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 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
2c004c063953ffa030013012b5313b887d2e19cc | subsection | 16 | 19 | Coherent structure frequency | The coherent structure frequency is another important parameter for the burner characterization. As the detection algorithm gives the coordinates of each detected vortex, it is possible to count the number of vortices that crosses a given interrogation area of the chamber during a given time (1 second for the present s... | {
"cite_spans": []
} | 0807.1871 | Characterization of the reactive flow field dynamics in a gas turbine
injector using high frequency PIV | [
"Séverine Barbosa",
"Philippe Scouflaire",
"Sébastien Ducruix"
] | [
"physics.flu-dyn"
] | 2,008 | en | Physics | [
241463,
45646,
12478,
944,
27771,
83,
15700,
5526,
171859,
96865,
56,
62816,
47691,
70,
149,
58994,
234873,
76199,
176866,
90,
111,
12638,
96391,
297,
1248,
24371,
4,
442,
7722,
54529,
14012,
24494,
7,
41421,
10,
34475,
101934,
1363,
16128,... | [
0.252685546875,
0.1968994140625,
0.1817626953125,
0.13818359375,
0.0614013671875,
0.033355712890625,
0.060455322265625,
0.12109375,
0.1968994140625,
0.257568359375,
0.2218017578125,
0.1875,
0.1134033203125,
0.033477783203125,
0.15625,
0.10394287109375,
0.2008056640625,
0.0417175292... | |
c21a299ec5f672cd106a50a93ad9a134658159fa | subsection | 17 | 19 | Coherent structure frequency | This last point is even more pronounced for the v spectrum of point A, where the signal is broadband in the range [0; 1000] HzIt can be noted that f_u and f_v are very close to the vortex frequency determined before and to the acoustic quater wave mode of the combustion chamber (paragraph REF ). One can imagine a strai... | {
"cite_spans": []
} | 0807.1871 | Characterization of the reactive flow field dynamics in a gas turbine
injector using high frequency PIV | [
"Séverine Barbosa",
"Philippe Scouflaire",
"Sébastien Ducruix"
] | [
"physics.flu-dyn"
] | 2,008 | en | Physics | [
3293,
4568,
6275,
3853,
1286,
97160,
309,
37534,
81,
235079,
62,
26073,
134744,
8262,
23,
37457,
2389,
4382,
50183,
831,
959,
1238,
454,
34,
136,
334,
621,
4552,
20903,
1248,
24371,
12478,
944,
27771,
83324,
8108,
70,
12600,
28692,
2799,
... | [
0.017547607421875,
0.09619140625,
0.130859375,
0.0196533203125,
0.047454833984375,
0.0517578125,
0.062042236328125,
0.014984130859375,
0.21484375,
0.179443359375,
0.1368408203125,
0.22119140625,
0.1219482421875,
0.1751708984375,
0.006378173828125,
0.1376953125,
0.0499267578125,
0.1... | |
d9d92376cac36728a4752d04e2b68ecc21432c8d | subsection | 18 | 19 | Concluding Remarks | The aim of the present study was to detail and evaluate the efficiency of a system of HFPIV operating at 12 kHz. This diagnostic is used to characterise the behavior of an experimental lean premixed swirl-stabilized burner representative of a gas turbine combustor, which may exhibit strong combustion instabilities unde... | {
"cite_spans": []
} | 0807.1871 | Characterization of the reactive flow field dynamics in a gas turbine
injector using high frequency PIV | [
"Séverine Barbosa",
"Philippe Scouflaire",
"Sébastien Ducruix"
] | [
"physics.flu-dyn"
] | 2,008 | en | Physics | [
464,
70,
13379,
35187,
22443,
151575,
13,
227066,
5426,
111,
572,
51006,
15583,
172852,
99,
427,
472,
93423,
52070,
83,
11814,
47,
62816,
3075,
123166,
195935,
65342,
10107,
425,
297,
91,
17084,
141,
9,
60625,
29367,
96865,
56,
99638,
10,... | [
0.1737060546875,
0.035675048828125,
0.056793212890625,
0.10009765625,
0.06195068359375,
0.078125,
0.03564453125,
0.23779296875,
0.19775390625,
0.03570556640625,
0.1453857421875,
0.208740234375,
0.3125,
0.1453857421875,
0.0355224609375,
0.2037353515625,
0.06060791015625,
0.265136718... | |
1302d9565d04d887297e59e6f005409f094c77cb | abstract | 0 | 6 | Abstract | We exhibit a triangulated category T having both products and coproducts, and
a triangulated subcategory S of T which is both localizing and colocalizing,
for which neither a Bousfield localization nor a colocalization exists. It
follows that neither the category S nor its dual satisfy Brown
representability. Our examp... | {
"cite_spans": []
} | 0807.1872 | Brown representability does not come for free | [
"Carles Casacuberta",
"Amnon Neeman"
] | [
"math.CT",
"math.AT"
] | 2,008 | en | Mathematics | [
1401,
80788,
1927,
66,
173857,
95487,
384,
19441,
15044,
38742,
136,
552,
57877,
1614,
33478,
6504,
53,
159,
111,
4000,
84382,
48370,
24707,
2725,
31649,
28394,
47691,
141,
32316,
70,
12488,
87758,
40407,
40218,
33636,
41159,
22929,
27781,
... | [
0.0264739990234375,
0.119873046875,
0.1453857421875,
0.1025390625,
0.1629638671875,
0.271484375,
0.138427734375,
0.031402587890625,
0.02947998046875,
0.1976318359375,
0.0201416015625,
0.10400390625,
0.1767578125,
0.13330078125,
0.194580078125,
0.2049560546875,
0.028564453125,
0.153... | |
48c7ad019e7b4b7be1405a2c45dcc82b1edbbe91 | subsection | 1 | 6 | Introduction | In recent years, several authors have proved remarkable generalizations
of Brown's representability theorem ; see, for example,
, , , . It therefore becomes important
to have an example of a triangulated category where Brown
representability fails. In this short note we produce such a category.There has also been consi... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 135,
"openalex_id": "",
"raw": "Edgar H. Brown, Cohomology theories, Annals of Math. 75 (1962), 467–484.",
"source_ref_id": "60341f52cb45776870ffbd444ff51979f105f499",
"start": 0
},
{
"arxiv_id": "",
... | 0807.1872 | Brown representability does not come for free | [
"Carles Casacuberta",
"Amnon Neeman"
] | [
"math.CT",
"math.AT"
] | 2,008 | en | Mathematics | [
17309,
5369,
42179,
77443,
4537,
47691,
40218,
33636,
41159,
70,
58391,
5526,
27781,
1927,
66,
173857,
95487,
35782,
16610,
27489,
150675,
103488,
4000,
12840,
13784,
53,
154453,
31649,
7,
28394,
2967,
502,
116311,
32628,
552,
29521,
25443,
3... | [
0.01434326171875,
0.0224761962890625,
0.01483154296875,
0.0401611328125,
0.065673828125,
0.1082763671875,
0.20654296875,
0.2459716796875,
0.137451171875,
0.04339599609375,
0.1378173828125,
0.0270538330078125,
0.0709228515625,
0.111328125,
0.09326171875,
0.1302490234375,
0.23815917968... | |
76b8d5dc4807336cc6f869bb410961c98b5dc7b0 | subsection | 2 | 6 | Description and proof | In his 1964 book ,
Freyd constructed an interesting abelian category. We briefly paraphrase
the construction. In this article, our foundational formalism for categories is that of
Mac Lane .Let I be the class of all small ordinals, and let R={\mathbb {Z}}[I] be the
polynomial ring freely generated by I. The ring R has ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 69,
"openalex_id": "",
"raw": "Peter Freyd, Abelian Categories, Harper and Row, New York, 1964.",
"source_ref_id": "be9f03aa7750fb5a2d470838091fd1dcc2a6378a",
"start": 0
},
{
"arxiv_id": "",
"doi": ... | 0807.1872 | Brown representability does not come for free | [
"Carles Casacuberta",
"Amnon Neeman"
] | [
"math.CT",
"math.AT"
] | 2,008 | en | Mathematics | [
38425,
12877,
21612,
11073,
64549,
49041,
10,
14473,
66,
95487,
59335,
121,
179665,
50961,
360,
903,
5582,
2446,
137374,
289,
23113,
8780,
100,
39283,
450,
111,
4727,
239,
86,
124480,
87,
186,
18507,
756,
19336,
23335,
8080,
2633,
627,
12... | [
0.1761474609375,
0.15283203125,
0.1962890625,
0.23388671875,
0.1689453125,
0.1453857421875,
0.1138916015625,
0.1923828125,
0.1580810546875,
0.289794921875,
0.0772705078125,
0.027862548828125,
0.10498046875,
0.12451171875,
0.033935546875,
0.001953125,
0.06500244140625,
0.00241088867... | |
5bcf31191eec33ffc83760a53c05fe0cd66fa2b9 | subsection | 3 | 6 | Description and proof | Each R-module is viewed as a chain complex concentrated in degree zero.
Let {\mathbf {A}}({A})\subset {\mathbf {K}}({A}) be the full subcategory of all acyclic
complexes. Both {\mathbf {K}}({A}) and
{\mathbf {A}}({A}) are triangulated categories with small {\text{\rm Hom}}-sets.In what follows, we refer to for the nece... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 355,
"openalex_id": "",
"raw": ", Triangulated Categories, Annals of Math. Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.",
"source_ref_id": "f744f4ef1354c0a70f8940422cc19c8ac838d33b",
"start": 279... | 0807.1872 | Brown representability does not come for free | [
"Carles Casacuberta",
"Amnon Neeman"
] | [
"math.CT",
"math.AT"
] | 2,008 | en | Mathematics | [
98423,
627,
9,
83279,
13,
83,
21455,
237,
121293,
27140,
142156,
79385,
45234,
10842,
125458,
150598,
284,
22144,
605,
4393,
1614,
33478,
6504,
756,
10,
187830,
238,
99115,
149766,
621,
1927,
66,
173857,
297,
39283,
19336,
27762,
3509,
2896... | [
0.141845703125,
0.207275390625,
0.048370361328125,
0.250244140625,
0.119873046875,
0.0260009765625,
0.1630859375,
0.08978271484375,
0.1781005859375,
0.2027587890625,
0.1934814453125,
0.140869140625,
0.1943359375,
0.004608154296875,
0.05084228515625,
0.164306640625,
0.1605224609375,
... | |
f26db23df23bfb1c0312ef39521980ca7ec1da93 | subsection | 4 | 6 | Description and proof | The inclusion functor i\colon {\mathbf {A}}({A})\longrightarrow {\mathbf {K}}({A}) has neither a right
adjoint nor a left adjoint.By , if a Bousfield
localization existed for {\mathbf {A}}({A})\subset {\mathbf {K}}({A}),
then the quotient category {A})={\mathbf {K}}({A})/{\mathbf {A}}({A})
would be equivalent to a full... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 512,
"openalex_id": "",
"raw": ", Triangulated Categories, Annals of Math. Studies, vol. 148, Princeton University Press, Princeton, NJ, 2001.",
"source_ref_id": "f744f4ef1354c0a70f8940422cc19c8ac838d33b",
"start": 130... | 0807.1872 | Brown representability does not come for free | [
"Carles Casacuberta",
"Amnon Neeman"
] | [
"math.CT",
"math.AT"
] | 2,008 | en | Mathematics | [
190440,
191,
7477,
18770,
17,
22796,
125458,
150598,
284,
10617,
54969,
118201,
605,
1556,
2725,
9319,
7108,
606,
513,
4288,
12488,
25737,
75358,
31649,
7,
28394,
4000,
47691,
13835,
41502,
95487,
183234,
4393,
1614,
33478,
6504,
36746,
1193,... | [
0.2205810546875,
0.10394287109375,
0.19482421875,
0.1937255859375,
0.16357421875,
0.0421142578125,
0.07305908203125,
0.1378173828125,
0.12646484375,
0.06536865234375,
0.01007080078125,
0.057647705078125,
0.1895751953125,
0.111572265625,
0.0977783203125,
0.0667724609375,
0.17651367187... | |
dcb97a66c7685b5ae680a656b27f9784376b80b4 | subsection | 5 | 6 | Description and proof | The composite{\mathbf {A}}({A}) @>i>> {\mathbf {K}}({A}) @>{\text{\rm Hom}}_{{\mathbf {K}}({A})}^{}({\mathbb {Z}},-)>> {A}bis a homological functor taking products
to products, and we assert that it is not representable
by any object of {\mathbf {A}}({A}).Suppose the contrary. If the composite (REF ) were representable... | {
"cite_spans": []
} | 0807.1872 | Brown representability does not come for free | [
"Carles Casacuberta",
"Amnon Neeman"
] | [
"math.CT",
"math.AT"
] | 2,008 | en | Mathematics | [
375,
77087,
13,
125458,
150598,
284,
47391,
605,
22829,
39,
27762,
5125,
1511,
9,
6454,
10,
7622,
109622,
7477,
18770,
35971,
38742,
47,
33657,
959,
18811,
2661,
2499,
36746,
2037,
78381,
2304,
1294,
4263,
11766,
919,
3542,
2806,
32316,
2... | [
0.1280517578125,
0.253173828125,
0.091064453125,
0.0704345703125,
0.189453125,
0.144775390625,
0.038421630859375,
0.0731201171875,
0.0175933837890625,
0.03277587890625,
0.145263671875,
0.157470703125,
0.08526611328125,
0.012908935546875,
0.1842041015625,
0.024566650390625,
0.171875,
... | |
3d5febbb8767bb5cfefa6f7210707fba7486a207 | abstract | 0 | 19 | Abstract | We discuss how various models of scale-free complex networks approach their
limiting properties when the size N of the network grows. We focus mainly on
equilibrated networks and their finite-size degree distributions. Our results
show that subleading corrections to the scaling of the position of the cutoff
are strong ... | {
"cite_spans": []
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
45252,
67842,
115774,
105994,
9,
32087,
27140,
33120,
51515,
17475,
214,
183871,
3229,
13267,
541,
55993,
7,
32153,
5201,
142613,
3674,
94418,
62539,
79385,
113068,
50339,
1614,
14507,
6238,
26785,
17514,
47,
117906,
19069,
59226,
16713,
37515,... | [
0.0621337890625,
0.0789794921875,
0.1978759765625,
0.1917724609375,
0.00238037109375,
0.185791015625,
0.2012939453125,
0.1925048828125,
0.135986328125,
0.2022705078125,
0.07574462890625,
0.1512451171875,
0.0355224609375,
0.1126708984375,
0.10693359375,
0.152587890625,
0.0433349609375... |
800e18faf83acb2324fa190d8b59f746d4754a0d | subsection | 1 | 19 | Introduction | Recent progress in understanding the structure and function of complex networks has been largely influenced by the application of statistical methods of modern physics. The statistical mechanics of networks , , , , , , , if restricted to structural properties, deals with two classes of problems.
In the first one, one c... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 169,
"openalex_id": "",
"raw": "R. Albert, and A.-L. Barabási, Rev. Mod. Phys. 74, 47 (2002); S. N. Dorogovtsev and J. F. F. Mendes, Adv. Phys. 51, 1079 (2002); M. E. J. Newman, SIAM Review 45, 167 (2003); S. Boccaletti, V. Latora... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
169549,
42658,
100094,
45646,
136,
32354,
27140,
33120,
7,
21334,
538,
79507,
71,
70,
38415,
111,
80835,
150624,
5744,
6,
34053,
27744,
289,
135969,
4,
2174,
173072,
297,
47,
118990,
141,
183871,
8,
16765,
678,
6626,
61112,
44402,
360,
16... | [
0.05511474609375,
0.1273193359375,
0.1513671875,
0.147705078125,
0.0310211181640625,
0.1329345703125,
0.187255859375,
0.289794921875,
0.07342529296875,
0.0740966796875,
0.030792236328125,
0.14501953125,
0.031158447265625,
0.030975341796875,
0.0592041015625,
0.035003662109375,
0.24951... |
244753f918e1f66801e270684de81d85d1707ee7 | subsection | 2 | 19 | Introduction | This approximation holds even for quite small networks of order 10^3 nodes because the convergence towards the limiting distribution is fast.The situation is not so clear for equilibrated networks, that is networks in which evolution is governed by rewiring of existing connections rather than by adding new nodes, and t... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 543,
"openalex_id": "",
"raw": "S. N. Dorogovtsev, J. F. F. Mendes, A. M. Povolotsky, and A. N. Samukhin, Phys. Rev. Lett. 95, 195701 (2005).",
"source_ref_id": "ed0586f85751aadd96db2788d0fb1c3b31c3c7c3",
"start": 375
... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
3293,
35707,
53950,
16401,
3853,
32233,
19336,
33120,
12989,
209,
363,
110,
988,
6637,
814,
98186,
17475,
214,
113068,
4271,
16648,
959,
221,
34735,
6,
142613,
3674,
28,
137089,
83,
23131,
297,
456,
17084,
111,
144573,
94878,
7,
43257,
39... | [
0.00152587890625,
0.08074951171875,
0.170166015625,
0.1031494140625,
0.042266845703125,
0.07464599609375,
0.10125732421875,
0.2181396484375,
0.13134765625,
0.0604248046875,
0.0870361328125,
0.1241455078125,
0.05718994140625,
0.0003662109375,
0.1241455078125,
0.078125,
0.1856689453125... |
2966adff3524236b605576aaa1f8f0bb3b4aa531 | subsection | 3 | 19 | Models and their properties in the thermodynamic limit | We shall start from defining three different models of equilibrated networks, whose finite-size properties will be further examined.
The word “equilibrated” means that networks (graphs in mathematical language) are maximally random under given constraints. These constraints are what defines the statistical ensemble of ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1781,
"openalex_id": "",
"raw": "S. N. Dorogovtsev, J. F. F. Mendes, and A. N. Samukhin, Nucl. Phys. B 666, 396 (2003).",
"source_ref_id": "846bcf88ce5022a92272f71674d0db3943ef8d81",
"start": 1627
},
{
"a... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
1401,
35299,
4034,
1295,
13204,
17262,
12921,
115774,
111,
6,
142613,
3674,
33120,
7,
94418,
13,
62539,
183871,
186,
160477,
71,
5,
581,
2565,
52,
63,
26950,
450,
41382,
23,
140363,
70760,
46876,
16,
621,
111340,
96759,
1379,
34475,
158,
... | [
0.0143280029296875,
0.060272216796875,
0.0712890625,
0.029876708984375,
0.17626953125,
0.1580810546875,
0.15869140625,
0.276611328125,
0.05450439453125,
0.03643798828125,
0.365478515625,
0.2401123046875,
0.26220703125,
0.08642578125,
0.1802978515625,
0.0587158203125,
0.1593017578125,... |
06f85665cd31a263da2dc6d1fac8ee261507904d | subsection | 4 | 19 | Models and their properties in the thermodynamic limit | It can be shown , , that the partition function of the system, being the sum over all configurations, assumes the form:Z(N,L) =
\sum _{k_1=0}^\infty \cdots \sum _{k_N=0}^\infty \frac{p(k_1)}{k_1!} \cdots \frac{p(k_N)}{k_N!} \delta _{2L,k_1+\dots +k_N},so it is equivalent to that of the balls-in-boxes model or the zero-... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 364,
"openalex_id": "",
"raw": "B. Waclaw, arXiv:0704.3702.",
"source_ref_id": "c9e0467c270507a4c09fab72cd9fce2d57377e09",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end": 364,
"openale... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
831,
186,
127887,
6,
2878,
1363,
32354,
5426,
8035,
10554,
645,
756,
180346,
7,
41591,
70,
3173,
1511,
839,
4,
866,
16,
11832,
24854,
92,
115187,
145407,
8152,
8353,
41872,
46632,
939,
238,
15464,
132076,
254,
132,
38,
454,
1743,
102,
... | [
0.055206298828125,
0.0280914306640625,
0.145263671875,
0.0279541015625,
0.2236328125,
0.1373291015625,
0.1695556640625,
0.1895751953125,
0.036956787109375,
0.1961669921875,
0.09918212890625,
0.1170654296875,
0.2069091796875,
0.028167724609375,
0.1319580078125,
0.0281524658203125,
0.1... |
884f87d55a8302f2389764acdf04f77d54cb6beb | subsection | 5 | 19 | Models and their properties in the thermodynamic limit | The purpose of choosing this particular distribution is that since finite-size effects in the GNR model are known , we can compare what happens if networks are equilibrated but have the same \pi _\infty (k) as growing ones. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 223,
"openalex_id": "",
"raw": "B. Waclaw and I. M. Sokolov, Phys. Rev. E 75, 056114 (2007).",
"source_ref_id": "b9e4ba8911ae96ac3206046dfca86d86398a7079",
"start": 0
}
]
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
581,
60042,
111,
218873,
903,
17311,
113068,
83,
450,
16792,
94418,
13,
62539,
93425,
527,
83752,
3299,
51529,
6,
831,
69101,
96276,
2174,
33120,
7,
621,
142613,
3674,
1284,
765,
5701,
1434,
46632,
939,
92,
237,
105925,
64333
] | [
0.04437255859375,
0.22900390625,
0.06463623046875,
0.2130126953125,
0.1302490234375,
0.18701171875,
0.265869140625,
0.09112548828125,
0.020111083984375,
0.013214111328125,
0.1820068359375,
0.0645751953125,
0.15673828125,
0.2274169921875,
0.075439453125,
0.25830078125,
0.223876953125,... |
2a1bec0d361883a8d8ec94680197d2ada08a6279 | subsection | 6 | 19 | Degree distribution for a finite network | In the previous section we discussed the behavior of the three models in the thermodynamic limit. Now we shall ask, how the degree distribution looks like for N<\infty .
Assume that \pi _N(k) and \pi _\infty (k) are degree distributions for finite N and N\rightarrow \infty ,
respectively.
It is convenient to write the ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 820,
"openalex_id": "",
"raw": "B. Waclaw and I. M. Sokolov, Phys. Rev. E 75, 056114 (2007).",
"source_ref_id": "b9e4ba8911ae96ac3206046dfca86d86398a7079",
"start": 550
},
{
"arxiv_id": "",
"doi": "... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
70,
40059,
123166,
17262,
115774,
182342,
242554,
17475,
35299,
26458,
3642,
79385,
113068,
33342,
1884,
100,
541,
16093,
41872,
46632,
939,
6,
62,
66596,
13,
1434,
839,
92,
16,
136,
94418,
118201,
4,
538,
142267,
33022,
62539,
10,
12996,
... | [
0.02294921875,
0.0160064697265625,
0.2091064453125,
0.09320068359375,
0.2293701171875,
0.15234375,
0.179443359375,
0.209716796875,
0.0198974609375,
0.05316162109375,
0.0445556640625,
0.261474609375,
0.26318359375,
0.1016845703125,
0.0748291015625,
0.052459716796875,
0.1575927734375,
... |
d4facabbd2fea3a5823054299286acf6e0bf6adf | subsection | 7 | 19 | Multigraphs | We shall start from multigraphs. As we said, we assume that the average degree \bar{k} is chosen to ensure \sum _k k \pi _\infty (k)=\bar{k}. This means that for a given number of nodes N, the number of links L=L(N) is fixed.
In case of the distribution (REF ), L=N.
The partition function (REF ) becomes a function of N... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 943,
"openalex_id": "",
"raw": "P. Bialas, Z. Burda, J. Jurkiewicz, and A. Krzywicki, Phys. Rev. E 67, 066106 (2003).",
"source_ref_id": "d19450e2c5cfd0f3d378f76cfbe7cd2b9d0b9e16",
"start": 559
}
]
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
35299,
4034,
1295,
6024,
41382,
7,
1301,
2804,
4,
41591,
83080,
79385,
6,
1299,
92,
19667,
19,
63284,
11832,
472,
41872,
1434,
46632,
939,
15,
16,
1369,
8152,
26950,
100,
10,
34475,
14012,
111,
110,
988,
541,
70,
22317,
339,
866,
839,... | [
0.058135986328125,
0.0806884765625,
0.0404052734375,
0.208251953125,
0.28564453125,
0.07379150390625,
0.00067138671875,
0.002716064453125,
0.025604248046875,
0.0948486328125,
0.195068359375,
0.25048828125,
0.0246429443359375,
0.16748046875,
0.1490478515625,
0.0992431640625,
0.0252075... |
df135f4fc33f9ef818f3482ab341aa0c63e20463 | subsection | 8 | 19 | Multigraphs | The highest peak shows data for N=100.]The theoretical value of \alpha , predicted for this model, should be 1/2 , the same value comes from the correspondence to the zero-range process . Therefore, for large N, plots of w(N,k) for different sizes should collapse into a single curve in the rescaled variable x=k/N^\alph... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 187,
"openalex_id": "",
"raw": "Z. Burda and A. Krzywicki, Phys. Rev. E 67, 046118 (2003).",
"source_ref_id": "59bcb21e2ddd172ee219701006000e68565d41d5",
"start": 39
},
{
"arxiv_id": "",
"doi": "",
... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
167375,
280,
344,
45831,
2053,
541,
1369,
3559,
70,
4524,
70760,
34292,
41872,
289,
14612,
6,
92054,
3299,
4,
5608,
22128,
5701,
32497,
1295,
42518,
6620,
45234,
94250,
9433,
21334,
23577,
111,
148,
839,
92,
16,
100,
12921,
13267,
7,
33... | [
0.201171875,
0.14990234375,
0.15185546875,
0.10009765625,
0.146484375,
0.2244873046875,
0.0272674560546875,
0.1856689453125,
0.028594970703125,
0.10693359375,
0.046722412109375,
0.1767578125,
0.050323486328125,
0.0975341796875,
0.225341796875,
0.027496337890625,
0.2093505859375,
0.... |
ca08b4a652f35f172e0d27bfcc321d3031a33cda | subsection | 9 | 19 | Multigraphs | Indeed, one can predict this scaling analytically, studying
the cutoff function w(N,k):w(N,k) \cong W(N,k)/W(N,0),whereW(N,k) = \oint \frac{dz}{2\pi i} z^{k-1-N\bar{k}} F^{N}(z),with \bar{k}=2 and F(z) given by Eq. (REF ). Following the lines of Sec. 6 from Ref. one can argue that the function under the integral is lo... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 373,
"openalex_id": "",
"raw": "M. R. Evans, S. N. Majumdar, and R. K. P. Zia, J. Stat. Phys. 123, 357 (2006).",
"source_ref_id": "69f125c799f75dee45642b4f9a4ada46067216b7",
"start": 223
},
{
"arxiv_id": ... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
102627,
297,
1632,
831,
92054,
903,
117906,
214,
6,
140815,
25958,
35187,
59226,
16713,
32354,
148,
839,
4,
92,
434,
16,
587,
449,
601,
132,
64,
1456,
77495,
2203,
4288,
41872,
132076,
24854,
8243,
8152,
304,
1434,
17,
97,
8353,
5759,
... | [
0.056365966796875,
0.0271759033203125,
0.03009033203125,
0.0853271484375,
0.247802734375,
0.0714111328125,
0.229736328125,
0.119140625,
0.02734375,
0.194580078125,
0.0271148681640625,
0.1092529296875,
0.2213134765625,
0.256103515625,
0.188232421875,
0.1556396484375,
0.1383056640625,
... |
a50fff799b2ac646ef42a938b81a316db20aef2d | subsection | 10 | 19 | Multigraphs | The assumed form of w(x) approximates the measured cutoff functions very well.
The D's obtained for different sizes N tend to some limiting values which we found to be D_{N\rightarrow \infty }=2.0 for \gamma =2.5, 1.9 for \gamma =3 and 3.1 for \gamma =3.5, with uncertainties of order 0.1. These values are in good agree... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 472,
"openalex_id": "",
"raw": "M. R. Evans, S. N. Majumdar, and R. K. P. Zia, J. Stat. Phys. 123, 357 (2006).",
"source_ref_id": "69f125c799f75dee45642b4f9a4ada46067216b7",
"start": 290
}
]
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
41591,
3173,
148,
132,
425,
35707,
5134,
120126,
72350,
59226,
16713,
32354,
4552,
5299,
391,
25,
113054,
12921,
13267,
541,
17660,
3060,
17475,
214,
142424,
14037,
454,
839,
54969,
118201,
46632,
939,
73011,
17705,
48413,
99420,
76067,
45151... | [
0.1427001953125,
0.1224365234375,
0.1844482421875,
0.0843505859375,
0.1329345703125,
0.1019287109375,
0.1412353515625,
0.05615234375,
0.15966796875,
0.1842041015625,
0.1949462890625,
0.1572265625,
0.0750732421875,
0.08349609375,
0.1220703125,
0.03277587890625,
0.041717529296875,
0.... |
1ca098061849241f1e1717f02c925a7d216977bd | subsection | 11 | 19 | Equilibrated trees | In Ref. , the partition function Z(N) for equilibrated trees is found to beZ(N) = \oint \frac{dz}{2\pi i} z^{-N-1} Z_{\rm GC}(z).Here Z_{\rm GC}(z) is a grand-canonical partition function obeying the equationZ_{\rm GC}(z) = z \tilde{F}(Z_{\rm GC}(z)),with\tilde{F}(z) = \sum _{k=0}^\infty \pi _\infty (k+1)z^k ,assuming ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 129,
"openalex_id": "",
"raw": "P. Bialas, Z. Burda, J. Jurkiewicz, and A. Krzywicki, Phys. Rev. E 67, 066106 (2003).",
"source_ref_id": "d19450e2c5cfd0f3d378f76cfbe7cd2b9d0b9e16",
"start": 0
},
{
"arxiv_... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
53295,
5,
70,
2878,
1363,
32354,
567,
132,
839,
100,
142613,
3674,
1360,
90,
14037,
47,
186,
1511,
16,
2203,
6,
31,
4288,
41872,
132076,
8243,
8152,
24854,
304,
1434,
17,
97,
9,
5759,
69657,
169,
25178,
42,
83,
9963,
38938,
19,
2153... | [
0.2293701171875,
0.025115966796875,
0.0234832763671875,
0.229248046875,
0.12353515625,
0.211181640625,
0.2003173828125,
0.02911376953125,
0.18505859375,
0.062286376953125,
0.302978515625,
0.103271484375,
0.228271484375,
0.10400390625,
0.09307861328125,
0.0231781005859375,
0.040405273... |
951ef7c5750af342f8baf38da7b6922bc724fa6d | subsection | 12 | 19 | Multicanonical simulations of simple graphs | So far we have considered equilibrated multi- and tree graphs. But real-world networks are usually simple graphs, that is they do not have multiple- and self connections, and have loops.
Unfortunately, for these reasons simple graphs are not accessible with the technique used before, because one does not know how to wr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 571,
"openalex_id": "",
"raw": "L. Bogacz, Z. Burda, W. Janke, and B. Waclaw, Comp. Phys. Comm. 173, 162 (2005).",
"source_ref_id": "d8692c2d98d0c615a336fabd8714fe9775934ac9",
"start": 458
},
{
"arxiv_id"... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
1061,
2060,
642,
90698,
142613,
3674,
6024,
136,
53201,
41382,
7,
4966,
2773,
32554,
33120,
621,
56104,
8781,
4,
83,
1836,
959,
48716,
15970,
94878,
765,
40956,
185397,
100,
81522,
70,
61353,
8108,
14602,
3714,
3642,
47,
33022,
2878,
1363... | [
0.00701904296875,
0.10076904296875,
0.011138916015625,
0.1043701171875,
0.285400390625,
0.081787109375,
0.1435546875,
0.032440185546875,
0.193115234375,
0.287109375,
0.0849609375,
0.0098876953125,
0.1876220703125,
0.169677734375,
0.2421875,
0.020477294921875,
0.10064697265625,
0.15... |
c351922cf11258237f57150fec02b34c2d777151 | subsection | 13 | 19 | Multicanonical simulations of simple graphs | The factor 1/\pi _\infty (k_1) gives additional advantage of flattening the distribution at no cost for k\ll k_{\rm cutoff}. To obtain the true distribution \pi _{\rm exp}(k) one multiplies \pi _{\rm flat}(k) by \pi _\infty (k)/r(k). Since all nodes are statistically equivalent in equilibrated graphs, this procedure mu... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 536,
"openalex_id": "",
"raw": "W. Janke, in: Computer Simulations of Surfaces and Interfaces, NATO Science Series, II. Mathematics, Physics and Chemistry – Vol. 114, edited by B. Dünweg, D. P. Landau, and A. I. Milchev (Kluwer, D... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
31461,
106,
64,
1434,
101,
46632,
939,
92,
115187,
16,
76199,
78301,
92940,
49878,
67,
592,
113068,
99,
110,
11034,
472,
1181,
39,
59226,
16713,
717,
113054,
29568,
6,
24854,
41872,
42,
14700,
8152,
132,
118126,
90,
390,
15,
756,
988,
... | [
0.206787109375,
0.10223388671875,
0.061279296875,
0.243408203125,
0.005523681640625,
0.201904296875,
0.137451171875,
0.1968994140625,
0.1710205078125,
0.0162506103515625,
0.07427978515625,
0.1220703125,
0.1827392578125,
0.1953125,
0.1875,
0.0496826171875,
0.27490234375,
0.013458251... |
594c6a04b07df38927e5abb0a98329d0166f0b0a | subsection | 14 | 19 | Multicanonical simulations of simple graphs | Therefore, simple graphs approach the thermodynamic limit even slower than equilibrated trees or degenerated graphs.Finally, we examined the large-x behavior of w(x) using the same method as for multigraphs. We found that the exponent \eta in Eq. (REF ) is not 1/(1-\alpha ) as for multigraphs but approximately 2.0 for ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 477,
"openalex_id": "",
"raw": "S. N. Dorogovtsev, J. F. F. Mendes, A. M. Povolotsky, and A. N. Samukhin, Phys. Rev. Lett. 95, 195701 (2005).",
"source_ref_id": "ed0586f85751aadd96db2788d0fb1c3b31c3c7c3",
"start": 344
... | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
228072,
8781,
41382,
7,
51515,
182342,
242554,
17475,
3853,
72803,
56,
3501,
142613,
1360,
90,
8,
48281,
3674,
25958,
4,
160477,
21334,
425,
123166,
111,
148,
132,
70,
5701,
55300,
6024,
14037,
1119,
54137,
41872,
4241,
864,
5,
15,
11766,... | [
0.01654052734375,
0.264892578125,
0.2978515625,
0.0906982421875,
0.1307373046875,
0.1796875,
0.172607421875,
0.1925048828125,
0.0093994140625,
0.1732177734375,
0.08612060546875,
0.013092041015625,
0.2142333984375,
0.146484375,
0.00714111328125,
0.08740234375,
0.14599609375,
0.00643... |
57988d25ac5121786246219cb36bbf957e616855 | subsection | 15 | 19 | Cutoffs and the distribution of maximal degree | So far we have studied the cutoffs in the degree distribution averaged over the ensemble of random graphs of a certain kind.
The value k_{\rm cutoff}\sim N^\alpha tells where the power law ends and \pi _N(k) starts to fall off rapidly.
One can also pose the question: how does the maximal degree, k_{\rm max}, scale with... | {
"cite_spans": []
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
1061,
2060,
22282,
59226,
16713,
7,
79385,
113068,
83080,
645,
70,
63304,
96759,
6,
41382,
10,
24233,
8562,
34292,
472,
454,
42,
39,
8152,
5072,
541,
8353,
289,
14612,
14192,
7440,
14537,
27165,
3564,
1434,
839,
92,
16,
4034,
6817,
5773... | [
0.013336181640625,
0.0926513671875,
0.1005859375,
0.18603515625,
0.257080078125,
0.027984619140625,
0.26171875,
0.2301025390625,
0.1907958984375,
0.0170135498046875,
0.0278472900390625,
0.1248779296875,
0.1724853515625,
0.027618408203125,
0.22265625,
0.0276336669921875,
0.04821777343... |
ec26d39ef8bc482dedd05a27541800c401bf22aa | subsection | 16 | 19 | Cutoffs and the distribution of maximal degree | One can also show that P(k_{\rm max}) for \alpha >1/(\gamma -1) is given by the Fréchet extreme value distribution: P(x)\propto x^{-\gamma } e^{-x^{-\gamma +1}}, while it approaches the Gumbel distribution: P(x) \propto e^{-x-e^{-x}} for \alpha <1/(\gamma -1), in the properly rescaled variable x=A+B k_{\rm max}.
Becaus... | {
"cite_spans": []
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
7639,
436,
92,
454,
42,
39,
18389,
16,
6,
289,
14612,
977,
50412,
17705,
192,
110218,
34475,
563,
4273,
65735,
120069,
34292,
113068,
425,
41872,
36290,
188,
1022,
24854,
9,
51912,
28,
8353,
57157,
47391,
4,
51515,
90,
62617,
4063,
132,... | [
0.103271484375,
0.1927490234375,
0.17919921875,
0.0765380859375,
0.11669921875,
0.1473388671875,
0.26611328125,
0.0296630859375,
0.0295257568359375,
0.03753662109375,
0.2120361328125,
0.008514404296875,
0.12890625,
0.173583984375,
0.0809326171875,
0.1434326171875,
0.1192626953125,
... |
ee5ac6155ebae8552f78b34b2ae337d3786cb047 | subsection | 17 | 19 | Cutoffs and the distribution of maximal degree | Plots for different N show good agreement of positions of the maximum. The distribution for \gamma =3.5 is also approximately Fréchet, which agrees with recent findings , the two for \gamma =2.5, \; 3 deviate slightly from Fréchet in the tail.
The perfect scaling of k_{\rm max} means that the thermodynamic limit for th... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 243,
"openalex_id": "",
"raw": "M. R. Evans and S. N. Majumdar, J. Stat. Mech. P05004 (2008).",
"source_ref_id": "0bdbc37a9bbf31c6fab1ed1417be83b0f6328a17",
"start": 71
}
]
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
168763,
100,
12921,
541,
7639,
4127,
106689,
19069,
38132,
113068,
6,
17705,
192,
72516,
189275,
563,
4273,
65735,
4,
53520,
7,
678,
17309,
90791,
70,
6626,
48413,
138,
30170,
2182,
161549,
46741,
15787,
117906,
214,
111,
472,
454,
42,
39... | [
0.1942138671875,
0.0120849609375,
0.17626953125,
0.131591796875,
0.0657958984375,
0.102294921875,
0.2303466796875,
0.155517578125,
0.2366943359375,
0.2474365234375,
0.0171356201171875,
0.1849365234375,
0.15478515625,
0.244873046875,
0.13525390625,
0.0355224609375,
0.1448974609375,
... |
e4e9b41f6e867a4af85ce4ba839ddc970103c7ef | subsection | 18 | 19 | Summary | In this paper we considered finite-size effects in the degree distribution of equilibrated networks.
We showed that the convergence towards the thermodynamic limit is very slow, thus in order to get reasonable estimation of the cutoff one has to consider subleading corrections to the scaling k_{\rm cutoff}\sim N^\alpha... | {
"cite_spans": []
} | 10.1103/PhysRevE.78.061125 | 0807.1874 | Approaching the thermodynamic limit in equilibrated scale-free networks | [
"B. Waclaw",
"L. Bogacz",
"W. Janke"
] | [
"cond-mat.stat-mech",
"cond-mat.dis-nn"
] | 2,008 | en | Physics | [
903,
15122,
90698,
94418,
13,
62539,
93425,
79385,
113068,
142613,
3674,
33120,
7,
168360,
158,
814,
110343,
98186,
182342,
242554,
17475,
83,
4552,
72803,
169022,
25902,
59226,
16713,
16916,
1614,
14507,
26785,
17514,
117906,
214,
472,
39,
5... | [
0.0032958984375,
0.067138671875,
0.0677490234375,
0.2203369140625,
0.10028076171875,
0.1796875,
0.22021484375,
0.199951171875,
0.2008056640625,
0.264892578125,
0.090576171875,
0.2091064453125,
0.002349853515625,
0.0305938720703125,
0.05712890625,
0.196533203125,
0.06103515625,
0.14... |
4ec78362a92519038421dd2a33e9f2ff633b987d | abstract | 0 | 17 | Abstract | The recent observation of an electronic cluster glass state composed of
random domains with unidirectional modulation of charge density and/or spin
density on Bi_{2}Sr_{2}CaCu_{2}O_{8+\delta} reinvigorates the debate of
existence of competing interactions and their importance in high temperature
superconductivity. By u... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
17309,
150556,
65133,
234737,
47589,
11341,
150350,
96759,
77758,
19844,
80581,
17055,
2320,
25534,
168,
7,
25927,
1843,
304,
20370,
33177,
670,
1019,
1328,
1743,
102,
19574,
686,
76312,
29865,
116311,
28813,
449,
182809,
131011,
11192,
52768,
... | [
0.0196685791015625,
0.1119384765625,
0.150634765625,
0.26953125,
0.247314453125,
0.1898193359375,
0.08349609375,
0.1458740234375,
0.1282958984375,
0.0679931640625,
0.1124267578125,
0.203125,
0.029998779296875,
0.138916015625,
0.16748046875,
0.026580810546875,
0.1361083984375,
0.097... |
f6075702bb125046813932be58b3144d69678b29 | subsection | 1 | 17 | Introduction | Since the discovery of high temperature superconductors (HTS) two
decades ago, many anomalous properties have been reported. One of
the most interesting properties is the possible existence of the
stripe state consisting of one dimensional charge-density modulation
coupled with spin ordering
, , . The first direct
expe... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 298,
"openalex_id": "",
"raw": "J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989).",
"source_ref_id": "a2b90d56249549b7ae81c6fe2639788610062102",
"start": 125
},
{
"arxiv_id": "",
"doi": "",... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
66016,
70,
103882,
53,
11192,
52768,
1601,
2271,
39367,
22230,
18544,
294,
16,
6626,
8,
23662,
7,
6650,
4,
5941,
2373,
2749,
10821,
183871,
765,
2809,
113771,
2684,
49041,
7722,
116311,
43613,
13,
11341,
58055,
214,
111,
1632,
6,
157955,
... | [
0.052093505859375,
0.013031005859375,
0.1468505859375,
0.01287841796875,
0.1378173828125,
0.2247314453125,
0.1871337890625,
0.134521484375,
0.19287109375,
0.07647705078125,
0.1995849609375,
0.190673828125,
0.0130615234375,
0.07720947265625,
0.07257080078125,
0.1192626953125,
0.013061... |
ece485292502bb71f8e5bb2ce4a6103329de2c25 | subsection | 2 | 17 | Introduction | There are
no other competing interactions and certainly no phase boundaries
between overdoped and underdoped regimes with different order
parameters to be worried.In this paper we will show that interactions represented by t and
J are actually competing with each other. This competition,
greatly enhanced by the strong ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1262,
"openalex_id": "",
"raw": "M. Fujita, H. Goka, K. Yamada, and M. Matsuda, Phys. Rev. Lett. 88, 167008 (2002).",
"source_ref_id": "2f7a8a709495f0183d2a3f7c7c35fffb2d3c6aec",
"start": 772
},
{
"arxiv_... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
8622,
621,
110,
3789,
28813,
449,
182809,
7,
68782,
93402,
6990,
10484,
17721,
645,
246,
20051,
136,
1379,
63647,
12921,
12989,
171859,
186,
203925,
4153,
903,
15122,
1221,
7639,
33636,
297,
390,
808,
821,
20653,
678,
12638,
3293,
130412,
... | [
0.025238037109375,
0.0252227783203125,
0.05401611328125,
0.1268310546875,
0.286376953125,
0.1173095703125,
0.244140625,
0.025299072265625,
0.039825439453125,
0.138671875,
0.07403564453125,
0.0251312255859375,
0.038177490234375,
0.07275390625,
0.149658203125,
0.0244140625,
0.149658203... |
383aa483d01ecaf3d8cb362d0aaeef8191295df6 | subsection | 3 | 17 | Introduction | However, a later VMC study of the t-t^{\prime }-J model
indicates that the stripe has about 1\% lower
energy than the uniform RVB based d-wave SC state for most of the
negative values of t^{\prime } except when -t^{\prime }/t is less than 0.1.
Similar results were also reported for the Hubbard model
. These results po... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 244,
"openalex_id": "",
"raw": "A. Himeda, T. Kato, and M. Ogata, Phys. Rev. Lett. 88, 117001 (2002).",
"source_ref_id": "9e6fc276dfcf43b628977732bf8e406ccc036636",
"start": 0
},
{
"arxiv_id": "",
"... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
33306,
310,
32557,
35187,
808,
18,
8353,
114654,
1375,
3299,
43613,
13,
1556,
3949,
92319,
48302,
61514,
75034,
35509,
104,
634,
272,
17715,
11341,
2684,
40907,
142424,
20,
64,
40715,
107754,
50339,
53465,
1299,
71,
19064,
8951,
2304,
19593... | [
0.004302978515625,
0.1231689453125,
0.124755859375,
0.0618896484375,
0.06597900390625,
0.06634521484375,
0.0282440185546875,
0.1949462890625,
0.1610107421875,
0.20263671875,
0.29833984375,
0.1881103515625,
0.0241851806640625,
0.0445556640625,
0.1661376953125,
0.202880859375,
0.15625,... |
5553d795ac15d7ed85ec07604ef74998c1be16da | subsection | 4 | 17 | The stripe-like states by the variational Monte Carlo method | We consider the extended t-J Hamiltonian,H=-\sum _{i,j,\sigma }t_{ij}\left(\tilde{c}_{i\sigma }^{\dag }\tilde{c}_{j\sigma }+h.c.\right)+J\sum _{<i,j>}\mathbf {S}_{i}\cdot \mathbf {S}_{j}which has been used to describe many physical systems in the
high-temperature superconductors . The hopping
amplitude t_{ij}=t, t^{\pr... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 281,
"openalex_id": "",
"raw": "K. Y. Yang et al., Phys. Rev. B 73, 224513 (2006).",
"source_ref_id": "7ff21f90b98407024b5b6ec5346775df2584981a",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"en... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
1401,
16916,
65042,
297,
808,
9,
1375,
94674,
3378,
841,
1369,
11832,
24854,
14,
170,
4,
41872,
20561,
192,
51912,
454,
13786,
8152,
133,
132,
3675,
112,
238,
8353,
6063,
1328,
127,
5,
54969,
16,
101,
2740,
125458,
150598,
10666,
294,
... | [
0.005096435546875,
0.101806640625,
0.1790771484375,
0.029083251953125,
0.1455078125,
0.0958251953125,
0.2392578125,
0.263916015625,
0.212646484375,
0.1287841796875,
0.05364990234375,
0.21826171875,
0.0170440673828125,
0.08294677734375,
0.17431640625,
0.01727294921875,
0.0169830322265... |
9e39eb1a8b100d45cacd9c5c139ff1e9aba171ec | subsection | 5 | 17 | The stripe-like states by the variational Monte Carlo method | For periodic stripes we assume
charge density \rho _{i} and staggered magnetization m_{i} are
anti-correlated, {\it i.e.} there are more holes at sites with
minimum staggered magnetization. For simplicity, we assume these
spatially varying functions with simple forms:\rho _{i}=\rho _{v}\cos [4\pi \delta \cdot (y_{i}-y_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 826,
"openalex_id": "",
"raw": "M. Raczkowski, D. Poilblanc, R. Fresard, and A. M. Oles, Phys. Rev. B 75, 094505 (2007); M. Raczkowski, M. Capello, D. Poilblanc, R. Fresard, and A. M. Oles, Phys. Rev. B 76, 140505(R) (2007).",
... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
1326,
120164,
43613,
90,
642,
41591,
25534,
168,
7,
2481,
6,
497,
24854,
14,
8152,
136,
1924,
21407,
297,
39411,
47691,
347,
621,
2874,
10517,
174822,
4,
217,
17,
5,
13,
1286,
108564,
15271,
15440,
134381,
5623,
118,
25958,
285,
38543,
... | [
0.036163330078125,
0.25,
0.30029296875,
0.18359375,
0.028839111328125,
0.134521484375,
0.192138671875,
0.209716796875,
0.112060546875,
0.06768798828125,
0.0200653076171875,
0.1666259765625,
0.020050048828125,
0.06298828125,
0.02008056640625,
0.019683837890625,
0.1405029296875,
0.19... |
1d40219ab084b4fe821ddf7d0d4c354f03a457b5 | subsection | 6 | 17 | The stripe-like states by the variational Monte Carlo method | Besides the antiphase stripe and AF-RVB stripes, we
could also have the AF stripe without both \Delta ^{M}_{v} and
\Delta ^{C}_{v} or the charge-density stripe without any staggered
magnetization but with pairing amplitude modulation.In general we have total seven variational parameters \mu _{v},
t^{\prime }_{v}, t^{\p... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1396,
"openalex_id": "",
"raw": "A. Himeda, T. Kato, and M. Ogata, Phys. Rev. Lett. 88, 117001 (2002).",
"source_ref_id": "9e6fc276dfcf43b628977732bf8e406ccc036636",
"start": 1008
}
]
} | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
86247,
2874,
153213,
43613,
13,
136,
28211,
9,
61293,
571,
90,
642,
5809,
2843,
765,
15490,
58598,
102,
594,
334,
441,
25534,
34609,
2481,
2499,
21407,
39411,
47691,
1284,
678,
80836,
217269,
17055,
5,
4537,
3622,
59671,
143834,
289,
1718... | [
0.00836181640625,
0.150390625,
0.1878662109375,
0.228515625,
0.078857421875,
0.03436279296875,
0.2265625,
0.03997802734375,
0.1385498046875,
0.11767578125,
0.072998046875,
0.08941650390625,
0.087646484375,
0.0093994140625,
0.1058349609375,
0.1016845703125,
0.1033935546875,
0.067260... |
f84166401af61224f4add2821aeb93afff883da1 | subsection | 7 | 17 | The stripe-like states by the variational Monte Carlo method | Then we formulate the wave function in the Hilbert
space with the fixed number of electrons N_{e},|\Phi \rangle =PP_{N_{e}}|\psi \rangle &=&PP_{N_{e}}\prod _{n}\gamma _{n}\bar{\gamma }^{\dag }_{n}|0\rangle \\&\propto &
P\left(\sum _{i,j}(\hat{U}^{-1}\hat{V})_{ij}c^{\dag }_{i\uparrow }c^{\dag }_{j\downarrow }\right)^{N_... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 540,
"openalex_id": "",
"raw": "S. Sorella, Phys. Rev. B 64, 024512 (2001).",
"source_ref_id": "9f70c4f40079fc54f70d32b277f2a868f386c44a",
"start": 394
},
{
"arxiv_id": "",
"doi": "",
"end": 7... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
26168,
259,
272,
32354,
23,
26370,
16466,
32628,
188347,
14012,
111,
77556,
1779,
541,
13,
45689,
14,
6,
41872,
5445,
133,
17487,
454,
24854,
47391,
58745,
15759,
619,
1369,
112348,
101,
19,
8152,
17705,
192,
51912,
8353,
6063,
2389,
1899... | [
0.1395263671875,
0.171142578125,
0.1358642578125,
0.1800537109375,
0.0245361328125,
0.132568359375,
0.2447509765625,
0.1683349609375,
0.1513671875,
0.04559326171875,
0.002777099609375,
0.13671875,
0.061004638671875,
0.02032470703125,
0.116943359375,
0.1158447265625,
0.003204345703125... |
57ac323f49cdf7b09a5aa57f8cc72752d8a0a5c7 | subsection | 8 | 17 | The stripe-like states by the variational Monte Carlo method | If we
reduce this strength by going beyond the simple trial wave functions
used in our discussion above we could push the phase separation
boundary to a much higher value of J/t. A simpler way to make this
adjustment is to introduce the hole-hole repulsion Jastrow
factor, , :P_{J}=\prod _{i<j}\left(1-(1-r^{\alpha }_{ij... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 586,
"openalex_id": "",
"raw": "C. S. Hellberg and E. J. Mele, Phys. Rev. Lett. 67, 2080 (1991).",
"source_ref_id": "7e15be2059b10d5a89ac99a78648f9b8db76e8cd",
"start": 179
},
{
"arxiv_id": "",
"doi... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
4263,
34390,
903,
90254,
107314,
70,
8781,
110324,
259,
272,
32354,
7,
11814,
23,
35107,
36917,
5809,
25944,
93402,
37451,
1363,
6,
99091,
6635,
47,
5045,
77546,
34292,
821,
64,
18,
42,
3917,
3249,
126596,
83,
65508,
108564,
9,
70919,
1... | [
0.038604736328125,
0.2164306640625,
0.04278564453125,
0.2283935546875,
0.0867919921875,
0.0204925537109375,
0.1064453125,
0.1749267578125,
0.1236572265625,
0.0845947265625,
0.13427734375,
0.020965576171875,
0.001007080078125,
0.020751953125,
0.049530029296875,
0.0244598388671875,
0.0... |
a3e6bfddc187facdf12d4639f47ed87e5637fc7e | subsection | 9 | 17 | The stripe-like states by the variational Monte Carlo method | Now the hole density has a much smaller variation
as shown by the empty circles in Fig.REF (a).It is surprising to find out that for all t^{\prime }/t, the AF-RVB stripe
state is almost degenerate in variational energy with the uniform
d-wave RVB state. This is quite remarkable as the two trial wave
functions are very ... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
14240,
108564,
168,
7,
2481,
1556,
5045,
164917,
143834,
237,
127887,
390,
70,
201505,
130370,
119895,
919,
15,
11,
83,
212615,
47,
100,
756,
808,
114654,
51912,
64,
18,
4,
28211,
9,
61293,
571,
43613,
13,
11341,
39555,
8,
48281,
67,
... | [
0.010650634765625,
0.2093505859375,
0.2030029296875,
0.1251220703125,
0.07330322265625,
0.0197296142578125,
0.07000732421875,
0.1387939453125,
0.2154541015625,
0.0197601318359375,
0.042022705078125,
0.019622802734375,
0.019775390625,
0.158447265625,
0.0908203125,
0.118896484375,
0.13... |
906d020cba3836135a7395d21fb3297eafbb73ea | subsection | 10 | 17 | The stripe-like states by the variational Monte Carlo method | If one of the interactions, like electron-lattice
interaction, becomes quite strong as observed in LBCO-1/8
, then the modulation will also become larger and
longer ranged.We have also investigated the pair-pair correlation function for the
optimized states with/without hole-hole repulsive Jastrow factors in
the case o... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 172,
"openalex_id": "",
"raw": "T. Valla et al., Science 314, 1914 (2006).",
"source_ref_id": "e935f4fb275ec6d7fbeeb0332ea1ec975eeef569",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end": 2477... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
4263,
1632,
70,
182809,
7,
4,
1884,
77556,
19,
2335,
24494,
24209,
32233,
37515,
139999,
71,
6,
48097,
9688,
5759,
23538,
17055,
2320,
1221,
150679,
51713,
37457,
32603,
80836,
9,
109637,
16106,
57860,
32354,
15572,
29367,
117249,
76228,
60... | [
0.049774169921875,
0.003936767578125,
0.014404296875,
0.2322998046875,
0.0113525390625,
0.011444091796875,
0.03155517578125,
0.1409912109375,
0.008453369140625,
0.08929443359375,
0.1455078125,
0.0733642578125,
0.056243896484375,
0.13916015625,
0.1121826171875,
0.011383056640625,
0.01... |
d02bbcd67f38c655fdb947a5994534b865439a71 | subsection | 11 | 17 | Density of states by the Gutzwiller approximation | According to the VMC calculation for the extended t-J model, it is
likely that there are a number of inhomogeneous states close in
energy to the uniform ground state. Then, some sort of small
perturbation may choose a particular stripe state as the ground
state. Assuming such a situation, here we regard a stripe state ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 861,
"openalex_id": "",
"raw": "N. Fukushima, cond-mat/0801.2280.",
"source_ref_id": "a4c2991293a9bc8cb16b778182c0887ca2c3e61e",
"start": 801
},
{
"arxiv_id": "",
"doi": "",
"end": 2441,
... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
129551,
47,
310,
32557,
74481,
65042,
808,
9,
1375,
3299,
4,
83,
47041,
10,
14012,
111,
23,
497,
432,
15292,
10821,
117249,
20903,
48302,
61514,
61585,
11341,
19336,
170950,
1363,
1543,
55076,
17311,
43613,
13,
237,
62,
66596,
214,
16648,... | [
0.0013427734375,
0.010467529296875,
0.1534423828125,
0.2325439453125,
0.186279296875,
0.1688232421875,
0.129638671875,
0.089599609375,
0.2269287109375,
0.21728515625,
0.01080322265625,
0.0283966064453125,
0.1258544921875,
0.01043701171875,
0.04638671875,
0.010589599609375,
0.01086425... |
76f47cc30e7d7174b104ae5542e9c98c75db4e04 | subsection | 12 | 17 | Density of states by the Gutzwiller approximation | (a) For\Delta _v^C\ne 0, we use parameters optimized by the VMC,t^{\prime }_v=-0.35, t^{\prime \prime }_v=0.16, m=0.15, \rho =0.03, \Delta _v^C=0.28, \Delta _v^M=0.02, but \mu = - 0.875t_v is adjusted to realize1/8 filling, in units of t_v. (b) The same parameters except for\Delta _v^C = 0.]Then, by taking the most dom... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 2436,
"openalex_id": "",
"raw": "S. Baruch and D. Orgad, cond-mat/0801.2436.",
"source_ref_id": "8cce16cf4067b0618c28627f1a2193fd0e81722e",
"start": 2360
}
]
} | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
11,
1326,
58598,
102,
101,
334,
8353,
441,
757,
642,
4527,
171859,
15572,
310,
32557,
24854,
41872,
114654,
51912,
454,
1369,
23320,
5,
5843,
4,
808,
6,
2485,
347,
1837,
497,
99929,
363,
3882,
594,
145407,
9550,
561,
2203,
20,
132208,
... | [
0.034576416015625,
0.005401611328125,
0.1893310546875,
0.14306640625,
0.014434814453125,
0.1658935546875,
0.0321044921875,
0.1253662109375,
0.126220703125,
0.015380859375,
0.06805419921875,
0.188232421875,
0.150390625,
0.1607666015625,
0.22314453125,
0.003570556640625,
0.003662109375... |
a6e7393ba414da694c19e2b7e8e5434df480e171 | subsection | 13 | 17 | Density of states by the Gutzwiller approximation | The
low energy spectra seem less influenced by the disorder than high
energy. This result shows that the node and the low energy V-shape
DOS are robust against this kind of inhomogeneity. This is possible
because nodal k-points do not have many states to mix with and
also the suppression of impurity scattering . The
su... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 313,
"openalex_id": "",
"raw": "Due to the projection operator, the impurity scattering matrix element is strongly renormalized with a factor proportional to the hole denstiy.",
"source_ref_id": "127304aab1e707bb56a821048cf8... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
27226,
48302,
48502,
1517,
48903,
40715,
79507,
171986,
11192,
16750,
45831,
110,
112,
136,
310,
9,
2420,
1081,
86559,
621,
60627,
26548,
903,
8562,
23,
497,
432,
15292,
2481,
7722,
2465,
472,
38496,
959,
5941,
117249,
17664,
15811,
48448,
... | [
0.15087890625,
0.1669921875,
0.0924072265625,
0.08740234375,
0.043121337890625,
0.126220703125,
0.102783203125,
0.2169189453125,
0.077880859375,
0.027008056640625,
0.012176513671875,
0.1810302734375,
0.1563720703125,
0.011199951171875,
0.2489013671875,
0.024078369140625,
0.2321777343... |
8aee9992db6129b04f28fd6bb2e9ddefdfc2607a | subsection | 14 | 17 | Density of states by the Gutzwiller approximation | Since A(k,\omega ) is regarded as the local DOS in the k-space,
Let us take the Fourier transform of renormalized u_R^n,
v_R^n, namely,( \tilde{u}_k^n , \tilde{v}_k^n ) \equiv \frac{1}{\sqrt{N_{\rm site}}} \sum _R e^{- i k R} \left( g^t_{R\uparrow } u_R^n,
g^t_{R\downarrow } v_R^n \right).Then, A_{\sigma }(k,\omega ) i... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
62,
132,
92,
306,
2765,
28601,
297,
237,
4000,
86559,
23,
472,
9,
65421,
5646,
65056,
6815,
27198,
456,
33176,
29367,
75,
1052,
8353,
19,
81,
3675,
112,
34,
334,
3181,
132076,
418,
864,
42,
1764,
11832,
17,
627,
2480,
118201,
454,
2... | [
0.2042236328125,
0.063232421875,
0.19189453125,
0.07720947265625,
0.253662109375,
0.1336669921875,
0.00433349609375,
0.0716552734375,
0.1715087890625,
0.2420654296875,
0.0236663818359375,
0.166015625,
0.034088134765625,
0.2271728515625,
0.009857177734375,
0.1646728515625,
0.224731445... |
abfdd537ccf6ef15ec3f34f403ee3922c5d72e73 | subsection | 15 | 17 | Conclusions | In summary, we have used a variational approach to examine the
possibility of having inhomogeneous ground states within the
extended t-J model with 1/8 doping. We considered states with
spatial modulation of charge density, staggered magnetization and
pairing amplitude. Besides the antiphase or inphase stripes
consider... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 633,
"openalex_id": "",
"raw": "Y. Kohsaka et al., Science 315, 1380 (2007).",
"source_ref_id": "651374f329f66374cf21266cf24b3f6c16d3bb37",
"start": 478
},
{
"arxiv_id": "",
"doi": "",
"end": ... | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
360,
177074,
4,
642,
11814,
143834,
289,
51515,
160477,
70,
207116,
19441,
23,
497,
432,
15292,
61585,
117249,
28032,
65042,
297,
808,
9,
1375,
3299,
678,
139893,
154917,
90698,
5623,
118,
17055,
2320,
111,
25534,
168,
7,
1924,
21407,
394... | [
0.0188140869140625,
0.043243408203125,
0.0189208984375,
0.000885009765625,
0.046142578125,
0.255615234375,
0.136474609375,
0.177978515625,
0.060333251953125,
0.018829345703125,
0.1339111328125,
0.052581787109375,
0.08465576171875,
0.123779296875,
0.1634521484375,
0.2086181640625,
0.2... |
bb2dae6fe866319fd9859b77f8b1958f5a695971 | subsection | 16 | 17 | Conclusions | In a realistic material,
other interactions such as impurity, disorder, and electron-lattice
interactions, etc., no doubt will help to determine the most
suitable local configuration of spins and holes but they will not
produce a globally ordered state unless there is a very strong and
dominant interaction like the ele... | {
"cite_spans": []
} | 10.1103/PhysRevB.78.134530 | 0807.1875 | The cluster glass state in the two-dimensional extended t-J model | [
"Chung-Pin Chou",
"Noboru Fukushima",
"Ting Kuo Lee"
] | [
"cond-mat.supr-con",
"cond-mat.str-el"
] | 2,008 | en | Physics | [
360,
61207,
1771,
4912,
3789,
182809,
566,
4717,
2481,
171986,
77556,
19,
2335,
24494,
92814,
1221,
4358,
83324,
2684,
202319,
4000,
180346,
111374,
136,
108564,
7,
1284,
959,
27489,
7964,
12989,
11341,
117934,
4552,
37515,
73944,
1884,
9,
... | [
0.036376953125,
0.216796875,
0.11865234375,
0.2198486328125,
0.0684814453125,
0.2359619140625,
0.022857666015625,
0.1171875,
0.012603759765625,
0.1585693359375,
0.14990234375,
0.03717041015625,
0.1107177734375,
0.18798828125,
0.075927734375,
0.0198516845703125,
0.12060546875,
0.136... |
f9addd916c522207cdf7e5f8159e61567f7bca99 | abstract | 0 | 12 | Abstract | We construct a metric simplicial complex which is an almost isometric model
of the moduli space M(S) of Riemann surfaces. We then use this model to compute
the "tangent cone at infinity" of M(S): it is the topological cone on the
quotient of the complex of curves C(S) by the mapping class group of S, endowed
with an ex... | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
1401,
64549,
186518,
134381,
289,
27140,
39555,
13882,
3299,
17055,
14,
32628,
276,
132,
294,
41419,
5761,
71579,
7068,
4527,
9969,
6743,
70,
14525,
2517,
158,
13,
99,
54241,
53,
2663,
109622,
98,
41502,
18750,
9709,
3132,
313,
291,
26783... | [
0.053619384765625,
0.135009765625,
0.225341796875,
0.2232666015625,
0.1148681640625,
0.207275390625,
0.094482421875,
0.05291748046875,
0.1519775390625,
0.1114501953125,
0.057220458984375,
0.09869384765625,
0.1304931640625,
0.05328369140625,
0.0994873046875,
0.1290283203125,
0.1671142... | |
36aba58d83160e0f8a521b77871ccb441ddbc47c | subsection | 1 | 12 | Introduction | Let S=S_{g,n} be a closed, orientable surface with genus g\ge 0
with n\ge 0 marked points, and let \operatorname{\operatorname{Teich}}(S) be the associated
Teichmüller space of marked conformal classes or (fixed area) constant curvature
metrics on S. Endow \operatorname{\operatorname{Teich}}(S) with the Teichmüller met... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1444,
"openalex_id": "",
"raw": "E. Leuzinger, Reduction theory for mapping class groups, preprint, Jan. 2008.",
"source_ref_id": "96a611cc6990d7e51500f5537c344106adda9406",
"start": 1395
}
]
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
10842,
159,
1369,
294,
177,
19,
155738,
23184,
71579,
107396,
706,
429,
757,
94419,
26847,
2633,
206469,
11627,
10476,
5548,
137272,
1413,
28103,
14155,
32628,
19911,
289,
61112,
55923,
53697,
130661,
6644,
27079,
357,
24293,
186518,
104,
857... | [
0.054840087890625,
0.139404296875,
0.0212249755859375,
0.1453857421875,
0.08660888671875,
0.025909423828125,
0.1431884765625,
0.1258544921875,
0.1654052734375,
0.1171875,
0.094482421875,
0.0748291015625,
0.109130859375,
0.15380859375,
0.083984375,
0.0292816162109375,
0.1368408203125,... | |
2fca8fc4a328856c524b1c3dac2665083f2fd556 | subsection | 2 | 12 | Introduction | In particular they deduce:\mbox{\bf Q}\mbox{-rank}(\Gamma )=\dim (\operatorname{Cone}(\Gamma \backslash G/K))Our first result is a determination of the metric space
\operatorname{Cone}(\operatorname{{\cal M}}(S)). The role of the
rational Tits building will be played by the complex of curves \operatorname{{\cal C}}(S)
... | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
17311,
1836,
8,
106357,
11728,
150598,
2396,
36467,
36557,
5771,
206469,
11935,
13,
12620,
5544,
527,
64,
605,
5117,
16750,
27354,
1363,
70,
186518,
32628,
11627,
6827,
276,
294,
31486,
168487,
46990,
7,
33976,
1221,
112730,
27140,
9709,
31... | [
0.06488037109375,
0.03955078125,
0.1375732421875,
0.1728515625,
0.10455322265625,
0.0802001953125,
0.121337890625,
0.123779296875,
0.1710205078125,
0.10540771484375,
0.1170654296875,
0.1939697265625,
0.1370849609375,
0.041046142578125,
0.067138671875,
0.05816650390625,
0.065795898437... | |
9d335cf0d91b1937f548bd1e6211f129022d4aca | subsection | 3 | 12 | Introduction | To endow S) with
the structure of a simplicial complex instead of an orbicomplex, we can simply replace
\operatorname{{\cal C}}(S) with its barycentric subdivision in the construction above.
}Our first result is that S) provides a simple and reasonably
accurate geometric model for \operatorname{{\cal M}}(S).Theorem 1
... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 866,
"openalex_id": "",
"raw": "Y. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. Jour. 83 (1996), no. 2, 249–286.",
"source_ref_id": "3f65bb1507c5defce63a645363d68fdc67bfe499",
... | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
717,
22,
24293,
159,
16,
678,
45646,
134381,
289,
27140,
64457,
707,
964,
277,
44974,
831,
42856,
91995,
6,
41872,
206469,
11627,
6827,
313,
47391,
294,
1909,
46899,
1771,
1614,
428,
25826,
23,
70,
50961,
36917,
5117,
16750,
87344,
8781,
... | [
0.0411376953125,
0.05938720703125,
0.1929931640625,
0.23486328125,
0.1968994140625,
0.04150390625,
0.14990234375,
0.221435546875,
0.0992431640625,
0.1790771484375,
0.0928955078125,
0.0916748046875,
0.0880126953125,
0.049041748046875,
0.1507568359375,
0.0209197998046875,
0.02882385253... | |
e447aee516c454e9dc0f038b31695b3e3fc00dc8 | subsection | 4 | 12 | Introduction | In contrast,
S) strongly exhibits aspects of positive curvature, since
even within the cone on a single simplex, any
two points x,y\in S) have whole families of distinct geodesics
between them, and these geodesics get arbitrarily far apart as
d(x,y)\rightarrow \infty . This is a basic property of the \sup metric on a q... | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
69822,
159,
16,
37515,
538,
80788,
128746,
24491,
130661,
6644,
3853,
28032,
158,
13,
11001,
8781,
425,
2499,
6626,
26847,
1022,
53,
28271,
87143,
117781,
20787,
988,
28021,
17721,
2046,
61799,
2060,
34955,
104,
118201,
46632,
939,
62822,
5... | [
0.161865234375,
0.1636962890625,
0.07958984375,
0.0927734375,
0.006011962890625,
0.1143798828125,
0.1060791015625,
0.1700439453125,
0.2061767578125,
0.111328125,
0.00244140625,
0.10089111328125,
0.173095703125,
0.06365966796875,
0.0682373046875,
0.132568359375,
0.08624267578125,
0.... | |
4c31bc1edaecc356978713b00037976fde086783 | subsection | 5 | 12 | Defining the map | We will define a map \tilde{\Psi }:\widetilde{(S)\rightarrow \operatorname{\operatorname{Teich}}(S) and
will prove that it is \operatorname{Mod}(S)-equivariant, and so descends to a map
\Psi :S)\rightarrow \operatorname{{\cal M}}(S). Let d=3g-3+n.
}The Collar Lemma in hyperbolic geometry gives that, for a fixed
topolog... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1431,
"openalex_id": "",
"raw": "Y. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. Jour. 83 (1996), no. 2, 249–286.",
"source_ref_id": "3f65bb1507c5defce63a645363d68fdc67bfe499",
... | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
1401,
1221,
61924,
10,
22288,
3675,
112,
683,
172,
51912,
113458,
132,
294,
16,
41872,
118201,
6,
206469,
11627,
24854,
10476,
5548,
47391,
23534,
450,
91616,
8152,
9,
13,
3181,
162591,
4,
136,
60887,
47,
54969,
172162,
6827,
276,
104,
... | [
0.0087890625,
0.011077880859375,
0.1944580078125,
0.00494384765625,
0.250244140625,
0.24072265625,
0.285400390625,
0.06591796875,
0.1646728515625,
0.004669189453125,
0.105224609375,
0.004547119140625,
0.16552734375,
0.004791259765625,
0.004425048828125,
0.004425048828125,
0.004333496... | |
6cee2f828a42143b8d9272591286115c35da149c | subsection | 6 | 12 | Defining the map | It is then immediate that the map \widetilde{\Psi } is an isometry from the \sup metric on each cone as above and this metric. Note that the factor of \frac{1}{4} leads to a factor of
\frac{1}{2} in the distance, and is consistent with the factor of \frac{1}{2} in the metric on the
Euclidean octant.\widetilde{\Psi } is... | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
83,
168894,
450,
22288,
6,
113458,
3675,
112,
683,
172,
142,
87739,
53,
1295,
2037,
186518,
98,
12638,
158,
13,
237,
36917,
903,
70,
31461,
111,
132076,
8152,
617,
37105,
7,
47,
10,
24854,
304,
23,
62488,
4,
136,
74729,
678,
41872,
... | [
0.16748046875,
0.09881591796875,
0.0079345703125,
0.190673828125,
0.008514404296875,
0.1964111328125,
0.260986328125,
0.25341796875,
0.1380615234375,
0.2066650390625,
0.010040283203125,
0.191650390625,
0.0804443359375,
0.032928466796875,
0.1334228515625,
0.142578125,
0.00851440429687... | |
527bc6c92549c30a7e89afa96f32b2b116d0e331 | subsection | 7 | 12 | Defining the map | As a consequence, the quotient space \operatorname{{\cal C}}^{\prime }(S)/\operatorname{Mod}(S) has a
simplicial structure so that the natural quotient map is simplicial. | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
1301,
179804,
6620,
41502,
18750,
32628,
206469,
11627,
6827,
313,
114654,
294,
64,
91616,
1556,
10,
134381,
289,
45646,
221,
450,
6083,
22288,
83
] | [
0.0133056640625,
0.09246826171875,
0.004913330078125,
0.1461181640625,
0.186279296875,
0.1956787109375,
0.1217041015625,
0.02008056640625,
0.168212890625,
0.1102294921875,
0.189453125,
0.12646484375,
0.059906005859375,
0.239501953125,
0.09112548828125,
0.030242919921875,
0.2509765625... | |
94eb3ba1e0286198f1aa54aa5e2af4b325a808f0 | subsection | 8 | 12 | Properties of | \Psi is almost onto: By a theorem of Bers, there is a constant C=C(g) such that every
X\in \operatorname{{\cal M}}(S) has a pants decomposition corresponding to a maximal simplex \sigma such that every curve of \sigma
has length at most C on X. With respect to these pants curves,
each of the twist coordinates is bound... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1838,
"openalex_id": "",
"raw": "Y. Minsky, Extremal length estimates and product regions in Teichmüller space, Duke Math. Jour. 83 (1996), no. 2, 249–286.",
"source_ref_id": "3f65bb1507c5defce63a645363d68fdc67bfe499",
... | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
41872,
683,
172,
83,
39555,
98,
188,
12,
3311,
70,
58391,
111,
2076,
7,
2685,
53697,
313,
1369,
441,
177,
16,
6044,
11907,
1193,
73,
6,
206469,
11627,
6827,
276,
47391,
294,
1556,
2652,
933,
8,
277,
40322,
42518,
214,
47,
10,
111340... | [
0.1138916015625,
0.1790771484375,
0.2470703125,
0.11083984375,
0.2587890625,
0.16552734375,
0.1929931640625,
0.0181732177734375,
0.020111083984375,
0.031219482421875,
0.1468505859375,
0.02789306640625,
0.185791015625,
0.10235595703125,
0.005035400390625,
0.17236328125,
0.093139648437... | |
ac9219ee6b78393cda7a8b3e3c346df163fce6c3 | subsection | 9 | 12 | Properties of | There is a constant C^{\prime \prime } such that if \Psi (x),\Psi (y) lie in the same simplex \Psi (\Delta ) of \operatorname{{\cal M}}(S).
then there is a (1,C^{\prime \prime }) quasi-geodesic \rho (x,y) in the metric d_{\operatorname{{\cal M}}(S)} joining \Psi (x) and \Psi (y) that stays in \Psi (\Delta ).Proof. [of ... | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
8622,
83,
10,
53697,
313,
8353,
114654,
6044,
2174,
683,
172,
425,
53,
28127,
23,
5701,
8781,
58598,
102,
111,
206469,
11627,
6827,
276,
294,
2685,
2858,
441,
12404,
429,
31,
988,
1771,
42,
497,
186518,
104,
33284,
24765,
10752,
4390,
... | [
0.044921875,
0.02374267578125,
0.02880859375,
0.228271484375,
0.10174560546875,
0.0226287841796875,
0.2110595703125,
0.01214599609375,
0.039764404296875,
0.0975341796875,
0.1707763671875,
0.130859375,
0.1124267578125,
0.10791015625,
0.05609130859375,
0.12548828125,
0.1953125,
0.202... | |
d53b64d38cb545b7af8145cbddf1a8f224ea81b8 | subsection | 10 | 12 | Properties of | By (\ref {eq:m1}) we then have |\rho |\ge \log (\ell _y(\gamma _1)/\ell _x(\gamma _1))-D^{\prime }\ge |\rho (x,y)|-2D^{\prime }.
Thus again we can assume \rho lies completely in simplices for which \gamma _1 is a vertex.
But now the conclusion again follows from (REF ).\diamondProof. [of Lemma REF ]
Suppose x\in \Delta... | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
3311,
29087,
13,
864,
418,
8152,
16,
642,
765,
6,
41872,
42,
497,
429,
4867,
15,
6796,
53,
132,
17705,
192,
17727,
64,
101,
425,
9,
397,
114654,
51912,
5428,
24854,
7,
13438,
831,
41591,
400,
90,
64557,
23,
112892,
5170,
83,
10,
4... | [
0.073974609375,
0.2178955078125,
0.092041015625,
0.1180419921875,
0.0924072265625,
0.006378173828125,
0.006561279296875,
0.01531982421875,
0.037322998046875,
0.005889892578125,
0.062286376953125,
0.0283050537109375,
0.254150390625,
0.08673095703125,
0.173095703125,
0.006317138671875,
... | |
e8883893f1c375331b875bf0f0b2fcc2af3e0b32 | subsection | 11 | 12 | Properties of | We now apply (REF ) to conclude that d_{S)}(x,y) is only larger by an additive constant.Dept. of Mathematics, University of Chicago5734 University Ave.Chicago, Il 60637E-mail: farb@math.uchicago.edu, masur@math.uic.edu | {
"cite_spans": []
} | 0807.1876 | Teichmuller geometry of moduli space, II: M(S) seen from far away | [
"Benson Farb",
"Howard Masur"
] | [
"math.GT",
"math.CV"
] | 2,008 | en | Mathematics | [
1401,
5036,
59911,
11766,
919,
47,
103876,
450,
104,
454,
294,
16,
8152,
425,
4,
53,
83,
4734,
150679,
390,
171793,
5844,
53697,
5,
4657,
6328,
111,
123426,
47148,
12535,
58823,
12243,
10289,
40168,
51379,
408,
519,
891,
1496,
196049,
2... | [
0.07440185546875,
0.1500244140625,
0.2076416015625,
0.1783447265625,
0.26123046875,
0.0064697265625,
0.1845703125,
0.03424072265625,
0.2083740234375,
0.1236572265625,
0.201171875,
0.0193939208984375,
0.0435791015625,
0.096923828125,
0.10260009765625,
0.156982421875,
0.1016845703125,
... | |
28c03ebde923ec56d86bd39ec688331fae77db6a | abstract | 0 | 11 | Abstract | We obtain novel nonlinear Schr\"{o}dinger-Pauli equations through a formal
non-relativistic limit of appropriately constructed nonlinear Dirac equations.
This procedure automatically provides a physical regularisation of potential
singularities brought forward by the nonlinear terms and suggests how to
regularise previ... | {
"cite_spans": []
} | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
1401,
113054,
21261,
351,
2256,
147,
8643,
42,
31,
6238,
56,
115043,
14,
28,
13722,
5256,
8305,
23113,
7962,
28728,
48242,
17475,
95307,
64549,
803,
20251,
3293,
50491,
191082,
87344,
72761,
20324,
15032,
38516,
67824,
31075,
91048,
40225,
... | [
0.042266845703125,
0.1729736328125,
0.13720703125,
0.12841796875,
0.2022705078125,
0.127685546875,
0.0579833984375,
0.038543701171875,
0.063232421875,
0.1473388671875,
0.118896484375,
0.1644287109375,
0.1376953125,
0.0689697265625,
0.2156982421875,
0.0911865234375,
0.03387451171875,
... | |
0e43695f659c3fed2bafaccb2b70a819a79b1ee6 | subsection | 1 | 11 | Introduction | Several nonlinear extensions of Schrödinger's equation have been constructed to probe the accuracy of quantum
linearity , , . For example, Weinberg proposed a class of equations which were then used
in several experimental tests, see , , , and references therein. The results
indicated that any potential non-linearity i... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 125,
"openalex_id": "",
"raw": "I. Bialynicki-Birula and J. Mycielski, Ann. Phys. 100, 62 (1976).",
"source_ref_id": "bcbe625c636b1aea91909008f56351eeb34dc746",
"start": 0
},
{
"arxiv_id": "",
"doi"... | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
48752,
351,
2256,
147,
111938,
8643,
14900,
6238,
56,
28,
5490,
765,
2809,
64549,
297,
47,
502,
70,
61689,
219,
110436,
192617,
2481,
6,
1326,
4,
61628,
4969,
26171,
10,
18507,
111,
13722,
5256,
3129,
3542,
11814,
23,
40368,
195935,
109... | [
0.03289794921875,
0.151611328125,
0.279052734375,
0.12890625,
0.180419921875,
0.0767822265625,
0.1893310546875,
0.170166015625,
0.1356201171875,
0.029205322265625,
0.18603515625,
0.021209716796875,
0.00439453125,
0.1502685546875,
0.021392822265625,
0.0211944580078125,
0.0346374511718... | |
da2c5826d0841534172f977f7b64b31be52425f8 | subsection | 2 | 11 | Introduction | The singularity resolution is discussed in Section and we end with a discussion in Section .We note in passing that nonlinear Schrödinger equations of other types have been constructed from Levy-Leblond's
“non-relativistic Dirac equation" which is itself the non-relativistic limit of the usual Dirac equation
, . | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 314,
"openalex_id": "",
"raw": "C. Duval, P. A. Horvathy and L. Palla, Phys. Rev. D52, 4700 (1995).",
"source_ref_id": "24a03614789817c41d6bc35b994e9a5ec84c069a",
"start": 93
},
{
"arxiv_id": "",
"d... | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
67824,
2481,
158839,
45252,
23,
140978,
3564,
35107,
20537,
351,
2256,
147,
8643,
14900,
6238,
56,
28,
13722,
5256,
3789,
52895,
64549,
1295,
636,
3033,
5267,
57893,
71,
6431,
7962,
28728,
48242,
803,
20251,
5490,
2320,
68034,
17475,
115723... | [
0.2235107421875,
0.119140625,
0.223388671875,
0.108642578125,
0.010711669921875,
0.1466064453125,
0.0802001953125,
0.109130859375,
0.05120849609375,
0.0947265625,
0.125732421875,
0.0274505615234375,
0.01348876953125,
0.1070556640625,
0.1297607421875,
0.0767822265625,
0.04415893554687... | |
fb172838964f37c2437078c2abbd4d7d9110430d | subsection | 3 | 11 | Non-Relativistic Limit | We start from nonlinear Dirac equations of the form\left(i\hbar \gamma ^\mu \partial _\mu -mc + \epsilon F \right) \psi =0 \, ,where F=F(\psi ,\bar{\psi })=fI and where we have made the small parameter \epsilon explicit. We demand that
F has certain properties so that desirable characteristics of the linear Dirac equat... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 729,
"openalex_id": "",
"raw": "C. Itzykson and J. B. Zuber, Quantum field theory (New York: McGraw-Hill International Book Co. 1980).",
"source_ref_id": "84a1b60d5a8be8d63f7dff4d6e915b8e005b375a",
"start": 569
},
... | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
1401,
4034,
1295,
351,
2256,
147,
803,
20251,
28,
13722,
5256,
3173,
14,
41872,
127,
1299,
6,
17705,
192,
561,
15866,
289,
20,
39,
238,
997,
13,
15759,
4759,
563,
54969,
16,
145407,
4,
919,
132,
24854,
51912,
1369,
568,
136,
7440,
7... | [
0.052215576171875,
0.1361083984375,
0.039154052734375,
0.1522216796875,
0.216064453125,
0.108154296875,
0.104248046875,
0.257080078125,
0.0419921875,
0.20751953125,
0.012725830078125,
0.1551513671875,
0.0157623291015625,
0.013153076171875,
0.0736083984375,
0.223388671875,
0.012756347... | |
c9df5632b86c60c332de0c35ac7b1fa456ada381 | subsection | 4 | 11 | Non-Relativistic Limit | From the lower component of (REF ) we have,\chi =\frac{i\hbar \mbox{$ \sigma $}\cdot \mbox{$ \nabla $}\varphi }{2mc}-\frac{i\hbar }{2mc^2}\frac{\partial \chi }{\partial t}+\frac{\epsilon f \chi }{2mc} \, .Let
\chi _0=\frac{i\hbar \mbox{$ \sigma $}\cdot \mbox{$ \nabla $}\varphi }{2mc}.
Then expanding (REF ) about \chi _... | {
"cite_spans": []
} | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
28090,
92319,
82761,
111,
11766,
919,
642,
765,
41872,
1861,
2203,
132076,
14,
127,
1299,
11728,
20561,
192,
15464,
76,
7119,
1961,
19379,
304,
238,
15866,
808,
1328,
15759,
4759,
1238,
2389,
71062,
113054,
670,
2480,
418,
363,
54969,
83,... | [
0.08648681640625,
0.1912841796875,
0.21142578125,
0.055145263671875,
0.1214599609375,
0.2177734375,
0.00543212890625,
0.0533447265625,
0.038330078125,
0.294921875,
0.08172607421875,
0.1781005859375,
0.0625,
0.09381103515625,
0.193359375,
0.122802734375,
0.12109375,
0.1419677734375,... | |
53cd8f757c5722b8ae9057dca2fb75df4c353a46 | subsection | 5 | 11 | Lorentz invariant | A Lorentz invariant f with one derivative and which is odd under the parity transformation isf_1= \epsilon \frac{\partial _\mu j^\mu _5}{\bar{\psi }\psi } \, ,where j^\mu _5 = \bar{\psi }\gamma ^\mu \gamma _5\psi is the usual chiral current. The non-relativistic limit isi\hbar \frac{\partial \varphi }{\partial t}\simeq... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 669,
"openalex_id": "",
"raw": "W. K. Ng and R. Parwani, SIGMA 5, 023 (2009).",
"source_ref_id": "98e23dae37f900b34bc94f6ea3c75f07f853f148",
"start": 481
}
]
} | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
62,
116166,
9136,
23,
162591,
1238,
678,
1632,
30057,
4935,
3129,
83,
70270,
1379,
366,
2481,
167201,
420,
115187,
1369,
6,
13,
15759,
4759,
41872,
132076,
15866,
289,
561,
1647,
758,
8152,
24854,
1299,
51912,
4,
136913,
2203,
17705,
192,... | [
0.053131103515625,
0.189453125,
0.220947265625,
0.092529296875,
0.306396484375,
0.2039794921875,
0.08343505859375,
0.1495361328125,
0.2205810546875,
0.136962890625,
0.036590576171875,
0.11962890625,
0.265625,
0.1427001953125,
0.1961669921875,
0.123291015625,
0.1907958984375,
0.1717... | |
67dc0ab9551a55aaf351f477ea1f441ecada8762 | subsection | 6 | 11 | Lorentz violating, parity even | Lorentz violating non-linear Dirac equations are of some interest , , , , . An example of such an
f with no derivatives and even under parity isf_3=A_\mu \frac{\bar{\psi }\gamma ^\mu \psi }{\bar{\psi }\psi }where A_{\mu } is a constant vector background field. The non-relativistic limit isi\hbar \frac{\partial \varphi ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 75,
"openalex_id": "",
"raw": "W. K. Ng and R. Parwani, SIGMA 5, 023 (2009).",
"source_ref_id": "98e23dae37f900b34bc94f6ea3c75f07f853f148",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end": 75... | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
116166,
9136,
54567,
1916,
351,
2256,
147,
803,
20251,
13722,
5256,
33946,
27781,
6044,
1238,
678,
110,
30057,
42991,
3853,
1379,
366,
2481,
83,
420,
363,
284,
561,
132076,
1299,
15759,
17705,
62,
53697,
173,
18770,
76615,
44457,
7962,
28... | [
0.1507568359375,
0.19091796875,
0.208984375,
0.037322998046875,
0.150390625,
0.1279296875,
0.0538330078125,
0.050018310546875,
0.1966552734375,
0.15966796875,
0.041839599609375,
0.107177734375,
0.1846923828125,
0.0572509765625,
0.1978759765625,
0.09423828125,
0.150634765625,
0.1704... | |
15d4302edd3afb9e099b7c94447bf5cea91eec37 | subsection | 7 | 11 | Lorentz violating, parity odd | A Lorentz violating f which is odd under parity isf_4=A_\mu \frac{\bar{\psi }\gamma _5\gamma ^\mu \psi }{\bar{\psi }\psi } \, .The non-relativistic equation isi\hbar \frac{\partial \varphi }{\partial t}&\simeq &-\frac{\hbar ^2\nabla ^2\varphi }{2m}-\frac{c\varphi ^\dag \mbox{$ A $}\cdot \mbox{$ \sigma $}\varphi }{|\var... | {
"cite_spans": []
} | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
62,
116166,
9136,
54567,
1916,
1238,
3129,
83,
70270,
1379,
366,
2481,
420,
454,
617,
1369,
284,
561,
132076,
1299,
15759,
17705,
192,
758,
351,
7962,
28728,
48242,
28,
5490,
2320,
16252,
127,
15866,
1961,
19379,
808,
1230,
13777,
864,
... | [
0.107421875,
0.1536865234375,
0.1983642578125,
0.2362060546875,
0.1197509765625,
0.14306640625,
0.0240936279296875,
0.122314453125,
0.247802734375,
0.18310546875,
0.1705322265625,
0.1094970703125,
0.2088623046875,
0.00689697265625,
0.192138671875,
0.029571533203125,
0.0712890625,
0... | |
9b63556e02bbdc834e6ab02922cc4e0b825319d4 | subsection | 8 | 11 | Apparent Singularities | From the above examples, we see the appearance of the following structures in the non-linear Schrödinger-Pauli
equations,X=\frac{\varphi ^\dag \mbox{$ \sigma $}\cdot \mbox{$ \nabla $}\varphi }{|\varphi |^2}\,\,\,,\,\,\,Y=\frac{(\mbox{$ \nabla $}\varphi ^\dag )\cdot (\mbox{$ \nabla $}\varphi )}{|\varphi |^2}\,\,\,,\,\,\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 682,
"openalex_id": "",
"raw": "W. K. Ng and R. Parwani, SIGMA 5, 023 (2009).",
"source_ref_id": "98e23dae37f900b34bc94f6ea3c75f07f853f148",
"start": 516
}
]
} | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
36917,
27781,
1957,
170894,
25632,
45646,
351,
2256,
147,
8643,
14900,
6238,
56,
115043,
14,
28,
13722,
5256,
1542,
132076,
1961,
19379,
6063,
11728,
20561,
76,
7119,
304,
1723,
1511,
110,
41872,
3173,
67824,
831,
71864,
31075,
6083,
803,
... | [
0.04583740234375,
0.1259765625,
0.039276123046875,
0.0997314453125,
0.05029296875,
0.2021484375,
0.1495361328125,
0.2191162109375,
0.1302490234375,
0.03167724609375,
0.1221923828125,
0.12109375,
0.0692138671875,
0.1336669921875,
0.1175537109375,
0.054595947265625,
0.2080078125,
0.0... | |
fe8c488f94bc6545c23371bdb3fe5527c90fee72 | subsection | 9 | 11 | Apparent Singularities | Singularities will appear in n\ge 2 classes of nonlinearities discussed in Ref, two examples
of which are given byV&=&Y^2=\frac{\left[(\mbox{$ \nabla $}\varphi ^\dag )\cdot (\mbox{$ \nabla $}\varphi )\right]\left[(\mbox{$ \nabla $}\varphi ^\dag )\cdot (\mbox{$ \nabla $}\varphi )\right]}{|\varphi |^2|\varphi |^2} \, , \... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 485,
"openalex_id": "",
"raw": "W. K. Ng and R. Parwani, SIGMA 5, 023 (2009).",
"source_ref_id": "98e23dae37f900b34bc94f6ea3c75f07f853f148",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end": 1... | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
53427,
35975,
31075,
1221,
108975,
23,
653,
429,
116,
61112,
351,
2256,
147,
45252,
53295,
6626,
27781,
856,
1230,
1723,
304,
132076,
2480,
11728,
76,
7119,
1961,
19379,
6063,
1511,
34735,
48302,
122925,
7,
67824,
69407,
117934,
20324,
1503... | [
0.1748046875,
0.2169189453125,
0.1541748046875,
0.045318603515625,
0.14794921875,
0.003082275390625,
0.0166015625,
0.11328125,
0.1285400390625,
0.19873046875,
0.1446533203125,
0.2266845703125,
0.153076171875,
0.0892333984375,
0.1680908203125,
0.0372314453125,
0.1627197265625,
0.105... | |
50de44d8eccc50535aac9271874fd01b742cc2ee | subsection | 10 | 11 | Discussion | We have illustrated how to obtain novel classes of nonlinear Schrödinger-Pauli equations starting from the
nonlinear Dirac equations constructed in Ref, the latter equations themselves being more general than
previous constructions , , . For example, we have cases where the time-derivatives appear
in the nonlinearity, ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 237,
"openalex_id": "",
"raw": "W. K. Ng and R. Parwani, SIGMA 5, 023 (2009).",
"source_ref_id": "98e23dae37f900b34bc94f6ea3c75f07f853f148",
"start": 0
},
{
"arxiv_id": "",
"doi": "",
"end": 2... | 0807.1877 | Nonlinear Schrodinger-Pauli Equations | [
"Wei Khim Ng",
"Rajesh R. Parwani"
] | [
"quant-ph",
"hep-th",
"nlin.PS"
] | 2,008 | en | Physics | [
58755,
3642,
113054,
21261,
61112,
351,
2256,
147,
8643,
14900,
6238,
56,
115043,
14,
28,
13722,
5256,
803,
20251,
64549,
23,
53295,
4,
70,
8035,
1286,
4537,
3501,
96362,
50961,
7,
6,
1326,
50218,
7440,
1733,
9,
820,
3984,
42991,
108975... | [
0.126953125,
0.042572021484375,
0.1107177734375,
0.12060546875,
0.218017578125,
0.1475830078125,
0.251953125,
0.13623046875,
0.059600830078125,
0.151611328125,
0.175048828125,
0.10699462890625,
0.1590576171875,
0.150390625,
0.029388427734375,
0.2149658203125,
0.058441162109375,
0.0... | |
d490593c949b600448dffdec90a561c7c4f9d29c | abstract | 0 | 76 | Abstract | The long-time asymptotics is analyzed for finite energy solutions of the 1D
Schr\"odinger equation coupled to a nonlinear oscillator; mathematically the
system under study is a nonlinear Schr\"odinger equation, whose nonlinear term
includes a Dirac delta. The coupled system is invariant with respect to the
phase rotati... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
4989,
6032,
237,
4650,
40934,
41637,
7968,
53,
94418,
48302,
51347,
106,
397,
8643,
42,
2199,
21228,
28,
5490,
2320,
24941,
351,
2256,
147,
362,
121226,
1290,
140363,
5426,
35187,
13579,
96853,
803,
20251,
40703,
71,
23,
162591,
15072,
93... | [
0.0853271484375,
0.15771484375,
0.04595947265625,
0.111572265625,
0.1746826171875,
0.08636474609375,
0.056671142578125,
0.013885498046875,
0.1617431640625,
0.1558837890625,
0.1756591796875,
0.0152587890625,
0.09002685546875,
0.04827880859375,
0.06396484375,
0.0955810546875,
0.2348632... | |
995f7875e23194af50c8c3990c34a27adb145d09 | subsection | 1 | 76 | Introduction | In this article we continue the study, initiated in , of large
time asymptotics for a model U(1)-invariant nonlinear Schrödinger
equationi\dot{\psi }(x,t)=
-\psi ^{\prime \prime }(x,t)-\delta (x)F(\psi (0,t)),\quad x\in {\mathbb {R}},Here \psi (x,t) is a
continuous complex-valued wave function and F is a continuous
fun... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 326,
"openalex_id": "",
"raw": "V.Buslaev, A.Komech, E.Kopylova, D.Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation, Comm. Partial Diff. Eqns 33 (2008), no. 4, 669-705.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
5582,
21342,
35187,
173969,
297,
6,
21334,
1733,
237,
4650,
40934,
41637,
3299,
345,
27750,
162591,
351,
2256,
8643,
14900,
6238,
56,
28,
5490,
15464,
15759,
51912,
132,
425,
4,
18,
16,
1369,
20,
41872,
24854,
114654,
9,
1743,
102,
15,
... | [
0.03076171875,
0.06903076171875,
0.07489013671875,
0.033721923828125,
0.003692626953125,
0.003509521484375,
0.1075439453125,
0.16650390625,
0.0750732421875,
0.1162109375,
0.1744384765625,
0.06793212890625,
0.194091796875,
0.1263427734375,
0.129638671875,
0.1883544921875,
0.1037597656... | |
cefc4602415358a026bc84acf716ed22c555de78 | subsection | 2 | 76 | Introduction | \end{equation}Then (\ref {SV}) is formally a Hamiltonian system with Hamiltonian
\begin{equation}
{\cal H}(\psi )=\frac{1}{2}\int |\psi ^{\prime }|^2 dx+U(\psi (0))
\end{equation}We assume that the potential U() satisfies the
inequality \begin{equation}
U(z)\ge A-B|z|^2 \quad {\rm with\; some}\quad A\in {\mathbb {R}},\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 922,
"openalex_id": "",
"raw": "A.I. Komech, A.A. Komech, Global well-posedness for the Schrödinger equation coupled to a nonlinear oscillator, Russ. J. Math. Phys. 14 (2007), no. 2, 164-173.",
"source_ref_id": "6e7a2d524a0b... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
3611,
5490,
2320,
41872,
29087,
43486,
83,
23113,
538,
10,
94674,
3378,
5426,
678,
6820,
6827,
572,
15759,
132076,
418,
304,
4288,
114654,
104,
425,
1328,
1062,
6649,
41591,
38516,
345,
132,
40407,
3387,
23,
13,
161789,
169,
429,
62,
9,... | [
0.0164794921875,
0.191162109375,
0.0308837890625,
0.018310546875,
0.26904296875,
0.287109375,
0.11474609375,
0.1517333984375,
0.07763671875,
0.1083984375,
0.251953125,
0.2088623046875,
0.2413330078125,
0.06622314453125,
0.01495361328125,
0.162353515625,
0.1292724609375,
0.193847656... | |
746c24a0899c5696f83179cc0b750a54e06f75fe | subsection | 3 | 76 | Solitary waves | Equation (REF ) admits finite energy solutions of type \psi _\omega (x)e^{i\omega t}, called
solitary waves or nonlinear eigenfunctions.
The frequency \omega and the amplitude \psi _\omega (x) solve the following
nonlinear eigenvalue problem:-\omega \psi _\omega (x)= -\psi _\omega ^{\prime \prime }(x)-\delta (x)F(\psi ... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 631,
"openalex_id": "",
"raw": "V.Buslaev, A.Komech, E.Kopylova, D.Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation, Comm. Partial Diff. Eqns 33 (2008), no. 4, 669-705.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
241,
5490,
2320,
11766,
919,
1388,
36456,
7,
94418,
13,
48302,
51347,
10644,
15759,
306,
2765,
425,
14,
808,
35839,
20245,
867,
53,
259,
3132,
707,
351,
2256,
147,
8518,
137175,
12478,
944,
27771,
136,
217269,
86869,
25632,
27494,
2967,
... | [
0.06561279296875,
0.2178955078125,
0.111572265625,
0.1395263671875,
0.1982421875,
0.002777099609375,
0.18994140625,
0.007568359375,
0.1959228515625,
0.08673095703125,
0.2059326171875,
0.203857421875,
0.119384765625,
0.214111328125,
0.09552001953125,
0.221923828125,
0.072265625,
0.0... | |
ea267159723d7f259fc4aa09c2de2bc007f29223 | subsection | 4 | 76 | Solitary waves | Briefly,the continuous spectrum coincides with
{\mathcal {C}}_+\cup {\mathcal {C}}_- where
{\mathcal {C}}_+=[i\omega , i\infty ), and
{\mathcal {C}}_-=(-i\infty ,-i\omega ];
the discrete spectrum always contains zero
on account of the circular symmetry of the problem, and there is a
corresponding generalized eigenspac... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
101936,
2347,
62005,
223,
235079,
60754,
90,
678,
125458,
6827,
441,
47391,
1328,
33874,
9,
14,
306,
2765,
17,
46632,
939,
74,
81604,
13,
11343,
70541,
45234,
15426,
115339,
954,
3019,
15123,
70,
2967,
42518,
4537,
29367,
8518,
65421,
6,
... | [
0.1051025390625,
0.0300445556640625,
0.1939697265625,
0.1002197265625,
0.2685546875,
0.1673583984375,
0.031097412109375,
0.0259246826171875,
0.00634765625,
0.09881591796875,
0.036773681640625,
0.00518798828125,
0.1324462890625,
0.193603515625,
0.07745361328125,
0.03125,
0.03756713867... | |
5b59b647aca5245b260eb9fe65e0215d1cff7148 | subsection | 5 | 76 | Statement of main theorem | Previously, in , we considered the case when
a^{\prime }\in (-\infty ,0)\cup (0,a/\sqrt{2}C^2).
In which case the operator {\bf C} has no discrete spectrum except zero.
In the present paper we will consider the case whena^{\prime }\in \big (a/\sqrt{2} C^2,~a\sqrt{2}(1+\sqrt{3})/4C^2\big ).In this case, there are, in ad... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 95,
"openalex_id": "",
"raw": "V.Buslaev, A.Komech, E.Kopylova, D.Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation, Comm. Partial Diff. Eqns 33 (2008), no. 4, 669-705.",
"source_ref_id": ... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
6422,
538,
90698,
7225,
3229,
10,
8353,
114654,
73,
9,
46632,
939,
6,
77495,
41872,
33874,
22085,
11,
64,
864,
3198,
304,
8152,
441,
10461,
39933,
150598,
313,
1556,
110,
81604,
235079,
40494,
45234,
70,
15122,
16916,
51912,
32976,
15,
... | [
0.014923095703125,
0.009613037109375,
0.081787109375,
0.15625,
0.1082763671875,
0.10595703125,
0.053863525390625,
0.266845703125,
0.1207275390625,
0.051361083984375,
0.1798095703125,
0.0885009765625,
0.00958251953125,
0.10528564453125,
0.009674072265625,
0.181884765625,
0.05331420898... | |
6ede4c4b3265b87c3662ed7fa8f13ff01905d903 | subsection | 6 | 76 | Statement of main theorem | In appendix E we express (REF ) in terms of C and a(C^2),
and hence show that
the Fermi Golden Rule holds generically for polynomial nonlinearity.Let us introduce the weighted Banach space L^p_{\beta } with the finite norm\Vert f\Vert _{L^p_{\beta }}=\Vert (1+|x|)^{\beta } f(x)\Vert _{L^p}Our main theorem is the follow... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
114689,
428,
425,
241,
36510,
11766,
919,
69407,
313,
136,
10,
441,
8353,
10461,
7639,
8002,
266,
43114,
139118,
16401,
7,
11212,
71407,
100,
35874,
1687,
15403,
351,
2256,
147,
2481,
65508,
57888,
297,
5458,
934,
32628,
339,
254,
59865,
... | [
0.06182861328125,
0.0560302734375,
0.0396728515625,
0.1011962890625,
0.1868896484375,
0.13623046875,
0.2308349609375,
0.0821533203125,
0.09368896484375,
0.06365966796875,
0.10504150390625,
0.0101318359375,
0.052276611328125,
0.1121826171875,
0.09130859375,
0.22998046875,
0.2775878906... | |
1eb95d01fa96af19db92cdb56043658ad2453ec0 | subsection | 7 | 76 | Linearization | In this section we summarize the spectral properties of the
operator \mathbf {C} and then give some estimates for the
linearized evolution.
The proof of these properties can be found in , with the
exception of proposition which is proved in appendix C. | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 253,
"openalex_id": "",
"raw": "V.Buslaev, A.Komech, E.Kopylova, D.Stuart, On asymptotic stability of solitary waves in a nonlinear Schrödinger equation, Comm. Partial Diff. Eqns 33 (2008), no. 4, 669-705.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
360,
903,
40059,
642,
29334,
14096,
13,
48502,
29088,
183871,
39933,
41872,
125458,
150598,
441,
8337,
3060,
25902,
1636,
192617,
29367,
28,
137089,
98869,
6097,
831,
186,
14037,
131527,
89261,
77443,
114689,
428,
425,
313
] | [
0.0217132568359375,
0.07196044921875,
0.103271484375,
0.031646728515625,
0.1439208984375,
0.120849609375,
0.042633056640625,
0.1553955078125,
0.154052734375,
0.20703125,
0.26318359375,
0.0543212890625,
0.0709228515625,
0.2012939453125,
0.166259765625,
0.03515625,
0.0201416015625,
0... | |
0d1b8fbf48daf8c5cfd2b045b7d1d8d781dcaa9b | subsection | 8 | 76 | Spectral properties | The linearized equation reads\dot{\chi }(x,t)={\bf C}\chi (x,t),~~~~~{\bf C}:=j^{-1}{\bf B}=
\left(
\begin{array}{rr}
0 & {\bf D}_2\\
-{\bf D}_1 & 0
\end{array}\right).Theorem REF generalizes to the equation (REF ):
the equation admits unique solution \chi (x,t)\in C_b({\mathbb {R}},H^1)
for every initial function \chi... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
581,
192617,
29367,
28,
5490,
2320,
12301,
7,
15464,
1861,
425,
4,
18,
1369,
150598,
313,
220802,
170,
5759,
335,
2480,
6820,
19305,
29161,
757,
391,
304,
115187,
619,
6,
3611,
53,
54969,
194,
3957,
58391,
9069,
919,
4537,
20650,
47,
... | [
0.0609130859375,
0.277099609375,
0.2423095703125,
0.1392822265625,
0.262939453125,
0.154052734375,
0.2186279296875,
0.05810546875,
0.1453857421875,
0.273681640625,
0.10430908203125,
0.0706787109375,
0.155517578125,
0.047149658203125,
0.1697998046875,
0.09674072265625,
0.0455322265625... | |
53be7fbc3cd7360fa0739eb55243b7fd3f84c443 | subsection | 9 | 76 | Spectral properties | Denote by e^{{\bf C}t} the dynamical group of equation (REF )
acting in the space H^1; for T>0 there exists c_T>0
such that\Vert e^{{\bf C}t}\chi _0\Vert _{H^1}\le c_T\Vert \chi _0\Vert _{H^1},\qquad |t|\le T.The resolvent {\bf R}(\lambda ):=({\bf C}-\lambda )^{-1} is an integral operator
with matrix valued integral ke... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
262,
48345,
390,
28,
150598,
313,
18,
84079,
289,
21115,
111,
5490,
2320,
11766,
919,
1030,
1916,
23,
32628,
572,
8353,
418,
100,
384,
2740,
2389,
32316,
501,
454,
618,
6044,
15896,
8152,
1861,
24854,
841,
41872,
6,
4,
91526,
58745,
1... | [
0.147216796875,
0.196533203125,
0.0908203125,
0.1763916015625,
0.211669921875,
0.093994140625,
0.15478515625,
0.21142578125,
0.1160888671875,
0.2130126953125,
0.009857177734375,
0.2308349609375,
0.09503173828125,
0.1407470703125,
0.21728515625,
0.1156005859375,
0.09442138671875,
0.... | |
6cfc1c2667a47b40db296b4b75bfc0e28c547907 | subsection | 10 | 76 | Spectral properties | The constants \alpha , \beta and D=D(\lambda ) are given by the formulas
\alpha =a+a^{\prime }C^2,\;\beta =a^{\prime }C^2,\;D=2i\alpha (k_++k_-)-4k_+k_-+\alpha ^2-\beta ^2.
In addition to this continuous spectrum, there is discrete spectrum, which
appears in this formalism as the set of
poles of \mathbf {R}(\lambda )... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
581,
53697,
7,
289,
14612,
59865,
136,
391,
1369,
397,
143,
6492,
85,
621,
34475,
390,
26168,
11,
1328,
114654,
441,
8353,
304,
2203,
55257,
14,
37223,
11565,
18504,
5442,
62005,
235079,
81604,
135179,
23113,
8780,
5423,
5664,
150598,
105... | [
0.04827880859375,
0.267333984375,
0.1138916015625,
0.06640625,
0.2039794921875,
0.2490234375,
0.06268310546875,
0.1644287109375,
0.06341552734375,
0.146240234375,
0.026519775390625,
0.196044921875,
0.141357421875,
0.0189056396484375,
0.14306640625,
0.027587890625,
0.195068359375,
0... | |
8457115e720f33ddea2547d604a2036c0a0a1ec4 | subsection | 11 | 76 | Spectral properties | In particular this operator can be applied to the Dirac measure \delta (x).
\end{}
Denote by X1 the eigensubspace corresponding to the two pure imaginary
eigenvalues, and by P1 a symplectic projection operator from L2(R) onto X1.
It may be represented by the formula
\begin{equation}
{\bf P}^1\psi =\frac{\langle \psi ,j... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
360,
17311,
903,
39933,
831,
190659,
47,
803,
20251,
72350,
6,
1743,
102,
425,
262,
48345,
390,
1193,
418,
8518,
22144,
65421,
42518,
214,
70,
6626,
34166,
114135,
53,
27494,
90,
4,
436,
954,
33209,
49086,
13452,
1830,
1295,
339,
304,
... | [
0.013397216796875,
0.054718017578125,
0.0794677734375,
0.287841796875,
0.0797119140625,
0.1575927734375,
0.0171051025390625,
0.123046875,
0.244384765625,
0.220703125,
0.0165863037109375,
0.12744140625,
0.1568603515625,
0.10186767578125,
0.0447998046875,
0.080810546875,
0.029998779296... | |
abadf36961cf9266ba4411b8113fb22fd28a8501 | subsection | 12 | 76 | Spectral properties | Then
for h\in {\cal M}_{\beta } with \beta \ge 2 the following bounds hold:
\begin{equation}
\Vert e^{{\bf C}t}({\bf C}\mp 2i\mu -0)^{-1}{\bf P}^ch\Vert _{L^{\infty }_{-\beta }}
+\Vert e^{{\bf C}t}({\bf C}\mp 2i\mu -0)^{-1}{\bf \Pi }^{\pm }h\Vert _{L^{\infty }_{-\beta }}
\le c(1+t)^{-3/2}\Vert h\Vert _{{\cal M}_{\beta ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
47009,
100,
1096,
73,
6827,
276,
59865,
678,
429,
116,
25632,
99091,
7,
16401,
6820,
5490,
15896,
28,
150598,
313,
2676,
14,
561,
5759,
46632,
29348,
26822,
127,
133,
501,
1328,
18,
8316,
12477,
23534,
89261,
114689,
428,
91616,
72403,
... | [
0.0166015625,
0.03167724609375,
0.1978759765625,
0.0712890625,
0.1968994140625,
0.145751953125,
0.26220703125,
0.0258331298828125,
0.1273193359375,
0.133544921875,
0.080078125,
0.2264404296875,
0.02996826171875,
0.1177978515625,
0.0155181884765625,
0.1884765625,
0.1146240234375,
0.... | |
652024277cf2c401b034adb49a02c45bb8bee80a | subsection | 13 | 76 | Spectral properties | Now we give a system of coupled equations
for \omega (t), \gamma (t), z(t) and f(x,t).Lemma 2.1 (cf.)
Given a solution of (REF ) in the form (REF ) with f\in X^c
as just described, the functions
\omega (t), \gamma (t), z(t) and f(x,t) satisfy the system\dot{\omega }=\frac{\langle {\bf P}^0{\bf Q}[\chi ],\Phi \rangle }{... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1120,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id"... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
14240,
8337,
5426,
111,
24941,
71,
28,
13722,
5256,
100,
41872,
306,
2765,
18,
247,
17705,
192,
97,
136,
1238,
425,
18023,
34513,
67466,
77878,
29806,
11766,
919,
3173,
678,
73,
1193,
8353,
238,
151552,
32354,
40407,
53,
15464,
132076,
... | [
0.047088623046875,
0.089599609375,
0.22119140625,
0.10040283203125,
0.2366943359375,
0.142578125,
0.09283447265625,
0.2393798828125,
0.1429443359375,
0.06048583984375,
0.0265045166015625,
0.1402587890625,
0.2294921875,
0.1326904296875,
0.0234832763671875,
0.182861328125,
0.1374511718... | |
d9c802e3efc3375afd3db88cb5a33e1a3c2b7af3 | subsection | 14 | 76 | Frozen spectral decomposition | The linear part of the evolution equation (REF ) for f is non-autonomous,
due to the dependence of the operator {\bf C} on \omega (t). In order to
make use of the dispersive properties obtained in §,
we introduce (following ) a small
modification of (REF ), which leads to an autonomous equation.
We fix an interval [0,T... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 296,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
192617,
2831,
28,
137089,
5490,
2320,
11766,
919,
100,
1238,
83,
351,
8913,
1687,
10821,
4743,
215131,
39933,
150598,
313,
98,
306,
2765,
18,
4527,
225202,
272,
183871,
113054,
5360,
642,
65508,
94015,
19336,
129344,
37105,
58963,
30022,
51... | [
0.2105712890625,
0.12841796875,
0.10406494140625,
0.2496337890625,
0.2249755859375,
0.05816650390625,
0.1368408203125,
0.24072265625,
0.01898193359375,
0.1611328125,
0.037872314453125,
0.1085205078125,
0.12744140625,
0.1414794921875,
0.07177734375,
0.01617431640625,
0.186279296875,
... | |
9c34bec570575f633321536256d24a8893013ecc | subsection | 15 | 76 | Frozen spectral decomposition | Denote {\cal R}_1(\omega )\!=\!{\cal R}(\Vert \omega -\omega _0\Vert _{C[0,T]}).Lemma 3.1
The function g is estimated in terms of h as follows:\Vert g\Vert _{L^{\infty }_{-\beta }}={\cal R}_1(\omega )|\omega -\omega _{T}|
\Vert h\Vert _{L^{\infty }_{-\beta }}Using the identities
{\bf P}^d(g+h)=0, {\bf P}_{T}^dg=g and ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
262,
48345,
6827,
627,
115187,
306,
2765,
1388,
38,
8152,
132,
15896,
6,
20,
41872,
2389,
18,
441,
618,
268,
18023,
45151,
581,
32354,
706,
83,
25902,
3674,
23,
69407,
111,
1096,
237,
28960,
7,
24854,
866,
8353,
46632,
51912,
454,
9,
... | [
0.09375,
0.15869140625,
0.1854248046875,
0.145263671875,
0.17236328125,
0.109130859375,
0.2384033203125,
0.00604248046875,
0.031097412109375,
0.00787353515625,
0.007232666015625,
0.105712890625,
0.006805419921875,
0.04644775390625,
0.00726318359375,
0.09381103515625,
0.0614013671875,... | |
67a2a7029072172a770bcc1be33c7ab6ed65a6b5 | subsection | 16 | 76 | Taylor expansion of dynamics | The preceding sections have provided a change of variables
\psi \mapsto (\omega ,\gamma ,z,h) under which (REF )
is mapped into the system comprising (REF )-(REF ) and (REF ).
Since we are interested in proving that for large
times z,h are small it is necessary to expand the inhomogeneous terms
in these equations in te... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
107671,
40059,
765,
62952,
15549,
77336,
41872,
15759,
62346,
2785,
306,
2765,
17705,
192,
169,
127,
1379,
11766,
919,
291,
48398,
3934,
5426,
48402,
9,
60892,
502,
6496,
21334,
20028,
97,
4,
621,
19336,
63559,
71062,
70,
23,
497,
432,
... | [
0.04376220703125,
0.1343994140625,
0.01275634765625,
0.07659912109375,
0.1651611328125,
0.224365234375,
0.0229644775390625,
0.20751953125,
0.1270751953125,
0.1671142578125,
0.069580078125,
0.1654052734375,
0.132080078125,
0.1033935546875,
0.1226806640625,
0.1756591796875,
0.110717773... | |
78145f4e19928d99c6b2e487bd382f0f043a5929 | subsection | 17 | 76 | Preliminaries | This section is devoted to some useful preliminary estimates.
We start with a bound for the denominator
\langle \partial _\omega \Psi -\partial _\omega {\bf P}^0\chi ,\Phi \rangle , where \Psi =\Psi _\omega ,
that appears in the equation of motion (REF )-(REF ).
We have, with \Delta = \langle \partial _\omega \Psi ,\Ps... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
3293,
40059,
83,
30396,
3674,
3060,
80234,
118276,
53,
25902,
1636,
1401,
4034,
678,
6,
99091,
100,
52240,
1290,
3066,
133,
15866,
289,
306,
2765,
683,
172,
20,
41872,
150598,
436,
8152,
2389,
1861,
45689,
14,
5445,
4,
7440,
135179,
23,... | [
0.065673828125,
0.139892578125,
0.005950927734375,
0.114990234375,
0.032562255859375,
0.0780029296875,
0.20751953125,
0.24365234375,
0.15966796875,
0.252197265625,
0.1595458984375,
0.0738525390625,
0.1514892578125,
0.026885986328125,
0.03515625,
0.2841796875,
0.0290985107421875,
0.... | |
9dbd61c819cb04463921330fa5bf762025f2dfc4 | subsection | 18 | 76 | Preliminaries | It is easy to check thatE_2[\chi ,\chi ]=\delta (x)\Bigl [a^{\prime }(C^2)|\chi |^2\Psi +2a^{\prime \prime }(C^2)(\Psi ,\chi )^2\Psi +2a^{\prime }(C^2)(\Psi ,\chi )\chi \Bigr ],\!\!\!\! E_3[\chi ,\chi ,\chi ]\!=\!\delta (x)\Bigl [a^{\prime }(C^2)|\chi |^2\chi +2a^{\prime \prime }(C^2)(\Psi ,\chi )^2\chi +2a^{\prime \pr... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
1650,
83,
23468,
47,
12765,
450,
647,
454,
304,
1065,
1861,
6,
4,
10114,
1369,
1743,
102,
425,
16,
129933,
141,
11,
24854,
41872,
114654,
51912,
132,
441,
10461,
58745,
8353,
683,
172,
94369,
1388,
42,
38,
241,
363,
15,
997,
132076,
... | [
0.042724609375,
0.06402587890625,
0.1729736328125,
0.060943603515625,
0.251953125,
0.1171875,
0.1907958984375,
0.134521484375,
0.245849609375,
0.0286865234375,
0.270751953125,
0.0552978515625,
0.0404052734375,
0.00897216796875,
0.1126708984375,
0.169677734375,
0.18115234375,
0.1352... | |
a3a4cf8fe92d889364ab63d063a8e343be377a08 | subsection | 19 | 76 | Preliminaries | Notice also that\langle E_2[X,Y],Z\rangle =\langle X, E_2[Y^*,Z]\ranglewhere X, Y, Z, are complex valued vector functions and
Y^*=(\overline{Y}_1,\overline{Y}_2).In the remaining part of the paper we shall prove the following
asymptotics:\Vert f(t)\Vert _{L^{\infty }_{-\beta }}\sim t^{-1}, \quad z(t)\sim t^{-1/2},\quad... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
438,
24494,
3066,
133,
241,
454,
304,
1542,
1723,
1511,
5445,
2203,
1193,
8353,
1639,
990,
567,
621,
27140,
34292,
173,
18770,
32354,
5465,
2256,
10461,
47143,
15122,
23534,
25632,
237,
4650,
40934,
41637,
15896,
1238,
18,
866,
46632,
939... | [
0.0352783203125,
0.0865478515625,
0.1624755859375,
0.0938720703125,
0.1480712890625,
0.044677734375,
0.1744384765625,
0.110107421875,
0.141357421875,
0.1627197265625,
0.15966796875,
0.033294677734375,
0.1195068359375,
0.03125,
0.05242919921875,
0.1015625,
0.1380615234375,
0.0130310... | |
27581dc866806dd0dd6f219ed3977b70d57cf643 | subsection | 20 | 76 | Equation for | Using the equality {\bf Q}[\chi ]=jE[\chi ], and the fact that
j({\bf P}^0)^*={\bf P}^0j (where ^* means adjoint with respect to
the Hermitian inner product \langle \,\cdot \, ,\,\cdot \,\rangle ),
we rewrite\langle {\bf P}^0{\bf Q}[\chi ],\Phi \rangle =\langle {\bf P}^0jE[\chi ],\Phi \rangle =-\langle E[\chi ],j({\bf ... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
345,
6953,
28,
161789,
150598,
2396,
1861,
170,
647,
15824,
1647,
436,
8353,
77495,
1639,
1369,
2389,
13331,
26950,
606,
513,
4288,
15072,
1840,
80611,
66,
75414,
12996,
3066,
133,
15464,
5445,
247,
642,
456,
434,
18781,
45689,
14,
241,
... | [
0.0987548828125,
0.067626953125,
0.1083984375,
0.2342529296875,
0.1873779296875,
0.18701171875,
0.2012939453125,
0.193359375,
0.1292724609375,
0.063232421875,
0.218994140625,
0.1123046875,
0.007293701171875,
0.1962890625,
0.06805419921875,
0.036285400390625,
0.1109619140625,
0.0153... | |
32883697266d202b915f71b6d322255ad1c3a3da | subsection | 21 | 76 | Equation for | Then equation (REF ) for \dot{\omega } can be expanded up to {\cal O}(t^{-3/2}),
assuming (REF ), as follows:\dot{\omega }\!\!\!&=&\!\!\!-\frac{1}{\Delta }
\Biggl [\langle E_2[{\rm w},{\rm w}]+2E_2[{\rm w},f]+E_3[{\rm w},{\rm w},{\rm w}],j\Psi \rangle +\langle E_2[{\rm w},{\rm w}], {\bf P}^0j{\rm w}\rangle \Biggr ]\\
\... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
47009,
28,
5490,
2320,
11766,
919,
100,
15464,
306,
2765,
831,
186,
71062,
297,
1257,
47,
6827,
180,
132,
18,
8353,
8316,
12477,
247,
66596,
214,
28960,
7,
38,
41872,
1230,
1369,
9,
132076,
24854,
418,
8152,
58598,
102,
51912,
6,
1299... | [
0.082763671875,
0.13720703125,
0.2802734375,
0.16943359375,
0.175537109375,
0.261962890625,
0.0770263671875,
0.24169921875,
0.162353515625,
0.302734375,
0.1468505859375,
0.10003662109375,
0.2607421875,
0.1595458984375,
0.1204833984375,
0.081298828125,
0.2388916015625,
0.23681640625... | |
18272af667bf07bd3f2c9b3b4948c5296d4275c5 | subsection | 22 | 76 | Equation for | First we compute the quadratic terms in (REF )
which are of order t^{-1}
according to (REF ): these are obtained from\langle E_2[{\rm w},{\rm w}],j\Psi \rangle =z^2\langle E_2[u,u],j\Psi \rangle +\overline{z}^2\langle E_2[u^*,u^*],j\Psi \rangle +2z\overline{z}\langle E_2[u^*,u],j\Psi \rangleTaking into account the defi... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
23972,
642,
9969,
6743,
70,
68587,
9523,
69407,
23,
15,
11766,
919,
1388,
3129,
621,
111,
12989,
808,
5759,
8152,
59499,
6,
2077,
6097,
113054,
297,
1295,
3066,
133,
241,
454,
304,
24854,
41872,
148,
4,
268,
170,
683,
172,
5445,
169,
... | [
0.0977783203125,
0.06353759765625,
0.1650390625,
0.176025390625,
0.040313720703125,
0.250244140625,
0.2193603515625,
0.288330078125,
0.146240234375,
0.0123291015625,
0.1580810546875,
0.229248046875,
0.037109375,
0.0123291015625,
0.0170135498046875,
0.05810546875,
0.1978759765625,
0... | |
998eec0f6ae33c41619e0f9f7e44acd7006855e3 | subsection | 23 | 76 | Equation for | Therefore we can substitute \Omega ^{\prime }_{10}
in (REF ) by their projection j{\bf P}^cj^{-1}\Omega ^{\prime }_{10}.Using again the equality {\bf Q}=jE we get\langle {\bf Q}[\chi ],j(\partial _\omega \Psi -\partial _\omega {\bf P}^0\chi )\rangle =\langle E[\chi ],\partial _\omega \Psi -\partial _\omega {\bf P}^0\ch... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
228072,
642,
831,
161740,
13,
6,
41872,
670,
87849,
114654,
963,
8152,
23,
11766,
919,
1388,
390,
2363,
13452,
1830,
1647,
24854,
150598,
436,
8353,
238,
170,
5759,
454,
13438,
70,
28,
161789,
2396,
1369,
647,
2046,
3066,
133,
10666,
10... | [
0.076904296875,
0.059814453125,
0.1312255859375,
0.254150390625,
0.1031494140625,
0.027740478515625,
0.09722900390625,
0.091064453125,
0.278564453125,
0.220703125,
0.2196044921875,
0.0284881591796875,
0.1090087890625,
0.173828125,
0.2548828125,
0.0285186767578125,
0.05084228515625,
... | |
1b18b12a8f98b11b85cff940472d247920911119 | subsection | 24 | 76 | Equation for | Note that{\bf C}-{\bf C}_{T}=j^{-1}(\omega -\omega _{T})+j^{-1}(V-V_{T}),\quad {\rm where}\quad V=-\delta (x)[a+bP_1].Also
{\bf P}_{T}^c{\bf P}^c={\bf P}_{T}^c[{\bf P}_{T}^c+{\bf P}_{T}^d-{\bf P}^d]
={\bf P}_{T}^c+{\bf P}_{T}^c[{\bf P}_{T}^d-{\bf P}^d].
Therefore, (REF ) becomes\dot{h}={\bf C}_{T}h+\sigma (t){\bf P}_{T... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
18622,
150598,
313,
8152,
9,
454,
618,
1369,
170,
24854,
5759,
132,
306,
2765,
20,
16,
1328,
8353,
856,
247,
41872,
91526,
10666,
39,
7440,
310,
1743,
102,
15,
425,
11,
683,
115187,
268,
6210,
436,
238,
1065,
71,
2203,
228072,
4,
11... | [
0.08880615234375,
0.265625,
0.17431640625,
0.021270751953125,
0.056396484375,
0.0215606689453125,
0.1446533203125,
0.02081298828125,
0.1259765625,
0.021453857421875,
0.160888671875,
0.0208587646484375,
0.06414794921875,
0.2030029296875,
0.020233154296875,
0.02093505859375,
0.11083984... | |
bbaad28650899d9231def8795a8b2380f890ef65 | subsection | 25 | 76 | Equation for | Hence, we denote{\bf C}_{M}(t)={\bf C}_{T}+i\sigma (t)({\bf \Pi }_{T}^{+}-{\bf \Pi }_{T}^{-})and rewrite (REF ) as\dot{h}={\bf C}_{M}(t)h+{\bf P}_{T}^cjE_2[w,w]+\tilde{H}_R,where\tilde{H}_R=H_R^{\prime }+\sigma (t)[{\bf P}_{T}^cj^{-1}-i({\bf \Pi }_{T}^{+}-{\bf \Pi }_{T}^{-})]hLemma 4.3
For h\in X_{T}^c we have\Vert [{... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
6620,
642,
8,
48345,
150598,
313,
8152,
454,
594,
18,
16,
1369,
24854,
618,
1328,
14,
41872,
20561,
192,
15,
6,
29348,
51912,
9,
456,
434,
18781,
11766,
919,
1388,
237,
15464,
127,
132,
436,
8353,
238,
170,
647,
304,
4,
268,
3675,
... | [
0.061492919921875,
0.05426025390625,
0.1748046875,
0.2491455078125,
0.2449951171875,
0.1610107421875,
0.0701904296875,
0.0552978515625,
0.2017822265625,
0.1707763671875,
0.01348876953125,
0.0213775634765625,
0.0139617919921875,
0.1749267578125,
0.1414794921875,
0.068603515625,
0.0143... | |
22ebf9039c080ede892d9d25f073b79cc3ffa27e | subsection | 26 | 76 | Canonical form | Our goal is to transform the evolution equations for (\omega ,\gamma , z,h)
to a more simple, canonical form. We will use the idea of normal coordinates,
trying to keep unchanged the estimates for the remainders as much as is
possible. | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
22929,
69236,
83,
47,
27198,
28,
137089,
13722,
5256,
100,
41872,
306,
2765,
17705,
192,
97,
127,
10,
1286,
8781,
74413,
21533,
3173,
5,
1401,
1221,
4527,
6528,
111,
3638,
176866,
90,
31577,
13695,
51,
152028,
70,
25902,
1636,
47143,
14... | [
0.169189453125,
0.275146484375,
0.177734375,
0.1092529296875,
0.2099609375,
0.130126953125,
0.2462158203125,
0.195068359375,
0.10345458984375,
0.047821044921875,
0.064453125,
0.0985107421875,
0.1845703125,
0.1309814453125,
0.10693359375,
0.146484375,
0.1380615234375,
0.036590576171... | |
28c67e061821467b5b2f81d970cda599da44c8ae | subsection | 27 | 76 | Canonical form of equation for | We expand out the middle term on the right hand side
of (REF ), obtaining\dot{h}={\bf C}_M(t) h+H_{20}z^2+H_{11}z\overline{z}+H_{02}\overline{z}^2+\tilde{H}_R.Here, the coefficients H_{ij} are defined byH_{20}={\bf P}^c_T jE_2[u,u],\quad H_{11}=2{\bf P}^c_T jE_2[u,u^*],
\quad H_{12}={\bf P}^c_T jE_2[u^*,u^*].We want to... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1388,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id"... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
1401,
71062,
1810,
86991,
13579,
7108,
3535,
5609,
11766,
919,
6,
113054,
15464,
127,
8152,
1369,
24854,
150598,
313,
594,
18,
16,
1096,
1328,
841,
454,
1549,
169,
8353,
304,
1662,
5465,
2256,
9550,
41872,
3675,
112,
1052,
13,
4,
552,
... | [
0.0280609130859375,
0.2027587890625,
0.040191650390625,
0.1295166015625,
0.19189453125,
0.039337158203125,
0.06207275390625,
0.0609130859375,
0.0792236328125,
0.1719970703125,
0.00750732421875,
0.0958251953125,
0.1485595703125,
0.1419677734375,
0.00811767578125,
0.00787353515625,
0.0... | |
f88ec675450e23916891092e956c81b3c542489c | subsection | 28 | 76 | Canonical form of equation for | Then we getH_{20}-2i\mu _Ta_{20}=-{\bf C}_Ta_{20},\quad \quad H_{11}=-{\bf C}_T a_{11},\quad \quad H_{02}+2i\mu _Ta_{02}=-{\bf C}_Ta_{02}and \hat{H}_R=\tilde{H}_R+H^{\prime }, where H^{\prime } is defined asH^{\prime }\!\!\!&=&\!\!\!\sum \partial _\omega a_{ij}{\cal R}(\omega ,|z|
+\Vert f\Vert _{L^{\infty }_{-\beta }}... | {
"cite_spans": []
} | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
47009,
642,
2046,
841,
1549,
8152,
5428,
14,
561,
9874,
9,
150598,
313,
91526,
572,
1662,
618,
10,
9550,
54651,
2943,
3675,
24854,
1328,
114654,
8353,
61924,
11832,
15866,
306,
2765,
13786,
6827,
627,
169,
15896,
1238,
46632,
59865,
304,
... | [
0.002685546875,
0.001708984375,
0.06903076171875,
0.14892578125,
0.2379150390625,
0.009674072265625,
0.1646728515625,
0.0997314453125,
0.15380859375,
0.1173095703125,
0.031097412109375,
0.16650390625,
0.048095703125,
0.06378173828125,
0.1253662109375,
0.1458740234375,
0.1337890625,
... | |
a67859d6bd9f2a14b1055882327a8810381ab175 | subsection | 29 | 76 | Canonical form of equation for | The remainder H^{\prime } can be written asH^{\prime }=\sum \limits _m({\bf C}_T-2i\mu _Tm-0)^{-1}A_m,\quad m\in \lbrace -1,0,1\rbracewith A_m\in X^c_T, satisfying the estimate\Vert A_{m}\Vert _{L^1_\beta }={\cal R}(\omega ,|z|+\Vert f\Vert _{L^{\infty }_{-\beta }})
|z|\Bigl (|z||\omega _T-\omega |+(|z|+\Vert f\Vert _{... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 509,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
47143,
820,
572,
8353,
114654,
831,
186,
59121,
237,
841,
1369,
11832,
6,
93343,
7,
39,
132,
24854,
41872,
150598,
313,
8152,
454,
618,
5428,
14,
561,
23320,
16,
5759,
284,
91526,
347,
73,
99407,
20,
206808,
418,
76228,
1193,
40407,
3... | [
0.2261962890625,
0.16943359375,
0.177978515625,
0.023345947265625,
0.269775390625,
0.0997314453125,
0.0445556640625,
0.20458984375,
0.06658935546875,
0.165771484375,
0.0279541015625,
0.213134765625,
0.006927490234375,
0.1702880859375,
0.00946044921875,
0.1075439453125,
0.002319335937... | |
d1387653da00b22b7379111d478eff8b16128002 | subsection | 30 | 76 | Canonical form of equation for | (cf. )&&\Omega _{20}+2i\mu b_{20}=0,\\
&&\Omega ^{\prime }_{10}+i\mu b^{\prime }_{10}+{\bf C}^*b^{\prime }_{10}=0,\\
&&\Omega _{21}+2Z_{11}b_{20}+i\mu b_{21}+2Z_{20}b_{02}+
\langle F_{11},b^{\prime }_{10}\rangle +\langle F_{20},b^{\prime }_{01}\rangle =0,\\
&&\Omega _{30}+2Z_{20}b_{20}+3i\mu b_{30}+\langle F_{20},b^{\... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 343,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id":... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
67466,
1230,
670,
87849,
1549,
8152,
54651,
14,
561,
876,
454,
24854,
145407,
4,
13273,
41872,
114654,
51912,
963,
1328,
8353,
150598,
313,
1639,
275,
619,
101,
3117,
1511,
9550,
3066,
133,
563,
1662,
5445,
997,
6746,
6,
1197,
85398,
70... | [
0.070556640625,
0.068115234375,
0.02606201171875,
0.206787109375,
0.203125,
0.00897216796875,
0.1763916015625,
0.0989990234375,
0.138916015625,
0.1361083984375,
0.02532958984375,
0.009002685546875,
0.1785888671875,
0.008636474609375,
0.00872802734375,
0.008758544921875,
0.2236328125,... | |
a1b425f6519dd72a33081ccf4b18ae4e7b32b379 | subsection | 31 | 76 | Canonical form of equation for | Substituting
(REF ) and (REF ) into (REF ) and putting the contribution of
g+h_1+k_1 in the remainder \tilde{Z}_R, we obtain\dot{z}&=&i\mu z+Z_{20}z^2+Z_{11}z\overline{z}+Z_{02}\overline{z}^2
+Z_{30}z^3+Z_{21}z^2\overline{z}+Z_{12}z\overline{z}^2+Z_{03}\overline{z}^3\\
&+&Z_{30}^{\prime }z^3+Z_{21}^{\prime }z^2\overlin... | {
"cite_spans": [
{
"arxiv_id": "",
"doi": "",
"end": 1248,
"openalex_id": "",
"raw": "V.S. Buslaev, C. Sulem, On asymptotic stability of solitary waves for nonlinear Schrödinger equations,Ann. Inst. Henri Poincaré, Anal. Non Linéaire 20 (2003), no.3, 419-475.",
"source_ref_id"... | 0807.1878 | On asymptotic stability of solitons for nonlinear Sch\"odinger equation | [
"A. I. Komech",
"E. A. Kopylova",
"D. Stuart"
] | [
"math-ph",
"math.AP",
"math.MP"
] | 2,008 | en | Physics | [
8273,
13480,
1916,
11766,
919,
1388,
136,
3934,
118620,
70,
127752,
111,
706,
1328,
127,
115187,
92,
23,
47143,
820,
6,
3675,
112,
1511,
1052,
642,
113054,
15464,
169,
8152,
1230,
1369,
14,
561,
97,
454,
24854,
1549,
8353,
304,
1662,
... | [
0.11962890625,
0.164306640625,
0.04034423828125,
0.1485595703125,
0.234130859375,
0.0268402099609375,
0.1353759765625,
0.05670166015625,
0.0870361328125,
0.0155487060546875,
0.195556640625,
0.0156402587890625,
0.0948486328125,
0.1500244140625,
0.109130859375,
0.1580810546875,
0.07379... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.