125,
0.053802490234375,
0.177... | |
5477989f944a996e3f5f0e1b10cc92d7f13a4624 | subsection | 19 | 22 | Quotients of ideals in | Then A_1=-\frac{1}{8}, A_2=0, so for any
a=a_1+a_2y_3=a_1e_1+a_2e_2 in \mathcal {A}, \varphi _T is given by
\varphi _T(a)=-\frac{a_1}{8}. Then the matrix of \Phi _T is
given by \left[ \begin{array}{cc} -\frac{1}{8} & 0
\\0 & -\frac{1}{8} \end{array} \right], so \operatorname{signature}\Phi _T=-2, and as a consequence o... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
62,
115187,
1369,
9,
132076,
418,
1019,
454,
304,
145407,
2499,
10,
11,
1328,
53,
363,
13,
125458,
6827,
284,
1961,
19379,
618,
34475,
50944,
425,
45689,
14,
101,
2480,
19305,
10060,
757,
2389,
54969,
206469,
11627,
137432,
5428,
179804,
... | [
0.0823974609375,
0.1708984375,
0.0249786376953125,
0.016998291015625,
0.1119384765625,
0.042236328125,
0.1622314453125,
0.013519287109375,
0.1348876953125,
0.1527099609375,
0.05316162109375,
0.05853271484375,
0.0237274169921875,
0.0845947265625,
0.0068359375,
0.235595703125,
0.039764... | |
b2da8f95f8e591c0a77e255e59fa17551452f137 | subsection | 20 | 22 | Quotients of ideals in | Of course \det [\Psi ]\ne 0, and as consequence of Theorem
\ref {nowe2} we get I(g|_{S^3(1)})\equiv 2+1+\frac{1}{2}(-1+1)
\equiv 1\ \mod {2} .
}
}Using similar methods we have computed some more difficult
examples:
\begin{}
h(x_1,x_2,x_3)==(2x_1x_2+x_2,2x_1x_3+4x_3,4x_3^2+5x_2,5x_2^2+4x_3)
is an immersion on the 2-dime... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
6619,
15411,
6,
41872,
3667,
683,
172,
86,
757,
179804,
581,
58391,
29087,
2600,
304,
642,
2046,
87,
177,
58745,
454,
294,
8353,
363,
27750,
8152,
16,
13,
3181,
116,
21748,
1328,
132076,
132,
5759,
334,
106,
13415,
51912,
21373,
150624,... | [
0.0396728515625,
0.1405029296875,
0.012786865234375,
0.058197021484375,
0.3134765625,
0.126220703125,
0.2298583984375,
0.07647705078125,
0.17236328125,
0.09051513671875,
0.05950927734375,
0.1990966796875,
0.2108154296875,
0.16162109375,
0.1392822265625,
0.031280517578125,
0.112670898... | |
d707eb6ebd1475dd44ac979598ac155669296982 | subsection | 21 | 22 | Quotients of ideals in | In that case
\dim _{\mathbb {R}}\mathcal {A}=18, and I(h|_{S^3(1)})\equiv 1\ \mod {2}.
\end{}
}}\begin{}{00}
\end{}\bibitem {BCRSz}E.~Becker, J.P.~Cardinal, M.F.~Roy, and Z.~Szafraniec,
Multivariate Bezutians, Kronecker symbol and Eisenbud-Levine
formula, Progress in Mathematics, vol.143, 1996, p. 79-104,
\bibitem {Cox... | {
"cite_spans": []
} | 0807.1838 | An algebraic formula for the intersection number of a polynomial
immersion | [
"Iwona Karolkiewicz",
"Aleksandra Nowel",
"Zbigniew Szafraniec"
] | [
"math.AG"
] | 2,008 | en | Mathematics | [
360,
450,
7225,
6,
41872,
5771,
24854,
125458,
5125,
1052,
47391,
284,
8152,
1369,
1819,
4,
136,
87,
127,
58745,
454,
294,
8353,
363,
27750,
16,
13,
3181,
334,
106,
13415,
10666,
304,
5,
372,
6820,
7049,
127872,
2982,
24318,
15396,
16... | [
0.0693359375,
0.133544921875,
0.2119140625,
0.022430419921875,
0.041717529296875,
0.263671875,
0.022674560546875,
0.140869140625,
0.1295166015625,
0.1358642578125,
0.0225677490234375,
0.16552734375,
0.022857666015625,
0.076416015625,
0.263916015625,
0.0836181640625,
0.155029296875,
... | |
a35ddc9b72b79bbfe830818271c986823547c9c5 | abstract | 0 | 7 | Abstract | Using the Vlasov-wave formalism, it is shown that self-consistency vanishes
in the plateau regime of the bump-on-tail instability if the plateau is broad
enough. This shows that, in contrast with the "turbulent trapping" Ansatz, a
renormalization of the Landau growth rate or of the quasilinear diffusion
coefficient is ... | {
"cite_spans": []
} | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
345,
310,
2512,
515,
9,
634,
272,
23113,
8780,
127887,
15970,
25553,
6892,
2408,
131,
4745,
90,
37385,
916,
63647,
373,
2676,
191,
46741,
23,
271,
41159,
2174,
134744,
20174,
45831,
69822,
987,
978,
22042,
87631,
214,
893,
38250,
456,
3... | [
0.003753662109375,
0.105224609375,
0.1981201171875,
0.25830078125,
0.006591796875,
0.1619873046875,
0.0858154296875,
0.2142333984375,
0.131103515625,
0.081298828125,
0.161865234375,
0.176025390625,
0.248046875,
0.09161376953125,
0.117919921875,
0.126953125,
0.05926513671875,
0.1873... | |
8f8c050eb0b5d7da6b50796506ea56419b518b54 | subsection | 1 | 7 | Introduction | The saturation of the bump-on-tail instability is a tough problem of kinetic plasma physics, which is still the subject of a controversy , . It was originally tackled in the frame of the Vlasov-Poisson formalism through the quasilinear approximation that neglects mode coupling , . However mode coupling was proved to be... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 140,
"openalex_id": "",
"raw": "Laval G and Pesme D 1999 Controversies about quasilinear theory Plasma Phys. Control. Fusion 41 A239–A246",
"source_ref_id": "1e965732bfeba799dccf03744bd2e886c2addf40",
"start": 0
},... | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
2134,
30494,
111,
373,
2676,
191,
46741,
23,
271,
41159,
83,
143033,
2967,
200,
86,
9523,
80796,
34053,
27744,
7464,
28368,
17340,
7864,
53,
7311,
36461,
123789,
310,
2512,
515,
7192,
14,
4503,
23113,
8780,
12404,
2256,
53950,
124789,
137... | [
0.228271484375,
0.160888671875,
0.0149688720703125,
0.1064453125,
0.2269287109375,
0.1982421875,
0.2978515625,
0.095703125,
0.2327880859375,
0.1759033203125,
0.0193328857421875,
0.16357421875,
0.1890869140625,
0.051116943359375,
0.0880126953125,
0.0902099609375,
0.2210693359375,
0.... | |
e873b664bbc3f089a41b355a6d017ed5527363f4 | subsection | 2 | 7 | Description of wave-particle self-consistency | The difficulty to describe the nonlinear regime of the Vlasov-Poisson system of equations, and the progress in the chaotic dynamics of Hamiltonian systems with a finite number of degrees of freedom were an incentive to tackle the description of the saturation regime with the so-called self-consistent Hamiltonian that d... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 517,
"openalex_id": "",
"raw": "Elskens Y and Escande D F 2003 Microscopic Dynamics of Plasmas and Chaos (Bristol: Institute of Physics Publishing)",
"source_ref_id": "9689a2ab8eb8b30972a4bc9c615b12db984efa08",
"start"... | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
34844,
53,
98363,
351,
2256,
147,
63647,
310,
2512,
515,
7192,
14,
4503,
5426,
111,
28,
13722,
5256,
70,
42658,
1608,
31,
9523,
84079,
7,
94674,
3378,
76519,
10,
94418,
13,
14012,
79385,
147452,
3542,
240804,
36461,
76811,
2134,
30494,
... | [
0.133544921875,
0.0175933837890625,
0.173583984375,
0.09222412109375,
0.1739501953125,
0.046875,
0.2427978515625,
0.046630859375,
0.105712890625,
0.1666259765625,
0.078125,
0.057525634765625,
0.156005859375,
0.162109375,
0.0313720703125,
0.114501953125,
0.20654296875,
0.06555175781... | |
230088e11f8e208e60b31d8d9aadb9a2e34f92ce | subsection | 3 | 7 | Description of wave-particle self-consistency | Equation () makes clear the link between Z_j = X_j + {\mathrm {i}}Y_j and the electric field of wave j.For the present paper, it is easier to describe the tail particles through a velocity distribution function f(x,p,t). This is possible through the so-called Vlasov-wave model that is obtained as a mean-field limit (li... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 489,
"openalex_id": "",
"raw": "Firpo M-C and Elskens Y 1998 Kinetic limit of N-body description of wave-particle self-consistent interaction J. Stat. Phys. 93 193–209",
"source_ref_id": "a2b0229db85773014c0b42ff5f7116bafd39... | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
241,
5490,
2320,
30482,
34735,
3126,
17721,
567,
454,
170,
2203,
1193,
997,
125458,
42,
39,
14,
1723,
39108,
44457,
111,
259,
272,
1647,
13379,
15122,
99156,
47,
98363,
70,
6,
46741,
2878,
66695,
8305,
191060,
939,
113068,
32354,
1238,
... | [
0.060546875,
0.2149658203125,
0.0484619140625,
0.0418701171875,
0.1358642578125,
0.166259765625,
0.055694580078125,
0.1468505859375,
0.06201171875,
0.199462890625,
0.0178680419921875,
0.0880126953125,
0.1104736328125,
0.032745361328125,
0.0290069580078125,
0.11279296875,
0.0289459228... | |
c3d1e8369faf849a33df2211954ca4d19043ea10 | subsection | 4 | 7 | Dynamics when the distribution is a plateau | This paper considers the dynamics defined by Eqs (REF -REF ), while starting at time t=0 from (i) a spectrum of Langmuir waves where all nearby waves are in resonance overlap and (ii) a spatially uniform plateau for the particle velocity distribution function over this overlap domain with a height f_0. We first start w... | {
"cite_spans": []
} | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
3293,
15122,
16916,
84079,
7,
61924,
71,
390,
241,
864,
11766,
919,
20,
6,
247,
12960,
72134,
99,
1733,
808,
145407,
1295,
15,
14,
16,
10,
235079,
15786,
561,
481,
259,
3132,
756,
43573,
1272,
621,
23,
3332,
191,
7154,
645,
6324,
15... | [
0.042022705078125,
0.1103515625,
0.111328125,
0.296630859375,
0.1531982421875,
0.18798828125,
0.0318603515625,
0.040130615234375,
0.07354736328125,
0.2490234375,
0.1630859375,
0.246826171875,
0.0318603515625,
0.0318603515625,
0.031646728515625,
0.0174560546875,
0.11865234375,
0.031... | |
c2eb6ad79db6ae79e4d58c1bb819051a5f336522 | subsection | 5 | 7 | Dynamics when the distribution is a plateau | The chaotic domain {\rm C} (t_0) in single-particle (Boltzmann or \mu ) phase space (x,p) defined by this frozen wave spectrum is bounded above and below in p by two KAM tori, respectively {\rm T_a} (t_0) and {\rm T_b} (t_0). The initial particle distribution function f(x,p,0) is assumed to be uniform on {\rm C} (0) ; ... | {
"cite_spans": []
} | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
1608,
31,
9523,
77758,
313,
18,
454,
77495,
23,
11001,
254,
26147,
136055,
9136,
5761,
707,
6,
561,
1388,
93402,
32628,
15,
425,
16,
61924,
903,
1238,
70463,
259,
272,
235079,
876,
167457,
36917,
136,
35064,
915,
390,
6626,
88139,
3826,... | [
0.223388671875,
0.1749267578125,
0.2015380859375,
0.21484375,
0.1170654296875,
0.06170654296875,
0.07080078125,
0.177734375,
0.006256103515625,
0.1617431640625,
0.1025390625,
0.1715087890625,
0.08673095703125,
0.0655517578125,
0.1751708984375,
0.019561767578125,
0.005706787109375,
... | |
0cd9c2c0d482e3325e151343a15800f1d201a838 | subsection | 6 | 7 | Conclusion | We have just shown that self-consistency vanishes in the plateau regime of the bump-on-tail instability if the plateau is broad enough. This means that the diffusion coefficient D(p) of particles with momentum p is that found for the dynamics of particles in a prescribed spectrum of Langmuir waves. Let D_{\rm QL}(p) be... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 448,
"openalex_id": "",
"raw": "Cary J R, Escande D F and Verga A D 1990 Non quasilinear diffusion far from the chaotic threshold Phys. Rev. Lett. 65 3132–3135",
"source_ref_id": "c98a4c32c8b0f47a5b2a53eb2fe86943557f9142",
... | 0807.1839 | Self-consistency vanishes in the plateau regime of the bump-on-tail
instability | [
"Dominique F. Escande",
"Yves Elskens"
] | [
"physics.plasm-ph",
"nlin.CD"
] | 2,008 | en | Physics | [
1660,
127887,
15970,
25553,
6892,
2408,
131,
4745,
90,
37385,
916,
63647,
373,
2676,
191,
46741,
23,
271,
41159,
2174,
83,
134744,
20174,
26950,
45755,
92105,
552,
13,
24500,
45964,
391,
132,
254,
16,
2878,
66695,
678,
3095,
316,
915,
1... | [
0.022796630859375,
0.09136962890625,
0.1834716796875,
0.190185546875,
0.25341796875,
0.093017578125,
0.1075439453125,
0.1253662109375,
0.05670166015625,
0.2110595703125,
0.2227783203125,
0.25830078125,
0.0677490234375,
0.19140625,
0.160400390625,
0.244873046875,
0.06451416015625,
0... | |
08ddd2adf70285a016a43f8ceacaf1121ae64f8d | abstract | 0 | 20 | Abstract | In this work we show that single horizon black hole behaves as a "laser". It
is in many aspects conceptually analogous to Corley and Jacobson work on the
two horizon black hole "laser". We started by proposition that circumference of
the black hole horizon holds the natural (integer) quantum number of
corresponding red... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
903,
4488,
7639,
11001,
156634,
22556,
108564,
186,
68991,
7,
237,
10,
2512,
56,
228113,
60223,
10821,
5631,
4293,
30492,
1681,
6626,
26859,
89261,
75723,
69988,
111,
16401,
6083,
6191,
110436,
14012,
34390,
15612,
1507,
259,
140909,
337,
7... | [
0.009552001953125,
0.06097412109375,
0.0894775390625,
0.2325439453125,
0.25927734375,
0.1363525390625,
0.2210693359375,
0.0386962890625,
0.1639404296875,
0.01141357421875,
0.10986328125,
0.0153045654296875,
0.1744384765625,
0.2161865234375,
0.093994140625,
0.073486328125,
0.010681152... | |
82c2623735db420c4460b085742d26f59dc07608 | subsection | 1 | 20 | Introduction | As it is known Corley and Jacobson [1] have shown that a two
horizon boson black hole can behave as a "laser" that amplifies
Hawking radiation. Precisely, Corley and Jacobson have shown that:
"High energy frequency spectrum of the Hawking radiation from a
single black hole horizon, whether the dispersion entails
sublum... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
1301,
442,
83,
51529,
5631,
4293,
136,
30492,
1681,
17550,
127887,
10,
6626,
6,
156634,
337,
22556,
108564,
831,
186,
68991,
237,
44,
2512,
56,
58,
70827,
1029,
96139,
214,
4567,
2320,
21286,
4,
163025,
48302,
12478,
944,
27771,
235079,
... | [
0.016937255859375,
0.033905029296875,
0.03399658203125,
0.0863037109375,
0.15625,
0.222412109375,
0.1029052734375,
0.2196044921875,
0.192138671875,
0.021026611328125,
0.1221923828125,
0.04010009765625,
0.2083740234375,
0.034027099609375,
0.270263671875,
0.135009765625,
0.192016601562... | |
908661b1d19100a499ae08c4c000df7130e92f61 | subsection | 2 | 20 | Introduction | Also ground state is
practically totally occupied while other states are practically
totally unoccupied that is a typical Bose condensation. Number of
the systems in this condensate (multiplied by Boltzmann constant)
represents black hole entropy which yields a simple explanation of
the black hole entropy. Obviously, f... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
22376,
61585,
11341,
138155,
112668,
95699,
34,
138518,
12960,
3789,
117249,
3269,
143414,
222201,
26854,
13,
188171,
1363,
103332,
76519,
903,
67,
297,
12981,
9136,
5761,
53697,
33636,
22556,
108564,
49478,
6493,
11180,
8781,
187136,
686,
7985... | [
0.03839111328125,
0.18798828125,
0.1591796875,
0.1561279296875,
0.1048583984375,
0.135009765625,
0.110595703125,
0.1094970703125,
0.01519775390625,
0.0254669189453125,
0.16748046875,
0.061614990234375,
0.1571044921875,
0.1175537109375,
0.1363525390625,
0.181640625,
0.18896484375,
0... | |
2bf1cbe4cc7042dd077219e52fd9b4f47b4a1c44 | subsection | 3 | 20 | A simple determination of the black hole thermodynamical characteristics
and a simple model of the black hole | Firstly, we shall shortly repeat our previous results [3]-[5].Analyze a Schwarzschild's black hole with mass M and
Schwarzschild's radiusR = \frac{2GM}{c^{2}}where G represents Newtonian gravitational constant and c -
speed of light.Introduce the following conditionm_{n}c R = n\frac{\hbar }{2p} \hspace{28.45274pt} {\rm... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
23972,
538,
642,
35299,
16610,
119140,
2446,
96362,
50339,
45963,
73026,
731,
10,
89383,
3232,
38472,
25,
7,
22556,
108564,
678,
46889,
276,
136,
4567,
223,
1052,
6,
41872,
132076,
24854,
304,
56948,
8152,
238,
8353,
136913,
527,
33636,
1... | [
0.06011962890625,
0.1446533203125,
0.03729248046875,
0.043548583984375,
0.0794677734375,
0.1302490234375,
0.0252685546875,
0.030914306640625,
0.1424560546875,
0.040191650390625,
0.1453857421875,
0.040313720703125,
0.06561279296875,
0.1954345703125,
0.15478515625,
0.2353515625,
0.0764... | |
c81278434bfa95f993927c3974c43ba97e7828d1 | subsection | 4 | 20 | A simple determination of the black hole thermodynamical characteristics
and a simple model of the black hole | But, for a
"microscopic" black hole, i.e. for M \le M_{P} it follows
m_{1}\ge M_{P}\ge M.Define now the following\sigma = \frac{M}{ m_{1}} = 4\pi \frac{GM^{2}}{\hbar c}that, after multiplication by Boltzmann constant k_{B}, yieldsS = k_{B}\sigma = 4\pi k_{B} \frac{GM^{2}}{\hbar c} = k_{B}\frac{c^{3}}{4G\hbar }A = k_{B}... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
4966,
100,
10,
187840,
12064,
18695,
22556,
108564,
276,
133,
683,
442,
28960,
347,
418,
429,
187423,
1212,
25632,
20561,
192,
132076,
594,
201,
1434,
56948,
304,
1299,
501,
7103,
127664,
12981,
9136,
5761,
53697,
472,
571,
11180,
19388,
... | [
0.035064697265625,
0.0310211181640625,
0.0202178955078125,
0.1053466796875,
0.105712890625,
0.0941162109375,
0.1640625,
0.2337646484375,
0.1162109375,
0.087646484375,
0.059722900390625,
0.01361083984375,
0.1474609375,
0.05499267578125,
0.058624267578125,
0.1298828125,
0.10400390625,
... | |
cc1bff4be56884b5ecfeda6efc0f614e50c7fc83 | subsection | 5 | 20 | A simple determination of the black hole thermodynamical characteristics
and a simple model of the black hole | Evidently, this temperature
is identical to Hawking black hole temperature.Further, according to (8)-(10) it followsdA = (32\pi \frac{G^{2}}{c^{3}}) M dMor, in a corresponding finite difference form\Delta A = (32\pi \frac{G^{2}}{c^{3}}) M \Delta M \hspace{28.45274pt} {\rm for} \hspace{28.45274pt} \Delta M \ll M .Now, a... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
156267,
538,
903,
52768,
83,
31943,
6827,
96139,
214,
22556,
108564,
9319,
4,
59499,
47,
10021,
90980,
16,
28960,
71,
284,
2203,
6460,
41872,
1434,
6,
132076,
24854,
724,
8353,
304,
47391,
363,
276,
104,
101398,
10,
42518,
94418,
13,
60... | [
0.08441162109375,
0.020538330078125,
0.1298828125,
0.258544921875,
0.0360107421875,
0.124755859375,
0.0789794921875,
0.1175537109375,
0.1651611328125,
0.1826171875,
0.2093505859375,
0.00103759765625,
0.021209716796875,
0.010009765625,
0.0210113525390625,
0.031951904296875,
0.00238037... | |
8c7c7aab1497306c264bb660a48085e485c4a8a7 | subsection | 6 | 20 | A simple determination of the black hole thermodynamical characteristics
and a simple model of the black hole | (But, as it has been
already pointed out, there is a principal difference in respect to
Bohr's atomic model.)In this way we have reproduced, i.e. determined exactly in a
mathematically and physically simple way, three most important
characteristics of Schwarzschild's black hole thermodynamics,
Bekenstein-Hawking entrop... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
82212,
237,
21771,
6275,
2685,
83,
10,
7893,
60212,
15072,
56352,
42,
34627,
1771,
3299,
903,
3917,
642,
765,
42238,
297,
83324,
71,
66161,
140363,
136,
72761,
8781,
17262,
2684,
5526,
62816,
48242,
89383,
3232,
38472,
22556,
108564,
182342... | [
0.037353515625,
0.019378662109375,
0.04510498046875,
0.058685302734375,
0.0208892822265625,
0.0189971923828125,
0.047821044921875,
0.09173583984375,
0.180908203125,
0.017364501953125,
0.08062744140625,
0.111328125,
0.111083984375,
0.0214385986328125,
0.209716796875,
0.0537109375,
0.1... | |
feea0effd9d70241026389d11bfd202abacc7fe6 | subsection | 7 | 20 | Single horizon black hole "laser" | Now we shall demonstrate that suggested model of the black hole
admits consistent theoretical description of the stimulated
emission of the radiation by black hole.Suppose that s has been initially in some lower quantum state
|E^{s}_{k}> and that higher quantum state |E^{s}_{n}> , for k
< n, has been initially "empty",... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
35299,
106804,
42459,
297,
3299,
111,
70,
22556,
108564,
36456,
74729,
4524,
76811,
142405,
3674,
28,
21150,
4567,
2320,
390,
2037,
78381,
91,
1556,
2809,
61475,
23,
3060,
92319,
110436,
11341,
647,
92,
77546,
100,
472,
4426,
653,
33548,
... | [
0.022705078125,
0.117431640625,
0.1243896484375,
0.015533447265625,
0.2333984375,
0.033660888671875,
0.03973388671875,
0.190673828125,
0.25634765625,
0.123046875,
0.1697998046875,
0.1192626953125,
0.1617431640625,
0.230712890625,
0.1085205078125,
0.106201171875,
0.17626953125,
0.19... | |
8a272d9d2bb780c271f2ec00791445906f6a6ec0 | subsection | 8 | 20 | Single horizon black hole "laser" | In this case L can be considered
as reservoir.Then, as it is well-known, statistical sum can be presented by
expressionZ = \sum _{n,N(n)=0}\exp [-N(n)\frac{E_{n}-\mu }{k_{B}T}] = \sum _{n=0} Z_{n} .Here \mu represents the chemical potential, N(n) - number of
the systems in quantum state of the individual system
|E^{s}_... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
360,
903,
7225,
339,
831,
186,
90698,
237,
74336,
17990,
4,
442,
83,
5299,
69723,
19,
80835,
289,
10554,
8121,
390,
125195,
1511,
2203,
6,
11832,
24854,
839,
16,
145407,
8152,
41872,
83613,
9,
132,
132076,
647,
454,
561,
51912,
92,
57... | [
0.0266265869140625,
0.086181640625,
0.1309814453125,
0.2099609375,
0.090087890625,
0.020965576171875,
0.1466064453125,
0.08294677734375,
0.1949462890625,
0.257080078125,
0.0209503173828125,
0.02105712890625,
0.02081298828125,
0.064208984375,
0.1220703125,
0.1171875,
0.266357421875,
... | |
cb8ada1b0435b32bcd51d281ee5190746e7b3713 | subsection | 9 | 20 | Single horizon black hole "laser" | It implies that right
hand of (29) can be approximated by first term in the sum. Namely,
according to (17), (24), it follows<N>_{n}n \simeq n \exp [-2n] \simeq 0
\hspace{28.45274pt} {\rm for}\hspace{5.69046pt} {\rm large} \hspace{5.69046pt} n \hspace{28.45274pt} i.e. \hspace{5.69046pt}n \gg 1 .For this reason practical... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
1650,
35388,
7108,
3535,
111,
77583,
831,
186,
35707,
53950,
27686,
390,
5117,
13579,
23,
10554,
538,
4,
43956,
58072,
28960,
839,
2740,
19,
6,
41872,
13777,
864,
653,
83613,
378,
5428,
268,
757,
127,
65421,
24854,
3882,
5,
170265,
1636... | [
0.0159149169921875,
0.11181640625,
0.163818359375,
0.2113037109375,
0.03570556640625,
0.2607421875,
0.0792236328125,
0.008087158203125,
0.1597900390625,
0.244873046875,
0.0411376953125,
0.018524169921875,
0.1541748046875,
0.2210693359375,
0.0279693603515625,
0.2265625,
0.007751464843... | |
80862de00d6de75e235990256bf8833464c25156 | subsection | 10 | 20 | Single horizon black hole "laser" | It can be effectively treated as a
Bose-Einstein condensation. Also, it points out unambiguously that
here inverse population cannot exist, even not approximately. In
other words, for a macroscopic black hole for one horizon there is
no stimulated emission of the radiation.Also, it can be pointed out that for a macrosc... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
1650,
831,
191984,
191607,
237,
10,
26854,
13,
9,
61061,
18055,
188171,
1363,
26847,
3688,
23,
37676,
43904,
53418,
32316,
189275,
111789,
12064,
18695,
22556,
108564,
100,
1632,
156634,
110,
142405,
28,
21150,
4567,
2320,
6275,
35431,
70264,... | [
0.0673828125,
0.048095703125,
0.0684814453125,
0.109619140625,
0.055328369140625,
0.031982421875,
0.126708984375,
0.138427734375,
0.01104736328125,
0.057861328125,
0.148193359375,
0.168701171875,
0.06243896484375,
0.0462646484375,
0.061004638671875,
0.048583984375,
0.2109375,
0.181... | |
4133eb14d0c1e5f9f19246be6f85df45b5709f30 | subsection | 11 | 20 | Single horizon black hole "laser" | In this way a microscopic bosonic black hole can be
(partially) considered as a "laser".Suppose now that black hole, precisely s, can be considered as a
great canonical statistical ensemble of the ideal, non-interacting
Fermi-Dirac quantum systems.In this case, as it is well-known, it followsZ_{n} = (1 + \exp [-\frac{E... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
903,
3917,
10,
11948,
12064,
18695,
337,
1681,
1771,
22556,
108564,
831,
186,
15866,
25958,
90698,
237,
44,
2512,
56,
2037,
5036,
134995,
91,
6782,
74413,
21533,
80835,
289,
63304,
70,
6397,
351,
9,
10433,
2263,
1916,
8002,
266,
14055,
... | [
0.031463623046875,
0.03106689453125,
0.07440185546875,
0.1170654296875,
0.131591796875,
0.104248046875,
0.1336669921875,
0.1522216796875,
0.145751953125,
0.153076171875,
0.23095703125,
0.093505859375,
0.049224853515625,
0.126953125,
0.0076904296875,
0.1767578125,
0.1068115234375,
0... | |
692b20788ebbcea442ea88d3fd9ca8f57214ca80 | subsection | 12 | 20 | A simple solution of the information loss paradox | In distinction from spontaneous emission of the radiation, that is
decoherent, stimulated emission of the radiation is coherent. It
represents the general characteristics of the quantum theory and
refers on the single horizon black hole too. Since by its Hawking
evaporation any macroscopic black hole turns out in a mic... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
149067,
19,
150189,
10821,
28,
21150,
111,
4567,
2320,
4,
83,
8,
587,
3334,
2517,
142405,
3674,
241463,
33636,
7,
70,
4537,
62816,
48242,
110436,
154453,
136,
15005,
98,
11001,
6,
156634,
22556,
108564,
5792,
390,
96139,
214,
70890,
4680,... | [
0.14697265625,
0.0274200439453125,
0.16845703125,
0.027557373046875,
0.1197509765625,
0.1993408203125,
0.027435302734375,
0.2269287109375,
0.152099609375,
0.02752685546875,
0.082275390625,
0.1591796875,
0.15771484375,
0.240478515625,
0.026123046875,
0.265869140625,
0.1397705078125,
... | |
708514d1c0f84ebc1a79bdbf4798867b4282376f | subsection | 13 | 20 | A simple solution of the information loss paradox | On the
other hand, SP as the sub-system of IN+SP, in respect to any
characteristics sub-systemic measurement realized outside black
hole horizon, is effectively described by mixture (51).In this way, before time moment of the total black hole
evaporation by means of Hawking radiation, according to usual
quantum mechani... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
2161,
70,
12047,
237,
1614,
16751,
5881,
1328,
9434,
23,
47,
2499,
62816,
48242,
7,
72350,
674,
185171,
50782,
22556,
108564,
156634,
4,
83,
191984,
151552,
390,
17664,
6644,
62634,
4153,
8108,
1733,
3095,
3622,
70890,
4680,
2320,
111,
96... | [
0.0158843994140625,
0.0164031982421875,
0.22314453125,
0.05029296875,
0.1988525390625,
0.1968994140625,
0.1722412109375,
0.21044921875,
0.173583984375,
0.0164794921875,
0.0163116455078125,
0.0209503173828125,
0.0947265625,
0.0284423828125,
0.0165252685546875,
0.1368408203125,
0.01629... | |
8404b03bd3ac1dd1dabe90bbdc200d42cf60aeef | subsection | 14 | 20 | A simple solution of the information loss paradox | Then
macroscopic black hole evaporation can be satisfactorily described
by (48) including effective description of Hawking radiation by
(51).But in the time moment when given black hole becomes microscopic
and ST non-neglecting, evaporation process must be described in
the more complex way, i.e. by a more complex than ... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
47009,
111789,
12064,
18695,
22556,
108564,
70890,
4680,
2320,
831,
64561,
538,
151552,
390,
72208,
60266,
76811,
96139,
214,
4567,
62634,
82212,
23,
70,
1733,
3095,
34475,
24209,
7,
11948,
16992,
351,
9,
8996,
1916,
9433,
8110,
186,
1286,
... | [
0.0133056640625,
0.11865234375,
0.124755859375,
0.07818603515625,
0.156005859375,
0.2174072265625,
0.187255859375,
0.229248046875,
0.132568359375,
0.05767822265625,
0.0950927734375,
0.0172882080078125,
0.203857421875,
0.0555419921875,
0.1915283203125,
0.1285400390625,
0.18408203125,
... | |
2b839ac461c0ad7e177539357a409a8c11b20ac8 | subsection | 15 | 20 | A simple solution of the information loss paradox | Super-system IN+SP+ST or, simplifying, SP+ST, is,
even after black hole, i.e. IN total evaporation, exactly
completely described by pure, correlated quantum state (63) or
(64), on the one hand. On the other hand, SP as a sub-system of
the super-system, in respect to any characteristics sub-systemic
measurement realized... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
4265,
9,
16751,
5881,
1328,
9434,
8545,
707,
112892,
18929,
12047,
83,
3853,
7103,
22556,
108564,
3622,
70890,
4680,
2320,
66161,
64557,
151552,
34166,
8231,
174822,
110436,
11341,
82036,
88235,
237,
1614,
1601,
62816,
72350,
185171,
191984,
... | [
0.233154296875,
0.0709228515625,
0.24072265625,
0.22265625,
0.25634765625,
0.220703125,
0.22705078125,
0.0325927734375,
0.1190185546875,
0.019012451171875,
0.232177734375,
0.083251953125,
0.0611572265625,
0.14599609375,
0.1461181640625,
0.2200927734375,
0.17578125,
0.1600341796875,... | |
5c43a9f726f42deb29fe808f35a3202bc9ffeeaa | subsection | 16 | 20 | Mass duality as T-duality | Previously suggested model of the black hole, according to which
large system is surrounded by (statistical ensemble of the) small
system on the horizon surface area, in some degree similar to
atomic nucleus by electronic shell in Bohr atomic theory, is
simple and intuitively clear. But this intuitive clearness,
strict... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
6422,
42459,
3299,
111,
22556,
108564,
59499,
21334,
5426,
613,
42,
167457,
59130,
63304,
19336,
156634,
71579,
16128,
21373,
34627,
315,
11030,
390,
65133,
128019,
56352,
154453,
83,
8781,
136,
100270,
34735,
4966,
7432,
81113,
214521,
4734,
... | [
0.0589599609375,
0.140380859375,
0.2330322265625,
0.019775390625,
0.1478271484375,
0.234130859375,
0.024566650390625,
0.094482421875,
0.1500244140625,
0.041961669921875,
0.063232421875,
0.0897216796875,
0.10150146484375,
0.135009765625,
0.1392822265625,
0.162841796875,
0.125366210937... | |
84aeaaa3fc3d60ce1897325ca3a07d493faafbbd | subsection | 17 | 20 | Mass duality as T-duality | Or we can say that (58) and (59) are mutually dual
in sense that (58) can be changed by (59) and vice versa by
changing of m_{1} by M and vice versa.Moreover, left hand of the condition (2) for n=1, according to
(1), can be transformed in the following waym_{1}c R = m_{1} \frac{2GM}{c^{2}} = Mc\frac{2Gm_{1}}{c^{2}}= Mc... | {
"cite_spans": []
} | 0807.1840 | Single Horizon Black Hole "Laser" and a Solution of the Information Loss
Paradox | [
"Vladan pankovic",
"Rade Glavatovic",
"Simo Ciganovic",
"Dusan Harper Petkovic",
"Lovro Loka Martinovic"
] | [
"gr-qc",
"astro-ph"
] | 2,008 | en | Physics | [
3347,
831,
5154,
71716,
136,
78255,
621,
199849,
538,
87758,
10422,
186,
98816,
390,
22925,
105274,
151134,
111,
347,
454,
418,
8152,
276,
4,
25737,
3535,
35431,
1737,
100,
653,
33000,
59499,
47,
798,
27198,
297,
23,
70,
25632,
3917,
39... | [
0.05194091796875,
0.11376953125,
0.07171630859375,
0.280517578125,
0.1116943359375,
0.259765625,
0.101318359375,
0.177734375,
0.1419677734375,
0.302001953125,
0.0782470703125,
0.0208282470703125,
0.216552734375,
0.03302001953125,
0.056976318359375,
0.0352783203125,
0.16357421875,
0... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.