{"qid": 0, "query": "Cooling a cup of coffee with help of a spoon", "score": 756, "views": 249628, "answer_pids": [1468504, 1468505, 1468511, 1468512, 1468523, 1468530, 1468532, 1468535, 1468536, 1468548, 1468554, 1468575, 1468577, 1468579, 1468614, 1521490, 1528127, 1533951, 1545044, 1615677, 1617214, 1644619], "question_author": ""}
{"qid": 1, "query": "Can I compute the mass of a coin based on the sound of its fall?", "score": 503, "views": 58422, "answer_pids": [1512470, 1512482, 1512859, 1513094, 1667753], "question_author": "Gergana Vandova"}
{"qid": 2, "query": "How does light bend around my finger tip?", "score": 452, "views": 55595, "answer_pids": [1509366, 1509370, 1509393, 1509406, 1509437, 1510569], "question_author": "GWW"}
{"qid": 3, "query": "How does gravity escape a black hole?", "score": 442, "views": 44747, "answer_pids": [1466074, 1466075, 1466076, 1466082, 1466083, 1466093, 1466095, 1466098, 1467142, 1467377, 1467381, 1468287, 1489056, 1524110, 1553907, 1568519, 1636541], "question_author": null}
{"qid": 4, "query": "Did the Big Bang happen at a point?", "score": 353, "views": 48575, "answer_pids": [1517817, 1517821, 1552539, 1554009, 1574857, 1575409], "question_author": "Nick T"}
{"qid": 5, "query": "Why are mirror images flipped horizontally but not vertically?", "score": 351, "views": 131342, "answer_pids": [1470002, 1470003, 1470004, 1472084, 1472086, 1472087, 1472089, 1472090, 1472095, 1472118, 1472173, 1473510, 1473518, 1481184, 1498163, 1502115, 1507521, 1534026, 1536590, 1541186, 1573557, 1621330, 1630816], "question_author": "Daniel Standage"}
{"qid": 6, "query": "Do we know why there is a speed limit in our universe?", "score": 313, "views": 46968, "answer_pids": [1546355, 1546370, 1546389, 1546391, 1546394, 1546414, 1546417, 1546424, 1546428, 1546438, 1546453, 1546462, 1546484, 1546522, 1546636, 1546839, 1546962, 1547666, 1547727, 1553579, 1560263], "question_author": "Daniel Standage"}
{"qid": 7, "query": "Book recommendations", "score": 307, "views": 105230, "answer_pids": [1471902], "question_author": "pablasso"}
{"qid": 8, "query": "What experiment would disprove string theory?", "score": 298, "views": 42935, "answer_pids": [1465535, 1465540, 1466218, 1467356, 1467357, 1467358, 1467360, 1467368, 1467527, 1488485], "question_author": "deceleratedcaviar"}
{"qid": 9, "query": "Why dont metals bond when touched together?", "score": 287, "views": 54936, "answer_pids": [1500819, 1500820, 1500821, 1500824, 1500827, 1500831, 1500842, 1500850, 1500852, 1510876], "question_author": "Nick T"}
{"qid": 10, "query": "Why do ballpoint pens write better on pages that have pages below them?", "score": 287, "views": 25719, "answer_pids": [1584400, 1584424], "question_author": "Kyra"}
{"qid": 11, "query": "What is Chirped Pulse Amplification, and why is it important enough to warrant a Nobel Prize?", "score": 284, "views": 25288, "answer_pids": [1603130, 1603416], "question_author": "Will Townes"}
{"qid": 12, "query": "Why does kinetic energy increase quadratically, not linearly, with speed?", "score": 273, "views": 68114, "answer_pids": [1465828, 1465829, 1465830, 1465831, 1465834, 1473107, 1473133, 1475306, 1484606, 1490455, 1493130, 1508792, 1509823, 1618970], "question_author": ""}
{"qid": 13, "query": "What really allows airplanes to fly?", "score": 266, "views": 74047, "answer_pids": [1465684, 1465688, 1465696, 1498114, 1498197, 1500054, 1530164, 1530418, 1538008, 1579744], "question_author": "Vebjorn Ljosa"}
{"qid": 14, "query": "Why are four-legged chairs so common?", "score": 256, "views": 48803, "answer_pids": [1546344, 1546351, 1546362, 1546371, 1546376, 1546377, 1546640, 1546650, 1546653, 1548541, 1579227], "question_author": ""}
{"qid": 15, "query": "If I sliced the universe in half, would the slice go through a star?", "score": 255, "views": 30706, "answer_pids": [1591587, 1591592, 1591615, 1591625, 1591680], "question_author": "Albert"}
{"qid": 16, "query": "Dont heavier objects actually fall faster because they exert their own gravity?", "score": 252, "views": 58938, "answer_pids": [1467565, 1467567, 1467570, 1467578, 1486685, 1499993, 1552774, 1594287], "question_author": "Gergana Vandova"}
{"qid": 17, "query": "How do towels stay on hooks?", "score": 238, "views": 37560, "answer_pids": [1589706, 1589708, 1589714, 1589762, 1589861, 1589863, 1589874], "question_author": null}
{"qid": 18, "query": "What exactly is a photon?", "score": 226, "views": 36024, "answer_pids": [1558757, 1558761, 1558782, 1558804, 1558820, 1558828, 1558836, 1558841, 1558846, 1558915, 1558916, 1558962, 1558967, 1559038, 1607025], "question_author": "Rowland Shaw"}
{"qid": 19, "query": "Does Earth really have two high-tide bulges on opposite sides?", "score": 225, "views": 36246, "answer_pids": [1512435, 1512441, 1512445, 1512447, 1512497], "question_author": "Albert"}
{"qid": 20, "query": "Strange ice found in my garden", "score": 206, "views": 17680, "answer_pids": [1591340], "question_author": "Chris S"}
{"qid": 21, "query": "Why is the detection of gravitational waves so significant?", "score": 204, "views": 19246, "answer_pids": [1547830, 1547833, 1547861, 1547889, 1547907, 1547911, 1547974, 1549134, 1549196], "question_author": null}
{"qid": 22, "query": "Surviving under water in air bubble", "score": 198, "views": 53281, "answer_pids": [1494341, 1494360, 1494741], "question_author": "Justin L."}
{"qid": 23, "query": "Why do we bend a book to keep it straight?", "score": 196, "views": 14957, "answer_pids": [1614736, 1614740, 1614752, 1614792, 1614818, 1614899, 1616235], "question_author": "ablmf"}
{"qid": 24, "query": "If photons have no mass, how can they have momentum?", "score": 194, "views": 245193, "answer_pids": [1466832, 1466834, 1472398, 1482524, 1487030, 1511112], "question_author": "LarsH"}
{"qid": 25, "query": "Why doesnt matter pass through other matter if atoms are 99.999% empty space?", "score": 190, "views": 70718, "answer_pids": [1513522, 1513604, 1523285], "question_author": ""}
{"qid": 26, "query": "Why is Googles quantum supremacy experiment impressive?", "score": 185, "views": 23722, "answer_pids": [1624932, 1624969, 1625005, 1625033], "question_author": "Rekar"}
{"qid": 27, "query": "Are units of angle really dimensionless?", "score": 183, "views": 17549, "answer_pids": [1552606, 1552607, 1552608, 1552610, 1552679, 1552686, 1552758, 1552760, 1552768, 1552857, 1552906, 1552996, 1553148, 1557239, 1586293, 1586298], "question_author": "jonsca"}
{"qid": 28, "query": "Why dont electrons crash into the nuclei they orbit?", "score": 182, "views": 79760, "answer_pids": [1475553, 1475663, 1493526, 1494364, 1514542, 1517768, 1534510, 1547010, 1619730], "question_author": null}
{"qid": 29, "query": "Why does space expansion not expand matter?", "score": 178, "views": 26191, "answer_pids": [1466784, 1467656, 1475081, 1483654, 1509582, 1520315, 1523731, 1531648, 1584643, 1594962, 1605690, 1623478], "question_author": "Daniel Standage"}
{"qid": 30, "query": "When separating an Oreo cookie, why does the cream stick to just one side only?", "score": 176, "views": 18027, "answer_pids": [1577757, 1577773], "question_author": ""}
{"qid": 31, "query": "Why does NASA use gold foil on equipment and gold-coated visors?", "score": 175, "views": 88724, "answer_pids": [1496900], "question_author": "shimonyk"}
{"qid": 32, "query": "How do moving charges produce magnetic fields?", "score": 174, "views": 211401, "answer_pids": [1493255, 1493257, 1493274, 1493275, 1506145, 1549759, 1611904], "question_author": "aqua"}
{"qid": 33, "query": "How does a knife cut things at the atomic level?", "score": 172, "views": 42286, "answer_pids": [1516706, 1516707, 1518908], "question_author": "Daniel Standage"}
{"qid": 34, "query": "Why do we not have spin greater than 2?", "score": 171, "views": 24313, "answer_pids": [1473315, 1556243], "question_author": ""}
{"qid": 35, "query": "Why do shadows from the sun join each other when near enough?", "score": 167, "views": 26233, "answer_pids": [1503585, 1503593, 1503604, 1503627, 1503630, 1503642], "question_author": "Seth"}
{"qid": 36, "query": "Gauge symmetry is not a symmetry?", "score": 167, "views": 19949, "answer_pids": [1472725, 1472726, 1480297, 1481487, 1529758], "question_author": ""}
{"qid": 37, "query": "Why do we actually see the sun?", "score": 166, "views": 13107, "answer_pids": [1495113, 1495116, 1495117, 1495120, 1495126, 1495135, 1514670], "question_author": "Fiktor"}
{"qid": 38, "query": "Whats the point of Hamiltonian mechanics?", "score": 165, "views": 44967, "answer_pids": [1501552, 1501554, 1501555, 1501561, 1501623, 1501673, 1502074, 1503656, 1542025, 1578607, 1603248, 1603268, 1615690], "question_author": "shigeta"}
{"qid": 39, "query": "Could Legolas actually see that far?", "score": 164, "views": 22248, "answer_pids": [1512829, 1512830, 1512831, 1512858, 1512865, 1512878, 1512989, 1607663], "question_author": ""}
{"qid": 40, "query": "How do you make more precise instruments while only using less precise instruments?", "score": 164, "views": 10996, "answer_pids": [1655073, 1655074, 1655079, 1655093, 1655097, 1655104, 1655107, 1655109, 1655110, 1655111, 1655121, 1655123, 1655124, 1655136, 1655137, 1655158, 1655179, 1655199, 1655324, 1656283], "question_author": ""}
{"qid": 41, "query": "Why does holding something up cost energy while no work is being done?", "score": 159, "views": 66624, "answer_pids": [1466699, 1466700, 1466703, 1466704, 1474611, 1500794, 1578658, 1600847], "question_author": "Sean Tilson"}
{"qid": 42, "query": "What makes a theory Quantum?", "score": 159, "views": 15328, "answer_pids": [1593593, 1593595, 1593598, 1593620, 1593647, 1593662, 1593663, 1593778, 1593783], "question_author": ""}
{"qid": 43, "query": "Given Newtons third law, why are things capable of moving?", "score": 158, "views": 132573, "answer_pids": [1486383, 1486384, 1486386, 1504722, 1523081, 1525219, 1564043, 1574858, 1575837, 1596615, 1596820, 1599741, 1603029], "question_author": ""}
{"qid": 44, "query": "Why does Stephen Hawking say black holes dont exist?", "score": 157, "views": 33895, "answer_pids": [1504011, 1504021, 1504169], "question_author": ""}
{"qid": 45, "query": "Why do sunbeams diverge even though the sun is much more than a few kilometers away?", "score": 157, "views": 33085, "answer_pids": [1523590, 1523693], "question_author": ""}
{"qid": 46, "query": "How do I explain to a six year old why people on the other side of the Earth dont fall off?", "score": 154, "views": 56341, "answer_pids": [1503678, 1503680, 1503685, 1503686, 1503688, 1503691, 1503694, 1503695, 1503697, 1503698, 1503700, 1503721, 1503805], "question_author": "jonsca"}
{"qid": 47, "query": "Simple check for the global shape of the Earth", "score": 154, "views": 22286, "answer_pids": [1603838, 1603839, 1603847, 1603850, 1603852, 1603853, 1603854, 1603858, 1603860, 1603864, 1603889, 1603898, 1603912, 1603983, 1603992, 1604061, 1604133, 1604152, 1604228, 1604302], "question_author": ""}
{"qid": 48, "query": "Why are the harmonics of a piano tone not multiples of the base frequency?", "score": 153, "views": 12436, "answer_pids": [1557512, 1557535, 1557564], "question_author": ""}
{"qid": 49, "query": "What is a field, really?", "score": 152, "views": 37439, "answer_pids": [1472336, 1472389, 1472411, 1472427, 1472754, 1472758, 1476385, 1504556, 1632292, 1632313, 1633509, 1640632], "question_author": null}
{"qid": 50, "query": "Why are the wet patches on these floor tiles circular?", "score": 151, "views": 23893, "answer_pids": [1591884, 1591887, 1591902, 1591903, 1591905, 1591951, 1591973, 1592009, 1592544], "question_author": "shigeta"}
{"qid": 51, "query": "Why is nuclear waste more dangerous than the original nuclear fuel?", "score": 150, "views": 25283, "answer_pids": [1565087, 1565089, 1565101, 1565103, 1565144, 1565196, 1565431], "question_author": "e.James"}
{"qid": 52, "query": "Is $\\pi^2 \\approx g$ a coincidence?", "score": 150, "views": 17476, "answer_pids": [1559643, 1559644, 1559658, 1559660, 1559695, 1559751], "question_author": "Rory M"}
{"qid": 53, "query": "Why does a mirror split my laser beam?", "score": 149, "views": 16878, "answer_pids": [1549273, 1549288], "question_author": ""}
{"qid": 54, "query": "Why is my hand not burned by the air in an oven at 200 \u00b0C?", "score": 148, "views": 49496, "answer_pids": [1569783, 1569792, 1569795, 1569805, 1569828, 1569877, 1569910, 1570004, 1570026, 1570099], "question_author": ""}
{"qid": 55, "query": "Why would spacetime curvature cause gravity?", "score": 146, "views": 49260, "answer_pids": [1506326, 1506328, 1506331, 1506360, 1531750], "question_author": ""}
{"qid": 56, "query": "Does someone falling into a black hole see the end of the universe?", "score": 142, "views": 31260, "answer_pids": [1499957, 1499968, 1499969, 1502641, 1505432, 1570967, 1590376, 1593220, 1593380], "question_author": "Emil H"}
{"qid": 57, "query": "Why are the windows of bridges of ships always inclined?", "score": 142, "views": 26681, "answer_pids": [1568392, 1568483, 1568510], "question_author": "IAE"}
{"qid": 58, "query": "Does the Planck scale imply that spacetime is discrete?", "score": 141, "views": 23201, "answer_pids": [1470737, 1477159], "question_author": "Damian Kao"}
{"qid": 59, "query": "Is temperature a Lorentz invariant in relativity?", "score": 140, "views": 11039, "answer_pids": [1500257, 1500261, 1502391, 1542832, 1619645, 1645031], "question_author": "David McGraw"}
{"qid": 60, "query": "Why do people categorically dismiss some simple quantum models?", "score": 139, "views": 20007, "answer_pids": [1482611, 1482649, 1482650, 1482702, 1482783], "question_author": ""}
{"qid": 61, "query": "If you view the Earth from far enough away can you observe its past?", "score": 138, "views": 76082, "answer_pids": [1471790, 1471793, 1471794, 1471795, 1471805], "question_author": ""}
{"qid": 62, "query": "Is time continuous or discrete?", "score": 138, "views": 40681, "answer_pids": [1483260, 1483261, 1483277, 1483278, 1483288, 1483322, 1497821, 1514826, 1621783], "question_author": "Cedric H."}
{"qid": 63, "query": "How can you weigh your own head in an accurate way?", "score": 138, "views": 25364, "answer_pids": [1495454, 1495455, 1495456, 1495462, 1495463, 1495465, 1495466, 1495468, 1495471, 1495479, 1495488, 1495556, 1495560, 1495570, 1495588, 1495589, 1495610, 1495611, 1498136, 1513034], "question_author": ""}
{"qid": 64, "query": "Why does my tea periodically alternate its rotational speed after stirring? (Link to video below)", "score": 138, "views": 11872, "answer_pids": [1630222, 1630225, 1630228, 1630238, 1630253, 1630266, 1630271, 1630288, 1630300], "question_author": "jonsca"}
{"qid": 65, "query": "How fast does gravity propagate?", "score": 136, "views": 18515, "answer_pids": [1468588, 1468589, 1468590, 1468610, 1468621, 1497162, 1525928, 1578035, 1662097], "question_author": ""}
{"qid": 66, "query": "How do I experimentally measure the surface area of a rock?", "score": 136, "views": 13926, "answer_pids": [1625445, 1625447, 1625448, 1625471, 1625476, 1625482, 1625484, 1625493, 1625501, 1625507, 1625526, 1625527, 1625543, 1625553, 1625562, 1625572], "question_author": ""}
{"qid": 67, "query": "Is the butterfly effect real?", "score": 135, "views": 37755, "answer_pids": [1546726, 1546729, 1546731, 1546732, 1546734, 1546747, 1546781, 1546787, 1546891, 1589764], "question_author": "bdonlan"}
{"qid": 68, "query": "Why doesnt water boil in the oven?", "score": 135, "views": 29924, "answer_pids": [1638086, 1638092, 1638207, 1638243], "question_author": "Thomas O"}
{"qid": 69, "query": "A list of inconveniences between quantum mechanics and (general) relativity?", "score": 134, "views": 25075, "answer_pids": [1465749, 1466147, 1466930, 1469211, 1502168, 1614235], "question_author": "arcyqwerty"}
{"qid": 70, "query": "Why are there only derivatives to the first order in the Lagrangian?", "score": 132, "views": 22892, "answer_pids": [1467874, 1467875, 1467876, 1467878, 1467888, 1562438], "question_author": ""}
{"qid": 71, "query": "Reading the Feynman lectures in 2012", "score": 131, "views": 30602, "answer_pids": [1480371, 1480373], "question_author": "nibot"}
{"qid": 72, "query": "Why is the vibration in my wire acting so oddly?", "score": 130, "views": 5673, "answer_pids": [1557041, 1557046, 1557069], "question_author": "Michael Kuhn"}
{"qid": 73, "query": "Why can I touch aluminum foil in the oven and not get burned?", "score": 129, "views": 45193, "answer_pids": [1590744, 1590760, 1590775, 1590778], "question_author": "Mad Scientist"}
{"qid": 74, "query": "Why does paper cut through things so well?", "score": 129, "views": 24132, "answer_pids": [1601810, 1601814, 1601827, 1601828, 1601972], "question_author": "Royi"}
{"qid": 75, "query": "What does it mean for two objects to touch?", "score": 128, "views": 32939, "answer_pids": [1477371, 1477372, 1477373, 1477383], "question_author": "Poshpaws"}
{"qid": 76, "query": "Why do spaceships heat up when entering earth but not when exiting?", "score": 125, "views": 29079, "answer_pids": [1588485, 1588495, 1588505, 1588513, 1588514, 1588515, 1588517, 1588518, 1588834], "question_author": ""}
{"qid": 77, "query": "Does a particle exert force on itself?", "score": 123, "views": 9914, "answer_pids": [1615894, 1615895, 1615900, 1615920, 1615950, 1617787], "question_author": "Pedery"}
{"qid": 78, "query": "Why does Newtons Third Law actually work?", "score": 122, "views": 24624, "answer_pids": [1580622, 1580623, 1580628, 1580655, 1580657, 1580679, 1580802], "question_author": "Jonathan."}
{"qid": 79, "query": "Why does ice cream get harder when colder?", "score": 122, "views": 12002, "answer_pids": [1587418, 1587533], "question_author": "Marton Trencseni"}
{"qid": 80, "query": "Why does the humidifier make a stoves flame orange?", "score": 121, "views": 31950, "answer_pids": [1605794, 1605806, 1605829, 1605830, 1605899], "question_author": ""}
{"qid": 81, "query": "How does mass leave the body when you lose weight?", "score": 120, "views": 231867, "answer_pids": [1467039, 1467040, 1467044, 1467045, 1467057, 1467184, 1470937, 1471224, 1473678, 1483276, 1496050], "question_author": "Giuseppe Accaputo"}
{"qid": 82, "query": "Is anti-matter matter going backwards in time?", "score": 119, "views": 84017, "answer_pids": [1465745, 1465746, 1465756, 1474518, 1484177], "question_author": null}
{"qid": 83, "query": "Calculus of variations -- how does it make sense to vary the position and the velocity independently?", "score": 119, "views": 12782, "answer_pids": [1466049, 1466106, 1467207, 1467214, 1469788, 1507625, 1628033], "question_author": "nibot"}
{"qid": 84, "query": "Why dont we use weights to store energy?", "score": 117, "views": 56031, "answer_pids": [1568935, 1568936, 1568952, 1568954, 1568956, 1568961, 1568964, 1568977, 1568979, 1568982, 1569162, 1569266], "question_author": ""}
{"qid": 85, "query": "Why do wet objects become darker?", "score": 117, "views": 23731, "answer_pids": [1480846, 1521363, 1525906], "question_author": "Thomas O"}
{"qid": 86, "query": "Why am I not burned by a strong wind?", "score": 117, "views": 12367, "answer_pids": [1504366, 1504367, 1504370, 1504373, 1504376], "question_author": ""}
{"qid": 87, "query": "The Role of Rigor", "score": 116, "views": 11363, "answer_pids": [1479541, 1479542, 1479543, 1480239], "question_author": ""}
{"qid": 88, "query": "On this infinite grid of resistors, whats the equivalent resistance?", "score": 115, "views": 81280, "answer_pids": [1466743, 1594482], "question_author": ""}
{"qid": 89, "query": "If all motion is relative, how does light have a finite speed?", "score": 115, "views": 36267, "answer_pids": [1535332, 1535333, 1535334, 1535335, 1535338, 1535344, 1535361, 1535394, 1535525], "question_author": "Justin L."}
{"qid": 90, "query": "Could we send a man safely to the Moon in a rocket without knowledge of general relativity?", "score": 115, "views": 14062, "answer_pids": [1528773, 1528774, 1528777, 1528801, 1528803], "question_author": "Phonon"}
{"qid": 91, "query": "Superfields and the Inconsistency of regularization by dimensional reduction", "score": 115, "views": 7125, "answer_pids": [1564847], "question_author": "johnny1bucket"}
{"qid": 92, "query": "Is it necessary to consume energy to perform computation?", "score": 114, "views": 5768, "answer_pids": [1495247], "question_author": null}
{"qid": 93, "query": "Toilet paper dilemma", "score": 113, "views": 17147, "answer_pids": [1655352, 1655453, 1655707, 1668148], "question_author": null}
{"qid": 94, "query": "Is it possible for information to be transmitted faster than light by using a rigid pole?", "score": 113, "views": 15114, "answer_pids": [1466802, 1466804, 1468207, 1471130, 1472406, 1472407, 1472408, 1474469, 1498064, 1498177, 1505209, 1531778, 1563353, 1564577, 1617622], "question_author": "DQdlM"}
{"qid": 95, "query": "Why do electrons, according to my textbook, exist forever?", "score": 113, "views": 15019, "answer_pids": [1531285, 1531286, 1531295, 1531566], "question_author": "ihuston"}
{"qid": 96, "query": "What happens to the energy when waves perfectly cancel each other?", "score": 112, "views": 79791, "answer_pids": [1477444, 1477445, 1477453, 1477458, 1477485, 1477640, 1521038, 1526747, 1602128], "question_author": ""}
{"qid": 97, "query": "Why does a helium filled balloon move forward in a car when the car is accelerating?", "score": 112, "views": 51418, "answer_pids": [1500704, 1500705, 1500708, 1500710, 1500711, 1500712, 1500770, 1500922], "question_author": "badp"}
{"qid": 98, "query": "What is the actual significance of the amplituhedron?", "score": 112, "views": 32395, "answer_pids": [1498117, 1498128, 1498132, 1498138, 1498151], "question_author": "Geodesic"}
{"qid": 99, "query": "Can the Heisenberg Uncertainty Principle be explained intuitively?", "score": 112, "views": 16783, "answer_pids": [1545872, 1545873, 1545877, 1545879, 1545880, 1545886, 1545892, 1545899, 1545901, 1545912, 1545935, 1546087, 1546090, 1546203, 1546207, 1580898], "question_author": "kmm"}
{"qid": 100, "query": "What, in simplest terms, is gauge invariance?", "score": 111, "views": 28886, "answer_pids": [1556962, 1556965, 1556969, 1556970, 1556982, 1556997, 1557005, 1557032, 1557034], "question_author": null}
{"qid": 101, "query": "Can Maxwells equations be derived from Coulombs Law and Special Relativity?", "score": 111, "views": 18962, "answer_pids": [1467614, 1467623, 1467645, 1467771, 1468239, 1469123, 1471969, 1480694, 1480697, 1515302, 1605929, 1608307, 1611606], "question_author": "Dragana Bozic"}
{"qid": 102, "query": "Is the universe fundamentally deterministic?", "score": 110, "views": 54356, "answer_pids": [1492654, 1492753, 1492759, 1519505, 1546039, 1617982], "question_author": "Jonathan."}
{"qid": 103, "query": "Why are most metals gray/silver?", "score": 110, "views": 53166, "answer_pids": [1496100, 1496101, 1496117, 1496158, 1497620, 1517953], "question_author": ""}
{"qid": 104, "query": "Is it possible to start fire using moonlight?", "score": 110, "views": 19334, "answer_pids": [1518919, 1518920, 1524183, 1545486, 1557148], "question_author": ""}
{"qid": 105, "query": "What are the justifying foundations of statistical mechanics without appealing to the ergodic hypothesis?", "score": 110, "views": 6391, "answer_pids": [1479392, 1479393, 1479394, 1479395, 1479396, 1488921], "question_author": ""}
{"qid": 106, "query": "Books for general relativity", "score": 109, "views": 111332, "answer_pids": [1465736, 1465844, 1475688, 1475689, 1480956, 1482994, 1498152, 1499444, 1516017, 1524402, 1543800, 1551194, 1551902, 1558710, 1579315, 1580900, 1619328], "question_author": ""}
{"qid": 107, "query": "Why is glass transparent?", "score": 109, "views": 100033, "answer_pids": [1469594, 1469595, 1477300, 1498442], "question_author": "Chris Wenham"}
{"qid": 108, "query": "How can anything ever fall into a black hole as seen from an outside observer?", "score": 109, "views": 28073, "answer_pids": [1476165, 1476178, 1476179, 1476196, 1476214, 1502918, 1611244], "question_author": null}
{"qid": 109, "query": "Whats inside a proton?", "score": 108, "views": 39232, "answer_pids": [1499408, 1499409, 1499410, 1499411, 1499419, 1499425, 1499439], "question_author": "Pellegrino"}
{"qid": 110, "query": "Why is oil a better lubricant than water?", "score": 108, "views": 25905, "answer_pids": [1554849, 1554850, 1554863, 1554871, 1555076, 1555224], "question_author": ""}
{"qid": 111, "query": "Why do tuning forks have two prongs?", "score": 108, "views": 21371, "answer_pids": [1488191, 1488192, 1488195, 1488205, 1488211, 1588160], "question_author": "AlanSE"}
{"qid": 112, "query": "Why do fusion and fission both release energy?", "score": 108, "views": 20259, "answer_pids": [1610238, 1610240, 1610242, 1610256, 1610260, 1610262, 1610270, 1610303, 1610313, 1610333, 1610521], "question_author": "Nick"}
{"qid": 113, "query": "Why is quantum entanglement considered to be an active link between particles?", "score": 108, "views": 18894, "answer_pids": [1467345, 1467347, 1467348, 1467502, 1467504, 1474481, 1491643], "question_author": "Jon Bringhurst"}
{"qid": 114, "query": "Why are some people are claiming that the Big Bang never happened?", "score": 108, "views": 18403, "answer_pids": [1526621, 1526628, 1526635], "question_author": ""}
{"qid": 115, "query": "How do we know that radioactive decay rates are constant over billions of years?", "score": 108, "views": 17534, "answer_pids": [1577271, 1577273, 1577274, 1577297, 1577341], "question_author": "Chris Wenham"}
{"qid": 116, "query": "How can I stand on the ground? EM or/and Pauli?", "score": 108, "views": 14294, "answer_pids": [1466164, 1466476, 1466489, 1473969, 1476611, 1488539, 1488547], "question_author": ""}
{"qid": 117, "query": "Is Angular Momentum truly fundamental?", "score": 108, "views": 12641, "answer_pids": [1466009, 1466012, 1466013, 1467196, 1467361, 1467419, 1467420, 1543497], "question_author": "Mad Scientist"}
{"qid": 118, "query": "What is $\\Delta t$ in the time-energy uncertainty principle?", "score": 107, "views": 32319, "answer_pids": [1488997, 1488998, 1489000, 1489002, 1515027, 1543897], "question_author": "Gordon Gustafson"}
{"qid": 119, "query": "Why the Principle of Least Action?", "score": 107, "views": 31349, "answer_pids": [1473638, 1473639, 1473647, 1473648, 1473649, 1473666, 1579670, 1579674, 1615743], "question_author": ""}
{"qid": 120, "query": "Why does the Suns (or other stars) nuclear reaction not use up all its fuel immediately?", "score": 107, "views": 19987, "answer_pids": [1515133, 1515134, 1515136, 1515144, 1515147, 1515237, 1515427, 1515664], "question_author": "Emlena.PhD"}
{"qid": 121, "query": "Intuitively, why are bundles so important in Physics?", "score": 107, "views": 13966, "answer_pids": [1497999, 1498001, 1563913, 1592624], "question_author": "shigeta"}
{"qid": 122, "query": "Why does a sticker slowly peel off, but if it is pulled quickly it tears?", "score": 107, "views": 12543, "answer_pids": [1621797, 1621860, 1621870, 1621888], "question_author": "kmm"}
{"qid": 123, "query": "What is known about the topological structure of spacetime?", "score": 107, "views": 8512, "answer_pids": [1466581, 1466583, 1466590, 1466591, 1466595, 1466639], "question_author": "Jamie Banks"}
{"qid": 124, "query": "Does the $\\frac{4}{3}$ problem of classical electromagnetism remain in quantum mechanics?", "score": 107, "views": 4319, "answer_pids": [1611121, 1611143, 1611352], "question_author": ""}
{"qid": 125, "query": "How can we see an atom now? What was the scale of this equipment?", "score": 106, "views": 18600, "answer_pids": [1590886, 1590888, 1590895, 1590911, 1590914, 1590916], "question_author": ""}
{"qid": 126, "query": "Is there such thing as imaginary time dilation?", "score": 105, "views": 7140, "answer_pids": [1553411, 1602895], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 127, "query": "Thought experiment - would you notice if you fell into a black hole?", "score": 105, "views": 7040, "answer_pids": [1533679, 1533680, 1533682, 1533990, 1558414], "question_author": "BBischof"}
{"qid": 128, "query": "Explain it to me like Im a physics grad: Greenhouse Effect", "score": 104, "views": 10270, "answer_pids": [1567193, 1567204, 1567226], "question_author": "Akhil Mathew"}
{"qid": 129, "query": "QM without complex numbers", "score": 103, "views": 25118, "answer_pids": [1481790, 1481791, 1481799, 1481807, 1481813, 1481962, 1481967, 1491693, 1500154, 1534358, 1538246, 1612173, 1638034, 1638035], "question_author": "pablasso"}
{"qid": 130, "query": "What Is Energy? Where did it come from?", "score": 102, "views": 43712, "answer_pids": [1467275, 1467276, 1467277, 1467279, 1475008, 1476994, 1592251, 1648336], "question_author": ""}
{"qid": 131, "query": "Why is filling a balloon from your mouth much harder initially?", "score": 102, "views": 26279, "answer_pids": [1512713, 1512721, 1512723, 1512733, 1512746, 1512754, 1512759, 1512768, 1565898], "question_author": ""}
{"qid": 132, "query": "If gravity isnt a force, then why do we learn in school that it is?", "score": 102, "views": 19581, "answer_pids": [1543066, 1543079, 1543092, 1543095, 1543127, 1543179, 1543357, 1543384, 1544381], "question_author": ""}
{"qid": 133, "query": "If I pull a metal bar for long enough with a constant small force, will it eventually break?", "score": 102, "views": 12545, "answer_pids": [1659870, 1659872, 1659883, 1659890, 1659891, 1659920, 1659925], "question_author": ""}
{"qid": 134, "query": "Which is stronger, a rope without knots or a rope with knots?", "score": 101, "views": 25945, "answer_pids": [1560502, 1560505, 1560508, 1560511, 1560517, 1560525, 1560529, 1560536, 1560539], "question_author": "kennytm"}
{"qid": 135, "query": "Would a pin head heated to 15 million degrees Celsius kill everyone in a 1000 mile radius?", "score": 100, "views": 34927, "answer_pids": [1576841, 1576843, 1576849, 1576918], "question_author": "bryn"}
{"qid": 136, "query": "Why is there a scarcity of lithium?", "score": 100, "views": 18721, "answer_pids": [1471423, 1471426, 1471431, 1471438, 1471439, 1472232, 1537029], "question_author": "Justin L."}
{"qid": 137, "query": "Why do most formulas in physics have integer and rational exponents?", "score": 99, "views": 15709, "answer_pids": [1526093, 1526101, 1526103, 1526124, 1526126, 1526128, 1526138, 1526156, 1526163, 1526184, 1526187, 1526302, 1526304, 1526308, 1526313], "question_author": ""}
{"qid": 138, "query": "Is there something similar to Noethers theorem for discrete symmetries?", "score": 99, "views": 14181, "answer_pids": [1470139, 1470140, 1470142, 1470147, 1470153, 1470155, 1470168, 1540384], "question_author": ""}
{"qid": 139, "query": "What is a good introductory book on quantum mechanics?", "score": 98, "views": 165000, "answer_pids": [1482172, 1485631, 1486169, 1488490, 1488733, 1488735, 1492558, 1493525, 1493919, 1513316, 1516724, 1537911, 1538605, 1585262, 1618395, 1647929], "question_author": null}
{"qid": 140, "query": "Best books for mathematical background?", "score": 98, "views": 53568, "answer_pids": [1465640, 1465643, 1465843, 1465853, 1467028, 1467029, 1467043, 1471191, 1488325, 1503990, 1506834, 1541187, 1580183], "question_author": ""}
{"qid": 141, "query": "What is the physical meaning of commutators in quantum mechanics?", "score": 98, "views": 24580, "answer_pids": [1515376, 1515383, 1515409, 1515434, 1515450, 1590112], "question_author": "Seamus"}
{"qid": 142, "query": "Differentiating Propagator, Greens function, Correlation function, etc", "score": 98, "views": 15670, "answer_pids": [1475925, 1615476, 1617834], "question_author": "Cedric H."}
{"qid": 143, "query": "What justifies dimensional analysis?", "score": 98, "views": 12009, "answer_pids": [1504669, 1504672, 1504675, 1504677, 1504680, 1504681, 1504718, 1538247, 1585350], "question_author": ""}
{"qid": 144, "query": "How is a quantum superposition different from a mixed state?", "score": 97, "views": 31843, "answer_pids": [1499130, 1499132, 1499135, 1499138, 1499145, 1548774, 1610951, 1639631], "question_author": ""}
{"qid": 145, "query": "Why dont miners get boiled to death at $4$ km deep?", "score": 97, "views": 20498, "answer_pids": [1563888, 1563892, 1563897, 1564385], "question_author": "Sklivvz"}
{"qid": 146, "query": "Trace of a commutator is zero - but what about the commutator of $x$ and $p$?", "score": 97, "views": 14578, "answer_pids": [1470966, 1470977, 1547513], "question_author": ""}
{"qid": 147, "query": "About the complex nature of the wave function?", "score": 96, "views": 39036, "answer_pids": [1469920, 1469926, 1469947, 1469956, 1470010, 1473519, 1473527, 1486573, 1589009, 1598036], "question_author": "jonsca"}
{"qid": 148, "query": "Why was carbon-12 chosen for the atomic mass unit?", "score": 96, "views": 26177, "answer_pids": [1576697, 1576748, 1579443], "question_author": ""}
{"qid": 149, "query": "Why does the shower curtain move towards me when I am taking a hot shower?", "score": 96, "views": 12014, "answer_pids": [1579222, 1579225, 1579251, 1579252], "question_author": ""}
{"qid": 150, "query": "Where is the flaw in this machine that decreases the entropy of a closed system?", "score": 96, "views": 6746, "answer_pids": [1538723], "question_author": "aqua"}
{"qid": 151, "query": "Why does the atmosphere rotate along with the earth?", "score": 95, "views": 99739, "answer_pids": [1466234, 1466235, 1466237, 1466532, 1474334, 1477434], "question_author": ""}
{"qid": 152, "query": "Why does hot water clean better than cold water?", "score": 95, "views": 56027, "answer_pids": [1510739, 1510740, 1510751, 1510769, 1510795, 1510836, 1511081], "question_author": ""}
{"qid": 153, "query": "What is the speed of sound in space?", "score": 95, "views": 33931, "answer_pids": [1525844, 1525845, 1525847, 1525858, 1525879, 1531401, 1549086], "question_author": ""}
{"qid": 154, "query": "How can a black hole produce sound?", "score": 95, "views": 19724, "answer_pids": [1487657, 1487658, 1487660, 1487667, 1487669, 1487677], "question_author": "Neil Mayhew"}
{"qid": 155, "query": "What is the proper way to explain the twin paradox?", "score": 95, "views": 13252, "answer_pids": [1549601, 1549613, 1549663, 1549732, 1550291, 1578732, 1606790, 1618070, 1623925, 1623974], "question_author": "Simon Nickerson"}
{"qid": 156, "query": "Why are radiators always placed under windows?", "score": 94, "views": 55193, "answer_pids": [1501058, 1501060, 1501064, 1501066, 1501079, 1501081, 1501085, 1501102, 1501124], "question_author": "Mad Scientist"}
{"qid": 157, "query": "Can we theoretically balance a perfectly symmetrical pencil on its one-atom tip?", "score": 94, "views": 20974, "answer_pids": [1530423, 1530424, 1530425, 1530441, 1530502, 1536290], "question_author": "walkytalky"}
{"qid": 158, "query": "If Earth had rings?", "score": 94, "views": 11983, "answer_pids": [1561553, 1561554, 1561622], "question_author": "Rory M"}
{"qid": 159, "query": "How can time dilation be symmetric?", "score": 94, "views": 11391, "answer_pids": [1590078, 1590089, 1590095, 1590101, 1590103, 1590115, 1590298], "question_author": "Jonathan."}
{"qid": 160, "query": "Quantum Entanglement - Whats the big deal?", "score": 93, "views": 13082, "answer_pids": [1489515, 1489524, 1489534, 1489538, 1489605, 1509225, 1509229], "question_author": "www0z0k"}
{"qid": 161, "query": "If we had a perfectly efficient computer and all the energy in the Milky-way available, what number could it count to?", "score": 93, "views": 11224, "answer_pids": [1554166], "question_author": ""}
{"qid": 162, "query": "Why does the LIGO observation disprove higher dimensions?", "score": 93, "views": 9836, "answer_pids": [1602808, 1602819], "question_author": "Justin L."}
{"qid": 163, "query": "Why does nature favour the Laplacian?", "score": 93, "views": 8148, "answer_pids": [1615333, 1615418, 1615447, 1615462], "question_author": "Poshpaws"}
{"qid": 164, "query": "What software programs are used to draw physics diagrams, and what are their relative merits?", "score": 92, "views": 169663, "answer_pids": [1465751, 1465752, 1465753, 1465755, 1465765, 1465820, 1466823, 1466824, 1467355, 1467364, 1472041, 1472053, 1477101, 1494453, 1495002, 1495245, 1496369, 1501737, 1502227], "question_author": "Larry Wang"}
{"qid": 165, "query": "Why do same/opposite electric charges repel/attract each other, respectively?", "score": 92, "views": 109428, "answer_pids": [1499275, 1505346], "question_author": "ablmf"}
{"qid": 166, "query": "What conservation law corresponds to Lorentz boosts?", "score": 92, "views": 17334, "answer_pids": [1472070, 1638434], "question_author": "Gergana Vandova"}
{"qid": 167, "query": "Why quantum mechanics?", "score": 92, "views": 12316, "answer_pids": [1486554, 1486557, 1486631, 1486646, 1486675, 1486679, 1486691, 1488606, 1488607, 1488913, 1497406, 1497407, 1497408, 1579675, 1600998], "question_author": "Mark Eichenlaub"}
{"qid": 168, "query": "Does a gun exert enough gravity on the bullet it fired to stop it?", "score": 92, "views": 11351, "answer_pids": [1539194, 1539195, 1539196, 1539203, 1539297], "question_author": "Mark Eichenlaub"}
{"qid": 169, "query": "Why does my wooden door disperse light into a rainbow color spectrum?", "score": 92, "views": 5539, "answer_pids": [1518243], "question_author": "Gergana Vandova"}
{"qid": 170, "query": "What is the relation between electromagnetic wave and photon?", "score": 91, "views": 50686, "answer_pids": [1502179, 1502184, 1502187, 1553140, 1605013, 1605017, 1605019], "question_author": ""}
{"qid": 171, "query": "Why is the $S_{z} =0$ state forbidden for photons?", "score": 91, "views": 19757, "answer_pids": [1486846, 1486852, 1486853, 1486859, 1556489, 1556490, 1556498], "question_author": "Rory M"}
{"qid": 172, "query": "Seeing something from only one angle means you have only seen (what?)% of its surface area at most?", "score": 91, "views": 17038, "answer_pids": [1581313, 1581314, 1581315, 1581366], "question_author": ""}
{"qid": 173, "query": "Are Newtons laws of motion laws or definitions of force and mass?", "score": 91, "views": 9252, "answer_pids": [1495228, 1495243, 1495249, 1523789, 1578531, 1578963], "question_author": "Dale"}
{"qid": 174, "query": "Does juggling balls reduce the total weight of the juggler and balls?", "score": 91, "views": 9186, "answer_pids": [1483483, 1483484, 1483485, 1483490, 1483499, 1483500, 1483515, 1644726], "question_author": ""}
{"qid": 175, "query": "Why does a remote car key work when held to your head/body?", "score": 90, "views": 184591, "answer_pids": [1505995, 1506016, 1506054, 1506117, 1506621, 1511000], "question_author": "kali281"}
{"qid": 176, "query": "Why does matter exist in 3 states (liquids, solid, gas)?", "score": 90, "views": 30163, "answer_pids": [1549707, 1549708, 1549710, 1549724, 1549725, 1549726, 1549751, 1549849, 1550285], "question_author": "kali281"}
{"qid": 177, "query": "How did Einstein know the speed of light was constant?", "score": 90, "views": 25645, "answer_pids": [1619149, 1619151, 1619155, 1619158, 1619242], "question_author": "Fomite"}
{"qid": 178, "query": "Why does a billiard ball stop when it hits another billiard ball head on?", "score": 90, "views": 21309, "answer_pids": [1602481, 1602482, 1602483, 1602562, 1602640, 1602678, 1616456], "question_author": "Fomite"}
{"qid": 179, "query": "Am I attracting Pluto?", "score": 90, "views": 16969, "answer_pids": [1537601, 1537622, 1537677, 1537705, 1537775, 1537848], "question_author": ""}
{"qid": 180, "query": "Why does a yellow object turn white under a yellow light? Shouldnt it turn yellow instead?", "score": 90, "views": 14497, "answer_pids": [1576705, 1576721, 1576743, 1576829], "question_author": ""}
{"qid": 181, "query": "Why is a laserpointer able to erase a glow-in-the-dark sticker?", "score": 90, "views": 7215, "answer_pids": [1498906, 1498996], "question_author": ""}
{"qid": 182, "query": "Can tin foil hats block anything?", "score": 89, "views": 34211, "answer_pids": [1539856, 1539858, 1539889], "question_author": "DQdlM"}
{"qid": 183, "query": "Why is a $5-60 mph$ time slower than a $0-60 mph$ time for some automobiles?", "score": 89, "views": 19662, "answer_pids": [1560014, 1560018, 1560025, 1560045], "question_author": "walkytalky"}
{"qid": 184, "query": "Why do scientists think that all the laws of physics that apply in our galaxy apply in other galaxies?", "score": 89, "views": 18740, "answer_pids": [1597994, 1597996, 1598001, 1598012, 1598040, 1598059, 1598101, 1598159, 1598167], "question_author": ""}
{"qid": 185, "query": "Why do phones land face down?", "score": 89, "views": 18015, "answer_pids": [1586028, 1586048, 1586051], "question_author": "Cedric H."}
{"qid": 186, "query": "What is a manifold?", "score": 89, "views": 17962, "answer_pids": [1566565, 1566566, 1566567, 1566568, 1566570, 1566581, 1566611], "question_author": "Poshpaws"}
{"qid": 187, "query": "Why cant $ i\\hbar\\frac{\\partial}{\\partial t}$ be considered the Hamiltonian operator?", "score": 89, "views": 13063, "answer_pids": [1474377, 1474378, 1474380, 1474381, 1474384, 1474849, 1474855, 1492365, 1562502, 1562566, 1567664, 1603698, 1612827, 1621708], "question_author": "Jamie Banks"}
{"qid": 188, "query": "How does this baby rattle work?", "score": 89, "views": 9618, "answer_pids": [1592905, 1592963, 1593211], "question_author": "walkytalky"}
{"qid": 189, "query": "Superluminal neutrinos", "score": 89, "views": 9008, "answer_pids": [1473227, 1473234, 1473237, 1473238, 1473249, 1473986, 1476173, 1509400], "question_author": "nmagerko"}
{"qid": 190, "query": "Why do metals only glow red, yellow and white and not through the full range of the spectrum?", "score": 88, "views": 22900, "answer_pids": [1576401, 1576403, 1576407, 1576414, 1576547, 1576562], "question_author": "Aleksandr Levchuk"}
{"qid": 191, "query": "What is time, does it flow, and if so what defines its direction?", "score": 88, "views": 11941, "answer_pids": [1547891, 1548860, 1582569, 1608779, 1631694, 1637929, 1656375], "question_author": "ths1104"}
{"qid": 192, "query": "How exactly do you avoid fooling yourself?", "score": 88, "views": 10367, "answer_pids": [1624796, 1624800, 1624805, 1624858], "question_author": null}
{"qid": 193, "query": "Why are differential equations for fields in physics of order two?", "score": 88, "views": 9738, "answer_pids": [1474884, 1485526, 1485664, 1485734, 1485735, 1485764, 1490732, 1523209, 1570270], "question_author": "jonsca"}
{"qid": 194, "query": "Quantum Field Theory from a mathematical point of view", "score": 87, "views": 28944, "answer_pids": [1479564, 1479565, 1479566, 1479567, 1479568, 1479569, 1479570, 1479571, 1479572, 1479573, 1502597], "question_author": "dsolimano"}
{"qid": 195, "query": "Why is light bent but not accelerated?", "score": 87, "views": 13625, "answer_pids": [1578065, 1578068, 1578069, 1578078, 1578091], "question_author": "Rory M"}
{"qid": 196, "query": "Why can Hiroshima be inhabited when Chernobyl cannot?", "score": 86, "views": 203742, "answer_pids": [1493096, 1493607, 1510371], "question_author": "Justin L."}
{"qid": 197, "query": "What makes running so much less energy-efficient than bicycling?", "score": 86, "views": 21914, "answer_pids": [1491649, 1491650, 1491652, 1491656, 1491663, 1491683, 1491702, 1618084], "question_author": "Justin L."}
{"qid": 198, "query": "Why is it bad taste to have a dimensional quantity in the argument of a logarithm or exponential function?", "score": 86, "views": 14675, "answer_pids": [1585254, 1585255, 1585256, 1585271], "question_author": "Justin L."}
{"qid": 199, "query": "What is spontaneous symmetry breaking in quantum systems?", "score": 86, "views": 13144, "answer_pids": [1480366, 1480367, 1480385, 1480389, 1480394, 1480463, 1480656, 1480767, 1480961, 1483949, 1541077], "question_author": ""}
{"qid": 200, "query": "Why does a full moon seem uniformly bright from earth, shouldnt it be dimmer at the border?", "score": 86, "views": 10898, "answer_pids": [1563657, 1563663, 1563675], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 201, "query": "Why is information indestructible?", "score": 85, "views": 24505, "answer_pids": [1480302, 1491834, 1495683, 1496290, 1593560, 1593569], "question_author": "Polynomial"}
{"qid": 202, "query": "Why dont fluorescent lights produce shadows?", "score": 85, "views": 22202, "answer_pids": [1525561, 1525587], "question_author": "ghchinoy"}
{"qid": 203, "query": "If you are vacuuming your carpet and you wrap the cord around your body do you become a magnet?", "score": 85, "views": 15600, "answer_pids": [1560936, 1560937, 1561044, 1561211], "question_author": ""}
{"qid": 204, "query": "When I walk down the stairs where does my potential energy go?", "score": 85, "views": 15475, "answer_pids": [1663166, 1663167, 1663168, 1663170, 1663318], "question_author": ""}
{"qid": 205, "query": "What is more fundamental, fields or particles?", "score": 85, "views": 13040, "answer_pids": [1512735, 1512736, 1512737, 1535446, 1635497], "question_author": ""}
{"qid": 206, "query": "Why does rainwater form moving waves on the ground? Is there a name for this effect?", "score": 85, "views": 6204, "answer_pids": [1573361, 1573362, 1573371], "question_author": "Vivi"}
{"qid": 207, "query": "List of freely available physics books", "score": 84, "views": 89403, "answer_pids": [1468951, 1468952, 1468953, 1468954, 1468955, 1468956, 1468957, 1468958, 1468959, 1468960, 1468972, 1468996, 1469007, 1469008, 1469009, 1469181, 1469570, 1469726, 1470357, 1479709, 1483662, 1484407, 1489568], "question_author": ""}
{"qid": 208, "query": "In the earths crust, why is there far more uranium than gold?", "score": 84, "views": 10523, "answer_pids": [1520288, 1520291], "question_author": "Pratik Deoghare"}
{"qid": 209, "query": "Could a living planet alter its own trajectory only by changing its shape?", "score": 84, "views": 10231, "answer_pids": [1555478, 1555490, 1555495, 1555496, 1555498, 1555508, 1555548], "question_author": ""}
{"qid": 210, "query": "Classical and quantum anomalies", "score": 84, "views": 8012, "answer_pids": [1482149], "question_author": ""}
{"qid": 211, "query": "What causes the water in this fountain to reverse direction?", "score": 84, "views": 7140, "answer_pids": [1589246, 1589250, 1589262, 1589289], "question_author": ""}
{"qid": 212, "query": "What is spin as it relates to subatomic particles?", "score": 83, "views": 15841, "answer_pids": [1465513, 1465514, 1482772, 1518124, 1594945], "question_author": ""}
{"qid": 213, "query": "If dark matter only interacts with gravity, why doesnt it all clump together in a single point?", "score": 83, "views": 10588, "answer_pids": [1541828, 1541829, 1541830, 1640452], "question_author": "Jonathan Fischoff"}
{"qid": 214, "query": "Why doesnt water actually perfectly wet glass?", "score": 83, "views": 8842, "answer_pids": [1608020, 1608091, 1657542], "question_author": null}
{"qid": 215, "query": "Can photons be detected without being absorbed?", "score": 83, "views": 8683, "answer_pids": [1581984, 1581985], "question_author": ""}
{"qid": 216, "query": "Is there a symmetry associated to the conservation of information?", "score": 83, "views": 7859, "answer_pids": [1485070, 1485085, 1485095, 1485139, 1497110, 1497120, 1577952], "question_author": ""}
{"qid": 217, "query": "What does one second after big bang mean?", "score": 83, "views": 6394, "answer_pids": [1571830, 1571864, 1571879, 1623438], "question_author": null}
{"qid": 218, "query": "What happened to David John Candlin?", "score": 83, "views": 5644, "answer_pids": [1495935], "question_author": null}
{"qid": 219, "query": "Why do travelling waves continue after amplitude sum = 0?", "score": 83, "views": 5077, "answer_pids": [1551027, 1551028, 1551134], "question_author": ""}
{"qid": 220, "query": "Will a hole cut into a metal disk expand or shrink when the disc is heated?", "score": 82, "views": 114927, "answer_pids": [1472091, 1472203, 1474779, 1476714, 1481933, 1483868, 1485194], "question_author": "Campadrenalin"}
{"qid": 221, "query": "Why does public mains power use 50-60 Hz and 100-240 V?", "score": 82, "views": 18587, "answer_pids": [1540698, 1540699, 1540704, 1540705, 1540718], "question_author": ""}
{"qid": 222, "query": "How can magnets be used to pick up pieces of metal when the force from a magnetic field does no work?", "score": 82, "views": 16991, "answer_pids": [1494277, 1494293, 1494296, 1494353, 1494354, 1496744, 1617122], "question_author": "jabal"}
{"qid": 223, "query": "How do laser tape measures work?", "score": 82, "views": 16064, "answer_pids": [1553740, 1553748, 1553764, 1553768, 1553790], "question_author": ""}
{"qid": 224, "query": "Why must a physical theory be mathematically self-consistent?", "score": 82, "views": 12375, "answer_pids": [1555606, 1555607, 1555608, 1555610, 1555612, 1555619, 1555620, 1555622, 1555630, 1555631, 1555641], "question_author": ""}
{"qid": 225, "query": "Why does a rubber band become a lighter color when stretched?", "score": 82, "views": 7022, "answer_pids": [1634291, 1634293, 1634323, 1634328, 1634362, 1634631], "question_author": ""}
{"qid": 226, "query": "Does centrifugal force exist?", "score": 81, "views": 71976, "answer_pids": [1508777, 1508778, 1508786, 1508800, 1508808, 1508809, 1508813, 1559891, 1623101], "question_author": ""}
{"qid": 227, "query": "Why does water stop boiling immediately after turning off the heat?", "score": 81, "views": 25042, "answer_pids": [1589404, 1589405, 1589407, 1589420, 1589601], "question_author": ""}
{"qid": 228, "query": "Why does fire make very little sound?", "score": 81, "views": 17470, "answer_pids": [1601462, 1601477, 1601485, 1601496, 1601505, 1601518, 1601626, 1601662], "question_author": "ghchinoy"}
{"qid": 229, "query": "Where does the extra kinetic energy come from in a gravitational slingshot?", "score": 81, "views": 13033, "answer_pids": [1516860, 1516865, 1516869, 1516897], "question_author": ""}
{"qid": 230, "query": "What was the major discovery on gravitational waves made March 17th, 2014, in the BICEP2 experiment?", "score": 80, "views": 6569, "answer_pids": [1506674, 1506685, 1506689, 1506765, 1506848, 1605952], "question_author": "walkytalky"}
{"qid": 231, "query": "Why doesnt the Moon fall onto the Earth?", "score": 79, "views": 196563, "answer_pids": [1470408, 1470409, 1470414, 1470416, 1470446, 1482139, 1489632], "question_author": ""}
{"qid": 232, "query": "How and why do accelerating charges radiate electromagnetic radiation?", "score": 79, "views": 81649, "answer_pids": [1493258, 1493295, 1570663, 1582612], "question_author": ""}
{"qid": 233, "query": "Visually stunning math concepts which are easy to explain", "score": 1420, "views": 527466, "answer_pids": [320567, 320584, 320677, 320684, 320763, 320938, 320976, 321177, 321199, 321705, 322029, 322060, 322177, 322303, 322516, 322615, 322621, 323039, 323215, 323371, 323892, 323904, 323934, 323957, 323959, 323988, 324053, 324054, 324083, 324122, 324215, 324294, 324326, 324337, 324594, 324787, 324832, 329242, 360642, 381776, 386343, 391190, 451007, 467217, 472868, 656974, 658671, 671034, 677859, 677885, 721730, 770324, 792682, 839955, 841815, 863322, 879105, 881871, 901172, 953871, 971918, 1028640, 1045182, 1070886, 1082478, 1135457, 1180633, 1197219, 1199455, 1299194, 1305746, 1313550, 1346867, 1346875, 1346876, 1346883, 1347243, 1347351, 1347751, 1349623, 1412462, 1418709, 1449201], "question_author": ""}
{"qid": 234, "query": "Is $\\frac{\\textrm{d}y}{\\textrm{d}x}$ not a ratio?", "score": 1169, "views": 117492, "answer_pids": [11699, 11700, 11701, 11740, 11771, 41529, 41598, 115437, 208214, 217621, 231765, 274935, 277622, 335713, 337148, 764123, 805886, 852082, 1125868, 1254046, 1272125, 1417757, 1418840, 1442483], "question_author": "Mark C"}
{"qid": 235, "query": "How long will it take Marie to saw another board into 3 pieces?", "score": 1048, "views": 145609, "answer_pids": [178845, 178846, 178885, 178912, 178941, 178945, 178948, 178977, 179031, 179086, 179157, 179298, 180152, 180164, 185771, 186630, 187374, 228019, 232101, 338624, 448985, 449002, 465859, 465867, 476121, 594459, 694395, 821377, 973698, 1025985], "question_author": ""}
{"qid": 236, "query": "Can I use my powers for good?", "score": 923, "views": 91662, "answer_pids": [37880, 37890, 37892, 37909, 37917, 37920, 37929, 37940, 37947, 37948, 37954, 37955, 37962, 37970, 37984, 37991, 38225, 38744, 39643, 39708, 39739, 50177, 50180, 59811, 169169, 401960, 427519, 441532, 531156], "question_author": "Nick"}
{"qid": 237, "query": "The staircase paradox, or why $\\pi\\ne4$", "score": 821, "views": 96254, "answer_pids": [7305, 7310, 7312, 7313, 7315, 12554, 12561, 21734, 22447, 22487, 78442, 170751, 241679, 671428, 671445, 714819, 718123, 742926, 762453, 877548, 1172930, 1318009], "question_author": "Justin L."}
{"qid": 238, "query": "How to study math to really understand it and have a healthy lifestyle with free time?", "score": 805, "views": 189279, "answer_pids": [23810, 23812, 23815, 23824, 23827, 23828, 23829, 23830, 23892, 24438, 24568, 24587, 39654, 39656, 39710, 65964, 65969, 66021, 107094, 120334, 129749, 225889, 229158, 363637, 442404, 539973, 614361], "question_author": ""}
{"qid": 239, "query": "Different methods to compute $\\sum\\limits_{k=1}^\\infty \\frac{1}{k^2}$ (Basel problem)", "score": 767, "views": 117351, "answer_pids": [4805, 4817, 4819, 4846, 4879, 4901, 5035, 5044, 24098, 30422, 30429, 34546, 55658, 101867, 134592, 154470, 154499, 268827, 276590, 278388, 290266, 400290, 418736, 446977, 454613, 554778, 562172, 624906, 641006, 655456, 706038, 706091, 707961, 717890, 759290, 759558, 871875, 871898, 876982, 876990, 880558, 898656, 1153106, 1247682, 1293712, 1366015, 1388323], "question_author": "Marta Cz-C"}
{"qid": 240, "query": "Whats an intuitive way to think about the determinant?", "score": 730, "views": 141916, "answer_pids": [400, 616, 655, 9813, 9817, 42931, 73886, 126427, 251230, 255177, 812381, 849583, 1142369, 1325723], "question_author": "Gergana Vandova"}
{"qid": 241, "query": "Does $\\pi$ contain all possible number combinations?", "score": 712, "views": 194021, "answer_pids": [107114, 107115, 107116, 107117, 107158, 107190, 107216, 107224, 107226, 107248, 107818, 173772, 718139], "question_author": ""}
{"qid": 242, "query": "Splitting a sandwich and not feeling deceived", "score": 670, "views": 68968, "answer_pids": [284832, 284834, 284835, 284839, 284859, 284864, 284865, 284879, 284882, 284919, 284926, 284944, 284949, 284985, 285006, 285038, 285137, 285147, 285166, 285191, 285343, 285597, 286104, 370421, 374595], "question_author": "Chris Wenham"}
{"qid": 243, "query": "What was the first bit of mathematics that made you realize that math is beautiful? (For childrens book)", "score": 663, "views": 53373, "answer_pids": [154091, 154092, 154093, 154095, 154096, 154097, 154098, 154099, 154100, 154101, 154102, 154104, 154107, 154108, 154112, 154113, 154115, 154119, 154120, 154129, 154131, 154142, 154143, 154146, 154150, 154152, 154155, 154157, 154158, 154167, 154168, 154175, 154176, 154178, 154181, 154187, 154189, 154190, 154196, 154197, 154200, 154212, 154213, 154220, 154225, 154226, 154227, 154230, 154241, 154258, 154263, 154266, 154269, 154273, 154278, 154282, 154283, 154284, 154286, 154296, 154300, 154309, 154328, 154330, 154335, 154337, 154344, 154346, 154347, 154361, 154366, 154389, 154394, 154404, 154417, 154430, 154435, 154445, 154476, 154487, 154493, 154537, 154677, 154695, 154702, 154714, 154716, 154769, 154818, 154823, 154836, 154846, 154864, 154886, 154894, 155066, 155105, 155254, 155268, 157491, 157620, 157840, 158276, 161288, 161478, 174623, 176583, 176595, 179691, 179732, 202992, 203506, 209615, 216493, 220230, 233405, 234053, 244731, 269262, 282980, 283173, 292348, 293635, 305676, 307182, 359865, 383365, 383371, 384018, 407842, 423207, 427062, 449075, 468340, 468733, 472965, 480022, 524327, 548595, 612581, 774506], "question_author": ""}
{"qid": 244, "query": "Why is $1 - \\frac{1}{1 - \\frac{1}{1 - \\ldots}}$ not real?", "score": 602, "views": 41113, "answer_pids": [653267, 653269, 653298, 653319, 653619, 653662, 856325], "question_author": ""}
{"qid": 245, "query": "Why can you turn clothing right-side-out?", "score": 599, "views": 82107, "answer_pids": [1645, 1646, 1647, 1651, 1672, 1779], "question_author": ""}
{"qid": 246, "query": "Examples of patterns that eventually fail", "score": 584, "views": 48864, "answer_pids": [57289, 57294, 57295, 57300, 57314, 57320, 57322, 57323, 57342, 57347, 57349, 57365, 57399, 57421, 57468, 57541, 57615, 65640, 73429, 114419, 136770, 168698, 209646, 232796, 257932, 332345, 383934, 384177, 384186, 424526, 450111, 519767, 687789, 923299, 993954, 1077064, 1337197, 1423492], "question_author": ""}
{"qid": 247, "query": "Mathematical difference between white and black notes in a piano", "score": 568, "views": 88368, "answer_pids": [6623, 6625, 6633, 6637, 6680, 6696, 6704, 6717, 6724, 6729, 6734, 6739, 6742, 6745, 6760, 6770, 6788, 6796, 6841, 6873, 846222], "question_author": ""}
{"qid": 248, "query": "Do complex numbers really exist?", "score": 513, "views": 67432, "answer_pids": [88, 89, 90, 91, 94, 95, 99, 103, 109, 132, 138, 412, 540, 1350, 1485, 1583, 1892, 2622, 2625, 5296, 5350, 5645, 5653, 5753, 6379, 7213, 7751, 8332, 9174, 57599, 61978, 62832, 325125, 884166], "question_author": ""}
{"qid": 249, "query": "What are imaginary numbers?", "score": 507, "views": 82197, "answer_pids": [99989, 99990, 99991, 100004, 100012, 100025, 100030, 100031, 100115, 100140, 100187, 100344, 100347, 101007, 101535, 101641, 202377, 288579, 448868, 507356, 1063471], "question_author": null}
{"qid": 250, "query": "How to prove that $\\lim\\limits_{x\\to0}\\frac{\\sin x}x=1$?", "score": 498, "views": 174465, "answer_pids": [39519, 39520, 39526, 39527, 39554, 78240, 185022, 209625, 390268, 393387, 478267, 482574, 496541, 729992, 752687, 770330, 780614, 836757, 873557, 969040, 1122879, 1172598, 1239463, 1239750, 1323932, 1440716, 1458587], "question_author": "AgCl"}
{"qid": 251, "query": "Integral $\\int_{-1}^1\\frac1x\\sqrt{\\frac{1+x}{1-x}}\\ln\\left(\\frac{2\\,x^2+2\\,x+1}{2\\,x^2-2\\,x+1}\\right) \\mathrm dx$", "score": 488, "views": 245723, "answer_pids": [255575, 256471, 258099, 296127, 299558, 603141, 633759, 725727], "question_author": "jonsca"}
{"qid": 252, "query": "The Egg: Bizarre behavior of the roots of a family of polynomials.", "score": 485, "views": 19572, "answer_pids": [103636, 104348, 104389, 104392, 104566, 150792, 385362], "question_author": "Noldorin"}
{"qid": 253, "query": "Is this Batman equation for real?", "score": 465, "views": 497350, "answer_pids": [28973, 28976, 28987, 28998, 29032, 29424, 29431, 29804, 77141, 301611], "question_author": ""}
{"qid": 254, "query": "Best Sets of Lecture Notes and Articles", "score": 461, "views": 58149, "answer_pids": [148124, 148147, 148213, 152190, 154540, 164099, 535457, 1036386, 1354629, 1361820], "question_author": ""}
{"qid": 255, "query": "My sons Sum of Some is beautiful! But what is the proof or explanation?", "score": 442, "views": 28323, "answer_pids": [190621, 190626, 190639, 191067, 191097, 191148, 191283, 191318, 191342, 674541], "question_author": "bryn"}
{"qid": 256, "query": "Proofs that every mathematician should know.", "score": 432, "views": 80596, "answer_pids": [90180, 90181, 90182, 90184, 90191, 90195, 90199, 90207, 90215, 90229, 90230, 90258, 90282, 90295, 90312, 90433, 90438, 93648, 95435, 95453, 108535, 157882, 157959], "question_author": ""}
{"qid": 257, "query": "Why does $1+2+3+\\cdots = -\\frac{1}{12}$?", "score": 431, "views": 55224, "answer_pids": [21223, 21226, 21246, 97201, 282877, 286582, 304635, 451507, 456421, 472711, 526177, 558110, 567537, 807364, 884640], "question_author": "Jamie Banks"}
{"qid": 258, "query": "What is the intuitive relationship between SVD and PCA?", "score": 423, "views": 219618, "answer_pids": [2308, 78408, 542272, 569527], "question_author": "Justin L."}
{"qid": 259, "query": "How can I evaluate $\\sum_{n=0}^\\infty(n+1)x^n$?", "score": 418, "views": 54434, "answer_pids": [16613, 16615, 16617, 16619, 16620, 42772, 110422, 110425, 247429, 250651, 285418, 352189, 373845, 438489, 516918, 661417, 824335, 989230, 1013557, 1325542, 1422720], "question_author": ""}
{"qid": 260, "query": "On familiarity (or How to avoid going down the Math Rabbit Hole?)", "score": 416, "views": 24756, "answer_pids": [276562, 276569, 276570, 276596, 276610, 276665, 276679, 276702, 276706, 276807, 276816, 276818, 277009, 277020, 277174, 277210, 277271, 277274, 277312, 277512, 279115], "question_author": ""}
{"qid": 261, "query": "Fourier transform for dummies", "score": 414, "views": 346933, "answer_pids": [608, 609, 623, 630, 635, 6314, 15164, 20232, 20236, 38188, 253207, 256031, 746803, 767378], "question_author": ""}
{"qid": 262, "query": "The Ring Game on $K[x,y,z]$", "score": 393, "views": 15341, "answer_pids": [1279218], "question_author": "c4il"}
{"qid": 263, "query": "Pedagogy: How to cure students of the law of universal linearity?", "score": 382, "views": 44265, "answer_pids": [281857, 281881, 281887, 281891, 281922, 281925, 281941, 281943, 281974, 281990, 281991, 282007, 282017, 282019, 282021, 282034, 282042, 282082, 282099, 282105, 282155, 282199, 282367, 282572, 282627, 282670, 282956, 283635, 283741, 393663], "question_author": ""}
{"qid": 264, "query": "Find five positive integers whose reciprocals sum to $1$", "score": 365, "views": 42610, "answer_pids": [139319, 139321, 139361, 139382, 139537, 139540, 139542, 139548, 139656, 139685, 140020, 140095, 144258, 537454, 583269, 748289, 788633, 925138], "question_author": "Daniel Standage"}
{"qid": 265, "query": "Can every proof by contradiction also be shown without contradiction?", "score": 360, "views": 22707, "answer_pids": [118525, 118526, 118527, 118528, 118532, 118533, 118567, 118572, 118578, 118661, 118708, 118936, 119453, 666788], "question_author": ""}
{"qid": 266, "query": "Obvious theorems that are actually false", "score": 355, "views": 74256, "answer_pids": [352400, 352403, 352404, 352405, 352406, 352409, 352417, 352422, 352423, 352429, 352430, 352437, 352450, 352464, 352466, 352519, 352534, 352545, 352637, 352639, 352643, 352694, 352698, 352702, 352703, 352719, 352721, 352731, 352765, 352794, 352799, 352972, 352997, 353003, 353004, 353009, 353018, 353041, 353108, 353162, 353186, 353544, 353558, 353728, 354238, 354288, 354537, 355017, 355248, 355281, 355353, 355680, 355983, 356507, 356595, 357547, 357549, 364581, 369674, 520858, 520866, 668632, 689960, 775631, 918152, 1026915, 1231289], "question_author": ""}
{"qid": 267, "query": "Zero to the zero power \u2013 is $0^0=1$?", "score": 354, "views": 41307, "answer_pids": [6340, 6341, 6342, 6343, 6370, 6405, 40070, 54167, 171260, 184762, 184770, 256333, 308101, 429973, 442662, 494121, 494153, 523644, 534747, 590399], "question_author": ""}
{"qid": 268, "query": "If $AB = I$ then $BA = I$", "score": 343, "views": 99054, "answer_pids": [2298, 2300, 2301, 2302, 2307, 2310, 2314, 2321, 2323, 2353, 2462, 2524, 125208, 299220, 346717, 359301, 645684, 657841, 691528, 852966, 876274, 944420, 966568, 977745, 1063362, 1144072, 1280686, 1322751, 1323082, 1344748], "question_author": "Andrey"}
{"qid": 269, "query": "Calculating the length of the paper on a toilet paper roll", "score": 342, "views": 44615, "answer_pids": [636683, 636702, 637013, 637206, 637408, 637468, 637669, 638181], "question_author": ""}
{"qid": 270, "query": "How can you prove that a function has no closed form integral?", "score": 332, "views": 41680, "answer_pids": [92, 116, 126, 1411, 1413, 3380], "question_author": "Casebash"}
{"qid": 271, "query": "A challenge by R. P. Feynman: give counter-intuitive theorems that can be translated into everyday language", "score": 331, "views": 28847, "answer_pids": [148, 149, 150, 151, 153, 155, 160, 213, 214, 499, 501, 509, 510, 549, 555, 600, 659, 922, 2612, 2865, 2908, 3063, 6805, 72135, 90375, 273992, 718703, 813338, 821313, 821675, 823843, 824655, 824674, 905864], "question_author": "Casebash"}
{"qid": 272, "query": "Surprising identities / equations", "score": 330, "views": 31146, "answer_pids": [234179, 234180, 234181, 234196, 234205, 234216, 234226, 234230, 234254, 234318, 234335, 234337, 234346, 234355, 234359, 234365, 234375, 234394, 234401, 234433, 234471, 234528, 234537, 234567, 234609, 234700, 234727, 234728, 234737, 234766, 234786, 235052, 235061, 235074, 235077, 235080, 235084, 235087, 235099, 235111, 235128, 235143, 235158, 235490, 235497, 235509, 235510, 235512, 235815, 235882, 235898, 235903, 236009, 236477, 236886, 236954, 237348, 259213, 270483, 271128, 271136, 271142, 282110, 298253, 305790, 305798, 347178, 361371, 378568, 393816, 395812, 412540, 428354, 428375, 428387, 448330, 448516, 450457, 501676, 508948, 508964, 565277, 589727, 819788, 831768, 922446, 931915, 942236, 942299, 951588, 1062828, 1082294, 1088019, 1117944, 1138030, 1228600, 1253346, 1310920, 1349541, 1377665, 1377667], "question_author": "Casebash"}
{"qid": 273, "query": "Is it true that $0.999999999\\ldots=1$?", "score": 327, "views": 50654, "answer_pids": [10, 11, 15, 16, 17, 20, 25, 42, 60, 63, 110, 322, 344762, 371845, 393581, 543182, 621607, 629234, 629679, 716851, 865299, 1005814, 1096352, 1104955, 1175706, 1241382, 1252199, 1359203, 1359818], "question_author": ""}
{"qid": 274, "query": "Why dont we define imaginary numbers for every impossibility?", "score": 325, "views": 13887, "answer_pids": [125600, 125602, 125604, 125609, 125610, 125622, 125623, 125626, 125634, 125665, 125692, 125729, 125742, 125917, 125936, 126139, 126505, 298961, 505760, 602385], "question_author": ""}
{"qid": 275, "query": "What is the importance of eigenvalues/eigenvectors?", "score": 318, "views": 195233, "answer_pids": [12797, 12800, 12802, 12821, 127297, 229771, 501629, 538160, 811569, 912540, 1258029], "question_author": "Edan Maor"}
{"qid": 276, "query": "Nice examples of groups which are not obviously groups", "score": 308, "views": 32506, "answer_pids": [171236, 171237, 171238, 171245, 171264, 171274, 171276, 171318, 171359, 171499, 171502, 171629, 171637, 171669, 171773, 171775, 171880, 172158, 225132, 282148, 326041, 460606, 460932, 460954, 462781, 469452, 500137, 657406, 831054, 840525], "question_author": ""}
{"qid": 277, "query": "Multiple-choice question about the probability of a random answer to itself being correct", "score": 306, "views": 88279, "answer_pids": [40223, 40226, 40227, 40239, 40250], "question_author": ""}
{"qid": 278, "query": "In Russian roulette, is it best to go first?", "score": 300, "views": 71587, "answer_pids": [49996, 50032, 50061, 50127, 575753], "question_author": ""}
{"qid": 279, "query": "One question to know if the number is 1, 2 or 3", "score": 294, "views": 55927, "answer_pids": [237315, 237316, 237317, 237324, 237327, 237331, 237335, 237345, 237365, 237371, 237382, 237385, 237398, 237400, 237407, 237415, 237431, 237442, 237461, 237469, 237471, 237484, 237506, 237596, 237666, 237681, 237692, 237698, 237719, 237773, 237834, 237865, 237927, 238082], "question_author": "Rob O'Callahan"}
{"qid": 280, "query": "Why does this innovative method of subtraction from a third grader always work?", "score": 292, "views": 35233, "answer_pids": [990242, 990326, 990351, 990651, 990658, 990673, 990677, 990700, 990701, 990801, 990936, 991321, 991452], "question_author": "Daniel Standage"}
{"qid": 281, "query": "Intuition for the definition of the Gamma function?", "score": 292, "views": 20924, "answer_pids": [2074, 2088, 2121, 11210, 137024, 302754, 370316, 643130], "question_author": ""}
{"qid": 282, "query": "Why can ALL quadratic equations be solved by the quadratic formula?", "score": 287, "views": 40102, "answer_pids": [26260, 26261, 26263, 26265, 26266, 26280, 26299, 26303, 101214, 236195, 292332, 302436, 327048, 373527, 413185, 463659, 646876, 655454, 656444, 717696, 908343, 1245859], "question_author": ""}
{"qid": 283, "query": "Math without pencil and paper", "score": 287, "views": 23323, "answer_pids": [705879, 705884, 705934, 705939, 706002, 706003, 706147, 706157, 706447, 706461, 706624, 706736, 706808, 707263, 707489, 1149730], "question_author": "bryn"}
{"qid": 284, "query": "V.I. Arnold says Russian students cant solve this problem, but American students can -- why?", "score": 286, "views": 43223, "answer_pids": [622624, 622637, 622642, 622709, 622733, 622953, 679829, 798662, 1333771], "question_author": ""}
{"qid": 285, "query": "Really advanced techniques of integration (definite or indefinite)", "score": 281, "views": 42647, "answer_pids": [397065, 397071, 397074, 397117, 397130, 397241, 397249, 397398, 405911, 679870, 725421, 789659, 839883, 891511, 1086604, 1252528, 1259441, 1443438, 1450986], "question_author": null}
{"qid": 286, "query": "Is mathematics one big tautology?", "score": 281, "views": 23498, "answer_pids": [181008, 181022, 181116, 181128, 181169, 181209, 181210, 181211, 181256, 181263, 181288, 181359, 181503, 181564, 182055, 187200, 188868, 437940, 560249], "question_author": "Innab"}
{"qid": 287, "query": "Help with a prime number spiral which turns 90 degrees at each prime", "score": 281, "views": 11506, "answer_pids": [789423, 789427, 789552, 789564, 789839, 789942, 790132, 790362, 790503, 790967, 792163, 792530, 804955], "question_author": "Mad Scientist"}
{"qid": 288, "query": "Is $10$ a magical number or I am missing something?", "score": 279, "views": 29198, "answer_pids": [84170, 84174, 84180, 84181, 84183, 84193, 84197, 84273, 84325, 84364, 604150], "question_author": ""}
{"qid": 289, "query": "Funny identities", "score": 278, "views": 22703, "answer_pids": [5068, 5071, 5073, 5074, 5075, 5076, 5077, 5078, 5079, 5109, 5110, 5112, 5188, 5192, 5618, 5972, 6485, 6825, 13165, 18027, 43254, 55907, 55934, 55938, 71585, 71588, 80861, 80863, 80867, 80886, 80888, 81060, 84003, 133428, 133431, 133437, 133458, 169663, 220018, 220019, 220033, 234244, 234248, 234262, 234380, 234390, 234391, 254810, 254857, 260723, 261097, 276573, 304273, 306534, 314919, 347179, 368323, 444516], "question_author": ""}
{"qid": 290, "query": "Cant argue with success? Looking for bad math that gets away with it", "score": 278, "views": 18947, "answer_pids": [126102, 126114, 126115, 126131, 126170, 126199, 126260, 126270, 126272, 126273, 126275, 126278, 126283, 126285, 126294, 126316, 126441, 126506, 126512, 128741, 128750, 142774, 188572, 216257, 251219, 301495, 305492, 334540, 394973, 394979, 856093, 884721, 916399, 917794, 1300842, 1304565, 1377053, 1397242], "question_author": "David Z"}
{"qid": 291, "query": "Is $7$ the only prime followed by a cube?", "score": 275, "views": 28667, "answer_pids": [297487, 297488, 297489, 297522, 297557], "question_author": "Daniel Standage"}
{"qid": 292, "query": "In the history of mathematics, has there ever been a mistake?", "score": 275, "views": 27316, "answer_pids": [70692, 70705, 70782, 70786, 70790, 70793, 70823, 70828, 70852, 70922, 72123, 72337, 176844, 176878, 363644, 555678, 578571, 578615, 578647, 779003, 1015874, 1268073, 1418529, 1420473], "question_author": ""}
{"qid": 293, "query": "The Mathematics of Tetris", "score": 274, "views": 32241, "answer_pids": [42091, 42092, 42093, 42356], "question_author": ""}
{"qid": 294, "query": "Given an infinite number of monkeys and an infinite amount of time, would one of them write Hamlet?", "score": 271, "views": 38186, "answer_pids": [9577, 9578, 9579, 9580, 9590, 9596, 9598, 9614, 9616, 9654, 9659, 9682, 9739], "question_author": ""}
{"qid": 295, "query": "Why do mathematicians use single-letter variables?", "score": 270, "views": 20515, "answer_pids": [13271, 13273, 13274, 13275, 13276, 13279, 13280, 13281, 13283, 13300, 13365, 16442, 45413, 45418, 51088, 51093, 55663, 58728, 67791, 67820, 192686, 192689, 951077, 985213, 985217, 1108630], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 296, "query": "How does one prove the determinant inequality $\\det\\left(6(A^3+B^3+C^3)+I_{n}\\right)\\ge 5^n\\det(A^2+B^2+C^2)$?", "score": 269, "views": 14280, "answer_pids": [793246, 964774, 1261078], "question_author": "Casebash"}
{"qid": 297, "query": "Is a matrix multiplied with its transpose something special?", "score": 266, "views": 503766, "answer_pids": [79896, 79897, 79898, 107732, 595572, 841197, 1047341, 1155554], "question_author": "e.James"}
{"qid": 298, "query": "How discontinuous can a derivative be?", "score": 260, "views": 21438, "answer_pids": [57636], "question_author": ""}
{"qid": 299, "query": "Too old to start math", "score": 258, "views": 42567, "answer_pids": [115428, 115429, 115433, 115448, 115449, 115458, 115505, 115524, 115559, 115591, 115592, 115706, 115710, 115711, 115718, 115722, 115801, 115826, 115911, 115916, 115941, 115984, 116003, 116010, 147540, 156582, 288951, 385551], "question_author": "Beofett"}
{"qid": 300, "query": "Evaluate $ \\int_{0}^{\\frac{\\pi}2}\\frac1{(1+x^2)(1+\\tan x)}\\:\\mathrm dx$", "score": 256, "views": 20297, "answer_pids": [370839, 550129, 1177926, 1179196, 1412718], "question_author": "Seamus"}
{"qid": 301, "query": "What is the maximum volume that can be contained by a sheet of paper?", "score": 254, "views": 16065, "answer_pids": [1052512, 1052545, 1052561, 1052828], "question_author": "Charles Stewart"}
{"qid": 302, "query": "Please explain the intuition behind the dual problem in optimization.", "score": 252, "views": 56922, "answer_pids": [110247, 114799, 279535, 285670, 744613, 1092869, 1228380, 1442955], "question_author": ""}
{"qid": 303, "query": "Evaluating the integral $\\int_0^\\infty \\frac{\\sin x} x \\,\\mathrm dx = \\frac \\pi 2$?", "score": 251, "views": 119865, "answer_pids": [3106, 3107, 3109, 3145, 3147, 3899, 4804, 18500, 54837, 54839, 81590, 82446, 93058, 156436, 201804, 216601, 216606, 489317, 509644, 520627, 711429, 815259, 874522, 1073340, 1074586, 1084977, 1153146, 1295236, 1455561], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 304, "query": "Norms Induced by Inner Products and the Parallelogram Law", "score": 250, "views": 64505, "answer_pids": [12004, 23343, 242654, 384612], "question_author": "Andrei"}
{"qid": 305, "query": "Fun but serious mathematics books to gift advanced undergraduates.", "score": 248, "views": 28450, "answer_pids": [127918, 127919, 127920, 127921, 127923, 127925, 127931, 127934, 127954, 127971, 127975, 127984, 127993, 128033, 128044, 128066, 128073, 128080, 128091, 128162, 128166, 128198, 128269, 128474, 128632, 128769, 128829, 128842, 128870, 128891, 129473, 129792, 129854, 130092, 134604, 208416, 361352], "question_author": "John Smith"}
{"qid": 306, "query": "Which answer in this list is the correct answer to this question?", "score": 247, "views": 128837, "answer_pids": [838880, 838884, 838891, 838906, 838910, 838967, 839219, 1237304, 1237335, 1237402, 1237604, 1237750, 1241672], "question_author": "Gergana Vandova"}
{"qid": 307, "query": "What is the practical difference between a differential and a derivative?", "score": 246, "views": 133348, "answer_pids": [13103, 13104, 153906, 834506, 1117658, 1143703], "question_author": "Tomer Vromen"}
{"qid": 308, "query": "Evaluating $\\lim\\limits_{n\\to\\infty} e^{-n} \\sum\\limits_{k=0}^{n} \\frac{n^k}{k!}$", "score": 244, "views": 25936, "answer_pids": [80958, 80961, 80996, 81120, 229466, 229485, 255898, 1104275, 1355856], "question_author": ""}
{"qid": 309, "query": "Does the square or the circle have the greater perimeter? A surprisingly hard problem for high schoolers", "score": 243, "views": 58345, "answer_pids": [482717, 482725, 482759, 482765, 482767, 482769, 482777, 482826, 482833, 482883, 482903, 482918, 483135, 483167, 483173, 483175, 483285, 484090, 485142, 485368, 485495, 485701, 514915, 943723, 1290600], "question_author": "Seamus"}
{"qid": 310, "query": "Can you answer my sons fourth-grade homework question: Which numbers are prime, have digits adding to ten and have a three in the tens place?", "score": 242, "views": 11207, "answer_pids": [34906, 34907, 35283, 181157, 239219, 943369, 1341119], "question_author": ""}
{"qid": 311, "query": "The Integral that Stumped Feynman?", "score": 240, "views": 24370, "answer_pids": [162402, 182846, 202066], "question_author": "Noldorin"}
{"qid": 312, "query": "In (relatively) simple words: What is an inverse limit?", "score": 240, "views": 22598, "answer_pids": [20539, 20540, 20541, 72807, 163768, 517933], "question_author": "Pratik Deoghare"}
{"qid": 313, "query": "What is the importance of the Collatz conjecture?", "score": 239, "views": 77925, "answer_pids": [1617, 1619, 1634, 6038, 6042, 18871, 18933], "question_author": ""}
{"qid": 314, "query": "What are some examples of when Mathematics accidentally discovered something about the world?", "score": 239, "views": 45347, "answer_pids": [696890, 696893, 696895, 696908, 696914, 696916, 696926, 696932, 696938, 697016, 697022, 697119, 697484, 697647, 697694, 697705, 697712, 698156, 699234, 700165, 741538, 744370, 788264, 833068, 960107, 1084005], "question_author": ""}
{"qid": 315, "query": "Derivative of sigmoid function $\\sigma (x) = \\frac{1}{1+e^{-x}}$", "score": 237, "views": 269467, "answer_pids": [41276, 41277, 494811, 861045, 918878, 1152103], "question_author": "gkadam"}
{"qid": 316, "query": "Conjectures that have been disproved with extremely large counterexamples?", "score": 237, "views": 41927, "answer_pids": [319, 320, 323, 325, 339, 667, 693, 1085, 1148, 89246, 160312, 172704, 724128, 986637, 1043927, 1191508, 1200200, 1244234, 1306626], "question_author": "Larry Wang"}
{"qid": 317, "query": "Best Fake Proofs? (A M.SE April Fools Day collection)", "score": 236, "views": 55811, "answer_pids": [164970, 164972, 164973, 164974, 164984, 164988, 164990, 165004, 165006, 165009, 165012, 165013, 165016, 165017, 165034, 165041, 165047, 165061, 165066, 165112, 165240, 165266, 165320, 165321, 165352, 165356, 165401], "question_author": "Sklivvz"}
{"qid": 318, "query": "Any open subset of $\\Bbb R$ is a countable union of disjoint open intervals", "score": 234, "views": 90033, "answer_pids": [151736, 151751, 151817, 152849, 154547, 186206, 225915, 375641, 469921, 473886, 554970, 571653, 748547, 751540, 868269, 936638], "question_author": ""}
{"qid": 319, "query": "Why study Algebraic Geometry?", "score": 234, "views": 59657, "answer_pids": [124626, 124661, 443878], "question_author": "Gabriel Fair"}
{"qid": 320, "query": "Why are rings called rings?", "score": 233, "views": 26575, "answer_pids": [32605], "question_author": "Gabriel Fair"}
{"qid": 321, "query": "What is the result of $\\infty - \\infty$?", "score": 232, "views": 176683, "answer_pids": [32240, 32246, 32247, 32248, 32254, 32261, 32282, 50986, 62166, 234088, 878006], "question_author": "Jonathan."}
{"qid": 322, "query": "Integral Milking", "score": 232, "views": 18938, "answer_pids": [1040744, 1040745, 1040756, 1040823, 1040845, 1041014, 1041029, 1042935, 1053171, 1081230], "question_author": ""}
{"qid": 323, "query": "A 1,400 years old approximation to the sine function by Mahabhaskariya of Bhaskara I", "score": 230, "views": 11635, "answer_pids": [408582, 408651, 408688], "question_author": ""}
{"qid": 324, "query": "How to read a book in mathematics?", "score": 229, "views": 54894, "answer_pids": [134120, 199701, 200222, 200225, 200257, 202246, 235349, 663615, 1129094, 1307448], "question_author": "Gordon Gustafson"}
{"qid": 325, "query": "Why do we care about dual spaces?", "score": 229, "views": 42659, "answer_pids": [2234, 2236, 2237, 2305, 662495, 767144, 1052431], "question_author": ""}
{"qid": 326, "query": "What are some counter-intuitive results in mathematics that involve only finite objects?", "score": 227, "views": 27193, "answer_pids": [778675, 778683, 778708, 778723, 778759, 778775, 778780, 778799, 778810, 778829, 778972, 779083, 779103, 779145, 779192, 779206, 779350, 779394, 779472, 779699, 779877, 779988, 780023, 780049, 780275, 780783, 780787, 781107, 781199, 781837, 783300, 783441, 933083, 1143596], "question_author": "Snowman"}
{"qid": 327, "query": "How many fours are needed to represent numbers up to $N$?", "score": 226, "views": 10026, "answer_pids": [149989, 244845, 992011], "question_author": ""}
{"qid": 328, "query": "Is there an elementary proof that $\\sum \\limits_{k=1}^n \\frac1k$ is never an integer?", "score": 225, "views": 37810, "answer_pids": [1638, 1642, 1643, 1650, 82324, 210967, 491782, 548569, 584093, 933256], "question_author": "John Smith"}
{"qid": 329, "query": "Your favourite application of the Baire Category Theorem", "score": 225, "views": 30171, "answer_pids": [83601, 83602, 83606, 83610, 83614, 83617, 83631, 83634, 83719, 83721, 83759, 83764, 86721, 164584, 170073, 194350, 194367, 412384, 416047, 847768, 956753, 1114241, 1116306, 1123396], "question_author": "Maxim V. Pavlov"}
{"qid": 330, "query": "What books must every math undergraduate read?", "score": 224, "views": 66731, "answer_pids": [49253, 49256, 49258, 144615, 1402159], "question_author": ""}
{"qid": 331, "query": "Proving you *cant* make $2011$ out of $1,2,3,4$: nice twist on the usual", "score": 223, "views": 16928, "answer_pids": [48076, 48402], "question_author": "Snowman"}
{"qid": 332, "query": "Does the open mapping theorem imply the Baire category theorem?", "score": 221, "views": 8813, "answer_pids": [564197], "question_author": ""}
{"qid": 333, "query": "Optimizing response times of an ambulance corp: short-term versus average", "score": 220, "views": 9459, "answer_pids": [113327, 113343, 113353, 113354, 113405, 113424, 113425, 113444, 113457, 113508, 113632, 113678], "question_author": ""}
{"qid": 334, "query": "Is there a 0-1 law for the theory of groups?", "score": 220, "views": 7576, "answer_pids": [739064], "question_author": ""}
{"qid": 335, "query": "How can a piece of A4 paper be folded in exactly three equal parts?", "score": 219, "views": 52352, "answer_pids": [321477, 321624, 322148, 322187, 322255, 322745, 323742, 323872, 328102, 509572, 509594, 864263, 1359085], "question_author": "Jonathan."}
{"qid": 336, "query": "Eigenvectors of real symmetric matrices are orthogonal", "score": 217, "views": 273141, "answer_pids": [43169, 43170, 190825, 356708, 1133796, 1185876], "question_author": "e.James"}
{"qid": 337, "query": "Do men or women have more brothers?", "score": 217, "views": 36573, "answer_pids": [691569, 691685, 691713, 691724, 691749, 691751, 691813, 691857, 691870, 691892, 692301, 692463, 692478, 692548, 693703], "question_author": "tim_hutton"}
{"qid": 338, "query": "In simple English, what does it mean to be transcendental?", "score": 216, "views": 19657, "answer_pids": [654645, 654646, 654648, 654651, 654655, 654660, 654687, 654873, 654878, 654879], "question_author": ""}
{"qid": 339, "query": "How to check if a point is inside a rectangle?", "score": 214, "views": 178280, "answer_pids": [95818, 95819, 95822, 95840, 95856, 95877, 95882, 95892, 95903, 95907, 95955, 95959, 95965, 95975, 95991, 96025, 96067, 163791, 554262, 621209, 654369], "question_author": "Gordon Gustafson"}
{"qid": 340, "query": "Good books and lecture notes about category theory.", "score": 213, "views": 57528, "answer_pids": [233, 236, 239, 260, 391, 605, 642, 681, 912, 928, 930, 931, 1783, 3430, 8931, 12002, 129938, 388459, 392426, 413525, 413529, 466034, 626033, 940789, 1200277], "question_author": "Chris Wenham"}
{"qid": 341, "query": "Why does this matrix give the derivative of a function?", "score": 212, "views": 14074, "answer_pids": [603956, 603959, 603983, 604102, 604163, 605388], "question_author": "Snowman"}
{"qid": 342, "query": "Is it faster to count to the infinite going one by one or two by two?", "score": 210, "views": 36567, "answer_pids": [293971, 293973, 294000, 294001, 294029, 294055, 294057, 294065, 294066, 294093, 294103, 294107, 294111, 294119, 294329, 294391, 294505, 294728, 294750, 294970, 295575, 295582], "question_author": ""}
{"qid": 343, "query": "What are the Differences Between a Matrix and a Tensor?", "score": 209, "views": 113854, "answer_pids": [193525, 193526, 193537, 193578, 193613, 836730, 839664, 948178], "question_author": ""}
{"qid": 344, "query": "How do I convince someone that $1+1=2$ may not necessarily be true?", "score": 209, "views": 32122, "answer_pids": [118195, 118196, 118199, 119804], "question_author": "Poshpaws"}
{"qid": 345, "query": "How do I sell out with abstract algebra?", "score": 209, "views": 31572, "answer_pids": [172623, 172639, 173643, 173646, 174806, 174876, 175229, 175404, 186489], "question_author": "Rory M"}
{"qid": 346, "query": "Why can a Venn diagram for $4+$ sets not be constructed using circles?", "score": 208, "views": 72143, "answer_pids": [929, 55233, 354121, 943574], "question_author": ""}
{"qid": 347, "query": "Identification of a quadrilateral as a trapezoid, rectangle, or square", "score": 208, "views": 16164, "answer_pids": [516800, 516802, 516818, 516852, 516859, 517091, 517107, 517136, 517280, 517329, 517358, 517498, 517565, 823773], "question_author": null}
{"qid": 348, "query": "Meaning of Rays in Polar Plot of Prime Numbers", "score": 207, "views": 20348, "answer_pids": [376408, 1231843], "question_author": ""}
{"qid": 349, "query": "What does $2^x$ really mean when $x$ is not an integer?", "score": 206, "views": 10268, "answer_pids": [67511, 67513, 67514, 67528, 67576, 67743, 67753, 67754, 197594], "question_author": ""}
{"qid": 350, "query": "What do modern-day analysts actually do?", "score": 205, "views": 19335, "answer_pids": [49149, 49549, 50962, 78220, 230640], "question_author": "nkassis"}
{"qid": 351, "query": "Proving $\\int_{0}^{\\infty} \\mathrm{e}^{-x^2} dx = \\frac{\\sqrt \\pi}{2}$", "score": 203, "views": 107144, "answer_pids": [5313, 5318, 5321, 6339, 6349, 62797, 65236, 74358, 84028, 117504, 117515, 183851, 244316, 507779, 554793, 645922, 679638, 700358, 1284031], "question_author": ""}
{"qid": 352, "query": "Is computer science a branch of mathematics?", "score": 203, "views": 57759, "answer_pids": [289247, 289250, 289251, 289286, 289302, 289314, 289326, 289359, 289370, 289373, 289506, 289507, 290790], "question_author": ""}
{"qid": 353, "query": "What is the difference between linear and affine function", "score": 200, "views": 139130, "answer_pids": [132818, 332587, 842839, 853681], "question_author": null}
{"qid": 354, "query": "Which one result in mathematics has surprised you the most?", "score": 198, "views": 55099, "answer_pids": [1752, 1753, 1754, 1755, 1761, 1762, 1768, 1771, 1772, 1773, 1778, 1784, 1787, 1790, 1799, 1800, 1832, 1843, 1844, 1869, 1870, 2279, 2280, 2282, 2283, 2291, 2318, 3677, 3685, 3698, 3703, 3707, 3709, 5185, 5189, 5193, 5195, 5203, 6197, 6201, 6208, 6213, 6226, 6229, 6244, 6285, 6311, 6326, 8605, 11403, 12073, 12080, 12087, 12105, 12114, 12115, 13406, 13420, 14687, 14688, 14692, 14819, 21434, 21469, 21470, 21471, 21473, 21475, 21476, 21478, 21490, 21492, 21496, 21521, 21522, 21523, 21525, 21528, 21529, 21531, 21544, 21545, 21546, 21548, 21564, 21565, 23677, 23687, 23745, 32906], "question_author": ""}
{"qid": 355, "query": "Books on Number Theory for Layman", "score": 197, "views": 104900, "answer_pids": [200, 205, 209, 211, 222, 301, 606, 810, 3253, 3259, 8918, 8920, 8921, 9212, 18852, 18864, 25782, 25799, 96402, 96406, 98025, 98032, 98034, 141337, 196232, 270992, 274000, 338938, 349307, 926673, 1456463], "question_author": ""}
{"qid": 356, "query": "Teaching myself differential topology and differential geometry", "score": 196, "views": 63843, "answer_pids": [7639, 7642, 7653, 7662, 7664, 7669, 12415, 125298, 415234, 1292256], "question_author": "Pratik Deoghare"}
{"qid": 357, "query": "List of interesting math videos / documentaries", "score": 196, "views": 39247, "answer_pids": [10474, 10476, 10479, 10481, 10485, 10492, 10494, 10507, 10510, 10514, 54814, 54817, 54819, 75091, 75380, 107481, 114346, 115626, 128594, 152295, 153549, 182050, 262049, 351444, 368100, 370827, 729851, 839946, 1355177], "question_author": "Justin L."}
{"qid": 358, "query": "Exterior Derivative vs. Covariant Derivative vs. Lie Derivative", "score": 196, "views": 24660, "answer_pids": [104635, 104822, 127410, 127942, 128159, 498750, 669791, 763301, 831251, 884334], "question_author": ""}
{"qid": 359, "query": "How to show $e^{e^{e^{79}}}$ is not an integer", "score": 196, "views": 13232, "answer_pids": [24271, 24323], "question_author": ""}
{"qid": 360, "query": "Are $14$ and $21$ the only interesting numbers?", "score": 194, "views": 22827, "answer_pids": [764008], "question_author": ""}
{"qid": 361, "query": "How could we define the factorial of a matrix?", "score": 193, "views": 12720, "answer_pids": [636945, 636963, 637298, 638449, 638621, 758810], "question_author": "vonjd"}
{"qid": 362, "query": "When can you switch the order of limits?", "score": 191, "views": 60017, "answer_pids": [8526, 8546, 8547, 24145, 916111, 943900], "question_author": "Jamie Banks"}
{"qid": 363, "query": "List of Interesting Math Blogs", "score": 190, "views": 70067, "answer_pids": [38, 50, 51, 172, 175, 216, 303, 307, 317, 514, 892, 18637, 18638, 18644, 18656, 46022, 46030, 46045, 46051, 138698, 175455, 207168], "question_author": "Rory M"}
{"qid": 364, "query": "Generalizing $\\int_{0}^{1} \\frac{\\arctan\\sqrt{x^{2} + 2}}{\\sqrt{x^{2} + 2}} \\, \\frac{\\operatorname dx}{x^{2}+1} = \\frac{5\\pi^{2}}{96}$", "score": 190, "views": 7087, "answer_pids": [262133], "question_author": "Vebjorn Ljosa"}
{"qid": 365, "query": "How do we prove that something is unprovable?", "score": 188, "views": 21246, "answer_pids": [774181, 774188, 774189, 774204, 774209, 774232, 775695, 912670], "question_author": "Sophie Alpert"}
{"qid": 366, "query": "How many sides does a circle have?", "score": 187, "views": 450795, "answer_pids": [17147, 21460, 21462, 37761, 37792, 213072, 505181, 572086, 572087, 1307880], "question_author": ""}
{"qid": 367, "query": "Whats the intuition behind Pythagoras theorem?", "score": 187, "views": 31167, "answer_pids": [299115, 299204, 299229, 299233, 299306, 299563, 299587, 299746, 299967, 301088, 301314, 320745, 339224, 1055601, 1305014], "question_author": ""}
{"qid": 368, "query": "Why is compactness so important?", "score": 187, "views": 21167, "answer_pids": [226333, 226334, 226335, 226336, 226340, 226347, 226354, 226360, 226378, 226638, 226914, 1135076], "question_author": "John Gietzen"}
{"qid": 369, "query": "Can a coin with an unknown bias be treated as fair?", "score": 187, "views": 16694, "answer_pids": [321590, 321591, 321592, 321680, 321697, 321699, 321713, 321740, 321800, 321855, 321997, 322003, 322043, 322169, 322170, 322227, 322545, 322551, 322572, 322581, 322670, 324014, 324313, 378289], "question_author": ""}
{"qid": 370, "query": "A Topology such that the continuous functions are exactly the polynomials", "score": 187, "views": 6525, "answer_pids": [205910, 207310, 1320342], "question_author": "Albert"}
{"qid": 371, "query": "What is the geometric interpretation of the transpose?", "score": 185, "views": 41013, "answer_pids": [19983, 19997, 1232581, 1256257, 1281367], "question_author": "mring"}
{"qid": 372, "query": "Counterintuitive examples in probability", "score": 185, "views": 34006, "answer_pids": [813345, 813348, 813365, 813367, 813374, 813375, 813397, 813527, 813568, 813592, 813609, 813679, 813681, 813745, 813827, 813844, 813862, 813931, 813975, 814256, 814262, 815419, 815494, 815550, 815724, 815736, 815740, 815750, 815827, 816364, 1267021], "question_author": ""}
{"qid": 373, "query": "What is a good complex analysis textbook, barring Ahlforss?", "score": 184, "views": 114952, "answer_pids": [16622, 16624, 16626, 16627, 16633, 16665, 16668, 16680, 22067, 22084, 22087, 22095, 22112, 22133, 22134, 38526, 80880, 80900, 80912, 126235, 301784, 301790, 551937, 644251, 813968, 1298063], "question_author": "Gordon Gustafson"}
{"qid": 374, "query": "What were some major mathematical breakthroughs in 2016?", "score": 184, "views": 14824, "answer_pids": [791056, 791065, 791214, 791233, 791596, 792065], "question_author": ""}
{"qid": 375, "query": "How to intuitively understand eigenvalue and eigenvector?", "score": 183, "views": 128315, "answer_pids": [118415, 118423, 118431, 194436, 365334, 770204, 1136234], "question_author": "bobtheowl2"}
{"qid": 376, "query": "Importance of Representation Theory", "score": 183, "views": 18857, "answer_pids": [377, 384, 1603, 1605, 1607, 1660, 1664, 1668, 1242618], "question_author": ""}
{"qid": 377, "query": "Advice to young mathematicians", "score": 183, "views": 13700, "answer_pids": [431107, 438859, 439033, 441322, 460689, 782637, 902542, 1184284, 1402737], "question_author": ""}
{"qid": 378, "query": "Why cant differentiability be generalized as nicely as continuity?", "score": 183, "views": 7268, "answer_pids": [508832, 508872, 508910, 509327, 511209, 1390784], "question_author": "David"}
{"qid": 379, "query": "How to define a bijection between $(0,1)$ and $(0,1]$?", "score": 182, "views": 50643, "answer_pids": [81186, 81187, 81191, 81196, 252922, 731061, 1074788, 1150160], "question_author": "nibot"}
{"qid": 380, "query": "Derivative of Softmax loss function", "score": 180, "views": 171689, "answer_pids": [398329], "question_author": ""}
{"qid": 381, "query": "Some users are mind bogglingly skilled at integration. How did they get there?", "score": 180, "views": 20718, "answer_pids": [333565, 438439, 744580, 781758], "question_author": "rdhs"}
{"qid": 382, "query": "How do people perform mental arithmetic for complicated expressions?", "score": 180, "views": 14482, "answer_pids": [553766, 553768, 553800, 553820, 553849, 553858, 554106, 554199, 554271, 554278, 554322, 554437, 554769, 554827, 555887, 556590, 557479, 847144], "question_author": ""}
{"qid": 383, "query": "Can we ascertain that there exists an epimorphism $G\\rightarrow H$?", "score": 180, "views": 5761, "answer_pids": [222748], "question_author": "Cedric H."}
{"qid": 384, "query": "What Does it Really Mean to Have Different Kinds of Infinities?", "score": 179, "views": 10461, "answer_pids": [2, 3, 4, 12, 21, 30, 318, 1484, 346732], "question_author": "nibot"}
{"qid": 385, "query": "How do you revise material that you already half-know, without getting bored and demotivated?", "score": 177, "views": 11220, "answer_pids": [896144, 896157, 896163, 896183, 896201, 896219, 896228, 896346, 896558, 896974, 897064, 897125, 897394, 1033648], "question_author": "magnetar"}
{"qid": 386, "query": "Are there any series whose convergence is unknown?", "score": 176, "views": 28694, "answer_pids": [11396, 26129, 302946, 897346, 1200422, 1269359, 1446924, 1451821], "question_author": "nibot"}
{"qid": 387, "query": "When can a sum and integral be interchanged?", "score": 175, "views": 87136, "answer_pids": [43767, 43775, 1117680], "question_author": "Gerard"}
{"qid": 388, "query": "A math contest problem $\\int_0^1\\ln\\left(1+\\frac{\\ln^2x}{4\\,\\pi^2}\\right)\\frac{\\ln(1-x)}x \\ \\mathrm dx$", "score": 175, "views": 10147, "answer_pids": [241056, 249001, 268576, 419708], "question_author": "harpalss"}
{"qid": 389, "query": "Symmetry of function defined by integral", "score": 174, "views": 8321, "answer_pids": [130247, 130986, 250688, 300150, 383059, 1246453], "question_author": "Gerard"}
{"qid": 390, "query": "Why $\\sqrt{-1 \\cdot {-1}} \\neq \\sqrt{-1}^2$?", "score": 173, "views": 14910, "answer_pids": [274, 300, 214332, 231632, 270120, 287912, 561074, 625484, 981388], "question_author": ""}
{"qid": 391, "query": "Self-Contained Proof that $\\sum\\limits_{n=1}^{\\infty} \\frac1{n^p}$ Converges for $p > 1$", "score": 172, "views": 24641, "answer_pids": [15988, 15989, 15993, 15996, 40900, 53140, 53564, 66449, 1065202], "question_author": "nico"}
{"qid": 392, "query": "There are apparently $3072$ ways to draw this flower. But why?", "score": 172, "views": 18514, "answer_pids": [671831, 671833, 671835, 672268, 672962, 672969, 673242, 673471, 673747], "question_author": "Justin L."}
{"qid": 393, "query": "Proving the identity $\\sum_{k=1}^n {k^3} = \\big(\\sum_{k=1}^n k\\big)^2$ without induction", "score": 172, "views": 15681, "answer_pids": [10316, 32590, 32591, 32592, 32598, 32602, 32606, 32612, 32616, 32630, 32738, 82650, 82655, 82656, 82669, 175440, 445595, 487054, 491714, 616977, 996589, 1004833, 1063275, 1203677, 1203724, 1254136], "question_author": "Fer"}
{"qid": 394, "query": "Proof that the trace of a matrix is the sum of its eigenvalues", "score": 171, "views": 167293, "answer_pids": [249903, 249904, 249908, 249945, 871436, 1095433], "question_author": ""}
{"qid": 395, "query": "Limit of $L^p$ norm", "score": 171, "views": 60796, "answer_pids": [118067, 331942], "question_author": ""}
{"qid": 396, "query": "Is there a categorical definition of submetry?", "score": 171, "views": 3069, "answer_pids": [1263476], "question_author": ""}
{"qid": 397, "query": "Deleting any digit yields a prime... is there a name for this?", "score": 170, "views": 27708, "answer_pids": [17820, 17826, 17847, 17864, 17897, 37347], "question_author": ""}
{"qid": 398, "query": "Inverse of the sum of matrices", "score": 169, "views": 279498, "answer_pids": [9905, 9906, 9907, 9908, 39648, 438443, 439827, 449871, 518962, 1169389, 1217659, 1251658], "question_author": "Justin L."}
{"qid": 399, "query": "What is $dx$ in integration?", "score": 168, "views": 72601, "answer_pids": [100336, 100337, 100339, 100340, 100341, 100342, 100356, 100425, 309070, 348069, 383891, 1055115], "question_author": "Larian LeQuella"}
{"qid": 400, "query": "Is $2048$ the highest power of $2$ with all even digits (base ten)?", "score": 168, "views": 16237, "answer_pids": [59554, 60547, 60764], "question_author": ""}
{"qid": 401, "query": "Why is Eulers Gamma function the best extension of the factorial function to the reals?", "score": 168, "views": 15104, "answer_pids": [947, 950, 953, 985, 988, 67724, 101335], "question_author": "Gabriel Fair"}
{"qid": 402, "query": "$L^p$ and $L^q$ space inclusion", "score": 167, "views": 92326, "answer_pids": [34930, 34934, 738894], "question_author": ""}
{"qid": 403, "query": "Do we have negative prime numbers?", "score": 167, "views": 54692, "answer_pids": [417513, 417515, 417520, 487356, 642275, 705031, 886912], "question_author": ""}
{"qid": 404, "query": "Why do people use it is easy to prove?", "score": 167, "views": 14562, "answer_pids": [28926, 28928, 28929, 29056, 39090, 39127, 64426, 298300], "question_author": ""}
{"qid": 405, "query": "Taking Seats on a Plane", "score": 166, "views": 89795, "answer_pids": [3306, 3438, 6246, 18008, 29553, 29784, 266529, 635074, 693869, 905005, 983117, 1002606, 1083515, 1212277, 1272450, 1320528, 1403189, 1456886], "question_author": "Casebash"}
{"qid": 406, "query": "Why is $1^{\\infty}$ considered to be an indeterminate form", "score": 166, "views": 26289, "answer_pids": [5975, 5977, 5985, 490427, 547375, 589460, 589465, 745572, 1451362], "question_author": "Larian LeQuella"}
{"qid": 407, "query": "Mathematical ideas that took long to define rigorously", "score": 166, "views": 19663, "answer_pids": [978823, 978833, 978837, 978838, 978862, 978870, 978925, 978966, 978972, 979067, 979118, 979285, 979389, 979784, 980790, 981686, 982118, 982874, 985685], "question_author": ""}
{"qid": 408, "query": "Open problems in General Relativity", "score": 166, "views": 16080, "answer_pids": [26930, 26937], "question_author": "Dom"}
{"qid": 409, "query": "How far can one get in analysis without leaving $\\mathbb{Q}$?", "score": 166, "views": 9158, "answer_pids": [182049, 182061, 182070, 182100, 182145, 193827, 194092, 562844], "question_author": "Dom"}
{"qid": 410, "query": "How can we sum up $\\sin$ and $\\cos$ series when the angles are in arithmetic progression?", "score": 165, "views": 72543, "answer_pids": [10008, 10010, 491320, 627306, 951800, 1373293], "question_author": null}
{"qid": 411, "query": "Rational roots of polynomials", "score": 165, "views": 4457, "answer_pids": [310215], "question_author": "Larry Wang"}
{"qid": 412, "query": "Lesser-known integration tricks", "score": 164, "views": 38227, "answer_pids": [37427, 37436, 37539, 63508, 196543, 304487, 304917, 1071686, 1261185], "question_author": "camurgo"}
{"qid": 413, "query": "A variation of Fermats little theorem in the form $a^{n-d}\\equiv a$ (mod $p$).", "score": 164, "views": 8253, "answer_pids": [533022], "question_author": ""}
{"qid": 414, "query": "Is there a homology theory that counts connected components of a space?", "score": 164, "views": 5280, "answer_pids": [994700], "question_author": "crskhr"}
{"qid": 415, "query": "Prove that $\\gcd(a^n - 1, a^m - 1) = a^{\\gcd(n, m)} - 1$", "score": 163, "views": 37584, "answer_pids": [4346, 4390, 515733, 578592, 589048, 948758, 998240], "question_author": "Justin L."}
{"qid": 416, "query": "An Introduction to Tensors", "score": 163, "views": 28266, "answer_pids": [5857, 5860, 5867, 5906, 5913, 1452422], "question_author": ""}
{"qid": 417, "query": "Transpose of inverse vs inverse of transpose", "score": 162, "views": 244965, "answer_pids": [161825, 291480, 790951, 795881, 879691, 905547], "question_author": "Rory M"}
{"qid": 418, "query": "How to prove Eulers formula: $e^{i\\varphi}=\\cos(\\varphi) +i\\sin(\\varphi)$?", "score": 162, "views": 142963, "answer_pids": [2093, 2094, 2095, 2141, 2149, 4954, 6663, 21089, 124878, 279748, 563623, 811341, 813790, 813795, 827059, 934384, 1020312], "question_author": "Chao Xu"}
{"qid": 419, "query": "Intuition behind Matrix Multiplication", "score": 162, "views": 59013, "answer_pids": [17111, 17113, 17115, 17122, 17138, 17140, 17154, 17163, 205232, 401951, 587266, 857995, 1115703], "question_author": "Uri"}
{"qid": 420, "query": "Striking applications of integration by parts", "score": 162, "views": 26427, "answer_pids": [11291, 11300, 11301, 11305, 11309, 11311, 11323, 11331, 11333, 11360, 11368, 13585, 18719, 18738, 22747, 37743, 37776, 248076, 551640, 1384396], "question_author": "Thomas O"}
{"qid": 421, "query": "Why is the Penrose triangle impossible?", "score": 162, "views": 7827, "answer_pids": [1325883, 1325886, 1326105, 1326144, 1326647, 1328758], "question_author": null}
{"qid": 422, "query": "Software for drawing geometry diagrams", "score": 161, "views": 174589, "answer_pids": [1221, 1223, 1224, 1226, 1227, 1229, 1232, 1235, 1242, 1243, 1248, 1251, 1264, 9997, 10005, 79482, 124800, 147505, 362093, 580915, 674789, 674883, 1044534, 1382657], "question_author": ""}
{"qid": 423, "query": "How to prove: if $a,b \\in \\mathbb N$, then $a^{1/b}$ is an integer or an irrational number?", "score": 161, "views": 39041, "answer_pids": [2654, 2655, 2656, 2661, 2667, 2669, 2680, 2725, 2729, 536574, 536578, 1462827], "question_author": "Justin L."}
{"qid": 424, "query": "Examples of mathematical discoveries which were kept as a secret", "score": 161, "views": 23121, "answer_pids": [418067, 418069, 418072, 418080, 418245, 418640, 418787, 418926, 418955, 419247, 419796, 419887, 420115, 420239, 420257, 420475, 421199, 421554, 422106, 422119, 422335, 422339], "question_author": "Andrey"}
{"qid": 425, "query": "What properties of busy beaver numbers are computable?", "score": 161, "views": 6743, "answer_pids": [844544], "question_author": "endolith"}
{"qid": 426, "query": "What happens when we (incorrectly) make improper fractions proper again?", "score": 161, "views": 5069, "answer_pids": [1325237, 1325394, 1325540, 1325620], "question_author": "Peter Mortensen"}
{"qid": 427, "query": "Can you provide me historical examples of pure mathematics becoming useful?", "score": 160, "views": 28113, "answer_pids": [134793, 134796, 134799, 134824, 134825, 134826, 134835, 134850, 134854, 134857, 134859, 134873, 134892, 134894, 134900, 134923, 134936, 134943, 135010, 135018, 135055, 135141, 135167, 135169, 135182, 135190, 135447, 135502, 135520, 135620, 148352, 211730, 216883], "question_author": "dF_"}
{"qid": 428, "query": "What are the most overpowered theorems in mathematics?", "score": 160, "views": 26800, "answer_pids": [253388, 253389, 253391, 253393, 253406, 253409, 253413, 253420, 253525, 253535, 253536, 253537, 253540, 253599, 253639, 253923, 254364, 254373, 254375, 254652, 254665, 256361, 257082, 298416, 298423, 374763, 374789, 374792, 375117, 375211, 470598, 589024, 589036], "question_author": ""}
{"qid": 429, "query": "Do we know if there exist true mathematical statements that can not be proven?", "score": 160, "views": 22726, "answer_pids": [279783, 279792, 279794, 279854, 279876, 279992, 280112, 280277, 282209, 323605], "question_author": "Saeid Kadkhodaei"}
{"qid": 430, "query": "Is there any integral for the Golden Ratio?", "score": 160, "views": 14770, "answer_pids": [643830, 643832, 643834, 643895, 643925, 644004, 644054, 644074, 644135, 644382, 646941, 667770, 671619, 672148, 678035, 678038, 680804, 680923, 681013, 681029, 681168, 681253, 682470, 683688, 684079, 696709, 702640, 806425, 806447, 819116, 838177, 871869, 1011303, 1091631, 1133762, 1134366, 1291874, 1297192, 1384938], "question_author": ""}
{"qid": 431, "query": "Using proof by contradiction vs proof of the contrapositive", "score": 158, "views": 81534, "answer_pids": [127136, 127169, 310128, 631662, 854269], "question_author": ""}
{"qid": 432, "query": "Evaluate $\\int_0^1 \\frac{\\log \\left( 1+x^{2+\\sqrt{3}}\\right)}{1+x}\\mathrm dx$", "score": 158, "views": 33576, "answer_pids": [200549, 200902, 283566], "question_author": "Mad Scientist"}
{"qid": 433, "query": "Induction on Real Numbers", "score": 158, "views": 24463, "answer_pids": [2490, 2504, 2516, 2630, 26708, 35584], "question_author": "nibot"}
{"qid": 434, "query": "Proof that ${\\left(\\pi^\\pi\\right)}^{\\pi^\\pi}$ (and now $\\pi^{\\left(\\pi^{\\pi^\\pi}\\right)}$) is a noninteger.", "score": 158, "views": 21599, "answer_pids": [187896, 1183631], "question_author": "bdonlan"}
{"qid": 435, "query": "The Hole in One Pizza", "score": 158, "views": 8003, "answer_pids": [918908, 919517, 920282], "question_author": ""}
{"qid": 436, "query": "Proof of $\\frac{1}{e^{\\pi}+1}+\\frac{3}{e^{3\\pi}+1}+\\frac{5}{e^{5\\pi}+1}+\\ldots=\\frac{1}{24}$", "score": 158, "views": 6758, "answer_pids": [182945, 183293, 183687, 220466, 657815, 1202579], "question_author": ""}
{"qid": 437, "query": "What functions can be made continuous by mixing up their domain?", "score": 158, "views": 2618, "answer_pids": [1223113], "question_author": "nibot"}
{"qid": 438, "query": "Is there a general formula for solving 4th degree equations (quartic)?", "score": 157, "views": 307497, "answer_pids": [467, 468, 487, 30592, 30626, 30694, 430283, 464684, 493045, 784667, 931490, 1221205], "question_author": "Wilhelm"}
{"qid": 439, "query": "Discontinuous derivative.", "score": 157, "views": 86674, "answer_pids": [140193, 198415], "question_author": "The Digital Gabeg"}
{"qid": 440, "query": "Monty hall problem extended.", "score": 157, "views": 54331, "answer_pids": [272877, 272879, 272883, 272993, 273023, 273099, 273133, 273178, 273209, 273280, 273896, 273897, 273996, 273999, 547581], "question_author": "Chao Xu"}
{"qid": 441, "query": "The sum of an uncountable number of positive numbers", "score": 157, "views": 32437, "answer_pids": [11448, 49458, 206766, 520046, 1064088], "question_author": ""}
{"qid": 442, "query": "Is Apple ipad / tablet good for mathematics students?", "score": 157, "views": 29830, "answer_pids": [97302, 102252, 102259, 102312, 104988, 123595, 124575], "question_author": "Claudiu"}
{"qid": 443, "query": "Stopping the Will I need this for the test question", "score": 157, "views": 14774, "answer_pids": [282324, 282334, 282344, 282347, 282406, 282407, 282431, 282432, 282465, 282467, 282479, 282480, 282483, 282484, 282489, 282519, 282523, 282527, 282576, 282629, 282634, 282668, 282673, 282695, 282727, 282790, 282836], "question_author": "Justin L."}
{"qid": 444, "query": "Are there any open mathematical puzzles?", "score": 157, "views": 11197, "answer_pids": [244748, 244749, 244752, 244754, 244768, 244769, 244770, 244774, 244776, 244850, 244971, 244990, 245266, 245621, 245662, 245721, 246068, 246253, 534355], "question_author": ""}
{"qid": 445, "query": "Find a real function $f:\\mathbb{R}\\to\\mathbb{R}$ such that $f(f(x)) = -x$?", "score": 156, "views": 21606, "answer_pids": [149001, 149016, 149024, 149028, 149181, 149526, 1258653], "question_author": "harpalss"}
{"qid": 446, "query": "How to distinguish between walking on a sphere and walking on a torus?", "score": 156, "views": 18607, "answer_pids": [364440, 364442, 364483, 364499, 364557, 364576, 364593, 364597, 364668, 364877, 364880, 364990, 365158, 365769, 366922, 367009, 782372, 782383, 846889], "question_author": ""}
{"qid": 447, "query": "Is there a quick proof as to why the vector space of $\\mathbb{R}$ over $\\mathbb{Q}$ is infinite-dimensional?", "score": 155, "views": 43107, "answer_pids": [3674, 3675, 3811, 35794, 871830, 1264269, 1266163, 1420738], "question_author": "kmm"}
{"qid": 448, "query": "Is there another simpler method to solve this elementary school math problem?", "score": 155, "views": 37491, "answer_pids": [222997, 223004, 223017, 223022, 223057, 223115, 223121, 223127, 223208, 223245, 223246, 223255, 223280, 223323, 223357, 223385, 223446, 236059, 1184856], "question_author": "Peter Smit"}
{"qid": 449, "query": "Most ambiguous and inconsistent phrases and notations in maths", "score": 155, "views": 15647, "answer_pids": [425041, 425162, 425300, 425363, 425371, 425383, 425403, 425830, 426106, 426458, 426493, 426848, 426876, 427087, 427217, 427784, 428236, 435140, 491357, 965977], "question_author": ""}
{"qid": 450, "query": "Why does factoring eliminate a hole in the limit?", "score": 155, "views": 12513, "answer_pids": [215857, 215858, 215860, 215908, 215936, 216057, 216072, 216261, 216418, 216753, 449282, 702316, 862581], "question_author": "Joe Clarke"}
{"qid": 451, "query": "Whats the point in being a skeptical learner", "score": 155, "views": 11207, "answer_pids": [644651, 644652, 644670, 644815, 644870, 644958, 645033, 645061], "question_author": ""}
{"qid": 452, "query": "Good book for self study of a First Course in Real Analysis", "score": 154, "views": 156691, "answer_pids": [32978, 32981, 32982, 33030, 33036, 112185, 115422, 115424, 115426, 284123, 284131, 296824, 296832, 346423, 435444, 435449, 441465, 622941, 623020, 643694, 800916, 867350, 894006, 1020468, 1186062, 1186070, 1295126], "question_author": "Gordon Gustafson"}
{"qid": 453, "query": "Overview of basic results about images and preimages", "score": 154, "views": 51800, "answer_pids": [170044, 170048, 170401, 170552], "question_author": ""}
{"qid": 454, "query": "How to determine with certainty that a function has no elementary antiderivative?", "score": 154, "views": 17108, "answer_pids": [128691], "question_author": "Davorak"}
{"qid": 455, "query": "What actually is a polynomial?", "score": 153, "views": 16932, "answer_pids": [828310, 828314, 828315, 828316, 828319, 828321, 828328, 828330, 828766, 828882, 828989, 829224, 829297, 829423, 829913, 830017, 1190447], "question_author": "Clark Gaebel"}
{"qid": 456, "query": "Sum of random decreasing numbers between 0 and 1: does it converge??", "score": 153, "views": 9315, "answer_pids": [810000, 810069, 810083, 810131], "question_author": ""}
{"qid": 457, "query": "Are there real-life relations which are symmetric and reflexive but not transitive?", "score": 152, "views": 51453, "answer_pids": [130115, 130116, 130117, 130120, 130122, 130148, 130151, 130169, 130206, 130509, 133245, 135224, 611996, 896718, 917812], "question_author": "futurebird"}
{"qid": 458, "query": "Calculate Rotation Matrix to align Vector A to Vector B in 3d?", "score": 151, "views": 287405, "answer_pids": [90925, 90928, 146778, 217245, 222175, 380948, 459158, 610446, 675054, 694543, 716179, 737281, 783517, 820309, 925112, 991820], "question_author": ""}
{"qid": 459, "query": "Whats new in higher dimensions?", "score": 151, "views": 20922, "answer_pids": [982577, 982632, 982755, 982785, 982846, 982861, 982894, 982926, 983025, 983336, 983450, 984420, 984946, 985688, 1037531, 1399410], "question_author": ""}
{"qid": 460, "query": "How do you describe your mathematical research in laymans terms?", "score": 151, "views": 18269, "answer_pids": [302244, 302258, 302292, 302329, 302378, 302416, 302423, 302469, 302472, 302502, 302527, 302584, 302587, 302619, 302765, 302844, 302853, 304030], "question_author": "bryn"}
{"qid": 461, "query": "Pythagorean triples that survive Eulers totient function", "score": 151, "views": 4835, "answer_pids": [1137704], "question_author": "Jonathan."}
{"qid": 462, "query": "Whats the significance of Tates thesis?", "score": 150, "views": 15611, "answer_pids": [13730], "question_author": "Noldorin"}
{"qid": 463, "query": "What is the difference between singular value and eigenvalue?", "score": 149, "views": 155078, "answer_pids": [65090, 65093, 271880, 369326, 1286388], "question_author": "kali281"}
{"qid": 464, "query": "Intuition of the meaning of homology groups", "score": 149, "views": 25247, "answer_pids": [21404, 21410, 21411, 21438, 73984, 563397, 864198, 1121927], "question_author": ""}
{"qid": 465, "query": "Why does the series $\\sum_{n=1}^\\infty\\frac1n$ not converge?", "score": 149, "views": 18658, "answer_pids": [154, 156, 27020, 27023, 27052, 76343, 250678, 250680, 250681, 250690, 251598, 277413, 277430, 311030, 311049, 461670, 466266, 473099, 745316, 750869, 1172199, 1184972, 1195383, 1195614, 1195627, 1298578, 1367082], "question_author": ""}
{"qid": 466, "query": "Why does an argument similiar to 0.999...=1 show 999...=-1?", "score": 149, "views": 18305, "answer_pids": [633236, 633243, 633253, 633262, 633328, 633385, 633388, 633411, 633588, 634379, 634389, 635170, 635429, 687635], "question_author": "Henry"}
{"qid": 467, "query": "Does a cubic matrix exist?", "score": 149, "views": 14793, "answer_pids": [38517, 38519, 38520, 38524, 38567, 38624], "question_author": ""}
{"qid": 468, "query": "Is the following matrix invertible?", "score": 149, "views": 11348, "answer_pids": [88462], "question_author": ""}
{"qid": 469, "query": "Do most mathematicians know most topics in mathematics?", "score": 148, "views": 19265, "answer_pids": [892295, 892336, 892451, 892511, 892542, 893223], "question_author": "Mehper C. Palavuzlar"}
{"qid": 470, "query": "Slice of pizza with no crust", "score": 148, "views": 17510, "answer_pids": [224463, 224533], "question_author": "LanceLafontaine"}
{"qid": 471, "query": "Has lack of mathematical rigour killed anybody before?", "score": 148, "views": 14132, "answer_pids": [493397, 493401, 493515, 493523, 493561, 493579, 493811, 493830, 493851, 493922, 493986, 493996, 494213, 494320], "question_author": ""}
{"qid": 472, "query": "Alternative notation for exponents, logs and roots?", "score": 147, "views": 41977, "answer_pids": [16282, 16306, 17963, 17966, 18000, 18020, 66925, 83369, 472534, 727825, 947661, 1138488, 1265628, 1307506], "question_author": ""}
{"qid": 473, "query": "Math and mental fatigue", "score": 147, "views": 37155, "answer_pids": [94126, 94281, 94304, 94310, 94318, 107888, 164587, 323156, 346770, 581722], "question_author": ""}
{"qid": 474, "query": "What is the Riemann-Zeta function?", "score": 147, "views": 22627, "answer_pids": [330, 340, 358, 495, 1059], "question_author": ""}
{"qid": 475, "query": "Why do both sine and cosine exist?", "score": 147, "views": 20873, "answer_pids": [562503, 562505, 562508, 562741, 562977, 562998, 563030, 563165, 563208, 563252, 563580, 563896, 564357], "question_author": "Aleksandra Zalcman"}
{"qid": 476, "query": "What is the best book to learn probability?", "score": 146, "views": 215324, "answer_pids": [17178, 17192, 22225, 54462, 54465, 277888, 367505, 500954, 925913], "question_author": "Rory M"}
{"qid": 477, "query": "The square roots of different primes are linearly independent over the field of rationals", "score": 146, "views": 21271, "answer_pids": [16597, 16603, 695103], "question_author": "Damian Kao"}
{"qid": 478, "query": "Can someone explain the math behind tessellation?", "score": 146, "views": 18757, "answer_pids": [19675, 20673, 20725, 458208, 1116217], "question_author": "Phil"}
{"qid": 479, "query": "Where to start learning Linear Algebra?", "score": 145, "views": 190946, "answer_pids": [2575, 2577, 2608, 2610, 2617, 90759, 90761, 90773, 95417, 111358, 202461, 248701, 306329, 347556, 379072, 494392], "question_author": ""}
{"qid": 480, "query": "Intuitively, what is the difference between Eigendecomposition and Singular Value Decomposition?", "score": 145, "views": 71591, "answer_pids": [152635, 152636, 172320, 203627, 512549, 1040128], "question_author": ""}
{"qid": 481, "query": "Looking for an intuitive explanation why the row rank is equal to the column rank for a matrix", "score": 145, "views": 46873, "answer_pids": [158462, 158478, 158754, 263297, 263317, 366356, 396324, 533673, 635699, 673827, 945517, 1096572, 1254093, 1269093, 1302457], "question_author": "jonsca"}
{"qid": 482, "query": "Example of infinite field of characteristic $p\\neq 0$", "score": 145, "views": 31049, "answer_pids": [30997, 344902], "question_author": "Douglas S. Stones"}
{"qid": 483, "query": "Is non-standard analysis worth learning?", "score": 145, "views": 18016, "answer_pids": [27398, 27408, 27410, 27418, 62218, 62232, 713416], "question_author": ""}
{"qid": 484, "query": "Connection between Fourier transform and Taylor series", "score": 144, "views": 33314, "answer_pids": [4242, 4245, 4248, 4285, 473544, 726233], "question_author": "Grisha Kirilin"}
{"qid": 485, "query": "Are we allowed to compare infinities?", "score": 144, "views": 9930, "answer_pids": [677753, 677756, 677757, 677784, 677808, 677895, 677919, 678019, 678085, 678113, 679524], "question_author": "arik-so"}
{"qid": 486, "query": "Online tool for making graphs (vertices and edges)?", "score": 143, "views": 202038, "answer_pids": [7786, 7787, 7800, 115451, 115452, 123070, 380622, 396828, 454124, 461308, 465055, 504028, 533122, 541573, 609763, 726831, 905849, 1419938], "question_author": "Fiktor"}
{"qid": 487, "query": "Whats 4 times more likely than 80%?", "score": 143, "views": 29199, "answer_pids": [217451, 217463, 217476, 217477, 421485], "question_author": "Alexander Galkin"}
{"qid": 488, "query": "Simple theorems that are instances of deep mathematics", "score": 143, "views": 20192, "answer_pids": [839132, 839137, 839147, 839149, 839161, 839185, 839188, 839328, 839387, 839524, 839628, 839717, 839729, 839870, 839934, 839957, 840344, 840612, 840627, 841229, 841875, 842638, 843779, 845562, 851824, 1015952, 1180371], "question_author": ""}
{"qid": 489, "query": "Do Arithmetic Mean and Geometric Mean of Prime Numbers converge?", "score": 143, "views": 6416, "answer_pids": [1074009, 1074010, 1074107, 1074611, 1074677, 1074745], "question_author": ""}
{"qid": 490, "query": "Partial derivative in gradient descent for two variables", "score": 142, "views": 86105, "answer_pids": [37372, 95652, 657845], "question_author": "Jonathan."}
{"qid": 491, "query": "Apparently sometimes $1/2 < 1/4$?", "score": 142, "views": 7077, "answer_pids": [835111, 835218, 835219, 835232, 835298, 835310, 835369], "question_author": ""}
{"qid": 492, "query": "Is the product of two Gaussian random variables also a Gaussian?", "score": 141, "views": 190458, "answer_pids": [52293, 52321, 186840, 386815, 925409], "question_author": ""}
{"qid": 493, "query": "Intuitive explanation of entropy", "score": 141, "views": 48552, "answer_pids": [157638, 164352, 176685, 253617, 333150, 382101, 670057, 753868, 810286, 825425, 930193, 1275786, 1352632], "question_author": "ptomato"}
{"qid": 494, "query": "The Best of Dover Books (a.k.a the best cheap mathematical texts)", "score": 141, "views": 29964, "answer_pids": [170081, 170083, 170085, 170087, 170089, 170091, 170092, 170096, 170098, 170100, 170164, 170173, 170198, 170316, 170528, 172380, 179611, 196297, 220872, 245153, 366763, 366774, 1363270], "question_author": "AAA"}
{"qid": 495, "query": "Are mathematical articles on Wikipedia reliable?", "score": 141, "views": 12904, "answer_pids": [324304, 324305, 324319, 324593, 324679, 324817, 325678, 325698, 325748, 387818], "question_author": ""}
{"qid": 496, "query": "What are the differences between rings, groups, and fields?", "score": 140, "views": 90025, "answer_pids": [37, 45, 105, 131665, 319455], "question_author": ""}
{"qid": 497, "query": "Why do units (from physics) behave like numbers?", "score": 140, "views": 12634, "answer_pids": [758616, 758625, 758691, 758776, 758851, 758890, 758946, 758988, 759378, 759602, 759662, 759673, 759920, 760301], "question_author": ""}
{"qid": 498, "query": "Studying Euclidean geometry using hyperbolic criteria", "score": 140, "views": 3298, "answer_pids": [836, 852, 901, 1727, 712217, 952702], "question_author": ""}
{"qid": 499, "query": "Examples of bijective map from $\\mathbb{R}^3\\rightarrow \\mathbb{R}$", "score": 139, "views": 36990, "answer_pids": [92417, 92418], "question_author": ""}
{"qid": 500, "query": "Identification of a curious function", "score": 139, "views": 5240, "answer_pids": [524922], "question_author": ""}
{"qid": 501, "query": "What is the difference between regression and classification?", "score": 138, "views": 203381, "answer_pids": [71596, 92227, 242202, 311072, 373495, 395018, 568067, 616663, 760191, 916579, 1215157], "question_author": "Hans-Peter Stricker"}
{"qid": 502, "query": "How to put 9 pigs into 4 pens so that there are an odd number of pigs in each pen?", "score": 138, "views": 35231, "answer_pids": [253293, 253294, 253295, 253296, 253336, 253340, 253390, 253419, 253472, 253474, 253490, 253532, 253608, 253655, 254209, 255524, 371938], "question_author": ""}
{"qid": 503, "query": "How do you respond to I was always bad at math?", "score": 138, "views": 11169, "answer_pids": [188901, 188906, 188923, 188924, 188925, 188933, 188935, 188937, 188978, 188984, 189007, 189012, 189030, 189032, 189050, 189055], "question_author": "Roy Maclean"}
{"qid": 504, "query": "What are the numbers before and after the decimal point referred to in mathematics?", "score": 136, "views": 226516, "answer_pids": [33933, 205369, 634785], "question_author": ""}
{"qid": 505, "query": "Can you explain the Axiom of choice in simple terms?", "score": 136, "views": 50664, "answer_pids": [3798, 3802, 3999, 500894, 500958, 504555, 635015], "question_author": "Charles Stewart"}
{"qid": 506, "query": "Why do we use the word scalar and not number in Linear Algebra?", "score": 136, "views": 48399, "answer_pids": [706379, 706383, 706388, 706392, 706404, 706488, 706582, 706593, 706662, 706899, 707131], "question_author": "Ahmad Farid"}
{"qid": 507, "query": "What is the difference between Fourier series and Fourier transformation?", "score": 135, "views": 190114, "answer_pids": [109232, 109260, 219069, 991128, 1303033], "question_author": "Theodor"}
{"qid": 508, "query": "What is the difference between a point and a vector?", "score": 135, "views": 59621, "answer_pids": [287811, 287813, 287821, 287823, 287824, 287829, 287833, 287854, 287868, 287971, 287991, 288019, 288022, 288115, 288254, 288352, 288376, 288949], "question_author": "Peter Mortensen"}
{"qid": 509, "query": "The direct sum $\\oplus$ versus the cartesian product $\\times$", "score": 135, "views": 30766, "answer_pids": [21271, 21274, 21276, 589288], "question_author": ""}
{"qid": 510, "query": "Is the blue area greater than the red area?", "score": 135, "views": 20844, "answer_pids": [1025653, 1025654, 1025655, 1025659, 1025679, 1025752, 1025833, 1028028, 1028135, 1126814], "question_author": "Rory M"}
{"qid": 511, "query": "Does $R[x] \\cong S[x]$ imply $R \\cong S$?", "score": 135, "views": 6862, "answer_pids": [7603, 7605, 7606, 7611], "question_author": ""}
{"qid": 512, "query": "Why does mathematical convention deal so ineptly with multisets?", "score": 134, "views": 7539, "answer_pids": [76895, 77325, 78122, 85070, 113264, 624738], "question_author": ""}
{"qid": 513, "query": "List of interesting math podcasts?", "score": 133, "views": 77819, "answer_pids": [0, 74, 602, 603, 148440, 156624, 163886, 250542, 289489, 430613, 829009, 886412, 897389, 935340, 941160, 1036098, 1101486, 1162234, 1217743, 1296880], "question_author": ""}
{"qid": 514, "query": "Can you raise a number to an irrational exponent?", "score": 133, "views": 47029, "answer_pids": [29259, 29261, 29263, 682229], "question_author": "leo"}
{"qid": 515, "query": "Examples of mathematical results discovered late", "score": 133, "views": 20257, "answer_pids": [307827, 307840, 307937, 307948, 307964, 308097, 308107, 308237, 308270, 308517, 308519, 308748, 308864, 308945, 308955, 309294, 309297, 309529, 309693, 309694, 310939, 313905, 319066, 319942, 319964, 320194, 406222, 421605, 674320, 674344], "question_author": ""}
{"qid": 516, "query": "$\\pi$ in arbitrary metric spaces", "score": 133, "views": 15310, "answer_pids": [123568, 123936, 124272, 127889, 218198, 218221, 218230], "question_author": ""}
{"qid": 517, "query": "Proving an alternating Euler sum: $\\sum_{k=1}^{\\infty} \\frac{(-1)^{k+1} H_k}{k} = \\frac{1}{2} \\zeta(2) - \\frac{1}{2} \\log^2 2$", "score": 133, "views": 9999, "answer_pids": [132979, 132981, 133411, 231997, 232009, 232170, 273404, 276834, 354320, 372870, 1174365, 1234877, 1240649, 1348759], "question_author": "Engineer"}
{"qid": 518, "query": "Is there any mathematical reason for this digit-repetition-show?", "score": 133, "views": 6927, "answer_pids": [811522, 811543], "question_author": "Akhil Mathew"}
{"qid": 519, "query": "Visually deceptive proofs which are mathematically wrong", "score": 132, "views": 26956, "answer_pids": [323905, 323938, 323941, 323955, 323975, 323980, 324029, 324044, 324052, 324098, 324186, 324386, 325118, 325164, 325637, 326766, 327178, 328132, 328440, 791338, 1011296, 1065020, 1118970], "question_author": "voix"}
{"qid": 520, "query": "Best book of topology for beginner?", "score": 131, "views": 110659, "answer_pids": [4362, 4369, 4370, 4374, 4375, 4385, 56102, 69798, 76148, 76150, 76504, 81476, 157997, 237874, 420554, 420574, 1278622], "question_author": ""}
{"qid": 521, "query": "Can someone explain G\u00f6dels incompleteness theorems in layman terms?", "score": 131, "views": 78484, "answer_pids": [211992, 212008, 212014, 212028, 212032, 212067, 212107, 212112, 212188, 212243, 212505, 212841, 256255], "question_author": ""}
{"qid": 522, "query": "Why is $1$ not a prime number?", "score": 131, "views": 76061, "answer_pids": [66, 85, 97, 180, 3389, 31340, 165880, 424417, 439679, 440695, 576129, 683714, 1006707, 1019351, 1250303], "question_author": "LanceLafontaine"}
{"qid": 523, "query": "Studying for the Putnam Exam", "score": 131, "views": 57946, "answer_pids": [147580, 148489, 148501, 166037, 166044, 188164], "question_author": "Piotr Migdal"}
{"qid": 524, "query": "Intuitive interpretation of the Laplacian", "score": 131, "views": 22330, "answer_pids": [26803, 26804, 26805, 26806, 26807, 547589, 1387168], "question_author": "Rory M"}
{"qid": 525, "query": "what is expected from a PhD student?", "score": 131, "views": 5958, "answer_pids": [215111, 215112, 215113, 215114, 215144, 215559, 215575, 215583, 215708, 292736], "question_author": ""}
{"qid": 526, "query": "Why is the eigenvector of a covariance matrix equal to a principal component?", "score": 130, "views": 102934, "answer_pids": [12952, 15875, 481009], "question_author": ""}
{"qid": 527, "query": "Sum of First $n$ Squares Equals $\\frac{n(n+1)(2n+1)}{6}$", "score": 130, "views": 70995, "answer_pids": [25636, 25638, 25639, 25641, 25644, 25645, 25677, 25718, 26976, 26978, 26982, 27036, 34681, 40374, 189928, 243642, 303276, 314958, 587154, 587175, 587583, 587658, 653157, 833514, 1131380, 1360594], "question_author": "Kevin Dente"}
{"qid": 528, "query": "Is it possible for a function to be in $L^p$ for only one $p$?", "score": 130, "views": 16232, "answer_pids": [29311, 29312, 29313, 384597], "question_author": ""}
{"qid": 529, "query": "What are some examples of notation that really improved mathematics?", "score": 130, "views": 13566, "answer_pids": [253627, 253630, 253638, 253641, 253643, 253644, 253660, 253670, 253673, 253678, 253686, 253705, 253738, 253758, 253806, 253826, 253849, 253851, 253856, 253902, 253936, 254099, 507855, 507880], "question_author": "Noah Snyder"}
{"qid": 530, "query": "Classification of prime ideals of $\\mathbb{Z}[X]$", "score": 129, "views": 22374, "answer_pids": [88039], "question_author": null}
{"qid": 531, "query": "Division by $0$", "score": 129, "views": 11058, "answer_pids": [14434, 14437, 32546, 32571, 32643, 42479, 116416, 296503, 298977, 334263, 455179, 556423], "question_author": "Noah Snyder"}
{"qid": 532, "query": "Application of Hilberts basis theorem in representation theory", "score": 129, "views": 3392, "answer_pids": [1323982], "question_author": "harpalss"}
{"qid": 533, "query": "Is $0$ a natural number?", "score": 128, "views": 81405, "answer_pids": [177, 178, 179, 183, 450733, 639604, 1043297], "question_author": "BBischof"}
{"qid": 534, "query": "Why is negative times negative = positive?", "score": 128, "views": 73300, "answer_pids": [5656, 5657, 5658, 5659, 5664, 5669, 5726, 5731, 5905, 178549, 181002, 309749, 327946, 427005, 512223, 528188, 584979, 596715, 596885, 596892, 596894, 596897, 596906, 596917, 596944, 596983, 596995, 596998, 597006, 597023, 597255, 597274, 748171, 1055688, 1120775], "question_author": "Tom Stephens"}
{"qid": 535, "query": "On Ph.D. Qualifying Exams", "score": 128, "views": 36781, "answer_pids": [129524, 129538, 130979, 1357032, 1357441], "question_author": "Generic Error"}
{"qid": 536, "query": "Elementary proof that $\\mathbb{R}^n$ is not homeomorphic to $\\mathbb{R}^m$", "score": 128, "views": 26044, "answer_pids": [13605, 13606, 13622, 250234, 779152, 884726, 1002903], "question_author": "Isaac"}
{"qid": 537, "query": "What are some examples of a mathematical result being counterintuitive?", "score": 128, "views": 23215, "answer_pids": [70794, 70796, 70800, 70801, 70809, 70825, 70833, 70834, 70837, 70840, 70850, 70917, 70926, 70930, 70931, 70933, 70953, 70956, 70966, 70970, 71138, 71243, 71244, 71303, 71305, 71309, 71319, 71690, 71720, 72118, 72122, 72232, 72930, 72934, 72951, 126890, 126904, 179718, 331666, 607937], "question_author": "Poshpaws"}
{"qid": 538, "query": "Can you be 1/12th Cherokee?", "score": 128, "views": 20525, "answer_pids": [455670, 455671, 455734, 455920, 455933, 456022, 456254, 456336, 456528, 456627, 456659, 456730], "question_author": ""}
{"qid": 539, "query": "Why did mathematicians take Russells paradox seriously?", "score": 128, "views": 16513, "answer_pids": [13422, 13436, 13437, 13462, 13476, 13478, 13485, 13486, 13551, 13696, 291160, 291625, 316951], "question_author": ""}
{"qid": 540, "query": "Values of $\\sum_{n=0}^\\infty x^n$ and $\\sum_{n=0}^N x^n$", "score": 127, "views": 54007, "answer_pids": [15762, 15763, 15770, 43708, 531032], "question_author": "BlueRaja - Danny Pflughoeft"}
{"qid": 541, "query": "How to find ${\\large\\int}_0^1\\frac{\\ln^3(1+x)\\ln x}x\\mathrm dx$", "score": 127, "views": 27708, "answer_pids": [384710, 384792, 384887, 385760, 386378, 394228, 1168757, 1205079, 1266632, 1354790], "question_author": "David Z"}
{"qid": 542, "query": "What is the probability that a point chosen randomly from inside an equilateral triangle is closer to the center than to any of the edges?", "score": 127, "views": 20537, "answer_pids": [655580, 656129, 661750, 661776], "question_author": "wrongusername"}
{"qid": 543, "query": "Why is learning modern algebraic geometry so complicated?", "score": 127, "views": 17332, "answer_pids": [185624, 237098, 364765, 603238], "question_author": "LanceLafontaine"}
{"qid": 544, "query": "What should be the intuition when working with compactness?", "score": 127, "views": 16960, "answer_pids": [175365, 175371, 175372, 175430, 175448, 218871, 474711, 474719, 474825, 475033, 714965, 956491, 969383, 1427074], "question_author": "Adam C."}
{"qid": 545, "query": "Motivation of irrationality measure", "score": 127, "views": 9001, "answer_pids": [1134979], "question_author": ""}
{"qid": 546, "query": "Show that the determinant of $A$ is equal to the product of its eigenvalues", "score": 126, "views": 170924, "answer_pids": [235086, 235088, 235105, 284849, 922843], "question_author": "Bugster"}
{"qid": 547, "query": "The median minimizes the sum of absolute deviations (the $ {\\ell}_{1} $ norm)", "score": 126, "views": 74642, "answer_pids": [58190, 58221, 58246, 425102, 608242, 717134, 889516, 931708], "question_author": ""}
{"qid": 548, "query": "Why is gradient the direction of steepest ascent?", "score": 126, "views": 64520, "answer_pids": [110161, 110162, 110164, 158869, 509782, 915775, 1004388, 1060370, 1168413, 1244558, 1307305, 1354917], "question_author": "Jacob"}
{"qid": 549, "query": "Learning Lambda Calculus", "score": 126, "views": 31092, "answer_pids": [589, 639, 759, 940, 16578, 107705, 407603, 445425, 1092192], "question_author": "Gabriel Fair"}
{"qid": 550, "query": "Is there a function that grows faster than exponentially but slower than a factorial?", "score": 126, "views": 18497, "answer_pids": [965948, 965978, 965994, 966079, 966554], "question_author": "Gabriel Fair"}
{"qid": 551, "query": "How to find solutions of linear Diophantine ax + by = c?", "score": 125, "views": 145899, "answer_pids": [11450, 11454, 11458, 709214, 1352062], "question_author": "Gergana Vandova"}
{"qid": 552, "query": "What is category theory useful for?", "score": 125, "views": 26988, "answer_pids": [149114, 149116, 149127, 412824], "question_author": ""}
{"qid": 553, "query": "Are half of all numbers odd?", "score": 125, "views": 23859, "answer_pids": [52606, 52609, 52610, 52611, 52613, 52627, 52792, 176024], "question_author": ""}
{"qid": 554, "query": "Can $x^{x^{x^x}}$ be a rational number?", "score": 125, "views": 21924, "answer_pids": [206231], "question_author": "Bugster"}
{"qid": 555, "query": "Prove that $C\\exp(x)$ is the only set of functions for which $f(x) = f(x)$", "score": 125, "views": 21011, "answer_pids": [30828, 30829, 30833, 30846, 30856, 30901, 83695, 575311], "question_author": ""}
{"qid": 556, "query": "Are if and iff interchangeable in definitions?", "score": 125, "views": 13022, "answer_pids": [256807, 256809, 256811, 256814, 256902, 256929, 257354, 257371, 257536, 325334], "question_author": "Martin"}
{"qid": 557, "query": "Is there an integral that proves $\\pi > 333/106$?", "score": 125, "views": 6175, "answer_pids": [1206, 1207, 65134, 621970, 839968, 839984], "question_author": ""}
{"qid": 558, "query": "Past open problems with sudden and easy-to-understand solutions", "score": 125, "views": 5896, "answer_pids": [619476, 619481, 619489, 619491, 619495, 619518, 619592, 619802, 619933, 619938, 620353, 620420, 620625, 620627, 621267, 621308, 621728, 622727, 622944], "question_author": ""}
{"qid": 559, "query": "How to prove $\\int_0^1\\tan^{-1}\\left[\\frac{\\tanh^{-1}x-\\tan^{-1}x}{\\pi+\\tanh^{-1}x-\\tan^{-1}x}\\right]\\frac{dx}{x}=\\frac{\\pi}{8}\\ln\\frac{\\pi^2}{8}?$", "score": 125, "views": 5291, "answer_pids": [269207, 301562], "question_author": "Gabriel Fair"}
{"qid": 560, "query": "What remains in a students mind", "score": 125, "views": 4327, "answer_pids": [73378, 80505, 80509, 196204, 196214, 196397, 461709], "question_author": "Sid"}
{"qid": 561, "query": "Proof $1+2+3+4+\\cdots+n = \\frac{n\\times(n+1)}2$", "score": 124, "views": 273228, "answer_pids": [1370, 1371, 1372, 1375, 1387, 1388, 1400, 18455, 18458, 18459, 18462, 18474, 18479, 34676, 44590, 102239, 178360, 293074, 373535, 374722, 400821, 447311, 447315, 774535, 776813, 795769, 804940, 882142, 1210220, 1308362], "question_author": null}
{"qid": 562, "query": "Why is $\\infty \\cdot 0$ not clearly equal to $0$?", "score": 124, "views": 182491, "answer_pids": [15715, 15716, 15718, 15720, 15728, 126290, 189579, 356391], "question_author": ""}
{"qid": 563, "query": "What are good books to learn graph theory?", "score": 124, "views": 90179, "answer_pids": [14961, 14966, 14976, 16811, 16839, 16842, 141964, 304314, 353174, 420333, 420340, 420346, 435854, 480947, 480951], "question_author": ""}
{"qid": 564, "query": "Can an irrational number raised to an irrational power be rational?", "score": 124, "views": 30118, "answer_pids": [53687, 53688, 53699, 54334, 1009232, 1413653], "question_author": ""}
{"qid": 565, "query": "Advantages of Mathematics competition/olympiad students in Mathematical Research", "score": 124, "views": 29245, "answer_pids": [2863, 2864, 2867, 36306, 36311, 94128, 119618, 266300, 386159, 665683, 1295295, 1422947, 1446714], "question_author": ""}
{"qid": 566, "query": "Whats the difference between predicate and propositional logic?", "score": 123, "views": 128720, "answer_pids": [5453, 5456, 270801, 535719, 899692], "question_author": "Steve"}
{"qid": 567, "query": "How can I find the surface area of a normal chicken egg?", "score": 123, "views": 38509, "answer_pids": [191166, 191171, 191219, 191225, 191246, 191257, 191376, 191431, 195477], "question_author": "InquilineKea"}
{"qid": 568, "query": "Why does LHopitals rule fail in calculating $\\lim_{x \\to \\infty} \\frac{x}{x+\\sin(x)}$?", "score": 123, "views": 11087, "answer_pids": [663045, 663048, 663200, 663488, 664085], "question_author": ""}
{"qid": 569, "query": "Why study linear algebra?", "score": 122, "views": 127210, "answer_pids": [124299, 124302, 124307, 124311, 124321, 124322, 124407, 124440, 124443, 124449, 124502, 278656], "question_author": ""}
{"qid": 570, "query": "Why is integration so much harder than differentiation?", "score": 122, "views": 41578, "answer_pids": [11409, 11410, 18269, 220488, 350875, 623523], "question_author": "jonsca"}
{"qid": 571, "query": "Why is $\\cos (90)=-0.4$ in WebGL?", "score": 122, "views": 15595, "answer_pids": [341987, 341988, 341990, 342100, 342206, 342412, 342783], "question_author": null}
{"qid": 572, "query": "Why \u201ccharacteristic zero\u201d and not \u201cinfinite characteristic\u201d?", "score": 122, "views": 9601, "answer_pids": [51058, 51059, 51062, 51066, 1383090], "question_author": ""}
{"qid": 573, "query": "Why are the solutions of polynomial equations so unconstrained over the quaternions?", "score": 122, "views": 7706, "answer_pids": [62871, 62876, 62897, 62904, 63183, 1216365], "question_author": "Moor Xu"}
{"qid": 574, "query": "Making Friends around a Circular Table", "score": 122, "views": 5857, "answer_pids": [591977, 802269], "question_author": "Paul"}
{"qid": 575, "query": "Is There An Injective Cubic Polynomial $\\mathbb Z^2 \\rightarrow \\mathbb Z$?", "score": 122, "views": 3717, "answer_pids": [484309], "question_author": "rubenvb"}
{"qid": 576, "query": "Real life applications of Topology", "score": 121, "views": 80103, "answer_pids": [38807, 38809, 38810, 38816, 52327, 52329, 52343, 52346, 92345, 93317, 201091, 255183, 460576, 460587, 460596, 460609, 705748, 716111, 1004629], "question_author": "mring"}
{"qid": 577, "query": "Whats your favorite proof accessible to a general audience?", "score": 121, "views": 16514, "answer_pids": [455266, 455270, 455273, 455275, 455276, 455277, 455278, 455279, 455282, 455290, 455295, 455304, 455310, 455314, 455324, 455326, 455338, 455350, 455422, 455425, 455866, 455871, 455880, 455896, 455903, 455915, 455930, 456032, 456134, 456169, 456188, 456196, 456859, 456911, 456912, 456929, 458346, 460050], "question_author": "Casebash"}
{"qid": 578, "query": "Pullback and Pushforward Isomorphism of Sheaves", "score": 121, "views": 7293, "answer_pids": [1386818], "question_author": ""}
{"qid": 579, "query": "Physical meaning of the null space of a matrix", "score": 120, "views": 146245, "answer_pids": [11663, 11664, 172142, 412660, 1411638], "question_author": ""}
{"qid": 580, "query": "How do you explain to a 5th grader why division by zero is meaningless?", "score": 120, "views": 27141, "answer_pids": [1062075, 1062076, 1062079, 1062091, 1062114, 1062117, 1062120, 1062242, 1062243, 1062290, 1062316, 1062531, 1062567, 1062711, 1063021, 1063044, 1063259], "question_author": ""}
{"qid": 581, "query": "Overview of basic facts about Cauchy functional equation", "score": 120, "views": 20936, "answer_pids": [198495], "question_author": ""}
{"qid": 582, "query": "Is 10 closer to infinity than 1?", "score": 120, "views": 17546, "answer_pids": [251841, 251842, 251844, 251845, 251847, 251861, 251883, 251985, 251987, 252162, 252277, 252281, 252330, 252351, 253870, 254061], "question_author": ""}
{"qid": 583, "query": "Examples of problems that are easier in the infinite case than in the finite case.", "score": 120, "views": 11122, "answer_pids": [725865, 725873, 725896, 725906, 725936, 725954, 725957, 725981, 725987, 726189, 726298, 726528, 726540, 726643, 726732, 726734, 726754, 726799, 727001, 727118, 727250, 727262, 727371, 728032, 797899, 952187], "question_author": ""}
{"qid": 584, "query": "Help find hard integrals that evaluate to $59$?", "score": 120, "views": 9056, "answer_pids": [84981, 84992, 84999, 85014, 85092], "question_author": "Felix Rehren"}
{"qid": 585, "query": "Is the inverse of a symmetric matrix also symmetric?", "score": 119, "views": 165813, "answer_pids": [154927, 154928, 154929, 154939, 270211, 282614, 891818, 1152157, 1180978], "question_author": ""}
{"qid": 586, "query": "When to learn category theory?", "score": 119, "views": 24059, "answer_pids": [11665, 11668, 11676, 11688, 11705, 11709, 11723, 44328], "question_author": ""}
{"qid": 587, "query": "In classical logic, why is $(p\\Rightarrow q)$ True if both $p$ and $q$ are False?", "score": 119, "views": 21212, "answer_pids": [25683, 25685, 25686, 25687, 25689, 25694, 25695, 25703, 25715, 25722, 25725, 25737, 36653, 50928, 220565, 305317, 423060, 628038, 638639, 968140], "question_author": ""}
{"qid": 588, "query": "Modular exponentiation by hand ($a^b\\bmod c$)", "score": 119, "views": 17745, "answer_pids": [42572, 42573, 42576, 42578, 700554, 700560, 710000, 1139722, 1282814, 1308074], "question_author": "Sami"}
{"qid": 589, "query": "How were old-school mathematics graphics created?", "score": 119, "views": 12431, "answer_pids": [362106, 362109, 362115, 362258, 362469], "question_author": "Claudiu"}
{"qid": 590, "query": "Infiniteness of non-twin primes.", "score": 119, "views": 11901, "answer_pids": [275092, 275093, 275094, 275095, 275177, 275249, 275371, 275413, 275522, 275812, 276388, 294596, 364155, 405313], "question_author": ""}
{"qid": 591, "query": "probability $2/4$ vs $3/6$", "score": 119, "views": 9363, "answer_pids": [300262, 300266, 300267, 300335, 300456, 300621], "question_author": ""}
{"qid": 592, "query": "Continuous projections on $\\ell_1$ with norm $>1$", "score": 119, "views": 3760, "answer_pids": [979744], "question_author": ""}
{"qid": 593, "query": "Is there a characterization of groups with the property $\\forall N\\unlhd G,\\:\\exists H\\leq G\\text{ s.t. }H\\cong G/N$?", "score": 119, "views": 3261, "answer_pids": [1180761], "question_author": "Jonathan Fischoff"}
{"qid": 594, "query": "All real numbers in $[0,2]$ can be represented as $\\sqrt{2 \\pm \\sqrt{2 \\pm \\sqrt{2 \\pm \\dots}}}$", "score": 119, "views": 2009, "answer_pids": [670089, 670113, 670577], "question_author": "Poshpaws"}
{"qid": 595, "query": "Olympiad Inequality $\\sum\\limits_{cyc} \\frac{x^4}{8x^3+5y^3} \\geqslant \\frac{x+y+z}{13}$", "score": 118, "views": 28051, "answer_pids": [685073, 688562, 689120, 689193, 711424, 719876, 1064735, 1065506, 1066359, 1112582, 1139040, 1316075, 1388680], "question_author": "John Gietzen"}
{"qid": 596, "query": "What is the difference between independent and mutually exclusive events?", "score": 117, "views": 582803, "answer_pids": [396636, 396638, 452813, 462651, 558530, 1318421], "question_author": ""}
{"qid": 597, "query": "What is the Jacobian matrix?", "score": 117, "views": 161685, "answer_pids": [8371, 8387, 8447, 19588, 72772, 461990, 492366, 693783, 976719], "question_author": ""}
{"qid": 598, "query": "If squaring a number means multiplying that number with itself then shouldnt taking square root of a number mean to divide a number by itself?", "score": 117, "views": 28403, "answer_pids": [626558, 626559, 626561, 626563, 626566, 626600, 626621, 626627, 626631, 626656, 626733, 626811, 627026, 627245, 627741, 628056, 628194, 628684, 628728, 629631, 629935, 632915, 988385, 1045159], "question_author": "VanJeer"}
{"qid": 599, "query": "Is there a known well ordering of the reals?", "score": 117, "views": 25760, "answer_pids": [3803, 3804], "question_author": ""}
{"qid": 600, "query": "Intuition behind Conditional Expectation", "score": 117, "views": 16611, "answer_pids": [12941, 12945, 12950, 12951, 12953, 1356220], "question_author": "Isaac"}
{"qid": 601, "query": "Is there an inverted dot product?", "score": 117, "views": 13863, "answer_pids": [970179, 970182, 970193, 970232, 970247, 970272, 970611, 970736, 970835, 970989, 971353, 971644], "question_author": "cpuguru"}
{"qid": 602, "query": "Find the average of $\\sin^{100} (x)$ in 5 minutes?", "score": 117, "views": 11656, "answer_pids": [13431, 13432, 13434, 13444, 13470, 210706, 210712, 210803, 343053], "question_author": ""}
{"qid": 603, "query": "How to find the Galois group of a polynomial?", "score": 116, "views": 40376, "answer_pids": [24456, 24460, 167989], "question_author": "Gerard"}
{"qid": 604, "query": "Strategies for Effective Self-Study", "score": 116, "views": 15699, "answer_pids": [21816, 21819, 21829, 21853, 21855, 243390], "question_author": "Gergana Vandova"}
{"qid": 605, "query": "What did Alan Turing mean when he said he didnt fully understand dy/dx?", "score": 116, "views": 12594, "answer_pids": [506275, 506283, 506340, 506374, 506540, 506836], "question_author": ""}
{"qid": 606, "query": "Is there a domain larger than (i.e., a supserset of) the complex number domain?", "score": 116, "views": 8658, "answer_pids": [450249, 450253, 450257, 450259, 450378, 450521, 450621, 450802, 450868, 478161], "question_author": ""}
{"qid": 607, "query": "Motivation for the rigour of real analysis", "score": 116, "views": 7808, "answer_pids": [836164, 836167, 836168, 836176, 836177, 836281, 836706, 837357, 837622], "question_author": "Gergana Vandova"}
{"qid": 608, "query": "Probability that a stick randomly broken in five places can form a tetrahedron", "score": 116, "views": 5620, "answer_pids": [169999, 170632, 174823, 237795, 281692], "question_author": ""}
{"qid": 609, "query": "The deep reason why $\\int \\frac{1}{x}\\operatorname{d}x$ is a transcendental function ($\\log$)", "score": 116, "views": 5326, "answer_pids": [818804, 818807, 818812, 818852, 818939, 819068, 820779, 922394, 922414], "question_author": ""}
{"qid": 610, "query": "Expected time to roll all 1 through 6 on a die", "score": 115, "views": 106796, "answer_pids": [15699, 15700, 959470], "question_author": "Am\u00e9rico Tavares"}
{"qid": 611, "query": "Proofs of AM-GM inequality", "score": 115, "views": 30714, "answer_pids": [305184, 305201, 305395, 305504, 355620, 355785, 355949, 355964, 454973, 525652, 647378, 647386, 647458, 811350, 909700, 987032, 1057783, 1221617, 1349840], "question_author": ""}
{"qid": 612, "query": "The math behind Warren Buffetts famous rule \u2013 never lose money", "score": 115, "views": 20647, "answer_pids": [1109306, 1109309, 1109310, 1109315, 1109317, 1109331, 1109837, 1110281, 1111490], "question_author": "Jonathan Fischoff"}
{"qid": 613, "query": "Compute $\\int_0^{\\pi/4}\\frac{(1-x^2)\\ln(1+x^2)+(1+x^2)-(1-x^2)\\ln(1-x^2)}{(1-x^4)(1+x^2)} x\\exp(\\frac{x^2-1}{x^2+1}) dx$", "score": 115, "views": 11502, "answer_pids": [350480, 350498, 757970], "question_author": "Yaroslav Bulatov"}
{"qid": 614, "query": "derivative of cost function for Logistic Regression", "score": 114, "views": 88336, "answer_pids": [222584, 673302, 845695, 946651, 1108290, 1151551, 1269296], "question_author": "Hans-Peter Stricker"}
{"qid": 615, "query": "Construction of a Borel set with positive but not full measure in each interval", "score": 114, "views": 19652, "answer_pids": [30434, 30483, 267538, 1143965], "question_author": "Mark Eichenlaub"}
{"qid": 616, "query": "Are the proofs by contradiction weaker than other proofs?", "score": 114, "views": 11451, "answer_pids": [141, 142, 143, 267, 1045, 27621], "question_author": ""}
{"qid": 617, "query": "Why are There No Triernions (3-dimensional analogue of complex numbers / quaternions)?", "score": 114, "views": 10996, "answer_pids": [687931, 687932, 687994, 688011, 688027, 688541], "question_author": "J. Musser"}
{"qid": 618, "query": "What parts of a pure mathematics undergraduate curriculum have been discovered since $1964?$", "score": 114, "views": 10117, "answer_pids": [445544, 445546, 445547, 445554, 445563, 445566, 445578, 445592, 445598, 445760, 445955, 445959, 445962, 445969, 446290, 446513, 446660], "question_author": "kali281"}
{"qid": 619, "query": "Is black hole pattern possible in Conways Game of Life that eats/clears everything?", "score": 114, "views": 9958, "answer_pids": [1447310, 1447324, 1447334, 1447433, 1447532, 1447979], "question_author": "kali281"}
{"qid": 620, "query": "A multiplication algorithm found in a book by Paul Erd\u0151s: how does it work?", "score": 114, "views": 9651, "answer_pids": [829918, 829972, 830462, 830564, 831553], "question_author": ""}
{"qid": 621, "query": "Are all limits solvable without LH\u00f4pital Rule or Series Expansion", "score": 114, "views": 7266, "answer_pids": [183668, 395074, 533617], "question_author": ""}
{"qid": 622, "query": "$n!$ is never a perfect square if $n\\geq2$. Is there a proof of this that doesnt use Chebyshevs theorem?", "score": 114, "views": 5957, "answer_pids": [587318], "question_author": "crasic"}
{"qid": 623, "query": "Convergence of $\\sum_{n=1}^{\\infty} \\frac{\\sin(n!)}{n}$", "score": 114, "views": 3451, "answer_pids": [1345277], "question_author": "back2dos"}
{"qid": 624, "query": "When is matrix multiplication commutative?", "score": 113, "views": 168471, "answer_pids": [85857, 85858, 85867, 85915, 529253, 922007, 1276390], "question_author": "Ebvirus"}
{"qid": 625, "query": "What is the difference and relationship between the binomial and Bernoulli distributions?", "score": 113, "views": 165138, "answer_pids": [358538, 358540, 358542, 487929], "question_author": "Tobias Kienzler"}
{"qid": 626, "query": "Is infinity a number?", "score": 113, "views": 78509, "answer_pids": [19406, 19423, 19437, 19472, 19581, 90870, 368069], "question_author": ""}
{"qid": 627, "query": "Am I just not smart enough?", "score": 113, "views": 23119, "answer_pids": [404281, 404284, 404306, 404310, 404311, 404330, 404344, 404345, 404360, 404542, 405257], "question_author": ""}
{"qid": 628, "query": "What is the smallest unknown natural number?", "score": 113, "views": 16541, "answer_pids": [525790, 525796, 525906, 526060, 526139, 526580, 526732, 526783, 527016, 527097, 527337, 527389, 527510, 527517], "question_author": ""}
{"qid": 629, "query": "Is there a positive definition for irrational numbers?", "score": 113, "views": 12493, "answer_pids": [736931, 736932, 736939, 736941, 736945, 736971, 737025, 737038, 737151, 737194, 737418, 737743, 737895, 738037, 738229, 738423, 738646, 743170, 960559], "question_author": "Tobias Kienzler"}
{"qid": 630, "query": "Solving Special Function Equations Using Lie Symmetries", "score": 113, "views": 2667, "answer_pids": [773498], "question_author": "Fiktor"}
{"qid": 631, "query": "Gross-Zagier formulae outside of number theory", "score": 113, "views": 2510, "answer_pids": [1028829], "question_author": "Akhil Mathew"}
{"qid": 632, "query": "Why is the derivative of a circles area its perimeter (and similarly for spheres)?", "score": 112, "views": 69936, "answer_pids": [378, 46286, 50191, 65936, 256040, 540502, 797101], "question_author": "Tom Stephens"}
{"qid": 633, "query": "Getting better at proofs", "score": 112, "views": 40688, "answer_pids": [4483, 4488, 4491, 4497, 4737], "question_author": ""}
{"qid": 634, "query": "Mathematical equivalent of Feynmans Lectures on Physics?", "score": 112, "views": 31952, "answer_pids": [32966, 32974, 32980, 33028, 33033, 33039, 33050, 33079, 33137, 33167, 33442, 35022, 40362, 184924, 604169, 868017], "question_author": "Cedric H."}
{"qid": 635, "query": "How often does it happen that the oldest person alive dies?", "score": 112, "views": 28284, "answer_pids": [167919, 170630, 182201, 182206, 182483, 183009], "question_author": ""}
{"qid": 636, "query": "Do mathematicians, in the end, always agree?", "score": 112, "views": 16469, "answer_pids": [404093, 404103, 404112, 404142, 404146, 404252, 404652, 404713, 404880, 404935, 405022, 405467, 405579, 406401, 413697, 416476], "question_author": "Charles Stewart"}
{"qid": 637, "query": "Can an infinite sum of irrational numbers be rational?", "score": 112, "views": 15894, "answer_pids": [641513, 641515, 641520, 641568, 641572, 641579, 641593, 641596, 641604, 641628, 641698, 641876, 642034, 642292, 642319, 642574, 643304, 643341, 643935, 772926, 936463], "question_author": ""}
{"qid": 638, "query": "Unexpected examples of natural logarithm", "score": 112, "views": 11531, "answer_pids": [882648, 882649, 882650, 882659, 882713, 882754, 882783, 882820, 883048, 883346, 883573, 883640, 883657, 883853, 883859, 883870, 883908, 889153, 889784, 1043862, 1209986, 1250025, 1250685, 1250690, 1436153], "question_author": ""}
{"qid": 639, "query": "What are the Axiom of Choice and Axiom of Determinacy?", "score": 112, "views": 10444, "answer_pids": [27977, 27984, 27990, 119011], "question_author": "robin girard"}
{"qid": 640, "query": "Least prime of the form $38^n+31$", "score": 112, "views": 4237, "answer_pids": [316990, 332328, 344901], "question_author": "jonsca"}
{"qid": 641, "query": "Different ways to prove there are infinitely many primes?", "score": 111, "views": 21009, "answer_pids": [26649, 26650, 26651, 26652, 26655, 26657, 26660, 26666, 26673, 26703, 44884, 129237, 202656, 273917, 286862, 288822, 650971, 814152, 1075905, 1366389], "question_author": ""}
{"qid": 642, "query": "Can you give an example of a complex math problem that is easy to solve?", "score": 111, "views": 17043, "answer_pids": [46206, 46216, 46223, 46239, 46242, 46340, 46631, 46650, 46710, 142517, 403451, 445506, 961701, 1208580], "question_author": "Jus12"}
{"qid": 643, "query": "Proof of Frullanis theorem", "score": 111, "views": 11662, "answer_pids": [32783, 129567, 210403, 531952, 659556, 793161, 890811], "question_author": ""}
{"qid": 644, "query": "More than 99% of groups of order less than 2000 are of order 1024?", "score": 111, "views": 10442, "answer_pids": [117438, 117439], "question_author": ""}
{"qid": 645, "query": "Find all functions $f$ such that if $a+b$ is a square, then $f(a)+f(b)$ is a square", "score": 111, "views": 3708, "answer_pids": [659066], "question_author": "Larry Wang"}
{"qid": 646, "query": "Does a four-variable analog of the Hall-Witt identity exist?", "score": 111, "views": 3492, "answer_pids": [370923, 374690], "question_author": ""}
{"qid": 647, "query": "How do I get the square root of a complex number?", "score": 110, "views": 278166, "answer_pids": [23639, 23641, 23643, 23693, 219360, 360050, 411559, 679317, 837266, 844803, 1253323], "question_author": "Hans-Peter Stricker"}
{"qid": 648, "query": "Prove that $||x|-|y||\\le |x-y|$", "score": 110, "views": 166766, "answer_pids": [65023, 659336, 1041996, 1070442, 1296740, 1315493, 1368767], "question_author": ""}
{"qid": 649, "query": "Prove that simultaneously diagonalizable matrices commute", "score": 110, "views": 72371, "answer_pids": [115303], "question_author": ""}
{"qid": 650, "query": "Could someone explain conditional independence?", "score": 110, "views": 63481, "answer_pids": [12690, 655351, 1001710, 1360537], "question_author": "dStulle"}
{"qid": 651, "query": "Good Book On Combinatorics", "score": 110, "views": 53635, "answer_pids": [8494, 8495, 8498, 8505, 8513, 8514, 8580, 8584, 14963, 14983, 14984, 16301, 18407, 18418, 43103, 635550, 682631, 713623], "question_author": "John Smith"}
{"qid": 652, "query": "A good way to retain mathematical understanding?", "score": 110, "views": 33788, "answer_pids": [325788, 325792, 325795, 325799, 325813, 325841, 325898, 326134, 326197, 326221, 326510, 326570, 326967, 328469, 354378, 698828, 864196], "question_author": ""}
{"qid": 653, "query": "Why is it important for a matrix to be square?", "score": 110, "views": 33701, "answer_pids": [1037624, 1037625, 1037631, 1037645, 1037771, 1037822, 1039150], "question_author": "Daniel Standage"}
{"qid": 654, "query": "Are there real world applications of finite group theory?", "score": 110, "views": 31546, "answer_pids": [154550, 154551, 154554, 154623, 154658, 154812, 154825, 154985], "question_author": ""}
{"qid": 655, "query": "Why cant calculus be done on the rational numbers?", "score": 110, "views": 16149, "answer_pids": [724032, 724042, 724045, 724046, 724053, 724072, 724088, 724131, 724392, 724527], "question_author": "qazwsx"}
{"qid": 656, "query": "Why is there no remainder in multiplication", "score": 110, "views": 12354, "answer_pids": [235181, 235184, 235190, 235195, 235205, 235209, 235240, 235250, 235261, 235263, 235269, 235274, 235383, 235422, 235435, 235466, 235467, 235478, 235533], "question_author": "qazwsx"}
{"qid": 657, "query": "What are the issues in modern set theory?", "score": 110, "views": 11334, "answer_pids": [13598, 13607, 13905, 13913, 13960, 619973, 1265129], "question_author": ""}
{"qid": 658, "query": "What makes a theorem fundamental?", "score": 110, "views": 6148, "answer_pids": [398873, 399288], "question_author": ""}
{"qid": 659, "query": "Why is an average of an average usually incorrect?", "score": 109, "views": 296316, "answer_pids": [49788, 49789, 760631], "question_author": "Tom Boardman"}
{"qid": 660, "query": "Finding a primitive root of a prime number", "score": 109, "views": 147910, "answer_pids": [67962, 336425, 448393, 1390887], "question_author": ""}
{"qid": 661, "query": "Do factorials really grow faster than exponential functions?", "score": 109, "views": 129571, "answer_pids": [166571, 166573, 166581, 166689, 166803, 171612, 171613, 171616, 196325, 304947, 654180, 701205], "question_author": "serg"}
{"qid": 662, "query": "What exactly is the difference between a derivative and a total derivative?", "score": 109, "views": 111371, "answer_pids": [87835, 271229, 502693, 517573, 582565, 935939, 1294419], "question_author": ""}
{"qid": 663, "query": "Nice proofs of $\\zeta(4) = \\frac{\\pi^4}{90}$?", "score": 109, "views": 46092, "answer_pids": [15400, 15405, 22193, 173913, 376541, 377490, 378028, 418842, 691884, 692267, 902728, 903788, 1092111, 1155690, 1293744], "question_author": ""}
{"qid": 664, "query": "Why, historically, do we multiply matrices as we do?", "score": 109, "views": 23328, "answer_pids": [131691, 131692, 351393], "question_author": "Justin L."}
{"qid": 665, "query": "Why do we still do symbolic math?", "score": 109, "views": 18221, "answer_pids": [374746, 374749, 374752, 374759, 374764, 374799, 374801, 374804, 374831, 374838, 374846, 374884, 375001, 375007, 375377, 375721, 376354, 376792, 376912, 377424, 380988, 439374, 444440, 508325], "question_author": ""}
{"qid": 666, "query": "Fastest way to meet, without communication, on a sphere?", "score": 109, "views": 10489, "answer_pids": [491123, 491128, 491213, 491214, 491262, 491326, 491341, 491515, 491525, 491673], "question_author": ""}
{"qid": 667, "query": "Grothendieck s question - any update?", "score": 109, "views": 7117, "answer_pids": [620776, 632865, 633247], "question_author": "nico"}
{"qid": 668, "query": "How to sum this series for $\\pi/2$ directly?", "score": 109, "views": 6595, "answer_pids": [40792, 40793, 40799, 40929, 1150680], "question_author": "Jamie Banks"}
{"qid": 669, "query": "In a family with two children, what are the chances, if one of the children is a girl, that both children are girls?", "score": 108, "views": 251571, "answer_pids": [8427, 8428, 8431, 8434, 8441, 8525, 8533, 8572, 180166, 180568, 292754, 584950, 694243, 694245, 694283, 818584, 1197538], "question_author": "Gabriel Fair"}
{"qid": 670, "query": "Can someone clearly explain about the lim sup and lim inf?", "score": 108, "views": 72571, "answer_pids": [229504, 229505, 229550], "question_author": ""}
{"qid": 671, "query": "How can I understand and prove the sum and difference formulas in trigonometry?", "score": 108, "views": 46490, "answer_pids": [793, 794, 796, 797, 798, 823, 280403, 764611, 974978, 1034479, 1454699], "question_author": "Gabriel Fair"}
{"qid": 672, "query": "Is zero odd or even?", "score": 108, "views": 37041, "answer_pids": [8695, 8697, 8713, 8760, 8768, 127395, 148445, 291349, 292189, 457515, 493176], "question_author": null}
{"qid": 673, "query": "Continuity of the roots of a polynomial in terms of its coefficients", "score": 108, "views": 15941, "answer_pids": [33507, 33508, 33514, 33515, 33519, 548233, 1333144, 1335669], "question_author": ""}
{"qid": 674, "query": "In calculus, which questions can the naive ask that the learned cannot answer?", "score": 108, "views": 14757, "answer_pids": [376457, 376461, 376483, 376485, 376490, 376501, 376609, 376646, 376675, 377016, 377306, 377342], "question_author": "Tom Stephens"}
{"qid": 675, "query": "Why are mathematical proofs that rely on computers controversial?", "score": 108, "views": 12061, "answer_pids": [282823, 282825, 282831, 283005, 283288, 283354, 283460, 283490, 284223, 285353], "question_author": "qazwsx"}
{"qid": 676, "query": "Getting Students to Not Fear Confusion", "score": 108, "views": 10545, "answer_pids": [92067, 92182, 92253, 936155], "question_author": "voix"}
{"qid": 677, "query": "Topology: The Board Game", "score": 108, "views": 10329, "answer_pids": [161630, 176659, 202963], "question_author": "John Smith"}
{"qid": 678, "query": "Fibonacci number that ends with 2014 zeros?", "score": 108, "views": 8497, "answer_pids": [370930, 370937, 371305, 371734, 1050877], "question_author": ""}
{"qid": 679, "query": "What is the smallest number of $45^\\circ-60^\\circ-75^\\circ$ triangles that a square can be divided into?", "score": 108, "views": 6541, "answer_pids": [37693, 131933, 140391, 367152], "question_author": "Albert"}
{"qid": 680, "query": "Generalization of Liouvilles theorem", "score": 108, "views": 3837, "answer_pids": [875724], "question_author": "Nicolas Rosewick"}
{"qid": 681, "query": "Complete course of self-study", "score": 107, "views": 59937, "answer_pids": [91715, 91727, 91729, 91732, 91735, 91742, 91748, 91754, 91762, 91784, 91787, 91796, 91817, 91820, 91923, 115730, 115733, 175054, 204673, 283177], "question_author": "Poshpaws"}
{"qid": 682, "query": "Motivation for Ramanujans mysterious $\\pi$ formula", "score": 107, "views": 25855, "answer_pids": [7960, 8090, 21657, 203385], "question_author": ""}
{"qid": 683, "query": "What is the most unusual proof you know that $\\sqrt{2}$ is irrational?", "score": 107, "views": 8840, "answer_pids": [524219, 524222, 524225, 524228, 524231, 524243, 524250, 524313, 524338, 527088, 528997, 531922, 532150, 541956, 545192, 753368, 795471, 795473, 1451289], "question_author": "Juan A. Navarro"}
{"qid": 684, "query": "How do we know an $ \\aleph_1 $ exists at all?", "score": 107, "views": 7824, "answer_pids": [24974], "question_author": "Daniel Standage"}
{"qid": 685, "query": "Connected metric spaces with disjoint open balls", "score": 107, "views": 5364, "answer_pids": [169200, 876202], "question_author": ""}
{"qid": 686, "query": "Why is the volume of a sphere $\\frac{4}{3}\\pi r^3$?", "score": 106, "views": 72063, "answer_pids": [96, 100, 106, 302, 791, 826, 2362, 2364, 15789, 15895, 30091, 98223, 146928, 238904, 530996, 544820, 566529, 1245710], "question_author": "Dale"}
{"qid": 687, "query": "Incremental averageing", "score": 106, "views": 68629, "answer_pids": [54967, 297409, 326655, 707085], "question_author": "Casebash"}
{"qid": 688, "query": "How to show that a set of discontinuous points of an increasing function is at most countable", "score": 106, "views": 37241, "answer_pids": [44319, 142705, 970468, 1211381], "question_author": ""}
{"qid": 689, "query": "If I flip a coin 1000 times in a row and it lands on heads all 1000 times, what is the probability that its an unfair coin?", "score": 106, "views": 32638, "answer_pids": [536977, 537004, 537016, 537020, 537089, 537452, 538073], "question_author": ""}
{"qid": 690, "query": "What made you choose your research field?", "score": 106, "views": 15181, "answer_pids": [61165, 61178, 61179, 61180, 61192, 124374, 356556], "question_author": ""}
{"qid": 691, "query": "How Do You Actually Do Your Mathematics?", "score": 106, "views": 6542, "answer_pids": [11819, 11821, 11831, 11833, 11835, 11838], "question_author": null}
{"qid": 692, "query": "Is this continuous analogue to the AM\u2013GM inequality true?", "score": 106, "views": 4060, "answer_pids": [848931], "question_author": ""}
{"qid": 693, "query": "Open mathematical questions for which we really, really have no idea what the answer is", "score": 106, "views": 3975, "answer_pids": [420561, 420565, 421422, 422033, 422131, 422138, 422153, 422258, 422278, 422336, 422341, 422367, 422470, 422993, 423006, 424249, 424259, 424262, 424269, 424281, 535535, 650073], "question_author": "Sophie Alpert"}
{"qid": 694, "query": "Is the derivative the natural logarithm of the left-shift?", "score": 106, "views": 3616, "answer_pids": [200740, 200744, 200988], "question_author": ""}
{"qid": 695, "query": "Has Prof. Otelbaev shown existence of strong solutions for Navier-Stokes equations?", "score": 105, "views": 55928, "answer_pids": [284567, 284900, 285645, 289224, 289620, 290743, 292705, 300406], "question_author": "Claus"}
{"qid": 696, "query": "lim sup and lim inf of sequence of sets.", "score": 105, "views": 52514, "answer_pids": [55554, 55555, 55565, 193506, 735423], "question_author": ""}
{"qid": 697, "query": "Why is it hard to prove whether $\\pi+e$ is an irrational number?", "score": 105, "views": 41469, "answer_pids": [80466, 458218], "question_author": "Tom Stephens"}
{"qid": 698, "query": "How do you go about learning mathematics?", "score": 105, "views": 25530, "answer_pids": [2255, 2256, 2259, 2276, 2281], "question_author": "qbi"}
{"qid": 699, "query": "Is $[0,1]$ a countable disjoint union of closed sets?", "score": 105, "views": 18866, "answer_pids": [3708, 3720, 82821, 107231, 505862], "question_author": "Daniel Standage"}
{"qid": 700, "query": "Chatting about mathematics (with real-time LaTeX rendering)", "score": 105, "views": 17459, "answer_pids": [42638, 42639, 42677, 42719, 76298, 407097, 453072, 487001, 487009, 487025, 525767, 544446, 900916, 919182, 1284642, 1284857, 1285411, 1286394, 1287160], "question_author": "Ultimate Gobblement"}
{"qid": 701, "query": "Calculating the integral $\\int_0^\\infty \\frac{\\cos x}{1+x^2}\\, \\mathrm{d}x$ without using complex analysis", "score": 105, "views": 12206, "answer_pids": [5375, 5376, 5404, 332306, 350434, 707445, 708926, 731074, 937130, 1043947, 1168499, 1393291], "question_author": "Justin L."}
{"qid": 702, "query": "Finding the limit of $\\frac {n}{\\sqrt[n]{n!}}$", "score": 105, "views": 11219, "answer_pids": [15468, 15472, 15476, 90115, 354598, 553309, 707564, 1358185], "question_author": ""}
{"qid": 703, "query": "What is the single most influential book every mathematician should read?", "score": 104, "views": 25944, "answer_pids": [157, 158, 159, 161, 162, 163, 169, 194, 225, 590, 944, 954, 961, 964, 969, 1015, 1020, 2507, 12165, 35188, 35191, 46261, 55158, 288528, 487980], "question_author": null}
{"qid": 704, "query": "Logic puzzle: Which octopus is telling the truth?", "score": 104, "views": 18834, "answer_pids": [299852, 299909, 299999, 300076, 300110, 300509, 300535, 300565, 300681, 300736, 301086], "question_author": "asanlua"}
{"qid": 705, "query": "Why is a geometric progression called so?", "score": 104, "views": 14972, "answer_pids": [513868, 513873, 513882, 513917, 536021, 536061], "question_author": "Justin L."}
{"qid": 706, "query": "Why, intuitively, is the order reversed when taking the transpose of the product?", "score": 104, "views": 12738, "answer_pids": [513153, 513174, 513283, 513350, 513359, 513498, 514076, 515819, 516710, 1153991], "question_author": ""}
{"qid": 707, "query": "Prove the theorem on analytic geometry in the picture.", "score": 104, "views": 6883, "answer_pids": [557408, 557412, 557423, 557453, 557738, 557747, 557830, 557866], "question_author": "Jonathan Fischoff"}
{"qid": 708, "query": "Good Physical Demonstrations of Abstract Mathematics", "score": 104, "views": 4786, "answer_pids": [295, 298, 834, 835, 897, 898, 1016, 1409, 3940, 5414, 5597, 5619, 5620, 26102, 26109, 75327, 285304], "question_author": ""}
{"qid": 709, "query": "Can I think of Algebra like this?", "score": 103, "views": 10924, "answer_pids": [591947, 591966, 591967, 591987, 591997, 592063, 592096, 592235, 592280, 592325, 593037, 593825], "question_author": "Mark Probst"}
{"qid": 710, "query": "Ways to evaluate $\\int \\sec \\theta \\, \\mathrm d \\theta$", "score": 103, "views": 9757, "answer_pids": [3904, 3908, 3909, 3917, 27610, 27629, 426625, 426627, 781656, 884285, 1114742, 1261356, 1321993, 1321997], "question_author": ""}
{"qid": 711, "query": "Closed form for $ \\int_0^\\infty {\\frac{{{x^n}}}{{1 + {x^m}}}dx }$", "score": 103, "views": 7336, "answer_pids": [56814, 56816, 56823, 59016, 88805, 184371, 243346, 265252, 266437, 970021], "question_author": ""}
{"qid": 712, "query": "Does this property characterize a space as Hausdorff?", "score": 103, "views": 4560, "answer_pids": [5641, 5747, 627417], "question_author": "Daniel Standage"}
{"qid": 713, "query": "All polynomials with no natural roots and integer coefficients such that $\\phi(n)|\\phi(P(n))$", "score": 103, "views": 3333, "answer_pids": [339411], "question_author": "kennytm"}
{"qid": 714, "query": "How to prove that eigenvectors from different eigenvalues are linearly independent", "score": 102, "views": 92447, "answer_pids": [15945, 15957, 593077, 657718, 1316299], "question_author": "Rory M"}
{"qid": 715, "query": "Lebesgue integral basics", "score": 102, "views": 41864, "answer_pids": [4325, 4327, 4328, 4329, 4330, 4333, 4336, 369685], "question_author": ""}
{"qid": 716, "query": "Why rationalize the denominator?", "score": 102, "views": 28047, "answer_pids": [446702, 446704, 446705, 446708, 446709, 446718, 446813, 446909, 446962, 447129, 447175, 447247, 447905, 448006], "question_author": "Albert"}
{"qid": 717, "query": "Mathematicians ahead of their time?", "score": 102, "views": 15442, "answer_pids": [334826, 334830, 334859, 334875, 334893, 334905, 334916, 334917, 334943, 334946, 335121, 335151, 335163, 335249, 335301, 335376, 335387, 335405, 336806, 338142, 366229, 366276, 366288, 439345, 691091, 832018, 1064016], "question_author": ""}
{"qid": 718, "query": "Are all algebraic integers with absolute value 1 roots of unity?", "score": 102, "views": 14537, "answer_pids": [2566, 2571, 2579, 25747, 453539, 1169377], "question_author": ""}
{"qid": 719, "query": "Can we remove any prime number with this strange process?", "score": 102, "views": 2241, "answer_pids": [1346414], "question_author": "Douglas S. Stones"}
{"qid": 720, "query": "Produce an explicit bijection between rationals and naturals?", "score": 101, "views": 51933, "answer_pids": [440051, 500124, 656688, 924346, 939694, 1064080, 1262458], "question_author": "magnetar"}
{"qid": 721, "query": "Is math built on assumptions?", "score": 101, "views": 20527, "answer_pids": [252594, 252595, 252596, 252597, 252601, 252602, 252610, 252615, 252621, 252666, 252708, 252796, 252826, 252865, 252970, 252974, 253055, 253257, 253276, 856882], "question_author": "TROLLHUNTER"}
{"qid": 722, "query": "Riddles that can be solved by meta-assumptions", "score": 101, "views": 13174, "answer_pids": [983427, 983432, 983453, 983460, 983517, 983542, 983710, 983770, 983786, 984366, 1087702], "question_author": ""}
{"qid": 723, "query": "Why are vector spaces not isomorphic to their duals?", "score": 101, "views": 12142, "answer_pids": [31074, 31090, 31213], "question_author": "endolith"}
{"qid": 724, "query": "What are some examples of mathematics that had unintended useful applications much later?", "score": 101, "views": 9954, "answer_pids": [226754, 226762, 226777, 226780, 226781, 226809, 226817, 226823, 226875, 226935, 226961, 227013, 227015, 227031, 227147, 227659, 227665, 228055, 370822], "question_author": ""}
{"qid": 725, "query": "Division by $0$ and its restrictions", "score": 101, "views": 8443, "answer_pids": [949433, 949434, 949446, 949457, 949458, 949464, 949649, 950656, 951056], "question_author": ""}
{"qid": 726, "query": "Theorems with an extraordinary exception or a small number of sporadic exceptions", "score": 101, "views": 7931, "answer_pids": [93794, 93797, 93799, 93801, 93803, 93805, 93807, 93808, 93813, 93828, 93833, 93842, 93846, 93847, 93848, 93849, 93859, 93902, 93946, 93948, 93950, 93951, 93955, 93958, 93969, 93988, 93989, 94387, 94388, 99412, 192105, 465924, 490504, 880594, 880659, 1061787, 1441980], "question_author": "Gordon Gustafson"}
{"qid": 727, "query": "True or false? $x^2\\ne x\\implies x\\ne 1$", "score": 101, "views": 5928, "answer_pids": [46144, 46145, 132869], "question_author": "Mark Eichenlaub"}
{"qid": 728, "query": "$\\int_{-\\infty}^{+\\infty} e^{-x^2} dx$ with complex analysis", "score": 100, "views": 25155, "answer_pids": [18651, 18654, 18687], "question_author": "Gergana Vandova"}
{"qid": 729, "query": "An integral involving Airy functions $\\int_0^\\infty\\frac{x^p}{\\operatorname{Ai}^2 x + \\operatorname{Bi}^2 x}\\mathrm dx$", "score": 100, "views": 17910, "answer_pids": [238701, 239907, 241059, 241860], "question_author": "Thomas Ingalls"}
{"qid": 730, "query": "Does notation ever become easier?", "score": 100, "views": 13211, "answer_pids": [772652, 772653, 772689, 772716, 772745, 772772, 773554, 773562, 773742, 773820], "question_author": ""}
{"qid": 731, "query": "Mathematician vs. Computer: A Game", "score": 100, "views": 11340, "answer_pids": [414002, 414004, 414331, 414420, 414440, 414812, 415072, 415304, 415821], "question_author": "InquilineKea"}
{"qid": 732, "query": "Non-textbook Math book recommendation to read to my kids", "score": 100, "views": 8701, "answer_pids": [939496, 939500, 939508, 939592, 939619, 939659, 939668, 939671, 939683, 939686, 939688, 939707, 939744, 939890, 939915, 940061, 940139, 940317, 940690], "question_author": "Nick T"}
{"qid": 733, "query": "Cardioid in coffee mug?", "score": 100, "views": 7036, "answer_pids": [606811, 606854, 607840], "question_author": "InquilineKea"}
{"qid": 734, "query": "Direct proof that the wedge product preserves integral cohomology classes?", "score": 100, "views": 5510, "answer_pids": [92281, 667776], "question_author": "Sami"}
{"qid": 735, "query": "How did Hermite calculate $e^{\\pi\\sqrt{163}}$ in 1859?", "score": 100, "views": 4410, "answer_pids": [52649, 52655, 52684, 1188692], "question_author": "Justin L."}
{"qid": 736, "query": "What does it mean to have a determinant equal to zero?", "score": 99, "views": 431870, "answer_pids": [168274, 168277, 168278, 168279, 168281, 288781, 831870, 1138766, 1159617], "question_author": ""}
{"qid": 737, "query": "Why is the volume of a cone one third of the volume of a cylinder?", "score": 99, "views": 169479, "answer_pids": [380, 382, 385, 16042, 119855, 238890, 529933, 571786, 898077, 1456376], "question_author": ""}
{"qid": 738, "query": "Dont see the point of the Fundamental Theorem of Calculus.", "score": 99, "views": 16596, "answer_pids": [438133, 438160, 438217, 438231, 438237, 438249, 438255, 438266, 438274, 439434, 439520], "question_author": "Ultimate Gobblement"}
{"qid": 739, "query": "Comparing $\\pi^e$ and $e^\\pi$ without calculating them", "score": 99, "views": 13088, "answer_pids": [4569, 4570, 4584, 5400, 5406, 197608, 456679, 616225, 895495, 1045728, 1265386, 1444372], "question_author": ""}
{"qid": 740, "query": "Is Bayes Theorem really that interesting?", "score": 99, "views": 10003, "answer_pids": [1401450, 1401482, 1401587, 1401657, 1401814, 1401856, 1403911], "question_author": "Poshpaws"}
{"qid": 741, "query": "Unconventional mathematics books", "score": 99, "views": 7326, "answer_pids": [588400, 588401, 588420, 588911, 588918, 589370, 589611, 591504, 591534, 591547, 592931, 592948, 593190, 671200, 671202, 716612, 764252, 764261, 788637], "question_author": "Greg Graviton"}
{"qid": 742, "query": "Does LH\u00f4pitals work the other way?", "score": 99, "views": 6614, "answer_pids": [260907, 260908, 260913, 260968, 262868, 262876], "question_author": "Isaac"}
{"qid": 743, "query": "Geometric interpretation of the Riemann-Roch for curves", "score": 99, "views": 3312, "answer_pids": [724952], "question_author": "Marcin Kotowski"}
{"qid": 744, "query": "Whats the difference between theorem, lemma and corollary?", "score": 98, "views": 53715, "answer_pids": [216383, 216384, 302684, 429825, 429830, 866618, 1079531], "question_author": ""}
{"qid": 745, "query": "Unusual mathematical terms", "score": 98, "views": 36971, "answer_pids": [453239, 453252, 453264, 453280, 453287, 453293, 453294, 453302, 453304, 453306, 453307, 453308, 453311, 453315, 453320, 453325, 453333, 453337, 453353, 453375, 453384, 453391, 453392, 453418, 453419, 453464, 453472, 453614, 453704, 453772, 453801, 453813, 454409, 454417, 454551, 455016, 455019, 456864], "question_author": "InquilineKea"}
{"qid": 746, "query": "Is infinity an odd or even number?", "score": 98, "views": 26821, "answer_pids": [26156, 26159, 26160, 26161, 26173], "question_author": ""}
{"qid": 747, "query": "How do you find the center of a circle with a pencil and a book?", "score": 98, "views": 19966, "answer_pids": [102971, 102990, 103018, 103087, 103089, 103140, 103162, 103180, 103256, 103539, 103554], "question_author": ""}
{"qid": 748, "query": "Using we have in maths papers", "score": 98, "views": 14818, "answer_pids": [296536, 296540, 296543, 296631, 296662, 296835, 296902, 296916, 296962, 297510, 299034, 646264], "question_author": ""}
{"qid": 749, "query": "How to learn from proofs?", "score": 98, "views": 11628, "answer_pids": [84612, 84617, 84622, 84630, 84660, 84862], "question_author": ""}
{"qid": 750, "query": "What are Some Tricks to Remember Fatous Lemma?", "score": 98, "views": 11321, "answer_pids": [118132, 118137, 118413, 118414, 423134, 426726, 678345, 846042, 861294], "question_author": "Chao Xu"}
{"qid": 751, "query": "Why dont analysts do category theory?", "score": 98, "views": 11260, "answer_pids": [85341, 85342, 85345, 85347, 85360, 85367], "question_author": ""}
{"qid": 752, "query": "Defining a manifold without reference to the reals", "score": 98, "views": 4724, "answer_pids": [28234, 28237, 30301, 30337], "question_author": "InquilineKea"}
{"qid": 753, "query": "Does a non-trivial solution exist for $f(x)=f(f(x))$?", "score": 98, "views": 4603, "answer_pids": [469195, 469605, 470280, 476268], "question_author": "Piotr Migdal"}
{"qid": 754, "query": "A community project: prove (or disprove) that $\\sum_{n\\geq 1}\\frac{\\sin(2^n)}{n}$ is convergent", "score": 98, "views": 4321, "answer_pids": [761109, 805706, 808246, 914732], "question_author": "Rory M"}
{"qid": 755, "query": "A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational", "score": 98, "views": 2576, "answer_pids": [440904, 440938, 441317, 1210376], "question_author": "Rory M"}
{"qid": 756, "query": "Expectation of the maximum of gaussian random variables", "score": 97, "views": 61932, "answer_pids": [46372, 46425], "question_author": "Nadja"}
{"qid": 757, "query": "Simplest or nicest proof that $1+x \\le e^x$", "score": 97, "views": 36048, "answer_pids": [233897, 233898, 233900, 233902, 233906, 233919, 233928, 233959, 233963, 233975, 233978, 233986, 234133, 234152, 234315, 548210, 557729, 589716, 938618, 938656, 977965, 1173422], "question_author": "David"}
{"qid": 758, "query": "What would base $1$ be?", "score": 97, "views": 28847, "answer_pids": [175383, 175386, 175425, 175460, 242369, 330006], "question_author": ""}
{"qid": 759, "query": "Should I put number combinations like 1111111 onto my lottery ticket?", "score": 97, "views": 22090, "answer_pids": [218282, 218283, 218284, 218285, 218286, 218294, 218303, 218323, 218337, 218365, 218377, 218503, 218579], "question_author": "Xinus"}
{"qid": 760, "query": "Continuous bijection from $(0,1)$ to $[0,1]$", "score": 97, "views": 22024, "answer_pids": [22554, 22555, 22558, 22559, 1230592], "question_author": "Larry Wang"}
{"qid": 761, "query": "How is a system of axioms different from a system of beliefs?", "score": 97, "views": 19180, "answer_pids": [91880, 91881, 91883, 94070, 368695], "question_author": "Dale"}
{"qid": 762, "query": "Why cant you add apples and oranges, but you can multiply and divide them?", "score": 97, "views": 14551, "answer_pids": [341955, 341968, 341976, 342103, 342232, 342632, 342696, 343873], "question_author": "Matt Cashatt"}
{"qid": 763, "query": "How can adding an infinite number of rationals yield an irrational number?", "score": 97, "views": 11498, "answer_pids": [59673, 59675, 59677, 59678, 59688, 59759, 550206], "question_author": "InquilineKea"}
{"qid": 764, "query": "Example of a very simple math statement in old literature which is (verbatim) a pain to understand", "score": 97, "views": 7944, "answer_pids": [727119, 727121, 727169, 727172, 727189, 727931, 727967, 728162, 728451, 728475, 728524, 729510], "question_author": "InquilineKea"}
{"qid": 765, "query": "Whats the largest possible volume of a taco, and how do I make one that big?", "score": 97, "views": 6610, "answer_pids": [47867], "question_author": ""}
{"qid": 766, "query": "Limit of sequence in which each term is defined by the average of preceding two terms", "score": 97, "views": 6521, "answer_pids": [858582, 858583, 858590, 858847, 858881, 858980, 858998, 859256, 859898, 861103, 861163, 939092, 956107], "question_author": "Casebash"}
{"qid": 767, "query": "Why are all the interesting constants so small?", "score": 97, "views": 6184, "answer_pids": [62212, 62219, 62424, 62576, 62608, 263793, 348393], "question_author": ""}
{"qid": 768, "query": "Help me put these enormous numbers in order: googol, googol-plex-bang, googol-stack and so on", "score": 97, "views": 4910, "answer_pids": [38409, 48139], "question_author": "Douglas S. Stones"}
{"qid": 769, "query": "$4494410$ and friends", "score": 97, "views": 4410, "answer_pids": [246091], "question_author": ""}
{"qid": 770, "query": "A semigroup $X$ is a group iff for every $g\\in X$, $\\exists! x\\in X$ such that $gxg = g$", "score": 97, "views": 3902, "answer_pids": [122924, 122927], "question_author": "SplashHit"}
{"qid": 771, "query": "Arithmetic-geometric mean of 3 numbers", "score": 97, "views": 3252, "answer_pids": [1355759], "question_author": "Vivi"}
{"qid": 772, "query": "Geometric way to view the truncated braid groups?", "score": 97, "views": 1476, "answer_pids": [1314418, 1393573], "question_author": ""}
{"qid": 773, "query": "Prove $\\operatorname{rank}A^TA=\\operatorname{rank}A$ for any $A\\in M_{m \\times n}$", "score": 96, "views": 65076, "answer_pids": [165656, 165756, 492174], "question_author": ""}
{"qid": 774, "query": "How to find a general sum formula for the series: 5+55+555+5555+.....?", "score": 96, "views": 46816, "answer_pids": [424987, 424988, 424992, 424993, 424997, 425475, 425863, 426438, 426456, 426725, 426750, 426807, 922091, 1018120], "question_author": ""}
{"qid": 775, "query": "What is the difference between a class and a set?", "score": 96, "views": 35746, "answer_pids": [70607, 70609, 70622, 80991, 179749, 285742], "question_author": "Matt Cashatt"}
{"qid": 776, "query": "Why is $\\frac{987654321}{123456789} = 8.0000000729?!$", "score": 96, "views": 32922, "answer_pids": [186133, 186140, 186151, 186323, 186578, 1034730, 1071082], "question_author": "Xinus"}
{"qid": 777, "query": "The Langlands program for beginners", "score": 96, "views": 29805, "answer_pids": [26213, 26215, 26225, 74191, 660003], "question_author": ""}
{"qid": 778, "query": "What makes elementary functions elementary?", "score": 96, "views": 15711, "answer_pids": [60526, 60538, 60593, 69593, 69597, 151190, 344091], "question_author": ""}
{"qid": 779, "query": "Why would I want to multiply two polynomials?", "score": 96, "views": 10606, "answer_pids": [6463, 6464, 6465, 6471, 6477, 6497, 6523, 6526, 6527, 6529, 6562, 6563, 10120], "question_author": ""}
{"qid": 780, "query": "If $x$, $y$, $x+y$, and $x-y$ are prime numbers, what is their sum?", "score": 96, "views": 9799, "answer_pids": [121563, 121564, 121565, 121745], "question_author": ""}
{"qid": 781, "query": "Mathematicians Tensors vs. Physicists Tensors", "score": 96, "views": 9542, "answer_pids": [975654, 975807, 975809, 976123, 976216], "question_author": "temi382"}
{"qid": 782, "query": "Is the determinant that shows up accidental?", "score": 96, "views": 4896, "answer_pids": [1002786, 1002811, 1002813, 1002816, 1002851], "question_author": "magnetar"}
{"qid": 783, "query": "Given a die, what is the probability that the second roll of a die will be less than the first roll?", "score": 95, "views": 97336, "answer_pids": [52030, 52032, 52033, 52034, 52051, 52070, 52071, 52099, 431052], "question_author": "Reinstate Monica - Goodbye SE"}
{"qid": 784, "query": "Why is the set of all sets a paradox, in Laymans terms?", "score": 95, "views": 48883, "answer_pids": [98, 104, 115, 123, 933, 1675, 1680, 118148, 385327, 385342, 508790, 954862], "question_author": ""}
{"qid": 785, "query": "Evaluate the integral: $\\int_{0}^{1} \\frac{\\ln(x+1)}{x^2+1} \\mathrm dx$", "score": 95, "views": 33999, "answer_pids": [78737, 78738, 78742, 159810, 193055, 797691, 1350934], "question_author": "Dale"}
{"qid": 786, "query": "What seemingly innocuous results in mathematics require advanced proofs?", "score": 95, "views": 15371, "answer_pids": [899727, 899740, 899750, 899767, 899824, 899836, 899879, 900258, 900346, 900403, 900459, 900501, 900821, 900877, 901134, 901408, 901454, 954110], "question_author": ""}
{"qid": 787, "query": "Prove $\\left(\\frac{2}{5}\\right)^{\\frac{2}{5}}<\\ln{2}$", "score": 95, "views": 12211, "answer_pids": [179754, 180031, 227231, 615693, 622024, 622035], "question_author": ""}
{"qid": 788, "query": "How to convince a math teacher of this simple and obvious fact?", "score": 95, "views": 12155, "answer_pids": [208146, 208149, 208150, 208152, 208235, 208244, 208282, 208283, 208348, 208479, 210080, 380355], "question_author": "InquilineKea"}
{"qid": 789, "query": "What kind of symmetry is the symmetric group about?", "score": 95, "views": 10314, "answer_pids": [24688, 24689, 24692, 24696, 24709, 24724], "question_author": "Vass"}
{"qid": 790, "query": "What are good math habits that have improved your mathematical practice?$ $", "score": 95, "views": 7759, "answer_pids": [845327, 845416, 845432, 845438, 845463, 846504, 898332, 1351702, 1353891], "question_author": null}
{"qid": 791, "query": "math fallacy problem: $-1= (-1)^3 = (-1)^{6/2} = \\sqrt{(-1)^6}= 1$?", "score": 95, "views": 6894, "answer_pids": [220378, 220379, 220381, 220383, 220385, 220613, 220711, 220878, 309947, 317933, 326837], "question_author": "Snowman"}
{"qid": 792, "query": "Cover of G\u00f6del, Escher, Bach", "score": 95, "views": 6640, "answer_pids": [897940, 897946, 898236, 1228541], "question_author": ""}
{"qid": 793, "query": "Problems that become easier in a more general form", "score": 95, "views": 4046, "answer_pids": [381517, 381531, 381540, 381542, 381545, 381618, 381623, 383368, 419318, 468713, 468743], "question_author": "Alexander"}
{"qid": 794, "query": "How prove this nice limit $\\lim\\limits_{n\\to\\infty}\\frac{a_{n}}{n}=\\frac{12}{\\log{432}}$", "score": 95, "views": 3027, "answer_pids": [227219, 231727, 282754], "question_author": "Vass"}
{"qid": 795, "query": "When does a sequence of rotated-and-circumscribed rectangles converge to a square?", "score": 95, "views": 1801, "answer_pids": [869628, 869629], "question_author": null}
{"qid": 796, "query": "Why is the area under a curve the integral?", "score": 94, "views": 107133, "answer_pids": [8549, 8550, 149405, 693853, 1025246], "question_author": null}
{"qid": 797, "query": "Good 1st PDE book for self study", "score": 94, "views": 64766, "answer_pids": [1682, 1683, 1684, 1685, 1686, 1691, 1692, 600899, 1016107], "question_author": "Poshpaws"}
{"qid": 798, "query": "Importance of matrix rank", "score": 94, "views": 59036, "answer_pids": [11647, 11651, 11655, 108681], "question_author": "InquilineKea"}
{"qid": 799, "query": "Much less than, what does that mean?", "score": 94, "views": 54876, "answer_pids": [595798, 595801, 595803, 596364, 597754, 598053, 854038], "question_author": "Claudiu"}
{"qid": 800, "query": "Simple beautiful math proof", "score": 94, "views": 38737, "answer_pids": [318508, 318510, 318513, 318514, 318516, 318524, 318532, 318553, 318578, 318584, 318589, 318763, 318832, 318867, 318994, 319003, 319015, 319029, 319053, 319063, 319202, 319220, 319224, 334213, 367349, 367387, 385918], "question_author": "Gordon Gustafson"}
{"qid": 801, "query": "Anecdotes about famous mathematicians or physicists", "score": 94, "views": 37366, "answer_pids": [15685, 15693, 15694, 15697, 15701, 15702, 15706, 15725, 15741, 24489, 27375, 46776, 161294, 443973, 473223, 473236, 509124, 525008, 763743], "question_author": "Gordon Gustafson"}
{"qid": 802, "query": "Why do bell curves appear everywhere?", "score": 94, "views": 24942, "answer_pids": [894355, 894356, 894358, 894377, 894424, 894474, 894568, 894574, 894599, 895966, 897626], "question_author": "Nick T"}
{"qid": 803, "query": "What are the differences between class, set, family, and collection?", "score": 94, "views": 24803, "answer_pids": [87178, 87186, 87194], "question_author": "Shane"}
{"qid": 804, "query": "What is the solution to Nashs problem presented in A Beautiful Mind?", "score": 94, "views": 21304, "answer_pids": [194674, 735779], "question_author": null}
{"qid": 805, "query": "Factorial and exponential dual identities", "score": 94, "views": 13006, "answer_pids": [11320, 13048, 60518, 621559], "question_author": ""}
{"qid": 806, "query": "The pepperoni pizza problem", "score": 94, "views": 11914, "answer_pids": [611918, 611940, 611966, 612070], "question_author": null}
{"qid": 807, "query": "How to evaluate $\\int_{0}^{\\infty} \\frac{x^{-\\mathfrak{i}a}}{x^2+bx+1} \\,\\mathrm{d}x$ using complex analysis?", "score": 94, "views": 8522, "answer_pids": [943489, 943529, 945394], "question_author": ""}
{"qid": 808, "query": "A Case Against the Math Gene", "score": 94, "views": 6099, "answer_pids": [38396, 38404, 38419, 38566, 39088, 78354, 464565], "question_author": ""}
{"qid": 809, "query": "Complexity class of comparison of power towers", "score": 94, "views": 5364, "answer_pids": [184240, 187054, 187357], "question_author": "Amelio Vazquez-Reina"}
{"qid": 810, "query": "Does $\\lfloor \\sqrt{p} \\rfloor$ generate all natural numbers?", "score": 94, "views": 2121, "answer_pids": [271382, 271384], "question_author": "Kit"}
{"qid": 811, "query": "Pointwise vs. Uniform Convergence", "score": 93, "views": 74113, "answer_pids": [268667, 268673, 387658, 914341], "question_author": ""}
{"qid": 812, "query": "What is the algebraic intuition behind Vieta jumping in IMO1988 Problem 6?", "score": 93, "views": 46792, "answer_pids": [730202, 730254, 730386, 733581, 734562, 935377, 1153172], "question_author": ""}
{"qid": 813, "query": "GRE Subject Test - Past Papers, Books, Advice", "score": 93, "views": 46666, "answer_pids": [739203, 739311], "question_author": ""}
{"qid": 814, "query": "Can the golden ratio accurately be expressed in terms of $e$ and $\\pi$", "score": 93, "views": 19092, "answer_pids": [212384, 212394, 212402, 212451, 212467, 212473, 212480, 212508, 212575, 212725, 212979, 213166, 634544, 639912, 933594, 1056750, 1296174], "question_author": "David Z"}
{"qid": 815, "query": "How do you explain the concept of logarithm to a five year old?", "score": 93, "views": 18649, "answer_pids": [65778, 65780, 65781, 65783, 65785, 65792, 65797, 65799, 65804, 65811, 65812, 65829, 65881, 65886, 66106, 66569, 483039, 483045, 483050], "question_author": "sigoldberg1"}
{"qid": 816, "query": "If $f_k \\to f$ a.e. and the $L^p$ norms converge, then $f_k \\to f$ in $L^p$", "score": 93, "views": 13678, "answer_pids": [20937, 27421], "question_author": ""}
{"qid": 817, "query": "What is a real-world metaphor for irrational numbers?", "score": 93, "views": 13592, "answer_pids": [787002, 787005, 787006, 787013, 787015, 787020, 787023, 787024, 787033, 787083, 787088, 787094, 787124, 787137, 787139, 787146, 787148, 787236, 787243, 787550, 787635, 788285, 788769, 791891], "question_author": "Amir Rezaei"}
{"qid": 818, "query": "Intuitive explanation of Cauchys Integral Formula in Complex Analysis", "score": 93, "views": 12658, "answer_pids": [2411, 2415, 2425, 2428, 2429, 2431, 2452, 2457, 4752, 4764, 10142, 17716, 534237, 626320], "question_author": ""}
{"qid": 819, "query": "Why should I believe in weak solutions to PDEs?", "score": 93, "views": 9999, "answer_pids": [1199800, 1199803, 1199812, 1199871, 1200006, 1200169, 1200197, 1200540, 1201769], "question_author": ""}
{"qid": 820, "query": "Explain homotopy to me", "score": 93, "views": 8559, "answer_pids": [642118, 642130, 642151, 642155, 642164, 642311, 642355, 642537, 642539, 642774], "question_author": "user701510"}
{"qid": 821, "query": "Prove elementarily that $\\sqrt[n+1] {(n+1)!} - \\sqrt[n] {n!}$ is strictly decreasing", "score": 93, "views": 5467, "answer_pids": [142860], "question_author": "Anixx"}
{"qid": 822, "query": "Root Calculation by Hand", "score": 93, "views": 3482, "answer_pids": [64991, 64993, 65003, 65005, 65018], "question_author": "Noldorin"}
{"qid": 823, "query": "What does the dot product of two vectors represent?", "score": 92, "views": 176401, "answer_pids": [346670, 346671, 346672, 659471, 961467, 977410, 1086566], "question_author": ""}
{"qid": 824, "query": "Whats the difference between simple induction and strong induction?", "score": 92, "views": 157018, "answer_pids": [238930, 727191], "question_author": "mring"}
{"qid": 825, "query": "What is the most elegant proof of the Pythagorean theorem?", "score": 92, "views": 60224, "answer_pids": [477, 484, 805, 807, 820, 6180, 6185, 38127, 122416, 125767, 125779, 213765, 214771, 252564, 271091, 548568, 549193], "question_author": ""}
{"qid": 826, "query": "Is the vector cross product only defined for 3D?", "score": 92, "views": 59834, "answer_pids": [93731, 93733, 93734, 93750, 203030, 216150], "question_author": ""}
{"qid": 827, "query": "Best Maths Books for Non-Mathematicians", "score": 92, "views": 49296, "answer_pids": [166, 167, 195, 203, 207, 221, 223, 227, 254, 641, 5299, 5302, 5304, 6812, 6813, 7076, 7078, 7079, 90432, 163240, 163264, 165207, 188742, 265628, 405890, 487762, 615497, 674650], "question_author": ""}
{"qid": 828, "query": "Self-studying real analysis \u2014 Tao or Rudin?", "score": 92, "views": 47167, "answer_pids": [372664, 372683, 560580, 885905], "question_author": ""}
{"qid": 829, "query": "What are differences between affine space and vector space?", "score": 92, "views": 44705, "answer_pids": [375934, 375939, 375945, 375946, 377276, 523830, 619241], "question_author": ""}
{"qid": 830, "query": "If $(a_n)\\subset[0,\\infty)$ is non-increasing and $\\sum a_n<\\infty$, then $\\lim{n a_n} = 0$", "score": 92, "views": 39488, "answer_pids": [2736, 2742, 37807, 44317, 228806, 228819, 334513, 460375, 718442, 826268, 827749, 912751, 1373936], "question_author": "Amir Rezaei"}
{"qid": 831, "query": "Is arrow notation for vectors not mathematically mature?", "score": 92, "views": 31802, "answer_pids": [450144, 450148, 450152, 450153, 450155, 450157, 450264, 450272, 450302, 450306, 450322, 450328, 450375, 450650, 450999, 452568, 454844], "question_author": "Anixx"}
{"qid": 832, "query": "I roll a die repeatedly until I get 6, and then count the number of 3s I got. Whats my expected number of 3s?", "score": 92, "views": 23793, "answer_pids": [565162, 565163, 565165, 565167, 565169, 565209, 565638, 566074, 566085, 566137, 566330, 566416, 566574, 566623], "question_author": "Poshpaws"}
{"qid": 833, "query": "What is the difference between a Hamel basis and a Schauder basis?", "score": 92, "views": 22658, "answer_pids": [281762, 281765, 281769, 281870, 281951], "question_author": ""}
{"qid": 834, "query": "Is there any conjecture that has been proved to be solvable/provable but whose direct solution/proof is not yet known?", "score": 92, "views": 13550, "answer_pids": [1061105, 1061111, 1061119, 1061128, 1061159, 1061278, 1061284, 1061375, 1061590, 1062555, 1063074], "question_author": ""}
{"qid": 835, "query": "Results that came out of nowhere.", "score": 92, "views": 12028, "answer_pids": [134064, 134065, 134067, 134069, 134155, 139369, 188504, 188519, 188526, 188557, 434902], "question_author": "stoicfury"}
{"qid": 836, "query": "Can the product of infinitely many elements from $\\mathbb Q$ be irrational?", "score": 92, "views": 8050, "answer_pids": [1004316, 1004317, 1004324, 1004325, 1005161, 1005357], "question_author": "Gergana Vandova"}
{"qid": 837, "query": "Divisibility by 7 rule, and Congruence Arithmetic Laws", "score": 92, "views": 7303, "answer_pids": [373796, 373797, 543986, 686877, 716066, 808756, 813809, 985997, 1050615], "question_author": "Gabriel Fair"}
{"qid": 838, "query": "Can you make a sphere out of a plane?", "score": 92, "views": 7035, "answer_pids": [1286417, 1286648, 1286649, 1286699, 1287147, 1287180], "question_author": "Nick T"}
{"qid": 839, "query": "Cutting sticks puzzle", "score": 92, "views": 6521, "answer_pids": [2413, 3270, 34540, 34803], "question_author": "Piotr Migdal"}
{"qid": 840, "query": "The Right Triangle Game", "score": 92, "views": 2916, "answer_pids": [838427], "question_author": ""}
{"qid": 841, "query": "Geometric interpretation of $\\det(A^T) = \\det(A)$", "score": 91, "views": 97350, "answer_pids": [284219, 284659, 798039, 818653, 971885], "question_author": "Roy Maclean"}
{"qid": 842, "query": "How to prove that exponential grows faster than polynomial?", "score": 91, "views": 73709, "answer_pids": [29472, 29475, 29476, 29479, 29480, 29483, 29484, 29501, 29889, 471220, 780830, 962763, 967311, 1012604, 1241759], "question_author": "Larian LeQuella"}
{"qid": 843, "query": "Alternative proof that $(a^2+b^2)/(ab+1)$ is a square when its an integer", "score": 91, "views": 55977, "answer_pids": [17206, 265665], "question_author": "Roy Maclean"}
{"qid": 844, "query": "Good book for self study of functional analysis", "score": 91, "views": 53696, "answer_pids": [4361, 4366, 4371, 4372, 4389, 4397, 4398, 4615, 15048, 40558, 55864, 65897, 65902, 301186, 372167, 1266335], "question_author": ""}
{"qid": 845, "query": "If $S$ is an infinite $\\sigma$ algebra on $X$ then $S$ is not countable", "score": 91, "views": 30513, "answer_pids": [152557, 152562, 332338, 402500, 409375, 884177, 1305833], "question_author": ""}
{"qid": 846, "query": "$x^y = y^x$ for integers $x$ and $y$", "score": 91, "views": 20201, "answer_pids": [5433, 5437, 6238, 6247, 41287, 373835], "question_author": ""}
{"qid": 847, "query": "100 blue-eyed islanders puzzle: 3 questions", "score": 91, "views": 20110, "answer_pids": [227782, 227792, 227803, 227805, 227828, 227862, 227911, 227937, 227953, 227966, 228010, 228070, 228111, 228186, 228282, 228380, 228417, 230231, 369363], "question_author": "bobthejoe"}
{"qid": 848, "query": "Surprise exam paradox?", "score": 91, "views": 19731, "answer_pids": [50472, 50480, 50504, 50530, 50532, 50551, 50678, 387170, 475109, 754135], "question_author": ""}
{"qid": 849, "query": "Demystify integration of $\\int \\frac{1}{x} \\mathrm dx$", "score": 91, "views": 16636, "answer_pids": [231448, 231458, 231464, 231479, 231493, 231516, 231532, 231543, 231622, 233804, 301442], "question_author": "FrankH"}
{"qid": 850, "query": "Your favourite maths puzzles", "score": 91, "views": 8393, "answer_pids": [348, 349, 353, 354, 355, 389, 727, 1373, 1381, 2447, 2510, 3610, 3613, 3650, 20339, 20342, 20344, 75966, 248075], "question_author": "Pratik Deoghare"}
{"qid": 851, "query": "Number of simple edge-disjoint paths needed to cover a planar graph", "score": 91, "views": 2554, "answer_pids": [357422], "question_author": ""}
{"qid": 852, "query": "Good abstract algebra books for self study", "score": 90, "views": 112663, "answer_pids": [29140, 29141, 29142, 29145, 29325, 203413, 404578, 650246, 1065990, 1151782, 1337731], "question_author": ""}
{"qid": 853, "query": "Does convergence in $L^p$ imply convergence almost everywhere?", "score": 90, "views": 50869, "answer_pids": [70002, 70003, 597138], "question_author": ""}
{"qid": 854, "query": "Is there any difference between mapping and function?", "score": 90, "views": 48486, "answer_pids": [49700, 49701, 49723, 291599, 378752, 650658], "question_author": "Thomas Ingalls"}
{"qid": 855, "query": "Normal subgroup of prime index", "score": 90, "views": 43139, "answer_pids": [82906, 82908, 129996, 143366, 672600, 1098559, 1103668, 1165565], "question_author": "Douglas S. Stones"}
{"qid": 856, "query": "Implies ($\\Rightarrow$) vs. Entails ($\\models$) vs. Provable ($\\vdash$)", "score": 90, "views": 32767, "answer_pids": [137336, 137340, 137434, 260294, 374837, 1280254], "question_author": ""}
{"qid": 857, "query": "Learning mathematics as if an absolute beginner?", "score": 90, "views": 28679, "answer_pids": [12385, 12390, 21948, 25643, 25692, 25756, 25780, 156807, 704766], "question_author": ""}
{"qid": 858, "query": "Why did no student correctly find a pair of $2\\times 2$ matrices with the same determinant and trace that are not similar?", "score": 90, "views": 18553, "answer_pids": [841783, 841847, 842063], "question_author": ""}
{"qid": 859, "query": "Will it become impossible to learn math?", "score": 90, "views": 10407, "answer_pids": [429086, 429133, 429218, 429301, 429308, 429747, 429751, 430138], "question_author": "Poshpaws"}
{"qid": 860, "query": "Should an undergrad accept that some things dont make sense, or study the foundation of mathematics to resolve this?", "score": 90, "views": 8751, "answer_pids": [311842, 311847, 311853, 311860, 311864, 311875, 311883, 311921, 311925, 311969, 312345, 312798, 315011, 798736], "question_author": ""}
{"qid": 861, "query": "Is $0! = 1$ because there is only one way to do nothing?", "score": 90, "views": 8411, "answer_pids": [710873, 710875, 710876, 710899, 710905, 710928, 710938, 711114, 711149, 711186], "question_author": "kieran"}
{"qid": 862, "query": "Conjecture $\\int_0^1\\frac{\\mathrm dx}{\\sqrt{1-x}\\ \\sqrt[4]x\\ \\sqrt[4]{2-x\\,\\sqrt3}}\\stackrel?=\\frac{2\\,\\sqrt2}{3\\,\\sqrt[8]3}\\pi$", "score": 90, "views": 5742, "answer_pids": [250089, 690544], "question_author": "Daniel Standage"}
{"qid": 863, "query": "Why are groups more important than semigroups?", "score": 90, "views": 3820, "answer_pids": [52476, 52493, 52546, 53050, 276353], "question_author": ""}
{"qid": 864, "query": "Can someone explain these strange properties of $10, 11, 12$ and $13$?", "score": 90, "views": 3606, "answer_pids": [826859, 826864, 826866, 827094, 827311], "question_author": ""}
{"qid": 865, "query": "What is the best book for studying discrete mathematics?", "score": 89, "views": 130575, "answer_pids": [945, 974, 977, 1401, 1427, 230007, 234796, 548156, 667467, 1081895], "question_author": ""}
{"qid": 866, "query": "Why cant you square both sides of an equation?", "score": 89, "views": 89638, "answer_pids": [257630, 257632, 257633, 257635, 257645, 257662, 257688, 257699, 257765, 257796, 257804, 257861, 257963, 452435], "question_author": "bobtheowl2"}
{"qid": 867, "query": "For any prime $p > 3$, why is $p^2-1$ always divisible by 24?", "score": 89, "views": 64148, "answer_pids": [511, 512, 515, 548, 562, 683, 1624, 9353, 265930, 265940, 303415, 343981, 477286, 649021, 663840, 1223156, 1223157, 1223158, 1223374], "question_author": "Stefano Borini"}
{"qid": 868, "query": "Is the notorious $n^2 + n + 41$ prime generator the last of its type?", "score": 89, "views": 33252, "answer_pids": [138830, 138847, 138866, 138880, 368348, 537710], "question_author": "ternaryOperator"}
{"qid": 869, "query": "Construct a function which is continuous in $[1,5]$ but not differentiable at $2, 3, 4$", "score": 89, "views": 22921, "answer_pids": [39134, 39149, 39150], "question_author": ""}
{"qid": 870, "query": "Easy math proofs or visual examples to make high school students enthusiastic about math", "score": 89, "views": 21942, "answer_pids": [533701, 533702, 533704, 533706, 533708, 533712, 533719, 533727, 533746, 533756, 533757, 533762, 533781, 533792, 533802, 533812, 533832, 533846, 533890, 533974, 533982, 534051, 534058, 534190, 534267, 534306, 534338, 534583, 534623, 534637, 535422], "question_author": "jonsca"}
{"qid": 871, "query": "Why do we define quotient groups for normal subgroups only?", "score": 89, "views": 20653, "answer_pids": [8018, 8021, 8036], "question_author": "jul"}
{"qid": 872, "query": "Formal proof for $(-1) \\times (-1) = 1$", "score": 89, "views": 20261, "answer_pids": [145271, 145272, 145273, 145275, 145276, 145277, 145280, 145286, 145308, 145596, 145774, 145847, 145864], "question_author": "Jonathan Fischoff"}
{"qid": 873, "query": "Mathematically, why was the Enigma machine so hard to crack?", "score": 89, "views": 18691, "answer_pids": [499551, 499752, 499767, 499912, 499924, 500412], "question_author": ""}
{"qid": 874, "query": "Is it possible to represent every huge number in abbreviated form?", "score": 89, "views": 12708, "answer_pids": [2045, 2046, 2067, 41356, 41379], "question_author": "Art"}
{"qid": 875, "query": "$1=2$ | Continued fraction fallacy", "score": 89, "views": 7215, "answer_pids": [195768, 195770, 195782, 195818, 195938, 200757, 1276189], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 876, "query": "How is a group made up of simple groups?", "score": 89, "views": 5673, "answer_pids": [13830, 13845, 29700, 29705], "question_author": ""}
{"qid": 877, "query": "Cute Determinant Question", "score": 89, "views": 3902, "answer_pids": [78465, 78475, 78548], "question_author": "InquilineKea"}
{"qid": 878, "query": "Modelling the Moving Sofa", "score": 89, "views": 3247, "answer_pids": [690586, 693174], "question_author": ""}
{"qid": 879, "query": "Graph theoretic proof: For six irrational numbers, there are three among them such that the sum of any two of them is irrational.", "score": 89, "views": 2992, "answer_pids": [722218, 722226, 722331, 722589, 722594, 722609], "question_author": "InquilineKea"}
{"qid": 880, "query": "Contest problem: Show $\\sum_{n = 1}^\\infty \\frac{n^2a_n}{(a_1+\\cdots+a_n)^2}0$, $\\sum_{n = 1}^\\infty \\frac{1}{a_n}<\\infty$", "score": 89, "views": 2673, "answer_pids": [101020, 101069, 101072, 101082, 117754, 117819], "question_author": "InquilineKea"}
{"qid": 881, "query": "Proof of every convex function is continuous", "score": 88, "views": 71991, "answer_pids": [129757, 275493, 275506, 381321, 857935, 927076, 1154964], "question_author": "InquilineKea"}
{"qid": 882, "query": "Are calculus and real analysis the same thing?", "score": 88, "views": 56854, "answer_pids": [17474, 385384, 385387, 638960, 733481, 1047624, 1119628, 1287887], "question_author": "DarenW"}
{"qid": 883, "query": "A goat tied to a corner of a rectangle", "score": 88, "views": 37788, "answer_pids": [745946], "question_author": "Hector"}
{"qid": 884, "query": "Predicting Real Numbers", "score": 88, "views": 28150, "answer_pids": [175612, 209118], "question_author": "Hector"}
{"qid": 885, "query": "Why is abuse of notation tolerated?", "score": 88, "views": 12967, "answer_pids": [128050, 128051, 128052, 128055, 128058, 128070, 128252, 352905], "question_author": "grizzly adam"}
{"qid": 886, "query": "Why are gauge integrals not more popular?", "score": 88, "views": 9827, "answer_pids": [15357, 15358, 15359, 15437], "question_author": "Keenan Pepper"}
{"qid": 887, "query": "Do numbers get worse than transcendental?", "score": 88, "views": 9692, "answer_pids": [958059, 958379, 958405, 959312], "question_author": "Hans-Peter Stricker"}
{"qid": 888, "query": "Whats so special about standard deviation?", "score": 88, "views": 8482, "answer_pids": [1122702, 1122787, 1122857, 1122886, 1123191, 1123372, 1123402, 1123466, 1124557], "question_author": ""}
{"qid": 889, "query": "What concept does an open set axiomatise?", "score": 88, "views": 8411, "answer_pids": [17218, 692294, 785112, 829888], "question_author": "Uli"}
{"qid": 890, "query": "Would you ever stop rolling the die?", "score": 88, "views": 7964, "answer_pids": [478477, 478492, 478555, 478714, 478755, 479180, 479604], "question_author": ""}
{"qid": 891, "query": "Geometric & Intuitive Meaning of $SL(2,R)$, $SU(2)$, etc... & Representation Theory of Special Functions", "score": 88, "views": 2396, "answer_pids": [855418], "question_author": ""}
{"qid": 892, "query": "References for multivariable calculus", "score": 87, "views": 52033, "answer_pids": [23083, 23117, 23139, 23147, 23163, 24517, 235581, 278874, 374949, 920215, 1204257, 1318275], "question_author": "jul"}
{"qid": 893, "query": "When is the closure of an open ball equal to the closed ball?", "score": 87, "views": 30703, "answer_pids": [55593, 126915, 1202845], "question_author": "Nick T"}
{"qid": 894, "query": "Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?", "score": 87, "views": 16538, "answer_pids": [89030, 89057, 89065, 89111, 89136, 89449], "question_author": "Douglas S. Stones"}
{"qid": 895, "query": "In what sense are math axioms true?", "score": 87, "views": 16225, "answer_pids": [64930, 64935, 64964, 64966, 65000, 730512, 1294944], "question_author": ""}
{"qid": 896, "query": "Identity for convolution of central binomial coefficients: $\\sum\\limits_{k=0}^n \\binom{2k}{k}\\binom{2(n-k)}{n-k}=2^{2n}$", "score": 87, "views": 9110, "answer_pids": [20282, 170533, 938242, 1038000], "question_author": ""}
{"qid": 897, "query": "Compute $\\int \\frac{\\sin(x)}{\\sin(x)+\\cos(x)}\\mathrm dx$", "score": 87, "views": 8997, "answer_pids": [91074, 91075, 91076, 91086, 151696, 554150, 607424, 608424], "question_author": "J. Musser"}
{"qid": 898, "query": "What are some conceptualizations that work in mathematics but are not strictly true?", "score": 87, "views": 8425, "answer_pids": [372462, 372465, 372512, 372514, 372523, 372547, 372554, 372587, 372613, 372616, 372739, 372766, 372776, 372788, 372815, 372834, 372927, 372977, 373012, 373030, 373040, 373096, 373165, 373172, 373843, 373865], "question_author": null}
{"qid": 899, "query": "Why do mathematicians sometimes assume famous conjectures in their research?", "score": 87, "views": 7535, "answer_pids": [334572, 334575, 334576, 334592, 334662, 334759, 335336, 335349, 336415], "question_author": ""}
{"qid": 900, "query": "Where are the axioms?", "score": 87, "views": 5321, "answer_pids": [710670, 710684, 710689, 710696, 710748, 711455], "question_author": null}
{"qid": 901, "query": "How to prove that $\\frac{\\zeta(2) }{2}+\\frac{\\zeta (4)}{2^3}+\\frac{\\zeta (6)}{2^5}+\\frac{\\zeta (8)}{2^7}+\\cdots=1$?", "score": 87, "views": 5127, "answer_pids": [444726, 444782, 444810, 705493], "question_author": ""}
{"qid": 902, "query": "How do we know that Cantors diagonalization isnt creating a different decimal of the same number?", "score": 87, "views": 5009, "answer_pids": [797507, 797730, 797762], "question_author": "Ultimate Gobblement"}
{"qid": 903, "query": "In categorical terms, why is there no canonical isomorphism from a finite dimensional vector space to its dual?", "score": 87, "views": 3461, "answer_pids": [278734], "question_author": "A B"}
{"qid": 904, "query": "Determining information in minimum trials (combinatorics problem)", "score": 87, "views": 2237, "answer_pids": [1419532], "question_author": "Rik Smith-Unna"}
{"qid": 905, "query": "What is the term for a factorial type operation, but with summation instead of products?", "score": 86, "views": 80110, "answer_pids": [32140, 32141, 32142, 32155], "question_author": "Rik Smith-Unna"}
{"qid": 906, "query": "Difference between gradient and Jacobian", "score": 86, "views": 54264, "answer_pids": [596617, 596625, 1068020, 1068029], "question_author": "Rik Smith-Unna"}
{"qid": 907, "query": "Why does Turn! Turn! Turn! equal 241217.524881?", "score": 86, "views": 19084, "answer_pids": [1457050], "question_author": "Gerard"}
{"qid": 908, "query": "Is there a size of rectangle that retains its ratio when its folded in half?", "score": 86, "views": 16409, "answer_pids": [467435, 467437, 467516, 467564, 467783, 468418, 468562, 469050], "question_author": "Douglas S. Stones"}
{"qid": 909, "query": "What is ultrafinitism and why do people believe it?", "score": 86, "views": 8178, "answer_pids": [327, 335, 3468], "question_author": ""}
{"qid": 910, "query": "Decidability of the Riemann Hypothesis vs. the Goldbach Conjecture", "score": 86, "views": 7276, "answer_pids": [868391, 868399, 868517], "question_author": "InquilineKea"}
{"qid": 911, "query": "Intuition in algebra?", "score": 86, "views": 7088, "answer_pids": [7828, 7829, 7841, 7869, 7881, 7920, 8012], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 912, "query": "A continuous, nowhere differentiable but invertible function?", "score": 86, "views": 6670, "answer_pids": [1051638, 1051640, 1051642, 1052714], "question_author": ""}
{"qid": 913, "query": "Conjecture $\\int_0^1\\frac{dx}{\\sqrt[3]x\\,\\sqrt[6]{1-x}\\,\\sqrt{1-x\\left(\\sqrt{6}\\sqrt{12+7\\sqrt3}-3\\sqrt3-6\\right)^2}}=\\frac\\pi9(3+\\sqrt2\\sqrt[4]{27})$", "score": 86, "views": 4643, "answer_pids": [247164, 305963], "question_author": ""}
{"qid": 914, "query": "Can someone explain this integration trick for log-sine integrals?", "score": 86, "views": 3343, "answer_pids": [1416955], "question_author": ""}
{"qid": 915, "query": "Polynomials such that roots=coefficients", "score": 86, "views": 3058, "answer_pids": [341245, 341263, 341364, 341476, 341483, 1251142, 1251214], "question_author": ""}
{"qid": 916, "query": "Does $X\\times S^1\\cong Y\\times S^1$ imply that $X\\times\\mathbb R\\cong Y\\times\\mathbb R$?", "score": 86, "views": 2391, "answer_pids": [192734], "question_author": "Sam Davies"}
{"qid": 917, "query": "Do groups, rings and fields have practical applications in CS? If so, what are some?", "score": 85, "views": 30731, "answer_pids": [209998, 210000, 210001, 210002, 210007, 210025, 210129, 210147, 210157, 210346, 210704, 211003], "question_author": "scrrr"}
{"qid": 918, "query": "What is special about the numbers 9801, 998001, 99980001 ..?", "score": 85, "views": 28126, "answer_pids": [53023, 123862, 126063, 185264, 198728, 839594], "question_author": "Qiaochu Yuan"}
{"qid": 919, "query": "How does Cantors diagonal argument work?", "score": 85, "views": 24237, "answer_pids": [20936, 20942, 21192], "question_author": ""}
{"qid": 920, "query": "Proof of $\\int_0^\\infty \\left(\\frac{\\sin x}{x}\\right)^2 \\mathrm dx=\\frac{\\pi}{2}.$", "score": 85, "views": 22396, "answer_pids": [7511, 7513, 7515, 7518, 7521, 7522, 180192, 234725, 368894, 697034, 936934, 1048440, 1111354, 1111707], "question_author": ""}
{"qid": 921, "query": "Why is a circle in a plane surrounded by 6 other circles?", "score": 85, "views": 21785, "answer_pids": [6062, 6078, 6111, 649732], "question_author": "johnny1bucket"}
{"qid": 922, "query": "Expected number of unpecked chicks - NYT article", "score": 85, "views": 21131, "answer_pids": [860698, 860702, 860707, 860922, 861020, 861382, 861611, 861642, 865254], "question_author": ""}
{"qid": 923, "query": "Whats the difference between $\\mathbb{R}^2$ and the complex plane?", "score": 85, "views": 18505, "answer_pids": [207944, 207950, 207953, 207955, 207957, 207963, 207970, 208061, 208065, 208162], "question_author": ""}
{"qid": 924, "query": "What is your favorite application of the Pigeonhole Principle?", "score": 85, "views": 15453, "answer_pids": [33158, 33166, 33169, 33171, 33183, 33188, 33190, 33204, 33214, 33216, 33217, 33218, 33237, 33256, 33350, 33537, 34159, 47905, 103745, 459009, 1047709], "question_author": "Incognito"}
{"qid": 925, "query": "Is mathematics just a bunch of nested empty sets?", "score": 85, "views": 13413, "answer_pids": [1021334, 1021340, 1021342, 1021421, 1021779, 1021783, 1021790, 1022275, 1022293], "question_author": "Gergely"}
{"qid": 926, "query": "Easy example why complex numbers are cool", "score": 85, "views": 12781, "answer_pids": [444164, 444165, 444166, 444167, 444168, 444178, 444231, 444298, 444313, 444315, 444318, 444356, 444360, 444369, 444378, 444448, 444452, 444456, 444853, 446185, 446738, 446783], "question_author": "Gabriel Fair"}
{"qid": 927, "query": "Why is the construction of the real numbers important?", "score": 85, "views": 11492, "answer_pids": [491813, 491815, 491843, 491855, 491993, 492139, 492389, 492471, 494927], "question_author": ""}
{"qid": 928, "query": "Any rectangular shape on a calculator numpad when divided by 11 gives an integer. Why?", "score": 85, "views": 10882, "answer_pids": [1075567, 1075573, 1075584, 1075679, 1075700, 1076642], "question_author": ""}
{"qid": 929, "query": "Calculating the volume of a restaurant take-away box that is circular on the bottom and square on the top", "score": 85, "views": 5448, "answer_pids": [1298499, 1298641, 1298695, 1298933, 1299806], "question_author": ""}
{"qid": 930, "query": "Paul Erdoss Two-Line Functional Analysis Proof", "score": 85, "views": 3315, "answer_pids": [126877, 127463, 145339], "question_author": ""}
{"qid": 931, "query": "Numbers $n$ such that the digit sum of $n^2$ is a square", "score": 85, "views": 3144, "answer_pids": [926813], "question_author": "Memming"}
{"qid": 932, "query": "How to solve these two simultaneous divisibilities : $n+1\\mid m^2+1$ and $m+1\\mid n^2+1$", "score": 85, "views": 3085, "answer_pids": [791248, 791812, 1115232], "question_author": null}
{"qid": 933, "query": "Do most numbers have exactly $3$ prime factors?", "score": 85, "views": 2872, "answer_pids": [1270282, 1271254], "question_author": ""}
{"qid": 934, "query": "Can a row of five equilateral triangles tile a big equilateral triangle?", "score": 85, "views": 2261, "answer_pids": [845492, 1253135], "question_author": ""}
{"qid": 935, "query": "Variance of sample variance?", "score": 84, "views": 141107, "answer_pids": [38428, 38480, 175960, 1269563, 1394673], "question_author": "Marta Cz-C"}
{"qid": 936, "query": "Union of two vector subspaces not a subspace?", "score": 84, "views": 122370, "answer_pids": [37876, 37879, 181313, 247697, 777557], "question_author": "vonjd"}
{"qid": 937, "query": "Probability density function vs. probability mass function", "score": 84, "views": 111484, "answer_pids": [12786, 12842, 38085, 536693], "question_author": "Konrad Rudolph"}
{"qid": 938, "query": "Determinant of a non-square matrix", "score": 84, "views": 91054, "answer_pids": [364360, 621138], "question_author": "Ky Leggiero"}
{"qid": 939, "query": "What is a simple example of an unprovable statement?", "score": 84, "views": 25516, "answer_pids": [434850, 434874, 434881, 434901, 434981, 435004, 435196, 435529, 435670, 436450, 437379], "question_author": "falagar"}
{"qid": 940, "query": "Mathematical precise definition of a PDE being elliptic, parabolic or hyperbolic", "score": 84, "views": 13830, "answer_pids": [11871, 775723], "question_author": "sharptooth"}
{"qid": 941, "query": "Does commutativity imply Associativity?", "score": 84, "views": 13521, "answer_pids": [81283, 81285, 81287, 81289, 607831, 607852, 756317, 917690, 917694, 917698, 917699, 941686, 1215111], "question_author": "Douglas S. Stones"}
{"qid": 942, "query": "What is the Tor functor?", "score": 84, "views": 13155, "answer_pids": [9126, 9148, 9162], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 943, "query": "100 Soldiers riddle", "score": 84, "views": 10934, "answer_pids": [52983, 52984, 52985, 52996, 53006, 53024, 53060, 53123, 53899], "question_author": ""}
{"qid": 944, "query": "Why does Friedberg say that the role of the determinant is less central than in former times?", "score": 84, "views": 6486, "answer_pids": [765089, 765099, 765100, 765103, 765115, 765120, 765125, 765302, 765490, 1402460], "question_author": "Justin L."}
{"qid": 945, "query": "Is Lagranges theorem the most basic result in finite group theory?", "score": 84, "views": 6259, "answer_pids": [20443, 1198567], "question_author": ""}
{"qid": 946, "query": "Is $\\sqrt {2 \\sqrt {3 \\sqrt {4 \\ldots}}}$ algebraic or transcendental?", "score": 84, "views": 3626, "answer_pids": [532449, 544987], "question_author": "Steve"}
{"qid": 947, "query": "$6!\\cdot 7!=10!$. Is there a natural bijection between $S_6\\times S_7$ and $S_{10}$?", "score": 84, "views": 1311, "answer_pids": [1376897, 1379775], "question_author": ""}
{"qid": 948, "query": "Show that the set of all finite subsets of $\\mathbb{N}$ is countable.", "score": 83, "views": 46408, "answer_pids": [100330, 100331, 100332, 100335, 100343, 100412, 385033], "question_author": ""}
{"qid": 949, "query": "$X$ is Hausdorff if and only if the diagonal of $X\\times X$ is closed", "score": 83, "views": 31911, "answer_pids": [69460, 69461, 69462, 69588], "question_author": "InquilineKea"}
{"qid": 950, "query": "Why are rotational matrices not commutative?", "score": 83, "views": 31035, "answer_pids": [770686, 770690, 770693, 771030, 771060, 771625, 772113, 775261], "question_author": ""}
{"qid": 951, "query": "How to use the Extended Euclidean Algorithm manually?", "score": 83, "views": 27674, "answer_pids": [44789, 44794, 122620, 313651, 1224379], "question_author": "Greg Slodkowicz"}
{"qid": 952, "query": "How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\\neq 0$?", "score": 83, "views": 21610, "answer_pids": [42743, 42745, 42773, 97382, 214316, 596560, 873176], "question_author": ""}
{"qid": 953, "query": "What are some interpretations of Von Neumanns quote?", "score": 83, "views": 21064, "answer_pids": [46441, 206573, 277280, 482234, 506959, 696345, 772820, 779341, 1026148, 1111904, 1117932, 1294872], "question_author": ""}
{"qid": 954, "query": "Why does the Mandelbrot set contain (slightly deformed) copies of itself?", "score": 83, "views": 10756, "answer_pids": [1629, 1712, 71337, 189831], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 955, "query": "If $a+b=1$ then $a^{4b^2}+b^{4a^2}\\leq1$", "score": 83, "views": 9924, "answer_pids": [827852, 829865, 830122, 831360, 833349, 1297063], "question_author": null}
{"qid": 956, "query": "General request for a book on mathematical history, for a VERY advanced reader.", "score": 83, "views": 9598, "answer_pids": [1028270, 1028274, 1028277, 1028285, 1028286, 1028307, 1028310, 1028349, 1028394, 1028409, 1028468, 1028470, 1028593, 1028774, 1028788, 1028813, 1028973, 1029062, 1029080, 1029219, 1029292, 1029504, 1029869, 1030670], "question_author": ""}
{"qid": 957, "query": "Terence Tao\u2013type books in other fields?", "score": 83, "views": 6259, "answer_pids": [1196214, 1196221, 1196227, 1196231, 1196372, 1196411, 1197648, 1198513, 1457275], "question_author": "shigeta"}
{"qid": 958, "query": "Alice and Bob play the determinant game", "score": 83, "views": 4341, "answer_pids": [110659, 1359856], "question_author": ""}
{"qid": 959, "query": "Is there a definitive guide to speaking mathematics?", "score": 83, "views": 3791, "answer_pids": [19687, 19802, 19809], "question_author": ""}
{"qid": 960, "query": "$\\frac{1}{n}$ as a difference of Egyptian fractions with all denominators $<n$", "score": 83, "views": 1842, "answer_pids": [1251537], "question_author": "Rory M"}
{"qid": 961, "query": "What are the practical applications of the Taylor Series?", "score": 82, "views": 163554, "answer_pids": [108069, 108070, 108071, 108072, 108073, 108074, 108089, 108162, 108173, 126803, 126810, 256262, 400318, 1349337], "question_author": "Mad Scientist"}
{"qid": 962, "query": "Expected Number of Coin Tosses to Get Five Consecutive Heads", "score": 82, "views": 126322, "answer_pids": [171900, 171940, 171989, 212939, 391198, 682636, 706417, 956367, 968758, 1275073], "question_author": ""}
{"qid": 963, "query": "Understanding of the theorem that all norms are equivalent in finite dimensional vector spaces", "score": 82, "views": 43756, "answer_pids": [30629, 30638, 30655, 630831, 983249, 1078194], "question_author": "Rik Smith-Unna"}
{"qid": 964, "query": "Why is $e^{\\pi \\sqrt{163}}$ almost an integer?", "score": 82, "views": 32866, "answer_pids": [2701, 2702, 2704], "question_author": ""}
{"qid": 965, "query": "$\\sqrt{c+\\sqrt{c+\\sqrt{c+\\cdots}}}$, or the limit of the sequence $x_{n+1} = \\sqrt{c+x_n}$", "score": 82, "views": 17956, "answer_pids": [59298, 59299, 362991, 1110379, 1267407], "question_author": ""}
{"qid": 966, "query": "Does mathematics require axioms?", "score": 82, "views": 15934, "answer_pids": [173126, 173145, 173151, 173840, 174420, 174446, 175479, 332676, 687298], "question_author": ""}
{"qid": 967, "query": "The myth of no prime formula?", "score": 82, "views": 11294, "answer_pids": [396379, 396382, 396385, 396452, 396714, 400989, 421086, 449236], "question_author": "Rik Smith-Unna"}
{"qid": 968, "query": "What is Bra and Ket notation and how does it relate to Hilbert spaces?", "score": 82, "views": 10752, "answer_pids": [686164, 686170, 686193, 686207, 686519, 686617, 686991], "question_author": "lamwaiman1988"}
{"qid": 969, "query": "Intuitive meaning of Exact Sequence", "score": 82, "views": 7965, "answer_pids": [196679, 196682, 196683, 196690, 196702, 196704, 196758, 196785], "question_author": ""}
{"qid": 970, "query": "Whats so natural about the base of natural logarithms?", "score": 82, "views": 7137, "answer_pids": [472, 473, 474, 482, 25030, 123431], "question_author": "Beta"}
{"qid": 971, "query": "What does surface area of a sphere actually mean (in terms of elementary school mathematics)?", "score": 82, "views": 7090, "answer_pids": [1297379, 1297381, 1297382, 1297385, 1297389, 1297553, 1297597, 1297653, 1297773, 1297830, 1298096, 1299422], "question_author": "Rik Smith-Unna"}
{"qid": 972, "query": "The closed form of $\\int_0^{\\pi/4}\\frac{\\log(1-x) \\tan^2(x)}{1-x\\tan^2(x)} \\ dx$", "score": 82, "views": 3989, "answer_pids": [1100245], "question_author": "Rik Smith-Unna"}
{"qid": 973, "query": "Can one deduce Liouvilles theorem (in complex analysis) from the non-emptiness of spectra in complex Banach algebras?", "score": 82, "views": 2148, "answer_pids": [265449], "question_author": "Rik Smith-Unna"}
{"qid": 974, "query": "How to get a reflection vector?", "score": 81, "views": 183088, "answer_pids": [7478, 7481, 1208792], "question_author": ""}
{"qid": 975, "query": "Prove that $i^i$ is a real number", "score": 81, "views": 76056, "answer_pids": [96572, 96573, 96576, 96609, 578376, 772475, 1460430], "question_author": "mring"}
{"qid": 976, "query": "Is there a bijective map from $(0,1)$ to $\\mathbb{R}$?", "score": 81, "views": 72615, "answer_pids": [100217, 100219, 100220, 100222, 100235, 100237, 100243, 100254, 100260, 1339306], "question_author": "DarenW"}
{"qid": 977, "query": "Is $\\mathbf{Q}(\\sqrt{2}, \\sqrt{3}) = \\mathbf{Q}(\\sqrt{2}+\\sqrt{3})$?", "score": 81, "views": 48313, "answer_pids": [48545, 48546, 79198, 190138, 377624, 1337416], "question_author": ""}
{"qid": 978, "query": "Cardinality of set of real continuous functions", "score": 81, "views": 34923, "answer_pids": [297, 131550, 131560, 173557, 1164793], "question_author": null}
{"qid": 979, "query": "A and B disjoint, A compact, and B closed implies there is positive distance between both sets", "score": 81, "views": 29979, "answer_pids": [26022, 26202], "question_author": ""}
{"qid": 980, "query": "Why is a full turn of the circle 360\u00b0? Why not any other number?", "score": 81, "views": 27859, "answer_pids": [72247, 72267, 72362, 184044, 381438, 389050], "question_author": "BlueRaja - Danny Pflughoeft"}
{"qid": 981, "query": "Is there any easy way to understand the definition of Gaussian Curvature?", "score": 81, "views": 25579, "answer_pids": [37055, 37057, 37062, 37094], "question_author": ""}
{"qid": 982, "query": "Are all infinities equal?", "score": 81, "views": 24532, "answer_pids": [91810, 91813, 91815, 91826, 91836], "question_author": "Rory M"}
{"qid": 983, "query": "How to generate a random number between 1 and 10 with a six-sided die?", "score": 81, "views": 24512, "answer_pids": [525363, 525365, 525367, 525392, 525540, 525575, 525658, 525776, 526118, 526121, 526174, 526282, 526287, 530453, 530777, 531131], "question_author": "sigoldberg1"}
{"qid": 984, "query": "What is $\\gcd(0,0)$?", "score": 81, "views": 20989, "answer_pids": [230155, 230156, 230158, 230160, 230269, 230293, 230296, 230297], "question_author": ""}
{"qid": 985, "query": "In Linear Algebra, what is a vector?", "score": 81, "views": 20690, "answer_pids": [744331, 744334, 744369, 744430, 744461, 744498, 745390], "question_author": ""}
{"qid": 986, "query": "Why isnt reflexivity redundant in the definition of equivalence relation?", "score": 81, "views": 18624, "answer_pids": [275, 276, 622051, 677121, 936265], "question_author": ""}
{"qid": 987, "query": "Do you prove all theorems whilst studying?", "score": 81, "views": 14714, "answer_pids": [60604, 60684, 338762], "question_author": ""}
{"qid": 988, "query": "Are all mathematicians human calculators?", "score": 81, "views": 13005, "answer_pids": [251817, 251818, 251819, 251824, 251831, 251833, 251857, 251928, 251934, 251969, 272646, 496288, 618569], "question_author": ""}
{"qid": 989, "query": "Compute $ \\lim\\limits_{n \\to \\infty }\\sin \\sin \\dots\\sin n$", "score": 81, "views": 9729, "answer_pids": [24127], "question_author": ""}
{"qid": 990, "query": "Does mathematics become circular at the bottom? What is at the bottom of mathematics?", "score": 81, "views": 9716, "answer_pids": [532581, 532583, 532584, 532585, 532589, 532610, 532612, 532767], "question_author": ""}
{"qid": 991, "query": "linear algebra over a division ring vs. over a field", "score": 81, "views": 9163, "answer_pids": [24010, 24012, 24023, 190397], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 992, "query": "Why is compactness in logic called compactness?", "score": 81, "views": 6745, "answer_pids": [503, 504, 505, 516, 518, 137526, 589600], "question_author": "jonsca"}
{"qid": 993, "query": "20 circles in the plane, all passing through the origin", "score": 81, "views": 6296, "answer_pids": [815414, 815415, 815529, 816008, 816269], "question_author": ""}
{"qid": 994, "query": "How to find this limit: $A=\\lim_{n\\to \\infty}\\sqrt{1+\\sqrt{\\frac{1}{2}+\\sqrt{\\frac{1}{3}+\\cdots+\\sqrt{\\frac{1}{n}}}}}$", "score": 81, "views": 3662, "answer_pids": [260393, 263074, 647325], "question_author": "lilinjn"}
{"qid": 995, "query": "Visual proof of $\\sum_{n=1}^\\infty \\frac{1}{n^4} = \\frac{\\pi^4}{90}$?", "score": 81, "views": 3082, "answer_pids": [514708, 674244], "question_author": "Stefano Borini"}
{"qid": 996, "query": "In $n>5$, topology = algebra", "score": 81, "views": 2409, "answer_pids": [683978], "question_author": null}
{"qid": 997, "query": "Conjectured formula for the Fabius function", "score": 81, "views": 1363, "answer_pids": [1263909], "question_author": ""}
{"qid": 998, "query": "Element-wise (or pointwise) operations notation?", "score": 80, "views": 153371, "answer_pids": [11298, 13379, 269971, 536934], "question_author": null}
{"qid": 999, "query": "Where exactly are complex numbers used in the real world?", "score": 80, "views": 151702, "answer_pids": [137072, 137080, 137088, 328698, 444709, 444716, 655360, 1057926, 1057937], "question_author": null}
{"qid": 1000, "query": "Is zero positive or negative?", "score": 80, "views": 139273, "answer_pids": [14568, 14569, 14590, 14612, 506406, 1152032], "question_author": ""}
{"qid": 1001, "query": "Prove that $\\lim \\limits_{n \\to \\infty} \\frac{x^n}{n!} = 0$, $x \\in \\Bbb R$.", "score": 80, "views": 77124, "answer_pids": [40762, 40764, 40765, 66676, 68188, 68190, 222223, 362859, 428662, 428669, 431926, 616854, 933248, 937383, 1072285], "question_author": "Gordon Gustafson"}
{"qid": 1002, "query": "Finding the Transform matrix from 4 projected points (with Javascript)", "score": 80, "views": 60804, "answer_pids": [142666, 142668, 161265, 421067], "question_author": null}
{"qid": 1003, "query": "Reference book on measure theory", "score": 80, "views": 30989, "answer_pids": [24634, 24635, 24640, 24651, 24663, 24669, 24675, 24687, 192347, 464939, 539200, 545925, 574291, 1042734], "question_author": ""}
{"qid": 1004, "query": "How to effectively and efficiently learn mathematics", "score": 80, "views": 27065, "answer_pids": [12565, 12566, 12567, 12568, 12570, 12591, 12601, 15194, 64221, 64224, 64238, 64239, 219826, 395500], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1005, "query": "Logic problem: Identifying poisoned wines out of a sample, minimizing test subjects with constraints", "score": 80, "views": 22334, "answer_pids": [672, 319776, 337221, 350417, 931624], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1006, "query": "What is a universal property?", "score": 80, "views": 17519, "answer_pids": [33481, 33489, 33568, 191871, 460955, 924963, 1196671], "question_author": ""}
{"qid": 1007, "query": "Why are differentiable complex functions infinitely differentiable?", "score": 80, "views": 14149, "answer_pids": [383, 485, 488, 500, 17835], "question_author": ""}
{"qid": 1008, "query": "Sheaf cohomology: what is it and where can I learn it?", "score": 80, "views": 11271, "answer_pids": [29099, 249529], "question_author": "Poshpaws"}
{"qid": 1009, "query": "Koch snowflake paradox: finite area, but infinite perimeter", "score": 80, "views": 9661, "answer_pids": [994612, 994634, 994655, 994780, 994872, 995661, 996410], "question_author": ""}
{"qid": 1010, "query": "I lost my love of math; Im getting it back. How can I determine if math is actually right for me?", "score": 80, "views": 8462, "answer_pids": [391779, 391850, 391919, 391922, 391948, 391980, 391991, 391992, 391994, 392010, 392062], "question_author": ""}
{"qid": 1011, "query": "Is there a function with a removable discontinuity at every point?", "score": 80, "views": 6745, "answer_pids": [2252, 2257, 2265], "question_author": ""}
{"qid": 1012, "query": "Direct proof that $\\pi$ is not constructible", "score": 80, "views": 6393, "answer_pids": [104493], "question_author": null}
{"qid": 1013, "query": "Generalized Euler sum $\\sum_{n=1}^\\infty \\frac{H_n}{n^q}$", "score": 80, "views": 5606, "answer_pids": [219094, 219263, 541676, 752683, 849637, 919096, 1216022], "question_author": ""}
{"qid": 1014, "query": "Rigorous nature of combinatorics", "score": 80, "views": 5457, "answer_pids": [874037, 874045, 874048, 874090, 874482, 874888], "question_author": "Sklivvz"}
{"qid": 1015, "query": "Integrals of $\\sqrt{x+\\sqrt{\\phantom|\\dots+\\sqrt{x+1}}}$ in elementary functions", "score": 80, "views": 3721, "answer_pids": [235065, 236235, 356737, 358354], "question_author": ""}
{"qid": 1016, "query": "Escaping infinitely many pursuers", "score": 80, "views": 3050, "answer_pids": [1441298, 1442318], "question_author": "Daniel"}
{"qid": 1017, "query": "Relations between p norms", "score": 79, "views": 71477, "answer_pids": [107935], "question_author": "Isaac"}
{"qid": 1018, "query": "Projection map being a closed map", "score": 79, "views": 30061, "answer_pids": [12472, 12509, 298357, 309604], "question_author": "Kristoffer Nolgren"}
{"qid": 1019, "query": "Math behind rotation in MS Paint", "score": 79, "views": 22248, "answer_pids": [25201, 25209, 25210, 25258, 1147780], "question_author": "crskhr"}
{"qid": 1020, "query": "The Monty Hall problem", "score": 79, "views": 21600, "answer_pids": [50229, 50230, 50260, 203881, 269884, 273027, 274116, 378653, 565239, 576643, 1201638, 1214989], "question_author": ""}
{"qid": 1021, "query": "Why is the M\u00f6bius strip not orientable?", "score": 79, "views": 14974, "answer_pids": [8722, 8819, 8823], "question_author": ""}
{"qid": 1022, "query": "Help me solve my fathers riddle and get my book back", "score": 79, "views": 11513, "answer_pids": [371696, 371709, 371712, 371714, 372041], "question_author": ""}
{"qid": 1023, "query": "Why did my friend lose all his money?", "score": 79, "views": 8922, "answer_pids": [94667, 94668, 389551], "question_author": ""}
{"qid": 1024, "query": "What is lost when we move from reals to complex numbers?", "score": 79, "views": 8906, "answer_pids": [1009845, 1009848, 1009849, 1009850, 1009853, 1010673, 1010706], "question_author": "LanceLafontaine"}
{"qid": 1025, "query": "Is linear algebra more \u201cfully understood\u201d than other maths disciplines?", "score": 79, "views": 8344, "answer_pids": [659667, 659672, 659674, 659675, 659684, 659840, 660452], "question_author": "Ultimate Gobblement"}
{"qid": 1026, "query": "Are these solutions of $2 = x^{x^{x^{\\:\\cdot^{\\:\\cdot^{\\:\\cdot}}}}}$ correct?", "score": 79, "views": 8219, "answer_pids": [45813, 45814, 45822, 45914], "question_author": "Alex Coplan"}
{"qid": 1027, "query": "How deep is the liquid in a half-full hemisphere?", "score": 79, "views": 8202, "answer_pids": [1216670, 1216671, 1216674, 1216697, 1217929, 1228773], "question_author": "kasia"}
{"qid": 1028, "query": "Why do books titled Abstract Algebra mostly deal with groups/rings/fields?", "score": 79, "views": 8023, "answer_pids": [9251, 9255, 9263, 9280, 21871, 22158, 84695], "question_author": ""}
{"qid": 1029, "query": "What exactly are eigen-things?", "score": 79, "views": 7755, "answer_pids": [143745, 143746, 143749, 143751, 144410], "question_author": ""}
{"qid": 1030, "query": "What is an example of a sequence which thins out and is finite?", "score": 79, "views": 7615, "answer_pids": [1236483, 1236485, 1236670, 1236672, 1236686, 1236693, 1236749, 1236780, 1236997, 1237003, 1237126, 1237578, 1237616], "question_author": "d4nt"}
{"qid": 1031, "query": "Why is $\\omega$ the smallest $\\infty$?", "score": 79, "views": 7376, "answer_pids": [5748, 5749, 5750, 14684, 24884], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1032, "query": "Besides proving new theorems, how can a person contribute to mathematics?", "score": 79, "views": 7059, "answer_pids": [448060, 448087, 448088, 448142, 448184, 448727, 448755, 448900, 450268], "question_author": "magnetar"}
{"qid": 1033, "query": "The Intuition behind lHopitals Rule", "score": 79, "views": 6760, "answer_pids": [263695, 263704, 263709, 263717, 263737, 412774], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1034, "query": "Motivation for spectral graph theory.", "score": 79, "views": 6744, "answer_pids": [147432, 147445, 148614, 149089, 150805, 227503], "question_author": "John Smith"}
{"qid": 1035, "query": "Mathematicians shocked(?) to find pattern in prime numbers", "score": 79, "views": 4455, "answer_pids": [658946, 659444, 665921], "question_author": "VanJeer"}
{"qid": 1036, "query": "Closed form for $\\int_0^\\infty\\ln\\frac{J_\\mu(x)^2+Y_\\mu(x)^2}{J_\\nu(x)^2+Y_\\nu(x)^2}\\mathrm dx$", "score": 79, "views": 2493, "answer_pids": [189369], "question_author": ""}
{"qid": 1037, "query": "Why cant the Polynomial Ring be a Field?", "score": 78, "views": 33526, "answer_pids": [1512, 1513, 1515, 1518, 311410], "question_author": "John Smith"}
{"qid": 1038, "query": "A path to truly understanding probability and statistics", "score": 78, "views": 31253, "answer_pids": [206323, 206330, 296544], "question_author": "Miriam"}
{"qid": 1039, "query": "Is there a shape with infinite area but finite perimeter?", "score": 78, "views": 27806, "answer_pids": [432665, 432714, 432792, 432868, 432878, 433042, 434174], "question_author": "Pratik Deoghare"}
{"qid": 1040, "query": "Infinite sets dont exist!?", "score": 78, "views": 14064, "answer_pids": [168529, 168531, 168534, 168687, 168729, 172528, 173741, 175381, 462962, 510337, 731332], "question_author": ""}
{"qid": 1041, "query": "Fastest way to check if $x^y > y^x$?", "score": 78, "views": 11206, "answer_pids": [238968, 238974, 1174190], "question_author": "Nikolaj-K"}
{"qid": 1042, "query": "What is integration by parts, really?", "score": 78, "views": 10019, "answer_pids": [354366, 354369, 354396, 354401, 354408, 355508, 355526, 1153412], "question_author": "David Z"}
{"qid": 1043, "query": "Conjectures (or intuitions) that turned out wrong in an interesting or useful way", "score": 78, "views": 7677, "answer_pids": [901272, 901273, 901274, 901277, 901283, 901295, 901333, 901349, 901357, 901423, 901461, 901580, 901638, 902220, 902526, 902616, 903353], "question_author": ""}
{"qid": 1044, "query": "What is the intuition behind uniform continuity?", "score": 78, "views": 7605, "answer_pids": [1175754, 1175761, 1175762, 1175772, 1176360, 1177893], "question_author": ""}
{"qid": 1045, "query": "Dividing 100% by 3 without any left", "score": 77, "views": 124836, "answer_pids": [320051, 320285, 320330, 320429, 320516, 320648, 320725, 320799, 321892, 449093, 1423365], "question_author": "LanceLafontaine"}
{"qid": 1046, "query": "Inscribing square in circle in just seven compass-and-straightedge steps", "score": 77, "views": 74187, "answer_pids": [619861, 789823], "question_author": ""}
{"qid": 1047, "query": "List of interesting integrals for early calculus students", "score": 77, "views": 33420, "answer_pids": [113761, 113771, 113773, 113777, 113778, 113779, 113811, 113813, 113816, 113822, 113835, 114030, 142573, 784787, 784789, 1106015], "question_author": ""}
{"qid": 1048, "query": "The Duals of $l^\\infty$ and $L^{\\infty}$", "score": 77, "views": 32767, "answer_pids": [25268, 25271], "question_author": "John Alexiou"}
{"qid": 1049, "query": "Finite subgroups of the multiplicative group of a field are cyclic", "score": 77, "views": 27108, "answer_pids": [31777, 159723, 457094], "question_author": ""}
{"qid": 1050, "query": "Whats a proof that the angles of a triangle add up to 180\u00b0?", "score": 77, "views": 24201, "answer_pids": [130678, 130680, 130681, 130683, 130684, 130687, 130752, 130849, 130907], "question_author": ""}
{"qid": 1051, "query": "Stirlings formula: proof?", "score": 77, "views": 17666, "answer_pids": [49562, 49573, 49944, 49948, 56135, 58878, 559994, 809109, 829316, 1184389, 1292872, 1410768, 1457458], "question_author": "\u00d6zg\u00fcr"}
{"qid": 1052, "query": "Is learning (theoretical) physics useful/important for a mathematician?", "score": 77, "views": 11963, "answer_pids": [18916, 19022, 21185, 21330, 21364, 21445, 21573, 21608, 22054, 213543], "question_author": ""}
{"qid": 1053, "query": "An example of a problem which is difficult but is made easier when a diagram is drawn", "score": 77, "views": 11797, "answer_pids": [452197, 452206, 452208, 452218, 452414, 452450, 452481, 452557, 452669, 452723, 452757, 452966, 452977, 453465, 453471, 453593, 453634, 453681, 453912, 454026, 455243], "question_author": ""}
{"qid": 1054, "query": "Multiple-choice: sum of primes below $1000$", "score": 77, "views": 10545, "answer_pids": [806564, 806565, 806567, 806568, 806607, 806701, 806730], "question_author": "EdoDodo"}
{"qid": 1055, "query": "Fake induction proofs", "score": 77, "views": 10316, "answer_pids": [485136, 485195, 487667, 487828, 487829, 487839, 489513, 489549, 489561, 544103, 1302483], "question_author": ""}
{"qid": 1056, "query": "Theorem that von Neumann proved in five minutes.", "score": 77, "views": 8436, "answer_pids": [390946], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1057, "query": "Why did mathematicians introduce the concept of uniform continuity?", "score": 77, "views": 6654, "answer_pids": [506306, 506338, 506344], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1058, "query": "Why is $i! = 0.498015668 - 0.154949828i$?", "score": 77, "views": 6333, "answer_pids": [101202, 101208, 101243], "question_author": "Maxim V. Pavlov"}
{"qid": 1059, "query": "Gerrymandering on a high-genus surface/can I use my powers for evil?", "score": 77, "views": 4857, "answer_pids": [141401], "question_author": ""}
{"qid": 1060, "query": "A strange integral: $\\int_{-\\infty}^{+\\infty} {dx \\over 1 + \\left(x + \\tan x\\right)^2} = \\pi.$", "score": 77, "views": 4789, "answer_pids": [422031, 422383, 900708, 901678], "question_author": "Thomas O"}
{"qid": 1061, "query": "Denesting radicals like $\\sqrt[3]{\\sqrt[3]{2} - 1}$", "score": 77, "views": 4321, "answer_pids": [475008, 664208, 1312518], "question_author": "Thomas O"}
{"qid": 1062, "query": "How are mathematicians taught to write with such an expository style?", "score": 77, "views": 3398, "answer_pids": [128912, 128924, 128931, 128975, 128985], "question_author": "LanceLafontaine"}
{"qid": 1063, "query": "Number of ways to write n as a sum of k nonnegative integers", "score": 76, "views": 137066, "answer_pids": [107720, 247227, 253629, 270373, 319013, 1427741], "question_author": ""}
{"qid": 1064, "query": "What is $\\sqrt{i}$?", "score": 76, "views": 127640, "answer_pids": [1983, 1993, 2001, 2021, 2026, 2039, 2159, 34003], "question_author": "bobtheowl2"}
{"qid": 1065, "query": "Teenager solves Newton dynamics problem - where is the paper?", "score": 76, "views": 79109, "answer_pids": [76009, 76233, 78327, 409948], "question_author": ""}
{"qid": 1066, "query": "Combinatorial proof of summation of $\\sum\\limits_{k = 0}^n {n \\choose k}^2= {2n \\choose n}$", "score": 76, "views": 75305, "answer_pids": [75073, 399022, 443970, 616146, 882761, 1425332], "question_author": "Gianpaolo R"}
{"qid": 1067, "query": "What is the meaning of the third derivative of a function at a point", "score": 76, "views": 60195, "answer_pids": [8313, 8314, 8315, 8317, 8318, 20819, 36184, 163268, 229257], "question_author": "Rajesh D"}
{"qid": 1068, "query": "Continuous mapping on a compact metric space is uniformly continuous", "score": 76, "views": 50108, "answer_pids": [56878, 626526, 1323334], "question_author": "Sami"}
{"qid": 1069, "query": "Prove that the set of all algebraic numbers is countable", "score": 76, "views": 49598, "answer_pids": [227729, 227730, 882203], "question_author": "Mononess"}
{"qid": 1070, "query": "Difference between metric and norm made concrete: The case of Euclid", "score": 76, "views": 31255, "answer_pids": [20602, 20603], "question_author": "Gergana Vandova"}
{"qid": 1071, "query": "(Theoretical) Multivariable Calculus Textbooks", "score": 76, "views": 26203, "answer_pids": [23710, 23712, 24125, 24362, 235580], "question_author": "J. Musser"}
{"qid": 1072, "query": "Quotient ring of Gaussian integers", "score": 76, "views": 21206, "answer_pids": [12826, 12829, 12830, 12831, 12912, 65801], "question_author": "J. Musser"}
{"qid": 1073, "query": "String Theory: What to do?", "score": 76, "views": 16122, "answer_pids": [19926, 19932, 19971], "question_author": "Marek"}
{"qid": 1074, "query": "What is the chance to get a parking ticket in half an hour if the chance to get a ticket is 80% in 1 hour?", "score": 76, "views": 14719, "answer_pids": [77550, 77552, 77559, 77561, 77621, 77645, 77707, 856267], "question_author": "Preece"}
{"qid": 1075, "query": "Why there is no sign of logic symbols in mathematical texts?", "score": 76, "views": 12418, "answer_pids": [545237, 545238, 545240, 545242, 545243, 545325, 545450, 545704, 546006, 546246, 546309, 586020], "question_author": "Fr\u00e9d\u00e9ric Grosshans"}
{"qid": 1076, "query": "Connections between metrics, norms and scalar products (for understanding e.g. Banach and Hilbert spaces)", "score": 76, "views": 11351, "answer_pids": [20573], "question_author": ""}
{"qid": 1077, "query": "Interesting and unexpected applications of $\\pi$", "score": 76, "views": 8483, "answer_pids": [304260, 304263, 304265, 304266, 304267, 304268, 304279, 304284, 304287, 304333, 304339, 304340, 304825, 305545, 306165, 306336, 306675, 446382, 460736, 558052, 558067, 569771, 641224, 723905, 808510, 882369, 1199353, 1261584, 1280883, 1408047], "question_author": "Kivanc Uyanik"}
{"qid": 1078, "query": "How to prove this identity $\\pi=\\sum\\limits_{k=-\\infty}^{\\infty}\\left(\\frac{\\sin(k)}{k}\\right)^{2}\\;$?", "score": 76, "views": 8286, "answer_pids": [157772, 157941, 161071, 164855, 175118, 262943, 694288], "question_author": ""}
{"qid": 1079, "query": "Thurstons 37th way of thinking about the derivative", "score": 76, "views": 6386, "answer_pids": [210389], "question_author": "AgCl"}
{"qid": 1080, "query": "Volumes of n-balls: what is so special about n=5?", "score": 76, "views": 3915, "answer_pids": [8756, 8840, 185243, 1069253], "question_author": "user542"}
{"qid": 1081, "query": "Ramanujan log-trigonometric integrals", "score": 76, "views": 3407, "answer_pids": [397260, 397532], "question_author": ""}
{"qid": 1082, "query": "Finding out the area of a triangle if the coordinates of the three vertices are given", "score": 75, "views": 273060, "answer_pids": [238462, 238463, 238464, 552549, 561802, 599998, 730319, 1198020], "question_author": ""}
{"qid": 1083, "query": "Factorial, but with addition", "score": 75, "views": 200039, "answer_pids": [266944, 266947, 266954, 266977], "question_author": ""}
{"qid": 1084, "query": "A simple explanation of eigenvectors and eigenvalues with big picture ideas of why on earth they matter", "score": 75, "views": 77309, "answer_pids": [19668, 29434, 943604, 1149812], "question_author": "Daniel Standage"}
{"qid": 1085, "query": "Zero probability and impossibility", "score": 75, "views": 55512, "answer_pids": [21925, 21926, 21927, 21930, 21931, 21935, 1388662], "question_author": ""}
{"qid": 1086, "query": "Whats the difference between stochastic and random?", "score": 75, "views": 43693, "answer_pids": [58753, 58771, 58772, 276320, 394424, 495368, 630656, 667893, 794657, 1255926], "question_author": "bobthejoe"}
{"qid": 1087, "query": "How many connected components does $\\mathrm{GL}_n(\\mathbb R)$ have?", "score": 75, "views": 14727, "answer_pids": [52219, 52220, 52224, 52264, 52274, 172421, 408666, 657395], "question_author": ""}
{"qid": 1088, "query": "Mathematical subjects you wish you learned earlier", "score": 75, "views": 8583, "answer_pids": [129, 130, 131, 133, 308, 310, 341, 619, 1546, 1557, 2862, 3208, 3232, 11322, 11334, 137147, 137161], "question_author": ""}
{"qid": 1089, "query": "Closed form for $\\sum \\frac{1}{n^n}$", "score": 75, "views": 6800, "answer_pids": [11766, 11767, 11773], "question_author": "John Smith"}
{"qid": 1090, "query": "Why cant you pick socks using coin flips?", "score": 75, "views": 5816, "answer_pids": [313927, 313928, 313929, 313990, 314008], "question_author": "John Smith"}
{"qid": 1091, "query": "Is it bad form to write mysterious proofs without explaining what one intends to do?", "score": 75, "views": 5639, "answer_pids": [788556, 788561, 788562, 788563, 788567, 788668, 788824, 789004, 789707], "question_author": "Stefano Borini"}
{"qid": 1092, "query": "Does $G\\cong G/H$ imply that $H$ is trivial?", "score": 75, "views": 4922, "answer_pids": [41883, 41898, 41907, 41910, 41924, 228453, 228466, 296712, 466056, 538026], "question_author": ""}
{"qid": 1093, "query": "What does the mysterious constant marked by C on a slide rule indicate?", "score": 75, "views": 4841, "answer_pids": [1105971], "question_author": "Maxim V. Pavlov"}
{"qid": 1094, "query": "Two curious identities on $x^x$,$e$,and $\\pi$", "score": 75, "views": 1875, "answer_pids": [118826, 119052], "question_author": "user547"}
{"qid": 1095, "query": "How to find perpendicular vector to another vector?", "score": 74, "views": 449721, "answer_pids": [69667, 69677, 105294, 150539, 325234, 990300, 990458, 991882, 1281563, 1281593, 1351306], "question_author": ""}
{"qid": 1096, "query": "How to calculate $\\,(a-b)\\bmod n\\,$ and $ {-}b \\bmod n$", "score": 74, "views": 311772, "answer_pids": [239811, 239812, 452446, 452456], "question_author": "J. Musser"}
{"qid": 1097, "query": "How was the normal distribution derived?", "score": 74, "views": 47249, "answer_pids": [181239, 677342, 1094415, 1113542, 1263105, 1296284], "question_author": ""}
{"qid": 1098, "query": "What is the proper way to study (more advanced) math?", "score": 74, "views": 29023, "answer_pids": [57494, 57501], "question_author": ""}
{"qid": 1099, "query": "What are the applications of functional analysis?", "score": 74, "views": 21338, "answer_pids": [608798, 608806, 608807, 608857, 608955, 609369, 609857, 609995, 612134, 612522, 763203, 1028427], "question_author": "J. Musser"}
{"qid": 1100, "query": "Very *mathematical* general physics book", "score": 74, "views": 11973, "answer_pids": [399972, 399981, 400036, 400189, 400221, 401991, 457316], "question_author": "J. Musser"}
{"qid": 1101, "query": "Why not include as a requirement that all functions must be continuous to be differentiable?", "score": 74, "views": 11548, "answer_pids": [1042169, 1042170, 1042172, 1042198, 1042208, 1042227, 1042262, 1042287, 1042350, 1042921, 1290137], "question_author": "J. Musser"}
{"qid": 1102, "query": "Bag of tricks in Advanced Calculus/ Real Analysis/Complex Analysis", "score": 74, "views": 9096, "answer_pids": [132933, 132939, 132953, 132961, 289750, 371458, 1021506], "question_author": "Martin Gales"}
{"qid": 1103, "query": "What would have been our number system if humans had more than 10 fingers? Try to solve this puzzle.", "score": 74, "views": 8529, "answer_pids": [215245, 215246, 215249, 215252, 215256, 215271, 215323, 215355, 215388, 215453, 215459, 215512], "question_author": "Poshpaws"}
{"qid": 1104, "query": "Does associativity imply commutativity?", "score": 74, "views": 6090, "answer_pids": [703053, 703061, 703067, 703108, 703374, 703391, 703769], "question_author": ""}
{"qid": 1105, "query": "Algebraic Intuition for Homological Algebra and Applications to More Elementary Algebra", "score": 74, "views": 6034, "answer_pids": [42582, 42586, 42588, 42589, 59498, 59582, 192791], "question_author": ""}
{"qid": 1106, "query": "A new imaginary number? $x^c = -x$", "score": 74, "views": 4753, "answer_pids": [44432, 44433, 44438, 44528, 918329], "question_author": "bobtheowl2"}
{"qid": 1107, "query": "About Euclids Elements and modern video games", "score": 74, "views": 4681, "answer_pids": [176190], "question_author": ""}
{"qid": 1108, "query": "A matrix and its transpose have the same set of eigenvalues/other version: $A$ and $A^T$ have the same spectrum", "score": 73, "views": 109028, "answer_pids": [63366, 63367, 373037], "question_author": "Zev Chonoles"}
{"qid": 1109, "query": "What is the math behind the game Spot It?", "score": 73, "views": 74038, "answer_pids": [19661, 19707, 39484, 45222, 129652, 1295463], "question_author": "Gordon Gustafson"}
{"qid": 1110, "query": "Are the eigenvalues of $AB$ equal to the eigenvalues of $BA$? (Citation needed!)", "score": 73, "views": 61330, "answer_pids": [63830, 701990, 735761, 741390], "question_author": "Piotr Migdal"}
{"qid": 1111, "query": "Why determinant of a 2 by 2 matrix is the area of a parallelogram?", "score": 73, "views": 59245, "answer_pids": [15820, 15825, 34804, 59314, 61236, 277246, 613209, 856464, 1137272], "question_author": "Ryan Dickherber"}
{"qid": 1112, "query": "Intuition behind using complementary CDF to compute expectation for nonnegative random variables", "score": 73, "views": 44540, "answer_pids": [34005, 34008, 34021, 34024], "question_author": "John Smith"}
{"qid": 1113, "query": "Proof that a Combination is an integer", "score": 73, "views": 40051, "answer_pids": [6587, 6588, 6590, 525202, 592074, 990317, 1149177], "question_author": "Richard Terrett"}
{"qid": 1114, "query": "A map is continuous if and only if for every set, the image of closure is contained in the closure of image", "score": 73, "views": 39628, "answer_pids": [58792, 58794, 58808, 58840, 58855, 58925, 288611], "question_author": "J. Musser"}
{"qid": 1115, "query": "Examples of finite nonabelian groups.", "score": 73, "views": 38915, "answer_pids": [156343, 156344, 156363, 156422, 156428, 156433, 156508, 156747, 168147, 461309], "question_author": "J. Musser"}
{"qid": 1116, "query": "Whats going on with compact implies sequentially compact?", "score": 73, "views": 26629, "answer_pids": [23926, 23927, 23928, 23932, 23969], "question_author": "J. Musser"}
{"qid": 1117, "query": "Best Algebraic Geometry text book? (other than Hartshorne)", "score": 73, "views": 25587, "answer_pids": [624, 656, 657, 658, 662, 716, 1014, 1786, 13380, 56469], "question_author": "eig"}
{"qid": 1118, "query": "Why does the symbol for the multiplication operation change shape?", "score": 73, "views": 22396, "answer_pids": [710175, 710180, 710186, 710194, 710239, 710468, 710682, 799943], "question_author": "lamwaiman1988"}
{"qid": 1119, "query": "Hanging a picture on the wall using two nails in such a way that removing any nail makes the picture fall down", "score": 73, "views": 20295, "answer_pids": [339751, 339753, 339761, 339762, 339788, 340164], "question_author": ""}
{"qid": 1120, "query": "Why does the google calculator give $\\tan 90^{\\circ} = 1.6331779e^{+16}$?", "score": 73, "views": 17930, "answer_pids": [246090, 246092, 246141, 246735], "question_author": null}
{"qid": 1121, "query": "Reference request: introduction to commutative algebra", "score": 73, "views": 14333, "answer_pids": [19976, 20002, 23116, 858164, 1280537, 1452310], "question_author": "aaronfarr"}
{"qid": 1122, "query": "When writing in math, do you use a comma or colon preceding an equation?", "score": 73, "views": 13222, "answer_pids": [837819, 837821, 837859, 837969, 838255], "question_author": ""}
{"qid": 1123, "query": "Is linear algebra laying the foundation for something important?", "score": 73, "views": 11841, "answer_pids": [658295, 658326, 658356, 658799, 658915, 659670, 659796, 660297], "question_author": ""}
{"qid": 1124, "query": "How come $32.5 = 31.5$? (The Missing Square puzzle.)", "score": 73, "views": 10310, "answer_pids": [173, 174, 176, 22870, 22925, 174240, 1117072], "question_author": ""}
{"qid": 1125, "query": "Evaluating $\\int_0^\\infty \\sin x^2\\, dx$ with real methods?", "score": 73, "views": 10016, "answer_pids": [94634, 94638, 94640, 95927, 220080, 699039, 1034764], "question_author": "Jonathan Fischoff"}
{"qid": 1126, "query": "Is there an easy way to show which spheres can be Lie groups?", "score": 73, "views": 9497, "answer_pids": [7059], "question_author": "Pierre"}
{"qid": 1127, "query": "How to solve fifth-degree equations by elliptic functions?", "score": 73, "views": 9361, "answer_pids": [247970], "question_author": "puri"}
{"qid": 1128, "query": "Intuition behind conjugation in group theory", "score": 73, "views": 9165, "answer_pids": [6794, 6795, 6797, 6798, 17913, 111870], "question_author": "Jonathan."}
{"qid": 1129, "query": "How to stop forgetting proofs - for a first course in Real Analysis?", "score": 73, "views": 9153, "answer_pids": [449458, 449459, 449505, 449542, 449578, 449841, 450083, 450286, 450337, 450415, 450448, 453567, 454194], "question_author": "DVK"}
{"qid": 1130, "query": "Polynomials irreducible over $\\mathbb{Q}$ but reducible over $\\mathbb{F}_p$ for every prime $p$", "score": 73, "views": 8840, "answer_pids": [81245, 81247, 723232], "question_author": "Poshpaws"}
{"qid": 1131, "query": "How to find the inverse modulo $m$?", "score": 72, "views": 182561, "answer_pids": [13870, 13871, 13872, 13873, 13877, 893601, 1352086], "question_author": ""}
{"qid": 1132, "query": "Finding a point along a line a certain distance away from another point!", "score": 72, "views": 158201, "answer_pids": [88634, 633758, 635686, 802833], "question_author": "bobthejoe"}
{"qid": 1133, "query": "Graph theory: adjacency vs incident", "score": 72, "views": 87858, "answer_pids": [96138, 96139, 96140, 337705], "question_author": "Yevgeny Simkin"}
{"qid": 1134, "query": "Space of bounded continuous functions is complete", "score": 72, "views": 57766, "answer_pids": [40204, 48274, 84817, 84826], "question_author": "Gergana Vandova"}
{"qid": 1135, "query": "Why is $\\Gamma\\left(\\frac{1}{2}\\right)=\\sqrt{\\pi}$?", "score": 72, "views": 55327, "answer_pids": [106680, 106683, 106685, 106693, 106695, 106699, 106700, 107989, 197961, 862177, 898009], "question_author": "nibot"}
{"qid": 1136, "query": "Simplest proof of Taylors theorem", "score": 72, "views": 54095, "answer_pids": [224516, 228929, 336961, 393667, 662877, 662915, 662928, 979314, 1197630, 1197641, 1454771], "question_author": "Calvin Cheng"}
{"qid": 1137, "query": "What algorithm is used by computers to calculate logarithms?", "score": 72, "views": 50167, "answer_pids": [32462, 32463, 32476], "question_author": "Douglas S. Stones"}
{"qid": 1138, "query": "Why do we need to learn integration techniques?", "score": 72, "views": 29579, "answer_pids": [95247, 95251, 95260, 95261, 95264, 95277, 95288, 95324, 95325, 95329, 95429, 95520, 126725, 652546, 1104555], "question_author": "Anixx"}
{"qid": 1139, "query": "Solving $DEF+FEF=GHH$, $KLM+KLM=NKL$, $ABC+ABC+ABC=BBB$", "score": 72, "views": 18181, "answer_pids": [823783, 823844, 823863, 823896, 823906, 823948, 824067, 824167, 824201, 824271, 824288, 824405, 824877, 825537, 826412], "question_author": ""}
{"qid": 1140, "query": "Why can a real number be defined as a Dedekind cut, that is, as a set of rational numbers?", "score": 72, "views": 13772, "answer_pids": [1049223, 1049225, 1049226, 1049235, 1049240, 1049275, 1049291, 1049318, 1049324, 1049368, 1049477, 1049535, 1049777, 1049797, 1049815], "question_author": ""}
{"qid": 1141, "query": "Mathematical research of Pok\u00e9mon", "score": 72, "views": 11561, "answer_pids": [355677, 355733, 356303], "question_author": ""}
{"qid": 1142, "query": "Why Zariski topology?", "score": 72, "views": 11265, "answer_pids": [81760, 81761, 81795, 81816, 83860, 579067], "question_author": "David Z"}
{"qid": 1143, "query": "Why should we prove obvious things?", "score": 72, "views": 9995, "answer_pids": [329854, 329858, 329883, 329897, 329902, 329918, 329959, 329969, 330090, 330127, 330448, 330904, 331245, 331535, 340249, 340257, 370715], "question_author": "John Alexiou"}
{"qid": 1144, "query": "Drunk man with a set of keys.", "score": 72, "views": 7843, "answer_pids": [779676, 779677, 781084], "question_author": ""}
{"qid": 1145, "query": "Will assuming the existence of a solution ever lead to a contradiction?", "score": 72, "views": 6845, "answer_pids": [960945, 960947, 960998, 961064, 961154, 961166, 961211, 961376, 961442, 961451, 962086], "question_author": "Anixx"}
{"qid": 1146, "query": "What is $x^y$? How to understand it?", "score": 72, "views": 5932, "answer_pids": [101992, 101994, 101999, 102015, 102019, 102084, 102087, 102205, 102226, 102280, 102432, 102684, 102697], "question_author": "Jack Kada"}
{"qid": 1147, "query": "Lebesgue measure theory vs differential forms?", "score": 72, "views": 5899, "answer_pids": [1053008], "question_author": "falagar"}
{"qid": 1148, "query": "Naturally occurring non-Hausdorff spaces?", "score": 72, "views": 5158, "answer_pids": [1337432, 1337435, 1337436, 1337441, 1337442, 1337443, 1337465, 1337560, 1337808, 1337991, 1338183, 1338193, 1338288, 1338376, 1340884], "question_author": ""}
{"qid": 1149, "query": "What are some mathematical topics that involve adding and multiplying pictures?", "score": 72, "views": 4753, "answer_pids": [998436, 998439, 998440, 998511, 998562, 998583, 998625, 998797, 999345, 999533, 999834, 1001663, 1147465], "question_author": ""}
{"qid": 1150, "query": "How to cut a cube out of a tree stump, such that a pair of opposing vertices are in the center?", "score": 72, "views": 4694, "answer_pids": [1395404, 1395433, 1395553, 1395649], "question_author": ""}
{"qid": 1151, "query": "Why are the last two numbers of this sequence never prime?", "score": 72, "views": 3587, "answer_pids": [744864, 744865], "question_author": "Mark Eichenlaub"}
{"qid": 1152, "query": "Is $ 0.112123123412345123456\\dots $ algebraic or transcendental?", "score": 72, "views": 3051, "answer_pids": [581047, 583941, 929346], "question_author": "bobthejoe"}
{"qid": 1153, "query": "Integral $\\int_1^\\infty\\frac{\\operatorname{arccot}\\left(1+\\frac{2\\pi}{\\operatorname{arcoth}x-\\operatorname{arccsc}x}\\right)}{\\sqrt{x^2-1}}\\mathrm dx$", "score": 72, "views": 2619, "answer_pids": [273217, 281324], "question_author": ""}
{"qid": 1154, "query": "Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$", "score": 72, "views": 2461, "answer_pids": [498252, 636044], "question_author": ""}
{"qid": 1155, "query": "Difference between Fourier transform and Wavelets", "score": 71, "views": 65879, "answer_pids": [134530, 134543, 134557, 144929, 579169, 623813], "question_author": "Santosh Linkha"}
{"qid": 1156, "query": "Is there a known mathematical equation to find the nth prime?", "score": 71, "views": 55094, "answer_pids": [774, 775, 779, 2560], "question_author": ""}
{"qid": 1157, "query": "Understanding Borel sets", "score": 71, "views": 49019, "answer_pids": [108832, 108835, 108849, 108850, 108859, 288892], "question_author": "Casebash"}
{"qid": 1158, "query": "$\\infty = -1 $ paradox", "score": 71, "views": 22927, "answer_pids": [19942, 19943, 19948, 19966, 20019, 20065], "question_author": ""}
{"qid": 1159, "query": "What do $\\pi$ and $e$ stand for in the normal distribution formula?", "score": 71, "views": 19131, "answer_pids": [15511, 15513, 15514, 15517, 29793, 29799, 29816, 29881], "question_author": "Neil G"}
{"qid": 1160, "query": "Completion of rational numbers via Cauchy sequences", "score": 71, "views": 15232, "answer_pids": [6764, 6772, 6809, 10923, 15464], "question_author": ""}
{"qid": 1161, "query": "Convergence of $\\sqrt{n}x_{n}$ where $x_{n+1} = \\sin(x_{n})$", "score": 71, "views": 7737, "answer_pids": [1919, 75560], "question_author": ""}
{"qid": 1162, "query": "Combinatorial proof that $\\sum \\limits_{k=0}^n \\binom{2k}{k} \\binom{2n-2k}{n-k} (-1)^k = 2^n \\binom{n}{n/2}$ when $n$ is even", "score": 71, "views": 6556, "answer_pids": [50927, 80574], "question_author": ""}
{"qid": 1163, "query": "Why does being holomorphic imply so much about a function?", "score": 71, "views": 6250, "answer_pids": [398802, 398806, 400107, 400181, 401533, 401604, 401627, 401634, 401875, 402433], "question_author": ""}
{"qid": 1164, "query": "Why should I care about adjoint functors", "score": 71, "views": 6043, "answer_pids": [13903, 13904, 13912, 13935, 35723], "question_author": "Gianpaolo R"}
{"qid": 1165, "query": "How did Euler prove the Mersenne number $2^{31}-1$ is a prime so early in history?", "score": 71, "views": 5720, "answer_pids": [160772, 160773, 160787], "question_author": "Stefano Borini"}
{"qid": 1166, "query": "Pseudo Proofs that are intuitively reasonable", "score": 71, "views": 5043, "answer_pids": [63225, 63229, 63230, 63247, 63269, 63271, 63273, 63351, 63352, 63353, 63357, 63399, 123135], "question_author": ""}
{"qid": 1167, "query": "Is $\\sqrt1+\\sqrt2+\\dots+\\sqrt n$ ever an integer?", "score": 71, "views": 4835, "answer_pids": [206929], "question_author": "Rory M"}
{"qid": 1168, "query": "Is this determinant identity known?", "score": 71, "views": 3596, "answer_pids": [942579], "question_author": ""}
{"qid": 1169, "query": "How was the Monsters existence originally suspected?", "score": 71, "views": 3372, "answer_pids": [92965], "question_author": ""}
{"qid": 1170, "query": "How to prove $\\int_{-\\infty}^{+\\infty} f(x)dx = \\int_{-\\infty}^{+\\infty} f\\left(x - \\frac{1}{x}\\right)dx?$", "score": 71, "views": 3223, "answer_pids": [213681, 213696, 213764, 404436, 577658, 773495], "question_author": "jp89"}
{"qid": 1171, "query": "Conjecture $_2F_1\\left(\\frac14,\\frac34;\\,\\frac23;\\,\\frac13\\right)=\\frac1{\\sqrt{\\sqrt{\\frac4{\\sqrt{2-\\sqrt[3]4}}+\\sqrt[3]{4}+4}-\\sqrt{2-\\sqrt[3]4}-2}}$", "score": 71, "views": 1413, "answer_pids": [265281, 296238], "question_author": "nico"}
{"qid": 1172, "query": "What exactly is Laplace transform?", "score": 70, "views": 111220, "answer_pids": [91281, 91287, 91331, 491107], "question_author": "Kashif Malik"}
{"qid": 1173, "query": "$\\sum k! = 1! +2! +3! + \\cdots + n!$ ,is there a generic formula for this?", "score": 70, "views": 89025, "answer_pids": [117903, 133845, 648753], "question_author": ""}
{"qid": 1174, "query": "Can a limit of an integral be moved inside the integral?", "score": 70, "views": 77123, "answer_pids": [122996, 123000, 1380450], "question_author": "Rajesh D"}
{"qid": 1175, "query": "If $G/Z(G)$ is cyclic, then $G$ is abelian", "score": 70, "views": 57421, "answer_pids": [33444, 894852], "question_author": "shigeta"}
{"qid": 1176, "query": "Prove if $n^2$ is even, then $n$ is even.", "score": 70, "views": 45843, "answer_pids": [189386, 189388, 189390, 189391, 189392, 189401, 189437, 189531, 189537], "question_author": "bobthejoe"}
{"qid": 1177, "query": "An Explanation of the Kalman Filter", "score": 70, "views": 40168, "answer_pids": [362211, 366435, 663821], "question_author": "fbrereto"}
{"qid": 1178, "query": "Geometric interpretation for complex eigenvectors of a 2\u00d72 rotation matrix", "score": 70, "views": 36293, "answer_pids": [117320, 117445, 242675, 884077], "question_author": ""}
{"qid": 1179, "query": "Proving that $1$- and $2D$ simple symmetric random walks return to the origin with probability $1$", "score": 70, "views": 36249, "answer_pids": [328, 329, 338, 345, 854, 14412, 22527, 832062], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1180, "query": "incremental computation of standard deviation", "score": 70, "views": 35530, "answer_pids": [53160, 53177, 549457, 549461, 886308, 906874, 1194222, 1260400], "question_author": "Xander Dunn"}
{"qid": 1181, "query": "Intuitive explanation of a definition of the Fisher information", "score": 70, "views": 30231, "answer_pids": [128709, 128711, 246400], "question_author": "bobthejoe"}
{"qid": 1182, "query": "Not every metric is induced from a norm", "score": 70, "views": 28277, "answer_pids": [83940, 83941, 85882, 384634, 643196, 1056911], "question_author": ""}
{"qid": 1183, "query": "Why does multiplying a number on a clock face by 10 and then halving, give the minutes? ${}{}$", "score": 70, "views": 22163, "answer_pids": [1000849, 1000855, 1000895, 1001483], "question_author": "Kip"}
{"qid": 1184, "query": "Equivalent Definitions of the Operator Norm", "score": 70, "views": 18224, "answer_pids": [86333, 86340, 561891, 971159, 1176721, 1256943], "question_author": "Dasuraga"}
{"qid": 1185, "query": "Is it morally right and pedagogically right to google answers to homework?", "score": 70, "views": 14412, "answer_pids": [96004, 96036, 96056, 96078, 96083, 96089, 96090, 96185, 96203], "question_author": ""}
{"qid": 1186, "query": "Why is the complex plane shaped like it is?", "score": 70, "views": 9817, "answer_pids": [709644, 709646, 709652, 709675, 709688, 709710, 709725, 709752, 709756, 709767, 710383], "question_author": "Mad Scientist"}
{"qid": 1187, "query": "What are some applications of elementary linear algebra outside of math?", "score": 70, "views": 8237, "answer_pids": [441731, 441732, 441734, 441735, 441742, 441744, 441745, 441747, 441763, 441774, 441789, 441796, 442164, 442221, 442475, 442512, 442562, 442597, 443200], "question_author": "Umber Ferrule"}
{"qid": 1188, "query": "How can one prove that $e<\\pi$?", "score": 70, "views": 6084, "answer_pids": [227918, 227924, 227925, 227931, 228053, 231300, 231588, 711248], "question_author": "fragant1996"}
{"qid": 1189, "query": "Why does this miracle method for matrix inversion work?", "score": 70, "views": 5288, "answer_pids": [556497, 556499, 556503, 556509, 556545, 624420], "question_author": ""}
{"qid": 1190, "query": "Is there an intuitive reason for a certain operation to be associative?", "score": 70, "views": 3594, "answer_pids": [986512, 991023], "question_author": "Anonymous Type"}
{"qid": 1191, "query": "Laplace, Legendre, Fourier, Hankel, Mellin, Hilbert, Borel, Z...: unified treatment of transforms?", "score": 70, "views": 3224, "answer_pids": [194620, 223487, 365377, 692318], "question_author": "Stefano Borini"}
{"qid": 1192, "query": "Are there infinitely many super-palindromes?", "score": 70, "views": 2852, "answer_pids": [116506], "question_author": "Everett"}
{"qid": 1193, "query": "Let, $A\\subset\\mathbb{R}^2$. Show that $A$ can contain at most one point $p$ such that $A$ is isometric to $A \\setminus \\{p\\}$.", "score": 70, "views": 1936, "answer_pids": [940125, 941318], "question_author": "claws"}
{"qid": 1194, "query": "Numerical phenomenon. Who can explain?", "score": 70, "views": 1809, "answer_pids": [1213318, 1213322], "question_author": ""}
{"qid": 1195, "query": "Probability of 3 people in a room of 30 having the same birthday", "score": 69, "views": 95134, "answer_pids": [14132, 14133, 481403, 678124, 686244, 805538], "question_author": ""}
{"qid": 1196, "query": "Difference between continuity and uniform continuity", "score": 69, "views": 69581, "answer_pids": [290664, 290665, 443745, 847714, 913691, 1052648, 1062418], "question_author": "Matt"}
{"qid": 1197, "query": "Compact sets are closed?", "score": 69, "views": 54887, "answer_pids": [116838, 116839, 171098, 187957], "question_author": ""}
{"qid": 1198, "query": "What is mathematical research like?", "score": 69, "views": 36014, "answer_pids": [144503, 144508, 144509, 144521, 421931, 1012411], "question_author": "nico"}
{"qid": 1199, "query": "Difference between complete and closed set", "score": 69, "views": 35449, "answer_pids": [3935, 157192, 820776, 1418859], "question_author": ""}
{"qid": 1200, "query": "When does L Hopitals rule fail?", "score": 69, "views": 32265, "answer_pids": [386451, 386462], "question_author": ""}
{"qid": 1201, "query": "In classical logic, why is $(p\\Rightarrow q)$ True if $p$ is False and $q$ is True?", "score": 69, "views": 30852, "answer_pids": [37314, 37316, 298001, 380956, 410748, 756840, 1246339], "question_author": "Andy Bale"}
{"qid": 1202, "query": "Whats a good place to learn Lie groups?", "score": 69, "views": 30585, "answer_pids": [97948, 97949, 97959, 97960, 97962, 98088, 98208, 98213, 155328, 215118, 724229, 1065717], "question_author": "A B"}
{"qid": 1203, "query": "Why do we use a Least Squares fit?", "score": 69, "views": 29405, "answer_pids": [33528, 33530, 33559, 136106, 136120, 256066, 368347, 391372, 563986, 1169095], "question_author": "Reinstate Monica - Goodbye SE"}
{"qid": 1204, "query": "Chance of meeting in a bar", "score": 69, "views": 19498, "answer_pids": [53180, 53181, 53183, 53306, 53367], "question_author": "A B"}
{"qid": 1205, "query": "What is the most expensive item I could buy with \u00a350?", "score": 69, "views": 17367, "answer_pids": [753596, 753597, 753602, 753635, 753661, 753688, 754055, 754174], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1206, "query": "Can someone please explain the Riemann Hypothesis to me... in English?", "score": 69, "views": 16115, "answer_pids": [4620, 4630, 4633, 4717, 108399, 108426, 163118], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1207, "query": "Prove $0! = 1$ from first principles", "score": 69, "views": 15168, "answer_pids": [11576, 11578, 14105, 35798, 35802, 35821, 76882, 103095, 116264, 226176, 250703, 323457, 323461, 450234, 697655, 720680], "question_author": ""}
{"qid": 1208, "query": "Why cant the second fundamental theorem of calculus be proved in just two lines?", "score": 69, "views": 12913, "answer_pids": [762526, 762527, 762530, 762541, 762591, 762611, 762711, 762791, 762909, 763180], "question_author": "Etienne Low-D\u00e9carie"}
{"qid": 1209, "query": "What is so interesting about the zeroes of the Riemann $\\zeta$ function?", "score": 69, "views": 10860, "answer_pids": [69244, 69253, 69374, 130631], "question_author": ""}
{"qid": 1210, "query": "Does every Abelian group admit a ring structure?", "score": 69, "views": 8641, "answer_pids": [48520, 48521, 48522], "question_author": ""}
{"qid": 1211, "query": "Are there an infinite number of prime numbers where removing any number of digits leaves a prime?", "score": 69, "views": 7884, "answer_pids": [682071, 682079, 682090, 682129, 683254], "question_author": "Casebash"}
{"qid": 1212, "query": "Japanese Temple Problem From 1844", "score": 69, "views": 7556, "answer_pids": [1102867, 1102874, 1102963, 1103123, 1103198, 1103399], "question_author": ""}
{"qid": 1213, "query": "The limit of truncated sums of harmonic series, $\\lim\\limits_{k\\to\\infty}\\sum_{n=k+1}^{2k}{\\frac{1}{n}}$", "score": 69, "views": 7429, "answer_pids": [38723, 38724, 38726, 38730, 38731, 62571, 78373, 320999, 724539, 753335, 960103], "question_author": "Andy Bale"}
{"qid": 1214, "query": "Understandable questions which are hard for non-mathematicians but easy for mathematicians", "score": 69, "views": 7263, "answer_pids": [991024, 991025, 991027, 991029, 991030, 991033, 991048, 991054, 991068, 991069, 991072, 991093, 991110, 991111, 991116, 991122, 991155, 991158, 991159, 991160, 991213, 991221, 991230, 991241, 991292, 991336, 991384, 991396, 991401, 991485], "question_author": "bobthejoe"}
{"qid": 1215, "query": "Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?", "score": 69, "views": 6776, "answer_pids": [1013531, 1013532, 1013697, 1014085, 1014264, 1016296], "question_author": null}
{"qid": 1216, "query": "Fractal behavior along the boundary of convergence?", "score": 69, "views": 6244, "answer_pids": [48494], "question_author": "\u00d6zg\u00fcr"}
{"qid": 1217, "query": "Are there any valid continuous Sudoku grids?", "score": 69, "views": 3967, "answer_pids": [1368243, 1368246], "question_author": "rory"}
{"qid": 1218, "query": "A nasty integral of a rational function", "score": 69, "views": 3310, "answer_pids": [129382, 266626, 780717], "question_author": ""}
{"qid": 1219, "query": "Show that $\\int_{0}^{\\pi/2}\\frac {\\log^2\\sin x\\log^2\\cos x}{\\cos x\\sin x}\\mathrm{d}x=\\frac14\\left( 2\\zeta (5)-\\zeta(2)\\zeta (3)\\right)$", "score": 69, "views": 3064, "answer_pids": [140019, 391062, 391101, 1173160], "question_author": ""}
{"qid": 1220, "query": "How much does symbolic integration mean to mathematics?", "score": 69, "views": 2457, "answer_pids": [20515, 20530, 20531, 1315506], "question_author": "shigeta"}
{"qid": 1221, "query": "Evaluating sums and integrals using Taylors Theorem", "score": 69, "views": 1442, "answer_pids": [1387165], "question_author": "bobthejoe"}
{"qid": 1222, "query": "How to show that $\\det(AB) =\\det(A) \\det(B)$?", "score": 68, "views": 148969, "answer_pids": [31980, 31983, 31988, 32014, 32033, 32101, 144467, 144644, 234849, 1248441, 1438413], "question_author": "Jerry Schirmer"}
{"qid": 1223, "query": "What is the equation for a 3D line?", "score": 68, "views": 130448, "answer_pids": [189889, 189890, 189892, 1177650], "question_author": "DarenW"}
{"qid": 1224, "query": "Density of sum of two independent uniform random variables on $[0,1]$", "score": 68, "views": 100792, "answer_pids": [169205, 1089581, 1095832], "question_author": "Kirby"}
{"qid": 1225, "query": "Use of without loss of generality", "score": 68, "views": 44881, "answer_pids": [65835, 65837, 65842, 65845, 202683], "question_author": ""}
{"qid": 1226, "query": "Why is $\\pi $ equal to $3.14159...$?", "score": 68, "views": 38006, "answer_pids": [9693, 9694, 9697, 9701, 186030, 188039, 188050, 188064, 224464], "question_author": ""}
{"qid": 1227, "query": "$\\sqrt{7\\sqrt{7\\sqrt{7\\sqrt{7\\sqrt{7\\cdots}}}}}$ approximation", "score": 68, "views": 35765, "answer_pids": [265429, 265431, 265433, 265435, 309941, 442493, 464701], "question_author": "Nav"}
{"qid": 1228, "query": "Limit $\\frac{x^2y}{x^4+y^2}$ is found using polar coordinates but it is not supposed to exist.", "score": 68, "views": 32791, "answer_pids": [327700, 327701, 327702, 682647], "question_author": "Rik Smith-Unna"}
{"qid": 1229, "query": "What is the largest eigenvalue of the following matrix?", "score": 68, "views": 27843, "answer_pids": [976990, 976992, 976994, 977001, 977161, 977419, 977479, 977627, 977773, 985696], "question_author": "Justin L."}
{"qid": 1230, "query": "Good books on Math History", "score": 68, "views": 23852, "answer_pids": [16764, 16768, 16795, 16810, 16856, 16862, 197297], "question_author": "Justin L."}
{"qid": 1231, "query": "Why is a circle 1-dimensional?", "score": 68, "views": 19438, "answer_pids": [463112, 463114, 463117, 463118, 463124, 463164, 463274, 463277, 463304, 463481, 463968, 463975, 463990, 464014, 464196, 505087], "question_author": ""}
{"qid": 1232, "query": "Overview of basic results on cardinal arithmetic", "score": 68, "views": 14030, "answer_pids": [66812], "question_author": ""}
{"qid": 1233, "query": "Is The empty set is a subset of any set a convention?", "score": 68, "views": 13848, "answer_pids": [749679, 749777, 749794, 749797, 749802, 750037, 750220], "question_author": ""}
{"qid": 1234, "query": "How do people apply the Lebesgue integration theory?", "score": 68, "views": 13240, "answer_pids": [28270, 28283, 95704, 927871], "question_author": "rgrig"}
{"qid": 1235, "query": "Why is radian so common in maths?", "score": 68, "views": 13050, "answer_pids": [692883, 692884, 692885, 693064, 693130, 693215, 693222, 693352, 693661, 693781, 693968, 694042, 694099], "question_author": "Graviton"}
{"qid": 1236, "query": "Why do engineers use derivatives in discontinuous functions? Is it correct?", "score": 68, "views": 11933, "answer_pids": [1347237, 1347244, 1347370, 1347446, 1352715], "question_author": "Pratik Deoghare"}
{"qid": 1237, "query": "Why are addition and multiplication commutative, but not exponentiation?", "score": 68, "views": 11568, "answer_pids": [19060, 19078, 19092, 138486, 180263, 180271, 182792, 183688, 183706, 183736, 196499, 933199, 1123042], "question_author": "Michael Haren"}
{"qid": 1238, "query": "$100$-th derivative of the function $f(x)=e^{x}\\cos(x)$", "score": 68, "views": 10255, "answer_pids": [677124, 677127, 677128, 677324, 677956, 678036, 679245], "question_author": ""}
{"qid": 1239, "query": "Is it generally accepted that if you throw a dart at a number line you will NEVER hit a rational number?", "score": 68, "views": 9760, "answer_pids": [78318, 78321, 78350, 78362, 78405, 78423], "question_author": ""}
{"qid": 1240, "query": "Is memory unimportant in doing mathematics?", "score": 68, "views": 8705, "answer_pids": [116370, 116373, 116374, 116375, 116391, 116412, 116415, 116590, 885075], "question_author": ""}
{"qid": 1241, "query": "Theorems names that dont credit the right people", "score": 68, "views": 8165, "answer_pids": [192625, 192627, 192631, 192633, 192634, 192638, 192642, 192650, 192652, 192769, 192866, 192870, 192876, 192906, 192915, 194300, 194563, 284078, 413759, 924180, 1226756], "question_author": "mbadawi23"}
{"qid": 1242, "query": "What is the oldest open problem in geometry?", "score": 68, "views": 7557, "answer_pids": [378239, 378257, 378312, 378324, 378465, 379023, 380137], "question_author": "Anixx"}
{"qid": 1243, "query": "Why the emphasis on Projective Space in Algebraic Geometry?", "score": 68, "views": 6314, "answer_pids": [44573, 44576, 87095, 1374100], "question_author": "stackErr"}
{"qid": 1244, "query": "Is there a great mathematical example for a 12-year-old?", "score": 68, "views": 6105, "answer_pids": [191792, 191796, 191806, 191809, 191816, 191836, 192051, 192124, 192192, 192208, 192209, 192210, 192255, 192306, 201686, 224851, 224877, 308051, 309227, 436863, 436893, 436908, 436909, 436916, 436940, 440397, 694492], "question_author": "bobthejoe"}
{"qid": 1245, "query": "Am I too young to learn more advanced math and get a teacher?", "score": 68, "views": 4954, "answer_pids": [84378, 120965, 120991, 121003, 144772], "question_author": "bobthejoe"}
{"qid": 1246, "query": "How to straighten a parabola?", "score": 68, "views": 4944, "answer_pids": [1457043, 1457070, 1457076, 1457196, 1457237, 1457335], "question_author": ""}
{"qid": 1247, "query": "Geometric intuition for the tensor product of vector spaces", "score": 68, "views": 4409, "answer_pids": [226523, 255231], "question_author": "bobthejoe"}
{"qid": 1248, "query": "How likely is it not to be anyones best friend?", "score": 68, "views": 2518, "answer_pids": [316125, 730609, 888785], "question_author": ""}
{"qid": 1249, "query": "What is a real world application of polynomial factoring?", "score": 67, "views": 160412, "answer_pids": [43824, 43825, 43826, 43831, 43840, 43870, 43872, 99491, 103355, 121456, 132038, 157552, 224015, 332068, 361389, 1387620], "question_author": "bobthejoe"}
{"qid": 1250, "query": "What is the proof that the total number of subsets of a set is $2^n$?", "score": 67, "views": 102819, "answer_pids": [250007, 250008, 250014, 250025, 514919], "question_author": "Boy S"}
{"qid": 1251, "query": "How to explain for my daughter that $\\frac {2}{3}$ is greater than $\\frac {3}{5}$?", "score": 67, "views": 26021, "answer_pids": [843241, 843245, 843248, 843249, 843253, 843269, 843274, 843282, 843299, 843312, 843327, 843356, 843398, 843417, 843422, 843433, 843465, 843486, 843551, 843586, 843599, 843656, 843770, 843997, 846187, 847079, 849028, 852224], "question_author": "Hans-Peter Stricker"}
{"qid": 1252, "query": "Where is the flaw in this proof that 1=2? (Derivative of repeated addition)", "score": 67, "views": 22207, "answer_pids": [664, 665, 670, 1284, 1290, 567902, 656390, 1240522, 1298039], "question_author": ""}
{"qid": 1253, "query": "How to write a good mathematical paper?", "score": 67, "views": 17342, "answer_pids": [91306, 91623, 137866, 371392, 1101512], "question_author": null}
{"qid": 1254, "query": "What are the names of numbers in the binary system?", "score": 67, "views": 11576, "answer_pids": [853511, 853514, 853517, 853526, 853652, 853678, 853759, 853786, 853794, 853805, 853817, 853831, 854237, 854590, 854792, 854834, 854925, 856201], "question_author": null}
{"qid": 1255, "query": "Why $9$ & $11$ are special in divisibility tests using decimal digit sums? (casting out nines & elevens)", "score": 67, "views": 10963, "answer_pids": [8959, 8960, 8961, 8962, 8970, 8971, 1189787], "question_author": ""}
{"qid": 1256, "query": "Is it technically incorrect to write proofs forward?", "score": 67, "views": 9165, "answer_pids": [913232, 913241, 913276, 913700, 913715, 913778, 913843, 913954, 914271, 914443, 914705], "question_author": ""}
{"qid": 1257, "query": "What Is Exponentiation?", "score": 67, "views": 8133, "answer_pids": [309065, 309080, 309081, 309089, 309098, 309177, 309185, 309298, 310007, 347820, 1302872], "question_author": "naughts"}
{"qid": 1258, "query": "Why do I get one extra wrong solution when solving $2-x=-\\sqrt{x}$?", "score": 67, "views": 7347, "answer_pids": [631393, 631394, 631395, 631397, 631414, 631419, 631422, 631425, 631428, 631793, 631801, 631840], "question_author": ""}
{"qid": 1259, "query": "Mathematicians dont quit, they fade away", "score": 67, "views": 7182, "answer_pids": [278011, 278048, 278098, 278103], "question_author": "Daniel Standage"}
{"qid": 1260, "query": "If I know the order of every element in a group, do I know the group?", "score": 67, "views": 5215, "answer_pids": [519113, 519115, 519117], "question_author": ""}
{"qid": 1261, "query": "Is it possible to find an infinite set of points in the plane where the distance between any pair is rational?", "score": 67, "views": 4720, "answer_pids": [758093, 758095, 758409], "question_author": ""}
{"qid": 1262, "query": "Have there been efforts to introduce non Greek or Latin alphabets into mathematics?", "score": 67, "views": 4691, "answer_pids": [83435, 83436, 83450, 83479, 83480, 83497, 83938, 309592, 309601, 380881, 790117], "question_author": ""}
{"qid": 1263, "query": "Algebra: Best mental images", "score": 67, "views": 4465, "answer_pids": [157643, 160757, 161999, 162333], "question_author": "Casebash"}
{"qid": 1264, "query": "What is the average rational number?", "score": 67, "views": 3804, "answer_pids": [694981, 694988], "question_author": "Anake"}
{"qid": 1265, "query": "Is this similarity to the Fourier transform of the von Mangoldt function real?", "score": 67, "views": 3699, "answer_pids": [549623, 612005], "question_author": ""}
{"qid": 1266, "query": "Closed form for $\\int_0^1\\log\\log\\left(\\frac{1}{x}+\\sqrt{\\frac{1}{x^2}-1}\\right)\\mathrm dx$", "score": 67, "views": 3232, "answer_pids": [183642, 370044, 1156817], "question_author": "Hristo Hristov"}
{"qid": 1267, "query": "Topological spaces admitting an averaging function", "score": 67, "views": 1545, "answer_pids": [559353, 559388, 559412, 621347], "question_author": "kiss my armpit"}
{"qid": 1268, "query": "How do I tell if matrices are similar?", "score": 66, "views": 81130, "answer_pids": [7921, 82192, 285454, 326659, 1067215], "question_author": "Gerard"}
{"qid": 1269, "query": "What isnt a vector space?", "score": 66, "views": 29928, "answer_pids": [920207, 920209, 920210, 920211, 920212, 920213, 920266, 920268, 920771, 920790, 921053, 921078, 921604], "question_author": "Dman"}
{"qid": 1270, "query": "Prove that $\\prod_{k=1}^{n-1}\\sin\\frac{k \\pi}{n} = \\frac{n}{2^{n-1}}$", "score": 66, "views": 25996, "answer_pids": [4830, 514771, 910627], "question_author": ""}
{"qid": 1271, "query": "Why is Euclids proof on the infinitude of primes considered a proof?", "score": 66, "views": 18280, "answer_pids": [282529, 282533, 282540, 282541, 282592, 282594, 282690, 282726, 282732, 313864, 363716], "question_author": "Jason DeVito"}
{"qid": 1272, "query": "When not to treat dy/dx as a fraction in single-variable calculus?", "score": 66, "views": 17416, "answer_pids": [733335, 733341, 733401, 733432, 733527], "question_author": "David LeBauer"}
{"qid": 1273, "query": "how to read a mathematical paper?", "score": 66, "views": 15990, "answer_pids": [7576, 7580, 7595], "question_author": "Gerard"}
{"qid": 1274, "query": "Why do people lose in chess?", "score": 66, "views": 15180, "answer_pids": [366364, 366369, 366378, 366478, 366479, 367242, 367707], "question_author": ""}
{"qid": 1275, "query": "Is it mathematically valid to separate variables in a differential equation?", "score": 66, "views": 13800, "answer_pids": [504002, 504009], "question_author": ""}
{"qid": 1276, "query": "How much Math do you REALLY do in your job?", "score": 66, "views": 13077, "answer_pids": [139012, 139026, 139992, 139997, 140002, 142106, 143045, 352317], "question_author": ""}
{"qid": 1277, "query": "What is the importance of the infinitesimal generator of Brownian motion?", "score": 66, "views": 12745, "answer_pids": [306134, 307220, 709977], "question_author": ""}
{"qid": 1278, "query": "Do Parabolic Trigonometric Functions exist?", "score": 66, "views": 12515, "answer_pids": [68514, 68526, 77347, 443571, 554347], "question_author": "Alex Stone"}
{"qid": 1279, "query": "Why is the tensor product important when we already have direct and semidirect products?", "score": 66, "views": 12436, "answer_pids": [10496, 10500, 10501, 10502, 10508, 10509, 10518, 78892, 94260, 1074610], "question_author": "Daniel Standage"}
{"qid": 1280, "query": "Nuking the Mosquito \u2014 ridiculously complicated ways to achieve very simple results", "score": 66, "views": 10033, "answer_pids": [343812, 343814, 343820, 343821, 343822, 343826, 343827, 343837, 343863, 343868, 343872, 343879, 343882, 344237, 344304, 344527, 344598, 1006473, 1028816, 1254935, 1260489], "question_author": "Daniel Standage"}
{"qid": 1281, "query": "Are proofs by contradiction really logical?", "score": 66, "views": 9433, "answer_pids": [648370, 648383, 648384, 648446, 648468, 648525, 648536, 648592, 648598, 649227, 649237, 1196312, 1398163], "question_author": "pointernil"}
{"qid": 1282, "query": "Is it possible to formulate category theory without set theory?", "score": 66, "views": 8620, "answer_pids": [596603, 596744, 596765, 597095, 597216, 597533], "question_author": ""}
{"qid": 1283, "query": "What is the logic/rationale behind the vector cross product?", "score": 66, "views": 8596, "answer_pids": [580827], "question_author": "Alex Jeffries"}
{"qid": 1284, "query": "Is there a simple function that generates the series; $1,1,2,1,1,2,1,1,2...$ or $-1,-1,1,-1,-1,1...$", "score": 66, "views": 8333, "answer_pids": [730741, 730745, 730746, 730750, 730757, 730797, 730810, 730838, 730845, 730851, 730872, 731057, 731062, 731138, 731213, 731228, 731471, 732309], "question_author": ""}
{"qid": 1285, "query": "Integral $\\int_0^1\\frac{\\ln\\left(x+\\sqrt2\\right)}{\\sqrt{2-x}\\,\\sqrt{1-x}\\,\\sqrt{\\vphantom{1}x}}\\mathrm dx$", "score": 66, "views": 7419, "answer_pids": [258510, 301466, 312952], "question_author": "Lucky"}
{"qid": 1286, "query": "Can $18$ consecutive integers be separated into two groups,such that their product is equal?", "score": 66, "views": 7252, "answer_pids": [285187, 285319], "question_author": "Paul Sonier"}
{"qid": 1287, "query": "Integration of forms and integration on a measure space", "score": 66, "views": 7052, "answer_pids": [26502, 26512], "question_author": ""}
{"qid": 1288, "query": "Simplicial homology of real projective space by Mayer-Vietoris", "score": 66, "views": 5348, "answer_pids": [270131], "question_author": ""}
{"qid": 1289, "query": "Evaluating the log gamma integral $\\int_{0}^{z} \\log \\Gamma (x) \\, \\mathrm dx$ in terms of the Hurwitz zeta function", "score": 66, "views": 4719, "answer_pids": [636708, 726179, 1173933], "question_author": "JeremiahLee"}
{"qid": 1290, "query": "Does the string of prime numbers contain all natural numbers?", "score": 66, "views": 4040, "answer_pids": [387111, 387113, 475723, 475741], "question_author": "David LeBauer"}
{"qid": 1291, "query": "Showing that $\\int\\limits_{-a}^a \\frac{f(x)}{1+e^{x}} \\mathrm dx = \\int\\limits_0^a f(x) \\mathrm dx$, when $f$ is even", "score": 66, "views": 3839, "answer_pids": [31855, 31895, 31907, 32437, 658554], "question_author": ""}
{"qid": 1292, "query": "Differential forms on fuzzy manifolds", "score": 66, "views": 2318, "answer_pids": [544224], "question_author": "Santosh Linkha"}
{"qid": 1293, "query": "Difference between NFA and DFA", "score": 65, "views": 216156, "answer_pids": [255855, 255874, 406409], "question_author": ""}
{"qid": 1294, "query": "Given this transformation matrix, how do I decompose it into translation, rotation and scale matrices?", "score": 65, "views": 94735, "answer_pids": [195978, 578385, 1273494], "question_author": "caeruleus"}
{"qid": 1295, "query": "How can a probability density be greater than one and integrate to one", "score": 65, "views": 86165, "answer_pids": [54344, 158850, 525904, 889035], "question_author": "Alex Stone"}
{"qid": 1296, "query": "Good books on mathematical logic?", "score": 65, "views": 39851, "answer_pids": [2475, 2477, 2478, 2486, 6747, 6748, 12124, 12126, 93500, 93523, 530355, 558819], "question_author": "Alex Stone"}
{"qid": 1297, "query": "There are 4 cups of liquid. Three are water and one is poison. If you were to drink 3 of the 4 cups, what is the probability of being poisoned?", "score": 65, "views": 38176, "answer_pids": [1026983, 1027002, 1027080, 1027362, 1027408, 1027475, 1027788, 1028017], "question_author": ""}
{"qid": 1298, "query": "How to convince a layperson that the $\\pi = 4$ proof is wrong?", "score": 65, "views": 31667, "answer_pids": [22977, 22980, 22981, 22983, 23016, 23018, 23020, 23081, 23082, 23138, 76454, 258552, 487759, 712972, 854267], "question_author": ""}
{"qid": 1299, "query": "Every subsequence of $x_n$ has a further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.", "score": 65, "views": 28924, "answer_pids": [186967, 896285, 938955, 1280511], "question_author": ""}
{"qid": 1300, "query": "Intuitive explanation of a positive semidefinite matrix", "score": 65, "views": 18382, "answer_pids": [5556, 5557, 5559, 14974, 1158062], "question_author": ""}
{"qid": 1301, "query": "Proving that the tensor product is right exact", "score": 65, "views": 16947, "answer_pids": [54294, 54298, 877303], "question_author": ""}
{"qid": 1302, "query": "Integrals of the form ${\\large\\int}_0^\\infty\\operatorname{arccot}(x)\\cdot\\operatorname{arccot}(a\\,x)\\cdot\\operatorname{arccot}(b\\,x)\\ dx$", "score": 65, "views": 15231, "answer_pids": [513044, 514375, 515022, 609126], "question_author": "Lazer"}
{"qid": 1303, "query": "Express 99 2/3% as a fraction? No calculator", "score": 65, "views": 14651, "answer_pids": [526809, 526810, 526812, 526813, 526817, 526846, 526869, 526902, 526928, 526952, 526956, 527224, 529511], "question_author": "Kirby"}
{"qid": 1304, "query": "What exactly is calculus?", "score": 65, "views": 14113, "answer_pids": [847581, 847583, 847586, 847592, 847740, 847779, 847781, 847783, 847863, 847885, 848135, 848213, 849448, 850223, 850391], "question_author": "InquilineKea"}
{"qid": 1305, "query": "Does it ever make sense NOT to go to the most prestigious graduate school you can get into?", "score": 65, "views": 11557, "answer_pids": [4977, 4978, 4980, 4985, 4986, 4990, 4992, 5080, 81312], "question_author": "jamiers"}
{"qid": 1306, "query": "Set of continuity points of a real function", "score": 65, "views": 10169, "answer_pids": [35739, 35909, 581022], "question_author": ""}
{"qid": 1307, "query": "Does every prime divide some Fibonacci number?", "score": 65, "views": 7783, "answer_pids": [306703, 306704, 306717, 306936, 308654], "question_author": "Robin Maben"}
{"qid": 1308, "query": "Does $\\sum _{n=1}^{\\infty } \\frac{\\sin(\\text{ln}(n))}{n}$ converge?", "score": 65, "views": 6764, "answer_pids": [282183, 298854], "question_author": ""}
{"qid": 1309, "query": "Understanding the Laplace operator conceptually", "score": 65, "views": 6339, "answer_pids": [349026, 349064, 349073, 349233, 351664], "question_author": ""}
{"qid": 1310, "query": "What is Cauchy Schwarz in 8th grade terms?", "score": 65, "views": 6137, "answer_pids": [596734, 596737, 596769, 596835, 597029, 597132], "question_author": "thejh"}
{"qid": 1311, "query": "Proving the existence of a proof without actually giving a proof", "score": 65, "views": 5812, "answer_pids": [639671, 639700, 639704, 639975, 639984, 641855], "question_author": "Sonny Ordell"}
{"qid": 1312, "query": "How and why does Grothendiecks work provide tools to attack problems in number theory?", "score": 65, "views": 5793, "answer_pids": [581453, 581721, 731533], "question_author": "JohnAllen"}
{"qid": 1313, "query": "Limits: How to evaluate $\\lim\\limits_{x\\rightarrow \\infty}\\sqrt[n]{x^{n}+a_{n-1}x^{n-1}+\\cdots+a_{0}}-x$", "score": 65, "views": 5105, "answer_pids": [16275, 16279, 16287, 16583, 32138], "question_author": ""}
{"qid": 1314, "query": "Does multiplying all a numbers roots together give a product of infinity?", "score": 65, "views": 4602, "answer_pids": [715513, 715526, 715619], "question_author": "Andreas"}
{"qid": 1315, "query": "What are some good ways to get children excited about math?", "score": 65, "views": 4221, "answer_pids": [43, 52, 53, 73, 108, 125, 128, 140, 144, 289, 754, 760, 1128, 1760, 1838, 4055, 156566], "question_author": "CptanPanic"}
{"qid": 1316, "query": "Why does $\\cos(x) + \\cos(y) - \\cos(x + y) = 0$ look like an ellipse?", "score": 65, "views": 3773, "answer_pids": [533315, 533321, 533335, 533366, 533400, 533436, 534111, 534313, 536704], "question_author": ""}
{"qid": 1317, "query": "When is an infinite product of natural numbers regularizable?", "score": 65, "views": 3671, "answer_pids": [765933], "question_author": "Andreas"}
{"qid": 1318, "query": "Why is it not true that $\\int_0^{\\pi} \\sin(x)\\; dx = 0$?", "score": 65, "views": 2489, "answer_pids": [586730, 586734, 586738, 586741, 586747, 586754, 586857, 586939, 586947, 590633, 595023], "question_author": ""}
{"qid": 1319, "query": "Evaluating $\\int_{0}^{1}\\cdots\\int_{0}^{1}\\bigl\\{\\frac{1}{x_{1}\\cdots x_{n}}\\bigr\\}^{2}\\:\\mathrm{d}x_{1}\\cdots\\mathrm{d}x_{n}$", "score": 65, "views": 2483, "answer_pids": [381731, 381814], "question_author": ""}
{"qid": 1320, "query": "Are the sums $\\sum_{n=1}^{\\infty} \\frac{1}{(n!)^k}$ transcendental?", "score": 65, "views": 1918, "answer_pids": [798321], "question_author": "Justin L."}
{"qid": 1321, "query": "Does $\\Bbb{CP}^{2n} \\mathbin{\\#} \\Bbb{CP}^{2n}$ ever support an almost complex structure?", "score": 65, "views": 1061, "answer_pids": [991028], "question_author": ""}
{"qid": 1322, "query": "Closed form solution for $\\sum_{n=1}^\\infty\\frac{1}{1+\\frac{n^2}{1+\\frac{1}{\\stackrel{\\ddots}{1+\\frac{1}{1+n^2}}}}}$.", "score": 65, "views": 969, "answer_pids": [452051, 452436, 455838], "question_author": ""}
{"qid": 1323, "query": "What is a good book for learning math, from middle school level?", "score": 64, "views": 141811, "answer_pids": [36465, 36490, 36495, 55378, 55431, 130732, 130741, 162896, 262704, 434423, 626724], "question_author": null}
{"qid": 1324, "query": "difference between dot product and inner product", "score": 64, "views": 78575, "answer_pids": [222355, 222356, 222357, 1158656], "question_author": "Tobias Kienzler"}
{"qid": 1325, "query": "Check if a point is within an ellipse", "score": 64, "views": 77985, "answer_pids": [40209, 40267, 118408, 691388], "question_author": ""}
{"qid": 1326, "query": "What does proving the Riemann Hypothesis accomplish?", "score": 64, "views": 47667, "answer_pids": [189970, 189971, 253846, 277988, 355595], "question_author": "Emmie MC"}
{"qid": 1327, "query": "Is Serge Langs Algebra still worth reading?", "score": 64, "views": 26801, "answer_pids": [167654, 168169], "question_author": "TROLLHUNTER"}
{"qid": 1328, "query": "Highest power of a prime $p$ dividing $N!$", "score": 64, "views": 24958, "answer_pids": [71506, 71528, 112045, 112486, 451426, 1315657, 1315673], "question_author": "bobthejoe"}
{"qid": 1329, "query": "Proof that Pi is constant (the same for all circles), without using limits", "score": 64, "views": 21814, "answer_pids": [1907, 1908, 1916, 142960, 171765, 219402, 765899], "question_author": "Jonathan Fischoff"}
{"qid": 1330, "query": "Interesting real life applications of serious theorems", "score": 64, "views": 21193, "answer_pids": [343443, 343449, 343486, 343491, 343496, 343605, 343686, 343709, 343754, 343842, 343847, 343986, 344037, 344040, 344514, 344858], "question_author": ""}
{"qid": 1331, "query": "How do I motivate myself to do math again?", "score": 64, "views": 18709, "answer_pids": [243530, 243541, 243546, 243562, 243568, 243655, 243656, 243660, 243730, 243768, 243869, 243917], "question_author": ""}
{"qid": 1332, "query": "Under what conditions the quotient space of a manifold is a manifold?", "score": 64, "views": 16195, "answer_pids": [230778, 230785, 498618], "question_author": ""}
{"qid": 1333, "query": "How to effectively study math?", "score": 64, "views": 15679, "answer_pids": [19664, 19677, 20131, 70548, 73287, 73293, 73294, 83215], "question_author": "Peter Smit"}
{"qid": 1334, "query": "Is there any branch of Mathematics which has no applications in any other field or in real world?", "score": 64, "views": 14615, "answer_pids": [138051, 138052, 138057, 352190], "question_author": "Boy S"}
{"qid": 1335, "query": "Nice expression for minimum of three variables?", "score": 64, "views": 14129, "answer_pids": [7473, 7475, 28606, 46700, 815999], "question_author": ""}
{"qid": 1336, "query": "What exactly is a number?", "score": 64, "views": 13993, "answer_pids": [368358, 368364, 368372, 368376, 368385, 368389, 368414, 368423, 368425, 368430, 368434, 368441, 368457, 368704, 368859, 371434], "question_author": "Emmie MC"}
{"qid": 1337, "query": "Dominoes and induction, or how does induction work?", "score": 64, "views": 11385, "answer_pids": [10810, 10811, 10823, 10829, 13655, 672891], "question_author": "jms"}
{"qid": 1338, "query": "Riemann hypothesis: is Bender-Brody-M\u00fcller Hamiltonian a new line of attack?", "score": 64, "views": 10811, "answer_pids": [839746, 840468, 841271, 843715, 845853], "question_author": "Abe"}
{"qid": 1339, "query": "Why do we not have to prove definitions?", "score": 64, "views": 9692, "answer_pids": [556508, 556511, 556513, 556514, 556520, 556552, 556599, 556727, 556734], "question_author": null}
{"qid": 1340, "query": "Optimal strategy for cutting a sausage?", "score": 64, "views": 6609, "answer_pids": [1061661, 1061663, 1061703], "question_author": "Graviton"}
{"qid": 1341, "query": "Symbol for probably equal to (barring pathology)?", "score": 64, "views": 6507, "answer_pids": [572462, 572470, 572473, 572545, 572688, 572851, 572942, 573100, 573336], "question_author": "Abe"}
{"qid": 1342, "query": "Is it not effective to learn math top-down?", "score": 64, "views": 6249, "answer_pids": [214924, 214926, 214928, 214932, 214934, 214935, 214940, 214947, 215558, 215699], "question_author": ""}
{"qid": 1343, "query": "If we randomly select 25 integers between 1 and 100, how many consecutive integers should we expect?", "score": 64, "views": 6181, "answer_pids": [612036, 612094, 612120, 612215, 612216, 612828], "question_author": "malexmave"}
{"qid": 1344, "query": "A new general formula for the quadratic equation?", "score": 64, "views": 5040, "answer_pids": [1291515, 1291520, 1291537, 1294072], "question_author": "Akhil Mathew"}
{"qid": 1345, "query": "Conways Murder Weapon", "score": 64, "views": 4496, "answer_pids": [918794, 918795], "question_author": "Emmie MC"}
{"qid": 1346, "query": "How to explain to the layperson what mathematics is, why its important, and why its interesting", "score": 64, "views": 3858, "answer_pids": [46440, 46471, 46479, 46493, 46527, 46540], "question_author": ""}
{"qid": 1347, "query": "Is the box topology good for anything?", "score": 64, "views": 3810, "answer_pids": [19820, 76754, 76789, 90363, 758251], "question_author": "David LeBauer"}
{"qid": 1348, "query": "Does the series $ \\sum\\limits_{n=1}^{\\infty} \\frac{1}{n^{1 + |\\sin(n)|}} $ converge or diverge?", "score": 64, "views": 3208, "answer_pids": [130850, 130851, 130969, 1252599], "question_author": "Garima Kushwaha"}
{"qid": 1349, "query": "Does Fermats Last Theorem hold for cyclotomic integers in $\\mathbb{Q(\\zeta_{37})}$?", "score": 64, "views": 3165, "answer_pids": [75767], "question_author": ""}
{"qid": 1350, "query": "In combinatorics, how can one verify that one has counted correctly?", "score": 64, "views": 2907, "answer_pids": [880865, 880873, 881195], "question_author": "Anil"}
{"qid": 1351, "query": "Algebraic Topology Challenge: Homology of an Infinite Wedge of Spheres", "score": 64, "views": 2328, "answer_pids": [82379, 83740, 558436], "question_author": ""}
{"qid": 1352, "query": "Fourier Transform of Derivative", "score": 63, "views": 130296, "answer_pids": [201844, 201857], "question_author": "qazwsx"}
{"qid": 1353, "query": "What is the relation between rank of a matrix, its eigenvalues and eigenvectors", "score": 63, "views": 77293, "answer_pids": [538252], "question_author": "jcollum"}
{"qid": 1354, "query": "What is the standard basis for fields of complex numbers?", "score": 63, "views": 76766, "answer_pids": [63130, 63134, 63138], "question_author": "Thunder"}
{"qid": 1355, "query": "Good books for a high schooler self-studying Abstract Algebra?", "score": 63, "views": 71935, "answer_pids": [6599, 6604, 6610, 82803], "question_author": ""}
{"qid": 1356, "query": "Why do the French count so strangely?", "score": 63, "views": 47091, "answer_pids": [183875, 183882, 183892, 183952, 184035, 184062, 189085, 189277], "question_author": ""}
{"qid": 1357, "query": "A comprehensive list of binomial identities?", "score": 63, "views": 40999, "answer_pids": [1886, 3693], "question_author": ""}
{"qid": 1358, "query": "What is a covector and what is it used for?", "score": 63, "views": 35794, "answer_pids": [117087, 117111, 117132, 397704, 959260], "question_author": ""}
{"qid": 1359, "query": "First-Order Logic vs. Second-Order Logic", "score": 63, "views": 28516, "answer_pids": [13041, 13042, 13153], "question_author": ""}
{"qid": 1360, "query": "Why is the determinant the volume of a parallelepiped in any dimensions?", "score": 63, "views": 28401, "answer_pids": [200383, 200405, 250428, 761282, 783169], "question_author": ""}
{"qid": 1361, "query": "Introductory texts on manifolds", "score": 63, "views": 26554, "answer_pids": [24785, 24786, 24787, 24806, 24807, 24810, 24813, 24834, 108580], "question_author": "Keenan Pepper"}
{"qid": 1362, "query": "Functions that are their own inverse.", "score": 63, "views": 22853, "answer_pids": [540559, 540567, 540568, 540571, 540802], "question_author": "Andreas"}
{"qid": 1363, "query": "How do I convince my students that the choice of variable of integration is irrelevant?", "score": 63, "views": 19817, "answer_pids": [288026, 288027, 288057, 288134, 288149, 288161, 288177, 288210, 288213, 288256, 288268, 288313, 288364, 288423, 288517, 288988], "question_author": ""}
{"qid": 1364, "query": "If a coin toss is observed to come up as heads many times, does that affect the probability of the next toss?", "score": 63, "views": 19678, "answer_pids": [717469, 717470, 717471, 717505, 717626, 717729, 717748, 717773, 717939], "question_author": "BBischof"}
{"qid": 1365, "query": "A way to find this shaded area without calculus?", "score": 63, "views": 17986, "answer_pids": [1105058, 1105064, 1105133, 1105175, 1105538, 1105900, 1122435, 1310538, 1390366], "question_author": "Gabriel Fair"}
{"qid": 1366, "query": "Learning Roadmap for Algebraic Topology", "score": 63, "views": 16945, "answer_pids": [46768, 46769, 46801, 46807], "question_author": "InquilineKea"}
{"qid": 1367, "query": "What is the limit of $n \\sin (2 \\pi \\cdot e \\cdot n!)$ as $n$ goes to infinity?", "score": 63, "views": 14308, "answer_pids": [40011, 425160, 898540], "question_author": "InquilineKea"}
{"qid": 1368, "query": "How does a non-mathematician go about publishing a proof in a way that ensures it to be up to the mathematical communitys standards?", "score": 63, "views": 13200, "answer_pids": [132750, 132755, 132756, 132876, 133148, 133188, 136786, 522294], "question_author": "InquilineKea"}
{"qid": 1369, "query": "When to give up on a hard math problem?", "score": 63, "views": 12342, "answer_pids": [352148, 352154, 352155, 352277, 352336, 352420, 352754, 352851, 355234], "question_author": "InquilineKea"}
{"qid": 1370, "query": "Simplicial Complex vs Delta Complex vs CW Complex", "score": 63, "views": 11761, "answer_pids": [599464], "question_author": ""}
{"qid": 1371, "query": "Why are primes considered to be the building blocks of the integers?", "score": 63, "views": 10019, "answer_pids": [452222, 452239, 452264, 452267, 452340, 452406, 452413, 452438, 452577, 453628], "question_author": "Gabriel Fair"}
{"qid": 1372, "query": "What are some mathematically interesting computations involving matrices?", "score": 63, "views": 9435, "answer_pids": [904703, 904710, 904729, 904731, 904743, 904756, 904777, 904779, 904783, 904796, 904800, 904847, 904852, 904859, 904861, 904919, 904961, 905049, 905087, 905137, 905219, 905264, 907089, 907190], "question_author": "Gabriel Fair"}
{"qid": 1373, "query": "Arranging numbers from $1$ to $n$ such that the sum of every two adjacent numbers is a perfect power", "score": 63, "views": 9397, "answer_pids": [476737], "question_author": ""}
{"qid": 1374, "query": "Given real numbers: define integers?", "score": 63, "views": 8020, "answer_pids": [127365, 127367, 127368, 127370, 127377, 127380, 127383, 127387, 127570], "question_author": ""}
{"qid": 1375, "query": "Arc length contest! Minimize the arc length of $f(x)$ when given three conditions.", "score": 63, "views": 7416, "answer_pids": [460460, 460474, 463854, 464078, 465116, 465178, 465289, 466431, 466908, 466913, 1289421, 1289490], "question_author": "RoflcoptrException"}
{"qid": 1376, "query": "Is $ \\sum\\limits_{n=1}^\\infty \\frac{|\\sin n|^n}n$ convergent\uff1f", "score": 63, "views": 7012, "answer_pids": [357712, 606387, 935024], "question_author": ""}
{"qid": 1377, "query": "What does strength refer to in mathematics?", "score": 63, "views": 6592, "answer_pids": [651660, 651664, 651665, 651666, 651994, 652216], "question_author": "Aaron Digulla"}
{"qid": 1378, "query": "Is there a reason it is so rare we can solve differential equations?", "score": 63, "views": 6557, "answer_pids": [1338826, 1338834, 1338838, 1339007, 1339157, 1344555], "question_author": "TCM"}
{"qid": 1379, "query": "Why it is important to write a function as sum of even and odd functions?", "score": 63, "views": 5210, "answer_pids": [1384657, 1384666, 1384832, 1384996], "question_author": ""}
{"qid": 1380, "query": "What is a proof?", "score": 63, "views": 5180, "answer_pids": [186972, 187001, 187026, 187495, 189052, 189460, 189462, 189582, 190660, 190808, 194471], "question_author": "Aaron Digulla"}
{"qid": 1381, "query": "Area covered by a constant length segment rotating around the center of a square.", "score": 63, "views": 5127, "answer_pids": [323643, 323675, 323691, 323722, 323882, 324230], "question_author": ""}
{"qid": 1382, "query": "Is it normal to treat Math Theorems as Black Boxes", "score": 63, "views": 4994, "answer_pids": [770441, 770519, 770671, 771086], "question_author": "Pratik Deoghare"}
{"qid": 1383, "query": "Good history of mathematical ideas book?", "score": 63, "views": 4803, "answer_pids": [650606, 650609, 650627, 650641, 650659, 650681, 650688, 650707, 650714, 650791, 650889, 651055, 651062, 651072, 651129, 651224, 651296], "question_author": "Kirby"}
{"qid": 1384, "query": "Abstract nonsense proof of snake lemma", "score": 63, "views": 4608, "answer_pids": [278268], "question_author": "Simon Verbeke"}
{"qid": 1385, "query": "How to find a total order with constrained comparisons", "score": 63, "views": 1422, "answer_pids": [1449497], "question_author": ""}
{"qid": 1386, "query": "Integers $n$ such that $i(i+1)(i+2) \\cdots (i+n)$ is real or pure imaginary", "score": 63, "views": 1394, "answer_pids": [275430, 275470, 277404], "question_author": "unsym"}
{"qid": 1387, "query": "Which Algebraic Properties Distinguish Lie Groups from Abstract Groups?", "score": 63, "views": 1029, "answer_pids": [1169870], "question_author": ""}
{"qid": 1388, "query": "I have learned that 1/0 is infinity, why isnt it minus infinity?", "score": 62, "views": 146371, "answer_pids": [65025, 65026, 65030, 65035, 65036, 65037, 65044, 65065, 65070, 65106, 65181, 79623, 694368], "question_author": null}
{"qid": 1389, "query": "Difference between axioms, theorems, postulates, corollaries, and hypotheses", "score": 62, "views": 133570, "answer_pids": [4470, 377334, 428899, 897975], "question_author": "mau"}
{"qid": 1390, "query": "Is the sum and difference of two irrationals always irrational?", "score": 62, "views": 130348, "answer_pids": [79391, 79392, 79438, 617843, 617916], "question_author": ""}
{"qid": 1391, "query": "What is the difference between the Frobenius norm and the 2-norm of a matrix?", "score": 62, "views": 72559, "answer_pids": [17793, 230807, 1025176, 1280434], "question_author": ""}
{"qid": 1392, "query": "How does a calculator calculate the sine, cosine, tangent using just a number?", "score": 62, "views": 66697, "answer_pids": [185878, 185885, 593319], "question_author": "unsym"}
{"qid": 1393, "query": "How do you compute negative numbers to fractional powers?", "score": 62, "views": 51727, "answer_pids": [151399, 151400, 151406, 656896, 656904], "question_author": "Steve314"}
{"qid": 1394, "query": "Proving $1^3+ 2^3 + \\cdots + n^3 = \\left(\\frac{n(n+1)}{2}\\right)^2$ using induction", "score": 62, "views": 43855, "answer_pids": [32964, 32965, 32972, 95057, 428599, 495318, 592356, 702641, 812520, 1015339, 1015342, 1227890, 1227900], "question_author": ""}
{"qid": 1395, "query": "Probability that random moves in the game 2048 will win", "score": 62, "views": 36369, "answer_pids": [318115, 318157, 318161, 318488, 319036, 322585, 396018, 493600], "question_author": ""}
{"qid": 1396, "query": "Why does $\\tan^{-1}(1)+\\tan^{-1}(2)+\\tan^{-1}(3)=\\pi$?", "score": 62, "views": 19695, "answer_pids": [98898, 98904, 98909, 98910, 123323], "question_author": ""}
{"qid": 1397, "query": "Proof of the Hockey-Stick Identity: $\\sum\\limits_{t=0}^n \\binom tk = \\binom{n+1}{k+1}$", "score": 62, "views": 15609, "answer_pids": [187324, 187368, 187511, 276001, 587120, 587135, 587143, 587147, 587160, 587404, 630722, 708027, 712162, 1199164, 1219715, 1280761], "question_author": "jp89"}
{"qid": 1398, "query": "How can you prove that the square root of two is irrational?", "score": 62, "views": 15419, "answer_pids": [1, 6, 1274, 9239, 9248, 9274, 113390, 330170, 412245, 528560, 657986, 920961, 1091952, 1368507, 1380961], "question_author": ""}
{"qid": 1399, "query": "Is the Law of Large Numbers empirically proven?", "score": 62, "views": 14705, "answer_pids": [461220, 461221, 461231, 461243, 461281, 461337, 461397, 461418, 461476, 461615, 461620, 461645, 461817, 461879, 461912, 462128, 464234], "question_author": ""}
{"qid": 1400, "query": "Cardinality of Borel sigma algebra", "score": 62, "views": 14272, "answer_pids": [37384, 1022435], "question_author": "Grigory M"}
{"qid": 1401, "query": "Motivation and methods for self-study", "score": 62, "views": 9992, "answer_pids": [91406, 91443, 91452], "question_author": "Grigory M"}
{"qid": 1402, "query": "Intuition on group homomorphisms", "score": 62, "views": 9115, "answer_pids": [117873, 117874, 117882, 117886, 117889, 1061989], "question_author": "Robin Maben"}
{"qid": 1403, "query": "Can someone explain the Yoneda Lemma to an applied mathematician?", "score": 62, "views": 8981, "answer_pids": [19859, 19861, 19864, 19879], "question_author": "harpalss"}
{"qid": 1404, "query": "Polynomial division: an obvious trick? [reducing mod $\\textit{simpler}$ multiples]", "score": 62, "views": 8758, "answer_pids": [1170599, 1170603, 1170605, 1170607, 1170653, 1170679, 1170730], "question_author": "harpalss"}
{"qid": 1405, "query": "How can I introduce complex numbers to precalculus students?", "score": 62, "views": 7940, "answer_pids": [122049, 122050, 122051, 122059, 122061, 122083, 122102, 122139, 122141, 122255, 125737], "question_author": "Sam"}
{"qid": 1406, "query": "$n$th derivative of $e^{1/x}$", "score": 62, "views": 7107, "answer_pids": [10182, 10210, 10263, 19589, 365531], "question_author": "Anil"}
{"qid": 1407, "query": "What should an amateur do with a proof of an open problem?", "score": 62, "views": 6991, "answer_pids": [381082, 381092, 381095, 381102, 381112, 381948], "question_author": ""}
{"qid": 1408, "query": "Why does an exponential function eventually get bigger than a quadratic", "score": 62, "views": 6806, "answer_pids": [1413045, 1413050, 1413089, 1413102, 1413213, 1413219, 1413277, 1413327, 1413371, 1413373, 1413461, 1413816, 1413987, 1413999, 1414911], "question_author": "Dalron"}
{"qid": 1409, "query": "Big List of Erd\u0151s elementary proofs", "score": 62, "views": 5354, "answer_pids": [712250, 712253, 712257, 712288, 712499, 712527, 712597, 712641, 713308, 714543], "question_author": ""}
{"qid": 1410, "query": "Continuity and the Axiom of Choice", "score": 62, "views": 5237, "answer_pids": [64374, 64377], "question_author": "Snowman"}
{"qid": 1411, "query": "Graphs for which a calculus student can reasonably compute the arclength", "score": 62, "views": 5137, "answer_pids": [1202165, 1202172, 1202180, 1202423, 1202580, 1202740, 1202937, 1204087], "question_author": ""}
{"qid": 1412, "query": "Conjugate subgroup strictly contained in the initial subgroup?", "score": 62, "views": 4517, "answer_pids": [55516, 55521, 198593, 219810], "question_author": ""}
{"qid": 1413, "query": "Rigour in mathematics", "score": 62, "views": 4037, "answer_pids": [217187, 217193, 217194, 217197, 217198, 217200, 217202, 217203, 217208, 217211, 217225, 217228, 217297, 217313, 218018], "question_author": "Michael Benfield"}
{"qid": 1414, "query": "Where does the word torsion in algebra come from?", "score": 62, "views": 3651, "answer_pids": [143944, 143945, 143973], "question_author": ""}
{"qid": 1415, "query": "Computation with a memory wiped computer", "score": 62, "views": 3316, "answer_pids": [588, 652], "question_author": "J. Musser"}
{"qid": 1416, "query": "Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.", "score": 62, "views": 2392, "answer_pids": [586165, 586186, 586230, 587802, 609090, 610290, 610359], "question_author": "Kaypro II"}
{"qid": 1417, "query": "Unexpected approximations which have led to important mathematical discoveries", "score": 62, "views": 2370, "answer_pids": [185836, 199475, 698703], "question_author": "Rajesh D"}
{"qid": 1418, "query": "On Ramanujans curious equality for $\\sqrt{2\\,(1-3^{-2})(1-7^{-2})(1-11^{-2})\\cdots} $", "score": 62, "views": 1683, "answer_pids": [772042, 1293010], "question_author": "bobthejoe"}
{"qid": 1419, "query": "Is $\\lfloor n!/e\\rfloor$ always even for $n\\in\\mathbb N$?", "score": 62, "views": 1270, "answer_pids": [593135, 593137, 593160], "question_author": "user4951"}
{"qid": 1420, "query": "How do I prove that a function is well defined?", "score": 61, "views": 93409, "answer_pids": [149367, 149368, 149369, 149412, 149426, 149485, 149518], "question_author": ""}
{"qid": 1421, "query": "Is the rank of a matrix the same of its transpose? If yes, how can I prove it?", "score": 61, "views": 80612, "answer_pids": [1403, 1404, 32625, 32632, 42895], "question_author": "Justin L."}
{"qid": 1422, "query": "Evaluating the indefinite integral $ \\int \\sqrt{\\tan x} ~ \\mathrm{d}{x}. $", "score": 61, "views": 63144, "answer_pids": [354752, 354757, 497240, 518268, 687332, 781407, 1340276], "question_author": "Eric O. Korman"}
{"qid": 1423, "query": "Proof that the irrational numbers are uncountable", "score": 61, "views": 62332, "answer_pids": [434, 54891, 55018, 333366], "question_author": "Simon Nickerson"}
{"qid": 1424, "query": "Can a complex number ever be considered bigger or smaller than a real number, or vice versa?", "score": 61, "views": 45585, "answer_pids": [457996, 457997, 457998, 458233, 458414, 458426, 594606], "question_author": "math4tots"}
{"qid": 1425, "query": "Prove every odd integer is the difference of two squares", "score": 61, "views": 42736, "answer_pids": [127276, 127279, 127324, 127617, 212105, 277141, 278137], "question_author": "graybrij"}
{"qid": 1426, "query": "how to be good at proving?", "score": 61, "views": 39253, "answer_pids": [96288, 96289, 96290, 96291, 96310, 96311, 96314], "question_author": ""}
{"qid": 1427, "query": "Are there any examples of non-computable real numbers?", "score": 61, "views": 36732, "answer_pids": [216111, 216128, 216129, 1350736, 1394220], "question_author": ""}
{"qid": 1428, "query": "What is a good book to study linear algebra?", "score": 61, "views": 36632, "answer_pids": [80840, 80841, 80843, 80862, 80971, 590441, 933234, 984813], "question_author": "J. Musser"}
{"qid": 1429, "query": "Why is empty set an open set?", "score": 61, "views": 35927, "answer_pids": [209156, 209158, 209161, 209162, 209163, 209220, 209227, 209446, 237919, 1172107], "question_author": "J. Musser"}
{"qid": 1430, "query": "How to prove and interpret $\\operatorname{rank}(AB) \\leq \\operatorname{min}(\\operatorname{rank}(A), \\operatorname{rank}(B))$?", "score": 61, "views": 26104, "answer_pids": [593, 594, 2303, 196985, 464409, 584315, 595616, 981404], "question_author": "Miriam"}
{"qid": 1431, "query": "Probability for the length of the longest run in $n$ Bernoulli trials", "score": 61, "views": 24372, "answer_pids": [31693, 31695, 31696, 31702, 728595], "question_author": "Qiaochu Yuan"}
{"qid": 1432, "query": "Proof of $(\\mathbb{Z}/m\\mathbb{Z}) \\otimes_\\mathbb{Z} (\\mathbb{Z} / n \\mathbb{Z}) \\cong \\mathbb{Z}/ \\gcd(m,n)\\mathbb{Z}$", "score": 61, "views": 24290, "answer_pids": [38096, 38106, 90242, 312197, 618028], "question_author": ""}
{"qid": 1433, "query": "What do prime ideals in $k[x,y]$ look like?", "score": 61, "views": 22994, "answer_pids": [30220, 30221, 860019], "question_author": ""}
{"qid": 1434, "query": "Characterizing units in polynomial rings", "score": 61, "views": 21089, "answer_pids": [10637, 184507], "question_author": "J. Musser"}
{"qid": 1435, "query": "Entire one-to-one functions are linear", "score": 61, "views": 19251, "answer_pids": [16143, 16144, 16160, 16162, 16174, 653277], "question_author": "romeovs"}
{"qid": 1436, "query": "Let $X$ be an infinite dimensional Banach space. Prove that every Hamel basis of X is uncountable.", "score": 61, "views": 18919, "answer_pids": [107819], "question_author": ""}
{"qid": 1437, "query": "How do you show monotonicity of the $\\ell^p$ norms?", "score": 61, "views": 18607, "answer_pids": [2430, 2441, 1267564, 1267575, 1429831], "question_author": "monn"}
{"qid": 1438, "query": "Strategies to denest nested radicals $\\sqrt{a+b\\sqrt{c}}$", "score": 61, "views": 18204, "answer_pids": [98794, 98800, 675474, 1138631, 1138632, 1312522], "question_author": "Daisy Sophia Hollman"}
{"qid": 1439, "query": "How would you explain to a 9th grader the negative exponent rule?", "score": 61, "views": 18088, "answer_pids": [281615, 281616, 281619, 281621, 281664, 281735, 281752, 281753, 281803, 281806, 281832, 281842, 281852, 281882, 281884, 281890, 281901, 281948, 281981, 282048, 282112, 282115, 282236, 365410], "question_author": "Dale"}
{"qid": 1440, "query": "Why learn to solve differential equations when computers can do it?", "score": 61, "views": 14624, "answer_pids": [380706, 380710, 380720, 380724, 380737, 380757, 380773, 380785, 380788, 380824, 380826, 380904, 380907, 380961, 381685], "question_author": ""}
{"qid": 1441, "query": "Big List of Fun Math Books", "score": 61, "views": 12522, "answer_pids": [84726, 84730, 84732, 84739, 84752, 84753, 84754, 84755, 84756, 84763, 84765, 84767, 84769, 84778, 84780, 84781, 84811, 84824, 84853, 84939, 84941, 84972, 84979, 85050, 85844, 90347], "question_author": "voix"}
{"qid": 1442, "query": "What is the definition of a set?", "score": 61, "views": 12160, "answer_pids": [574646, 574659, 574660, 574674, 574896, 574901, 575332, 575912, 575914, 575932, 575956, 576830, 596976], "question_author": "Mac Cowell"}
{"qid": 1443, "query": "Is there such a thing as proof by example (not counter example)", "score": 61, "views": 11001, "answer_pids": [160187, 160189, 160193, 160194, 160199, 160214, 160252, 160273, 160292, 160301, 160419, 160548, 182460, 480411, 618854], "question_author": ""}
{"qid": 1444, "query": "Can there be two distinct, continuous functions that are equal at all rationals?", "score": 61, "views": 10731, "answer_pids": [312, 313, 13191, 223694], "question_author": ""}
{"qid": 1445, "query": "Whats the goal of mathematics?", "score": 61, "views": 9975, "answer_pids": [15564, 15565, 15567, 15568, 15575, 15579, 15582, 15590, 15636, 15644, 15683, 16278, 48958, 48979, 632505], "question_author": "Bravo"}
{"qid": 1446, "query": "How to debug math?", "score": 61, "views": 9943, "answer_pids": [682881, 682883, 682886, 682887, 682892, 683242, 683385, 683583], "question_author": "Gabriel Fair"}
{"qid": 1447, "query": "What does communicated by mean in math papers?", "score": 61, "views": 8840, "answer_pids": [22338, 22339], "question_author": "Gabriel Fair"}
{"qid": 1448, "query": "Using Gr\u00f6bner bases for solving polynomial equations", "score": 61, "views": 8597, "answer_pids": [2117, 2119, 2124, 2125, 2132, 199989, 384399], "question_author": "harpalss"}
{"qid": 1449, "query": "Which is larger? $20!$ or $2^{40}$?", "score": 61, "views": 8511, "answer_pids": [255378, 255379, 255381, 255382, 255383, 255384, 255385, 255401, 255528, 255579, 255638, 255917, 255965, 559130, 1002487, 1414954], "question_author": "David LeBauer"}
{"qid": 1450, "query": "Unsolved Problems due to Lack of Computational Power", "score": 61, "views": 8469, "answer_pids": [1203575, 1203580, 1203697, 1203860, 1203882, 1203943, 1203958, 1203962, 1203969, 1203978, 1204095, 1204372, 1204396, 1204561, 1204698, 1422504, 1462924], "question_author": ""}
{"qid": 1451, "query": "Whats wrong with this reasoning that $\\frac{\\infty}{\\infty}=0$?", "score": 61, "views": 7932, "answer_pids": [779594, 779597, 779599, 779604, 779617, 779693, 779710, 779793, 780164], "question_author": ""}
{"qid": 1452, "query": "Are there theoretical applications of trigonometry?", "score": 61, "views": 7839, "answer_pids": [682245, 682248, 682253, 682264, 682339, 682353, 682360, 682477, 682587, 683110, 683337, 684005], "question_author": "Christian"}
{"qid": 1453, "query": "The last digit of $2^{2006}$", "score": 61, "views": 7683, "answer_pids": [136767, 136771, 136772, 136773, 136775, 136839, 136853, 1341757], "question_author": "Casebash"}
{"qid": 1454, "query": "Intuition for the Importance of Modular Forms", "score": 61, "views": 7245, "answer_pids": [155426, 158790, 168787, 200285, 270158], "question_author": "Casebash"}
{"qid": 1455, "query": "Linear Algebra Versus Functional Analysis", "score": 61, "views": 6757, "answer_pids": [729659, 729661, 729667, 729673, 729679, 729732, 729736, 730285, 730310], "question_author": ""}
{"qid": 1456, "query": "Whats the point of studying topological (as opposed to smooth, PL, or PDiff) manifolds?", "score": 61, "views": 6606, "answer_pids": [11325, 11327, 11328, 28443, 131640, 166952, 170204], "question_author": ""}
{"qid": 1457, "query": "Find three non-constant, pairwise unequal functions $f,g,h:\\mathbb R\\to \\mathbb R$...", "score": 61, "views": 5732, "answer_pids": [885637, 885642, 885681, 885910, 886184, 886495], "question_author": ""}
{"qid": 1458, "query": "Surprisingly elementary and direct proofs", "score": 61, "views": 4688, "answer_pids": [189851, 189857, 189860, 189867, 189912, 190164, 190177], "question_author": "Anil"}
{"qid": 1459, "query": "Closed Form for $~\\int_0^1\\frac{\\text{arctanh }x}{\\tan\\left(\\frac\\pi2~x\\right)}~dx$", "score": 61, "views": 3759, "answer_pids": [524445, 550025, 562668], "question_author": "Seamus"}
{"qid": 1460, "query": "Is $n \\sin n$ dense on the real line?", "score": 61, "views": 3512, "answer_pids": [109183, 109226, 109228], "question_author": ""}
{"qid": 1461, "query": "When are nonintersecting finite degree field extensions linearly disjoint?", "score": 61, "views": 2120, "answer_pids": [1102711], "question_author": "Chao Xu"}
{"qid": 1462, "query": "The relation between trace and determinant of a matrix", "score": 60, "views": 61685, "answer_pids": [793664, 793672, 793675, 793678, 793766, 793801, 794689], "question_author": "math4tots"}
{"qid": 1463, "query": "How to prove every closed interval in R is compact?", "score": 60, "views": 34784, "answer_pids": [173706, 173719, 173720, 173722, 173728, 173765, 173838, 173872, 173894, 183777, 513488, 607792], "question_author": ""}
{"qid": 1464, "query": "Is there a step by step checklist to check if a multivariable limit exists and find its value?", "score": 60, "views": 31849, "answer_pids": [152141, 152146], "question_author": ""}
{"qid": 1465, "query": "Is an automorphism of the field of real numbers the identity map?", "score": 60, "views": 22628, "answer_pids": [210156, 210173, 210385, 274159, 703531, 835167], "question_author": "Unreasonable Sin"}
{"qid": 1466, "query": "What do eigenvalues have to do with pictures?", "score": 60, "views": 21526, "answer_pids": [47890, 47898], "question_author": "Anixx"}
{"qid": 1467, "query": "Why cant you flatten a sphere?", "score": 60, "views": 18074, "answer_pids": [163932, 163933, 163936, 163937, 163941, 163980, 163987, 164010, 164015, 164063, 164078, 164081, 164174, 166443], "question_author": "Kirk Woll"}
{"qid": 1468, "query": "List of problem books in undergraduate and graduate mathematics", "score": 60, "views": 17751, "answer_pids": [146401, 146404, 146408, 146429, 146432, 146436, 434065, 794943, 1186646], "question_author": ""}
{"qid": 1469, "query": "What is the importance of Calculus in todays Mathematics?", "score": 60, "views": 16256, "answer_pids": [30836, 30841, 30902], "question_author": "Pete"}
{"qid": 1470, "query": "Why does $\\frac{1}{x} < 4$ have two answers?", "score": 60, "views": 16077, "answer_pids": [733005, 733006, 733007, 733008, 733153, 733157, 733206, 734157, 735129, 735270, 746531, 746543, 746789], "question_author": "J. Musser"}
{"qid": 1471, "query": "Explain this mathematical meme (Geometers bird interrupting Topologists bird)", "score": 60, "views": 15600, "answer_pids": [1035826, 1041829], "question_author": ""}
{"qid": 1472, "query": "Coin flipping probability game ; 7 flips vs 8 flips", "score": 60, "views": 14033, "answer_pids": [292442, 292444, 292445, 292449, 292559, 292587, 292602, 292650, 292651, 292663, 292725, 292757, 292833, 293132, 293200, 293841, 294091, 295685, 423659], "question_author": "J. Musser"}
{"qid": 1473, "query": "Finite Groups with exactly $n$ conjugacy classes $(n=2,3,...)$", "score": 60, "views": 12978, "answer_pids": [27875, 27912, 27983, 30136], "question_author": ""}
{"qid": 1474, "query": "Help understanding Algebraic Geometry", "score": 60, "views": 12435, "answer_pids": [130464, 130476, 130480, 130494], "question_author": ""}
{"qid": 1475, "query": "7 fishermen caught exactly 100 fish and no two had caught the same number of fish. Then there are three who have together captured at least 50 fish.", "score": 60, "views": 11496, "answer_pids": [1080078, 1080087, 1080503, 1080806, 1080823, 1080844, 1081211, 1081463, 1083579], "question_author": "Larry Wang"}
{"qid": 1476, "query": "Can a number have infinitely many digits before the decimal point?", "score": 60, "views": 11134, "answer_pids": [397773, 397774, 397777, 397781, 397784, 397845, 398000], "question_author": ""}
{"qid": 1477, "query": "How can I show that $\\sqrt{1+\\sqrt{2+\\sqrt{3+\\sqrt\\ldots}}}$ exists?", "score": 60, "views": 10977, "answer_pids": [204711, 204721, 204723, 204883, 353252, 533689, 1190012], "question_author": ""}
{"qid": 1478, "query": "Whats the intuition with partitions of unity?", "score": 60, "views": 10427, "answer_pids": [208227, 208229, 208240, 208251, 1075151], "question_author": ""}
{"qid": 1479, "query": "Center-commutator duality", "score": 60, "views": 9551, "answer_pids": [14328, 14329, 14330, 136473], "question_author": "Kirby"}
{"qid": 1480, "query": "In how many different ways can a 9-panel comic grid be used?", "score": 60, "views": 9501, "answer_pids": [862520, 862566, 862821, 863230, 879954], "question_author": ""}
{"qid": 1481, "query": "I need mathematical proof that the distance from zero to 1 is the equal to the distance from 1 to 2", "score": 60, "views": 8853, "answer_pids": [25438, 25439, 25441, 25448, 25455, 25461, 25462, 25482, 25492, 25516, 25518, 25519], "question_author": ""}
{"qid": 1482, "query": "I almost quit self-studying mathematics, but should I continue?", "score": 60, "views": 8706, "answer_pids": [258619, 258620, 258624, 258625, 258633, 258636, 258666, 258672, 258696, 259322, 259687, 260752], "question_author": "Vaolter"}
{"qid": 1483, "query": "What is Trinity Hall Prime number?", "score": 60, "views": 8552, "answer_pids": [909107, 909109, 909160, 909566, 930312], "question_author": "harpalss"}
{"qid": 1484, "query": "Examples of morphisms of schemes to keep in mind?", "score": 60, "views": 8318, "answer_pids": [295728, 295973, 296864, 297901], "question_author": "Kirby"}
{"qid": 1485, "query": "Review of my T-shirt design", "score": 60, "views": 7477, "answer_pids": [787705, 787706, 787995], "question_author": "bobthejoe"}
{"qid": 1486, "query": "Determinant of a rank $1$ update of a scalar matrix, or characteristic polynomial of a rank $1$ matrix", "score": 60, "views": 7236, "answer_pids": [383366, 1327263], "question_author": "David Z"}
{"qid": 1487, "query": "Is A New Kind of Science a new kind of science?", "score": 60, "views": 6887, "answer_pids": [495982, 496218, 496266], "question_author": "user634"}
{"qid": 1488, "query": "What am I doing when I separate the variables of a differential equation?", "score": 60, "views": 6783, "answer_pids": [14933, 14938, 14951], "question_author": "Poshpaws"}
{"qid": 1489, "query": "Does every set have a group structure?", "score": 60, "views": 6542, "answer_pids": [54337, 54338, 54347, 54364], "question_author": "J. Musser"}
{"qid": 1490, "query": "Why do we use trig functions in Fourier transforms, and not other periodic functions?", "score": 60, "views": 6432, "answer_pids": [1011581, 1011585, 1011702, 1011999, 1012200, 1012249, 1012342], "question_author": "Agnel Kurian"}
{"qid": 1491, "query": "How to evaluate $I=\\int_0^{\\pi/2}x^2\\ln(\\sin x)\\ln(\\cos x)\\ \\mathrm dx$", "score": 60, "views": 5260, "answer_pids": [157188, 157196, 157294, 157297, 157686, 158066, 299972], "question_author": "J. Musser"}
{"qid": 1492, "query": "Is it possible to simplify $\\frac{\\Gamma\\left(\\frac{1}{10}\\right)}{\\Gamma\\left(\\frac{2}{15}\\right)\\ \\Gamma\\left(\\frac{7}{15}\\right)}$?", "score": 60, "views": 5230, "answer_pids": [190674], "question_author": "please delete me"}
{"qid": 1493, "query": "Can squares of infinite area always cover a unit square?", "score": 60, "views": 3945, "answer_pids": [958260, 958264, 958265, 958297, 958561], "question_author": "Thomas O"}
{"qid": 1494, "query": "Whats the probability that a sequence of coin flips never has twice as many heads as tails?", "score": 60, "views": 3894, "answer_pids": [31849, 31856, 31877, 32035], "question_author": ""}
{"qid": 1495, "query": "$5^n+n$ is never prime?", "score": 60, "views": 3222, "answer_pids": [2444, 2445, 2446], "question_author": "user4951"}
{"qid": 1496, "query": "What does it take to divide by $2$?", "score": 60, "views": 2856, "answer_pids": [634130, 1013728], "question_author": ""}
{"qid": 1497, "query": "If a two variable smooth function has two global minima, will it necessarily have a third critical point?", "score": 60, "views": 1867, "answer_pids": [1408158, 1409475, 1410426, 1413561, 1413944], "question_author": ""}
{"qid": 1498, "query": "Subgroups as isotropy subgroups and regular orbits on tuples", "score": 60, "views": 1494, "answer_pids": [1332948], "question_author": "J. Musser"}
{"qid": 1499, "query": "A question about Sylow subgroups and $C_G(x)$", "score": 60, "views": 1088, "answer_pids": [1211913], "question_author": "J. Musser"}
{"qid": 1500, "query": "How to know if a point is inside a circle?", "score": 59, "views": 220053, "answer_pids": [99566, 99568, 99573, 177373], "question_author": ""}
{"qid": 1501, "query": "Poisson Distribution of sum of two random independent variables $X$, $Y$", "score": 59, "views": 158510, "answer_pids": [109188, 109189, 144573, 211313, 261964, 1008372], "question_author": ""}
{"qid": 1502, "query": "Explanation on arg min", "score": 59, "views": 92332, "answer_pids": [111914, 369177, 818854], "question_author": "Gabriel Fair"}
{"qid": 1503, "query": "Product of two Gaussian PDFs is a Gaussian PDF, but Product of two Gaussian Variables is not Gaussian", "score": 59, "views": 89266, "answer_pids": [456925, 940698, 977695, 1108625], "question_author": "Sergio"}
{"qid": 1504, "query": "Is the product of symmetric positive semidefinite matrices positive definite?", "score": 59, "views": 72861, "answer_pids": [58497, 225524, 1221210], "question_author": ""}
{"qid": 1505, "query": "Matrices commute if and only if they share a common basis of eigenvectors?", "score": 59, "views": 38990, "answer_pids": [3680, 3681, 705409, 1369627], "question_author": "Zigu"}
{"qid": 1506, "query": "What are the formal names of operands and results for basic operations?", "score": 59, "views": 36210, "answer_pids": [408288, 671989], "question_author": "bobthejoe"}
{"qid": 1507, "query": "Differentiating an Inner Product", "score": 59, "views": 34498, "answer_pids": [49961, 49964], "question_author": "Marek"}
{"qid": 1508, "query": "How to show that the commutator subgroup is a normal subgroup", "score": 59, "views": 33120, "answer_pids": [71829, 1289877], "question_author": "Rajesh D"}
{"qid": 1509, "query": "Nobody told me that self teaching could be so damaging...", "score": 59, "views": 28970, "answer_pids": [431592, 431603, 431604, 431612, 431623, 431648, 431649, 433273, 433653, 433668, 433689, 433797], "question_author": "bobthejoe"}
{"qid": 1510, "query": "(undergraduate) Algebraic Geometry Textbook Recommendations", "score": 59, "views": 24914, "answer_pids": [1089, 1093, 1096, 1098, 2223, 2242, 13377, 13384, 541895], "question_author": "Mad Scientist"}
{"qid": 1511, "query": "Why is $\\mathbb{Z}[\\sqrt{-n}], n\\ge 3$ not a UFD?", "score": 59, "views": 20406, "answer_pids": [37428, 467872], "question_author": ""}
{"qid": 1512, "query": "Dot Product Intuition", "score": 59, "views": 17060, "answer_pids": [165226, 214725, 304897, 667506, 668055, 668916, 669914, 670048, 670093, 670133, 670296, 670537], "question_author": "maxymoo"}
{"qid": 1513, "query": "Divisor -- line bundle correspondence in algebraic geometry", "score": 59, "views": 16999, "answer_pids": [1200], "question_author": "Thomas O"}
{"qid": 1514, "query": "How to tell if Im good enough for graduate school?", "score": 59, "views": 14500, "answer_pids": [229663, 229674, 229725, 229731, 230088, 235428], "question_author": "Covi"}
{"qid": 1515, "query": "How is the derivative truly, literally the best linear approximation near a point?", "score": 59, "views": 13634, "answer_pids": [687969, 687970, 687977, 687980, 687991, 688166, 688362, 688482, 688660], "question_author": "Patrick87"}
{"qid": 1516, "query": "Why is the Continuum Hypothesis (not) true?", "score": 59, "views": 13374, "answer_pids": [95494, 95496, 95497, 95501, 95556, 95580], "question_author": ""}
{"qid": 1517, "query": "Good examples of double induction", "score": 59, "views": 13283, "answer_pids": [10269, 10270, 10276, 10281, 10282, 10284, 16091, 86019, 113332], "question_author": ""}
{"qid": 1518, "query": "A very general method for proving inequalities. Too good to be true?", "score": 59, "views": 9930, "answer_pids": [852753, 852793, 852805], "question_author": "oceanhug"}
{"qid": 1519, "query": "Is it possible to have three real numbers that have both their sum and product equal to $1$?", "score": 59, "views": 8977, "answer_pids": [787461, 787487, 787497, 787529, 787541, 787559, 787595, 787619, 787626, 787854, 787869, 788289, 788297, 788382, 790037], "question_author": "Daniel Standage"}
{"qid": 1520, "query": "What was the book that opened your mind to the beauty of mathematics?", "score": 59, "views": 8693, "answer_pids": [356131, 356134, 356135, 356145, 356186, 356190, 356203, 356226, 356230, 356248, 356251, 356252, 356253, 356254, 356257, 356258, 356259, 356526, 356528, 356550, 356553, 356575, 356726, 356848, 356852, 356853, 356855, 356996, 357004, 357007, 357240, 357245, 357283, 357388, 357524, 357527, 357575, 357585, 357587, 357590, 359917, 376769, 486544], "question_author": ""}
{"qid": 1521, "query": "Research done by high-school students", "score": 59, "views": 8267, "answer_pids": [87705, 87706, 87707, 87708, 87717, 87729, 87745, 87766, 87840, 159342, 159367, 167971], "question_author": "gremau"}
{"qid": 1522, "query": "Why does the discriminant in the Quadratic Formula reveal the number of real solutions?", "score": 59, "views": 8150, "answer_pids": [923288, 923293, 923294, 923306, 923376, 923635, 924746, 924825], "question_author": ""}
{"qid": 1523, "query": "Understanding the intuition behind math", "score": 59, "views": 6996, "answer_pids": [13045, 13051, 13069, 13136, 778886], "question_author": "F. P."}
{"qid": 1524, "query": "A circle rolls along a parabola", "score": 59, "views": 6871, "answer_pids": [17569, 17575, 17641, 17717], "question_author": "Boy S"}
{"qid": 1525, "query": "Why is $\\varphi$ called the most irrational number?", "score": 59, "views": 6802, "answer_pids": [186033, 186036], "question_author": "bobthejoe"}
{"qid": 1526, "query": "How can I intuitively understand complex exponents?", "score": 59, "views": 6598, "answer_pids": [935089, 935090, 935091, 935093, 935111, 935261, 935785], "question_author": "bobthejoe"}
{"qid": 1527, "query": "It looks straightforward, but actually it isnt", "score": 59, "views": 6538, "answer_pids": [194899, 194910, 194917, 194968, 195754, 197590, 197599, 197843], "question_author": ""}
{"qid": 1528, "query": "Refuting the Anti-Cantor Cranks", "score": 59, "views": 5317, "answer_pids": [242679, 242684, 242837, 242880, 301445, 510448, 674807, 681697], "question_author": "Jaime Soto"}
{"qid": 1529, "query": "Could I be using proof by contradiction too much?", "score": 59, "views": 5105, "answer_pids": [152488, 152490, 152492, 152501, 152503, 152513, 152659], "question_author": null}
{"qid": 1530, "query": "Is there any geometric intuition for the factorials in Taylor expansions?", "score": 59, "views": 4959, "answer_pids": [1329216, 1329229, 1329307, 1329414, 1329628, 1329653, 1329692], "question_author": "Jaime Soto"}
{"qid": 1531, "query": "Why do differential forms have a much richer structure than vector fields?", "score": 59, "views": 4243, "answer_pids": [355298, 355304, 445054, 963487, 1348807], "question_author": ""}
{"qid": 1532, "query": "A Challenging Logarithmic Integral $\\int_0^1 \\frac{\\log(x)\\log(1-x)\\log^2(1+x)}{x}dx$", "score": 59, "views": 3729, "answer_pids": [334261, 1333867], "question_author": ""}
{"qid": 1533, "query": "Quadratic reciprocity via generalized Fibonacci numbers?", "score": 59, "views": 3505, "answer_pids": [9388], "question_author": "Daisy Sophia Hollman"}
{"qid": 1534, "query": "Penroses remark on impossible figures", "score": 59, "views": 2377, "answer_pids": [621396], "question_author": "Jonathan."}
{"qid": 1535, "query": "Why are asymptotically one half of the integer compositions gap-free?", "score": 59, "views": 1629, "answer_pids": [452133], "question_author": "mring"}
{"qid": 1536, "query": "Mirror algorithm for computing $\\pi$ and $e$ - does it hint on some connection between them?", "score": 59, "views": 1372, "answer_pids": [701349, 750584], "question_author": "fuzzytron"}
{"qid": 1537, "query": "Difference between \u2248, \u2243, and \u2245", "score": 58, "views": 198479, "answer_pids": [368062, 368063, 1191680, 1340330, 1357320], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1538, "query": "What is the difference between only if and iff?", "score": 58, "views": 107583, "answer_pids": [36073, 36074, 36078, 36081, 297877, 533973, 841910, 842602, 1083595], "question_author": ""}
{"qid": 1539, "query": "Gradient of squared $2$-norm", "score": 58, "views": 105467, "answer_pids": [375282, 435516, 1409727], "question_author": ""}
{"qid": 1540, "query": "Distance/Similarity between two matrices", "score": 58, "views": 94467, "answer_pids": [235390, 526394], "question_author": "Santosh Linkha"}
{"qid": 1541, "query": "Are all eigenvectors, of any matrix, always orthogonal?", "score": 58, "views": 63396, "answer_pids": [72205, 795984, 817766, 1454627], "question_author": "Garima Kushwaha"}
{"qid": 1542, "query": "Area of a square inside a square created by connecting point-opposite midpoint", "score": 58, "views": 39197, "answer_pids": [162523, 162532, 162537, 162542, 162584, 162609, 939954, 939958, 939962, 939968, 940122, 940203, 940402, 940573], "question_author": null}
{"qid": 1543, "query": "How to generate random points on a sphere?", "score": 58, "views": 28160, "answer_pids": [619056, 619059, 619064, 619066, 619134, 619288], "question_author": ""}
{"qid": 1544, "query": "Number of monic irreducible polynomials of prime degree $p$ over finite fields", "score": 58, "views": 24072, "answer_pids": [21753, 21791], "question_author": "Robert Filter"}
{"qid": 1545, "query": "Should I be worried that I am doing well in analysis and not well in algebra?", "score": 58, "views": 21084, "answer_pids": [168190, 168192, 168196, 168835, 173283, 450224], "question_author": "Boy S"}
{"qid": 1546, "query": "Negative versus Minus", "score": 58, "views": 19632, "answer_pids": [79332, 79333, 79334, 79335, 79336, 79338, 79339, 79340, 79341, 79358, 79360, 79374, 79385, 79390, 79393, 79781, 79848, 79953, 80021, 80657, 113783, 225508, 345997, 1369661], "question_author": ""}
{"qid": 1547, "query": "Discontinuous linear functional", "score": 58, "views": 19401, "answer_pids": [51338, 51351, 52029], "question_author": ""}
{"qid": 1548, "query": "Geometric interpretation for sum of fourth powers", "score": 58, "views": 18154, "answer_pids": [212621, 482167], "question_author": null}
{"qid": 1549, "query": "The two-daughter-problem", "score": 58, "views": 16752, "answer_pids": [816378, 816386, 816389, 816439, 816531, 816579, 816635, 816828], "question_author": ""}
{"qid": 1550, "query": "Has there ever been an application of dividing by $0$?", "score": 58, "views": 15012, "answer_pids": [682768, 682769, 682775, 682817, 683178, 683256, 683434, 683732, 684411], "question_author": "Michael Haren"}
{"qid": 1551, "query": "Why do we miss 8 in the decimal expansion of 1/81, and 98 in the decimal expansion of 1/9801?", "score": 58, "views": 13290, "answer_pids": [414636, 414639, 415221], "question_author": ""}
{"qid": 1552, "query": "Why does the Cauchy-Schwarz Inequality even have a name?", "score": 58, "views": 11721, "answer_pids": [653030, 653032, 653037, 653156, 653165], "question_author": ""}
{"qid": 1553, "query": "What use is the Yoneda lemma?", "score": 58, "views": 8062, "answer_pids": [49690, 49691, 49703], "question_author": "Snowman"}
{"qid": 1554, "query": "Is there a function whose antiderivative can be found but whose derivative cannot?", "score": 58, "views": 7685, "answer_pids": [615057, 615065, 615152, 615279, 615382, 615388], "question_author": ""}
{"qid": 1555, "query": "Similar matrices and field extensions", "score": 58, "views": 7495, "answer_pids": [30398, 30406, 31405, 33272, 270354, 281839], "question_author": "R. Zurschmitten"}
{"qid": 1556, "query": "Are there mathematical concepts that exist in dimension $4$, but not in dimension $3$?", "score": 58, "views": 7188, "answer_pids": [1210045, 1210190, 1210215, 1210235, 1210236, 1210300, 1210347, 1210408, 1210454, 1211908, 1212365, 1212422], "question_author": "Rory M"}
{"qid": 1557, "query": "Is there a definition of determinants that does not rely on how they are calculated?", "score": 58, "views": 7116, "answer_pids": [11907, 11908, 11909, 12471, 12474], "question_author": "Snowman"}
{"qid": 1558, "query": "Unexpected use of topology in proofs", "score": 58, "views": 6297, "answer_pids": [1013775, 1013777, 1013781, 1013789, 1013828, 1013843, 1013966, 1014289, 1014389, 1014534, 1014563, 1014569, 1014575, 1014632], "question_author": ""}
{"qid": 1559, "query": "Does this Fractal Have a Name?", "score": 58, "views": 6223, "answer_pids": [692006, 692102, 692213, 692234], "question_author": "Nick T"}
{"qid": 1560, "query": "How is the G\u00f6dels Completeness Theorem not a tautology?", "score": 58, "views": 6170, "answer_pids": [334474, 334476, 334479, 334481, 334536, 334733, 334800, 334845], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1561, "query": "How many 7-note musical scales are possible within the 12-note system?", "score": 58, "views": 6130, "answer_pids": [697593, 697594, 697975, 724869, 1248977], "question_author": ""}
{"qid": 1562, "query": "Approximating a $\\sigma$-algebra by a generating algebra", "score": 58, "views": 5937, "answer_pids": [112325, 745473, 1219483], "question_author": ""}
{"qid": 1563, "query": "How do different definitions of degree coincide?", "score": 58, "views": 5629, "answer_pids": [112288], "question_author": null}
{"qid": 1564, "query": "What is the size of each side of the square?", "score": 58, "views": 5230, "answer_pids": [1310976, 1310989, 1310993, 1311156], "question_author": "Gabriel Fair"}
{"qid": 1565, "query": "A non-losing strategy for tic-tac-toe $\\times$ tic-tac-toe", "score": 58, "views": 4265, "answer_pids": [205931, 303209], "question_author": ""}
{"qid": 1566, "query": "Can there be an injective function whose derivative is equivalent to its inverse function?", "score": 58, "views": 3938, "answer_pids": [734492, 734493, 734517, 737051], "question_author": "JohnAllen"}
{"qid": 1567, "query": "Evaluating $\\int_0^\\infty \\frac{dx}{\\sqrt{x}[x^2+(1+2\\sqrt{2})x+1][1-x+x^2-x^3+...+x^{50}]}$", "score": 58, "views": 3396, "answer_pids": [409291], "question_author": ""}
{"qid": 1568, "query": "Trigonometric sums related to the Verlinde formula", "score": 58, "views": 3146, "answer_pids": [172701], "question_author": "Mad Scientist"}
{"qid": 1569, "query": "A bestiary about adjunctions", "score": 58, "views": 3083, "answer_pids": [24920, 24924, 24938, 252658, 733074, 733077, 733087], "question_author": ""}
{"qid": 1570, "query": "Is OEIS A248049 an integer sequence?", "score": 58, "views": 2736, "answer_pids": [1402237], "question_author": "masfenix"}
{"qid": 1571, "query": "Prove that $\\int_{0}^{1}\\sin{(\\pi x)}x^x(1-x)^{1-x}\\,dx =\\frac{\\pi e}{24} $", "score": 58, "views": 2647, "answer_pids": [403058, 870565], "question_author": ""}
{"qid": 1572, "query": "Golden Number Theory", "score": 58, "views": 2560, "answer_pids": [10370, 10444, 13438, 81203], "question_author": "Kit"}
{"qid": 1573, "query": "Explicit norm on $\\mathcal{C}^0(\\mathbb{R},\\mathbb{R})$", "score": 58, "views": 2278, "answer_pids": [210214, 273370, 423604], "question_author": "Gabriel Fair"}
{"qid": 1574, "query": "An illusionist and their assistant are about to perform the following magic trick", "score": 58, "views": 1976, "answer_pids": [1340459], "question_author": "Gabriel Fair"}
{"qid": 1575, "query": "The most effective windshield-wiper setup. (Packing a square with sectors)", "score": 58, "views": 1754, "answer_pids": [989579, 1233644], "question_author": "Sonny Ordell"}
{"qid": 1576, "query": "How prove this inequality $\\sin{\\sin{\\sin{\\sin{x}}}}\\le\\frac{4}{5}\\cos{\\cos{\\cos{\\cos{x}}}}$", "score": 58, "views": 1633, "answer_pids": [269017, 269938], "question_author": "Steven Stadnicki"}
{"qid": 1577, "query": "Period of the sum/product of two functions", "score": 57, "views": 87237, "answer_pids": [82897, 1039717], "question_author": "user786"}
{"qid": 1578, "query": "Example of Partial Order thats not a Total Order and why?", "score": 57, "views": 62347, "answer_pids": [173456, 173457, 173458, 173459, 173460, 173493, 174884, 1169492], "question_author": ""}
{"qid": 1579, "query": "What does curly (curved) less than sign $\\succcurlyeq$ mean?", "score": 57, "views": 57747, "answer_pids": [296715, 296718, 296727, 560500], "question_author": "oshirowanen"}
{"qid": 1580, "query": "Why we consider log likelihood instead of Likelihood in Gaussian Distribution", "score": 57, "views": 38361, "answer_pids": [379054, 379064, 1412396], "question_author": "ProfSmiles"}
{"qid": 1581, "query": "Is there an equation to describe regular polygons?", "score": 57, "views": 37992, "answer_pids": [22378, 22381, 22386, 36856, 317921, 520376, 1444345], "question_author": "Simon Verbeke"}
{"qid": 1582, "query": "Eigenvalues of the rank one matrix $uv^T$", "score": 57, "views": 32893, "answer_pids": [29306, 29308, 268006], "question_author": ""}
{"qid": 1583, "query": "Under what condition we can interchange order of a limit and a summation?", "score": 57, "views": 31443, "answer_pids": [12672, 31757, 1220585], "question_author": ""}
{"qid": 1584, "query": "Calculating pi manually", "score": 57, "views": 28267, "answer_pids": [483022, 483025, 483044, 483096, 483115, 483265, 706323, 706328, 913937, 914061], "question_author": ""}
{"qid": 1585, "query": "Difference between basis and subbasis in a topology?", "score": 57, "views": 21904, "answer_pids": [210232, 210233, 210242, 210243], "question_author": "bobthejoe"}
{"qid": 1586, "query": "Visualizing the 4th dimension.", "score": 57, "views": 21667, "answer_pids": [861932, 861934, 861937, 861941, 862310, 862493, 862720], "question_author": null}
{"qid": 1587, "query": "Why does the derivative of sine only work for radians?", "score": 57, "views": 21090, "answer_pids": [534318, 534319, 534320, 534322, 534323, 534324, 534325, 534329, 534369, 534569, 534655, 534761, 534834, 535244, 537531, 537532], "question_author": ""}
{"qid": 1588, "query": "Origin of the dot and cross product?", "score": 57, "views": 17723, "answer_pids": [33031, 33035, 33037, 33059, 931515, 1455909], "question_author": "zeller"}
{"qid": 1589, "query": "What is the arithmetic mean of no numbers?", "score": 57, "views": 15455, "answer_pids": [385205, 385221, 385229, 385248, 385260, 385421, 385477, 385483, 385490], "question_author": "Mad Scientist"}
{"qid": 1590, "query": "The $\\sigma$-algebra of subsets of $X$ generated by a set $\\mathcal{A}$ is the smallest sigma algebra including $\\mathcal{A}$", "score": 57, "views": 13996, "answer_pids": [28785, 28787, 28872], "question_author": ""}
{"qid": 1591, "query": "Would a proof to the Riemann Hypothesis affect security?", "score": 57, "views": 13381, "answer_pids": [36711, 36712, 36713, 36719], "question_author": "J. Musser"}
{"qid": 1592, "query": "What lies beyond the Sedenions", "score": 57, "views": 10072, "answer_pids": [45097, 208319, 490450], "question_author": ""}
{"qid": 1593, "query": "Is it possible to determine if you were on a M\u00f6bius strip?", "score": 57, "views": 9545, "answer_pids": [908329, 908330, 908331, 908338, 908370, 908391, 908445, 908477, 908520, 908606, 908621, 908657, 908771, 908816, 909011, 909063, 909108], "question_author": ""}
{"qid": 1594, "query": "Big O Notation is element of or is equal", "score": 57, "views": 9390, "answer_pids": [787008, 787017, 787027, 787177, 787484, 793679], "question_author": ""}
{"qid": 1595, "query": "Is there a slowest rate of divergence of a series?", "score": 57, "views": 9177, "answer_pids": [211339, 211340, 211348, 211511, 523485], "question_author": "Seamus"}
{"qid": 1596, "query": "Any smart ideas on finding the area of this shaded region?", "score": 57, "views": 9131, "answer_pids": [721713, 721715, 721728, 721748, 721937, 722056, 722136, 722162, 722186, 725600, 821886, 847641], "question_author": "Tony The Lion"}
{"qid": 1597, "query": "Can we calculate $ i\\sqrt { i\\sqrt { i\\sqrt { \\cdots } } }$?", "score": 57, "views": 7419, "answer_pids": [991833, 991834, 991836, 991838, 991850, 992118, 992163], "question_author": ""}
{"qid": 1598, "query": "Is it possible that A counter-example exists but it cannot be found", "score": 57, "views": 6919, "answer_pids": [328392, 328396, 328433, 328622, 328636, 329092, 329133, 329688], "question_author": ""}
{"qid": 1599, "query": "Applications of complex numbers to solve non-complex problems", "score": 57, "views": 6915, "answer_pids": [790511, 790517, 790520, 790554, 790605, 790625, 790662, 790671, 790720, 790721, 790732, 790758, 790761, 790907, 790975, 791040, 791064, 791150, 791181, 791201, 791376, 791444, 791791, 791813, 792049, 792071], "question_author": "Robert Filter"}
{"qid": 1600, "query": "Is it an abuse of language to say *the* integers, *the* rational numbers, or *the* real numbers, etc.?", "score": 57, "views": 6364, "answer_pids": [700661, 700662, 700664, 700665, 700667, 700676, 700691, 700927, 701032, 703182], "question_author": ""}
{"qid": 1601, "query": "Paradox: increasing sequence that goes to $0$?", "score": 57, "views": 6248, "answer_pids": [61, 356, 705, 1495, 1498, 1503, 21553, 21914], "question_author": "Mad Scientist"}
{"qid": 1602, "query": "Is it possible to prove a mathematical statement by proving that a proof exists?", "score": 57, "views": 4228, "answer_pids": [133822, 135306], "question_author": "Mad Scientist"}
{"qid": 1603, "query": "Is $6.12345678910111213141516171819202122\\ldots$ transcendental?", "score": 57, "views": 3879, "answer_pids": [13775, 13776], "question_author": "Sean Tilson"}
{"qid": 1604, "query": "Big list of guided discovery books", "score": 57, "views": 3565, "answer_pids": [454044, 454198, 454225, 455787, 455975, 455982, 457591, 458458, 603599, 611373, 1456795], "question_author": "Harold Cavendish"}
{"qid": 1605, "query": "Are there any Turing-undecidable problems whose undecidability is independent of the Halting problem?", "score": 57, "views": 3439, "answer_pids": [530017, 530018], "question_author": "SIMEL"}
{"qid": 1606, "query": "Funny double infinite sum", "score": 57, "views": 3427, "answer_pids": [753426, 753436, 753461], "question_author": "dmckee --- ex-moderator kitten"}
{"qid": 1607, "query": "Geometry problem involving infinite number of circles", "score": 57, "views": 3225, "answer_pids": [382975, 383094], "question_author": ""}
{"qid": 1608, "query": "What proportion of positive integers have two factors that differ by 1?", "score": 57, "views": 3124, "answer_pids": [1112013, 1112090, 1113472], "question_author": "graybrij"}
{"qid": 1609, "query": "What are Different Approaches to Introduce the Elementary Functions?", "score": 57, "views": 2528, "answer_pids": [609596, 609598, 623159, 623620, 623625, 624673, 625204, 625777, 625781, 626642], "question_author": "zeller"}
{"qid": 1610, "query": "Finding triplets $(a,b,c)$ such that $\\sqrt{abc}\\in\\mathbb N$ divides $(a-1)(b-1)(c-1)$", "score": 57, "views": 2488, "answer_pids": [506077, 508933, 554696], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1611, "query": "How to think of the group ring as a Hopf algebra?", "score": 57, "views": 2323, "answer_pids": [248887, 1173498], "question_author": "Rory M"}
{"qid": 1612, "query": "Whats the value of this Vi\u00e8te-style product involving the golden ratio?", "score": 57, "views": 2021, "answer_pids": [4884], "question_author": "Dokkat"}
{"qid": 1613, "query": "Is $1+x+\\frac{x^2}2+\\dots+\\frac{x^n}{n!}$ irreducible?", "score": 57, "views": 1911, "answer_pids": [66681], "question_author": ""}
{"qid": 1614, "query": "How many values of $2^{2^{2^{.^{.^{.^{2}}}}}}$ depending on parenthesis?", "score": 57, "views": 1336, "answer_pids": [851665], "question_author": "Chris"}
{"qid": 1615, "query": "Looking for a definitive source about Dirichlet finally proving the Unit Theorem in the Sistine Chapel", "score": 57, "views": 1293, "answer_pids": [1186340], "question_author": "Rajesh D"}
{"qid": 1616, "query": "Whats the sign of $\\det\\left(\\sqrt{i^2+j^2}\\right)_{1\\le i,j\\le n}$?", "score": 57, "views": 1178, "answer_pids": [1386929, 1387127, 1387180], "question_author": ""}
{"qid": 1617, "query": "Is there a good reason why $\\left\\lfloor \\frac{n!}{11e}\\right\\rfloor$ is always even?", "score": 57, "views": 847, "answer_pids": [756631], "question_author": null}
{"qid": 1618, "query": "What does E mean in 9.0122222900391E-5?", "score": 56, "views": 438002, "answer_pids": [3687, 3688, 51710], "question_author": "Nate Eldredge"}
{"qid": 1619, "query": "What\u2019s the difference between analytical and numerical approaches to problems?", "score": 56, "views": 212308, "answer_pids": [394641, 394659, 394665, 394669, 733286], "question_author": "quant_dev"}
{"qid": 1620, "query": "How to find the multiplicity of eigenvalues?", "score": 56, "views": 164443, "answer_pids": [154625, 154629], "question_author": ""}
{"qid": 1621, "query": "Is the power set of the natural numbers countable?", "score": 56, "views": 85860, "answer_pids": [40818, 40819, 894182], "question_author": "Rajesh D"}
{"qid": 1622, "query": "Finding $\\int x^xdx$", "score": 56, "views": 84668, "answer_pids": [71577, 71611, 448515, 779626, 975959, 996374, 1361780], "question_author": "Amelio Vazquez-Reina"}
{"qid": 1623, "query": "Calculate on which side of a straight line is a given point located?", "score": 56, "views": 70523, "answer_pids": [132540, 132547, 1170627], "question_author": ""}
{"qid": 1624, "query": "Generating correlated random numbers: Why does Cholesky decomposition work?", "score": 56, "views": 56744, "answer_pids": [82548], "question_author": ""}
{"qid": 1625, "query": "product distribution of two uniform distribution, what about 3 or more", "score": 56, "views": 53480, "answer_pids": [292999, 535485, 1037850], "question_author": ""}
{"qid": 1626, "query": "Dantzigs unsolved homework problems", "score": 56, "views": 50585, "answer_pids": [244997, 1181500], "question_author": "quant_dev"}
{"qid": 1627, "query": "Interesting math-facts that are visually attractive", "score": 56, "views": 49896, "answer_pids": [228198, 228200, 228201, 228206, 228209, 228219, 228266, 228337, 228397, 228399, 228808, 230865, 231548, 1065260], "question_author": "Kirby"}
{"qid": 1628, "query": "How does TREE(3) grow to get so big? (Laymen explanation)", "score": 56, "views": 39854, "answer_pids": [152613], "question_author": "Rajesh D"}
{"qid": 1629, "query": "Order of general- and special linear groups over finite fields.", "score": 56, "views": 31426, "answer_pids": [38947, 1007039], "question_author": "bobthejoe"}
{"qid": 1630, "query": "Why are two permutations conjugate iff they have the same cycle structure?", "score": 56, "views": 31403, "answer_pids": [25666, 25667, 25672, 69503], "question_author": ""}
{"qid": 1631, "query": "What is an operator in mathematics?", "score": 56, "views": 28258, "answer_pids": [84936, 454746, 454750, 533970, 645051], "question_author": "Rajesh D"}
{"qid": 1632, "query": "Does $ \\int_0^{\\infty}\\frac{\\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral?", "score": 56, "views": 27374, "answer_pids": [35519, 35521, 35522, 50074, 53965], "question_author": ""}
{"qid": 1633, "query": "Are there any differences between tensors and multidimensional arrays?", "score": 56, "views": 26934, "answer_pids": [464551, 464566, 464567], "question_author": ""}
{"qid": 1634, "query": "Is it possible to write a sum as an integral to solve it?", "score": 56, "views": 19135, "answer_pids": [418047, 418061, 418558, 522104, 1134052], "question_author": "Snowman"}
{"qid": 1635, "query": "Why divide by $2m$", "score": 56, "views": 18606, "answer_pids": [376019, 376020, 1046395, 1105183], "question_author": "LanceLafontaine"}
{"qid": 1636, "query": "Can math be subjective?", "score": 56, "views": 18080, "answer_pids": [539784, 539788, 539789, 539796, 539915, 539926, 540090, 540553, 540677, 541159, 541176], "question_author": ""}
{"qid": 1637, "query": "Is there an easy way to see associativity or non-associativity from an operations table?", "score": 56, "views": 17286, "answer_pids": [85074, 85075, 85076, 85084, 239546, 1351426], "question_author": "L\u00e9o L\u00e9opold Hertz \uc900\uc601"}
{"qid": 1638, "query": "Matrix is conjugate to its own transpose", "score": 56, "views": 16780, "answer_pids": [33168, 268320, 679967, 1059899], "question_author": "Yevgeny Simkin"}
{"qid": 1639, "query": "Is there a third dimension of numbers?", "score": 56, "views": 16090, "answer_pids": [17293, 17295, 17298, 17301, 25916], "question_author": null}
{"qid": 1640, "query": "Proving that $m+n\\sqrt{2}$ is dense in $\\mathbb R$", "score": 56, "views": 15886, "answer_pids": [38577, 38579, 38598, 667074], "question_author": "J. Musser"}
{"qid": 1641, "query": "What is the antiderivative of $e^{-x^2}$", "score": 56, "views": 15840, "answer_pids": [241349, 241352, 241433, 241475, 241564, 393427], "question_author": ""}
{"qid": 1642, "query": "Is there a math expression equivalent to the conditional ternary operator?", "score": 56, "views": 15752, "answer_pids": [1143885, 1143886, 1143887, 1143891, 1143938, 1143996, 1144039, 1144461, 1144463, 1144802, 1144901, 1145387, 1147046, 1241805], "question_author": "Christopher Jon Mankowski"}
{"qid": 1643, "query": "Understanding G\u00f6dels Incompleteness Theorem", "score": 56, "views": 13802, "answer_pids": [9152, 9153, 9154, 9166, 9180], "question_author": ""}
{"qid": 1644, "query": "Do harmonic numbers have a \u201cclosed-form\u201d expression?", "score": 56, "views": 12755, "answer_pids": [27970, 27979, 179939, 634370, 1008106], "question_author": ""}
{"qid": 1645, "query": "Is $\\{0\\}$ a field?", "score": 56, "views": 12367, "answer_pids": [200152, 200153, 200154, 200155, 200178, 331673], "question_author": ""}
{"qid": 1646, "query": "How hard is the proof of $\\pi$ or $e$ being transcendental?", "score": 56, "views": 12321, "answer_pids": [7285, 7288, 7297, 7298, 21942, 193377], "question_author": "Keenan Pepper"}
{"qid": 1647, "query": "Interesting results easily achieved using complex numbers", "score": 56, "views": 12225, "answer_pids": [2933, 2934, 2935, 2936, 2937, 2939, 2940, 2944, 2960, 3002, 3003, 3004, 3008, 3019, 3164, 3166, 4628, 137091, 137093, 983603], "question_author": "didxga"}
{"qid": 1648, "query": "All natural numbers are equal.", "score": 56, "views": 9491, "answer_pids": [162771, 162772, 162773, 162785, 162800, 162813, 162845, 162884, 162903], "question_author": "mring"}
{"qid": 1649, "query": "Geometrical interpretation of Ricci curvature", "score": 56, "views": 9403, "answer_pids": [218914, 218920, 218950], "question_author": "Memming"}
{"qid": 1650, "query": "Why are real numbers useful?", "score": 56, "views": 7408, "answer_pids": [399130, 399131, 399132, 399137, 399144, 399151, 399223, 399298, 399301, 399380, 399414, 399713, 399921, 405808], "question_author": ""}
{"qid": 1651, "query": "$x^p-c$ has no root in a field $F$ if and only if $x^p-c$ is irreducible?", "score": 56, "views": 7356, "answer_pids": [189668], "question_author": "Rox"}
{"qid": 1652, "query": "What kind of work do modern day algebraists do?", "score": 56, "views": 7306, "answer_pids": [91642, 246567], "question_author": ""}
{"qid": 1653, "query": "Choice of $q$ in Baby Rudins Example 1.1", "score": 56, "views": 7157, "answer_pids": [8383, 8403, 52731, 71787, 71848, 237792, 719603, 792624, 792639, 793344, 1190296, 1226773, 1302618], "question_author": ""}
{"qid": 1654, "query": "Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?", "score": 56, "views": 7080, "answer_pids": [216771, 216772, 216774, 216775, 216776, 216962, 217236], "question_author": ""}
{"qid": 1655, "query": "Is there any conjecture that we know is provable/disprovable but we havent found a proof of yet?", "score": 56, "views": 6340, "answer_pids": [857621, 857778, 858114], "question_author": "zeller"}
{"qid": 1656, "query": "Gaps or holes in rational number system", "score": 56, "views": 6336, "answer_pids": [1299788, 1299789, 1300019, 1300045, 1300250, 1327186], "question_author": null}
{"qid": 1657, "query": "What is $-i$ exactly?", "score": 56, "views": 6056, "answer_pids": [186244, 186245, 186248, 186249, 186253, 186256, 186257, 186319, 186332, 186349, 187609, 189023, 189149, 192125, 192710, 539439], "question_author": ""}
{"qid": 1658, "query": "What knot is this?", "score": 56, "views": 5924, "answer_pids": [804346, 804354, 804385], "question_author": "Nick"}
{"qid": 1659, "query": "Doesnt the unprovability of the continuum hypothesis prove the continuum hypothesis?", "score": 56, "views": 5608, "answer_pids": [917319, 917324, 917406, 917418, 918095], "question_author": "Gabriel Fair"}
{"qid": 1660, "query": "What is the fastest/most efficient algorithm for estimating Eulers Constant $\\gamma$?", "score": 56, "views": 5207, "answer_pids": [66118, 66130, 66132, 66134, 66137, 66185, 66997, 67005, 67038, 88002, 645791, 913167], "question_author": "SIMEL"}
{"qid": 1661, "query": "What is the crime of l\u00e8se-Bourbaki?", "score": 56, "views": 4943, "answer_pids": [1062633, 1062637, 1062893], "question_author": "SIMEL"}
{"qid": 1662, "query": "Natural example of cosets", "score": 56, "views": 4845, "answer_pids": [661753, 661755, 661756, 661930, 661941, 661951, 661964, 662110, 662154, 662158, 662159, 662161, 662209, 699656], "question_author": "J. Musser"}
{"qid": 1663, "query": "Is Complex Analysis equivalent Real Analysis with $f:\\mathbb R^2 \\to \\mathbb R^2$?", "score": 56, "views": 4836, "answer_pids": [95438, 95439, 95440, 95441, 95448, 95509], "question_author": "LanceLafontaine"}
{"qid": 1664, "query": "If $e^A$ and $e^B$ commute, do $A$ and $B$ commute?", "score": 56, "views": 4734, "answer_pids": [165415, 165448, 165490, 237682], "question_author": ""}
{"qid": 1665, "query": "A discrete math riddle", "score": 56, "views": 4132, "answer_pids": [147146, 743612], "question_author": "Rory M"}
{"qid": 1666, "query": "How does one give a mathematical talk?", "score": 56, "views": 3954, "answer_pids": [76760, 76777, 76802, 76821], "question_author": ""}
{"qid": 1667, "query": "Probability that a quadratic equation has real roots", "score": 56, "views": 3915, "answer_pids": [1350486, 1350489, 1350491], "question_author": ""}
{"qid": 1668, "query": "How to identify surfaces of revolution", "score": 56, "views": 3728, "answer_pids": [270481, 271408, 271423, 271828], "question_author": "ironcyclone"}
{"qid": 1669, "query": "Is there an abstract definition of a matrix being upper triangular?", "score": 56, "views": 3717, "answer_pids": [1429022, 1429025, 1429030, 1429035, 1429268, 1429431, 1429626], "question_author": ""}
{"qid": 1670, "query": "Self-learning mathematics - help needed!", "score": 56, "views": 3609, "answer_pids": [386760, 386814, 386836, 386957], "question_author": ""}
{"qid": 1671, "query": "What is this pattern called?", "score": 56, "views": 3018, "answer_pids": [503691, 503696], "question_author": ""}
{"qid": 1672, "query": "The $100$th derivative of $(x^2 + 1)/(x^3 - x)$", "score": 56, "views": 2849, "answer_pids": [92468], "question_author": ""}
{"qid": 1673, "query": "Evaluate $\\int_0^1\\left(\\frac{1}{\\ln x} + \\frac{1}{1-x}\\right)^2 \\mathrm dx$", "score": 56, "views": 2608, "answer_pids": [135220, 135227, 135307, 135320, 135435, 185344, 339558, 646558], "question_author": "Santosh Linkha"}
{"qid": 1674, "query": "Infinite Series $\\sum\\limits_{n=1}^\\infty\\frac{(H_n)^2}{n^3}$", "score": 56, "views": 2216, "answer_pids": [256226, 257255, 257671, 873753], "question_author": "Mykie"}
{"qid": 1675, "query": "In which ordered fields does absolute convergence imply convergence?", "score": 56, "views": 1933, "answer_pids": [58474], "question_author": "John C"}
{"qid": 1676, "query": "Crazy pattern in the simple continued fraction for $\\sum_{k=1}^\\infty \\frac{1}{(2^k)!}$", "score": 56, "views": 1768, "answer_pids": [733862, 734238], "question_author": "Philipp"}
{"qid": 1677, "query": "A sequence that avoids both arithmetic and geometric progressions", "score": 56, "views": 1373, "answer_pids": [1118814, 1120433], "question_author": "Sam"}
{"qid": 1678, "query": "Proof of triangle inequality", "score": 55, "views": 168156, "answer_pids": [146660, 146674, 146690, 459539, 759742, 860002], "question_author": ""}
{"qid": 1679, "query": "How to find the factorial of a fraction?", "score": 55, "views": 125770, "answer_pids": [186490, 186491, 473351, 473397, 753406], "question_author": ""}
{"qid": 1680, "query": "What is the general equation of the ellipse that is not in the origin and rotated by an angle?", "score": 55, "views": 114432, "answer_pids": [199726, 199727, 199732, 199744, 203499, 1096716], "question_author": "bobthejoe"}
{"qid": 1681, "query": "Prove that the union of countably many countable sets is countable.", "score": 55, "views": 91774, "answer_pids": [270702, 270709, 270714, 271065, 377588, 405162, 594432, 1453811], "question_author": "bobthejoe"}
{"qid": 1682, "query": "Symbol for elementwise multiplication of vectors", "score": 55, "views": 65957, "answer_pids": [27971, 27972, 27991, 876944], "question_author": ""}
{"qid": 1683, "query": "Difference between supremum and maximum", "score": 55, "views": 55236, "answer_pids": [81052, 81053, 81054, 81055], "question_author": null}
{"qid": 1684, "query": "What is difference between a ring and a field?", "score": 55, "views": 51850, "answer_pids": [71531, 71532, 71534], "question_author": ""}
{"qid": 1685, "query": "Picking random points in the volume of sphere with uniform probability", "score": 55, "views": 41594, "answer_pids": [45483, 45484, 45486, 353248], "question_author": "Andy Bale"}
{"qid": 1686, "query": "Necessity/Advantage of LU Decomposition over Gaussian Elimination", "score": 55, "views": 38572, "answer_pids": [128927], "question_author": "jp89"}
{"qid": 1687, "query": "Introductory Group theory textbook", "score": 55, "views": 34399, "answer_pids": [13933, 13939, 13941, 13942, 13950, 33269, 128308, 148800], "question_author": null}
{"qid": 1688, "query": "Does non-symmetric positive definite matrix have positive eigenvalues?", "score": 55, "views": 32315, "answer_pids": [155084, 1158896], "question_author": "bobthejoe"}
{"qid": 1689, "query": "Difference between Analytic and Holomorphic function", "score": 55, "views": 24102, "answer_pids": [259783], "question_author": ""}
{"qid": 1690, "query": "Evaluating the nested radical $ \\sqrt{1 + 2 \\sqrt{1 + 3 \\sqrt{1 + \\cdots}}} $.", "score": 55, "views": 21803, "answer_pids": [4187, 4195, 646585], "question_author": "bobthejoe"}
{"qid": 1691, "query": "Do matrices $ AB $ and $ BA $ have the same minimal and characteristic polynomials?", "score": 55, "views": 19715, "answer_pids": [148512, 148516, 148521, 369599, 1068381, 1243806, 1326541], "question_author": null}
{"qid": 1692, "query": "What does it take to get a job at a top 50 math program in the U.S.?", "score": 55, "views": 16843, "answer_pids": [49274, 49279, 49368, 49380, 49385, 293708, 318043], "question_author": ""}
{"qid": 1693, "query": "Is an irrational number odd or even?", "score": 55, "views": 16246, "answer_pids": [512891, 512895, 512897, 512904, 512921, 513119, 513179], "question_author": ""}
{"qid": 1694, "query": "Definitions of Hessian in Riemannian Geometry", "score": 55, "views": 16235, "answer_pids": [42579, 42585], "question_author": "Kevin H. Lin"}
{"qid": 1695, "query": "How to calculate the pullback of a $k$-form explicitly", "score": 55, "views": 15252, "answer_pids": [260565, 891074, 1101823], "question_author": ""}
{"qid": 1696, "query": "How to prove that $\\log(x)1$?", "score": 55, "views": 15182, "answer_pids": [620494, 620495, 620496, 620497, 620499, 620504, 620651, 620668, 620847, 620851, 704117, 1165273], "question_author": ""}
{"qid": 1697, "query": "Exam with $12$ yes/no questions (half yes, half no) and $8$ correct needed to pass, is it better to answer randomly or answer exactly 6 times yes?", "score": 55, "views": 14332, "answer_pids": [1023396, 1023397, 1023427, 1023545, 1023915, 1023999], "question_author": ""}
{"qid": 1698, "query": "Distinguishing probability measure, function and distribution", "score": 55, "views": 14171, "answer_pids": [442242], "question_author": ""}
{"qid": 1699, "query": "False proof: $\\pi = 4$, but why?", "score": 55, "views": 12073, "answer_pids": [553166, 553167, 553446, 553570], "question_author": ""}
{"qid": 1700, "query": "Books about the Riemann Hypothesis", "score": 55, "views": 11403, "answer_pids": [142266, 500285], "question_author": ""}
{"qid": 1701, "query": "Why is $\\text{Hom}(V,W)$ the same thing as $V^* \\otimes W$?", "score": 55, "views": 10935, "answer_pids": [300664, 1445778], "question_author": "Jan"}
{"qid": 1702, "query": "What is exactly the difference between a definition and an axiom?", "score": 55, "views": 10856, "answer_pids": [566463, 566467, 566476, 566483, 566491, 566540, 566604, 566840, 566846, 566854, 567950], "question_author": ""}
{"qid": 1703, "query": "Non-associative operations", "score": 55, "views": 9841, "answer_pids": [1065218, 1065223, 1065224, 1065247, 1065251, 1065270, 1065297, 1065302, 1065310, 1065321, 1065472, 1065474, 1065676, 1065809, 1065824, 1065826, 1065936, 1065942, 1065972, 1065976, 1066037, 1066117, 1091873], "question_author": ""}
{"qid": 1704, "query": "Is $\\sqrt{64}$ considered $8$? or is it $8,-8$?", "score": 55, "views": 9119, "answer_pids": [573408, 573411, 573450, 573457, 573556, 573573, 573776, 573999, 576360, 690258, 1407022], "question_author": "Ch Zh"}
{"qid": 1705, "query": "What is $\\int_0^1\\frac{x^7-1}{\\log(x)}\\mathrm dx$?", "score": 55, "views": 8764, "answer_pids": [256770, 256776, 556498, 890712, 890790, 1056955, 1360350], "question_author": ""}
{"qid": 1706, "query": "Mathematical explanation behind a picture posted (lifted from facebook)", "score": 55, "views": 8357, "answer_pids": [38955, 38988], "question_author": "David Z"}
{"qid": 1707, "query": "Path-connected and locally connected space that is not locally path-connected", "score": 55, "views": 8285, "answer_pids": [360132, 589986], "question_author": ""}
{"qid": 1708, "query": "Can I search for factors of $\\ (11!)!+11!+1\\ $ efficiently?", "score": 55, "views": 7391, "answer_pids": [1251521, 1256146], "question_author": ""}
{"qid": 1709, "query": "Solutions to the matrix equation $\\mathbf{AB-BA=I}$ over general fields", "score": 55, "views": 6930, "answer_pids": [51329, 51331, 51402, 51458], "question_author": ""}
{"qid": 1710, "query": "Square root confusion: Why am I getting an answer if it doesnt work?", "score": 55, "views": 6831, "answer_pids": [703761, 703768, 703773, 703843, 703881, 704220, 704370, 704399, 704623], "question_author": "Guillermo Phillips"}
{"qid": 1711, "query": "Geometric understanding of differential forms.", "score": 55, "views": 6388, "answer_pids": [206281, 206291, 206297, 966083, 1435834], "question_author": ""}
{"qid": 1712, "query": "Why are the only division algebras over the real numbers the real numbers, the complex numbers, and the quaternions?", "score": 55, "views": 5577, "answer_pids": [333, 914, 1356442, 1409501], "question_author": ""}
{"qid": 1713, "query": "What is the formula for pi used in the Python decimal library?", "score": 55, "views": 5088, "answer_pids": [1108914, 1108915, 1109721], "question_author": ""}
{"qid": 1714, "query": "Intuition for why the difference between $\\frac{2x^2-x}{x^2-x+1}$ and $\\frac{x-2}{x^2-x+1}$ is a constant?", "score": 55, "views": 4890, "answer_pids": [1021887, 1021888, 1021890, 1021891, 1021894, 1021899, 1021903, 1021937, 1021941, 1021968, 1022237, 1022247, 1022373, 1022484, 1022576, 1024199], "question_author": ""}
{"qid": 1715, "query": "Axiom of choice and automorphisms of vector spaces", "score": 55, "views": 3743, "answer_pids": [15672, 15991], "question_author": ""}
{"qid": 1716, "query": "Show that floating point $\\sqrt{x \\cdot x} \\geq x$ for all long $x$.", "score": 55, "views": 3272, "answer_pids": [116304], "question_author": "xport"}
{"qid": 1717, "query": "How to entertain a crowd with mathematics?", "score": 55, "views": 3164, "answer_pids": [175762, 175763, 175764, 175772, 175773, 175781, 175800, 175818, 175820, 175826, 175830, 175841], "question_author": "Justin L."}
{"qid": 1718, "query": "Colliding Bullets", "score": 55, "views": 3154, "answer_pids": [599094, 599880, 605501, 750295, 908186], "question_author": ""}
{"qid": 1719, "query": "False beliefs about Lebesgue measure on $\\mathbb{R}$", "score": 55, "views": 2980, "answer_pids": [48743, 48745, 48749, 48755, 53103, 629759, 629762], "question_author": ""}
{"qid": 1720, "query": "Evaluating $\\int P(\\sin x, \\cos x) \\text{d}x$", "score": 55, "views": 2693, "answer_pids": [16254, 16255, 702251], "question_author": "Lagerbaer"}
{"qid": 1721, "query": "Is there a geometric idea behind Sylows theorems?", "score": 55, "views": 2361, "answer_pids": [240161, 711027, 750929], "question_author": ""}
{"qid": 1722, "query": "Is $29$ the only prime of the form $p^p+2$?", "score": 55, "views": 2209, "answer_pids": [1168896, 1168916, 1202089], "question_author": ""}
{"qid": 1723, "query": "Is Stokes Theorem natural in the sense of category theory?", "score": 55, "views": 2146, "answer_pids": [495648, 496188], "question_author": ""}
{"qid": 1724, "query": "Trying to define $\\mathbb{R}^{0.5}$ topologically", "score": 55, "views": 1796, "answer_pids": [156388, 156394, 163720], "question_author": "Gordon Gustafson"}
{"qid": 1725, "query": "Conjectured value of a harmonic sum $\\sum_{n=1}^\\infty\\left(H_n-\\,2H_{2n}+H_{4n}\\right)^2$", "score": 55, "views": 1508, "answer_pids": [562136, 568806, 1281876], "question_author": "skywaddler"}
{"qid": 1726, "query": "Elementary proof that the derivative of a real function is continuous somewhere", "score": 55, "views": 1389, "answer_pids": [792292], "question_author": "user1271772"}
{"qid": 1727, "query": "Can $\\sqrt{p}^{\\sqrt{p}^{\\sqrt{p}}}$ be an integer, if $p$ is a non-square positive integer?", "score": 55, "views": 1214, "answer_pids": [343965], "question_author": "Jacob"}
{"qid": 1728, "query": "How to tell if a set of vectors spans a space?", "score": 54, "views": 437271, "answer_pids": [29832, 29834, 29836], "question_author": ""}
{"qid": 1729, "query": "Easy way of memorizing values of sine, cosine, and tangent", "score": 54, "views": 393661, "answer_pids": [608031, 608036, 608041, 608068, 608071, 608477, 609002], "question_author": ""}
{"qid": 1730, "query": "What does \u2208 mean?", "score": 54, "views": 282508, "answer_pids": [361418, 878294, 878299], "question_author": "bobthejoe"}
{"qid": 1731, "query": "Recognizable vs Decidable", "score": 54, "views": 98101, "answer_pids": [14098, 579634, 845004], "question_author": "719016"}
{"qid": 1732, "query": "Why have we chosen our number system to be decimal (base 10)?", "score": 54, "views": 73974, "answer_pids": [5028, 5029, 5030, 5045, 5069, 91824], "question_author": ""}
{"qid": 1733, "query": "How the product of two integrals is iterated integral? $\\int\\cdot \\int = \\iint$", "score": 54, "views": 50293, "answer_pids": [251395], "question_author": ""}
{"qid": 1734, "query": "How to take the gradient of the quadratic form?", "score": 54, "views": 40528, "answer_pids": [110000, 1150290, 1213888, 1265643, 1278389], "question_author": "bobthejoe"}
{"qid": 1735, "query": "Why cant a set have two elements of the same value?", "score": 54, "views": 35455, "answer_pids": [394291, 394312, 394336, 394456, 394607, 397477, 846574], "question_author": "balaka"}
{"qid": 1736, "query": "Order of finite fields is $p^n$", "score": 54, "views": 35250, "answer_pids": [38374, 376043, 496477, 1049037, 1121346, 1137426], "question_author": "balaka"}
{"qid": 1737, "query": "Why is $A^TA$ invertible if $A$ has independent columns?", "score": 54, "views": 34940, "answer_pids": [708826, 708827, 708828], "question_author": "Daisy Sophia Hollman"}
{"qid": 1738, "query": "Proof that the largest eigenvalue of a stochastic matrix is $1$", "score": 54, "views": 33639, "answer_pids": [21527, 21537, 1043370, 1083451, 1155788, 1307692], "question_author": ""}
{"qid": 1739, "query": "how does expectation maximization work?", "score": 54, "views": 32194, "answer_pids": [13734, 367028], "question_author": "user1202664"}
{"qid": 1740, "query": "Are functions of independent variables also independent?", "score": 54, "views": 30314, "answer_pids": [5026, 5034, 429175, 1167247], "question_author": ""}
{"qid": 1741, "query": "What is an intuitive explanation for $\\operatorname{div} \\operatorname{curl} F = 0$?", "score": 54, "views": 27745, "answer_pids": [14659, 14664, 14675, 14730, 14731], "question_author": "BCS"}
{"qid": 1742, "query": "What is the difference between projected gradient descent and ordinary gradient descent?", "score": 54, "views": 26231, "answer_pids": [259085, 1340558], "question_author": ""}
{"qid": 1743, "query": "Functions which are Continuous, but not Bicontinuous", "score": 54, "views": 24045, "answer_pids": [36330, 36333, 36336, 36354, 36557, 36558], "question_author": "Pete"}
{"qid": 1744, "query": "Choose a random number between $0$ and $1$ and record its value. Keep doing it until the sum of the numbers exceeds $1$. How many tries do we need?", "score": 54, "views": 23249, "answer_pids": [57232, 57234, 57241, 1300880, 1374897], "question_author": "endolith"}
{"qid": 1745, "query": "How I can prove that the sequence $\\sqrt{2} , \\sqrt{2\\sqrt{2}}, \\sqrt{2\\sqrt{2\\sqrt{2}}}$ converges to 2?", "score": 54, "views": 20113, "answer_pids": [100348, 100350, 100351, 100360, 100386, 195637, 325040], "question_author": "Joseph O'Rourke"}
{"qid": 1746, "query": "Difference between bijection and isomorphism?", "score": 54, "views": 17917, "answer_pids": [28816, 28818, 28819, 28824, 28838], "question_author": ""}
{"qid": 1747, "query": "$\\lim\\limits_{n \\to{+}\\infty}{\\sqrt[n]{n!}}$ is infinite", "score": 54, "views": 14283, "answer_pids": [69323, 69324, 69326, 69349, 69357, 69395, 69396, 69436, 70802, 79947, 205009], "question_author": "claws"}
{"qid": 1748, "query": "An easy example of a non-constructive proof without an obvious fix?", "score": 54, "views": 13763, "answer_pids": [458990, 458994, 458997, 459071, 459093, 459132, 459133, 459413, 459443, 459446, 459481, 459745, 460313, 617941], "question_author": null}
{"qid": 1749, "query": "Why doesnt induction extend to infinity? (re: Fourier series)", "score": 54, "views": 12941, "answer_pids": [50818, 50827, 50841, 266253, 270680, 643248, 1281922], "question_author": ""}
{"qid": 1750, "query": "A really complicated calculus book", "score": 54, "views": 12939, "answer_pids": [146945, 146948, 146966, 146972, 146978, 146981, 146986, 147084, 150121, 327506], "question_author": "bobthejoe"}
{"qid": 1751, "query": "Why isnt several complex variables as fundamental as multivariable calculus?", "score": 54, "views": 12233, "answer_pids": [138884, 138897, 138952], "question_author": "bobthejoe"}
{"qid": 1752, "query": "Parabola is an ellipse, but with one focal point at infinity", "score": 54, "views": 12191, "answer_pids": [336820, 336822, 336836, 336990], "question_author": "balaka"}
{"qid": 1753, "query": "Formal definition of conditional probability", "score": 54, "views": 10173, "answer_pids": [231445], "question_author": ""}
{"qid": 1754, "query": "Why is it worth spending time on type theory?", "score": 54, "views": 10116, "answer_pids": [257113, 257909, 257927], "question_author": "gkadam"}
{"qid": 1755, "query": "Why is cross product defined in the way that it is?", "score": 54, "views": 8278, "answer_pids": [745554, 745555, 745582, 745597, 745666, 745672, 745681, 746150, 746163, 746523, 746907], "question_author": ""}
{"qid": 1756, "query": "What are the rules for equals signs with big-O and little-o?", "score": 54, "views": 7575, "answer_pids": [44912, 44916], "question_author": "balaka"}
{"qid": 1757, "query": "What book can bridge high school math and the more advanced topics?", "score": 54, "views": 7551, "answer_pids": [29408, 29409, 29413, 29415, 29416, 29417, 29450, 29452, 29454, 29481, 29489, 29555], "question_author": ""}
{"qid": 1758, "query": "Foundation for analysis without axiom of choice?", "score": 54, "views": 7517, "answer_pids": [53512, 53514, 53526, 53539, 337815], "question_author": "bobthejoe"}
{"qid": 1759, "query": "Intuition behind ideal", "score": 54, "views": 7108, "answer_pids": [33863, 33864, 33865, 33866, 33888, 63346, 63355, 63378], "question_author": ""}
{"qid": 1760, "query": "How far is the list of known primes known to be complete?", "score": 54, "views": 7106, "answer_pids": [713055, 713059, 713061, 713239], "question_author": ""}
{"qid": 1761, "query": "Why the real and imaginary parts of a complex analytic function are not independent?", "score": 54, "views": 7024, "answer_pids": [1230898, 1230905, 1230925, 1231079, 1231083, 1231316, 1231553], "question_author": ""}
{"qid": 1762, "query": "Do circles divide the plane into more regions than lines?", "score": 54, "views": 6948, "answer_pids": [96247, 96252, 96258, 96267], "question_author": "J. Musser"}
{"qid": 1763, "query": "On the functional square root of $x^2+1$", "score": 54, "views": 6731, "answer_pids": [2389, 5055, 57009, 57011, 217146, 704790, 1286354], "question_author": "skywaddler"}
{"qid": 1764, "query": "$n!+1$ being a perfect square", "score": 54, "views": 6527, "answer_pids": [4598, 4599, 4600], "question_author": ""}
{"qid": 1765, "query": "Does $\\zeta(3)$ have a connection with $\\pi$?", "score": 54, "views": 6110, "answer_pids": [18952, 18954, 20297, 20312, 1111737], "question_author": ""}
{"qid": 1766, "query": "Why do we need so many trigonometric definitions?", "score": 54, "views": 6094, "answer_pids": [1004902, 1004908, 1004909, 1004915, 1004916, 1004976, 1005569], "question_author": "Andy Bale"}
{"qid": 1767, "query": "Prove that the sum of pythagorean triples is always even", "score": 54, "views": 5933, "answer_pids": [820729, 820732, 820734, 820739, 821737, 821978], "question_author": "pikk"}
{"qid": 1768, "query": "What is the explanation for this visual proof of the sum of squares?", "score": 54, "views": 5643, "answer_pids": [769925, 769926, 769927, 770241], "question_author": "Draco"}
{"qid": 1769, "query": "Why are topological spaces interesting to study?", "score": 54, "views": 4966, "answer_pids": [213310, 213483, 213487, 215159, 215175, 219866], "question_author": ""}
{"qid": 1770, "query": "Good math books to discover stuff by yourself", "score": 54, "views": 4549, "answer_pids": [792980, 792982, 793350, 793488, 794999, 795005, 796988], "question_author": ""}
{"qid": 1771, "query": "Integral $\\int_{-1}^{1} \\frac{1}{x}\\sqrt{\\frac{1+x}{1-x}} \\log \\left( \\frac{(r-1)x^{2} + sx + 1}{(r-1)x^{2} - sx + 1} \\right) \\, \\mathrm dx$", "score": 54, "views": 4369, "answer_pids": [257784, 440954], "question_author": "thomasrive"}
{"qid": 1772, "query": "Why is it that $\\mathbb{Q}$ cannot be homeomorphic to _any_ complete metric space?", "score": 54, "views": 4357, "answer_pids": [8790, 8791, 8792, 8811], "question_author": "Mark Rogers"}
{"qid": 1773, "query": "Why is the Laplacian important in Riemannian geometry?", "score": 54, "views": 4079, "answer_pids": [294076, 396505, 783432, 1296944], "question_author": "Polynomial"}
{"qid": 1774, "query": "MIT 2015 Integration Question", "score": 54, "views": 4077, "answer_pids": [812793, 814966, 816135], "question_author": "Covi"}
{"qid": 1775, "query": "Is there a name for this type of polygon?", "score": 54, "views": 4051, "answer_pids": [610834, 610835, 610836], "question_author": "serg"}
{"qid": 1776, "query": "Intuition behind Snake Lemma", "score": 54, "views": 4047, "answer_pids": [92000, 92003, 99981], "question_author": ""}
{"qid": 1777, "query": "Is it possible for a function to be smooth everywhere, analytic nowhere, yet Taylor series at any point converges in a nonzero radius?", "score": 54, "views": 3875, "answer_pids": [282004], "question_author": "Nick T"}
{"qid": 1778, "query": "Proof that a trigonometric function of a rational angle must be non-transcendental", "score": 54, "views": 3378, "answer_pids": [742948, 742949, 742981], "question_author": null}
{"qid": 1779, "query": "Does $|n^2 \\cos n|$ diverge to $+\\infty$?", "score": 54, "views": 3308, "answer_pids": [65191], "question_author": "nibot"}
{"qid": 1780, "query": "What are exact sequences, metaphysically speaking?", "score": 54, "views": 3025, "answer_pids": [71516, 71517], "question_author": ""}
{"qid": 1781, "query": "What structure does the alternating group preserve?", "score": 54, "views": 2585, "answer_pids": [3076, 3110, 3342, 3347, 22213, 1184131, 1276563], "question_author": "Vortico"}
{"qid": 1782, "query": "About the integral $\\int_{0}^{+\\infty}\\sin(x\\,\\log x)\\,dx$", "score": 54, "views": 2431, "answer_pids": [617860], "question_author": ""}
{"qid": 1783, "query": "Effective Research Notes", "score": 54, "views": 2283, "answer_pids": [444976, 450483], "question_author": ""}
{"qid": 1784, "query": "A new continued fraction for Ap\u00e9rys constant, $\\zeta(3)$?", "score": 54, "views": 1782, "answer_pids": [71575], "question_author": "Gordon Gustafson"}
{"qid": 1785, "query": "Two questions about weakly convergent series related to $\\sin(n^2)$ and Weyls inequality", "score": 54, "views": 1404, "answer_pids": [1041167], "question_author": ""}
{"qid": 1786, "query": "What is the integral of 1/x?", "score": 53, "views": 578291, "answer_pids": [103003, 143269], "question_author": ""}
{"qid": 1787, "query": "How does one denote the set of all positive real numbers?", "score": 53, "views": 107755, "answer_pids": [15213, 15219, 15310, 15352, 551964, 560174], "question_author": "Eric"}
{"qid": 1788, "query": "Prove: If a sequence converges, then every subsequence converges to the same limit.", "score": 53, "views": 82099, "answer_pids": [105774, 105777, 629794], "question_author": ""}
{"qid": 1789, "query": "How many prime numbers are known?", "score": 53, "views": 56562, "answer_pids": [132126, 142417, 1145407], "question_author": "wrongusername"}
{"qid": 1790, "query": "What books are prerequisites for Spivaks Calculus?", "score": 53, "views": 52131, "answer_pids": [37517, 37522, 37535, 133404, 277911], "question_author": ""}
{"qid": 1791, "query": "Create unique number from 2 numbers", "score": 53, "views": 46791, "answer_pids": [12891, 12892, 12899, 310552, 889832], "question_author": "wrongusername"}
{"qid": 1792, "query": "What is the main difference between a vector space and a field?", "score": 53, "views": 44509, "answer_pids": [406266, 406267, 406282, 545004, 579501, 784782, 836945], "question_author": ""}
{"qid": 1793, "query": "If eigenvalues are positive, is the matrix positive definite?", "score": 53, "views": 42313, "answer_pids": [2574, 2576, 2584, 18589, 87213], "question_author": ""}
{"qid": 1794, "query": "$\\epsilon$-$\\delta$ proof that $\\lim\\limits_{x \\to 1} \\frac{1}{x} = 1$.", "score": 53, "views": 38549, "answer_pids": [196505, 196510, 196519, 415477, 1048902], "question_author": "Aengus"}
{"qid": 1795, "query": "Finding Value of the Infinite Product $\\prod \\Bigl(1-\\frac{1}{n^{2}}\\Bigr)$", "score": 53, "views": 35647, "answer_pids": [10122, 10123, 10124, 10126, 930728, 1223881], "question_author": "jqp"}
{"qid": 1796, "query": "Pen, pencils and paper to write math", "score": 53, "views": 35281, "answer_pids": [75176, 75177, 75178, 75179, 75181, 75187, 75188, 75191, 75192, 75193, 75200, 75202, 75206, 75223, 75231, 75256], "question_author": ""}
{"qid": 1797, "query": "Video lectures on Real Analysis?", "score": 53, "views": 32417, "answer_pids": [149062, 149105, 149108, 151964, 154865], "question_author": "unsym"}
{"qid": 1798, "query": "If $a^3 =a$ for all $a$ in a ring $R$, then $R$ is commutative.", "score": 53, "views": 21595, "answer_pids": [56817, 139752, 139754, 139772, 170618, 397327, 938874], "question_author": ""}
{"qid": 1799, "query": "Which of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6} , 7^{1/7}$ is largest, and how to find out without calculator?", "score": 53, "views": 21076, "answer_pids": [531250, 531251, 531253, 531255, 531265, 531458, 531817, 532156, 532160, 534300], "question_author": "Andrei"}
{"qid": 1800, "query": "Zero divided by zero must be equal to zero", "score": 53, "views": 20191, "answer_pids": [608329, 608334, 608357, 608494, 608545, 608803, 608878, 609693, 610020, 610095], "question_author": ""}
{"qid": 1801, "query": "Proof that every number \u2265 $8$ can be represented by a sum of fives and threes.", "score": 53, "views": 18400, "answer_pids": [480209, 480211, 480214, 480218, 480237, 480506, 480554, 493184], "question_author": "serg"}
{"qid": 1802, "query": "What is the difference between a Ring and an Algebra?", "score": 53, "views": 15038, "answer_pids": [178965, 178972, 178999, 179001], "question_author": "Brennan Vincent"}
{"qid": 1803, "query": "Intuitive understanding of the derivatives of $\\sin x$ and $\\cos x$", "score": 53, "views": 14756, "answer_pids": [246, 248, 250, 255, 256, 334, 523, 661, 1297, 1318, 4854, 4878, 4889, 11293, 206790, 1359389], "question_author": "bobthejoe"}
{"qid": 1804, "query": "A linear operator commuting with all such operators is a scalar multiple of the identity.", "score": 53, "views": 14725, "answer_pids": [15135, 15136, 15137, 15147, 15149, 15151, 654575], "question_author": ""}
{"qid": 1805, "query": "Prove that if $(ab)^i = a^ib^i \\forall a,b\\in G$ for three consecutive integers $i$ then G is abelian", "score": 53, "views": 13569, "answer_pids": [21863, 21864, 21865, 21872], "question_author": null}
{"qid": 1806, "query": "Is $\\pi$ equal to $180^\\circ$?", "score": 53, "views": 13468, "answer_pids": [469961, 469963, 470083, 470108, 470475, 470487, 470599, 470611, 471172, 519560, 558880], "question_author": ""}
{"qid": 1807, "query": "Is every Lebesgue measurable function on $\\mathbb{R}$ the pointwise limit of continuous functions?", "score": 53, "views": 13301, "answer_pids": [8444, 8446, 32500, 268751], "question_author": ""}
{"qid": 1808, "query": "Seeking a laymans guide to Measure Theory", "score": 53, "views": 13077, "answer_pids": [37652, 37673, 37678, 37683, 37688, 37691, 123418, 127584, 127593, 127610], "question_author": "AndreKR"}
{"qid": 1809, "query": "What are the Laws of Rational Exponents?", "score": 53, "views": 11590, "answer_pids": [634973, 634981, 636191, 637194, 638534, 638899, 721134, 1215569], "question_author": ""}
{"qid": 1810, "query": "An intuitive approach to the Jordan Normal form", "score": 53, "views": 9102, "answer_pids": [283987, 284043, 284356, 286245], "question_author": ""}
{"qid": 1811, "query": "Intuitive Understanding of the constant $e$", "score": 53, "views": 8279, "answer_pids": [14213, 14214, 14215, 14217, 14220, 28895, 28901, 28941, 29596, 29597, 139008, 164502, 198774, 198789, 415592, 527223, 791881, 1269726], "question_author": ""}
{"qid": 1812, "query": "How far can one see over the ocean?", "score": 53, "views": 8047, "answer_pids": [382245, 382249, 382323, 382761], "question_author": ""}
{"qid": 1813, "query": "Every power series is the Taylor series of some $C^{\\infty}$ function", "score": 53, "views": 7983, "answer_pids": [33424, 33426, 182363], "question_author": "harpalss"}
{"qid": 1814, "query": "Is the axiom of choice really all that important?", "score": 53, "views": 7297, "answer_pids": [504475, 505610], "question_author": "Jake"}
{"qid": 1815, "query": "On average, how many friends would I need to have to have at least one friends birthday every day?", "score": 53, "views": 6989, "answer_pids": [566412], "question_author": ""}
{"qid": 1816, "query": "Is there a purely algebraic proof of the Fundamental Theorem of Algebra?", "score": 53, "views": 6899, "answer_pids": [83753, 1017907], "question_author": "Bala"}
{"qid": 1817, "query": "What books should I get to self study beyond Calculus for someone about to start undergrad mathematics?", "score": 53, "views": 6838, "answer_pids": [664604, 664611, 664619, 664620, 664627, 664674, 664691, 664738, 664749, 664771, 664776, 664787, 664794, 664918, 664935, 664995, 665042, 665241, 665572, 665579, 665627, 666003, 666409, 668165, 674430], "question_author": "Akhil Mathew"}
{"qid": 1818, "query": "Are there any objects which arent sets?", "score": 53, "views": 6825, "answer_pids": [662581, 662585, 662588, 662718, 662816, 663176], "question_author": "pogibas"}
{"qid": 1819, "query": "What is the theme of analysis?", "score": 53, "views": 5648, "answer_pids": [1180931, 1180933, 1180994, 1181037, 1181102, 1181329, 1181797], "question_author": "Sergio"}
{"qid": 1820, "query": "Is there any geometric way to characterize $e$?", "score": 53, "views": 5246, "answer_pids": [80661, 80662, 80663, 80670, 80673, 80683, 105186, 1432051], "question_author": ""}
{"qid": 1821, "query": "Why and How do certain manipulations in indefinite integrations just work?", "score": 53, "views": 5192, "answer_pids": [1459856, 1459858, 1459859, 1459861, 1459865, 1459944, 1460393, 1460438, 1460498], "question_author": ""}
{"qid": 1822, "query": "History of the Concept of a Ring", "score": 53, "views": 4612, "answer_pids": [226, 553, 1103], "question_author": "J. Musser"}
{"qid": 1823, "query": "Simplest way to get the lower bound $\\pi > 3.14$", "score": 53, "views": 4400, "answer_pids": [929724, 929726, 929729, 929757, 929772, 929796, 929804, 929824, 929986, 930077, 930267, 931270, 937997, 1036219], "question_author": ""}
{"qid": 1824, "query": "What to answer when people ask what I do in mathematics", "score": 53, "views": 4358, "answer_pids": [466146, 466156, 466169, 466205, 466344, 466384], "question_author": ""}
{"qid": 1825, "query": "How many super imaginary numbers are there?", "score": 53, "views": 4236, "answer_pids": [962598, 962607, 962672], "question_author": ""}
{"qid": 1826, "query": "Fibonacci infinite sum resulting in $\\pi$", "score": 53, "views": 4026, "answer_pids": [565203, 565233, 565297], "question_author": ""}
{"qid": 1827, "query": "Is $i = \\sqrt{e^{\\pi\\sqrt{e^{\\pi\\sqrt\\ldots}}}}$?", "score": 53, "views": 3435, "answer_pids": [419491, 419493], "question_author": "Abel"}
{"qid": 1828, "query": "Example of a compact set that isnt the spectrum of an operator", "score": 53, "views": 3044, "answer_pids": [22869], "question_author": ""}
{"qid": 1829, "query": "Why is one $\\infty$ number enough for complex numbers?", "score": 53, "views": 3020, "answer_pids": [16817, 16819, 16820, 16822, 16825, 16889], "question_author": "Dale"}
{"qid": 1830, "query": "How do the Catalan numbers turn up here?", "score": 53, "views": 2662, "answer_pids": [19279, 19281], "question_author": "Kirby"}
{"qid": 1831, "query": "What is so special about $\\alpha=-1$ in the integral of $x^\\alpha$?", "score": 53, "views": 2319, "answer_pids": [34131, 34144, 139823, 139828, 767455, 884925], "question_author": ""}
{"qid": 1832, "query": "To evaluate $\\int_0^{+\\infty} \\frac{\\;\\mathrm dx}{\\sqrt[3]{x^3+a^3}\\sqrt[3]{x^3+b^3}\\sqrt[3]{x^3+c^3}}$", "score": 53, "views": 2048, "answer_pids": [276946], "question_author": null}
{"qid": 1833, "query": "On a long proof", "score": 53, "views": 1944, "answer_pids": [173594, 173595], "question_author": "Richard Terrett"}
{"qid": 1834, "query": "Why do all the Platonic Solids exist?", "score": 53, "views": 1412, "answer_pids": [1252608], "question_author": ""}
{"qid": 1835, "query": "Bijection $f\\colon\\mathbb{N}\\to\\mathbb{N}$ with $f(0)=0$ and $|f(n)-f(n-1)|=n$", "score": 53, "views": 1177, "answer_pids": [1044788, 1046152, 1270598], "question_author": "bobthejoe"}
{"qid": 1836, "query": "How to solve an nth degree polynomial equation", "score": 52, "views": 139795, "answer_pids": [100457, 100458, 100462, 444128, 459939, 557197, 610495, 954719], "question_author": "pikk"}
{"qid": 1837, "query": "Number of onto functions", "score": 52, "views": 138531, "answer_pids": [159165, 159167, 509066, 513456], "question_author": ""}
{"qid": 1838, "query": "What does := mean?", "score": 52, "views": 95235, "answer_pids": [13778, 15172], "question_author": "pikk"}
{"qid": 1839, "query": "Adding two polar vectors", "score": 52, "views": 90214, "answer_pids": [544184, 544263, 544282, 544434, 909641], "question_author": "Mateen Ulhaq"}
{"qid": 1840, "query": "Sum of the alternating harmonic series $\\sum_{k=1}^{\\infty}\\frac{(-1)^{k+1}}{k} = \\frac{1}{1} - \\frac{1}{2} + \\cdots $", "score": 52, "views": 63095, "answer_pids": [423, 424, 427, 22773, 171691, 493322, 725859, 1089706, 1178456, 1408212], "question_author": "Manan"}
{"qid": 1841, "query": "Average Distance Between Random Points on a Line Segment", "score": 52, "views": 54005, "answer_pids": [98354, 98358, 98359, 98374, 557302, 565994, 603389, 784693, 1298805], "question_author": "mring"}
{"qid": 1842, "query": "Understanding Dot and Cross Product", "score": 52, "views": 46794, "answer_pids": [39, 40, 802, 11131], "question_author": "harpalss"}
{"qid": 1843, "query": "What are good books/other readings for elementary set theory?", "score": 52, "views": 45488, "answer_pids": [6356, 6357, 6358, 6360, 6381, 6402], "question_author": "harpalss"}
{"qid": 1844, "query": "Best Algebraic Topology book/Alternative to Allen Hatcher free book?", "score": 52, "views": 42854, "answer_pids": [44115, 45684, 66241, 183082, 387101, 726184, 778482, 829303, 1124574], "question_author": "grokus"}
{"qid": 1845, "query": "What is the implicit function theorem?", "score": 52, "views": 37865, "answer_pids": [14306, 14307, 14326, 84392, 324224], "question_author": "student"}
{"qid": 1846, "query": "What does it mean to integrate with respect to the distribution function?", "score": 52, "views": 37688, "answer_pids": [179248, 179253, 179388], "question_author": "Hans-Peter Stricker"}
{"qid": 1847, "query": "Very good linear algebra book.", "score": 52, "views": 33530, "answer_pids": [346210, 346213, 346216, 346230, 346247, 346289, 346298, 365811, 501625, 933226, 963030, 1118631], "question_author": "student"}
{"qid": 1848, "query": "Can a piece of A4 paper be folded so that its thick enough to reach the moon?", "score": 52, "views": 31964, "answer_pids": [318796, 318797, 318830, 318954, 319096, 319280, 319394], "question_author": ""}
{"qid": 1849, "query": "Every nonzero element in a finite ring is either a unit or a zero divisor", "score": 52, "views": 29804, "answer_pids": [32338, 32339, 32360, 229119, 500734], "question_author": ""}
{"qid": 1850, "query": "Projection is an open map", "score": 52, "views": 26766, "answer_pids": [120174, 409398, 413398, 1313281], "question_author": ""}
{"qid": 1851, "query": "Can someone explain Cayleys Theorem step by step?", "score": 52, "views": 23888, "answer_pids": [174385, 1193018], "question_author": "mivilar"}
{"qid": 1852, "query": "Lemma/Proposition/Theorem, which one should we pick?", "score": 52, "views": 21409, "answer_pids": [14005, 14014, 14027, 407214, 1416474], "question_author": "Shinta"}
{"qid": 1853, "query": "Game theory - self study", "score": 52, "views": 20494, "answer_pids": [40238, 40465, 44348, 44363, 91575, 119458, 190778], "question_author": "Emmie MC"}
{"qid": 1854, "query": "What are the required backgrounds of Robin Hartshornes Algebraic Geometry book?", "score": 52, "views": 17918, "answer_pids": [101597, 105139, 105166, 118712], "question_author": ""}
{"qid": 1855, "query": "Semi-direct v.s. Direct products", "score": 52, "views": 16377, "answer_pids": [54799], "question_author": "gkadam"}
{"qid": 1856, "query": "Why is it that if I count years from 2011 to 2014 as intervals I get 3 years, but if I count each year separately I get 4 years?", "score": 52, "views": 15761, "answer_pids": [373983, 373984, 373985, 374025, 374027, 374034, 374226, 374233, 374339, 374489, 679038], "question_author": ""}
{"qid": 1857, "query": "What exactly is infinity?", "score": 52, "views": 14665, "answer_pids": [126201, 126202, 126203, 126204, 126206, 126210, 126259, 126312, 126721, 196637], "question_author": ""}
{"qid": 1858, "query": "How do we know the ratio between circumference and diameter is the same for all circles?", "score": 52, "views": 12064, "answer_pids": [681848, 681851, 683325, 683637, 1148036], "question_author": "Eric"}
{"qid": 1859, "query": "Good introductory book on geometric algebra", "score": 52, "views": 11843, "answer_pids": [43972, 43973, 43975, 44978, 81458, 162816, 658024, 745689, 1441847], "question_author": "sharptooth"}
{"qid": 1860, "query": "Jacobi identity - intuitive explanation", "score": 52, "views": 11164, "answer_pids": [58453, 58460, 58461, 58463, 400679], "question_author": null}
{"qid": 1861, "query": "Why does the volume of the unit sphere go to zero?", "score": 52, "views": 11040, "answer_pids": [35435, 35436, 35438, 35439, 35520, 220937], "question_author": ""}
{"qid": 1862, "query": "Is it possible to place 26 points inside a rectangle that is 20 cm by 15 cm so that the distance between every pair of points is greater than 5 cm?", "score": 52, "views": 10761, "answer_pids": [873116, 873158, 873964], "question_author": "Vass"}
{"qid": 1863, "query": "What does it mean to solve an equation?", "score": 52, "views": 9575, "answer_pids": [628653, 628657, 628667, 628670, 628678, 628692, 629214], "question_author": ""}
{"qid": 1864, "query": "Isomorphic quotients by isomorphic normal subgroups", "score": 52, "views": 8512, "answer_pids": [21802], "question_author": ""}
{"qid": 1865, "query": "How can this function have two different antiderivatives?", "score": 52, "views": 8363, "answer_pids": [565538, 565539, 565540, 566214, 566364, 587889], "question_author": ""}
{"qid": 1866, "query": "Why arent integration and differentiation inverses of each other?", "score": 52, "views": 7728, "answer_pids": [445639], "question_author": "Bravo"}
{"qid": 1867, "query": "Is 128 the only multi-digit power of 2 such that each of its digits is also a power of 2?", "score": 52, "views": 7548, "answer_pids": [436724, 437117, 437244, 439240], "question_author": ""}
{"qid": 1868, "query": "Do we really need reals?", "score": 52, "views": 7431, "answer_pids": [444157, 444159, 444169, 444189, 444226, 444250, 444372, 444381, 444383, 444979, 445318, 445389], "question_author": ""}
{"qid": 1869, "query": "How can someone reject a math result if everything has to be proved?", "score": 52, "views": 6983, "answer_pids": [1333592, 1333607, 1333672, 1333692, 1333786, 1334010, 1334181], "question_author": "Amu"}
{"qid": 1870, "query": "What are some things we can prove they must exist, but have no idea what they are?", "score": 52, "views": 6895, "answer_pids": [574785, 574793, 574806, 574826, 574840, 574847, 574899, 575112, 575114, 575165, 575183, 575276, 575680, 576251, 576293, 577265, 577976], "question_author": "mring"}
{"qid": 1871, "query": "I dont understand G\u00f6dels incompleteness theorem anymore", "score": 52, "views": 6824, "answer_pids": [1333686, 1333699, 1333844, 1333874, 1333903], "question_author": ""}
{"qid": 1872, "query": "Where to begin with foundations of mathematics", "score": 52, "views": 6595, "answer_pids": [71264, 71265, 71275, 71286, 71490, 71493, 71515, 73700, 252737], "question_author": "J. M. ain't a mathematician"}
{"qid": 1873, "query": "Development of the Idea of the Determinant", "score": 52, "views": 6385, "answer_pids": [55482, 55491], "question_author": "Graviton"}
{"qid": 1874, "query": "Do we need to formally teach the Greek Alphabet?", "score": 52, "views": 6380, "answer_pids": [308909, 308911, 308914, 308989, 309042, 309233, 309333, 309340, 309351, 309455, 309560, 309567, 310021], "question_author": null}
{"qid": 1875, "query": "The Meaning of the Fundamental Theorem of Calculus", "score": 52, "views": 6241, "answer_pids": [158737, 158861, 159827, 161007, 412346], "question_author": "occam98"}
{"qid": 1876, "query": "A conjecture involving prime numbers and circles", "score": 52, "views": 5828, "answer_pids": [1063064, 1063410, 1063714, 1063973], "question_author": "Vass"}
{"qid": 1877, "query": "Is the percentage symbol a constant?", "score": 52, "views": 5755, "answer_pids": [1139603, 1139604, 1139605, 1139622, 1139650, 1139677, 1139801, 1139819, 1140024], "question_author": "Vaughn Climenhaga"}
{"qid": 1878, "query": "Is a proof still valid if only the writer understands it?", "score": 52, "views": 5393, "answer_pids": [346942, 346943, 346948, 347031, 347042, 347086, 347089, 347243, 347384, 347840, 348243, 348273], "question_author": ""}
{"qid": 1879, "query": "Set Theoretic Definition of Numbers", "score": 52, "views": 5219, "answer_pids": [8303, 8312, 8647], "question_author": ""}
{"qid": 1880, "query": "Why cant Antoines necklace fall apart?", "score": 52, "views": 4841, "answer_pids": [1296042, 1296051], "question_author": "Douglas S. Stones"}
{"qid": 1881, "query": "Areas versus volumes of revolution: why does the area require approximation by a cone?", "score": 52, "views": 4599, "answer_pids": [4067, 4070], "question_author": ""}
{"qid": 1882, "query": "Designing an Irrational Numbers Wall Clock", "score": 52, "views": 4263, "answer_pids": [8995, 9002, 9004, 9006, 9011, 9015, 13023, 13025], "question_author": ""}
{"qid": 1883, "query": "Must eigenvalues be numbers?", "score": 52, "views": 4228, "answer_pids": [612507, 612508, 612627, 637503], "question_author": ""}
{"qid": 1884, "query": "How does the divisibility graphs work?", "score": 52, "views": 4210, "answer_pids": [193912, 193918, 193922, 194012], "question_author": "sharptooth"}
{"qid": 1885, "query": "Axiom of Choice: Where does my argument for proving the axiom of choice fail? Help me understand why this is an axiom, and not a theorem.", "score": 52, "views": 4173, "answer_pids": [708460, 708461, 708462, 709008], "question_author": ""}
{"qid": 1886, "query": "The identity cannot be a commutator in a Banach algebra?", "score": 52, "views": 4067, "answer_pids": [28906, 28915], "question_author": ""}
{"qid": 1887, "query": "If the field of a vector space werent characteristic zero, then what would change in the theory?", "score": 52, "views": 3957, "answer_pids": [881563, 881565, 881592, 881598, 881604, 881631, 881711, 881726, 1050281], "question_author": ""}
{"qid": 1888, "query": "What are some math books written in dialogue or story form, e.g., a teacher explaining to a student?", "score": 52, "views": 3784, "answer_pids": [620804, 620829, 620836, 620844, 620846, 620880, 620928, 620946, 621019, 621154, 621236, 621443, 621488, 622081, 625736, 633067, 633069], "question_author": "Jonathan."}
{"qid": 1889, "query": "A zero sum subset of a sum-full set", "score": 52, "views": 3737, "answer_pids": [1482, 1130804], "question_author": ""}
{"qid": 1890, "query": "Why cant we define more elementary functions?", "score": 52, "views": 3719, "answer_pids": [357899, 357903, 357904, 357915, 814003], "question_author": ""}
{"qid": 1891, "query": "A conjectured closed form of $\\int\\limits_0^\\infty\\frac{x-1}{\\sqrt{2^x-1}\\ \\ln\\left(2^x-1\\right)}dx$", "score": 52, "views": 3674, "answer_pids": [240164, 243032, 345617, 792828], "question_author": ""}
{"qid": 1892, "query": "Sheldon Cooper Primes", "score": 52, "views": 3584, "answer_pids": [598260, 1059929, 1194755], "question_author": "Justin Smith"}
{"qid": 1893, "query": "Notations that are mnemonic outside of English", "score": 52, "views": 3226, "answer_pids": [217849, 217850, 217851, 217852, 217854, 217857, 217858, 217860, 217870, 217910, 217949, 217950, 217967, 321781, 376051, 443245, 464772], "question_author": ""}
{"qid": 1894, "query": "Hidden patterns in $\\sin(a x^2)$", "score": 52, "views": 3098, "answer_pids": [847341, 849511], "question_author": "pikk"}
{"qid": 1895, "query": "AM-GM-HM Triplets", "score": 52, "views": 2990, "answer_pids": [48338, 48344, 48464, 89159, 752542], "question_author": ""}
{"qid": 1896, "query": "Why is the Galois Correspondence intuitively plausible?", "score": 52, "views": 2818, "answer_pids": [1440700, 1440721, 1440756], "question_author": "justausr"}
{"qid": 1897, "query": "Generalizing the sum of consecutive cubes $\\sum_{k=1}^n k^3 = \\Big(\\sum_{k=1}^n k\\Big)^2$ to other odd powers", "score": 52, "views": 2489, "answer_pids": [251500, 253237, 254733, 254939, 255196], "question_author": "Casebash"}
{"qid": 1898, "query": "A curious equality of integrals involving the prime counting function?", "score": 52, "views": 2186, "answer_pids": [1141542], "question_author": "dan_waterworth"}
{"qid": 1899, "query": "Xmas Maths 2015", "score": 52, "views": 1660, "answer_pids": [619722, 619723], "question_author": "LanceLafontaine"}
{"qid": 1900, "query": "How to evaluate $\\int_0^\\infty\\operatorname{erfc}^n x\\ \\mathrm dx$?", "score": 52, "views": 1386, "answer_pids": [307456, 307936, 312822], "question_author": ""}
{"qid": 1901, "query": "Time to reach a final state in a random dynamical system (answer known, proof unknown)", "score": 52, "views": 1347, "answer_pids": [789940, 790215, 790239, 790299, 790317], "question_author": "bobthejoe"}
{"qid": 1902, "query": "What numbers can be created by $1-x^2$ and $x/2$?", "score": 52, "views": 1217, "answer_pids": [984253, 984306, 1086921], "question_author": "bobthejoe"}
{"qid": 1903, "query": "Is there a way to get trig functions without a calculator?", "score": 51, "views": 196906, "answer_pids": [232784, 232789, 232793, 521477, 1065653, 1239004, 1293370], "question_author": ""}
{"qid": 1904, "query": "Sample Standard Deviation vs. Population Standard Deviation", "score": 51, "views": 144031, "answer_pids": [8453], "question_author": ""}
{"qid": 1905, "query": "Why does an integral change signs when flipping the boundaries?", "score": 51, "views": 79820, "answer_pids": [526104, 526110, 526113, 526114, 526250, 526251, 526381, 526569, 528300], "question_author": ""}
{"qid": 1906, "query": "Why is the absolute value function not differentiable at $x=0$?", "score": 51, "views": 78548, "answer_pids": [413725], "question_author": "Greg Graviton"}
{"qid": 1907, "query": "Effect of elementary row operations on determinant?", "score": 51, "views": 75010, "answer_pids": [283854], "question_author": "ArchitectofAges"}
{"qid": 1908, "query": "Do all square matrices have eigenvectors?", "score": 51, "views": 72764, "answer_pids": [291141, 291142, 291143, 291244, 291264, 291311, 291712, 375084, 682269], "question_author": "Jonathan."}
{"qid": 1909, "query": "Is zero a prime number?", "score": 51, "views": 55393, "answer_pids": [247230, 247235, 247240, 247374, 249172, 417507, 574035], "question_author": "bobthejoe"}
{"qid": 1910, "query": "How to find a basis for the intersection of two vector spaces in $\\mathbb{R}^n$?", "score": 51, "views": 53643, "answer_pids": [13859, 13860, 13861, 703929, 1202421, 1262591], "question_author": "Andrei"}
{"qid": 1911, "query": "Subgroup of index $2$ is Normal", "score": 51, "views": 48973, "answer_pids": [44220, 1134480, 1182774, 1375528], "question_author": "Rory M"}
{"qid": 1912, "query": "The generating function for the Fibonacci numbers", "score": 51, "views": 44536, "answer_pids": [161140, 161143, 161145, 161528], "question_author": ""}
{"qid": 1913, "query": "Examples of infinite groups such that all their respective elements are of finite order.", "score": 51, "views": 39974, "answer_pids": [78506, 78507, 78508, 78509, 78510, 78534, 78710, 204991, 381812, 464918, 760121], "question_author": "yohay kaplan"}
{"qid": 1914, "query": "Why are $3D$ transformation matrices $4 \\times 4$ instead of $3 \\times 3$?", "score": 51, "views": 32134, "answer_pids": [206, 208, 210], "question_author": "amal antony"}
{"qid": 1915, "query": "Show that $\\langle 2,x \\rangle$ is not a principal ideal in $\\mathbb Z [x]$", "score": 51, "views": 31809, "answer_pids": [19335, 19336, 19338, 19376, 20726, 146359, 412297, 1427842], "question_author": "Sklivvz"}
{"qid": 1916, "query": "Every linear mapping on a finite dimensional space is continuous", "score": 51, "views": 29634, "answer_pids": [58060, 356845, 924393], "question_author": "Marek"}
{"qid": 1917, "query": "difference between maximal element and greatest element", "score": 51, "views": 28953, "answer_pids": [23847, 23848, 23849, 23850, 23857, 76153, 87168], "question_author": "Ricardo Altamirano"}
{"qid": 1918, "query": "How can a set contain itself?", "score": 51, "views": 24401, "answer_pids": [432936, 432947, 433704, 591588, 905887], "question_author": "arsenm"}
{"qid": 1919, "query": "Sum of two closed sets in $\\mathbb R$ is closed?", "score": 51, "views": 21466, "answer_pids": [63469, 67169, 101541, 945684, 1384571], "question_author": "Jack Schmidt"}
{"qid": 1920, "query": "How unique are $U$ and $V$ in the Singular Value Decomposition?", "score": 51, "views": 21407, "answer_pids": [287360, 1215549, 1455376], "question_author": "Anna"}
{"qid": 1921, "query": "Are normal subgroups transitive?", "score": 51, "views": 20865, "answer_pids": [123708, 123725, 1083731], "question_author": ""}
{"qid": 1922, "query": "Why is the ring of matrices over a field simple?", "score": 51, "views": 18098, "answer_pids": [12435, 24009], "question_author": "Robert Whitley"}
{"qid": 1923, "query": "In set theory, how are real numbers represented as sets?", "score": 51, "views": 16386, "answer_pids": [33319, 33321, 33324, 33327, 33345, 33674], "question_author": "Steven Jeuris"}
{"qid": 1924, "query": "Is it possible to plot a graph of any shape?", "score": 51, "views": 15387, "answer_pids": [566691, 566694, 566695, 566859, 567218], "question_author": ""}
{"qid": 1925, "query": "Proving $\\frac{\\sin x}{x} =\\left(1-\\frac{x^2}{\\pi^2}\\right)\\left(1-\\frac{x^2}{2^2\\pi^2}\\right) \\left(1-\\frac{x^2}{3^2\\pi^2}\\right)\\cdots$", "score": 51, "views": 11755, "answer_pids": [79455, 739456], "question_author": ""}
{"qid": 1926, "query": "Subgroups of finitely generated groups are not necessarily finitely generated", "score": 51, "views": 11435, "answer_pids": [4571, 4572, 12176, 147078, 336523], "question_author": null}
{"qid": 1927, "query": "$|G|>2$ implies $G$ has non trivial automorphism", "score": 51, "views": 10718, "answer_pids": [4835, 4836, 1067640], "question_author": "Aslan986"}
{"qid": 1928, "query": "There exists a power of 2 such that the last five digits are all 3s or 6s. Find the last 5 digits of this number", "score": 51, "views": 10642, "answer_pids": [282091, 282092, 282096, 282205, 282225, 282254], "question_author": "harpalss"}
{"qid": 1929, "query": "Does a Fourier transformation on a (pseudo-)Riemannian manifold make sense?", "score": 51, "views": 10622, "answer_pids": [7852, 1290326], "question_author": ""}
{"qid": 1930, "query": "$\\sin 1^\\circ$ is irrational but how do I prove it in a slick way? And $\\tan(1^\\circ)$ is .....", "score": 51, "views": 10283, "answer_pids": [49068, 49069, 49070, 49132, 49148, 49215], "question_author": ""}
{"qid": 1931, "query": "How many fair dice exist?", "score": 51, "views": 9793, "answer_pids": [960571, 960573, 960576, 960617, 960651, 960745, 960956, 961380, 961492], "question_author": ""}
{"qid": 1932, "query": "How to start a math blog?", "score": 51, "views": 9454, "answer_pids": [163075, 429245], "question_author": "Charles Stewart"}
{"qid": 1933, "query": "Modus moron rule of inference?", "score": 51, "views": 9372, "answer_pids": [824865, 824866, 824918, 824969, 825004, 825188, 825357, 884890, 931446], "question_author": null}
{"qid": 1934, "query": "precise official definition of a cell complex and CW-complex", "score": 51, "views": 9082, "answer_pids": [22406], "question_author": ""}
{"qid": 1935, "query": "Alternative proofs that $A_5$ is simple", "score": 51, "views": 8697, "answer_pids": [156663, 156830, 158048, 928810, 1416573], "question_author": null}
{"qid": 1936, "query": "Can you give me some concrete examples of magmas?", "score": 51, "views": 8451, "answer_pids": [337282, 337284, 337295, 337437, 337787, 341083, 881625, 883609, 1298512], "question_author": ""}
{"qid": 1937, "query": "Does Monty Hall logic apply to this real world situation?", "score": 51, "views": 7887, "answer_pids": [1032251, 1032270, 1032315, 1032343, 1032432, 1032516], "question_author": "719016"}
{"qid": 1938, "query": "Are there dictionaries in math?", "score": 51, "views": 7298, "answer_pids": [720563, 720629, 720637, 720654, 720860], "question_author": "Kirby"}
{"qid": 1939, "query": "How did early mathematicians make it without Set theory?", "score": 51, "views": 6913, "answer_pids": [765337, 765342, 765353, 765370, 765394, 765629, 765821], "question_author": "Evan Carroll"}
{"qid": 1940, "query": "Sharing a pepperoni pizza with your worst enemy", "score": 51, "views": 6666, "answer_pids": [635155, 635242, 636047], "question_author": "tel"}
{"qid": 1941, "query": "Is it wrong to tell children that $1/0 =$ NaN is incorrect, and should be $\u221e$?", "score": 51, "views": 6184, "answer_pids": [73133, 73135, 73159, 73160, 73170, 73178, 73191, 73221, 73259, 313250], "question_author": ""}
{"qid": 1942, "query": "Why is the Daniell integral not so popular?", "score": 51, "views": 6042, "answer_pids": [88683, 88705, 88762], "question_author": "Cam"}
{"qid": 1943, "query": "Say $a=b$. Is Do the same thing to both sides of an equation, and it still holds an axiom?", "score": 51, "views": 5744, "answer_pids": [1036603, 1036612, 1036705, 1036722, 1036770], "question_author": "tel"}
{"qid": 1944, "query": "Why is there antagonism towards extended real numbers?", "score": 51, "views": 5371, "answer_pids": [303468, 303482, 303493, 303636, 303707], "question_author": null}
{"qid": 1945, "query": "What is a topological space good for?", "score": 51, "views": 4950, "answer_pids": [692091, 692205, 692229, 692245, 692886, 924981], "question_author": ""}
{"qid": 1946, "query": "Are rational points dense on every circle in the coordinate plane?", "score": 51, "views": 4931, "answer_pids": [1089603, 1089636, 1090135, 1091580], "question_author": null}
{"qid": 1947, "query": "What do mathematicians mean by equipped?", "score": 51, "views": 4697, "answer_pids": [455402, 455404, 455405, 455503, 455764, 456197], "question_author": null}
{"qid": 1948, "query": "Category-theoretic limit related to topological limit?", "score": 51, "views": 4643, "answer_pids": [33285, 158432], "question_author": null}
{"qid": 1949, "query": "Notation for an interval when you dont know which bound is greater", "score": 51, "views": 4519, "answer_pids": [1037834, 1037840, 1037849, 1037864, 1037870, 1037882, 1037889, 1037890, 1037937, 1038254, 1038255, 1038471, 1038901], "question_author": "Martin Gales"}
{"qid": 1950, "query": "Why can we use induction when studying metamathematics?", "score": 51, "views": 4497, "answer_pids": [894548, 894635, 894749, 896701], "question_author": ""}
{"qid": 1951, "query": "Finding the value of $\\sqrt{1+2\\sqrt{2+3\\sqrt{3+4\\sqrt{4+5\\sqrt{5+\\dots}}}}}$", "score": 51, "views": 4328, "answer_pids": [204075, 204082, 204090, 204183, 206550], "question_author": "gkadam"}
{"qid": 1952, "query": "What if $\\pi$ was an algebraic number? (significance of algebraic numbers)", "score": 51, "views": 3973, "answer_pids": [218781, 218782, 218784, 218786, 218787, 218801, 218811, 218814, 218853, 220480, 239047], "question_author": "Malabarba"}
{"qid": 1953, "query": "Solution to the equation of a polynomial raised to the power of a polynomial.", "score": 51, "views": 3864, "answer_pids": [1082158, 1082167, 1082339, 1082548], "question_author": "Mikola"}
{"qid": 1954, "query": "Are commutative C*-algebras really dual to locally compact Hausdorff spaces?", "score": 51, "views": 3841, "answer_pids": [97739, 104621, 112587, 244849, 514352], "question_author": ""}
{"qid": 1955, "query": "Is $dx\\,dy$ really a multiplication of $dx$ and $dy$?", "score": 51, "views": 3832, "answer_pids": [309341, 310158, 310257, 310280, 374766, 393123, 394552], "question_author": "yahoo301503"}
{"qid": 1956, "query": "Proving $\\sum_{n=-\\infty}^\\infty e^{-\\pi n^2} = \\frac{\\sqrt[4] \\pi}{\\Gamma\\left(\\frac 3 4\\right)}$", "score": 51, "views": 3553, "answer_pids": [231310, 693354, 693377], "question_author": null}
{"qid": 1957, "query": "How do you prove Gautschis inequality for the gamma function?", "score": 51, "views": 3194, "answer_pids": [53178, 796057, 949524], "question_author": "Xenon"}
{"qid": 1958, "query": "Function that is the sum of all of its derivatives", "score": 51, "views": 2951, "answer_pids": [639957, 640081, 645433], "question_author": "Joy"}
{"qid": 1959, "query": "Identity for simple 1D random walk", "score": 51, "views": 2909, "answer_pids": [282477], "question_author": ""}
{"qid": 1960, "query": "Prove $\\int\\limits_{0}^{\\pi/2}\\frac{dx}{1+\\sin^2{(\\tan{x})}}=\\frac{\\pi}{2\\sqrt{2}}\\bigl(\\frac{e^2+3-2\\sqrt{2}}{e^2-3+2\\sqrt{2}}\\bigr)$", "score": 51, "views": 2859, "answer_pids": [319519, 446529, 490747], "question_author": "InquilineKea"}
{"qid": 1961, "query": "How to maintain enthusiasm and joy in teaching when the material grows stale", "score": 51, "views": 2850, "answer_pids": [285622, 285624, 285625, 285682, 285755, 285794, 288252, 294644], "question_author": "Pwninstein"}
{"qid": 1962, "query": "An exotic sequence", "score": 51, "views": 2740, "answer_pids": [310362, 310384, 310407, 310529, 313727], "question_author": ""}
{"qid": 1963, "query": "Making trigonometric substitutions rigorous", "score": 51, "views": 2732, "answer_pids": [32067, 32082, 394438], "question_author": ""}
{"qid": 1964, "query": "A new kind of fractal?", "score": 51, "views": 2501, "answer_pids": [106591, 106616], "question_author": "J. Musser"}
{"qid": 1965, "query": "How do you go about doing mathematics on a day to day basis?", "score": 51, "views": 2484, "answer_pids": [5070, 313617], "question_author": "Joy"}
{"qid": 1966, "query": "$C^{k}$-manifolds: how and why?", "score": 51, "views": 2403, "answer_pids": [418746], "question_author": "Joy"}
{"qid": 1967, "query": "Why do I only breathe out of one nostril?", "score": 295, "views": 280646, "answer_pids": [1675171, 1675172, 1675175], "question_author": ""}
{"qid": 1968, "query": "Why are so few foods blue?", "score": 201, "views": 31923, "answer_pids": [1683833, 1683840, 1683841], "question_author": "Ben"}
{"qid": 1969, "query": "Does DNA have the equivalent of IF-statements, WHILE loops, or function calls? How about GOTO?", "score": 159, "views": 17808, "answer_pids": [1677061, 1677062, 1677066, 1677067, 1677072, 1682779], "question_author": ""}
{"qid": 1970, "query": "How many times did terrestrial life emerge from the ocean?", "score": 153, "views": 6806, "answer_pids": [1668665, 1669401], "question_author": "Gabriel Fair"}
{"qid": 1971, "query": "Do bacteria die of old age?", "score": 134, "views": 16688, "answer_pids": [1675131], "question_author": null}
{"qid": 1972, "query": "Is there a reason why human eyesight and plants make use of the same wavelength of light?", "score": 121, "views": 5827, "answer_pids": [1668475, 1668690, 1674252, 1674741], "question_author": "Faheem Mitha"}
{"qid": 1973, "query": "Why is thymine rather than uracil used in DNA?", "score": 116, "views": 115502, "answer_pids": [1668322, 1668323, 1669880], "question_author": ""}
{"qid": 1974, "query": "Are male and female brains physically different from birth?", "score": 108, "views": 24657, "answer_pids": [1686793, 1686796, 1686805], "question_author": ""}
{"qid": 1975, "query": "What is the effect of non-vaccinated people on vaccinated people?", "score": 108, "views": 10902, "answer_pids": [1688597, 1688600, 1688620, 1690802], "question_author": "David LeBauer"}
{"qid": 1976, "query": "Why do plants have green leaves and not red?", "score": 107, "views": 20115, "answer_pids": [1668447, 1668453, 1668496, 1670017, 1674078, 1678111, 1683493, 1683926], "question_author": ""}
{"qid": 1977, "query": "Can HIV be transmitted via mosquitos?", "score": 102, "views": 12599, "answer_pids": [1681060, 1681101], "question_author": "Gordon Gustafson"}
{"qid": 1978, "query": "Whats the evidence against SARS-CoV-2 being engineered by humans?", "score": 98, "views": 16224, "answer_pids": [1691807, 1691831, 1692330], "question_author": ""}
{"qid": 1979, "query": "Do beneficial viruses exist? If so, what examples are there?", "score": 97, "views": 22265, "answer_pids": [1688536, 1688538, 1688540, 1688544, 1688545], "question_author": ""}
{"qid": 1980, "query": "Does the string ...CATCAT... appear in the DNA of Felis catus?", "score": 94, "views": 16059, "answer_pids": [1682659, 1682662, 1682666, 1684697], "question_author": "yoyostein"}
{"qid": 1981, "query": "Why did the process of sleep evolve in many animals? What is its evolutionary advantage?", "score": 91, "views": 6640, "answer_pids": [1668392, 1668393, 1668620, 1691132], "question_author": "cpx"}
{"qid": 1982, "query": "Are humans the only species who drink milk as adults?", "score": 89, "views": 30554, "answer_pids": [1685874, 1685876, 1685880, 1685883, 1685888, 1685907], "question_author": "Alex Stone"}
{"qid": 1983, "query": "How could humans have interbred with Neanderthals if were a different species?", "score": 86, "views": 29787, "answer_pids": [1679418, 1679419, 1679757, 1683266], "question_author": null}
{"qid": 1984, "query": "Why do some bad traits evolve, and good ones dont?", "score": 85, "views": 19438, "answer_pids": [1678380, 1678381, 1678384, 1678388, 1678431, 1684613, 1686248, 1686249], "question_author": "SMUsamaShah"}
{"qid": 1985, "query": "Death because of distilled water consumption", "score": 83, "views": 125155, "answer_pids": [1669263, 1670092, 1670603, 1674428], "question_author": "Lucky"}
{"qid": 1986, "query": "Why is the heart not in the middle of the body?", "score": 82, "views": 27436, "answer_pids": [1685214, 1685292], "question_author": ""}
{"qid": 1987, "query": "Why 20 amino acids instead of 64?", "score": 82, "views": 19958, "answer_pids": [1668533, 1668534, 1668542], "question_author": "Alex Stone"}
{"qid": 1988, "query": "Is there an RGB equivalent for smells?", "score": 82, "views": 14699, "answer_pids": [1683514, 1683519, 1683529], "question_author": "Alex Stone"}
{"qid": 1989, "query": "Whats up with this leaf?", "score": 81, "views": 8111, "answer_pids": [1690547, 1690556], "question_author": "cpx"}
{"qid": 1990, "query": "How does the brains energy consumption depend on mental activity?", "score": 75, "views": 15982, "answer_pids": [1668621, 1668624, 1669830], "question_author": ""}
{"qid": 1991, "query": "Do animals exhibit handedness (paw-ness?) preference?", "score": 75, "views": 5906, "answer_pids": [1684150, 1684158], "question_author": ""}
{"qid": 1992, "query": "Why do Humans not produce Vitamin C like other mammals?", "score": 74, "views": 53256, "answer_pids": [1668427], "question_author": ""}
{"qid": 1993, "query": "Why shouldnt dogs eat chocolate?", "score": 74, "views": 14273, "answer_pids": [1685186, 1685189, 1685190], "question_author": ""}
{"qid": 1994, "query": "Why are there no wheeled animals?", "score": 71, "views": 8268, "answer_pids": [1670917, 1670918, 1670919, 1677377], "question_author": "LanceLafontaine"}
{"qid": 1995, "query": "What is the evolutionary advantage of red-green color blindness?", "score": 67, "views": 22321, "answer_pids": [1680821, 1680823], "question_author": "Gergana Vandova"}
{"qid": 1996, "query": "Do ants or other insects sleep, and if so why?", "score": 65, "views": 9168, "answer_pids": [1675147, 1675150], "question_author": ""}
{"qid": 1997, "query": "What could cause a forest of bent trees?", "score": 65, "views": 6198, "answer_pids": [1678421], "question_author": ""}
{"qid": 1998, "query": "Is there any evidence that sexual selection may lead to extinction of species?", "score": 65, "views": 3711, "answer_pids": [1668348, 1671193], "question_author": "unsym"}
{"qid": 1999, "query": "Why does evolution not make our life longer?", "score": 64, "views": 11014, "answer_pids": [1673758, 1673763, 1673764, 1673766, 1673767, 1673768, 1673773, 1681363], "question_author": "Gianpaolo R"}
{"qid": 2000, "query": "Why does it hurt the next day after doing significant exercise?", "score": 64, "views": 7717, "answer_pids": [1685305, 1685308], "question_author": "Daisy Sophia Hollman"}
{"qid": 2001, "query": "Can a woman give birth to twins with different fathers?", "score": 63, "views": 10627, "answer_pids": [1675762, 1675779], "question_author": "Boy S"}
{"qid": 2002, "query": "Are there organisms with fewer than 1000 neurons?", "score": 63, "views": 8455, "answer_pids": [1684216, 1684217, 1684220, 1684236], "question_author": "J. Musser"}
{"qid": 2003, "query": "How come large herbivores have such thin legs?", "score": 61, "views": 15324, "answer_pids": [1686364, 1686368, 1686371], "question_author": ""}
{"qid": 2004, "query": "A new species of small bird?", "score": 61, "views": 6699, "answer_pids": [1682103], "question_author": ""}
{"qid": 2005, "query": "Why dont mammals have more than 4 limbs?", "score": 60, "views": 53159, "answer_pids": [1675391, 1675394, 1675398, 1675415], "question_author": ""}
{"qid": 2006, "query": "Why do smaller mammals move intermittently?", "score": 60, "views": 9604, "answer_pids": [1678033, 1678043, 1678060], "question_author": "Fomite"}
{"qid": 2007, "query": "Are the social-distancing measures implemented against SARS-CoV-2 also suppressing the spread of other viruses?", "score": 59, "views": 11848, "answer_pids": [1692337, 1692343], "question_author": ""}
{"qid": 2008, "query": "Why do the fastest runners tend to be black?", "score": 58, "views": 38219, "answer_pids": [1682359, 1682361, 1682364], "question_author": ""}
{"qid": 2009, "query": "Life without DNA?", "score": 56, "views": 11482, "answer_pids": [1668290, 1668291, 1668294, 1668298, 1677038], "question_author": "A T"}
{"qid": 2010, "query": "What is the benefit of fever during infections?", "score": 56, "views": 11040, "answer_pids": [1684347, 1684371], "question_author": "Justin L."}
{"qid": 2011, "query": "Why isnt a virus alive?", "score": 55, "views": 10197, "answer_pids": [1671520, 1671525, 1671526, 1671530, 1671533, 1671545], "question_author": "Chris K"}
{"qid": 2012, "query": "Why do men have nipples?", "score": 55, "views": 7916, "answer_pids": [1669214, 1669343, 1676770, 1679454], "question_author": ""}
{"qid": 2013, "query": "Why are there no organisms with metal body parts, like weapons, bones, and armour? (Or are there?)", "score": 54, "views": 14225, "answer_pids": [1671512, 1671515, 1671522, 1677113, 1678280, 1680845, 1683486, 1683735, 1685113, 1690609], "question_author": "Javier"}
{"qid": 2014, "query": "Human perception of time depending on age", "score": 54, "views": 2558, "answer_pids": [1668556, 1668809, 1668834, 1669539], "question_author": null}
{"qid": 2015, "query": "Do large animals experience a meaningful delay when moving their most distant appendages?", "score": 52, "views": 6447, "answer_pids": [1689903], "question_author": "Robert Filter"}
{"qid": 2016, "query": "Why does cracking a joint make noise?", "score": 52, "views": 4033, "answer_pids": [1674526, 1674530, 1677617], "question_author": null}