files / problems / simon-ionization-excess / compute / q1 / certs / nam_beta.json
{
"arxiv": "1009.2367v3",
"urls_opened": [
"https://arxiv.org/abs/1009.2367",
"https://arxiv.org/html/1009.2367v3"
],
"dps": 80,
"definition": "\u03b2 = inf \u222c (x\u00b2+y\u00b2)/(2|x-y|) d\u03c1 d\u03c1 / \u222b |x| d\u03c1 over probability measures",
"printed_interval": [
"0.8218",
"0.8705"
],
"printed_interval_valid": true,
"lower_bound": {
"method": "Nam g(\u03bb) analytic minorant, \u03bb \u2265 0.8",
"g_formula": "g(\u03bb) = \u03bb - (\u221a((\u03bb+2)/3 - 2\u221a(\u03bb(1-\u03bb))) - \u221a((2/3)(1-\u03bb)))\u00b2",
"g_at_0.843": "0.8218039259820358042774446446719606156168",
"g_at_0.843_interval": [
"0.8218039259820358042774446446719606156168",
"0.8218039259820358042774446446719606156168"
],
"g_at_0.843_gt_0.8218": true,
"g_at_0.8435": "0.8218066192347422322265295524256739599865",
"g_at_0.8435_interval": [
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"0.8218066192347422322265295524256739599865"
],
"lambda_max": "0.8434764011484514594075847820711715804375",
"g_max": "0.8218066258637801575911113101687651067908",
"g_max_interval_at_truncated_lambda": [
"0.8218066258637801575911113101687651067908",
"0.8218066258637801575911113101687651067908"
],
"Nam_printed_g_0.843": "0.821804...",
"Nam_printed_g_max": "0.8218066...",
"Nam_printed_lambda_max": "0.843476...",
"equality_geometry_at_lambda_max": {
"lambda_prime": "0.02166977528467130181647347190240647364664",
"a": "1",
"b": "0.4557040842637063855384667733663603025491",
"c": "1.057863120135131651466273697450701299790",
"triangle_ok": true,
"W_ratio": "0.8218066258637801575911113101687651067908",
"g": "0.8218066258637801575911113101687651067908",
"W_equals_g": true
},
"inf_W_equals_g_at_lambda_max": true,
"grid_is_not_a_lower_bound": true,
"grid_bc_at_lambda_max": {
"min": 0.8218066727609882,
"at_b_c": [
0.4561403563157894,
1.0583036545666842
],
"note": "grid min \u2265 true inf; used only as a check"
},
"grid_bc_at_0.843": {
"min": 0.8218041219768617
},
"grid_r_s_angle_at_0.8434764": {
"min": 0.8218098241611241,
"at_r_s_mu_dist": [
1.0,
0.4492759130434782,
0.12359550561797761,
1.0444098231881376
]
}
},
"upper_bound": {
"nam_trial": {
"density": "(3/4) r^{-3/2} 1_{[1,9]}(r) dr",
"closed_form": "115/81 - (1/2) ln 3",
"value": "0.8704469420856982407221304679584902340960",
"value_from_antiderivatives": "0.8704469420856982407221304679584902340960",
"interval": [
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],
"lt_0.8705": true,
"Nam_printed": "0.8704..."
},
"second_trial_atomic_radial": {
"description": "64 point masses on a geometric grid [1,10], masses optimised from a Nam-like r^{-3/2} start, then frozen as 18-digit decimals and re-evaluated in mpmath",
"n": 64,
"rmax": 10.0,
"value": "0.8702205069317059741384184186706383236382",
"float64_optimiser": 0.8702205069317062,
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"radii": [
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],
"Nam_printed_beta_rad_numeric": "0.8702"
}
},
"one_over_beta": {
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"Nam_rounds_to": "1.22",
"1.2168_lt_1.22": true
},
"enclosures": {
"beta_gt_0.8218": true,
"beta_lt_0.8705": true,
"beta_lt_nam_trial": true,
"beta_lt_second_trial": true
}
}