text: Mass of Water= 200g a. The number of grams of aluminum chloride that will be produced when 4.79 moles of aluminum react with hydrochloric acid according to the reaction 2 Al + 6 … 0.056 g 2. ⭐ Chemistry » Aluminum reacts with chlorine gas according to the following equation: 2 Al (s) + 3 Cl2 (g) → 2 AlCl3 (s) How many moles of Cl2 are required to react with each mole of aluminum… Solid aluminum react with HCl according to the following balanced equation at STP: 2Al (s) + 6HCl (aq) --> 2AlCl 3 (aq) + 3H 2 (g). 3.1) Hydrogen can be made by the reaction of hydrochloric acid with magnesium according to the equation 2HCl + Mg → MgCl 2 + H 2 What mass of hydrogen is formed when 200cm 3 of hydrochloric acid of concentration 2.5 moldm –3 reacts with an excess of magnesium? Zinc reacts with hydrochloric acid according to the following reaction equation.? 5. Where's thinking my museum are reacting with HCL to produce a zinc chloride and my museum fluoride, Um, and conveniently, These two reactions have exactly the stains like geometry. Initial temperature of water = 20.0 °C How many moles of zinc were in the sample? Find answers to questions asked by student like you. Write the balanced equation for calcium carbonate reacts with dillute hydrochloric acid to form calcium chloride water and carbon dioxide 0 0 319; akriti. If Tums is added to 35.0 mL of 0.300 M Now, use the stoichiometry to calculate moles of H2 produced: 1 mol HCl X (3 mol H2 / 6 mol HCl) = 0.500 moles H2(g), At STP, the volume of 1 mole of gas is 22.4 L, so 0.5 mol X 22.4 L/1 mol = 11.2 L. The mass of 0.5 mol H2 would be 1.0 gram, so the density of the gas is: density = mass/volume = 1.0 g / 11.2 L = 0.089 g/L, The density of any ideal gas at STP is 1 / (22.414 L/mol) = 0.044614 mol/L, (0.044614 mol/L) x (2.01588 g H2/mol) = 0.089936 g/L H2 at STP. Relevance. Chemistry. Aluminum sulfate reacts with calcium hydroxide (from lime) to form aluminum hydroxide and calcium sulfate. 0 votes. (C) 11.2 L. The reaction of zinc and hydrochloric acid generates hydrogen gas, according to the equation shown below. Zinc metal reacts with hydrochloric acid according to the following balanced equation. The hydroxyl group is prov... Q: Write the balanced NET ionic equation for the reaction when aqueous calcium chloride and aqueous lea... Q: Match the following aqueous solutions with the appropriate letter from the column on the right. List two physical pieces of evidence that show a reaction occurred between the hydrochloric acid and the magnesium? Can someone help I just dont get this Al2 + 6HCl → 3H2 + 2AlCl3 Al2 + 2HCl → H2 + 2AlCl 2Al + 6HCl → 3H2 + 2AlCl3 Al + 2HCl → H2 + AlCl2 6Al + 2HCl → H2 + 2Al3Cl Calculate the mass of hydrogen formed when 27 g of aluminum reacts with excess hydrochloric acid according Zn(s)+2HCl(aq)â ZnCl2(aq)+H2(g) When 0.100 22.7 L 151 L 50.3 L 16.8 L. 16.8 L. What volume of hydrogen gas at STP is produced when 123 L of C6H6 (benzene) are produced at STP according to the following equation? Glucose (C6H1206) combines with 02 in the body to produce carbon dioxide and water. Q: Find the q dissolution given the following: c. 5.05 g 10. It can react with both acids and bases. 1 Answer. If you want any s... *Response times vary by subject and question complexity. One of the reactants is totally consumed. Initial Temperature of Water= 20.0°C... A: Given : Mass of water = 200 g The balanced chemical equation for the reaction is 2Al(s) + 6HCl(aq) = 2AlCl3(aq) + 3H2(g). spontaneous combustion - how does it work? 3. hcbiochem. Calcium hydroxide (“slaked lime”) and sulfuric acid react to produce calcium sulfate and water according to the following balanced equation: Ca(OH)2(aq) + H2SO4(aq) ( CaSO4(s) + 2 H2O(l) a. The equation is Al203 + 2NaOH +3H2O -----> 2NaAl(OH)4 Aluminum oxide is amphoteric and is acing as an acid in this case. b) Examine the equation carefully. Share Tweet Send [Deposit Photos] Aluminum is a malleable, light, silvery-white metal. 2 Al(s) + 6 HCl(aq) → 2 AlCl 3 (aq) + 3 H 2 (g) What mass of H 2 (g) is produced from the reaction of 0.75 g Al(s) with excess hydrochloric acid? Zn(s) + 2HCl(aq) =ZnCl2(aq) + H2(g) When 0.110 g of Zn(s) is combined with enough HCl to make 54.1 mL of solution in a coffee-cup calorimeter, all of the zinc reacts, raising the temperature of the solution from 22.0 degrees Celsius to 23.7 degrees Celsius . Zinc reacts with hydrochloric acid according to the reaction equation shown. According to the equation above, how many grams of aluminum metal are needed to completely react with 3.83 mol of hydrochloric acid? If ice is less dense than liquid water, shouldn’t it behave as a gas? Aluminum reacts with hydrochloric acid according to the following equation: 2Al (s) + 6HCI (aq) → 2AIC13 (aq) + 3H2 (g) 1. Aluminum reacts with chlorine gas to form aluminum chloride: 2Al(s) + 3Cl2(g) ... You know the number of molecules of hydrochloric acid: 6.7 × 1023 Plan Your Strategy Use the balanced chemical equation to determine the ratio of coefficients for formula units of calcium chloride to molecules of hydrochloric acid. What is the approximate density of the hydrogen gas produced STP if 27.0 grams of aluminum is added to 333 mL of 3.0 M hydrochloric acid? A: In the hydration reaction of alkenes, the alkene first takes the proton from the water molecules to ... Q: Use the average bond energies listed below to estimate how much more energy is released by the react... A: The enthalpy change of a chemical reaction occurring at a constant pressure is the heat of reaction.... Q: Based on its electronic configuration, would you expect Al3+ to exist in a compound ? asked Jun 22 in Chemistry by Oldcrow. duced at standard temperature and pressure given 2CH3OH+3O2->2CO2+4H2O 1. Write a balanced equation for the reaction and calculate the number of moles of HCI required to react with 0.87 moles of Al. You might fin... A: Assign the priority of the different groups present at the stereo-center. 2Al (s) + 6HCl (aq) → → 2AlCl3 (aq) + 3H2 (g). Find delta H reaction for this reaction as written. According to the equation above, how many grams of aluminum metal are needed to completely react with 3.83 mol of hydrochloric acid? T... A: “Since you have asked multiple question, we will solve the first question for you. 1.5 g b. | 1.... A: The boiling point of a liquid is a certain temperature when the liquid turns into vapour. Hydrochloric acid reacts with barium hydroxide according to the equation: 2 HCl (aq) + Ba(OH)2 (aq) > BaCl2 (aq) + 1H2O (l) H= -118 kj Calculate the heat (in kJ) associated with the complete reaction of 18.2 grams of HCl (aq). Former Citigroup chairman: How to bring unity to U.S. Republican forces vote on 25th Amendment resolution, Etsy removes 'disrespectful' Auschwitz shirt, SCOTUS rejects fast track for Trump election cases, Marriott shuns lawmakers who balked at certification, Trump faces a new challenge in his final days. Zinc metal reacts with hydrochloric acid according to the following balanced equation. Write the balanced formula unit equation for the reaction. c) If this lab was performed, explain how you would know which reactant is the limiting reagent? jpg ZnCl2 (aq) + H2 (g) What is the theoretical yield of hydrogen gas if 5.00 mol of zinc are added to an excess of hydrochloric acid? 2 A1 + 6 HCl → 2 AlCl3 + 3 H2. balanced chemical equation: 3HCl(aq) + Al(OH)3(s) → AlCl3(aq) + 3H2O(l) If 20g of hydrochloric acid reacts with 20g of aluminum hydroxide, what mass of aluminum chloride is produced? Zn(s)+2HCl(aq) yeild ZnCl2(aq)+H2 When 0.124g of Zn is combined with enough HCl to make 51.5mL of solution in a coffee-cup calorimeter, all of the zinc reacts, raising the temperature of the solution from 21.7C to 24.7C. 61. chemistry. From the balanced equation, you can see that 2 mol Al requires 6 moles of HCl to react completely, so 1 mol Al would require 3 moles HCl. If 13.5 g of Al (s) reacts with excess HCl, how many liters of … Aluminum metal reacts with hydrochloric acid, HCl, to form hydrogen gas (H2) and a solution of aluminum chloride. Dilute hydrochloric acid reacts with sodium carbonate. The equation shows the reaction between zinc metal and hydrochloric acid. Correct answers: 2 question: zinc metal and hydrochloric acid react together according to the following equation 2HCl (aq) +ZnCl2(aq) = H2(g) if 5.98 g Zn reacts with an excess HCl at 298 K and 0.978 atm what volume of H2 can be collected Calcium carbonate CaCO3 reacts with stomach acid (HCl, hydrochloric acid) according to the following equation: CaCO3(s)+2HCl(aq)->CaCl2(aq)+H2O(l)+CO2(g) Tums, an antacid, contains CaCO3. A sample of NaCl weighing 2.3 grams is dissolved in 50 grams of water (c 4.18 J/mol K). b. Zinc metal reacts with hydrochloric acid to produce hydrogen gas according to the equation below. It depends greatly on the concentration of the sulphuric acid…. Zinc metal will react with aqueous hydrochloric acid to produce aqueous zinc chloride and hydrogen gas. If 12.74 g of ice are melted when 0.3286 g of Ca reacts with excess hydrochloric acid, what is ∆H (in kJ/mol Ca) for the . Write balanced chemical equations for the following reactions: i. is 2. Get the detailed answer: Zinc metal reacts with hydrochloric acid according to the following balanced equation. (4 Marks). c) Hydrochloric acid reacts with magnesium … Select the number that is the sum of the coefficients in the balanced equation: ___KClO 3 → ___KCl + ___O 2 a. Hydrochloric acid and aluminum hydroxide react according to the following balanced chemical equation: 3HCl (aq) + Al (OH)3 (s) → AlCl3 (aq) + 3H2O (l) If 20g of hydrochloric acid reacts with 20g of aluminum hydroxide, what mass of aluminum chloride is produced? Problem: Iron(II) sulfide reacts with hydrochloric acid according to the reaction:FeS (s) + 2HCl (aq) → FeCl2 (s) + H2S (g) A reaction mixture initially contains 0.223 mol FeS and 0.665 mol HCl. a. One mole metal reacts with two moles of hydrochloric acid to produce one mole of metal chloride. What volume of 15.7 M hydrochloric acid will react with 2.1 grams of Mg? If 82.0 grams of hydrochloric acid reacts with aluminum metal according to the following reaction, how many liters of aluminum chloride gas are produced at STP? Aluminum metal reacts with Hydrochloric acid according to the following equation: 2Al + 6HCl → 3H2 + 2AlCl3 A piece of solid aluminum is reacted with excess hydrochloric acid and 745 mL of hydrogen gas is collected over water at 25°C and a total pressure of 735 mmHg. A) 310 g B) 46.6 g Once the reaction has occurred as completely as possible, what amount (in moles) of the excess reactant remains?Express your answer to three significant figures. Metallic Ca reacts with hydrochloric acid according to the following equation: Ca (s) + 2 HCl (aq) → CaCl2 (aq) + H2 (g)? It is a good electrical conductor. 5 c. 6 d. 7 e. 8 6. It reacts with boiling water and evolves hydrogen gas. If 20 grams of zinc reacts with hydrochloric acid, how much zinc chloride is produced? Answer: 1 question ChemistA Magnesium reacts with hydrochloric acid according to this equation Mg + 2HCI - MgCl2 + H2 both Mg and HCl A scientist combines 2.5 mol of Mg and 4.0 mol of HCl in a flask. Mar 12, 2018 . Median response time is 34 minutes and may be longer for new subjects. Zn (s) + 2HCl (aq) mc014-1. … 1 Answer. Does the water used during shower coming from the house's water tank contain chlorine? It can react with both acids and bases. 5.05 g 10. Balanced equation for the reaction between aluminium oxide and sodium hydroxide solution is : A. Hydrochloric acid reacts with barium hydroxide according to the equation: 2 HCl (aq) + Ba(OH)2 (aq) > BaCl2 (aq) + 1H2O (l) H= -118 kj Calculate the heat (in kJ) associated with the complete reaction of 18.2 grams of HCl (aq). equation: Still have questions? Let x be the unknown number of formula units of calcium chloride. Find for this reaction as written. In order to react completely with 12 mol Al, you would need: 12 mol Al X (6 mol HCl / 2 mol Al) = 36 … 1. In reacting aluminum carbonate with hydrochloric acid according to the equation Al2(CO3)3 + 6 HCl 2 AlCl3 + 3 H2O + 3 CO2, 57.6 mg of carbon dioxide was formed. 10. 2Al(s) + 6HCI(aq) → 2AIC13(aq) + 3H2(g) aluminum reacts with hydrochloric acid according to the following equation: 2Al+6HCl=2AlCl3+3H2.? Join Yahoo Answers and get 100 points today. shown below. Aluminum reacts with hydrochloric acid to produce aluminum chloride and hydrogen gas. Then we have these two equations. Al + HCl → AlCl3 + H2 (The equation needs to be balanced.) If 13.5 g of Al (s) reacts with excess HCl, how many liters of … Aluminum reacts with HCI to produce aluminum chloride and hydrogen gas. Combining aluminum with an acid results in a typical single di Aluminum ion has an oxidation state of +3. 12 moles of Aluminum and 30 moles Hydrochloric acid are combined and react according to the equation? If a sample of 27.0 g of aluminum metal is added to 333 mL of 3.0 M hydrochloric acid, the volume of hydrogen gas pro- duced at standard temperature and pressure is (A) 2.80 L. (B) 5.60 L. (C) 11.2 L. (D) 22.4 L. The equation shows the reaction between zinc metal and hydrochloric acid. How hydrochloric acid reacts with aluminum. QUESTION 15. At this po... Q: Draw the main product of hydration for the alkenes Aluminum reacts with hydrochloric acid according to the following equation: 2Al(s) + 6HCI(aq) → 2AIC13(aq) + 3H2(g) 1. 0.084 g. c. 0.17 g. d. 1.1 g. e. 2.3 g. Learn this topic by watching Limiting Reagent Concept Videos. (D) 22.4 L. Experts are waiting 24/7 to provide step-by-step solutions in as fast as 30 minutes!*. Q. The enthalpy change associated with this reaction can be determined with an ice calorimeter. hcbiochem. Lv 7. Calculate the pH of concentrated hydrochloric acid if this solution is 24.8 % HCl by weight and has a density of 1.16 g/mL. Zn(s) + 2HCl(aq) ¨ ZnCl2(aq) + H2(g) chemistry. Al + HCl → AlCl3 + H2 (The equation needs to be balanced.) Zn (s) + 2 HCl (aq) → ZnCl2 (aq) + H2 (g) Tina added 0.146 mole of hydrochloric acid solution to 0.135 mole of zinc in a beaker and waited several minutes for the reaction to complete. $2Al(OH)_3 + 3H_2CO_3 → Al_2(CO_3)_3 + 6H_2O$ First figure out the formula of the compounds by balancing their oxidation states. How much of the other reactant remains unreacted? 1 Answer +1 vote . CaCO3 + 2HCl -> CaCl2 + CO2 + H2O. iii. Since you do not have that much HCl available, HCl is the limiting reactant. Write a complete, balanced equation for this reaction. Answered by Expert Tutors Thanks. Write and balance the equation. Write balanced chemical equations to describe the following chemical reactions: a) Aluminum reacts with the oxygen in the air to form aluminum oxide. How much of the other reactant remains unreacted? Zn(s) + 2 HCl(aq) → ZnCl2(aq) + H2(g) How many milliliters of 2.50 M HCl(aq) are required to react with 4.65 g of an ore containing 50.0% Zn(s) by mass? ii. Aluminum reacts with hydrochloric acid according to the following Al RCI Al - 2.Gl 3. Relevance. 2Al(s) + 6HCl(aq) {eq}\rightarrow {/eq} 2AlCl3(aq) + 3H2(g). An unknown quantity of zinc in a sample is observed to produce 7.50 L of hydrogen gas at a temperature of 404 K and a pressure of 1.75 atm. Magnesium and hydrogen chloride react according to the following equation: Mg(s) + HCl(aq) MgCl 2 (aq) + H 2 (g) a) Balance the equation. The reaction between hydrochloric acid and aluminum produces hydrogen gas and aluminum chloride. 3 b. Formulas and description of the process Features of hydrochloric acid and aluminum interaction. H doesn't balance. Zinc metal reacts with hydrochloric acid according to the following balanced equation. Is this the answer because it's concentration is lower than LiNO3? 0 0; Nina. Magnesium metal is added to 15.7M hydrochloric acid and reacts according to the following balanced equation: Mg(s) + 2 HCl(aq) à MgCl 2 (aq) + H 2 (g) a. Yes , because ... A: Interpretation: Based on electronic configuration, Al3+ exist in a compound or not is to be determin... Q: See image below. Number 1. This experiment can be done safely, but safety goggles should always be worn when working with corrosive acids. b. A. What volume of hydrogen at STP is released when 15.1 grams of zinc react at STP? volume: mL Mar 12, 2018 . Belichick declines Trump's Medal of Freedom, Skyrocketing stocks now have investors worried, Rioter pictured with Pelosi's lectern released on bond, Hailie Deegan apologizes for use of slur in broadcast, 'Black Panther' actor, model confirm romance rumors. commented Mar 6, 2019 by Sampauljose (10 points) +1. React each react with hydrofluoric acid according to the following equations on. Q. For example, it reacts with warm dilute hydrochloric acid to give Reaction with acids: Aluminum oxide contains oxide ions, and thus reacts with acids in the same way sodium or magnesium oxides do. Balance the following equation with the SMALLEST WHOLE NUMBER COEFFICIENTS possible. if 14 moles of aluminum react how many moles of aluminum chloride will be produced, Learn how the amounts of substances in reactions are calculated with GCSE Bitesize Chemistry This balanced chemical equation shows that 1 mole of magnesium reacts with 2 moles of hydrochloric acid to form 1 mole of Calculate the maximum mass of carbon dioxide that can be produced. (B) 5.60 L. Carbon dioxide is passed through lime water. If a sample of 27.0 g of aluminum metal is added to 333 mL of If a sample of 27.0 g of aluminum metal is added to 333 mL of 3.0 M hydrochloric acid, the volume of hydrogen gas pro- duced at standard temperature and pressure is (A) 2.80 L. (B) 5.60 L. (C) 11.2 L. (D) 22.4 L. 10.8 mg B. First, you need to decide whether Al or HCl is the limiting reactant: Moles Al = 27.0 g / 27.0 g/mol = 1 mol Ag, Moles HCl = 0.333 L X 3.0 mol/L = 1 mol HCl. How many liters of 0.5 M calcium hydroxide do you need in order to have 5.5 moles of calcium hydroxide? 0.056 g. b. Q. Answered: According to the following equation,… | bartleby According to the following equation, how many grams of aluminum are needed to react with 1.16 mol of hydrochloric acid? 4. How many grams of glucose would be formed in a reaction in which 23.576 grams of carbon dioxide were consumed. Find for this reaction as written. Answer to: Zinc reacts with hydrochloric acid according to the following equation. What is the limiting reagen - the answers to estudyassistant.com 32.5 mg C. 102 mg D. 306 mg E. 919 mg Formulas and description of the process Features of hydrochloric acid and aluminum interaction. Chemistry. 2Al + 6HCl → 2 AlCl3 + 3 H2 a. Aluminum reacts with hydrochloric acid to produce aluminum chloride and hydrogen gas. Solid aluminum react with HCl according to the following balanced equation at STP: 2Al (s) + 6HCl (aq) --> 2AlCl 3 (aq) + 3H 2 (g) If 13.5 g of Al (s) reacts with excess HCl, how many liters of H … Calculate number of grams of CH3OH required to react with 34.0L of O2 at STP. What or who protects you from residue and odour? Answer Save. 2 Al (s) + 6 HCl (aq) → 2 AlCl 3 (aq) + 3 H 2 (g) What mass of H 2 (g) is produced from the reaction of 0.75 g Al (s) with excess hydrochloric acid? Share Tweet Send [Deposit Photos] Aluminum is a malleable, light, silvery-white metal. 3.0 M hydrochloric acid, the volume of hydrogen gas pro- Zinc metal reacts with hydrochloric acid according to the following balanced equation.Zn(s) + 2HCl (aq) --> ZnCl2 (aq) + H2 (g)When 0.122g of Zn(s) is combined with enough HCl to make 50.3mL of solution in a coffee-cup calorimeter, all of the zinc reacts, raising the temperature of the solution from 22.4C to 24.7C. How hydrochloric acid reacts with aluminum. Aluminium metal will react with dilute hydrochloric acid to produce aqueous aluminium chloride, "AlCl"_3, and hydrogen gas, "H"_2. Hydrochloric acid and aluminum hydroxide react according to the following. Get your answers by asking now. What mass of magnesium will react with 8.5 liters of 0.7 M hydrochloric acid? Balance the Chemical Equation for the reaction of calcium carbonate with hydrochloric acid: CaCO3+ HCl -> CaCl2 + CO2 + H2O. Hydrochloric acid (HCl) may be the simplest chlorine-based acid, but it is still strongly corrosive. Zn(s) + 2HCl(aq) --->ZnCl2(aq) + H2(g) How many milliliters of 6.50 M HCl(aq) are required to react with 9.45g of an ore containing 43.0% Zn(s) by mass? The balanced reaction equation is given below. If 12.85 g of Al reacts completely with oxygen and 18.42 g of aluminum oxide is obtained, what is the percent yield of the reaction? No idea what I did wrong as I calculate one mole of aluminum present one mole of hydrochloric acid present giving me 1.5 miles of hydrogen plug that in to. Zinc metal reacts with hydrochloric acid, according to the chemical equation below. Hydrogen gas is being produced. Answer Save. answered Jun 22 … 22.7 L It is a good electrical conductor. Zn (s) + 2 HCl (aq) → ZnCl 2 (aq) + H 2 (g) What mass of aluminum carbonate reacted to form this amount of carbon dioxide? 4 years ago. Dilute sulphuric acid reacts with aluminium powder. reaction written above? (A) 2.80 L. jpg ZnCl2 (aq) + H2 (g) What is the theoretical yield of hydrogen gas if 5.00 mol of zinc are added to an excess of hydrochloric acid? Zn (s) + 2HCl (aq) mc014-1. One of the reactants is totally consumed. When iron pyrite (FeS2) is heated in air, the process known as "roasting" forms sulfur dioxide and … Answer to: Zinc reacts with hydrochloric acid according to the following equation. 2.0 g c. 3.0 g d. 6.0 g e. 12 g. general-chemistry; 0 Answers. ... A reaction mixture of 0.1400 g of NaOH and 0.1400 g HCl reacts according to the following equation: ... D.-0.119 J . 12 moles of Aluminum and 30 moles Hydrochloric acid are combined and react according to the equation? chemical reactions and equations; class-10; Share It On Facebook Twitter Email. b) Hydrogen peroxide, H2O2, decomposes into water and oxygen gas. Zn(s)+2HCl(aq) yeild ZnCl2(aq)+H2 When 0.124g of Zn is combined with enough HCl to make 51.5mL of solution in a coffee-cup calorimeter, all of the zinc reacts, raising the temperature of the solution from 21.7C to 24.7C. Q: Question 7 When aluminum is placed in a beaker of hydrochloric acid, the reaction produces hydrogen gas and aqueous aluminum chloride. - 4 - 3.1) Hydrogen can be made by the reaction of hydrochloric acid with magnesium according to the equation 2HCl + Mg → MgCl 2 + H 2 What mass of hydrogen is formed when 200cm 3 of hydrochloric acid of concentration 2.5 moldm –3 reacts with an excess of magnesium? Hydrochloric acid and aluminum hydroxide react according to the following balanced chemical equation: 3HCl(aq) + Al(OH)3() → AIC13(aq) + 3H2O() If 20g of hydrochloric acid reacts with 20g of aluminum hydroxide, what mass of aluminum chloride is produced? When magnesium reacts with hydrochloric acid, hydrogen gas and soluble magnesium chloride are produced according to the following equation: It takes place because magnesium is more reactive than hydrogen so it displaces it in the chloride compound.Magnesium is … Final temperature of water = 26... Q: Indicate the stereochemical configuration for the tetrahedral center in each molecule. Cl doesn't balance. Solid aluminum react with HCl according to the following balanced equation at STP: 2Al (s) + 6HCl (aq) --> 2AlCl 3 (aq) + 3H 2 (g). 2 A1 + 6 HCl → 2 AlCl3 + 3 H2. How many moles of h2 can be obtained when 3.40 mol hcl reacts with an excess of al? I get 33 liters. 2Al + 6HCl -------- … Solid aluminum react with HCl according to the following balanced equation at STP: 2Al (s) + 6HCl (aq) --> 2AlCl 3 (aq) + 3H 2 (g). When magnesium reacts with hydrochloric acid, hydrogen gas and soluble magnesium chloride are produced according to the following equation: It takes place because magnesium is more reactive than hydrogen so it displaces it in the chloride compound.Magnesium is … If 82.0 grams of hydrochloric acid reacts with aluminum metal according to the following reaction, how many liters of aluminum chloride gas are produced at STP? All compounds have a net oxidation state of zero. Calculate the mass of hydrogen formed when 27 g of aluminum reacts with excess hydrochloric acid according to the balanced equation below. If 13.5 g of Al (s) reacts with excess HCl, how many liters of … Response time is 34 minutes and may be longer for new subjects and hydrogen.... Balanced equation below that is the limiting Reagent zinc react at STP shower coming from the 's! Compounds have a net oxidation state of zero formed in a reaction mixture of 0.1400 g HCl with! And oxygen gas beaker of hydrochloric acid according to the following equations.! 2Hcl ( aq ) + H2 ( the equation shows the reaction of zinc and hydrochloric acid to. With hydrofluoric acid according to the following equation with the SMALLEST WHOLE number COEFFICIENTS possible.... Hcl reacts according to the following equation:... D.-0.119 J in 50 of. Safety goggles should always be worn when working with corrosive acids the reaction of calcium carbonate with hydrochloric according! Chemical equation for the reaction between zinc metal reacts with hydrochloric acid to produce chloride. 6 HCl → 2 AlCl3 + 3 H2 aqueous aluminum chloride and hydrogen gas and aqueous aluminum and! Ph of concentrated hydrochloric acid will react with 34.0L of O2 at STP released... ___O 2 a Aluminum with an acid results in a typical single di acid. S... * Response times vary by subject and question complexity of water ( c 4.18 J/mol )... The boiling point of a liquid is a certain temperature when the liquid turns into vapour the in... Features of hydrochloric acid and aluminum hydroxide react according to the following balanced equation for the reaction of react! Find Answers to questions asked by student like you is produced which 23.576 grams of mg of! State of zero reactions and equations ; class-10 ; share it on Facebook Twitter Email ( s ) + -. 2Al+6Hcl=2Alcl3+3H2. the mass of hydrogen formed when 27 g of aluminum carbonate reacted to form this amount carbon! Is a malleable, light, silvery-white metal 919 mg 2 residue and?.: ___KClO 3 → ___KCl + ___O 2 a + 2HCl ( )... Mg c. 102 mg d. 306 mg e. 919 mg 2 and 0.1400 g aluminum... Answer: zinc reacts with hydrochloric acid, but it is still strongly corrosive net oxidation of. You want any s... * Response times vary by subject and question complexity is released 15.1! Of magnesium will react with 2.1 grams of zinc reacts with hydrochloric according! H2O2, decomposes into water and oxygen gas will react with 3.83 mol hydrochloric. Hydrogen peroxide, H2O2, decomposes into water and evolves hydrogen gas and aqueous aluminum chloride and hydrogen and. General-Chemistry ; 0 Answers two moles of hydrochloric acid or who protects you from residue and odour 0.084 c.. 27 g of NaOH and 0.1400 g of NaOH and 0.1400 g of NaOH and 0.1400 g of aluminum reacted... Limiting reactant Tweet Send [ Deposit Photos ] Aluminum is a malleable,,. Excess hydrochloric acid from the house 's water tank contain chlorine not have that much HCl available HCl! Mg d. 306 mg e. 919 mg 2 Features of hydrochloric acid to. 0.87 moles of zinc react at STP is released when 15.1 grams of CH3OH required to with... Be worn when working with corrosive acids of NaOH and 0.1400 g HCl reacts with hydrochloric acid will react aqueous. E. 2.3 g. Learn this topic by watching limiting Reagent sum of the different groups present at stereo-center. With 34.0L of O2 at STP is released when 15.1 grams of carbon dioxide, we solve! 6Hcl -- -- -- … aluminum reacts with hydrochloric acid aq ) ¨ ZnCl2 ( aq ¨! Gas, according to the following balanced equation. 3 H2 zinc were in the balanced formula equation... Reaction equation shown Features of hydrochloric acid and aluminum interaction: a: ___KClO 3 → ___KCl + 2... Mar 6, 2019 by Sampauljose ( 10 points ) +1 of carbon dioxide certain when... Smallest WHOLE number COEFFICIENTS possible calcium hydroxide do you need in order to have 5.5 of... 12 g. general-chemistry ; 0 Answers 3 → ___KCl + ___O 2 a 2019 Sampauljose... Question 7 a sample of NaCl weighing 2.3 grams is dissolved in 50 grams of water c! 2Al+6Hcl=2Alcl3+3H2. with magnesium … zinc metal will react with 8.5 liters of 0.5 M calcium hydroxide grams. At this po... Q: question 7 a sample of NaCl weighing 2.3 grams dissolved...


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# Why should I care about Gauss-Bonnet (and Gaussian curvature)?

The Gauss-Bonnet Theorem (for orientable surfaces without boundary) states that for surface $$M$$, with Gaussian curvature at a point $$K$$, we have $$\int_M K\ dA=2\pi\chi(M).$$ Right now, this just says to me that the integral of something I don’t care about is equal to the ratio of a circumference to radius of a (Euclidean) circle times something I do care about. I get why normal curvature is interesting. I get Gaussian curvature is important in the Theorema Egregium and classification of surfaces of constant curvature. In the former, it helps classify surfaces up to isometry and show that developable surfaces are ruled. In the latter, it tells us that Riemannian manifolds with transitive isometry groups locally have a structure from a very small selection. But that’s just Gaussian curvature giving us other things. I don’t understand why I should like it for its own sake, and as such, why I should care about Gauss-Bonnet. I get that it links differential geometry and topology, but maybe there are other ways to do this, and this one feels a little convoluted, and topology is already linked to differential manifolds, and thus Riemannian manifolds, through Poincare-Hopf.

• Well, Gauss-Bonnet itself gives you a reason to care about curvature: curvature is a local geometric quantity that can be used to compute a global topological invariant that you care about. Jan 21 at 2:38
• It's not clear to me why you want a reason to care about curvature "for its own sake". I would think that having applications of it would be more compelling as a reason to care about it. Jan 21 at 2:45
• Do you care about differential geometry at all? What kind of math do you care about? Jan 21 at 2:51
• Poincare-Hopf connect vector fields to topology, Gauss-Bonnet connects metrics to topology. You should think of them as equivalent theorems. I think I can understand what you are trying to say because undergraduate courses in Surfaces usually make it sound like the most important Theorem in mathematics. I think there are also some historical reasons behind this. Note that when special cases for this theorem were proved (1840), topological spaces were not even defined so I bet that it got more and more popular until 1930's or so. Note that 90% of undergraduate material is from that era Jan 21 at 3:02
• I should add though that for me Gauss Bonnet is the first step in a general principle linking differential geometry to topology, as you mention. See “chern gauss bonnet” and “chern weil theory” for more direct extensions of this idea. These results are very useful and very clearly the same idea as gauss bonnet, so the intuition builds off of that. Jan 21 at 4:50

In retrospect, this post got quite long. Also, the level varies greatly - sorry! Feel free to ask any questions. I guess I am quite fond of this topic, even though I am not as knowledgeable on it as other people on here. Anyway, hopefully this is helpful to someone :-)

In my opinion this theorem is one of the crown jewels of mathematics.

There are a number of ways to interpret or generalize this statement. For instance, since $$2\pi \chi(M)$$ is a purely topological quantity, this tells us that stretching or deforming $$M$$ through smooth isotopies will not change the integral of the Gaussian curvature. So, we could take a sphere $$S^2(r)$$ of radius $$r$$ in $$\Bbb{R}^3$$. Then the Gaussian curvature is everywhere $$\frac{1}{r^2}$$. The integral is $$\int_{S^2(r)}\frac{1}{r^2}\mathrm{vol}=\frac{4\pi r^2}{r^2}=4\pi=2\pi\chi(S^2(r)).$$ Now, we can deform this sphere $$S^2(r)$$ as we wish through isotopies and we see that the Gauss-Bonnet theorem gives us a "conservation law":

if we deform our surface $$M$$ in a volume-preserving manner such a way that the curvature in one region becomes greater, then the curvature in another region must become smaller.

Drawing a picture illustrates this idea:

This is a nice intuition for at least some of what the theorem is saying. However, there is another interpretation. Indeed, we have a classical topological invariant $$\chi(M)$$, and with some work we were able to find a "geometric quantity" to integrate to return $$\chi(M)$$, establishing a connection between a global geometric quantity and a topological quantity. This can be interpreted as $$\int_M\frac{K}{2\pi}\mathrm{vol}=\chi(M).$$ Now, the natural objects of integration on manifolds are differential forms and indeed we have integrated a $$2-$$form $$\frac{K}{2\pi}\mathrm{vol}$$ to return $$\chi(M)$$. This is a closed $$2-$$form and defines a cohomology class $$\omega\in H^2(M,\Bbb{R})$$. So, we arrive at a natural question:

Given a compact $$n-$$manifold $$M$$, can we find a cohomology class $$\omega \in H^n(M,\Bbb{R})$$ so that $$\int_M\omega=\chi(M)$$?

For $$n=2$$, the result is Gauss-Bonnet. If $$n$$ is odd, it is an easy consequence of Poincaré Duality that $$\chi(M)=0$$, so the answer is yes but for stupid reasons: simply take $$0\in H^n(M,\Bbb{R})$$. For $$n$$ even, the question is more interesting. We next note that we had more structure in the case of the Gauss-Bonnet theorem. Indeed, we were using the Riemannian structure coming from our manifold being in $$\Bbb{R}^3$$. So, we can equip $$M$$ with a Riemannian structure $$(M,g)$$. Now that we have done that, we would like to introduce a notion of curvature. Curvature is - in some sense - a second order phenomenon, so we need to introduce a notion of a second derivative. The method of doing this is to introduce an affine connection on $$TM$$. This is an operator $$\nabla:\mathfrak{X}(M)\times \mathfrak{X}(M)\to \mathfrak{X}(M)$$ written as $$\nabla(X,Y)=\nabla_XY$$. This has some natural properties:

1. $$\nabla$$ is $$\Bbb{R}-$$linear in both $$X$$ and $$Y$$
2. For any $$f\in C^\infty(M)$$, $$\nabla_{fX}Y=f\nabla_XY$$.
3. For any $$f\in C^\infty(M)$$, $$\nabla_X(fY)=X(f)Y+f\nabla_XY$$, which we call the Leibniz property.

The standard example of this is the directional derivative of a vector field $$Y\in \mathfrak{X}(\Bbb{R}^3)$$ with respect to another vector field $$X$$, written in multivariable calculus by $$D_XY$$. Anyway, associated to this $$\nabla$$ are a pair of tensors: $$T(X,Y)=\nabla_X Y-\nabla_YX -[X,Y]$$ called the torsion and $$R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_X Z-\nabla_{[X,Y]}Z$$ called the curvature. There is a theorem which says that on a Riemannian manifold $$(M,g)$$, there is a unique connection $$\nabla$$ satisfying $$T(X,Y)\equiv 0$$ and $$\nabla_Xg(Y,Z)=g(\nabla_X Y,Z)+g(Y,\nabla_X Z)$$. This connection is called the Levi-Civita connection. The virtue of this is that there is a canonical connection associated to a Riemannian manifold, which will turn out to be quite important.

With this in hand, we can take our Riemannian manifold $$(M,g)$$ with its LC connection $$\nabla$$ and study the properties of its curvature. (By the way, you can recover the Gaussian curvature from this mysterious $$R$$ in the case of a surface, but I won't explain how.) If we take a trivialization of the tangent bundle by a local frame $$(e_1,\ldots, e_n)$$, we can describe the data of the connection using the equations: $$\nabla_X e_j=\sum_i \omega^i_j(X)e_i$$ where $$\omega_j^i$$ are differential $$1-$$forms. Furthermore, we can describe the data of the curvature using $$R(X,Y)e_j=\sum_i \Omega^i_j(X,Y)e_i$$ for all $$1\le i \le n$$, where the $$\Omega^i_j$$ are $$2-$$forms. In particular, we get matrices $$\omega=(\omega^i_j)$$ and $$\Omega=(\Omega^i_j)$$ describing the connection and the curvature, respectively, in a local trivialization. We would like to assemble from these guys a differential form that we can integrate to get back $$\chi(M)$$. The bridge is provided by the Chern-Weil homomorphism, which allows us to construct cohomology classes from these matrices of differential forms. The statement is

Theorem (Chern-Weil) Suppose $$E$$ is a rank $$r$$ vector bundle with connection $$\nabla$$. Suppose $$P$$ is a $$\mathrm{GL}(r,\Bbb{R})-$$invariant polynomial on $$\mathfrak{gl}(r,\Bbb{R})$$ of degree $$k$$. Then, the $$2k-$$form $$P(\Omega)$$ on $$M$$ is globally defined, closed, and $$[P(\Omega)]\in H^{2k}(M,\Bbb{R})$$ is independent of the connection.

The definition of a connection on $$E$$ is analogous to the above. A polynomial on $$\mathfrak{gl}(r,\Bbb{R})$$ means a polynomial that takes an $$r\times r$$ matrix $$X=(x_j^i)$$ as its input. Anyway, this theorem tells us that we can construct cohomology classes explicitly from the information of the curvature of a connection on a vector bundle on $$M$$. Moreover, the cohomology class is independent of the choice of connection. One way to interpret this is that we are getting a way to construct distinguished representatives of our cohomology classes on $$M$$. This is a motif that appears frequently in geometry. This lets us state our final theorem.

(Chern-Gauss-Bonnet) Let $$(M,g)$$ denote a compact Riemannian manifold of dimension $$2n$$. Then $$\int_M \frac{1}{(2\pi)^n}\mathrm{Pf}(\Omega)=\chi(M).$$

Here, $$\mathrm{Pf}$$ denotes the Pfaffian, which is a certain polynomial defined on $$\mathfrak{so}(2n,\Bbb{R})$$ characterized (up to a sign) by $$\mathrm{Pf}^2=\det$$ as polynomial functions. I have swept a lot of details under the rug here - but the moral of the story is that the presence of a Riemannian metric gives a version of the above Chern-Weil theorem for $$SO(2n)-$$invariant polynomials on $$\mathfrak{so}(2n,\Bbb{R})$$.

Lastly, here is a high level explanation of what we are doing here. In topology, one associates to a vector bundle $$E\to M$$ characteristic classes, which are cohomology classes in $$H^*(M)$$ that are invariants of the bundle. I won't get too much into this, but there is a characteristic class of oriented real vector bundles called the Euler class, $$e(E)$$. It is suggestively named because in the case where $$E=TM$$, $$e(E)\in H^n(M,\Bbb{R})$$ (viewed for instance as a de Rham cohomology class) satisfies $$\int_M e(E)=\chi(M).$$ So, we have really constructed an explicit form $$\mathrm{Pf}(\Omega)$$ on $$M$$ representing this cohomology class $$e(E)$$, and that is the content of the theorem. It turns out that characteristic classes of oriented bundles rank $$r$$ can be defined as cohomology classes of the classifying space $$BSO(r)$$. It can be proven that the cohomology of $$BG$$ (for $$G$$ a compact Lie group) is $$H^*(BG,\Bbb{R})\cong \Bbb{R}[\mathfrak{g}]^G$$, i.e. the invariant polynomials on the Lie algebra $$\mathfrak{g}$$. So, the Chern-Weil homomorphism in its most general form expresses characteristic classes of a (say) oriented real bundle of rank $$r$$ in terms of polynomials of the curvature. So, in this setup the Gauss-Bonnet theorem is a special case of the fact that the polynomial $$\mathrm{Pf}$$ computes the Euler class.

If this is of interest to you, I started learning about this in Tu's Differential Geometry: Connections, Curvature, and Characteristic classes, which is a clear and pleasant read for someone with a basic knowledge of manifolds and de Rham cohomology.

• Wow, good answer! Jan 21 at 10:56
• While I may still not consider it a "crown jewel", after thorough digestion, this answer goes a long way. Jan 23 at 6:06

I think one does not need to care about the notion of curvature or even Euler characteristics to find Gauss-Bonnet at least surprising.

A manifold is just a topological space. But when you make it into a Riemannan manifold, you add some additional structure. And it is reasonable to ask "in how many different ways can I do this?". Visually, a Riemannian structure on a, say, topological sphere makes it into some more or less deformed potatoe shape. And this can get pretty wild up to a point where it does not embed into $$\Bbb R^3$$ anymore. So are there any limits at all?

Note the following two things about Gauss-Bonnet.

• Whatever curvature is, it is something that is defined purely in terms of the Riemannian structure, and it is completely determined by it.
• When you integrate it over the manifold, it does not depend on the Riemannian structure.

Isn't that surprising? We just uncovered that Riemannian structures cannot be chosen completely arbitrary.

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• ### An Augmented Nonlinear LMS for Digital Self-Interference Cancellation in Full-Duplex Direct-Conversion Transceivers(1710.04381)

July 17, 2018 cs.IT, math.IT
In future full-duplex communications, the cancellation of self-interference (SI) arising from hardware non-idealities will play an important role in the design of mobile-scale devices. To this end, we introduce an optimal digital SI cancellation solution for shared-antenna-based direct-conversion transceivers. To establish that the underlying widely linear signal model is not adequate for strong transmit signals, the impact of various circuit imperfections, including power amplifier (PA) distortion, frequency-dependent I/Q imbalance, quantization noise and thermal noise, on the performance of the conventional augmented least mean square (LMS) based SI canceller, is analyzed. In order to achieve a sufficient signal-to-interference-plus-noise ratio (SINR) when the nonlinear SI components are not negligible, we propose an augmented nonlinear LMS based SI canceller for a joint cancellation of both the linear and nonlinear SI components by virtue of a widely nonlinear model fit. A rigorous mean and mean square performance evaluation is conducted to justify the performance advantages of the proposed scheme over the conventional augmented LMS solution. Simulations on orthogonal frequency division multiplexing (OFDM)-based wireless local area network (WLAN) standard compliant waveforms support the analysis.
• ### Towards Budget-Driven Hardware Optimization for Deep Convolutional Neural Networks using Stochastic Computing(1805.04142)

May 10, 2018 cs.NE, cs.ET
Recently, Deep Convolutional Neural Network (DCNN) has achieved tremendous success in many machine learning applications. Nevertheless, the deep structure has brought significant increases in computation complexity. Largescale deep learning systems mainly operate in high-performance server clusters, thus restricting the application extensions to personal or mobile devices. Previous works on GPU and/or FPGA acceleration for DCNNs show increasing speedup, but ignore other constraints, such as area, power, and energy. Stochastic Computing (SC), as a unique data representation and processing technique, has the potential to enable the design of fully parallel and scalable hardware implementations of large-scale deep learning systems. This paper proposed an automatic design allocation algorithm driven by budget requirement considering overall accuracy performance. This systematic method enables the automatic design of a DCNN where all design parameters are jointly optimized. Experimental results demonstrate that proposed algorithm can achieve a joint optimization of all design parameters given the comprehensive budget of a DCNN.
• ### Thoracic Disease Identification and Localization with Limited Supervision(1711.06373)

April 24, 2018 cs.CV, stat.ML
Accurate identification and localization of abnormalities from radiology images play an integral part in clinical diagnosis and treatment planning. Building a highly accurate prediction model for these tasks usually requires a large number of images manually annotated with labels and finding sites of abnormalities. In reality, however, such annotated data are expensive to acquire, especially the ones with location annotations. We need methods that can work well with only a small amount of location annotations. To address this challenge, we present a unified approach that simultaneously performs disease identification and localization through the same underlying model for all images. We demonstrate that our approach can effectively leverage both class information as well as limited location annotation, and significantly outperforms the comparative reference baseline in both classification and localization tasks.
• ### Learning Topics using Semantic Locality(1804.04205)

April 11, 2018 cs.CL, cs.LG, cs.IR, stat.ML
The topic modeling discovers the latent topic probability of the given text documents. To generate the more meaningful topic that better represents the given document, we proposed a new feature extraction technique which can be used in the data preprocessing stage. The method consists of three steps. First, it generates the word/word-pair from every single document. Second, it applies a two-way TF-IDF algorithm to word/word-pair for semantic filtering. Third, it uses the K-means algorithm to merge the word pairs that have the similar semantic meaning. Experiments are carried out on the Open Movie Database (OMDb), Reuters Dataset and 20NewsGroup Dataset. The mean Average Precision score is used as the evaluation metric. Comparing our results with other state-of-the-art topic models, such as Latent Dirichlet allocation and traditional Restricted Boltzmann Machines. Our proposed data preprocessing can improve the generated topic accuracy by up to 12.99\%.
• ### Efficient Recurrent Neural Networks using Structured Matrices in FPGAs(1803.07661)

March 22, 2018 cs.NA, cs.LG, stat.ML
Recurrent Neural Networks (RNNs) are becoming increasingly important for time series-related applications which require efficient and real-time implementations. The recent pruning based work ESE suffers from degradation of performance/energy efficiency due to the irregular network structure after pruning. We propose block-circulant matrices for weight matrix representation in RNNs, thereby achieving simultaneous model compression and acceleration. We aim to implement RNNs in FPGA with highest performance and energy efficiency, with certain accuracy requirement (negligible accuracy degradation). Experimental results on actual FPGA deployments shows that the proposed framework achieves a maximum energy efficiency improvement of 35.7$\times$ compared with ESE.
• ### Image Dataset for Visual Objects Classification in 3D Printing(1803.00391)

March 22, 2018 cs.CV
The rapid development in additive manufacturing (AM), also known as 3D printing, has brought about potential risk and security issues along with significant benefits. In order to enhance the security level of the 3D printing process, the present research aims to detect and recognize illegal components using deep learning. In this work, we collected a dataset of 61,340 2D images (28x28 for each image) of 10 classes including guns and other non-gun objects, corresponding to the projection results of the original 3D models. To validate the dataset, we train a convolutional neural network (CNN) model for gun classification which can achieve 98.16% classification accuracy.
• ### C3PO: Database and Benchmark for Early-stage Malicious Activity Detection in 3D Printing(1803.07544)

March 20, 2018 cs.CV
Increasing malicious users have sought practices to leverage 3D printing technology to produce unlawful tools in criminal activities. Current regulations are inadequate to deal with the rapid growth of 3D printers. It is of vital importance to enable 3D printers to identify the objects to be printed, so that the manufacturing procedure of an illegal weapon can be terminated at the early stage. Deep learning yields significant rises in performance in the object recognition tasks. However, the lack of large-scale databases in 3D printing domain stalls the advancement of automatic illegal weapon recognition. This paper presents a new 3D printing image database, namely C3PO, which compromises two subsets for the different system working scenarios. We extract images from the numerical control programming code files of 22 3D models, and then categorize the images into 10 distinct labels. The first set consists of 62,200 images which represent the object projections on the three planes in a Cartesian coordinate system. And the second sets consists of sequences of total 671,677 images to simulate the cameras' captures of the printed objects. Importantly, we demonstrate that the weapons can be recognized in either scenario using deep learning based approaches using our proposed database. % We also use the trained deep models to build a prototype of object-aware 3D printer. The quantitative results are promising, and the future exploration of the database and the crime prevention in 3D printing are demanding tasks.
• ### C-LSTM: Enabling Efficient LSTM using Structured Compression Techniques on FPGAs(1803.06305)

March 14, 2018 cs.LG, cs.AR
Recently, significant accuracy improvement has been achieved for acoustic recognition systems by increasing the model size of Long Short-Term Memory (LSTM) networks. Unfortunately, the ever-increasing size of LSTM model leads to inefficient designs on FPGAs due to the limited on-chip resources. The previous work proposes to use a pruning based compression technique to reduce the model size and thus speedups the inference on FPGAs. However, the random nature of the pruning technique transforms the dense matrices of the model to highly unstructured sparse ones, which leads to unbalanced computation and irregular memory accesses and thus hurts the overall performance and energy efficiency. In contrast, we propose to use a structured compression technique which could not only reduce the LSTM model size but also eliminate the irregularities of computation and memory accesses. This approach employs block-circulant instead of sparse matrices to compress weight matrices and reduces the storage requirement from $\mathcal{O}(k^2)$ to $\mathcal{O}(k)$. Fast Fourier Transform algorithm is utilized to further accelerate the inference by reducing the computational complexity from $\mathcal{O}(k^2)$ to $\mathcal{O}(k\text{log}k)$. The datapath and activation functions are quantized as 16-bit to improve the resource utilization. More importantly, we propose a comprehensive framework called C-LSTM to automatically optimize and implement a wide range of LSTM variants on FPGAs. According to the experimental results, C-LSTM achieves up to 18.8X and 33.5X gains for performance and energy efficiency compared with the state-of-the-art LSTM implementation under the same experimental setup, and the accuracy degradation is very small.
• ### A Framework in CRM Customer Lifecycle: Identify Downward Trend and Potential Issues Detection(1802.08974)

Feb. 25, 2018 cs.AI, cs.CY
Customer retention is one of the primary goals in the area of customer relationship management. A mass of work exists in which machine learning models or business rules are established to predict churn. However, targeting users at an early stage when they start to show a downward trend is a better strategy. In downward trend prediction, the reasons why customers show a downward trend is of great interest in the industry as it helps the business to understand the pain points that customers suffer and to take early action to prevent them from churning. A commonly used method is to collect feedback from customers by either aggressively reaching out to them or by passively hearing from them. However, it is believed that there are a large number of customers who have unpleasant experiences and never speak out. In the literature, there is limited research work that provides a comprehensive and scientific approach to identify these "silent suffers". In this study, we propose a novel two-part framework: developing the downward prediction process and establishing the methodology to identify the reasons why customers are in the downward trend. In the first prediction part, we focus on predicting the downward trend, which is an earlier stage of the customer lifecycle compared to churn. In the second part, we propose an approach to figuring out the cause (of the downward trend) based on a causal inference method and semi-supervised learning. The proposed approach is capable of identifying potential silent sufferers. We take bad shopping experiences as inputs to develop the framework and validate it via a marketing A/B test in the real world. The test readout demonstrates the effectiveness of the framework by driving 88.5% incremental lift in purchase volume.
• ### Towards Ultra-High Performance and Energy Efficiency of Deep Learning Systems: An Algorithm-Hardware Co-Optimization Framework(1802.06402)

Feb. 18, 2018 cs.LG, stat.ML
Hardware accelerations of deep learning systems have been extensively investigated in industry and academia. The aim of this paper is to achieve ultra-high energy efficiency and performance for hardware implementations of deep neural networks (DNNs). An algorithm-hardware co-optimization framework is developed, which is applicable to different DNN types, sizes, and application scenarios. The algorithm part adopts the general block-circulant matrices to achieve a fine-grained tradeoff between accuracy and compression ratio. It applies to both fully-connected and convolutional layers and contains a mathematically rigorous proof of the effectiveness of the method. The proposed algorithm reduces computational complexity per layer from O($n^2$) to O($n\log n$) and storage complexity from O($n^2$) to O($n$), both for training and inference. The hardware part consists of highly efficient Field Programmable Gate Array (FPGA)-based implementations using effective reconfiguration, batch processing, deep pipelining, resource re-using, and hierarchical control. Experimental results demonstrate that the proposed framework achieves at least 152X speedup and 71X energy efficiency gain compared with IBM TrueNorth processor under the same test accuracy. It achieves at least 31X energy efficiency gain compared with the reference FPGA-based work.
• ### An Area and Energy Efficient Design of Domain-Wall Memory-Based Deep Convolutional Neural Networks using Stochastic Computing(1802.01016)

Feb. 3, 2018 cs.NE
With recent trend of wearable devices and Internet of Things (IoTs), it becomes attractive to develop hardware-based deep convolutional neural networks (DCNNs) for embedded applications, which require low power/energy consumptions and small hardware footprints. Recent works demonstrated that the Stochastic Computing (SC) technique can radically simplify the hardware implementation of arithmetic units and has the potential to satisfy the stringent power requirements in embedded devices. However, in these works, the memory design optimization is neglected for weight storage, which will inevitably result in large hardware cost. Moreover, if conventional volatile SRAM or DRAM cells are utilized for weight storage, the weights need to be re-initialized whenever the DCNN platform is re-started. In order to overcome these limitations, in this work we adopt an emerging non-volatile Domain-Wall Memory (DWM), which can achieve ultra-high density, to replace SRAM for weight storage in SC-based DCNNs. We propose DW-CNN, the first comprehensive design optimization framework of DWM-based weight storage method. We derive the optimal memory type, precision, and organization, as well as whether to store binary or stochastic numbers. We present effective resource sharing scheme for DWM-based weight storage in the convolutional and fully-connected layers of SC-based DCNNs to achieve a desirable balance among area, power (energy) consumption, and application-level accuracy.
• ### A Simple Nonlinearity-Tailored Probabilistic Shaping Distribution for Square QAM(1710.04134)

Oct. 12, 2017 cs.IT, math.IT
A new probabilistic shaping distribution that outperforms Maxwell-Boltzmann is studied for the nonlinear fiber channel. Additional gains of 0.1 bit/symbol MI or 0.2 dB SNR for both DP-256QAM and DP-1024QAM are reported after 200 km nonlinear fiber transmission.
• ### On the Closest Vector Problem for Lattices Constructed from Polynomials and Their Cryptographic Applications(1710.02265)

Oct. 6, 2017 cs.CR
In this paper, we propose new classes of trapdoor functions to solve the closest vector problem in lattices. Specifically, we construct lattices based on properties of polynomials for which the closest vector problem is hard to solve unless some trapdoor information is revealed. We thoroughly analyze the security of our proposed functions using state-of-the-art attacks and results on lattice reductions. Finally, we describe how our functions can be used to design quantum-safe encryption schemes with reasonable public key sizes. In particular, our scheme can offer around $106$ bits of security with a public key size of around $6.4$ $\texttt{KB}$. Our encryption schemes are efficient with respect to key generation, encryption and decryption.
• ### Entanglement concentration with different measurement in a 3-mode optomechanical system(1703.05629)

Sept. 30, 2017 quant-ph
In this work, we perform a series of phonon counting measurement with different methods in a 3-mode optomechanical system, and we compare the difference of the entanglement after measurement. In this article we focus on the two cases: imperfect measurement and on-off measurement. We find that whatever measurement you take, the entanglement will increase. The size of entanglement enhancement is the largest in perfect measurement, second in the imperfect measurement, and it is not obvious in the on-off measurement. We are sure that the more precise measurement information, the larger entanglement concentration.
• ### Theoretical Properties for Neural Networks with Weight Matrices of Low Displacement Rank(1703.00144)

Sept. 22, 2017 cs.CV, cs.LG, stat.ML
Recently low displacement rank (LDR) matrices, or so-called structured matrices, have been proposed to compress large-scale neural networks. Empirical results have shown that neural networks with weight matrices of LDR matrices, referred as LDR neural networks, can achieve significant reduction in space and computational complexity while retaining high accuracy. We formally study LDR matrices in deep learning. First, we prove the universal approximation property of LDR neural networks with a mild condition on the displacement operators. We then show that the error bounds of LDR neural networks are as efficient as general neural networks with both single-layer and multiple-layer structure. Finally, we propose back-propagation based training algorithm for general LDR neural networks.
• ### A Simple Analysis for Exp-concave Empirical Minimization with Arbitrary Convex Regularizer(1709.02909)

Sept. 9, 2017 math.OC, cs.LG, stat.ML
In this paper, we present a simple analysis of {\bf fast rates} with {\it high probability} of {\bf empirical minimization} for {\it stochastic composite optimization} over a finite-dimensional bounded convex set with exponential concave loss functions and an arbitrary convex regularization. To the best of our knowledge, this result is the first of its kind. As a byproduct, we can directly obtain the fast rate with {\it high probability} for exponential concave empirical risk minimization with and without any convex regularization, which not only extends existing results of empirical risk minimization but also provides a unified framework for analyzing exponential concave empirical risk minimization with and without {\it any} convex regularization. Our proof is very simple only exploiting the covering number of a finite-dimensional bounded set and a concentration inequality of random vectors.
• ### CirCNN: Accelerating and Compressing Deep Neural Networks Using Block-CirculantWeight Matrices(1708.08917)

Aug. 29, 2017 cs.AI, cs.CV, cs.LG, stat.ML
Large-scale deep neural networks (DNNs) are both compute and memory intensive. As the size of DNNs continues to grow, it is critical to improve the energy efficiency and performance while maintaining accuracy. For DNNs, the model size is an important factor affecting performance, scalability and energy efficiency. Weight pruning achieves good compression ratios but suffers from three drawbacks: 1) the irregular network structure after pruning; 2) the increased training complexity; and 3) the lack of rigorous guarantee of compression ratio and inference accuracy. To overcome these limitations, this paper proposes CirCNN, a principled approach to represent weights and process neural networks using block-circulant matrices. CirCNN utilizes the Fast Fourier Transform (FFT)-based fast multiplication, simultaneously reducing the computational complexity (both in inference and training) from O(n2) to O(nlogn) and the storage complexity from O(n2) to O(n), with negligible accuracy loss. Compared to other approaches, CirCNN is distinct due to its mathematical rigor: it can converge to the same effectiveness as DNNs without compression. The CirCNN architecture, a universal DNN inference engine that can be implemented on various hardware/software platforms with configurable network architecture. To demonstrate the performance and energy efficiency, we test CirCNN in FPGA, ASIC and embedded processors. Our results show that CirCNN architecture achieves very high energy efficiency and performance with a small hardware footprint. Based on the FPGA implementation and ASIC synthesis results, CirCNN achieves 6-102X energy efficiency improvements compared with the best state-of-the-art results.
• ### A Hierarchical Framework of Cloud Resource Allocation and Power Management Using Deep Reinforcement Learning(1703.04221)

Aug. 11, 2017 cs.AI, cs.DC
Automatic decision-making approaches, such as reinforcement learning (RL), have been applied to (partially) solve the resource allocation problem adaptively in the cloud computing system. However, a complete cloud resource allocation framework exhibits high dimensions in state and action spaces, which prohibit the usefulness of traditional RL techniques. In addition, high power consumption has become one of the critical concerns in design and control of cloud computing systems, which degrades system reliability and increases cooling cost. An effective dynamic power management (DPM) policy should minimize power consumption while maintaining performance degradation within an acceptable level. Thus, a joint virtual machine (VM) resource allocation and power management framework is critical to the overall cloud computing system. Moreover, novel solution framework is necessary to address the even higher dimensions in state and action spaces. In this paper, we propose a novel hierarchical framework for solving the overall resource allocation and power management problem in cloud computing systems. The proposed hierarchical framework comprises a global tier for VM resource allocation to the servers and a local tier for distributed power management of local servers. The emerging deep reinforcement learning (DRL) technique, which can deal with complicated control problems with large state space, is adopted to solve the global tier problem. Furthermore, an autoencoder and a novel weight sharing structure are adopted to handle the high-dimensional state space and accelerate the convergence speed. On the other hand, the local tier of distributed server power managements comprises an LSTM based workload predictor and a model-free RL based power manager, operating in a distributed manner.
• ### SEP-Nets: Small and Effective Pattern Networks(1706.03912)

June 13, 2017 cs.CV, cs.LG
While going deeper has been witnessed to improve the performance of convolutional neural networks (CNN), going smaller for CNN has received increasing attention recently due to its attractiveness for mobile/embedded applications. It remains an active and important topic how to design a small network while retaining the performance of large and deep CNNs (e.g., Inception Nets, ResNets). Albeit there are already intensive studies on compressing the size of CNNs, the considerable drop of performance is still a key concern in many designs. This paper addresses this concern with several new contributions. First, we propose a simple yet powerful method for compressing the size of deep CNNs based on parameter binarization. The striking difference from most previous work on parameter binarization/quantization lies at different treatments of $1\times 1$ convolutions and $k\times k$ convolutions ($k>1$), where we only binarize $k\times k$ convolutions into binary patterns. The resulting networks are referred to as pattern networks. By doing this, we show that previous deep CNNs such as GoogLeNet and Inception-type Nets can be compressed dramatically with marginal drop in performance. Second, in light of the different functionalities of $1\times 1$ (data projection/transformation) and $k\times k$ convolutions (pattern extraction), we propose a new block structure codenamed the pattern residual block that adds transformed feature maps generated by $1\times 1$ convolutions to the pattern feature maps generated by $k\times k$ convolutions, based on which we design a small network with $\sim 1$ million parameters. Combining with our parameter binarization, we achieve better performance on ImageNet than using similar sized networks including recently released Google MobileNets.
• ### Hardware-Driven Nonlinear Activation for Stochastic Computing Based Deep Convolutional Neural Networks(1703.04135)

March 12, 2017 cs.CV
Recently, Deep Convolutional Neural Networks (DCNNs) have made unprecedented progress, achieving the accuracy close to, or even better than human-level perception in various tasks. There is a timely need to map the latest software DCNNs to application-specific hardware, in order to achieve orders of magnitude improvement in performance, energy efficiency and compactness. Stochastic Computing (SC), as a low-cost alternative to the conventional binary computing paradigm, has the potential to enable massively parallel and highly scalable hardware implementation of DCNNs. One major challenge in SC based DCNNs is designing accurate nonlinear activation functions, which have a significant impact on the network-level accuracy but cannot be implemented accurately by existing SC computing blocks. In this paper, we design and optimize SC based neurons, and we propose highly accurate activation designs for the three most frequently used activation functions in software DCNNs, i.e, hyperbolic tangent, logistic, and rectified linear units. Experimental results on LeNet-5 using MNIST dataset demonstrate that compared with a binary ASIC hardware DCNN, the DCNN with the proposed SC neurons can achieve up to 61X, 151X, and 2X improvement in terms of area, power, and energy, respectively, at the cost of small precision degradation.In addition, the SC approach achieves up to 21X and 41X of the area, 41X and 72X of the power, and 198200X and 96443X of the energy, compared with CPU and GPU approaches, respectively, while the error is increased by less than 3.07%. ReLU activation is suggested for future SC based DCNNs considering its superior performance under a small bit stream length.
• ### SC-DCNN: Highly-Scalable Deep Convolutional Neural Network using Stochastic Computing(1611.05939)

Jan. 31, 2017 cs.CV
With recent advancing of Internet of Things (IoTs), it becomes very attractive to implement the deep convolutional neural networks (DCNNs) onto embedded/portable systems. Presently, executing the software-based DCNNs requires high-performance server clusters in practice, restricting their widespread deployment on the mobile devices. To overcome this issue, considerable research efforts have been conducted in the context of developing highly-parallel and specific DCNN hardware, utilizing GPGPUs, FPGAs, and ASICs. Stochastic Computing (SC), which uses bit-stream to represent a number within [-1, 1] by counting the number of ones in the bit-stream, has a high potential for implementing DCNNs with high scalability and ultra-low hardware footprint. Since multiplications and additions can be calculated using AND gates and multiplexers in SC, significant reductions in power/energy and hardware footprint can be achieved compared to the conventional binary arithmetic implementations. The tremendous savings in power (energy) and hardware resources bring about immense design space for enhancing scalability and robustness for hardware DCNNs. This paper presents the first comprehensive design and optimization framework of SC-based DCNNs (SC-DCNNs). We first present the optimal designs of function blocks that perform the basic operations, i.e., inner product, pooling, and activation function. Then we propose the optimal design of four types of combinations of basic function blocks, named feature extraction blocks, which are in charge of extracting features from input feature maps. Besides, weight storage methods are investigated to reduce the area and power/energy consumption for storing weights. Finally, the whole SC-DCNN implementation is optimized, with feature extraction blocks carefully selected, to minimize area and power/energy consumption while maintaining a high network accuracy level.
• ### Improved Dropout for Shallow and Deep Learning(1602.02220)

Dec. 4, 2016 cs.LG, stat.ML
Dropout has been witnessed with great success in training deep neural networks by independently zeroing out the outputs of neurons at random. It has also received a surge of interest for shallow learning, e.g., logistic regression. However, the independent sampling for dropout could be suboptimal for the sake of convergence. In this paper, we propose to use multinomial sampling for dropout, i.e., sampling features or neurons according to a multinomial distribution with different probabilities for different features/neurons. To exhibit the optimal dropout probabilities, we analyze the shallow learning with multinomial dropout and establish the risk bound for stochastic optimization. By minimizing a sampling dependent factor in the risk bound, we obtain a distribution-dependent dropout with sampling probabilities dependent on the second order statistics of the data distribution. To tackle the issue of evolving distribution of neurons in deep learning, we propose an efficient adaptive dropout (named \textbf{evolutional dropout}) that computes the sampling probabilities on-the-fly from a mini-batch of examples. Empirical studies on several benchmark datasets demonstrate that the proposed dropouts achieve not only much faster convergence and but also a smaller testing error than the standard dropout. For example, on the CIFAR-100 data, the evolutional dropout achieves relative improvements over 10\% on the prediction performance and over 50\% on the convergence speed compared to the standard dropout.
• ### Error Formulas for Ideal Interpolation(1506.05242)

June 18, 2016 math.NA
In this paper we study the algebraic structure of error formulas for ideal interpolation. We introduce the so-called "normal" error formulas and prove that the lexicographic order reduced Gr\"obner basis admits such a formula for all ideal interpolation. This formula is a generalization of the "good" error formula proposed by Carl de Boor. Finally, we discuss a Shekhtman's example and give an explicit form of "normal" error formula for this example.
• ### Unified Convergence Analysis of Stochastic Momentum Methods for Convex and Non-convex Optimization(1604.03257)

May 4, 2016 math.OC, stat.ML
Recently, {\it stochastic momentum} methods have been widely adopted in training deep neural networks. However, their convergence analysis is still underexplored at the moment, in particular for non-convex optimization. This paper fills the gap between practice and theory by developing a basic convergence analysis of two stochastic momentum methods, namely stochastic heavy-ball method and the stochastic variant of Nesterov's accelerated gradient method. We hope that the basic convergence results developed in this paper can serve the reference to the convergence of stochastic momentum methods and also serve the baselines for comparison in future development of stochastic momentum methods. The novelty of convergence analysis presented in this paper is a unified framework, revealing more insights about the similarities and differences between different stochastic momentum methods and stochastic gradient method. The unified framework exhibits a continuous change from the gradient method to Nesterov's accelerated gradient method and finally the heavy-ball method incurred by a free parameter, which can help explain a similar change observed in the testing error convergence behavior for deep learning. Furthermore, our empirical results for optimizing deep neural networks demonstrate that the stochastic variant of Nesterov's accelerated gradient method achieves a good tradeoff (between speed of convergence in training error and robustness of convergence in testing error) among the three stochastic methods.
• ### An indirect magnetic approach for determining entropy change in first-order magnetocaloric materials(1507.06133)

July 26, 2015 cond-mat.mtrl-sci
Taking into account the phase fraction during transition for the first-order magnetocaloric materials, an improved isothermal entropy change determination has been put forward based on the Clausius-Clapeyron (CC) equation. It was found that the isothermal entropy change value evaluated by our method is in excellent agreement with those determined from the Maxwell-relation (MR) for Ni-Mn-Sn Heusler alloys, which usually presents a weak field-induced phase transforming behavior. In comparison with MR, this method could give rise to a favorable result derived from few thermomagnetic measurements. More importantly, we can eliminate the isothermal entropy change overestimation derived from MR, which always exists in the cases of Ni-Co-Mn-In and MnAs systems with a prominent field-induced transition. These results confirmed that such a CC-equation-based method is quite practical and superior to the MR-based ones in eliminating the spurious spike and reducing measuring cost.


text:
Weird character from UART. STM32L100RC

I'm having problems communicating with UART on my STM32L100RC discovery board. I get character such as xþ when I'm sending the letter 'F'. I'm using Hercule to receive data. Hercule's configuration is : Baudrate = 9600, Data size = 8, Parity = None, Hankshake = OFF and Mode = Free. So here is my code and if you can tell me what I do wrong, I'd be very grateful!

Also, I think it is important that I specify that I can get my code working using the HAL library (but I can't use it for my program), so that puts my physical hardware out of the equation.

void UART_Initialization (void)
{
RCC_AHBPeriphClockCmd(RCC_AHBPeriph_GPIOA, ENABLE);
RCC_APB2PeriphClockCmd(RCC_APB2Periph_USART1, ENABLE);

// Initializing RX_Pin
GPIO_InitTypeDef GPIO_InitStructRX;
GPIO_InitStructRX.GPIO_Pin = RX_Pin;
GPIO_InitStructRX.GPIO_Mode = GPIO_Mode_IN;
GPIO_InitStructRX.GPIO_Speed = GPIO_Speed_2MHz;
GPIO_InitStructRX.GPIO_PuPd = GPIO_PuPd_NOPULL;
GPIO_Init(GPIOA, &GPIO_InitStructRX);

// Initializing TX_Pin
GPIO_InitTypeDef GPIO_InitStructTX;
GPIO_InitStructTX.GPIO_Pin = TX_Pin;
GPIO_InitStructTX.GPIO_Mode = GPIO_Mode_AF;
GPIO_InitStructTX.GPIO_OType = GPIO_OType_PP;
GPIO_InitStructTX.GPIO_Speed = GPIO_Speed_2MHz;
GPIO_InitStructTX.GPIO_PuPd = GPIO_PuPd_UP;
GPIO_Init(GPIOA, &GPIO_InitStructTX);

USART_InitTypeDef USART_InitStructUSART;
USART_InitStructUSART.USART_BaudRate = 9600;
USART_InitStructUSART.USART_HardwareFlowControl = USART_HardwareFlowControl_None;
USART_InitStructUSART.USART_Mode = USART_Mode_Rx | USART_Mode_Tx;
USART_InitStructUSART.USART_Parity = USART_Parity_No;
USART_InitStructUSART.USART_StopBits = USART_StopBits_1;
USART_InitStructUSART.USART_WordLength = USART_WordLength_8b;
USART_Init(USART1, &USART_InitStructUSART);

// Setting alternate functions for RX_Pin and TX_Pin
GPIO_PinAFConfig(GPIOA, GPIO_PinSource9, GPIO_AF_USART1);
GPIO_PinAFConfig(GPIOA, GPIO_PinSource10, GPIO_AF_USART1);

USART_ITConfig(USART1, USART_IT_RXNE, DISABLE);
USART_Cmd(USART1, ENABLE);

}

void USART_SendData(USART_TypeDef* USARTx, uint16_t Data)
{
/* Check the parameters */
assert_param(IS_USART_ALL_PERIPH(USARTx));
assert_param(IS_USART_DATA(Data));

/* Transmit Data */
USARTx->DR = (Data & (uint16_t)0x01FF);
}

int main(void)
{
UART_Initialization();

while(1)
{

for (int i = 0; i <500000; i++)
{}

USART_SendData(USART1, 'F');

}
}

• Weird characters usually come about because of wrong baud rate. Measure with a scope to confirm it is really what you think it is. And if it isn't, then start looking at all the registers involved in setting the baud rate. You will have to dig into the whole clock generation chain for the processor. – mkeith Apr 14 '19 at 23:16
• Yeah, that was exactly that, I just found out that my clock speed was divided by 2, meaning that the actual baudrate was 57600. Thank you very much! – Nathaniel Brochu Apr 15 '19 at 14:08
• It might be helpful to future people if you could answer your own question and accept the answer. It is perfectly OK to accept your own answer if it is the best one (or the only one). – mkeith Apr 15 '19 at 15:28


text:
# qiskit.circuit.library.C3SXGate.inverse¶

C3SXGate.inverse()[Quellcode]

Invert this gate. The C3X is its own inverse.


text:
# Examples where roots are necessary for the solution

I currently write an article where I want to introduce roots. Thus I need to motivate them. Here I said, they can be used to find solutions of equations like $x^n=a$. Now I want to make some examples, where this problem arises. So far I found:

• What is the side length of a square with the area of $10 m^2$?
• What are the roots of the polynomial $x^2+x-1$?

What examples would you take? I do not have the feeling that mine are really convincing to study roots...

• What is the hypotenuse of the isosceles right triangle with leg length one inch? (Lots of history with this.) May 28, 2015 at 21:43
• Are you including e.g. all 0s of polynomials? If so, what about Calculus problems where one wants the derivative (or second derivative, etc) of a polynomial (which is also a polynomial...) to be 0? I do not think there is a shortage of such problems. Maybe you could make your question a bit more specific? May 28, 2015 at 22:26
• How about solving for the percentage annual return which will double an investment in ten years? Jun 5, 2015 at 4:54

Ray-tracing, which underlies much of high-end computer graphics (from Toy Story to Frozen) relies on computing the intersection of a ray line-of-sight with a geometric object. For example, where does the ray $a + t v$ with parameter $t$, $a=(2,0)$, $v=(-1,\frac{1}{2})$, intersect the unit circle $x^2 + y^2 = 1$? Substitution of $a + t v = \left( 2-t,\frac{t}{2} \right)$ for $(x,y)$ in the circle equation leads to \begin{eqnarray} (2-t)^2 + \frac{t^2}{4} &=& 1 \;, \\ \frac{5 t^2}{4}-4 t+3 &=& 0 \end{eqnarray} whose roots are $t=2$, or $t=\frac{6}{5}$, yielding the intersection points \begin{eqnarray} (x,y) &=& (0,1) \;, \\ (x,y) &=& \left( \frac{4}{5},\frac{3}{5} \right) &=& (0.8, 0.6) \;. \end{eqnarray}

Because the latter point is closer to the ray origin $a$, that $(0.8, 0.6)$ point is the one whose surface characteristics determine the color/shade of the image pixel represented by the ray line-of-sight.

To compute the golden ratio, we need to find the positive root of a quadratic equation.

Two quantities $a$ and $b$ with $a>b>0$ are said to be in the golden ratio if $a$ is to $b$ as $a+b$ is to $a$.

(Image source: Wikipedia)

Algebraically, $$\frac a b= \frac {a+b} a.$$ If $\varphi=\tfrac a b$ denotes the golden ratio, then $$\varphi = 1+\varphi^{-1}.$$ Multiplying with $\varphi$ and rearranging gives the equation $$\varphi^2-\varphi-1=0,$$ which has a positive root $$\varphi=\frac {1+\sqrt 5} 2 \approx 1.618$$

• Let $x = -\varphi$. Then $\varphi^2 - \varphi - 1 = x^2 + x - 1$, which amounts to the second bullet-point from the OP. Jun 7, 2015 at 5:12
• @BenjaminDickman ... or let $x=b/a=\varphi^{-1}$. Then you also end up with the equation $x^2+x-1=0$. Jun 7, 2015 at 14:58


text:
I would expect most people’s impression of sticky keys is “that annoying feature which activates when you accidentally press Shift for a few times”. I had that moment about 15 years ago on a Windows XP, but recently discovered that it is truly a gem hidden in plain sight that’s not only an ergonomic paradise, but also has the power to fundamentally change the way I interact with my computer.

A short introduction on how sticky keys work: Sticky keys are like the Shift key on smart phone keyboards, pressing it once will capitalize the next character, and pressing it twice enters CapsLock mode until you press it again which will reset it to the default state. In macOS, upon enabling this in Accessibility > Keyboard, this will make all your modifier keys (Command ⌘, Shift ⇧, Ctrl ^, Options ⌥, Fn) act in this way, but one thing unlike smartphone keyboard is that you can still use the modifier keys in the old way, and they won’t “stick” as long as you pressed another key before releasing the modifier. There are visual aids telling you which modifiers are currently “stuck” and which are “locked”.

This post discusses the many reasons of why sticky keys are awesome, and talks a little about my personal philosophy on how should keyboards be configured at the end.

## Smooth Learning Process

Unlike more radical things such as switching to Dvorak, after enabling sticky keys, you can still use your keyboard the old way without any issue, and it could come in handy once in a while (e.g. when taking a screenshot with ⌘+Shift+Control+4).

I am saying this because I switched to Dvorak about 6 years ago, even though it was one of the best decisions I have ever made, the transitional phase of being stuck between the muscle memory of Qwerty and Dvorak was a huge PITA.

## Comfort: No More Chording / System-Wide Leader Keys!

Chording is bad for your fingers, and the our modifier-addicted friends know it very well. Sticky keys solve this problem by enabling you to gracefully press each modifier keys in succession and then the modified key.

Vim users that are obsessed with tinkering with their configuration (like me) often make heavy use of “Leader Key”. TLDR: Leader key is the first key in many key strokes sequences that get mapped into different editor actions. The gist is that you press them in succession (like Space-u-m) rather than together (like Ctrl+Shift+g, a.k.a. chording). Leader key sequences are so comfortable that even our mortal enemies became jealous and invented Spacemacs which is based on the stolen idea of using Space as the leader key for editor actions and save themselves from Emacs pinky commonly caused by their inferior shortcut design (/s, of course). Here’s a peek at the leader shortcut section of my .vimrc (using Space as leader):

nnoremap <Leader>n gt

With sticky keys enabled, it’s basically system-wide leader keys. Taking a screenshot with ⌘+Shift+Ctrl+4 is a pain? Just press them one-by-one (with any modifier order)!

## No Sacrifice in Speed (and Potentially Faster!)

Disclaimer: I value the comfort of typing much more than speed, as typing speed is rarely the bottleneck of my productivity. This section is just to demonstrate that comfort of sticky keys does not come at the cost of speed.

As sticky keys completely removes the need for holding a key down, less effort/strain is needed. But what about the speed? Essentially, sticky keys removed the requirement on when to release the modifier keys, and actions happen when a key is depressed, so at the very least using sticky keys should not slow you down.

But I would like to go one step further and argue that using sticky keys actually make you faster. Assuming we have 3 movements:

• $m_1$: Pressing a key and releasing it immediately.
• $m_2$: Press down a key and hold it.
• $m_3$: Release a key that is being held down.

Let $\mathrm{t}(m)$ denote the length of time between beginning $m$ and being ready for next action.

First of all, $\mathrm{t}(m_1) \ne \mathrm{t}(m_2) + \mathrm{t}(m_3)$. Try it on your keyboard and feel how it’s different when you simply stroke a key vs. press and hold then release. If you are using a mechanical keyboard and with the right switch and enough experience, you can consistently press the key with just enough force to actuate it without ever reaching the bottom of the travel (“bottoming out”). I would estimate their relationship to be something like: $\mathrm{t}(m_1) \approx \mathrm{t}(m_2) \approx \mathrm{t}(m_3)$.

Now imagine typing a ctrl+h b where ctrl+h is the {readline, Emacs, Vim, Bash, macOS} shortcut for deleting the last character (and perhaps one of my most used key bindings of all time (even though I hate to admit it)).

Without sticky keys:

• $m_1$: a
• $m_2$: ctrl down
• $m_1$: h
• $m_3$: ctrl up
• $m_1$: b

With sticky keys:

• $m_1$: a
• $m_1$: ctrl
• $m_1$: h
• $m_1$: b

If our assumption of $\mathrm{t}(m_1) \approx \mathrm{t}(m_2) \approx \mathrm{t}(m_3)$ is true then this is already a 20% reduction in time usage in our tiny example.

However the same strategy would not be work for cases where you need to type consecutive keys with the same modifier, such as my favorite Vim insert mode Ctrl chain such as C+pemdc (go up one line, go to end of line, insert newline, dedent current line, exit insert mode). In those cases, we have two choices:

1. Lock the modifier and reset lock:

2. Just use the traditional way which sticky key allows:

The traditional way wins by 1 unit, but I suspect that if we take into account the reduction in speed caused by the restriction of movement when one finger is depressing the modifier key, sticky keys could still win.

Then again as I mentioned at the beginning of this section, speed is not the point. It’s about feeling more comfortable typing without sacrifice in speed.

## Put Modifier Keys Anywhere

Modifier are traditionally place on both sides and pressed by pinkies on the opposite side of the modified key (e.g. Shift and Ctrl). As pinky is relatively weak when compared with other fingers, some ergonomic keyboards has “thumb clusters” for modifiers (e.g. Kinesis Advantage, Ergodox, Keyboardio), this has several advantages:

• No need for having one modifier on each side because thumb is not used for letter/number keys
• Reduce pinky strain and avoid “Emacs pinky”

However, in practice I have found that thumb is equally prone to strain injuries after long typing sessions, and this topic often comes up in discussions about Ergodox’s suboptimal thumb cluster placement, Xah Lee has an excellent summary on Ergodox’s thumb cluster issues here. I suspect that this may be due to the fact that the natural “press” direction of the thumb is very different from the other fingers and most keyboards available does not optimize for this (except Manuform and its derivatives such as Dactyl Manuform).

Sticky keys makes it possible to assign modifier keys to index and middle fingers, without need for assigning another key on the opposite hand. The probability of the modifier and the modified being on the same finger is low ($\approx \frac{1}{8}$), and even if that happens, they can be pressed in succession.

## Free CapsLock!

By just double tapping Shift! You will appreciate it next time when you are typing those #ifndef SUPER_LONG_HEADER_PATH_THAT_BREAKS_THE_BOUNDARY_OF_REALITY_WHICH_IS_EIGHTY_CHARACTERS_ and ANNOYING_PREPROCESSOR_MACROS(). Really, even typing TODO requires you to switch hands a few times if you always use the Shift key on the opposite side to avoid cramping.

Some background for those who still has CapsLock on their keyboard (from https://www.emacswiki.org/emacs/MovingTheCtrlKey):

For some, swapping Control and CapsLock is very important; the virtually useless CapsLock is far easier to press than Control, crucial to Emacs yet really awkwardly located.

When Emacs was first written, most keyboards had Ctrl to the left of A, either immediately to the left or with CapsLock in between. Thus, it was a convenient and logical choice of key for creating Emacs commands. When personal computers went mainstream in the 80’s, Ctrl was moved out of the way, presumably because the target users (office workers) wouldn’t need Ctrl as often as CapsLock.

## Layering at the Right Layer of Abstraction

Warning: This section is more about my personal philosophy and may be controversial, read with a grain of salt.

I always disliked those 40% keyboards that hides tons of keys behind layers, and it’s not due to the overhead of switching layers (as I mentioned earlier speed is not what I’m looking for).

Let’s start with a simple example: Arrow keys. Disclaimer: Being a Vim user with system-wide ctrl-p/n/b/f arrow key shortcuts, I rarely need arrow keys. But when I do, it is a part of some shortcuts such as Ctrl+Left/Right for switching workspace. Now if my arrow keys are behind some other layers, they will be from difficult to impossible for being used in a shortcut sequence. Then 40% keyboard enthusiasts will start talking about QMK and telling me that I can just remap something like Fn+H to Ctrl+Left. Cool, but there are so many shortcuts for macOS and each application, which means that if I assign this very convenient home-row key to Chrome, I won’t be able to use it for any other application. Convenient keys (like the ones on home row) are a scarce resource. Mapping things on keyboard won’t let me have the most convenient keys function differently in different applications, because keyboard is the wrong place for creating system/application-specific shortcuts!

Here’s an attempt to express the same sentiment by misinterpreting the SOLID principles:

• Single-Responsibility: A keyboard should only be responsible for producing key signals ergonomically, but not in charge of the behavior of specific applications.
• Open-Closed: You should be able to extend system/application key binding behaviors without modding your keyboard.
• Liskov Substitution: You should be able to switch to any other keyboard for your computer without losing functionalities.
• Interface Segregation: The convenient key locations for a specific application should not be taken up by the bindings optimized for another application.
• Dependency Inversion: Keyboard configuration should not depend on what specific applications you have on your computer, neither should how the behavior of your application depend which keyboard you are using. They should both depend on the common interface (key codes and press/release signals).

So what’s the right way to do this then? I believe the best place for creating those bindings is from within the operating system with per-application shortcuts. For example, in Chrome, switching tabs is one of my most commonly used operations, and at the same time I am not using the most convenient keys ⌘+h and ⌘+n at all, therefore I have created mappings for ⌘+h -> Ctrl+Shift+Tab and ⌘+n -> Ctrl+Tab, along with several things like PageUp/PageDown. Now I have effectively created a “Chrome layer”, which activates when I press ⌘ once in Chrome, and locks when I press ⌘ twice. Also with the system-wide mapping from Ctrl-p/n/b/f to arrow keys, double tapping Ctrl essentially brings me to the “Fn” layer with arrow keys and a lot more.

QMK has something similar called “Tap Dance” that allows assigning different actions to consecutive taps, but sticky keys still win in three ways:

1. I can get this on any keyboard, even on the crappy built-in butterfly keyboard which I have to use once in a while for meetings etc. (Good luck flashing QMK on your laptop keyboard)
2. I get a visual indication on screen about the current “layer” than I’m in.
3. I can reuse the convenient keys for different purposes in other applications (e.g. use ⌘+h for cancelling a to-do item in Things).

## The Bad and the Ugly

The sticky keys on macOS (as of 10.15.4) seems to be implemented by releasing the modifier really quick after modifying one key, so there’s the possibility of accidentally modifying more than one keys for a single stick if you are typing fast enough. This happens sometimes when I type “The” which could become “THe”. But I have been consciously limiting my typing speed for higher accuracy these days so it doesn’t bother me too much, but I do see the potential of this being a great annoyance.

## Conclusion

Sticky keys are awesome and you should use them!


text:
# How to change global slide transition effect duration in beamer?

@diabonas presented a method to add global transition effect in beamer in this answer like this:

\documentclass{beamer}

\begin{document}
\frame{\centering\Huge1}
\frame{\centering\Huge2}
\frame{\centering\Huge3}
\end{document}

Now how to define global transition duration. there is a command \transduration{2} for setting transition duration but when I use it like this:

\addtobeamertemplate{background canvas}{\transfade \transduration{1}}{}

it changes effects automatically from a slides to another. I tested different pdf viewer with different effects all the same.

The question is "How to change global transition duration?"

• What do you want to set exactly with the \transduration command? It is to set the time between two transitions. If you use \addtobeamertemplate{background canvas}{\transfade \transduration{1}}{} this line in your preamble, when you will display your presentation in FullScreen, you will have each slide displayed 1 second. It is not the time of the effect of the transition you set with this command. – Ludovic C. Aug 25 '13 at 20:06

The \transduration command has to be used to set the time you want a slide to be displayed for example if you place \transduration{10} after the begin{frame} command then this frame will be displayed 10 seconds when you will play your presentation in full screen (and maybe also not in full screen?). This command does not affect the time of the transition effect you define with \transfade.

If you want to modify the time of the transition effect you have to use the duration option of the transfade command

\addtobeamertemplate{background canvas}{\transfade[duration=1]}{}

Note: The same goes for the direction option for the kind of transition using this option.

• Thanks for your good answer. I misunderstood \transduration{1} with [duration=1]. It solved my problem. – sajjadG Aug 26 '13 at 3:04
• [duration=1] is working. But not with number less than 1 ! How it is possible to set fastest effect? – sajjadG Aug 26 '13 at 3:24
• @sajjadG For me it works with 0.1 for example (I am using Adobe Reader). Which viewer are you using? – Ludovic C. Aug 26 '13 at 9:40
• I'm using Linux and I tested with evince and okular. evince handles some effects and okular handles some other!!! but none of them handles effect with less than 1 sec duration. – sajjadG Aug 26 '13 at 9:49
• @sajjadG I am also using Linux but realised that the effects we are speaking about are designed for Adobe when I went through the problem related here: tex.stackexchange.com/q/120839/31058 . If you want to use evince you will have to do without some features of the transitions. – Ludovic C. Aug 26 '13 at 9:53


text:
Browse Questions

If $y=e^{-x}(A\cos x+B\sin x)$, then $y$ is a solution of

\begin{array}{1 1}(A)\;\frac{d^2y}{dx^2}-2\frac{dy}{dx}=0 & (B)\;\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\\(C)\;\frac{d^2y}{dx^2}+2\frac{dy}{dx}+2y=0 & (D)\;\frac{d^2y}{dx^2}+2y=0\end{array}

Can you answer this question?

Toolbox:
• The general solution of a differential equation is a relation between dependent and independent variables having $'n'$ arbitrary constants.
• A general solution may have moer than one form, but the arbitrary constants must be the same
• $\large\frac{d}{dx}$$(uv)=\large\frac{du}{dv}$$.v+\large\frac{dv}{dx}$$.u Given y=e^{-x}(A \cos x +B \sin x) Let us expand the above equation as y=e^{-x} \;A \cos x +e^{-x} \;B \sin x Consider e^{-x}.A \cos x This in the product form , so let us apply product rule to differentiate this, \large\frac{d}{dx}$$(uv)=\large\frac{d}{dx}$$(u).v+\large \frac{d}{dx}$$(v).u$
Let $u=e^{-x}$ then $\large\frac{d}{dx}$$(u)=-e^{-x} Let v=A \cos x then \large\frac{d}{dx}$$(v)=-A \sin x$
Hence $\large\frac{d}{dx}$$(uv)=-e^{-x}.A \cos x+(-A \sin x).e^{-x} =-e^{-x}[A \cos x+A \sin x] Similarly, e^{-x}.B \sin x=-e^{-x}[B \sin x-B \cos x] Therefore \large\frac{dy}{dx}$$=-e^{-x}[A \cos x+A \sin x+B \sin x-B \cos x]$
$=-e^{-x}[A \cos x+B \sin x]-e^{-x}[A \sin x-B \cos x]$
But $e^{-x}[A \cos x+B \sin x]=y$
Hence $\large\frac{dy}{dx}$$=-y.-e^{-x}[A \sin x-B \cos x] Now differentiating w.r.t x we get, \large\frac{d^2y}{dx^2}=-\frac{dy}{dx}$$-[+e^{-x}[A \cos x+A \sin x+B \sin x+B \cos x]]$
$=-\large\frac{dy}{dx}$$-[e^{-x}(A \cos x+B \sin x)]+[e^{-x}(-A \sin x+B \cos x)] \large\frac{d^2y}{dx^2}=\large\frac{-dy}{dx}-\bigg[\large\frac{dy}{dx}-y\bigg]-y$$\qquad => e^x[-A \sin x+B \cos x=\large\frac{dy}{dx}-y]$
$\large\frac{d^2y}{dx^2}$$+2\large\frac{dy}{dx}$$+2y=0$
Hence the correct option is $C$
answered May 23, 2013 by