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mmiguel

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Everything posted by mmiguel

  1. I was attempting to recreate a problem from an EE final I had 1.5 years ago based on memory, but I realized after I submitted that I had some minor problems in the statement. One which causes the actual calculation to be slightly more difficult than I anticipated (harder integral), and now.... units. I would argue that the differential volume elements have shape and are objects, but this would not be helpful, because integration involves more than just summing stuff together - it also involves a change in units due to multiplication of differential elements of whatever units belong to the variable of integration. Thus if you intend to use something as part of an integrand, such that after integration it becomes what you originally had, what's in the integrand is actually a density of the original quantity vs. being portions of the original quantity. In essence, bonanova is correct.
  2. (1) The Former (2) Pulling out any element from any list is one operation (including the main list). For sorting, I didn't really specify a way of identifying competitors in order to sort. Say for example that, you uniquely assign a number from 0 to N-1 for each competitor. In the main list, you can access any element in constant time (O(1)). You can say the xth slot corresponds to competitor x. For the sublists, you can assume whatever sorting you want as long as you consistently assume it is the same way the whole algorithm (unless of course you spend operations to change it in some way).
  3. A number of bats are in a cave. • 4563 bats can see out of their left eye. • 3645 bats can see out of the right eye. • At least 5436 bats cannot see out of their left eye. • At least 6354 bats cannot see out of their right eye. What is the smallest number of bats that can be in the cave?
  4. I generated the 1000 numbers in the following manner: While I have less than 1000 numbers{ Pick a random whole number between 1 and 100000 If I don't already have this number, add it to my list } In this case, the longest increasing subsequence was length 55. Challenge: Can you determine the probability distribution of the length of longest increasing subsequences for M distinct numbers ranging from 1 to N (generated in the way described above)? (Cumulative or probability mass function would be good). I have no idea what the answer to this might look like, but I'm curious if it would turn out to be a common distribution, or perhaps a very uncommon one. If we can't come up with a good theory, we can get some empirical data. Perhaps from empirical data, we could come up with a theory!!
  5. I don't think that a paradox defies any law that governs our universe. There are no physical laws that say that statements we make or thoughts we conceive are true or false. There is no physical law saying that a statement has to be either true or false. A paradox does not defy anything greater than itself --- it defies itself. i.e. it is a contradiciton. It implies something, then implies something else that cannot be true if the first thing it implied were true. The result is either the first thing is true, or the second thing is, but we don't know which one, and due to symmetry in the way this was asserted, we can make no meaningful conclusion. Example: A number, x, is equal to one, and it is equal to zero. This is a paradox, but it is not an interesting one, since it is obvious where the self-inconsistency is. Most likely, instead of thinking of this paradox as a universe destroying statement of pure contradiction, we would write it off as nonsense. What attracts people to other paradoxes is that they are less obvious, the implications sound more familiar, and practical thinking builds up the confidence of the person thinking about the paradox, confidence that they understand what is going on. Then at the end, they come across an inconsistency, and are not sure what to believe. There's my two cents.
  6. I have spent some time thinking about this idea before. Essentially, the belief that everything in the future is determined by the state of the universe at any point in the past is called determinism. This is a philosophical debate that no one can really prove or disprove. I believe in determinism, but if you think about it, it shouldn't impact how you feel about the universe and carrying out your life. It is different than the Greek concept of destiny. The reason being, that no one can possibly predict the future without knowing everything at one point in time. No physical creature could ever know everything at one point in time, because they would need a physical body to support the mental processes going on to model the external world --- and it would not be possible for them to model all of the information contained within such a physical body by storing that information in the same physical body. Therefore, no one can tell you it is your destiny to fail or succeed or blah blah blah. Another thing, is that the phrase "free will" refers to multiple things. Under determinism, you can still make decisions, it's just that any decision you make is part of the unfolding of the future, and you made them for the deterministic reasons that influenced your mind, which may have been pre-determined, but so what. What you cannot do, is make a decision which was not caused by anything. You cannot do something with no motive, no impetus, no influencing factor to make you do that. Essentially, you cannot be truly random. Does that prevent you from making decisions you actually care about? No. You can still choose to vote however you want, to buy what you want, to act like you want etc. All you should realize is that there are reasons for you doing those things, be it derived from the way you were raised, your experiences in life, the culture around where you were born, etc. That isn't so depressing, and in many senses of the word "free will", you still have it. You are free to be what you are, even if what you are is a deterministic, evolving process of incredible complexity (just like all life). Think about it, even these thoughts, and the conclusions you draw from them are part of that evolving process, and your conclusions may influence future decisions. I think that this was determined at the start of the universe, but unknowable by any part of the universe until now. Essentially, I think that believing in determinism means that randomness doesn't really exist. What we mistake for randomness, is actually just complexity. Both of them are explanations for unpredictability, but I think randomness is more of a superstitious one (and false). Some folks might say quantum mechanics and stuff proves randomness, but I'm not really convinced - all it shows is that a model that ideally assumes randomness is consistent with experimentation to a high degree of accuracy. It doesn't mean that an explanation that unveils a further degree of complexity in a deterministic way does not exist. So yeah.
  7. p is radius from x axis R(p) is resistance in ohms as a function of radius from x-axis
  8. So which treatment deserves further investment by the charities supporting people with this disease?
  9. Nice! That is a different one than what I had in mind, but it is the same length. Did you use a general purpose algorithm?
  10. You are a doctor in charge of a large hospital, and you have to decide which treatment should be used for a particular disease. You have the following data from last month: there were 390 patients with the disease. Treatment A was given to 160 patients of whom 100 were men and 60 were women; 20 of the men and 40 of the women recovered. Treatment B was given to 230 patients of whom 210 were men and 20 were women; 50 of the men and 15 of the women recovered. A.) Which treatment is better for men? B.) Which treatment is better for women? C.) Which treatment is better for any person, regardless of gender?
  11. A giant jelly weighs 100 pounds and is 99% water by weight. It is set out in the sun and water evaporates until it is 98% water by weight. How much does the jelly now weigh?
  12. This may need the help of a computer. Find the longest increasing subsequence of the following 1000 distinct integers. For a sequence of distinct integers, x_1, x_2, ... x_n, A subsequence of length k is an ordered group of integers from the above sequence x_i_1, x_i_2, ... x_i_k, which retains order from original sequence, i.e. i_1 < i_2 < ... < i_k. Note this does not require anything to be consecutive. An increasing subsequence is a subsequence that increases i.e. x_i_1 < x_i_2 < ... < x_i_k. The longest increasing subsequence is the increasing subsequence with the longest length. Good luck! 75004, 56991, 47082, 19635, 42, 89416, 84822, 18366, 8364, 40699, 23120, 7706, 33866, 37458, 86632, 81435, 52136, 3743, 99783, 54643, 62785, 53908, 22213, 85048, 58919, 16044, 71291, 11096, 59245, 56845, 74161, 56308, 23503, 19719, 29656, 86735, 44249, 50773, 14885, 34940, 35646, 3160, 60678, 41299, 59937, 63207, 29450, 89997, 58911, 72711, 64902, 68401, 16899, 4501, 1318, 39683, 66480, 38689, 35095, 80701, 44054, 92075, 33988, 84671, 77537, 8839, 79523, 97433, 8391, 92759, 24585, 89033, 92547, 96080, 26073, 25745, 87630, 73072, 3266, 93177, 93474, 28759, 41164, 2305, 13755, 75280, 2259, 79239, 15564, 27736, 54375, 46184, 3033, 68422, 206, 83226, 8360, 78954, 17911, 37711, 92022, 39688, 66587, 16693, 2339, 92043, 58894, 94182, 93325, 55418, 59121, 57753, 83252, 53228, 68183, 23544, 89165, 57872, 57644, 24952, 22655, 23713, 61634, 68070, 58226, 29503, 71317, 70499, 61070, 9764, 62770, 13522, 36014, 65168, 20298, 43841, 34994, 25063, 86785, 52194, 70311, 81444, 11232, 88436, 83271, 50301, 18761, 29847, 6141, 57699, 61747, 65283, 79193, 80955, 36628, 70822, 66120, 77500, 80812, 18704, 24394, 62520, 15265, 68518, 57144, 69129, 89394, 63356, 5355, 78864, 43417, 22656, 27494, 77278, 19023, 96999, 73467, 68113, 76376, 70153, 95341, 26452, 60308, 25357, 808, 87635, 11781, 92324, 9737, 5709, 2360, 9393, 26000, 5525, 66538, 46059, 11299, 68012, 94219, 20545, 49775, 12230, 38787, 42187, 13954, 79224, 38397, 12641, 18491, 27646, 60193, 43950, 80949, 90488, 93327, 29476, 69194, 83723, 18315, 9124, 82261, 56942, 49866, 14000, 90370, 3396, 77977, 85948, 68664, 60297, 84844, 53267, 67547, 54998, 82753, 3496, 38235, 8761, 3958, 69268, 69156, 24947, 78750, 46293, 23810, 52238, 19946, 44903, 81118, 76939, 82285, 84902, 74667, 26038, 63344, 26064, 51882, 62396, 92332, 71340, 57031, 68813, 21653, 11169, 45501, 3257, 76192, 97946, 20814, 35897, 14289, 91788, 71387, 44558, 83180, 2647, 93610, 96951, 47531, 27440, 47218, 11286, 52915, 93140, 3246, 3433, 39289, 33040, 87405, 79325, 71823, 50094, 30705, 17990, 85064, 81540, 59265, 27807, 62779, 55603, 18992, 59418, 23469, 88774, 23189, 45086, 60996, 10533, 33384, 31671, 42096, 39391, 96782, 84041, 66203, 25541, 37887, 97948, 42245, 93682, 8423, 70414, 53152, 69561, 66511, 28690, 73268, 45783, 59991, 43532, 79262, 3537, 11824, 61333, 94830, 64856, 53646, 59199, 89931, 11691, 86182, 91830, 67544, 51331, 77435, 68866, 44869, 44133, 87167, 22813, 59897, 18498, 46753, 56559, 25556, 86163, 7, 99422, 92240, 88075, 36062, 94790, 63981, 31121, 93567, 21313, 18638, 70913, 10363, 81772, 89573, 19218, 82300, 67325, 68429, 94999, 47454, 271, 78843, 7864, 37438, 13609, 43965, 80713, 56972, 48789, 8213, 52201, 27, 99917, 90747, 85131, 85191, 88148, 68718, 21861, 48539, 76163, 61870, 72247, 6328, 77510, 85161, 25671, 72616, 1494, 69915, 6554, 545, 95768, 37429, 66219, 86730, 47812, 69958, 15057, 95984, 82951, 85727, 4871, 49055, 2670, 87984, 52839, 1390, 24351, 34921, 28927, 8073, 4050, 70171, 85043, 3888, 84466, 27613, 52344, 81876, 41264, 75722, 13727, 78878, 2151, 31297, 19708, 58176, 25846, 47132, 58684, 15779, 43179, 96985, 94527, 50770, 48819, 64608, 51588, 4126, 2349, 11697, 40969, 23669, 18253, 33731, 48139, 88237, 70099, 27141, 55890, 61625, 71161, 73823, 7413, 84892, 8336, 20848, 68198, 44231, 67412, 52152, 99178, 42452, 35439, 63439, 90645, 99836, 39082, 20436, 10443, 7534, 7308, 56095, 78778, 80929, 50848, 4734, 81855, 71427, 49395, 25020, 66462, 13295, 24367, 24077, 29723, 79883, 95739, 51912, 8202, 55787, 71631, 90593, 7549, 52322, 28105, 75281, 94227, 85491, 86745, 14201, 71018, 33535, 35597, 93171, 16611, 39463, 32185, 3766, 41114, 10756, 89390, 48987, 19558, 78031, 61062, 99330, 66976, 51138, 96108, 4417, 13389, 19419, 9359, 97062, 26579, 72817, 12401, 83340, 46928, 20740, 54991, 3744, 28769, 64999, 9641, 17251, 1117, 4656, 37723, 38410, 68597, 72258, 19024, 52105, 75443, 67881, 41368, 39165, 42547, 82381, 9538, 40546, 7034, 56061, 26456, 50629, 53180, 1819, 22409, 3833, 11855, 58669, 27778, 10663, 79657, 38833, 12714, 46197, 21632, 10778, 21247, 38994, 25892, 44301, 55788, 3232, 91229, 75892, 50093, 47105, 59594, 32454, 18488, 25106, 43564, 97242, 28572, 25663, 70039, 91706, 97061, 23416, 29455, 47052, 10566, 12143, 26216, 76453, 29371, 52863, 67569, 58398, 19473, 80787, 57301, 6821, 19269, 14382, 85683, 55428, 1711, 87501, 13629, 33064, 69783, 77117, 43023, 64628, 20502, 93203, 82528, 74892, 26203, 88761, 88038, 46496, 35584, 94159, 64423, 61958, 66594, 53304, 10172, 76705, 56290, 35347, 53466, 15360, 47748, 76041, 32612, 56078, 54272, 78187, 39832, 9834, 16764, 65969, 20091, 66682, 98199, 86121, 8412, 33017, 61921, 63269, 51421, 82766, 39295, 81130, 5731, 46723, 38277, 46094, 45392, 26417, 77154, 92349, 3104, 28788, 97644, 86701, 77673, 93956, 10113, 6253, 46268, 75770, 77218, 7318, 82262, 43967, 64407, 78133, 46480, 12561, 22850, 16789, 16129, 66381, 23275, 86098, 78141, 96486, 22257, 46046, 43321, 81526, 39600, 70788, 75764, 31406, 9363, 94082, 68919, 73381, 12959, 67194, 42059, 95390, 56476, 91971, 47480, 7679, 951, 47318, 31896, 79076, 40517, 83064, 45687, 76070, 98892, 75615, 30432, 44716, 42029, 92085, 11992, 47877, 31085, 48856, 62469, 25805, 58986, 41969, 66607, 9650, 19730, 60892, 1891, 45233, 64014, 27531, 50628, 58999, 25691, 22573, 18814, 56087, 16281, 43278, 70701, 88716, 72662, 8998, 37160, 57163, 53597, 75790, 47438, 13103, 18399, 16465, 85715, 84749, 36760, 59481, 68031, 73626, 39941, 99556, 48286, 631, 22187, 20672, 60292, 13294, 55728, 42836, 60868, 57542, 65793, 67511, 80985, 22778, 3118, 67635, 62144, 537, 60887, 36394, 18717, 1329, 15206, 51603, 76202, 70066, 89487, 58899, 16514, 14226, 83030, 75993, 57830, 6729, 12395, 60987, 6978, 78289, 69405, 81122, 65066, 38164, 4680, 23383, 68997, 39291, 74106, 79572, 66198, 45202, 63754, 314, 61671, 85182, 29085, 70065, 79251, 51533, 27482, 21828, 24281, 98349, 17770, 47250, 87577, 73692, 68513, 11779, 84333, 22388, 92362, 41332, 20732, 42300, 17794, 39118, 98820, 27239, 57836, 61284, 7438, 94360, 21918, 92132, 16495, 76257, 72770, 98012, 56210, 94963, 13159, 64244, 47544, 86650, 85170, 73094, 84045, 87091, 11321, 30070, 24693, 30661, 10754, 65326, 95425, 71842, 86623, 69595, 18271, 53478, 66871, 65051, 51952, 61675, 56902, 38295, 93632, 1422, 67653, 3823, 39370, 37089, 85086, 21651, 34279, 50938, 21553, 66385, 22009, 80771, 49446, 78005, 98473, 24097, 60795, 39608, 48406, 8878, 23676, 93051, 40222, 48557, 76866, 92950, 14235, 47887, 72266, 17844, 84939, 56057, 70925, 25003, 87227, 49456, 51275, 73926, 75899, 21522, 27319, 27803, 37639, 48036, 13770, 64662, 50456, 47995, 2545, 4404, 99938, 27323, 49744, 50772, 61467, 3524, 85916, 3089, 11310, 73870, 61462, 32734, 14100, 55427, 6236, 97491, 61637, 62180, 18231, 10939, 40795, 39273, 37057, 80598, 97684, 38541, 5465, 68441, 31373, 45814, 10120, 20062, 51881, 74221, 2725, 23346, 35000, 61254, 77930, 273, 50381
  13. Consider the following list of numbers. Your job is to erase as few of those numbers as possible such that the remaining numbers appear in increasing order. 9 44 32 12 7 42 34 92 35 37 41 8 20 27 83 64 61 28 39 93 29 17 13 14 55 21 66 72 23 73 99 1 2 88 77 3 65 83 84 62 5 11 74 68 76 78 67 75 69 70 22 71 24 25 26
  14. There are a set of N competitors in a tournament. For every possible pairing of competitors, a battle was fought, with one competitor triumphing over another. A tournament king is a competitor for which it is true that when comparing to any other competitor, the king either triumphed over that competitor directly, or the king directly triumphed over another competitor who directly triumphed over that competitor. Given such a tournament, give an efficient algorithm to quickly find any single tournament king. You may choose one of three data storage methods: Method I You have at your disposal a look-up table which given any pair of competitors, will tell you who won. Such a look-up is considered one operation. You may also remove entries from this table, and each removal is considered an operation. You may simultaneously look-up and remove the entry that is looked up, and that is considered a single operation. You can make a copy of the initial table, which is N*(N-1)/2 operations (roughly N^2). Method II You have a list of N smaller lists, where each of the smaller lists corresponds to a competitor, and the contents of the smaller lists are those the competitor defeated. Looking up/modifying/creating an element from any list is 1 operation. Method III You have a list of N smaller lists, where each of the smaller lists corresponds to a competitor, and the contents of the smaller lists are those the competitor was defeated by. Looking up/modifying/creating an element from any list is 1 operation. What is the time complexity of operations in your algorithm (an estimate of the number of operations as a function of N, where any proportional constant on the whole expression, or constant terms can be ignored)? What about finding all tournament kings? Who can give the algorithms with the lowest time complexity?
  15. mmiguel

    2 Aces

    Remove all cards form a standard deck except for the Aces and Kings. Randomly select 2 cards from this and deal them to a friend. She looks at the two cards and says: "After counting the number of red Aces in my hand, I can tell you that this number is greater than zero." What is the probability that both of her cards are Aces of any color?
  16. Let me know if you want any hints
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