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Deponent Forms of the Subjunctive in Latin

by 10:50 AM
Deponents are verbs existing in all four conjugations which are passive in form and active in meaning. These kinds of verbs only have three principal parts for the rules are slightly altered (as shown in the chart below).




The perfect system is very similar to the regular subjunctive formations. You use the 3rd principal part which is analogous to the 4th for both tenses. For the perfect tense you include the present active subjunctive form of the verb of being (sum, esse, fui, futurum). For the pluperfect tense you use the imperfect active form of the verb of being.


The present subjunctive is also formed in a similar. You start with the stem from the second principal part, use we fear a liar, and then add on the P.E. (passive present).
For the imperfect subjunctive you take the 2nd PP (passive infinitive) and recreate the active infinitive). Then you add on your passive P.E. (as you did with regular subjunctive mood imperfect tense verbs).

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Independent Uses of the Subjunctive in Latin

by 10:54 AM
Salvete omnes! Today I will be discussing the Independent Uses of Subjunctive Mood Verbs.

The jussive and the hortatory subjunctive are two related independent uses and is used to provide suggestions. They are both translated as let [person] [verb]. The difference lies in the person. The hortatory subjunctive uses the first person and is therefore translated as Let me (sg. form) or Let us (pl. form). The jussive subjunctive describes the third person usage and translates to Let him (sg. form) or Let them (pl. form).

Hortatory Example: petamus - Let us seek (present subjunctive) 

BTW: If you are unsure about how to form any of these verbs, then I would highly recommend checking out my previous video on subjunctive tense verb forms.

Jussive Example: Dicat - Let him speak (present subjunctive) 

A deliberative subjunctive is usually found in a first person subjunctive mood verb and is used to indicate a decision-making process or doubt with an established decision. In English a deliberative subjunctive is translated as “should.”

Deliberative Subjunctive Example: ubi moverem - Where was I moving to? (imperfect subjunctive) 

A main subjunctive can also express a possibility. This is known as a potential subjunctive. The translation is “would.”

Example: Iuvem - I should help.

*Please note that you will never have individual subjunctive verbs. These should always be interpreted in context.

The final use is the optative subjunctive and is the most differentiated based on time. The helper word utinam signals the use of an Optative Subjunctive. These kind of subjunctive expresses a wish (whether possible or impossible in context). A present subjunctive is used to express a wish for the future and is translated as “If only ..., may [verb] ...”. An imperfect subjunctive expresses an impossible wish in the present time and is translated as “If only ..., things were [verb] ...”. An pluperfect subjunctive expresses an impossible wish in the past time and is translated as “If only ..., things had [verb] ...”. If you wish to express a negative wish (I do not wish ...), then you would use the helper word ne to signal this intention.

Example: Di videant - May the Gods see. (pres. subjunctive) 

 Example: Di viderent - If only the Gods would see. (imp. subjunctive) 

 Example: Di vidissent - If only the Gods had seen. (plu. subjunctive)
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Chebyshev Bias

by 7:56 AM
Chebyshev's Bias is a phenomenon which notes that the number of primes whose remainder is three when divided by four exceeds the number of primes whose remainder is one after such a division. This can be easily visualized with the following simple graph. 
Chebyshev's Bias to n=14. Credit: MakeTheBrainHappy
The x-axis scale denotes the list of the first n primes. So the number 8 in the example would take into account the first eight primes (2, 3, 5, 7, 11, 13, 17, 19). The y-axis measures the difference between the number of primes in this list with remainders of 3 (after dividing each prime by four) and those with remainders if one. For this example, it means that four out of the eight primes have a remainder of three and three out of the eight primes have a remainder of one. 2 is the only number that is not accounted for by this bias because it's remainder when divided by four is neither 1 or 3. 

Each time the 4k+1 primes (primes with remainder one when divided by four) equals to number of 4k+3 (primes with remainder three when divided by four). This happens fairly often when the window is this small but it is still evident that there is a skew towards the 4k+3 primes. 
Chebyshev's Bias to n=100,000. Credit: MakeTheBrainHappy
Next Question: Are there any sets of primes at which the number of 4k+1 primes is greater than the number of 4k+3 primes? Yes, but the first one is at the set of the first 2946 primes. The graph above shows that the ratio becomes more skewed as larger sets of primes are considered. We can postulate that their are three points on the graph where the 4k+1 primes are either equal or take the lead (near 0 (i.e. the first graph), near 3000, and around 50000). 

Here the bias was examined up to the first 100,000,000 primes with some intermediate graphs and new terms. Note: The number of crossings around a particular point increases as the terms increase. 

Chebyshev's Bias to n=1,000,000. Credit: MakeTheBrainHappy
Chebyshev's Bias to n=10,000,000. Credit: MakeTheBrainHappy
Chebyshev's Bias to n=100,000,000. Credit: MakeTheBrainHappy
Sets of primes (number n on the x-axis) where the number of 4k+3 primes equals the number of 4k+1 primes: [1, 3, 7, 13, 89, 2943, 2945, 2947, 2949, 2951, 2953, 50371, 50375, 50377, 50379, 50381, 50393, 50413, 50423, 50425, 50427, 50429, 50431, 50433, 50435, 50437, 50439, 50445, 50449, 50451, 50503, 50507, 50515, 50517, 50821, 50843, 50853, 50855, 50857, 50859, 50861, 50865, 50893, 50899, 50901, 50903, 50905, 50907, 50909, 50911, 50913, 50915, 50917, 50919, 50921, 50927, 50929, 51119, 51121, 51123, 51127, 51151, 51155, 51157, 51159, 51161, 51163, 51177, 51185, 51187, 51189, 51195, 51227, 51261, 51263, 51285, 51287, 51289, 51291, 51293, 51297, 51299, 51319, 51321, 51389, 51391, 51395, 51397, 51505, 51535, 51537, 51543, 51547, 51551, 51553, 51557, 51559, 51567, 51573, 51575, 51577, 51595, 51599, 51607, 51609, 51611, 51615, 51617, 51619, 51621, 51623, 51627, 806807, 806809, 806817, 806819, 806839, 806841, 806935, 806937, 806963, 806965, 807049, 807051, 807053, 807059, 807071, 807147, 807149, 807151, 807153, 807155, 807157, 807167, 807169, 807181, 807183, 807185, 807199, 807203, 807205, 807209, 807211, 807231, 807233, 807235, 807237, 807239, 807241, 807243, 807245, 807251, 807283, 807285, 807327, 807329, 807331, 807339, 807341, 807353, 807493, 807495, 807501, 807505, 807519, 807573, 807575, 807577, 807579, 807581, 807671, 807673, 807675, 807677, 807679, 807691, 807699, 807701, 807709, 807757, 807759, 807761, 807763, 807765, 807767, 807819, 807821, 807829, 807831, 807833, 807835, 807853, 807855, 807891, 807895, 807897, 807899, 807921, 807927, 807929, 807957, 807963, 807965, 807977, 807979, 807981, 807983, 807985, 807987, 807989, 807991, 807999, 808001, 808007, 808009, 808139, 808141, 808145, 808147, 808157, 808163, 808167, 808169, 808171, 808173, 808175, 808179, 808181, 808183, 808187, 808189, 808285, 808291, 808463, 808465, 808477, 808479, 808481, 808577, 808579, 808587, 808589, 808591, 808593, 808595, 809473, 809479, 809487, 809569, 810139, 810141, 810155, 810157, 810167, 810169, 810171, 810197, 810201, 810203, 810211, 810231, 811209, 811211, 811215, 811223, 811401, 811413, 811415, 811421, 811423, 811425, 811427, 811519, 811523, 811527, 811529, 811537, 811549, 811551, 811557, 811559, 811561, 814061, 814069, 814089, 48517583, 48517585, 48517591, 48517601, 48517603, 48517715, 48517717, 48517719, 48517723, 48517737, 48517739, 48517743, 48517745, 48517747, 48517751, 48517761, 48517765, 48517767, 48517769, 48517773, 48517775, 48517777, 48517783, 48517785, 48517823, 48517825, 48517827, 48517857, 48517861, 48517863, 48517865, 48517867, 48517871, 48517873, 48517879, 48517883, 48517885, 48517887, 48517889, 48517891, 48517903, 48517915, 48517917, 48517921, 48517927, 48517933, 48517935, 48517937, 48517939, 48517941, 48517943, 48517945, 48517947, 48517949, 48517967, 48517975, 48517977, 48518027, 48518029, 48518031, 48518033, 48518055, 48518103, 48518105, 48518107, 48518145, 48518149, 48518151, 48518153, 48518155, 48518163, 48518171, 48518183, 48518185, 48518187, 48518189, 48518191, 48518193, 48518273, 48518275, 48518277, 48518279, 48518281, 48518283, 48518285, 48518541, 48518543, 48518545, 48518577, 48518579, 48518581, 48518583, 48518585, 48518587, 48518589, 48518613, 48518615, 48518617, 48518663, 48518665, 48518667, 48518673, 48518675, 48518677, 48518687, 48518689, 48518691, 48518693, 48518713, 48518715, 48518717, 48518719, 48518721, 48518727, 48518731, 48518733, 48518737, 48518741, 48518743, 48518749, 48518755, 48518757, 48518759, 48518773, 48518775, 48518779, 48518787, 48518789, 48518795, 48518835, 48518837, 48518839, 48518841, 48518889, 48518891, 48518893, 48518929, 48518931, 48518933, 48518939, 48518941, 48518943, 48518963, 48520127, 48520171, 48520177, 48520239, 48520251, 48520253, 48520301, 48520303, 48520325, 48520347, 48520349, 48520353, 48520355, 48520357, 48520527, 48520567, 48520569, 48520573, 48520575, 48520577, 48520579, 48520593, 48520601, 48520607, 48520615, 48522631, 48522633, 48522635, 48522647, 48522651, 48522653, 48522741, 48522753, 48522755, 48522757, 48522761, 48522763, 48522765, 48522767, 48522769, 48522771, 48522773, 48522785, 48522787, 48522789, 48522791, 48522793, 48522795, 48522801, 48522881, 48522889, 48522895, 48522897, 48522899, 48522901, 48522903, 48522907, 48522941, 48522963, 48522965, 48522973, 48522975, 48522989, 48522995, 48522997, 48522999, 48523001, 48523003, 48523009, 48523013, 48523029, 48523031, 48523035, 48523037, 48523039, 48523041, 48523323, 48523325, 48523327, 48523329, 48523331, 48523341, 48523353, 48523417, 48523419, 48523425, 48523427, 48523429, 48523439, 48523441, 48523443, 48523445, 48523471, 48523501, 48523517, 48523523, 48523525, 48523545, 48523547, 48523549, 48523555, 48523557, 48523563, 48523569, 48523809, 48523811, 48523813, 48523835, 48523837, 48523875, 48523895, 48523897, 48523899, 48523901, 48523913, 48523915, 48523921, 48523923, 48523927, 48523931, 48523933, 48524287, 48524293, 48524295, 48524303, 48524305, 48524307, 48524309, 48524311, 48524313, 48524315, 48524317, 48524319, 48524321, 48524323, 48524325, 48524327, 48524331, 48524333, 48524343, 48524345, 48524347, 48524349, 48524351, 48524353, 48524375, 48524377, 48524425, 48524427, 48524429, 48524431, 48524435, 48524437, 48524439, 48524447, 48524457, 48524459, 48524461, 48524463, 48524465, 48524467, 48524469, 48524471, 48524611, 48524761, 48524763, 48524769, 48525159, 48525161, 48537809, 48537811, 48537813, 48537815, 48537817, 48537823, 48537837, 48537839, 48537841, 48537899, 48537951, 48537953, 48537981, 48537983, 48538065, 48538075, 48538077, 48538079, 48538081, 48538083, 48538085, 48538087, 48538091, 48538093, 48538103, 48538105, 48538107, 48538125, 48538145, 48538147, 48538149, 48538151, 48538159, 48538161, 48538163, 48538165, 48538169, 48538173, 48538197, 48538199, 48538207, 48538211, 48538213, 48538217, 48538221, 48538543, 48538555, 48538603, 48538605, 48538607, 48538625, 48538655, 48538931, 48538933, 48538945, 48538949, 48538951, 48538953, 48538955, 48538957, 48538959, 48538969, 48538971] 

Sets of primes (number n on the x-axis) where the number of 4k+3 primes is less than the number of 4k+1 primes: [2946, 50378, 50380, 50382, 50383, 50384, 50385, 50386, 50387, 50388, 50389, 50390, 50391, 50392, 50414, 50415, 50416, 50417, 50418, 50419, 50420, 50421, 50422, 50424, 50426, 50428, 50430, 50436, 50438, 50446, 50447, 50448, 50450, 50822, 50823, 50824, 50825, 50826, 50827, 50828, 50829, 50830, 50831, 50832, 50833, 50834, 50835, 50836, 50837, 50838, 50839, 50840, 50841, 50842, 50844, 50845, 50846, 50847, 50848, 50849, 50850, 50851, 50852, 50854, 50856, 50858, 50862, 50863, 50864, 50866, 50867, 50868, 50869, 50870, 50871, 50872, 50873, 50874, 50875, 50876, 50877, 50878, 50879, 50880, 50881, 50882, 50883, 50884, 50885, 50886, 50887, 50888, 50889, 50890, 50891, 50892, 50902, 50904, 50906, 50908, 50916, 50918, 50920, 50928, 51120, 51122, 51152, 51153, 51154, 51158, 51160, 51178, 51179, 51180, 51181, 51182, 51183, 51184, 51186, 51188, 51196, 51197, 51198, 51199, 51200, 51201, 51202, 51203, 51204, 51205, 51206, 51207, 51208, 51209, 51210, 51211, 51212, 51213, 51214, 51215, 51216, 51217, 51218, 51219, 51220, 51221, 51222, 51223, 51224, 51225, 51226, 51262, 51264, 51265, 51266, 51267, 51268, 51269, 51270, 51271, 51272, 51273, 51274, 51275, 51276, 51277, 51278, 51279, 51280, 51281, 51282, 51283, 51284, 51286, 51294, 51295, 51296, 51300, 51301, 51302, 51303, 51304, 51305, 51306, 51307, 51308, 51309, 51310, 51311, 51312, 51313, 51314, 51315, 51316, 51317, 51318, 51392, 51393, 51394, 51552, 51558, 51568, 51569, 51570, 51571, 51572, 51574, 51576, 51578, 51579, 51580, 51581, 51582, 51583, 51584, 51585, 51586, 51587, 51588, 51589, 51590, 51591, 51592, 51593, 51594, 51596, 51597, 51598, 51600, 51601, 51602, 51603, 51604, 51605, 51606, 51612, 51613, 51614, 51622, 806808, 806810, 806811, 806812, 806813, 806814, 806815, 806816, 806818, 806820, 806821, 806822, 806823, 806824, 806825, 806826, 806827, 806828, 806829, 806830, 806831, 806832, 806833, 806834, 806835, 806836, 806837, 806838, 806840, 806938, 806939, 806940, 806941, 806942, 806943, 806944, 806945, 806946, 806947, 806948, 806949, 806950, 806951, 806952, 806953, 806954, 806955, 806956, 806957, 806958, 806959, 806960, 806961, 806962, 806964, 807050, 807054, 807055, 807056, 807057, 807058, 807060, 807061, 807062, 807063, 807064, 807065, 807066, 807067, 807068, 807069, 807070, 807072, 807073, 807074, 807075, 807076, 807077, 807078, 807079, 807080, 807081, 807082, 807083, 807084, 807085, 807086, 807087, 807088, 807089, 807090, 807091, 807092, 807093, 807094, 807095, 807096, 807097, 807098, 807099, 807100, 807101, 807102, 807103, 807104, 807105, 807106, 807107, 807108, 807109, 807110, 807111, 807112, 807113, 807114, 807115, 807116, 807117, 807118, 807119, 807120, 807121, 807122, 807123, 807124, 807125, 807126, 807127, 807128, 807129, 807130, 807131, 807132, 807133, 807134, 807135, 807136, 807137, 807138, 807139, 807140, 807141, 807142, 807143, 807144, 807145, 807146, 807148, 807152, 807168, 807182, 807186, 807187, 807188, 807189, 807190, 807191, 807192, 807193, 807194, 807195, 807196, 807197, 807198, 807212, 807213, 807214, 807215, 807216, 807217, 807218, 807219, 807220, 807221, 807222, 807223, 807224, 807225, 807226, 807227, 807228, 807229, 807230, 807232, 807236, 807238, 807240, 807244, 807246, 807247, 807248, 807249, 807250, 807284, 807286, 807287, 807288, 807289, 807290, 807291, 807292, 807293, 807294, 807295, 807296, 807297, 807298, 807299, 807300, 807301, 807302, 807303, 807304, 807305, 807306, 807307, 807308, 807309, 807310, 807311, 807312, 807313, 807314, 807315, 807316, 807317, 807318, 807319, 807320, 807321, 807322, 807323, 807324, 807325, 807326, 807340, 807342, 807343, 807344, 807345, 807346, 807347, 807348, 807349, 807350, 807351, 807352, 807354, 807355, 807356, 807357, 807358, 807359, 807360, 807361, 807362, 807363, 807364, 807365, 807366, 807367, 807368, 807369, 807370, 807371, 807372, 807373, 807374, 807375, 807376, 807377, 807378, 807379, 807380, 807381, 807382, 807383, 807384, 807385, 807386, 807387, 807388, 807389, 807390, 807391, 807392, 807393, 807394, 807395, 807396, 807397, 807398, 807399, 807400, 807401, 807402, 807403, 807404, 807405, 807406, 807407, 807408, 807409, 807410, 807411, 807412, 807413, 807414, 807415, 807416, 807417, 807418, 807419, 807420, 807421, 807422, 807423, 807424, 807425, 807426, 807427, 807428, 807429, 807430, 807431, 807432, 807433, 807434, 807435, 807436, 807437, 807438, 807439, 807440, 807441, 807442, 807443, 807444, 807445, 807446, 807447, 807448, 807449, 807450, 807451, 807452, 807453, 807454, 807455, 807456, 807457, 807458, 807459, 807460, 807461, 807462, 807463, 807464, 807465, 807466, 807467, 807468, 807469, 807470, 807471, 807472, 807473, 807474, 807475, 807476, 807477, 807478, 807479, 807480, 807481, 807482, 807483, 807484, 807485, 807486, 807487, 807488, 807489, 807490, 807491, 807492, 807494, 807496, 807497, 807498, 807499, 807500, 807502, 807503, 807504, 807520, 807521, 807522, 807523, 807524, 807525, 807526, 807527, 807528, 807529, 807530, 807531, 807532, 807533, 807534, 807535, 807536, 807537, 807538, 807539, 807540, 807541, 807542, 807543, 807544, 807545, 807546, 807547, 807548, 807549, 807550, 807551, 807552, 807553, 807554, 807555, 807556, 807557, 807558, 807559, 807560, 807561, 807562, 807563, 807564, 807565, 807566, 807567, 807568, 807569, 807570, 807571, 807572, 807580, 807672, 807674, 807676, 807762, 807764, 807820, 807832, 807834, 807836, 807837, 807838, 807839, 807840, 807841, 807842, 807843, 807844, 807845, 807846, 807847, 807848, 807849, 807850, 807851, 807852, 807854, 807898, 807928, 807930, 807931, 807932, 807933, 807934, 807935, 807936, 807937, 807938, 807939, 807940, 807941, 807942, 807943, 807944, 807945, 807946, 807947, 807948, 807949, 807950, 807951, 807952, 807953, 807954, 807955, 807956, 807958, 807959, 807960, 807961, 807962, 807966, 807967, 807968, 807969, 807970, 807971, 807972, 807973, 807974, 807975, 807976, 807982, 807984, 808000, 808002, 808003, 808004, 808005, 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48538954, 48538956, 48538958, 48538970]

The following program code was used to test the different values up to the first 100,000,000 primes. 

This screenshot check's & plots Chebyshev's Bias up to 1,000,000 primes. Changing the while(nthPrime <= _________) term allowed the program to test the first 100,000,000 primes. Credit: MakeTheBrainHappy

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Scratch 101: Space Slideshow

by 12:22 PM
Hello and welcome to Scratch 101! In this tutorial we will be learning to create a slideshow in multiple ways with different levels of abstraction.  

Objective:
Learn how to create a slideshow. Key topics include loops, conditionals, and input devices. 

Final Project:
Space Slideshow! Example Image/Preview:
Source: https://kids.nationalgeographic.com/explore/space/milky-way/#milky-way-2.jpg
Graphical Setup:
The graphical template is located here: https://scratch.mit.edu/projects/253424201/
Scripts:
Lesson Notes:
Notice that their are two ways to create the slideshow effect. The first script checks at regular intervals to see if their was a key press as opposed to the second script which is more abstract and knows exactly when the space bar was pressed. If you are unfamiliar with abstraction, then make sure to check out this tutorial.

 Extension: 

  • Change the number of seconds in the wait block. What happens? 
  • What happens if both scripts are present in your program at the same time?
  • Add program documentation (comments) explaining the purpose of each segment of code. 
  • Research the 2D Cartesian System and how Scratch utilizes this for the stage feature. 

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