text: Author: Created by mrlowe. Examples of compounds are water (H 2 O), carbon dioxide (CO 2), hydrogen chloride (HCl), sulphuric acid (H 2 SO 4), potassium hydroxide (KOH), methane (CH 4), acetic acid (CH 3 COOH). Multiply this by 100 and you get 89%. OCR Gateway Chemistry Revision sheets C1,C2,C3, Entire OCR A-Level Chemistry Course Powerpoint, Atoms, Elements and Compounds (Activate KS3). of atoms of the element in the molecule is inversely proportional to it's atomic weight which doesn't really make sense. For instance, the percent composition of oxygen in water is 89%. What was the computer's original price? Percentage composition by mass in a compound Type in an answer and click CHECK, no need for % sign, lower case only. 40.1% C. 32.0% D. 25.6% This quiz is designed to test your knowledge on the chemical elements and as a result calculation of percent composition of chemical compounds. *Response times vary by subject and question complexity. 5. BROWSE BY DIFFICULTY. Select the best answer from the choices. A powerpoint that shows how to calculate %Composition and sets it in the context of fertilisers. To link to this page, copy the following code to your site: Percent composition is used to find the percentage of elements in a compound. 9 pages. These spreadsheets developed by enthusiasts will make your job much more easier, alowing you to shorten the time used for endless calculations of power cables, voltage drop, power factor, circuit breakers, capacitors, cable size, power transformers etc. Q: A solution is made by adding 0.978 mol sucrose to 250.0 g water. Step 3: Find the mass percent of each atom. One must know the molar mass of the elements and the compound in order to get percent composition. So, it consists of atoms of different elements. You will need a calculator and a periodic table. All these compounds contain at least two atoms of elements. One must know the molar mass of the elements and the compound in order to get percent composition. questions: C = 12, Cl = 35.5, Fe = 56, H = 1, Mg = 24, N = 14, Na = 23, O = 16, S = 32, By now I assume you can do formula mass calculations and read formula without any trouble, so ALL the detail of such calculations is NOT shown, just the bare essentials! But it does ask for the mass percentage of all the elements in our formula. View US version . Determine The Mass Of Air Actually Supplied Per Kg Of Coal, And The Percentage … Read more. The atomic masses of Cu and S are 63.5 g/mol and 32.1 g/mol respectively. This is the aptitude questions and answers section on "Percentage" with explanation for various interview, competitive examination and entrance test. For example id 2 have 2 pass and 2 fail therefore percentage pass = … Solved examples with detailed answer description, explanation are given and it would be easy to understand. 10. 253g of zinc = 253 grams of zinc. Created: Jul 1, 2013. So I was studying Stoichiometry and there was a whole section about calculating molecular formulas using JUST the percentage composition of the elements. And the mass percentages of the elements are. Percentage Questions: These questions are somewhat harder, but give them your best shot! registered in England (Company No 02017289) with its registered office at 26 Red Lion The molar mass for question #9 was determined by experiment to be 60.0 g/mol. convert gold lb to mass. To find the percent composition of red M&M’s in a bag of 200 M&M’s, if there are 26 red, you would use this formula. they divided the percentage of the element by it's atomic weight which basically suggests that no. The vanillin molecule is the primary molecule present in vanilla extract. Determine the empirical formula of acetic acid. Preview and details Files included (1) pptx, 65 KB. What is the percent of composition … About this resource. 4b. Practice Problems: Chemical Formulas (Answer Key) H 3 PO 4, Phosphoric acid, is used in detergents, fertilizers, toothpastes and flavoring in carbonated beverages.Calculate the percent composition by mass to two decimal places of H, P and O in this compound. Mass of Sample x 100 = Percent Composition. Free. This workbook goes through the theory of percentage and mass composition as well as several worked examples, which are included in the teacher version. The percent composition is the percent by mass of each element in a compound. Knowledge application - use your knowledge to answer questions about how to calculate percent composition and … Other percentage mass composition calculations including % of any component in a compound or a mixture. MS Excel Spreadsheets (XLS, XLSX) This section is dedicated to tools every electrical engineer can use in daily work. Preview. 16 divided by 18 is.89. Percent composition: %X = (mass of X)/(mass of mixture) * 100%. Solution for What is the percent composition of oxygen by mass to one decimal place in CuSO4 . A compound of lead has the following percentage composition: Pb = 90.66%, O = 9.34%. On the surface, it looks like this question isn’t looking for percentage composition. mass % = (mass of component/mass of total) x 100. This leaves hydrogen making up 11% of the water compound. Percent composition is the term used to describe the percent by mass of each element in a compound. Chemical compounds are made up of chemical elements which have different numbers attached to them. What is the molecular formula? 4.7 3 customer reviews. Atomic masses used for 4b. Conditions. 1 lb is 453 grams on Earth)..225lb * (453 grams/lb) = 102 grams of gold. The ratio is multiplied by 100 … I'm wondering that after you calculate the percentage composition of a compound, is it okay that sometimes its a few decimals away from 100? Answer. 16 divided by 18 is .89. calculate the percent composition of nitrogen in aluminum nitrate Al(NO3). Multiply this by 100 and you get 89%. Like for Cu MnO4, after my calculations, its percentage sum is 100.0226951% and for Sn(SO4)2, its 100.0139624%. The Result can either be Pass or Fail. g E represents the total amount of element E present in the compound and g T represents the total amount of all the elements present in the compound. For example, 50\% of a number means 50 parts of it out of a total of 100, and since 50 is one half of 100, 50\% means half of the total amount. S will weigh 2(63.5)+32.1 =159.1 g. The mass of Cu = 2×63.5 =127 g. The percent composition of Cu = 159.1127×100. The first thing we should do is condense down the formula into its chemical formula listing the number of atoms of each element only. Ian earns £420 a week after a 5% rise. The compound, when combusted in oxygen, provided 0.220 g CO2 and 0.239 g water. It is calculated in a similar way that we just indicated for the peanut butter. Result must be the following table based on formulae: For each id, count Pass / Total Counts for given id. 5H2O? GCSE/IGCSE/AS Chemistry On-line Calculation Questions Quiz 4. Calculating percentage composition When using chemical formula it is possible to calculate the percentage composition of the chemical. Loading... Save for later. The basic equation = mass of element / mass of compound X 100% For instance, if you had a 80.0 g sample of a compound that was 20.0 g element X and 60.0 g element y then the percent composition of each element would be: Determine the percent composition of Ca 3 (PO 4) 2: 4. A powerpoint that shows how to calculate %Composition and sets it in the context of fertilisers. What is the percentage composition of this compound? This quiz will cover basic percent composition problems. . Percentage composition of compounds Having worked out the Mr of a compound, the percentage of that compound that is made from a particular element can now be worked out. For carbon: mass % C = (mass of 1 mol of carbon/mass of 1 mol of CO 2 ) x 100. mass % C = (12.01 g / 44.01 g) x … London WC1R 4HQ. A. The percentage composition of a given element is expressed using the following formula, $$\%C_{E}=\frac{g^{E}}{g^{T}}\times 100$$ Here, %C E is the percentage composition of the element E, that is to be calculated. Example 2: Find the percent composition of copper (Cu) in CuBr2. The molar mass of water is 18g/mol, and oxygen has a molar mass of 16g/mol. 57.7% B. 11. Total mass in the sample. It also contains mapped diagrams showing all possible questions on this topic and the pathways for approaching each type of problem. An Iyonix PC is sold for £1200 at a reduction of 20% on its recommended retail price. In one of the steps. Question: 14) The Percentage Composition By Mass Of A Certain Fuel Is C 88%, H2 3.6%, O2 4.8% And Ash 3.6%. Question 9 . For instance, the percent composition of oxygen in water is 89%. Tes Global Ltd is All answers included. The drug known as LSD has the formula C 20 H 25 N 3 O. (Pb =207,O=16 ) Asked … What was his pay before? So, in the M&M’s problem: 26 red x 100 = 13%. EASY. The percentage composition of a material is generally taken to mean the percentage by mass of the elements within a substance. A sample of a liquid with a mass of 8.657 grams was decomposed into its elements and gave 5.217 grams of carbon, 0.9620 grams of hydrogen, and 2.478 grams of oxygen. . This website and its content is subject to our Terms and This is … The percentage composition of acetic acid is found to be 39.9% C, 6.7% H, and 53.4% O. Calculate the empirical formula of the compound. Percentages Percentage means “number of parts per one hundred” and is denoted by the \bf{\%} sign. A compound is a pure substance made up of different elements. Note that lb is a measure of force and not mass, but it can be converted to mass if given gravity (ie. Important questions on Percentage Composition. The Percentage Composition By Volume Of The Flue Gases Are CO2 10.9%, CO 1%, 02 7.1%, And N2 81%. Percent composition in chemistry typically refers to the percent each element is of the compound's total mass. For each of the following questions or statements, select the most appropriate response and click its letter: Start Congratulations - you have completed Quiz #2-3 PRACTICE: Molar Masses & Percent Composition . What is the molecular formula of vanillin? Example: Calcium carbonate - CaCO 3 … Percentage Composition. What is the mass percent composition of carbon in a 0.100 g sample of a carbon-hydrogen-oxygen compound? This resource is designed for UK teachers. 75mL * (13.6g/mL) = 1020 grams of mercury It is typically found using the molar mass values for both the elements in the compound and the compound itself. I want to calculate the percentage pass for each id. The molecular mass of vanillin is 152.08 grams per mole and contains 63.18% carbon, 5.26% hydrogen, and 31.56% oxygen. Median response time is 34 minutes and may be longer for new subjects. Square Composition. 200 total. Percent Composition Formula Questions: Answer the following questions: 9. Make sure you are … easy 41 Questions medium 280 Questions hard 125 Questions. The molar mass of water is 18g/mol, and oxygen has a molar mass of 16g/mol. (10.9.1) % by mass = mass of element mass of compound × 100 % Percent Composition from Mass Data 1 mole of C u2. Type of problem gravity ( ie tools every electrical engineer can use in daily work each element a. 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G = D4×C5⋊2C8order 320 = 26·5

Direct product of D4 and C5⋊2C8

Series: Derived Chief Lower central Upper central

 Derived series C1 — C10 — D4×C5⋊2C8
 Chief series C1 — C5 — C10 — C20 — C2×C20 — C2×C5⋊2C8 — C22×C5⋊2C8 — D4×C5⋊2C8
 Lower central C5 — C10 — D4×C5⋊2C8
 Upper central C1 — C2×C4 — C4×D4

Generators and relations for D4×C52C8
G = < a,b,c,d | a4=b2=c5=d8=1, bab=a-1, ac=ca, ad=da, bc=cb, bd=db, dcd-1=c-1 >

Subgroups: 286 in 134 conjugacy classes, 77 normal (33 characteristic)
C1, C2 [×3], C2 [×4], C4 [×2], C4 [×2], C4 [×3], C22, C22 [×4], C22 [×4], C5, C8 [×5], C2×C4 [×3], C2×C4 [×2], C2×C4 [×4], D4 [×4], C23 [×2], C10 [×3], C10 [×4], C42, C22⋊C4 [×2], C4⋊C4, C2×C8 [×8], C22×C4 [×2], C2×D4, C20 [×2], C20 [×2], C20 [×3], C2×C10, C2×C10 [×4], C2×C10 [×4], C4×C8, C22⋊C8 [×2], C4⋊C8, C4×D4, C22×C8 [×2], C52C8 [×2], C52C8 [×3], C2×C20 [×3], C2×C20 [×2], C2×C20 [×4], C5×D4 [×4], C22×C10 [×2], C8×D4, C2×C52C8 [×2], C2×C52C8 [×2], C2×C52C8 [×4], C4×C20, C5×C22⋊C4 [×2], C5×C4⋊C4, C22×C20 [×2], D4×C10, C4×C52C8, C203C8, C20.55D4 [×2], C22×C52C8 [×2], D4×C20, D4×C52C8
Quotients: C1, C2 [×7], C4 [×4], C22 [×7], C8 [×4], C2×C4 [×6], D4 [×2], C23, D5, C2×C8 [×6], C22×C4, C2×D4, C4○D4, Dic5 [×4], D10 [×3], C4×D4, C22×C8, C8○D4, C52C8 [×4], C2×Dic5 [×6], C22×D5, C8×D4, C2×C52C8 [×6], D4×D5, D42D5, C22×Dic5, C22×C52C8, D4×Dic5, D4.Dic5, D4×C52C8

Smallest permutation representation of D4×C52C8
On 160 points
Generators in S160
(1 40 137 22)(2 33 138 23)(3 34 139 24)(4 35 140 17)(5 36 141 18)(6 37 142 19)(7 38 143 20)(8 39 144 21)(9 42 134 104)(10 43 135 97)(11 44 136 98)(12 45 129 99)(13 46 130 100)(14 47 131 101)(15 48 132 102)(16 41 133 103)(25 123 151 63)(26 124 152 64)(27 125 145 57)(28 126 146 58)(29 127 147 59)(30 128 148 60)(31 121 149 61)(32 122 150 62)(49 156 106 118)(50 157 107 119)(51 158 108 120)(52 159 109 113)(53 160 110 114)(54 153 111 115)(55 154 112 116)(56 155 105 117)(65 73 83 92)(66 74 84 93)(67 75 85 94)(68 76 86 95)(69 77 87 96)(70 78 88 89)(71 79 81 90)(72 80 82 91)
(1 5)(2 6)(3 7)(4 8)(9 13)(10 14)(11 15)(12 16)(17 39)(18 40)(19 33)(20 34)(21 35)(22 36)(23 37)(24 38)(25 147)(26 148)(27 149)(28 150)(29 151)(30 152)(31 145)(32 146)(41 99)(42 100)(43 101)(44 102)(45 103)(46 104)(47 97)(48 98)(49 110)(50 111)(51 112)(52 105)(53 106)(54 107)(55 108)(56 109)(57 61)(58 62)(59 63)(60 64)(65 87)(66 88)(67 81)(68 82)(69 83)(70 84)(71 85)(72 86)(73 77)(74 78)(75 79)(76 80)(89 93)(90 94)(91 95)(92 96)(113 117)(114 118)(115 119)(116 120)(121 125)(122 126)(123 127)(124 128)(129 133)(130 134)(131 135)(132 136)(137 141)(138 142)(139 143)(140 144)(153 157)(154 158)(155 159)(156 160)
(1 94 132 63 113)(2 114 64 133 95)(3 96 134 57 115)(4 116 58 135 89)(5 90 136 59 117)(6 118 60 129 91)(7 92 130 61 119)(8 120 62 131 93)(9 125 153 139 77)(10 78 140 154 126)(11 127 155 141 79)(12 80 142 156 128)(13 121 157 143 73)(14 74 144 158 122)(15 123 159 137 75)(16 76 138 160 124)(17 112 146 43 88)(18 81 44 147 105)(19 106 148 45 82)(20 83 46 149 107)(21 108 150 47 84)(22 85 48 151 109)(23 110 152 41 86)(24 87 42 145 111)(25 52 40 67 102)(26 103 68 33 53)(27 54 34 69 104)(28 97 70 35 55)(29 56 36 71 98)(30 99 72 37 49)(31 50 38 65 100)(32 101 66 39 51)
(1 2 3 4 5 6 7 8)(9 10 11 12 13 14 15 16)(17 18 19 20 21 22 23 24)(25 26 27 28 29 30 31 32)(33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48)(49 50 51 52 53 54 55 56)(57 58 59 60 61 62 63 64)(65 66 67 68 69 70 71 72)(73 74 75 76 77 78 79 80)(81 82 83 84 85 86 87 88)(89 90 91 92 93 94 95 96)(97 98 99 100 101 102 103 104)(105 106 107 108 109 110 111 112)(113 114 115 116 117 118 119 120)(121 122 123 124 125 126 127 128)(129 130 131 132 133 134 135 136)(137 138 139 140 141 142 143 144)(145 146 147 148 149 150 151 152)(153 154 155 156 157 158 159 160)

G:=sub<Sym(160)| (1,40,137,22)(2,33,138,23)(3,34,139,24)(4,35,140,17)(5,36,141,18)(6,37,142,19)(7,38,143,20)(8,39,144,21)(9,42,134,104)(10,43,135,97)(11,44,136,98)(12,45,129,99)(13,46,130,100)(14,47,131,101)(15,48,132,102)(16,41,133,103)(25,123,151,63)(26,124,152,64)(27,125,145,57)(28,126,146,58)(29,127,147,59)(30,128,148,60)(31,121,149,61)(32,122,150,62)(49,156,106,118)(50,157,107,119)(51,158,108,120)(52,159,109,113)(53,160,110,114)(54,153,111,115)(55,154,112,116)(56,155,105,117)(65,73,83,92)(66,74,84,93)(67,75,85,94)(68,76,86,95)(69,77,87,96)(70,78,88,89)(71,79,81,90)(72,80,82,91), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,145)(32,146)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,61)(58,62)(59,63)(60,64)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(153,157)(154,158)(155,159)(156,160), (1,94,132,63,113)(2,114,64,133,95)(3,96,134,57,115)(4,116,58,135,89)(5,90,136,59,117)(6,118,60,129,91)(7,92,130,61,119)(8,120,62,131,93)(9,125,153,139,77)(10,78,140,154,126)(11,127,155,141,79)(12,80,142,156,128)(13,121,157,143,73)(14,74,144,158,122)(15,123,159,137,75)(16,76,138,160,124)(17,112,146,43,88)(18,81,44,147,105)(19,106,148,45,82)(20,83,46,149,107)(21,108,150,47,84)(22,85,48,151,109)(23,110,152,41,86)(24,87,42,145,111)(25,52,40,67,102)(26,103,68,33,53)(27,54,34,69,104)(28,97,70,35,55)(29,56,36,71,98)(30,99,72,37,49)(31,50,38,65,100)(32,101,66,39,51), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160)>;

G:=Group( (1,40,137,22)(2,33,138,23)(3,34,139,24)(4,35,140,17)(5,36,141,18)(6,37,142,19)(7,38,143,20)(8,39,144,21)(9,42,134,104)(10,43,135,97)(11,44,136,98)(12,45,129,99)(13,46,130,100)(14,47,131,101)(15,48,132,102)(16,41,133,103)(25,123,151,63)(26,124,152,64)(27,125,145,57)(28,126,146,58)(29,127,147,59)(30,128,148,60)(31,121,149,61)(32,122,150,62)(49,156,106,118)(50,157,107,119)(51,158,108,120)(52,159,109,113)(53,160,110,114)(54,153,111,115)(55,154,112,116)(56,155,105,117)(65,73,83,92)(66,74,84,93)(67,75,85,94)(68,76,86,95)(69,77,87,96)(70,78,88,89)(71,79,81,90)(72,80,82,91), (1,5)(2,6)(3,7)(4,8)(9,13)(10,14)(11,15)(12,16)(17,39)(18,40)(19,33)(20,34)(21,35)(22,36)(23,37)(24,38)(25,147)(26,148)(27,149)(28,150)(29,151)(30,152)(31,145)(32,146)(41,99)(42,100)(43,101)(44,102)(45,103)(46,104)(47,97)(48,98)(49,110)(50,111)(51,112)(52,105)(53,106)(54,107)(55,108)(56,109)(57,61)(58,62)(59,63)(60,64)(65,87)(66,88)(67,81)(68,82)(69,83)(70,84)(71,85)(72,86)(73,77)(74,78)(75,79)(76,80)(89,93)(90,94)(91,95)(92,96)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)(153,157)(154,158)(155,159)(156,160), (1,94,132,63,113)(2,114,64,133,95)(3,96,134,57,115)(4,116,58,135,89)(5,90,136,59,117)(6,118,60,129,91)(7,92,130,61,119)(8,120,62,131,93)(9,125,153,139,77)(10,78,140,154,126)(11,127,155,141,79)(12,80,142,156,128)(13,121,157,143,73)(14,74,144,158,122)(15,123,159,137,75)(16,76,138,160,124)(17,112,146,43,88)(18,81,44,147,105)(19,106,148,45,82)(20,83,46,149,107)(21,108,150,47,84)(22,85,48,151,109)(23,110,152,41,86)(24,87,42,145,111)(25,52,40,67,102)(26,103,68,33,53)(27,54,34,69,104)(28,97,70,35,55)(29,56,36,71,98)(30,99,72,37,49)(31,50,38,65,100)(32,101,66,39,51), (1,2,3,4,5,6,7,8)(9,10,11,12,13,14,15,16)(17,18,19,20,21,22,23,24)(25,26,27,28,29,30,31,32)(33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48)(49,50,51,52,53,54,55,56)(57,58,59,60,61,62,63,64)(65,66,67,68,69,70,71,72)(73,74,75,76,77,78,79,80)(81,82,83,84,85,86,87,88)(89,90,91,92,93,94,95,96)(97,98,99,100,101,102,103,104)(105,106,107,108,109,110,111,112)(113,114,115,116,117,118,119,120)(121,122,123,124,125,126,127,128)(129,130,131,132,133,134,135,136)(137,138,139,140,141,142,143,144)(145,146,147,148,149,150,151,152)(153,154,155,156,157,158,159,160) );

G=PermutationGroup([(1,40,137,22),(2,33,138,23),(3,34,139,24),(4,35,140,17),(5,36,141,18),(6,37,142,19),(7,38,143,20),(8,39,144,21),(9,42,134,104),(10,43,135,97),(11,44,136,98),(12,45,129,99),(13,46,130,100),(14,47,131,101),(15,48,132,102),(16,41,133,103),(25,123,151,63),(26,124,152,64),(27,125,145,57),(28,126,146,58),(29,127,147,59),(30,128,148,60),(31,121,149,61),(32,122,150,62),(49,156,106,118),(50,157,107,119),(51,158,108,120),(52,159,109,113),(53,160,110,114),(54,153,111,115),(55,154,112,116),(56,155,105,117),(65,73,83,92),(66,74,84,93),(67,75,85,94),(68,76,86,95),(69,77,87,96),(70,78,88,89),(71,79,81,90),(72,80,82,91)], [(1,5),(2,6),(3,7),(4,8),(9,13),(10,14),(11,15),(12,16),(17,39),(18,40),(19,33),(20,34),(21,35),(22,36),(23,37),(24,38),(25,147),(26,148),(27,149),(28,150),(29,151),(30,152),(31,145),(32,146),(41,99),(42,100),(43,101),(44,102),(45,103),(46,104),(47,97),(48,98),(49,110),(50,111),(51,112),(52,105),(53,106),(54,107),(55,108),(56,109),(57,61),(58,62),(59,63),(60,64),(65,87),(66,88),(67,81),(68,82),(69,83),(70,84),(71,85),(72,86),(73,77),(74,78),(75,79),(76,80),(89,93),(90,94),(91,95),(92,96),(113,117),(114,118),(115,119),(116,120),(121,125),(122,126),(123,127),(124,128),(129,133),(130,134),(131,135),(132,136),(137,141),(138,142),(139,143),(140,144),(153,157),(154,158),(155,159),(156,160)], [(1,94,132,63,113),(2,114,64,133,95),(3,96,134,57,115),(4,116,58,135,89),(5,90,136,59,117),(6,118,60,129,91),(7,92,130,61,119),(8,120,62,131,93),(9,125,153,139,77),(10,78,140,154,126),(11,127,155,141,79),(12,80,142,156,128),(13,121,157,143,73),(14,74,144,158,122),(15,123,159,137,75),(16,76,138,160,124),(17,112,146,43,88),(18,81,44,147,105),(19,106,148,45,82),(20,83,46,149,107),(21,108,150,47,84),(22,85,48,151,109),(23,110,152,41,86),(24,87,42,145,111),(25,52,40,67,102),(26,103,68,33,53),(27,54,34,69,104),(28,97,70,35,55),(29,56,36,71,98),(30,99,72,37,49),(31,50,38,65,100),(32,101,66,39,51)], [(1,2,3,4,5,6,7,8),(9,10,11,12,13,14,15,16),(17,18,19,20,21,22,23,24),(25,26,27,28,29,30,31,32),(33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48),(49,50,51,52,53,54,55,56),(57,58,59,60,61,62,63,64),(65,66,67,68,69,70,71,72),(73,74,75,76,77,78,79,80),(81,82,83,84,85,86,87,88),(89,90,91,92,93,94,95,96),(97,98,99,100,101,102,103,104),(105,106,107,108,109,110,111,112),(113,114,115,116,117,118,119,120),(121,122,123,124,125,126,127,128),(129,130,131,132,133,134,135,136),(137,138,139,140,141,142,143,144),(145,146,147,148,149,150,151,152),(153,154,155,156,157,158,159,160)])

80 conjugacy classes

 class 1 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D 4E ··· 4L 5A 5B 8A ··· 8H 8I ··· 8T 10A ··· 10F 10G ··· 10N 20A ··· 20H 20I ··· 20X order 1 2 2 2 2 2 2 2 4 4 4 4 4 ··· 4 5 5 8 ··· 8 8 ··· 8 10 ··· 10 10 ··· 10 20 ··· 20 20 ··· 20 size 1 1 1 1 2 2 2 2 1 1 1 1 2 ··· 2 2 2 5 ··· 5 10 ··· 10 2 ··· 2 4 ··· 4 2 ··· 2 4 ··· 4

80 irreducible representations

 dim 1 1 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 2 2 4 4 4 type + + + + + + + + + - - + - + - image C1 C2 C2 C2 C2 C2 C4 C4 C4 C8 D4 D5 C4○D4 D10 Dic5 Dic5 D10 Dic5 C8○D4 C5⋊2C8 D4×D5 D4⋊2D5 D4.Dic5 kernel D4×C5⋊2C8 C4×C5⋊2C8 C20⋊3C8 C20.55D4 C22×C5⋊2C8 D4×C20 C5×C22⋊C4 C5×C4⋊C4 D4×C10 C5×D4 C5⋊2C8 C4×D4 C20 C42 C22⋊C4 C4⋊C4 C22×C4 C2×D4 C10 D4 C4 C4 C2 # reps 1 1 1 2 2 1 4 2 2 16 2 2 2 2 4 2 4 2 4 16 2 2 4

Matrix representation of D4×C52C8 in GL4(𝔽41) generated by

 40 0 0 0 0 40 0 0 0 0 40 22 0 0 26 1
,
 40 0 0 0 0 40 0 0 0 0 40 0 0 0 26 1
,
 10 0 0 0 29 37 0 0 0 0 1 0 0 0 0 1
,
 32 10 0 0 32 9 0 0 0 0 14 0 0 0 0 14
G:=sub<GL(4,GF(41))| [40,0,0,0,0,40,0,0,0,0,40,26,0,0,22,1],[40,0,0,0,0,40,0,0,0,0,40,26,0,0,0,1],[10,29,0,0,0,37,0,0,0,0,1,0,0,0,0,1],[32,32,0,0,10,9,0,0,0,0,14,0,0,0,0,14] >;

D4×C52C8 in GAP, Magma, Sage, TeX

D_4\times C_5\rtimes_2C_8
% in TeX

G:=Group("D4xC5:2C8");
// GroupNames label

G:=SmallGroup(320,637);
// by ID

G=gap.SmallGroup(320,637);
# by ID

G:=PCGroup([7,-2,-2,-2,-2,-2,-2,-5,56,219,136,12550]);
// Polycyclic

G:=Group<a,b,c,d|a^4=b^2=c^5=d^8=1,b*a*b=a^-1,a*c=c*a,a*d=d*a,b*c=c*b,b*d=d*b,d*c*d^-1=c^-1>;
// generators/relations

׿
×
𝔽

format: plain


text:
### Home > CC2 > Chapter 1 > Lesson 1.2.7 > Problem1-124

1-124.

What is the probability of getting either blue or green on a spinner that is $\frac { 3 } { 10 }$ green and $\frac { 1 } { 5 }$ blue? Show your work.

Refer to the Math Notes in Lesson 1.2.8 to change the fraction of blue parts to an equivalent fraction so that both fractions have the same denominator.

Add both fractions to get the probability of getting blue or green on the spinner.


text:
# Why is using cross-sectional data to infer / predict longitudinal changes a Bad Thing?

I'm looking for a paper which I hope exists, but don't know if it does. It could be a set of case studies, and / or an argument from probability theory, about why using cross-sectional data to infer / predict longitudinal changes may be a Bad Thing (i.e. is not necessarily so, but can be).

I've seen the mistake made in a couple of big ways: inferences were made that because richer people in Britain travel more, then as society gets richer, the population as a whole will travel more. That inference turned out to be untrue for an extended period - more than a decade. And a similar pattern with domestic electricity use: cross-sectional data implies big increases with income, that don't manifest over time.

There are several things going on, including cohort effects and supply-side constraints.

It would be very useful to have a single reference that compiled case studies like that; and / or used probability theory to illustrate why it is that using cross-sectional data to infer / predict longitudinal changes can be very very misleading.

Does such a paper exist, and if so, what is it?

• I believe economists would think about these phenomena as a kind of general equilibrium effect. Stats people call this a violation of Stable Unit Treatment Value Assumption. I think the panel vs cross section issue is a bit of a red herring. – Dimitriy V. Masterov Feb 27 '15 at 2:51

You partially answer your own question by asking for "longitudinal" changes. Cross-section data are called because they take a snap shot in time, literally a cross-section sliced out of a time-evolving society with its many relationship. Therefore, the best inference you can hope to do is under the assumption that whatever it is you are studying is time-invariant, or at least has concluded its evolution.

On the other, the data you are looking for are longitudinal data or panel data for Economists.

A good reference that explains mostly methods but also highlights two prominent examples from Economics is here. Example 2.1 has company investment rates.

Section 3 is a little more theoretical but carries a lot of insight: a panel data model can be \begin{eqnarray} y_{i,t} = \alpha y_{i,t-1} + x_{i,t} \gamma + \eta_{i} + v_{i,t}. \end{eqnarray}

Now, this type of model can capture state dependence, which is (next to unobserved heterogeneity) a common explanation for why people behave differently. Therefore, if you only observe people traveling at a given point in time, your $\alpha$ will be unidentified, meaning you are not aware on how much their travel yesterday has influenced their decision to travel again.

Now, shut down time dependence for a moment but keep in mind that this equation may likely have been the true model.

In a cross section model now, you would drop the subscript $t$ entirely because you only have data in one period. Therefore, you also have no possibility of accounting for the fact that each individual in your data set may have wildly different $\eta_{i}'s$, which will bias your regressions upward generally, at least when true model is dynamic. This is likely the reason of the overestimate, because of an unobserved individual effect (can be common, too), that you did not measure but that was reflected in your cross-section study.

Now, enter panel data again. What we can do is subtract the mean over time of each variable which, given the mean of $\eta_i$ is constant over time, would eliminate this term. This transformation (others are possible) allows you to focus only on the dynamics (and in fact you'd lose any time-invariant regressors).

Now, this is the main difference between cross-section and panel data. The fact that you can eliminate the time-invariant effect because you have that time variation allows you to remove certain biases that cross-section estimation doesn't allow you to detect. Therefore, before you contemplate a policy change such as a higher tax on traveling because you expect people to travel and you want more government revenue, it is more useful to have seen the phenomenon over a few years so you can be sure that you are not capturing unobserved heterogeneity in your sample which you interpret as a propensity to travel.

To estimate these models, it is best to go through the reference. But beware: different assumptions about people's behavior will make different estimations procedures admissible or not.

I hope this helps!

It sounds very much like the definition of a non-ergodic process (measures over realizations not being equal to measures over time). Sadly, very few interesting real-world phenomena are ergodic. I guess this could be a case for finer-scale sampling and inference, where certain simplifications might be carried out. I'm thinking for examples of small time- or spatial scales, where chaotic behaviour is not observed so predictors can be linearized. But I'm just rambling here.. I'm afraid I can't help you with specific literature on the topic, either. Sorry :/ But interesting question nonetheless


text:
# American Institute of Mathematical Sciences

December  2013, 18(10): 2689-2704. doi: 10.3934/dcdsb.2013.18.2689

## Numerical simulation of chemotaxis models on stationary surfaces

 1 Institut für Angewandte Mathematik, TU Dortmund, 44227 Dortmund, Germany, Germany, Germany

Received  November 2012 Revised  April 2013 Published  October 2013

In this paper we present an implicit finite element method for a class of chemotaxis models, where a new linearized flux-corrected transport (FCT) algorithm is modified in such a way as to keep the density of on-surface living cells nonnegative. Level set techniques are adopted for an implicit description of the surface and for the numerical treatment of the corresponding system of partial differential equations. The presented scheme is able to deliver a robust and accurate solution for a large class of chemotaxis-driven models. The numerical behavior of the proposed scheme is tested on the blow-up model on a sphere and an ellipsoid and on the pattern-forming dynamics model of Escherichia coli on a sphere.
Citation: Andriy Sokolov, Robert Strehl, Stefan Turek. Numerical simulation of chemotaxis models on stationary surfaces. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2689-2704. doi: 10.3934/dcdsb.2013.18.2689
##### References:

show all references

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# Symbols

 A differential equation symbols cartoon by Nick Kim. [49]
In thermodynamics, symbols or "notations" are means of representing measurable quantities or formulaic representations of these quantities by means of letters, modifications of letters, e.g. italics, bold, hats, cross-bars, etc., or other shapes, figurines, or operators.

Philosophy is written in that great book that lies before our gaze—I mean the universe—but we cannot understand it if we do not first learn the language and grasp the symbols in which it is written.”
Galileo Galilei (1623), The Assayer [52]

“I know several men who see all nature in symbols, and express themselves conformably whether in Quintics or Quantics, Invariants or Congruents.”
James Maxwell (1863), letter (Ѻ) to (someone)

“The first time I heard about chemical thermodynamics was when a second-year undergraduate brought me the news early in my freshman year. He told me a spine-chilling story of endless lectures with almost three-hundred numbered equations, all of which, it appeared, had to be committed to memory and reproduced in exactly the same form in subsequent examinations. Not only did these equations contain all the normal algebraic symbols but in addition they were liberally sprinkled with stars, daggers, and circles so as to stretch even the most powerful of minds.”
Brian Smith (1973), Basic Chemical Thermodynamics [1]
Table
The following is an (under-construction) table of the various symbols used in thermodynamics and its connective fields. Some symbols are linked to as separate article, as listed in the adjacent navigation box. Curiously, of note, it seems that smaller letter, e.g. p, t, v, were preferred at the turn 19th century, whereas capital letters, e.g. P, T, V, were preferred at the turn of the 20th century and onward, for the most part. In the meaning column (two), names shown in brackets are what people tend to call the symbol in the modern sense; if no brackets are shown, then it is the name used according to the reference given:

 Symbol Meaning Formula Date Person Δ (link) Fire or Heat 450BC Egyptians, Greeks, or Democritus [53]Empedocles. [14] + (Plus sign) 14891518 Johannes Widmann (Ѻ) Henricus Grammateus [43] – (Minus sign) 14891518 Johannes Widmann (Ѻ) Henricus Grammateus [43] = (link) (Equals sign) 1557 Robert Recorde. [42] >< c.1600 Thomas Harriot [51] Log. Logarithm 1616 Appears as an abbreviation for logarithm in A Description of the Admirable Table of Logarithmes (1616), an English translation by Edward Wright of John Napier's work. [47] ∞ Infinity c.1655 John Wallis [54] ≤≥ 1670 John Wallis [51] $\int$ Integral 1675 Gottfried Leibniz: according to Leibniz's notebooks, a critical breakthrough occurred on 11 November 1675, when he employed integral calculus for the first time to find the area under a function y=ƒ(x). He introduced several notations used to this day, for instance the integral sign ∫ representing an elongated S, from the Latin word summa. [32] $d \,$ Derivative 1675 Gottfried Leibniz: in the same notebook, Leibniz employed the use of d for differentials, from the Latin word differentia. [32] ϕ (phi) Phlogiston 1704 Georg Stahl: [24] T Temperature ? V Volume ? P Pressure or weight of the overlying atmosphere 1738 Daniel Bernoulli: “the weight P holding down the piston in [a given] position is the same as the weight of the overlying atmosphere, which we shall designate P in what follows.”[34] n Number of particles 1738 Daniel Bernoulli: [34] f(x) Function 1755 Leonhard Euler: was the first to write f(x) to denote the function f applied to the argument x. [44] i Imaginary number 1755 Leonhard Euler: [44] e Natural log base 1755 Leonhard Euler: [44] ∑ Summation 1755 Leonhard Euler: [44] → Dart(Reaction arrow) 1757 William Cullen: “the dart → between them expresses the elective attraction; when I put a dart with the tail to one substance and the point to another, I mean that the substance to which the tail is directed unites with the one to which the point is directed more strongly than it does with the one united to it in the crotchet {” . [9] { Crotchet(Bonding bracket) equates to: AB or A≡B (modern) 1757 William Cullen: “by the mark { I mean them united to another” [9] ∂ Partial differential "curly d" 1770 Marquis de Condorcet AB Chemical union AB 1775 Torbern Bergman T Vis viva (kinetic energy) $\tfrac{1}{2} mv^2$ 1788 Joseph Lagrange: [6] $\int_{a}^{b} \,$ Definite integral 1822 Joseph Fourier, in his Analytical Theory of Heat, introduced the symbol, and stated: “We refer in general by the sign (adjacent) as the Integral symbol with a and b as the Limits of the full integration that begins when the variable is equivalent to a, and is complete when the variable is equal to b.” [46] s (or e) Heat 1824 Sadi Carnot: "[s is] the heat liberated (set free) or absorbed" when "a gas varies in volume without change in temperature."; "let e be the quantity of heat employed to maintain the temperature of the gas constant during its dilation." [21] r (or u) Motive power (work) 1824 Sadi Carnot: [21] Q Heat 1834 Emile Clapeyron: "Q is an absolute quantity of heat which the gas possesses." [21] The "Q" supposedly is short for small "quantity"; the usage may date earlier, possibly to Joseph Fourier, and his Analytical Theory of Heat (1822)? R Gas constant 1834 Emile Clapeyron: (possibly used earlier?) [21][23] U Force function 1835 William Hamilton: "the function which has been here called U may be named the force-function of a system." [7] ∇ Quaternion operatorNabla (Maxwell, c.1873)Del (Gibbs and Wilson, 1901) 1837 William HamiltonCoined by Peter Tait, in dispute with Maxwell (link); or by Maxwell in honor of Tait, in his circa 1873/74 poem "Chief Musician upon Nabla" (link) (link) T Vis viva $\sum \frac{1}{2}mv^2$ ? Rudolf Clausius: J Ergal $J_0 - \int_{t_0}^{t} \sum (Xdx +Ydy + Zdz)$ ? Rudolf Clausius: [26] U Energy T + J ? Rudolf Clausius: "[U is the] "energy of the system"; [according to the conservation of energy] the sum of the vis viva T and the ergal J [which] remains constant during its motion." [26] W Work F (xyz) + const. ? Rudolf Clausius: "[W is] the work done whenever a body moves under the influence of a force" and "the product of the force and distance moved through is the mechanical work which the force performs during the motion." S Entropy $\frac{dQ}{T}$ 1865 Rudolf Clausius: [27] U (Internal energy) 1865 Francois Massieu: [3][12] U' (Heat content) 1865 Francois Massieu: [3][12] Ψ (psi) Characteristic function $~ \frac{-U+tS}{t} ~$or$\frac{-\psi}{t} \,$ 1869 Francois Massieu: [12] Ψ' (psi prime) $~ \frac{-U+tS-pv}{t} ~$or$\frac{-\zeta}{t} \,$ 1869 Francois Massieu: [12] E Entropy 1872 Ludwig Boltzmann: [36] H Heat 1873 Willard Gibbs: "heat received by the body in passing from one state to another". [5] ε (epsilon) Energy 1873 Willard Gibbs: "the energy of body in a given state". [5] η (eta) Entropy 1873 Willard Gibbs: "the entropy of a body in a given state". [5] $\bar{d} \,$(d-hat) Inexact differential 1875 Carl Neumann: [28] χ (chi) (Heat content) ε + pv 1876 Willard Gibbs: "heat function at constant pressure" [3] Ψ (psi) (Helmholtz function) ε – tη 1876 Willard Gibbs: “the force function for constant temperature” [11] ζ (zeta) (Gibbs function) ε – tη + pv 1876 Willard Gibbs: μ (mu) Potential(Chemical potential) 1876 Willard Gibbs: "if to any homogeneous mass in a state of hydrostatic stress we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and it entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered." [8] P Potential difference (of a galvanic cell) or Pressure 1877 Hermann Helmholtz: [20] ℰ (Electromotive force) $~ P_k - P_a ~$ 1877 Hermann Helmholtz: "P is the potential difference (a and k referring to the anode and cathode, respectively) in a concentration cell." [20] J Mechanical equivalent of heat 1882 Hermann Helmholtz: [20] θ Absolute temperature "temperature reckoned from –273° C" 1882 " " S Entropy 1882 Hermann Helmholtz: [20] U Total energy 1882 Hermann Helmholtz: "the total energy (gesammt-energie) of the system [19] , F Free energy U – JTS 1882 Hermann Helmholtz: "the free energy (frieie energie) of the system." [20] , B Bound energy JTS(J=1 in modern joule units) 1882 Hermann Helmholtz: "the latent energy (gebundene energie) of the system." [19] $\leftrightarrows \,$ Reversible reaction 1884 Jacobus van't Hoff [45] A–BA=B Uniting dashes(chemical bond) represent the "force lines" associated with atomic valencies. 1893 Walther Nernst ln Natural logarithm 1893 Used by Irving Stringham (1847-1909) in Uniplanar Algebra (Cajori vol. 2, page 107). [47] U Energy 1897 Max Planck: "the energy of a body or system of bodies." [19] Φ (phi) Entropy 1897 Max Planck: [19] F Free energy U – TΦ 1897 Max Planck: “The function F, thus bearing the same relation to the external work that the energy U does to the sum of the external heat and work, has been called by Helmholtz the free energy (frieie energie) of the system (it should rather be called the ‘free energy for isothermal processes’).” [19] Ψ (psi) (Gibbs energy) $~ \Phi - \frac{U + pV}{T} ~$or$~ -\frac{G}{T} ~$ 1897 Max Planck: “multiplying Φ by – T, we get the thermodynamic potential at constant pressure U + pV – TΦ” [19][2] U (Internal energy) 1897 Pierre Duhem: [18][3] – (Heat content) 1897 Pierre Duhem: [3] $~ \mathfrak{F} ~$ (Helmholtz free energy) 1897 Pierre Duhem: [18][3] Φ (phi) (Gibbs potential) 1897 Pierre Duhem: [18][3] h Universal constant(Planck constant) $6.55 \cdot 10^{-27} erg \cdot sec \$ 1900 Max Planck: "universal constant"[38] k Universal constant(Boltzmann constant) $1.35 \cdot 10^{-16} \frac{erg}{deg} \$ 1900 Max Planck: "universal constant" [38] A Reaction energy 1905 Fritz Haber: [31] Q Reaction heat 1905 Fritz Haber: [31] q Latent heat of reaction 1905 Fritz Haber: "here we use minus q to designate the heat which is used up or in other words, becomes latent (or bound?)." [31] S Entropy $\frac{q}{T}$ 1905 Fritz Haber: [31] U Total energy 1905 Fritz Haber: [31] N Avogadro's constant $70.5 \cdot 10^{22} \$ 1909 Jean Perrin: “the invariable number N is a universal constant, which may appropriately be designated Avogadro’s constant.” [37] H Enthalpy E + PV 1909 Heike Kamerlingh-Onnes: coined the name enthalpy from the Greek εν (en) ‘in’ and θαλπος (thalpos) ‘to heat’, to define the word enthalpos, to warm within. [4] U Energy 1912 Otto Sackur: [17] H Heat content U + pv 1912 Otto Sackur: “the heat of reaction at constant pressure is equal to the change in the function H. For this reason H is called the heat content of the system, or the heat function for constant pressure.” [17] Ψ (psi) Free energy function U – TS 1912 Otto Sackur: [17] ζ (zeta) Thermodynamic potential U – TS + pv 1912 Otto Sackur: “some authors use the term free energy for the function ζ, although this term was invented by Helmholtz for the function ψ (See G.N. Lewis, J. Am. Chem. Soc. 35: 14 (1913)).” [17] U Energy content 1917 Walther Nernst: [33] Q Heat flow 1917 " " A Work German “Arbeit”, meaning work, 1917 " " S Entropy 1917 " " $\oint \$ Contour integral signLine integralPath integralCurve integral 1917 Arnold Sommerfeld [50] E Internal energy 1923 Lewis and Randall: "[E is] the energy contained within a system, or its internal energy, a property of the system." [10] A Helmholtz free energy E – TS 1923 " " F Free energy (E + PV) – TS H – TS 1923 " " A Work 1924 James Partington: [15] d Ordinary differential " " θ (theta) or T Temperature 1924 [" "]: “the symbol θ is used throughout to denote temperature measured on any scale, unless the scale is specified.” [15] U Intrinsic energy 1924 " " H Enthalpy; Total heat; Heat function U + pV 1924 [" "]: “the function H was called by Gibbs the heat function at constant pressure, and denoted by χ; the name enthalpy has been proposed by Kamerling Onnes (τό θαλπος = heat). It is usually called by engineers the total heat, or heat content, but these names refer to an obsolete period in the theory of heat.” [15] F Free energy U – TS 1924 " " Z Thermodynamic potential U – TS + pV 1924 [" "]: “Z is called the thermodynamic potential by analogy with the potential function in dynamics: ϕ1 – ϕ2 = work” [15] Cycle integral ∮(circle integral)closed path integral $\sum \frac{Q}{T} \,$ 1924 Used by Partington (1924) in a thermodynamic sense; may have been used by Arnold Sommerfeld in 1917, if not sooner? Δ Change "excess of final over initial value"(e.g. $\Delta G = G_f - G_i \,$) 1933 Edward Guggenheim - defined in formulaic terms [11]; symbols used previously by Gilbert Lewis (1923). H Heat content E + PV 1933 " " F Helmholtz free energy E – TS 1933 [" "]: “the function F is due to Helmholtz, and was named by him the ‘free energy’. It is sometimes called the ‘work function’. We shall call F the ‘Helmholtz free energy’.” [11] G Gibbs free energy H – TS 1933 Edward Guggenheim: “the function G is due to Gibbs, and is often referred to by modern writers as ‘free energy’. We shall call G the ‘Gibbs free energy’.” [11] U Internal energy 1934 Theophile de Donder: [16] E Internal energy 1936 Theophile de Donder: [3] A Affinity $~ - \left ( {\partial G\over \partial \xi} \right )_{p,T} ~$ 1936 " " Ξ or B (?) Exergy 1956 Zoran Rant: [39] S Entropy $\frac{d Q_{rev}}{T}$ 1968 Clement Adkins: "[the rev subscript is used] to emphasize that the equality holds for reversible changes only." [41] E Total exergy 2004 Norio Sato: “we shall use the symbol E (capital epsilon) to express total exergy. [40] ε Molar exergy 2004 Norio Sato: “we shall use the symbol ε (small epsilon) to express molar (or specific exergy). [40]

Undated
The following needed to be dated with reference to the actual person who first used the symbol (starting point references are shown in brackets):

 $~ c_p ~$ Specific heat at constant pressure [19] $~ c_v ~$ Specific heat at constant volume [19] γ (gamma) $~ \frac{c_p}{c_v} ~$ 1897 Max Planck (?): [19] J Massieu function $\frac{-F}{t} \,$ 1960 Herbert Callen (?): [13] Y Planck function $\frac{-G}{t} \,$ 1960 Herbert Callen (?): [13]

Discussion
The following is a discussion section on some of the symbols.

 The Greek alphabet: used greatly in thermodynamics; capital delta Δ, for example, refers to heat (in chemistry) or change (in thermodynamics), as in before minus after of some variable, small delta δ represents an inexact differential, as in δW or δQ, as does đ (d-crossbar), and d refers to an exact differential, as in dU or dS.

Some common symbol examples include, the symbol for heat or fire, θ∆ics the shorthand symbol for thermo-dynamics, U for internal energy, G for Gibbs free energy, A of F for Helmholtz free energy, S for entropy, T for temperature, P for pressure, V for volume, etc.

Etymologies
The etymologies of many symbols are very difficult to track down. Belgian chemist Theophile de Donder's 1936 book A Thermodynamic Theory of Affinity contains one of the first symbol tables for the main thermodynamic functions used by various pioneering authors up to that year. [3] One of the few articles on symbol etymology history is the 1997 "A Brief History of Thermodynamics Notations" by American chemical thermodynamicist Rubin Battino who commented, in motivation prior to writing his article, that: [2]

“I was recently asked, ‘I understand the use of H for enthalpy because that is heat related, but where does S for entropy come from?’ I was stumped and decided to do some historical research to track down not only the origin of S but also the other principal thermodynamic terminology and notation.”

This semi error filled article (e.g. attributing enthalpy, H, to Clapeyron, or the statement "Clausius in 1834 [age 12] was using Q for heat"), however, fails in its essential mission to determine the origin of H to enthalpy (which was assigned in 1909 by Dutch physicist Heike Kamerlingh-Onnes), and the origin of S to entropy, assigned by German physicist Rudolf Clausius in 1865. The origin of S as entropy is arguably assigned either (a) on the following of Sadi Carnot's 1824 use of the small letter

“[s represents] the heat liberated (set free) or absorbed”

during an isothermal expansion of a body of gas; or (b) in honor of S. Carnot's first name "Sadi". [21] Both of these tables are shown below:

 De Donder's 1936 Symbol Table Battino's 2001 Symbol Table Belgian chemist Theophile de Donder’s 1936 thermodynamic potential notation table. [3] American chemical thermodynamicist Rubin Battino's 2001 thermodynamic symbol etymology table. [2]

Jensen's symbol research
A few interesting symbol etymology articles come from American chemistry historian William Jensen, who runs a semi-annual "Ask the Historian" column of the Journal of Chemical Education, who takes email queries on "historical origins of symbols". In his 2003 “The Universal Gas Constant R”, a response to a reader's inquiry, which “traces the history of the gas constant R and the probable reason for its representation by the letter R", Jensen tracks the first use of the constant to French engineer Emile Clapeyron's use of R in his 1834 Memoir on the Motive Power of Fire. [22][23]

In his 2010 "Why Are q and Q Used to Symbolize Heat?", Jensen attempts to outline the historical usages of "q" and "Q" in the works of Clapeyron, Rudolf Clausius, August Horstmann, Hermann Helmholtz, Fritz Haber, up through Gilbert Lewis. [48]

Characteristic function
Characteristic function table

 IUPAC's 2007 Chemical Thermodynamics Symbol Table Table of common chemical thermodynamics symbols from the 2007 (3rd ed) of Quantities, Units and Symbols in Physical Chemistry. [30]

References
1. Smith, Brian E. (1973). Basic Chemical Thermodynamics. Imperial College Press.
2. (a) Battino, Rubin, Strong, L.E., and Wood, S.E. (1997). "A Brief History of Thermodynamics Notation", Journal of Chemical Education, 74: 304-305.
(b) Battino, Rubin. (2001). "Ch. 4: A Brief History of Thermodynamics Notation", in: Drug-Receptor Thermodynamics: Introduction and Application by Robert Raffa. Wiley.
(c) Rubin Battino (homepage) – RubinBattino.com.
3. De Donder, T. (1936). Thermodynamic Theory of Affinity: A Book of Principles (pg. xvi-xvii). Oxford: Oxford University Press.
4. Laidler, Keith J. (1993). The World of Physical Chemistry (pg. 110). Oxford University Press.
5. Gibbs, J. Willard. (1873). "Graphical Methods in the Thermodynamics of Fluids", Transactions of the Connecticut Academy (quantities). I. pp. 309-342, April-May.
6. Lagrange, Joseph. (1788). Analytical Mechanics. Publisher.
7. (a) Hamilton, W.R. (1834). “On a general method in dynamics by which the study of the motions of all free systems of attracting or repelling points is reduced to the search and differentiation of one central relation, or characteristic function.” Philos. Trans. R. Soc. London, 124:247-308.
(b) Hamilton, W.R. (1835). “A second essay on a general method in dynamics.” Philos. Trans. R. Soc. London, 125:95-144.
8. Gibbs, Willard. (1876). "On the Equilibrium of Heterogeneous Substances", Transactions of the Connecticut Academy, III. pp. 108-248, Oct., 1875-May, 1876, and pp. 343-524, may, 1877-July, 1878.
9. (a) Crosland, M. P. (1959). “The use of diagrams as chemical ‘equations’ in the lecture notes of William Cullen and Joseph Black.” Annals of Science, Vol 15, Num 2, June.
(b) Thims, Libb. (2007). Human Chemistry (Volume Two), (preview), (pgs. 385-388). Morrisville, NC: LuLu.
10. Lewis, Gilbert N. and Randall, Merle. (1923). Thermodynamics and the Free Energy of Chemical Substances (pgs. 155-59). McGraw-Hill Book Co., Inc.
11. Guggenheim, Edward, A. (1933). Modern Thermodynamics by the Methods of Willard Gibbs (pg. 11). London: Methuen & Co.
12. (a) Massieu, Francois. (1869). “Sur les Fonctions Caracteristiques des Divers Fluids (On the Various Functions Characteristic of Fluids)”, Comptes Rendus 69, pgs. 858-62, 1057-61.
(b) Gibbs refers to Massieu’s notation in 1875 and 1876 (See: Sci. Pap. 1, pg. 86-87, 358).
(c) Aris, Rutherford, Davis, Howard T., and Stuewer, Roger H. (1983). Springs of Scientific Creativity (ch. 5: The Scientific Style of Josiah Willard Gibbs, by Martin J. Klein, pgs 142-62, esp. pg. 161). University of Minnesota Press.
13. (a) Perrot, Pierre. (1998). A to Z of Thermodynamics (Massieu function, pg. 190). Oxford University Press.
(b) Callen, Herbert B. (1960). Thermodynamics: an Introduction to Physical Theories of Equilibrium Thermostatics and Irreversible Thermodynamics (pg. 101). Wiley.
14. Greek philosopher Plato mentions the elements [fire (), earth (), air (), water ()] as of pre-Socratic origin, a list created by the Ionian philosopher Empedocles (ca. 450 BC).
15. Partington, James R. (1924). Chemical Thermodynamics: An Introduction to General Thermodynamics and its Applications to Chemistry. D. Van Nostrand.
16. De Donder, Theophile. (1934). L’Affinitie (Part I, Part II, Part III). Paris: Gauthier-Villars.
17. Sackur, Otto. (1912). Lehrbuch der Thermochemie und Thermodynamik (A Textbook of Thermo-chemistry and Thermodynamics (U, S: pg. 142, H: pgs. 171, 175, Ψ: pg. 172, ζ: 174, pg. 180), 1917 Eng. trans. by George Gibson). MacMillan.
18. Duhem, P. (1897). Traite Elementaire de Mecanique Chimique (Treatise on Elementary Chemical Mechanics). Paris: Hermann.
19. Planck, Max. (1897). Treatise on Thermodynamics (U: pg. 45, Φ pg. 89, F: pg. 113). Longmans, Green and Co.
20. Cahan, David (1993). Hermann von Helmholtz and the Foundations of Nineteenth-Century Science (Contributors: Helge Kragh, pg. xvii; ch. 10: "Between Physics and Chemistry - Helmholtz's Route to a Theory of Chemical Thermodynamics" by Helge Kragh). University of California Press.
21. Carnot, Sadi. (1824). Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power ("s being the heat", pg. 82 (32)). Paris: Chez Bachelier, Libraire, Quai Des Augustins, No. 55.
22. Clapeyron, Emile. (1834). “Memoir on the Motive Power of Heat”, Journal de l’Ecole Polytechnique. XIV, 153 (and Poggendorff's Annalender Physick, LIX, [1843] 446, 566).
23. (a) Jensen, William B. (2003). “The Universal Gas Constant R” (abstract: “this column traces the history of the gas constant R and the probable reason for its representation by the letter R.), J. Chem. Edu. 80: 731.
(b) William B. Jensen (faculty) – Department of Chemistry, University of Cincinnati.
24. Henri-Victor Regnault – NNDB.
25. Partington, J.R. (1937). A Short History of Chemistry (ϕ: pg. 148). Dover.
26. Clausius, Rudolf. (1879). The Mechanical Theory of Heat (section: Mathematical introduction, pgs. 1-20). London: Macmillan & Co.
27. (a) Clausius, Rudolf. (1865). “On Several Forms of the Fundamental Equations of the Mechanical Theory of Heat (Ninth Memoir).” Read at the Philosophical Society of Zurich on the 24th of April, 1865, published in the Vierteljahrsschrift of this society, Bd. x. S. 1.; Pogg. Ann. July, 1865, Bd. cxxv. S. 353; Journ. de Liouville, 2e ser. t. x. p. 361.
(b) Clausius, R. (1865). The Mechanical Theory of Heat – with its Applications to the Steam Engine and to Physical Properties of Bodies (pg. 357) (Ninth Memoir). London: John van Voorst, 1 Paternoster Row. MDCCCLXVII.
(c) See: Entropy (etymology).
28. (a) Neumann, Carl. (1875). Lectures on the Mechanical Theory of Heat (Vorlesungen über die mechanische Theorie der Wärme), Germany.
(b) Laider, Keith, J. (1993). The World of Physical Chemistry. Oxford University Press.
29. Mills I, Cvitas T, Homann K, et al. (1988). Quantities, and Units and Symbols in Physical Chemistry. IUPAC. Oxford: Blackwell Scientific Publishing.
30. Cohen, E. Richard, Cvitas, Tomislav. (2007). Quantities, Units and Symbols in Physical Chemistry, 3rd ed. (Chemical Thermodynamics Symbol Table, pg. 56). Royal Society of Chemistry.
31. Haber, Fritz. (1905). Thermodynamics of Technical Gas Reactions, (Translator’s Preface, 1907, pg. vii). Longmans, Green, and Co.
32. (a) This ingenious and suggestive notation (integral sign ∫ for integration and d for differentiation) for calculus is probably his most enduring mathematical legacy. Leibniz did not publish anything about his calculus until 1684.
(b) For an English translation of Leibniz's paper, see Struik (1969: 271–84), who also translates parts of two other key papers by Leibniz on the calculus.
(c) Struik, D. J. (1969). A Source Book in Mathematics, 1200–1800. Harvard University Press.
33. Nernst, Walther. (1918). The New Heat Theorem: Its Foundations in Theory and Experiment (Die theoretischen und experimentellen Grundlagen des neuen Wärmesatzes), [tr. 1926]. E.P. Dutton & Co.
34. (a) Bernoulli, Daniel. (1738). “On the Properties and Motions of Elastic Fluids, Especially Air” (Hydrodynamica, Section 10) in: The Kinetic Theory of Gases of Gases (pgs. 57-65), 2003, by Stephen G. Brush, Nancy S. Hall. Imperial College Press.
(b) Bernoulli, Daniel. (1738). Hydrodynamica, Sive Vivibus et Motimus Fluidorum Commentarii. Sectio Decima: “De affectionibus atque botimus fluidorum elasticorum, praecipue autem aeris.” (pgs. 200-204). Argentorati, Sumptibus Johannes Reinholdi Dulseckeri.
35. Thomson, William. (1849). “An Account of Carnot’s Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault’s Experiments on Steam”, (127-203) Transactions of the Edinburgh Royal Society, xiv.; Annales de Chime, xxxv. 1852.
36. Boltzmann, Ludwig. (1872). "Further Studies on the Thermal Equilibrium of Gas Molecules" (“Weitere Studien über das Wärmegleichgewicht unter Gasmolekülen”), in Sitzungsberichte der Akademie der Wissenschaften, Mathematische-Naturwissenschaftliche Klasse (pgs. 275-370; term: "entropy" symbol "E", pgs. 263, 308, 346, etc), Bd. 66, Dritte Heft, Zweite Abteilung, Vienna: Gerold.
37. (a) Perrin, Jean. (1909). “Brownian Motion and Molecular Reality” ("Mouvement brownien et réalité moléculaire"), Annales de Chimie et de Physique, 18: 1–114.
(b) Engl. Trans. by Frederick Soddy (London: Taylor and Francis, 1910) [Excerpt: sections 1-6 complete (from: David M. Knight, ed., Classical Scientific Papers: Chemistry (New York: American Elsevier, 1968) and the abridgment reprinted in Henry A. Boorse & Lloyd Motz, The World of the Atom, Vol. 1 (New York: Basic Books, 1966)].
[38] (a) Planck, Max. (1900). “On Irreversible Radiation Processes” (“Über Irreversible Strahlungsvorgänge”), Annalen der Physik. Vol. 1, pg. 69, 99.
(b) Planck, Max. (1901). “On the Law of Distribution of Energy in the Normal Spectrum”, Annalen der Physik, Vol. 4, pg. 553 ff.
(c) Hoffmann, Dieter. (2008). Max Planck: Annalen Papers (Section: Max Planck’s Vita, pg. viii; section: Max Planck on Entropy and Irreversibility by Werner Ebeling, pgs 29-38, etc.). Wiley-VCH.
39. Rant, Zoran. (1956). "Exergie, ein neues Wort fur "Technische Arbeitsfahigkeit" (Exergy, a new word for "technical available work")". Forschung auf dem Gebiete des Ingenieurwesens, 22: 36–37.
40. Sato, Norio. (2004). Chemical Energy and Exergy: an Introduction to Chemical Thermodynamics for Engineers (pg. 99). Elsevier.
41. Adkins, Clement J. (1967). Equilibrium Thermodynamics (section 5.2: Entropy, pgs. 71-). McGraw-Hill.
42. Recorde, Robert. (1557). The Whetstone of Witte: whiche is the seconde parte of Arithmeteke: containing the extraction of rootes; the cossike practise, with the rule of equation; and the workes of Surde Nombers. Publisher.
43. (a) Grammateus, Henricus. (1518). A New Skill Book (Ayn new Kunstlich Buech). Vienna.
(b) Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.
(c) Plus and minus signs – Wikipedia.
44. Boyer, Carl B.; Uta C. Merzbach (1991). A History of Mathematics (pp.439–445). John Wiley & Sons.
45. (a) Nernst, Walther. (1895). Theoretical Chemistry: from the Standpoint of Avogadro’s Rule & Thermodynamics ($\leftrightarrows \,$, pg. 358). MacMillan and Co.
(b) The exact publication of Van't Hoff likely being his 1884 Studies in Chemical Dynamics.
46. Miller, Jeff. (2009). “Earliest Uses of Symbols of Calculus”, Jeff560.tripod.com.
47. Cajori, Florian. (2008). “Earliest Uses of Function Symbols.”, Jeff560.tripod.com.
48. Jensen, William B. (2010). “Why Are q and Q Used to Symbolize Heat?”, Journal of Chemical Education, 87, 1142.
49. Kim, Nick D. (c.2006). “Differential Equations”, NearingZero.net (WayBack).
50. (a) Table of mathematical symbols by introduction date – Wikipedia.
(b) Line integral – Wikipedia.
51. (a) Stedall, Jacqueline. (2003). The Greate Invention of Algebra: Thomas Harriot’s Treatise on Equations (symbols: =, <, >, pg. 8; modern inequality signs, pg. 302). Oxford University Press.
(b) Stedall, Jacqueline. (2012). The History of Mathematics: A Very Short Introduction. Oxford University Press.
(c) Table of mathematical symbols by introduction date – Wikipedia.
52. Crick, Francis. (1994). The Astonishing Hypothesis: the Scientific Search for the Soul (pg. 203) (Ѻ). Simon and Schuster.
53. Thims, Libb. (2019). Human Chemical Thermodynamics: Chemical Thermodynamics Applied to the Humanities Sociology, Economics, History, Philosophy, Ethics,
Government, Politics, Business, Religion, and Relationship (pdf). Publisher.
54. Inwood, Stephen. (2003). The Man Who Knew Too Much: the Strange and Inventive Life of Robert Hooke 1653-1703 (pg. 374). Pan MacMillan.


text:
# Primary Grammar Phrases

Chief Justice Roberts just isn’t alleged by my supply to be one of many two alleged sticklers on this level. (“The Cottar’s Saturday Night”)In colloquial speech, the development got here to get pleasure from widespread use. Viewing a number of the commonest irregular verbs, along with the straightforward previous in addition to their previous participle varieties, can be useful in understanding how they are fashioned. This is an occasion of the well-known imperfective paradox.

Nevertheless, in plenty of such sentences it might be potential to locate the adverb to higher benefit elsewhere within the sentence. The pronouns, buy descriptive essay verb varieties, and determiners that are used to speak to someone, as an example, you, your, you slept. The pronouns, verb varieties, and determiners which are used by a speaker to identify himself or herself, or to discuss with a gaggle including himself or herself, for instance, I, we, my, we had been, I went. Auxiliary verbs are used to form tenses or passive forms of other verbs. See also modal verb.Learn more about auxiliary verbs.

The historical origin of break up ergativity in Iranian is well-attested. As shown in Table 4, this precept had a passive orientation within the case of transitive verbs, with the agent expressed either by a genitive noun phrase (4.i), or a genitive clitic pronoun (4.ii). The participle is also accompanied by a type of the copula verb âbaâ agreeing with the passive subject (4.iii). A FEELING THAT IT IS WRONG NO RATIONAL BASIS PINKER FOWLER Any speaker who has not been brainwashed by the split-verb fantasy can sense that these corrections go towards the rhythm and logic of English phrasing.

The Victorians decided that splitting an infinitive was a grammatical mistake, and some folks still agree with them. The simple present tense is conjugated the same regardless of whether the verb is regular or irregular. This contains including an â-sâ or â-esâ for the third-person singular. Be conscious that sure exceptions, like the verb âbe,â have special present tense varieties as nicely.

The equal of the present participle in Spanish (called a âgerundâ or âgerundioâ) usually ends with both -ando or -iendo in Spanish and -ing in English. The past participle (called the âparticipioâ) is very commonly used as an adjective or along side the auxiliary (âhelpingâ) verb haber, when forming previous tense varieties. Another dual-system principle, the declarative-procedural mannequin , could also be extra amenable to a deeper connection between irregular morphology and semantics. According to the declarative-procedural mannequin, irregular morphology depends on temporal-lobe constructions that underlie âdeclarativeâ reminiscence, together with each episodic and semantic memory. While sustaining a clear distinction between regular and irregular morphology like different dual-system theories, the declarative-procedural model suggests that the psychological lexicon could additionally be grounded in the semantic representations of declarative memory.

See extra examples of comparative and superlative adjectives. The further emphasis used when pronouncing a specific word or syllable. For occasion, in the word class, the primary syllable (cat-) is careworn. An adjective that describes the qualities of an individual or factor (e.g. an expensive automotive, a slender woman).

For instance if we now have two http://asu.edu distinct occasions of ingesting a bottle of wine we get an occasion of ingesting two bottles of wine. For verbal predicates Krifka assumes that they are occasion denoting and moreover that thematic relations like agent, theme, and so forth are represented as in (31-b). Thematic roles of verbs are crucial since they decide which verbs give raise to the drink wine versus drink a bottle of winedistinction.

A construction, as a outcome of Russell, Wiener and Kamp, exhibits that the time line may be constructed from occasions. We will briefly sketch this development earlier than we transfer on to lexical and grammatical aspect. In English, verb forms are usually used to signal the time when an motion or event occurs or a state holds. Thus in , Johnâs promise was given earlier than the current time, within the promising is simultaneous with the current time and says that the studentâs work on his thesis will occur at some time after the current. It ought to be noted that the verb forms used to express temporal info may be used to signal info that is not purely temporal.

An event construction $$\langle E, P, O\rangle$$ merely consists of a set of occasions $$E$$ and the relations $$P$$ and $$O$$ and is characterised by axioms A1â7. Kamp took up Russellâs concepts and slightly modified them. The construction takes event constructions as primitives from which the construction of time, i.e., a complete order, can be derived. Sentence presents the writing of a letter as a single entire and it’s clear that this activity was completed when the letter was dropped at the publish workplace. By distinction, focuses on an ongoing exercise of letter writing in the past which isn’t completed due to the interruption.

Updated: May 11, 2022 — 12:10 am


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Order KöMaL! Competitions Portal

P. 4389. In order to make a flat coil, which have 200 turns, a piece of wire of negligible resistance is wound around a rectangular frame of area 50 cm2. The frame is rotated about one of the symmetry axis of the rectangle, which is in the plane of the rectangle. The frame is in magnetic field of induction 0,1 T, which is perpendicular to the axis of rotation. A bulb rated at 6,3 V, and 3 W, and a coil of resistance 10 , and of inductance 50 mH, are connected to the ends of the rotating coil in series.

a) What should the number of revolutions of the coil be, in order to gain 6.3 V voltage across the bulb.

b) At what phase angle does the current lag behind the voltage across the 50 mH coil?

(5 points)

Deadline expired on 12 December 2011.

Google Translation (Sorry, the solution is published in Hungarian only.)

Megoldás. $\displaystyle a)$ 26 Hz-nél.

$\displaystyle b)$ $\displaystyle 39{,}7^\circ.$

Statistics on problem P. 4389.
 32 students sent a solution. 5 points: Bingler Arnold, Bolgár Dániel, Filep Gábor, Kollarics Sándor, Koncz Gabriella, Kovács 444 Áron, Nagy Lajos, Nárai Ádám, Sárvári Péter, Virágh Anna. 4 points: Czipó Bence, Fülep Andrea , Garami Anna, Horváth Dániel, Jenei Márk, Juhász Péter, Laczkó Zoltán Balázs, Mázik László, Papp Roland, Pataki Bálint Ármin, Szabó 928 Attila, Szilágyi 585 Dezsõ, Tilk Bence, Váncsa Szilárd, Ürge László. 3 points: 5 students. 2 points: 2 students.

• Problems in Physics of KöMaL, November 2011

•  Our web pages are supported by: ELTE


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# Tetrahedron with Orthogonal Perpendiculars to Opposite Edges

Tetrahedron in which common perpendiculars of the opposite edges are pairwise orthogonal is isosceles; and vice versa.

### Proof

Let $AB$ and $CD$ be a pair of opposite edges of a tetrahedron $ABCD.$ Let $\alpha$ be the plane through $AB$ parallel to $CD,$ and $\gamma$ the plane through $CD$ parallel to $AB.$ The common perpendicular, say $h,$ to $AB$ and $CD$ is perpendicular to both of these planes.

These two planes define the opposite faces of the parallelepiped associated with the tetrahedron. Similar arguments apply to the other two pairs of opposite faces.

As a general rule, the normals to two planes are perpendicular only if the planes are perpendicular. Thus the common perpendiculars of the opposite edges are pairwise orthogonal iff this is true of the pairs of opposite faces of the parallelepiped associated with the tetrahedron.

It follows that the parallelepiped is a cuboid, which is equivalent to its bing isosceles.

This problem has been given in 1984 at a Romanian selection test for the Balkan Mathematical Olympiad.

### References

1. T. Andreescu, R. Gelca, Mathematical Olympiad Challenges, Birkhäuser, 2004, 5th printing, 1.8 (p. 25)

 62645580


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# In tikz, how to pass named coordinates to a named pic?

I have found problems when passing a named argument to a pic, but only when the pic is also named.

Follows the illustrative code and the problem occurs in the line tagged with %NOT OK. That line gives an error because a name to the pic is given (MyLink 3); but if the pic name is omitted (line tagged %OK 2) the code works as expected. On the other hand, keeping the pic named, if using the explicit value for the coordinate (instead of name P1 as shown), the code works without errors (line tagged %OK 1).

Just a remark on the output that may give a clue to the solution but that is escaping me: if the coordinates are passed explicitly, they are considered in a local reference frame of the pic; if, on the other hand, they are passed as a defined name (after the \coordinate command) they are considered in the global reference frame of the entire tikzpicture!

Any help for this problem (or a workaround) is welcome. Thanks in advance.

\documentclass[]{standalone}
\usepackage{tikz}
\tikzset{
code={
\coordinate (-start) at (0,0);
\coordinate (-end)   at (#1);
\draw [thick] (-start) -- (-end);
\draw [fill=white] (-start) circle (3pt);
\draw [fill=white] (-end)   circle (3pt);
}
}
}
\begin{document}
\begin{tikzpicture}
\draw[step=5pt,help lines] grid (40pt,40pt);

\coordinate (P1) at (-10:25pt);

\end{tikzpicture}

\end{document}

• The commented line works fine with the latest version of TikZ (3.1.5b). – Torbjørn T. Sep 14 '20 at 22:06
• Thanks a lot. Indeed, I tested in version 3.0.1a. I need to upgrade my version! – vsantos Sep 15 '20 at 9:56

[This is really a long comment...that is too long for a comment]

Like Torbjørn, the code in the OP compiles for me so I wonder if the problem is really that the OP does not seem to be using the correct syntax for the \pic command. According to the manual, the syntax is:

\path ... pic ⟨foreach statements⟩ [⟨options⟩] (⟨prefix⟩) at(⟨coordinate⟩) :⟨animation attribute⟩={⟨options⟩} {⟨pic type⟩} ...;

In particular, when it is supplied the prefix should come before the coordinates. As such the MWE should look more like:

\documentclass[border=2mm,tikz]{standalone}
\tikzset{
code={
\coordinate (-start) at (0,0);
\coordinate (-end)   at (#1);
\draw [thick] (-start) -- (-end);
\draw [fill=white] (-start) circle (3pt);
\draw [fill=white] (-end)   circle (3pt);
}
}
}
\begin{document}

\begin{tikzpicture}
\draw[step=5pt,help lines] grid (40pt,40pt);

\coordinate (P1) at (-10:25pt);

\end{tikzpicture}

\end{document}

I have changed the the third pic to use MyLink 2. With the code in the OP, the red link is just drawn over the top of the green link.

This produces:

What is actually happening here is that:

• the first (black) pick draws a black link from the origin to the point (60:30pt)
• the second pic draws a blue link from the end of the black link to the relative coordinate that is -10 degrees and 25pt away
• the third pic draws a green link from the end of the blue link to the absolute coordinate P1.
• the fourth pic draws a link from the end of the first black link to the absolute coordinate P1
• Thanks for the complete answer. Is still fails on my setup, but I believe that the problem is due to my older Tikz version, as Torbjørn stated in the comment above. – vsantos Sep 15 '20 at 10:02


text:
# Addition and Subtraction of Surds

In addition and subtraction of surds we will learn how to find the sum or difference of two or more surds only when they are in the simplest form of like surds.

For addition and subtraction of surds, we have to check the surds that if they are similar surds or dissimilar surds.

Follow the following steps to find the addition and subtraction of two or more surds:

Step I: Convert each surd in its simplest mixed form.

Step II: Then find the sum or difference of rational co-efficient of like surds.

Step III: Finally, to get the required sum or difference of like surds multiply the result obtained in step II by the surd-factor of like surds.

Step IV: The sum or difference of unlike surds is expressed in a number of terms by connecting them with positive sign (+) or negative (-) sign.

If the surds are similar, then we can sum or subtract rational coefficients to find out the result of addition or subtraction.

$$a\sqrt[n]{x}\pm b\sqrt[n]{x} = (a\pm b)\sqrt[n]{x}$$

The above equation shows the rule of addition and subtraction of surds where irrational factor is $$\sqrt[n]{x}$$ and a, b are rational coefficients.

Surds firstly need to be expressed in their simplest form or lowest order with minimum radicand, and then only we can find out which surds are similar. If the surds are similar, we can add or subtract them according to the rule mentioned above.

For example we need to find the addition of $$\sqrt[2]{8}$$, $$\sqrt[2]{18}$$.

Both surds are in same order. Now we need find express them in their simplest form.

So $$\sqrt[2]{8}$$ = $$\sqrt[2]{4\times 2}$$ = $$\sqrt[2]{2^{2}\times 2}$$ = $$2\sqrt[2]{2}$$

And $$\sqrt[2]{18}$$ = $$\sqrt[2]{9\times 2}$$ = $$\sqrt[2]{3^{2}\times 2}$$ = $$3\sqrt[2]{2}$$.

As both surds are similar, we can add their rational co-efficient and find the result.

Now $$\sqrt[2]{8}$$ + $$\sqrt[2]{18}$$ = $$2\sqrt[2]{2}$$ + $$3\sqrt[2]{2}$$ = $$5\sqrt[2]{2}$$.

Similarly we will find out subtraction of $$\sqrt[2]{75}$$, $$\sqrt[2]{48}$$.

$$\sqrt[2]{75}$$= $$\sqrt[2]{25\times 3}$$= $$\sqrt[2]{5^{2}\times 3}$$= $$5\sqrt[2]{3}$$

$$\sqrt[2]{48}$$ = $$\sqrt[2]{16\times 3}$$ = $$\sqrt[2]{4^{2}\times 3}$$= $$4\sqrt[2]{3}$$

So $$\sqrt[2]{75}$$ - $$\sqrt[2]{48}$$ = $$5\sqrt[2]{3}$$ - $$4\sqrt[2]{3}$$ = $$\sqrt[2]{3}$$.

But if we need to find out the addition or subtraction of $$3\sqrt[2]{2}$$ and $$2\sqrt[2]{3}$$, we can only write it as $$3\sqrt[2]{2}$$ + $$2\sqrt[2]{3}$$ or $$3\sqrt[2]{2}$$ - $$2\sqrt[2]{3}$$. As the surds are dissimilar, further addition and subtraction are not possible in surd forms.

Examples of Addition and Subtraction of Surds:

1. Find the sum of √12 and √27.

Solution:

Sum of √12 and √27

= √12 + √27

Step I: Express each surd in its simplest mixed form;

= $$\sqrt{2\cdot 2\cdot 3}$$ + $$\sqrt{3\cdot 3\cdot 3}$$

= 2√3 + 3√3

Step II: Then find the sum of rational co-efficient of like surds.

= 5√3

2. Simplify $$3\sqrt[2]{32}$$ + $$6\sqrt[2]{45}$$ - $$\sqrt[2]{162}$$ - $$2\sqrt[2]{245}$$.

Solution:

$$3\sqrt[2]{32}$$ + $$6\sqrt[2]{45}$$ - $$\sqrt[2]{162}$$ - $$2\sqrt[2]{245}$$

= $$3\sqrt[2]{16\times 2}$$ + $$6\sqrt[2]{9\times 5}$$ - $$\sqrt[2]{81\times 2}$$ - $$2\sqrt[2]{49\times 5}$$

= $$3\sqrt[2]{4^{2}\times 2}$$ + $$6\sqrt[2]{3^{2}\times 5}$$ - $$\sqrt[2]{9^{2}\times 2}$$ - $$2\sqrt[2]{7^{2}\times 5}$$

= $$12\sqrt[2]{2}$$ + $$18\sqrt[2]{5}$$ - $$9\sqrt[2]{2}$$ - $$14\sqrt[2]{5}$$

= $$3\sqrt[2]{2}$$ + $$4\sqrt[2]{5}$$

3. Subtract 2√45 from 4√20.

Solution:

Subtract 2√45 from 4√20

= 4√20 - 2√45

Now convert each surd in its simplest form

= 4$$\sqrt{2\cdot 2\cdot 5}$$ - 2$$\sqrt{3\cdot 3\cdot 5}$$

= 8√5 - 6√5

Clearly, we see that 8√5 and 6√5 are like surds.

Now find the difference of rational co-efficient of like surds

= 2√5.

4. Simplify $$7\sqrt[3]{128}$$ + $$5\sqrt[3]{375}$$ - $$2\sqrt[3]{54}$$ - $$2\sqrt[3]{1029}$$.

Solution:

$$7\sqrt[3]{128}$$ + $$5\sqrt[3]{375}$$ - $$2\sqrt[3]{54}$$ - $$2\sqrt[3]{1029}$$

= $$7\sqrt[3]{64\times 2}$$ + $$5\sqrt[3]{125\times 3}$$ - $$\sqrt[3]{27\times 2}$$ - $$2\sqrt[3]{343\times 3}$$

= $$7\sqrt[3]{4^{3}\times 2}$$ + $$5\sqrt[3]{5^{3}\times 3}$$ - $$\sqrt[3]{3^{3}\times 2}$$ - $$2\sqrt[3]{7^{3}\times 3}$$

= $$28\sqrt[3]{2}$$ + $$25\sqrt[3]{3}$$ - $$3\sqrt[3]{2}$$ - $$14\sqrt[3]{3}$$

= $$25\sqrt[3]{2}$$ + $$11\sqrt[3]{3}$$.

5. Simplify: 5√8 - √2 + 5√50 - 2$$^{5/2}$$

Solution:

5√8 - √2 + 5√50 - 2$$^{5/2}$$

Now convert each surd in its simplest form

= 5$$\sqrt{2\cdot 2\cdot 2}$$ - √2 + 5$$\sqrt{2\cdot 5\cdot 5}$$ - $$\sqrt{2^{5}}$$

= 5$$\sqrt{2\cdot 2\cdot 2}$$ - √2 + 5$$\sqrt{2\cdot 5\cdot 5}$$ - $$\sqrt{2\cdot 2\cdot 2\cdot 2\cdot 2}$$

= 10√2 - √2 + 25√2 - 4√2

Clearly, we see that 8√5 and 6√5 are like surds.

Now find the sum and difference of rational co-efficient of like surds

= 30√2

6. Simplify $$24\sqrt[3]{3}$$ + $$5\sqrt[3]{24}$$ - $$2\sqrt[2]{28}$$ - $$4\sqrt[2]{63}$$.

Solution:

$$24\sqrt[3]{3}$$ + $$5\sqrt[3]{24}$$ - $$2\sqrt[2]{28}$$ - $$4\sqrt[2]{63}$$

= $$24\sqrt[3]{3}$$ + $$5\sqrt[3]{8\times 3}$$ - $$2\sqrt[2]{4\times 7}$$ - $$4\sqrt[2]{9\times 7}$$

=  $$24\sqrt[3]{3}$$ + $$5\sqrt[3]{2^{3}\times 3}$$ - $$2\sqrt[2]{2^{2}\times 7}$$ - $$4\sqrt[2]{3^{2}\times 7}$$

= $$24\sqrt[3]{3}$$ + $$10\sqrt[3]{3}$$ - $$4\sqrt[2]{7}$$ - $$12\sqrt[2]{7}$$

= $$34\sqrt[3]{3}$$ - $$16\sqrt[2]{7}$$.

7. Simplify: 2∛5 - ∛54 + 3∛16 - ∛625

Solution:

2∛5 - ∛54 + 3∛16 - ∛625

Now convert each surd in its simplest form

= 2∛5 - $$\sqrt[3]{2\cdot 3\cdot 3\cdot 3}$$ + 3$$\sqrt[3]{2\cdot 2\cdot 2\cdot 2}$$ - $$\sqrt[3]{5\cdot 5\cdot 5\cdot 5}$$

= 2∛5 - 3∛2 + 6∛2 - 5∛5

= (6∛2 - 3∛2) + (2∛5 - 5∛5), [Combining the like surds]

Now find the difference of rational co-efficient of like surds

= 3∛2 - 3∛5

8. Simplify $$5\sqrt[2]{7}$$ + $$3\sqrt[2]{20}$$ - $$2\sqrt[2]{80}$$ - $$3\sqrt[2]{84}$$.

Solution:

$$5\sqrt[2]{7}$$ + $$3\sqrt[2]{20}$$ - $$2\sqrt[2]{80}$$ - $$3\sqrt[2]{84}$$

= $$5\sqrt[2]{7}$$ + $$3\sqrt[2]{4\times 5}$$ - $$2\sqrt[2]{16\times 5}$$ - $$3\sqrt[2]{16\times 6}$$

= $$5\sqrt[2]{7}$$ + $$3\sqrt[2]{2^{2}\times 5}$$ - $$2\sqrt[2]{4^{2}\times 2}$$ - $$3\sqrt[2]{4^{2}\times 6}$$

= $$5\sqrt[2]{7}$$ + $$6\sqrt[2]{5}$$ - $$8\sqrt[2]{5}$$ - $$12\sqrt[2]{6}$$

= $$5\sqrt[2]{7}$$ - $$2\sqrt[2]{5}$$ - $$12\sqrt[2]{6}$$.

Note:

√x + √y ≠ $$\sqrt{x + y}$$ and

√x - √y ≠ $$\sqrt{x - y}$$

Surds