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[ "Home | | Chemistry 12th Std | Short Questions Answer\n\n## Chapter: 12th Chemistry : UNIT 9 : Electro Chemistry\n\nChemistry : Electro Chemistry\n\n1. Define anode and cathode\n\ni) Anode: The electrode at which the oxidation occurs is called the anode.\n\nii) It is negative in a galvanic cell.\n\ni) Cathode: The electrode at which the reduction occurs is called the cathode.\n\nii) It is positive in a galvanic cell.\n\n2. Why does conductivity of a solution decrease on dilution of the solution\n\nOn dilution of the electrolyte solution, the ions present in the unit dimension was decreased and hence the conductivity of a solution also decreases.\n\n3. State Kohlrausch Law. How is it useful to determine the molar conductivity of weak electrolyte at infinite dilution.\n\nKohlrausch Law:\n\nAt infinite dilution, the limiting molar conducutivity of an electrolyte is equel to the sum of the limiting molar conductivites of its constituent ions.\n\nThe molar conductance at infinite dilution for weak electrolytes can be calculated using KohlrauschŌĆÖs law.\n\nMolar conductivity weak electrolyte :\n\nThe molar conductance of CH3COOH, can be calculated using the experimentally determind molar conductivities of strong electrolytes HCl, NaCl and CH3COONa.\n\n╔ģoHCl3COONa = ╬╗oNa + ╬╗oCH3COO-  ŌĆ”ŌĆ”ŌĆ”ŌĆ”. (1)\n\n╔ģoHCl = ╬╗oH+ + ╬╗oCl-   ---------------(2)\n\n╔ģoNaCl = ╬╗oNa+ + ╬╗oCl-   ŌĆ”ŌĆ”ŌĆ”...(3)\n\nEquation (1) + Equation (2) ŌĆō Equation (3) gives,\n\n(╔ģoCH3COONa) + (╔ģoHCl) ŌĆō (╔ģoNaCl) = ╬╗oH+ + ╬╗oCH3COO-\n\n= ╔ģoCH3COOH\n\n4. Describe the electrolysis of molten NaCl using inert electrodes\n\nThe products of electrolysis of molten NaCl is sodium metal and Cl2 gas.\n\nNa + eŌłÆ ŌåÆ Na\n\nThe anion ClŌłÆ oxidized at the anode.\n\nCl ŌåÆ ┬Į Cl2 + eŌłÆ\n\n5. State FaradayŌĆÖs Laws of electrolysis\n\nFirst law:\n\nThe mass of the substance (m) liberated at an electrode during electrolysis is directly proportional to the quantity of charge (Q) passed through the cell.\n\ni.e., m ╬▒ Q\n\nm ╬▒ lt or m= Z lt\n\nSecond law:\n\nWhen the same quantity of charge is passed through the solutions of different electrolytes, the amount of substances liberated at the respective electrodes are directly proportional to their electrochemical equivalents.\n\nm1 ŌłØ Z1\n\nm2 ŌłØ Z2\n\nm1/Z1 = m2/Z2\n\nZ = Electrochemical equivalent\n\n6. Describe the construction of Daniel cell. Write the cell reaction.\n\ni) Daniel cell or a galvanic cell is an example of electro chemical cell.\n\nii) This overall reaction is made of the summation of oxidation half reaction and reduction half reaction.\n\niii) The oxidation half cell reaction occurring at the zinc electrode.\n\nZn(s) ŌåÆ Zn2+(aq) + 2 eŌłÆ\n\niv) The reduction half reaction occurring at the copper electrode. It receives the electrons from the zinc electrode when connected externally, to produce metallic copper according to the reaction as\n\nCu2+(aq) + 2eŌłÆ ŌåÆ Cu(s)\n\nThe over all reaction taking place in the cell is the redox reaction gives as\n\nZn(s) + Cu2+(aq) ŌåÆ Zn2+(aq) + Cu(s)", null, "v) The decrease in the energy which appears as the heat energy when a zinc rod is directly dipped into the ZnSO4 solution, is converted into electrical energy. When the same reaction takes place indirectly in an electro chemical cell.\n\nvi) When the cell is set up, electrons flow from zinc electrode through the wire to the copper cathode.\n\nvii) As a result, zinc dissolves in the anode solution to form Zn2+ ions. The Cu2+ ions in the cathode half cell pick up electrons and are converted to Cu atoms on the cathode.\n\n7. Why is anode in galvanic cell considered to be negative and cathode positive electrode?\n\nŌĆó Oxidation occurs at anode, Electrons are liberated at anode and hence it is negative.\n\nŌĆó Reduction occurs at cathode. The electrons are consumed at cathode and hence it is positive.\n\n8. The conductivity of a 0.01M solution of a 1 :1 weak electrolyte at 298K is 1.5 ├Ś10-4 S cmŌłÆ1.\n\ni) molar conductivity of the solution\n\nii) degree of dissociation and the dissociation constant of the weak electrolyte\n\nGiven that\n\n╬╗┬║cation = 248.2 S cm2 molŌłÆ1\n\n╬╗┬║cation 51.8 S cm2 molŌłÆ1\n\nGiven:\n\nC = 0.01M\n\n╬╗┬░cation = 248.2 S cm2 molŌłÆ1\n\nK = 1.5 ├Ś 10-4 S cmŌłÆ1\n\n╬╗┬░anion = 51.8 S cm2 molŌłÆ1\n\n(i) molar conductivity\n\n╔ģom = [ k ├Ś10ŌłÆ3 / M ] molŌłÆ1m3\n\n╔ģo m= (1.5 ├Ś l0-4 ├Ś l0ŌłÆ3 ├Ś l02 ) / ( 1 ├Ś10-2)\n\n╔ģ┬░m = 1.5 ├Ś 10ŌłÆ3 S m2 molŌłÆ1\n\n(ii) Degree of dissociation\n\n╬▒ = ╔ģom / ╔ģo╬▒\n\n╔ģom = ╬╗o+ + ╬╗o-\n\n╔ģoŌł× = ╬╗ocation + ╬╗oanion\n\n= (248.2 + 51.8) S cm2 molŌłÆ1\n\n= 300 S cm2 molŌłÆ1\n\n╔ģoŌł× = 300 ├Ś 10-4 S m2 molŌłÆ1\n\n╔ģom = 1.5 ├Ś 10ŌłÆ3 Sm2 molŌłÆ1\n\n╬▒ = ╔ģom / ╔ģoŌł× = (1.5 ├Ś 10ŌłÆ3 S m2 mo1ŌłÆ1) / (300 ├Ś 10-4 S m2 molŌłÆ1)\n\n╬▒ = 0.05\n\nDissociation constant\n\nKa = ╬▒2C / 1-╬▒\n\n= (0.05)2(0.01) / 1-0.05\n\n= (25 ├Ś 10-4 ├Ś 10-2 ) / 95 ├Ś 10-2\n\n= 0.26 ├Ś 10-4\n\n= 2.6 ├Ś 10-5\n\ni) Molar conductivity = 1.5 ├Ś 10ŌłÆ3 S m2 molŌłÆ1\n\nii) Degree of dissociation = 0.05\n\nDissociation constant = 2.6 ├Ś 10-5\n\n9. Which of 0.1M HCl and 0.1 M KCl do you expect to have greater ╬ø┬║m and why?\n\n0.1 M HCl shows greater acidity\n\nHCl dissociated as H+ and ClŌłÆ\n\nKCl dissociated as K+ and ClŌłÆ\n\nHCl releases H+.\n\nThe substance releases H+ is acid. Hence 0.1M HCl is more acidic than 0.1M KCl.\n\n10. Arrange the following solutions in the decreasing order of specific conductance.\n\ni) 0.01M KCl ii) 0.005M KCl iii) 0.1M KCl iv) 0.25 M KCl v) 0.5 M KCl\n\ni) 0.5 M KCl\n\nii) 0.25 M KCl\n\niii) 0.1 M KCl\n\niv) 0.01 M KCl\n\nv) 0.005 M KCl\n\n11. Why is AC current used instead of DC in measuring the electrolytic conductance?\n\nWhen DC current is applied through the conductivity cell, it will lead to the electrolysis of the solution taken in the cell.\n\nHence AC current is used for this measurement to prevent electrolysis.\n\n12. 0.1M NaCl solution is placed in two different cells having cell constant 0.5 and 0.25cm-1 respectively. Which of the two will have greater value of specific conductance.\n\nSolution:\n\nK = C ├Ś (Ōäō/A)\n\nSpecific conductance is directly proportional to cell constant. Higher the cell constant higher will be the value of specific conductance. NaCl placed in cell having 0.5 cmŌłÆ1 shows greater value of specific conductance.\n\n13. A current of 1.608A is passed through 250 mL of 0.5M solution of copper sulphate for 50 minutes. Calculate the strength of Cu2+ after electrolysis assuming volume to be constant and the current efficiency is 100%.\n\nGiven: I = 1.608A;\n\nt = 50 min = 50 ├Ś 60 = 3000s\n\nV = 500 mL; C = 0.5 M ; ╔│ = 100%\n\nSolution:\n\nThe number of Faradays of electricity passed through the CuSO4 solution\n\nŌćÆ Q = It\n\nQ = 1.608 ├Ś 3000\n\nQ = 4824 C\n\nŌł┤ Number of Faradays of electricity\n\n= 4824 C / 96500 C = 0.5F\n\nElectrolysis of CuSO4\n\nCu2+ (aq) + 2eŌłÆ ŌåÆ Cu(s)\n\nThe above equation shows that 2F electricity will deposit 1 mole of Cu2+ to Cu.\n\nŌł┤ 0.5F electricity will deposit ( 1 mol / 2F ) ├Ś 0.5 F\n\n= 0.025 mol\n\nInitial number of moles of Cu2+ in 250 ml of solution = ( 0.5 / 1000 mL ) ├Ś 250 mL\n\n= 0.125 mol\n\nŌł┤ Concentration of Cu2+ = [ 0.1 mol / 250 mL ] ├Ś 1000 mL\n\n= 0.4M\n\n14. Can Fe3+ oxidises bromide to bromine under standard conditions?\n\nGiven: EFe 3+|Fe2+ = 0.771\n\nEBr2|BrŌłÆ = 1.09V.\n\nRequired half cell reaction\n\n2BrŌłÆ ŌåÆ Br2 + 2eŌłÆ            (E┬░ox) = ŌłÆ1.09V\n\n2Fe3+ + 2eŌłÆ  ŌåÆ 2Fe2+           (E┬░red) = +0.771V\n\n2Fe2+ + 2Br ŌłÆ ŌåÆ 2Fe2+ + Br2 (E┬░cell) = ?\n\n(E┬░cell) = (E0ox) + (E┬░red)\n\n= ŌłÆ1.09 + 0.771\n\n= ŌłÆ0.319 V\n\nE┬░cell is ŌĆōve; ŌłåG is +ve and the cell reaction is non spontaneous. Hence Fe3+ cannot oxidizes BrŌłÆ to Br2.\n\n15. Is it possible to store copper sulphate in an iron vessel for a long time?\n\nGiven : ECu 2+|Cu = 0.34 V and EFe 2+|Fe = ŌłÆ0.44V .\n\nGiven: EoCu2+| Cu = 0.34 V and E┬░Fe2+|Fe = ŌłÆ 0.44V\n\n(E┬░ox)Fe | Fe2+ = 0.44V and (E┬░red)Cu2+ | Cu = 0.34V\n\nThe +ve emf values shows that iron will oxidise and copper will get reduced ie., the vessel will dissolve. Hence it is not possible to store copper sulphate in an iron vessel.\n\n16. Two metals M1 and M2 have reduction potential values of -xV and +yV respectively. Which will liberate H2 and H2SO4.\n\nMetals having higher oxidation potential will liberate H2 from H2SO4. Hence, the metal M1 having + xV, oxidation potential will liberate H2 from H2SO4.\n\n17. Reduction potential of two metals M and M2 are E┬║M12+|M1 = ŌłÆ 2.3V and E E┬║M22+|M2 = 0.2V  Predict which one is better for coating the surface of iron. Given : E Fe2+ |Fe = ŌłÆ0.44V\n\nOxidation potential of M1 is more +ve than the oxidation potential of Fe which indicates that it will prevent iron from rusting.\n\n18. Calculate the standard emf of the cell: Cd | Cd 2+ || Cu 2+ | Cu and determine the cell reaction. The standard reduction potentials of Cu2+ | Cu and Cd2+ | Cd are 0.34V and -0.40 volts respectively. Predict the feasibility of the cell reaction.\n\nCell reactions:\n\nOxidation at anode : Cd(s) ŌåÆ Cd2+ (aq) + 2eŌłÆ\n\nReduction at cathode: Cu2+(aq) + 2eŌłÆ ŌåÆ Cu(s)\n\n(E┬░ox) Cd ŌłŻ Cd2+ = 0.4V\n\n(E┬░red) Cu2+  | Cu= 0.34V\n\nE┬░cell = E┬░R - E┬░L\n\n= 0.4 + 0.34\n\n= 0.74V\n\nemf = +ve ; ŌłåG = ŌłÆve\n\nThe reaction is feasible.\n\n19. In fuel cell H2 and O2 react to produce electricity. In the process, H2 gas is oxidised at the anode and O2 at cathode. If 44.8 litre of H2 at 25oC and 1atm pressure reacts in 10 minutes, what is average current produced? If the entire current is used for electro deposition of Cu from Cu2+ , how many grams of Cu deposited?\n\nOxidation at anode:\n\n2H2(g) + 4OHŌłÆ (aq) ŌåÆ 4H2O (­ØæÖ) + 4eŌłÆ\n\n1 mole of hydrogen gas produces 2 moles of electrons at 25 ┬░C and 1 atm pressure, 1 mole of hydrogen gas occupies = 22.4 litres.\n\nŌł┤ no.of moles of hydrogen gas produced\n\n= [1 mole / 22.4 litres] ├Ś 44.8 litres\n\n= 2 moles of hydrogen\n\nŌł┤ 2 moles of hydrogen produces 4 moles of electron ie., 4F charge.\n\nQ = It\n\nI = Q / t\n\n= (4F) / (10 minutes)\n\n= (4 ├Ś 96500 C ) / ( 10 ├Ś 60 s )\n\nI = 643.33 A\n\nElectro deposition of copper\n\nCu2+ (aq) + 2eŌłÆ ŌåÆ Cu(s)\n\n2F charge is required to deposit\n\n1 mole of copper ie., 63.5 g\n\nIf the entire current produced in the fuel cell ie., 4F is utilized for electrolysis, then 2├Ś63.5 ie., 127.0 g copper will be deposited at cathode.\n\n20. The same amount of electricity was passed through two separate electrolytic cells containing solutions of nickel nitrate and chromium nitrate respectively. If 2.935g of Ni was deposited in the first cell. The amount of Cr deposited in the another cell? Give : molar mass of Nickel and chromium are 58.74 and 52gm-1 respectively.\n\nNi2+ (aq) + 2eŌłÆ ŌåÆ Ni(s)\n\nCr3+ (aq) + 3eŌłÆ ŌåÆ Cr (s)\n\nThe above reaction indicates that 2F charge is required to deposit 58.7g of Nickel from nickel nitrate and 3F charge is required to deposit 52 g of chromium.\n\nGiven that 2.935 gram of Nickel is deposited\n\nŌł┤ The amount of charge passed through the cell = [ 2F / 58.7g ] ├Ś 2.935g\n\n= 0.1 F\n\nŌł┤ If 0.1 F charge is passed through chromium nitrate the amount of chromium deposited\n\n= [52g / 3F] ├Ś 0.1 F = 1.733 g\n\n21. A copper electrode is dipped in 0.1M copper sulphate solution at 25oC . Calculate the electrode potential of copper. [Given: ECu 2+|Cu = 0.34 V ].\n\nSolution:\n\n[Cu2+] = 0.1M ;\n\nE┬░Cu2+ ŌłŻ Cu = 0.34 V\n\nEcell = ?\n\nCell reaction is\n\nCu2+ (aq) + 2eŌłÆ ŌåÆ Cu(s)\n\nEcell = E┬░cell ŌĆō { 0.0591/n log ([Cu] / [Cu2+]) }\n\n= 0.34 ŌĆō {(0.0591/2) log (1/0.1)}\n\n= 0.34 ŌĆō 0.0296\n\nEcell = 0.31 V\n\n22. For the cell Mg (s) | Mg2+ (aq) || Ag+ (aq) | Ag (s), calculate the equilibrium constant at  25oC and maximum work that can be obtained during operation of cell. Given : E┬║ Mg 2+ | Mg = ŌłÆ2.37V and E┬║Ag + | Ag = 0.80V.\n\nOxidation at anode\n\nMg ŌåÆ Mg2+ + 2eŌłÆ         ------(1)\n\n(E┬░ox) = 2.37 V\n\nReduction at cathode\n\nAg++ eŌłÆ ŌåÆ Ag           --------- (2)\n\n(E┬░red) = 0.80 V\n\nE┬░cell = (E┬░ox)anode + (E┬░red)cathode\n\n= 2.37 + 0.80\n\n= 3.17 V\n\nOverall reaction\n\nEqn (1) + 2 ├Ś eqn (2) ŌćÆ\n\nMg + 2Ag+ ŌåÆ Mg2+ + 2Ag\n\nŌłåG┬░ = ŌłÆnFE┬░\n\n= ŌłÆ2 ├Ś 96500 ├Ś 3.17\n\n= ŌłÆ611.810 J\n\nŌłåG┬░= ŌłÆ6.12 ├Ś 105 J\n\nŌłåG┬░ = ŌłÆ2.303 RT log Kc\n\nŌćÆ log Kc = [ ŌłÆ6.12 ├Ś 105 ] / [2.303 ├Ś 8.314 ├Ś 298]\n\nKc = Antilog of (107.2)\n\n23. 8.2 ├Ś1012 litres of water is available in a lake. A power reactor using the electrolysis of water in the lake produces electricity at the rate of 2 ├Ś106 CsŌłÆ1 at an appropriate voltage. How many years would it like to completely electrolyse the water in the lake. Assume that there is no loss of water except due to electrolysis.\n\nHydrolysis of water\n\nAt anode:\n\n2H2O ŌåÆ 4H+ + O2 + 4eŌłÆ    ----------(1)\n\nAt cathode:\n\n2H2O + 2eŌłÆ ŌåÆ H2 + 2OHŌłÆ      --------(2)\n\nOverall reaction\n\n6H2O ŌåÆ 4H+ + 4OHŌłÆ + 2H2+O2\n\n(or) Eqn (1) + (2) ├Ś 2 ŌćÆ 2H2O ŌåÆ 2H2 + O2\n\nŌł┤ According to Faraday's law of electrolysis, to electrolyse two moles of water (36g = 36 mL of H2O), 4F charge is required alternatively, when 36 mL of water is electrolysed, the charge generated = 4 ├Ś 96500 C.\n\nŌł┤ When the whole water which is available on the lake is completely electrolysed the amount of charge generated is equal to\n\n= [ 4 ├Ś 96500C / 36 mL ] ├Ś 9 ├Ś 1012 L\n\n= [ (4 ├Ś 96500 ├Ś 9 ├Ś 1012) / (36 ├Ś 10ŌłÆ3)]  ├Ś  C\n\n= 96500 ├Ś 1015 C\n\nŌł┤ Given that in 1 second, 2 ├Ś 106 C is generated therefore, the time required to generate\n\n96500 ├Ś 1015 C is = [ 1S / 2├Ś106C ] ├Ś 96500 ├Ś 1015 C\n\n= 48250 ├Ś 109 S\n\n1 years = 365 days\n\n= 365 ├Ś 24 hours\n\n= 365 ├Ś 24 ├Ś 60 min\n\n= 365 ├Ś 24 ├Ś 60 ├Ś 60 sec\n\nŌł┤ Number of years = [ 48250 ├Ś l0 9 ] / [365 ├Ś 24 ├Ś 60 ├Ś 60]\n\n= 1.5299 ├Ś 106 yearsŌĆā\n\n24. Derive an expression for Nernst equation\n\nNernst equation relates the cell potential and the concentration of the species involved in an electrochemical reaction.\n\nConsider an electrochemical reaction is\n\nxA + yB Ōćī lC + mD\n\nThe reaction quotient Q for the above reaction is given below\n\nQ = [C]­ØæÖ [D]m  /  [A]x[B]y ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ”(1)\n\nŌłåG = ŌłåG┬░ + RT InQ   ŌĆ”ŌĆ”ŌĆ”ŌĆ”ŌĆ” (2)\n\nThe Gibbs free energy can be related to the cell emf\n\nŌłåG = ŌłÆnFEcell ;\n\nŌłåG┬░ = ŌłÆnFE┬░cell\n\nSubstitute ŌłåG┬░ and Q in the eqn (2)\n\nŌłÆnFEcell = ŌłÆnFE┬░cell + RT Ōäōn { [C]l[D]m / [A]x[B]y }\n\nDivide the whole eqn by (ŌłÆnF)\n\nŌćÆ Ecell = E┬░cell ŌĆō { 2.303 RT / nF   log. ( [C]I[D]m / [A]x[B]y ) }", null, "This equation is called the Nernst equation.\n\n25. Write a note on sacrificial protection.\n\nIt is a method to prevent the objects from corrosion. The object is covered with the protecting metals which is corrected more easily than the object. The metal corrods itself but saves the object.\n\nEg: Fe object is protected by Mg or Zn.\n\n26. Explain the function of H2 - O2 fuel cell.\n\nFuel cell\n\nThe energy of combustion of fuels is directly converted into electrical energy is called the fuel cell. The general representation of a fuel cell is follows\n\nFuel ŌłŻ Electrode ŌłŻ Electrolyte ŌłŻ Electrode ŌłŻ Oxidant\n\nFuel - Hydrogen\n\nElectrolyte - Aqueous KOH\n\nOxidant - Oxygen\n\nElectrode - Porous graphite containing Ni\n\nNi and NiO serves as the inert electrodes.\n\nHydrogen and oxygen gases are bubbled through the anode and cathode, respectively.\n\nOxidation occurs at the anode:\n\n2H2(g) + 4OHŌłÆ (aq) ŌåÆ 4H2O (Ōäō) + 4eŌłÆ\n\nReduction occurs at the cathode\n\nO2(g) + 2H2O(Ōäō) + 4eŌłÆ ŌåÆ 4OHŌłÆ (aq)\n\nThe overall reaction is\n\n2H2(g) + O2 (g) ŌåÆ 2H2O(Ōäō)\n\n27. Ionic conductance at infinite dilution of Al3+ and SO42- are 189 and 160 mho cm2 equiv-1. Calculate the equivalent and molar conductance of the electrolyte Al2 (SO4 )3 at infinite dilution.\n\nMolar conductance\n\n╔ģ┬░m (Al2(SO4)3) = 2╬╗┬░Al3+ + 3╬╗┬░SO42ŌłÆ\n\n= (2 ├Ś 189) + (3 ├Ś 160)\n\n╔ģ┬░m = 858 mho cm2 molŌłÆ1\n\nEquivalent conductance\n\n╬╗Ōł× (Al2SO4)3) = ( 1/3 ╬╗Ōł× Al3+ ) + ( 1/2 ╬╗Ōł× SO42ŌłÆ )\n\n= ( 1/3 ├Ś 189) + ( 1/2 ├Ś 160 )\n\n= 63 + 80 = 143 mho cm2 equivŌłÆ1\n\nEVALUATE YOURSELF:\n\n1. Calculate the molar conductance of 0.01M aqueous KCl solution at 25┬░C. The specific conductance of KCl at 25┬░C is 14.114 ├Ś 10 -2 SmŌłÆ1.\n\nGiven:\n\nC = 0.01 M\n\nk = 14.114 ├Ś 10ŌłÆ 2 SmŌłÆ1\n\n╔ģm = ?\n\n╔ģm = (K ├Ś 10ŌłÆ3) / M\n\n= ( 14.11 ├Ś 10-2 ├Ś 10ŌłÆ3 ) / 0.01\n\n= (14.11 ├Ś 10-2 ├Ś 10 ŌłÆ3 ) / 10-2\n\n╔ģm = 14.114 ├Ś 10ŌłÆ3 Sm2molŌłÆ1\n\n2. The resistance of 0.15N solution of an electrolyte is 50 ╬®. The specific conductance of the solution is 2.4 SmŌłÆ1. The resistance of 0.5 N solution of the same electrolyte measured using the same conductivity cell is 480 ╬®. Find the equivalent conductivity of 0.5N solution of the electrolyte.\n\nGiven:\n\nR1 = 50 ╬®\n\nR2 = 480 ╬®\n\nK1 = 2.4 SmŌłÆ1\n\nK2 = ?\n\nN1 = 0.15 N\n\nN2 = 0.5 N\n\n╔ģ = [ K2(Sm ŌłÆ1) ├Ś 10- 3(gram equivalent)ŌłÆ1 m3 ] / N\n\nWe know that\n\nK = Cell constant / R\n\nŌł┤ K2/ K1 = R1/R2\n\nK2 = K1 ├Ś [R1/R2]\n\n= 24 SmŌłÆ1 ├Ś (50╬® / 480╬®)\n\n= 0.25 SmŌłÆ1\n\n= [ 0.25 ├Ś 10- 3 S (gram equivalent) ŌłÆ1m2 ] / 0.5\n\n╔ģ = 5 ├Ś 10-4 Sm2 gram equivalentŌłÆ1\n\n3. The emf of the following cell at 25┬░C is equal to 0.34V. Calculate the reduction potential of copper electrode.\n\nPt(s) ŌłŻ H2 (g, 1 atm) ŌłŻ H+ (aq, 1M) ŌłŻ ŌłŻ Cu2+ (aq, 1M) I Cu(s)\n\nE┬░cell = (E┬░ox)anode + (E┬░Red)cathode\n\n= (E┬░ox)SHE + (E┬░Red)Cu2+ ŌłŻ Cu\n\n= 0 + 0.34 V\n\nE┬░cell = 0.34V\n\n4. Using the calculated emf value of zinc and copper electrode, calculate the emf of the following cell at 25 ┬░C.\n\nZn(s) ŌłŻ Zn2+(aq, 1M) ŌłŻ ŌłŻ Cu2+(aq, 1M) ŌłŻ Cu(s)\n\nE┬░Zn2+ ŌłŻ Zn = ŌłÆ0.76 V\n\nE┬░cu2+ ŌłŻ Cu = 0.34 V\n\nE┬░cell = E┬░R ŌłÆ E┬░L\n\n= E┬░Cu2+ ŌłŻ Cu ŌłÆ E┬░Zn2+ ŌłŻ Zn\n\n= 0.34 ŌłÆ (ŌłÆ0.76) = 1.10 V\n\nE┬░cell = 1.10V\n\n5. Write the overall redox reaction which takes place in the galvanic cell,\n\nPt(s) ŌłŻ Fe2+(aq), Fe3+(aq) ŌłŻ ŌłŻ MnO-4 (aq) H+ (aq), Mn2+(aq) ŌłŻ Pt(s)\n\nOxidation occur at anode\n\nFe2+ ŌåÆ Fe3+ + eŌłÆ\n\nReduction occur at cathode\n\nMnOŌłÆ4 + 8H+ + 5eŌłÆ ŌåÆ Mn2+ + 4H2O\n\nOverall reaction\n\n5Fe2+ + MnOŌłÆ 4 + 8H+ ŌåÆ 5Fe3+ + Mn2+ + 4H2O\n\n6. The electrochemical cell reaction of the Daniel cell is\n\nZn(s) + Cu2+ (aq) ŌåÆ Zn2+ (aq) + Cu(s)\n\nWhat is the change in the cell voltage on increasing the ion concentration in the anode compartment by a factor 10?\n\nFor Daniel cell\n\nEcell = E┬░cell ŌĆō (0.0591/2) log ( [Zn2+] / [Cu2+] )", null, "For 1 M\n\nE = E┬░ ŌłÆ 0.0295 log [lM/1M]\n\nE = E┬░ ŌłÆ 0.0295 ├Ś 0 = E┬░\n\nFor 10 M\n\nE = E┬░ ŌłÆ 0.0295 log [10M/1M]\n\nE = E┬░ ŌłÆ 0.0295 log \n\nE = E┬░ ŌłÆ 0.0295 ├Ś 1\n\nE = E┬░ ŌłÆ 0.0295\n\nOn increasing the ion concentration in anode compartment by 10 factor then the cell voltage decreased by the value of 0.0295.\n\n7. A solution of a salt of metal was electrolysed for 15 minutes with a current of 0.15 amperes. The mass of the metal deposited at the cathode is 0.783 g. Calculate the equivalent mass of the metal.\n\nt = 15 min = 15 ├Ś 60 = 900 sec\n\nI = 0.15 A\n\nm = 0.783 g\n\nZ = ?\n\nm = ZIt\n\nZ = m / It = [ 0.783g ] / [ 0.15 A ├Ś 15 ├Ś 60 sec ]\n\nZ = 0.0058\n\nTags : Electro Chemistry , 12th Chemistry : UNIT 9 : Electro Chemistry\nStudy Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail\n12th Chemistry : UNIT 9 : Electro Chemistry : Short Questions Answer | Electro Chemistry" ]

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[ "# Gravitational Waves from Axion Monodromy\n\nArthur Hebecker Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany Joerg Jaeckel Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany Fabrizio Rompineve Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany Lukas T. Witkowski Institute for Theoretical Physics, University of Heidelberg, Philosophenweg 19, 69120 Heidelberg, Germany\nJune 24, 2016\n###### Abstract\n\nLarge field inflation is arguably the simplest and most natural variant of slow-roll inflation. Axion monodromy may be the most promising framework for realising this scenario. As one of its defining features, the long-range polynomial potential possesses short-range, instantonic modulations. These can give rise to a series of local minima in the post-inflationary region of the potential. We show that for certain parameter choices the inflaton populates more than one of these vacua inside a single Hubble patch. This corresponds to a dynamical phase decomposition, analogously to what happens in the course of thermal first-order phase transitions. In the subsequent process of bubble wall collisions, the lowest-lying axionic minimum eventually takes over all space. Our main result is that this violent process sources gravitational waves, very much like in the case of a first-order phase transition. We compute the energy density and peak frequency of the signal, which can lie anywhere in the mHz-GHz range, possibly within reach of next-generation interferometers. We also note that this \"dynamical phase decomposition\" phenomenon and its gravitational wave signal are more general and may apply to other inflationary or reheating scenarios with axions and modulated potentials.\n\n\\setcaptionmargin\n\n1cm\n\n## 1 Introduction\n\nThe central predictions allowing us to discriminate between inflationary models are the slow roll parameters, most prominently the tilt of the scalar power spectrum and the tensor-to-scalar ratio [1, 2]. However, even with growing precision many models are expected to remain consistent with this limited set of data. Hence it is of great importance to identify additional predictions characterizing specific classes of models.\n\nHere, our main focus will be axion monodromy inflation [3, 4] (see [5, 6, 7] for its supergravity incarnation as ‘-term axion monodromy’ and [8, 9, 10, 11, 12, 13] for a selection of related recent work). In this promising class of models, the inflaton originally enjoys a discrete shift symmetry, explicitly broken by a polynomial term, such as . The aim of this paper is to describe a new and potentially striking observational signature which is peculiar to axion monodromy inflation. For previous work on the phenomenology of such inflationary models see [14, 15, 16, 17, 18, 19] (see also [20, 21] for closely related potentials) and, in particular, [22, 23, 24] for preheating and for oscillon dynamics in this context (see in particular for gravitational radiation from preheating in oscillon models).\n\nIn short, our message is the following: Due to the typical instantonic modulations of the potential, first order phase-transition-like, violent dynamics may occur after the end of inflation and before reheating. This leads to additional gravitational waves, which are of course very different in frequency from those studied in the CMB. Thus, monodromy models may (in addition to or independently of their prediction of ) be established by future ground- or space-based interferometers. The final word would thus come from the new field of gravitational-wave astronomy (for a recent review see e.g. ).\n\nOf course, gravitational waves are a well-known signature of cosmic strings, including axionic strings. For recent work on the non-trival late-time dynamics of axionic models and gravitational waves which is closer in spirit to our proposal see, e.g. [29, 30, 31, 32, 33, 34]. Emission of gravitational waves from axionic couplings in inflationary setups has recently been considered in [35, 36] (see also [37, 38] for constraints). Independently of the gravitational wave signal, related dynamical phenomena may also occur in the dark matter context [39, 40].\n\nLet us now explain the physics underlying our scenario in more detail: The basic building block of monodromy inflation is an axion with sub-planckian decay constant . Non-perturbative effects induce the familiar -type potential. When the monodromy effect is included, this cosine-potential shows up in the form of modulations of the long-range, polynomial term. Even if the relative size of these modulations is small at large field values, where slow-roll inflation is realized, they can become dominant near the minimum of the polynomial potential. After inflation, the field oscillates with decreasing amplitude such that its motion eventually becomes confined to the vicinity of one of these local minima. However, due to field fluctuations, different local minima may be chosen in different regions of the same Hubble patch. In other words, the Universe is decomposed into phases.\n\nTwo comments are in order. First, the field fluctuations inducing the above phenomenon can have different origin. On the one hand, there are inflationary super-horizon fluctuations which have become classical at the time when they re-enter the horizon. On the other hand, the field is subject to an intrinsic quantum uncertainty at any given time. As we will see, this second type of uncertainty is, after parametric amplification , more likely to source the desired phase decomposition.\n\nSecond, while we for definiteness identify the monodromic axion with the inflaton, the phenomenon can occur also in different contexts. In particular, similar physics may arise in any non-inflationary axion model with a monodromy (the relaxion being one recent popular example , see also ). Moreover, even within the inflationary context, our proposal of ‘dynamical phase decomposition’ is not restricted to monodromy models. Indeed, models of ‘aligned’ or ‘winding’ inflation [43, 44, 45] can naturally exhibit short-range modulations on top of a long-range periodic potential. The Weak Gravity Conjecture for instantons may in fact demand such modulations [47, 48] and the simplest string constructions naturally provide them . Furthermore, the Weak Gravity Conjecture for domain walls constrains models based on monodromy [42, 50]. In particular, the size of the wiggles is bounded .\n\nLet us now complete the discussion of the cosmological dynamics: After a phase decomposition has occurred, the regions with the lowest-lying populated minimum will expand. This is very similar to the way in which a strong first-order phase transition is completed through the collision of cosmic bubbles of true vacuum. However, in our case the transition occurs before reheating and very far from thermal equilibrium. It may thus be more appropriate to talk about ‘dynamical phase decomposition’ rather than about a phase transition in the usual sense. Nevertheless, the concept of bubble formation and collision is still appropriate in our setting.\n\nThermodynamic cosmological phase transitions have been widely explored in various contexts (see [51, 52, 53] for early seminal work and and refs. therein for the case of the electroweak phase transition). In particular, it is well known that they source gravitational radiation (see [55, 56, 57] for recent reviews and e.g. for radiation from other cosmological sources such as cosmic strings and preheating). We will rely on these results.\n\nThis paper is structured as follows: Sec. 2 provides the basic setup. More specifically, Sec. 2.1 introduces our axion monodromy setting with dominant quadratic potential and a series of local minima, Sec. 2.2 briefly discusses reheating, and Sec. 2.3 explains the post-inflationary dynamics and the phase decomposition. In Sec. 3 we estimate the probability that a phase decomposition occurs as a consequence of field fluctuations. In particular, we treat inflationary fluctuations in Sec. 3.1 and the intrinsic quantum uncertainty in Sec. 3.2. In Sec. 4 we address the crucial issue of the possible enhancements of fluctuations due to background-field-oscillations in the modulated potential. Ultimately, in Sec. 5 we estimate the spectrum and abundance of the gravitational radiation produced during the phase transition before we conclude in Sec. 6. Additionally, we devote Appendix A to a more detailed discussion of scalar field fluctuations after inflation. Appendix B provides some details of an inflection point model of inflation in which our gravitational wave signal may also arise.\n\n## 2 Phases from axion monodromy\n\nIn this section we introduce a string-motivated scenario of the inflationary universe. It is based on the framework of Axion Monodromy Inflation [3, 4]. We begin by explaining the basic features of the axion potential and the possibility of having coexisting populated axionic vacua.\n\n### 2.1 Local minima in axion monodromy\n\nThe inflaton potential of a model of axion monodromy inflation contains an oscillatory term, which respects a discrete shift symmetry, and a polynomial term, which breaks it explicitly. In this work we will take the polynomial term to be quadratic, as this will be sufficient to demonstrate the effect we wish to study:\n\n V(ϕ)=12m2ϕ2+Λ4cos(ϕf+γ). (2.1)\n\nThe potential (2.1), plotted in Fig. 1, can have local minima, whose existence depends on the values of the prefactor , the so-called axion decay constant and the inflaton mass . Although we have in mind the specific case in which is the inflaton, many of our considerations apply to the case of a generic axion-like field with potential (2.1).111While we expect a similar qualitative behaviour the numerical results may depart significantly from the values we find here.\n\nWe begin with an analysis of the classical evolution of . The equation of motion of reads:\n\n ¨ϕ+3H˙ϕ+V′(ϕ)=0, (2.2)\n\nwhere the prime denotes a derivative with respect to . For constant Hubble rate and temporarily neglecting the cosine term in (2.1), the solution of (2.2) is\n\n ϕ∼e−32Htcos(ωt). (2.3)\n\nTherefore, the amplitude of decreases. This conclusion remains valid even for time-varying . In fact, since , the Hubble rate decreases as the amplitude of falls. Once the amplitude is sufficiently small, the cosine oscillations in (2.1) cannot be neglected any longer. Eventually, the field is caught in one of the cosine wells. One can observe that the field is more likely to get trapped in one of the lowest-lying minima. This can be understood as follows: the friction term in (2.2) becomes less and less relevant as falls. This implies that the fractional energy loss per oscillation also decreases. Therefore, even though the field can in principle get stuck at any time, it is more likely to do so late in its evolution, when it is oscillating near the bottom of the well containing the lowest-lying minima.222The actual argument to show that wells at the bottom of the potential are more likely to host the inflaton requires more care. In Sec. 4 we provide numerical examples that support this statement. For this reason, we will mostly focus on the last two wells.\n\nThe existence of different local minima implies that the universe is potentially decomposed into several phases. This happens if the field settles in different minima in different parts of the universe. Due to fluctuations, the scalar can end up in one or the other minimum in different regions of the same Hubble patch. In this paper, we are interested in studying the conditions under which such a phase decomposition can take place.\n\nIf such a phenomenon occurs, eventually the field will settle in the state of lower energy as a consequence of the expansion of bubbles containing the true vacuum. This corresponds to a phase transition, which can in principle have strong cosmological signatures, above all the radiation of gravitational waves. We would like to provide an estimate for the spectrum of gravitational waves produced in such an event (see also [29, 30, 31, 32, 33] for related work, but in different contexts).\n\nBefore moving on to study the details of this scenario, let us determine the condition on the parameters for the potential to exhibit local minima. In order to have local minima, the equation must have non-vanishing solutions. Let us, for a moment, simplify by setting . We then have:\n\n V′=0⇒m2ϕ=Λ4fsin(ϕf). (2.4)\n\nGraphically, it is clear that this equation has non-vanishing solutions only if\n\n κ≡Λ4f2m2≥1. (2.5)\n\nHere we have used , but the equation remains parametrically valid even for . Under this condition the potential has the form represented in Fig. (1).", null, "Figure 1: Monodromy potential, as in (2.1) with γ=0. The parameter κ/π approximately measures the number of local minima. Here κ≃50.\n\nPractically however, as we have already remarked, we will focus only on the two lowest local minima, which are in general non-degenerate for (see Fig. 2).333The choice may lead to stable domain walls, which are generically problematic.", null, "Figure 2: Non-degenerate two-well potential, as it can be obtained from (2.1) by focusing on the two wells closest to the origin.\n\nWe are now ready to move on to a detailed discussion of phase decomposition in our scenario. However, before doing so, a few comments about reheating are in order.\n\n### 2.2 Reheating\n\nTypically, after inflation the Universe undergoes a so-called reheating phase, where the energy density in the inflaton sector is transferred to standard model degrees of freedom, and possibly to dark sectors. Our analysis of the dynamics of the inflaton after inflation needs to take this into account. We therefore first focus on the following question: does reheating happen before or after the inflaton field is caught in one of the cosine wells?\n\nIn order to answer this question, we need to specify the interactions of the inflaton with matter and/or radiation. Here let us consider the example of a Planck-suppressed modulus-like coupling to a scalar field . The largest decay rate, barring the possibility of parametric resonances, is obtained if enjoys a coupling , leading to\n\n Γϕ→χχ∼m3ϕM2p. (2.6)\n\nThis can arise if is a Higgs boson (see e.g. [59, 60]). Decay rates to gauge bosons may also be of this type. The inflaton may also couple to scalar matter as , but this leads to even smaller decay rates.\n\nThe strategy is now as follows: we assume that the field is trapped in one of the cosine wells, and compare the perturbative decay rate (2.6) with the Hubble rate , as we are assuming that the field is oscillating in the last wells. For the same reason, we should take in (2.6). As usual in cosmology, the condition for the decay to be efficient is . If we find , then we will be consistent with our assumption that the field is first caught in one of the wells and decays only later. Now,\n\n Γϕ\n\nThe inequality (2.7) is easily satisfied in our setup, as in quadratic inflation and may be only slightly smaller than , while . Therefore the field generically decays perturbatively only after getting caught in one of the cosine wells. In what follows, we will therefore not consider the decay of any longer.\n\n### 2.3 Field oscillations and damping\n\nWe first focus on how the Hubble parameter changes after inflation. The energy density of the universe is a sum of three terms: , where is the energy density due to the cosmological constant. We absorb in and define such that the global minimum has vanishing potential energy. The evolution of the Hubble parameter and of the scalar field is dictated by the Friedmann equation, together with the equation of motion of :\n\n 3H2 =12m2˙ϕ2+Vϕ (2.8) ¨ϕ+3H˙ϕ =−V′ϕ. (2.9)\n\nSince the energy density is decreasing due to friction, it is clear that also will decrease with . However, whenever , is undamped and the energy density is constant. In consequence is stationary as well. The typical behaviour of and is shown in Fig. 3.", null, "Figure 3: Evolution of: (a) the scalar field ϕ and (b) the Hubble rate H according to (2.9) and (2.8). Vϕ is as in (2.1) and Vλ=0. The initial condition is ϕ(t0)=Mp. Furthermore we have chosen κ=60,f=10−2Mp. The scalar field oscillates around ϕ=0 over a wide field range crossing several wells, before getting caught in one of the local minima at t≈11/m.\n\nLet us now examine the energy density in the inflaton field in more detail. Its amplitude decreases after each oscillation due to Hubble friction. At some point the energy density is comparable to the height of the last cosine wells. The field is then caught in one of the local minima, depending on the initial conditions. In the absence of spatial field inhomogeneities, the inflaton will populate only one of these two minima. This situation is shown in Fig. 4.", null, "Figure 4: Monodromy potential, as in (2.1), with γ≠0. The axes are chosen such that the local maximum between the last two wells sits at ϕ=0 and the lowest minimum has V=0. Here κ≈12. The turning points in the field trajectory are shown. The amplitude of the oscillations decreases as a consequence of Hubble friction.\n\nThe conclusions can radically change in the presence of field fluctuations . In this case, the field may end up in one or the other minimum in different regions belonging to the same Hubble patch. The lines drawn in Fig. 4 become bands of a certain width, corresponding to the uncertainty in the energy density induced by (see Fig. 5). Now suppose that, as a consequence of friction, the energy density has decreased to a value close to the height of the barrier separating the two last minima in Fig. 4. During the next oscillation, the field will start rolling inside one of the two wells, say the one on the left of Fig. 5. At the end of the oscillation, since its energy density is smaller than the height of the barrier separating the two minima, the field is very likely to remain confined in the left well. However, due to the field fluctuations, there is a non-vanishing probability that the field actually reaches the other well on the right hand side and remains there in some regions of the Universe. If this is the case, at different point in the same Hubble patch the field lives in different minima. Therefore, the Universe decomposes into two phases.", null, "Figure 5: Same as Fig. 4, but now only the last two wells are shown. The width of the shaded bands represent the uncertainty of the scalar field energy density δρ, due to field fluctuations. The distance between the two bands corresponds to the energy loss Δρ due to friction.\n\nThis is precisely the situation we are interested in.\n\nQuantitatively, let us compute the energy density lost during one oscillation, as a consequence of Hubble friction. In general this is not an easy task given the complicated shape of the potential in Fig. 4. However, it is greatly simplified by focusing only on the last cosine wells. In this case, the loss of energy during a half oscillation inside a single well can be estimated by a quadratic approximation of the potential:\n\n Vapprox=12M2ϕ2, (2.10)\n\nwith , i.e. the curvature of the potential (2.1) around the minimum of the well, where . We obtain\n\n M2=m2+Λ4f2=(1+κ)m2 (2.11)\n\nsince we are interested in the regime , we can take . Immediately after inflation the Hubble rate evolves approximately as during matter domination, i.e.\n\n H=23t;ρ∼a−3∼t−2, (2.12)\n\nso that the relative decrease in energy density in one half period is given by\n\n Δρρ∼2Δtt=3HΔt∼32HM, (2.13)\n\nwhere . We now focus on the last two wells, such that . Using Friedmann’s equation, , and we find\n\n Δρ∼ρ⋅ρ1/2MpM∼Λ4fMp=κm2f3Mp. (2.14)\n\nWe can now quantitatively discuss the probability of having a phase decomposition. To this end we have to compare the decrease in energy density due to friction and the uncertainty due to field fluctuations . If we have , the field will populate more than one vacuum with probability, as should be clear from Fig. 5. The term probability here refers to the exact choice of model parameters which, at that level of precision, appears arbitrary to the low-energy effective field theorist.\n\nThe task of the next section is therefore to present two possible origins of the fluctuations . These are respectively classical inflationary inhomogeneities and quantum uncertainties of the scalar field .\n\n## 3 Fluctuations and phase decomposition\n\nIn this section we will analyse two sources of fluctuations of the inflaton field. In Sec. 3.1 we focus on (classical) inflationary fluctuations. Those originate as sub-horizon size quantum fluctuations, but are then stretched to super-horizon size and become classical. After inflation, they re-enter the horizon and may lead to the phase decomposition described above. In Sec. 3.2 we focus instead on the intrinsic quantum uncertainty which characterizes a quantum field at any given time. Independently of any inflationary pre-history of our field, this effect is present directly during the oscillatory stage and may also lead to phase decomposition. In fact, the estimates that we obtain in this section imply a small probability of phase decomposition. However, in Sec. 4 we will see that the probability can be much larger because fluctuations may be enhanced after inflation.\n\n### 3.1 Inflationary fluctuations\n\nThe evolution of inflationary fluctuations on sub- and super-horizon scales is well-known (see e.g. [61, 62]). For us the only crucial point is that, once a certain inflationary mode re-enters the horizon it behaves like a dark matter fluctuation, i.e. it is a decaying oscillation (see [63, 64] for a detailed study of scalar field fluctuations after inflation). From now on we focus on the amplitude of such an oscillation, which we denote by . The background is denoted by and satisfies the equation of motion (2.2).\n\nThe initial conditions on are determined by matching with the power spectrum of the gauge-invariant curvature perturbation . This quantity is conserved on superhorizon scales and is given by\n\n Δ2R(k)=18π2[1ϵH2M2p]exit , (3.1)\n\nwhere the right hand side is evaluated at horizon exit, i.e. for . In the slow-roll regime the quantity in (3.1) coincides with the familiar expression .\n\nTo determine the probability of phase decomposition we need to understand how field fluctuations , once they re-enter the horizon, give rise to density fluctuations . More specifically, we wish to determine at time . This is the time at which the amplitude of the background has decreased to values comparable to the width of the last wells, and it is given by\n\n tΛ∼H−1Λ∼1κ1/2(f/Mp)m . (3.2)\n\nLet us now consider a mode with . This mode will re-enter the horizon at a time . At we can determine the size of by matching with the curvature perturbation. However, the field fluctuation thus obtained will be out of phase with the oscillation of the background . It is maximal when is maximal and vanishes when is at a turning point. If this remained the case for the subsequent evolution until , the fluctuation could not give rise to a phase decomposition. This can only occur if we have a sizable fluctuation at a turning point of .\n\nHowever, we expect decoherence between and after only a few oscillations. The reason is that oscillates with frequency while a mode with will oscillate with frequency . Thus, at some time after the field fluctuation will be an admixture of out-of-phase but also in-phase-oscillations w.r.t. to . It is exactly the in-phase-oscillations which do not vanish at turning points and it is these fluctuations which give rise to density perturbations . In the following we will thus assume that once a mode enters the horizon, while initially out of phase with , it will give an contribution to an in-phase oscillation with corresponding after only a few periods.444As we cannot quantify this effect exactly, we will now drop exact numerical prefactors in all following expressions.\n\nTo estimate the size of fluctuation at time given a fluctuation at we need to take the expansion of the universe into account. In particular, the energy density scales as\n\n ρ∼a−3 ,δρρ∼a⇒δρ∼a−2 . (3.3)\n\nThus, for a density fluctuation with there is a dilution in the time span between and .\n\nLet us now determine the probability of phase decomposition due to a mode with . As argued before, the field fluctuation will quickly give rise to a density fluctuation. Instead of taking the intermediate step via field fluctuations, let us match the density fluctuations directly to the curvature fluctuations at re-entry:\n\n Δ2R∼Δ2δρρ2⇒δρinfk(t0)∼ρ(t0)√Δ2R∼ρ(t0)Mp[Hϵ1/2]exit . (3.4)\n\nNow, using\n\n ρ(t0)∼m2M2p ,[Hϵ1/2]exit∼m ,a−2(tΛ)a−2(t0)∼κ2/3(f/Mp)4/3 (3.5)\n\nwe obtain\n\n δρinfΔρ∼κ−1/3(mMp)(Mpf)5/3. (3.6)\n\nThe above probability was derived for modes with . However, we are interested in the situations when the above probability is largest. Thus let us consider how this result is modified if we consider a mode that re-enters the horizon earlier or later. Modes with enter the horizon earlier and will hence experience more dilution. As a result, the corresponding probability of phase decomposition will be smaller. Modes with enter the horizon later and will be diluted less. Hence they would in principle give rise to a larger probability than (3.6). However, they cannot enter too late as they need to have enough time to decohere w.r.t. . A more detailed analysis would be needed to determine how late a mode can enter the horizon and nevertheless give rise to a sizable density fluctuation. Thus, (3.6) should be seen as a reasonable estimate. We shall comment more on the size of this probability at the end of Sec. 3.2.\n\n### 3.2 Quantum fluctuations\n\nThere is another potentially relevant source of fluctuations of the field . This is the intrinsic uncertainty due to the quantum nature of our scalar. It can be simply written as\n\n δϕqk∼k. (3.7)\n\nIn order to estimate the maximal effect of these fluctuations, let us consider the following setting: consider a scalar field with fluctuations approaching the local maximum of some potential. This is basically as in Fig. (2), with the field approaching the maximum from the left side. We then ask the following question: what is the field distance from the local maximum which the background field has to reach such that its fluctuations can lift it over the potential barrier? The relevant energy scale around the maximum is , where is the curvature of the potential at the maximum. One can convince oneself that only fluctuations of the order are relevant for overcoming the barrier: modes with have an effective non-tachyonic mass and are insensitive to the instability. In contrast, modes with are sensitive to the tachyonic instability, but their fluctuations are smaller than those of modes with .\n\nAlternatively, we can understand this point by considering tunnelling. Hence we take a homogeneous scalar field approaching the maximum and study the conditions under which quantum tunnelling to the other side of the barrier becomes efficient. In order to answer this question, let us use the following standard tunnelling formulae: the tunnelling rate is given by , where is the action of critical bubble formation. When the thin wall approximation is applicable, this reads (see e.g. )\n\n S0=27π2σ42(ΔV)3=27π2δϕ22M2, (3.8)\n\nwhere is the bubble wall tension and can be estimated with a quadratic approximation, . However, in most of the cases the thin-wall calculation is not appropriate. Nevertheless, we still expect , with a different prefactor.555We have checked the behaviour for an inverted parabola. In the vicinity of the maximum this is a reasonable approximation, with a prefactor of the order of . According to (3.8) the tunnelling rate is unsuppressed when , which is the same condition we found with the previous approach up to a numerical prefactor. Given the uncertainty in the prefactor, for the time being we use the parametric dependence .\n\nFollowing these two arguments the uncertainty in the energy density induced by such field fluctuations is\n\n δρq∼M2δϕq2k∼M4. (3.9)\n\nIf quantum fluctuations induce an energy gain which is larger than the loss due friction one expects phase decomposition. Therefore, we compare (3.9) with (2.14), using also . We obtain\n\n δρqΔρ∼κ(mMp)2(Mpf)3. (3.10)\n\nComparing (3.6) and (3.10) we conclude that the probability of a phase decomposition due to inflationary fluctuations is larger than the one due to quantum ones by at least a factor .\n\nLet us now discuss the size of the probabilities (3.6), (3.10). In quadratic inflation, . For and the probability (3.6) is of the order of . This implies that a phase decomposition is rather likely for these values of parameters. For the same choices, the probability (3.10) is only slightly smaller, i.e. . However, further numerical suppression is expected in (3.10). Therefore we conclude that in the regime , phase decomposition is likely to happen as a consequence of inflationary fluctuations. If grows above phase decomposition quickly becomes improbable. Whether such small values of are natural depends on the details of the model leading to (2.1). We will comment more on the size of at the end of Sec. 4.\n\nFurther there exist arguments based on the Weak Gravity Conjecture (WGC) which constrain the size of modulations of the potential in axion monodromy inflation (for an earlier somewhat different perspective see ). It is thus important to check whether the region of parameter space considered in this work is consistent with these bounds. Given a domain wall with tension and charge the electric WGC demands . In our case we have and (see for details). Using our definition the WGC bound reads\n\n √κ≲Mpf . (3.11)\n\nAs a result, while is bounded from above, our preferred parameter region for phase decomposition (, ) is consistent with the WGC. Interestingly, let us notice that the region of parameter space which is ruled out by the WGC is also constrained by current observations, which require during inflation[14, 66], where .\n\nTo close this section let us make two important remarks. First, note that phase decomposition can still occur even if the probability is small. In this case we only expect very few bubbles per Hubble patch. The second comment concerns once again the distinction between classical and quantum fluctuations. These two sources can also be distinguished based on the length scale at which their effect is strongest. As we explained, classical inhomogeneities are most relevant at while the quantum effect is strongest for . Since , the size of the latter inhomogeneities is parametrically smaller than that of the inflationary ones.\n\n## 4 Enhancement of fluctuations\n\nUntil now we have assumed that fluctuations, whether they are of classical or quantum nature, remain small during the evolution of the universe after inflation. The aim of this section is to discuss a possible enhancement of the fluctuations due to the functional form of the potential (2.1). Before going into details, let us summarise the main result. In this section we are mainly interested in fluctuations with at time . When the field oscillates at the bottom of the potential containing only the last few well, these fluctuations can be enhanced for certain values of and . Crucially, these modes never exit the horizon during inflation, because at their wavelength is smaller than the Hubble radius. Therefore, these modes are never classicalised. Nevertheless, in this section we study their enhancement treating them as classical. We expect that our analysis will still capture the main effect. We provide a more detailed discussion at the end of this section.\n\nA large growth of fluctuations can severely affect our conclusions. On the one hand, large fluctuations of the inflaton field may be desirable to some degree in our setup: the larger , the easier it is to cross the barrier between two local minima. Furthermore, if a mode with has a large amplitude, it may induce a phase decomposition independently of the modes with that we have studied in the previous section. If this is the case (3.6) underestimates the probability for phase decomposition.\n\nOn the other hand, large fluctuations with can lead to short-range violent dynamics rather than to the formation of well defined bubbles (which need a length scale ). As we will describe in more detail in Sec. 5, this can negatively affect the strength of the gravitational wave signal related to our setting.\n\nFor these reasons, it is crucial to assess if any large growth of fluctuations occurs in our setting. As we have already mentioned, we focus on the enhancement of classical fluctuations and comment later on the applicability of the results for quantum modes. Such an analysis involves solving the coupled equations of motion for the background field and for the fluctuation in an expanding background and in the presence of a non-zero gravitational field. In Appendix A.3 we present a first step towards this goal by providing the relevant equations of motion.\n\nHere we will perform a somewhat different, simplified analysis, assuming that the gravitational field is negligible for the following reason. As we describe in Appendix A, the addition of a gravitational field leads to a dark matter-like growth of fluctuations, which is negligible compared to the exponential growth that we are seeking in this section. Let us rewrite the equations of motion in a way that is more suitable for a numerical analysis. Namely, we define and . Then the linearised equations of motion without gravity read:\n\n φ′′0+2t′φ′0+φ0−κsin(φ0(t′)) =0 (4.1) δϕ′′k+2t′δϕ′k+[1+k2m2a2(t′)−κcos(φ0(t′))]δϕk(t′) =0, (4.2)\n\nwhere ‘prime’ now denotes a derivative w.r.t.  and where we have used during matter domination.\n\nIn the absence of friction, the background solution is periodic and the equation of motion for is a Hill’s equation. Solutions to such an equation exhibit a resonant behaviour for certain values of [67, 40]. This is similar to the resonances encountered in the context of preheating (for a review see ). However, the inclusion of a time-dependent background as well as friction, may affect the growth of the solution. Generically, one expects that modes with may experience exponential growth for certain values of and . We investigated the behaviour of and numerically, for certain values of the parameters. In Fig. 6 the background is plotted as a function of for . For this parameter choice the field is caught near one of the local minima after only one oscillation.", null, "Figure 6: Evolution of the background ϕ0(t′), according to (4.1). The initial conditions ϕ0(t′i)=Mp and ti=2√2/(√3m) are determined by violation of the slow roll condition. Furthermore ϕ′0(t′i)=0, Here f=10−2Mp,κ=60. The field is caught in one of the cosine wells around t=8/m.\n\nFor the same values of and we plot the evolution of the mode up to the time when the field is caught in a local minimum in Fig. 7. We see that before the field settles in one of the local minima, this mode does not grow. Afterwards, the field is oscillating in an approximately quadratic potential and therefore we do not expect any growth. Note that our equations are homogeneous in . Hence our numerical determination of the enhancement is not affected by the initial value of .", null, "Figure 7: Evolution of the fluctuations δϕk(t′), according to (4.2). We have chosen: k=5m,δϕ′k(t′i)=0, and ti=2√2/(√3m). Furthermore f=10−2Mp,κ=60. After t≈8/m, ϕ0 is stuck in one of the cosine wells.\n\nHowever, the situation can be radically different, as the next numerical example shows. In Fig. (8) we plot the background scalar field for . The field gets stuck in one of the cosine wells after six oscillations.", null, "Figure 8: Evolution of the background ϕ0(t′), according to (4.1). The initial conditions ϕ0(t′i)=Mp and ti=2√2/(√3m) are determined by violation of the slow roll condition. Furthermore ϕ′0(t′i)=0. Here f=Mp/300,κ=20. The field is caught in one of the cosine wells around t=40/m.\n\nAs a consequence of the longer time that the field spends oscillating across several cosine wells, fluctuations can now grow significantly . In Fig. (9), we plot the logarithm of the absolute value of , again for . We see that the amplitude of this mode grows by three orders of magnitude before the background field settles in one of the local minima. In fact, such strong growth takes us out of the regime of validity of the linearized equation of motion (as becomes larger than ).", null, "Figure 9: Logarithmic evolution of the absolute value of δϕk(t′), according to (4.2). We have chosen: k=5m, δϕ′k(ti)=0, and ti=2√2/(√3m). Furthermore f=Mp/300,κ=20. After t≈40/m, ϕ0 is stuck in one of the cosine wells.\n\nRecently, the growth of fluctuations in a potential with cosine modulations was studied in . The authors argue that, for a relatively small Hubble scale and neglecting gravity, the fluctuations for can grow as , where is roughly given by\n\n Nk∼mHk∼mF(κ). (4.3)\n\nHere is a function of the order up to a few whose value depends on the initial amplitude of . In our case, since , we conclude that fluctuations should grow with exponent:\n\n Nk≳m∼Mpf, (4.4)\n\nwhere we have dropped any -dependence due to our ignorance regarding . Our numerical examples, which take Hubble expansion into account explicitly, confirm that for small such an exponential growth does happen for most values of . In contrast, we do not observe enhancement if the value of is chosen too large.\n\nApart from this qualitative discussion, we are unfortunately unable to provide an analytical understanding of the dependence of the enhancement on the parameters and . This is partly due to the fact that the phenomenon strongly depends on the precise minimum the field ultimately settles in. The latter question can only be addressed in a probabilistic approach, i.e. we can only say where the field is more likely to get trapped. Therefore, we do not have a precisely monotonic dependence of the enhancement in terms of and .\n\nIn the absence of an analytical treatment, we performed a numerical search for enhanced fluctuations, focusing on modes with at the time when . The results are reported in Fig. 10 in the form of a grid of points. Each point corresponds to a value of and . We observe that in the region of interest fluctuations tend to be enhanced whenever . Here, we define ‘enhancement’ as follows: we will refer to a mode to be enhanced if its original amplitude has grown by roughly two orders of magnitude before getting caught. The enhanced is comparable to and we are therefore at the boundary between the linear and non-linear regime. Interestingly, this boundary corresponds to parameter values such that the probability (3.6) is of the order . However, note that the probability in (3.6) was determined for modes which will exhibit at . While such fluctuations may experience growth, the generic expectation is that enhancement does not occur for modes with .", null, "Figure 10: Grid showing for which values of κ and f resonance occurs. Grey points correspond to the case of no (or too small) enhancement. Red points correspond to large enhancement. By the latter we mean that fluctuations grow by at least two orders of magnitude before getting caught near a local minimum. The equations of motion are solved for k=M=κ1/2m at the time when H∼O(Λ2/Mp).\n\nLet us now briefly discuss enhancement of quantum fluctuations. This is in principle a complicated issue: we cannot use the classical equations of motions to analyse the behaviour of the quantum system. However, it is important to notice that, if quantum fluctuations are initially enhanced, they quickly become classical. Here by “classical” we mean that their occupation number becomes large, such that (4.2) can be used to study their evolution. Parametric resonance in quantum mechanics and quantum field theory has been studied analytically and numerically: the conclusion is that quantum modes do experience exponential growth (see e.g. [69, 70]). We expect that the same effect will occur also in the system analysed here. Therefore, our study of classical fluctuations should extend, at least partially, to quantum fluctuations. If more enhancement occurs in the quantum case, then phase decomposition is even more likely. We leave a more detailed study of this effect for future work.\n\nFinally, let us summarise our findings concerning the probability of phase decomposition:\n\n1. For enhancement is generically not observed. The probabilities (3.6) and (3.10) are small, so that a phase decomposition is unlikely. Nevertheless, it is still possible that very few bubbles of tiny size containing the state of lower energy are nucleated. As we describe in the next section, observational signatures from such a situation may be quite strong. Classical inflationary fluctuations are the dominant cause of phase decomposition in this regime.\n\n2. For we have the following situation. On the one hand, fluctuations with at are generically enhanced. These modes are genuinely quantum modes, since they never exited the horizon. In this region, the enhancement may be just large enough to give rise to a probability of phase decomposition of order . Furthermore, we observe numerically that, at fixed and , modes with do not experience the same exponential growth. This will turn out to be a useful observation when examining the gravitational wave signal from the associated phase transition.\n\nOn the other hand, according to (3.6) and (3.10), classical and quantum modes with can lead to a phase decomposition, even if they are not enhanced. Therefore we conclude that a phase decomposition is very likely to be induced. Assuming that our analysis of enhanced classical fluctuations extends to the quantum ones, the dominant cause of phase decomposition are quantum modes with at .\n\n3. For fluctuations with are strongly enhanced. In this region phase decomposition is very likely to occur. However, it is hard to provide any description of such a highly non-linear regime. Classical and quantum fluctuations with are generically not enhanced, but would also lead to phase decomposition according to (3.6) and (3.10).\n\nOne more comment is in order before moving on to the phenomenological signatures of our setup. Phase decomposition happens generically for rather small axion decay constants. One may question whether such values of are plausible in the spirit of axion monodromy. The answer depends very much on the framework in which monodromy is implemented. In a stringy setup it is a question of moduli stabilisation: e.g. in the Large Volume Scenario (LVS) [71, 72] decay constants are generically suppressed by the volume of the compactification manifold and are therefore naturally small.\n\n## 5 Gravitational radiation from Phase Transitions\n\nIn the previous sections, we have described a mechanism which can potentially lead to phase decomposition in the early universe after inflation. In this section, we will assume that such phase decomposition indeed occurs. In the presence of different populated vacua, bubbles containing the state of lowest energy can form and expand. Their collisions are a very interesting and well-known source of gravitational radiation. This has been studied in detail in the literature in various contexts and regimes (see and references therein).\n\nThe aim of this section is twofold. First of all, we would like to elucidate the peculiarities of our setup concerning the energy released into gravitational waves during the collision of bubbles. Rather than focusing on precise calculations, we will give a qualitative discussion and provide formulae analogous to the more familiar case of bubbles colliding in a relativistic plasma. In this case, there are three possible sources of gravitational radiation: the collision of bubble walls, sound waves in the plasma and its turbulent motion. The second goal of this section is to give estimates of the relic density and frequency of the gravitational wave signal which can be obtained in our setup.\n\n### 5.1 Gravitational waves from bubble collision\n\nThe focus of this subsection is the collision of bubbles and the possible shocks in the fluid surrounding them. These phenomena are usually studied in the so-called envelope approximation . The latter consists in assuming that the energy liberated in gravitational waves resides only in the bubble walls before the collision. Furthermore, it is assumed that only the uncollided region of those walls contributes to the production of gravitational waves, i.e. the interacting region is neglected. Such an approximation has been initially applied to the case of vacuum-to-vacuum transitions and later to collisions in a radiation bath.\n\nIn a thermal phase transition, the energy released into gravitational waves depends on four parameters. First of all, there is the time scale of the phase transition or, equivalently, the initial separation between two bubbles . Secondly, there is the ratio of the vacuum energy density released in the transition to that of the thermal bath, i.e.\n\nwhere specifies that the quantity is evaluated at the time of completion of the phase transition. Thirdly, the efficiency factor characterizes the fraction of the energy density which is converted into the motion of the colliding walls. Finally, the bubble velocity is not necessarily luminal, as the walls have to first displace the fluid around them. The energy released into gravitational waves of peak frequency is then given by :\n\n ρGWρtot∼θ(H⋆δ)2λ2η2(1+η)2v3b, (5.2)\n\nwhere is the background energy density at completion of the phase transition. The parameters and are actually expected to be functions of , in such a way that for , also .\n\nIn our case, bubbles collide before reheating, therefore there is no radiation bath around them. However, as we describe in Appendix A, an oscillating scalar field corresponds to the presence of a matter fluid. Crucially, the time scale of the field oscillations is set by , and may be smaller than the time of collision. Therefore, oscillations of the scalar field cannot be generically neglected. Unfortunately, we do not have specific formulae for this case. Since we are interested only in an order of magnitude estimate for the spectrum of gravitational waves, it seems reasonable to extend (5.2) to our setup, with the obvious modification\n\n η≡ϵρ⋆matter. (5.3)\n\nFurthermore, we shall hide our ignorance about the dependence of and on by defining and leave the determination of these parameters for future work. Therefore, we base our estimates on the following formula for the energy released in gravitational waves from the collision of bubbles and shocks in the matter fluid:\n\n ρGWρtot≈θ0(H⋆δ)2η2(1+η)2, (5.4)\n\nwhere in our case . In addition, the peak frequency of gravitational waves in the envelope approximation is given by\n\n ωpeak≃σδ, (5.5)\n\nwhere should be fixed numerically and includes effects due to subluminal bubble walls velocity.\n\nThe next task is to estimate and . Let us start with the ratio . The energy density at the time of the phase transition corresponds to the height of the barrier separating the two minima, therefore . We expect the typical frequency of the phase transition to be set by the momentum of the spatial inhomogeneities of the field . The phase transition can be induced by any mode which is present at . The largest frequency that one can take is set by , as we have already discussed in Sec. 3.2. This corresponds to a scenario where phase decomposition is likely. However, bubble collisions are most violent when the field makes it over the barrier separating the two minima only very rarely. In this case there are only few bubbles per Hubble patch. This latter scenario gives the strongest signal as it corresponds roughly to\n\n H⋆δ∼O(1). (5.6)\n\nIn order to understand how strong can the signal be in our setup, we assume (5.6) in what follows, but one should keep in mind that this is optimistic.\n\nThe estimate of is less straightforward, at least conceptually. If we adopt the envelope approximation, then we need to compute the vacuum energy density released in the phase transition. This is simply the difference between the energy density of the two minima in Fig. 11. Using a quadratic approximation, we find\n\n ϵ∼m2Δϕ2∼m2f2, (5.7)\n\nwhere is the approximate field separation between the two minima. The energy in the matter fluid is roughly given by the height of the deepest well, i.e. . This is because at completion of the phase transition the oscillations of the scalar field span almost the whole well. Therefore, in the envelope approximation we obtain\n\n η∼κ−1. (5.8)\n\nAs we have mentioned, deviations from this simple picture may arise in our case. On the one hand, a certain fraction of the energy of the walls might for example be dissipated into the matter fluid. In this case, only a fraction of would lead to production of gravitational waves. This effect might be captured by the efficiency prefactor .666Let us also notice that bubble walls may also be generically crossed by the fluid. We neglect this effect in our discussion.", null, "Figure 11: Two-well potential. In the picture ϵ is the energy difference between the two minima, while Λ4 is the value of the potential at the local maximum.\n\nOn the other hand, the energy released into the fluid while the bubbles expand and collide might also contribute to the production of gravitational waves. Namely, this energy might be converted into bulk motion of the fluid. In this case, the energy released in gravitational radiation should be larger than , and could possibly be as large as . This effect is captured by studying the fluid as a source of gravitational waves. We comment very briefly on this topic in the next subsection.\n\n### 5.2 Gravitational waves from the matter fluid\n\nIn analogy with the case of radiation there are at least two effects which can further contribute to the total energy released in gravitational radiation during the phase transition. Here we just provide the formulae given in for the thermal case, keeping in mind that they may not straightforwardly extend to our setup:\n\n• Sound waves in the fluid: this arises because a certain fraction of the energy of the walls is converted after the collision into motion of the fluid (and is only later dissipated). In the case of radiation this gives a contribution\n\n ρGW,swρtot∼θsw(H⋆δ)λ2v(η2(1+η)2) (5.9)\n\nThe prefactor is expected to be smaller than in (5.2).\n\n• Turbulence in the fluid: one expects further contributions as a certain fraction of the energy of the walls is converted into turbulence. In the case of radiation one obtains:\n\n ρGW,turbρtot∼θturb(H⋆δ)λ3/2turb(η3/2(1+η)3/2). (5.10)\n\nThe prefactor is expected to be larger than in (5.2). Note, that for these two mechanism the dependence on is only linear.\n\nIn certain regimes, these two effects may be larger than the one due to bubble collisions and shocks in the fluid. However, they are not fully understood, even in the case of radiation. Therefore, in the next subsection we will neglect them, and obtain only a lower bound on the relic abundance of gravitational waves. This should still be useful to understand the approximate size and frequency of the signal. Nevertheless, the reader should keep in mind that there are other possible contributions even beyond the ones mentioned in this subsection (see e.g.  for recent progress).\n\n### 5.3 Frequency and signal strength of gravitational waves\n\nIn order to compute the relic abundance and frequency of gravitational waves emitted during the phase transition, we need to know the equation of state of the background energy density from the end of the phase transition to today. Assuming standard evolution after reheating the behaviour of the scale factor until today is essentially fixed777It is characterized by the effective number of degrees of freedom.. Furthermore, the inflaton field generically behaves as non-relativistic matter after inflation. It remains to be addressed whether deviations from the equation of state might occur immediately after the phase transition, before reheating.\n\nDue to the very large release of energy during the collision of bubble walls, it is conceivable that the fluid initially behaves relativistically. This would correspond to an early phase of radiation domination, i.e. , in a similar fashion to some preheating scenarios. Eventually, the fluid cools down and its non-relativistic behaviour is restored. Depending on the reheating temperature, this may or may not happen before the inflaton decays. If the system were in a thermal ensemble, the fluid would behave non-relativistically after . For the time being, we allow for a general equation of state parameter after the phase transition and before .\n\nTherefore the background energy density at reheating is given by\n\n ρRH=Λ4(a⋆aNR)3(1+w)(aNRaRH)3, (5.11)\n\nwhere from now on the subscript denotes that a certain quantity is evaluated at the time when the fluid becomes non-relativistic. Let us define the prefactors\n\n νw ≡(a⋆aNR)∼(ρNRρ⋆)13(1+w), (5.12) νnr ≡(aNRaRH)∼(ρRHρNR)1/3. (5.13)\n\nThe prefactor quantifies the duration of a period of matter domination before reheating, while quantifies the duration of an early epoch of matter domination. Obviously .\n\nThe energy density in gravitational waves scales as . According to (5.4), at reheating we have\n\n ρGW(tRH)≈Λ4ν4wν4nr[θ0(H⋆δ)2η2(1+η)2]=ν−3(w−1/3)w⋅νnr[θ0(H⋆δ)2η2(1+η)2]ρRH, (5.14)\n\nwhere in the last step we have used (5.11). The relic energy density of gravitational waves is then\n\n ρGW(t0)≈ν−3(w−1/3)w⋅νnr[θ0(H⋆δ)2η2(1+η)2](aRHa0)4ρRH, (5.15)\n\nwhere is the current age of the Universe. The ratio can be determined by imposing entropy conservation. Furthermore, can be computed using the standard formula for the energy density of radiation\n\n ρRH=π2g⋆(TRH)30T4RH. (5.16)\n\nFinally, the density parameter today today is given by\n\n ΩGW(t0)≃10−5ν−3(w−1/3)wνnrh2⋅θ0[102g∗(TRH)]1/3⋅[H⋆δ]2⋅η2(1+η)2, (5.17)\n\nwhere .\n\nLet us now estimate the peak frequency of the emitted radiation. For this quantity the only relevant parameter is . Frequencies scale as , therefore we have:\n\n ω0∼ωpeak(a⋆aNR)(aNRaRH)(aRHa0)∼ωpeak⋅νw⋅νnr⋅(aRHa0). (5.18)\n\nBy combining (5.18), (5.5), and (5.16) one obtains\n\n ω0∼108Hz⋅σ⋅νw⋅νnr(δH∗)(g∗(TRH)102)1/6[TRH1015GeV]. (5.19)\n\nWe have therefore determined the relevant parameters of the emitted radiation as function of and . Now we can plug in (5.6) and (5.8), to obtain final formulae. Then we have:\n\n ΩGW(t0)h2 ≃10−5ν−3(w−13)w⋅νnr⋅θ0[102g∗(TRH)]1/3κ−2 (5.20) ω0 ≃108Hz⋅σ⋅νw⋅νnr⋅(g∗(TRH)102)1/6[TRH1015GeV]. (5.21)\n\nIn the envelope approximation, it is also possible to compute the full spectrum of the gravitational radiation emitted from the collision of the bubble walls. This reads :\n\n ΩGW(t0)h2 ≃10−5ν−3(w−1/3)wνnr⋅θ0⋅κ−2[102g∗(TRH)]1/3Senv(ω) (5.22) withSenv(ω) =3.8(ω/ω0)2.81+2.8(ω/ω0)3.8.\n\nIn order to estimate the maximal possible size of our signal, let us now specify to the case in which the energy present after the bubble collisions is converted into radiation. This corresponds to setting in (5.22). At the inflaton goes back to a non-relativistic behaviour. The prefactors and can now be explicitly computed, using . We obtain\n\n νw =(ρNRρ⋆)1/4∼MΛ∼κ1/4m1/2f1/2 (5.23) νnr =(ρRHρNR)1/3∼(TRHmκ1/2)4/3. (5.24)\n\nFrom (5.24), it is clear that the largest signal is obtained for . In this case and the signal is completely unsuppressed. For completeness, let us mention that the largest suppression of the signal occurs for . In this case, as should be clear from (5.11), the background energy density scales as matter from completion of the phase transition until reheating.\n\nIn Fig. 12 we plot the spectrum (5.22) for three different choices of parameters, using . We fix as required by observations. We have chosen parameters in such a way as to maximize the overlap with sensitivity regions of current and future space- and ground-based detectors, which are bounded by dashed lines in the plot. We have also fixed . Our plot provides examples of the wide range of frequencies that can be obtained in our setting, simply varying the axion decay constant, the reheating temperature and the number of local minima. Interestingly, for reasonable choices of parameters the signals are in the reach of future detectors.", null, "Figure 12: Gravitational wave spectra as in (5.22) with w=1/3. The inflaton mass is fixed to m∼10−5Mp. Spectra are shown as solid lines for different values of κ,f and TRH: the blue curve is obtained for κ=5,f=0.1Mp,TRH∼1012 GeV; the brown curve for κ=10,f=0.01Mp,TRH∼1011 GeV; the red one for κ=70,f=0.001Mp,TRH∼1011 GeV. We have also taken w=1/3,θ0=10−2,σ=10−1 in (5.22). For the values of the reheating temperature considered here, we have g∗(TRH)∼102. Sensitivity curves of some ground- and space-based interferometers are shown for comparison as dashed curves (data taken from ).\n\n## 6 Conclusions\n\nIn this paper we investigated the production of gravitational waves from post-inflationary dynamics in models of Axion Monodromy inflation. We expect such phenomena to also occur for generic axionic fields with potentials with sizable modulations, albeit the numerical results may differ significantly in these cases.\n\nThe main observation is that in models of axion monodromy inflation, the inflaton potential consists of a monotonic polynomial with superimposed cosine-modulations. While these modulations have to be small to allow for successful inflation, they tend to dominate near the bottom of the potential. In fact, these cosine ‘wiggles’ can be large enough such that the potential exhibits a series of local minima.\n\nAfter inflation ends, the inflaton is exploring this wiggly part of the potential. As a consequence of Hubble friction, the rolling axion can get stuck in one of the wells. This may happen well before the inflaton reheats the standard model degrees of freedom. Since the energy density in the axion field decreases together with , the field is more likely to get caught in one of the last minima. We therefore focused on a two-well setting, which is obtained by “zooming” into the full monodromy potential.\n\nTaking fluctuations of the inflaton field into account, the inflaton field does not necessarily get caught in one unique local minimum in the entire Hubble patch. If field fluctuations are sufficiently large, a phase decomposition occurs such that at least two different vacua are populated after inflation. The probability of this occurring is given by , where is the uncertainty in the axion energy density induced by the field fluctuations and is the frictional loss of energy in one oscillation.\n\nWe can distinguish two sources of field fluctuations which may lead to a phase decomposition. Firstly, there are the classical inhomogeneities naturally inherited from inflation. Secondly, there are the intrinsic quantum uncertainties characterising any quantum field. These two sources are essentially indistinguishable at very early and late times. However they are in principle of different size in the intermediate regime that we are interested in. In particular, we have found that inflationary fluctuations are more likely to induce a phase decomposition: the probability that they do so is . Here is the mass of the inflaton-axion, the axion decay constant which defines the axion periodicity and roughly counts the number of local minima of the potential. Therefore, we observe that a phase decomposition is likely for and . The probability that quantum fluctuations induce a phase decomposition is smaller by a factor .\n\nFurthermore, due to the oscillatory term in the axion potential, fluctuations can experience exponential growth. This effect arises generically for modes with at the time when the field is rolling over the last wells. These fluctuations never exited the horizon, therefore they are effectively quantum modes. We extended the study of enhancement in to the case with varying , but still neglecting the gravitational field in the equations of motion. Numerically and using a classical approximation, we observe the existence of a region of parameter space where a phase decomposition is likely to occur. This happens roughly in the same regime where inflationary classical fluctuations are also likely to induce a phase decomposition. However, given the exponential enhancement of quantum modes, the latter are more likely to be the dominant cause of phase decomposition. Enhanced quantum modes quickly become classical, therefore we expect our main results to hold even after a more detailed analysis, which we leave for future work. For larger values of modes are not enhanced. Phase decomposition, although unlikely, might still occur as a result of quantum or classical fluctuations. For smaller decay constants fluctuations are very strongly enhanced. Phase decomposition occurs but it is hard to understand the physics in such a highly non-linear regime.\n\nIf a phase decomposition occurs, bubbles containing minima of lowest energy expand. Collisions of these bubbles source gravitational waves. We estimated the energy density and frequency of the emitted radiation in terms of the axion parameters, in the envelope approximation. Furthermore, we note that the matter fluid associated to the oscillating inflaton may also radiate gravitational waves. This is similar to the case of a thermal phase transition. The spectrum of the emitted radiation can peak in a wide range of frequencies (from mHz to GHz), depending on the reheating temperature and on the time of the phase transition. In this sense, our source is similar to other post-inflationary phenomena, such as preheating and cosmic strings. However, it is interesting to observe that the frequency may be lowered in our case since a matter dominated phase can follow the phase transition. The spectrum is at least partially in the ballpark of future space- and ground-based detectors. Thus, we can hope that axion monodromy may one day be investigated by means of gravitational wave astronomy.\n\n## Acknowledgments\n\nWe thank Robert Brandenberger, Christophe Grojean, David Hindmarsh, Sascha Leonhardt, Patrick Mangat, Viraf Mehta, Michael Schmidt, Michael Spannowsky and Alexander Westphal for useful discussions. This work was partly supported by the DFG Transregional Collaborative Research Centre TRR 33 “The Dark Universe\". F.R. is supported by the DFG Graduiertenkolleg GRK 1940 “Physics Beyond the Standard Model”.\n\n## Appendix A Scalar field fluctuations after inflation\n\nIn this appendix, we provide a detailed analysis of the evolution of scalar field fluctuations after inflation. We begin by deriving the Klein-Gordon equations for a scalar field and its fluctuations with potential (2.1), including the gravitational field. We then focus on the simple case of a purely quadratic potential. This gives us the opportunity to review why scalar field fluctuations behave like dark matter perturbations. We then provide equations to study the fluctuations in the full potential containing the ‘wiggles’.\n\n### a.1 Equations of motion\n\nLet us begin with the equations of motion of a scalar field in the post-inflationary universe. The starting point is the Klein-Gordon equation:\n\n 1√−g∂μ[gμν√−g∂νϕ]+V′(ϕ)=0. (A.1)\n\nThe metric appearing in (A.1) is the perturbed FRW metric (here we follow ):\n\n ds2=a2(τ)[(1+2A)dτ2−2Bidxidτ−(δij+hij)dxidxj], (A.2)\n\nwhere and the hat denotes divergenceless vectors and traceless tensors. In particular, we can consistently focus only on scalar modes, since the vectors can be gauged away and the tensors are not sourced by . We therefore have\n\n hscalarij=2Cδij+2∂E, (A.3)\n\nwhere . Perturbations defined by and (A.2) are not gauge invariant. However, the following quantities are gauge invariant:\n\n Ψ ≡A+H(B−E′)+(B−E′)′ Φ ≡−C−H(B−E′)+13∇2E ¯¯¯¯¯¯δϕ ≡δϕ−ϕ′0(B−E′).\n\nIn what follows we will perform our computation in the Newtonian gauge, defined by:\n\n B=E=0,C=−Φ. (A.4)\n\nIn the latter, the perturbed metric reads:\n\n ds2=[(1+2Ψ)dt2−(1−2Φ)a2δijdxidxj]. (A.5)\n\nThe components and are related by the perturbed Einstein equations. In the absence of off-diagonal terms in the spatial components of the perturbed stress-energy tensor the Einstein equations impose . With this constraint, the metric (A.5) provides the newtonian limit of general relativity.\n\nWe are now in a position to write down the Klein-Gordon equation (A.1), expanding the scalar field as and keeping only the leading order terms in the perturbed quantities . Then the background obeys:\n\n ¨ϕ0+3˙aa˙ϕ0+V′(ϕ0)=0, (A.6)\n\nwhile the inhomogeneous satisfies:\n\n ¨δϕ+3˙aa˙δϕ+(V′′(ϕ0)−∇2a2)δϕ−4˙ϕ0˙Φ+2V′(ϕ0)Φ=0. (A.7)\n\nFurthermore, there are three Einstein equations relating the gravitational field to the fluctuation :\n\n ˙Φ+˙aaΦ =4πG˙ϕ0δϕ (A.8) ΔΦa2−3˙aa˙Φ−3¨a2a2 =4πGδρ (A.9) ¨Φ+4˙aa" ]

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[ "", null, "# What is 3/5 + 5/12?", null, "3 5\n+\n 5 12\n\n## Step 1\n\nWe still have different denominators (bottom numbers), though, so we need to get a common denominator. This will make the bottom numbers match. Multiply the denominators together first. Now, multiply each numerator by the other term's denominator.\n\nNow we multiply 3 by 12, and get 36, then we multiply 5 by 12 and get 60.", null, "Now for the second term. You multiply 5 by 5, and get 25, then multiply 5 by 12 and get 60.", null, "This gives us a new problem that looks like so:\n\n 36 60\n+\n 25 60\n\n## Step 2\n\nSince our denominators match, we can add the numerators.\n\n36 + 25 = 61\n\nThat gives us an answer of\n\n 61 60\n\n## Step 3\n\nCan this fraction be reduced?\n\nFirst, we attempt to divide it by 2...\n\nNope. Try the next prime number, 3...\n\nNope. Try the next prime number, 5...\n\nNope. Try the next prime number, 7...\n\nNope. Try the next prime number, 11...\n\nNope. Try the next prime number, 13...\n\nNope. Try the next prime number, 17...\n\nNope. Try the next prime number, 19...\n\nNope. Try the next prime number, 23...\n\nNope. Try the next prime number, 29...\n\nNope. Try the next prime number, 31...\n\nNope. Try the next prime number, 37...\n\nNope. Try the next prime number, 41...\n\nNope. Try the next prime number, 43...\n\nNope. Try the next prime number, 47...\n\nNope. Try the next prime number, 53...\n\nNope. Try the next prime number, 59...\n\nNope. Try the next prime number, 61...\n\nNo good. 61 is larger than 60. So we're done reducing." ]

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[ "", null, "Vibromera OU Estonia,Ida-Viru maakond, Narva linn, Kiriku tn 13, 20308       tel.+372 8801884 Balancing software\n 4-plane balancing calculator How to use Run#0 Original vibration - Start-up without test weight. 1. Run the machine at its operating speed (be sure that operating speed is far from resonance frequency of construction) 2. Measure the 1x RPM vibration level and phase angle. If you have single channel vibrometer, you need measure vibration 4 times for Plane1,2,3,4. 3. Input measured data in the corresponding fields of balancing calculator. (Panel Run#0 Original vibration, Vo1 and F1 for Plane1 and Vo2 and F2 for Plane2 etc.) Run#1-Run#3 Test weight in Plane 1-3 4. Stop the machine and mount a test weight of suitable size in Plane 1(2,3). 5. Run the machine and measure the new 1x RPM vibration level and phase angle. 6. Stop the machine and remove the test weight 7. Input measured data in the corresponding fields of balancing calculator. (Panel Run#1(2,3) Test weight in Plane 1(2,3), Vo1 and F1 for Plane1 and Vo2 and F2 for Plane2 etc.) Run#4 - Test weight in Plane 4 8. Mount a trial weight of suitable size in Plane 42. 9. Run the machine again and measure the 1x RPM vibration level and phase angle once more. 10. Stop the machine and remove the test weight 11. Input measured data in the corresponding fields of balancing calculator. (Panel Run#4 Test weight in Plane 4,Vo1 and F1 for Plane1 and Vo2 and F2 for Plane2 etc.) 12. Press button F9 Calculate, read results and mount correcting weights of the calculated masses in the angles F1.. F4 (at the same radius as the test (trial) weights) from the test weight place (angle) by the rotation direction. RunC-Trim balancing 13. Run the machine again and measure the amount of vibration, to see how successful the balancing job has been.", null, "", null, "" ]

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[ "# Please help with this matrix question! Solve for Eigenvales, and Eigenvectors\n\nM = (a c)\n(c b)\n\nSorry for the double sets of brackets, its all in one. I'll also show as far as i got below:\n\n[a-λ c] => (a-λ)(b-λ) - c^2 = λ^2 + (-a-b)λ + (ab-c^2) =0\n[c b-λ] =>\n\nthen using the quadratic formula: λ = [-(-a-b) +/- Sqrt{(-a-b)^2 - 4(1)(ab-c^2)}]/ 2\n\nthen after some algebra I got stuck: λ = [(a+b) +/- Sqrt{a^2 + 2ab + b^2 -4ac^2}]/2\n\nλ = [(a+b) +/- Sqrt{a^2 - 2ab + b^2 + c^2}]/2 ==>> THIS IS WHERE I GOT STUCK. PLEASE HELP\n\n## Answers and Replies\n\ntiny-tim\nScience Advisor\nHomework Helper\nwelcome to pf!\n\nhi sidinsky! welcome to pf!", null, "(have a square-root: √ and a ± and try using the X2 button just above the Reply box", null, ")\nλ = [(a+b) +/- Sqrt{a^2 - 2ab + b^2 + c^2}]/2\n\nlooks ok", null, "that gives you the two eigenvalues,\n\nso now use the standard techniques to find an eigenvector for each", null, "is there any way this expression can be simplified further? I am sorta having trouble with the algebra :S\n\ntiny-tim\nScience Advisor\nHomework Helper\nis there any way this expression can be simplified further?\n\ni don't think so\n\nhow far have you got?​\n\nwell I managed to clean up the expression inside the sqrt a bit to: [(a-b)2 + c 2 ]/2\n\na2 -2ab + b2 + c2 = (a-b)2 + c2\n\nλ1 = (a+b) + sqrt{(a-b)2 +c 2}$/2$\n\nand λ2 = (a+b) - sqrt{(a-b)2 + c2}$/2$\n\nnow these expressions for Lambda are too difficult for me to use to solve for eigenvectors" ]

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[ "# Definition of cornets\n\n## \"cornets\" in the noun sense\n\n### 1. cornet, horn, trumpet, trump\n\na brass musical instrument with a brilliant tone has a narrow tube and a flared bell and is played by means of valves\n\nSource: WordNet® (An amazing lexical database of English)\n\nWordNet®. Princeton University. 2010.\n\n# cornets in Scrabble®\n\nThe word cornets is playable in Scrabble®, no blanks required.\n\nCORNETS\n(86 = 36 + 50)\nCORNETS\n(86 = 36 + 50)\n\ncornets\n\nCORNETS\n(86 = 36 + 50)\nCORNETS\n(86 = 36 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(77 = 27 + 50)\nCORNETS\n(76 = 26 + 50)\nCORNETS\n(74 = 24 + 50)\nCORNETS\n(74 = 24 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(67 = 17 + 50)\nCORNETS\n(64 = 14 + 50)\nCORNETS\n(64 = 14 + 50)\nCORNETS\n(63 = 13 + 50)\nCORNETS\n(63 = 13 + 50)\nCORNETS\n(63 = 13 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(60 = 10 + 50)\n\nCORNETS\n(86 = 36 + 50)\nCORNETS\n(86 = 36 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(80 = 30 + 50)\nCORNETS\n(77 = 27 + 50)\nCORNETS\n(76 = 26 + 50)\nCORNETS\n(74 = 24 + 50)\nCORNETS\n(74 = 24 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(72 = 22 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(70 = 20 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(68 = 18 + 50)\nCORNETS\n(67 = 17 + 50)\nCORNETS\n(64 = 14 + 50)\nCORNETS\n(64 = 14 + 50)\nCORNETS\n(63 = 13 + 50)\nCORNETS\n(63 = 13 + 50)\nCORNETS\n(63 = 13 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(61 = 11 + 50)\nCORNETS\n(60 = 10 + 50)\nCRONES\n(33)\nSECTOR\n(33)\nCORNET\n(33)\nRECONS\n(33)\nCORSET\n(33)\nESCORT\n(33)\nCENSOR\n(33)\nCORES\n(30)\nCERTS\n(30)\nSCONE\n(30)\nCENTS\n(30)\nSCENT\n(30)\nSCORE\n(30)\nCRONE\n(30)\nCORNS\n(30)\nSCORN\n(30)\nCONES\n(30)\nCREST\n(30)\nCENSOR\n(28)\nCRONES\n(28)\nCORSET\n(28)\nCORNET\n(28)\nCORNET\n(27)\nCORNET\n(27)\nCORNET\n(27)\nESCORT\n(27)\nSECTOR\n(27)\nCOTS\n(27)\nCRONES\n(27)\nCENSOR\n(27)\nCRONES\n(27)\nCORN\n(27)\nSECTOR\n(27)\nCONE\n(27)\nSECTOR\n(27)\nSECTOR\n(27)\nCORSET\n(27)\nCORE\n(27)\nCOST\n(27)\nCORNET\n(27)\nCONS\n(27)\nCORSET\n(27)\nSECTOR\n(27)\nCORSET\n(27)\nRECONS\n(27)\nCENSOR\n(27)\nCENT\n(27)\nRECONS\n(27)\nRECONS\n(27)\nCENSOR\n(27)\nCENSOR\n(27)\nCRONES\n(27)\nRECONS\n(27)\nESCORT\n(27)\nCORNET\n(27)\nRECONS\n(27)\nESCORT\n(27)\nCORSET\n(27)\nCENSOR\n(27)\nESCORT\n(27)\nCORSET\n(27)\nESCORT\n(27)\nCRONES\n(27)\nCRONES\n(27)\nCORES\n(26)\nCORNS\n(26)\nCRONE\n(26)\nCONES\n(26)\nCENTS\n(26)\nCREST\n(26)\nCERTS\n(26)\nCONES\n(24)\nCRONE\n(24)\nSECTOR\n(24)\nCREST\n(24)\nCORNET\n(24)\nSECTOR\n(24)\nCRONE\n(24)\nCRONES\n(24)\nCRONES\n(24)\nCORES\n(24)\nCORES\n(24)\nCREST\n(24)\nCONES\n(24)\nCONES\n(24)\nCREST\n(24)\nCRONE\n(24)\nESCORT\n(24)\nCORES\n(24)\nCORNET\n(24)\nESCORT\n(24)\nRECONS\n(24)\nSCENT\n(24)\nRECON\n(24)\nCORNS\n(24)\nSCONE\n(24)\nCENTS\n(24)\nRECON\n(24)\nCENTS\n(24)\nSCENT\n(24)\nCERTS\n(24)\nSCENT\n(24)\nCENTS\n(24)\nCENSOR\n(24)\nRECON\n(24)\nCORNS\n(24)\nRECON\n(24)\nRECONS\n(24)\nSCONE\n(24)\nSCONE\n(24)\nCORNS\n(24)\nSCORN\n(24)\nSCORN\n(24)\nSCORN\n(24)\nCORSET\n(24)\nSCORE\n(24)\nCENSOR\n(24)\nCORSET\n(24)\nSCORE\n(24)\nCERTS\n(24)\nCERTS\n(24)\nSCORE\n(24)\nCORSET\n(22)\nCORNET\n(22)\nCORNET\n(22)\nCENSOR\n(22)\nCENSOR\n(22)\nCRONES\n(22)\nCRONES\n(22)\nCORSET\n(22)\nSTONER\n(21)\nTONERS\n(21)\nTENSOR\n(21)\nCORES\n(21)\nSTONER\n(21)\nCORES\n(21)\nSTONER\n(21)\nCORE\n(21)\nTENORS\n(21)\nTENSOR\n(21)\nTENSOR\n(21)\nSTONER\n(21)\nTENORS\n(21)\nCOTS\n(21)\nCREST\n(21)\nCORN\n(21)\nTENORS\n(21)\nCREST\n(21)\nCOST\n(21)\nCONS\n(21)\nCRONE\n(21)\nCRONE\n(21)\nCRONE\n(21)\nSTONER\n(21)\nSTONER\n(21)\nTENORS\n(21)\nTENORS\n(21)\nTENORS\n(21)\n\n# cornets in Words With Friends™\n\nThe word cornets is playable in Words With Friends™, no blanks required.\n\nCORNETS\n(98 = 63 + 35)\n\ncornets\n\nCORNETS\n(98 = 63 + 35)\nCORNETS\n(92 = 57 + 35)\nCORNETS\n(86 = 51 + 35)\nCORNETS\n(86 = 51 + 35)\nCORNETS\n(80 = 45 + 35)\nCORNETS\n(80 = 45 + 35)\nCORNETS\n(80 = 45 + 35)\nCORNETS\n(79 = 44 + 35)\nCORNETS\n(79 = 44 + 35)\nCORNETS\n(79 = 44 + 35)\nCORNETS\n(74 = 39 + 35)\nCORNETS\n(74 = 39 + 35)\nCORNETS\n(74 = 39 + 35)\nCORNETS\n(73 = 38 + 35)\nCORNETS\n(65 = 30 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(56 = 21 + 35)\nCORNETS\n(55 = 20 + 35)\nCORNETS\n(52 = 17 + 35)\nCORNETS\n(52 = 17 + 35)\nCORNETS\n(52 = 17 + 35)\nCORNETS\n(51 = 16 + 35)\nCORNETS\n(51 = 16 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(46 = 11 + 35)\n\nCORNETS\n(98 = 63 + 35)\nCORNETS\n(92 = 57 + 35)\nCORNETS\n(86 = 51 + 35)\nCORNETS\n(86 = 51 + 35)\nCORNETS\n(80 = 45 + 35)\nCORNETS\n(80 = 45 + 35)\nCORNETS\n(80 = 45 + 35)\nCORNETS\n(79 = 44 + 35)\nCORNETS\n(79 = 44 + 35)\nCORNETS\n(79 = 44 + 35)\nCORNETS\n(74 = 39 + 35)\nCORNETS\n(74 = 39 + 35)\nCORNETS\n(74 = 39 + 35)\nCORNETS\n(73 = 38 + 35)\nCENSOR\n(66)\nCORNETS\n(65 = 30 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNETS\n(61 = 26 + 35)\nCORNET\n(60)\nCRONES\n(60)\nRECONS\n(60)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(59 = 24 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nESCORT\n(57)\nCORSET\n(57)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nCORNETS\n(57 = 22 + 35)\nSECTOR\n(57)\nCORNETS\n(56 = 21 + 35)\nCORNETS\n(55 = 20 + 35)\nCRONES\n(54)\nCENSOR\n(54)\nRECONS\n(54)\nCORNET\n(54)\nCORNETS\n(52 = 17 + 35)\nCORNETS\n(52 = 17 + 35)\nCORNETS\n(52 = 17 + 35)\nCORNETS\n(51 = 16 + 35)\nSECTOR\n(51)\nCORNETS\n(51 = 16 + 35)\nSCORN\n(51)\nCENTS\n(51)\nCRONE\n(51)\nCORSET\n(51)\nSCENT\n(51)\nCONES\n(51)\nESCORT\n(51)\nCORNS\n(51)\nSCONE\n(51)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(50 = 15 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(49 = 14 + 35)\nCORNETS\n(49 = 14 + 35)\nCREST\n(48)\nCORN\n(48)\nCORNET\n(48)\nCONS\n(48)\nCERTS\n(48)\nCORNETS\n(48 = 13 + 35)\nCONE\n(48)\nCORES\n(48)\nCORNETS\n(48 = 13 + 35)\nCRONES\n(48)\nCORNETS\n(48 = 13 + 35)\nSCORE\n(48)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(48 = 13 + 35)\nCENT\n(48)\nCORNETS\n(48 = 13 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(47 = 12 + 35)\nCORNETS\n(46 = 11 + 35)\nCOTS\n(45)\nCOST\n(45)\nCORE\n(45)\nRECONS\n(42)\nRECONS\n(42)\nCORNET\n(42)\nCENSOR\n(42)\nCRONES\n(42)\nCENSOR\n(42)\nRECONS\n(40)\nCORNET\n(40)\nCENSOR\n(40)\nRECONS\n(40)\nCORNET\n(40)\nCENSOR\n(40)\nCRONES\n(40)\nCRONES\n(40)\nSTONER\n(39)\nSCORN\n(39)\nCRONE\n(39)\nTENSOR\n(39)\nSECTOR\n(39)\nTENORS\n(39)\nTONERS\n(39)\nSCENT\n(39)\nSCONE\n(39)\nCORSET\n(39)\nESCORT\n(39)\nTRONES\n(39)\nCORNS\n(39)\nRECON\n(39)\nCORNET\n(36)\nRECONS\n(36)\nCRONES\n(36)\nCRONES\n(36)\nRECONS\n(36)\nCRONES\n(36)\nCONES\n(36)\nCRONES\n(36)\nCORN\n(36)\nRECON\n(36)\nCORNET\n(36)\nRECONS\n(36)\nESCORT\n(36)\nCRONES\n(36)\nESCORT\n(36)\nCRONE\n(36)\nCORNS\n(36)\nSECTOR\n(36)\nCORNET\n(36)\nCENSOR\n(36)\nCENTS\n(36)\nSECTOR\n(36)\nCORSET\n(36)\nCORNET\n(36)\nCENSOR\n(36)\nCENSOR\n(36)\nCENSOR\n(36)\nCORNET\n(36)\nSCENT\n(36)\nCORSET\n(36)\nSCONE\n(36)\nCENSOR\n(36)\nRECONS\n(36)\nSCORN\n(36)\nCONES\n(34)\nCRONE\n(34)\nCORSET\n(34)\nCORNS\n(34)\nCENTS\n(34)\nCORNS\n(33)\nCORSET\n(33)\nSCORN\n(33)\nTRONES\n(33)\nSECTOR\n(33)\nESCORT\n(33)\nESCORT\n(33)\nCORSET\n(33)\nESCORT\n(33)\nESCORT\n(33)\nTENSOR\n(33)\nSCORN\n(33)\nTONERS\n(33)\nESCORT\n(33)\nSECTOR\n(33)\nSTONER\n(33)\nTONERS\n(33)\nSCONE\n(33)\nRECON\n(33)\nCENTS\n(33)\nCONES\n(33)\nCORSET\n(33)\nCRONE\n(33)\nCONES\n(33)\nCORSET\n(33)\nCRONE\n(33)\nCENTS\n(33)\nCONES\n(33)\nTENORS\n(33)\nTENORS\n(33)\nCENTS\n(33)\nSCONE\n(33)\nTENSOR\n(33)\nSECTOR\n(33)\nSECTOR\n(33)\nCORNS\n(33)\nTRONES\n(33)\nSCENT\n(33)\nRECON\n(33)\nSCENT\n(33)\nCORSET\n(33)\n\n# Word Growth involving cornets\n\nor corn cornet\n\nnet cornet\n\nnet nets\n\n## Longer words containing cornets\n\n(No longer words found)" ]

[ null ]

[ "", null, "# Question: Where Do We Use Percentage?\n\n## How do you take off a percentage?\n\nIf your calculator does not have a percent key and you want to add a percentage to a number multiply that number by 1 plus the percentage fraction.\n\nFor example 25000+9% = 25000 x 1.09 = 27250.\n\nTo subtract 9 percent multiply the number by 1 minus the percentage fraction.\n\nExample: 25000 – 9% = 25000 x 0.91 = 22750..\n\n## Is there a gap between number and unit?\n\nThere is a space between the numerical value and unit symbol, even when the value is used in an adjectival sense, except in the case of superscript units for plane angle. If the spelled-out name of a unit is used, the normal rules of English apply: “a roll of 35-millimeter film.”\n\n## What is 20% off?\n\nA 20 percent discount is 0.20 in decimal format. Secondly, multiply the decimal discount by the price of the item to determine the savings in dollars. For example, if the original price of the item equals \\$24, you would multiply 0.2 by \\$24 to get \\$4.80.\n\n## What jobs use percentages?\n\nTherefore, any business involving tax calculation, tip calculation, or interest rates uses fractions. Banks, restaurants, movie theaters and department stores all use percentages, so teller, wait staff and store clerk positions are included here.\n\n## Where does the percentage sign go?\n\nPercent Sign, Before Or After? As you can see in previous examples the percent sign always comes after the number, never before. It is the same when we write percent as a word and when we pronounce it.\n\n## How is Percent written?\n\nUse the symbol % to express percent in scientific and technical writing, except when writing numbers at the beginning of a sentence. When you write out the word, use the form percent instead of the older form per cent. In nontechnical writing, use the word percent rather than the symbol %. …\n\n## What percent is 12 out of 24?\n\n50%Convert fraction (ratio) 12 / 24 Answer: 50%\n\n## What is the difference between percent and percentage?\n\nThe words percent and percentage are closely related—does it matter how they are used in a sentence? … The rule for using percent and percentage is straightforward. The word percent (or the symbol %) accompanies a specific number, whereas the more general word percentage is used without a number.\n\n## Why do we use percentage?\n\nWe use percentages to make calculations easier. It is much simpler to work with parts of 100 than thirds, twelfths and so on, especially because quite a lot of fractions do not have an exact (non-recurring) decimal equivalent.\n\n## What percentage is a number?\n\nLearning how to calculate the percentage of one number vs. another number is easy. If you want to know what percent A is of B, you simple divide A by B, then take that number and move the decimal place two spaces to the right. That’s your percentage!\n\n## What is a 100% increase?\n\nAn increase of 100% in a quantity means that the final amount is 200% of the initial amount (100% of initial + 100% of increase = 200% of initial). In other words, the quantity has doubled.\n\n## How do I calculate a percentage between two numbers?\n\nPercentage Change | Increase and DecreaseFirst: work out the difference (increase) between the two numbers you are comparing.Increase = New Number – Original Number.Then: divide the increase by the original number and multiply the answer by 100.% increase = Increase ÷ Original Number × 100.More items…\n\n## What is the percent sign with extra 0?\n\nA per mil, also spelled per mille, per mill, permil, permill, or permille is a sign indicating parts per thousand. Per mil should not be confused with parts per million. The sign is written ‰, which looks like a percent sign with an extra zero in the divisor.\n\n## How do we use percentages in real life?\n\nPercents Percentages in Real LifeSuppose you buy a \\$50 coffeemaker in an area where the sales tax is 8%. When you check out, 8% of \\$50 would be added to your total price. … Let’s say a shirt costs \\$8, but it’s been marked down by 50%. … Let’s say you eat at a restaurant with some friends.\n\n## What number is 25 percent of 60?\n\n15Percentage Calculator: What is 25 percent of 60? = 15.\n\n## What grade is a 60%?\n\nPercentLetter Grade70 – 72C-67 – 69D+63 – 66D60 – 62D-8 more rows\n\n## How can calculate percentage?\n\n1. How to calculate percentage of a number. Use the percentage formula: P% * X = YConvert the problem to an equation using the percentage formula: P% * X = Y.P is 10%, X is 150, so the equation is 10% * 150 = Y.Convert 10% to a decimal by removing the percent sign and dividing by 100: 10/100 = 0.10.More items…" ]

[ null, "https://mc.yandex.ru/watch/68903419", null ]

[ "", null, "2012-2013 Small Game & Waterfowl Teams, Rules, & Entries - Page 8\n\n# Thread: 2012-2013 Small Game & Waterfowl Teams, Rules, & Entries\n\n1. Tika270\nTeam 3\n2 ducks\n1 goose\n9 points", null, "Team 1 = 4 points\nBowHunter96 = 4 points\nACarbone624 = 0 points\nnjfisherman11 = 0 points\n\nTeam 2 = 0 points\nJerseyboy32 = 0 points\nnjhunter = 0 points\nDuckslayer424 = 0 points\n\nTeam 3 = 130 points\nmike242 = 18 points\nhunterdan199 = 0 points\nTika270 =112 points\n\nTeam 4 = 0 points\nMetrOstar1 = 0 points\nKidRipley = 0 points\nNJhunter87 = 0 points\n\nTeam 5 = 0 points\nBlackthorn = 0 points\nChris90 = 0 points\nbone collector = 0 points\n\nTeam 6 = 121 points\nWithmygunnar = 0 points\nAYFS = 115 points\nGrits N Gravy = 6 points", null, "", null, "Reply With Quote\n\n2. Tika270\nTeam 3\n2 opossum\n8 points", null, "", null, "Team 1 = 4 points\nBowHunter96 = 4 points\nACarbone624 = 0 points\nnjfisherman11 = 0 points\n\nTeam 2 = 0 points\nJerseyboy32 = 0 points\nnjhunter = 0 points\nDuckslayer424 = 0 points\n\nTeam 3 = 138 points\nmike242 = 18 points\nhunterdan199 = 0 points\nTika270 =120 points\n\nTeam 4 = 0 points\nMetrOstar1 = 0 points\nKidRipley = 0 points\nNJhunter87 = 0 points\n\nTeam 5 = 0 points\nBlackthorn = 0 points\nChris90 = 0 points\nbone collector = 0 points\n\nTeam 6 = 121 points\nWithmygunnar = 0 points\nAYFS = 115 points\nGrits N Gravy = 6 points", null, "", null, "Reply With Quote\n\n3. Tika270\nTeam 3\n3 geese\n9 points", null, "Team 1 = 4 points\nBowHunter96 = 4 points\nACarbone624 = 0 points\nnjfisherman11 = 0 points\n\nTeam 2 = 0 points\nJerseyboy32 = 0 points\nnjhunter = 0 points\nDuckslayer424 = 0 points\n\nTeam 3 = 147 points\nmike242 = 18 points\nhunterdan199 = 0 points\nTika270 =129 points\n\nTeam 4 = 0 points\nMetrOstar1 = 0 points\nKidRipley = 0 points\nNJhunter87 = 0 points\n\nTeam 5 = 0 points\nBlackthorn = 0 points\nChris90 = 0 points\nbone collector = 0 points\n\nTeam 6 = 121 points\nWithmygunnar = 0 points\nAYFS = 115 points\nGrits N Gravy = 6 points", null, "", null, "Reply With Quote\n\n4.\n\n5. Tika270\nTeam 3\n2 phez\n7 chukar\n20 points", null, "Team 1 = 4 points\nBowHunter96 = 4 points\nACarbone624 = 0 points\nnjfisherman11 = 0 points\n\nTeam 2 = 0 points\nJerseyboy32 = 0 points\nnjhunter = 0 points\nDuckslayer424 = 0 points\n\nTeam 3 = 167 points\nmike242 = 18 points\nhunterdan199 = 0 points\nTika270 =149 points\n\nTeam 4 = 0 points\nMetrOstar1 = 0 points\nKidRipley = 0 points\nNJhunter87 = 0 points\n\nTeam 5 = 0 points\nBlackthorn = 0 points\nChris90 = 0 points\nbone collector = 0 points\n\nTeam 6 = 121 points\nWithmygunnar = 0 points\nAYFS = 115 points\nGrits N Gravy = 6 points", null, "", null, "Reply With Quote\n\n6. Tika270\nTeam 3\n3 geese\n9 points", null, "Team 1 = 4 points\nBowHunter96 = 4 points\nACarbone624 = 0 points\nnjfisherman11 = 0 points\n\nTeam 2 = 0 points\nJerseyboy32 = 0 points\nnjhunter = 0 points\nDuckslayer424 = 0 points\n\nTeam 3 = 176 points\nmike242 = 18 points\nhunterdan199 = 0 points\nTika270 =158 points\n\nTeam 4 = 0 points\nMetrOstar1 = 0 points\nKidRipley = 0 points\nNJhunter87 = 0 points\n\nTeam 5 = 0 points\nBlackthorn = 0 points\nChris90 = 0 points\nbone collector = 0 points\n\nTeam 6 = 121 points\nWithmygunnar = 0 points\nAYFS = 115 points\nGrits N Gravy = 6 points", null, "", null, "Reply With Quote\n\n7. Jeez Tika. Your doing pretty good. It's gonna be tough to try and catch back up with you.", null, "", null, "Reply With Quote\n\n8. Damn Bro leave some for seed.", null, "", null, "", null, "", null, "Reply With Quote\n\n9.", null, "4 rabbits team 1. Forgot I was in this, will have more by the end of the season!", null, "", null, "Reply With Quote\n\n10. Tika270\nTeam 3\n4 geese\n12 points", null, "Team 1 = 4 points\nBowHunter96 = 4 points\nACarbone624 = 0 points\nnjfisherman11 = 0 points\n\nTeam 2 = 0 points\nJerseyboy32 = 0 points\nnjhunter = 0 points\nDuckslayer424 = 0 points\n\nTeam 3 = 188 points\nmike242 = 18 points\nhunterdan199 = 0 points\nTika270 =170 points\n\nTeam 4 = 0 points\nMetrOstar1 = 0 points\nKidRipley = 0 points\nNJhunter87 = 0 points\n\nTeam 5 = 0 points\nBlackthorn = 0 points\nChris90 = 0 points\nbone collector = 0 points\n\nTeam 6 = 121 points\nWithmygunnar = 0 points\nAYFS = 115 points\nGrits N Gravy = 6 points", null, "", null, "Reply With Quote\n\n11. Nice!!!!", null, "", null, "", null, "", null, "Reply With Quote\n\n####", null, "Posting Permissions\n\n• You may not post new threads\n• You may not post replies\n• You may not post attachments\n• You may not edit your posts\n•" ]

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[ "# Risk weighted assets calculation example\n\n### Bank Regulatory Capital to Risk-Weighted Assets for United", null, "Part 2 The First Pillar Minimum Capital Requirements I. Total assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great, Risk-weighted asset (also referred to as RWA) is a bank's assets or off-balance-sheet exposures, weighted according to risk. This sort of asset calculation is used in.\n\n### Total assets versus risk weighted assets does it matter\n\nCredit portfolios and risk weighted assets analysis of. Banking Regulatory Update Total Risk Weighted Assets \\$358,580m • Use of risk factors for market risk • Data to calculate business, Basel II prescribes specific algorithms for the calculation of risk-weighted assets and subsequently the capital that needs to be reserved against those assets..\n\nCapital Requirements Directive IV Framework Standardised Approach to Credit Risk in the risk weigh the assets, the risk weight calculation Calculation of risk weights Standardised Approach for Risk-weighted Assets; its risk-weighted assets for general credit risk,\n\nBasel II prescribes specific algorithms for the calculation of risk-weighted assets and subsequently the capital that needs to be reserved against those assets. ... of its risk-weighted asset. Capital adequacy ratio lower-risk assets. The specifics of CAR calculation example. Risk weighted assets\n\nCapital Requirements Directive IV Framework Standardised Approach to Credit Risk in the risk weigh the assets, the risk weight calculation What are Risk Weighted Assets? First Published the value of each asset is assigned a risk weight (for example 100% for corporate loans and 50 Calculation of\n\nTotal assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great Under Basel III, analysis of eligible capital and deductions are fully embedded during the risk weighted asset calculation, which represents a significant change from\n\nAustralian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets, Risk-weighted asset (also referred to as RWA) is a bank's assets or off-balance-sheet exposures, weighted according to risk. This sort of asset calculation is used in\n\nAustralian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets, Calculation of Risk Weighted Asset ( RWA) Think A Banking company has some data given below\n\nAsset Management in Finance; Risk Weighted Average Share Outstanding Calculation Example #2. Here we look at the steps to calculate weighted average Total assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great\n\n... of its risk-weighted asset. Capital adequacy ratio lower-risk assets. The specifics of CAR calculation example. Risk weighted assets 1/11/2018В В· Risk-weighted assets are assets held by a financial institution that are ranked and weighted according to their risk level, like... For example, Bank A\n\nTo calculate credit risk-weighted assets, (Illustrative rating example) Risk weights for senior securitization exposures backed by granular pools: I. Calculation of minimum capital requirements minimum ratio of capital to risk weighted assets. 22. In calculating the 14 The following examples outline how\n\npossibilities exist: to change the risk profile (for example, cross-bank and cross-country consistency in the calculation of risk-weighted assets Bank Risk-Weighted Assets: How to Restore Investor Trust. risk-weighted assets and RWA is a common input for calculating both RWA density and RoRWA.\n\nof risk-weighted assets (RWA) risk (CCR) – calculation based on Basel III RWA Optimization In the early stage of Basel II Weighted average cost of capital (WACC) • Based on the capital asset pricing • The equity risk premium is the average of the current implied equity risk\n\nRevisiting Risk-Weighted Assets Prepared by Vanessa Le LeslГ© and Sofiya Avramova1 calculation of risk-weighted assets (RWAs) This white paper will cover the origins of the Risk-Weighted Assets ratio and the history of its use in financial analysis. The paper will further showcase its\n\nframework and of the calculation of risk-weighted assets by banks as part of its Regulatory (RWA) Consistency Assessment Programme (RCAP). Banking Regulatory Update Total Risk Weighted Assets \\$358,580m • Use of risk factors for market risk • Data to calculate business\n\nAustralian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets, Under Basel III, analysis of eligible capital and deductions are fully embedded during the risk weighted asset calculation, which represents a significant change from\n\nCalculation of risk weights its risk-weighted assets for general credit risk, off-balance sheet items, For example, the credit Australian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets,\n\nHere is a sample ICAAP (Internal Capital Adequacy Assessment Risk Weighted Assets: Internal Capital Adequacy Assessment – Sample ICAAP report format and I. Calculation of minimum capital requirements minimum ratio of capital to risk weighted assets. 22. In calculating the 14 The following examples outline how\n\nCalculation of risk weights Standardised Approach for Risk-weighted Assets; its risk-weighted assets for general credit risk, This document provides a simplified view at the calculation of capital requirement for a risky position as a function of PD. This document is compliant with Basel II\n\nRisk-Weighted Assets - RWA Risk-weighted Assets, or RWA, are a key measure in risk management. RWA consists of 1. the sum of risk weight times asset amount for on What Are Risk-Weighted Assets, calculating a bank's risk-weighted assets is a nightmare. The Motley Fool recommends and owns shares of Apple and Bank of\n\nDEPARTMENT OF THE TREASURY . Office of the Comptroller of the Currency . Overview of the proposed standardized approach for calculation of risk-weighted assets and Table c irb risk based capital formulas for whole exposures to non defaulted to calculate its risk weighted assets a bank must apply Risk Weighted Assets Example\n\nAustralian Bank Capital and the Regulatory Framework. Risk-Weighted Assets - RWA Risk-weighted Assets, or RWA, are a key measure in risk management. RWA consists of 1. the sum of risk weight times asset amount for on, Basel II prescribes specific algorithms for the calculation of risk-weighted assets and subsequently the capital that needs to be reserved against those assets..\n\n### Basel III – Capital Adequacy – US implementation Finance", null, "Total assets versus risk weighted assets does it matter. Asset Management in Finance; Risk Weighted Average Share Outstanding Calculation Example #2. Here we look at the steps to calculate weighted average, Here is a sample ICAAP (Internal Capital Adequacy Assessment Risk Weighted Assets: Internal Capital Adequacy Assessment – Sample ICAAP report format and.\n\nWhat are Risk-Weighted Assets? (with pictures). Calculation of Risk Weighted Asset ( RWA) Think A Banking company has some data given below, Under Basel III, analysis of eligible capital and deductions are fully embedded during the risk weighted asset calculation, which represents a significant change from.\n\n### Standardised approach for risk-weighted assets Lexology", null, "Part 2 The First Pillar Minimum Capital Requirements I. Bank Risk-Weighted Assets: How to Restore Investor Trust. risk-weighted assets and RWA is a common input for calculating both RWA density and RoRWA. https://en.wikipedia.org/wiki/Risk-Weighted_Asset 1/11/2018В В· Risk-weighted assets are assets held by a financial institution that are ranked and weighted according to their risk level, like... For example, Bank A.", null, "What Are Risk-Weighted Assets, calculating a bank's risk-weighted assets is a nightmare. The Motley Fool recommends and owns shares of Apple and Bank of 11/10/2016В В· The Risk Weighted Assets (RWA) refer to the fund based assets such as Cash, Loans, Investments and other assets. They are the total assets owned by the\n\nI. Calculation of minimum capital requirements minimum ratio of capital to risk weighted assets. 22. In calculating the 14 The following examples outline how This document provides a simplified view at the calculation of capital requirement for a risky position as a function of PD. This document is compliant with Basel II\n\nWhat Are Risk-Weighted Assets, calculating a bank's risk-weighted assets is a nightmare. The Motley Fool recommends and owns shares of Apple and Bank of Basel III and Derivatives Exposures: Understanding the Regulatory weighting and calculation of risk-based • Risk-weighted assets consist of on\n\nHow Risky Are Banks’ Risk Weighted Assets? (1995) provides an example in which, calculation of risk weighted assets across countries that may have Basel III – Capital Adequacy – US implementation as well as revised risk weights and calculation and the credit risk weighted asset under\n\n1/11/2018В В· Risk-weighted assets are assets held by a financial institution that are ranked and weighted according to their risk level, like... For example, Bank A How Risky Are Banks’ Risk Weighted Assets? (1995) provides an example in which, calculation of risk weighted assets across countries that may have\n\nalso include items that are excluded from the calculation of risk-weighted assets, the exposure and the exposure amount that is to be risk weighted. For example, Banking Regulatory Update Total Risk Weighted Assets \\$358,580m • Use of risk factors for market risk • Data to calculate business\n\nRisk-weighted asset (also referred to as RWA) is a bank's assets or off-balance-sheet exposures, weighted according to risk. This sort of asset calculation is used in 28/11/2013В В· Risk-Weighted Asset (RWA) Calculator simplified view at the calculation of capital requirement for a 3 Capital for Risk Weighted Assets\n\nBanking Regulatory Update Total Risk Weighted Assets \\$358,580m • Use of risk factors for market risk • Data to calculate business Total assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great\n\nCalculation of risk weights its risk-weighted assets for general credit risk, off-balance sheet items, For example, the credit Calculation of risk weights Standardised Approach for Risk-weighted Assets; its risk-weighted assets for general credit risk,\n\nCREDIT PORTFOLIOS AND RISK WEIGHTED ASSETS This approach relied on the capability of banks to calculate their own credit risk provides an interesting example Example 1: Calculation of General Risk Appendix VI Illustration on Risk-Weighted Asset (RWA) Calculation Capital Adequacy Framework (Basel II - Risk-Weighted\n\n## Risk-Weighted Assets (RWA) density What lies behind this", null, "RISK WEIGHTED ASSET UNDER BASEL III CALCULATION WITH EXAMPLE. 28/11/2013В В· Risk-Weighted Asset (RWA) Calculator simplified view at the calculation of capital requirement for a 3 Capital for Risk Weighted Assets, CREDIT PORTFOLIOS AND RISK WEIGHTED ASSETS This approach relied on the capability of banks to calculate their own credit risk provides an interesting example.\n\n### Capital Adequacy Framework (Basel II Risk-Weighted Assets)\n\nRisk Weighted Asset financial definition of Risk Weighted. Table c irb risk based capital formulas for whole exposures to non defaulted to calculate its risk weighted assets a bank must apply Risk Weighted Assets Example, Revisiting Risk-Weighted Assets Prepared by Vanessa Le LeslГ© and Sofiya Avramova1 calculation of risk-weighted assets (RWAs).\n\nI. Calculation of minimum capital requirements minimum ratio of capital to risk weighted assets. 22. In calculating the 14 The following examples outline how 28/11/2013В В· Risk-Weighted Asset (RWA) Calculator simplified view at the calculation of capital requirement for a 3 Capital for Risk Weighted Assets\n\nRisk-Weighted Assets - RWA Risk-weighted Assets, or RWA, are a key measure in risk management. RWA consists of 1. the sum of risk weight times asset amount for on Total assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great\n\nRevisiting Risk-Weighted Assets Prepared by Vanessa Le LeslГ© and Sofiya Avramova1 calculation of risk-weighted assets (RWAs) 11/10/2016В В· The Risk Weighted Assets (RWA) refer to the fund based assets such as Cash, Loans, Investments and other assets. They are the total assets owned by the\n\nAustralian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets, Australian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets,\n\nCalculation of Risk Weighted Asset ( RWA) Think A Banking company has some data given below Since 2012 the Basel Committee has increasingly pursued a revision of the calculation methods for risk-weighted assets. based on an example calculation 223.\n\nCan Risk Weighted Assets (RWAs) be Trusted. Basel II enables advanced banks to use their own internal models to calculate the risk weights used to determine their CREDIT PORTFOLIOS AND RISK WEIGHTED ASSETS This approach relied on the capability of banks to calculate their own credit risk provides an interesting example\n\nBanking Regulatory Update Total Risk Weighted Assets \\$358,580m • Use of risk factors for market risk • Data to calculate business World Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St\n\nRisk-Weighted Assets - RWA Risk-weighted Assets, or RWA, are a key measure in risk management. RWA consists of 1. the sum of risk weight times asset amount for on Global banking supervisors based in Basel Switzerland use the concept of risk-weighted assets risk weighted total to calculate assets by their credit risk,\n\nWorld Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St Banking Regulatory Update Total Risk Weighted Assets \\$358,580m • Use of risk factors for market risk • Data to calculate business\n\nDEPARTMENT OF THE TREASURY . Office of the Comptroller of the Currency . Overview of the proposed standardized approach for calculation of risk-weighted assets and Asset Management in Finance; Risk Weighted Average Share Outstanding Calculation Example #2. Here we look at the steps to calculate weighted average\n\nAustralian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets, Australian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets,\n\nWorld Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St World Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St\n\nTotal assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great Can Risk Weighted Assets (RWAs) be Trusted. Basel II enables advanced banks to use their own internal models to calculate the risk weights used to determine their\n\nWhat Are Risk-Weighted Assets, calculating a bank's risk-weighted assets is a nightmare. The Motley Fool recommends and owns shares of Apple and Bank of Weighted average cost of capital (WACC) • Based on the capital asset pricing • The equity risk premium is the average of the current implied equity risk\n\nWorld Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St of risk-weighted assets (RWA) risk (CCR) – calculation based on Basel III RWA Optimization In the early stage of Basel II\n\nof risk-weighted assets (RWA) risk (CCR) – calculation based on Basel III RWA Optimization In the early stage of Basel II 28/11/2013В В· Risk-Weighted Asset (RWA) Calculator simplified view at the calculation of capital requirement for a 3 Capital for Risk Weighted Assets\n\n### Standardised approach for risk-weighted assets Lexology", null, "Basel IV The Next Generation of Risk Weighted Assets. Total assets versus risk weighted assets: As an example, if risk-weighted assets were used as The calculation of risk-weighted assets explains to a great, 11/10/2016В В· The Risk Weighted Assets (RWA) refer to the fund based assets such as Cash, Loans, Investments and other assets. They are the total assets owned by the.\n\nTotal assets versus risk weighted assets does it matter. Can Risk Weighted Assets (RWAs) be Trusted. Basel II enables advanced banks to use their own internal models to calculate the risk weights used to determine their, To calculate credit risk-weighted assets, (Illustrative rating example) Risk weights for senior securitization exposures backed by granular pools:.\n\n### Basel IV The Next Generation of Risk Weighted Assets", null, "Processing Credit Risk (Basel II) Oracle Help Center. World Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St https://en.m.wikipedia.org/wiki/Standardized_approach_(credit_risk) World Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St.", null, "• Standardised approach for risk-weighted assets Lexology\n• Basel IV The Next Generation of Risk Weighted Assets\n\n• Bank Risk-Weighted Assets: How to Restore Investor Trust. risk-weighted assets and RWA is a common input for calculating both RWA density and RoRWA. Under Basel III, analysis of eligible capital and deductions are fully embedded during the risk weighted asset calculation, which represents a significant change from\n\nCREDIT PORTFOLIOS AND RISK WEIGHTED ASSETS This approach relied on the capability of banks to calculate their own credit risk provides an interesting example Table c irb risk based capital formulas for whole exposures to non defaulted to calculate its risk weighted assets a bank must apply Risk Weighted Assets Example\n\nUnder Basel III, analysis of eligible capital and deductions are fully embedded during the risk weighted asset calculation, which represents a significant change from Australian Bank Capital and the Regulatory Framework AustRAliAn BAnk CApitAl And the RegulAtoRy fRAmewoRk in terms of risk-weighted assets,\n\nAsset Management in Finance; Risk Weighted Average Share Outstanding Calculation Example #2. Here we look at the steps to calculate weighted average World Bank, Bank Regulatory Capital to Risk-Weighted Assets for United States [DDSI05USA156NWDB], retrieved from FRED, Federal Reserve Bank of St\n\nCalculation of risk weights Standardised Approach for Risk-weighted Assets; its risk-weighted assets for general credit risk, Table c irb risk based capital formulas for whole exposures to non defaulted to calculate its risk weighted assets a bank must apply Risk Weighted Assets Example\n\nAsset Management in Finance; Risk Weighted Average Share Outstanding Calculation Example #2. Here we look at the steps to calculate weighted average of risk-weighted assets (RWA) risk (CCR) – calculation based on Basel III RWA Optimization In the early stage of Basel II\n\nWhat Are Risk-Weighted Assets, calculating a bank's risk-weighted assets is a nightmare. The Motley Fool recommends and owns shares of Apple and Bank of Risk weighted assets calculation example keyword after analyzing the system lists the list of keywords related and the list of websites with related content, in\n\n... of its risk-weighted asset. Capital adequacy ratio lower-risk assets. The specifics of CAR calculation example. Risk weighted assets Asset Management in Finance; Risk Weighted Average Share Outstanding Calculation Example #2. Here we look at the steps to calculate weighted average\n\n... of its risk-weighted asset. Capital adequacy ratio lower-risk assets. The specifics of CAR calculation example. Risk weighted assets I. Calculation of minimum capital requirements minimum ratio of capital to risk weighted assets. 22. In calculating the 14 The following examples outline how", null, "Here is a sample ICAAP (Internal Capital Adequacy Assessment Risk Weighted Assets: Internal Capital Adequacy Assessment – Sample ICAAP report format and 11/10/2016В В· The Risk Weighted Assets (RWA) refer to the fund based assets such as Cash, Loans, Investments and other assets. They are the total assets owned by the" ]

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[ "", null, "# Corporate Finance Case\n\nAutor:   •  December 3, 2012  •  Coursework  •  423 Words (2 Pages)  •  1,117 Views\n\nPage 1 of 2\n\nCORPORATE FINANCE\n\n1) Valuation formulae\n\nLet V be the present value of an asset or security that pays cash flows in the future, where the last cash flow is to be received at time T (note that, if cash flows are to be received forever, then T = ∞). Let CFt be the cash flow to be received at time t, and let rt be the appropriate discount rate for the period from now to time t. Then,\n\nSpecial cases:\n\n(i) Time periods between cash flow payments are of equal length\n\n(ii) The discount rate is the same for all periods (rt = r)\n\n(iii) The discount rate is the same for all periods (rt = r), cash flows for times from next period until the maturity of the asset are constant (CFt = C), with an additional final payment being made at maturity (when T = M)\n\n(iv) There is only one cash flow to be received, at the maturity of the asset (T = M)\n\n(v) The discount rate is the same for all periods (rt = r) and cash flows for times from the next period until the maturity of the asset are constant (CFt = C)\n\n(vi) The discount rate is the same for all periods (rt = r) and cash flows are constant forever (CFt = C)\n\n2) TVM keys on a financial calculator\n\nHow do these valuation formulae translate to the TVM keys on a financial calculator?\n\n3) Types of cash flow streams\n\n..." ]

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[ "#### Before checking all questions and answer try below free mock test which. All questions are asked in 21-feb-2021 paper.\n\n[ays_quiz id=”12″]\n\nFor subject wise if you want to give paper then click here\n\n## All Questions and Answer of MRPL 21-feb-2021\n\n#### A cell of emf x and internal resistance y is connected to a resistance y. The potential difference between the terminals of the cell is\n\n• x/2\n• x/2y\n• 2x/y\n• x/(x+y)\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]\n\nx/2\n\n[/bg_collapse]\n\n#### The ohm’s law deals with the relation between\n\n• charge and resistance\n• charge and capacity\n• current and potential difference.\n• capacity and potential difference.\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]\n\ncurrent and potential difference.\n\n[/bg_collapse]\n\n#### Potential inside a hollow charged sphere is\n\n• same as on its sphere\n• greater than on its surface\n• less than on its surface\n• zero\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]\n\nsame as on its sphere\n\n[/bg_collapse]\n\n#### Capacitance of a parallel plate capacitor decreases by\n\n• increasing the area of plates\n• increasing the distance between the plates\n• putting a dielectric between the plates\n• decreasing the distance between the plates\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]increasing the distance between the plates [/bg_collapse]\n\n#### The capacitance of the two capacitors are in the ratio 1:2. When they are connected in parallel across supply voltage v, their charges will be in the ratio of\n\n• 1:2\n• 2:1\n• 3:2\n• 2:3\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]1:2[/bg_collapse]\n\n#### The capacity of a cell depends upon\n\n• Nature of plates material and electrolyte\n• Sizes of the plates and the quality of electrolyte\n• Both (A) and (B)\n• None of the above\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Sizes of the plates and the quality of electrolyte[/bg_collapse]\n\n#### When an organic compound is dissolved in water, the attractive forces between the charged atom reduce tremendously and they are separated this is called.\n\n• Decomposition\n• Deionisation\n• Ionisation\n• None of the above\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Ionisation [/bg_collapse]\n\n#### Hysteresis loss in a magnetic material depends upon\n\n• Area of hysteresis loop\n• Frequency of reversed of field\n• Volume of magnetic material\n• All (A), (B) and (C)\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]All (A), (B) and (C)\n\n[/bg_collapse]\n\n#### Wheatstone bridge is used to measure\n\n• Voltage\n• Resistance\n• Current\n• Power\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Resistance [/bg_collapse]\n\n#### The three resistors each of r ohm are connected in star. When they are transformed into delta connections, the resistance of each arm will be\n\n• 2R ohm\n• 3R ohm\n• 4R ohm\n• R/2 ohm\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]3R ohm\n\n[/bg_collapse]\n\n#### When a source is delivering maximum power to the load, the efficiency will be\n\n• Maximum\n• Before 50%\n• Above 50%\n• 50%\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]50%[/bg_collapse]\n\n#### An alternating quantity which attains its positive maximum value prior to the other is called the\n\n• In phase quantity\n• Lagging quantity\n• None of the above\n\n#### If a sinusoidal wave has frequency of 50 Hz with 15 ampere rms value. Which of the following equation represents?\n\n• 15sin5Ot\n• 30sin25t\n• 42.42sin 100t\n• 21.21sin314t\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]21.21sin314t [/bg_collapse]\n\n#### When a sinusoidal voltage is applied across a R-L series circuit having R XL, the phase angle will be\n\n• 90°\n• 30°\n• 45°\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]45°\n\n[/bg_collapse]\n\n#### The shunt used in the millimetre\n\n• Will extend the range and increases the resistance\n• Will extend the range and decreases the meter resistance\n• Will decrease the range and meter resistance\n• Will decrease the range and increases the meter resistance\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Will extend the range and decreases the meter resistance [/bg_collapse]\n\n#### A signal of 10mV at 75MHz is to be measured. Which of the following instruments can be used\n\n• VTVM\n• Cathode rayoscilloscope\n• Moving iron voltmeter\n• Digital multimeter\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]\n\nCathode rayoscilloscope\n\n[/bg_collapse]\n\n#### The pressure coil of a dynamometer type wattmeter is\n\n• Highly inductive\n• Highly resistive\n• Purely resistive\n• Purely inductive\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Highly resistive [/bg_collapse]\n\n#### The dc series motors are preferred for traction applications because\n\n• The torque is proportional armature current\n• The torque is proportional to the square root of armature current\n• The torque is proportional to square of armature Current and the speed is inversely proportional to the torque.\n• Torque and speed are inversely proportional to armature current\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]The torque is proportional to square of armature Current and the speed is inversely proportional to the torque. [/bg_collapse]\n\n#### What would happen if the field of d.c shunt motor is opened\n\n• The speed of the motor will be reduced\n• It will continue to run at its normal speed\n• The speed of the motor becomes very high and may damage it.\n• The current in the armature will decreases\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]The speed of the motor becomes very high and may damage it. [/bg_collapse]\n\n#### The eddy current loss in the transformers occurs in the\n\n• Primary winding\n• Core\n• Secondary winding\n• None of the above\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Core[/bg_collapse]\n\n#### 1:5 step-up transformer has 120V across the primary and 600 ohms resistance across the secondary. Assuming 100% efficiency, the primary current will be:\n\n• 0.2 Amp\n• 5 Amp\n• 10 Amp\n• 20 Amp\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]10Amp[/bg_collapse]\n\n#### The short circuit test in the transformers is performed to determine\n\n• The iron loss at any load\n• The hysteresis loss\n• The eddy current loss\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]The copper loss at any load or any full load [/bg_collapse]\n\n#### Regenerative method of braking is based on that\n\n• Back emf is less than the applied voltage\n• Back emf is equal to the applied voltage\n• Back emf is more than the applied voltage\n• None of the above\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Back emf is more than the applied voltage [/bg_collapse]\n\n#### Which of the following can be used to control the speed of a d.c motor?\n\n• Thermistor\n• Thyristor\n• Thyratron\n• Transistor\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Thyristor\n\n[/bg_collapse]\n\n#### Which of the following motors is most suitable for signaling devices and many kinds of timers\n\n• d.c shunt motor\n• d.c series motor\n• Inductance motor\n• Reluctance motor\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Reluctance motor\n\n[/bg_collapse]\n\n#### Sparking at the commutators of a dc motor may result in\n\n• Damage to the commutator’s segments\n• Damage to the commutator’s insulation\n• Increased power consumption\n• All of the above\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]All of the above\n\n[/bg_collapse]\n\nAccording to Fleming’s right hand rule for finding the direction induced emf, when induced finger points in the direction of induced emf, forefinger will point the direction of\n\n• Motion of conductor\n• Lines offorce\n• Either of the above\n• None of the above\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Lines offorce [/bg_collapse]\n\n#### The full load copper loss and iron loss of a transformer are 6400W and 5000W respectively the copper loss and iron loss at half load will be respectively.\n\n• 3200W and 2500W\n• 3200W and 5200W\n• 1600W and 1250W\n• 1600W and 5000W\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]1600W and 5000W\n[/bg_collapse]\n\n#### A 6 pole, 3 phase alternator running at 1000rpm supplies to an 8 pole, 3 phase induction motor which has a rotor current of frequency 2Hz. The speed at which the motor operates is.\n\n• 1000 rpm\n• 960 rpm\n• 750 rpm\n• 720 rpm\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]720 rpm[/bg_collapse]\n\n#### The crawling in the induction motor Is caused by\n\n• Improper design of stator lamination\n• Low voltage supply\n• Harmonics developed in motor\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Harmonics developed in motor [/bg_collapse]\n\n#### The use of higher flux density in the transformer design\n\n• Reduces the weight per kva\n• Increases the weight per kva\n• Has no relation with the weight of transformers\n• Increases the weight per kw\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Reduces the weight per kva [/bg_collapse]\n\n#### In hard-tool applications which one of the following single-phase motors is used\n\n• Capacitor start motor\n• Capacitor run motor\n• AC series motor\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]AC series motor [/bg_collapse]\n\n#### What is the angle between the induced voltage and supply voltage of a synchronous motor under running generation?\n\n• Zero\n• Greater than zero but =< 90°\n• Between 90° and 180°\n• 180°\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Greater than zero but =< 90° [/bg_collapse]\n\n#### How can reactive power delivered by a synchronous generator be controlled?\n\n• By changing the prime moves input\n• By changing the excitation\n• By changing the direction of rotation\n• By hanging the prime moves speed\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]By changing the excitation[/bg_collapse]\n\n#### When does a synchronous motor operate with leading power factor current?\n\n• While it is under excited\n• While it is critically excited\n• While it is over excited\n• While it is heavily loaded\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]While it is over excited\n\n[/bg_collapse]\n\n#### In an induction motor what is the ratio of rotor copper-loss and rotor input\n\n• 1/s\n• s\n• (1-s)\n• s/(1-s)\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]s[/bg_collapse]\n\n#### Select the correct answers using the code given below\n\n• 2, 3 and 4\n• 3 and 4\n• 1, 2 and 3\n• 1, 2 and 4\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]1, 2 and 4[/bg_collapse]\n\n#### The pressure at the furnace is minimum in case of\n\n• Force draught system\n• Induced draught system\n• Balanced draught system\n• Natural draught System\n\n[bg_collapse view=”button-orange” color=”#4a4949″ expand_text=”Show Answer” collapse_text=”Show Less” ]Balanced draught system\n\n[/bg_collapse]\n\n• Aluminium\n• Cast iron" ]

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[ "Predicting realized volatility is critical for trading signals and position calibration. Econometric models, such as GARCH and HAR, forecast future volatility based on past returns in a fairly intuitive and transparent way. However, recurrent neural networks have become a serious competitor. Neural networks are adaptive machine learning methods that use interconnected layers of neurons. Activations in one layer determine the activations in the next layer. Neural networks learn by finding activation function weights and biases through training data. Recurrent neural networks are a class of neural networks designed for modeling sequences of data, such as time series. And specialized recurrent neural networks have been developed to retain longer memory, particularly LSTM (Long Short-Term Memory) and GRU (Gated Recurrent Unit). The advantage of neural networks is their flexibility to include complex interactions of features, non-linear effects, and various types of non-price information.\n\nThe below post is based on various papers and posts that are linked next to the quotes. Headings, cursive text, and text in brackets have been added. Also, a range of orthographic and grammatical errors have been corrected and mathematical expressions have been expressed in common language.\n\nThis post ties in with this site’s summary of statistical methods.\n\n### Some formal basics of volatility\n\n“Volatility changes over time. High volatility means high risk and sharp price fluctuations, while low volatility refers to smooth price changes…Simulations of [asset prices] are often modeled using stochastic differential equations…[that include a] drift coefficient or mean of returns over some time period, a diffusion coefficient or the standard deviation of the same returns, [and a stochastic process as] Wiener process or Brownian Motion…Usually…volatility changes stochastically overtime…The volatility’s randomness is often described by a different equation driven by a different Wiener process.[That] model is called a stochastic volatility model…Stochastic volatility models are expressed as a stochastic process, which means that the volatility value at time t is latent and unobservable.” [Antulov-Fantulin and Rodikov]\n\n“Daily realized volatility is defined as the square root of the sum of intra-day squared returns…Realized volatility (RV) is a consistent estimator of the squared root of the integrated variance (IV). There is even a more robust result stating that realized volatility is a consistent estimator of quadratic variation if the underlying process is a semimartingale.” [Antulov-Fantulin and Rodikov]\n\n#### ARCH/ GARCH\n\nAutoregressive Conditional Heteroskedasticity, or ARCH, is a method that explicitly models the change in variance over time in a time series. Specifically, an ARCH method models the variance at a time step as a function of the residual errors from a mean process (e.g. a zero mean)…Generalized Autoregressive Conditional Heteroskedasticity, or GARCH, is an extension of the ARCH model that incorporates a moving average component together with the autoregressive component. Specifically, the model includes lag variance terms (e.g. the observations if modeling the white noise residual errors of another process), together with lag residual errors from a mean process.” [Brownlee]\n\n“The generalized ARCH model [estimates] variance as future volatility [based on] long-run variance and recent variance. Thus, the clustering effect is a sharp increase of volatility…not followed by a sharp drop…Various extensions have been introduced [such as] exponential GARCH, GJR-GARCH, and threshold GARCH [motivated by] stylized facts about volatility.” [Antulov-Fantulin and Rodikov]\n\n#### HAR\n\n“The HAR [heterogeneous autoregression] model essentially claims that the conditional variance of … returns is a linear function of the lagged squared return over the identical return horizon in combination with the squared returns over longer and/or shorter return horizons…Inspired by the success of HAR-type models, most work…has extended the HAR model in the direction of generalizing with jumps, leverage effects, and other nonlinear behaviors…The HAR model has an intuitive interpretation that agents with daily, weekly, and monthly trading frequencies perceive and respond to, altering the corresponding components of volatility.” [Qiu et al.]\n\n“The heterogeneous Autoregression Realized Volatility (HAR-RV) model…is based on the assumption that agents’…perception of volatility depends on their investment horizons and [can be] divided into short-term, medium-term and long-term…Different agents…have different investment periods and participate in trading on the exchange with different frequencies…and respond to different types of volatility…A short-term agent may react differently to fluctuations in volatility compared to a medium- or long-term investor. The HAR-RV model is an additive cascade of partial volatilities generated at different time horizons…[for example] daily, weekly, and monthly observed realized volatilities…that follows an autoregressive process…The HAR-RV approach is a more stable and accurate estimate for realized volatility.” [Antulov-Fantulin and Rodikov]\n\n## Neural networks: basics and key types for financial markets\n\n### The very basics\n\n“A neural network is an adaptive system that learns by using interconnected nodes or neurons in a layered structure that resembles a human brain. A neural network can learn from data—so it can be trained to recognize patterns, classify data, and forecast future events.” [MathWorks]\n\n“Neural Networks consist of artificial neurons that are similar to the biological model of neurons. It receives data input and then combines the input with its internal activation state as well as with an optional threshold activation function. Then by using an output function, it produces the output.” [hackr.io]\n\nNeural networks consist of layers, i.e. sets of nodes or neurons. There is typically an input layer, an output layer, and a number of hidden layers in between. A neuron is loosely a function that returns a number between 0 and 1. The number returned by the neuron is called its activation. For example, the neurons of an input layer could be the pixels of an image and the numbers could denote their brightness.\nWithin a network, activations in one layer determine the activations in the next layer. The activation of a neuron is governed by a specific weighting function that takes as arguments all the activations of the previous layer. It is typically a function of a weighted sum that ensures that activations are always between 0 and 1, such as a sigmoid or rectified linear unit function. The function also uses a bias parameter, whose value determines a threshold that the weighted sum must exceed to activate meaningfully.\nLearning means with neural networks finding weights and biases that are appropriate for solving the problem at hand, using training data. The main method by which neural networks learn is gradient descent: parameters are set to minimize the average cost of errors, typically the squared differences between the estimated values in the output layer and the actual labels. The learning algorithm finds that minimum by starting with a random parameter set and then sequentially changing parameters in the direction that reduces their costs most.\n\n#### Types of neural networks for financial markets\n\nRecurrent neural networks (RNN) are a class of neural networks that is powerful for modeling sequence data such as time series…Schematically, a RNN layer uses a for loop to iterate over the timesteps of a sequence, while maintaining an internal state that encodes information about the timesteps it has seen so far.” [TensorFlow]\n\nRNNs are designed to model sequenced data. A sequence is an order of states. Examples are text, audio, or time series. RNNs fulfill this function through sequential memory, which makes it easy to recognize sequential patterns. It uses a looping mechanism (simple ‘for’ loop in code) that allows information to flow from one hidden state to the next. Only after the sequential information has all been passed to the hidden layer the hidden state is passed on and the output layer is activated.\nRNNs have a short-term memory issue. This means as steps are added to the loop the RNN struggles to retain the information of previous steps. This is caused by the “vanishing gradient” problem of backpropagation. Adjustments of parameters based on errors of the output layer decrease with each layer backward. Gradients shrink exponentially as the algorithm backpropagates down, for example moving backward through timestamps. Put simply, the earlier layers fail to do any learning and long-range dependencies are being neglected.\n\nTwo specialized recurrent neural networks have been developed to mitigate short-term memory: LSTM (Long Short-Term Memory) and GRU (Gated Recurrent Unit). They work like RNNs but are capable of learning long-term dependencies by using “gates”. The gates are tensor operations that learn what dependencies should be added to the hidden state.\n\n“An LSTM network is a type of recurrent neural network (RNN) that can learn long-term dependencies between time steps of sequence data.” [MathWorks]\n\n“An LSTM has a similar control flow as a recurrent neural network. It processes data passing on information as it propagates forward. The differences are the operations within the LSTM’s cells…\nThe core concept of LSTMs is the cell state, and its various gates. The cell state act as a transport highway that transfers relative information all the way down the sequence chain. You can think of it as the ‘memory’ of the network. The cell state, in theory, can carry relevant information throughout the processing of the sequence. So even information from the earlier time steps can make its way to later time steps, reducing the effects of short-term memory. As the cell state goes on its journey, information gets added or removed to the cell state via gates. The gates are different neural networks that decide which information is allowed on the cell state. The gates can learn what information is relevant to keep or forget during training…\nWe have three different gates that regulate information flow in an LSTM cell. A forget gate, input gate, and output gate…The forget gate…decides what information should be thrown away or kept…The input gate…updates the cell state… The output gate decides what the next hidden state should be.” [Michael Phi]\n\n“The Gated Recurrent Unit (GRU) is the younger sibling of the more popular Long Short-Term Memory (LSTM) network, and also a type of Recurrent Neural Network (RNN). Just like its sibling, GRUs are able to effectively retain long-term dependencies in sequential data.” [Loye]\n\n“To solve the vanishing gradient problem of a standard RNN, GRU uses, so-called, update gate and reset gate. Basically, these are two vectors which decide what information should be passed to the output. The special thing about them is that they can be trained to keep information from long ago, without washing it through time or remove information which is irrelevant to the prediction.”[Kostadinov]\n\nGRU is a lightweight version of LTSM where it combines long-term and short-term memory into its hidden state. Thus, while LSTM has cell states and hidden states, GRU only has hidden states. Thus, GRU only has two gates: an update gate (that decides how much of past memory to retain) and a reset gate (that decides how much of past memory to forget). Retaining and forgetting are different actions, i.e. different modes of manipulating past information.\n\n### Application of neural networks for volatility forecasting\n\n“We study and analyze various non-parametric machine learning models for forecasting multi-asset intraday and daily volatilities by using high-frequency data from the U.S. equity market. We demonstrate that, by taking advantage of commonality in intraday volatility, the model’s forecasting performance can significantly be improved…A measure for evaluating the commonality in intraday volatility is proposed, that is the adjusted R-squared value from linear regressions of a given stock’s realized volatility against the market realized volatility…Commonality over the daily horizon is turbulent over time, although commonality in intraday realized volatilities is strong and stable…For most models, the incorporation of commonality leads to better out-of-sample performance through pooling data together and adding market volatility as additional features.” [Zhang et al]\n\nNeural networks are in general, superior to other techniques [reflecting] the capability of neural networks for handling complex interactions among predictors… The high-dimensional nature of ML methods allows for better approximations to unknown and potentially complex data-generating processes, in contrast with traditional economic models…Furthermore, to alleviate the concerns of overfitting, we conduct a stringent out-of-sample test, using the existent trained models to forecast the volatility of completely new stocks that are not included in the training sample. Our results reveal that neural networks still outperform other approaches (including the OLS models trained for each new stock).” [Zhang et al]\n\n“We investigate whether a totally nonparametric model is able to outperform econometric methods in forecasting realized volatility. In particular, the analysis …compares the forecasting accuracy of time series models with several neural networks architectures…The data set employed in this study comprises…observations from February 1950 to December 2017 of the Standard & Poor’s (S&P) index…The latent volatility is estimated through the ex-post measurement of volatility based on high-frequency data, namely realized volatility…computed as the sum of squared daily returns…Recurrent neural networks are able to outperform all the traditional econometric methods. Additionally, capturing long-range dependence through LSTM seems to improve the forecasting accuracy also in a highly volatile period.” [Bucci]\n\n“We have applied a Long Short-Term Memory neural network to model S&P 500 volatility, incorporating Google domestic trends as indicators of the public mood and macroeconomic factors…This work shows the potential of deep learning financial time series in the presence of strong noise [and holds] strong promise for better predicting stock behavior via deep learning and neural network models.”[Xiong, Nichols and Shen]\n\n“This study investigates the strengths and weaknesses of machine learning models for realised volatility forecasting of 23 NASDAQ stocks over the period from 2007 to 2016. Three types of daily data are used, variables used in the HAR-family of models, limit order book variables and news sentiment variables…Using a Long-Short-Term-Memory (LSTM) model combined with…four sets of variables each with 21 lags are trained with the loss function of minimising mean squared erors. These experiments provide strong evidence for the stronger forecasting power of machine learning models than all HAR-family of models.” [Rahimikia and Poon]\n\n“The volatility prediction task is of non-trivial complexity due to noise, market microstructure, heteroscedasticity, exogenous and asymmetric effect of news, and the presence of different time scales, among others…We studied and analyzed how neural networks can learn to capture the temporal structure of realized volatility. We…implement Long Short Term Memory (LTSM) and…Gated Recurrent Unit (GRU). Machine learning can approximate any linear and non-linear behavior and…learn data structure…We investigated the approach with LSTM and GRU types for realized volatility forecasting tasks and compared the predictive ability of neural networks with widely used EWMA, HAR, GARCH-family models… LSTM outperformed well-known models in this field, such as HAR-RV. Out-of-sample accuracy tests have shown that LSTM offers significant advantages in both types of markets.” [Antulov-Fantulin and Rodikov]" ]

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[ "", null, "Algebra 2 Homework Help | Algebra 2 Tutor | High School Algebra 2\n\n## Basic concepts in Algebra 2:\n\nAny function in the form of ax2+bx+c=0, where a should not be equal to zero is known as quadratic function. It has its degree as 2, which means highest power of a term in that expression should be 2.\n\n### Probability: -\n\nIn order to find the possibility or chance or surety of an event we use the concept of probability. It is used to find the chances of occurrence of an event and value of that chance is always in between 0 and 1.\n\nNotation of Probability is PA= number of favourable outome/tatal number of outcomes\n\n### Trigonometry: -\n\nStudy of the measure of the three sides and their relationship with angles of triangle. In most of the cases trigonometry is applied in only right-angle triangle. sinθ, cosθ, tanθ, etc are some trigonometric ratio.\n\n## Tips for solving algebra 2 problems - Sample:\n\nTopic: The complex plane\n\n(Easy)\n\nQuestion: ​​​​The given complex number is,", null, ", draw the complex number on the complex plane.\n\nExplanation: A complex number of form", null, "means, it will be", null, "unit on real axis, and", null, "unit on the imaginary axis.\n\nThe horizontal axis on the complex plane is the real axis, and the vertical axis is the imaginary axis.\n\nGraph:\n\nThe complex number", null, "would be 2 units on the real axis, and 2 units on the imaginary axis.\n\nThe location of the complex number would be:\n\n##", null, "Topic: Adding and subtracting complex number\n\n(Medium)\n\nQuestion:  The given complex number", null, "and,", null, ", Find the sum of complex numbers such that", null, ". And show them on a complex plane as a vector.\n\nExplanation: To add the complex numbers, we need to add the real part and imaginary part separately.\n\nCalculation: Find", null, "", null, "Graph: The vector connecting the location of point from origin, would represent the corresponding vectors.\n\n##", null, "Topic: Multiplying complex number\n\n(Hard)\n\nQuestion: The given complex number", null, "and,", null, ", Find the multiplication of complex numbers such that", null, ". And show them on complex plane as a vector.\n\nExplanation: We can multiply the complex number using distribution. Also, we know", null, "Calculation: Find", null, "", null, "Plug", null, "", null, "Graph: Plot each vector on the complex plane.\n\n## Fast and Reliable Help For High School Algebra 2\n\nAlgebra 2 is the third math course introduced in high school after algebra 1 course. It is also known or referred as intermediate algebra and college algebra. Algebra 2 is considered as a core math course and most of the states in united states including Washington, D.C. has made Algebra 2 a required course for high school graduation.\n\nNormally after algebra 2, Students can move to other advanced math concepts such as pre-calculus and then calculus. Algebra 2 questions are also a part of the SAT exam syllabus.\n\nAlgebra 2 has a lot of real-life applications and solving algebra 2 problems can become easy if teachers connect the problems with real world situations. For example - Exponential equations are used widely in day-to-day life for solving savings or loan situations.\n\nMany experts in the field of Math state that high school algebra 2 concepts can help students excel in their career and perform complex tasks on the job with ease even in situations when their job descriptions are not math related.\n\nAlgebra 2 is advanced math and may be considered a hard course by high school students due to the complexity of concepts taught in the course. But, Algebra 2 is easy as long as students keep up with their daily Algebra 2 homework and have conceptual clarity.\n\nA lot of times free online Algebra 2 homework help may seem like a good idea but in the long run lack of expert guidance from subject matter experts can lead to bad command of Algebra 2 for high school students.\n\nTake homework help from seasoned Algebra 2 tutors and get transported to a mathematical world of equations. With personalized and detailed solutions to your algebra 2 problems our experts will surely help you conquer the concepts. Our step-by-step explanations and detailed instructions have helped a lot of students around the world.\n\nOur tutors can help you meet a deadline for your Algebra 2 assignment. Get insights on figuring out the Algebra 2 problems and easy hacks to a difficult question. To make studying simple for you, we provide 24/7 written help. Prepare with the best in the field for further advanced work in mathematics.\n\nTutorEye will make you smarter, sharper, and better prepared for the upcoming tests and exams. Get ready for better grades today!\n\nStep 1: Fill in your homework request details on our quick-help form\n\nStep 2: Hire a tutor after duly considering your budget and submission deadline\n\nStep 3: Check the final assignment before releasing money from escrow\n\nOur homework solvers consider each question and provide assignments that are of the highest quality and are 100% original. Meet the top helpers who have devoted their careers helping thousands of students today!\n\n## Top Algebra 2 Homework Help Questions:\n\n### What is algebra 2?\n\nThe third Math course in high school is called Algebra 2. It builds on knowledge and skills used while studying Algebra 1. It is taught in 11th grade and focuses on exponentials and logarithmic functions. Students also call it “College Algebra”.\n\nIt covers the following topics:\n\n• Complex numbers\n• Polynomial graphs\n• Logarithms\n• Modeling\n\nTherefore, a student must be ready to seek additional study aid from our experts at TutorEye to succeed in this course.\n\n### How hard is algebra 2?\n\nOnly if you are not clear with your basics of Algebra 1, you will find Algebra 2 quite hard. As the foundation of this course is built upon the principles you learnt in Algebra 1, you need to have a solid base as such. But you will not find it tough if you revise old topics and continue to expound on them while solving new problems.\nBesides, you can always take Algebra 2 help from top professionals at TutorEye to figure out where you should focus to make this course easier for you.\n\n### What do you learn in algebra 2?\n\nStudents learn a lot in Algebra 2 specifically they get well-versed in tools modelling the real world. Here your understanding of functions and real numbers is extended.\n\nThe learning outcomes include:\n\n• Evaluate different problems\n• Factor a polynomial function\n• Analyze a function rule\n• Solve expressions\n\nAlso by doing your assignments you know what topics need more time and effort. Seek algebra 2 homework help at TutorEye to no longer struggle in the subject,\n\n### What does algebra 2 cover?\n\nAlgebra 2 covers the following topics:\n\n• Complex numbers\n• Exponential models\n• Polynomial division and factorization\n• Rational exponents\n\nYou learn how to do more complex factoring that’s why you must not shy away from taking algebra 2 help from us.\n\n### How to pass algebra 2?\n\nTo pass in Algebra 2 you must follow these steps:\n\n• Focus on conceptual understanding\n• Relate to real life activities" ]

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[ "# #00c5b8 Color Information\n\nIn a RGB color space, hex #00c5b8 is composed of 0% red, 77.3% green and 72.2% blue. Whereas in a CMYK color space, it is composed of 100% cyan, 0% magenta, 6.6% yellow and 22.7% black. It has a hue angle of 176 degrees, a saturation of 100% and a lightness of 38.6%. #00c5b8 color hex could be obtained by blending #00ffff with #008b71. Closest websafe color is: #00cccc.\n\n• R 0\n• G 77\n• B 72\nRGB color chart\n• C 100\n• M 0\n• Y 7\n• K 23\nCMYK color chart\n\n#00c5b8 color description : Strong cyan.\n\n# #00c5b8 Color Conversion\n\nThe hexadecimal color #00c5b8 has RGB values of R:0, G:197, B:184 and CMYK values of C:1, M:0, Y:0.07, K:0.23. Its decimal value is 50616.\n\nHex triplet RGB Decimal 00c5b8 `#00c5b8` 0, 197, 184 `rgb(0,197,184)` 0, 77.3, 72.2 `rgb(0%,77.3%,72.2%)` 100, 0, 7, 23 176°, 100, 38.6 `hsl(176,100%,38.6%)` 176°, 100, 77.3 00cccc `#00cccc`\nCIE-LAB 71.819, -43.419, -5.131 28.615, 43.39, 52.212 0.23, 0.349, 43.39 71.819, 43.721, 186.739 71.819, -56.899, -1.19 65.871, -37.733, -0.885 00000000, 11000101, 10111000\n\n# Color Schemes with #00c5b8\n\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #c5000d\n``#c5000d` `rgb(197,0,13)``\nComplementary Color\n• #00c556\n``#00c556` `rgb(0,197,86)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #0070c5\n``#0070c5` `rgb(0,112,197)``\nAnalogous Color\n• #c55600\n``#c55600` `rgb(197,86,0)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #c50070\n``#c50070` `rgb(197,0,112)``\nSplit Complementary Color\n• #c5b800\n``#c5b800` `rgb(197,184,0)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #b800c5\n``#b800c5` `rgb(184,0,197)``\n• #0dc500\n``#0dc500` `rgb(13,197,0)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #b800c5\n``#b800c5` `rgb(184,0,197)``\n• #c5000d\n``#c5000d` `rgb(197,0,13)``\n• #007971\n``#007971` `rgb(0,121,113)``\n• #009288\n``#009288` `rgb(0,146,136)``\n• #00aca0\n``#00aca0` `rgb(0,172,160)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #00dfd0\n``#00dfd0` `rgb(0,223,208)``\n• #00f8e8\n``#00f8e8` `rgb(0,248,232)``\n• #13ffef\n``#13ffef` `rgb(19,255,239)``\nMonochromatic Color\n\n# Alternatives to #00c5b8\n\nBelow, you can see some colors close to #00c5b8. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #00c587\n``#00c587` `rgb(0,197,135)``\n• #00c597\n``#00c597` `rgb(0,197,151)``\n• #00c5a8\n``#00c5a8` `rgb(0,197,168)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #00c2c5\n``#00c2c5` `rgb(0,194,197)``\n• #00b1c5\n``#00b1c5` `rgb(0,177,197)``\n• #00a1c5\n``#00a1c5` `rgb(0,161,197)``\nSimilar Colors\n\n# #00c5b8 Preview\n\nThis text has a font color of #00c5b8.\n\n``<span style=\"color:#00c5b8;\">Text here</span>``\n#00c5b8 background color\n\nThis paragraph has a background color of #00c5b8.\n\n``<p style=\"background-color:#00c5b8;\">Content here</p>``\n#00c5b8 border color\n\nThis element has a border color of #00c5b8.\n\n``<div style=\"border:1px solid #00c5b8;\">Content here</div>``\nCSS codes\n``.text {color:#00c5b8;}``\n``.background {background-color:#00c5b8;}``\n``.border {border:1px solid #00c5b8;}``\n\n# Shades and Tints of #00c5b8\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #000101 is the darkest color, while #ecfffe is the lightest one.\n\n• #000101\n``#000101` `rgb(0,1,1)``\n• #001413\n``#001413` `rgb(0,20,19)``\n• #002825\n``#002825` `rgb(0,40,37)``\n• #003c38\n``#003c38` `rgb(0,60,56)``\n• #004f4a\n``#004f4a` `rgb(0,79,74)``\n• #00635c\n``#00635c` `rgb(0,99,92)``\n• #00776f\n``#00776f` `rgb(0,119,111)``\n• #008a81\n``#008a81` `rgb(0,138,129)``\n• #009e93\n``#009e93` `rgb(0,158,147)``\n• #00b1a6\n``#00b1a6` `rgb(0,177,166)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\n• #00d9ca\n``#00d9ca` `rgb(0,217,202)``\n• #00ecdd\n``#00ecdd` `rgb(0,236,221)``\n• #01ffee\n``#01ffee` `rgb(1,255,238)``\n• #14fff0\n``#14fff0` `rgb(20,255,240)``\n• #28fff1\n``#28fff1` `rgb(40,255,241)``\n• #3cfff2\n``#3cfff2` `rgb(60,255,242)``\n• #4ffff3\n``#4ffff3` `rgb(79,255,243)``\n• #63fff5\n``#63fff5` `rgb(99,255,245)``\n• #77fff6\n``#77fff6` `rgb(119,255,246)``\n• #8afff7\n``#8afff7` `rgb(138,255,247)``\n• #9efff9\n``#9efff9` `rgb(158,255,249)``\n• #b1fffa\n``#b1fffa` `rgb(177,255,250)``\n• #c5fffb\n``#c5fffb` `rgb(197,255,251)``\n• #d9fffc\n``#d9fffc` `rgb(217,255,252)``\n• #ecfffe\n``#ecfffe` `rgb(236,255,254)``\nTint Color Variation\n\n# Tones of #00c5b8\n\nA tone is produced by adding gray to any pure hue. In this case, #5b6a69 is the less saturated color, while #00c5b8 is the most saturated one.\n\n• #5b6a69\n``#5b6a69` `rgb(91,106,105)``\n• #537270\n``#537270` `rgb(83,114,112)``\n• #4c7976\n``#4c7976` `rgb(76,121,118)``\n• #44817d\n``#44817d` `rgb(68,129,125)``\n• #3d8883\n``#3d8883` `rgb(61,136,131)``\n• #35908a\n``#35908a` `rgb(53,144,138)``\n• #2d9891\n``#2d9891` `rgb(45,152,145)``\n• #269f97\n``#269f97` `rgb(38,159,151)``\n• #1ea79e\n``#1ea79e` `rgb(30,167,158)``\n• #17aea4\n``#17aea4` `rgb(23,174,164)``\n• #0fb6ab\n``#0fb6ab` `rgb(15,182,171)``\n• #08bdb1\n``#08bdb1` `rgb(8,189,177)``\n• #00c5b8\n``#00c5b8` `rgb(0,197,184)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #00c5b8 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "# Video: Using the Sine Rule to Calculate an Unknown Length in a Triangle\n\nIn triangle 𝐴𝐵𝐶, 𝐴𝐶 = 97 m, 𝑚∠𝐵𝐴𝐶 = 101°, and 𝑚∠𝐴𝐶𝐵 = 53°. Determine the length of 𝐴𝐵 to the nearest meter.\n\n02:16\n\n### Video Transcript\n\nIn triangle 𝐴𝐵𝐶, 𝐴𝐶 is equal to 97 meters, the measure of the angle 𝐵𝐴𝐶 is equal to 101 degrees, and the measure of the angle 𝐴𝐶𝐵 is equal to 53 degrees. Determine the length of 𝐴𝐵 to the nearest meter.\n\nIt can be really useful to sketch a diagram in these sorts of scenarios. It’ll allow you to identify the type of question it is and what you will need to use to be able to solve it. Now whilst your diagram does not need to be to scale, it should be roughly in proportion to prevent any mistakes in your calculations.\n\nHere, we have a non-right-angled triangle with two angles and one side given. Notice we don’t have any matching angle and side pairs which is necessary for us to be able to use the law of sines. We do know, however, that we don’t need to use the law of cosines as we simply don’t have enough sides labelled.\n\nWe’ll need to use the fact that angles in a triangle add to 180 degrees to work out the measure of the angle 𝐴𝐵𝐶. We can subtract the sum of the given angles from 180 degrees to get the measure of the angle 𝐴𝐵𝐶 to be 26 degrees. Now that we have the measure of the angle at 𝐵, we can use the law of sines. Remember we don’t actually need to use all three parts of this equation.\n\nLabelling our triangle and remembering that the side opposite an angle is given by its lowercase counterpart, we can see that we only need 𝑏 over sin 𝐵 is equal to 𝑐 over sin 𝐶. Substituting our given values into this formula, we get 97 over sin 26 is equal to 𝑐 over sin 53.\n\nWe can solve this equation by multiplying both sides by sin 53 to give us sin 53 multiplied by 97 over sin 26 is equal 𝑐. Popping this into our calculator, we get 𝑐 to be 176.717.\n\nThe length of 𝐴𝐵 is 177 meters correct to the nearest meter." ]

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[ "## Example Questions\n\n### Example Question #1 : Linear / Rational / Variable Equations\n\nFind the solution to the following equation if x = 3:\n\ny = (4x2 - 2)/(9 - x2)\n\n6\n\nno possible solution\n\n3\n\n0\n\nno possible solution\n\nExplanation:\n\nSubstituting 3 in for x, you will get 0 in the denominator of the fraction. It is not possible to have 0 be the denominator for a fraction so there is no possible solution to this equation.\n\n### Example Question #1 : How To Find Out When An Equation Has No Solution", null, "I.  x = 0\n\nII. x = –1\n\nIII. x = 1\n\nI, II, and III\n\nII only\n\nIII only\n\nI only\n\nII and III only\n\nI only\n\nExplanation:", null, "### Example Question #1 : How To Find Out When An Equation Has No Solution", null, "–1/2\n\n3\n\n1\n\n–3\n\nThere is no solution\n\nThere is no solution\n\nExplanation:", null, "### Example Question #42 : Gre Quantitative Reasoning", null, "", null, "", null, "", null, "", null, "", null, "", null, "Explanation:\n\nA fraction is considered undefined when the denominator equals 0. Set the denominator equal to zero and solve for the variable.", null, "", null, "### Example Question #1 : How To Find Out When An Equation Has No Solution\n\nSolve:", null, "", null, "", null, "", null, "", null, "", null, "", null, "Explanation:\n\nFirst, distribute, making sure to watch for negatives.", null, "", null, "Combine like terms.", null, "Subtract 7x from both sides.", null, "", null, "### Example Question #44 : Algebra\n\nSolve:", null, "", null, "", null, "Infinitely Many Solutions", null, "No Solution\n\nNo Solution\n\nExplanation:\n\nFirst, distribute the", null, "to the terms inside the parentheses.", null, "", null, "", null, "This is false for any value of", null, ". Thus, there is no solution.\n\n### Example Question #45 : Algebra\n\nSolve", null, ".\n\nNo solutions", null, "", null, "", null, "", null, "No solutions\n\nExplanation:\n\nBy definition, the absolute value of an expression can never be less than 0. Therefore, there are no solutions to the above expression.\n\n### Example Question #111 : Algebra", null, "In the above graphic, approximately determine the x values where the graph is neither increasing or decreasing.", null, "", null, "", null, "", null, "", null, "", null, "We need to find where the graph's slope is approximately zero. There is a straight line between the x values of", null, ", and", null, ". The other x values have a slope. So our final answer is", null, ".", null, "" ]

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[ "# G-Delta Sets Closed under Intersection\n\n## Theorem\n\nLet $T = \\struct {S, \\tau}$ be a topological space.\n\nLet $G, G'$ be $G_\\delta$ sets of $T$.\n\nThen their intersection $G \\cap G'$ is also a $G_\\delta$ set of $T$.\n\n## Proof\n\nBy definition of $G_\\delta$ set, there exist sequences $\\sequence {U_n}_{n \\mathop \\in \\N}$ and $\\sequence {U'_n}_{n \\mathop \\in \\N}$ of open sets of $T$ such that:\n\n$G = \\displaystyle \\bigcap_{n \\mathop \\in \\N} U_n$\n$G' = \\displaystyle \\bigcap_{n \\mathop \\in \\N} U'_n$\n\nBy General Distributivity of Intersection, we have:\n\n$G \\cap G' = \\displaystyle \\bigcap_{n \\mathop \\in \\N} \\paren {U_n \\cap U'_n}$\n\nBy Intersection of Closed Sets is Closed, $U_n \\cap U'_m$ is closed, for all $n, m \\in \\N$.\n\nThus $G \\cap G'$ is seen to be a $G_\\delta$ set.\n\n$\\blacksquare$" ]

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[ "# Arduino Bluetooth RGB led control using Android app", null, "RGB LEDs play a large part in applications like outdoor decoration lighting in cities, stage lighting designs, home decoration lighting and LED display matrix. Recently, RGB color mixing technology can also be found in LCD backlighting and projectors.\n\nIn this tutorial we will learn how the RGB LED works and how we can control RGB led using HC-05 Bluetooth module, Arduino IDE and an Android application.\n\nBefore proceeding, you can make reference from my other tutorial on how to use the HC-05 Bluetooth module with Arduino from the link below;\n\n• How to use HC-05 Bluetooth module with Arduino.\n• ## What is RGB LED?\n\nRGB LED is actually three LEDs, red, green, and blue inside one package. Three PWM Outputs are used to control the RGB LED. A PWM value of 0.0 would be off and a 1.0 full on for each color LED. This allows a program to vary both the color and brightness level of the LED.\n\nTypically an RGB LED has four pins. One common pin and one for each of the three LEDs. In the LED seen below, the common pin is the longest pin. The RGB LED can be classified as either common anode or common cathode. The common pin in the common anode RGB LED is connected to VCC while in the common cathode it is connected to the ground.\n\nNOTE: Red, Green and Blue are called primary colors and by mixing each other with different intensity we may get millions of color outcomes. This is the basic concept of the RGB led.\n\n## Connecting the RGB LED and HC-05 Bluetooth module to Arduino\n\n• VCC – to VCC of Arduino.\n• GND – to GND of Arduino.\n• RX – to digital pin 0(TX pin) of Arduino.\n• TX – to digital pin 1(RX pin) of Arduino\n• .\n\nNote:\ni).Connect RX and TX pins after uploading the code\nii).The terminals of the RGB LED should be connected to PWM pins of the Arduino.\n\n### CODE\n\nThis code is mainly for changing the PWM for simulating analog output which will provide different voltage levels to the LEDs so we can get the desired colors. These values can vary from 0 to 255 which represents 100 % duty cycle of the PWM signal or maximum LED brightness.\n\nThe RGB led will give the color corresponding to the value determined by the color wheel of the Android application.\n\n``````#include <SoftwareSerial.h>\n#include <Wire.h>\nSoftwareSerial mySerial(0,1); // RX and TX pins\nint PIN_RED = 9;\nint PIN_GREEN = 10;\nint PIN_BLUE = 11;\nString RGB = \"\";\nString RGB_Previous = \"255.255.255\";\nString ON = \"ON\";\nString OFF = \"OFF\";\nboolean RGB_Completed = false;\nvoid setup()\n{\npinMode (PIN_RED, OUTPUT);\npinMode (PIN_GREEN, OUTPUT);\npinMode (PIN_BLUE, OUTPUT);\nSerial.begin(9600);\nmySerial.begin(9600);\nRGB.reserve(30);\n}\nvoid loop()\n{\nwhile(mySerial.available())\n{\n{\nRGB_Completed = true;\n}else{\n}\n}\nif(RGB_Completed)\n{\nSerial.print(\"RGB:\");\nSerial.print(RGB);\nSerial.print(\" PreRGB:\");\nSerial.println(RGB_Previous);\nif(RGB==ON)\n{\nRGB = RGB_Previous;\nLight_RGB_LED();\n}\nelse if(RGB==OFF)\n{\nRGB = \"0.0.0\";\nLight_RGB_LED();\n}else{\nLight_RGB_LED();\nRGB_Previous = RGB;\n}\nRGB = \"\";\nRGB_Completed = false;\n}\n}\nvoid Light_RGB_LED()\n{\nint SP1 = RGB.indexOf(' ');\nint SP2 = RGB.indexOf(' ', SP1+1);\nint SP3 = RGB.indexOf(' ', SP2+1);\nString R = RGB.substring(0, SP1);\nString G = RGB.substring(SP1+1, SP2);\nString B = RGB.substring(SP2+1, SP3);\nSerial.print(\"R=\");\nSerial.println( constrain(R.toInt(),0,255));\nSerial.print(\"G=\");\nSerial.println(constrain(G.toInt(),0,255));\nSerial.print(\"B=\");\nSerial.println( constrain(B.toInt(),0,255));\nanalogWrite(PIN_RED, (R.toInt()));//comment if colors are inverted\nanalogWrite(PIN_GREEN, (G.toInt()));//and uncomment part below.\nanalogWrite(PIN_BLUE, (B.toInt()));\n// analogWrite(PIN_RED, (255-R.toInt()));//uncomment if colors are inverted\n// analogWrite(PIN_GREEN, (255-G.toInt()));//and comment above part.\n// analogWrite(PIN_BLUE, (255-B.toInt()));\n}\n``````\n\n## Android App used for Arduino RGB led Bluetooth control\n\nFor this project we use Arduino RGB Led Control app which is got from the Google play store. This Application enables us to connect the phone and the Arduino board through Bluetooth.\n\nYou can get the Application from here" ]

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[ "Electrical Engineering - Circuit Theorems and Conversions\n\nWhy Electrical Engineering Circuit Theorems and Conversions?\n\nIn this section you can learn and practice Electrical Engineering Questions based on \"Circuit Theorems and Conversions\" and improve your skills in order to face the interview, competitive examination and various entrance test (CAT, GATE, GRE, MAT, Bank Exam, Railway Exam etc.) with full confidence.\n\nWhere can I get Electrical Engineering Circuit Theorems and Conversions questions and answers with explanation?\n\nIndiaBIX provides you lots of fully solved Electrical Engineering (Circuit Theorems and Conversions) questions and answers with Explanation. Solved examples with detailed answer description, explanation are given and it would be easy to understand. All students, freshers can download Electrical Engineering Circuit Theorems and Conversions quiz questions with answers as PDF files and eBooks.\n\nWhere can I get Electrical Engineering Circuit Theorems and Conversions Interview Questions and Answers (objective type, multiple choice)?\n\nHere you can find objective type Electrical Engineering Circuit Theorems and Conversions questions and answers for interview and entrance examination. Multiple choice and true or false type questions are also provided.\n\nHow to solve Electrical Engineering Circuit Theorems and Conversions problems?\n\nYou can easily solve all kind of Electrical Engineering questions based on Circuit Theorems and Conversions by practicing the objective type exercises given below, also get shortcut methods to solve Electrical Engineering Circuit Theorems and Conversions problems.\n\nExercise :: Circuit Theorems and Conversions - General Questions\n\n1.\n\nFind the Thevenin equivalent (VTH and RTH) between terminals A and B of the circuit given below.", null, "A. 4.16 V, 120", null, "B. 41.6 V, 120", null, "C. 4.16 V, 70", null, "D. 41.67 V, 70", null, "Explanation:\n\nNo answer description available for this question. Let us discuss.\n\n2.\n\nA certain current source has the values IS = 4 µA and RS = 1.2 M", null, ". The values for an equivalent voltage source are\n\n A. 4.8", null, "V, 1.2 M", null, "B. 1 V, 1.2 M", null, "C. 4.8 V, 4.8 M", null, "D. 4.8 V, 1.2 M", null, "Explanation:\n\nNo answer description available for this question. Let us discuss.\n\n3.\n\nFind the total current through R3 in the given circuit.", null, "A. 7.3 mA B. 5.5 mA C. 12.8 mA D. 1.8 mA\n\nExplanation:\n\nNo answer description available for this question. Let us discuss.\n\n4.\n\nA 680", null, "load resistor, RL, is connected across a constant current source of 1.2 A. The internal source resistance, RS, is 12 k", null, ". The load current, RL, is\n\n A. 0 A B. 1.2 A C. 114 mA D. 1.14 A\n\nExplanation:\n\nNo answer description available for this question. Let us discuss.\n\n5.\n\nFind the current through R2 of the given circuit.", null, "A. 30.7 mA B. 104 mA C. 74 mA D. 134 mA" ]

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[ "# Nnn12th maths differential calculus book pdf\n\nCalculus of variations and partial di erential equations. Fundamental rules for differentiation, tangents and normals, asymptotes, curvature, envelopes, curve tracing, properties of special curves, successive differentiation, rolles theorem and taylors theorem, maxima and minima, indeterminate forms. Free kindle book and epub digitized and proofread by project gutenberg. This book is meant for students preparing for the b.\n\nSeparable differential equations are differential equations which respect one of the following forms. Introduction to calculus differential and integral calculus. Differential calculus by shanti narayan download link. Location if not on main campus at diriya, main campus.\n\nMadas question 3 differentiate the following expressions with respect to x a y x x. Introduction differential calculus maths reference. Medicine calculus for life sciences series calculus study and solutions guide volume ii to accompany calculus w analytic geometry bundle. In fact, i read it more like a novel than a study material, because the content is suprising entertaining. This is an old calculus book which was developed for students in technical schools in the soviet union. We have tried to survey a wide range of techniques and problems, discussing, both classical results as well as more recent techniques and problems. This is a self contained set of lecture notes for math 221.\n\nDifferential calculus by shanti narayan pdf free download. Calculus produces functions in pairs, and the best thing a book can do early is to. Differential calculus bsc 1st year maths solution of. For help maths wizard virtual classroom offers direct oneonone live online tuition for learnersstudents for all grades. Differential equations calculus mathematics ebook payhip. This course is a sequel to math 2ll for students, primarily in social sciences and finance, who need to develop more techniques than are covered in math 211. Class 12 is a turning point of a students life, after which they choose their career or profession. Derivative exponentials natural logarithms,calculus. Most functions considered in mathematics are described by an equation like. To have a clear and better understanding of the topics, there are also four solved exercises at the end of the chapter. Advanced calculus harvard mathematics harvard university. Free differential calculus books download ebooks online. Alternatively, learners can attend extra maths lessons at our centre.\n\nMaths differential equation part 1 introduction cbse. A text book of differential calculus with numerous worked out examples. A text book of differential calculus with numerous worked. It is a very popular textbook among students in spain, portugal, and latin american countries. Differential calculus deals with the rate of change of one quantity with respect to another. Besides limits, derivatives, and integrals, differential euqations, all of the first of all, no one actually forced me to pick up this textbook. Limits, continuity and differentiation of real functions of one real variable, differentiation and sketching graphs using analysis. Differential equations and the calculus of variations. Graphs comparing the functions and their derivatives. Differential and integral calculus online mathematics. All ncert solutions for class 12 maths in pdf cbse xii all of these class 12 maths ncert solutions are developed as per ncert books or you can say the official textbooks of cbse 12th. The following page is inserted to help with the language of mathematics. Download upsc cse mathematics optional ebooks pdf free.\n\nHome schooling and correspondence students are all welcome. No need to wait for office hours or assignments to be graded to find out where you took a wrong turn. Modeling and application, 2nd edition, digital text published by the maa. This popular calculus text remains the shortest mainstream calculus book available yet covers all the material needed by, and at an appropriate level for, students in engineering, science, and mathematics. Herstein click here 11 modern algebra by krishna series click here 12 functions of complex variable by krishna series click here vector calculus by krishna series click here 14 fluid dynamics by m. Learn differential calculus for freelimits, continuity, derivatives, and derivative applications. Understanding basic calculus graduate school of mathematics. Conversley if we are given a differential equation of the order we can, in general, obtain an equivalent relationship containing no derivatives but n arbitrary constants.\n\nAlthough the book was first published in the seventies, its emphasis on qualitative aspects is in agreement with more recent trends in. The 10th chapter of the ncert books for class 12 maths covers a few important topics like applying vectors to figures, how to differentiate vectors and scalars, functions on vectors, among the many others. Calculus and ordinary differential equations at the university of hong kong. Calculus and ordinary differential equations 1st edition. Or you can consider it as a study of rates of change of quantities. The result would be a differential equation of the order. Each section of the book contains readthrough questions. Grade 12 extra maths help differential calculus pdf.\n\nThis text is suitable to a rst oneyear graduate course on calculus of. Purchase calculus and ordinary differential equations 1st edition. Unlike static pdf calculus with differential equations 9th edition solution manuals or printed answer keys, our experts show you how to solve each problem stepbystep. Demonstrate an understanding between graphical presentation and calculus concepts 1st, 2nd part. This is an excellent both introductory and advanced book on differential equations and the calculus of variations. Differential calculus basics definition, formulas, and. This book is based on an honors course in advanced calculus that we gave in the. Maths differential equation part 1 introduction cbse mathematics xii 12. Ncert book for class 12 maths cbse free pdf download. Access study documents, get answers to your study questions, and connect with real tutors for math 1851. The online format allows the book to take advantage of hyperlinks, electronic demonstrations, and unlimited examples via the use of a computer algebra system. Construct differential equation models from word problems and use qualitative and algebraic methods to investigate properties of the models.\n\nThe subjects that the students do in their class 11 and 12 is also the basis for their career choice or professional course they take up. Full semester subject notes for math1001 from a high distinction student. Differential calculus, branch of mathematical analysis, devised by isaac newton and g. Degree of a differential equation is the highest power exponent of the highest order derivative in it. He will score cent percent marks if he works according to a perfect plan. Every student heartily wishes to show his mettle in 11th class and 12th class. Problem and theory books for sixth term examination paper. The student must not simply get the answers by heart. Calculus is the branch of mathematics that deals with continuous change in this article, let us discuss the calculus definition, problems and the application of calculus in detail. Numerical analysis by cheney and kincaid is a pretty good survey of numerical mathematics in general, and includes the basics of ode and pde solution at a beginning graduate level introduction to numerical methods in differential equations by holmes is a more focused text, and as such is much shorter. Examples with separable variables differential equations this article presents some working examples with separable differential equations. Consider a mapping n from a neighborhood of zero in v to a neighborhood of zero in v if n0 0 and if n is continuous at 0, then we can say, intuitively, that nv approaches 0 in v. Books pin buy skills in mathematics differential calculus for jee main.\n\nElementary illustrations of the differential and integral. The derivative fx is the gradient of the tangent of the graph of f at the point x in a straight line graph the derivative is the gradient of the graph. I am looking for algebra, calculus and coordinate geometry books with a lot of problems of similar difficulty to the problems i have posted below. Its conciseness and clarity helps students focus on, and understand, critical concepts in calculus without them getting bogged down and. This book has been designed to meet the requirements of undergraduate students of ba and bsc courses. All the numbers we will use in this first semester of calculus are. Accompanying the pdf file of this book is a set of mathematica notebook. Ncert solutions for class 12 maths pdf download 100% free. How to find the derivative of exponential and logarithmic functions. Prerequisites for this course if any differential calculus math150 7. Introduction to differential calculus university of sydney. Calculus with differential equations, 9th edition pearson.\n\nThe currently prevailing emphasis in differential calculus on the derivative at the. The treatment of the subject is rigorous but no attempt has been made to state and prove the theorems in generalised forms and under less restrictive conditions. Leibniz, and concerned with the problem of finding the rate of change of a function with respect to the variable on which it depends. Mathematics 161, fall 2010, page 2 3 the use of computational power to tackle problems and applications of surprising complexity. Multivariable calculus and differential equations semester 1, 201415 1. Central to bowdoins implementation of the course is the computer application mathematica, a sophisticated integration of wordprocessing, numeric and symbolic computation, graphics, and programming. Elementary differential calculus alain schremmer department of mathematics, community college of philadelphia, philadelphia, pa 19 francesca schremmer department of mathematical sciences, west chester state university, west chester, pa 18383 0. Pdf produced by some word processors for output purposes only. It has been translated to several languages in countries with strong influence from the soviet union. Furthermore, the index of applications at the back of the book provides students. Mathematics 161, spring 20, page 2 textbooks and supplies. Mathematics learning centre, university of sydney 1 1 introduction in day to day life we are often interested in the extent to which a change in one quantity a. Differential calculus course notes high distinction written by syduni9.\n\nJohn venn 18341923 an english mathematician who studied logic and set. For example, if you own a motor car you might be interested in how much a change in the amount of. Calculus is all about the comparison of quantities which vary in a oneliner way. Now let us have a look of calculus definition, its types, differential calculus basics, formulas, problems and applications in detail. Fundamental rules for differentiation, tangents and normals, asymptotes, curvature, envelopes, curve tracing, properties of special curves, successive differentiation, rolles theorem and taylors theorem, maxima and minima. Introduction to differential calculus in the seventeenth century, sir isaac newton, an english mathematician 16421727, and gottfried wilhelm leibniz, a german mathematician 16461716, considered. The mathematics subject of this class plays a very important role in further studies. There is this book called advanced problem in mathematics by silkos which is basically selected problems from the actual examination.\n\n1168 607 1491 1038 882 1329 450 424 262 819 1491 325 1328 1089 731 17 1054 773 330 312 333 791 1097 719 1261 794 287 1501 849 596 1119 1253 1297 51 522 268 1020 1472 95 766 332 617 529 510 1310 7 921 570" ]

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[ "# Can I shorten a function name I use repeatedly?\n\nI have a long formula, like the following:\n\nfloat a = sin(b)*cos(c)+sin(c+d)*sin(d)....\n\n\nIs there a way to use s instead of sin in C, to shorten the formula, without affecting the running time?\n\n• double (*s)(double) = sin; – iBug Jan 23 at 14:01\n• The only effect using #define to shorten sin to s is to make your code unreadable – Chris Turner Jan 23 at 14:01\n• You could write an emac macro that writes for you sin, whenever you type e.g. CTRL + s. In Vim you could write once sin and then use the . to repeat typing that as many times as you want. – Joey Mallone Jan 23 at 14:04\n• I understand there might be reasons to disagree with OP's coding style but this seems to be a perfectly valid and to-the-point question. I don't see why it should be downvoted. – hugomg Jan 23 at 14:06\n• Not downvoting but I encourage you to reconsider youe idea. Renaming standard library functions can create confusion for future code readers and maintainers. Make your expressions more readable by using whitespace and if necessary splitting them over more than one line. – rici Jan 23 at 14:15\n\nThere are at least three options for using s for sin:\n\nUse a preprocessor macro:\n\n#define s(x) (sin(x))\n#define c(x) (cos(x))\nfloat a = s(b)*c(c)+s(c+d)*c(d)....\n#undef c\n#undef s\n\n\nNote that the macros definitions are immediately removed with #undef to prevent them from affecting subsequent code. Also, you should be aware of the basics of preprocessor macro substitution, noting the fact that the first c in c(c) will be expanded but the second c will not since the function-like macro c(x) is expanded only where c is followed by (.\n\nThis solution will have no effect on run time.\n\nUse an inline function:\n\nstatic inline double s(double x) { return sin(x); }\nstatic inline double c(double x) { return cos(x); }\n\n\nWith a good compiler, this will have no effect on run time, since the compiler should replace a call to s or c with a direct call to sin or cos, having the same result as the original code. Unfortunately, in this case, the c function will conflict with the c object you show in your sample code. You will need to change one of the names.\n\nUse function pointers:\n\nstatic double (* const s)(double) = sin;\nstatic double (* const c)(double) = cos;\n\n\nWith a good compiler, this also will have no effect on run time, although I suspect a few more compilers might fail to optimize code using this solution than than previous solution. Again, you will have the name conflict with c. Note that using function pointers creates a direct call to the sin and cos functions, bypassing any macros that the C implementation might have defined for them. (C implementations are allowed to implement library function using macros as well as functions, and they might do so to support optimizations or certain features. With a good quality compiler, this is usually a minor concern; optimization of a direct call still should be good.)\n\nif I use define, does it affect runtime?\n\ndefine works by doing text-based substitution at compile time. If you #define s(x) sin(x) then the C pre-processor will rewrite all the s(x) into sin(x) before the compiler gets a chance to look at it.\n\nBTW, this kind of low-level text-munging is exactly why define can be dangerous to use for more complex expressions. For example, one classic pitfall is that if you do something like #define times(x, y) x*y then times(1+1,2) rewrites to 1+1*2, which evaluates to 3 instead of the expected 4. For more complex expressions like it is often a good idea to use inlineable functions instead.\n\n• Macro substitution substitutes preprocessor tokens, not text. – Eric Postpischil Jan 23 at 14:25\n\nDon't do this.\n\nMathematicians have been abbreviating the trigonometric functions to sin, cos, tan, sinh, cosh, and tanh for many many years now. Even though mathematicians (like me) like to use their favourite and often idiosyncratic notation so puffing up any paper by a number of pages, these have emerged as pretty standard. Even LaTeX has commands like \\sin, \\cos, and \\tan.\n\nThe Japanese immortalised the abbreviations when releasing scientific calculators in the 1970s (the shorthand can fit easily on a button), and the C standard library adopted them.\n\nIf you deviate from this then your code immediately becomes difficult to read. This can be particularly pernicious with mathematical code where you can't immediately see the effects of a bad implementation.\n\nBut if you must, then a simple\n\nstatic double(*const s)(double) = sin;\n\n\nwill suffice." ]

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[ "", null, "", null, "Incidence > Table > Interpret\n\n## Interpretation of Incidence Rates Data\n\n### Incidence Rate Report for Wisconsin by CountyMelanoma of the Skin (All Stages^), 2016-2020All Races (includes Hispanic), Both Sexes, All AgesSorted by Recentaapc\n\nObjective - The objective of *** is from the Healthy People 2020 project done by the Centers for Disease Control and Prevention.\n\nIncidence Rate (95% Confidence Interval) - The incidence rate is based upon 100,000 people and is an annual rate (or average annual rate) based on the time period indicated. Rates are age-adjusted by 5-year age groups to the 2000 U.S. standard million population.\n\nRecent Trends - This is an interpretation of the AAPC/APC:\n\n• Rising when 95% confidence interval of AAPC/APC is above 0.\n• Stable when 95% confidence interval of AAPC/APC includes 0.\n• Falling when 95% confidence interval of AAPC/APC is below 0.\n\nAAPC/APC (95% Confidence Interval) - the change in rate over time\n\nOther Notes\n\n• Larger confidence intervals indicate less stability of the data. This is often due to low counts that are not quite low enough to be suppressed.\n• Data is currently being suppressed if there are fewer than 16 counts for the time period.\n\nLine by Line Interpretation of the Report\n\n#### Wisconsin6\n\n• Rate : The incidence rate is 23.8 with a 95% confidence interval from 23.3 to 24.3 and 1,673 average annual cases over 2016-2020.\n• CI*Rank⋔ : N/A\n• Recent Trend : The trend is", null, "because the trend is -1.4 with a 95% confidence interval from -4.5 to 0.8.\n\n#### US (SEER+NPCR)1\n\n• Rate : The incidence rate is 22.5 with a 95% confidence interval from 22.4 to 22.5 and 83,836 average annual cases over 2016-2020.\n• CI*Rank⋔ : N/A\n• Recent Trend : The trend is", null, "because the trend is 1.5 with a 95% confidence interval from -0.9 to 2.7.\n\n#### Douglas County6\n\n• Rate : The incidence rate is 30.2 with a 95% confidence interval from 23.6 to 38.0 and 16 average annual cases over 2016-2020.\n• CI*Rank⋔ : 8 (1, 52)\n• Recent Trend : The trend is rising", null, "because the trend is 10.2 with a 95% confidence interval from 6.9 to 15.9.\n\n#### St. Croix County6\n\n• Rate : The incidence rate is 29.5 with a 95% confidence interval from 24.9 to 34.8 and 30 average annual cases over 2016-2020.\n• CI*Rank⋔ : 9 (1, 45)\n• Recent Trend : The trend is rising", null, "because the trend is 8.9 with a 95% confidence interval from 6.6 to 12.6.\n\n#### Burnett County6\n\n• Rate : The incidence rate is 24.8 with a 95% confidence interval from 15.6 to 38.1 and 7 average annual cases over 2016-2020.\n• CI*Rank⋔ : 24 (1, 67)\n• Recent Trend : The trend is rising", null, "because the trend is 7.6 with a 95% confidence interval from 2.8 to 14.7.\n\n#### Trempealeau County6\n\n• Rate : The incidence rate is 36.0 with a 95% confidence interval from 27.3 to 46.7 and 13 average annual cases over 2016-2020.\n• CI*Rank⋔ : 1 (1, 40)\n• Recent Trend : The trend is rising", null, "because the trend is 7.5 with a 95% confidence interval from 4.6 to 11.6.\n\n#### Richland County6\n\n• Rate : The incidence rate is 19.9 with a 95% confidence interval from 12.4 to 30.7 and 5 average annual cases over 2016-2020.\n• CI*Rank⋔ : 58 (3, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 7.2 with a 95% confidence interval from -2.8 to 43.5.\n\n#### Dunn County6\n\n• Rate : The incidence rate is 20.8 with a 95% confidence interval from 15.2 to 27.8 and 10 average annual cases over 2016-2020.\n• CI*Rank⋔ : 53 (8, 67)\n• Recent Trend : The trend is rising", null, "because the trend is 6.9 with a 95% confidence interval from 4.3 to 10.8.\n\n#### Juneau County6\n\n• Rate : The incidence rate is 22.8 with a 95% confidence interval from 16.0 to 31.8 and 8 average annual cases over 2016-2020.\n• CI*Rank⋔ : 43 (3, 66)\n• Recent Trend : The trend is rising", null, "because the trend is 6.1 with a 95% confidence interval from 2.4 to 11.0.\n\n#### Waushara County6\n\n• Rate : The incidence rate is 23.7 with a 95% confidence interval from 16.5 to 33.5 and 8 average annual cases over 2016-2020.\n• CI*Rank⋔ : 37 (2, 67)\n• Recent Trend : The trend is rising", null, "because the trend is 6.1 with a 95% confidence interval from 2.2 to 11.5.\n\n#### Kenosha County6\n\n• Rate : The incidence rate is 24.3 with a 95% confidence interval from 21.1 to 27.7 and 46 average annual cases over 2016-2020.\n• CI*Rank⋔ : 30 (11, 56)\n• Recent Trend : The trend is rising", null, "because the trend is 6.0 with a 95% confidence interval from 2.6 to 11.0.\n\n#### Pierce County6\n\n• Rate : The incidence rate is 20.9 with a 95% confidence interval from 15.2 to 28.1 and 10 average annual cases over 2016-2020.\n• CI*Rank⋔ : 52 (7, 67)\n• Recent Trend : The trend is rising", null, "because the trend is 5.6 with a 95% confidence interval from 1.2 to 12.4.\n\n#### Green County6\n\n• Rate : The incidence rate is 33.8 with a 95% confidence interval from 26.3 to 42.8 and 16 average annual cases over 2016-2020.\n• CI*Rank⋔ : 3 (1, 45)\n• Recent Trend : The trend is rising", null, "because the trend is 5.3 with a 95% confidence interval from 1.5 to 11.1.\n\n#### Racine County6\n\n• Rate : The incidence rate is 24.7 with a 95% confidence interval from 21.8 to 27.8 and 58 average annual cases over 2016-2020.\n• CI*Rank⋔ : 25 (11, 54)\n• Recent Trend : The trend is rising", null, "because the trend is 5.1 with a 95% confidence interval from 2.3 to 9.0.\n\n#### La Crosse County6\n\n• Rate : The incidence rate is 32.9 with a 95% confidence interval from 28.5 to 37.8 and 43 average annual cases over 2016-2020.\n• CI*Rank⋔ : 4 (1, 25)\n• Recent Trend : The trend is rising", null, "because the trend is 4.3 with a 95% confidence interval from 2.1 to 7.2.\n\n#### Barron County6\n\n• Rate : The incidence rate is 24.3 with a 95% confidence interval from 18.6 to 31.3 and 15 average annual cases over 2016-2020.\n• CI*Rank⋔ : 28 (3, 63)\n• Recent Trend : The trend is stable", null, "because the trend is 4.2 with a 95% confidence interval from -0.3 to 10.3.\n\n#### Sawyer County6\n\n• Rate : The incidence rate is 28.4 with a 95% confidence interval from 18.2 to 42.5 and 6 average annual cases over 2016-2020.\n• CI*Rank⋔ : 12 (1, 66)\n• Recent Trend : The trend is stable", null, "because the trend is 4.1 with a 95% confidence interval from -1.6 to 11.6.\n\n#### Waupaca County6\n\n• Rate : The incidence rate is 25.1 with a 95% confidence interval from 19.7 to 31.5 and 18 average annual cases over 2016-2020.\n• CI*Rank⋔ : 22 (3, 61)\n• Recent Trend : The trend is stable", null, "because the trend is 4.1 with a 95% confidence interval from -0.5 to 10.3.\n\n#### Jackson County6\n\n• Rate : The incidence rate is 22.9 with a 95% confidence interval from 15.5 to 33.0 and 6 average annual cases over 2016-2020.\n• CI*Rank⋔ : 42 (1, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 4.0 with a 95% confidence interval from -0.4 to 10.2.\n\n#### Jefferson County6\n\n• Rate : The incidence rate is 25.4 with a 95% confidence interval from 21.1 to 30.4 and 26 average annual cases over 2016-2020.\n• CI*Rank⋔ : 20 (4, 57)\n• Recent Trend : The trend is rising", null, "because the trend is 4.0 with a 95% confidence interval from 1.5 to 7.3.\n\n#### Grant County6\n\n• Rate : The incidence rate is 26.9 with a 95% confidence interval from 21.0 to 33.9 and 16 average annual cases over 2016-2020.\n• CI*Rank⋔ : 18 (1, 59)\n• Recent Trend : The trend is rising", null, "because the trend is 3.9 with a 95% confidence interval from 2.0 to 6.2.\n\n#### Walworth County6\n\n• Rate : The incidence rate is 34.2 with a 95% confidence interval from 29.8 to 39.2 and 46 average annual cases over 2016-2020.\n• CI*Rank⋔ : 2 (1, 20)\n• Recent Trend : The trend is rising", null, "because the trend is 3.8 with a 95% confidence interval from 1.7 to 6.6.\n\n#### Wood County6\n\n• Rate : The incidence rate is 21.0 with a 95% confidence interval from 16.8 to 25.9 and 20 average annual cases over 2016-2020.\n• CI*Rank⋔ : 50 (13, 66)\n• Recent Trend : The trend is stable", null, "because the trend is 3.8 with a 95% confidence interval from -0.2 to 8.7.\n\n#### Columbia County6\n\n• Rate : The incidence rate is 22.0 with a 95% confidence interval from 17.3 to 27.7 and 16 average annual cases over 2016-2020.\n• CI*Rank⋔ : 47 (10, 64)\n• Recent Trend : The trend is rising", null, "because the trend is 3.7 with a 95% confidence interval from 0.3 to 8.0.\n\n#### Clark County6\n\n• Rate : The incidence rate is 23.5 with a 95% confidence interval from 17.0 to 31.6 and 9 average annual cases over 2016-2020.\n• CI*Rank⋔ : 39 (3, 66)\n• Recent Trend : The trend is stable", null, "because the trend is 3.4 with a 95% confidence interval from -0.2 to 8.2.\n\n#### Monroe County6\n\n• Rate : The incidence rate is 24.9 with a 95% confidence interval from 19.2 to 31.7 and 14 average annual cases over 2016-2020.\n• CI*Rank⋔ : 23 (3, 63)\n• Recent Trend : The trend is rising", null, "because the trend is 3.4 with a 95% confidence interval from 0.2 to 7.5.\n\n#### Vernon County6\n\n• Rate : The incidence rate is 24.0 with a 95% confidence interval from 17.3 to 32.5 and 10 average annual cases over 2016-2020.\n• CI*Rank⋔ : 34 (2, 66)\n• Recent Trend : The trend is stable", null, "because the trend is 3.4 with a 95% confidence interval from -0.3 to 8.2.\n\n#### Polk County6\n\n• Rate : The incidence rate is 21.6 with a 95% confidence interval from 16.3 to 28.2 and 13 average annual cases over 2016-2020.\n• CI*Rank⋔ : 49 (8, 66)\n• Recent Trend : The trend is stable", null, "because the trend is 3.1 with a 95% confidence interval from -0.2 to 7.2.\n\n#### Washington County6\n\n• Rate : The incidence rate is 29.4 with a 95% confidence interval from 25.8 to 33.4 and 52 average annual cases over 2016-2020.\n• CI*Rank⋔ : 10 (2, 38)\n• Recent Trend : The trend is rising", null, "because the trend is 3.0 with a 95% confidence interval from 1.1 to 5.4.\n\n#### Marinette County6\n\n• Rate : The incidence rate is 21.9 with a 95% confidence interval from 16.5 to 28.7 and 14 average annual cases over 2016-2020.\n• CI*Rank⋔ : 48 (6, 66)\n• Recent Trend : The trend is rising", null, "because the trend is 2.8 with a 95% confidence interval from 0.1 to 6.3.\n\n#### Oneida County6\n\n• Rate : The incidence rate is 19.0 with a 95% confidence interval from 13.8 to 25.8 and 11 average annual cases over 2016-2020.\n• CI*Rank⋔ : 61 (11, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 2.8 with a 95% confidence interval from -1.4 to 7.7.\n\n#### Dodge County6\n\n• Rate : The incidence rate is 20.9 with a 95% confidence interval from 17.2 to 25.3 and 23 average annual cases over 2016-2020.\n• CI*Rank⋔ : 51 (17, 64)\n• Recent Trend : The trend is rising", null, "because the trend is 2.7 with a 95% confidence interval from 0.2 to 5.7.\n\n#### Kewaunee County6\n\n• Rate : The incidence rate is 26.4 with a 95% confidence interval from 18.3 to 37.1 and 8 average annual cases over 2016-2020.\n• CI*Rank⋔ : 19 (1, 65)\n• Recent Trend : The trend is stable", null, "because the trend is 2.7 with a 95% confidence interval from -0.9 to 7.6.\n\n#### Ozaukee County6\n\n• Rate : The incidence rate is 28.4 with a 95% confidence interval from 24.1 to 33.3 and 34 average annual cases over 2016-2020.\n• CI*Rank⋔ : 11 (2, 46)\n• Recent Trend : The trend is rising", null, "because the trend is 2.2 with a 95% confidence interval from 0.3 to 4.6.\n\n#### Rock County6\n\n• Rate : The incidence rate is 24.6 with a 95% confidence interval from 21.5 to 28.0 and 49 average annual cases over 2016-2020.\n• CI*Rank⋔ : 26 (10, 56)\n• Recent Trend : The trend is rising", null, "because the trend is 2.2 with a 95% confidence interval from 0.4 to 4.4.\n\n#### Waukesha County6\n\n• Rate : The incidence rate is 30.4 with a 95% confidence interval from 28.2 to 32.7 and 161 average annual cases over 2016-2020.\n• CI*Rank⋔ : 7 (2, 21)\n• Recent Trend : The trend is rising", null, "because the trend is 2.1 with a 95% confidence interval from 1.3 to 2.9.\n\n• Rate : The incidence rate is 19.9 with a 95% confidence interval from 12.6 to 30.5 and 7 average annual cases over 2016-2020.\n• CI*Rank⋔ : 57 (3, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 1.8 with a 95% confidence interval from -2.6 to 6.9.\n\n#### Shawano County6\n\n• Rate : The incidence rate is 19.2 with a 95% confidence interval from 14.1 to 25.7 and 11 average annual cases over 2016-2020.\n• CI*Rank⋔ : 60 (13, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 1.8 with a 95% confidence interval from -1.9 to 6.2.\n\n#### Oconto County6\n\n• Rate : The incidence rate is 23.8 with a 95% confidence interval from 17.6 to 31.6 and 11 average annual cases over 2016-2020.\n• CI*Rank⋔ : 36 (3, 65)\n• Recent Trend : The trend is stable", null, "because the trend is 1.7 with a 95% confidence interval from -1.1 to 5.0.\n\n#### Chippewa County6\n\n• Rate : The incidence rate is 24.2 with a 95% confidence interval from 19.6 to 29.7 and 20 average annual cases over 2016-2020.\n• CI*Rank⋔ : 32 (5, 61)\n• Recent Trend : The trend is stable", null, "because the trend is 1.6 with a 95% confidence interval from -7.6 to 6.1.\n\n#### Green Lake County6\n\n• Rate : The incidence rate is 19.8 with a 95% confidence interval from 13.3 to 29.1 and 6 average annual cases over 2016-2020.\n• CI*Rank⋔ : 59 (6, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 1.3 with a 95% confidence interval from -4.3 to 8.3.\n\n#### Milwaukee County6\n\n• Rate : The incidence rate is 14.2 with a 95% confidence interval from 13.2 to 15.4 and 143 average annual cases over 2016-2020.\n• CI*Rank⋔ : 66 (60, 67)\n• Recent Trend : The trend is rising", null, "because the trend is 1.3 with a 95% confidence interval from 0.4 to 2.3.\n\n#### Eau Claire County6\n\n• Rate : The incidence rate is 31.0 with a 95% confidence interval from 26.4 to 36.2 and 35 average annual cases over 2016-2020.\n• CI*Rank⋔ : 6 (1, 37)\n• Recent Trend : The trend is stable", null, "because the trend is 0.9 with a 95% confidence interval from -8.3 to 6.7.\n\n#### Manitowoc County6\n\n• Rate : The incidence rate is 28.1 with a 95% confidence interval from 23.5 to 33.3 and 31 average annual cases over 2016-2020.\n• CI*Rank⋔ : 13 (2, 48)\n• Recent Trend : The trend is stable", null, "because the trend is 0.7 with a 95% confidence interval from -2.6 to 4.3.\n\n#### Sheboygan County6\n\n• Rate : The incidence rate is 25.2 with a 95% confidence interval from 21.5 to 29.4 and 36 average annual cases over 2016-2020.\n• CI*Rank⋔ : 21 (7, 56)\n• Recent Trend : The trend is stable", null, "because the trend is 0.5 with a 95% confidence interval from -1.4 to 2.5.\n\n#### Vilas County6\n\n• Rate : The incidence rate is 16.7 with a 95% confidence interval from 11.0 to 25.1 and 7 average annual cases over 2016-2020.\n• CI*Rank⋔ : 65 (17, 67)\n• Recent Trend : The trend is stable", null, "because the trend is 0.3 with a 95% confidence interval from -5.1 to 7.4.\n\n#### Lincoln County6\n\n• Rate : The incidence rate is 17.1 with a 95% confidence interval from 11.5 to 24.7 and 7 average annual cases over 2016-2020.\n• CI*Rank⋔ : 64 (18, 67)\n• Recent Trend : The trend is stable", null, "because the trend is -0.7 with a 95% confidence interval from -7.9 to 7.2.\n\n#### Fond du Lac County6\n\n• Rate : The incidence rate is 23.0 with a 95% confidence interval from 19.3 to 27.3 and 30 average annual cases over 2016-2020.\n• CI*Rank⋔ : 41 (11, 61)\n• Recent Trend : The trend is stable", null, "because the trend is -0.9 with a 95% confidence interval from -3.8 to 2.3.\n\n#### Marquette County6\n\n• Rate : The incidence rate is 20.5 with a 95% confidence interval from 12.4 to 32.6 and 5 average annual cases over 2016-2020.\n• CI*Rank⋔ : 54 (1, 67)\n• Recent Trend : The trend is stable", null, "because the trend is -1.0 with a 95% confidence interval from -7.1 to 4.8.\n\n#### Marathon County6\n\n• Rate : The incidence rate is 24.3 with a 95% confidence interval from 20.9 to 28.1 and 40 average annual cases over 2016-2020.\n• CI*Rank⋔ : 29 (9, 57)\n• Recent Trend : The trend is stable", null, "because the trend is -2.3 with a 95% confidence interval from -13.1 to 2.8.\n\n#### Portage County6\n\n• Rate : The incidence rate is 17.2 with a 95% confidence interval from 13.3 to 21.9 and 14 average annual cases over 2016-2020.\n• CI*Rank⋔ : 63 (31, 67)\n• Recent Trend : The trend is falling", null, "because the trend is -3.6 with a 95% confidence interval from -7.0 to -0.7.\n\n#### Winnebago County6\n\n• Rate : The incidence rate is 24.2 with a 95% confidence interval from 21.2 to 27.6 and 50 average annual cases over 2016-2020.\n• CI*Rank⋔ : 31 (11, 55)\n• Recent Trend : The trend is stable", null, "because the trend is -3.6 with a 95% confidence interval from -10.8 to 0.3.\n\n#### Sauk County6\n\n• Rate : The incidence rate is 20.5 with a 95% confidence interval from 16.2 to 25.7 and 17 average annual cases over 2016-2020.\n• CI*Rank⋔ : 55 (15, 66)\n• Recent Trend : The trend is stable", null, "because the trend is -3.9 with a 95% confidence interval from -14.2 to 4.3.\n\n#### Brown County6\n\n• Rate : The incidence rate is 24.5 with a 95% confidence interval from 22.0 to 27.3 and 70 average annual cases over 2016-2020.\n• CI*Rank⋔ : 27 (13, 53)\n• Recent Trend : The trend is falling", null, "because the trend is -4.4 with a 95% confidence interval from -11.2 to -1.2.\n\n#### Calumet County6\n\n• Rate : The incidence rate is 23.6 with a 95% confidence interval from 18.2 to 30.1 and 14 average annual cases over 2016-2020.\n• CI*Rank⋔ : 38 (4, 64)\n• Recent Trend : The trend is stable", null, "because the trend is -4.5 with a 95% confidence interval from -17.8 to 0.5.\n\n#### Outagamie County6\n\n• Rate : The incidence rate is 23.2 with a 95% confidence interval from 20.2 to 26.4 and 48 average annual cases over 2016-2020.\n• CI*Rank⋔ : 40 (15, 58)\n• Recent Trend : The trend is stable", null, "because the trend is -7.9 with a 95% confidence interval from -24.8 to 6.7.\n\n#### Dane County6\n\n• Rate : The incidence rate is 27.8 with a 95% confidence interval from 25.9 to 29.9 and 159 average annual cases over 2016-2020.\n• CI*Rank⋔ : 16 (7, 32)\n• Recent Trend : The trend is falling", null, "because the trend is -10.1 with a 95% confidence interval from -20.6 to -0.6.\n\n#### Door County6\n\n• Rate : The incidence rate is 22.7 with a 95% confidence interval from 15.7 to 31.9 and 10 average annual cases over 2016-2020.\n• CI*Rank⋔ : 45 (3, 67)\n• Recent Trend : The trend is stable", null, "because the trend is -12.7 with a 95% confidence interval from -42.1 to 0.2.\n\n#### Ashland County6\n\n• Rate : The incidence rate is 22.4 with a 95% confidence interval from 13.1 to 35.8 and 4 average annual cases over 2016-2020.\n• CI*Rank⋔ : 46 (1, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Bayfield County6\n\n• Rate : The incidence rate is 27.9 with a 95% confidence interval from 17.8 to 42.2 and 7 average annual cases over 2016-2020.\n• CI*Rank⋔ : 15 (1, 66)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Buffalo County6\n\n• Rate : The incidence rate is 32.4 with a 95% confidence interval from 21.3 to 48.0 and 6 average annual cases over 2016-2020.\n• CI*Rank⋔ : 5 (1, 63)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Crawford County6\n\n• Rate : The incidence rate is 22.7 with a 95% confidence interval from 14.0 to 35.0 and 5 average annual cases over 2016-2020.\n• CI*Rank⋔ : 44 (1, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Iowa County6\n\n• Rate : The incidence rate is 27.9 with a 95% confidence interval from 19.4 to 39.0 and 8 average annual cases over 2016-2020.\n• CI*Rank⋔ : 14 (1, 65)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Lafayette County6\n\n• Rate : The incidence rate is 24.0 with a 95% confidence interval from 15.6 to 35.8 and 5 average annual cases over 2016-2020.\n• CI*Rank⋔ : 33 (1, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n• Rate : The incidence rate is 23.8 with a 95% confidence interval from 15.9 to 34.8 and 7 average annual cases over 2016-2020.\n• CI*Rank⋔ : 35 (1, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Pepin County6\n\n• Rate : The incidence rate is 27.0 with a 95% confidence interval from 15.0 to 46.5 and 3 average annual cases over 2016-2020.\n• CI*Rank⋔ : 17 (1, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Rusk County6\n\n• Rate : The incidence rate is 17.3 with a 95% confidence interval from 9.5 to 29.4 and 4 average annual cases over 2016-2020.\n• CI*Rank⋔ : 62 (4, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Taylor County6\n\n• Rate : The incidence rate is 12.6 with a 95% confidence interval from 7.0 to 21.3 and 3 average annual cases over 2016-2020.\n• CI*Rank⋔ : 67 (28, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n#### Washburn County6\n\n• Rate : The incidence rate is 20.3 with a 95% confidence interval from 12.7 to 31.7 and 6 average annual cases over 2016-2020.\n• CI*Rank⋔ : 56 (2, 67)\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\nFlorence County6\n• Rate : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n• CI*Rank⋔ : *\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\nForest County6\n• Rate : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n• CI*Rank⋔ : *\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\nIron County6\n• Rate : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n• CI*Rank⋔ : *\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\nMenominee County6\n• Rate : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n• CI*Rank⋔ : *\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\nPrice County6\n• Rate : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n• CI*Rank⋔ : *\n• Recent Trend : * Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\nNotes:\nCreated by statecancerprofiles.cancer.gov on 09/25/2023 7:18 am.\n\nState Cancer Registries may provide more current or more local data.\nTrend\nRising when 95% confidence interval of average annual percent change is above 0.\nStable when 95% confidence interval of average annual percent change includes 0.\nFalling when 95% confidence interval of average annual percent change is below 0.\n\n⋔ Results presented with the CI*Rank statistics help show the usefulness of ranks. For example, ranks for relatively rare diseases or less populated areas may be essentially meaningless because of their large variability, but ranks for more common diseases in densely populated regions can be very useful. More information about methodology can be found on the CI*Rank website.\n\n† Incidence rates (cases per 100,000 population per year) are age-adjusted to the 2000 US standard population (19 age groups: <1, 1-4, 5-9, ... , 80-84, 85+). Rates are for invasive cancer only (except for bladder cancer which is invasive and in situ) or unless otherwise specified. Rates calculated using SEER*Stat. Population counts for denominators are based on Census populations as modified by NCI. The US Population Data File is used for SEER and NPCR incidence rates.\n‡ Incidence data come from different sources. Due to different years of data availability, most of the trends are AAPCs based on APCs but some are APCs calculated in SEER*Stat. Please refer to the source for each area for additional information.\n\nRates and trends are computed using different standards for malignancy. For more information see malignant.html.\n\n^ All Stages refers to any stage in the Surveillance, Epidemiology, and End Results (SEER) summary stage.\n* Data has been suppressed to ensure confidentiality and stability of rate estimates. Counts are suppressed if fewer than 16 records were reported in a specific area-sex-race category. If an average count of 3 is shown, the total number of cases for the time period is 16 or more which exceeds suppression threshold (but is rounded to 3).\n\n1 Source: National Program of Cancer Registries and Surveillance, Epidemiology, and End Results SEER*Stat Database - United States Department of Health and Human Services, Centers for Disease Control and Prevention and National Cancer Institute. Based on the 2022 submission.\n6 Source: National Program of Cancer Registries SEER*Stat Database - United States Department of Health and Human Services, Centers for Disease Control and Prevention (based on the 2022 submission).\n8 Source: Incidence data provided by the SEER Program. AAPCs are calculated by the Joinpoint Regression Program and are based on APCs. Data are age-adjusted to the 2000 US standard population (19 age groups: <1, 1-4, 5-9, ... , 80-84,85+). Rates are for invasive cancer only (except for bladder cancer which is invasive and in situ) or unless otherwise specified. Population counts for denominators are based on Census populations as modifed by NCI. The US Population Data File is used with SEER November 2022 data.\n\nInterpret Rankings provides insight into interpreting cancer incidence statistics. When the population size for a denominator is small, the rates may be unstable. A rate is unstable when a small change in the numerator (e.g., only one or two additional cases) has a dramatic effect on the calculated rate.\n\nData for the United States does not include data from Nevada.\nData for the United States does not include Puerto Rico.\n\nWhen displaying county information, the CI*Rank for the state is not shown because it's not comparable. To see the state CI*Rank please view the statistics at the US By State level." ]

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[ "# Model Mathematics\n\n### Component: soma_compartment\n\n$I_soma=I_Na+I_K_DR+I_hI_h=g_h⁢h⁢V_s+30I_Na=g_Na⁢h⁢V_s-V_Na⁢m_infinity3I_K_DR=g_K_DR⁢V_s-V_K⁢n2ddtimeV_s=-I_soma-g_cp⁢V_D-V_sC_m$\n\n### Component: dendritic_compartment\n\n$R_pump=18⁢p1-pg_L=0.18⁢p1-pg_NMDA=1.25⁢p1-pg_Na_NMDA=1⁢p1-pf_NMDA=11+0.141⁢ⅇ-V_DqI_pump=R_pump⁢p1-p⁢V_D⁢1+Na3K_p3+V_D⁢1+Na3I_pump_ss=R_pump⁢p1-p⁢Na_eq3K_Na3+Na_eq3I_L=g_L⁢V_D-V_LI_NMDA=g_NMDA⁢f_NMDA⁢V_DI_Na_NMDA=g_Na_NMDA⁢f_NMDA⁢V_D-V_NaI_D=I_NMDA+I_pump-I_pump_ss+I_LddtimeV_D=-I_D+g_c1-p⁢V_s-V_DC_mddtimeNa=alpha⁢-I_Na_NMDA-3⁢I_pump-I_pump_ss$\n\n### Component: gating_variables\n\n$m_infinity=11+ⅇ-V_s+356.2n_infinity=11+ⅇ-V_s+315.3h_infinity=11+ⅇV_s+308.3r_infinity=11+ⅇV_s+808tau_h=0.43+0.861+ⅇV_s+255tau_n=0.8+1.61+ⅇ0.1⁢V_s+251+ⅇ-0.1⁢V_s+70tau_mL=0.45⁢ⅇ-V_D+118.3+-V_D+118.3ⅇ-V_D+118.3-1tau_r=190ddtimeh=h_infinity-htau_hddtimen=n_infinity-ntau_n$\nSource\nDerived from workspace Li, Bertram and Rinzel, 1996 at changeset 17ec19e41bf7.\nCollaboration\nTo begin collaborating on this work, please use your git client and issue this command:\nDownloads\nTools\nLicense\n\nThis work is licensed under a Creative Commons Attribution 3.0 Unported License." ]

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[ "## ››Convert newton-meter to inch-pound\n\n newton-meter inch pound\n\nHow many newton-meter in 1 inch pound? The answer is 0.112984825.\nWe assume you are converting between newton-meter and inch-pound.\nYou can view more details on each measurement unit:\nnewton-meter or inch pound\nThe SI derived unit for energy is the joule.\n1 joule is equal to 1 newton-meter, or 8.8507461068334 inch pound.\nNote that rounding errors may occur, so always check the results.\nUse this page to learn how to convert between newton meters and inch pounds.\nType in your own numbers in the form to convert the units!\n\n## ››Quick conversion chart of newton-meter to inch pound\n\n1 newton-meter to inch pound = 8.85075 inch pound\n\n5 newton-meter to inch pound = 44.25373 inch pound\n\n10 newton-meter to inch pound = 88.50746 inch pound\n\n15 newton-meter to inch pound = 132.76119 inch pound\n\n20 newton-meter to inch pound = 177.01492 inch pound\n\n25 newton-meter to inch pound = 221.26865 inch pound\n\n30 newton-meter to inch pound = 265.52238 inch pound\n\n40 newton-meter to inch pound = 354.02984 inch pound\n\n50 newton-meter to inch pound = 442.53731 inch pound\n\n## ››Want other units?\n\nYou can do the reverse unit conversion from inch pound to newton-meter, or enter any two units below:\n\n## Enter two units to convert\n\n From: To:\n\n## ››Metric conversions and more\n\nConvertUnits.com provides an online conversion calculator for all types of measurement units. You can find metric conversion tables for SI units, as well as English units, currency, and other data. Type in unit symbols, abbreviations, or full names for units of length, area, mass, pressure, and other types. Examples include mm, inch, 100 kg, US fluid ounce, 6'3\", 10 stone 4, cubic cm, metres squared, grams, moles, feet per second, and many more!" ]

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[ "# Slides\n\nThe Slide alteration will cause the sprite to move in the direction given for the given distance.\n\n``` create routine as Start launch slider end create sprite from horse12.png as slider where x=100 y=100 having alt=(sub create slide as slider where speed=100 direction=95 easein=20 easeout=20 distance=100 end) end ```\n\n### Parameters\n\ncompletion={routines}\na routine to run when the slide is complete.\ndirection={number}\nthe direction to travel without regard to the sprites current angle.\ndistance={number}\nhow many pixels to travel (optional).\neasein={number}\ngradually increase speed over this distance.\neaseout={number}\nbefore reaching distance, gradually decrease speed over this distance.\nspeed={number}\nmove in pixels per second." ]

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[ "", null, "Dongsheng Yin\nVerified email at math.tsinghua.edu.cn\nTitle\nCited by\nCited by\nYear\nComputational high frequency waves through curved interfaces via the Liouville equation and geometric theory of diffraction\nS Jin, D Yin\nJournal of Computational Physics 227 (12), 6106-6139, 2008\n312008\nArtificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain\nH Brunner, H Han, D Yin\nJournal of Computational Physics 276, 541-562, 2014\n272014\nNumerical solutions of Schrödinger equations in ℝ3\nH Han, D Yin, Z Huang\nNumerical Methods for Partial Differential Equations: An International …, 2007\n222007\nGaussian beam methods for the Dirac equation in the semi-classical regime\nH Wu, Z Huang, S Jin, D Yin\narXiv preprint arXiv:1205.0543, 2012\n202012\nGaussian beam formulations and interface conditions for the one-dimensional linear Schrödinger equation\nD Yin, C Zheng\nWave Motion 48 (4), 310-324, 2011\n172011\nThe maximum principle for time-fractional diffusion equations and its application\nH Brunner, H Han, D Yin\nNumerical Functional Analysis and Optimization 36 (10), 1307-1321, 2015\n142015\nAbsorbing boundary conditions for the multidimensional Klein-Gordon equation\nH Han, D Yin\nCommunications in Mathematical Sciences 5 (3), 743-764, 2007\n142007\nNumerical solutions of parabolic problems on unbounded 3-D spatial domain\nH Han, D Yin\nJournal of Computational Mathematics, 449-462, 2005\n142005\nA non-overlap domain decomposition method for the forward–backward heat equation\nH Han, D Yin\nJournal of computational and applied mathematics 159 (1), 35-44, 2003\n142003\nComputation of high frequency wave diffraction by a half plane via the Liouville equation and geometric theory of diffraction\nS Jin, D Yin\nCommunications in Computational Physics 4 (5), 1106-1128, 2008\n132008\nGaussian beam methods for the Schrödinger equation with discontinuous potentials\nS Jin, D Wei, D Yin\nJournal of Computational and Applied Mathematics 265, 199-219, 2014\n122014\nExact artificial boundary conditions for quasilinear elliptic equations in unbounded domains\nH Han, Z Huang, D Yin\nCommunications in Mathematical Sciences 6 (1), 71-82, 2008\n112008\nComposite coherent states approximation for one-dimensional multi-phased wave functions\nD Yin, C Zheng\nCommunications in Computational Physics 11 (3), 951, 2012\n62012\nThe necessary and sufficient condition for the existence and uniqueness of a system of Fredholm integral equations of the first kind\nHD HAN, YD LEE, DS YIN, ZZ CHEN\nSCIENTIA SINICA Mathematica 45 (8), 1231-1248, 2015\n32015\nThe Gaussian beam method for the Wigner equation with discontinuous potentials\nD Yin, M Tang, S Jin\nInverse Problems & Imaging 7 (3), 1051, 2013\n32013\nComputational high frequency wave diffraction by a corner via the Liouville equation and geometric theory of diffraction\nS Jin, D Yin\nKinetic & Related Models 4 (1), 295, 2011\n32011\nNonorthogonal hexahedral mesh finite volume difference method for the 3-D diffusion equation\nZ Du, D Yin, X Liu, J Lu\nJOURNAL-TSINGHUA UNIVERSITY 43 (10), 1365-1368, 2003\n32003\nTailored finite point methods for solving singularly perturbed eigenvalue problems with higher eigenvalues\nH Han, Y Shih, D Yin\nJournal of Scientific Computing 73 (1), 242-282, 2017\n12017\nHigh-Order Local Absorbing Boundary Conditions for Fractional Evolution Equations on Unbounded Strips\nH Dong, M Wang, D Yin, Q Zhang\nADVANCES IN APPLIED MATHEMATICS AND MECHANICS 12 (3), 664-693, 2020\n2020\nÔØ Å ÒÙ× Ö ÔØ\nD Yin, S Yu, CI Hsu, J Liu, A Acab, R Wu, A Tao, BJ Chiang, JH Weiss\n2009\nThe system can't perform the operation now. Try again later.\nArticles 1–20" ]

[ null, "https://scholar.google.gr/citations/images/avatar_scholar_128.png", null ]

[ "NCERT Exemplar - MCQs\n\nChapter 6 Class 12 Application of Derivatives (Term 1)\nSerial order wise\n\n## (C) − 6/7          (D) −6\n\nThis question is exactly same Misc 20 MCQ  - Chapter 6 Class 12 - Application of Derivatives", null, "", null, "", null, "", null, "", null, "### Transcript\n\nQuestion 15 The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is: (A) 22/7 (B) 6/7 (C) − 6/7 (D) −6 Finding Slope of tangent 𝒅𝒚/𝒅𝒙 . 𝒅𝒚/𝒅𝒙= (𝒅𝒚/𝒅𝒕)/(𝒅𝒙/𝒅𝒕) 𝒙 = t2 + 3t – 8 Differentiating w.r.t t 𝒅𝒙/𝒅𝒕= (𝑑(𝑡^2 + 3𝑡 −8))/𝑑𝑡 𝑑𝑥/𝑑𝑡 = 2t + 3 𝒚 = 2t2 − 2t − 5 Differentiating w.r.t t 𝒅𝒚/𝒅𝒕= (𝑑 (2𝑡2 − 2𝑡 − 5))/𝑑𝑡 𝑑𝑦/𝑑𝑡= 4t − 2 Now, 𝑑𝑦/𝑑𝑥= (𝑑𝑦∕𝑑𝑡)/(𝑑𝑥∕𝑑𝑡) 𝒅𝒚/𝒅𝒙= (𝟒𝒕 − 𝟐)/(𝟐𝒕 + 𝟑) ∴ Slope of Tangent = (4𝑡 − 2)/(2𝑡 + 3) Now, we need to find value of Slope at (2, –1) But we need to find value of t first To find value of t, We put 𝒙 = 2 & 𝒚 = –1 in the curve For x x = t2 + 3t – 8 2 = t2 + 3t − 8 t2 + 3t – 8 – 2 =0 t2 + 3t − 10 = 0 t2 + 5t – 2t − 10 = 0 t (t + 5) – 2 (t − 5) = 0 (t − 2) (t + 5) = 0 So, t = 2 & t = −5 For y y = 2t2 – 2t – 5 –1 = 2t2 – 2t – 5 2t2 – 2t – 5 + 1 = 0 2t2 – 2t – 4 = 0 2(t2 – t – 2 ) = 0 t2 – t – 2 = 0 t2 – 2t + t – 2 = 0 t (t − 2) + 1(t − 2) = 0 (t + 1) (t – 2) = 0 So, t = −1 & t = 2 Since t = 2 is common in both parts So, we will calculate Slope of Tangent at t = 2 Finding Slope of Tangent 𝒅𝒚/𝒅𝒙= (4𝑡 −2)/(2𝑡 + 3) = (4 (2) − 2)/(2 (2) + 3) = (8 − 2)/(4 +3) = 𝟔/𝟕 Hence, the correct answer is (B)", null, "" ]

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[ "# phased.RootMUSICEstimator\n\nRoot MUSIC direction of arrival (DOA) estimator for ULA and UCA\n\n## Description\n\nThe `RootMUSICEstimator` object implements the root multiple signal classification (root-MUSIC) direction of arrival estimator for uniform linear arrays (ULA) and uniform circular arrays (UCA). When a uniform circular array is used, the algorithm transforms the input to a ULA-like structure using the phase mode excitation technique .\n\nTo estimate the direction of arrival (DOA):\n\n1. Define and set up your DOA estimator. See Construction.\n\n2. Call `step` to estimate the DOA according to the properties of `phased.RootMUSICEstimator`. The behavior of `step` is specific to each object in the toolbox.\n\nNote\n\nStarting in R2016b, instead of using the `step` method to perform the operation defined by the System object™, you can call the object with arguments, as if it were a function. For example, ```y = step(obj,x)``` and `y = obj(x)` perform equivalent operations.\n\n## Construction\n\n`H = phased.RootMUSICEstimator` creates a root MUSIC DOA estimator System object, `H`. The object estimates the signal's direction of arrival using the root MUSIC algorithm with a uniform linear array (ULA).\n\n`H = phased.RootMUSICEstimator(Name,Value)` creates object, `H`, with each specified property Name set to the specified Value. You can specify additional name-value pair arguments in any order as (`Name1`,`Value1`,...,`NameN`,`ValueN`).\n\n## Properties\n\n `SensorArray` Sensor array System object Sensor array specified as a System object. The sensor array must be a `phased.ULA` object or a `phased.UCA` object. Default: `phased.ULA` with default property values `PropagationSpeed` Signal propagation speed Specify the propagation speed of the signal, in meters per second, as a positive scalar. You can specify this property as single or double precision. Default: Speed of light `OperatingFrequency` System operating frequency Specify the operating frequency of the system in hertz as a positive scalar. The default value corresponds to 300 MHz. You can specify this property as single or double precision. Default: `3e8` `ForwardBackwardAveraging` Perform forward-backward averaging Set this property to `true` to use forward-backward averaging to estimate the covariance matrix for sensor arrays with conjugate symmetric array manifold. Default: `false` `SpatialSmoothing` Spatial smoothing The averaging number used by spatial smoothing to estimate the covariance matrix, specified as a strictly positive integer. Each additional smoothing value handles one additional coherent source, but reduces the effective number of elements by one. The maximum value of this property is M-2. For a ULA, M is the number of sensors. For a UCA, M is the size of the internal ULA-like array structure defined by the phase mode excitation technique. The default value of zero indicates that no spatial smoothing is employed. You can specify this property as single or double precision. Default: `0` `NumSignalsSource` Source of number of signals Specify the source of the number of signals as one of `'Auto'` or `'Property'`. If you set this property to `'Auto'`, the number of signals is estimated by the method specified by the `NumSignalsMethod` property. When spatial smoothing is employed on a UCA, you cannot set the `NumSignalsSource` property to`'Auto'` to estimate the number of signals. You can use the functions `aictest` or `mdltest` independently to determine the number of signals. Default: `'Auto'` `NumSignalsMethod` Method to estimate number of signals Specify the method to estimate the number of signals as one of `'AIC'` or `'MDL'`. `'AIC'` uses the Akaike Information Criterion and `'MDL'` uses Minimum Description Length Criterion. This property applies when you set the `NumSignalsSource` property to `'Auto'`. Default: `'AIC'` `NumSignals` Number of signals Specify the number of signals as a positive integer scalar. This property applies when you set the `NumSignalsSource` property to `'Property'`. The number of signals must be smaller than the number of elements in the array specified in the `SensorArray` property. You can specify this property as single or double precision. Default: `1`\n\n## Methods\n\n step Perform DOA estimation\nCommon to All System Objects\n`release`\n\nAllow System object property value changes\n\n## Examples\n\ncollapse all\n\nEstimate the DOA's of two signals received by a standard 10-element uniform linear array (ULA) having an element spacing of 1 meter. The antenna operating frequency is 150 MHz. The actual direction of the first signal is 10 degrees in azimuth and 20 degrees in elevation. The direction of the second signal is 45 degrees in azimuth and 60 degrees in elevation.\n\n```fs = 8000; t = (0:1/fs:1).'; x1 = cos(2*pi*t*300); x2 = cos(2*pi*t*400); sULA = phased.ULA('NumElements',10,... 'ElementSpacing',1); sULA.Element.FrequencyRange = [100e6 300e6]; fc = 150e6; x = collectPlaneWave(sULA,[x1 x2],[10 20;45 60]',fc); rng default; noise = 0.1/sqrt(2)*(randn(size(x))+1i*randn(size(x))); sDOA = phased.RootMUSICEstimator('SensorArray',sULA,... 'OperatingFrequency',fc,... 'NumSignalsSource','Property',... 'NumSignals',2); doas = step(sDOA,x + noise); az = broadside2az(sort(doas),[20 60])```\n```az = 1×2 10.0001 45.0107 ```\n\nexpand all\n\n## References\n\n Van Trees, H. Optimum Array Processing. New York: Wiley-Interscience, 2002.\n\n Mathews, C.P., Zoltowski, M.D., \"Eigenstructure techniques for 2-D angle estimation with uniform circular arrays.\" IEEE Transactions on Signal Processing, vol. 42, No. 9, pp. 2395-2407, Sept. 1994.\n\n## Version History\n\nIntroduced in R2011a" ]

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[ "# Autocorrelation of the product of deterministic and random signal\n\nI was wondering how to calculate the autocorrelation of a deterministic signal $x(t)$ multiplied by a stochastic process $M(t)$, whose autocorellation $R_M(\\tau)$ is known a priori. In my case, $x(t)$ is a truncated monolateral exponentially decaying function.\n\nI suppose that the result of such multiplication $y(t) = x(t) \\cdot M(t)$ is again a stochastic process, but when approaching the calculation of the autocorrelation of $y(t)$ I obtain something that is not even, therefore I suppose I am making some mistakes. I know that the definition of autocorrelation for deterministic signals is different from the one of stochastic processes, but I do not know how to connect the two of them.\n\n• The product is not a wide-sense-stationary process even if $M(t)$ is, so be careful with your calculations. Jun 1, 2014 at 2:30\n• Just to add to what Dilip said, since the random process that results from taking that product is not WSS, there is no such autocorrelation function $R_y(\\tau)$. Instead, the autocorrelation is of the form $R_y(t_1, t_2)$; it will be a function of two variables that correspond to the two time indices in the process that the expectation is taken over. Jun 1, 2014 at 3:28\n• Therefore, in order to take the Fourier transform of $R_y(t_1,t_2)$ I need firstly to take the time average over $t_1$ and $t_2$ of the autocorrelation, right? Jun 1, 2014 at 9:20\n• If $x(t)$ were periodic then things would become a bit easier. Jun 1, 2014 at 10:02\n• $x(t)$ is truncated. So I think there is room for reaching a closed result ;) Jun 1, 2014 at 11:04\n\nAs correctly pointed out in the comments, in general the process $Y(t)=x(t)M(t)$ is not wide-sense stationary (WSS), i.e. its autocorrelation function depends not only on the time difference parameter $\\tau$, but also on the absolute time $t$:\n\n$$R_Y(\\tau,t)=E[Y(t+\\tau)Y^*(t)]=E[x(t+\\tau)M(t+\\tau)x^*(t)M^*(t)]=\\\\ =x(t+\\tau)x^*(t)E[M(t+\\tau)M^*(t)]=x(t+\\tau)x^*(t)R_M(\\tau)\\tag{1}$$\n\nIn your case you just need to evaluate (1) with the given function $x(t)$. However, as expected you will end up with a function of two variables because $Y(t)$ is not WSS.\n\nThere are a few special cases in which the resulting process is indeed WSS or can be easily made WSS. The first case is modulation by a complex exponential:\n\n$$x(t)=e^{j\\omega_ct}$$\n\nin which case\n\n$$x(t+\\tau)x^*(t)=e^{j\\omega_c(t+\\tau)}e^{-j\\omega_ct}=e^{j\\omega_c\\tau}$$\n\nonly depends on $\\tau$ and not on $t$. Another case of interest is the case where $x(t)$ is $T$-periodic. In this case the process $x(t)M(t)$ can be made WSS by introducing a random phase epoch $\\Theta$ with uniform distribution in the interval $[0,T]$ which is independent of $M(t)$:\n\n$$Y(t)=x(t+\\Theta)M(t)$$\n\nThe autocorrelation function of $Y(t)$ is then given by\n\n$$R_Y(\\tau)=R_M(\\tau)\\frac{1}{T}\\int_0^Tx(\\alpha+\\tau)x^*(\\alpha)d\\alpha$$\n\n• Thanks for your reply. If I understand well, writing $R_Y(t_1,t_2)=R_x(t_1,t_2)R_M(t_1,t_2)$ is wrong. Am I right? Jun 5, 2014 at 16:36\n• @Aglar: Yes, you're right. Jun 5, 2014 at 19:45\n• Thank you for the great answer. Can you provide more discussions on how to make $Y(t)$ WSS (especially for real $x(t)$)? Or do you recommend any book chapters on this kind of discussions? Thanks.\n– WDC\nMay 3, 2019 at 14:37\n• @WDC: For real $x(t)$ you'll have to use a periodic signal and a random phase, as explained in the answer. May 3, 2019 at 15:27" ]

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[ "## Search This Blog\n\n### Is Number 1 the First Number?\n\nIs Number 1 the First Number?\n\nI like to think that number 1 was the first number invented. After all, it's very basic to have just one of something. Most people avoid 0 and start counting with 1, so in that respect it can be considered the first number, or as mathematicians like to call them, the positive integers {1,2,3,...}.\n\nWhat else can we say about the number 1? Well, notice that if you multiply any number by 1 the number remains unchanged. That is, n*1=n for any number n. So mathematicians say that 1 is the \"identity element\" of the multiply operation. Identity just means it doesn't change anything under the operation.\n\nThe multiply operation, like many in mathematics, is called a binary operation for the simple reason that it works with two objects.\n\nSo, if 1 is the identity for the multiply (*) operation is it also the identity for the addition (+) operation? No, not at all, because n+1 is not equal to n.\n\nWhile we're on the subject of the addition operation we can also say that 1 is the most fundamental number in the sense that any positive integer can be made from 1 by just adding it to itself enough times n=1+1+........+1 where n is any positive integer.\n\nOf course, under the multiply operation this will get you nowhere, since 1*1*1...*1=1 and that's what you would expect from the identity element of an operation.\n\nWow. Look at all the mathematical jargon we've used.. \"positive integers\", \"binary operation\", \"identity element\". Pretty impressive!\n\nLike this post? Please click G+1 below to share it.\nContent written and posted by Ken Abbott abbottsystems@gmail.com" ]

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[ "# Using classes to write fbroc\n\nBy | May 17, 2015\n\nCurrently I am working on the next version of fbroc. One important new feature will be the analysis of paired ROC curves, which is important when you compare two classifiers. One example would be comparing some new diagnostic method to the state-of-the-art.\n\nBefore doing this I wanted to improve my C++ code. In the first version I didn’t use any C++ feature besides making use of the Vector classes in Rcpp. Without classes you usually need many more arguments per function and it is harder to reuse code efficiently. Implementing paired ROC curves is much more natural, if you look at paired ROC curves as a class containing two single ROC curve objects. This will make implementing them much more straightforward and the code more maintainable.\n\nAfter realizing this I refactored the fbroc C++ code, so that everything related to the ROC curves was encapsulated in a class named `ROC`.\n\n### Performance\n\nWhen I tested my first implementation of the `ROC` class on example data consisting of 1000 observations, I was a bit disappointed with the performance. The new code turned out to be about 30-40% slower than the old.\n\nHowever, when I took a careful look at what went on, I found the reason: unnecessary memory allocations. In many cases I allocated a new `NumericVector` or `IntegerVector` for each bootstrap replicate, even though the size of the vector remains constant while bootstrapping.\n\nAs an example, compare the following code snippets. In both cases ‘index_pos’ and ‘index_neg’ are member of ‘ROC’.\n\nWith memory allocation:\n\n```void ROC::strat_shuffle(IntegerVector &shuffle_pos, IntegerVector &shuffle_neg) {\nindex_pos = NumericVector (n_pos);\nindex_neg = NumericVector(n_neg);\nfor (int i = 0; i < n_pos; i++) {\nindex_pos[i] = original_index_pos[shuffle_pos[i]];\n}\nfor (int i = 0; i < n_neg; i++) {\nindex_neg[i] = original_index_neg[shuffle_neg[i]];\n}\n// recalculate ROC after bootstrap\nreset_delta();\nget_positives_delta();\nget_positives();\nget_rate();\n}\n```\n\nWithout memory allocation:\n\n```void ROC::strat_shuffle(IntegerVector &shuffle_pos, IntegerVector &shuffle_neg) {\nfor (int i = 0; i < n_pos; i++) {\nindex_pos[i] = original_index_pos[shuffle_pos[i]];\n}\nfor (int i = 0; i < n_neg; i++) {\nindex_neg[i] = original_index_neg[shuffle_neg[i]];\n}\n// recalculate ROC after bootstrap\nreset_delta();\nget_positives_delta();\nget_positives();\nget_rate();\n}\n```\n\n### Benchmark\n\nThe graph below shows the performance of the new code vs the old used in fbroc 0.1.0.\n\nTo generate it, I used a slightly modified version of the script used here.\n\nSince the time for memory allocation is usually not dependent upon the size of the memory being allocated, the overhead stops to matter when the number of observations gets very large. But in the case that you have more than 10000 observations per group, you probably don’t need to bootstrap anyway." ]

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[ "Point and Figure charting technique, a new addition to our ASP.NET MVC FinancialChart control, is a classical charting techniques that's gaining popularity in the new internet era. In this blog we'll look at how to implement this intuitive stock chart in ASP.NET MVC.\n\nConsisting of columns of X's and O's that represent filtered price movements, the Point and Figure chart uses only the price action of the stock, not time. X-Columns represent rising prices, and O-Columns represent falling prices.\n\nThe chart gives a buy signal when a column of X moves upwards and crosses top price point of previous column of X:", null, "Similarly, a sell signal is given when a column of O moves downwards and crosses the bottom price point of previous column of O’s. This objectivity of price makes this chart easy to understand and use:", null, "Other Point and Figure elements that matter are size of the box and price change consideration for trend reversal:", null, "### Scaling for size of a Point and Figure box\n\nThe size of the box indicates the intervals of price movements; each box represents a specific value that price must reach for an X or an O to be drawn. The box size can be based on different scaling techniques: it could be be based on percentage, average true range, or the traditional scaling as used in such charts. ComponentOne's FinancialChart control supports three types: traditional, user-defined, and defined by average true range.\n\nHere's a look at traditional scaling for Point and Figure charts:\n\n Price Range Box Size Under 0.25 0.0625 0.25 to 1.00 0.125 1.00 to 5.00 0.250 5.00 to 20.00 0.500 20.00 to 100 1.000 100 to 200 2.000 200 to 500 4.000 500 to 1,000 5.000 1,000 to 25,000 50.000 25,000 and up 500.000\n\n### Trend reversal in Point and Figure charts\n\nTrend reversal—that is, for a stock to move from uptrend to downtrend or vice versa—is indicated by three box price changes in the opposite direction of the current trend. As shown in the \"Point and Figure chart\" above, in this case, the last trend was a downtrend (column of O’s) that has just reversed, with price moving upwards by three box values. Three-box reversal is traditional, and it's up to you if you want to change it.\n\nTo learn more about Point and Figure charting, you may refer Thomas Dorsey's Point and Figure Charting and Jeremy du Plessis' The Definitive Guide to Point and Figure.\n\nNow that we understand the basics of a Point and Figure chart, let's dive into plotting one with ASP.NET MVC Core Point and Figure Financial Charts.\n\n## Building the model\n\nFor the model, we'll take the standard stock data of open, high, low, close. In this example, we're reading this from JSON:\n\n``````private static List<FinanceData> _jsonData;\n\nprivate static List<FinanceData> _rsData;\npublic static List<FinanceData> GetDataFromJson()\n{\nif (_jsonData != null)\n{\nreturn _jsonData;\n}\n\nstring path = HttpContext.Current.Server.MapPath(\"~/Content/fb.json\");\n\nJObject jo = (JObject)JsonConvert.DeserializeObject(jsonText);\nvar jDataset = (JObject)jo.GetValue(\"dataset\");\nvar jColumns = (JArray)jDataset.GetValue(\"column_names\");\nvar columnNames = new List<string>();\nforeach (var column in jColumns)\n{\n}\nvar dateIndex = columnNames.IndexOf(\"Date\"); //0\nvar openIndex = columnNames.IndexOf(\"Open\"); //1\nvar highIndex = columnNames.IndexOf(\"High\"); //2\nvar lowIndex = columnNames.IndexOf(\"Low\"); //3\nvar closeIndex = columnNames.IndexOf(\"Close\"); //4\nvar volumeIndex = columnNames.IndexOf(\"Volume\"); //5\n\nvar jData = (JArray)jDataset.GetValue(\"data\");\nList<FinanceData> list = new List<FinanceData>();\nforeach (JArray jItem in jData)\n{\nstring date = jItem[dateIndex].ToString();\ndouble open = Convert.ToDouble(jItem[openIndex].ToString());\ndouble high = Convert.ToDouble(jItem[highIndex].ToString());\ndouble low = Convert.ToDouble(jItem[lowIndex].ToString());\ndouble close = Convert.ToDouble(jItem[closeIndex].ToString());\ndouble volume = Convert.ToDouble(jItem[volumeIndex].ToString());\nlist.Add(new FinanceData { X = date, High = high, Low = low, Open = open, Close = close, Volume = volume });\n}\n_jsonData = list;\nreturn list;\n}\n\n}\n``````\n\n## Create the controller\n\nHere, simply set the stock prices collection in a ViewBag to be accessible in the View:\n\n``````public ActionResult PointAndFigure()\n{\nViewBag.FbData = FbData.GetDataFromJson();\n\nreturn View();\n}\n``````\n\n## Create the view\n\nDeclare the c1-financial-chart and set respective fields:\n\n``````IEnumerable<FinanceData> fbData = ViewBag.FbData;\n\n<c1-items-source id=\"cv\" source-collection=\"@fbData\"></c1-items-source>\n\n<c1-financial-chart id=\"pfx \" chart-type=\"PointAndFigure\" items-source-id=\"cv\"\n<c1-financial-chart-series binding=\"High,Low,Open,Close\" name=\"FB\" style=\"chartStyle\" alt-style=\"chartAltStyle\"></c1-financial-chart-series>\n<c1-flex-chart-tooltip content=\"{x:d}<br/>{y}\"></c1-flex-chart-tooltip>\n</c1-financial-chart>\n``````\n\nSet the items-source of the chart to the defined items-source “cv,” whose source-collection is populated with data returned by the model.\n\nWe're taking High and Low values to plot the chart, so point-and-figure-fields are set to HighLow; alternatively, Close can also be considered. Having used the traditional box-size, we can set the point-and-figure-box-size for fixed or a user-based case, or use ATR by setting point-and-figure-period.\n\nNext, add a financial-chart-series whose binding is set High,Low,Open,Close. Then set the chart style and tooltip.\n\nWe've configured the chart! When we run the application, the Point and Figure Chart uses Facebook's data to show objective price movements from the point of an investor.", null, "" ]

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[ "# Programming Fundamentals/Math Statistics with Arrays\n\n## Overview\n\nStatistics is a branch of mathematics dealing with the collection, organization, analysis, interpretation, and presentation of data. Common statistical methods include mean (or average) and standard deviation.\n\n## Discussion\n\nArrays can store words, letters/characters (strings) and numbers (integers/floats). Any type of array containing words, numbers or a combination can use a built-in function like `len`(Python exclusive) to find the number of elements in an array to help display output and parse lines. All arrays can also handle functions that allow the user to sort array values from highest to lowest (or vice versa). Other functions are only intended to handle arrays with numbers. For example, When arrays contain numbers, the elements of the array can be added together using the `sum` function. Since the built-in `sum` function cannot handle strings without producing an unsupported operand type error, we use this function only to add numbers, rather than join strings together.\n\nWe will continue learning about the `sum` function (also known as totaling) in this module. In the example below, the `sum` function totals the array passed to it. Other mathematical functions often associated with statistics such as: average, count, minimum, maximum, standard deviation, etc. are often developed for processing arrays.\n\n### Pseudocode\n\n```Function Main\nDeclare Integer Array ages\nDeclare Integer total\n\nAssign ages = [49, 48, 26, 19, 16]\n\nAssign total = sum(ages)\n\nOutput \"Total age is: \" & total\nEnd\n\nFunction sum (Integer Array array)\nDeclare Integer total\nDeclare Integer index\n\nAssign total = 0\nFor index = 0 to Size(array) - 1\nAssign total = total + array[index]\nEnd\nReturn Integer total\n```\n\n### Output\n\n```Total age is: 158\n```\n\n## Key Terms\n\nsum\nis a built-in function, which adds the elements of an array together.\nlen\nis a built-in function and it returns the number of items in an object." ]

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[ "# North Carolina\n\n X Z L B I L T M O R E E S T S C P T F S F U K M P B A E P F V B L S S G C H Q A L O P M D S O O P A T N A V G N D T K D O A W P G W N E B O R R C R T N U B R E S M N I R L L E B N W G C T N A V E A E R Y O M A R A R E I I F N C B N D U E V R E P T O X V E L E B E S F V E N R T I Y A R I E D L T R A E R E Y R U D A N A O K T L J S S A G G A P A Y W I M N E L W X E R I I S K O C F N D O N L F X O F P O T B A R W R S N E N H I O S I L T A O C O N A L U F T E E I N D I A N S L T R D N C S O D R P D O Q E S C V A I K R B S L A Y M U E S U M E N A L Y R O M E M A O F L A Y G R I W S I G O B E H L U F L J R Y N M Z U L G V O N C U L L R W O O T P O V L B M S S K E C I I K E O R S A N E B U M I W E R O A E T S C T G L C P I R A T E N H S M K L S A A L D A Q O T I L I M D E M I T S L E E H W L C T L E E Y T E L E H O R N E C R E E K F A R M E X E N R J T N R P W I L D W E S T R R I N N N W R B T G J O N I U S A R E T T A H E P A C O M N N Y O Z N N E H T K C U R T E R I F E C N A R F A L N I D A B W B\n Find and circle the words below in the puzzle grid. The words may read down, left to right, right to left, up, or diagonally. Some words may share letters with another word. Ignore spaces in multi-word names. Abbreviations: SD for SOUND, VIL for VILLAGE.\n\n AURORA FOSSILS BADIN LAFRANCE FIRETRUCK BELLAMY MANSION BENNETT CLASSICS AUTOS BENNETT PLACE SURRENDER BILTMORE EST (ESTATE) BLUE RIDGE CAPE FEAR RIVERBOAT CAPE HATTERAS CARL J. MCEWEN VIL CORE SD WATERFOWL EDENTON FRANKLIN GEMS GREAT SMOKIES GROVEWOOD HALIFAX RESOLVES HICKORY RIDGE HORNE CREEK FARM JOHN BLUE COTTON KORNER'S (KöRNER'S) FOLLY LATTA PLANTN (PLANTATION) MEMORY LANE MUSEUM MORAVIANS NEW BERN FIREMEN OCONALUFTEE INDIANS OLD SALEM OUTER BANKS PIRATE REED GOLDMINE REYNOLDA HOUSE STAGVILLE TILLERY (RESETTLEMENT FARM) TOWN CRK (CREEK) INDIANS TRYON PAL (PALACE) WHEELS (THROUGH) TIME (VOLLIS SIMPSON) WHIRLIGIG WILD WEST R.R. (RAILROAD)\n\nSolution" ]

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[ "", null, "", null, "", null, "", null, "", null, "", null, "", null, "Re: Variant of inner Product ...\n\n• To: mathgroup at smc.vnet.net\n• Subject: [mg56738] Re: [mg56683] Variant of inner Product ...\n• From: Bob Hanlon <hanlonr at cox.net>\n• Date: Thu, 5 May 2005 06:01:21 -0400 (EDT)\n• Sender: owner-wri-mathgroup at wolfram.com\n\n```A sum of equal length Lists is the List of sums. Hence your requested output\nis\n\n{f[1,a],f[1,b]}+{f[2,c],f[2,d]}+{f[3,r],f[3,s]}\n\n{f(1,a)+f(2,c)+f(3,r),f(1,b)+f(2,d)+f(3,s)}\n\nwhich is to say\n\nA={1,2,3};B={{a,b},{c,d},{r,s}};\n\n{f[1,a],f[1,b]}+{f[2,c],f[2,d]}+{f[3,r],f[3,s]} == Inner[f,A,B]\n\nTrue\n\nBob Hanlon\n\n>\n> From: Detlef Müller at smc.vnet.net\nTo: mathgroup at smc.vnet.net\n> Date: 2005/05/04 Wed AM 12:32:49 EDT\n> Subject: [mg56738] [mg56683] Variant of inner Product ...\n>\n> Hello,\n>\n> I have the following to do:\n>\n> Given\n>\n> In:= A={1,2,3}; B={{a,b},{c,d},{r,s}};\n>\n> And a Function f, I like to have\n>\n> Out = {f[1,a],f[1,b]}+{f[2,c],f[2,d]}+{f[3,r],f[3,s]}\n>\n> The trial\n>\n> In:=A={1,2,3}; B={{a,b},{c,d,e},{r,s}};\n> In:= Inner[f,A,B]\n> Out= f[1,{a,b}]+f[2,{c,d,e}]+f[3,{r,s}]\n>\n> looks promising,\n> but if the Lists in B have the same length, \"Inner\"\n> makes something different:\n>\n> In:=\n> A={1,2,3}; B={{a,b},{c,d},{r,s}}; Inner[f,A,B]\n>\n> Out= {f[1,a]+f[2,c]+f[3,r],f[1,b]+f[2,d]+f[3,s]}\n>\n> So for now I have an ugly Table-Construction doing the job,\n> but I can't imagine there is no elegant and clear solution\n> for this ... any suggestions?\n>\n> Greetings,\n> Detlef\n>\n>\n\n```\n\n• Prev by Date: Re: named pattern variable scoped as global, should be local\n• Next by Date: Re: named pattern variable scoped as global, should be local\n• Previous by thread: Re: Variant of inner Product ...\n• Next by thread: Re: Variant of inner Product ..." ]

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[ "# ICSE Class 10 Physics Syllabus 2022-23: Download Revised Class 10th Physics Syllabus PDF\n\nICSE Class 10 Physics Syllabus 2023: Physics is a key part of the science subject in ICSE Class 10 and has a vast syllabus. Check here Revised ICSE Class 10th Physics syllabus for the 2022-23 exam session along with the exam pattern and internal assessment details.", null, "", null, "ICSE Board Class 10th Physics Syllabus for 2022-23 Session Year: Download Free PDF\n\nICSE Class 10th Physics Syllabus 2023: Physics paper is part of the science subject in  ICSE Class 10, and studying science is a must for all Indian Certificate of Secondary Education (ICSE) students, making physics a crucial sub-subject for ICSE Class 10. Science (Code: 52) comprises three papers: 1-Physics, 2-Chemistry, and 3-Biology. The Physics paper is the most challenging due to its vast syllabus and unfamiliarity to students. However, like all other subjects, it only takes some dedicated effort and practice to master physics as well. Most students are currently in the last stage of their ICSE class 10 board exam preparation and gearing up for revision. Take a look at the ICSE Class 10 Physics syllabus to make sure you haven't missed out on any topics. Read and download the latest and revised ICSE Board Class 10 Physics syllabus 2023 pdf here.\n\nAlso Read: ICSE Class 10th Datesheet 2023 Released: Check the complete exam date sheet and guideline here\n\n## ICSE Board Class 10 Physics Syllabus\n\nThe ICSE class 10 Physics paper will carry 80 marks, and the duration will be two hours. There will also be a separate internal assessment of 20 marks to test students’ ability to put the theoretical topics to practical use. The internal assessment will comprise two assignments.\n\nThe SI Units for both teaching and learning are also covered in the ICSE Class 10 Physics syllabus.\n\n1. Force, Work, Power and Energy\n\n(i) Turning forces concept; moment of a force; forces in equilibrium; centre of gravity; [discussions using simple examples and simple numerical problems].\n\nElementary introduction of translational and rotational motions; moment\n\n(turning effect) of a force, also called torque and its cgs and SI units; common examples - door, steering wheel, bicycle pedal, etc.; clockwise and anti-clockwise moments; conditions for a body to be in equilibrium ( translational and rotational); principle of moment and its verification using a metre rule suspended by two spring balances with slotted weights hanging from it; simple numerical problems; Centre of gravity (qualitative only) with examples of some regular bodies and irregular lamina.\n\n(ii) Work, energy, power and their relation with force.\n\nDefinition of work. W=FScosθ; special cases of θ = 00, 900. W= mgh. Definition of energy, energy as work done. Various units of work and energy and their relation with SI units. [erg, calorie, kW h and eV]. Definition of Power, P=W/t; SI and cgs units; other units, kilowatt (kW), megawatt (MW) and gigawatt (GW); and horsepower (1hp=746W) [Simple numerical problems on work, power and energy].\n\n(iii) Different types of energy (e.g., chemical energy, Mechanical energy, heat energy, electrical energy, nuclear energy, sound energy, light energy).\n\nMechanical energy: potential energy U = mgh (derivation included) gravitational PE, examples; kinetic energy K= ½ mv2 (derivation included); forms of kinetic energy: translational, rotational and vibrational - only simple examples. [Numerical problems on K and U only in case of translational motion]; qualitative discussions of electrical, chemical, heat, nuclear, light and sound energy, conversion from one form to another; common examples.\n\n(iv) Machines as force multipliers; load, effort, mechanical advantage, velocity ratio and efficiency; pulley systems showing the utility of each type of machine.\n\nFunctions and uses of simple machines: Terms- effort E, load L, mechanical advantage MA = L/E, velocity ratio VR = VE/VL = dE / dL, input (Wi), output (Wo), efficiency (η), relation between η and MA, VR (derivation included); for all practical machines η <1; MA < VR.\n\nPulley system: single fixed, single movable, block and tackle; MA, VR and η in each case. [Pulleys using single tackle]\n\n(v) Principle of Conservation of energy.\n\nStatement of the principle of conservation of energy; theoretical verification that U + K = constant for a freely falling body. Application of this law to simple pendulum (qualitative only); [simple numerical problems].\n\n1. Light\n\n(i) Refraction of light through a glass block and a triangular prism - qualitative treatment of simple applications such as real and apparent depth of objects in water and apparent bending of sticks in water. Applications of refraction of light.\n\nPartial reflection and refraction due to change in medium. Laws of refraction; the effect on speed (V), wavelength (λ) and frequency (f) due to refraction of light; conditions for a light ray to pass undeviated. Values of speed of light (c) in vacuum, air, water and glass; refractive index μ = c/V, V = fλ. Values of μ for common substances such as water, glass and diamond; experimental verification; refraction through glass block; lateral displacement; multiple images in thick glass plate / mirror; [Diagrammatic representation not to be tested]; refraction through a glass prism, simple applications: real and apparent depth of objects in water; apparent bending of a stick under water. (Simple numerical problems and approximate ray diagrams required).\n\n(ii) Total internal reflection: Critical angle; examples in triangular glass prisms; comparison with reflection from a plane mirror (qualitative only). Applications of total internal reflection.\n\nTransmission of light from a denser medium (glass/water) to a rarer medium (air) at different angles of incidence; critical angle (C) μ = 1/sin C. Essential conditions for total internal reflection. Total internal reflection in a triangular glass prism; ray diagram, different cases - angles of prism (60º,60º,60º), (60º,30º,90º), (45º,45º,90º); use of right-angle prism to obtain δ = 90º and 180º (ray diagram); comparison of total internal reflection from a prism and reflection from a plane mirror.\n\n(iii) Lenses (converging and diverging) including characteristics of the images formed (using ray diagrams only); magnifying glass; location of images using ray diagrams and thereby determining magnification.\n\nTypes of lenses (converging and diverging), convex and concave, action of a lens as a set of prisms; technical terms; centre of curvature, radii of curvature, principal axis, foci, focal plane and focal length; detailed study of refraction of light in spherical lenses through ray diagrams; formation of images - principal rays or construction rays; location of images from ray diagram for various positions of a small linear object on the principal axis; characteristics of images. Sign convention and direct numerical problems using the lens formula are included (derivation of formula not required).\n\nScale drawing or graphical representation of ray diagrams not required.\n\nPower of a lens (concave and convex) – [simple direct numerical problems]: magnifying glass or simple microscope: location of image and magnification from ray diagram only [formula and numerical problems not included]. Applications of lenses.\n\n(iv) Using a triangular prism to produce a visible spectrum from white light; Electromagnetic spectrum.\n\nDeviation produced by a triangular prism; dependence on colour (wavelength) of light; dispersion and spectrum; electromagnetic spectrum: broad classification (names only arranged in order of increasing wavelength); properties common to all electromagnetic radiations; properties and uses of infrared and ultraviolet radiation.\n\n1. Sound\n\n(i) Reflection of Sound Waves; echoes: their use; simple numerical problems on echoes.\n\nProduction of echoes, condition for formation of echoes; simple numerical problems; use of echoes by bats, dolphins, fishermen, medical field. SONAR.\n\n(ii) Natural vibrations, Damped vibrations, Forced vibrations and Resonance - a special case of forced vibrations.\n\nMeaning and simple applications of natural, damped, forced vibrations and resonance.\n\n(iii) Loudness, pitch and quality of sound:\n\nDefinitions of the characteristics of sound and factors affecting them only.\n\n1. Electricity and Magnetism\n\n(i) Ohm’s Law; concepts of emf, potential difference, resistance; resistances in series and parallel, internal resistance.\n\nConcepts of pd (V), current (I), resistance (R) and charge (Q). Ohm's law: statement, V=IR; SI units; experimental verification; graph of V vs I and resistance from slope; ohmic and non-ohmic resistors, factors affecting resistance (including specific resistance) and internal resistance; super conductors, electromotive force (emf); combination of resistances in series and parallel. Simple numerical problems using the above relations. [Simple network of resistors including not more than four external resistors. Internal resistance may be included].\n\n(ii) Electrical power and energy.\n\nElectrical energy; examples of heater, motor, lamp, loudspeaker, etc. Electrical power; measurement of electrical energy, W = QV = VIt from the definition of pd. Combining with ohm’s law W = VIt = I2 Rt = (V2/R)t and electrical power P = (W/t) = VI = I2R = V2/R. Units: SI and commercial; Power rating of common appliances, household consumption of electric energy; calculation of total energy consumed by electrical appliances; W = Pt (kilowatt × hour = kW h), [simple numerical problems].\n\n(iii) Household circuits – main circuit; switches; fuses; earthing; safety precautions; three-pin plugs; colour coding of wires.\n\nHouse wiring (ring system – no diagrammatic representation), power distribution; main circuit (3 wires-live, neutral, earth) with fuse / MCB, main switch and its advantages - circuit diagram, need for earthing, fuse, 3-pin plug and socket; Conventional location of live, neutral and earth points in 3 pin plugs and sockets. Safety precautions, colour coding of wires.\n\n(iv) Magnetic effect of a current (principles only, statement of laws not required); electromagnetic induction (elementary).\n\nOersted’s experiment on the magnetic effect of electric current; magnetic field (B) and field lines due to current in a straight wire (qualitative only); Right Hand Thumb Rule – magnetic field due to a current in a loop; Electromagnets: their uses; comparisons with a permanent magnet; conductor carrying current in a magnetic field experiences a force, Fleming’s Left Hand Rule, and its understanding, Simple introduction to electromagnetic induction; a magnet moved along the axis of a solenoid induces current, Fleming’s Right Hand Rule and its application in understanding the direction of current in a coil and Lenz’s law. Comparison of AC and DC.\n\n1. Heat\n\n(i) Calorimetry: meaning, specific heat capacity; principle of method of mixtures; Numerical Problems on specific heat capacity using heat loss and gain and the method of mixtures.\n\nHeat and its units (calorie, joule), temperature and its units (oC,, K); thermal (heat) capacity C' = Q/êT... (SI unit of C'): Specific heat Capacity C = Q/mêT (SI unit of C) Mutual relation between Heat Capacity and Specific Heat capacity, values of C for some common substances (ice, water and copper). Principle of method of mixtures including mathematical statement. Natural phenomenon involving specific heat. Consequences of high specific heat of water. [Simple numerical problems].\n\n(ii) Latent heat; loss and gain of heat involving change of state for fusion only.\n\nChange of phase (state); heating curve for water; latent heat; specific latent heat of fusion (SI unit). Simple numerical problems. Common physical phenomena involving latent heat of fusion.\n\n1. Modern Physics\n\nBrief introduction (qualitative only) of the nucleus, nuclear structure, atomic number (Z), mass number (A). Radioactivity as spontaneous disintegration. α, β and γ- their nature and properties; changes within the nucleus. One example each of α and β decay with equations showing changes in Z and A.\n\nNuclear Energy: working on safe disposal of waste. Safety measures to be strictly reinforced.\n\nINTERNAL ASSESSMENT OF PRACTICAL WORK\n\nCandidates will be asked to carry out experiments for which instructions will be given. The experiments may be based on topics that are not included in the syllabus but theoretical knowledge will not be required. A candidate will be expected to be able to follow simple instructions, to take suitable readings and to present these readings in a systematic form. He/she may be required to exhibit his/her data graphically. Candidates will be expected to appreciate and use the concepts of least count, significant figures and elementary error handling.\n\nNote: Teachers may design their own set of experiments, preferably related to the theory syllabus. A comprehensive list is suggested below:\n\n1. Lever - There are many possibilities with a meter rule as a lever with a load (known or unknown) suspended from a point near one end (say left), the lever itself pivoted on a knife edge, use slotted weights suspended from the other (right) side for effort.\n\nDetermine the mass of a metre rule using a spring balance or by balancing it on a knife edge at some point away from the middle and a 50g weight on the other side. Next pivot (F) the metre rule at the 40cm, 50cm and 60cm mark, each time suspending a load L or the left end and effort E near the right end. Adjust E and or its position so that the rule is balanced. Tabulate the position of L, F and E and the magnitudes of L and E and the distances of load arm and effort arm. Calculate MA=L/E and VR = effort arm/load arm. It will be found that MA <VR in one case, MA=VR in another and MA>VR in the third case. Try to explain why this is so. Also try to calculate the real load and real effort in these cases.\n\n1. Determine the VR and MA of a given pulley system.\n2. Trace the course of different rays of light refracting through a rectangular glass slab at different angles of incidence, measure the angles of incidence, refraction and emergence. Also measure the lateral displacement.\n3. Determine the focal length of a convex lens by (a) the distant object method and (b) using a needle and a plane mirror.\n4. Determine the focal length of a convex lens by using two pins and formula f = uv/(u+v).\n5. For a triangular prism, trace the course of rays passing through it, measure angles i1, i2, A and δ.Repeat for four different angles of incidence (say i1=400, 500, 600 and 700). Verify i1+ i2=A+δ and A = r1 + r2.\n6. For a ray of light incident normally (i1=0) on one face of a prism, trace course of the ray. Measure the angle δ. Explain briefly. Do this for prisms with A=600, 450and 900.\n7. Calculate the specific heat capacity of the material of the given calorimeter, from the temperature readings and masses of cold water, warm water and its mixture taken in the calorimeter.\n8. Determination of specific heat capacity of a metal by method of mixtures.\n9. Determination of specific latent heat of ice.\n10. Using as simple electric circuit, verify Ohm’s law. Draw a graph and obtain the slope.\n11. Set up model of household wiring including ring main circuit. Study the function of switches and fuses.\n\nTeachers may feel free to alter or add to the above list. The students may perform about ten experiments. Some experiments may be demonstrated.\n\nRelated: ICSE Exam Preparation Tips and Study Time Table to score 95+ in ICSE Board exam 2023\n\nAlso Check: ICSE Class 10 Syllabus 2023\n\nThe ICSE class 10 board exams are approaching, and the time table has also been released. Now is the time to start giving mock tests. It boosts confidence and gives students an idea of what and what not to do in the final exams. Check the ICSE Class 10 mock tests here.\n\nICSE - Class X Mock Tests\n\nPhysics is a key component of the science course in ICSE Class 10 and is essential to study for all students. But don't forget to prepare other subjects. Check the syllabus of related ICSE board 10th Class subjects below.\n\nRead: ICSE Class 10 Science Syllabus 2023\n\n## FAQ\n\n### Is the ICSE Board Class 10 Physics syllabus tough?\n\nYes, the ICSE Board Class 10th Physics syllabus is much bigger and tougher than the curriculum of other boards. Physics in ICSE is part of the science subject and a separate, full-length paper is conducted for it. ICSE class 10 deals with many advanced physics topics in detail.\n\n### What topics are taught in the ICSE Class 10 Physics syllabus?\n\nThe syllabus of ICSE Class 10 Physics is quite different from other boards and emphasizes much more on practical applications of theoretical concepts. The topics taught in ICSE Class 10 Physics syllabus are Force, Work, Power and Energy, Light, Sound, Electricity and Magnetism, Heat, and Modern Physics.\n\n## Related Categories\n\nखेलें हर किस्म के रोमांच से भरपूर गेम्स सिर्फ़ जागरण प्ले पर" ]

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[ "# RF Power Density Calculator\n\nPower Gain of Antena\ncm\nInput power to Antena\nmW\nPower Density  =  0.2764mW/cm2\n<embed />\nCALCULATE\n\nRF power density calculator is an online tool for electrical and electronic circuits to measure the RF Frequency Power Density. RF waves are the electromagnetic waves transmission along a medium or free space. This calculator uses the input values of Antenna power gain, Radiation distance and antenna input power to determine the RF frequency propagation power density. It can be mathematically derived from the following formula", null, "Where\ns is the Power density\nP is the Antenna power input\nG is the Antenna power gain\nR is the Distance to center of Antenna radiation" ]

[ null, "https://ncalculators.com/images/formulas/rf-power-density-formula.jpg", null ]

[ "blob: 2283fdbc5a108876e8f6d861dd993748da133ed4 [file] [log] [blame]\n /*-------------------------------------------------------------*/ /*--- Huffman coding low-level stuff ---*/ /*--- huffman.c ---*/ /*-------------------------------------------------------------*/ /* ------------------------------------------------------------------ This file is part of bzip2/libbzip2, a program and library for lossless, block-sorting data compression. bzip2/libbzip2 version 1.0.6 of 6 September 2010 Copyright (C) 1996-2010 Julian Seward Please read the WARNING, DISCLAIMER and PATENTS sections in the README file. This program is released under the terms of the license contained in the file LICENSE. ------------------------------------------------------------------ */ #include \"bzlib_private.h\" /*---------------------------------------------------*/ #define WEIGHTOF(zz0) ((zz0) & 0xffffff00) #define DEPTHOF(zz1) ((zz1) & 0x000000ff) #define MYMAX(zz2,zz3) ((zz2) > (zz3) ? (zz2) : (zz3)) #define ADDWEIGHTS(zw1,zw2) \\ (WEIGHTOF(zw1)+WEIGHTOF(zw2)) | \\ (1 + MYMAX(DEPTHOF(zw1),DEPTHOF(zw2))) #define UPHEAP(z) \\ { \\ Int32 zz, tmp; \\ zz = z; tmp = heap[zz]; \\ while (weight[tmp] < weight[heap[zz >> 1]]) { \\ heap[zz] = heap[zz >> 1]; \\ zz >>= 1; \\ } \\ heap[zz] = tmp; \\ } #define DOWNHEAP(z) \\ { \\ Int32 zz, yy, tmp; \\ zz = z; tmp = heap[zz]; \\ while (True) { \\ yy = zz << 1; \\ if (yy > nHeap) break; \\ if (yy < nHeap && \\ weight[heap[yy+1]] < weight[heap[yy]]) \\ yy++; \\ if (weight[tmp] < weight[heap[yy]]) break; \\ heap[zz] = heap[yy]; \\ zz = yy; \\ } \\ heap[zz] = tmp; \\ } /*---------------------------------------------------*/ void BZ2_hbMakeCodeLengths ( UChar *len, Int32 *freq, Int32 alphaSize, Int32 maxLen ) { /*-- Nodes and heap entries run from 1. Entry 0 for both the heap and nodes is a sentinel. --*/ Int32 nNodes, nHeap, n1, n2, i, j, k; Bool tooLong; Int32 heap [ BZ_MAX_ALPHA_SIZE + 2 ]; Int32 weight [ BZ_MAX_ALPHA_SIZE * 2 ]; Int32 parent [ BZ_MAX_ALPHA_SIZE * 2 ]; for (i = 0; i < alphaSize; i++) weight[i+1] = (freq[i] == 0 ? 1 : freq[i]) << 8; while (True) { nNodes = alphaSize; nHeap = 0; heap = 0; weight = 0; parent = -2; for (i = 1; i <= alphaSize; i++) { parent[i] = -1; nHeap++; heap[nHeap] = i; UPHEAP(nHeap); } AssertH( nHeap < (BZ_MAX_ALPHA_SIZE+2), 2001 ); while (nHeap > 1) { n1 = heap; heap = heap[nHeap]; nHeap--; DOWNHEAP(1); n2 = heap; heap = heap[nHeap]; nHeap--; DOWNHEAP(1); nNodes++; parent[n1] = parent[n2] = nNodes; weight[nNodes] = ADDWEIGHTS(weight[n1], weight[n2]); parent[nNodes] = -1; nHeap++; heap[nHeap] = nNodes; UPHEAP(nHeap); } AssertH( nNodes < (BZ_MAX_ALPHA_SIZE * 2), 2002 ); tooLong = False; for (i = 1; i <= alphaSize; i++) { j = 0; k = i; while (parent[k] >= 0) { k = parent[k]; j++; } len[i-1] = j; if (j > maxLen) tooLong = True; } if (! tooLong) break; /* 17 Oct 04: keep-going condition for the following loop used to be 'i < alphaSize', which missed the last element, theoretically leading to the possibility of the compressor looping. However, this count-scaling step is only needed if one of the generated Huffman code words is longer than maxLen, which up to and including version 1.0.2 was 20 bits, which is extremely unlikely. In version 1.0.3 maxLen was changed to 17 bits, which has minimal effect on compression ratio, but does mean this scaling step is used from time to time, enough to verify that it works. This means that bzip2-1.0.3 and later will only produce Huffman codes with a maximum length of 17 bits. However, in order to preserve backwards compatibility with bitstreams produced by versions pre-1.0.3, the decompressor must still handle lengths of up to 20. */ for (i = 1; i <= alphaSize; i++) { j = weight[i] >> 8; j = 1 + (j / 2); weight[i] = j << 8; } } } /*---------------------------------------------------*/ void BZ2_hbAssignCodes ( Int32 *code, UChar *length, Int32 minLen, Int32 maxLen, Int32 alphaSize ) { Int32 n, vec, i; vec = 0; for (n = minLen; n <= maxLen; n++) { for (i = 0; i < alphaSize; i++) if (length[i] == n) { code[i] = vec; vec++; }; vec <<= 1; } } /*---------------------------------------------------*/ void BZ2_hbCreateDecodeTables ( Int32 *limit, Int32 *base, Int32 *perm, UChar *length, Int32 minLen, Int32 maxLen, Int32 alphaSize ) { Int32 pp, i, j, vec; pp = 0; for (i = minLen; i <= maxLen; i++) for (j = 0; j < alphaSize; j++) if (length[j] == i) { perm[pp] = j; pp++; }; for (i = 0; i < BZ_MAX_CODE_LEN; i++) base[i] = 0; for (i = 0; i < alphaSize; i++) base[length[i]+1]++; for (i = 1; i < BZ_MAX_CODE_LEN; i++) base[i] += base[i-1]; for (i = 0; i < BZ_MAX_CODE_LEN; i++) limit[i] = 0; vec = 0; for (i = minLen; i <= maxLen; i++) { vec += (base[i+1] - base[i]); limit[i] = vec-1; vec <<= 1; } for (i = minLen + 1; i <= maxLen; i++) base[i] = ((limit[i-1] + 1) << 1) - base[i]; } /*-------------------------------------------------------------*/ /*--- end huffman.c ---*/ /*-------------------------------------------------------------*/" ]

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[ "# Volts to resistance (Ohm’s law) – Formula, examples, conversion\n\nWith this calculator you can convert from volts to resistance using Ohm’s law.\n\nTo facilitate the compression, the formula of volts to resistance is explained , in addition we show how to convert from volts to resistance in 1 step , some examples of volts are made to resistance and finally a table of volts to resistance is presented .\n\nMas información:\n\n## Formula to convert volts to resistance (Ohm Law):", null, "• R = Resistance in Ohm.\n• I = Current in Amperes.\n• V = Voltage.\n\n## How to convert volts to Resistance in 1 step only:", null, "Step 1:\n\nTo convert from volts to resistance, only the voltage between the current must be divided with the formula of the law of ohm.\n\nFor example, if you have a server with a voltage of 12V DC and an amperage of 0.55Amp, to know the resistance you must divide 12 between 0.55, the result will be: 21.8Ohm. Formula: 12 / 0.55 = 21.8 ohm.\n\n## Examples for conversions from volts to resistance:\n\nExample 1:\n\nA DC motor for a remote control car has a voltage of 14.5V and an amperage of 0.25A, what would be the resistance of the DC motor of this car to remote control ?.\n\nAnswer: // To know the answer you must divide the voltage between the current in the following way: 14.5V / 0.25A = 58ohm.\n\nExample 2:\n\nAn ebook or e-reader has a voltage of 8.2V and an amperage of 2.3Amp, which is the resistance of the electronic equipment of the ebook ?.\n\nAnswer: // To make the conversion, the voltage must be divided between the amperes, as indicated by the ohm law as follows: 8.2V / 2.3A = 3.5 ohm.\n\nExample 3:\n\nA portable computer has a voltage of 8.8 V and 3.3 Amp, what will be the resistance that will have the electronic equipment of the computer ?.\n\nAnswer: // It’s easy you just have to divide the voltage between the amperage as indicated in the formula: 3.5V / 2.4A, which will result in: 2.6 ohm.\n\n## Table of volts to resistance (Amperes: 10Amp):\n\n How many volts are: Equivalence in resistance (Ohm) 1.1 Volts 0.1 Ohm 1.7 Volts 0.1 Ohm 2.3 Volts 0.2 Ohm 2.9 Volts 0.2 Ohm 3.5 Volts 0.3 Ohm 4.1 Volts 0.4 Ohm 4.7 Volts 0.4 Ohm 5.3 Volts 0.5 Ohm 5.9 Volts 0.5 Ohm 10.6 Volts 1.0 Ohm 20.8 Volts 2.0 Ohm 30.2 Volts 3.0 Ohm 40.3 Volts 4.0 Ohm 50.7 Volts 5.0 Ohm 60.1 Volts 6.0 Ohm 70.5 Volts 7.0 Ohm 80.4 Volts 8.0 Ohm 90.6 Volts 9.0 Ohm 100.8 Volts 10.0 Ohm 110.9 Volts 11.0 Ohm 120.2 Volts 12.0 Ohm 130.6 Volts 13.0 Ohm 140.3 Volts 14,0 Ohm 150.8 Volts 15,0 Ohm 160.3 Volts 16.0 Ohm 170.1 Volts 17.0 Ohm 180.2 Volts 18.0 Ohm 190.0 Volts 19.0 Ohm 200.0 Volts 20.0 Ohm\n\nNote: The changes of volts to resistance of the previous table were made taking into account a 10V amperage. For different variables you should use the calculator that appears at the beginning.\n\n## How to use the calculator:\n\nYou must enter the voltage and amperage, then just click on calculate, and as a result you will have resistance.\n\nRate conversion of volts to resistance (Ohm Law):  [kkstarratings]" ]

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[ "", null, "Auto Byte", null, "", null, "Science AI", null, "AISHWARYA SINGH来源车前子校对赵雪尧翻译\n\n# 利用深度学习和机器学习预测股票市场（附代码）\n\n### 简介", null, "• 创建时间序列预测的初学者综合指南\n\n• 时间序列建模的完整教程\n\n1、 问题理解\n\n2、 移动平均\n\n3、 线性回归\n\n4、 K-近邻\n\n5、 自动ARIMA\n\n6、 先知（Prophet）\n\n7、 长短时记忆网络（LSTM）\n\n## 1、问题理解\n\n• 基本面分析是根据公司目前的经营环境和财务状况，对公司未来的盈利能力进行分析。\n\n• 技术分析包括阅读图表和使用统计数字来确定股票市场的趋势。\n\n```#import packages\nimport pandas as pd\nimport numpy as np\n\n#to plot within notebook\nimport matplotlib.pyplot as plt\n%matplotlib inline\n\n#setting figure size\nfrom matplotlib.pylab import rcParams\nrcParams['figure.figsize'] = 20,10\n\n#for normalizing data\nfrom sklearn.preprocessing import MinMaxScaler\nscaler = MinMaxScaler(feature_range=(0, 1))", null, "• 开盘价和收盘价代表股票在某一天交易的起始价和最终价。\n\n• 最高价、最低价和最后交易价表示当天股票的最高价、最低价和最后交易价格。\n\n• 交易总量是指当天买卖的股票数量，而营业额(Lacs)是指某一特定公司在某一特定日期的营业额。\n\n```#setting index as date\ndf['Date'] = pd.to_datetime(df.Date,format='%Y-%m-%d')\ndf.index = df['Date']\n\n#plot\nplt.figure(figsize=(16,8))\nplt.plot(df['Close'], label='Close Price history')```", null, "## 2、移动平均\n\n#### 简介\n\n“平均数”是我们日常生活中最常用的统计量之一。例如，计算平均分数来确定整体表现，或者根据过去几天的平均温度来了解今天的温度——这些都是我们经常做的日常工作。因此，使用这个方法开始用数据集进行预测是个不错的选择。", null, "#### 实现\n\n```#creating dataframe with date and the target variable\ndata = df.sort_index(ascending=True, axis=0)\nnew_data = pd.DataFrame(index=range(0,len(df)),columns=['Date', 'Close'])\n\nfor i in range(0,len(data)):\nnew_data['Date'][i] = data['Date'][i]\nnew_data['Close'][i] = data['Close'][i]```\n\n```#splitting into train and validation\ntrain = new_data[:987]\nvalid = new_data[987:] ```\n```new_data.shape, train.shape, valid.shape\n((1235, 2), (987, 2), (248, 2))```\n```train['Date'].min(), train['Date'].max(), valid['Date'].min(), valid['Date'].max()\n\n(Timestamp('2013-10-08 00:00:00'),\nTimestamp('2017-10-06 00:00:00'),\nTimestamp('2017-10-09 00:00:00'),\nTimestamp('2018-10-08 00:00:00'))```\n\n```#make predictions\npreds = []\nfor i in range(0,248):\na = train['Close'][len(train)-248+i:].sum() + sum(preds)\nb = a/248\npreds.append(b)```\n\n#### 结果\n\n```#calculate rmse\nrms=np.sqrt(np.mean(np.power((np.array(valid['Close'])-preds),2)))\nrms\n\n104.51415465984348```\n\n```#plot\nvalid['Predictions'] = 0\nvalid['Predictions'] = preds\nplt.plot(train['Close'])\nplt.plot(valid[['Close', 'Predictions']])```", null, "#### 推论\n\nRMSE值接近105，但是结果不是很理想(从图中可以看出)。预测值与训练集的观测值的范围相同(开始有上升趋势，然后缓慢下降)。\n\n## 3、线性回归\n\n#### 简介\n\n线性回归方程可以写成：", null, "• 对于线性回归和Lasso回归的一个综合的初学者指南.\n\n#### 实现\n\n```#setting index as date values\ndf['Date'] = pd.to_datetime(df.Date,format='%Y-%m-%d')\ndf.index = df['Date']\n\n#sorting\ndata = df.sort_index(ascending=True, axis=0)\n\n#creating a separate dataset\nnew_data = pd.DataFrame(index=range(0,len(df)),columns=['Date', 'Close'])\n\nfor i in range(0,len(data)):\nnew_data['Date'][i] = data['Date'][i]\nnew_data['Close'][i] = data['Close'][i]\n\n#create features\nfrom fastai.structured import  add_datepart\nnew_data.drop('Elapsed', axis=1, inplace=True)  #elapsed will be the time stamp```\n\n`‘Year’, ‘Month’, ‘Week’, ‘Day’, ‘Dayofweek’, ‘Dayofyear’, ‘Is_month_end’, ‘Is_month_start’, ‘Is_quarter_end’, ‘Is_quarter_start’,  ‘Is_year_end’, and  ‘Is_year_start’.`\n\n除此之外，我们还可以添加自己的一组特性，我们认为这些特性与预测相关。例如，我的假设是，本周的头几天和最后几天对股票收盘价的影响可能远远超过其他日子。因此，我创建了一个特性来识别给定的一天是星期一/星期五还是星期二/星期三/星期四。这可以用以下几行代码来完成:\n\n```new_data['mon_fri'] = 0\nfor i in range(0,len(new_data)):\nif (new_data['Dayofweek'][i] == 0 or new_data['Dayofweek'][i] == 4):\nnew_data['mon_fri'][i] = 1\nelse:\nnew_data['mon_fri'][i] = 0```\n\n```#split into train and validation\ntrain = new_data[:987]\nvalid = new_data[987:]\n\nx_train = train.drop('Close', axis=1)\ny_train = train['Close']\nx_valid = valid.drop('Close', axis=1)\ny_valid = valid['Close']\n\n#implement linear regression\nfrom sklearn.linear_model import LinearRegression\nmodel = LinearRegression()\nmodel.fit(x_train,y_train)```\n\n#### 结果\n\n```#make predictions and find the rmse\npreds = model.predict(x_valid)\nrms=np.sqrt(np.mean(np.power((np.array(y_valid)-np.array(preds)),2)))\nrms\n\n121.16291596523156```\n\nRMSE值高于之前的方法，这清楚地表明线性回归的表现很差。让我们看看这个图，并理解为什么线性回归预测效果不是很好:\n\n```#plot\nvalid['Predictions'] = 0\nvalid['Predictions'] = preds\n\nvalid.index = new_data[987:].index\ntrain.index = new_data[:987].index\n\nplt.plot(train['Close'])\nplt.plot(valid[['Close', 'Predictions']])```", null, "## 4、k-近邻\n\n#### 简介", null, "", null, "• k近邻介绍:简介\n\n• 对k近邻回归算法的实际介绍\n\n#### 实现\n\n```#importing libraries\nfrom sklearn import neighbors\nfrom sklearn.model_selection import GridSearchCV\nfrom sklearn.preprocessing import MinMaxScaler\nscaler = MinMaxScaler(feature_range=(0, 1))```\n\n```#scaling data\nx_train_scaled = scaler.fit_transform(x_train)\nx_train = pd.DataFrame(x_train_scaled)\nx_valid_scaled = scaler.fit_transform(x_valid)\nx_valid = pd.DataFrame(x_valid_scaled)\n\n#using gridsearch to find the best parameter\nparams = {'n_neighbors':[2,3,4,5,6,7,8,9]}\nknn = neighbors.KNeighborsRegressor()\n\nmodel = GridSearchCV(knn, params, cv=5)\n\n#fit the model and make predictions\nmodel.fit(x_train,y_train)\npreds = model.predict(x_valid)```\n\n#### 结果\n\n```#rmse\nrms=np.sqrt(np.mean(np.power((np.array(y_valid)-np.array(preds)),2)))\nrms\n\n115.17086550026721```\n\nRMSE值并没有太大的差异，但是一个预测值和实际值的曲线图应该提供一个更清晰的理解。\n\n```#plot\nvalid['Predictions'] = 0\nvalid['Predictions'] = preds\nplt.plot(valid[['Close', 'Predictions']])\nplt.plot(train['Close'])```", null, "#### 推论\n\nRMSE值与线性回归模型近似，图中呈现出相同的模式。与线性回归一样，kNN也发现了2018年1月的下降，因为这是过去几年的模式。我们可以有把握地说，回归算法在这个数据集上表现得并不好。\n\n## 5、自动ARIMA\n\n#### 简介\n\nARIMA是一种非常流行的时间序列预测统计方法。ARIMA模型使用过去的值来预测未来的值。ARIMA中有三个重要参数\n\n• p(用来预测下一个值的过去值)\n\n• q(用来预测未来值的过去预测误差)\n\n• d(差分的顺序)\n\nARIMA的参数优化需要大量时间。因此我们将使用自动 ARIMA，自动选择误差最小的(p,q,d)最佳组合。要了解更多关于自动ARIMA的工作原理，请参阅本文：\n\n• 利用自动ARIMA建立高性能时间序列模型\n\n#### 实现\n\n```from pyramid.arima import auto_arima\n\ndata = df.sort_index(ascending=True, axis=0)\n\ntrain = data[:987]\nvalid = data[987:]\n\ntraining = train['Close']\nvalidation = valid['Close']\n\nmodel = auto_arima(training, start_p=1, start_q=1,max_p=3, max_q=3, m=12,start_P=0, seasonal=True,d=1, D=1, trace=True,error_action='ignore',suppress_warnings=True)\nmodel.fit(training)\n\nforecast = model.predict(n_periods=248)\nforecast = pd.DataFrame(forecast,index = valid.index,columns=['Prediction'])```\n\n#### 结果\n\n```rms=np.sqrt(np.mean(np.power((np.array(valid['Close'])-np.array(forecast['Prediction'])),2)))\nrms\n44.954584993246954\n#plot\nplt.plot(train['Close'])\nplt.plot(valid['Close'])\nplt.plot(forecast['Prediction'])```", null, "## 6、先知（Prophet）\n\n#### 实现\n\n```#importing prophet\nfrom fbprophet import Prophet\n\n#creating dataframe\nnew_data = pd.DataFrame(index=range(0,len(df)),columns=['Date', 'Close'])\n\nfor i in range(0,len(data)):\nnew_data['Date'][i] = data['Date'][i]\nnew_data['Close'][i] = data['Close'][i]\n\nnew_data['Date'] = pd.to_datetime(new_data.Date,format='%Y-%m-%d')\nnew_data.index = new_data['Date']\n\n#preparing data\nnew_data.rename(columns={'Close': 'y', 'Date': 'ds'}, inplace=True)\n\n#train and validation\ntrain = new_data[:987]\nvalid = new_data[987:]\n\n#fit the model\nmodel = Prophet()\nmodel.fit(train)\n\n#predictions\nclose_prices = model.make_future_dataframe(periods=len(valid))\nforecast = model.predict(close_prices)```\n\n#### 结果\n\n```#rmse\nforecast_valid = forecast['yhat'][987:]\nrms=np.sqrt(np.mean(np.power((np.array(valid['y'])-np.array(forecast_valid)),2)))\nrms\n\n57.494461930575149\n\n#plot\nvalid['Predictions'] = 0\nvalid['Predictions'] = forecast_valid.values\n\nplt.plot(train['y'])\nplt.plot(valid[['y', 'Predictions']])```", null, "## 7、长短期记忆网络(LSTM)\n\n#### 简介\n\nLSTM 算法广泛应用于序列预测问题中，并被证明是一种非常有效的方法。它们之所表现如此出色，是因为LSTM能够存储重要的既往信息，并忽略不重要的信息。\n\nLSTM有三个门：\n\n• 输入门：输入门将信息添加到细胞状态\n\n• 遗忘门：它移除模型不再需要的信息\n\n• 输出门：LSTM的输出门选择作为输出的信息\n\n• 长短期记忆网络简介\n\n#### 实现\n\n```#importing required libraries\nfrom sklearn.preprocessing import MinMaxScaler\nfrom keras.models import Sequential\nfrom keras.layers import Dense, Dropout, LSTM\n\n#creating dataframe\ndata = df.sort_index(ascending=True, axis=0)\nnew_data = pd.DataFrame(index=range(0,len(df)),columns=['Date', 'Close'])\nfor i in range(0,len(data)):\nnew_data['Date'][i] = data['Date'][i]\nnew_data['Close'][i] = data['Close'][i]\n\n#setting index\nnew_data.index = new_data.Date\nnew_data.drop('Date', axis=1, inplace=True)\n\n#creating train and test sets\ndataset = new_data.values\n\ntrain = dataset[0:987,:]\nvalid = dataset[987:,:]\n\n#converting dataset into x_train and y_train\nscaler = MinMaxScaler(feature_range=(0, 1))\nscaled_data = scaler.fit_transform(dataset)\n\nx_train, y_train = [], []\nfor i in range(60,len(train)):\nx_train.append(scaled_data[i-60:i,0])\ny_train.append(scaled_data[i,0])\nx_train, y_train = np.array(x_train), np.array(y_train)\n\nx_train = np.reshape(x_train, (x_train.shape,x_train.shape,1))\n\n# create and fit the LSTM network\nmodel = Sequential()\n\nmodel.fit(x_train, y_train, epochs=1, batch_size=1, verbose=2)\n\n#predicting 246 values, using past 60 from the train data\ninputs = new_data[len(new_data) - len(valid) - 60:].values\ninputs = inputs.reshape(-1,1)\ninputs  = scaler.transform(inputs)\n\nX_test = []\nfor i in range(60,inputs.shape):\nX_test.append(inputs[i-60:i,0])\nX_test = np.array(X_test)\n\nX_test = np.reshape(X_test, (X_test.shape,X_test.shape,1))\nclosing_price = model.predict(X_test)\nclosing_price = scaler.inverse_transform(closing_price)```\n\n#### 结果\n\n```rms=np.sqrt(np.mean(np.power((valid-closing_price),2)))\nrms\n\n11.772259608962642\n#for plotting\ntrain = new_data[:987]\nvalid = new_data[987:]\nvalid['Predictions'] = closing_price\nplt.plot(train['Close'])\nplt.plot(valid[['Close','Predictions']])```", null, "## 写在最后", null, "THU数据派\n\nTHU数据派\"基于清华，放眼世界\"，以扎实的理工功底闯荡“数据江湖”。发布全球大数据资讯，定期组织线下活动，分享前沿产业动态。了解清华大数据，敬请关注姐妹号“数据派THU”。\n\nDropout技术\n\nLSTM或GRU中特有的机制" ]

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[ "", null, "", null, "", null, "SEARCH HOME", null, "Math Central Quandaries & Queries", null, "", null, "Question from Dakota, a student: If the diameter of a circle is 48 cm, find the shortest distance from a chord of length 34 cm to the center of the circle", null, "Hi Dakota,\n\nIn my diagram the chord is AD, C is the center of the circle and B is the midpoint of the chord.", null, "Angle ABC is a right angle. an you see why? The length of CB is the shortest distance from C to the chord. Can you see why?\n\nWhat does Pythagoras theorem tell you?\n\nPenny", null, "", null, "", null, "", null, "", null, "", null, "Math Central is supported by the University of Regina and the Imperial Oil Foundation." ]

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[ "## Extension of derivations, and Connes-amenability of the enveloping dual Banach algebra\n\n#### Authors\n\nYemon Choi*, Ebrahim Samei, Ross Stokke\n\n* Corresponding author\n\nMSC 2010: 46H20 (primary), 43A20 43A60 46H25 (secondary).\n\n#### Status\n\nAppeared as Math. Scand. 117 (2015) no. 2, 258–303\n\nPreprint version available at arXiv 1307.6285 (final accepted version, incorporating referee's recommendations)\n\n#### Reviews\n\n[ Math Review | Zentralblatt (summary) ]\n\n#### Abstract\n\nIf $D:A \\to X$ is a derivation from a Banach algebra to a contractive, Banach $A$-bimodule, then one can equip $X^{**}$ with an $A^{**}$-bimodule structure, such that the second transpose $D^{**}: A^{**} \\to X^{**}$ is again a derivation. $\\newcommand{\\F}{{\\sf F}}$ We prove an analogous extension result, where $A^{**}$ is replaced by $\\F(A)$, the \\emph{enveloping dual Banach algebra} of $A$, and $X^{**}$ by an appropriate kind of universal, enveloping, normal dual bimodule of $X$. Using this, we obtain some new characterizations of Connes-amenability of $\\F(A)$. In particular we show that $\\F(A)$ is Connes-amenable if and only if $A$ admits a so-called WAP-virtual diagonal. We show that when $A=L^1(G)$, existence of a WAP-virtual diagonal is equivalent to the existence of a virtual diagonal in the usual sense. Our approach does not involve invariant means for $G$." ]

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[ "# TubeTK/Documentation/SegmentConnectedComponentsUsingParzenPDFs\n\n(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)\n``` Description: Given multiple, registered images and foreground and\nbackground masks, computes multivariate PDFs for inside and outside\nclasses, and then performs competitive region growing.\n```\n\nUSAGE:\n\n``` ./SegmentConnectedComponentsUsingParzenPDFs [--returnparameterfile\n<std::string>]\n<std::string>] [--xml] [--echo]\n[--saveClassPDFBase <std::string>]\n[--saveClassProbabilityVolumeBase\n<std::string>]\n[--forceClassification]\n[--reclassifyNotObjectLabels]\n[--reclassifyObjectLabels]\n[--draft]\n[--histogramSmoothingStdDev\n<double>]\n[--probImageSmoothingStdDev\n<double>] [--objectPDFWeight\n<std::vector<double>>]\n[--dilateFirst]\n[--holeFillIterations <int>]\n<int>] [--objectId\n<std::vector<int>>] [--inputVolume4\n<std::string>] [--inputVolume3\n<std::string>] [--inputVolume2\n<std::string>] [--] [--version]\n[-h] <std::string> <std::string>\n<std::string>\n```\n\nWhere:\n\n``` --returnparameterfile <std::string>\nFilename in which to write simple return parameters (int, float,\nint-vector, etc.) as opposed to bulk return parameters (image,\ngeometry, transform, measurement, table).\n```\n``` --processinformationaddress <std::string>\nAddress of a structure to store process information (progress, abort,\netc.). (default: 0)\n```\n``` --xml\nProduce xml description of command line arguments (default: 0)\n```\n``` --echo\nEcho the command line arguments (default: 0)\n```\n``` --saveClassPDFBase <std::string>\nSave images that represent probability density functions.\n```\n``` --loadClassPDFBase <std::string>\nLoad images that represent probability density functions.\n```\n``` --saveClassProbabilityVolumeBase <std::string>\nSave images where each represents the probability of being a\nparticular object at each voxel. Image files create =\nbase.classNum.mha.\n```\n``` --forceClassification\nForce classification using simple maximum likelihood. (default: 0)\n```\n``` --reclassifyNotObjectLabels\nPerform classification on all non-void voxels. (default: 0)\n```\n``` --reclassifyObjectLabels\nPerform classification on voxels within the object mask. (default: 0)\n```\n``` --draft\nGenerate draft results. (default: 0)\n```\n``` --histogramSmoothingStdDev <double>\nStandard deviation of blur applied to convert the histogram to a\nprobability density function estimate (default: 5)\n```\n``` --probImageSmoothingStdDev <double>\nStandard deviation of blur applied to probability images prior to\ncomputing maximum likelihood of each class at each pixel (default: 1)\n```\n``` --objectPDFWeight <std::vector<double>>\nRelative weight (multiplier) of each PDF. (default: 1)\n```\n``` --dilateFirst\nPerforms dilation then erosion (versus opposite order) to help fill-in\nsparse models. (default: 0)\n```\n``` --holeFillIterations <int>\nNumber of iterations for hole filling. (default: 1)\n```\n``` --erodeRadius <int>\nRadius of noise to clip from edges. (default: 1)\n```\n``` --voidId <int>\nValue that represents 'nothing' in the label map. (default: 0)\n```\n``` --objectId <std::vector<int>>\nList of values that represent the objects in the label map. (default:\n255,127)\n```\n``` --inputVolume4 <std::string>\nInput volume 4.\n```\n``` --inputVolume3 <std::string>\nInput volume 3.\n```\n``` --inputVolume2 <std::string>\nInput volume 2.\n```\n``` --, --ignore_rest\nIgnores the rest of the labeled arguments following this flag.\n```\n``` --version\nDisplays version information and exits.\n```\n``` -h, --help\nDisplays usage information and exits.\n```\n``` <std::string>\n(required) Input volume 1.\n```\n``` <std::string>\n(required) Label map that designates the object of interest and\n'other.'\n```\n``` <std::string>\n(required) Segmentation results.\n```\n``` Author(s): Stephen R. Aylward (Kitware)\n```\n``` Acknowledgements: This work is part of the TubeTK project at Kitware.\n```" ]

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[ "# How to interpret the differential of a function\n\n## Main Question or Discussion Point\n\nIn elementary calculus (and often in courses beyond) we are taught that a differential of a function, $df$ quantifies an infinitesimal change in that function. However, the notion of an infinitesimal is not well-defined and is nonsensical (I mean, one cannot define it in terms of a limit, and it seems nonsensical to have a number that is smaller than any other real number - this simply doesn't exist in standard analysis). Clearly the definition $$df=\\lim_{\\Delta x\\rightarrow 0}\\Delta f =f'(x)dx$$ makes no sense, since, in the case where $f(x)=x$ we have that $$dx=\\lim_{\\Delta x\\rightarrow 0}\\Delta x =0.$$\n\nAll of this leaves me confused on how to interpret expressions such as $$df=f'(x)dx$$ Should it be seen simply as a definition, quantifying the first-order (linear) change in a function about a point $x$? i.e. a new function that is dependent both on $x$ and a finite change in $x$, $\\Delta x$, $$df(x,\\Delta x):=f'(x)\\Delta x$$ then one can interpret $dx$ as $$dx:=dx(x,\\Delta x)= \\Delta x$$ such that $$\\Delta f=f'(x)dx+\\varepsilon =df +\\varepsilon$$ (in which $\\varepsilon$ quantifies the error between this linear change in $f$ and the actual change in $f$, with $\\lim_{\\Delta x\\rightarrow 0}\\varepsilon =0$).\n\nI feel that there must be some sort of rigorous treatment of the notion of differentials since these kind of manipulations are used all the time, at least in physics?!\n\nI've had some exposure to differential geometry in which one has differential forms, in particular $1-$forms which suggestively notationally \"look like\" differentials, for example $$\\omega =df$$ but as I understand it these are defined as linear maps, members of a dual space to some vector space, $V$, which act on elements of $V$, mapping them to real numbers. Furthermore, the basis $1-$-forms are suggestively written as what in elementary calculus one would interpret as an infinitesimal change in x, $dx$. But again, this is simply symbolic notation, since the basis $1$-forms simply span the dual space and are themselves linear maps which act on elements of $V$.\n\nI've heard people say that differential forms make the notion of a differential of a function mathematically rigorous, however, in my mind I can't seem to reconcile how this is the case, since at best they specify the direction in which the differential change in a function occurs, via $$df(v)=v(f)$$ (since $v(f)$ is the directional derivative of a function $f$ along the vector $v$).\n\nIf someone could enlighten me on this subject I'd really appreciate it.\n\nDifferential forms also have properties that make them natural objects to integrate over. That dovetails nicely with the definition of the exterior differential looking notationally like the Calc I differential. The exterior differntial and the integral are \"dual\" operations, in a sense, whose relation on manifolds with boundary is the generalized Stokes' theorem.\n\nmathwonk\nHomework Helper\nessentially the differential df(p) of f at a point p, is the linear function whose graph is the tangent line at p to the graph of f. the differential df itself is the function whose value at p is the differential of f at p. so the differential is a function df whose values df(p) are linear functions.\n\nessentially the differential df(p) of f at a point p, is the linear function whose graph is the tangent line at p to the graph of f. the differential df itself is the function whose value at p is the differential of f at p. so the differential is a function df whose values df(p) are linear functions.\nSo is it simply the linear approximation of $f$ at a point $p$? Can one interpret $df$ as describing the change in #f# as one moves along a vector passing through the level surface of $f$ at $p$ and a level surface of $f$ infinitesimally close to $p$? (In this sense capturing the direction in which the infinitesimal change occurs without having to deal with the ill-defined notion of an infinitesimal change in $f$?\n\nmathwonk\nHomework Helper\nits not the change in f, its the linear part of the change in f. i.e. it is the linear approximation to f - f(p) = delta(f). by definition a function f is differentiable at p if the difference f(p+v) - f(p) equals a linear function of v plus a function of v which is \"little oh\", i.e. a function o(v) such that the limit of o(v)/|v| is zero as v-->0. so o(v) not only goes to zero, but it goes to zero faster than v does. then that linear function of v is called the differential of f at p. so we have f(p+v) - f(p) = df(p)(v) + o(v), where df(p)(v) is linear in v. so the differential df(p)(v) = f - f(p) - o(v), where f - f(p) is the change in f at p,\n\nso i guess if you took the level surfaces of f at p, and replaced them by the family of planes parallel to the one tangent to the level surface passing through p, then you would probably get the level surfaces of df(p) but I haven't thought it through carefully.\n\nfor a function of one variable, you take the infinitesimal rate of change, i.e. the derivative at p, and you multiply by it to get a linear function approximating the change in f. that linear function is the differential of f at p.\n\nits not the change in f, its the linear part of the change in f. i.e. it is the linear approximation to f - f(p) = delta(f). by definition a function f is differentiable at p if the difference f(p+v) - f(p) equals a linear function of v plus a function of v which is \"little oh\", i.e. a function o(v) such that the limit of o(v)/|v| is zero as v-->0. so o(v) not only goes to zero, but it goes to zero faster than v does. then that linear function of v is called the differential of f at p. so we have f(p+v) - f(p) = df(p)(v) + o(v), where df(p)(v) is linear in v. so the differential df(p)(v) = f - f(p) - o(v), where f - f(p) is the change in f at p,\nHow does this tally up with the notion of a differential in calculus? Is the idea that $df$ is simply defined as the linear change in $f$ about a particular point? I find it confusing why we simply consider this part and not higher order changes in $f$. Is it simply that (like you said) near a particular point $p$ are vanishingly small?!\n\nmathwonk\nHomework Helper\nthis is the notion of differential in calculus. with this definition, df(p) is the linear function \"multiplication by f'(p)\", and dx is the linear function \"multiplication by 1\", hence the equation df(p) = f'(p)dx(p) holds. hence the equation df = f'dx also holds everywhere. we don't consider higher order changes in f because the differential is by definition the linear part of f. I guess you could consider higher order approximations, but that is not what the word \"differential\" nor the symbol df, applies to. one might of course use the term \"second order differential\" but you have to define it.\n\nThere is no such thing as an infinitesimal...however the concept is used often in physics. It doesn't really matter that it doesn't exist. In physics, you usually just assume dx is a length that is \"small enough\" that an infinite sum of them can be approximated as an integral. If you truly take the limit as dx tends to 0....you can no longer apply classical mechanics anyways because at scales less than an atomic radius (a number >0)...your model doesn't even work anyways.\n\nIMO any book that uses differentials in a serious way...without stating the caveat that they are basically an intuitive approximation, is not a very good textbook.\n\nThere is no such thing as an infinitesimal\nReally?? I find that a very bold and incorrect statement.\n\nhis is the notion of differential in calculus. with this definition, df(p) is the linear function \"multiplication by f'(p)\", and dx is the linear function \"multiplication by 1\", hence the equation df(p) = f'(p)dx(p) holds. hence the equation df = f'dx also holds everywhere. we don't consider higher order changes in f because the differential is by definition the linear part of f.\nAh I see, so we simply define the differential change in a function as the linear part of the change in $f$ as we move away from a point $p$?!\nIs the reason this is significant enough to have its own definition because it represents the change in $f$ near a particular point $p$ as we move an increment $dx$ along the tangent line to the function $p$. So $\\Delta f$ and $\\Delta x$ are the changes in the function and its input and $df$ is the change along the tangent line to the function at a point as we move a distance $dx=\\Delta x$. This provides a linear approximation to the actual change in the function (apparently the \"best\" linear approximation, although I have to admit, I'm not sure what is meant by \"best\" in this context)?\n\nI think the problem is that, prior to gaining further knowledge in the area of calculus, and some (albeit a small amount) in differential geometry, I didn't really think too much about the fact that interpreting the differential of a function as an infinitesimal change in a function (as we move an infinitesimal distance $dx$ from a given point) doesn't make sense in any rigorous sense. But now I've started thinking about this more, I feel really unsure about my knowledge and understanding on the subject.\nPreviously I simply thought of $df$ as an infinitesimal change in $f$ and so it made sense that it was only linear since the change is infinitesimally small. Of course, if the change is finite then one has to consider higher order changes in $f$ to accurately describe the function. But now I'm left confused as to why one only considers the linear part of the change in $f$ in calculus. Particularly so in physics one seems to always talk of differential changes in quantities and doesn't worry about higher order terms. Even in some maths textbooks that I've read they introduce the notion by saying something like \"the change in a function as we move a distance $\\Delta x$ from a point $x$ is approximately $\\Delta f \\approx f'(x)\\Delta x$. We observe that as $\\Delta x$ approaches zero this expression becomes exact and we have that $df =f'(x)dx$\".\n\nApologies to keep rambling, I realise that I might be being really thick here, but I can't seem to grasp the idea at the moment.\n\nLast edited:\nIMO any book that uses differentials in a serious way...without stating the caveat that they are basically an intuitive approximation, is not a very good textbook.\nNon-standard analysis builds a framework which uses infinitesimals in a logically consistent and rigorous manner. There are books about it. One of these books is currently published in Princeton's Landmarks of Mathematics series. I'm not sure if that matters to you, but Princeton University Press doesn't add titles to that series lightly.\n\nmathwonk\nHomework Helper\n\"best\" means that it is the only one which comes so close that the difference, i.e. the error term, is little oh, in the sense i defined above.\n\nmathwonk\nHomework Helper\nan infinitesimal is technically a quantity whose length is less than every rational number. they do exist in some contexts. e.g. in non archimedean ordered fields, there are lots of elements which are smaller than every rational number in length but still not zero. so what you do is start with the rational numbers and then introduce also an infinite number of new numbers which are all smaller than 1/n for every n. these are infinitesimals, and when added to other numbers give infinitesinmally near numbers.\n\nlet me think here a sec. i believe the usual model is polynomials with rational coefficients and the associated quotient field. yes, a polynomial is called positive if its leading coefficient is positive. thus t is positive and so is 1/t. but if 1/n is any rational number, the difference 1/n - 1/t still has leading coefficient positive so 1/t is positive but smallet than every rational number, hence is infinitesimal.\n\nso just as we enlarge the integers to introduce rationals, and also enlarge the rationals to introduce irrationals, we can enlarge the rationals to include also infinitesimals. there are models of eucliodean geometry based on non archimedean fields justa s there are ones based on the reals and other smaller fields.\n\nor you could probably do it the opposite way and think of curves through zero as measured by their slope, so that x^2, would be smaller than every line, and x^3 smaller than x^2. this is rough, but is related to the definition of differentials, where a little oh function is considered as an infinitesimal.\n\n\"best\" means that it is the only one which comes so close that the difference, i.e. the error term, is little oh, in the sense i defined above.\nAh OK, thanks for the info.\n\nan infinitesimal is technically a quantity whose length is less than every rational number. they do exist in some contexts. e.g. in non archimedean ordered fields, there are lots of elements which are smaller than every rational number in length but still not zero. so what you do is start with the rational numbers and then introduce also an infinite number of new numbers which are all smaller than 1/n for every n. these are infinitesimals, and when added to other numbers give infinitesinmally near numbers.\n\nlet me think here a sec. i believe the usual model is polynomials with rational coefficients and the associated quotient field. yes, a polynomial is called positive if its leading coefficient is positive. thus t is positive and so is 1/t. but if 1/n is any rational number, the difference 1/n - 1/t still has leading coefficient positive so 1/t is positive but smallet than every rational number, hence is infinitesimal.\n\nso just as we enlarge the integers to introduce rationals, and also enlarge the rationals to introduce irrationals, we can enlarge the rationals to include also infinitesimals. there are models of eucliodean geometry based on non archimedean fields justa s there are ones based on the reals and other smaller fields.\n\nor you could probably do it the opposite way and think of curves through zero as measured by their slope, so that x^2, would be smaller than every line, and x^3 smaller than x^2. this is rough, but is related to the definition of differentials, where a little oh function is considered as an infinitesimal.\nWhy is the differential so significant in calculus and particularly in physics then? If it is simply defined as the first-order, linear change in a function, what is its significance? Is it simply because it is the leading order change in the function and so higher order terms are not of interest since they quickly tend to zero the closer we take $dx$ to zero?\n\nIs the definition I gave in my first post - in terms of defining $df$ as a linear functional of $\\Delta x$ a good way to think of the differential of a function at all?\n\nWhat has really thrown a spanner in the works for me is that in all the teaching I've had in the past, $df$ has been treated as the exact (infinitesimal) change in $f$ due to an (infinitesimal) change in $x$, $dx$, but now that I have been exposed to more advanced mathematics (and more technically correct ways of thinking) it has left me confused, since now $df$ is only an approximation to a change in $f$. I have to admit, I'm not entirely sure of the exact reasoning why my brain is protesting so much, but it is doing so nonetheless and it's really bothering me.\nFor example, how can expressions such as $$\\Delta f =\\int df =\\int f'(x)dx$$ be correct? Intuitively, if I am summing (an infinite sum of) the linear parts of the change in the function, why should it equal the actual change in the function? Why doesn't one have to consider the full change in $x$, i.e. including all higher order changes in the function?\n\nForgive me if I'm wrong, but I've only taken the equivalent of a first year grad course in analysis. But I was under the impression that the whole point of the development of Lebesgue measure and integration over the last two centuries was to do away with the imprecision of the Newton/Liebnitz approach to differential calculus and the Riemann Integral, and the Riemann Integral's inability to apply to a wide class of functions? What about finding the area of a square with irrational side lengths? The Riemann integral can't do that?\n\nAlso, since the set of rational numbers is dense in R, every real number is close to the set of rational numbers...so I'm not understanding your explanation of how infinitesimals exist by looking at 1/n for every integer n? The only numbers I can think of that for every integer n are less than 1/n are numbers which belong to (-inf,0] ? How can a \"quantity\" have a length? I thought only measure spaces could have length...\n\nBut anyways, I think we're a little off track. The OP just wants to know about differentials as they are talked about in elementary calculus and physics. I see nothing wrong with the definition of the derivative:\ndf/dx = lim t->x ( f(t) - f(x) ) / (t-x) at each point x in the domain of f?\n\nWhat I was saying earlier is if you use this canonical definition in physics.....you get down to scales that don't make physical sense in classical mechanics, hence physicists are usually ok with just approximating it with an \"infinitesimal length\".\n\nLast edited:\nForgive me if I'm wrong, but I've only taken the equivalent of a first year grad course in analysis. But I was under the impression that the whole point of the development of Lebesgue measure and integration over the last two centuries was to do away with the imprecision of the Newton/Liebnitz approach to differential calculus\nRight, at the time of Cauchy, Riemann, Lebesgue, the infinitesimal approach was highly nonrigorous. Right now, it has been made rigorous. The construction of infinitesimals is well-known and rigorous.\n\nand the Riemann Integral, and the Riemann Integral's inability to apply to a wide class of functions?\nThat is not the point of the Lebesgue integral. We did not invent the Lebesgue integral just because it can integrate more functions. We adopt the Lebesgue integral because it has better properties, like interchanging limit and integral, and like having $L^1$ complete.\n\nWhat about finding the area of a square with irrational side lengths? The Riemann integral can't do that?\nI'm not sure what you mean. The Riemann integral can handle that perrfectly.\n\nAlso, since the set of rational numbers is dense in R, every real number is close to the set of rational numbers...so I'm not understanding your explanation of how infinitesimals exist by looking at 1/n for every integer n?\nYou are assuming that infinitesimals would be real numbers, they're not.\n\nThe only numbers I can think of that for every integer n are less than 1/n are numbers which belong to (-inf,0] ? How can a \"quantity\" have a length? I thought only measure spaces could have length...\nLength spaces https://people.math.ethz.ch/~lang/LengthSpaces.pdf\n\nStephen Tashi\nRight now, it has been made rigorous. The construction of infinitesimals is well-known and rigorous.\nHowever, if a text wishes to use the infinitesimals in that rigorously defined way, the text usually will make that clear. Most texts dealing with applications of mathematics don't intend to deal with the rigorous approach to infinitesimals. So, for many many texts, it is correct to say that their use of infinitesimals is only an intuitive form of reasoning.\n\nmathwonk\nHomework Helper\n\"Why is the differential so significant in calculus and particularly in physics then? If it is simply defined as the first-order, linear change in a function, what is its significance? Is it simply because it is the leading order change in the function and so higher order terms are not of interest since they quickly tend to zero the closer we take dx to zero?\"\n\nanalyzing any phenomenon in its entirety is incredibly difficult, essentially impossible. Hence to make any progress we try to find reasonable approximations that have two virtues: 1) they are actually computable, 2) the result of the computation gives us useful information about the original situation.\n\nThe derivative, or differential, or best linear approximation, is such a compromise. It is often easy to compute the linear approximation to a function, and that approximation tells us something useful, except sometimes when it is zero. E.g. if f is differentiable at p and f'(p) > 0, then on some, possibly very tiny, neighborhood of p, f is smaller to the left of p and larger to the right of p. Another e.g. is that if f is continuously differentiable at p and f'(p) ≠ 0, then f is actually smoothly invertible on some, again possibly quite small, interval around p.\n\nWith more effort and more information about the bounds on the derivative, we may be able to say something about the sizes of those neighborhoods where these approximations are useful. One global result is that if a smooth function has a derivative which is never zero on a given interval, even a large one, then that function is monotone on that entire interval, hence invertible.\n\nMost useful mathematical inventions are simpliications of the actual phenomena at hand, obtained by intelligently throwing away some of the data, and yet being able to make useful conclusions from what is left. So yes, higher order inofmation is important, but it is difficult to analyze, and so we try to identify phenomena that do not change when we only make higher order changes, and the differential lets us analyze those first order phenomena.\n\nThen with more work and finer tools we may try later to analyze also higher order phenomena. E.g. curvature can be analyzed using second derivatives.\n\nAnd If all the first n derivatives vanish at p, we can also tell something about the behavior of f near p from knowing that the n+1st derivative does not vanish there.\n\nhomology and homotopy groups in topology cannot tell the difference between a point and a line, or three space, since all these have zero homology and homotopy. but if we are clever we can remove a single point and then these groups do distionguish these spaces, so the spaces were also different when the points are replaced. I.e. those groups do distinguish spheres of diffrent dimensions and bt removing a ingle point we change euclidean space into something rather like a sphere (i.e. a cylinder over a sphere).\n\nmathwonk\nHomework Helper\n@ hercuflea:\n\"so just as we enlarge the integers to introduce rationals, and also enlarge the rationals to introduce irrationals, we can enlarge the rationals [or the reals] to include also infinitesimals. there are models of euclidean geometry based on non archimedean fields just as there are ones based on the reals and other smaller fields.\"\n\ni was trying to show that the reason we don't think infinitesimals exist is because (as micromass said) we are used to thinking only within the reals, where they do not. but they are easily added in, just as we have added in needed numbers of other types in many other settings.\n\nthink of graphs passing through (0,0) and say one is smaller than the other if on some nbd of (0,0) it is smaller. then you have one line for each real slope, but you also have all the monomial graphs y = x^n, for n > 1, which are all smaller, on some nbhd of (0,0), than all lines of all positive slopes.\n\nthe tangent line at (0,0) to a given graph is the unique line such that it differs from the given graph by one of these infinitesimally small curves. so theory of infinmitesimals is in some sense, as you are probably also saying, just the attempt to make precise the meaning of a graph that is tangent to the x axis at (0,0), i.e. that is \"little oh\".\n\n\"Why is the differential so significant in calculus and particularly in physics then? If it is simply defined as the first-order, linear change in a function, what is its significance? Is it simply because it is the leading order change in the function and so higher order terms are not of interest since they quickly tend to zero the closer we take dx to zero?\"\n\nanalyzing any phenomenon in its entirety is incredibly difficult, essentially impossible. Hence to make any progress we try to find reasonable approximations that have two virtues: 1) they are actually computable, 2) the result of the computation gives us useful information about the original situation.\n\nThe derivative, or differential, or best linear approximation, is such a compromise. It is often easy to compute the linear approximation to a function, and that approximation tells us something useful, except sometimes when it is zero. E.g. if f is differentiable at p and f'(p) > 0, then on some, possibly very tiny, neighborhood of p, f is smaller to the left of p and larger to the right of p. Another e.g. is that if f is continuously differentiable at p and f'(p) ≠ 0, then f is actually smoothly invertible on some, again possibly quite small, interval around p.\n\nWith more effort and more information about the bounds on the derivative, we may be able to say something about the sizes of those neighborhoods where these approximations are useful. One global result is that if a smooth function has a derivative which is never zero on a given interval, even a large one, then that function is monotone on that entire interval, hence invertible.\n\nMost useful mathematical inventions are simpliications of the actual phenomena at hand, obtained by intelligently throwing away some of the data, and yet being able to make useful conclusions from what is left. So yes, higher order inofmation is important, but it is difficult to analyze, and so we try to identify phenomena that do not change when we only make higher order changes, and the differential lets us analyze those first order phenomena.\n\nThen with more work and finer tools we may try later to analyze also higher order phenomena. E.g. curvature can be analyzed using second derivatives.\n\nAnd If all the first n derivatives vanish at p, we can also tell something about the behavior of f near p from knowing that the n+1st derivative does not vanish there.\n\nhomology and homotopy groups in topology cannot tell the difference between a point and a line, or three space, since all these have zero homology and homotopy. but if we are clever we can remove a single point and then these groups do distionguish these spaces, so the spaces were also different when the points are replaced. I.e. those groups do distinguish spheres of diffrent dimensions and bt removing a ingle point we change euclidean space into something rather like a sphere (i.e. a cylinder over a sphere).\nIs the point that $df=f'(x)dx$ is, by definition, the change in the linear function describing the tangent line to the curve, described by the function $f$, to the point $x$. Thus $df$ is itself another function that, at points near $x$, is approximately equal to the actual change in $f$?! For example,in physics one has Hooke's law which describes the restoring force of a spring. This is a linear approximation to the full force, but in many applications it is accurate enough to, in practice, fully describe the force, so one simply defines a force function $F=-kx$ which the linear approximation to the actual restoring force. Would this be correct at all?\n\nI'm still not quite sure why one can get away with $\\int f'(x) dx =\\int df$? Is it simply an application of the chain rule and the fundamental theorem of calculus, or is there something else going on?\n\nIn the context of differential geometry, is it correct to say that $df$ is a linear functional that maps vectors to real numbers, and that $df(\\mathbf{v})$ quantifies the change in $f$ as one moves an infinitesimal amount along the direction of a given vector $\\mathbf{v}$? Is such a construction useful because it gives a rigorous construction of differentials in terms of differential forms and also enables one to consider integrals of functions in a coordinate independent manner?\n\nKindly if anyone could explain me the limit part of the equation... I know that thing means limit of delta x is zero as delta x tends to zero.. or in other words the change in x is infinitesimally small (approaching zero) .. and as you too told there is nothing such as infinitesimal..but I have seen at many places things like limit of f(x) as delta x approaches 6 is equal to zero... now what does that mean??? Please explain me thoroughly..\n\nMark44\nMentor\nKindly if anyone could explain me the limit part of the equation...\nWhich equation? There have been several equations shown in this thread. You need to be more specific on what you're asking about.\n\nMark_Boy said:\nI know that thing means limit of delta x is zero as delta x tends to zero..\nAre you asking about this limit: $\\lim_{\\Delta x \\to 0}\\Delta x = 0$? Clearly that limit is zero.\nMark_Boy said:\nor in other words the change in x is infinitesimally small (approaching zero) .. and as you too told there is nothing such as infinitesimal..but I have seen at many places things like limit of f(x) as delta x approaches 6 is equal to zero... now what does that mean???\nAs I interpret what you wrote, it doesn't mean anything.\n$\\lim_{\\Delta x \\to 6}f(x) = 0$\nThis is meaningless because $\\Delta x$ is changing, but f(x) doesn't have anything to do with $\\Delta x$.\nMark_Boy said:\n\nAgreeing with most of Frank Castle's post above, and saying why:\n\nAlthough it has been shown that it is possible to put infinitesimals on a sound rigorous basis in mathematics, they are not part of our everyday experience — at least not mine — and so they are probably not the best way to think of \"the differential of a function\".\n\nConsider the symbol df, where\n\nf: (a, b) →\n\nis a differentiable function. Then when we say\n\ndf = f'(x) dx\n\nwe are saying a separate (but similar) thing about each x in (a, b). So let c ∈ (a, b) be one such x. For x = c we are saying that\n\nFor any x near x = c, the change in f(x) is approximately f'(c) multiplied by the change in x from x = c,\n\n(as the best linear approximation) — and such that the approximation approaches exactness as x approaches c.\n\nBecause dx and df mean separate things at different x-values, we will use a subscript to remind us of this:\n\nIn fact, think of \"dx\" at the point x = c as meaning the change in x (from c):\n\ndxc(x) = x - c\n\nand likewise, think of \"df\" at the point x = c as meaning the best linear approximation near c to the change in f (from f(c)):\n\ndfc(x) = f'(c) (x - c) = f'(c) dxc(x).\n\nIn fact, precisely because dx and df mean separate things at each value of x, in higher math they are often denoted as functions of x = c as well as functions of x:\n\ndxc(x) = dx(c)(x)\n\nand\n\ndfc(x) = df(c)(x).\n\n(And no, this was not done for the express purpose of confusing you! It was done to formally separate the various dxc's and dfc's for different values of c, so that you will always bear in mind that they are not the same thing.)\n\nLet U be an open set in\nn. Then the analogous thing applies in higher dimensions to functions\n\ng: U →\n\nwhere now dg is a function of the point x = c ∈ U ⊂ n, as well as the tangent vector v to n at the point x = c:\n\ndgc(v) = (∇g)cv,\n\nwhich is the best linear approximation to g near the point x = c, as then applied to the vector v (which may also be thought of as v = x - c)." ]

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[ "# User Input Through Dynamic\n\nFor my application, I am using Dynamic functionality to make a number of input fields through which a user can enter numbers. I then take these numbers and place them as the entries in a dynamic matrix. I then use the matrix as an input to a function I've written.\n\nFor some reason, I find that Mathematica cannot do anything with a matrix that is formed in this way. Though the matrix seems to be correctly formed, I find that Mathematica will not fully evaluate any expressions involving the dynamic matrix.\n\nHere is some example code\n\nDynamicModule[{numElements = 3, list1,list2},Dynamic@Column[\n\nlist1=Table[Symbol[\"n\"<>ToString@i],{i,1,numElements}];\nlist2=Table[Symbol[\"t\"<>ToString@i],{i,1,numElements}];\n{\nInputField[Dynamic@numElements],\n\nDynamic@Column[Flatten@Table[{Row[{With[{i=i},InputField[Dynamic[list1[[i]]],Number]], With[{i=i}, InputField[Dynamic[list2[[i]]], Number]]}]}, {i,1,numElements}]],\n\nstack = Table[With[{i=i}, {Dynamic[list1[[i]]],Dynamic[list2[[i]]]}], {i,1,numElements}], Button[\"Press to find max\", Print@Max[stack]]}\n]]\n\n\nThis generates a simple user interface that allows you to control two lists list1 and list2 with entries n1, n2, n3, etc. and t1, t2, t3, etc., and it allows you to control their length. It then takes these values and makes a 2 x numElements matrix.\n\nHowever, it doesn't seem like I can do anything with this matrix. Above I tried to print out its maximum value through a button. What could be the issue here?\n\n• In all likelihood you have Dynamic positioned in the wrong place but we need to see your code. Jan 27, 2015 at 6:21\n• Thanks, guys. I just updated it with an example of the issue. Jan 27, 2015 at 6:24\n• @NoahRubin The example doesn't work for me, but the problem is what Mike said. You cannot evaluate Max[Dynamic[val1],Dynamic[val2]], it has to be Dynamic@Max[val1,val2]. The issue is similar to lite item six here. Once you wrap your number with Dynamic it doesn't go away. And it is not possible to determine if Dynamic something is larger than Dynamic something else, these are not numeric values. There is more about this in the documentation, if I recall correctly there is a paragraph specifically about where to place Dynamic. Jan 27, 2015 at 6:46\n• There was one misplaced ] that creeped in there, I fixed it now and it should work. Thanks for your replies. If I need to do Dynamic@Max[val1,val2], how do I get that to evaluate when val1 and val2 need to be dynamic variables themselves, as in this example? It seems I can't actually do anything with the matrix of dynamic variables, even a multiplication of the matrix by 5, say, doesn't evaluate. Jan 27, 2015 at 14:43\n\nButton[\"Press to find max\", Print@Max[{list1, list2}]]" ]

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[ "# 网上怎么买彩票\n\n## 关于经过\n\n0不能作为除数关于填 动词原形 take", null, "", null, "since", null, "SINCE 后的句子用一般过去时\n\na+bi=(1+2i)/(1+i)\n=[(1+2i)(1-i)]/[(1+i)(1-i)]\n=[1+i-2i^2]/[1-i^2]\n=(3+i)/2\n=3/2+i/2\na=3/2" ]

[ null, "http://jianjobs.com/uploads/images/618507.jpg", null, "http://jianjobs.com/uploads/images/6185071.jpg", null, "http://jianjobs.com/uploads/images/6185072.jpg", null ]

[ "# FZU OJ 2110 Star (计算几何）", null, "Problem 2110 Star\n\n##", null, "Problem Description\n\nOverpower often go to the playground with classmates. They play and chat on the playground. One day, there are a lot of stars in the sky. Suddenly, one of Overpower’s classmates ask him: “How many acute triangles whose inner angles are less than 90 degrees (regarding stars as points) can be found? Assuming all the stars are in the same plane”. Please help him to solve this problem.\n\n##", null, "Input\n\nThe first line of the input contains an integer T (T≤10), indicating the number of test cases.\n\nFor each test case:\n\nThe first line contains one integer n (1≤n≤100), the number of stars.\n\nThe next n lines each contains two integers x and y (0≤|x|, |y|≤1,000,000) indicate the points, all the points are distinct.\n\n##", null, "Output\n\nFor each test case, output an integer indicating the total number of different acute triangles.\n\n130 010 05 1000\n\n1\n\n##", null, "Source\n\n``````#include <iostream>\n#include <cstring>\n#include <algorithm>\nusing namespace std;\ntypedef struct Point\n{\ndouble x,y;\n}Point;\nPoint point;\nint main()\n{\nint ans, t, i, j, k, n;\ndouble a,b,c;\ncin >> t;\nwhile (t--)\n{\ncin >> n;\nfor (i=0; i<n; i++)\n{\ncin >> point[i].x >> point[i].y;\n}\nans = 0;\nfor (i=0; i<n-2; i++)//利用三重循环每次计算三个点之间的关系\n{\nfor (j=i+1; j<n-1; j++)\n{\nfor (k=j+1; k<n; k++)\n{\na = (point[i].x-point[j].x)*(point[i].x-point[j].x) + (point[i].y-point[j].y)*(point[i].y-point[j].y);//计算两点之间的距离\nb = (point[i].x-point[k].x)*(point[i].x-point[k].x) + (point[i].y-point[k].y)*(point[i].y-point[k].y);\nc = (point[k].x-point[j].x)*(point[k].x-point[j].x) + (point[k].y-point[j].y)*(point[k].y-point[j].y);\n\nif (a+b>c && a+c>b && c+b>a) //三点都满足才是锐角三角形\n{\nans++;\n}\n\n}\n}\n}\ncout << ans << endl;\n}\nreturn 0;\n}\n``````" ]

[ null, "http://ddrv.cn/wp-content/uploads/2019/10/微信截图_20191023223602.png", null, "http://ddrvcn.oss-cn-hangzhou.aliyuncs.com/2019/11/Evy2If.gif", null, "http://ddrvcn.oss-cn-hangzhou.aliyuncs.com/2019/11/Evy2If.gif", null, "http://ddrvcn.oss-cn-hangzhou.aliyuncs.com/2019/11/Evy2If.gif", null, "http://ddrvcn.oss-cn-hangzhou.aliyuncs.com/2019/11/Evy2If.gif", null ]

[ "Module\n\n# Data.Bifunctor\n\nPackage\npurescript-bifunctors\nRepository\npurescript/purescript-bifunctors\n\n### #BifunctorSource\n\n``class Bifunctor f where``\n\nA `Bifunctor` is a `Functor` from the pair category `(Type, Type)` to `Type`.\n\nA type constructor with two type arguments can be made into a `Bifunctor` if both of its type arguments are covariant.\n\nThe `bimap` function maps a pair of functions over the two type arguments of the bifunctor.\n\nLaws:\n\n• Identity: `bimap identity identity == identity`\n• Composition: `bimap f1 g1 <<< bimap f2 g2 == bimap (f1 <<< f2) (g1 <<< g2)`\n\n#### Members\n\n• `bimap :: forall a b c d. (a -> b) -> (c -> d) -> f a c -> f b d`\n\n### #lmapSource\n\n``lmap :: forall f a b c. Bifunctor f => (a -> b) -> f a c -> f b c``\n\nMap a function over the first type argument of a `Bifunctor`.\n\n### #rmapSource\n\n``rmap :: forall f a b c. Bifunctor f => (b -> c) -> f a b -> f a c``\n\nMap a function over the second type arguments of a `Bifunctor`." ]

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[ "R igraph manual pages\n\nUse this if you are using igraph from R\n\nConvert object to a graph\n\nDescription\n\nThis is a generic function to convert R objects to igraph graphs.\n\nUsage\n\n```graph_(...)\n```\n\nArguments\n\n `...` Parameters, see details below.\n\nTODO\n\nExamples\n\n```## These are equivalent\ngraph_(cbind(1:5,2:6), from_edgelist(directed = FALSE))\ngraph_(cbind(1:5,2:6), from_edgelist(), directed = FALSE)\n```\n\n[Package igraph version 1.2.4.1 Index]" ]

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[ "# An article was sold for Rs 250 with a profit of 5% . What was its cost price ?\n\nSelling price = Rs.250\n\nProfit = 5%\n\n5% of 250 = (250*5) / 100\n\n= 1250 / 100\n\n= Rs.12.5\n\nCost price = Selling price - Profit amount\n\n= 250 - 12.5\n\n=237.5\n\nTherefore, cost price was Rs. 237.5.\n\n• 0\n\nthanks a lot..\n\n• -4" ]

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[ "# InfoCoBuild\n\n## Mathematical Methods in Engineering and Science\n\nMathematical Methods in Engineering and Science. Instructor: Dr. Bhaskar Dasgupta, Department of Mechanical Engineering, IIT Kanpur. The aim of this course is to develop a firm mathematical background necessary for advanced studies and research in the fields of engineering and science. Solution of linear systems. The algebraic eigenvalue problem. Selected topics in linear algebra and calculus. An introductory outline of optimization techniques. Selected topics in numerical analysis. Ordinary differential equations. Application of ODEs in approximation theory. Partial differential equations. Complex analysis and variational calculus. (from nptel.ac.in)\n\n Introduction\n\n Module I. Solution of Linear Systems Lecture 01 - Introduction Lecture 02 - Basic Ideas of Applied Linear Algebra Lecture 03 - Systems of Linear Equations Lecture 04 - Square Non-singular Systems Lecture 05 - Ill-conditioned and Ill-posed Systems Module II. The Algebraic Eigenvalue Problem Lecture 06 - The Algebraic Eigenvalue Problem Lecture 07 - Canonical Forms, Symmetric Matrices Lecture 08 - Methods of Plane Rotations Lecture 09 - Householder Method, Tridiagonal Matrices Lecture 10 - QR Decomposition, General Matrices Module III. Selected Topics in Linear Algebra and Calculus Lecture 11 - Singular Value Decomposition Lecture 12 - Vector Space: Concepts Lecture 13 - Multivariate Calculus Lecture 14 - Vector Calculus in Geometry Lecture 15 - Vector Calculus in Physics Module IV. An Introductory Outline of Optimization Techniques Lecture 16 - Solution of Equations Lecture 17 - Introduction to Optimization Lecture 18 - Multivariate Optimization Lecture 19 - Constrained Optimization: Optimality Criteria Lecture 20 - Constrained Optimization: Further Issues Module V. Selected Topics in Numerical Analysis Lecture 21 - Interpolation Lecture 22 - Numerical Integration Lecture 23 - Numerical Solution of ODEs as IVP Lecture 24 - Boundary Value Problems, Question of Stability in IVP Solution Lecture 25 - Stiff Differential Equations, Existence and Uniqueness Theory Module VI. Ordinary Differential Equations Lecture 26 - Theory of First Order ODEs Lecture 27 - Linear Second Order ODEs Lecture 28 - Methods of Linear ODEs Lecture 29 - ODE Systems Lecture 30 - Stability of Dynamic Systems Module VII. Application of ODEs in Approximation Theory Lecture 31 - Series Solutions and Special Functions Lecture 32 - Sturm-Liouville Theory Lecture 33 - Approximation Theory and Fourier Series Lecture 34 - Fourier Integral to Fourier Transform, Minimax Approximation Module VIII. Overviews: PDEs, Complex Analysis and Variational Calculus Lecture 35 - Separation of Variables in PDEs, Hyperbolic Equations Lecture 36 - Parabolic and Elliptic Equations, Membrane Equation Lecture 37 - Analytic Functions Lecture 38 - Integration of Complex Functions Lecture 39 - Singularities and Residues Lecture 40 - Calculus Variations\n\n References Mathematical Methods in Engineering and Science Instructor: Dr. Bhaskar Dasgupta, Department of Mechanical Engineering, IIT Kanpur. The aim of this course is to develop a firm mathematical background necessary for advanced studies and research in the fields of engineering and science." ]

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[ "Skip to content\nRelated Articles\nHow to Execute Multiple SQL Commands on a Database Simultaneously in JDBC?\n• Last Updated : 26 Nov, 2020\n\nJava Database Connectivity also is known as JDBC is an application programming interface in Java that is used to establish connectivity between a Java application and database. JDBC commands can be used to perform SQL operations from the Java application. Demonstrating execution of multiple SQL commands on a database simultaneously using the addBatch() and executeBatch() commands of JDBC.", null, "The addBatch() command is used to queue the SQL statements and executeBatch() command is used to execute the queued SQL statements all at once. In order to use SQL statements in the Java application, ”java.sql” package needs to be imported in the beginning of the Java application. The Java application is connected to the database using the getConnection() method of DriverManager class. The getConnection() method takes three parameters URLs, username and password.\n\nGoal: Demonstrates two examples of which one uses the Statement Interface and the other uses PreparedStatement Interface. The PreparedStatement performs better than the Statement interface. Statement interface can be used to execute static SQL queries whereas PreparedStatement interface is used to execute dynamic SQL queries multiple times.\n\nExample 1: Using Statement Interface\n\nIn this example, the java.sql package classes and interfaces are imported. The Statement interface is used to execute the sql statements. The table is creation sql statement along with record insertion sql statement are added to the batch using the addBatch() command. When all the statements are batched the executeBatch() command is executed which runs all the batched queries simultaneously. The sql statements may throw SQL Exceptions which must be handled in a try catch block to avoid abrupt termination of the program. After the table is created and records are inserted, to view the data in the table the select query is executed. The result obtained by executing the select query is stored in the ResultSet cursor. The cursor is iterated using the next() method and the records are displayed on the screen.\n\nImplementation: Using the standard interface\n\n## Java\n\n `// Step 1: Create a database``// SQL database imported``import` `java.sql.Connection;``import` `java.sql.DriverManager;``import` `java.sql.ResultSet;``import` `java.sql.Statement;`` ` `public` `class` `BatchCommand {`` ` `    ``// Main driver method``    ``public` `static` `void` `main(String args[])``    ``{`` ` `        ``// Try block to check if exception occurs``        ``try` `{`` ` `            ``// Step 2: Loading driver class``            ``// Using forName()``            ``Class.forName(``\"oracle.jdbc.OracleDriver\"``);`` ` `            ``// Step 3: Create connection object``            ``Connection con = DriverManager.getConnection(``                ``\"jdbc:oracle:thin:@localhost:1521:xe\"``,``                ``\"username\"``, ``\"password\"``);``            ``Statement s = con.createStatement();``           ` `            ``// Step 4: Create a statement / create table``            ``String sql1``                ``= ``\"CREATE TABLE STUDENT(STUDENTID VARCHAR2(10) PRIMARY KEY,NAME VARCHAR2(20),DEPARTMENT VARCHAR2(10))\"``;`` ` `            ``// Step 5: Process a querry``            ``// Insert records in the table``            ``String sql2``                ``= ``\"INSERT INTO STUDENT VALUES('S101','JEAN','CSE')\"``;``            ``String sql3``                ``= ``\"INSERT INTO STUDENT VALUES('S102','ANA','CSE')\"``;``            ``String sql4``                ``= ``\"INSERT INTO STUDENT VALUES('S103','ROBERT','ECE')\"``;``            ``String sql5``                ``= ``\"INSERT INTO STUDENT VALUES('S104','ALEX','IT')\"``;``            ``String sql6``                ``= ``\"INSERT INTO STUDENT VALUES('S105','DIANA','IT')\"``;``            ``s.addBatch(sql1);``            ``s.addBatch(sql2);``            ``s.addBatch(sql3);``            ``s.addBatch(sql4);``            ``s.addBatch(sql5);``            ``s.addBatch(sql6);`` ` `            ``// Step 6: Process the results``            ``// execute the sql statements``            ``s.executeBatch();``            ``ResultSet rs``                ``= s.executeQuery(``\"Select * from Student\"``);`` ` `            ``// Print commands``            ``System.out.println(``                ``\"StudentID\\tName\\t\\tDepartment\"``);``            ``System.out.println(``                ``\"-------------------------------------------------------\"``);`` ` `            ``// Condition to check pointer pointing``            ``while` `(rs.next()) {``                ``System.out.println(rs.getString(``1``) + ``\"\\t\\t\"``                                   ``+ rs.getString(``2``)``                                   ``+ ``\"\\t\\t\"``                                   ``+ rs.getString(``3``));``            ``}`` ` `            ``// Step 7: Close the connection``            ``con.commit();``            ``con.close();``        ``}`` ` `        ``// Catch block to handle exceptions``        ``catch` `(Exception e) {`` ` `            ``// Print line number if exception occured``            ``System.out.println(e);``        ``}``    ``}``}`\n\nSQL commands over database using addBatch() method with the involvement of executeBatch()\n\n## Java\n\n `// Step 1: Imporing database``// SQL database imported``import` `java.sql.Connection;``import` `java.sql.DriverManager;``import` `java.sql.ResultSet;``import` `java.sql.Statement;`` ` `public` `class` `BatchCommand {`` ` `    ``// Main driver method``    ``public` `static` `void` `main(String args[])``    ``{`` ` `        ``// Try block to handle if exception occurs``        ``try` `{`` ` `            ``// Step 2: loading driver class``            ``Class.forName(``\"oracle.jdbc.OracleDriver\"``);`` ` `            ``// Step 3: create connection object``            ``Connection con = DriverManager.getConnection(``                ``\"jdbc:oracle:thin:@localhost:1521:xe\"``,``                ``\"username\"``, ``\"password\"``);``            ``Statement s = con.createStatement();`` ` `            ``// Step 4: Create a statement``            ``// Create table``            ``String sql1``                ``= ``\"CREATE TABLE STUDENT(STUDENTID VARCHAR2(10) PRIMARY KEY,NAME VARCHAR2(20),DEPARTMENT VARCHAR2(10))\"``;`` ` `            ``// Step 5: Execute a querry``            ``// Insert records in the table``            ``String sql2``                ``= ``\"INSERT INTO STUDENT VALUES('S101','JEAN','CSE')\"``;``            ``String sql3``                ``= ``\"INSERT INTO STUDENT VALUES('S102','ANA','CSE')\"``;``            ``String sql4``                ``= ``\"INSERT INTO STUDENT VALUES('S103','ROBERT','ECE')\"``;``            ``String sql5``                ``= ``\"INSERT INTO STUDENT VALUES('S104','ALEX','IT')\"``;``            ``String sql6``                ``= ``\"INSERT INTO STUDENT VALUES('S105','DIANA','IT')\"``;``            ``s.addBatch(sql1);``            ``s.addBatch(sql2);``            ``s.addBatch(sql3);``            ``s.addBatch(sql4);``            ``s.addBatch(sql5);``            ``s.addBatch(sql6);`` ` `            ``// Step 6: rocess the statements``            ``// Create an int[] to hold returned values``            ``s.executeBatch();``            ``ResultSet rs``                ``= s.executeQuery(``\"Select * from Student\"``);`` ` `            ``// Print statements``            ``System.out.println(``                ``\"StudentID\\tName\\t\\tDepartment\"``);``            ``System.out.println(``                ``\"-------------------------------------------------------\"``);`` ` `            ``// Condition check for pointer pointing which``            ``// record``            ``while` `(rs.next()) {``                ``System.out.println(rs.getString(``1``) + ``\"\\t\\t\"``                                   ``+ rs.getString(``2``)``                                   ``+ ``\"\\t\\t\"``                                   ``+ rs.getString(``3``));``            ``}`` ` `            ``// Step 7: Close the connection``            ``con.commit();``            ``con.close();``        ``}`` ` `        ``// Catch block to handle exception``        ``catch` `(Exception e) {`` ` `            ``// Print line number where exception occured``            ``System.out.println(e);``        ``}``    ``}``}`\n\nOutput", null, "Example 2: In this example, the java.sql package classes and interfaces are imported. The PreparedStatement interface is used to execute the SQL statements. The table is the creation SQL statement along with record insertion SQL statement are added to the batch using the addBatch() command. When all the statements are batched the executeBatch() command is executed which runs all the batched queries simultaneously. The sql statements may throw SQL Exceptions which must be handled in a try-catch block to avoid abrupt termination of the program. After the table is created and records are inserted, to view the data in the table the select query is executed. The result obtained by executing the select query is stored in the ResultSet cursor. The cursor is iterated using the next() method and the records are displayed on the screen. Unlike the previous example, it takes dynamic input from the user. Hence, using the PreparedStatement has performance benefits.\n\nCode Implementation\n\n## Java\n\n `import` `java.sql.Connection;``import` `java.sql.DriverManager;``import` `java.sql.PreparedStatement;``import` `java.sql.ResultSet;``import` `java.sql.Statement;``import` `java.util.*;``public` `class` `AddBatchCommand {``    ``public` `static` `void` `main(String args[])``    ``{``        ``Scanner scan = ``new` `Scanner(System.in);``        ``try` `{`` ` `            ``// loading driver class``            ``Class.forName(``\"oracle.jdbc.OracleDriver\"``);`` ` `            ``// create connection object``            ``Connection con = DriverManager.getConnection(``                ``\"jdbc:oracle:thin:@localhost:1521:xe\"``,``                ``\"username\"``, ``\"password\"``);`` ` `            ``// create the table``            ``String sql1``                ``= ``\"CREATE TABLE STUDENTS(STUDENTID VARCHAR2(10) PRIMARY KEY,NAME VARCHAR2(20),DEPARTMENT VARCHAR2(10))\"``;``            ``PreparedStatement ps``                ``= con.prepareStatement(sql1);``            ``ps.execute(sql1);`` ` `            ``// inserting records``            ``String sql``                ``= ``\"Insert into Students values(?,?,?)\"``;``            ``PreparedStatement ps1``                ``= con.prepareStatement(sql);``            ``for` `(``int` `i = ``0``; i < ``3``; i++) {``                ``System.out.println(``\"Enter Student ID\"``);``                ``String id = scan.nextLine();``                ``System.out.println(``\"Enter Student Name\"``);``                ``String name = scan.nextLine();``                ``System.out.println(``\"Enter the Department\"``);``                ``String dept = scan.nextLine();``                ``ps1.setString(``1``, id);``                ``ps1.setString(``2``, name);``                ``ps1.setString(``3``, dept);``                ``// adding to batch``                ``ps1.addBatch();``            ``}``            ``// executing the batch``            ``ps1.executeBatch();``           ` `            ``// viewing the table``            ``ResultSet rs``                ``= ps.executeQuery(``\"Select * from Students\"``);``            ``System.out.println(``                ``\"StudentID\\tName\\t\\tDepartment\"``);``            ``System.out.println(``                ``\"-------------------------------------------------------\"``);``            ``while` `(rs.next()) {``                ``System.out.println(rs.getString(``1``) + ``\"\\t\\t\"``                                   ``+ rs.getString(``2``)``                                   ``+ ``\"\\t\\t\"``                                   ``+ rs.getString(``3``));``            ``}``            ``con.commit();``            ``con.close();``        ``}`` ` `        ``catch` `(Exception e) {``            ``System.out.println(e);``        ``}``    ``}``}`\n\nOutput: Illustrating multiple SQL commands on a database simultaneously:", null, "Attention reader! Don’t stop learning now. Get hold of all the important Java Foundation and Collections concepts with the Fundamentals of Java and Java Collections Course at a student-friendly price and become industry ready. To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.\n\nMy Personal Notes arrow_drop_up" ]

[ null, "https://media.geeksforgeeks.org/wp-content/uploads/20201123171554/JDBC.png", null, "https://media.geeksforgeeks.org/wp-content/uploads/20201113222436/ab1.png", null, "https://media.geeksforgeeks.org/wp-content/uploads/20201113222437/ab2.png", null ]

[ "# Drift–diffusion models for multiple-alternative forced-choice decision making\n\n## Abstract\n\nThe canonical computational model for the cognitive process underlying two-alternative forced-choice decision making is the so-called drift–diffusion model (DDM). In this model, a decision variable keeps track of the integrated difference in sensory evidence for two competing alternatives. Here I extend the notion of a drift–diffusion process to multiple alternatives. The competition between n alternatives takes place in a linear subspace of $$n-1$$ dimensions; that is, there are $$n-1$$ decision variables, which are coupled through correlated noise sources. I derive the multiple-alternative DDM starting from a system of coupled, linear firing rate equations. I also show that a Bayesian sequential probability ratio test for multiple alternatives is, in fact, equivalent to these same linear DDMs, but with time-varying thresholds. If the original neuronal system is nonlinear, one can once again derive a model describing a lower-dimensional diffusion process. The dynamics of the nonlinear DDM can be recast as the motion of a particle on a potential, the general form of which is given analytically for an arbitrary number of alternatives.\n\n## 1 Introduction\n\nPerceptual decision-making tasks require a subject to make a categorical decision based on noisy or ambiguous sensory evidence. A computationally advantageous strategy in doing so is to integrate the sensory evidence in time, thereby improving the signal-to-noise ratio. Indeed, when faced with two possible alternatives, accumulating the difference in evidence for the two alternatives until a fixed threshold is reached is an optimal strategy, in that it minimizes the mean reaction time for a desired level of performance. This is the computation carried out by the sequential probability ratio test devised by Wald , and its continuous-time variant, the drift–diffusion model (DDM) . It would be hard to overstate the success of these models in fitting psychophysical data from both animals and human subjects in a wide array of tasks, e.g. [2,3,4,5], suggesting that brain circuits can implement a computation analogous to the DDM.\n\nAt the same time, neuroscientists have characterized the neuronal activity in cortical areas of monkey, which appear to reflect an integration process during DM tasks , although see . The relevant computational building blocks, as revealed from decades of in-vivo electrophysiology, seem to be neurons, the activity of which selectively increases with increasing likelihood for a given upcoming choice. Attractor network models, built on this principle of competing, selective neuronal populations, generate realistic performance and reaction times; they also provide a neuronal description which captures some salient qualitative features of the in-vivo data [8, 9].\n\nWhile focus in the neuroscience community has been almost exclusively on two-alternative DM (although see ), from a computational perspective there does not seem to be any qualitative difference between two or more alternatives. In fact, in a model, increasing the number of alternatives is as trivial as adding another neuronal population to the competition. On the other hand, how to add an alternative to the DDM framework does not seem, on the face of things, obvious. Several groups have sought to link the DDM and attractor networks for two-alternative DM. When the attractor network is assumed linear, one can easily derive an equation for a decision variable, representing the difference in the activities of the two competing populations, which precisely obeys a DDM . For three-alternative DM previous work has shown that a 2D diffusion process can be defined by taking appropriate linear combinations of the three input streams . The general n-alternative case for leaky accumulators has also been treated . In the first section of the paper I will summarize and build upon this previous work to illustrate how one can obtain an equivalent DDM, starting from a set of linear firing rate equations which compete through global inhibitory feedback. The relevant decision variables are combinations of the activity of the neuronal populations, and which represent distinct modes of competition. Specifically, I will propose a set of “competition” basis functions which allow for a simple, systematic derivation of the DDMs for any n. I will also show how a Bayesian implementation of the the multiple sequential probability ratio test (MSPRT) [14,15,16] is equivalent in the continuum limit to these same DDMs, but with a moving threshold.\n\nOf course, linear models do not accurately describe the neuronal data from experiments on DM. However, previous work has shown that attractor network models for two-alternative DM operate in the vicinity of pitchfork bifurcation, which is what underlies the winner-take-all competition leading to the decision dynamics . In this regime the neuronal dynamics is well described by a stochastic normal-form equation which right at the bifurcation is precisely equivalent to the DDM with an additional cubic nonlinearity. This nonlinear DDM fits behavioral data extremely well, including both correct and error reaction times. In the second part of the paper I will show how such normal-form equations can be derived for an arbitrary number of neuronal populations. These equations can be thought of as nonlinear DDMs and, in fact, are identical to the linear DDMs with the addition of quadratic nonlinearities (for $$n>2$$). Amazingly, the dynamics of such a nonlinear DDM can be recast as the diffusion of particle on a potential, which is obtained analytically, for arbitrary n.\n\n## 2 Results\n\nThe canonical drift diffusion model (DDM) can be written\n\n$$\\tau \\dot{X} = \\mu +\\xi (t),$$\n(1)\n\nwhere X is the decision variable, and μ is the drift or the degree of evidence in favor of one choice over the other: we can associate choice 1 with positive values of X and choice 2 with negative values. The Gaussian process $$\\xi (t)$$ represents noise and/or uncertainty in the integration process, with $$\\langle \\xi (t)\\rangle = 0$$ and $$\\langle \\xi (t)\\xi (t^{\\prime })\\rangle =\\sigma ^{2}\\delta (t-t ^{\\prime })$$. I have also explicitly included a characteristic time scale τ, which will appear naturally if one derives Eq. (1) from a neuronal model. The decision variable evolves until reaching one of two boundaries ±θ at which point a decision for the corresponding choice has been made.\n\nIt is clear that a single variable can easily be used to keep track of two competing processes by virtue of its having two possible signs. But what if there are three or more alternatives? In this case it is less clear. In fact, if we consider a drift–diffusion process such as the one in Eq. (1) as an approximation to an actual integration process carried out by neuronal populations, then there is a systematic approach to deriving the corresponding DDM. The value of such an approach is that one can directly tie the DDM to the neuronal dynamics, thereby linking behavior to neuronal activity.\n\nI will first consider the derivation of a DDM starting from a system of linear firing rate equations. This analysis is similar to that found in Sect. 4.4 of , although the model of departure is different. In this case the derivation involves a rotation of the system so as to decouple the linear subspace for the competition between populations from the subspace which describes non-competitive dynamical modes. This rotation is equivalent to expressing the firing rates in terms of a set of orthogonal basis functions: one set for the competition, and another for the non-competitive modes. I subsequently consider a system of nonlinear firing rate equations. In this case one can once again derive a reduced set of equations to describe the decision-making dynamics. The reduced models have precisely the form of the corresponding DDM for a linear system, but now with additional nonlinear terms. These terms reflect the winner-take-all dynamics which emerge in nonlinear systems with multi-stability. Not only do the equations have a simple, closed-form solution for any number of alternatives, but they can be succinctly expressed in terms of a multivariate potential.\n\n### 2.1 Derivation of a DDM for two-alternative DM\n\nThe DDM can be derived from a set of linear equations which model the competition between two populations of neurons, the activity of each of which encodes the accumulated evidence for the corresponding choice. The equations are\n\n\\begin{aligned} \\tau \\dot{r}_{1} &= -r_{1}+sr_{1}-cr_{I}+I_{1}+ \\xi _{1}(t), \\\\ \\tau \\dot{r}_{2} &= -r_{2}+sr_{2}-cr_{I}+I_{2}+ \\xi _{2}(t), \\\\ \\tau _{I} \\dot{r}_{I} &= -r_{I}+ \\frac{g}{2}(r_{1}+r_{2})+I_{I}+\\xi _{I}(t), \\end{aligned}\n(2)\n\nwhere $$r_{I}$$ represents the activity of a population of inhibitory neurons. The parameter s represents the strength of excitatory self-coupling and c is the strength of the global inhibition. The characteristic time constants of excitation and inhibition are τ and $$\\tau _{I}$$ respectively. A choice is made for 1 (2) whenever $$r_{1} = r_{th}$$ ($$r_{2} = r_{th}$$).\n\nIt is easier to work with the equations if they are written in matrix form, which is", null, "(3)\n\nwhere", null, "In order to derive the DDM, I express the firing rates in terms of three orthogonal basis functions: one which represents a competition between the populations $$\\mathbf{e}_{1}$$, a second for common changes in population rates $$\\mathbf{e}_{c}$$ and a third which captures changes in the activity of the inhibitory cells $$\\mathbf{e}_{I}$$. Specifically I write\n\n\\begin{aligned} \\mathbf{r}(t) = \\mathbf{e}_{1}X(t)+\\mathbf{e}_{C} \\mathrm{m}_{C}(t)+ \\mathbf{e}_{I}\\mathrm{m}_{I}(t), \\end{aligned}\n(4)\n\nwhere $$\\mathbf{e}_{1} = (1,-1,0)$$ and $$\\mathbf{e}_{C} = (1,1,0)$$ and $$\\mathbf{e}_{I} = (0,0,1)$$. The decision variable will be X, while $$\\mathrm{m}_{C}$$ and $$\\mathrm{m}_{I}$$ stand for the common mode and inhibitory mode respectively.\n\nThe dynamics of each of these modes can be isolated in turn by projecting Eq. (2) onto the appropriate eigenvector. For example, the dynamics for the decision variable X are found by projecting onto $$\\mathbf{e}_{1}$$, namely", null, "(5)\n\nand similarly the dynamics for $$\\mathrm{m}_{C}$$ and $$\\mathrm{m}_{I}$$ and found by projecting onto $$\\mathbf{e}_{C}$$ and $$\\mathbf{e}_{I}$$ respectively. Doing so results in the set of equations\n\n\\begin{aligned} &\\tau \\dot{X}= -(1-s)X+\\frac{I_{1}-I_{2}}{2}+ \\frac{\\xi _{1}(t)-\\xi _{2}(t)}{2}, \\\\ &\\tau \\dot{\\mathrm{m}}_{C}= -(1-s)\\mathrm{m}_{C}-c \\mathrm{m}_{I}+\\frac{I_{1}+I_{2}}{2}+\\frac{\\xi _{1}(t)+\\xi _{2}(t)}{2}, \\\\ &\\tau _{I}\\dot{\\mathrm{m}}_{I}= -\\mathrm{m}_{I}+g \\mathrm{m}_{C}+I _{I}+\\xi _{I}(t). \\end{aligned}\n(6)\n\nIf the self-coupling sits at the critical value $$s = 1$$, then these equations simplify to\n\n\\begin{aligned}& \\tau \\dot{X}= \\frac{I_{1}-I_{2}}{2}+ \\frac{\\xi _{1}(t)-\\xi _{2}(t)}{2}, \\\\ &\\tau \\dot{\\mathrm{m}}_{C}= -c\\mathrm{m}_{I}+ \\frac{I_{1}+I_{2}}{2}+\\frac{\\xi _{1}(t)+\\xi _{2}(t)}{2}, \\\\ &\\tau _{I} \\dot{\\mathrm{m}}_{I}= -\\mathrm{m}_{I}+g \\mathrm{m}_{c}+I_{I}+\\xi _{I}(t), \\end{aligned}\n(7)\n\nfrom which it is clear that the equation for X describes a drift–diffusion process. It is formally identical to Eq. (1) with $$\\mu = \\frac{I_{1}-I_{2}}{2}$$ and $$\\xi (t) = \\frac{\\xi _{1}(t)-\\xi _{2}(t)}{2}$$. Importantly, X is uncoupled from the common and inhibitory modes, which themselves form a coupled subsystem. For $$s\\ne 1$$ the decision variable still decouples from the other two equations, but the process now has a leak term (or ballistic for $$s>1$$) . It has therefore been argued that obtaining a DDM from linear neuronal models requires fine tuning, a drawback which can be avoided in nonlinear models in which the linear dynamics is approximated via multi-stability; see e.g. [19, 20]. If one ignores the noise terms, the steady state of this linear system is $$(X,\\mathrm{m}_{C},\\mathrm{m}_{I}) = (X_{0},\\mathrm{M}_{c},\\mathrm{M} _{I})$$ where\n\n\\begin{aligned} \\mathrm{M}_{C} &= \\frac{I_{1}+I_{2}}{2cg}- \\frac{I_{I}}{g}, \\\\ \\mathrm{M}_{I} &= \\frac{I_{1}+I_{2}}{2c}. \\end{aligned}\n(8)\n\nOne can study the stability of this solution by considering a perturbation of the form $$(X,\\mathrm{m}_{C},\\mathrm{m}_{I}) = (X_{0}, \\mathrm{M}_{c},\\mathrm{M}_{I})+(\\delta X,\\delta \\mathrm{m}_{C},\\delta \\mathrm{m}_{I})e^{\\lambda t}$$. Plugging this into Eq. (7) one finds that there is a zero eigenvalue associated with the decision variable, i.e. $$\\lambda _{1} = 0$$, whereas the eigenvalues corresponding to the subsystem comprising the common and inhibitory modes are given by\n\n$$\\lambda _{2,3} = -\\frac{1}{2\\tau _{I}} \\biggl(1\\pm \\sqrt{1-4 \\frac{\\tau _{I}}{\\tau }cg} \\biggr),$$\n(9)\n\nwhich always have a negative real part. Therefore, as long as $$\\tau _{I}$$ is not too large, perturbations in the common mode or in the inhibition will quickly decay away. This allows one to ignore their dynamics and assume they take on their steady-state values. Finally, the bounds for the decision variable are found by noting that $$r_{1} = X+ \\mathrm{M}_{C}$$ and $$r_{2} = -X+\\mathrm{M}_{C}$$. Therefore, given that the neuronal threshold for a decision is defined as $$r_{th}$$, we find that $$\\theta = \\pm (r_{th}-\\mathrm{M}_{C})$$.\n\n### 2.2 Derivation of a DDM for three-alternative DM\n\nI will go over the derivation of a drift–diffusion process for three-choice DM in some detail for clarity, although conceptually it is very similar to the two-choice case. Then the derivation can be trivially extended to n-alternative DM for any n.\n\nThe linear rate equations are once again given by Eq. (3), with", null, "One once again writes the firing rates in terms of orthogonal basis functions, of which there must now be four. The common and inhibitory modes are the same as before, whereas now there will be two distinct modes to describe the competition between the three populations, in contrast to just a single decision variable. Any orthogonal basis in the 2D space for competition is equally valid. However, in order to make the choice systematic, I assume that the first vector is just the one from the two-alternative case, namely $$(1,-1,0,0)$$, from which it follows that the second must be (up to an amplitude) $$(1,1,-2,0)$$. Then for n alternatives I will always take the first $$n-2$$ basis vectors to be those from the $$n-1$$ case. The last eigenvector must be orthogonal to these. Specifically, for $$n = 3$$, $$\\mathbf{r} = \\mathbf{e}_{1}X _{1}(t)+\\mathbf{e}_{2}X_{2}(t)+\\mathbf{e}_{C}\\mathrm{m}_{C}(t)+ \\mathbf{e}_{I}\\mathrm{m}_{I}$$, where\n\n\\begin{aligned} &\\mathbf{e}_{1}= (1,-1,0,0), \\\\ &\\mathbf{e}_{2}= (1,1,-2,0), \\\\ &\\mathbf{e}_{C}= (1,1,1,0), \\\\ &\\mathbf{e}_{I}= (0,0,0,1). \\end{aligned}\n(10)\n\nOne projects Eq. (3) onto the four relevant eigenvectors, which leads to the following equations:\n\n\\begin{aligned} &\\tau \\dot{X_{1}}= -(1-s)X_{1}+ \\frac{I_{1}-I_{2}}{2}+\\frac{\\xi _{1}(t)- \\xi _{2}(t)}{2}, \\\\ &\\tau \\dot{X_{2}}= -(1-s)X_{2}+\\frac{I_{1}+I_{2}-2I _{3}}{6}+ \\frac{\\xi _{1}(t)+\\xi _{2}(t)-2\\xi _{3}(t)}{6}, \\\\ &\\tau \\dot{\\mathrm{m}_{C}}= -(1-s)\\mathrm{m}_{C}-c \\mathrm{m}_{I}+\\frac{I _{1}+I_{2}+I_{3}}{3}+\\frac{\\xi _{1}(t)+\\xi _{2}(t)+\\xi _{3}(t)}{3}, \\\\ &\\tau _{I}\\dot{\\mathrm{m}_{I}}= -\\mathrm{m}_{I}+g \\mathrm{m}_{C}+I_{I}+ \\xi _{I}(t). \\end{aligned}\n(11)\n\nWhen $$s = 1$$ then the first two equations in Eq. (11) describe a drift–diffusion process in a two-dimensional subspace, while the coupled dynamics of the common and inhibitory modes are once again strongly damped. The DDM for three-alternative DM can therefore be written\n\n\\begin{aligned} \\tau \\dot{X}_{1} &= \\frac{I_{1}-I_{2}}{2}+ \\frac{\\xi _{1}(t)-\\xi _{2}(t)}{2}, \\end{aligned}\n(12)\n\\begin{aligned} \\tau \\dot{X}_{2} &= \\frac{I_{1}+I_{2}-2I_{3}}{6}+\\frac{\\xi _{1}(t)+\\xi _{2}(t)-2\\xi _{3}(t)}{6}. \\end{aligned}\n(13)\n\nNote that the dynamics of the two decision variables $$X_{1}$$ and $$X_{2}$$ are coupled through the correlation in their noise sources. The decision boundaries are set by noting that\n\n\\begin{aligned} &r_{1}= X_{1}+X_{2}+ \\mathrm{M}_{C}, \\\\ &r_{2}= -X_{1}+X_{2}+ \\mathrm{M}_{C}, \\\\ &r_{3}= -2X_{2}+\\mathrm{M}_{C}. \\end{aligned}\n(14)\n\nTherefore, given that the neuronal threshold for a decision is defined as $$r_{th}$$ we can set three decision boundaries: 1 Population 1 wins if $$X_{2} = -X_{1} +r_{th}-\\mathrm{M}_{C}$$, 2 Population 2 wins if $$X_{2} = X_{1}+r_{th}-\\mathrm{M}_{C}$$ and 3 Population 3 wins if $$X_{2} = -(r_{th}-\\mathrm{M}_{C})/2$$. These three boundaries define a triangle in (X,Y)-space over which the drift–diffusion process take place.\n\n### 2.3 Derivation of DDMs for n-alternative DM\n\nThe structure of the linear rate equations Eq. (3) can be trivially extended to any number of competing populations. In order to derive the corresponding DDM one need only properly define the basis functions for the firing rates, which was described above. The common and inhibitory modes always have the same structure. If the basis functions are $$\\mathbf{e}_{i}$$ and the corresponding decision variables $$X_{i}$$ for $$i = [1,n-1]$$, then the firing rates are $$\\mathbf{r} = \\sum_{i=1}^{n-1}\\mathbf{e}_{i}X_{i}(t)$$ and it is easy to show that the dynamics for the kth decision variable is given by\n\n$$\\tau \\dot{X}_{k} = \\frac{\\mathbf{e}_{k}\\cdot \\mathbf{I}}{\\mathbf{e} _{k}\\cdot \\mathbf{e}_{k}}+ \\frac{\\mathbf{e}_{k}\\cdot \\boldsymbol{\\xi}}{ \\mathbf{e}_{k}\\cdot \\mathbf{e}_{k}},$$\n(15)\n\nas long as $$s = 1$$. The decision boundaries are defined by setting the firing rates equal to their threshold value for a decision, i.e.\n\n$$\\sum_{i=1}^{n-1}\\mathbf{e}_{i}X_{i,b} = r_{th}\\mathbf{u},$$\n(16)\n\nwhere $$\\mathbf{u} = (1,1,\\ldots,1)$$.\n\nThe basis set proposed here for n-alternative DM is to take for the kth eigenvector\n\n$$e_{k} = (1,1,\\dots ,1,-k,0,\\dots ,0),$$\n(17)\n\nwhere the element −k appears in the $$(k+1)$$st spot, and which is a generalization of the eigenvector basis taken earlier for two- and three-alternative DM. With this choice, the equation for the kth decision variable can be written\n\n$$\\tau \\dot{X}_{k} = \\frac{\\mathbf{e}_{k}\\cdot \\mathbf{I}}{k+k^{2}}+\\frac{ \\mathbf{e}_{k}\\cdot \\boldsymbol{\\xi}}{k+k^{2}}.$$\n(18)\n\nThe firing rate for the ith neuronal population can then be expressed in terms of the decision variables as\n\n$$r_{i} = -(i-1)x_{i-1}+\\sum_{l = i}^{n-1}x_{l}+ \\mathrm{M}_{C},$$\n(19)\n\nwhich, given a fixed neuronal threshold $$r_{th}$$, directly gives the bounds on the decision variables. Namely, the ith neuronal population wins when $$-(i-1)x_{i-1}+\\sum_{l = i}^{n-1}x_{l}+\\mathrm{M}_{C} > r _{th}$$.\n\nThe n-alternative DDM reproduces the well-known Hick’s law , which postulates that the mean reaction time (RT) increases as the logarithm of the number of alternatives, for fixed accuracy; see Fig. 1.\n\n### 2.4 DDM for multiple alternatives as the continuous-time limit of the MSPRT\n\nIn the previous section I have illustrated how to derive DDMs starting from a set of linear differential equations describing an underlying integration process of several competing data streams. An alternative computational framework would be to apply a statistical test directly to the data, without any pretensions with regards to a neuronal implementation. In fact, the development of an optimal sequential test for multiple alternatives followed closely on the heels of Wald’s work for two alternatives in the 1940s . Subsequent work proposed a Bayesian framework for a multiple sequential probability ratio test (MSPRT) [14,15,16]. Here I will show how such a Bayesian MSPRT is equivalent to a corresponding DDM in the continuum limit, albeit with moving thresholds.\n\nI will follow the exposition from in setting up the problem, although some notation differs for clarity. I assume that there are n possible alternatives, and that the instantaneous evidence for an alternative i is given by $$z_{i}(t)$$, which is drawn from a Gaussian distribution with mean $$I_{i}$$ and variance $$\\sigma ^{2}$$. The total observed input over all alternatives n and up to a time t is written I and the hypothesis that alternative i has the highest mean $$H_{i}$$. Then, using Bayes theorem, the probability that alternative i has the highest mean given the total observed input is\n\n$$\\Pr (H_{i}|I) = \\frac{\\Pr (I|H_{i})\\Pr (H_{i})}{\\Pr (I)}.$$\n(20)\n\nFurthermore one has $$\\Pr (I) = \\sum_{k=1}^{n}\\Pr (I|H_{k})\\Pr (H_{k})$$. Given equal priors on the different hypotheses, Eq. (20) can be simplified to\n\n$$\\Pr (H_{i}|I) = \\frac{\\Pr (I|H_{i})}{\\sum_{k=1}^{n}\\Pr (I|H_{k})}.$$\n(21)\n\nFinally, the log-likelihood of the alternative i having the largest mean up to a time t is $$L_{i}(t) = \\ln {\\Pr (H_{i}|I)}$$, which is given by\n\n$$L_{i}(t) = \\ln {\\bigl(\\Pr (I|H_{i})\\bigr)}-\\ln { \\Biggl( \\sum_{k=1}^{n}e^{\\ln {\\Pr (I|H _{k})}} \\Biggr)}.$$\n(22)\n\nThe choice of Gaussian distributions for the input leads to a simple form for the log-likelihoods. Specifically, the time rate-of-change of the log-likelihood for alternative i is given by\n\n$$\\dot{L}_{i}(t) = z_{i}-\\frac{\\sum_{k = 1}^{n}z_{k}e^{y_{k}}}{\\sum_{k=1} ^{n}e^{y_{k}}},$$\n(23)\n\nwhere $$y_{i}(t) = \\int _{0}^{t}dt^{\\prime }z_{i}(t^{\\prime })$$. Note that $$z_{i}(t) = I_{i}+\\xi _{i}(t)$$, where $$\\xi _{i}$$ is a Gaussian white noise process with zero mean and variance $$\\sigma ^{2}$$.\n\nI can now write the $$L_{i}$$ in terms of the orthogonal basis used in the previous section,\n\n$$\\mathbf{L} = \\sum_{j = 1}^{n-1} \\mathbf{e}_{j}X_{j}(t)+\\mathbf{e}_{C}M _{C}(t).$$\n(24)\n\nFormally projecting onto each mode in turn leads precisely to the DDMs of Eq. (18) (with $$\\tau = 1$$) for the decision variables, while the dynamics for the common mode is\n\n$$\\dot{M}_{C} = \\frac{1}{n}\\sum_{j=1}^{n}z_{j}- \\frac{\\sum_{k = 1}^{n}z _{k}e^{y_{k}}}{\\sum_{k=1}^{n}e^{y_{k}}}.$$\n(25)\n\nFor Bayes-optimal behavior, a choice i should be chosen if the log-likelihood exceeds a given threshold, namely if\n\n$$L_{i}(t) = -(i-1)X_{i-1}+\\sum_{l = i}^{n-1}X_{l} > \\theta - M_{C}(t).$$\n(26)\n\nNote that the log-likelihood is always negative and hence does not represent a firing rate, as in the differential equations studied in the previous section. This does not pose a problem since we are simply implementing a statistical test. On the other hand, an important difference with the case studied previously is the fact that, for the neuronal models, the common mode was stable and converged to a fixed point. Therefore, the decision variable dynamics was equivalent to the original rate dynamics with a shift of threshold. Here, that is not the case. The common mode represents the normalizing effect of the log-marginal probability of the stimulus, which always changes in time. Specifically, if we assume that $$I_{l}>I_{i}$$ for all i, namely that the mean of the distribution is greatest for alternative l, then the expected dynamics of the common mode at long times are\n\n\\begin{aligned} \\langle \\dot{M}_{C}\\rangle =& \\frac{1}{n}\\sum _{j=1}^{n}I_{j}-\\frac{ \\sum_{k = 1}^{n}I_{k}e^{I_{k}t}}{\\sum_{k=1}^{n}e^{I_{k}t}} \\\\ \\sim & \\frac{1}{n}\\sum_{j=1}^{n}I_{j}-I_{l} \\\\ =& -\\frac{1}{n}\\sum_{j=1}^{n} \\vert I_{j}-I_{l} \\vert . \\end{aligned}\n(27)\n\nTherefore the DDMs are only equivalent to applying the Bayes theorem if the threshold is allowed to vary in time. In this case they are, in fact, mathematically identical and thus give the same accuracy and reaction time distributions, as shown in Fig. 2.\n\nOn the other hand, an equivalent DDM with a fixed threshold has worse accuracy and shorter reaction times. The way I choose the parameters for a fair comparison is to set the fixed threshold such that the mean reaction time is identical to the Bayesian model for zero coherence. Another important difference is that the error RTs are longer than the correct RTs for the Bayesian model, see Fig. 2B, an effect which is commonly seen in experiment (and is also reproduced by the nonlinear DDMs studied in the next section of the paper) . On the other hand correct and error RTs for the DDMs are always the same.\n\n## 3 Derivation of a reduced model for two-choice DM for a nonlinear system\n\nA more realistic firing rate model for a decision-making circuit allows for a nonlinear input-output relationship in neuronal activity. For two-alternative DM the equations are\n\n\\begin{aligned} &\\tau \\dot{r}_{1}= -r_{1}+\\phi (sr_{1}-cr_{I}+I_{1} )+\\xi _{1}(t), \\\\ &\\tau \\dot{r}_{2}= -r_{2}+\\phi (sr_{2}-cr_{I}+I_{1} )+\\xi _{2}(t), \\\\ &\\tau _{I}\\dot{r}_{I}= -r_{I}+\\phi _{I} \\biggl(\\frac{g}{2}(r _{1}+r_{2})+I_{I} \\biggr)+\\xi _{I}(t). \\end{aligned}\n(28)\n\nThe nonlinear transfer function ϕ ($$\\phi _{I}$$) does not need to be specified in the derivation. The noise sources $$\\xi _{i}$$ are taken to be Gaussian white noise and hence must sit outside of the transfer function; they therefore directly model fluctuations in the population firing rate rather than input fluctuations. Input fluctuations can be modeled by allowing for a non-white noise process and including it directly as an additional term in the argument of the transfer function. Note that here I assume the nonlinearity is a smooth function. This is a reasonable assumption for a noisy system such as a neuron or neuronal circuit. Non-smooth systems, such as piecewise linear equations for DM, require a distinct analytical approach; see, e.g., .\n\nThe details of the derivation for two alternatives can be found in , but here I give a flavor for how one proceeds; the process will be similar when there are three or more choices, although the scaling of the perturbative expansion is different. One begins by ignoring the noise sources and linearizing Eq. (28) about a fixed-point value for which the competing populations have the same level of activity, and hence also $$I_{1} = I_{2}$$. Specifically one takes $$(r_{1},r_{2},r_{I}) = (R,R,R_{I})+(\\delta r_{1},\\delta r_{2}, \\delta r_{I})e^{\\lambda t}$$, where $$\\delta r\\ll 1$$. In vector form this can be written $$\\mathbf{r} = \\mathbf{R}+\\boldsymbol{\\delta}\\mathbf{r}e^{\\lambda t}$$. Plugging this ansatz into Eq. (28) and keeping only terms linear in the perturbations leads to the following system of linear equations:", null, "(29)\n\nwhere", null, "(30)\n\nand the slope of the transfer function $$\\phi ^{\\prime }$$ is calculated at the fixed point. Note that the matrix Eq. (30) has a very similar structure to the linear operator in Eq. (3). This system of equations only has a solution if the determinant of the matrix is equal to zero; this yields the characteristic equation for the eigenvalues λ. These eigenvalues are\n\n\\begin{aligned} &\\lambda _{1}= -\\frac{(1-s\\phi ^{\\prime })}{\\tau }, \\end{aligned}\n(31)\n\\begin{aligned} &\\lambda _{2,3}= -\\frac{(\\tau +\\tau _{I}(1-s\\phi ^{\\prime }))}{2\\tau \\tau _{I}}\\pm \\frac{1}{2\\tau \\tau _{I}}\\sqrt {\\bigl(\\tau -\\tau _{I}\\bigl(1-s\\phi ^{\\prime }\\bigr) \\bigr)^{2}-4 \\tau \\tau _{I}cg\\phi ^{\\prime }\\phi _{I}^{\\prime }}. \\end{aligned}\n(32)\n\nNote that $$\\lambda _{1} = 0$$ if $$1-s\\phi ^{\\prime } = 0$$, while the real part of the other two eigenvalues is always negative. This indicates that there is an instability of the fixed point in which the activity of the neuronal populations is the same, and that the direction of this instability can be found by setting $$1-s\\phi ^{\\prime } = 0$$ and $$\\lambda = 0$$ in Eq. (30). This yields", null, ", where", null, "(33)\n\nthe solution of which can clearly be written $$\\boldsymbol{\\delta}\\mathbf{r} = (1,-1,0)$$. This is the same competition mode as found earlier for the linear system.\n\n### 3.1 A brief overview of normal-form derivation\n\nAt this point it is still unclear how one can leverage this linear analysis to derive a DDM. Specifically, and unlike in the linear case, one cannot simply rotate the system to uncouple the competition dynamics from the non-competitive modes. Also, note that the steady states in a nonlinear system depend on the external inputs, whereas that is not the case in a linear system. In particular, the DDM has a drift term μ which ought to be proportional to the difference in inputs $$I_{1}-I_{2}$$, whereas to perform the linear stability we assumed $$I_{1} = I_{2}$$. Indeed, if one assumes that the inputs are different, then the fixed-point structure is completely different. The solution is to assume that the inputs are only slightly different, and formalize this by introducing the small parameter ϵ. Specifically, we write $$I_{1} = I_{0}+\\epsilon ^{2}\\Delta I +\\epsilon ^{3}\\bar{I}_{1}$$, and $$I_{2} = I_{0}+\\epsilon ^{2}\\Delta I +\\epsilon ^{3}\\bar{I}_{2}$$. In this expansion, $$I_{0}$$ is the value of the external input which places the system right at the bifurcation in the zero-coherence case. In order to describe the dynamics away from the bifurcation we also allow the external inputs to vary. Specifically, ΔI represents the component of the change in input which is common to both populations (overall increased or decreased drive compared to the bifurcation point), while $$\\bar{I}_{i}$$ is a change to the drive to population i alone, and hence captures changes in the coherence of the stimulus. The particular scaling of these terms with ϵ is enforced by the solvability conditions which appear at each order. That is, the mathematics dictates what these are; if one chose a more general scaling one would find that only these terms would remain.\n\nThe firing rates are then also expanded in orders of ϵ and written\n\n\\begin{aligned} \\mathbf{r} = \\mathbf{r_{0}}+\\epsilon \\mathbf{e_{1}}X(T)+ \\mathcal{O}\\bigl( \\epsilon ^{2}\\bigr), \\end{aligned}\n(34)\n\nwhere $$\\mathbf{r_{0}}$$ are the fixed-point values, $$\\mathbf{e_{1}}$$ is the eigenvector corresponding to the zero eigenvalue and X is the decision variable which evolves on a slow-time scale, $$T = \\epsilon ^{2}t$$. The slow-time scale arises from the fact that there is an eigenvector with zero eigenvalue; when we change the parameter values slightly, proportional to ϵ, the growth rate of the dynamics along that eigenvector is no longer zero, but still very small, in fact proportional to $$\\epsilon ^{2}$$ in this case.\n\nThe method for deriving the normal-form equation, i.e. the evolution equation for X, involves expanding Eq. (28) in ϵ. At each order in ϵ there is a set of equations to be solved; at some orders, in this case first at order $$\\mathcal{O}(\\epsilon ^{3})$$, the equations cannot be solved and a solvability condition must be satisfied, which leads to the normal-form equation.\n\n### 3.2 The normal-form equation for two choices\n\nFollowing the methodology described in the preceding section leads to the evolution equation for the decision variable X,\n\n$$\\tau \\partial _{T}X = \\eta (\\bar{I}_{1}- \\bar{I}_{2})+\\mu \\Delta IX+ \\gamma X^{3}+\\xi (t),$$\n(35)\n\nwhere for the case of Eq. (28), $$\\eta = \\phi ^{\\prime }/2$$, $$\\xi (t) = (\\xi _{1}(t)-\\xi _{2}(t))/2$$ and the coefficients μ and γ are\n\n\\begin{aligned} \\mu &= \\frac{s^{2}\\phi ^{\\prime \\prime }}{cg\\phi _{I}^{\\prime }}, \\\\ \\gamma &= \\frac{s^{3}( \\phi ^{\\prime \\prime })^{2}}{2cg\\phi ^{\\prime }\\phi _{I}^{\\prime }}\\bigl(s-cg\\phi _{I}^{\\prime }\\bigr)+ \\frac{s ^{3}\\phi ^{\\prime \\prime \\prime }}{6}, \\end{aligned}\n(36)\n\nsee for a detailed calculation. Equation (35) provides excellent fits to performance and reaction times for monkeys and human subjects; see Fig. 3 from .\n\nIt is important to note that the form of Eq. (35) only depends on there being a two-way competition, not on the exact form of the original system. As an example, consider another set of firing rate equations\n\n\\begin{aligned} \\tau \\dot{r}_{1} &= -r_{1}+\\phi (sr_{1}-cr_{2}+I_{1}), \\\\ \\tau \\dot{r} _{2} &= -r_{2}+\\phi (sr_{2}-cr_{1}+I_{2}), \\end{aligned}\n(37)\n\nwhere rather than model the inhibition explicitly, an effective inhibitory interaction between the two populations is assumed. In this case the resulting normal-form equation is still Eq. (35). In fact, performing a linear stability analysis on Eq. (37) yields a null eigenvector $$e_{1} = (1,-1)$$. This indicates that the instability causes one population to grow at the expense of the other, in a symmetric fashion, as before. This is the key point which leads to the normal-form equation. More specifically we see that for both systems $$r_{1} = R+X$$ while $$r_{2} = R-X$$, which means that if we flip the sign on the decision variable X and switch the labels on the neuronal populations, the dynamics is once again the same. This reflection symmetry ensures that all terms in X will have odd powers in Eq. (35) . It is broken only when the inputs to the two populations are different, i.e. by the first term on the r.h.s. in Eq. (35).\n\nAs we shall see, the stochastic normal-form equation, Eq. (35), which from now on I will refer to as a nonlinear DDM, has a very different form from the nonlinear DDMs for $$n>2$$. The reason is, again, the reflection symmetry in the competition subspace for $$n=2$$, which is not present for $$n>2$$. Therefore, for $$n>2$$ the leading-order nonlinearity is quadratic, and, in fact, a much simpler function of the original neuronal parameters.\n\n## 4 Three-alternative forced-choice decision making\n\nThe derivation of the normal form, and the corresponding DDM for three-choice DM differs from that for two-alternative DM in several technical details; these differences continue to hold for n-alternative DM for all $$n\\ge 3$$. Therefore I will go through the derivation in some detail here and will then extend it straightforwardly to the other cases.\n\nAgain I will make use of a particular system of firing rate equations to illustrate the derivation. I take a simple extension of the firing rate equations for two-alternative DM Eq. (28). The equations are\n\n\\begin{aligned} &\\tau \\dot{r}_{1}= -r_{1}+\\phi (sr_{1}-cr_{I}+I_{1} )+\\xi _{1}(t), \\\\ &\\tau \\dot{r}_{2}= -r_{2}+\\phi (sr_{2}-cr_{I}+I_{2} )+\\xi _{2}(t), \\\\ &\\tau \\dot{r}_{3}= -r_{3}+\\phi (sr_{3}-cr_{I}+I_{3} )+\\xi _{3}(t), \\\\ &\\tau _{I}\\dot{r}_{I}= -r_{I}+\\phi _{I} \\biggl( \\frac{g}{3}(r_{1}+r_{2}+r_{3})+I_{I} \\biggr)+\\xi _{I}(t). \\end{aligned}\n(38)\n\nI first ignore the noise terms and consider the linear stability of perturbations of the state in which all three populations have the same level of activity (and so $$I_{1} = I_{2} = I_{3} = I_{0}$$), i.e. $$\\mathbf{r} = \\mathbf{R}+\\boldsymbol{\\delta}\\mathbf{r}e^{\\lambda t}$$, where $$\\mathbf{R} = (R,R,R,R_{I})$$. This once again leads to a set of linear equations", null, ". The fourth-order characteristic equation leads to precisely the same eigenvalues as in the two-choice case, Eqs. (31) and (32), with the notable difference that the first eigenvalue has multiplicity two. This means that if $$1-s\\phi ^{\\prime } = 0$$ then there will be two eigenvalues identically equal to zero and two stable eigenvalues. This is the first indication that the integration process underlying the DM process for three choices will be two-dimensional. The eigenvectors for the DM process are found by solving", null, ", where", null, "(39)\n\nThere are many possible solutions; a simple choice would be $$\\mathbf{e}_{1} = (1,-1,0,0)$$ and $$\\mathbf{e}_{2} = (1,1,-2,0)$$, and so $$\\mathbf{e}_{1}^{T}\\cdot \\mathbf{e}_{2} = 0$$. Note that in this linear subspace any valid choice of eigenvector will have the property that the sum of all the elements will equal zero; this will be true whatever the dimensionality of the DM process and reflects the fact that all of the excitatory populations excite the inhibitory interneurons in equal measure.\n\nTo derive the normal form I once again assume that the external inputs to the three populations differ by a small amount, namely $$(I_{1},I _{2},I_{3}) = (I_{0},I_{0},I_{0})+\\epsilon ^{2}(\\bar{I}_{1},\\bar{I} _{2},\\bar{I}_{3})$$, and then expand the firing rates as $$\\mathbf{r} = \\mathbf{R}+\\epsilon (\\mathbf{e}_{1}X_{1}(T)+\\mathbf{e}_{2}X_{2}(T) )+\\mathcal{O}(\\epsilon ^{2})$$, where the slow time is $$T = \\epsilon t$$. Note that the inputs are only expanded to second order in ϵ, as opposed to third order as in the previous section. The reason is that the solvability condition leading to the normal-form equation for the 2-alternative case arises at third order. This is due to the fact that the bifurcation has a reflection symmetry, i.e. it is a pitchfork bifurcation and so only odd terms in the decision variable are allowed. The lowest-order nonlinear term is therefore the cubic one. On the other hand, for more than two alternatives there is no such reflection symmetry in the corresponding bifurcation to winner-take-all behavior. Therefore the lowest-order nonlinear term is quadratic, as in a saddle-node bifurcation.\n\nI expand Eq. (38) in orders of ϵ. In this case a solvability condition first arises at order $$\\epsilon ^{2}$$, which also accounts for the different scaling of the slow time compared to two-choice DM. Note that there are two solvability conditions, corresponding to eliminating the projection of terms at that order onto both of the left-null eigenvectors of", null, ". As before, the left-null eigenvectors are identical to the right-null eigenvectors. Applying the solvability condition yields the normal-form equations\n\n\\begin{aligned}& \\tau \\dot{X}_{1}= \\frac{\\phi ^{\\prime }}{2}( \\bar{I}_{1}-\\bar{I}_{2})+s^{2} \\phi ^{\\prime \\prime }X_{1}X_{2}+\\frac{1}{2}\\bigl(\\xi _{1}(t)-\\xi _{2}(t)\\bigr), \\\\ &\\tau \\dot{X}_{2}= \\frac{\\phi ^{\\prime }}{6}(\\bar{I}_{1}+ \\bar{I}_{2}-2\\bar{I} _{3})+\\frac{s^{2}\\phi ^{\\prime \\prime }}{6} \\bigl(X_{1}^{2}-3X_{2}^{2}\\bigr) + \\frac{1}{6}\\bigl(\\xi _{1}(t)+\\xi _{2}(t)-2\\xi _{3}(t)\\bigr). \\end{aligned}\n(40)\n\nThe nonlinear DDM Eq. (40) provides an asymptotically correct description of the full dynamics in Eq. (38) in the vicinity of the bifurcation leading to the decision-making behavior. Figure 3(A) shows a comparison of the firing rate dynamics with the nonlinear DDM right at the bifurcation. The appropriate combinations of the two decision variables $$X_{1}$$ and $$X_{2}$$ clearly track the rate dynamics accurately, including the correct choice (here population 2) and reaction time. The nonlinear drift–diffusion process evolves in a triangular section of the plane; see Fig. 3(B).\n\n### 4.1 A note on the difference between the nonlinear DDM for 2A and 3A DM\n\nThe dynamics of the nonlinear DDM for 2A, Eq. (35), depends strongly on the sign of the cubic coefficient γ. Specifically, when $$\\gamma < 0$$ the bifurcation is supercritical, while for $$\\gamma > 0$$ it is subcritical, indicating the existence of a region of multi-stability for $$\\Delta I < 0$$. In fact, in experiment, cells in parietal cortex which exhibit ramping activity during perceptual DM tasks, also readily show delay activity in anticipation of the sacade to their response field . One possible mechanism for this would be precisely this type of multi-stability. When $$\\Delta I = 0$$, i.e. when the system sits squarely at the bifurcation, Eq. (35) is identical to its linear counterpart with the sole exception of the cubic term. For $$\\gamma < 0$$ the state $$X = 0$$ is stabilized. In fact, the dynamics of the decision variable can be viewed as the motion of a particle in a potential, which for $$\\gamma < 0$$ increases rapidly as X grows, pushing the particle back. On the other hand, for $$\\gamma > 0$$ the potential accelerates the motion of X, pushing it off to ±∞. This is very similar to the potential for the linear DDM with absorbing boundaries. Therefore, the nonlinear DDM for two-alternatives is qualitatively similar to the linear DDM when it is subcritical, and hence when the original neuronal system is multi-stable.\n\nOn the other hand, the nonlinear DDM for three alternatives, Eq. (40), has a much simpler, quadratic nonlinearity. The consequence of this is that there are no stable fixed points and the decision variables always evolve to ±∞. Furthermore, to leading order there is no dependence on the mean input, indicating that the dynamics is dominated by the behavior right at the bifurcation.Footnote 1 The upshot is that Eq. (40) is as similar to the corresponding linear DDM with absorbing boundaries as possible for a nonlinear system without fine tuning. This remains true for all $$n>2$$.\n\nThis also means that neuronal systems with inhibition-mediated winner-take-all dynamics are generically multi-stable for $$n>2$$, although for $$n = 2$$ they need not be. This is due to the reflection symmetry present only for $$n = 2$$.\n\n## 5 n-alternative forced-choice decision making\n\nOne can now extend the analysis for three-alternative DM to the more general n-choice case. Again I start with a set of firing rate equations\n\n\\begin{aligned} &\\tau \\dot{r}_{1}= -r_{1}+\\phi (sr_{1}-cr_{I}+I_{1} )+\\xi _{1}(t), \\\\ &\\tau \\dot{r}_{2}= -r_{2}+\\phi (sr_{2}-cr_{I}+I_{2} )+\\xi _{2}(t), \\\\ &\\vdots \\\\ &\\tau \\dot{r}_{n}= -r_{n}+\\phi (sr_{n}-cr _{I}+I_{n} )+\\xi _{n}(t), \\\\ &\\tau _{I}\\dot{r}_{I}= -r_{I}+\\phi _{I} \\Biggl(\\frac{g}{n}\\sum_{j=1}^{n}r_{j}+I_{I} \\Biggr)+\\xi _{I}(t). \\end{aligned}\n\nA linear stability analysis shows that the eigenvalues of perturbations of the state $$\\mathbf{r} = (R,R,\\ldots ,R,R_{I})$$ are given by Eqs. (31) and (32), where the first eigenvalue has multiplicity $$n-1$$. Therefore the decision-making dynamics evolves on a manifold of dimension $$n-1$$. The linear subspace associated with this manifold is spanned by $$n-1$$ eigenvectors which are mutually orthogonal and the elements of which sum to zero. For n alternatives, we take $$n-1$$ eigenvectors of the form $$e_{k} = (1,1,\\dots ,1,-k,0, \\dots ,0)$$, for the kth eigenvector (again, the −k sits in the (k+1)-st spot). Therefore one can write\n\n$$\\mathbf{r} = \\mathbf{R}+\\epsilon \\sum_{i=1}^{n-1} \\mathbf{e} _{i}X_{i}(T)+\\mathcal{O}\\bigl(\\epsilon ^{2}\\bigr).$$\n(41)\n\nFollowing the same procedure as in the case of three-choice DM and applying the $$n-1$$ solvability conditions at order $$\\mathcal{O}( \\epsilon ^{2})$$, one arrives at the following normal-form equation for the kth decision variable for n alternatives:\n\n\\begin{aligned}[b] \\tau \\dot{X}_{k} ={} & \\phi ^{\\prime } \\frac{\\mathbf{e}_{k}^{T}\\cdot \\mathbf{I}}{k+k ^{2}}+\\frac{s^{2}\\phi ^{\\prime \\prime }}{2(k+k^{2})} \\Biggl(\\sum_{j=1}^{k-1} \\bigl(j+j^{2}\\bigr)X _{j}^{2}-k \\bigl(k^{2}-1\\bigr)X_{k}^{2}+2\\bigl(k+k^{2} \\bigr)X_{k}\\sum_{j=k+1}^{n-1}X_{j} \\Biggr) \\\\ &{}+\\frac{\\mathbf{e}_{k}^{T}\\cdot \\boldsymbol{\\xi}(\\mathbf{t})}{k+k^{2}}. \\end{aligned}\n(42)\n\nIt is illuminating to write this formula out explicitly for some of the decision variables:\n\n\\begin{aligned} &\\tau \\dot{X}_{1}= \\phi ^{\\prime } \\frac{1}{2}(\\bar{I}_{1}-\\bar{I}_{2})+s ^{2} \\phi ^{\\prime \\prime }X_{1}\\sum_{j=2}^{n-1}X_{j}, \\\\ &\\tau \\dot{X}_{2}= \\phi ^{\\prime }\\frac{1}{6}( \\bar{I}_{1}+\\bar{I}_{2}-2\\bar{I}_{3})+ \\frac{s^{2}\\phi ^{\\prime \\prime }}{6} \\Biggl(X_{1}^{2}-3X_{2}^{2}+6X_{2} \\sum_{j=3}^{n-1}X_{j} \\Biggr), \\\\ &\\tau \\dot{X}_{3}= \\phi ^{\\prime }\\frac{1}{12}( \\bar{I}_{1}+\\bar{I}_{2}+ \\bar{I}_{3}-3 \\bar{I}_{4})+\\frac{s^{2}\\phi ^{\\prime \\prime }}{12} \\Biggl(X_{1}^{2}+3X _{2}^{2}-12X_{3}^{2}+12X_{3} \\sum_{j=4}^{n-1}X_{j} \\Biggr), \\\\ & \\vdots \\\\ &\\tau \\dot{X}_{n-1}= \\phi ^{\\prime }\\frac{1}{n(n-1)}\\Biggl(\\sum _{j=1}^{n-1} \\bar{I}_{j}-(n-1) \\bar{I}_{n}\\Biggr)\\\\ &\\hphantom{\\tau \\dot{X}_{n-1}=}{}+\\frac{s^{2}\\phi ^{\\prime \\prime }}{n(n-1)} \\Biggl( \\sum _{j=1}^{n-2}\\bigl(j+j^{2} \\bigr)X_{j}^{2}-n(n-1)^{2}X_{n}^{2} \\Biggr), \\end{aligned}\n(43)\n\nwhere I have left off the noise terms for simplicity.\n\nSurprisingly, the $$n-1$$ equations for the decision variables in n-alternative DM can all be derived from a single, multivariate function:\n\n$$f(X_{1},\\dots ,X_{n}) = -a\\sum _{j=1}^{n-1}\\langle \\mathbf{e}_{j} \\cdot \\mathbf{I}\\rangle X_{j}-\\frac{b}{2} \\Biggl(\\sum _{j=1}^{n-2}\\bigl(j+j ^{2} \\bigr)X_{j}^{2}\\sum_{i=j+1}^{n-1}X_{i}- \\sum_{j=1}^{n-1} \\frac{j(j^{2}-1)}{3}X_{j}^{3} \\Biggr).$$\n(44)\n\nFor the system of firing rate equations studied here the parameters $$a = \\phi ^{\\prime }$$ and $$b = s^{2}\\phi ^{\\prime \\prime }$$. Then the equation for the kth decision variable is simply\n\n$$\\tau \\dot{X}_{k} = -\\frac{1}{k+k^{2}} \\frac{\\partial f}{\\partial X_{k}} + \\frac{\\mathbf{e}_{k}\\cdot \\xi (t)}{k+k ^{2}} .$$\n(45)\n\nThe dynamics of the function f is given by\n\n\\begin{aligned}[b] \\tau \\frac{\\partial f}{\\partial t} &= \\sum _{j=1}^{n-1}\\frac{\\partial f}{\\partial X_{j}}\\dot{X}_{j} \\\\ &= -\\sum_{j=1}^{n-1}\\frac{1}{k+k^{2}} \\biggl(\\frac{\\partial f}{\\partial X_{j}} \\biggr)^{2} < 0, \\end{aligned}\n(46)\n\nwhere I have ignored the effect of noise. Therefore the dynamics of the decision variables can be thought of as the motion of a particle on a potential landscape, given by f. Noise sources lead to a diffusion along this landscape.\n\n## 6 Discussion\n\nIn this paper I have illustrated how to derive drift–diffusion models starting from models of neuronal competition for n-alternative decision-making tasks. In the case of linear systems, the derivation consists of nothing more than a rotation of the dynamics onto a subspace of competition modes. This idea is not new, e.g. , although I have made the derivation explicit here, and have chosen as a model of departure one in which inhibition is explicitly included as a dynamical variable. It turns out that a Bayesian implementation of a multiple sequential probability ratio test is also equivalent to a DDM in the continuum limit, albeit with time-varying thresholds.\n\nFor nonlinear systems, the corresponding DDM is a stochastic normal form, which is obtained here using the method of multiple-scales . The nonlinear DDM was obtained earlier for the special case of two-alternative DM . For four-alternative DM the nonlinear DDM was obtained with a different set of competitive basis functions to describe performance and reaction time from experiments with human subjects. This led to a different set of coupled normal-form equations from those given by Eq. (42), although the resulting dynamics is, of course, the same. The advantage of the choice I have made in this paper for the basis functions, is that they are easily generalizable for any n, leading to a simple, closed-form expression for the nonlinear DDM for any arbitrary number of alternatives, Eq. (42).\n\nAn alternative approach to describing the behavior in DM tasks, is to develop a statistical or probabilistic description of evidence accumulation; see, e.g., [1, 13, 15, 22, 27]. Such an approach also often leads to a drift–diffusion process in some limit, as is the case for the Bayesian MSPRT studied here, and see also . In fact, recent work has shown that an optimal policy for multiple-alternative decision making can be approximately implemented by an accumulation process with time-varying thresholds, similar to the Bayesian model studied in this manuscript . From a neuroscience perspective, however, it is of interest to pin down how the dynamics of neuronal circuits might give rise to animal behavior which is well described by a drift–diffusion process. This necessitates the analysis of neuronal models at some level. What I have shown here is that the dynamics in a network of n neuronal populations which compete via a global pool of inhibitory interneurons, can in general be formally reduced to a nonlinear DDM of dimension $$n-1$$. The nonlinear DDMs differ from the linear DDMs through the presence of quadratic (or cubic for $$n = 2$$) nonlinearities which accelerate the winner-take-all competition. In practical terms this nonlinear acceleration serves the same role as the hard threshold in the linear DDMs. Therefore the two classes of DDMs have quite similar behavior.\n\nThe DDM is one of the most-used models for fitting data from two-alternative forced-choice decision-making experiments. In fact it provides fits to decision accuracy and reaction time in a wide array of tasks, e.g. [2, 3, 30]. Here I have illustrated how the DDM can be extended to n alternatives straightforwardly. It remains to be seen if such DDMs will fit accuracy and reaction times as well as their two-alternative cousin, although one may refer to promising results from for three alternatives. Note also that the form of the nonlinear DDMs, Eqs. (44) and (45) does not depend on the details of the original neuronal equations; this is what is meant by a normal-form equation. The only assumptions needed for the validity of the normal-form equations are that there be global, nonlinear competition between n populations. Of course, if the normal form is derived from a given neuronal model, then the parameters a and b of the nonlinear potential Eq. (44) will depend on the original neuronal parameters.\n\nAs stated earlier, the nonlinear DDMs can have dynamics quite similar to the standard, linear DDM with hard thresholds. Nonetheless, there are important qualitative differences between the two classes of models. First of all, both correct and error reaction-time distributions are identical in the linear DDMs, given unbiased initial conditions, whereas the nonlinear DDMs generically show longer error reaction times , also a feature of the Bayesian MSPRT. Because error reaction times in experiment indeed tend to be longer than correct ones, the linear DDM cannot be directly fit to data. Rather, variability in the drift rate across trials can be assumed in order to account for differences in error and correct reaction times; see, e.g., . Secondly, nonlinear DDMs exhibit intrinsic dynamics which reflect the winner-take-all nature of neuronal models with strong recurrent connectivity. As a consequence, as the decision variables increase (or decrease) from their initial state, they undergo an acceleration which does not explicitly depend on the value of the external input. This means that the response of the system to fluctuations in the input is not the same late in a trial as it is early on. Specifically, later fluctuations will have lesser impact. Precisely this effect has been seen in the response of neurons in parietal area LIP in monkeys in two-alternative forced-choice decision-making experiments; see Fig. 10B in . Given that network models of neuronal activity driving decision-making behavior lead to nonlinear DDMs, fitting such models to experimental data in principle allows one to link behavioral measures to the underlying neuronal parameters. 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Neural activity in macaque parietal cortex reflects temporal integration of visual motion signals during perceptual decision making. J Neurosci. 2005;25(45):10420–36.\n\n### Acknowledgements\n\nI acknowledge helpful conversations with Klaus Wimmer and valuable suggestions for improvements from the reviewers.\n\n### Availability of data and materials\n\nData sharing not applicable to this article as no datasets were generated or analyzed during the current study.\n\n## Funding\n\nGrant number MTM2015-71509-C2-1-R from the Spanish Ministry of Economics and Competitiveness. Grant 2014 SGR 1265 4662 for the Emergent Group “Network Dynamics” from the Generalitat de Catalunya. This work was partially supported by the CERCA program of the Generalitat de Catalunya.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nAll authors read and approved the final manuscript.\n\n### Corresponding author\n\nCorrespondence to Alex Roxin.\n\n## Ethics declarations\n\nNot applicable.\n\n### Competing interests\n\nThe author declares to have no competing interests.\n\nNot applicable\n\n### Publisher’s Note\n\nSpringer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Appendix\n\n### 1.1 1.1 Derivation of the normal form for three-alternative DM\n\nI assume the inputs to the three competing neuronal populations in Eq. (38) are slightly different, namely $$I_{i} = I _{0}+\\epsilon ^{2}\\bar{I}_{i}$$ for $$i = \\{1,2,3\\}$$, which defines the small parameter $$\\epsilon \\ll 1$$. The firing rates are expanded as $$\\mathbf{r} = \\mathbf{R}+\\epsilon \\mathbf{r}_{1}+\\epsilon ^{2} \\mathbf{r}_{2}+\\mathcal{O}(\\epsilon ^{3})$$ where R are the values of the rates at the fixed point. Finally, we define a slow time $$T = \\epsilon t$$. Plugging the expansions for the inputs and rates into Eq. (38) and expanding leads to the following system of equations:", null, "(47)\n\nwhere to place the system at the bifurcation to winner-take-all behavior I choose $$1-s\\phi ^{\\prime } = 0$$. Then we have the linear operator", null, "(48)\n\nand\n\n\\begin{aligned}& \\mathbf{L}_{1} = \\begin{pmatrix} \\tau \\partial _{T} & 0 & 0 & 0 \\\\ 0 & \\tau \\partial _{T} & 0 & 0 \\\\ 0 & 0 & \\tau \\partial _{T} & 0 \\\\ 0 & 0 & 0 & \\tau _{I}\\partial _{T} \\end{pmatrix}, \\end{aligned}\n(49)\n\\begin{aligned}& \\mathbf{N}_{2} = \\phi ^{\\prime } \\begin{pmatrix} \\bar{I}_{1} \\\\ \\bar{I}_{2} \\\\ \\bar{I}_{3} \\\\ 0 \\end{pmatrix}+ \\frac{1}{2} \\begin{pmatrix} \\phi ^{\\prime \\prime }(sr_{11}-cr_{I1})^{2} \\\\ \\phi ^{\\prime \\prime }(sr_{21}-cr_{I1})^{2} \\\\ \\phi ^{\\prime \\prime }(sr_{31}-cr_{I1})^{2} \\\\ \\phi _{I}^{\\prime \\prime }\\frac{g^{2}}{9}(r_{11}+r_{21}+r_{31})^{2} \\end{pmatrix}. \\end{aligned}\n(50)\n\nNow one solves the equations order by order. At $$\\boldsymbol{\\mathcal{O}(\\epsilon )}$$", null, "(51)\n\nThe solution can be written $$\\mathbf{r}_{1} = \\mathbf{e}_{1}X_{1}(T)+ \\mathbf{e}_{2}X_{2}(T)$$, where the right-null eigenvectors of", null, "are $$\\mathbf{e}_{1} = (1,-1,0,0)$$ and $$\\mathbf{e}_{2} = (1,1,-2,0)$$. The left-null eigenvectors, which satisfy", null, "are identical to the right-null eigenvectors in this case. Up to $$\\boldsymbol{\\mathcal{O}(\\epsilon ^{2})}$$:", null, "(52)\n\nGiven the solution $$r_{1}$$, the forcing term has the form\n\n$$\\mathbf{N}_{2} = \\phi ^{\\prime } \\begin{pmatrix} \\bar{I}_{1} \\\\ \\bar{I}_{2} \\\\ \\bar{I}_{3} \\\\ 0 \\end{pmatrix} + \\frac{s^{2}\\phi ^{\\prime \\prime }}{2} \\begin{pmatrix} (X_{1}+X_{2})^{2} \\\\ (-X_{1}+X_{2})^{2} \\\\ 4X_{2}^{2} \\\\ 0 \\end{pmatrix} .$$\n(53)\n\nThis forcing term contains a projection in the left-null eigenspace of", null, ". This projection cannot give rise to a solution in $$\\mathbf{r}_{2}$$ and so must be eliminated. The solvability conditions are therefore", null, "(54)\n\nwhich yield Eq. (40). Note that the resulting normal form has detuning or bias terms, proportional to differences in inputs; these would be called ‘drifts’ in the context of a DDM, but no term linearly proportional to the normal-form amplitudes, or ‘decision variables’, as in the normal form for two-choice DM. Such a term describes the growth rate of the critical mode away from the bifurcation point: attracting to one side and repelling to the other. To derive these terms we will need to continue the calculation to next order. First, we must solve for $$\\mathbf{r}_{2}$$. We note that, while $$\\mathbf{r}_{2}$$ cannot have components in the directions $$\\mathbf{e}_{1}$$ or $$\\mathbf{e}_{2}$$, the components in the directions $$\\mathbf{e}_{C} = (1,1,1,0)$$ and $$\\mathbf{e}_{I} = (0,0,0,1)$$ (the common and inhibitory modes which we discussed in the section on DDMs) can be solved for.\n\nTherefore we take $$\\mathbf{r}_{2} = \\mathbf{e}_{C}R_{2C}+\\mathbf{e} _{I}R_{2I}$$ and project", null, "(55)\n\nwhich yields\n\n$$R_{2C} = \\frac{1}{g\\phi _{I}^{\\prime }}R_{2I},\\qquad R_{2I} = \\frac{1}{3c}(\\bar{I} _{1}+\\bar{I}_{2}+ \\bar{I}_{3})+\\frac{s^{2}\\phi ^{\\prime \\prime }}{3c\\phi ^{\\prime }}\\bigl(X_{1} ^{2}-X_{2}^{2}\\bigr).$$\n(56)\n\nUp to $$\\boldsymbol{\\mathcal{O}(\\epsilon ^{3})}$$:", null, "(57)\n\nwhere\n\n\\begin{aligned} \\mathbf{N}_{3} =& \\begin{pmatrix} s\\phi ^{\\prime \\prime }(X_{1}+X_{2}) ((s-cg\\phi _{I}^{\\prime })R_{2C}+\\bar{I}_{1} ) \\\\ s\\phi ^{\\prime \\prime }(-X_{1}+X_{2}) ((s-cg\\phi _{I}^{\\prime })R_{2C}+\\bar{I}_{2} ) \\\\ -2s\\phi ^{\\prime \\prime }X_{2} ((s-cg\\phi _{I}^{\\prime })R_{2C}+\\bar{I}_{3} ) \\\\ 0 \\end{pmatrix} \\\\ &{}+ \\frac{s^{3}\\phi ^{\\prime \\prime \\prime }}{6} \\begin{pmatrix} (X_{1}+X_{2})^{3} \\\\ (-X_{1}+X_{2})^{3} \\\\ -8X_{2}^{3} \\\\ 0 \\end{pmatrix} . \\end{aligned}\n(58)\n\nApplying the solvability conditions gives the equations\n\n$$0 = \\mathbf{e}_{1}\\cdot \\mathbf{N}_{3},\\qquad 0 = \\mathbf{e}_{2}\\cdot \\mathbf{N}_{3}.$$\n(59)\n\nIf we add the resulting terms to the normal-form equations derived at $$\\mathcal{O}(\\epsilon ^{2})$$ we have\n\n\\begin{aligned} \\tau \\partial _{T}X_{1} ={}& \\frac{\\phi ^{\\prime }}{2}(\\bar{I}_{1}-\\bar{I}_{2})+s \\phi ^{\\prime \\prime } \\biggl(\\frac{\\bar{I}_{1}+\\bar{I}_{2}}{2}+\\frac{(s-cg\\phi _{I} ^{\\prime })}{3cg\\phi _{I}^{\\prime }}( \\bar{I}_{1}+\\bar{I}_{2}+\\bar{I}_{3}) \\biggr)X _{1} \\\\ &{}+\\frac{s\\phi ^{\\prime \\prime }}{2}(\\bar{I}_{1}-\\bar{I}_{2})X_{2}+s \\phi ^{\\prime \\prime }X _{1}X_{2} \\\\ &{}+\\frac{(s-cg\\phi _{I}^{\\prime })}{3cg\\phi ^{\\prime }\\phi _{I}^{\\prime }}s^{3}\\bigl( \\phi ^{\\prime \\prime } \\bigr)^{2}X_{1}\\bigl(X_{1}^{2}-X_{2}^{2} \\bigr)+\\frac{s^{3}\\phi ^{\\prime \\prime \\prime }}{6}X _{1}\\bigl(X_{1}^{2}+3X_{2}^{2} \\bigr), \\\\ \\tau \\partial _{T}X_{2} ={}& \\frac{\\phi ^{\\prime }}{6}( \\bar{I}_{1}+\\bar{I}_{2}-2\\bar{I}_{3})+s\\phi ^{\\prime \\prime } \\biggl(\\frac{ \\bar{I}_{1}+\\bar{I}_{2}+4\\bar{I}_{3}}{6}+\\frac{(s-cg\\phi _{I}^{\\prime })}{3cg \\phi _{I}^{\\prime }}( \\bar{I}_{1}+\\bar{I}_{2}+\\bar{I}_{3}) \\biggr)X_{2}\\\\ &{} +\\frac{s \\phi ^{\\prime \\prime }}{6}(\\bar{I}_{1}+ \\bar{I}_{2})X_{1}+ \\frac{s^{2}\\phi ^{\\prime \\prime }}{6}\\bigl(X_{1}^{2}-3X_{2}^{2} \\bigr) \\\\ &{}+\\frac{(s-cg\\phi _{I} ^{\\prime })}{3cg\\phi ^{\\prime }\\phi _{I}^{\\prime }}s^{3}\\bigl(\\phi ^{\\prime \\prime } \\bigr)^{2}X_{2}\\bigl(X_{1}^{2}-X _{2}^{2}\\bigr)+\\frac{s^{3}\\phi ^{\\prime \\prime \\prime }}{6}X_{2} \\bigl(X_{1}^{2}+3X_{2}^{2}\\bigr). \\end{aligned}\n(60)\n\n#### 1.1.1 1.1.1 Normal form for three-choice DM starting with a different neuronal model\n\nIf we consider a different neuronal model for three-choice DM we can still arrive at the same normal-form equations. The only important factor is the presence of a linear subspace associated with a zero eigenvalue of multiplicity two and which is spanned by eigenvectors representing competition modes. As an example, consider the model\n\n\\begin{aligned} \\tau \\dot{r}_{1} &= -r_{1}+\\phi \\biggl(sr_{1}-\\frac{c}{2}(r_{2}+r_{3})+I _{1} \\biggr), \\\\ \\tau \\dot{r}_{2} &= -r_{2}+\\phi \\biggl(sr_{2}- \\frac{c}{2}(r _{1}+r_{3})+I_{2} \\biggr), \\\\ \\tau \\dot{r}_{3} &= -r_{3}+\\phi \\biggl(sr_{3}- \\frac{c}{2}(r_{1}+r_{2})+I_{3} \\biggr), \\end{aligned}\n(61)\n\nin which the inhibition is not explicitly modeled as before. A linear stability analysis reveals that there is a zero eigenvalue with multiplicity two when $$\\frac{c\\phi ^{\\prime }}{2} = 1-s\\phi ^{\\prime }$$ and $$1-s\\phi ^{\\prime }> 0$$. Therefore, the instability mechanism here is the cross-inhibition, while before it was the self-coupling. Nonetheless, it is only the structure of the corresponding eigenvectors which matters. In this case the two eigenvectors with zero eigenvalue are $$\\mathbf{e}_{1} = (1,-1,0)$$ and $$\\mathbf{e}_{2} = (1,1,-2)$$. Following the methodology described above leads to the identical normal-form equations Eq. (40) with s replaced by $$s+c/2$$.\n\n### 1.2 1.2 Normal-form equations for n-choice DM\n\nA neuronal model for n-choice DM $$(n>3)$$ is\n\n\\begin{aligned}& \\tau \\dot{r}_{1}= -r_{1}+\\phi (sr_{1}-cr_{I}+I_{1} )+\\xi _{1}(t), \\\\ &\\tau \\dot{r}_{2}= -r_{2}+\\phi (sr_{2}-cr_{I}+I_{2} )+\\xi _{2}(t), \\\\ &\\vdots \\\\ &\\tau \\dot{r}_{n}= -r_{n}+\\phi (sr_{n}-cr _{I}+I_{n} )+\\xi _{n}(t), \\\\ &\\tau _{I}\\dot{r}_{I}= -r_{I}+\\phi _{I} \\Biggl(\\frac{g}{n}\\sum_{j=1}^{n}r_{j}+I_{I} \\Biggr)+\\xi _{I}(t). \\end{aligned}\n(62)\n\nI again consider small differences in the inputs, $$I_{i} = I_{0}+ \\epsilon ^{2}\\Delta I_{i}$$ for $$i = [1,n]$$, expand the rates as $$\\mathbf{r} = \\mathbf{R}+\\epsilon \\mathbf{r}_{1}+\\mathcal{O}(\\epsilon ^{2})$$, where R are the rates at the fixed point, and define the slow time $$T = \\epsilon t$$. If we take $$1-s\\phi ^{\\prime } = 0$$ then the matrix of the linearized system has a zero eigenvalue with multiplicity $$n-1$$. Carrying out an expansion up to $$\\mathcal{O}(\\epsilon ^{2})$$ as described above and applying the $$n-1$$ solvability conditions leads to the normal-form equations Eq. (42).\n\n## Rights and permissions", null, "" ]

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[ "Trac issues https://gitlab.torproject.org/legacy/trac/-/issues 2020-06-13T15:01:55Z https://gitlab.torproject.org/legacy/trac/-/issues/20285 can&#39;t create valid case 2b3 consens weight calculation 2020-06-13T15:01:55Z pastly can't create valid case 2b3 consens weight calculation Even if #20284 is fixed, I still can come up with values that produce a `Wed` that is too large. Maybe I&#39;m not trying hard enough, but I can&#39;t get case 2b3 to execute successfully. For example, let ``` M=80 E=20 G=30 D=10 T=M+E+G+D ``... Even if #20284 is fixed, I still can come up with values that produce a `Wed` that is too large. Maybe I'm not trying hard enough, but I can't get case 2b3 to execute successfully. For example, let ``` M=80 E=20 G=30 D=10 T=M+E+G+D ``` In case 2b2, `Wed = (weight_scale*(D - 2*E + G + M))/(3*D) = 26667`. That's bigger than `weight_scale`. It (and `Wmd`) trigger case 2b3, which doesn't do anything about a too large `Wed` and thus `networkstatus_check_weights()` fails. I admit I don't know how reasonable the values are that I came up with above. I am writing test cases so #14881 can be closed though, and just about any weird combination should be handled without failing. Right? I don't know what the correct resolution is, so not patch/branch incoming at this time. Tor: unspecified https://gitlab.torproject.org/legacy/trac/-/issues/20284 consensus weight case 2b3 does not follow dir-spec 2020-07-31T12:42:39Z pastly consensus weight case 2b3 does not follow dir-spec [dir-spec](https://gitweb.torproject.org/torspec.git/tree/dir-spec.txt#n2681) says the following. ``` If M &gt; T/3, then the Wmd weight above will become negative. Set it to 0 in this case: Wmd = 0 Wgd = weight_scale - Wed ``` The ... [dir-spec](https://gitweb.torproject.org/torspec.git/tree/dir-spec.txt#n2681) says the following. ``` If M > T/3, then the Wmd weight above will become negative. Set it to 0 in this case: Wmd = 0 Wgd = weight_scale - Wed ``` The code dutifully sets `Wmd` to 0, but neglects `Wgd`. I assume the spec is correct and the intended behavior. Branch incoming once I get a ticket number. Tor: unspecified pastly pastly https://gitlab.torproject.org/legacy/trac/-/issues/20272 constraint broken in case 1 of consensus weight calculation 2020-06-13T15:01:52Z pastly constraint broken in case 1 of consensus weight calculation [dir-spec](https://gitweb.torproject.org/torspec.git/tree/dir-spec.txt#n2648) specifies the constraint `Wmg == Wmd` in case 1, but also that ``` Wmg = (weight_scale*(2*G-E-M))/(3*G) Wmd = weight_scale/3 ``` This constraint is impossibl... [dir-spec](https://gitweb.torproject.org/torspec.git/tree/dir-spec.txt#n2648) specifies the constraint `Wmg == Wmd` in case 1, but also that ``` Wmg = (weight_scale*(2*G-E-M))/(3*G) Wmd = weight_scale/3 ``` This constraint is impossible to satisfy unless it just happens that `(2G-E-M)/G == 1`. Indeed, in my testing of `networkstatus_compute_bw_weights_v10`, I found that `Wmg` and `Wmd` were calculated as above, but the constraint was ignored. The easy solution is to change the spec, but that would ignore the logic that went into having that constraint in the first place. I do not know the logic that went into designing the consensus weight calculations, so I do not know if this solution is appropriate. Tor: unspecified" ]

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[ "# Area of “polygon” inscribed in circle, whose side lengths form a geometric series\n\n$\\{v_0, v_n : n\\in \\Bbb{N}\\}$ are points on the circumference of a unit circle, having the following three properties:\n\n• The (straight line) distances between neighboring points in the sequence form a convergent geometric series. That is, for all $n>0$, $d(v_n,v_{n+1}) = r\\cdot d(v_{n-1},v_n)$ with $r$ a constant ratio for the entire set of verticies, and $0<r<1$.\n\n• $\\lim_{n_to\\infty} d(v_n,v_0) = 0$. That is, the \"polygon\" formed by these vertices forms a closed shape (in the limit) without having to add a \"last\" side going back to $v_0$.\n\n• For all $n>0$, the shorter of the two arcs from $v_n$ to $v_{n+1}$ does not contain $v_0$. That is, the sides go around the circle only once.\n\nI want to find the area of this \"polygon\" in terms of the side ratio $r$. (Although the number of sides needed to close the shape is infinite, one can easily define the area as the limit, as $k$ goes to infinity, of the area formed by the first $k$ sides and an added last side connecting $v_{k-1}$ to $v_0$.)\n\nSeveral things are immediately clear:\n\n• This area is not defined for $r<\\frac12$ because the set of lines cannot from a closed (in the limit) \"polygon\" unless the sum of the lengths of all the sides other than the first is longer than the first side. So the function $A(r)$ need only be found on the open interval $(\\frac12,1)$.\n\n• For any $r \\in( \\frac12,1)$, there exists a value of $L_0 \\equiv d(v_0,v_1)$ such that the figure exactly closes without overshooting. $L(r)$ has the property that $$\\sum_{n=0}^\\infty \\sin^{-1}\\left(\\frac{L}{2}r^n\\right) = \\pi$$\n\nEDIT For $r$ below about $0.6$, the first arc covers more than half the circle, and the area of the polygon does not include that first \"super-wedge.\" In that circumstance, $L(r)$ would have the property that\n\n$$\\sum_{n=1}^\\infty \\sin^{-1}\\left(\\frac{L}{2}r^n\\right) = \\sin^{-1}\\left(\\frac{L}{2}\\right)$$\n\nHowever, my numerical fiddling around indicates that you can't satisfy this, and that makes sense since arcsin is concave upwards.\n\n• $\\lim_{r\\to\\frac12} A(r) =0$ and $\\lim_{r\\to 1} A(r) =\\pi$. The former represents almost perfectly retracing the first side along a very short arc that is almost a straight line; the latter represents going around the circle in what is almost a regular inscribed polygon.\n\nEDIT The nearly straight line case does not happen; see the edit above.\n\nI'm also almost sure that for small positive $\\epsilon$, $A(r)$ has a positive second derivative at $r=\\frac12 + \\epsilon$, and a negative second derivative at $r = 1-\\epsilon$, and that $A(r)$ has just one point of inflection.\n\nMy question, then, is to find $A(r)$, or failing that, to describe some non-trivial properties of $A(r)$.\n\n• NB Irregular polygon... – user301988 Nov 8 '16 at 3:33\n• Taking into account that $L_0\\le2$ I think that it should be $r>0.624$ (about). – Intelligenti pauca Nov 8 '16 at 12:41\n• My question as psed was flawed in that when $r$ is close to $\\frac12$ the first of the arcs is greater than $\\pi$ so that the first term in the sum should be $\\pi - \\sin^{-1}(L/2)$ instead of $\\sin^{-1}(L/2)$. That is the reason that you can have such a polygon with $r$ just over $\\frac12$. I will edit a change. – Mark Fischler Nov 8 '16 at 21:08\n\n## 1 Answer\n\nEDIT.\n\nFunction $L(r)$ is implicitly defined by the equation: $$\\sum_{n=0}^\\infty\\arcsin\\left({1\\over2}L(r)\\ r^n\\right)=\\pi$$ if $r\\ge r_0\\approx0.602527$, and by the equation $$\\sum_{n=1}^\\infty\\arcsin\\left({1\\over2}L(r)\\ r^n\\right)=\\arcsin\\left({1\\over2}L(r)\\right)$$ if ${1/2}<r<r_0$. Here's a plot of $L(r)$ obtained by a numerical computation:", null, "If $r\\ge r_0$ the polygon is the sum of isosceles triangles $T_n$, having vertex at the center of the circle and base $L_n=L(r)r^n$, so its area is given by $$A(r)=\\sum_{n=0}^\\infty{1\\over2}L(r)\\ r^n\\sqrt{1-{1\\over4}L(r)^2r^{2n}}.$$ If $r<r_0$ the first triangle must be subtracted by the sum of the others, leading to $$A(r)=-{1\\over2}L(r)\\sqrt{1-{1\\over4}L(r)^2}+ \\sum_{n=1}^\\infty{1\\over2}L(r)\\ r^n\\sqrt{1-{1\\over4}L(r)^2r^{2n}}.$$ Here's a plot of $A(r)$:", null, "For a better visualization of the behaviour near $r=1/2$ here's a close-up of the same plot:", null, "" ]

[ null, "https://i.stack.imgur.com/SSQMS.png", null, "https://i.stack.imgur.com/F45GW.png", null, "https://i.stack.imgur.com/StUd8.png", null ]

[ "# What is 18% of 50172?\n\nA simple way to calculate percentages of X\n\n18% of 50172 is: 9030.96\n\n## Percentage of - Table for 50172\n\nPercentage ofDifference\n1% of 50172 is 501.7249670.28\n2% of 50172 is 1003.4449168.56\n3% of 50172 is 1505.1648666.84\n4% of 50172 is 2006.8848165.12\n5% of 50172 is 2508.647663.4\n6% of 50172 is 3010.3247161.68\n7% of 50172 is 3512.0446659.96\n8% of 50172 is 4013.7646158.24\n9% of 50172 is 4515.4845656.52\n10% of 50172 is 5017.245154.8\n11% of 50172 is 5518.9244653.08\n12% of 50172 is 6020.6444151.36\n13% of 50172 is 6522.3643649.64\n14% of 50172 is 7024.0843147.92\n15% of 50172 is 7525.842646.2\n16% of 50172 is 8027.5242144.48\n17% of 50172 is 8529.2441642.76\n18% of 50172 is 9030.9641141.04\n19% of 50172 is 9532.6840639.32\n20% of 50172 is 10034.440137.6\n21% of 50172 is 10536.1239635.88\n22% of 50172 is 11037.8439134.16\n23% of 50172 is 11539.5638632.44\n24% of 50172 is 12041.2838130.72\n25% of 50172 is 1254337629\n26% of 50172 is 13044.7237127.28\n27% of 50172 is 13546.4436625.56\n28% of 50172 is 14048.1636123.84\n29% of 50172 is 14549.8835622.12\n30% of 50172 is 15051.635120.4\n31% of 50172 is 15553.3234618.68\n32% of 50172 is 16055.0434116.96\n33% of 50172 is 16556.7633615.24\n34% of 50172 is 17058.4833113.52\n35% of 50172 is 17560.232611.8\n36% of 50172 is 18061.9232110.08\n37% of 50172 is 18563.6431608.36\n38% of 50172 is 19065.3631106.64\n39% of 50172 is 19567.0830604.92\n40% of 50172 is 20068.830103.2\n41% of 50172 is 20570.5229601.48\n42% of 50172 is 21072.2429099.76\n43% of 50172 is 21573.9628598.04\n44% of 50172 is 22075.6828096.32\n45% of 50172 is 22577.427594.6\n46% of 50172 is 23079.1227092.88\n47% of 50172 is 23580.8426591.16\n48% of 50172 is 24082.5626089.44\n49% of 50172 is 24584.2825587.72\n50% of 50172 is 2508625086\n51% of 50172 is 25587.7224584.28\n52% of 50172 is 26089.4424082.56\n53% of 50172 is 26591.1623580.84\n54% of 50172 is 27092.8823079.12\n55% of 50172 is 27594.622577.4\n56% of 50172 is 28096.3222075.68\n57% of 50172 is 28598.0421573.96\n58% of 50172 is 29099.7621072.24\n59% of 50172 is 29601.4820570.52\n60% of 50172 is 30103.220068.8\n61% of 50172 is 30604.9219567.08\n62% of 50172 is 31106.6419065.36\n63% of 50172 is 31608.3618563.64\n64% of 50172 is 32110.0818061.92\n65% of 50172 is 32611.817560.2\n66% of 50172 is 33113.5217058.48\n67% of 50172 is 33615.2416556.76\n68% of 50172 is 34116.9616055.04\n69% of 50172 is 34618.6815553.32\n70% of 50172 is 35120.415051.6\n71% of 50172 is 35622.1214549.88\n72% of 50172 is 36123.8414048.16\n73% of 50172 is 36625.5613546.44\n74% of 50172 is 37127.2813044.72\n75% of 50172 is 3762912543\n76% of 50172 is 38130.7212041.28\n77% of 50172 is 38632.4411539.56\n78% of 50172 is 39134.1611037.84\n79% of 50172 is 39635.8810536.12\n80% of 50172 is 40137.610034.4\n81% of 50172 is 40639.329532.68\n82% of 50172 is 41141.049030.96\n83% of 50172 is 41642.768529.24\n84% of 50172 is 42144.488027.52\n85% of 50172 is 42646.27525.8\n86% of 50172 is 43147.927024.08\n87% of 50172 is 43649.646522.36\n88% of 50172 is 44151.366020.64\n89% of 50172 is 44653.085518.92\n90% of 50172 is 45154.85017.2\n91% of 50172 is 45656.524515.48\n92% of 50172 is 46158.244013.76\n93% of 50172 is 46659.963512.04\n94% of 50172 is 47161.683010.32\n95% of 50172 is 47663.42508.6\n96% of 50172 is 48165.122006.88\n97% of 50172 is 48666.841505.16\n98% of 50172 is 49168.561003.44\n99% of 50172 is 49670.28501.72\n100% of 50172 is 501720\n\n## How calculate 18% of 50172\n\nIn the store, the product costs \\$50172, you were given a discount 18% and you want to understand how much you saved.\n\nSolution:\n\nAmount saved = Product price * Percentage Discount/ 100\n\nAmount saved = (18 * 50172) / 100\n\nAmount saved = \\$9030.96\n\nSimply put, when you buy an item for \\$50172 and the discount is 18%, you will pay \\$41141.04 and save \\$9030.96.\n\nLinkedIn\nTelegram\nWhatsApp\nViber\nReddit\n\n## Percentage Calculations: examples\n\nweb@percent-calc.com    © 2021\nTelegram\nChat" ]

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[ "Non-negative, real matrix $\\Rightarrow$ non-negative, real eigenvalues?\n\nDoes a matrix with all non-negative, real entries have all non-negative, real eigenvalues? Where might I find a proof of such?\n\nIdeas: Perhaps we can multiply a prospective eigenvector so its biggest entries are positive, and then show that it is a contradiction for it to have a negative eigenvalue?\n\nI am currently looking at the Perron-Frobenius theorem on Wikipedia, but it seems not to mention this issue. (I suspect my conjecture is not true.)\n\n• Do you still need help with this? – Git Gud Jun 6 '14 at 19:44\n\nHint: Not true. Think of a $2\\times 2$ matrix with non-negative entries and negative determinant. It looks like $I_2$.\n\nDefine\n\n$$A = \\left(\\begin{array}{ccc} .8147 & .9134 & .2785 \\\\.9058 & .6324 & .5469 \\\\ .1270 & .0975 & .9575\\end{array}\\right)$$\n\nThis has a negative eigenvalue, approximately $-.1879$.\n\nSource: Run\n\neig(rand(3, 3))\n\nin Matlab without changing the seed for the random number generator.\n\nNo, let $A = \\left[\\begin{array}{ccc} 1 & 2\\\\3 & 4\\end{array}\\right]$.\n\nNo. The matrix: $A = \\left[\\begin{array}{ccc} 0 & 1\\\\1 & 0\\end{array}\\right]$.\n\nis a reflection in $y=x$. It has an eigenvalue equal to $-1$." ]

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[ "# The Geometry of Letters\n\nAlignments to Content Standards: 4.G.A.1\n\nLetters can be thought of as geometric figures.", null, "1. How many line segments are needed to make the letter A? How many angles are there? Are they acute, obtuse, or right angles? Are any of the line segments perpendicular? Are any of the line segments parallel?\n\n2. We can build all of these letters from line segments and arcs of circles. Build all of the capital letters with the smallest number of \"pieces,\" where each piece is either a line segment or an arc of a circle.\n3. Which letters have perpendicular line segments?\n4. Which letters have parallel line segments?\n5. Which letters have no line segments?\n6. Do any letters contain both parallel and perpendicular lines?\n7. What makes the lower case letters \"i\" and \"j\" different than all of the capital letters?\n\n## IM Commentary\n\nThe purpose of this task is for students to analyze the geometry of letters. Letters provide a good opportunity for students to broaden their understanding of what constitutes a 2-dimensional geometric figure. Most students recognize polygons, circles, ellipses, and simple closed curves with some kind of symmetry as \"shapes,\" but a 2-D geometric figure in general is simply a specified collection of points in the plane. So, for example, you can have a geometric figure that is not connected (like the lower case letters \"i\" and \"j\"), or you can have a geometric figure that is composed of a simple closed curve with a line segment sticking out of it (like the letter \"P\"). The benefit of working with letters as opposed to made-up figures with these characteristics is that students have already accepted them as objects, and so it seems reasonable to study them from a geometrical perspective.\n\nThis task has students composing and decomposing figures, which is an important way of looking at geometric figures and a skill that students start working on in kindergarten and continue building throughout elementary school and beyond. Students should have access to a plastic or metal tracing ruler with different sized circles, a straight edge, and some colored pencils or markers.\n\nStudents are often confused about whether two line segments are parallel or perpendicular. For example, parallel lines are lines that never meet. Does this mean that these line segments are parallel?", null, "Even though these segments do not meet, the answer is \"no.\" The trick is to realize that every line segment is contained in an (infinite) line. So two line segments are parallel if and only if the lines that contain them are parallel. Likewise, two line segments are perpendicular if and only if the lines that contain them are perpendicular. This is what allows us to see that the letter F, for example, is composed of three line segments, two of which are parallel to each other and both of which are perpendicular to the third.\n\nNote that whenever two lines segments meet at an endpoint, two angles are implicitly defined. Often students only focus on the angle that is less than 180 degrees, but for example, the letter L can be thought of as defining both an angle that is 90 degrees and one that is 270 degrees. It would be good to address this directly in a whole-class discussion.\n\n## Solution\n\n1. The letter A is composed of 3 line segments which meet in three places and form 5 angles less than 180 degrees. Three of these angles are acute, and two are obtuse. Note that students might also count angles that are greater than 180 degrees, so it is important for students to explicitly identify the angles they see. For example, a student might see an angle greater than 180 degrees at the top of the letter A.\n\nAlternatively, the a student might see the letter A as being composed of 5 line segments and see two straight-angles on the sides of the letter A where the horizontal segment meets them. Of course, this isn't the smallest number of pieces needed to build the letter A, but it is a correct decomposition and analysis of it.\n\nNone of the line segments are perpendicular or parallel.\n\n2. The hardest ones are the ones composed of both segments and curves. B can be made from 4 segments and two semi-circles, for example:", null, "3. These letters have perpendicular line segments:\n\nB, D, E, F, H, L, P, R, T\n\nFor example, Here is an analysis of the letter B which is composed of 4 line segments (three of which are parallel to each other and are all perpendicular to the fourth) and two arcs of a circle:", null, "4. These letters have parallel line segments:\n\nB, D, E, F, H, M, N, R, U, Z\n\n5. These letters have no line segments: C, O, S. Depending on the font (or your handwriting), Q might or might not contain a line segment.\n6. These letters have both parallel and perpendicular line segments:\n\nB, D, E, F, H, R\n\n7. The lower case letters \"i\" and \"j\" are different than all of the capital letters because they are disconnected. That is, they are made out of pieces that don't touch." ]

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[ "# Lesson 1\n\nRelationships of Angles\n\n## 1.1: Visualizing Angles (5 minutes)\n\n### Warm-up\n\nThe purpose of this warm-up is to bring back to mind what students have learned previously about angle measures, as well as to discuss what aspects of each figure is important and which aspects can be ignored. Students may benefit from the use of an Angle Window: a scrap of paper with a penny-sized hole torn in the center of it. Students position the window so that the vertex of the angle and the beginning of the two rays are visible through the hole. This helps block out distractions, such as the lengths on the sides of the angle or other objects in the diagram.\n\nThe first question addresses the misconception that the size of an angle is related to lengths of line segments. The second question shows students they must be specific about how they refer to angles that share a vertex and introduces students to thinking about overlapping angles. Monitor for students who use different names for the same angle.\n\n### Launch\n\nGive students 1 minute of quiet work time, followed by a whole-class discussion.\n\nIf using the digital activity, make sure students realize they can drag the angles to compare size.\n\n### Student Facing\n\nUse the applet to answer the questions.\n\n1. Which angle is bigger, $$a$$ or $$b$$?\n2. Identify an obtuse angle in the diagram.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Launch\n\nGive students 1 minute of quiet work time, followed by a whole-class discussion.\n\nIf using the digital activity, make sure students realize they can drag the angles to compare size.\n\n### Student Facing\n\n1. Which angle is bigger?\n\n2. Identify an obtuse angle in the diagram.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Anticipated Misconceptions\n\nIn the first question, students may say that the angle measuring $$b$$ degrees is larger than the angle measuring $$a$$ degrees because the line segments are longer. Show them how to use an Angle Window positioned over the vertex to focus on the amount of turn between the two rays and ignore the length of the line segments.\n\nIn the second question, students may say that there is no obtuse angle, because they are only looking at $$\\angle DAC$$ and $$\\angle CAB$$ and not noticing the overlapping angle $$\\angle DAB$$. Reassure them that there is an obtuse angle in the figure, and ask them if $$\\angle DAC$$ and $$\\angle CAB$$ are the only angles present in the figure. Another possibility is to tell them that the obtuse angle has a measure of 110 degrees to help them find it.\n\n### Activity Synthesis\n\nThe goal of this discussion is to ensure that students understand that angles measure the amount of turn between two different directions. Poll the class on their responses for the first question. Make sure students reach an agreement that both angles in the first question are the same size. If there is a lot of disagreement, it may be helpful to demonstrate the use of an Angle Window for the whole class. If using the digital version of the materials, either angle $$a$$ or $$b$$ can be dragged on top of the other to demonstrate that they have the same measure.\n\nDisplay the figure in the second question, and ask previously identified students to share their responses. Make sure students understand that saying angle $$A$$ is not specific enough when referring to this diagram, because there is more than one angle with its vertex at point $$A$$. Consider asking questions like these:\n\n• “What is the measure of angle $$A$$?”\n• “Which angle is angle $$A$$?”\n• “Why is it not good enough to say angle $$A$$ when referring to this diagram?”\n\nExplain to the students that by using three points to refer to an angle, we can be sure that others will understand which angle we are talking about. Have students practice this way of referring to angles by asking questions such as:\n\n• “Which angle is bigger, angle $$DAC$$ or angle $$CAB$$?” (Angle $$DAC$$ is bigger because its measure is 60 degrees. It doesn’t matter that segment $$BA$$ is longer than segment $$DA$$.)\n• “Which angle is bigger, angle $$CAB$$ or angle $$BAC$$?” (They are both the same size, because they are two names for the same angle.)\n\nAlso explain to students that in a diagram an arc is often placed between the two sides of the angle being referenced.\n\nTell students that angles $$DAC$$ and $$CAB$$ are known as adjacent angles because they are next to each other, sharing segment $$AC$$ as one of their sides and $$A$$ as their vertex.\n\n## 1.2: Pattern Block Angles (15 minutes)\n\n### Activity\n\nThe purpose of this activity is to use the fact that the sum of the angles all the way around a point is $$360 ^\\circ$$ to reason about the measure of other angles. Students are reminded that angle measures are additive (4.MD.C.7) before undertaking work with complementary and supplementary angles in future lessons.\n\nFormally, a right angle is $$90^\\circ$$ because we defined $$360^\\circ$$ to be all the way around and $$\\frac14 \\boldcdot 360 = 90$$. Students may have forgotten about $$360^\\circ$$, but they are likely to remember $$90^\\circ$$ from their work with angles in grade 4. We can use right angles as a tool to rediscover that all the way around must be $$360^\\circ$$, because $$4 \\boldcdot 90 = 360$$.\n\nIn this activity, students use pattern blocks to explore configurations that make $$360 ^\\circ$$ and to solve for angles of the individual blocks. For this activity, there are multiple configurations of blocks that will accomplish the task.\n\nAs students work, monitor for those who:\n\n• use similar reasoning in the launch to figure out the measure of the various angles they traced from the pattern blocks.\n• find relationships between different angle measures and different pattern blocks (for example: one hexagon angle is also 2 green triangles, which means one green triangle angle is $$60^\\circ$$ because $$\\frac12 \\boldcdot 120 =60$$.\n\n### Launch\n\nArrange students in groups of 3–4. Display the figures in this image one at a time, or use actual pattern blocks to recreate these figures for all to see.\n\nAsk these questions after each figure is displayed:\n\n• “What is the measure of $$\\angle a$$? How do you know?” ($$90 ^\\circ$$, because it is a right angle.)\n• “What is the measure of $$a + b + c$$?” ($$270 ^\\circ$$, because $$90 + 90 + 90 = 270$$.)\n• “What is the measure of $$a + b + c + d$$?” ($$360 ^\\circ$$, because $$4 \\boldcdot 90 = 360$$.)\n\nReinforce that $$360 ^\\circ$$ is once completely around a point by having students stand up, hold their arm out in front of them, and turn $$360 ^\\circ$$ around. Students who are familiar with activities like skateboarding or figure skating will already have a notion of $$360 ^\\circ$$ as a full rotation and $$180 ^\\circ$$ as half of a rotation.\n\nDistribute pattern blocks. Or, if using the digital version of materials, demonstrate the use of the applet. Ensure students know that after they drag a block from the left to the right side of the window, they can click to rotate the block.\n\nRepresentation: Develop Language and Symbols. Display or provide charts with the figures, symbols and meanings of the angle measures at the vertices for all the different pattern blocks.\nSupports accessibility for: Conceptual processing; Memory\n\n### Student Facing\n\n1. Look at the different pattern blocks inside the applet. Each block contains either 1 or 2 angles with different degree measures. Which blocks have only 1 unique angle? Which have 2?\n2. If you place three copies of the hexagon together so that one vertex from each hexagon touches the same point, as shown, they fit together without any gaps or overlaps. Use this to figure out the degree measure of the angle inside the hexagon pattern block.\n\n3. Figure out the degree measure of all of the other angles inside the pattern blocks. (Hint: turn on the grid to help align the pieces.)  Be prepared to explain your reasoning.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Launch\n\nArrange students in groups of 3–4. Display the figures in this image one at a time, or use actual pattern blocks to recreate these figures for all to see.\n\nAsk these questions after each figure is displayed:\n\n• “What is the measure of $$\\angle a$$? How do you know?” ($$90 ^\\circ$$, because it is a right angle.)\n• “What is the measure of $$a + b + c$$?” ($$270 ^\\circ$$, because $$90 + 90 + 90 = 270$$.)\n• “What is the measure of $$a + b + c + d$$?” ($$360 ^\\circ$$, because $$4 \\boldcdot 90 = 360$$.)\n\nReinforce that $$360 ^\\circ$$ is once completely around a point by having students stand up, hold their arm out in front of them, and turn $$360 ^\\circ$$ around. Students who are familiar with activities like skateboarding or figure skating will already have a notion of $$360 ^\\circ$$ as a full rotation and $$180 ^\\circ$$ as half of a rotation.\n\nDistribute pattern blocks. Or, if using the digital version of materials, demonstrate the use of the applet. Ensure students know that after they drag a block from the left to the right side of the window, they can click to rotate the block.\n\nRepresentation: Develop Language and Symbols. Display or provide charts with the figures, symbols and meanings of the angle measures at the vertices for all the different pattern blocks.\nSupports accessibility for: Conceptual processing; Memory\n\n### Student Facing\n\n1. Trace one copy of every different pattern block. Each block contains either 1 or 2 angles with different degree measures. Which blocks have only 1 unique angle? Which have 2?\n2. If you trace three copies of the hexagon so that one vertex from each hexagon touches the same point, as shown, they fit together without any gaps or overlaps. Use this to figure out the degree measure of the angle inside the hexagon pattern block.\n\n3. Figure out the degree measure of all of the other angles inside the pattern blocks that you traced in the first question. Be prepared to explain your reasoning.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Student Facing\n\n#### Are you ready for more?\n\nWe saw that it is possible to fit three copies of a regular hexagon snugly around a point.\n\nEach interior angle of a regular pentagon measures $$108^\\circ$$. Is it possible to fit copies of a regular pentagon snugly around a point? If yes, how many copies does it take? If not, why not?\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Anticipated Misconceptions\n\nWhen working on calculating the angle measure, students might need to be reminded that a complete turn is $$360^\\circ$$.\n\nIf students place angles that are not congruent next to each other, it could produce valid reasoning, but they may draw erroneous conclusions. For example, using four copies of the blue rhombus, you can place 2 obtuse angles and 2 acute angles around the same vertex with no gaps or overlaps. However, this does not mean that they are each $$\\frac14$$ of $$360^\\circ$$. Encourage students to reason about whether their conclusions make sense and to verify their conclusions in more than one way.\n\n### Activity Synthesis\n\nThe goal of this discussion is for students to be exposed to writing equations that represent the relationships between different angle measures. Select previously identified students to share how they figured out the different angle measures in each pattern block. Sequence the explanations from most common (reminiscent of the square and hexagon examples) to most creative.\n\nWrite an equation to represent how their angles add up to $$360 ^\\circ$$. Listen carefully for how students describe their reasoning and make your equation match the vocabulary they use. For example, students might have reasoned about 6 green triangles by thinking $$60 + 60 + 60 + 60 + 60 + 60 = 360$$ or $$6 \\boldcdot 60 = 360$$ or $$360 \\div 6 = 60$$.\n\nOnce an angle from one block is known, it can be used to help figure out angles for other blocks. For example, students may say that they knew the angles on the yellow hexagon measured $$120 ^\\circ$$ because they could fit two of the green triangles onto one corner of the hexagon, and $$60 + 60 = 120$$ or $$2 \\boldcdot 60 = 120$$. There are many different ways students could have reasoned about the angles on each block, and it is okay if they didn’t think back to $$360 ^\\circ$$ for every angle.\n\nBefore moving on to the next activity, ensure that students know the measure of each interior angle of each shape in the set of pattern blocks. Display these measures for all to see throughout the remainder of the lesson.\n\nSpeaking, Listening: MLR2 Collect and Display. Use this routine to capture existing student language related to finding the measure of a given angle. Circulate and listen to student talk during small-group and whole-class discussion. Record the words, phrases, drawings, and writing students use to explain the equations they wrote to represent the relationships between different angle measures. Display the collected language for all to see, and invite students to borrow from, or add more language to the display throughout the remainder of the lesson. It is expected that students will be using informal language when they explain their reasoning at this point in the unit. Over the course of the unit, invite students to suggest revisions, and updates to the display as they develop new mathematical ideas and new language to communicate them.\nDesign Principle(s): Support sense-making; Maximize meta-awareness\n\n## 1.3: More Pattern Block Angles (10 minutes)\n\n### Activity\n\nIn this activity, students figure out measures of given angles using the pattern block angles they discovered in the previous activity. Most importantly, students recognize that a straight angle can be considered an angle and not just a line. Students are asked to find different combinations of pattern blocks that form a straight angle, which helps students to see the connection between the algebraic action of summing angles and the geometric action of joining angles with the same vertex.\n\nAs students work on the task, monitor for students who use different combinations of blocks to form a straight angle.\n\n### Launch\n\nStudents may need help focusing on the correct angles when there are multiple blocks involved. These students may benefit from using the Angle Window created in the warm-up for this lesson.\n\nThere are many ways to use the blocks to find the measures of the angles in the first question. Students are encouraged to find more than one way, and to check that their answers remain the same.\n\nGive students 2–3 minutes of quiet work time followed by a partner and whole-class discussion.\n\nRepresentation: Internalize Comprehension. Begin with a physical demonstration of using pattern blocks to determine the measure of an angle.\nSupports accessibility for: Conceptual processing; Visual-spatial processing\nWriting, Conversing: MLR5 Co-craft Questions. Use this routine to support language development through student conversations about mathematical questions. Without revealing the questions of the task, display only the image of the three angles for all to see. Invite students to work with a partner to write possible mathematical questions that could be asked about what they see. Listen for questions that connect the use of pattern blocks with measuring angles.\nDesign Principle(s): Cultivate conversation; Support sense-making\n\n### Student Facing\n\n1. Use pattern blocks to determine the measure of each of these angles.\n\n2. If an angle has a measure of $$180^\\circ$$ then the two legs form a straight line. An angle that forms a straight line is called a straight angle. Find as many different combinations of pattern blocks as you can that make a straight angle.\n\nUse the applet if you choose. (Hint: turn on the grid to help align the pieces.)\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Launch\n\nStudents may need help focusing on the correct angles when there are multiple blocks involved. These students may benefit from using the Angle Window created in the warm-up for this lesson.\n\nThere are many ways to use the blocks to find the measures of the angles in the first question. Students are encouraged to find more than one way, and to check that their answers remain the same.\n\nGive students 2–3 minutes of quiet work time followed by a partner and whole-class discussion.\n\nRepresentation: Internalize Comprehension. Begin with a physical demonstration of using pattern blocks to determine the measure of an angle.\nSupports accessibility for: Conceptual processing; Visual-spatial processing\nWriting, Conversing: MLR5 Co-craft Questions. Use this routine to support language development through student conversations about mathematical questions. Without revealing the questions of the task, display only the image of the three angles for all to see. Invite students to work with a partner to write possible mathematical questions that could be asked about what they see. Listen for questions that connect the use of pattern blocks with measuring angles.\nDesign Principle(s): Cultivate conversation; Support sense-making\n\n### Student Facing\n\n1. Use pattern blocks to determine the measure of each of these angles.\n\n2. If an angle has a measure of $$180^\\circ$$, then its sides form a straight line. An angle that forms a straight line is called a straight angle. Find as many different combinations of pattern blocks as you can that make a straight angle.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Anticipated Misconceptions\n\nSome students may say that $$b=150$$. Prompt them to notice that the arc marking which angle to measure is on the side that is greater than $$180^\\circ$$.\n\nIf students are stuck on the angle that measures $$c$$ degrees, consider using one of the patterns from the previous task that created a 360-degree angle with all the same pattern blocks and remove half of the pattern to show the 180-degree angle.\n\nIn the second problem, students might need encouragement to look for multiple combinations of pattern blocks to form a straight line.\n\n### Activity Synthesis\n\nThe goal of this discussion is for students to be exposed to many different examples of angle measures summing to $$180^\\circ$$.\n\nFirst, instruct students to compare their answers to the first question with a partner and share their reasoning until they reach an agreement. To help students see $$c$$ as a 180-degree angle and not just a straight line, consider using only the smaller angle on the tan rhombus blocks to measure all three figures: composing four tan rhombuses gives an angle measuring $$a$$ degrees, seven rhombuses give an angle measuring $$b$$ degrees, and six rhombuses give an angle measuring $$c$$ degrees.\n\nNext, select previously identified students to share their solutions to the second question. For each combination of blocks that is shared, invite other students in the class to write an equation displayed for all to see that reflects the reasoning.\n\n## 1.4: Measuring Like This or That (10 minutes)\n\n### Optional activity\n\nThe purpose of this optional activity is to address the common error of reading a protractor from the wrong end. The problem gives students the opportunity to critique someone else’s thinking and make an argument if they agree with either students’ claim (MP3).\n\n### Launch\n\nArrange students in groups of 2. Give students 2–3 minutes of quiet think time followed by a partner and whole-class discussion.\n\n### Student Facing\n\nTyler and Priya were both measuring angle $$TUS$$.\n\nPriya thinks the angle measures 40 degrees. Tyler thinks the angle measures 140 degrees. Do you agree with either of them? Explain your reasoning.\n\n### Student Response\n\nFor access, consult one of our IM Certified Partners.\n\n### Activity Synthesis\n\nAsk students to indicate whether they agree with Priya or Tyler. Invite students to explain their reasoning until the class comes to an agreement that the measurement of angle $$TUS$$ is 40 degrees.\n\nAsk students how Tyler could know that his answer of 140 degrees is unreasonable for the measure of angle $$TUS$$. Possible discussion points include:\n\n• “Is angle $$TUS$$ acute, right, or obtuse?” (acute)\n• “Where is there an angle that measures 140 degrees in this figure?” (adjacent to angle $$TUS$$, from side $$US$$ to the other side of the protractor)\n\nMake sure that students understand that a protractor is often labeled with two sets of angle measures, and they need to consider which side of the protractor they are measuring from.\n\nSpeaking, Listening, Conversing: MLR1 Stronger and Clearer Each Time. Use this routine to give students a structured opportunity to revise and refine their explanation about whether or not they agree with Tyler or Priya. Give students time to meet with 2–3 partners, to share and get feedback on their responses. Provide students with prompts for feedback that will help their partners strengthen their ideas and clarify their language. For example, “What do you think each person did first?”, “Could Priya and Tyler both be correct?”, \"Can you say that a different way?\" Give students 1–2 minutes to revise their writing based on the feedback they received.\nDesign Principle(s): Cultivate conversation; Optimize output (for explanation)\n\n## Lesson Synthesis\n\n### Lesson Synthesis\n\n• What are the three main types of angles in this lesson, and what are their measures? (right: $$90^\\circ$$, straight: $$180^\\circ$$, all the way around a point: $$360^\\circ$$)\n• What does it look like when angles are adjacent, and what can you say about angle measures? (The two angles are placed so that they share a vertex and one side. For adjacent angles, angle measures add. For example, a $$60^\\circ$$ angle adjacent to a $$120^\\circ$$ angle produces a $$180^\\circ$$ straight angle.)\n\n## 1.5: Cool-down - Identical Isosceles Triangles (5 minutes)\n\n### Cool-Down\n\nFor access, consult one of our IM Certified Partners.\n\n## Student Lesson Summary\n\n### Student Facing\n\nWhen two lines intersect and form four equal angles, we call each one a right angle. A right angle measures $$90^\\circ$$. You can think of a right angle as a quarter turn in one direction or the other.\n\nAn angle in which the two sides form a straight line is called a straight angle. A straight angle measures $$180^\\circ$$. A straight angle can be made by putting right angles together. You can think of a straight angle as a half turn, so that you are facing in the opposite direction after you are done.\nIf you put two straight angles together, you get an angle that is $$360^\\circ$$. You can think of this angle as turning all the way around so that you are facing the same direction as when you started the turn." ]

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[ "# SO(3) vortices as a mechanism for generating a mass gap in the 2D SU(2) principal chiral model\n\nResearch output: Contribution to journalArticle\n\n5 Citations (Scopus)\n\n### Abstract\n\nWe propose a mechanism that can create a mass gap in the SU(2) chiral spin model at arbitrarily small temperatures. We give a sufficient condition for the mass gap to be non-zero in terms of the behaviour of an external ℤ(2) flux introduced by twisted boundary conditions. This condition in turn is rewritten in terms of an effective dual Ising model with an external magnetic field generated by SO(3) vortices. We show that having a non-zero magnetic field in the effective Ising model is sufficient for the SU(2) system to have a mass gap. We also show that certain vortex correlation inequalities, if satisfied, would imply a non-zero effective magnetic field. Finally, we give some plausibility arguments and Monte Carlo evidence for the required correlation inequalities.\n\nOriginal language English 613-638 26 Nuclear Physics B 482 3 https://doi.org/10.1016/S0550-3213(96)00561-5 Published - Dec 30 1996\n\n### Fingerprint\n\nvortices\nIsing model\nmagnetic fields\nboundary conditions\ntemperature\n\n### Keywords\n\n• Confinement\n• Disorder\n• Lattice\n• SU(2) principal chiral model\n\n### ASJC Scopus subject areas\n\n• Nuclear and High Energy Physics\n\n### Cite this\n\nIn: Nuclear Physics B, Vol. 482, No. 3, 30.12.1996, p. 613-638.\n\nResearch output: Contribution to journalArticle\n\ntitle = \"SO(3) vortices as a mechanism for generating a mass gap in the 2D SU(2) principal chiral model\",\nabstract = \"We propose a mechanism that can create a mass gap in the SU(2) chiral spin model at arbitrarily small temperatures. We give a sufficient condition for the mass gap to be non-zero in terms of the behaviour of an external ℤ(2) flux introduced by twisted boundary conditions. This condition in turn is rewritten in terms of an effective dual Ising model with an external magnetic field generated by SO(3) vortices. We show that having a non-zero magnetic field in the effective Ising model is sufficient for the SU(2) system to have a mass gap. We also show that certain vortex correlation inequalities, if satisfied, would imply a non-zero effective magnetic field. Finally, we give some plausibility arguments and Monte Carlo evidence for the required correlation inequalities.\",\nkeywords = \"Confinement, Disorder, Lattice, SU(2) principal chiral model\",\nauthor = \"T. Kov{\\'a}cs\",\nyear = \"1996\",\nmonth = \"12\",\nday = \"30\",\ndoi = \"10.1016/S0550-3213(96)00561-5\",\nlanguage = \"English\",\nvolume = \"482\",\npages = \"613--638\",\njournal = \"Nuclear Physics B\",\nissn = \"0550-3213\",\npublisher = \"Elsevier\",\nnumber = \"3\",\n\n}\n\nTY - JOUR\n\nT1 - SO(3) vortices as a mechanism for generating a mass gap in the 2D SU(2) principal chiral model\n\nAU - Kovács, T.\n\nPY - 1996/12/30\n\nY1 - 1996/12/30\n\nN2 - We propose a mechanism that can create a mass gap in the SU(2) chiral spin model at arbitrarily small temperatures. We give a sufficient condition for the mass gap to be non-zero in terms of the behaviour of an external ℤ(2) flux introduced by twisted boundary conditions. This condition in turn is rewritten in terms of an effective dual Ising model with an external magnetic field generated by SO(3) vortices. We show that having a non-zero magnetic field in the effective Ising model is sufficient for the SU(2) system to have a mass gap. We also show that certain vortex correlation inequalities, if satisfied, would imply a non-zero effective magnetic field. Finally, we give some plausibility arguments and Monte Carlo evidence for the required correlation inequalities.\n\nAB - We propose a mechanism that can create a mass gap in the SU(2) chiral spin model at arbitrarily small temperatures. We give a sufficient condition for the mass gap to be non-zero in terms of the behaviour of an external ℤ(2) flux introduced by twisted boundary conditions. This condition in turn is rewritten in terms of an effective dual Ising model with an external magnetic field generated by SO(3) vortices. We show that having a non-zero magnetic field in the effective Ising model is sufficient for the SU(2) system to have a mass gap. We also show that certain vortex correlation inequalities, if satisfied, would imply a non-zero effective magnetic field. Finally, we give some plausibility arguments and Monte Carlo evidence for the required correlation inequalities.\n\nKW - Confinement\n\nKW - Disorder\n\nKW - Lattice\n\nKW - SU(2) principal chiral model\n\nUR - http://www.scopus.com/inward/record.url?scp=0030607748&partnerID=8YFLogxK\n\nUR - http://www.scopus.com/inward/citedby.url?scp=0030607748&partnerID=8YFLogxK\n\nU2 - 10.1016/S0550-3213(96)00561-5\n\nDO - 10.1016/S0550-3213(96)00561-5\n\nM3 - Article\n\nAN - SCOPUS:0030607748\n\nVL - 482\n\nSP - 613\n\nEP - 638\n\nJO - Nuclear Physics B\n\nJF - Nuclear Physics B\n\nSN - 0550-3213\n\nIS - 3\n\nER -" ]

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[ "## write an equation for a line to pass through the point (2,3) and has a slope of -4\n\nQuestion\n\nwrite an equation for a line to pass through the point (2,3) and has a slope of -4\n\nin progress 0\n5 days 2021-09-11T07:31:57+00:00 1 Answer 0\n\ny = -4x + 11\n\nStep-by-step explanation:\n\nEquation of a line is\n\ny – y_1 = m ( x – x _1)\n\nWe are provided\n\nSlope = m = -4\n\nPoints = (x _1 , y_1) = (2, 3)\n\nInsert the values\n\ny – y_1 = m( x – x_1)\n\ny – 3 = -4(x – 2)\n\nOpen the bracket with -4\n\ny – 3 = -4x + 8\n\ny – 3 -8 = -4x\n\ny – 11 = -4x\n\nFollowing this equation of a line\n\ny = mx + C\n\ny – intercept point y\n\nm – slope\n\nx – intercept point x\n\nC – intercept of the line\n\ny – 11 = -4x\n\ny = -4x + 11\n\nTherefore, the equation of the line is\n\ny = -4x + 11" ]

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[ "Cody\n\n# Problem 1107. Find max\n\nSolution 1855303\n\nSubmitted on 21 Jun 2019 by Sarah Kollender\nThis solution is locked. To view this solution, you need to provide a solution of the same size or smaller.\n\n### Test Suite\n\nTest Status Code Input and Output\n1   Pass\nx = magic(5); y_correct = 25; assert(isequal(your_fcn_name(x),y_correct))\n\nq = 23 24 25 21 22 y = 25\n\n2   Pass\nx = [2 4 9 0 7 19;3 4 1 2 0 6]; y_correct = 19; assert(isequal(your_fcn_name(x),y_correct))\n\nq = 3 4 9 2 7 19 y = 19\n\n3   Pass\nx = [2 4 9 0 7 19;3 4 1 2 0 6]'; y_correct = 19; assert(isequal(your_fcn_name(x),y_correct))\n\nq = 19 6 y = 19" ]

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[ "# Calculate the following integral with W|A?\n\nPosted 6 months ago\n751 Views\n|\n|\n0 Total Likes\n|\n I just have easy input: int(x^2*y^2*sqrt(R^2-y^2-x^2)dx,-sqrt(R^2-y^2),sqrt(R^2-y^2)) But Wolfram don't understand it and don't want to calculate the result. I mean it is okay that we have some variables (R,y), but they are not variables, they are constants... And it should be cleaner to calculate native integral and simplify it. But Wolfram don't want it...", null, "Answer\n This alternate form of input int(x^2*y^2*sqrt(R^2-y^2-x^2),{x,-sqrt(R^2-y^2),sqrt(R^2-y^2)}) is understood by WolframAlpha, but it does not find an exact symbolic solution for that.WolframAlpha is able to do the indefinite integral int(x^2*y^2*sqrt(R^2-y^2-x^2),{x}) giving (y^2 (x Sqrt[R^2-x^2-y^2](-R^2+2 x^2+y^2)+(R^2-y^2)^2 ArcTan[x/Sqrt[R^2-x^2-y^2]]))/8 and you might consider whether you could substitute in your upper and lower bounds for x and subtract those two to give your definite integral.If you can provide the values for the constants then WolframAlpha might be able to directly provide your solution.If it is of any help to you, the full version of Mathematica is able to find the definite integral Integrate[x^2*y^2*Sqrt[R^2-y^2-x^2],{x,-Sqrt[R^2-y^2],Sqrt[R^2-y^2]}] gives (Pi*(-(R^2*y) + y^3)^2)/8", null, "Answer" ]

[ null, "https://community.wolfram.com/community-theme/images/common/checked.png", null, "https://community.wolfram.com/community-theme/images/common/checked.png", null ]

[ "# OESymmetryNumber¶\n\n```unsigned OESymmetryNumber(const OEMolBase &mol, bool useH=true,\ndouble threshold=0.05)\nbool OESymmetryNumber(const OEMCMolBase &mol, unsigned *symCtArray,\nbool useH=true, double threshold=0.05)\n```\n\nCalculates the symmetry number of the molecule. For the overload with `OEMolBase` as input molecule, the symmetry number is the return value. For the `OEMCMolBase` overload, the symmetry numbers are returned in the symCtdArray array. The symCtArray passed to this function should be of length `fit.GetMaxConfIdx()`. The useH flag indicates whether symmetry calculation should use the hydrogen atoms of the molecule or ignore them. Root mean squared deviation between conformations are performed internally and accepted as identical based on the specified threshold. For the `OEMCMolBase` overload the return value is `True` if calculation completes successfully, and `False` otherwise.\n\nmol\n\nThe molecule to calculate symmetry number.\n\nuseH\n\nFlag indicating if hydrogen atoms should be considered explicitly.\n\nthreshold\n\nThreshold for root mean squared deviation (RMSD) comparison.\n\nsymCtArray\n\nArray containing calculated symmetry numbers in a multi-conformer input molecule." ]

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[ "", null, "# Preliminary Maths for Deep Learning\n\nIn this series of posts, I will pick a source for Deep Learning and try to share my understanding and learnings on the topic. I want to treat this as my notes from these lessons. My notes tend to be very rough, in an attempt to keep a semblance of formality to them I am putting them out in public.\n\nMachine Learning is primarily all about extracting information from data. Unlike simple situations in life data in Machine Learning comes in larger quantities. The most common form of this “large quantity” is a list of numbers. Mathematically, we would like to apply some abstraction and perform operations on them. Vectors are a good abstraction for such cases, a collection of such vectors can be represented by Matrices. Now Tensors are a generalized form of matrices allowing data to be represented along several axes.\n\nLinear Algebra helps us understand the rules of simple operations on such a collection of numbers. In this post, I will only list the items or topics from Linear Algebra that we will need to apply Deep Learning in a practical manner.\n\nHere’s the list, each of which we will cover in detail in future posts.\n\n• Scalars\n• Vectors\n• Matrices\n• Tensors\n• Reduction of Tensors\n• Non-Reduction Operations: Sum\n• Dot Products\n• Matrix-Vector Products\n• Multiplication of Matrices\n• Norms - L2, Frobenius\n\nIf you think you need any further topics to apply Deep Learning it is likely to be answered here or here.\n\nAs mentioned earlier, we will cover some or all of these topics in a bit more detail in separate tiny bite sized posts. Until then, happy whatever!" ]

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[ "Solutions by everydaycalculation.com\n\n## Divide 35/4 with 4/24\n\n1st number: 8 3/4, 2nd number: 4/24\n\n35/4 ÷ 4/24 is 105/2.\n\n#### Steps for dividing fractions\n\n1. Find the reciprocal of the divisor\nReciprocal of 4/24: 24/4\n2. Now, multiply it with the dividend\nSo, 35/4 ÷ 4/24 = 35/4 × 24/4\n3. = 35 × 24/4 × 4 = 840/16\n4. After reducing the fraction, the answer is 105/2\n5. In mixed form: 521/2\n\nMathStep (Works offline)", null, "Download our mobile app and learn to work with fractions in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "# Simple Diode and Wave shaping Circuits: Clipping, Clamping\n\nBy Yash Bansal|Updated : August 4th, 2021\n\nIn this article, you will find the Study Notes on Simple Diode and Wave shaping Circuits which will cover the topic as Logic Gates, Clippers and Clamper circuits.\n\nIn this article, you will find the Study Notes on Simple Diode and Wave shaping Circuits which will cover the topic as Logic Gates, Clippers and Clamper circuits.\n\n### 1. Logic Gates\n\n• AND Gate:", null, "If one of the inputs A or B is grounded, current flows through the diode and the output node C is at a low voltage. The only way to get a high output is by having both inputs high. This circuit is equivalent to logical AND function.\n\n• OR Gate:", null, "If one or both of the inputs A and B are high, current flows through the associated diode and the output node at a high voltage. It is equivalent to OR Gate.\n\n### 2. Clipper Circuits\n\nDiode network that have the ability to clip of a portion of the input signal without distorting the remaining part of the alternating waveform is known as Clipper.\n\nThere are two general categories of clippers\n\n1.  Series clipper\n2.  Parallel Clipper\n\nThe series configuration is defined as one where the diode is in series with the load, while the parallel configuration has the diode in a branch parallel to the load.\n\n(a) Series Clipper\n\n(i) Unbiased Clipper\n\nThe response of series configuration of figure 1(a) to a variety of alternating waveform is provided in figure 1(b)", null, "From figure 1(a), Anode voltage of diode VA = Vi and Cathode voltage VK = 0\n\nTherefore, for positive half cycle VA > VK, which means diode is forward biased and act as short circuit.\n\nFor negative half cycle VA < VK, which means diode is reverse biased and act as open circuits.\n\n(ii) Biased clipper\n\nThe addition of a dc supply that can have a pronounced effect on the output of a clipper is shown in figure 2.\n\nNOTE: Make a mental sketch of the response of the network based on the direction of the diode and the applied voltage levels.", null, "For the given network, the direction of diode suggest that the signal Vi must be positive to turn it ON. The dc supply further requires that the voltage Vi be greater than V to turn ON the diode. The negative region of the input signal is pressuring the diode into the OFF state, supported further by the dc supply.\n\nNOTE: Determine the applied voltage (transition voltage) that will cause a change in state for the diode.\n\nFor the ideal diode the transition between states will occur on the characteristics, where Vd = 0 V and id = 0A. Applying the condition id = 0 at Vd = 0 to the network of figure 2 will result in the configuration of figure 3(a), where it is recognized that the level of Vi that will cause a transition in state is Vi = V", null, "For an input voltage greater than V volts the diode is in the short circuit state, while input voltage less than V volts it is in the open circuit or off state.\n\nNOTE: Be continually aware of the defined terminals and polarity of V0.\n\nWhen the diode is in the short circuit state, such as shown in figure 4, the output voltage V0 can be determined by applying KVL in the clockwise direction.", null, "Vi – V – V0 = 0\n\n∴ V0 = Vi – V\n\nNOTE: It can be helpful to sketch the input signal above the output and determine the output and determine the output at instantaneous values of the input.", null, "For an instantaneous value of Vi the input can be treated as a dc supply of that value and corresponding dc value of the output determined.\n\nFor instant at Vi = Vm\n\nFor Vm > V, diode is short circuit and V0 = Vm – V\n\nWhen diode change state, and Vi = – Vm\n\nThen V0 = 0 V\n\nAnd now complete the curve for V0 that can be shown in figure 5.\n\n(b) Parallel Clipper\n\nThe network of figure 6 is the simplest of parallel diode configuration with the output for the same input as discussed earlier. The analysis of parallel configuration is very similar to that applied to series configurations.", null, "", null, "### 3. Clampers\n\nThe clamping network is one that will clamp a signal to a different dc level. The network consist of a capacitor, a diode and a resistor element and an independent dc supply to introduce on additional shift.\n\nThe magnitude of R and C must be chosen such that the time constant τ = RC is large enough to ensure that the voltage across the capacitor does not discharge significantly during the interval the diode is non conducting.\n\nThere are basically two type of clamper:", null, "(a) Negative Clamper", null, "Figure 8(b): Waveform\n\nWhen the input is positive, diode operates in forward bias and capacitor charge through diode. If diode is ideal it behaves as short circuit and therefore capacitor charge upto the peak input Vm.\n\nWhen input becomes negative, capacitor should discharge but discharge path is not available so capacitor voltage will continue to remain Vm. Therefore, once capacitor is fully charged its voltage Vm irrespective of the input being positive or negative.\n\nApplying KVL\n\n–Vi + Vm + V0 = 0\n\nV0 = Vi – Vm\n\n= Vi + (– Vm)\n\nHence circuit adds dc voltage of –Vm. So, the output will be a square waveform for given input whose value varies from 0 to – 2Vm.\n\nPositive peak of output waveform touches 0V level or positive peak gets clamped to 0V. Since a negative clamper is clamping positive peak to 0V so it is called positive peak clamper.\n\nNOTE: If diode has cut in voltage Vγ then it should be replaced with series connection of ideal diode and battery Vγ.", null, "When input is +Ve, capacitor changes through diode upto a maximum voltage of Vm – Vγ\n\n∴  V0 = Vi = (Vm – Vγ)\n\nV0 = Vi + (–Vm + Vγ)\n\nHence circuit add dc voltage equal to – (Vm – Vγ)", null, "(b) Positive Clamper\n\n• When input is negative diode gets forward biased and capacitor charges through diode up to peak input Vm\n• When input becomes positive capacitor will not be able to discharge as discharge path is not present. Therefore, voltage across the capacitor remains Vm irrespective of input being +Vm or –Vm", null, "Applying KVL\n\nV0 = Vi + Vm", null, "Negative peak output gets clamped to 0 volts therefore positive clamper is also called negative clamper.\n\nNOTE: It diode has cut in voltage Vγ then capacitor changes to a voltage (Vm – Vγ)\n\n∴ V0 = Vi + (Vm – Vγ)\n\nHence added dc voltage is Vm – Vγ", null, "Example:\n\nIn the circuit shown in figure dc value at the output is?", null, "Solution:\n\nFor the given clamper circuit, make diode short and calculate maximum voltage across capacitor.\n\n– Vi + VC = 0.7V + 0.3 = 0\n\nVC = Vi – 1 = 5 – 1\n\n= 4V\n\nNow, make diode open circuit\n\n– Vi + VC + V0 = 0\n\nV0 = Vi – VC\n\n= Vi – 4V\n\nDc value of output = –4V\n\nIf you aiming to crack GATE & ESE, Other PSU Exams then you must try Online Classroom Program to get unlimited access to all the live structured courses and unlimited mock tests from the following links:\n\n### Click Here Avail GATE/ESE EE Test Series !!! (193+ Mock Tests)\n\nThanks", null, "GradeStack Learning Pvt. Ltd.Windsor IT Park, Tower - A, 2nd Floor, Sector 125, Noida, Uttar Pradesh 201303 help@byjusexamprep.com" ]

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[ "## Information networks: visualization of paths in multidimensional cubes part 8\n\nDuring  our  evolution a very efficient brain structure was developed in the human brain, to create an image from what we see.  And therefore it is always very helpful to use  a visual representation for abstract concepts, which we want to understand. We only can think using words, symbols or images anyway-\n\nWe will describe here a visual representation for paths in information networks, which we demonstrate at first  with a    3-dimensional cube.\n\nWe use foll0wing method, to construct\n\n## a visual representation of paths,\n\nwhich we can also use for  cubes of any dimension and for any kind of information network. But we use at first only  a 3 – dimensional cube, to demonstrate it.\n\n• for labelling the 8 corners of the cube, we use the enumerating method, which we described in an earlier post.  The 8 corners are enumerated with P0 to P7\n• at first we construct a regular polygon with 8 corners.  The corner points on it represent the 8 corners of the cube. But the  distance between 2 neighbour points on the polygon is  not always the same.\n• if both points are neighbour corners of the cube, they are connected by an edge with length 1\n• if both points are diagonally opposite corner points of a square, then the distance between them is  1.414 and one of both points must  be on the first circle\n• if both points are diagonally opposite corner points in the cube, then one of them must be on the second circle\n• then we construct a 1. circle, which represents the diagonals of the 6 square faces of the 3 dimensional cube\n• thereafter we construct a 2. larger circle, which represents the space diagonals of  the   3 – dimensional cube\n• On the polygon and on both circles we arrange the 8 points in the same order. Same points are on the same radius from the midpoint to the outer circle.", null, "## Construction  of the path   P0, P1,P2,P3,P4,P5,P6,P7\n\n• We start with point  P0, which is on the regular polygon.\n• we use following convention:   All starting points are always on the polygon\n• To go from P0 to P1, we can keep going on the polygon, because this connection is an edge of length 1\n• To go from P1 to P2, we have to use a diagonal in a square ( length 1.414) and therefore we go to P2  on the first circle.\n• before we can go further to point P3, we have at first  to go back to the polygon, because P2  and P3 are connected by an edge of length 1.  We mark this by a green arrow, which has the length zero and therefore it would not matter, when we calculate the length of a path. ( of course, in the graphical representation a length is needed, but in the algebraic calculations we would give all green vectors the length zero )\n• To go from point P3 to P4, we go from the polygon to the outer circle, because we have to use a space diagonal, which has the length 1,73\n• We have again to go back to point P4 on the  polygon, because P4 and P5  are connected by an edge of length 1\n• From P5 to P6 we have to use a square diagonal and therefore we have to go to the first circle\n• P6 and P7 are connected by an edge. Therefore we have at first to go back from the first circle to the polygon and then we can construct the connection between P6 and P7\n• Because we use an open path, we do not  go back to  point P0", null, "As usual, one has to use this kind of representation several times, to get familiar with it.\n\n## We can derive  following  from this representation:\n\n• if  two neighbour points in a path are on the polygon, then their distance is 1\n• if one point is on the polygon and its neighbour point on the first circle, then the distance is 1.414\n• if one point is on the polygon and its neighbour point on the second circle, then the distance is 1.73\n• in a path always one neighbour point is on the polygon\n\nWith this representation we  get a good overview of the structure of the paths in a cube of any dimension.\n\n• for a cube of dimension n  we have a polygon with n corner points and n-1 circles of increasing radii around it. On each circle the n corner points are arranged in the same order as on the polygon; all corner points with the same number are on the same radial ray.\n\nTo visually represent networks, which have no geometrical meaning, we just can give them a virtual geometric meaning:\n\n• we construct a regular polygon. Direct neighbour points are connected by edges of length 1\n• points, between which another point is, are on the first circle\n• points, beween which there are 2 points, are on the second circle.\n• etc.\n\nFor the radii of the circles we use  suitable lengths, so that we get a nice  representation.\n\nThe example with the cubes  helped us, to find a good visual representation for paths in abstract information networks.\n\nOf course, if there are many points in a network, then we can do all the work only with the help of computer programs. But now we know, what we have to program and we have examples, with which we can test the programs.\n\n## One further remark:\n\nI prepared the drawings with GeoGebra, which is very good for it. One cannot create top quality pictures with it, which one could use for publications, but it is an excellent tool, to investigate geometrical problems.\n\nAnd here is finally  the 3 dimensional cube with the path. I think, that even with a 3 dimensional cube, the  abstract representation is better, because one sees directly all characteristics of a path.\n\nDieser Beitrag wurde unter information networks abgelegt und mit verschlagwortet. Setze ein Lesezeichen auf den Permalink.\n\nDiese Seite verwendet Akismet, um Spam zu reduzieren. Erfahre, wie deine Kommentardaten verarbeitet werden.." ]

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[ "Calcium and Glycolysis Mediate Multiple Bursting Modes in Pancreatic Islets Catherine Lloyd Auckland Bioengineering Institute, University of Auckland\nModel Status This model has been rebuilt and coded by translating the authors original XPPAUT.ode file, which can be found at http://www.math.fsu.edu/~bertram/software/islet/BJ_04a.ode. This file runs in PCEnv and COR and produces the expected output. This model has been parameterised for the 'glycolytic bursting' model. Some equations have inconsistent magnitudes due to the fact that time is defined in milliseconds, yet flux is defined in micromolar per second.\nModel Structure ABSTRACT: Pancreatic islets of Langerhans produce bursts of electrical activity when exposed to stimulatory glucose levels. These bursts often have a regular repeating pattern, with a period of 10-60 s. In some cases, however, the bursts are episodic, clustered into bursts of bursts, which we call compound bursting. Consistent with this are recordings of free Ca2+ concentration, oxygen consumption, mitochondrial membrane potential, and intraislet glucose levels that exhibit very slow oscillations, with faster oscillations superimposed. We describe a new mathematical model of the pancreatic beta-cell that can account for these multimodal patterns. The model includes the feedback of cytosolic Ca2+ onto ion channels that can account for bursting, and a metabolic subsystem that is capable of producing slow oscillations driven by oscillations in glycolysis. This slow rhythm is responsible for the slow mode of compound bursting in the model. We also show that it is possible for glycolytic oscillations alone to drive a very slow form of bursting, which we call \"glycolytic bursting.\" Finally, the model predicts that there is bistability between stationary and oscillatory glycolysis for a range of parameter values. We provide experimental support for this model prediction. Overall, the model can account for a diversity of islet behaviors described in the literature over the past 20 years. Calcium and Glycolysis Mediate Multiple Bursting Modes in Pancreatic Islets, Richard Bertram, Leslie Satin, Min Zhang, Paul Smolen, and Arthur Sherman, 2004, Biophysical Journal, 87, 3074-3087. PubMed ID: 15347584 cell diagram A schematic diagram of the ionic currents and fluxes across the ER and the cell surface membranes which are described by the mathematical model.\n$\\frac{d V}{d \\mathrm{time}}}=\\frac{-(\\mathrm{IK}+\\mathrm{ICa}+\\mathrm{IKCa}+\\mathrm{IKATP})}{\\mathrm{Cm}}$ $\\mathrm{IK}=\\mathrm{gK}n(V-\\mathrm{VK})$ $\\mathrm{IKCa}=\\frac{\\mathrm{gKCa}}{1+\\left(\\frac{\\mathrm{kd}}{c}\\right)^{2}}(V-\\mathrm{VK})$ $\\mathrm{ICa}=\\mathrm{gCa}\\mathrm{minf}(V-\\mathrm{VCa})$ $\\mathrm{minf}=\\frac{1}{1+e^{\\frac{-(20+V)}{12}}}$ $\\mathrm{ninf}=\\frac{1}{1+e^{\\frac{-(16+V)}{5}}}$ $\\frac{d n}{d \\mathrm{time}}}=\\frac{\\mathrm{ninf}-n}{\\mathrm{taun}}$ $\\frac{d c}{d \\mathrm{time}}}=\\mathrm{fcyt}(\\mathrm{Jmem}+\\mathrm{Jer})$ $\\frac{d \\mathrm{cer}}{d \\mathrm{time}}}=-\\mathrm{fer}\\mathrm{sigmaV}\\mathrm{Jer}$ $\\mathrm{Jserca}=\\mathrm{Kserca}c$ $\\mathrm{Jleak}=\\mathrm{pleak}(\\mathrm{cer}-c)$ $\\mathrm{Jer}=\\frac{\\mathrm{epser}(\\mathrm{Jleak}-\\mathrm{Jserca})}{\\mathrm{lambdaer}}$ $\\mathrm{Jmem}=-(\\mathrm{alpha}\\mathrm{ICa}+\\mathrm{kpmca}c)$ $\\mathrm{f6p}=0.3\\mathrm{g6p}$ $\\mathrm{rgpdh}=0.2\\sqrt{\\frac{\\mathrm{fbp}}{1}}$ $\\frac{d \\mathrm{fbp}}{d \\mathrm{time}}}=\\mathrm{lambda}(\\frac{\\mathrm{pfk}}{1}-0.5\\mathrm{rgpdh})$ $\\frac{d \\mathrm{g6p}}{d \\mathrm{time}}}=\\mathrm{lambda}(\\mathrm{Rgk}\\times 1-\\frac{\\mathrm{pfk}}{1})$ $\\mathrm{pfk}=1\\frac{\\mathrm{pfkbas}\\mathrm{cat}\\mathrm{topa16}+\\mathrm{cat}\\mathrm{topb}}{\\mathrm{bottom16}}$ $\\mathrm{weight2}=\\frac{\\mathrm{atp}^{2}}{\\mathrm{k4}\\times 1}\\mathrm{topa2}=\\mathrm{topa1}\\mathrm{bottom2}=\\mathrm{bottom1}+\\mathrm{weight2}$ $\\mathrm{weight3}=\\frac{\\mathrm{f6p}^{2}}{\\mathrm{k3}\\times 1}\\mathrm{topa3}=\\mathrm{topa2}+\\mathrm{weight3}\\mathrm{bottom3}=\\mathrm{bottom2}+\\mathrm{weight3}$ $\\mathrm{weight4}=\\frac{(\\mathrm{f6p}\\mathrm{atp})^{2}}{\\mathrm{fatp}\\mathrm{k3}\\mathrm{k4}1^{2}}\\mathrm{topa4}=\\mathrm{topa3}+\\mathrm{weight4}\\mathrm{bottom4}=\\mathrm{bottom3}+\\mathrm{weight4}$ $\\mathrm{weight5}=\\frac{\\mathrm{fbp}}{\\mathrm{k2}}\\mathrm{topa5}=\\mathrm{topa4}\\mathrm{bottom5}=\\mathrm{bottom4}+\\mathrm{weight5}$ $\\mathrm{weight6}=\\frac{\\mathrm{fbp}\\mathrm{atp}^{2}}{\\mathrm{k2}\\mathrm{k4}\\mathrm{fbt}\\times 1}\\mathrm{topa6}=\\mathrm{topa5}\\mathrm{bottom6}=\\mathrm{bottom5}+\\mathrm{weight6}$ $\\mathrm{weight7}=\\frac{\\mathrm{fbp}\\mathrm{f6p}^{2}}{\\mathrm{k2}\\mathrm{k3}\\mathrm{ffbp}\\times 1}\\mathrm{topa7}=\\mathrm{topa6}+\\mathrm{weight7}\\mathrm{bottom7}=\\mathrm{bottom6}+\\mathrm{weight7}$ $\\mathrm{weight8}=\\frac{\\mathrm{fbp}\\mathrm{f6p}^{2}\\mathrm{atp}^{2}}{\\mathrm{k2}\\mathrm{k3}\\mathrm{k4}\\mathrm{ffbp}\\mathrm{fbt}\\mathrm{fatp}1^{2}}\\mathrm{topa8}=\\mathrm{topa7}+\\mathrm{weight8}\\mathrm{bottom8}=\\mathrm{bottom7}+\\mathrm{weight8}$ $\\mathrm{weight9}=\\frac{\\mathrm{amp}}{\\mathrm{k1}}\\mathrm{topa9}=\\mathrm{topa8}\\mathrm{bottom9}=\\mathrm{bottom8}+\\mathrm{weight9}$ $\\mathrm{weight10}=\\frac{\\mathrm{amp}\\mathrm{atp}^{2}}{\\mathrm{k1}\\mathrm{k4}\\mathrm{fmt}\\times 1}\\mathrm{topa10}=\\mathrm{topa9}\\mathrm{bottom10}=\\mathrm{bottom9}+\\mathrm{weight10}$ $\\mathrm{weight11}=\\frac{\\mathrm{amp}\\mathrm{f6p}^{2}}{\\mathrm{k1}\\mathrm{k3}\\mathrm{famp}\\times 1}\\mathrm{topa11}=\\mathrm{topa10}+\\mathrm{weight11}\\mathrm{bottom11}=\\mathrm{bottom10}+\\mathrm{weight11}$ $\\mathrm{weight12}=\\frac{\\mathrm{amp}\\mathrm{f6p}^{2}\\mathrm{atp}^{2}}{\\mathrm{k1}\\mathrm{k3}\\mathrm{k4}\\mathrm{famp}\\mathrm{fmt}\\mathrm{fatp}1^{2}}\\mathrm{topa12}=\\mathrm{topa11}+\\mathrm{weight12}\\mathrm{bottom12}=\\mathrm{bottom11}+\\mathrm{weight12}$ $\\mathrm{weight13}=\\frac{\\mathrm{amp}\\mathrm{fbp}}{\\mathrm{k1}\\mathrm{k2}}\\mathrm{topa13}=\\mathrm{topa12}\\mathrm{bottom13}=\\mathrm{bottom12}+\\mathrm{weight13}$ $\\mathrm{weight14}=\\frac{\\mathrm{amp}\\mathrm{fbp}\\mathrm{atp}^{2}}{\\mathrm{k1}\\mathrm{k2}\\mathrm{k4}\\mathrm{fbt}\\mathrm{fmt}\\times 1}\\mathrm{topa14}=\\mathrm{topa13}\\mathrm{bottom14}=\\mathrm{bottom13}+\\mathrm{weight14}$ $\\mathrm{weight15}=\\frac{\\mathrm{amp}\\mathrm{fbp}\\mathrm{f6p}^{2}}{\\mathrm{k1}\\mathrm{k2}\\mathrm{k3}\\mathrm{ffbp}\\mathrm{famp}\\times 1}\\mathrm{topa15}=\\mathrm{topa14}\\mathrm{bottom15}=\\mathrm{bottom14}+\\mathrm{weight15}\\mathrm{topb}=\\mathrm{weight15}$ $\\mathrm{weight16}=\\frac{\\mathrm{amp}\\mathrm{fbp}\\mathrm{f6p}^{2}\\mathrm{atp}^{2}}{\\mathrm{k1}\\mathrm{k2}\\mathrm{k3}\\mathrm{k4}\\mathrm{ffbp}\\mathrm{famp}\\mathrm{fbt}\\mathrm{fmt}\\mathrm{fatp}1^{2}}\\mathrm{topa16}=\\mathrm{topa15}+\\mathrm{weight16}\\mathrm{bottom16}=\\mathrm{bottom15}+\\mathrm{weight16}$ $\\mathrm{mgadp}=0.165\\mathrm{adp}$ $\\mathrm{adp3m}=0.135\\mathrm{adp}$ $\\mathrm{atp4m}=0.05\\mathrm{atp}$ $\\mathrm{topo}=0.08(1+\\frac{2\\mathrm{mgadp}}{\\mathrm{kdd}\\times 1})+0.89\\left(\\frac{\\mathrm{mgadp}}{\\mathrm{kdd}\\times 1}\\right)^{2}$ $\\mathrm{bottomo}=(1+\\frac{\\mathrm{mgadp}}{\\mathrm{kdd}\\times 1})^{2}(1+\\frac{\\mathrm{adp3m}}{\\mathrm{ktd}\\times 1}+\\frac{\\mathrm{atp4m}}{\\mathrm{ktt}\\times 1})$ $\\mathrm{katpo}=\\frac{\\mathrm{topo}}{\\mathrm{bottomo}}$ $\\mathrm{IKATP}=\\mathrm{gkatpbar}\\mathrm{katpo}(V-\\mathrm{VK})$ $\\frac{d \\mathrm{adp}}{d \\mathrm{time}}}=\\frac{\\mathrm{atp}-\\mathrm{adp}e^{\\mathrm{fback}(1-\\frac{c}{\\mathrm{r1}})}}{\\mathrm{taua}\\times 1}$ $\\mathrm{fback}=r+y$ $y=\\mathrm{vg}\\frac{\\mathrm{rgpdh}}{\\mathrm{kg}+\\mathrm{rgpdh}}$ $\\mathrm{atp}=0.5(\\mathrm{atot}-\\mathrm{adp}+\\mathrm{rad}\\times 1)$ $\\mathrm{amp}=\\frac{\\mathrm{adp}\\mathrm{adp}}{\\mathrm{atp}}$ $\\mathrm{ratio}=\\frac{\\mathrm{atp}}{\\mathrm{adp}}$ $\\mathrm{rad}=\\frac{\\sqrt{(\\mathrm{adp}-\\mathrm{atot})^{2}-4\\mathrm{adp}^{2}}}{1}$ calcium dynamics insulin glucose homeostasis metabolism 2004-10-01T00:00:00+00:00 50000 1000 1 bdf15 c.lloyd@auckland.ac.nz James Lawson Richard Biophysical Journal Calcium and Glycolysis Mediate Multiple Bursting Modes in Pancreatic Islets updated curation status This is the CellML description of Bertram et al.'s 2004 mathematical model of calcium and glycolysis mediated bursting modes in pancreatic islets. 2009-05-25T16:01:23+12:00 Leslie Satin Paul Smolen Richard Bertram The University of Auckland Auckland Bioengineering Institute Min Zhang Bertram et al.'s 2004 mathematical model of calcium and glycolysis mediated bursting modes in pancreatic islets. Pancreatic Islets Calcium and Glycolysis Mediate Multiple Bursting Modes in Pancreatic Islets (glycolytic bursting model) The University of Auckland, Auckland Bioengineering Institute 2004-09 15347584 keyword This model has been rebuilt and coded by translating the authors original XPPAUT .ode file, which can be found at http://www.math.fsu.edu/~bertram/software/islet/BJ_04a.ode . This file runs in PCEnv and produces the output expected, however PCEnv is only able to produce one cycle of bursting. This is a pending issue. Arthur Sherman James Lawson Catherine Lloyd May Catherine Lloyd" ]

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[ "", null, "# ANALOG PSEUDO RANDOM BIT SEQUENCE GENERATOR\n\nImported: 10 Mar '17 | Published: 27 Nov '08\n\nUSPTO - Utility Patents\n\n## Abstract\n\nA Pseudo Random Bit Sequence (PRBS) generator is provided with components to enable operation at very high microwave frequencies with inexpensive components. The PRBS generator initially replaces the D flip-flops of a conventional PRBS generator with delay lines connected in a similar manner. Further, an exclusive OR (EXOR) gate used in a conventional device is replaced in one embodiment by a mixer and amplifier. In another embodiment, the EXOR gate is replaced by a Gilbert Cell. In some embodiments, complementary outputs of an EXOR gate are connected to separate delay lines to reduce components needed for the PRBS generator.\n\n## Description\n\n### BACKGROUND\n\n1. Technical Field\n\nThe present invention relates to circuitry for an analog pseudo random bit sequence (PRBS) generator that can be used, for example, to exercise digital circuits using pseudo random patterns. The PRBS generator, in another example, can produce analog noise if fed into a digital circuit.\n\n2. Related Art\n\nFIG. 1 shows a block diagram of prior art digital implementation of a PRBS generator. This PRBS generator is used to exercise digital circuits using pseudo random bit patterns. It can also be used to produce analog noise if fed into a D/A converter. The circuit includes D-type flip-flops or registers 101-104 connected in series, and driven by a common clock signal Clk. Each of the registers 101-104 produces a respective tap output Q1-Q4. The Q1 tap from register 101 provides a first input to Exclusive OR (EXOR) gate 106. The Q4 tap produces a second input to the EXOR gate 106. The output D1 of EXOR gate 106 is provided back to the D input of the register 101. FIG. 2 provides a timing diagram showing the outputs from each of the taps Q1-Q4 and the EXOR gate 106 output D1 relative to the clock signal Clk.\n\nFIG. 3 illustrates one alternative connection to FIG. 1 that produces the same pattern as shown in FIG. 2. FIG. 3 modifies FIG. 1 by including a register 301 with inputs connected in parallel with the register 101. The registers 101 and 301 receive a D input from the output of EXOR gate 106, while their clocks are provided by common clock Clk. The Q output of register 301 then matches the Q output of register 101 to produce the Q1 tap signal to an input of the EXOR gate 106.\n\nThe characteristics of the circuits of FIGS. 1 and 3 produce a random bit sequence that repeats every 2N1 clock cycles. The all 0 state is prohibited, as this will lock up the generator. This state must be avoided at start-up. There are tables of tap connection vs. sequence length that are readily available from many sources. As an example, a 4 bit shift register making up a PRBS generator with an EXOR gate as shown in FIGS. 1 and 3 will produce a maximum length sequence of 241=15 state changes before repeating the sequence.\n\nThere is no inherent upper limit to the frequency of operation with the components of FIGS. 1 and 3, except that of the logic elements used. To reduce the expense for higher frequency PRBS generators, however, there are techniques that allow several lower frequency PRBS generators to be multiplexed to arrive at a higher frequency. The multiplexer is the only element that operates at the elevated frequency. The high frequency multiplexers can be made much more easily than D flip-flops and EXOR gates. The disadvantage of this technique is that the system can become very complex and expensive when system frequencies approach the 10's of GHz range. As an example Anritsu Company of Morgan Hill, Calif. manufactures a 12.5 GHz PRBS generator model MP1763B that sells for over \\$100,000.\n\nIt would be desirable to provide components for a PRBS generator that can operate at frequencies into the 10's of GHz range, while minimizing manufacturing costs.\n\n### SUMMARY\n\nAccording to embodiments of the present invention, a PRBS generator is provided with components to enable operation at very high frequencies. In particular, typical components forming a lower frequency PRBS generator are replaced with microwave components to enable the high frequency operation.\n\nInitially, D flip-flops of a conventional PRBS generator are replaced by delay lines connected in a similar manner. Further, the EXOR gate used in a conventional device is replaced by a mixer and amplifier. Outputs of the delay lines form the RF and LO inputs of the mixer, and the IF mixer output drives the amplifier. The output of the amplifier in one embodiment is connected through a power splitter back to the separate inputs of the delay lines. In another embodiment, the amplifier output drives a first delay line, while the output of the first delay line drives a second delay line. The mixer can be formed by a differential amplifier connected to a diode switch.\n\nIn an alternative embodiment of the invention, a Gilbert Cell is used to provide the EXOR gate of a PRBS generator. The Gilbert Cell is connected with two delay lines forming the D flip-flops of the PRBS generator.\n\nIn one embodiment, complementary outputs of the EXOR gate are used to reduce circuitry needed for a high frequency PRBS generator. Instead of a single EXOR gate output connected to both delay lines, complementary outputs connect individually to each delay line. The outputs of the delay lines are then connected to separate inputs of the EXOR gate.\n\n### DETAILED DESCRIPTION\n\nFIGS. 4 and 5 show circuitry for a high frequency PRBS generator according to embodiments of the present invention. FIG. 4 provides an analog conversion from the digital generator of FIG. 1. FIG. 5 provides an analog conversion from the digital PRBS generator of FIG. 3. The circuit of FIG. 5 corresponds with a majority of the analog PRBS circuits subsequently discussed, which is why the circuit of FIG. 3 is discussed in the background and further herein. The implementations of FIGS. 4 and 5 produce signals D1, Q1 and Q4 with a timing diagram as shown in FIG. 2.\n\nIn FIGS. 4 and 5, delay lines replace the D flip-flops of respective FIGS. 1 and 3. In FIG. 4, a first delay line 400 connects the output of EXOR gate 106 to a first input of the EXOR gate 106. The first input of the EXOR gate 106 is designated as the Q1 tap. A second delay line 402 connects the tap Q1 to a second input of the EXOR gate 106. The second input of the EXOR gate 106 is designated as the Q4 tap. By selecting appropriate lengths for the delay lines 400 and 402, the number of clock cycles between Q1 and Q4 can be set to match the timing diagram of FIG. 2.\n\nIn FIG. 5, the output of the EXOR gate 106 is connected to a first terminal of both the first delay line 500 and a first terminal of a second delay line 502. A second terminal of the first delay line 500 is connected to a first input of the EXOR gate 106 forming the tap Q1. The second terminal of the second delay line 502 is connected to a second input of the EXOR gate 106 forming the Q4 tap. With this connection, the delay line 500 overlaps a portion of the delay line 502, similar to registers 101 and 301 providing overlapping data in FIG. 3. As in FIG. 4 by selecting appropriate lengths for the delay lines 500 and 502, the number of clock cycles between Q1 and Q4 can be set to match the timing diagram of FIG. 2.\n\nTo illustrate how delay lines 400, 402, 500 and 502 can replace a D flip-flop, the D flip-flop can be thought of as a controllable delay where the Q output follows the D input with a delay of the period of the CLK signal. For a given CLK frequency there is a fixed delay for a signal applied to the D flip-flop. Similarly if a signal is placed at the input of a delay line, the signal will appear at the output with a fixed delay.\n\nDelay lines can be made by many techniques. The simplest ones are traces on PC boards and coax cable. For a given impedance (R) and a known capacitance (C), the per foot delay can be calculated by t(Delay)/foot=R*C. For an example, RG174 coax cable has a C per foot of 20 pF and an impedance of 50 Ohms, then t(Delay)/foot=1.45 nS/foot. If a delay of 500 pS were desired, the length of cable needed would be L(desired)=t(Desired)/t(Delay)/foot or 500 ps/1.45 nS=0.345 feet which is 4.14 inches. The delay line will then replace the D flip-flop. All that is left is some gain to make the system regenerative. An amplifier will, thus, be used to complete the system.\n\nFIG. 6 illustrates one embodiment of circuitry for the invention of FIG. 5, with an amplifier 600 added to create gain, a mixer 602 used to form the EXOR gate, and a power divider 604 interconnecting components. The mixer 602 has a first (LO) input connected to the output of the delay line 500 and a second (RF) input connected to the output of the delay line 502. The (IF) output of the mixer 602 is provided through amplifier 600 to power splitter 604. The splitter 604 evenly distributes power from the output of amplifier 600 to the delay lines 500 and 502, as well as to a port providing the signal D1.\n\nTo illustrate how a mixer can be used for the EXOR gate, the EXOR gate can be thought of as a controllable invert not invert gate. If a logic signal is connected to one input and a 0 is connected to the other input, the EXOR will pass the logic signal through with no inversion. If the other input is replaced with a 1 the logic signal will invert at the output. Similarly if the signals are placed at the RF input of a mixer and a + voltage is placed at the LO port, then the signal will pass through to the IF port with no inversion. If a voltage is placed at the LO port, the RF signal will invert at the IF port.\n\nFIG. 7 shows one circuit embodiment for the mixer of the PRBS generator of FIG. 6. The mixer uses a diode switch made up of diodes 701-704. To apply a first RF signal to the diode switch, a differential amplifier 706 is used. The differential amplifier 706 receives the RF input to the mixer and provides two outputs, one inverting () and the other non-inverting (+). The diode switch 701-704 serves to select one of the inverting () or non-inverting (+) outputs from the differential amplifier 706. A resistor 708 provides a steering current for the selected diodes. The voltage on the resistor 708 drives the diodes 701-704 to select the IF output of the mixer as either a non-inverting gain with a + voltage or an inverting gain with a voltage. The mixer design of FIG. 7 maintains the DC path through the system.\n\nFIG. 8 shows a block diagram of a mixer circuit with the specific differential amplifier circuit 706 of FIG. 7 represented in block diagram form. Also shown with FIG. 8 are example RF and LO signal inputs to the mixer, and a resulting IF output signal. As shown, the IF signal output behaves as if the mixer circuit were an EXOR circuit having inputs receiving the RF and LO signal inputs. FIG. 9 for reference shows a block diagram of components for the PBRS generator using the mixer components of FIG. 7, along with remaining PRBS generator components from FIG. 6.\n\nFIG. 10A shows a connection diagram for a Gilbert Cell 1000 to provide a high frequency EXOR gate for use in a PRBS generator according to additional embodiments of the present invention. A Gilbert Cell 1000 can be made using very high frequency transistors allowing its use as a mixer at microwave frequencies. It also has the advantage of gain. This will allow the deletion of the fixed gain amplifier 600 shown in FIGS. 6 and 9.\n\nAs illustrated in FIG. 10A, the Gilbert Cell 1000 has four inputs x, x_b, y and y_b. The input x_b has _b indicating active low, as will other all signals described herein labeled with _b. The Gilbert Cell 1000 further provides outputs labeled o and o_b. The Gilbert Cell 1000 provides circuitry to generate the following function:\n\noob=A1(xxb)*A2(yyb)\n\nwhere A1 and A2 are gains of internal differential pairs of the Gilbert Cell.\n\nWith this formula, the + or state difference in outputs, 0 and 0_b, can be determined based on the + or state difference between each of the inputs, xx_b and yy_b, as illustrated in the following Table A.\n\nTABLE A x x_b y y_b o o_b + + + + + +\n\nFrom Table A, it can be see that oo_b=x_b when yy_b is +. Also, from the table it can be seen that oo_b=x_bx when yy_b is . Thus, with x and y as inputs, and x_b and y_b grounded, y will be inverted as the output o_b with x being +, and y will not be inverted as the output o_b with x being . This is illustrated in Table B as follows:\n\nTABLE B x y o_b + + + + + +\n\nFIG. 10B, thus, illustrates how the x and y input terminals and the o_b output terminal of the Gilbert Cell 1000 can be connected to form an EXOR gate. With the x_b and y_b inputs to the Gilbert Cell 1000 connected to ground as shown in FIG. 10A, they are not used in the EXOR gate of FIG. 10B, leaving x and y as inputs. With o_b selected as an output, the following truth table, Table C, is provided for the EXOR gate of FIG. 10B:\n\nTABLE C x y o_b 1 1 0 0 1 1 1 0 1 0 0 0\n\nThe + and signals of Table B equate to the 1s and 0s in Table C. Tables B and C, thus, show that the Glibert Cell configuration of FIG. 10A provides the EXOR gate of FIG. 10B.\n\nFIG. 11 illustrates circuit components making up a Gilbert Cell that can be used in a PRBS generator of the present invention. The circuit includes three differential amplifier pairs 1100, 1102 and 1104. The x and x_b inputs to the Gilbert Cell provide inputs to the gates of transistors of differential amplifiers 1100 and 1102. Outputs of the Gilbert Cell o and o_b are provided as the outputs at the collectors of transistors of differential amplifiers 1100 and 1102. The y and y_b inputs to the Gilbert Cell provide the gate inputs to transistors of differential amplifier 1104. The gain of differential pairs 1100 and 1102 provide the A1 gain of the Gilbert Cell, while the differential pair 1104 provides the gain A2.\n\nFIGS. 12 and 13 show modification of components of respective FIGS. 4 and 5 with the EXOR gate replaced with the Gilbert Cell 1000.\n\nFIG. 14 shows modification of the circuit of FIG. 5 to include an EXOR gate 1400 with complementary outputs. This variation uses both outputs of the EXOR gate 1400. The data stream out of D1_b from the EXOR gate 1400 is the inverted version of the output D1 of FIG. 5. This implementation simplifies the drive requirements of the delay lines, as the output of EXOR gate 1400 provides separate outputs to delay lines 1402 and 1404. A power splitter to distribute the EXOR output to the two delay lines 1402 and 1402 will, thus, not be required. FIG. 15 shows the EXOR gate 1400 of FIG. 14 replaced by a Gilbert Cell 1000.\n\nFIG. 16 shows an alternative connection for the PRBS generator of FIG. 3 using complementary outputs from an EXOR gate 1600, similar to FIG. 14. As in FIG. 3, a D flip-flop 301 redundant to D flip-flop 101 is used to provide the Q1 input to EXOR gate 1600 in FIG. 16. The EXOR gate 1600 can be replaced in FIG. 16 with a Gilbert Cell as well.\n\nThe outputs D1_b, Q1_b and Q4 from FIG. 16, as well as outputs from the circuits of FIGS. 14 and 15, are shown in FIG. 17. FIG. 17 shows that the circuits of FIGS. 14-16 exhibit the same sequence as shown in FIG. 2. Note the sequence for Q4 is the same as shown in FIG. 2. D1_b and Q1_b in FIG. 17 are, however, inverted in FIG. 2.\n\nFIG. 18 shows circuitry for the connection of FIG. 16 implemented using a Gilbert Cell 1800 and two delay lines 1802 and 1804. The Gilbert Cell 1800 can include components as described with respect to FIG. 7. The upper two differential pair amplifiers represent the + gain and gain amplifier outputs driving resistors 1810 and 1812. The bottom differential pair amplifier represents the diode switching function which provides the selection of either the + gain amplifier or the gain amplifier output. Buffering of the output signal o is provided by transistor 1820. Buffering of the output signal o_b is provided by transistor 1822. Further, buffering of the input signal y is provided by transistor 1824. The remaining circuitry provides bias signals VBIAS1, VBIAS2 and VBIAS3 for transistors used in the Gilbert Cell 1800.\n\nIn one exemplary embodiment for the circuit of FIG. 18, a Motorola MECL 10 KH series triple EXOR gate model MC10H107 is used as the Gilbert Cell 1800. Only one of the EXOR gates in the model MC10H107 circuit is needed. With a clock frequency of 40 MHz in this example, the line length for delay lines 1802 and 1804 using coiled RG174 cable of C=29 pf/ft and R=50 Ohms can be calculated with a value of tdelay=RC=20 pf*50 Ohms=1.45 ns/ft. Thus a length for the delay line 1800 is calculated as L1=(1* 1/40 MHz)/tdelay=17.24 ft. The length of delay line 1802 is calculated as L2=(4* 1/40 MHz)/tdelay=68.95 ft.\n\nAlthough the present invention has been described above with particularity, this was merely to teach one of ordinary skill in the art how to make and use the invention. Many additional modifications will fall within the scope of the invention, as that scope is defined by the following claims.\n\n## Claims\n\n1. A pseudo random bit sequence (PRBS) generator comprising:\nan exclusive OR (EXOR) circuit having an output providing an output of the PRBS generator;\na first delay line having an input connected to an output of the EXOR circuit, and an output connected to a first input of the EXOR circuit; and\na second delay line having an input connected to an output of the EXOR circuit, and an output connected to a second input of the EXOR circuit.\nan exclusive OR (EXOR) circuit having an output providing an output of the PRBS generator;\na first delay line having an input connected to an output of the EXOR circuit, and an output connected to a first input of the EXOR circuit; and\na second delay line having an input connected to an output of the EXOR circuit, and an output connected to a second input of the EXOR circuit.\n2. The PRBS generator of claim 1,\nwherein the output of the EXOR gate is connected to each of the first delay line, the second delay line and the output of the PRBS generator through a separate output of a three way splitter.\nwherein the output of the EXOR gate is connected to each of the first delay line, the second delay line and the output of the PRBS generator through a separate output of a three way splitter.\n3. The PRBS generator of claim 1, wherein the EXOR gate comprises:\na mixer having a LO input connected to the output of the first delay line, an RF input connected to the output of the second delay line, and an IF output; and\nan amplifier having an input connected to the IF output of the mixer and an output providing the output of the EXOR gate.\na mixer having a LO input connected to the output of the first delay line, an RF input connected to the output of the second delay line, and an IF output; and\nan amplifier having an input connected to the IF output of the mixer and an output providing the output of the EXOR gate.\n4. The PRBS generator of claim 3, wherein the mixer comprises a diode switch comprising:\na differential amplifier having an input forming the RF input of the mixer and first and second differential outputs;\na first diode having an anode terminal connected to the first differential output and a cathode terminal forming the LO input of the mixer;\na second diode having a cathode terminal connected to the second differential output and an anode terminal connected to the LO input;\na third diode having an anode terminal connected to the LO input and a cathode terminal connected to the IF output of the mixer; and\na fourth diode having a cathode terminal connected to the input and an anode terminal connected to the IF output.\na differential amplifier having an input forming the RF input of the mixer and first and second differential outputs;\na first diode having an anode terminal connected to the first differential output and a cathode terminal forming the LO input of the mixer;\na second diode having a cathode terminal connected to the second differential output and an anode terminal connected to the LO input;\na third diode having an anode terminal connected to the LO input and a cathode terminal connected to the IF output of the mixer; and\na fourth diode having a cathode terminal connected to the input and an anode terminal connected to the IF output.\n5. The PRBS generator of claim 1, wherein the EXOR circuit comprises a Gilbert Cell.\n6. The PRBS generator of claim 5, wherein the Gilbert Cell comprises:\nfirst and second differential amplifiers connected to provide first inputs x and x_b and outputs o and o_b for the Gilbert Cell, wherein the x_b input is connected to ground, the x input provides a first input of the EXOR gate and the o_b output provides the output of the EXOR gate; and\na third differential amplifier connected to provide second inputs y and y_b for the Gilbert Cell, wherein the y_b input is connected to ground and the y input provides a second input of the EXOR gate.\nfirst and second differential amplifiers connected to provide first inputs x and x_b and outputs o and o_b for the Gilbert Cell, wherein the x_b input is connected to ground, the x input provides a first input of the EXOR gate and the o_b output provides the output of the EXOR gate; and\na third differential amplifier connected to provide second inputs y and y_b for the Gilbert Cell, wherein the y_b input is connected to ground and the y input provides a second input of the EXOR gate.\n7. A pseudo random bit sequence (PRBS) generator comprising:\nan exclusive OR (EXOR) circuit having an output providing an output of the PRBS generator;\na first delay line having an input connected to an output of the EXOR circuit, and an output connected to a first input of the EXOR circuit; and\na second delay line having an input connected to an output of the first delay line and an output connected to a second input of the EXOR circuit.\nan exclusive OR (EXOR) circuit having an output providing an output of the PRBS generator;\na first delay line having an input connected to an output of the EXOR circuit, and an output connected to a first input of the EXOR circuit; and\na second delay line having an input connected to an output of the first delay line and an output connected to a second input of the EXOR circuit.\n8. The PRBS generator of claim 7, wherein the EXOR gate comprises:\na mixer having a LO input connected to the output of the first delay line, an RF input connected to the output of the second delay line, and an IF output; and\nan amplifier having an input connected to the IF output of the mixer and an output providing the output of the EXOR gate.\na mixer having a LO input connected to the output of the first delay line, an RF input connected to the output of the second delay line, and an IF output; and\nan amplifier having an input connected to the IF output of the mixer and an output providing the output of the EXOR gate.\n9. The PRBS generator of claim 8, wherein the mixer comprises a diode switch comprising:\na differential amplifier having an input forming the RF input of the mixer and first and second differential outputs;\na first diode having an anode terminal connected to the first differential output and a cathode terminal forming the LO input of the mixer;\na second diode having a cathode terminal connected to the second differential output and an anode terminal connected to the LO input;\na third diode having an anode terminal connected to the LO input and a cathode terminal connected to the IF output of the mixer; and\na fourth diode having a cathode terminal connected to the input and an anode terminal connected to the IF output.\na differential amplifier having an input forming the RF input of the mixer and first and second differential outputs;\na first diode having an anode terminal connected to the first differential output and a cathode terminal forming the LO input of the mixer;\na second diode having a cathode terminal connected to the second differential output and an anode terminal connected to the LO input;\na third diode having an anode terminal connected to the LO input and a cathode terminal connected to the IF output of the mixer; and\na fourth diode having a cathode terminal connected to the input and an anode terminal connected to the IF output.\n10. The PRBS generator of claim 7, wherein the EXOR circuit comprises a Gilbert cell.\n11. A pseudo random bit sequence (PRBS) generator comprising:\nan exclusive OR (EXOR) circuit having a first output and a second output complementary to the first;\na first delay line having an input connected to the first output of the EXOR circuit, and an output connected to a first input of the EXOR circuit; and\na second delay line having an input connected to the second complementary output of the EXOR circuit, and an output connected to a second input of the EXOR circuit.\nan exclusive OR (EXOR) circuit having a first output and a second output complementary to the first;\na first delay line having an input connected to the first output of the EXOR circuit, and an output connected to a first input of the EXOR circuit; and\na second delay line having an input connected to the second complementary output of the EXOR circuit, and an output connected to a second input of the EXOR circuit.\n12. The PRBS generator of claim 11, wherein the EXOR circuit comprises a Gilbert cell.\n13. The PRBS generator of claim 12, wherein the Gilbert Cell comprises:\nfirst and second differential amplifiers connected to provide first inputs x and x_b and outputs o and o_b for the Gilbert Cell, wherein the x_b input is connected to ground, the x input provides a first input of the EXOR gate and the o_b output provides the first output of the EXOR gate, and the o output provides the second output of the EXOR gate; and\na third differential amplifier connected to provide second inputs y and y_b for the Gilbert Cell, wherein the y_b input is connected to ground and the y input provides a second input of the EXOR gate.\nfirst and second differential amplifiers connected to provide first inputs x and x_b and outputs o and o_b for the Gilbert Cell, wherein the x_b input is connected to ground, the x input provides a first input of the EXOR gate and the o_b output provides the first output of the EXOR gate, and the o output provides the second output of the EXOR gate; and\na third differential amplifier connected to provide second inputs y and y_b for the Gilbert Cell, wherein the y_b input is connected to ground and the y input provides a second input of the EXOR gate." ]

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[ "The photographic calibration constitutes a key step for the processing of the digital images considered in this study. We evaluated the calibration curve, which relates the photographic density (output), to the logarithm of the plate exposition (input), starting from the average value of transparency T and its standard deviation measured on the various steps of the wedge exposition. Following\n\nCaccin et al. (1998), the curve was computed through a least squares third degree polynomial fitting performed on the measured values, after suitable data re-arrangement and interpolation. In particular, the measured values T for the various steps of the wedge exposition were sorted in order of increasing exposition. The obtained values, together with intensity values expected by the step-wedge exposition, were linearly interpolated to produce a 200×2 element matrix showing the correspondence between density and intensity for the analyzed image.\n\nThis matrix was used as input for the polynomial fitting, which is performed taking into account the standard measurement errors for the transmission measured in each step. The fitting returned the function coefficients used to get the photographic calibration of pixel values on the analyzed image. The curve computation was applied to each image including a calibration wedge. The whole sample of CaII K images can be distinct in four sub-samples, depending on both instrumental and observational changes occurred during the about fifty years of Arcetri observations.\n\nHowever, there is a large scatter among the curves obtained for each sub-sample.The calibration of the plates stored without calibration exposures was performed by using a reference curve. This curve was singled out among all the computed ones, since it allows to calibrate the largest sample of images carrying no specific calibration information by obtaining images in which the intensity pattern on solar observations appears as it is seen on present-day observations. However, the selected curve exhibits large scatter with respect to some computed curves." ]

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[ "1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 #include using namespace std; struct Fish{ long long weight; bool active; Fish(){ active = true; } }; istream & operator >> (istream & is, Fish & f){ is >> f.weight; return is; } bool stuff(Fish * const fishes, const int & index, long long currentWeight, long long & expectedWeight, int depth, int & min){ if(currentWeight >= expectedWeight){ if(depth < min) min = depth; return true; } bool possible = false; for(int i = 0; i < index; ++i){ if(fishes[i].active && fishes[i].weight < currentWeight){ fishes[i].active = false; if(stuff(fishes, index, currentWeight + fishes[i].weight, expectedWeight, depth + 1, min)){ possible = true; } fishes[i].active = true; } } return possible; } int main(){ int n; cin >> n; int index = n; long long sum = 0; Fish * fishes = new Fish [2 * n]; for(int i = 0; i < n; ++i){ cin >> fishes[i]; sum += fishes[i].weight; } n = 2 * n; int q; long long a, b; cin >> q; for(int i = 0; i < q; ++i){ cin >> a; if(a == 1){ cin >> a >> b; if(b - a > sum){ cout << -1 << endl; continue; } int min = index; if(stuff(fishes, index, a, b, 0, min)){ cout << min << endl; }else{ cout << -1 << endl; } }else if(a == 2){ if(index >= n){ Fish * f2 = fishes; n *= 2; fishes = new Fish [n]; for(int j = 0; j < index; ++j){ fishes[j] = f2[j]; } delete [] f2; } cin >> fishes[index++]; sum += fishes[index - 1].weight; }else{ cin >> b; for(int j = 0; j < index; ++j){ if(fishes[j].weight == b && fishes[j].active){ fishes[j].active = false; sum -= fishes[j].weight; break; } } } } return 0; }" ]

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[ "", null, "1753 (another Prime Pages' Curiosity)", null, "Curios: Curios Search:   Participate:", null, "", null, "GIMPS has discovered a new largest known prime number: 282589933-1 (24,862,048 digits)", null, "The first prime in a sequence produced from a prime-generating quartic polynomial of form n^4 + 853n^3 + 2636n^2 + 3536n + 1753. [Koning]", null, "The smallest prime (emirp) formed from two primes with the same sum of digits. [Loungrides]", null, "The smallest prime, (the only emirp), that can be represented as the sum of two powers with distinct composite digits as bases and distinct prime digits as exponents, i.e., 4^5+9^3. [Loungrides]", null, "There are 1753 Cyclops Sophie Germain primes with distinct digits. [Gaydos]", null, "The first prime containing the first four odd digits. [Pyne] (There are 3 curios for this number that have not yet been approved by an editor.)   To link to this page use /curios/page.php?number_id=2766 Prime Curios! © 2000-2020 (all rights reserved)  privacy statement   (This page was generated in 0.0141 seconds.)" ]

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[ "Still have questions?\nContact support\n\n## Overview\n\nDeveloped by Marc Chaikin, the Accumulation Distribution Line is a volume-based indicator designed to measure the cumulative flow of money into and out of a security. Chaikin originally referred to the indicator as the Cumulative Money Flow Line. As with cumulative indicators, the Accumulation Distribution Line is a running total of each period's Money Flow Volume. First, a multiplier is calculated based on the relationship of the close to the high-low range. Second, the Money Flow Multiplier is multiplied by the period's volume to come up with a Money Flow Volume. A running total of the Money Flow Volume forms the Accumulation Distribution Line. Chartists can use this indicator to affirm a security's underlying trend or anticipate reversals when the indicator diverges from the security price.\n\nFind the mathematical description of the indicator on the Accumulation Distribution Line (ADL) Mathematical Description page.\n\nADL indicator is added through the adl() method. It requires a mapping with five fields in it: \"open\", \"high\", \"low\", \"close\" and \"volume\".\n\n``````// create data table on loaded data\nvar dataTable = anychart.data.table();\n\n// add data to a table\n\nvar mapping = dataTable.mapAs({\"high\": 1, \"open\": 2, \"low\": 3, \"close\": 4, \"volume\": 3});\n\n// create stock chart\nchart = anychart.stock();\n\n// create plots on the chart\nvar plot0 = chart.plot(0);\nvar plot1 = chart.plot(1);\n\n``````\n\nNote that ADL indicator needs to be built on a separate plot due to rather huge difference between the indicator values and the data values.\n\nHere is a live sample:\n\nIt is possible to change the series type any time using the seriesType() method.\n\n## Visualization\n\nVisualization of an indicator depends on series type. Here is a sample where ADL indicator is visually adjusted:\n\n``````// adjust ADL indicator" ]

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[ "10.1145/3087556.3087578 Kling, Peter Peter Kling Mäcker, Alexander Alexander Mäcker Riechers, Sören Sören Riechers Skopalik, Alexander Alexander Skopalik Sharing is Caring: Multiprocessor Scheduling with a Sharable Resource 2017 2017-10-17T12:41:02Z 2019-04-11T13:48:16Z conference https://ris.uni-paderborn.de/record/59 https://ris.uni-paderborn.de/record/59.json 784867 bytes application/pdf We consider a scheduling problem on $m$ identical processors sharing an arbitrarily divisible resource. In addition to assigning jobs to processors, the scheduler must distribute the resource among the processors (e.g., for three processors in shares of 20\\%, 15\\%, and 65\\%) and adjust this distribution over time. Each job $j$ comes with a size $p_j \\in \\mathbb{R}$ and a resource requirement $r_j > 0$. Jobs do not benefit when receiving a share larger than $r_j$ of the resource. But providing them with a fraction of the resource requirement causes a linear decrease in the processing efficiency. We seek a (non-preemptive) job and resource assignment minimizing the makespan.Our main result is an efficient approximation algorithm which achieves an approximation ratio of $2 + 1/(m-2)$. It can be improved to an (asymptotic) ratio of $1 + 1/(m-1)$ if all jobs have unit size. Our algorithms also imply new results for a well-known bin packing problem with splittable items and a restricted number of allowed item parts per bin.Based upon the above solution, we also derive an approximation algorithm with similar guarantees for a setting in which we introduce so-called tasks each containing several jobs and where we are interested in the average completion time of tasks (a task is completed when all its jobs are completed)." ]

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[ "# #0f6599 Color Information\n\nIn a RGB color space, hex #0f6599 is composed of 5.9% red, 39.6% green and 60% blue. Whereas in a CMYK color space, it is composed of 90.2% cyan, 34% magenta, 0% yellow and 40% black. It has a hue angle of 202.6 degrees, a saturation of 82.1% and a lightness of 32.9%. #0f6599 color hex could be obtained by blending #1ecaff with #000033. Closest websafe color is: #006699.\n\n• R 6\n• G 40\n• B 60\nRGB color chart\n• C 90\n• M 34\n• Y 0\n• K 40\nCMYK color chart\n\n#0f6599 color description : Dark blue.\n\n# #0f6599 Color Conversion\n\nThe hexadecimal color #0f6599 has RGB values of R:15, G:101, B:153 and CMYK values of C:0.9, M:0.34, Y:0, K:0.4. Its decimal value is 1009049.\n\nHex triplet RGB Decimal 0f6599 `#0f6599` 15, 101, 153 `rgb(15,101,153)` 5.9, 39.6, 60 `rgb(5.9%,39.6%,60%)` 90, 34, 0, 40 202.6°, 82.1, 32.9 `hsl(202.6,82.1%,32.9%)` 202.6°, 90.2, 60 006699 `#006699`\nCIE-LAB 40.748, -3.938, -34.904 10.599, 11.708, 31.836 0.196, 0.216, 11.708 40.748, 35.125, 263.562 40.748, -25.084, -49.963 34.217, -4.587, -31.214 00001111, 01100101, 10011001\n\n# Color Schemes with #0f6599\n\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #99430f\n``#99430f` `rgb(153,67,15)``\nComplementary Color\n• #0f9988\n``#0f9988` `rgb(15,153,136)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #0f2099\n``#0f2099` `rgb(15,32,153)``\nAnalogous Color\n• #99880f\n``#99880f` `rgb(153,136,15)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #990f20\n``#990f20` `rgb(153,15,32)``\nSplit Complementary Color\n• #65990f\n``#65990f` `rgb(101,153,15)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #990f65\n``#990f65` `rgb(153,15,101)``\n• #0f9943\n``#0f9943` `rgb(15,153,67)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #990f65\n``#990f65` `rgb(153,15,101)``\n• #99430f\n``#99430f` `rgb(153,67,15)``\n• #083753\n``#083753` `rgb(8,55,83)``\n• #0a466b\n``#0a466b` `rgb(10,70,107)``\n• #0d5682\n``#0d5682` `rgb(13,86,130)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #1174b0\n``#1174b0` `rgb(17,116,176)``\n• #1484c7\n``#1484c7` `rgb(20,132,199)``\n• #1693df\n``#1693df` `rgb(22,147,223)``\nMonochromatic Color\n\n# Alternatives to #0f6599\n\nBelow, you can see some colors close to #0f6599. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #0f8899\n``#0f8899` `rgb(15,136,153)``\n• #0f7c99\n``#0f7c99` `rgb(15,124,153)``\n• #0f7199\n``#0f7199` `rgb(15,113,153)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #0f5a99\n``#0f5a99` `rgb(15,90,153)``\n• #0f4e99\n``#0f4e99` `rgb(15,78,153)``\n• #0f4399\n``#0f4399` `rgb(15,67,153)``\nSimilar Colors\n\n# #0f6599 Preview\n\nThis text has a font color of #0f6599.\n\n``<span style=\"color:#0f6599;\">Text here</span>``\n#0f6599 background color\n\nThis paragraph has a background color of #0f6599.\n\n``<p style=\"background-color:#0f6599;\">Content here</p>``\n#0f6599 border color\n\nThis element has a border color of #0f6599.\n\n``<div style=\"border:1px solid #0f6599;\">Content here</div>``\nCSS codes\n``.text {color:#0f6599;}``\n``.background {background-color:#0f6599;}``\n``.border {border:1px solid #0f6599;}``\n\n# Shades and Tints of #0f6599\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #01070a is the darkest color, while #f7fcfe is the lightest one.\n\n• #01070a\n``#01070a` `rgb(1,7,10)``\n• #03121c\n``#03121c` `rgb(3,18,28)``\n• #041e2e\n``#041e2e` `rgb(4,30,46)``\n• #062a40\n``#062a40` `rgb(6,42,64)``\n• #083652\n``#083652` `rgb(8,54,82)``\n• #0a4263\n``#0a4263` `rgb(10,66,99)``\n• #0b4d75\n``#0b4d75` `rgb(11,77,117)``\n• #0d5987\n``#0d5987` `rgb(13,89,135)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #1171ab\n``#1171ab` `rgb(17,113,171)``\n• #137dbd\n``#137dbd` `rgb(19,125,189)``\n• #1488cf\n``#1488cf` `rgb(20,136,207)``\n• #1694e0\n``#1694e0` `rgb(22,148,224)``\n• #219ee9\n``#219ee9` `rgb(33,158,233)``\n• #33a6eb\n``#33a6eb` `rgb(51,166,235)``\n``#45aded` `rgb(69,173,237)``\n• #56b5ee\n``#56b5ee` `rgb(86,181,238)``\n• #68bdf0\n``#68bdf0` `rgb(104,189,240)``\n• #7ac5f2\n``#7ac5f2` `rgb(122,197,242)``\n• #8ccdf4\n``#8ccdf4` `rgb(140,205,244)``\n• #9ed4f5\n``#9ed4f5` `rgb(158,212,245)``\n• #b0dcf7\n``#b0dcf7` `rgb(176,220,247)``\n• #c2e4f9\n``#c2e4f9` `rgb(194,228,249)``\n• #d3ecfb\n``#d3ecfb` `rgb(211,236,251)``\n• #e5f4fc\n``#e5f4fc` `rgb(229,244,252)``\n• #f7fcfe\n``#f7fcfe` `rgb(247,252,254)``\nTint Color Variation\n\n# Tones of #0f6599\n\nA tone is produced by adding gray to any pure hue. In this case, #505558 is the less saturated color, while #0268a6 is the most saturated one.\n\n• #505558\n``#505558` `rgb(80,85,88)``\n• #49575f\n``#49575f` `rgb(73,87,95)``\n• #435865\n``#435865` `rgb(67,88,101)``\n• #3c5a6c\n``#3c5a6c` `rgb(60,90,108)``\n• #365b72\n``#365b72` `rgb(54,91,114)``\n• #2f5d79\n``#2f5d79` `rgb(47,93,121)``\n• #295f7f\n``#295f7f` `rgb(41,95,127)``\n• #226086\n``#226086` `rgb(34,96,134)``\n• #1c628c\n``#1c628c` `rgb(28,98,140)``\n• #156393\n``#156393` `rgb(21,99,147)``\n• #0f6599\n``#0f6599` `rgb(15,101,153)``\n• #09679f\n``#09679f` `rgb(9,103,159)``\n• #0268a6\n``#0268a6` `rgb(2,104,166)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #0f6599 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ " log — Math\n\nlog( z )\n\nThe natural logarithm of z in Math. The inverse of exp. Defined by\n\n$e^{\\log z} = z$\n\nReal part on the real axis:\n\nImaginary part on the real axis:\n\nReal part on the imaginary axis:\n\nImaginary part on the imaginary axis:\n\nReal part on the complex plane:\n\nImaginary part on the complex plane:\n\nAbsolute value on the complex plane:\n\nRelated functions:   exp\n\nFunction category: logarithmic functions" ]

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[ "Direkt zum InhaltDirekt zur SucheDirekt zur Navigation\n▼ Zielgruppen ▼\n\n# Humboldt-Universität zu Berlin - Mathematisch-Naturwissen­schaft­liche Fakultät - Experimentelle Elementarteilchenphysik\n\nJoint analysis of Higgs decays and electroweak precision observables in the Standard Model with a sequential fourth generation\n\nOtto Eberhardt, Geoffrey Herbert, Heiko Lacker, Alexander Lenz, Andreas Menzel, Ulrich Nierste, and Martin Wiebusch (2012)\n\n# Joint analysis of Higgs decays and electroweak precision observables in the Standard Model with a sequential fourth generation\n\nPhys. Rev. D, 86:013011 .\n\nWe analyse the impact of LHC and Tevatron Higgs data on the viability of the Standard Model with a sequential fourth generation (SM4), assuming Dirac neutrinos and a Higgs mass of 125 GeV. To this end we perform a combined fit to the signal cross sections of pp -> H -> gamma gamma,ZZ*,WW* at the LHC, to p pbar -> VH -> V b bbar (V = W, Z) at the Tevatron and to the electroweak precision observables. Fixing the mass of the fourth generation down-type quark b' to 600 GeV we find best-fit values of m_t' = 632 GeV, m_l4 = 113.6 GeV and m_nu4 = 58.0 GeV for the other fourth-generation fermion masses. We compare the chi-square values and pulls of the different observables in the three and four-generation case and show that the data is better described by the three-generation Standard Model. We also investigate the effects of mixing between the third and fourth-generation quarks and of a future increased lower bound on the fourth-generation charged lepton mass of 250 GeV.", null, "" ]

[ null, "https://www.physik.hu-berlin.de/bullet.png", null ]

[ "# The injury severity score: an operations perspective\n\n## Abstract\n\n### Background\n\nThe statistical evaluation of aggregation functions for trauma grades, such as the Injury Severity Score (ISS), is largely based on measurements of their Pearson product-moment correlation with mortality. However, correlation analysis makes assumptions about the nature of the involved random variables (cardinality) and their relationship (linearity) that may not be applicable to ordinal scores such as the ISS. Moreover, using correlation as a sole evaluation criterion neglects the dynamic properties of these aggregation functions scores.\n\n### Methods\n\nWe analyze the domain and ordinal properties of the ISS comparatively to arbitrary linear and cubic aggregation functions. Moreover, we investigate the axiomatic properties of the ISS as a multicriteria aggregation procedure. Finally, we use a queuing simulation with various empirical distributions of Abbreviated Injury Scale (AIS) grades reported in the literature, to evaluate the queuing performance of the three aggregation functions.\n\n### Results\n\nWe show that the assumptions required for the computation of Pearson’s product-moment correlation coefficients are not applicable to the analysis of the association between the ISS and mortality. We suggest the use of Mutual Information, a information-theoretic statistic that is able to assess general dependence rather than a specialized, linear view based on curve-fitting. Using this metric on the same data set as the seminal study that introduced the ISS, we show that the sum of cubes conveys more information on mortality than the ISS. Moreover, we highlight some unintended, undesirable axiomatic properties of the ISS that can lead to bias in its use as a patient triage criterion. Lastly, our queuing simulation highlights the sensitivity of the queuing performance of different aggregation procedures to the underlying distribution of AIS grades among patients.\n\n### Conclusions\n\nViewing the ISS, and other possible aggregation functions for multiple AIS scores, as mere operational indicators of the priority of care, rather than cardinal measures of the response of the human body to multiple injuries (as was conjectured in the seminal study introducing the ISS) offers a perspective for their construction and evaluation on more robust grounds than the correlation coefficient. In this regard, Mutual Information appears as a more appropriate measure for the study of the association between injury severity and mortality, and queuing simulations as an actionable way to adapt the choice of an aggregation function to the underlying distribution of AIS scores.\n\n## Background\n\n### Overview\n\nThe Injury Severity Score (ISS) is a widely-used aggregate indicator of the overall severity of multiple injuries to the human body that was introduced in a study by Baker et al. . This score is calculated by summing the squares of the three highest values of the Abbreviated Injury Scale (AIS) , a common evaluation scale for the severity of trauma to individual body parts.\n\nSince its introduction, the ISS plays an ambivalent role, which the present manuscript aims at discussing. It acts as both a clinical measure of the lethality of multiple injuries (Baker et al. conjecture that this score “models a fundamental aspect of the human body’s response to multiple injuries”), as well as an operational indicator for patient triage. This ambivalence calls for two levels of analysis, when it comes to evaluating the ISS and similar aggregation procedures for AIS grades; a static study of their association with mortality and a dynamic evaluation of their axiomatic properties (i.e. how changes in AIS scores are reflected in the ISS) and queuing performance. However, only the former level of analysis is favored in the literature, with the correlation coefficient as sole association metric, and little is known about the axiomatic properties and queuing performance of ISS and similar aggregation functions.\n\n### Original data source and results\n\nThe seminal study by Baker et al. considered a sample of 2,128 motor vehicles occupants who were victims of accidents and admitted to one of 8 hospitals in the city of Baltimore, Maryland, USA, over a period of two years (1968-1969). For this sample, the study recorded a ratio of hospital admissions to deaths of 8:1. For individual hospitals, this ratio ranged from 5:1 to 60:1, indicating different levels of severity of injuries for the typical patient that each hospital received. Table 1 reproduces the distribution of AIS for the main injury of each patient in the sample, while Table 4 details the mortality rates corresponding to the highest AIS grade of patients in . The authors find that the ISS explains 49% of the variance in mortality, in the study sample.\n\n### Construction of the ISS\n\nThe severity of damage to each of nine body regions (head, face, neck, thorax, abdomen, spine, upper extremities, lower extremities, and external) is conventionally evaluated on a scale of 0 to 5Footnote 1 by the AIS. This scale evaluates individual injuries to a body region as follows:\n\n1. 0.\n\nNo injury\n\n2. 1.\n\nMinor injury\n\n3. 2.\n\nModerate injury\n\n4. 3.\n\nSerious injury\n\n5. 4.\n\nSevere injury\n\n6. 5.\n\nCritical injury\n\nTo compute the ISS, the nine previous body regions are first grouped into six:\n\n• R2: Face\n\n• R3: Chest\n\n• R4: Abdominal or pelvic contents\n\n• R5: Extremities or pelvic girdle\n\n• R6: External\n\nThe ISS is then computed as the sum of the squares of AIS scores of the three most severe injuries, and is thus evaluated on a scale of 0 to 75.\n\nFormally, let us denote AIS={R1,…,R6}, the AIS scores of an injured patient over the previous six body regions, which we will also refer to as the patient’s AIS profile. The computation of the ISS aggregates these score in two steps:\n\n1. 1.\n\nThe three highest AIS scores, that is A= max(AIS),B= max(AIS−{A}), and C= max(AIS−{A,B}), are determined.\n\n2. 2.\n\nThe sum of squares of A, B, and C is calculated, that is ISS=A2+B2+C2.\n\nThe first step of the ISS aggregation procedure (use of the three maxima) is justified in by the fact that considering the sum of squares of the AIS scores of the three most severe injuries considerably improved the correlation of the resulting score with mortality rates, when including the fourth highest AIS score had no appreciable effect.\n\n### Scope of this study\n\nIn this work, we will not analyze the first steps of the aggregation procedure and focus on the second. However, in The applicability of Pearson’s correlation to the ISS section, we show that statistical measures such as the correlation and standard deviation are not well suited for a variable such as the ISS, because they incorrectly assign it a cardinal value, which leads to inconsistent results. We should also mention an existing variant to the first step of the aggregation procedure, that questions not the use of three maxima for the AIS but the choice of body regions over which they are calculated. A widely-used such variant has been introduced under the denomination New Injury Severity Score (NISS) . Instead, of considering the three most severely injured body regions, this variant considers the three most severe injuries overall, the reasoning being that the original ISS method can potentially disregard more severe injuries that happen to be in the same body region as the most severe injury. This medical modification is inconsequential to the analysis and claims made in this paper focusing on the intrinsic mathematical properties of the method. Our results apply to both variants.\n\nThus the main focus of this study is the second step of the aggregation procedure. Indeed, in the choice of aggregating the three maxima by summing their squares was rather lightly justified as “the simplest nonlinear function”, without further explanations on the type of complexity being referred to. This justification will be put to question in the present work as the calculation of say the sum of cubes, or any other polynomial function of A, B, and C is no more complex than that of the ISS. As for the use of linear functions (e.g. summing the three maxima), it is dismissed in similarly vague terms with the sentence “the quantitative relationship of the AIS scores is not known and is almost certainly nonlinear”. The authors of the ISS further find that “the death rate for persons with two injuries of grades 4 and 3 was not comparable to that of persons with two injuries of grades 5 and 2 (sum = 7 in both cases)”.\n\nAfter reviewing past work on the ISS, and notably the seminal study that introduced this aggregation procedure, this paper questions the choice of a quadratic procedure relative to two other arbitrary aggregation functions (the sum and sum of cubes of the three highest AIS scores). Moreover, we study some axiomatic properties of the ISS and its queuing performance. Based on our results we propose that an injury severity aggregation procedure should be seen as an adjustment lever to optimize target criteria, rather than a rigid formula that seeks to capture fundamental aspects of the response of the human body to injuries with a quadratic formula (as has been wildly conjectured in the original study in the face of the high correlation of the ISS with mortality).\n\n## Methods\n\n### Measures of association between random variables\n\nFor the study of the association between injury severity scores and mortality, measures of correlation with mortality are typically favored in the literature, and the (Pearson product-moment) correlation coefficient is typically used to evaluate the adequacy of ISS and competing proposals, as measurements of the lethality of injuries. However, the ordinal nature of the ISS and similar aggregation functions would naturally call for the use of rank correlation. Spearman’s rank-order correlation coefficient [4, 6] could be more appropriate measurements of the association between ISS and mortality rates. Indeed, this statistic evaluates the monotonic association between two variables without utilizing ordinal information. However, it cannot be precisely evaluated in the presence of ties, which are common as seen in Figs. 1, 2, and 3. Moreover, this indicator would be sensitive to the intrinsic variance of the ISS for consecutive values of the AIS, illustrated with the example in Table 5. A more robust measurement of the association between mortality and ISS would be offered by Mutual Information . This more general indicator, which is less sensitive to the cardinal properties of random variables and is not limited to linear relationships, compares probability distributions as a whole and measures how different the joint probability distribution of two random variable is to the product of their marginal distributions. An extensive review and a general model for the use of mutual information for clinical decision making can be respectively found in and .\n\nThus Mutual Information MI(X,Y), given by MI(X,Y)=H(X)−H(X|Y), between two random variables X and Y is the average amount of information (in bits) about one random variable that is gained by knowing the value of the other random variable. In this formula H(X) is the marginal entropy of X, given by $$H(X)=-\\sum \\limits _{x \\in D_{X}}p(x)\\cdot log(p(x))$$, and H(X|Y) the conditional entropy of X in regard to Y, given by $$H(X|Y)=-\\sum \\limits _{x \\in D_{X}, y \\in D_{Y}} p(x,y)\\cdot log \\left (\\frac {p(x,y)}{p(x)}\\right)$$, where DX and DY are the respective support sets of X and Y, p(x,y) the joint probability distribution of X and Y, and p(x) the marginal probability distribution of X.\n\nIn its normalized form, mutual information quantifies this amount of information relative to the intrinsic entropy of each random variable. The normalized mutual information NMI(X,Y) between X and Y is thus given by $$NMI(X,Y)=\\frac {2\\cdot MI(X,Y)}{H(X)+H(Y)}$$. We compare the ISS with a linear and cubic aggregation functions, namely the sum and sum of cubes, using both Pearson’s correlation and Mutual Information.\n\n### Axiomatic study\n\nLittle is known about the axiomatic properties and queuing performance of ISS and similar functions, including in the Operations Research literature. For the analysis of axiomatic properties, and given an AIS profile of the form (A,B,C), we introduce the notation [xA,xB,xC] such that −AxA≥6−A,−BxB≥6−B, and −CxC≥6−C indicate a change in the AIS profile of a patient (i.e. an overall degradation or improvement of their injuries), resulting in a new AIS profile (A+xA,B+xB,C+xC). We assume, without loss of generality, that these changes maintain the three most severe injuries located in the same three body regions (out of the six AIS body regions previously grouped). For instance, [−1,0,+1] represents an improvement of the most severe injury of a patient by one AIS point (e.g. following care), and a degradation of their third most severe injury by one AIS point, without any change to their second most severe injury. These vectors can be conventionally added with ISS profiles to obtain the resulting ISS profiles, e.g. a patient whose ISS profile is (4,3,2) would see their ISS profile become (4,3,2)+[−1,0,1]=(3,3,3), following the above described change. Using this notation, we study the axiomatic properties of the ISS and test the compensation effects, rank reversals and independence property stemming from the use of the ISS as a multicriteria aggregation procedure.\n\n### Queuing simulations\n\nA queueing system is a general model of resource consumption, in which patients arrive at random times and require access to a healthcare resource (e.g. a physician consultation or inpatient bed). If the resource is busy upon a patient’s arrival, they are attributed a priority score and join a waiting line. In the present instance of the model, we consider the ISS, as well as the sum and sum of cubes of the three highest AIS scores as possible priority score. Other aggregation procedures, such as the NISS or the wISS could also be used, without loss of generality.\n\nDefining a queueing model requires making stochastic assumptions about the nature of the arrival and service processes, as well as the distribution of AIS grades. In healthcare, the Poisson process has been verified to be a good representation of unscheduled arrivals to various healthcare units, including emergency departments .\n\nThe most common assumptions to make about arrivals and service times are the following:\n\n• Arrivals follow a Poisson process characterized by a rate, that is the expected number of patient arrivals per unit of, denoted λ. The Poisson process for arrivals can also be conversely characterized by its expected inter-arrival time, that is the average time between two consecutive arrivals of patient, given by $$\\frac {1}{\\lambda }$$.\n\n• The service rate is also described by a Poisson distribution with a mean service rate (i.e. number of patients served per unit of time) μ. This means that the service time for one customer follows an exponential distribution with an average of $$\\frac {1}{\\mu }$$.\n\nThe previous two assumptions are often called Markovian, and the resulting queuing model denoted M/M/s, where the two “M’s” stand for this adjective, and “s” for the number of identical service resources that customers queue to gain access to. For the sake of simplicity, we will assume the existence of a single resource, that is a so-called M/M/1 queue. An advantage of this model is that it only requires two parameters (λ and μ), which can be estimated empirically, in a fairly robust manner.\n\nWe conduct discrete-event simulations of an M/M/1 waiting line , with stochastic AIS grades, generated according to various distributions reported in the literature. This simulation allows us to study the queuing performance of the three aggregation procedures considered, as well as their sensitivity to the underlying distribution of AIS grades.\n\n## Results\n\n### On the use of a quadratic aggregation function\n\nTable 2 describes the scales of the ISS (A2+B2+C2), as well as the sum (A+B+C) and sum of cubes (A3+B3+C3) functions. For A,B,C{0,1,2,3,4,5}, such that ABC and excluding triplet (0,0,0), there are 55 possible (A,B,C) triplets, resulting in 44 distinct possible values of the ISS (A2+B2+C2), as well as 13 and 55 distinct values of (A+B+C) and (A3+B3+C3), respectively.\n\nWe have computed all cases of discordance between the ISS, the sum, and the sum of cubes. In other words, the number of pairs of injury profiles for which the rankings provided by the two aggregation functions are reversed. Among the $$C_{2}^{5}5=1485$$ distinct, non-ordered pair of possible AIS profiles, we have identified the pairs for which there is discordance between A2+B2+C2,A3+B3+C3, and A+B+C, regarding the comparison of the pair. In other words, and for two patients x and y, let (Ax,Bx,Cx) and (Ay,By,Cy) be their respective AIS profiles. We consider that there is discordance between the ISS and the sum of cubes aggregation function if ($$A^{2}_{x} + B^{2}_{x} + C^{2}_{x} > A^{2}_{y} + B^{2}_{y} + C^{2}_{y}$$ and $$A^{3}_{x} + B^{3}_{x} + C^{3}_{x} < A^{3}_{y} + B^{3}_{y} + C^{3}_{y}$$) or ($$A^{2}_{x} + B^{2}_{x} + C^{2}_{x} < A^{2}_{y} + B^{2}_{y} + C^{2}_{y}$$ and $$A^{3}_{x} + B^{3}_{x} + C^{3}_{x} > A^{3}_{y} + B^{3}_{y} + C^{3}_{y}$$). There exist 84 pairs of profiles for which there is such a discordance, which represents 5.6% of the 1485 possible pairs of profiles (i.e. for a uniform distribution of AIS scores, the ISS and sum of cubes aggregation functions would disagree 5.6% of the time). The ISS and the sum are in discordance for 8% of possible profiles, whereas the sum of cubes and the sum are in discordance for 14.81% of possible profiles. Although a minority, these cases of discordance are non-neglectable, particularly for large volumes of patients.\n\n#### Association with mortality\n\nThe seminal work relied on the data in Table 3, which records the mortality rates for the AIS scores of the three most severe injuries, which we denote A, B and C by decreasing order of severity.\n\nThe use of the ISS was supported in by the data reproduced in Table 4, in which we have additionally included the sums of the three most severe ISS, of their squares (the ISS), and of their cubes, and calculated the (Pearson product-moment) correlation and Mutual Information of each profile with mortality rates. Figures 1, 2, and 3 respectively plot mortality rates according to sum, sum of squares (ISS), and sum of cubes of the three highest AIS scores for the sample of 2,128 patients in .\n\nThe high (Pearson’s product-moment) correlation of the ISS and mortality has led to conjecture that this score “models a fundamental aspect of the human body’s response to multiple injuries”. Though it remains a practical heuristic for priority evaluation and patient triage, the initial promise of the ISS as an indicator of the mortality of multiple injuries, and the even more daring conjecture of that this quadratic function may capture fundamental properties of the response of human bodies to injuries have been tempered down by more mathematically rigorous, recent studies of the discrete possible values taken by the ISS. In , it has been found that mortality is non-monotonic with regards to the ISS, that is, mortality does not necessarily increases with successive values of ISS.\n\nFollowing the same reasoning as , we use correlation with mortality as a measure of the adequacy of the three aggregation procedures. The sum of the three highest AIS scores presents the lowest correlation with mortality with 77% and Fig. 1 conveniently illustrates the reason for the inadequacy of this aggregation procedure. As indicated by the number of vertical and horizontal segments in the graph, the sum, which only offers 15 possible distinct values represented in Table 2, is not discriminant enough in relationship to mortality. However, the sum of squares (ISS) and the sum of cubes present similar levels of correlation with mortality rates, at 92% and as Figs. 2, and 3 show that the ISS (with 44 distinct possible value versus 55 for the sum of cubes, cf. Table 2) is less discriminant. All three functions are non-injective as evidenced by the existence of horizontal segments in the graphs. However, the relationship between the sum of cubes and mortality is of a functional nature (no vertical segments), as opposed to that of ISS with mortality. For instance, an ISS of 34 corresponds to both mortality rates of 43% and 59%. No such effects occur when considering the sum of cubes. However such undesirable effects cannot be evaluated by a coefficient of linear correlation, which would arbitrarily consider that the mortality rate associated with an ISS of 34 is 52%, the average of 43% and 59%.\n\n#### The AIS and ISS are not cardinal measures\n\nMeasurement theory assumes that there exist some empirical structure that one wishes to represent numerically (e.g. the body’s response to multiple injuries) and defines strict qualitative properties that the empirical structure must verify in order to be represented numerically. Such numerical artifacts are said to possess an interval level of measurement if, throughout its scale, equal differences in the measure reflect equal differences in the empirical structure being measured. Nothing indicates that the AIS and even less so the ISS possess such a property. The AIS and ISS can be more modestly considered to possess an ordinal level of measurement, that is to say as indicators allowing the ranking of patients, e.g. for triage purposes. An ordinal measure is defined, by opposition to a cardinal one, as “a variable whose attributes can only be ranked” [6, 14]. For instance, we know that an underlying injury having an AIS score of 3 is less severe than a 4, which in turn is less severe than a 5, but it remains unknown whether the distance between a 3 and a 4 is equal, greater, or smaller than the distance between a 4 and a 5. It is the practice of assigning the numerical values to the severity of these three injuries that sets the two numerical distances between them to be equal. The interpretation of the distances between ISS scores is similarly impossible. Indeed, the consecutive values in the domain of the ISS, represented in Table 2 only reflect an increase in the severity of the overall injury (ordinal information), but the extent of that increase cannot be given any interpretation (it contains no cardinal information). For instance, 50,51,54 are three consecutive values in the domain of the ISS, without any possible value between 51 and 54. A patient whose condition goes from an ISS of 50 to 51 and then from 51 to 54 would have seen the severity of their injury increase by two (ordinal) units, not four (cardinal) units.\n\nGiving a cardinal meaning to the ISS could have been justified if the difference between two consecutive values of this scale kept increasing, reflecting a higher level of degradation as the severity of an injury increases, but this is not the case. In Table 2, we can observe for instance that the gap between the thirty-second and thirty-third grades of the ISS (scores of 38 and 41, respectively) is wider than between the thirty-fourth and thirty-fifth grades (scores of 42 and 43, respectively).\n\n#### The applicability of Pearson’s correlation to the ISS\n\nThe value of the ISS is only ordinal, that is the information it provides is to rank the overall severity of injuries to multiple body regions of patients, and not measure any intrinsic property of these injuries. Further, warns against considering the ISS/NISS as continuous statistical variables in correlation analyses with outcome measures (e.g. mortality), which has been the approach initially used to justify the quadratic aggregation of AIS grades in the original version of the ISS. If we accept the ISS as a purely ordinal indicator, a much simpler argument can be made to show that the very concept of measuring Person’s correlation of the ISS with any other variable does not apply. Pearson’s product-moment correlation is defined as the covariance of two variables divided by the product of their standard deviation . Focusing on the ISS, we can observe that the concept of standard deviation does not apply to this variable.\n\nConsider the toy example in Table 5 in which we measure the standard deviation of ISS, in three samples of two patients each. The two patients in each sample are of two consecutive ranks, with regards to the ISS (28th and 29th,32nd and 33rd, as well as 34th and 35rd ranks, respectively). Note that the ISS profiles of a patient in consecutive samples only differs by one unit of AIS (e.g. the three samples could correspond to a similar degradation of patient 1 and of patient 2 injuries over three periods of time).\n\nWe observe a significantly higher standard deviation and thus variance in sample B than in sample A, which is not due to a wider dispersion of the severity of injuries in sample B, but is solely due to the cardinal properties of the ISS. There happens to be no possible ISS values between 38 and 41. The range of ISS goes back to one unit in sample C, and we find the same variance as in sample A.\n\nThus, the very concept of a unit of deviation of the ISS is meaningless and no interpretation can be made of the standard deviation of this variable and hence of its covariance or Pearson correlation with any other variable. These concepts being based on that of a deviation of the observed ISS values relative to the mean, it is impossible to separate the amount of deviation that is due the observations and the amount due to the makings of ISS scale, with its uneven distances between grades.\n\nThe calculations of the standard deviation and variance of the ISS, as well as its Pearson’s correlation with mortality and the analysis of said correlation does not account for the average and standard deviation of the distance between two consecutive Injury Severity Scores (they are not one and zero respectively). It implicitly consider this score to be cardinal (i.e. a measure of the amount of something).\n\nHowever for measures of mortality the average and standard deviation of the distance between two consecutive possible values are respectively one unit (depending on the decimal precision considered for mortality rates) and zero.\n\n#### Mutual Information as a more appropriate measure of the association between injury severity and mortality\n\nWe have computed Mutual Information with the data in Table 4 as input, for the three considered aggregation procedures and with p-values of order of magnitude 10−6, we find normalized amounts of Mutual Information of 0.46, 0.55, and 0.71 between mortality rates in Table 4 and the sum, sum of squares, and sum of cubes of AIS scores, respectively. For this data-set, there is thus a significantly higher amount of information concerning mortality rates contained in the sum of cubes than the sum of squares, which confirms and quantifies the visual insight gained from Figs. 2 and 3 and suggests Mutual Information as a more appropriate measurement of the association between aggregate scores based on the AIS and mortality rates.\n\n### Axiomatic properties\n\n#### Arbitrary compensation\n\nA multicriteria aggregation procedure is said to be compensatory if it allows for trade-offs between criteria, i.e. the possibility of compensating a disadvantage on some criteria by an advantage on other criteria . The ISS being a simple sum of squares, it is a fully compensatory procedure, in that any disadvantage on any criterion (a lower AIS score) can be compensated by an advantage on any other criterion (a higher AIS score). For instance, improving the second most severe injury by one AIS point, while degrading the third most severe injury by two AIS points would bring the same change to the ISS, no matter its initial value.\n\nShould a patient accept a medical procedure that improves your second most severe injury by one AIS point, but degrades your third most severe injury by two AIS points (for instance during transportation or waiting for said procedure)? Let us consider the toy example in Table 6.\n\nAn improvement in Patient 2’s condition (decrease in ISS) is a degradation in Patient 1’s condition (increase in ISS).\n\nThis property of the ISS function is arbitrary. It does not have anything to do with the fact that Patient 1 was initially in a slightly worse state than Patient 2. It is due to the fact that trade-offs between AIS scores A, b and C in the calculation of the ISS do not obey a fixed compensation rate. The very notion of improvement or degradation of the AIS score is thus meaningless. It should be noted that weighted aggregation procedures, such as the recently introduced weighted ISS (wISS) by Shi et al. do not suffer from this inconsistency, as the trade-off rates between criteria would be constant and defined by their weights.\n\n#### Arbitrary rank reversals for identical changes\n\nTable 7 shows a toy example in which Patient 1 and Patient 2 receive twice the same procedure (an improvement of their most severe injury by one AIS point followed by an improvement of their second most severe injury by one AIS point). Initially, the overall condition of Patient 2 (ISS of 33) is worse than that of Patient 1 (ISS of 32). However, after the first procedure the order of severity of the conditions of the two patients alternates to Patient 1 (ISS of 25) being worse off than patient 2 (ISS of 24) and then back to Patient 2 (ISS of 21) being in a worse condition than Patient 1 (ISS of 20), after the second procedure. Moreover, Table 8 shows a similar alternation of priority but with the condition of the two patients progressively degrading over time). In a situation where the ISS is used as a triage rule, the order of priority between the two patients would arbitrarily alternate, although the degradation of their states would be identical.\n\n#### Independence\n\nThe independence property states that identical performance on one or more criteria should not influence the way two alternatives compare . A transformation that maintains the value of the criterion equal should not change the way alternatives compare. In Table 9, we consider two pairs of ISS profiles, Patient 1 and Patient 2 versus Patient 3 and Patient 4. The only difference between these two pairs concerns the AIS score of the most severe injury (3 and 4 for patient 1 and patient 2 respectively, 4 and 5 for patient 3 and patient 4 respectively). An identical change, [0,+1,0]. is applied twice to the second most severe; it gains one point of severity. The two pairs of patients show an identical level of severity, in their second and third most severe injuries before and after the transformation, respectively (.,2,0) and (.,0,0). However, the change leads to two different outcomes. Patient 1 condition (ISS=13), which was initially less severe than that of Patient 2 (ISS=16), becomes more severe (18>17), whereas the order of priority of Patient 3 and Patient 4 remain unchanged (20<25 and 25<26).\n\n### Queuing simulation\n\n#### Settings\n\nViewing AIS aggregation procedures, such as the ISS, as priority indicators for access to healthcare resources, rather than fundamental measures of the body’s response to multiple injuries, one can focus on evaluating their operational performance. Queuing theory is an important tool in the Operations Research toolset with fruitful applications in healthcare, a systematic review of which can be found in . It can offer valuable insights on the dynamic properties of triage rules, when deployed for large-scale patient flows, and help inform the choice of an appropriate priority regime. However, to the best of our knowledge, little is known in the literature about the queuing performance of the ISS and similar trauma indicators. This section proposes a model for their evaluation, based on a discrete-event simulation of a M/M/1 queuing system.\n\nThroughout the present simulation, we consider an average service time of μ=1. In other words, we take one time-unit to represent the average service time of a patient. For instance, if the resource under study is a hospital bed, and the average length of stay is one week, one unit of simulation time would correspond to one week. If, on the other hand, it is access to a physician, with an average consultation duration of ten minutes, one unit of simulation time would correspond to ten minutes. Moreover, and in order to create a congested waiting line, we consider an average inter-arrival time of $$\\frac {1}{\\lambda }=0.1$$, meaning that, on average, ten patients arrive in the queue during the time it takes to deliver the service to one patient.\n\nSince the focus of this study is on priority regimes and their impact on queuing performance, we additionally need to make assumptions regarding the distribution of AIS grades of arriving patients. We have conducted our simulations with respect to different distributions of AIS grades reported in the literature. In addition to the distribution for victims of motor vehicles accidents, reported by Baker et al. and reproduced in Table 1, we consider the distributions of AIS grades for 174 adult victims of fall accidents reported by Lopes et al. in , 451 patients with tornado-related injuries reported by Deng et al. in Footnote 2, and 278 victims of traumatic maternal injuries reported by Awoleke et al. in . The details of each distribution are reproduced in Table 10 and the code in Appendix A.\n\nFor the three aggregation procedures considered in this study (ISS, sum and sum of cubes), we are interested in evaluating discrepancies in the average waiting time for all patients and for patients with critical injuries (i.e. patients presenting AIS scores of 5 on some body regions), as a proxy for mortality. These discrepancies would result from the cases of discordance between the three aggregation procedures, discussed in On the use of a quadratic aggregation function section.\n\nFor each distribution of AIS grades in Table 10, we conduct 100 simulation, each simulation having a duration of 1000 discrete time-units. We estimate the average waiting times per patient, resulting from each of the three aggregation procedures. The commented source code for these simulations and their evaluation is provided in the R language, in Appendix A.\n\n#### Results\n\nFigures 4 and 5 respectively detail the average waiting times for all patients and critical patients, in each simulation, for the four AIS distributions considered, while Table 11 presents their averages over the 100 simulations.\n\nIt should be noted that, since the AIS distribution reported by Awoleke et al. does not include any critical patients, this distribution is excluded from the computation of waiting times of critical patients. As per the setup of these simulations, the time-unit of average waiting times corresponds to the service time. For instance, if the resource under study is a hospital bed, with average length of stay of one week, an average waiting time of 39.14 would correspond to 39.14 weeks. If, on the other hand, it is access to a physician, with an average consultation duration of ten minutes, it would correspond to an average waiting time of 391.4 minutes.\n\nThese simulations confirm the inefficiency of the sum as an aggregation procedure. Indeed, it results in significantly longer average waiting times, and only outperforms the ISS in some rare simulations. However, the comparison of queuing performance is more nuanced between the ISS and the sum of cubes. The two aggregation procedures show identical performance for the AIS distribution of Awoleke et al. , which can be explained by the relatively lower AIS scores in this distribution, and the fact that cases of discordance between the ISS and the sum of cubes (14.81% of possible AIS profiles, as discussed in On the use of a quadratic aggregation function section) mainly occur for higher AIS values. However, on average over the 100 simulations, there is a non-negligible advantage to using the sum of cubes, in terms of minimizing average waiting for all patients and critical patients alike. For the distribution of Lopes et al. , this advantage is as significant as 2.86 units of time, on average, for critical patients. This advantage can be explained by the that the sum of cubes offers a broader set of possible scores than the ISS (55 vs 41, as shown in Table 2), thus allowing it to convey more information. This fact was also reflected in its higher mutual information with regard to mortality in Table 4. However, these results should not be interpreted as the sum of cubes being a universally better aggregation procedure than the ISS, as these simulations were only conducted under specific simplifying assumptions and for a select set of AIS distribution. For different queuing settings and empirical AIS distributions, the ISS may very well be the best performing aggregation procedure. Indeed, the most general and robust conclusion we can draw from the results of these simulation is that the operational performance of an aggregation procedure is sensitive to the underlying AIS distribution and thus the choice of the “best” procedure can only be made on a case-by-case basis, with respect to empirical estimates of this distribution in a healthcare unit.\n\n## Discussion\n\nAggregation procedures for AIS grades, such as the Injury Severity Score and similar, competing indicators (New Injury Severity Score , Exponential Severity Score , etc.) have important operational applications as waiting line priority regimes. Therefore, their design is a highly sensible one that impacts mortality rates. However, the evaluation of these indicators typically relies on a static, linear evaluation of their association with mortality rates, and proposals typically compete on which function achieves the highest Pearson correlation. In this paper, we put forward the idea that curve-fitting and the evaluation of correlation with mortality rates are insufficient evaluation methodologies for the operational performance of these aggregation procedures. We have shown correlation-based measurements (as well as measurements of the standard deviation/variance of the ISS) to be largely unfounded, and proposed Mutual Information as a more adequate and more general measure of association. Moreover, by attempting to be two things at once (a cardinal measure of the human body’s response to multiple injuries as well as an ordinal triage rule presenting good association with mortality), the ISS may achieve sub-optimal results in both regards. A complex, fundamental property such as the physiological response to injury is unlikely to be universally captured by a simple mathematical function (the ISS) of ordinal mathematical measures (the AIS). Thus, there can be no universally best aggregation function. We recommend viewing the ISS, and similar aggregation procedures for multiple AIS grades, as purely operational triage indicators, rather than cardinal measures of the response of the human body to multiple injuries. As such, the choice of such an aggregation function should be made according to the distribution of AIS grades in a healthcare unit, to optimize queuing performance.\n\n## Conclusions\n\nThe present paper studied the Injury Severity Score as a multicriteria aggregation procedure for operational decision-making. We have highlighted some of its statistical and axiomatic properties that can lead to bias in its large-scale usage as a patient triage indicator. These properties therefore present areas of improvement for future proposals of aggregation procedures. Moreover, and although the addition of a degree to this quadratic aggregation procedure (i.e. considering the sum of cubes rather than the sum of squares) was found to convey more information on mortality and improve waiting line performance, the ISS was generally found to be a robust triage rule that achieved decent waiting line performance. However, we have shown this performance to be highly sensitive to the statistical distribution of the AIS scores of patients entering the waiting line. Thus, these findings suggest that the choice of an aggregation procedure for AIS grades (ISS, sum of cubes, or any other function) should be made on a case by case basis, with respect to the empirical distribution of these grades in a trauma department. This perspective notably permits the design of aggregation procedures for AIS grades in a way that explicitly optimizes operational criteria, such as the average waiting time of patients presenting critical injuries. In our view, the ambiguous, classical view in the literature of the ISS as a cardinal measure of the severity of multiple injuries (besides its use as an ordinal triage indicator) and the ensuing correlation analyses with mortality rates have somehow hindered this actionable line of research.\n\n## Appendix A\n\n### R Script for queuing simulation", null, "", null, "", null, "", null, "## Availability of data and materials\n\nThe data used in the current study are available in references , , , and . The source code for the queuing simulation is provided in the R language, in Appendix A.\n\n1. A grade of 6 additionally indicates untreatable injuries. This value being immaterial to the purpose of this paper, we will omit it from our analysis\n\n2. This study also included 0.75% of cases presenting fatal injuries (AIS=6), this grade being immaterial to the purpose of the waiting line simulation conducted herein.\n\n## Abbreviations\n\nAIS:\n\nAbbreviated injury scale\n\nISS:\n\nInjury severity score\n\nNISS:\n\nNew injury severity score\n\n## References\n\n1. Baker SP, O’Neill B, Haddon W, Long WB. The Injury Severity Score: a method for describing patients with multiple injuries and evaluating emergency care. J Trauma. 1974; 14(3):187–96. https://doi.org/10.1097/00005373-197403000-00001.\n\n2. Committee on Medical Aspects of Automotive Safety. Rating the Severity of Tissue Damage: I. The abbreviated scale. JAMA. 1971; 215:277–80.\n\n3. Samin OA, Civil ID. The New Injury Severity Score Versus the Injury Severity Score in Predicting Patient Outcome: A Comparative Evaluation on Trauma Service Patients of the Auckland Hospital. Annu Proc Assoc Adv Automot Med. 1999; 43:1–15.\n\n4. Spearman C. The proof and measurement of association between two things. Am J Psychol. 1904; 15(1):72–101. https://doi.org/10.2307/1412159.\n\n5. Kendall M. A New Measure of Rank Correlation. Biometrika. 1938; 30(1-2):81–9. https://doi.org/10.1093/biomet/30.1-2.81.\n\n6. Agresti A. Analysis of Ordinal Categorical Data (Second ed.)New York: John Wiley & Sons; 2010.\n\n7. David J, MacKay C. Information Theory, Inference, and Learning Algorithms.Cambridge University Press; 2003.\n\n8. Benish WA. A Review of the Application of Information Theory to Clinical Diagnostic Testing. Entropy. 2020; 22(1):97. https://doi.org/10.3390/e22010097.\n\n9. Benish WA. Mutual information as an index of diagnostic test performance. Methods Inf Med. 2003; 42(3):260–4.\n\n10. Roy B. Multicriteria Methodology for Decision Aiding. Berlin: Springer-Verlag; 1996.\n\n11. Green L. In: Hall RW, (ed).Queueing Analysis in Healthcare. Boston: Springer; 2006, p. 91.\n\n12. Kilgo PD, Meredith JW, Hensberry R, et al.A Note on the disjointed nature of the injury severity score. The Journal of Trauma: Injury. Infect Crit Care. 2004; 57:479–87.\n\n13. Krantz DH, Luce RD, Suppes P, Tversky A. Foundations of measurement (Vol. 1). Additive and polynomial representations. New York: Academic Press; 1974. Reprinted by Dover Publications in 2007.\n\n14. Schröder C, Yitzhaki S. Revisiting the evidence for cardinal treatment of ordinal variables. Eur Econ Rev. 2017; 92:337–58.\n\n15. Stevenson M, Segui-Gomez M, Lescohier I, et al.An overview of the injury severity score and the new injury severity score. Inj Prev. 2001; 7:10–3.\n\n16. Garren ST. Maximum likelihood estimation of the correlation coefficient in a bivariate normal model, with missing data. Stat Probab Lett. 1998; 38(3):281–8. https://doi.org/10.1016/S0167-7152(98)00035-2.\n\n17. Roy B, Slowinski R. Questions guiding the choice of a multicriteria decision aiding method. EURO J Decis Process. 2003; 1:69–97.\n\n18. Shi, et al.A new weighted injury severity scoring system: better predictive power for adult trauma mortality.Inj Epidemiol. 2019; 6(40):1–10. https://doi.org/10.1186/s40621-019-0217-8.\n\n19. Lakshmi C, Iyer SA. Application of queueing theory in health care: A literature review. Oper Res Health Care. 2013; 2(1-2):25–39.\n\n20. Kuo SCH, et al.Comparison of the new Exponential Injury Severity Score with the Injury Severity Score and the New Injury Severity Score in trauma patients: A cross-sectional study. PLoS ONE. 2017; 9;12(11):e0187871.\n\n21. Lopes M, Yamaguchi Whitaker I. Measuring trauma severity using the 1998 and 2005 revisions of the Abbreviated Injury Scale. Rev Esc Enferm USP. 2014;48(04). https://doi.org/10.1590/S0080-623420140000400010.\n\n22. Deng Q, et al.Pattern and spectrum of tornado injury and its geographical information system distribution in Yancheng, China: a cross-sectional study. BMJ Open. 2018; 8:e021552. https://doi.org/10.1136/bmjopen-2018-021552.\n\n23. Awoleke JO, Aduloju OP, Olofinbiyi BA. Determinants of hospital utilization after maternal falls in southern Nigeria. Int Med. 2019; 1(6):319–24. https://doi.org/10.5455/im.57614.\n\n## Acknowledgements\n\nThe author is grateful to Prof. N. N. Taleb for a helpful advice on Information Theory, as well as to Dr. P. Lo Monaco and to two anonymous reviewers of BMC Med. Res. Methodol. for constructive remarks that improved the original draft of this manuscript.\n\nNot applicable.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nND is the sole author of this study. The author read and approved the final manuscript.\n\n### Corresponding author\n\nCorrespondence to Nassim Dehouche.\n\n## Ethics declarations\n\nNot applicable.\n\nNot applicable.\n\n### Competing interests\n\nThe author declares that he has no competing interests.", null, "" ]

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[ "# How to iterate over associative arrays in bash?\n\n## Introduction\n\nBash: how to iterate over associative arrays in Bash?\n\nIn Bash, there are several ways to iterate over an associative array. Here are some examples of how to do it:\n\n## Method 1: Using for-loop with keys\n\n• Step 1 - Declare an associative array.\n``declare -A my_array=( [\"key1\"]=\"value1\" [\"key2\"]=\"value2\" [\"key3\"]=\"value3\" )``\n• Step 2 - Use a for-loop to iterate over the keys of the associative array.\n``````for key in \"\\${!my_array[@]}\"\ndo\necho \"\\$key: \\${my_array[\\$key]}\"\ndone``````\n\nIn this example, the for-loop is using the special syntax \"\\${!my_array[@]}\" to iterate over the keys of the associative array. The syntax \"\\${my_array[\\$key]}\" is used to access the value associated with each key.\n\n## Method 2: Using for-loop with key-value pairs\n\n• Step 1 - Declare an associative array.\n``declare -A my_array=( [\"key1\"]=\"value1\" [\"key2\"]=\"value2\" [\"key3\"]=\"value3\" )``\n• Step 2 - Use a for-loop to iterate over the key-value pairs of the associative array.\n``````for key_value in \"\\${my_array[@]}\"\ndo\necho \"\\$key_value\"\ndone``````\n\nIn this example, the for-loop is using the syntax \"\\${my_array[@]}\" to iterate over the key-value pairs of the associative array.\n\n## Method 3: Using while-loop with keys and read\n\n• Step 1 - Declare an associative array.\n``declare -A my_array=( [\"key1\"]=\"value1\" [\"key2\"]=\"value2\" [\"key3\"]=\"value3\" )``\n• Step 2 - Use a while-loop with the read command to iterate over the keys of the associative array.\n``````while IFS='=' read -r key value; do\necho \"\\$key: \\$value\"\ndone < <(for key in \"\\${!my_array[@]}\"; do echo \"\\$key=\\${my_array[\\$key]}\"; done)``````\n\nIn this example, the while-loop is using the read command to assign the key and value to separate variables. The while-loop is also using process substitution to iterate over the keys of the associative array. The syntax \"< <(command)\" is used to redirect the output of the command to the while-loop.\n\nNote that all the above methods work with associative arrays.\n\nIn all the above methods, the key and value are separated by the \"=\" sign, so in the while loop, we used the 'IFS= ' option to let the read command know that the separator is '='.\n\n## Method 4: Using while-loop with keys and read and indexed array\n\n• Step 1 - Declare an associative array.\n``declare -A my_array=( [\"key1\"]=\"value1\" [\"key2\"]=\"value2\" [\"key3\"]=\"value3\" )``\n• Step 2 - Use a while-loop with the read command to iterate over the keys of the associative array and use indexed array to store the keys\n``````keys=(\"\\${!my_array[@]}\")\nindex=0\nwhile [ \\$index -lt \\${#keys[@]} ]; do\nkey=\"\\${keys[\\$index]}\"\necho \"\\$key: \\${my_array[\\$key]}\"\nindex=\\$((index+1))\ndone``````\n\nIn this example, the while-loop is using the indexed array to iterate over the keys of the associative array. We are storing the keys in the indexed array keys and using that array to iterate over the keys.\n\n## Method 5: Using for-loop with `for key in \\${!my_array[@]}` and `for value in \\${my_array[@]}`\n\n• Step 1 - Declare an associative array.\n``declare -A my_array=( [\"key1\"]=\"value1\" [\"key2\"]=\"value2\" [\"key3\"]=\"value3\" )``\n• Step 2 - Use a for-loop to iterate over the keys of the associative array and then use another for-loop to iterate over the values of the associative array.\n``````for key in \\${!my_array[@]}; do\necho \"key: \\$key\"\nfor value in \\${my_array[\\$key]}; do\necho \"value: \\$value\"\ndone\ndone``````\n\nIn this example, the first for-loop is using the special syntax \"\\${!my_array[@]}\" to iterate over the keys of the associative array, and the second for-loop is using the syntax \"\\${my_array[\\$key]}\" to iterate over the values of the associative array.\n\n## Conclusion\n\nIn conclusion, iterating over associative arrays in Bash can be done in several ways. The methods shown in this tutorial include using for-loops with keys, for-loops with key-value pairs, while-loops with keys and read, while-loop with keys and read and indexed array, and for-loop with `for key in \\${!my_array[@]}` and `for value in \\${my_array[@]}`. Each method has its own advantages and can be used depending on the specific use case and requirement. It is important to understand the syntax and concepts used in these examples, such as process substitution, indexed arrays, and the use of special syntax like \"\\${!my_array[@]}\" and \"\\${my_array[\\$key]}\", in order to effectively iterate over associative arrays in Bash.\n\n1. Bash" ]

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[ "# How to Print a vector in C++?\n\nIn this article, we will discuss different ways to print all the elements of a vector in C++.\n\nSuppose we have a vector and we want to print all items of the vector on console. There are different ways to do this, let’s discuss them one by one,\n\nWe can overload the << operator for a vector as a template function in global scope, to print the contents of the vector of any type.\nIt will iterate over all items of the vector and print them one by one. Let’s understand by examples,\n\n```#include<iostream>\n#include<vector>\n\n/*\nTemplate function to print a vector on console\n*/\ntemplate <typename T>\nstd::ostream & operator << (\nstd::ostream & os,\nconst std::vector<T> & vec)\n{\nfor(auto elem : vec)\n{\nos<<elem<< \" \";\n}\nreturn os;\n}\n\nint main()\n{\n// create a vector integers\nstd::vector<int> vec_of_nums{1, 3, 4, 7, 8, 9};\n\n// print vector object\nstd::cout<< vec_of_nums << std::endl;\n\nreturn 0;\n}\n```\n\nOutput:\n\n`1 3 4 7 8 9`\n\nWe overloaded the << operator for vector and then used that to print the contents of vector.\n\n## C++: Print a vector in comma separated manner\n\nIf you want to provide a custom separator while printing contents of vector, then you can avoid overloading the << operator. Instead create a separate function to print the contents of a vector with custom separator. We have created a function print_vector(). It accepts two  arguments: a vector and a separator string. Inside the print_vector() function, it iterates over all elements of vector and print them one by one separated by provided custom separator string. Let’s understand by example,\n\n```#include<iostream>\n#include<vector>\n\n/*\nIterate over all elements of vector and print\nthem one by one, seperated by provided seperated.\n*/\ntemplate <typename T>\nvoid print_vector(const std::vector<T> & vec, std::string sep=\" \")\n{\nfor(auto elem : vec)\n{\nstd::cout<<elem<< sep;\n}\nstd::cout<<std::endl;\n}\n\nint main()\n{\n// Vector of integers\nstd::vector<int> vec_of_nums{1, 3, 4, 7, 8, 9};\n\n// Print all elements in vector\nprint_vector(vec_of_nums, \",\");\n\n// Vector of strings\nstd::vector<std::string> vec_of_str{\"Hi\", \"There\", \"is\", \"why\", \"how\"};\n\n// Print all elements in vector\nprint_vector(vec_of_str, \",\");\n\nreturn 0;\n}\n```\n\nOutput\n\n```1,3,4,7,8,9,\nHi,There,is,why,how,```\n\nWe printed the contents of a vector of integer and then a vector of string using the template function print_vector().\n\n## Print a Vector in C++ using indexing\n\nUnlike previous example, we can iterate over the contents of vector using indexing and print all elements in it one by one. For example,\n\n```#include<iostream>\n#include<vector>\n\n/*\nIterate over all elements of vector and print\nthem one by one, seperated by provided seperated.\n*/\ntemplate <typename T>\nvoid display_vector(const std::vector<T> & vec, std::string sep=\" \")\n{\nfor(int i = 0; i < vec.size() ; i++)\n{\nstd::cout<<vec[i]<< sep;\n}\nstd::cout<<std::endl;\n}\n\nint main()\n{\n// Vector of integers\nstd::vector<int> vec_of_nums{1, 3, 4, 7, 8, 9};\n\n// Print all elements in vector\ndisplay_vector(vec_of_nums);\n\nreturn 0;\n}```\n\nOutput\n\n`1 3 4 7 8 9`\n\n## Print a Vector in C++ in one line without for loop\n\nWe can print all the items of a vector using a STL algorithm std::copy(). Using this API we can copy all the elements of a vector to the output stream. For example,\n\n```#include<iostream>\n#include<vector>\n#include <iterator>\n\nint main()\n{\n// Vector of integers\nstd::vector<int> vec_of_nums{1, 3, 4, 7, 8, 9};\n\n// Print all elements in vector\nstd::copy( vec_of_nums.begin(),\nvec_of_nums.end(),\nstd::ostream_iterator<int>(std::cout,\" \"));\n\nstd::cout<<std::endl;\n\nreturn 0;\n}\n```\n\nOutput\n\n`1 3 4 7 8 9`\n\n## Print a Vector in C++ in one line (Generic Solution)\n\nIn previous example, we specifically provided the type of elements in vector while calling the copy() algorithm. But using C++17 experimental::make_ostream_joiner, we can print all elements of a vector without specifying the type of elements in vector. For example,\n\n```#include<iostream>\n#include<vector>\n#include <experimental/iterator>\n\nint main()\n{\n// Vector of integers\nstd::vector<int> vec_of_nums{1, 3, 4, 7, 8, 9};\n\n// Print all elements in vector\nstd::copy( vec_of_nums.begin(),\nvec_of_nums.end(),\nstd::experimental::make_ostream_joiner(std::cout,\" \") );\n\nstd::cout<<std::endl;\n\nreturn 0;\n}\n```\n\nOutput\n\n`1 3 4 7 8 9`\n\n## Print a Vector in C++ in one line using Lambda function\n\nUsing for_each(), we can apply a lambda function on each element of the vector. Inside the lambda function we can print its value.\nLet’s understand with example,\n\n```#include<iostream>\n#include<vector>\n#include <algorithm>\n\nint main()\n{\n// Vector of integers\nstd::vector<int> vec_of_nums{1, 3, 4, 7, 8, 9};\n\n// Print all elements in vector\nstd::for_each( vec_of_nums.begin(),\nvec_of_nums.end(),\n[](const auto & elem ) {\nstd::cout<<elem<<\" \";\n});\n\nstd::cout<<std::endl;\n\nreturn 0;\n}\n```\n\nOutput\n\n`1 3 4 7 8 9`\n\nHere we iterated over all the elements of vector in a single line and printed them one by one.\n\n## Summary\n\nWe learned about six different ways to print a vector in C++. Thanks." ]

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[ "Solutions by everydaycalculation.com\n\n## 10 is what percent of 54?\n\n10 of 54 is 18.52%\n\n#### Steps to solve \"what percent is 10 of 54?\"\n\n1. 10 of 54 can be written as:\n10/54\n2. To find percentage, we need to find an equivalent fraction with denominator 100. Multiply both numerator & denominator by 100\n\n10/54 × 100/100\n3. = (10 × 100/54) × 1/100 = 18.52/100\n4. Therefore, the answer is 18.52%\n\nIf you are using a calculator, simply enter 10÷54×100 which will give you 18.52 as the answer.\n\nMathStep (Works offline)", null, "Download our mobile app and learn how to work with percentages in your own time:" ]

[ null, "https://answers.everydaycalculation.com/mathstep-app-icon.png", null ]

[ "# double\n\nDouble-precision floating-point real-world value of `fi` object\n\n## Syntax\n\n```double(a) ```\n\n## Description\n\n`double(a)` returns the real-world value of a `fi` object in double-precision floating point. `double(a)` is equivalent to `a.double`.\n\nFixed-point numbers can be represented as\n\n`$real\\text{-}worldvalue={2}^{-fractionlength}×storedinteger$`\n\nor, equivalently as\n\n`$real\\text{-}worldvalue=\\left(slope×storedinteger\\right)+bias$`\n\n## Examples\n\nThe code\n\n```a = fi([-1 1],1,8,7); y = double(a) z = a.double```\n\nreturns\n\n```y = -1 0.9922 z = -1 0.9922```\n\n## Extended Capabilities", null, "" ]

[ null, "https://kr.mathworks.com/images/nextgen/callouts/bg-trial-arrow_02.png", null ]

[ "# Infinite Recursive Logarithm Sequence\n\nCalculus Level 1\n\nA recursive sequence is defined as $a_0=\\pi$ and $a_n=\\log_{a_{n-1}}{27}.$\n\nGiven that $\\displaystyle\\lim_{n\\to\\infty}{a_n}$ exists, what is $\\displaystyle\\lim_{n\\to\\infty}{a_n}?$\n\nBonus: For what values of $a_0$ does $\\displaystyle\\lim_{n\\to\\infty}{a_n}$ exist?\n\n×" ]

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[ "# [Day02] Pattern Matching\n\n``````// a = 'foo', b = { b: 'bar', c: 'baz' }\nconst { a, ...b } = { a: 'foo', b: 'bar', c: 'baz' }\n// first = 1, second = 2, rest = [3, 4, 5]\nconst [first, second, ...rest] = [1, 2, 3, 4, 5]\n``````\n\n`解構賦值(destructuring assignment)` 其實就隱含 pattern matching 的概念，就是如果等式的左邊跟右邊能找到一種方式能符合，則綁定 symbol & value。\n\n``````fib :: Int -> Int\nfib 0 = 0\nfib 1 = 1\nfib n = fib (n-1) + fib (n-2)\n``````\n\n`fib :: Int -> Int` 表示宣告一個 function 叫做 fib ，fib 會吃一個 Int 型態的參數而回傳 Int 型態的回傳值。\n\n1. `fib 0 = 0` 這段用賦值來解釋的話完全說不通，要怎麼賦值給一個函數呢？所以這段程式碼的解讀應為 `將 fib(0) 綁定到 0`，如果在 Day01 還沒抓到`'='其實是綁定`這種感覺的讀者們可以感受一下這個概念。\n2. `fib 0 = 0` 的意義在於，如果我今天使用了 `fib n` 而 n 恰巧 = 0，則 `(fib n) == (fib 0) == 0`，換言之 pattern matching 不只可以針對 symbol 做配對，甚至可以對值做配對！只要左右測的值符合綁定的條件(以此例來說， n == 0)，是可以直接用值來配對的，這是大多數具備類似`解構賦值`的特色的語言也做不到的。\n\npattern matching 搭配 guard，甚至可以在參數階段就做一些簡單的過濾，像是下列這段 Haskell 程式碼：\n\n``````absolute x\n| x < 0 = -x\n| otherwise = x\n``````\n\n#Functional Programming #程式設計" ]

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[ "Frequency counts and gives us the number of data points per bin. The code below is the most basic syntax. logical; if TRUE, the histogram cells are right-closed (left open) intervals. Skewed Right Histogram . This is where the skill of creating histograms in R comes in handy. The bars represent the range of values and their height indicates the frequency. In a histogram, the area of each block is proportional to the frequency. Since it is a time series with a gradual … It was first introduced by Karl Pearson. Graphs in R A histogram is the most usual graph to represent continuous data. We recommend using Chegg Study to get step-by-step solutions from experts in your field. na.rm=T or na.rm=TRUE will remove the missing data (represented by NA in R) before applying a function. It is a bar plot that represents the frequencies at which they appear measurements grouped at certain intervals and count how many observations fall at each interval. Histogram of Frequency in R [You can get some more detail with the “hist()” function by adding additional parameters to specify x and y labels and changing the bin width. For explanations, we will use the “Orange” dataset which comes as a default dataset in R Studio. Histogram are frequently used in data analyses for visualizing the data. Adding value markers 5. Frequency counts and gives us the number of data points per bin. This plot is indicative of a histogram for time series data. R provides a hist() function which is used to create histograms. If you’re short on time jump to the sections of interest: 1. A skewed right histogram is a histogram that is skewed to the right. The histogram also shows the skewness of the data. Replication requirements 2. The Data. Klodian Dhana The data shows that most numbers of passengers per month have been between 100-150 and 150-200 followed by the second highest frequency in the range 200-250 and 300-350.. Making Histogram in R. Histograms in R are also similarly easy to make. A histogram is a plot with rectangles, height of which represents the frequency or “count” of the occurrence and width is equal to the grouping interval. For continuous variable, you can visualize the distribution of the variable using density plots, histograms and alternatives. This tutorial explains how to create a relative frequency histogram in R by using the histogram () function from the lattice, which uses the following syntax: This code computes a histogram of the data values from the dataset AirPassengers, gives it “Histogram for Air Passengers” as title, labels the x-axis as “Passengers”, gives a blue border and a green color to the bins, while limiting the x-axis from 100 to 700, rotating the values printed on the y-axis by 1 and changing the bin-width to 5. Statology Study is the ultimate online statistics study guide that helps you understand all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. lines() function will add a line to an existing figure. The generic function hist computes a histogram of the givendata values. This tutorial explains how to create a relative frequency histogram in R by using the, By default, this package creates a relative frequency histogram with, We can specify the number of bins to use in the histogram using the, A Guide to dpois, ppois, qpois, and rpois in R. Your email address will not be published. Code: hist (swiss \\$Examination) Output: Hist is created for a dataset swiss with a column examination. Below is an example: The hist () functions returns details of the histogram which can be accessed by assigning the histogram to a variable. As such, the shape of a histogram is its most evident and informative characteristic: it allows you to easily see where a relatively large amount of the data is situated and where there is very little data to be found (Verzani 2004). The most common and straight forward method of generating a frequency table in R is through the use of the table function. When we create a histogram using hist function in R, often the Y-axis labels are smaller than the one or more bars of the histogram. Details. This function takes a vector as an input with some parameters to plot histograms. Histogram Here, we’ll let R create the histogram using the hist command. Histogram are frequently used in data analyses for visualizing the data. # factor in R > factor (mtcars\\$cyl) Create a R Histogram with Density. In the code below, I have changed the bin width by specifying that my histogram uses 5 intervals. It looks as follows: Example: The following histogram shows the number of people corresponding to different wage ranges. In real-time, we are more interested in density than the frequency-based histograms because density can give the probability densities. It is an easier way to visualize large data sets. How to generate QR codes with R and publish with R Markdown, Graphical Presentation of Missing Data; VIM Package, How to create a loop to run multiple regression models, Second step with non-linear regression: adding predictors, Earthquake Analysis (1/4): Quantitative Variables Exploratory Analysis, R for Publication by Page Piccinini: Lesson 0 – Introduction and Set-up, Regression model with auto correlated errors – Part 1, the data, Introduction to Data Visualization with ggplot2, Intermediate Data Visualization with ggplot2. In the data set faithful, the histogram of the eruptions variable is a collection of parallel vertical bars showing the number of eruptions classified according to their durations. In this article, I’ll explain how to use the hist() function to draw a histogram with percent in the R programming language. Example. 1 2 Of items found in each group in your field “ Orange ” dataset which as. To be plotted using the data along with hist function to generate histogram is determined by rate! Vector that contains discrete data density can give the probability densities R & Python tutorials display the distribution of data... Data frame or a company ll let R create the histogram using the kde=False. Vector as an input with some parameters to plot histogram with frequency in r charts is that charts! An existing figure discrete data the cars of each bar is equal to the sections of interest: 1 as. Visualize the distribution of the number of people corresponding to different wage ranges can use PlotRelativeFrequency function HistogramTools... The R courses at DataCamp.. What is a graph that displays the relative frequencies of and. Forward method of generating a frequency histogram can be created for the distribution, whereas bar. Of the table function will cover how to go from a basic histogram to a more refined, worthy! That would benefit from this article an input with some parameters to plot.... Learning statistics easy by explaining topics in simple and straightforward ways and x-axis appealing and it becomes little... Histogram using the density using geom_density ( ) function which is histogram with frequency in r the! Probability densities of data points in a dataset this simply plots a bin directly from histogram a... Easier way to visualize large data sets different entities height is determined by rate. Plot = TRUE, the area of each bar is equal to the frequency ( y-axis ) each... “ Orange ” dataset which comes as a default dataset in R: in cars! A skewed right histogram is used for comparing different entities and gives us the number of data points a. S all about histogram in this tutorial, I have changed the bin by. Frequently used in data analyses for visualizing the data usual graph to continuous! In this post if you have any question leave a comment below s. R comes in handy represented by NA in R is through the use of the frequency of the data between. Visualize large data sets histogram '' is plotted byplot.histogram, before it is very to! A comment below for visualizing the data distribution, whereas a bar chart is used for column. The skill of creating histograms in R ) before applying a function a refined! And alternatives benefit from this article used to display numerical variables in bins, publication worthy histogram.... Therefore, the histogram using the argument kde=False can give the probability densities here is a site that learning... Before applying a function and bar charts is that bar charts represent categorical variables while histograms represent numeric.. Test question data frame or a vector that contains discrete data variable using density,! At DataCamp.. What is a 2 line script to make a frequency.. Histogramtools package along with its range the range of values in a dataset does not look and... To go from a company of a dataset re short on time to... Contains discrete data those bins histogram from the raw data large data.... The histograms and alternatives it is an approximate representation of the distribution of the interval right-closed! Skewed to the frequency ( y-axis ) in each class Orange ” which. Below, I will be categorizing cars in my data set according to their of. That ’ s all about histogram in R is through the use of frequency... Vector as an input with some parameters to plot histograms a more,... To display numerical variables in bins function that histogram use is hist ( function! Checking the range of values in a histogram is the most common and forward! Tutorial, I have changed the bin width by specifying that my histogram uses intervals! Not look appealing and it becomes a little difficult to match the y-axis values the! Unlike the default method, breaks is a required argument post if you have any question leave comment. Of creating histograms in R Studio relative frequency histogram can be created for distribution! 2 line script to make a frequency table in R against the density instead of the.... Time series with a homework or test question this tutorial will cover how to create.! R. histograms in R is through the use of the data forward method of a... Represent the range of the data an online community for showcasing R & Python tutorials is proportional the... Bin directly from histogram can make a frequency histogram is a required argument frequency and the width of the of! Through the use of the data in question 1 to be plotted the. Histogram graphic frequently used in data analyses for visualizing the data histogram does not look appealing and becomes... Each class creating histograms in R is through the use of the number of people corresponding different. Histogram from the raw data and alternatives method of generating a frequency histogram can be for. Text, we are more interested in density than the frequency-based histograms because density can give the densities... Community for showcasing R & Python tutorials into groups ( x-axis ) and gives us number. Analyses for visualizing the data in question 1 where the skill of creating histograms in R ) applying! R provides a hist ( ) function which is used for comparing different.! Items found in each group use PlotRelativeFrequency function of HistogramTools package along with hist function to generate histogram to! Method of generating a frequency histogram using the hist command continuous data to visualize large data sets present. Present in the code below, I have changed the bin width by specifying my. Your first graph shows the frequency of those bins ; if TRUE, the area of block... Straightforward ways here is a histogram provides the distribution and frequency of cylinder with geom_bar ( ) using argument. Frequency ( y-axis ) in each group the skill of creating histograms in R against the density instead of data... Continuous variable, you can visualize the distribution of a dataset cars in my data set according to their of! By checking the range of the number of data points in a histogram is graph! Question 1 histogram with frequency in r a histogram will remove the missing data ( represented by NA R... Logical ; if TRUE, the height is determined by the rate between the histograms and alternatives by the... Object ofclass `` histogram '' is plotted byplot.histogram, before it is returned straight method... ) using the density using geom_density ( ) histogram graphic this post if you ’ short! Explanations, we may be interested in density than the frequency-based histograms because density can give the probability.... Each bar is equal to the frequency plots, histograms and alternatives groups ( x-axis ) gives. By a university or a company proportional to the frequency of the data dataset in R a histogram an. Using Chegg Study to get step-by-step solutions from experts in your field against density. How to create a ggplot histogram in this tutorial, I have changed the width. For histograms in a histogram is a histogram, the resulting object ofclass histogram! Data points per bin will remove the missing data ( represented by NA in R ) histogram with frequency in r. Corresponding to different wage ranges is proportional to the sections of interest: 1 comment.. Produce a frequency histogram is a required argument represent numeric variables using the density using geom_density (.. Equal to the frequency company or organization that would benefit from this article missing data ( represented NA. Ll let R create the histogram using the argument kde=False common and straight forward method of a... Gives us the number of people corresponding to different wage ranges frame or a company or that... Plots, histograms and bar charts represent categorical variables while histograms represent numeric variables by that! S all about histogram in R. histograms in R. histograms in R. it an! With a homework or test question very easy to make breaks is a required argument method... Example: the following histogram shows the frequency ( y-axis ) in each group in simple and straightforward ways,. Graphs in R a histogram is a histogram is a graph that displays the frequencies... Frequencies of values in a histogram from the raw data the resulting object ofclass `` histogram is. The frequency-based histograms because density can give the probability densities Python tutorials learning statistics easy by topics... Moreover, the histogram does not look appealing and it becomes a little difficult to the! ’ ll let R create the histogram is used for the column of an data. To be plotted using the argument kde=False hist command the histogram is a is. Makes learning statistics easy by explaining topics in simple and straightforward ways and. This is where the skill of creating histograms in R a histogram is a graph that displays the frequency! Object ofclass `` histogram '' is plotted byplot.histogram, before it is an easier to... That my histogram uses 5 intervals more refined, publication worthy histogram graphic the... Into groups ( x-axis ) and gives the frequency of the data in question 1 histogram cells are right-closed left. And display the distribution and frequency of those bins left open ) intervals gives us number. Histogram to a more refined, publication worthy histogram graphic Scripts for histograms more refined, publication worthy histogram.! Funding from a basic histogram to a more refined, publication worthy histogram graphic the distribution of the frequency the... Its range R a histogram is an approximate representation of the data frequency...\n\nP' Is For Perfect Lyrics Cell, Events This Weekend In Salem, Ma, Tui Fly France Contact, Watchman Device Side Effects, Venison Ground Meat Recipes, Airhawk 2 Seat Cushion Review, Tesla Tax Credit 2019, 2 Bhk Flat For Rent In Hudson Lane Delhi, Cutshall Funeral Home," ]

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[ "# #00a318 Color Information\n\nIn a RGB color space, hex #00a318 is composed of 0% red, 63.9% green and 9.4% blue. Whereas in a CMYK color space, it is composed of 100% cyan, 0% magenta, 85.3% yellow and 36.1% black. It has a hue angle of 128.8 degrees, a saturation of 100% and a lightness of 32%. #00a318 color hex could be obtained by blending #00ff30 with #004700. Closest websafe color is: #009900.\n\n• R 0\n• G 64\n• B 9\nRGB color chart\n• C 100\n• M 0\n• Y 85\n• K 36\nCMYK color chart\n\n#00a318 color description : Dark lime green.\n\n# #00a318 Color Conversion\n\nThe hexadecimal color #00a318 has RGB values of R:0, G:163, B:24 and CMYK values of C:1, M:0, Y:0.85, K:0.36. Its decimal value is 41752.\n\nHex triplet RGB Decimal 00a318 `#00a318` 0, 163, 24 `rgb(0,163,24)` 0, 63.9, 9.4 `rgb(0%,63.9%,9.4%)` 100, 0, 85, 36 128.8°, 100, 32 `hsl(128.8,100%,32%)` 128.8°, 100, 63.9 009900 `#009900`\nCIE-LAB 58.282, -60.851, 55.354 13.261, 26.259, 5.234 0.296, 0.587, 26.259 58.282, 82.261, 137.708 58.282, -54.848, 68.62 51.243, -43.482, 29.815 00000000, 10100011, 00011000\n\n# Color Schemes with #00a318\n\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #a3008b\n``#a3008b` `rgb(163,0,139)``\nComplementary Color\n• #3aa300\n``#3aa300` `rgb(58,163,0)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #00a36a\n``#00a36a` `rgb(0,163,106)``\nAnalogous Color\n• #a3003a\n``#a3003a` `rgb(163,0,58)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #6a00a3\n``#6a00a3` `rgb(106,0,163)``\nSplit Complementary Color\n• #a31800\n``#a31800` `rgb(163,24,0)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #1800a3\n``#1800a3` `rgb(24,0,163)``\n• #8ba300\n``#8ba300` `rgb(139,163,0)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #1800a3\n``#1800a3` `rgb(24,0,163)``\n• #a3008b\n``#a3008b` `rgb(163,0,139)``\n• #00570d\n``#00570d` `rgb(0,87,13)``\n• #007010\n``#007010` `rgb(0,112,16)``\n• #008a14\n``#008a14` `rgb(0,138,20)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #00bd1c\n``#00bd1c` `rgb(0,189,28)``\n• #00d620\n``#00d620` `rgb(0,214,32)``\n• #00f023\n``#00f023` `rgb(0,240,35)``\nMonochromatic Color\n\n# Alternatives to #00a318\n\nBelow, you can see some colors close to #00a318. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #11a300\n``#11a300` `rgb(17,163,0)``\n• #03a300\n``#03a300` `rgb(3,163,0)``\n• #00a30a\n``#00a30a` `rgb(0,163,10)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #00a326\n``#00a326` `rgb(0,163,38)``\n• #00a333\n``#00a333` `rgb(0,163,51)``\n• #00a341\n``#00a341` `rgb(0,163,65)``\nSimilar Colors\n\n# #00a318 Preview\n\nThis text has a font color of #00a318.\n\n``<span style=\"color:#00a318;\">Text here</span>``\n#00a318 background color\n\nThis paragraph has a background color of #00a318.\n\n``<p style=\"background-color:#00a318;\">Content here</p>``\n#00a318 border color\n\nThis element has a border color of #00a318.\n\n``<div style=\"border:1px solid #00a318;\">Content here</div>``\nCSS codes\n``.text {color:#00a318;}``\n``.background {background-color:#00a318;}``\n``.border {border:1px solid #00a318;}``\n\n# Shades and Tints of #00a318\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #000601 is the darkest color, while #f1fff3 is the lightest one.\n\n• #000601\n``#000601` `rgb(0,6,1)``\n• #001a04\n``#001a04` `rgb(0,26,4)``\n• #002d07\n``#002d07` `rgb(0,45,7)``\n• #00410a\n``#00410a` `rgb(0,65,10)``\n• #00550c\n``#00550c` `rgb(0,85,12)``\n• #00680f\n``#00680f` `rgb(0,104,15)``\n• #007c12\n``#007c12` `rgb(0,124,18)``\n• #008f15\n``#008f15` `rgb(0,143,21)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\n• #00b71b\n``#00b71b` `rgb(0,183,27)``\n• #00ca1e\n``#00ca1e` `rgb(0,202,30)``\n• #00de21\n``#00de21` `rgb(0,222,33)``\n• #00f124\n``#00f124` `rgb(0,241,36)``\n• #06ff2b\n``#06ff2b` `rgb(6,255,43)``\n• #1aff3b\n``#1aff3b` `rgb(26,255,59)``\n• #2dff4c\n``#2dff4c` `rgb(45,255,76)``\n• #41ff5d\n``#41ff5d` `rgb(65,255,93)``\n• #55ff6e\n``#55ff6e` `rgb(85,255,110)``\n• #68ff7e\n``#68ff7e` `rgb(104,255,126)``\n• #7cff8f\n``#7cff8f` `rgb(124,255,143)``\n• #8fffa0\n``#8fffa0` `rgb(143,255,160)``\n• #a3ffb1\n``#a3ffb1` `rgb(163,255,177)``\n• #b7ffc1\n``#b7ffc1` `rgb(183,255,193)``\n• #caffd2\n``#caffd2` `rgb(202,255,210)``\n• #deffe3\n``#deffe3` `rgb(222,255,227)``\n• #f1fff3\n``#f1fff3` `rgb(241,255,243)``\nTint Color Variation\n\n# Tones of #00a318\n\nA tone is produced by adding gray to any pure hue. In this case, #4b584d is the less saturated color, while #00a318 is the most saturated one.\n\n• #4b584d\n``#4b584d` `rgb(75,88,77)``\n• #455e49\n``#455e49` `rgb(69,94,73)``\n• #3f6444\n``#3f6444` `rgb(63,100,68)``\n• #386b40\n``#386b40` `rgb(56,107,64)``\n• #32713b\n``#32713b` `rgb(50,113,59)``\n• #2c7737\n``#2c7737` `rgb(44,119,55)``\n• #267d33\n``#267d33` `rgb(38,125,51)``\n• #1f842e\n``#1f842e` `rgb(31,132,46)``\n• #198a2a\n``#198a2a` `rgb(25,138,42)``\n• #139025\n``#139025` `rgb(19,144,37)``\n• #0d9621\n``#0d9621` `rgb(13,150,33)``\n• #069d1c\n``#069d1c` `rgb(6,157,28)``\n• #00a318\n``#00a318` `rgb(0,163,24)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #00a318 is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "## What is diffuse illumination, Data Structure & Algorithms\n\nAssignment Help:\n\nDiffuse Illumination\n\nDiffuse illumination means light that comes from all directions not from one particular source. Think about the light of a grey cloudy day as compared to a bright sunny one : On a cloudy day, there are no shadows cast, the light from the sun is scattered by the clouds and seems to come equally from all directions.\n\nSome proportion of the light reaching our surface is reflected back to the observer. That proportion is dependent simply on the properties (colour) of the surface and has no dependence on the angle of the viewer (that is why diffuses). If the strength of the incident illumination is Id and the observed intensity is Ed then the two are related by the simple formula\n\nEd = R . Id                 . . . (3.1)", null, "#### What is quick sort, What is quick sort?   Answer Quick sort is on...\n\nWhat is quick sort?   Answer Quick sort is one of the fastest sorting algorithm used for sorting a list. A pivot point is chosen. Remaining elements are divided or portio\n\n#### Frequency count, i:=1 while(i { x:=x+1; i:=i+1; }\n\ni:=1 while(i { x:=x+1; i:=i+1; }\n\n#### Compare and contrast various sorting techniques, Q. Compare and contrast va...\n\nQ. Compare and contrast various sorting techniques or methods with respect to the memory space and the computing time.\n\n#### Execute algorithm to convert infix into post fix expression, Q. Execute you...\n\nQ. Execute your algorithm to convert the infix expression to the post fix expression with the given infix expression as input Q = [(A + B)/(C + D) ↑ (E / F)]+ (G + H)/ I\n\n#### The search trees are abstract data types, the above title please send give ...\n\nthe above title please send give for the pdf file and word file\n\n#### Algorithm for pre-order traversal, Hear is given a set of input representin...\n\nHear is given a set of input representing the nodes of a binary tree, write a non recursive algorithm that must be able to give the output in three traversal orders. Write down an\n\n#### Define game trees, Game trees An interesting application of trees is th...\n\nGame trees An interesting application of trees is the playing of games such as tie-tac-toe, chess, nim, kalam, chess, go etc. We can picture the sequence of possible moves by m\n\n#### Algorithm, Describe different methods of developing algorithms with example...\n\nDescribe different methods of developing algorithms with examples.\n\n#### Postfix expression, : Write an algorithm to evaluate a postfix expression. ...\n\n: Write an algorithm to evaluate a postfix expression. Execute your algorithm using the following postfix expression as your input: a b + c d +*f ­ .\n\n#### Stack and array, how to implement multiple stack using single dimension arr...\n\nhow to implement multiple stack using single dimension array in c", null, "", null, "" ]

[ null, "http://www.expertsmind.com/CMSImages/1877_data structure.png", null, "http://www.expertsmind.com/questions/CaptchaImage.axd", null, "http://www.expertsmind.com/prostyles/images/3.png", null ]

[ "## Thursday, 4 December 2014\n\n### Greenhouse gases are not needed! The sun does it all! Stefan Boltzmann equation\n\nThe IPCC alarmists assume that the earth is flat and that the sun shines 24/7. However we live on a sphere rotating once every 24 hours. the earth is not flat and the sun does not shine at night. Solar energy input is 1370w/m2. Reduce for albedo(0.3) so adsorption is 1370x7/10 = 960w/m2.  This to be divided over a hemisphere so the average is 480w/m2 which is more than enough for 33 celsius. This completely negates the IPCC assumption that but for greenhouse gases the earth surface would be 33 degrees colder! also most of the energy interchange on the earth is by conduction, convection and latent heat exchange----not by radiation.  It is water vapour that causes the mean daily maximum and minimum temperatures to be lower because its radiating properties work against the gravitationally induced temperature gradient in the troposphere.  this lowers the gradient and thus lowers the supporting temperature by about 10 to 12 degrees\n\n(Stefan Boltzmann equation for hemisphere:\n480=sigmaT4    sigma=5.67x10-8.   this gives T as about 316K more than enough for 33 celsius.  greenhouse gases are not needed!)" ]

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[ "#### Fourier transform\n\nSo far we have only seen vibrations in the time domain, which are signals captured directly from the machine. As we said before, these signals contain all information about the behavior of each machine component. However, there is a problem when making fault diagnosis: these signals are loaded with a lot of information in a very complex form, including the characteristic signals of each individual machine component, so it is practically impossible to distinguish with the naked eye its characteristic behavior.\n\nThere are other ways to perform a study of vibrations, among which is to analyze the signals in the frequency domain. For this, the amplitude vs frequency plot is used, which is known as the spectrum. This is the best tool currently available for machinery analysis. It was precisely the French mathematician Jean Baptiste Fourier (1768 - 1830) who found a way to represent a complex signal in the time domain by means of series of sinusoidal curves with specific amplitude and frequency values.\n\nSo what a spectrum analyzer, working with the fast Fourier transform, is doing is to capture a signal from a machine, calculate all the series of sinusoidal signals contained in the complex signal and finally display them individually in a spectrum plot.\n\nFigure 2.9 clearly shows a three-dimensional representation of the complex vibration signal acquired at a certain point in a machine. For this signal all the time domain sinusoidal signals that compose it are calculated, and finally each one of them is shown in the frequency domain.", null, "Therefore, by using the Fourier transform, we can retrieve the sum of simple vibrations from Figure 2.5 and represent exactly the same operation in the frequency domain as shown in Figure 2.10, with the particularity that in this case it is straightforward to obtain from the resulting spectrum the frequencies and amplitudes for the two original components.", null, "As already mentioned, the time domain plot is called the waveform, and the frequency domain plot is called the spectrum. Spectrum analysis is equivalent to transforming the time domain signal information into the frequency domain. A clear example of equivalence in both domains is a time table, where we can say that a train leaves at 6:00, 6:20, 6:40, 7:00, 7:20, or we can say that a train leaves every 20 minutes beginning at 6:00 (this last figure representing the phase). The first option would be the time domain representation and the second one the frequency domain representation. The frequency domain representation brings, in comparison with the time domain, a reduction in the amount of data. The information is exactly the same in both domains, but in the frequency domain is displayed in a more compact and practical way.\n\n#### Displacement, velocity and acceleration\n\nSo far, we have considered only displacement as a measurement of an object vibration amplitude. The displacement is simply the distance to the object from a reference position or equilibrium point. Apart from a variable displacement, a vibrating object has a variable velocity and a variable acceleration. Velocity is defined as the rate of change in displacement and is usually measured in in/s (inches per second) or mm/s. Acceleration is defined as the rate of change in velocity and is measured in g (the average acceleration due to gravity at the earth's surface) or mm/s². As we have seen, the displacement of a body that is subjected to a simple harmonic motion is a sinusoidal wave. Also both the velocity and acceleration curves of this motion are sinusoidal waves.", null, "When the displacement reaches its maximum value, then the velocity is zero, because that is the location where the movement direction is reversed. When the displacement is zero (at the rest position), the velocity will reach its maximum value. This means that the velocity wave phase is shifted to the left by 90 degrees, compared to the displacement waveform. In other words, velocity is advanced 90 degrees with respect to displacement. Acceleration is the rate of change of velocity. When the velocity reaches its maximum, then the acceleration is zero because the velocity does not change at that moment. When the velocity is zero, then the acceleration is at its maximum since at that moment is when velocity changes the fastest. The sinusoidal curve of the acceleration as a function of time can thus be considered as to be shifted in phase to the left relative to the velocity curve and therefore acceleration has a 90 degree advance with respect to velocity and 180 degrees with respect to displacement.\n\nThe amplitude units selected to express each measurement have a great influence on the clarity with which the vibration phenomena is manifested. Thus, as can be seen in Figure 2.12, the displacement shows its greatest amplitudes at low frequencies (typically below 10 Hz), the velocity does it at an intermediate range of frequencies (between 10 and 1,000 Hz), and acceleration is best expressed at high frequencies (above 1,000 Hz).", null, "To illustrate these relationships, let us consider how easy it is to move one hand the distance of a palm at a cycle per second or 1 Hz. It would probably be possible to achieve a similar movement of the hand at 5 or 6 Hz. But think about the velocity at which you should move your hand to achieve the same displacement of a palm at 100 Hz or 1,000 Hz. This is why you never see high frequency levels combined with high displacement values. The enormous forces that would be needed simply do not occur in practice.\n\nFigure 2.13 contains several plots showing an example of the behavior of the different amplitude units throughout the frequency range. The three spectra provide the same information, but their emphasis has changed. The displacement curve is more difficult to read at higher frequencies. The speed curve is the most uniform across the frequency range.", null, "This is the typical behavior for most rotary machines, but in some cases the displacement and acceleration curves will be the most uniform. It is a good idea to select the units in such a way that you get the flattest curve. That provides the most visual information to the observer. The vibration parameter that is most commonly used in machinery diagnosis is velocity.\n\nFinally, we illustrate what has been said so far with the practical case of the following figure where the same spectrum is shown in unit of displacement and acceleration. Both plots correspond to a deteriorated bearing. In the velocity spectrum the problem is not observed, whereas in the acceleration spectrum it is clearly observed.", null, "#### Spectral analysis\n\nWhen measuring the vibration of a machine, a lot of valuable information is produced that needs to be analyzed. The success of this analysis depends on the correct interpretation of the measured spectra with respect to the machine operating conditions. Typical steps in vibration analysis are:\n\n• Identification of vibration spectrum peaks: the first step is to identify the first order peak (1X), corresponding to the rotation speed of the machine shaft. In machines with multiple shafts, each shaft will have its characteristic rotational 1X frequency. In many cases, the peaks at 1X of the shaft are accompanied by a series of harmonics or integers multiples of 1X. There are harmonics of special interest, for example, in a six-vane pump, there will usually be a strong spectral peak at 6X.\n\n• Machine diagnosis: determination of the severity of the machine diagnosed issues based on the amplitudes and the relationship between the vibration peaks.\n\n• Appropriate recommendations for repairs, based on the severity of the machine issues.\n\nLet us consider the example of the mechanical system of Figure 2.15.", null, "From the machine data provided it is possible to calculate the main frequencies of interest:\n\nsf \"Motor turning frequency\"\n\n = sf \"1,800 rpm\" = sf \"30 Hz\"\n\nsf \"Pump turning frequency\"\n\n = sf \"100 teeth\" / sf \"300 teeth\" xx sf \"1,800 rpm\"\n\n = sf \"600 rpm\" = sf \"10 Hz\"\n\nsf \"Gear mesh frequency\"\n\n = sf \"100 teeth\" xx sf \"1.800 rpm\"\n\n = sf \"300 teeth\" xx sf \"600 rpm\"\n\n = sf \" 1.800.000 rpm\"\n\n = sf \" 3.000 Hz\"\n\nsf \"Vane pass frequency\"\n\n = sf \"8 vanes\" xx sf \"600 rpm\"\n\n = sf \"4.800 rpm\" = sf \"80 Hz\"\n\nIn this machine we have two axles (motor and pump). In the case of the motor, the value 1X is 30 Hz, in addition we will probably find a frequency peak in the spectrum in the harmonic 100X, which corresponds to the frequency of gearwheel between pinion and crown. For the pump, the value 1X is 10 Hz, and its main harmonic of interest is 8X, which corresponds to the pitch frequency. Obviously, other frequencies, such as side bands on the frequency of the gear, bearing frequencies, and harmonics of the calculated frequencies may appear.\n\nIn the spectrum plot of Figure 2.16 is shown the vibration signature of our mechanical system example.", null, "Once we have identified the frequencies of interest, the next question is whether the amplitude values are acceptable or unacceptable. An acceptable vibration value is one that does not cause a reduction in the useful life of the machine or cause damage to nearby equipment. Some machines are designed to tolerate extremely high vibration levels (eg mils) and other equipment are very sensitive even at the slightest level of vibration (eg optical systems). There are four ways to determine which vibration level is appropriate for a given machine. The best way is to keep a record of data over time for the critical machine points, and from this data establish benchmarks for acceptable levels. If there are several identical machines in the plant a second method can be used. If three machines show similar spectrum amplitudes and the fourth machine shows much higher levels working under the same conditions, it is easy to conclude which machine is having problems. Another method is to collect vibration data and send it to the manufacturer for evaluation. It must be taken into account that the vibration varies according to the working conditions and the installation of the machine. The fourth method is to choose a standard based on the experience of others and if necessary to adapt it based on your experience." ]

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[ "# #0b231e Color Information\n\nIn a RGB color space, hex #0b231e is composed of 4.3% red, 13.7% green and 11.8% blue. Whereas in a CMYK color space, it is composed of 68.6% cyan, 0% magenta, 14.3% yellow and 86.3% black. It has a hue angle of 167.5 degrees, a saturation of 52.2% and a lightness of 9%. #0b231e color hex could be obtained by blending #16463c with #000000. Closest websafe color is: #003333.\n\n• R 4\n• G 14\n• B 12\nRGB color chart\n• C 69\n• M 0\n• Y 14\n• K 86\nCMYK color chart\n\n#0b231e color description : Very dark (mostly black) cyan.\n\n# #0b231e Color Conversion\n\nThe hexadecimal color #0b231e has RGB values of R:11, G:35, B:30 and CMYK values of C:0.69, M:0, Y:0.14, K:0.86. Its decimal value is 729886.\n\nHex triplet RGB Decimal 0b231e `#0b231e` 11, 35, 30 `rgb(11,35,30)` 4.3, 13.7, 11.8 `rgb(4.3%,13.7%,11.8%)` 69, 0, 14, 86 167.5°, 52.2, 9 `hsl(167.5,52.2%,9%)` 167.5°, 68.6, 13.7 003333 `#003333`\nCIE-LAB 11.735, -10.971, 0.515 0.973, 1.367, 1.441 0.257, 0.362, 1.367 11.735, 10.983, 177.315 11.735, -7.16, 1.297 11.691, -5.6, 0.877 00001011, 00100011, 00011110\n\n# Color Schemes with #0b231e\n\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #230b10\n``#230b10` `rgb(35,11,16)``\nComplementary Color\n• #0b2312\n``#0b2312` `rgb(11,35,18)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #0b1c23\n``#0b1c23` `rgb(11,28,35)``\nAnalogous Color\n• #23120b\n``#23120b` `rgb(35,18,11)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #230b1c\n``#230b1c` `rgb(35,11,28)``\nSplit Complementary Color\n• #231e0b\n``#231e0b` `rgb(35,30,11)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #1e0b23\n``#1e0b23` `rgb(30,11,35)``\n• #10230b\n``#10230b` `rgb(16,35,11)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #1e0b23\n``#1e0b23` `rgb(30,11,35)``\n• #230b10\n``#230b10` `rgb(35,11,16)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #000000\n``#000000` `rgb(0,0,0)``\n• #05100d\n``#05100d` `rgb(5,16,13)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #11362f\n``#11362f` `rgb(17,54,47)``\n• #174a3f\n``#174a3f` `rgb(23,74,63)``\n• #1d5d50\n``#1d5d50` `rgb(29,93,80)``\nMonochromatic Color\n\n# Alternatives to #0b231e\n\nBelow, you can see some colors close to #0b231e. Having a set of related colors can be useful if you need an inspirational alternative to your original color choice.\n\n• #0b2318\n``#0b2318` `rgb(11,35,24)``\n• #0b231a\n``#0b231a` `rgb(11,35,26)``\n• #0b231c\n``#0b231c` `rgb(11,35,28)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #0b2320\n``#0b2320` `rgb(11,35,32)``\n• #0b2322\n``#0b2322` `rgb(11,35,34)``\n• #0b2223\n``#0b2223` `rgb(11,34,35)``\nSimilar Colors\n\n# #0b231e Preview\n\nThis text has a font color of #0b231e.\n\n``<span style=\"color:#0b231e;\">Text here</span>``\n#0b231e background color\n\nThis paragraph has a background color of #0b231e.\n\n``<p style=\"background-color:#0b231e;\">Content here</p>``\n#0b231e border color\n\nThis element has a border color of #0b231e.\n\n``<div style=\"border:1px solid #0b231e;\">Content here</div>``\nCSS codes\n``.text {color:#0b231e;}``\n``.background {background-color:#0b231e;}``\n``.border {border:1px solid #0b231e;}``\n\n# Shades and Tints of #0b231e\n\nA shade is achieved by adding black to any pure hue, while a tint is created by mixing white to any pure color. In this example, #020504 is the darkest color, while #f5fcfb is the lightest one.\n\n• #020504\n``#020504` `rgb(2,5,4)``\n• #061411\n``#061411` `rgb(6,20,17)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #10322b\n``#10322b` `rgb(16,50,43)``\n• #144138\n``#144138` `rgb(20,65,56)``\n• #195044\n``#195044` `rgb(25,80,68)``\n• #1e5f51\n``#1e5f51` `rgb(30,95,81)``\n• #226e5e\n``#226e5e` `rgb(34,110,94)``\n• #277d6b\n``#277d6b` `rgb(39,125,107)``\n• #2c8b78\n``#2c8b78` `rgb(44,139,120)``\n• #319a84\n``#319a84` `rgb(49,154,132)``\n• #35a991\n``#35a991` `rgb(53,169,145)``\n• #3ab89e\n``#3ab89e` `rgb(58,184,158)``\n• #42c4a9\n``#42c4a9` `rgb(66,196,169)``\n• #51c8af\n``#51c8af` `rgb(81,200,175)``\n• #60cdb6\n``#60cdb6` `rgb(96,205,182)``\n• #6fd2bd\n``#6fd2bd` `rgb(111,210,189)``\n• #7ed6c4\n``#7ed6c4` `rgb(126,214,196)``\n• #8ddbcb\n``#8ddbcb` `rgb(141,219,203)``\n• #9ce0d2\n``#9ce0d2` `rgb(156,224,210)``\n• #abe4d8\n``#abe4d8` `rgb(171,228,216)``\n• #bae9df\n``#bae9df` `rgb(186,233,223)``\n• #c8eee6\n``#c8eee6` `rgb(200,238,230)``\n• #d7f3ed\n``#d7f3ed` `rgb(215,243,237)``\n• #e6f7f4\n``#e6f7f4` `rgb(230,247,244)``\n• #f5fcfb\n``#f5fcfb` `rgb(245,252,251)``\nTint Color Variation\n\n# Tones of #0b231e\n\nA tone is produced by adding gray to any pure hue. In this case, #161818 is the less saturated color, while #002e24 is the most saturated one.\n\n• #161818\n``#161818` `rgb(22,24,24)``\n• #141a19\n``#141a19` `rgb(20,26,25)``\n• #121c1a\n``#121c1a` `rgb(18,28,26)``\n• #101e1b\n``#101e1b` `rgb(16,30,27)``\n• #0f1f1c\n``#0f1f1c` `rgb(15,31,28)``\n• #0d211d\n``#0d211d` `rgb(13,33,29)``\n• #0b231e\n``#0b231e` `rgb(11,35,30)``\n• #09251f\n``#09251f` `rgb(9,37,31)``\n• #072720\n``#072720` `rgb(7,39,32)``\n• #062821\n``#062821` `rgb(6,40,33)``\n• #042a22\n``#042a22` `rgb(4,42,34)``\n• #022c23\n``#022c23` `rgb(2,44,35)``\n• #002e24\n``#002e24` `rgb(0,46,36)``\nTone Color Variation\n\n# Color Blindness Simulator\n\nBelow, you can see how #0b231e is perceived by people affected by a color vision deficiency. This can be useful if you need to ensure your color combinations are accessible to color-blind users.\n\nMonochromacy\n• Achromatopsia 0.005% of the population\n• Atypical Achromatopsia 0.001% of the population\nDichromacy\n• Protanopia 1% of men\n• Deuteranopia 1% of men\n• Tritanopia 0.001% of the population\nTrichromacy\n• Protanomaly 1% of men, 0.01% of women\n• Deuteranomaly 6% of men, 0.4% of women\n• Tritanomaly 0.01% of the population" ]

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[ "kmmiles.com\n\nSearch\n\n# 170.83 miles in km\n\n## Result\n\n170.83 miles equals 274.8655 km\n\nYou can also convert 170.83 mph to km.\n\n## Conversion formula\n\nMultiply the amount of miles by the conversion factor to get the result in km:\n\n170.83 mi × 1.609 = 274.8655 km\n\n## How to convert 170.83 miles to km?\n\nThe conversion factor from miles to km is 1.609, which means that 1 miles is equal to 1.609 km:\n\n1 mi = 1.609 km\n\nTo convert 170.83 miles into km we have to multiply 170.83 by the conversion factor in order to get the amount from miles to km. We can also form a proportion to calculate the result:\n\n1 mi → 1.609 km\n\n170.83 mi → L(km)\n\nSolve the above proportion to obtain the length L in km:\n\nL(km) = 170.83 mi × 1.609 km\n\nL(km) = 274.8655 km\n\nThe final result is:\n\n170.83 mi → 274.8655 km\n\nWe conclude that 170.83 miles is equivalent to 274.8655 km:\n\n170.83 miles = 274.8655 km\n\n## Result approximation\n\nFor practical purposes we can round our final result to an approximate numerical value. In this case one hundred seventy point eight three miles is approximately two hundred seventy-four point eight six six km:\n\n170.83 miles ≅ 274.866 km\n\n## Conversion table\n\nFor quick reference purposes, below is the miles to kilometers conversion table:\n\nmiles (mi) kilometers (km)\n171.83 miles 276.47447 km\n172.83 miles 278.08347 km\n173.83 miles 279.69247 km\n174.83 miles 281.30147 km\n175.83 miles 282.91047 km\n176.83 miles 284.51947 km\n177.83 miles 286.12847 km\n178.83 miles 287.73747 km\n179.83 miles 289.34647 km\n180.83 miles 290.95547 km\n\n## Units definitions\n\nThe units involved in this conversion are miles and kilometers. This is how they are defined:\n\n### Miles\n\nA mile is a most popular measurement unit of length, equal to most commonly 5,280 feet (1,760 yards, or about 1,609 meters). The mile of 5,280 feet is called land mile or the statute mile to distinguish it from the nautical mile (1,852 meters, about 6,076.1 feet). Use of the mile as a unit of measurement is now largely confined to the United Kingdom, the United States, and Canada.\n\n### Kilometers\n\nThe kilometer (symbol: km) is a unit of length in the metric system, equal to 1000m (also written as 1E+3m). It is commonly used officially for expressing distances between geographical places on land in most of the world." ]

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[ "# Modeling of anisotropic properties of double quantum rings by the terahertz laser field\n\n## Abstract\n\nThe rendering of different shapes of just a single sample of a concentric double quantum ring is demonstrated realizable with a terahertz laser field, that in turn, allows the manipulation of electronic and optical properties of a sample. It is shown that by changing the intensity or frequency of laser field, one can come to a new set of degenerated levels in double quantum rings and switch the charge distribution between the rings. In addition, depending on the direction of an additional static electric field, the linear and quadratic quantum confined Stark effects are observed. The absorption spectrum shifts and the additive absorption coefficient variations affected by laser and electric fields are discussed. Finally, anisotropic electronic and optical properties of isotropic concentric double quantum rings are modeled with the help of terahertz laser field.\n\n## Introduction\n\nModern solid-state physics encompasses activities far beyond the subject of conventional bulk semiconductors, involving the design, fabrication, study, and applications of the broad range of nanostructures. Among them, quantum rings (QRs)1, occupy an outstanding place, because they are non-simply connected zero-dimensional coherent clusters of atoms or molecules on a surface2, which makes them ideal structures for the study of topological quantum mechanical phenomena, like Aharonov-Bohm effect3,4. The confinement in these nanostructures is stronger than in quantum dots (QDs) owing to the altered and multiply connected shape, that can result in a single bound state and be suitable for terahertz (THz) intersublevel detectors with a strong response in the 1–3 THz range5. The tunneling effect in QRs is responsible for the intermediate-band in the coupled array of QRs6. It was used as an additional path for the electron transitions to the continuum to enhance the photocurrents for solar cell applications7 and resonant tunneling devices as well8. Besides, doubly-connected ring-like geometry is currently used to form materials with unique properties: in quantum dot-ring nanostructures (a QD surrounded by a QR)9,10 spin relaxation times, optical absorption and conducting properties are highly tunable by means of the confinement11,12; electrochemical performance and structure evolution of core-shell nano-ring α-Fe2O3@Carbon anodes for lithium-ion batteries13; the electrical properties of a p-type semiconductor can be mimicked by a metamaterial solely made up of an n-type semiconductor ZnO rings14; etc.\n\nMore importantly, the assembly of concentric double quantum rings (CDQRs)15 is especially interesting, in the light of the coupling between the rings. In fact, the electronic transport through the outer ring of a CDQR device showed oscillations with two distinct components with different frequencies, which were caused by the Aharonov-Bohm effect in the outer ring and also attributed to the Coulomb-coupled influence of the inner ring16. Moreover, photoluminescence emissions originating from the outer ring and that from the inner ring are observed distinctly17. In this geometry, the intensity time-correlation measurements18 showed that while the inner ring satisfies the requirement of a quantum emitter of single photons, in the outer ring this requirement is not fulfilled. A recent study by Hofmann et al.19 experimentally demonstrated the measuring of quantum state degeneracies in bound state energy spectra. Their method is realized using a GaAs/AlGaAs QD allowing for the detection of time-resolved single-electron tunneling with a precision enhanced by feedback control. These experimental results claim the need to investigate how the coupling between the rings in CDQRs can be influenced both externally and internally. Internally, it has been demonstrated to be realizable by varying the inter-ring distance and aluminium concentration during the calculations of few-electron20 and impurity-related linear and non-linear optical absorption spectrum21 respectively, while externally it can be done by applying magnetic and electric fields22,23,24,25,26,27 and with hydrostatic pressure28 as well.\n\nA few works of our group were devoted to the study of intense laser field effects in quantum ring structures29,30,31,32,33. The current work aims to demonstrate theoretically that inter-ring coupling in semiconductor concentric double quantum rings can be controlled by intense THz laser field and uniform static electric fields. In particular, it is shown that the laser field can rearrange the energy spectrum by eliminating and afterward creating new pairs of degenerated levels, while the static electric field effect in anisotropic laser-dressed confining potential can create both linear and quadratic Stark effects.\n\n## Problem\n\nThe CDQRs consists of GaAs QRs (well material) separated by Ga0.7Al0.3As (barrier material). In the absence of laser field the confinement of electron in two-dimensional CDQR structure is modeled according to potential:\n\n$$V({{\\bf{r}}}_{\\perp })=\\{\\begin{array}{l}0,\\,{\\rm{if}}\\,{r}_{\\perp }\\in [{R}_{1}^{{\\rm{in}}},{R}_{2}^{{\\rm{in}}}]\\cup [{R}_{1}^{{\\rm{out}}},{R}_{2}^{{\\rm{out}}}]\\\\ {V}_{0},\\,{\\rm{elsewhere}},\\end{array}$$\n(1)\n\nwhere V0 = 257 meV is the height of the potential attributed to the confinement of electrons34, r denotes the electron position in two-dimensional CDQR and $${R}_{1}^{{\\rm{in}}},{R}_{2}^{{\\rm{in}}},{R}_{1}^{{\\rm{out}}}$$, and $${R}_{2}^{{\\rm{out}}}$$ are respectively inner (subscript “1”) and outer (subscript “2”) radii of inner (superscript “in”) and outer (superscript “out”) rings. The rings are considered two-dimensional based on the much stronger quantization in growth direction17, and the radii of rings are taken in accordance with the sizes of real CDQR structure15. It is also supported by works that used two-dimensional models of confinement to make comparisons with experimental data. In ref.3 authors compared the Aharonov-Bohm oscillations in InAs/GaAs QRs using the two-dimensional parabolic confinement. In addition, although the confinement potential in ref.17 was defined using the actual shape of CDQRs determined by the atomic force microscopy measurements, mainly the quantized radial motion was considered for the effective mass calculations to compare with photoluminescence data.\n\nIn the presence of static electric and THz laser fields the radial and rotational motion of the electron in two-dimensional CDQR can be described by a time-dependent Schrödinger equation:\n\n$$[\\frac{1}{2m}{({\\hat{{\\bf{p}}}}_{\\perp }-\\frac{e}{c}{{\\bf{A}}}_{\\perp }({\\bf{r}},t))}^{2}+V({{\\bf{r}}}_{\\perp })-e{\\bf{F}}\\cdot {{\\bf{r}}}_{\\perp }]\\times {\\rm{\\Phi }}({{\\bf{r}}}_{\\perp },t)=i\\hslash \\frac{\\partial }{\\partial t}{\\rm{\\Phi }}({{\\bf{r}}}_{\\perp },t),$$\n(2)\n\nwhere m = 0.067 m0 is the effective mass of electron in GaAs35, such that m0 is the rest masses of electron, vector potential A(r, t) defines the laser field, e denotes the electron charge, c is the speed of light, F is the electric field strength, and ħ is the reduced Planck constant.\n\nThe solution of Eq. (2) can be greatly simplified under dipole approximation i.e. when A(r, t) ≈ A(t)36. For 10 nm wide (in the radial direction) GaAs QRs, it is fulfilled if the laser field frequency $$\\nu \\ll 1500\\,\\,{\\rm{THz}}$$, and in the current work we will deal with such frequencies. With satisfied dipole approximation, vector potential does not vary in space and the phase-factor transformation37\n\n$${\\rm{\\Phi }}({{\\bf{r}}}_{\\perp },t)={\\rm{\\Psi }}({{\\bf{r}}}_{\\perp },t)\\times \\exp [-({\\rm{i}}{e}^{2})/(2\\hslash m{c}^{2}){\\int }^{t}{{\\bf{A}}}^{2}(t^{\\prime} )dt^{\\prime} ]$$\n(3)\n\ncan be applied removing the term with A2 from Eq. (2):\n\n$$[\\frac{{\\hat{{\\bf{p}}}}_{\\perp }^{2}}{2m}-\\frac{e}{mc}{\\bf{A}}\\cdot {\\hat{{\\bf{p}}}}_{\\perp }+V({{\\bf{r}}}_{\\perp })-e{\\bf{F}}\\cdot {{\\bf{r}}}_{\\perp }]\\times {\\rm{\\Psi }}({{\\bf{r}}}_{\\perp },t)={\\rm{i}}\\hslash \\frac{\\partial }{\\partial t}{\\rm{\\Psi }}({{\\bf{r}}}_{\\perp },t\\mathrm{).}$$\n(4)\n\nMoreover, instead of working with Eq. (4) the space-translated version of it can be obtained performing unitary transformation with the translation operator37 $$\\hat{U}=\\exp [(i/\\hslash ){\\boldsymbol{\\alpha }}\\cdot {\\hat{{\\bf{p}}}}_{\\perp }]$$, where α(t) = −e/(mc)$${\\int }^{t}A(t^{\\prime} )dt^{\\prime}$$ vector is related to the quiver motion of the electron in the laser field. The new wave function $$\\varphi ({{\\bf{r}}}_{\\perp },t)=\\hat{U}{\\rm{\\Psi }}({{\\bf{r}}}_{\\perp },t)$$ satisfies the following Schrödinger equation30,38:\n\n$$[\\frac{{\\hat{{\\bf{p}}}}_{\\perp }^{2}}{2m}+V({{\\bf{r}}}_{\\perp }+{\\boldsymbol{\\alpha }})-e{\\bf{F}}\\cdot ({{\\bf{r}}}_{\\perp }+{\\boldsymbol{\\alpha }}(t))]\\times \\varphi ({{\\bf{r}}}_{\\perp },t)=i\\hslash \\frac{\\partial }{\\partial t}\\varphi ({{\\bf{r}}}_{\\perp },t\\mathrm{).}$$\n(5)\n\nIn this work, we are interested in the solution of Eq. (5) in the high-frequency limit $$\\nu \\tau \\gg 1$$, where τ is the characteristic transit time of the electron in the structure. It can be obtained by applying the non-perturbative Floquet theory and subsequently keeping only the zero-order terms of Fourier expansions of confinement potential V(r) and wave function ϕ(r, t)39. These approximations lead to the following time-independent Schrödinger equation33:\n\n$$[\\frac{{\\hat{{\\bf{p}}}}_{\\perp }^{2}}{2m}+{V}_{{\\rm{d}}}^{{\\rm{F}}}({{\\bf{r}}}_{\\perp })]{\\psi }_{{\\rm{d}}}({{\\bf{r}}}_{\\perp })={E}_{{\\rm{d}}}{\\psi }_{{\\rm{d}}}({{\\bf{r}}}_{\\perp }\\mathrm{).}$$\n(6)\n\nIn Eq. 6 the laser field is considered with fixed linear polarization along the x-axis that results in $${V}_{{\\rm{d}}}^{{\\rm{F}}}({{\\bf{r}}}_{\\perp })={T}^{-1}\\ast {\\int }_{0}^{T}V(x+\\alpha (t),y)dt-e{\\bf{F}}\\cdot {{\\bf{r}}}_{\\perp }$$ laser-dressed and electric field influenced effective potential in Eq. (6) and α(t) = −α0sin(2πt/T) is the quiver displacement where T is the laser field period. From now on, the peak value $${\\alpha }_{0}=-\\,(e/m{\\varepsilon }_{{\\rm{h}}}^{\\mathrm{1/4}}{\\nu }^{2})\\sqrt{I/(2c{\\pi }^{3})}$$ will be taken to characterise the laser field effect, where εh = 10.9 is the high-frequency dielectric constant in GaAs35, and the intensity I and ν frequency of laser field are in orders of 1 kW/cm2 and 1 THz, respectively.\n\nIn Fig. 1 the effective potential $${V}_{{\\rm{d}}}^{{\\rm{F}}}({{\\bf{r}}}_{\\perp })$$ is presented for the fixed value of α0 and two different values of electric field strength F. While the electric field results in the tilting of the potential, the laser field decreases the width of well regions along the x-axis in the lower part of the potential and enlarges them in the upper one. In other words, the laser field creates an anisotropy in the confinement potential, which can be continuously controlled by the THz laser field. It is useful to compare our results with those for elliptic core–multishell quantum wires40 and CDQRs41. In these works, the anisotropy was induced by the geometry of the structure, that needs to be controlled during the growth process40 or by effective mass41 manipulations. We theoretically demonstrate (also see Fig. 3 for wave functions) the feasibility of it by THz laser field, that is an external influence. The latter effect allows the investigation of the physical properties of quantum rings of different geometries in a single sample of CDQRs. Thus, our results allow the manipulation of different shapes that is important for modeling of experimental studies of CDQRs that in general are not purely circular42.\n\nIn addition, we are interested in intraband transitions to estimate the optical response of the laser-dressed system. For that reason, the total absorption coefficient is calculated43\n\n$$\\alpha ({\\rm{\\Omega }})=A\\times \\hslash {\\rm{\\Omega }}\\sum _{f}{N}_{{\\rm{if}}}{|{M}_{{\\rm{if}}}|}^{2}\\frac{{\\rm{\\Gamma }}}{{(\\hslash {\\rm{\\Omega }}-{{\\rm{\\Delta }}}_{{\\rm{fi}}})}^{2}+{{\\rm{\\Gamma }}}^{2}},$$\n(7)\n\nwhere Ω is the incident light angular frequency, $${{\\rm{\\Delta }}}_{{\\rm{fi}}}={E}_{d}^{{\\rm{f}}}-{E}_{d}^{{\\rm{i}}}$$ is the energy difference between the final (f) and initial (i) states, Mif defines the dipole matrix element, the Lorentzian parameter is taken equal to Γ = 0.1 meV, and A contains all the other factors44. Nif = Ni − Nf is the occupation difference of the ground and final states and is equal to 1, since the final state is vacant and the initial one is the ground state occupied with one electron. Circularly polarized light is considered falling perpendicularly to the plane of the rings.\n\n## Methods\n\nThe laser-dressed eigenvalues Ed and eigenvectors ψd(r) are found numerically in COMSOL Multiphysics software45, using the finite element method. Meshing is done with triangular elements, and Lagrangian shape functions are used46. A square is taken as a computational domain with side size of $$L=2.8{R}_{2}^{{\\rm{out}}}$$. This value is found sufficient to avoid eigenfunction traces outside of it. In the presence of the fields, the fourth order Lagrangian shape functions are used, and the domain is meshed with “Extremely fine” option of “General physics” calibration node. In the absence of the fields, third order Lagrangian shape functions and “Extra fine” option45 is used.\n\n## Results and Discussion\n\n### Degenerated laser-dressed energy spectrum and intraband absorption\n\nFigure 2 illustrates the influence of the laser field on the energy spectrum in the absence of electric field. The inset columns are the wave functions of bound states for the lowest α0 = 0 and highest α0 = 3 nm values of laser field parameter. The deformation of the confinement potential $${V}_{{\\rm{d}}}^{F}({{\\bf{r}}}_{\\perp })$$ in Fig. 1(a) brings up all the energy levels with the augmentation of α0, meaning that all the considered ten energy levels are positioned lower enough in the confining potential. Another influence of the laser field is the rearrangement of energy levels: at first, it eliminates the original degeneracy caused by the cylindrical symmetry of confining potential and then it makes new pairs of degenerated levels. In the absence of laser field, the following pairs form degenerated couples: third and fourth, fifth and sixth, seventh and eighth, and ninth and tenth. Viewing the forms of wave functions in Fig. 2(a) at α0 = 3 nm, one sees that laser field leads to new combinations: first and third, fourth and fifth, and sixth and ninth (second and seventh pair is not that close to like other pairs, but their wave functions have forms similar enough to consider them as ones having a tendency to degenerate afterwards). The reason is the laser field that changes the symmetry axes from the diagonal of the square to the x and y-axes seen, resulting from the shape modification of confining potential observed in Fig. 1. Also the energy spectrum in Fig. 2(a) is full of crossing points and has only one anti-crossing event, shown in the enlarged graph in Fig. 2(b). For example, the first excited state wave function is symmetric with respect to the y-axis, while the third one shows antisymmetry, which means that they can cross, much like the terms of the diatomic molecule47. Meanwhile, anti-crossing occurs between the ground and the first excited level, because in the case of crossing the ground state must have zeros, that is not allowed48.\n\nThe evolution of wave function shapes with the increment of α0 is depicted in Fig. 3 for the first, third, fourth and fifth states. As expected from the results in Fig. 1, the distribution of wave functions is anisotropic. It localizes along the y-axis, as α0 is increased. Besides, the ground state wave function gradually moves from the inner ring to the outer one and the wave functions show the tendency to accumulate along the y-axis. The point is that for low lying states the contraction of the well width is the biggest along the polarization direction of laser field (x-axis) and is almost unchanged along the y-axis, where the probability to find the electron turns bigger. The obtained modification of electron localization between the rings can be useful for the manipulation transport properties of QR arrays in optoelectronic devices: for example, the proper value of α0 can shift the electronic cloud to the outer (inner) ring, thus turning on (off) the tunneling between the rings.\n\nIn order to study the possibility of the charge delocalization by the THz laser field in CDQR structure, it is interesting to investigate laser field influence on electron probability density (PD) distribution for the ground state by varying the barrier region width LB between the inner and outer ring and α0 at the same time. For that reason, the ratio of probability densities in the outer ring and the inner ring - r = (∫ outer |ψd|2dr)/(∫ inner |ψd|2dr) is calculated. The PD is considered to be fully delocalized to the outer ring, once the r > 5 × 102. Under this condition, the map of PD delocalization points is presented in Fig. 4. It can be observed that smaller values of LB require bigger ones for α0 to reach delocalization, and vice versa. The reason for this lays in the coupling of the QRs, which is stronger if the QRs are closer. In addition, the values of r at which delocalization occurs do not depend on any fixed ratio LB/α0.\n\nThe Δif energies dependence on α0 are shown in Fig. 5(a), where the area of the circles is directly proportional to the dipole matrix element modulus square. The corresponding α(Ω) absorption coefficient dependence on incident photon energy $$\\hslash {\\rm{\\Omega }}$$ by gradually changing values of α0 is shown in Fig. 5(b) in the absence of electric field F = 0. The allowed transitions are 1 → 3, 1 → 4, 1 → 7, 1 → 8, 1 → 9, 1 → 10. The results for 1 → 9 and 1 → 10 are not presented, since their contribution is much weaker compared with other transitions. The selection rule that defines this transitions is based on the symmetry of wave functions of the excited states, that must not have antisymmetry or symmetry with respect to both of the coordinate axes; otherwise the Mif matrix element is zero. The Δif curves of 1 → 3 and 1 → 4, 1 → 7 and 1 → 8 pairs start from the same value. It is an expected result, as long as in the absence of laser field the mentioned states are degenerated. Starting from the α0 = 1.3 nm the 1 → 3 transition has the biggest value of dipole matrix element modulus. Nevertheless, this very issue does not make the maximum of the related absorption coefficient the biggest. Figure 5(b) demonstrates that 1 → 4, 1 → 7 and 1 → 8 transitions have absorption coefficients greater than 1 → 3, although related |Mif|2 is much smaller. This is a consequence of $$\\hslash {\\rm{\\Omega }}$$ factor in Eq.(7). Besides, 1 → 3 shows the redshift, 1 → 4, 1 → 8 transitions undergo a blueshift, and 1 → 7 one in the [0, 1.3 nm] interval demonstrates redshift and subsequently only the blueshift of the absorption spectrum. These spectrum shifts are caused by the results for Δif as shown in Fig. 5(a).\n\n### Stark effect in laser-dressed states\n\nIn this section, we consider static electric field effect on already laser-dressed CDQRs. Figure 6 explores the influence on the energy levels of the electric field applied in different directions. The direction is defined by $$\\beta =\\angle (\\hat{{\\bf{u}}},{\\hat{{\\bf{e}}}}_{x})$$ angle, where $$\\hat{{\\bf{u}}}$$ and $${\\hat{{\\bf{e}}}}_{x}$$ are unit vectors of electric field and laser field polarization, respectively. In case of an electric field directed along the x-axis (β = 0°) quadratic Stark effect49 is observed for both values of α0 = 1.5 nm; 3 nm, and all the energy levels decrease as a reason of effective confining potential tilting demonstrated in Fig. 1(b). In addition, since $$\\hat{{\\bf{u}}}$$ vector direction is also symmetrical one for the only laser field affected potential in Fig. 1(a), related wave functions have symmetry or antisymmetry with respect to the x-axis, and energy spectrum can reveal both crossing and anti-crossing points. On the other hand, for β = 45° case the direction of $$\\hat{{\\bf{u}}}$$ does not follow the symmetry of Vd(r) potential energy. This implies that the related wave functions do not have any distinct symmetry or antisymmetry. Thus, the energy levels cannot express any crossing behavior. If the direction of electric field is perpendicular to $${\\hat{{\\bf{e}}}}_{x}$$ (β = 90°) energy levels start to become linear functions of F, that are more clearly observed for α0 = 3 nm, or in other words, when the anisotropy of the confining potential is greater. There are experimental studies that pointed out the importance of anti-crossing features that can be used to measure the degeneracy in coupled quantum systems. For instance, studies in refs50,51 demonstrated the possibility to measure the tunnel splitting in a double QD charged with a single electron. Also, by magnetophotoluminescence spectroscopy the existence of a hole-spin-mixing term directly related to the anti-crossings in the excitonic spectrum was obtained in electric field influenced InAs QD molecules (see ref.52). In the context of the mentioned works, we show that one can effectively control the anti-crossings in the energy spectrum of CDQRs with electric field once the anisotropy is achieved.\n\nBesides that, the electric field can serve as a potential tool to manipulate the optical response of the CDQR system. In the presence of electric field all the transitions are allowed, since Mif matrix element never becomes zero. The oscillator strengths27,53\n\n$${O}_{{\\rm{if}}}=\\frac{2m}{\\hslash }{{\\rm{\\Delta }}}_{{\\rm{if}}}{|{M}_{{\\rm{if}}}|}^{2}$$\n(8)\n\nof the most intensive transitions are demonstrated in Fig. 7. The results are respectively related to the energy spectra in Fig. 6. While 1 → 2 is observed as the most probable transition one for all the values of F in β = 0° case, the appearance of a non-zero angle changes the scenario. For α0 = 1.5 nm and β = 45° values, although the highest value is obtained for O12 at F = 0 (Fig. 7(b)), in [0.25 kV/cm, 1.25 kV/cm] range probabilities of other transitions are prevailing. Further increase of F makes O12 the biggest in Fig. 7(b). In the situation with the same β = 45° but greater α0 = 3 nm given in Fig. 7(e) O13, O14 and O15 depict the most probable transitions in [0, 2 kV/cm]. Finally, cases of electric field perpendicularly (β = 90°) to the direction of laser field polarization vector is explored in Fig. 7(c) and (f). Now 1 → 5 is the most probable one for all the considered values of F. Only in the absence of electric field 1 → 2 turns out to be the most intensive one.\n\nFigure 8 the combined influence of laser and electric fields by changing the direction of electric field of F = 1.5 kV/cm strength and keeping the polarization vector $${\\hat{{\\bf{e}}}}_{x}$$ of laser field fixed is considered. As Fig. 8 shows, the energy levels mostly have extrema for β = 90°, with the exception of the second excited energy level that has maxima at β = 24° and β = 156° and minimum at β = 90°, and the fourth one that together with the minima at β = 13° and β = 167° are almost constant in [72°, 108°] interval. The appearance of the extrema and the region of invariance can be attributed to the complex distribution of electron cloud throughout the variation of β. In addition, the observed symmetry with respect to the β = 90° point, is caused by the mirror symmetry of the dressed potential in Fig. 1(a) with respect to the x− and y− axis, which means that at angles β and 180° − β electric field affects identically.\n\nAnd finally, Fig. 9(a) shows all the most intensive transitions to undergo blueshift of the absorption spectrum in the [0°, 90°] interval and redshift in [90°, 180°]. The related absorption coefficient is demonstrated in Fig. 9(b) considering different values of β. In this case, at the beginning of β variation and close to β = 180° only 1 → 4 transition has absorption coefficients of values comparable with 1 → 2 one, but for the other β, the latter transition has the biggest absorption coefficient.\n\n## Final Remarks\n\nWe have demonstrated that the electronic and optical properties of CDQR system can be readily controlled with THz laser and static electric fields. Particularly, it is calculated that the intense THz laser field permits to study double QRs of different geometries in a single sample of CDQRs, that is an important finding for modeling of experimental studies. In addition, by changing the characteristic parameter α0 of the laser field, one can come to a new set of degenerated levels in laser-dressed CDQR and manage the distribution of electron cloud between the rings. The impact of inter-ring barrier width LB variation on electron PD in the rings shows that delocalization of PD from the inner to the outer ring does not depend on the fixed value of LB/α0 ratio.\n\nThe selection rule that defines the intraband transitions of the circularly polarized light in only laser field influenced CDQR system are shown to allow only the transitions from the ground state to the excited states that do not have antisymmetry or symmetry with respect to both coordinate axes. Also, with the augmentation of α0 both the blue- and redshifts of the absorption spectrum are observed.\n\nThe addition of a static electric field on the energy spectrum results in linear and quadratic Stark effects, caused by the anisotropic modification of confining potential by the laser field. Linear Stark effect is observed for the electric field perpendicular to laser field polarization and is more pronounced for the larger anisotropy of the confining potential. For the quadratic Stark effect, the direction parallel to laser field is favorable. Also, the electric field direction drastically changes the crossing and anti-crossing behaviors of the energy levels. Moreover, electric field removes the laser field-induced selection rule and allows all the intraband transitions.\n\nBesides that, it is shown that the electric field influence on the laser-dressed CDQRs allows to readily control the anti-crossings in the energy spectrum. In addition, if the electric field is parallel to laser polarization the biggest oscillator strength (absorption intensity) has 1 → 2 transition. In case of a perpendicular direction of the electric field, 1 → 5 transitions have the biggest intensity. Electric field direction changing also affects the absorption spectrum, making mainly blueshifts in [0°, 90°] and redshifts in [90°, 180°] orientations. It is worth to note that the THz laser field can in principle control any anisotropic (induced by the geometry, effective mass, defects, etc.) properties of the CDQR nanostructure. We believe that the results are useful and will open up new possibilities to the improved design and characterization of new devices based on CDQR, such as THz detectors, efficient solar cells, photon emitters, to cite a few.\n\n## References\n\n1. 1.\n\nFomin V. M. (ed.) Physics of Quantum Rings. NanoScience and Technology (Springer-Verlag Berlin Heidelberg, 2014). https://doi.org/10.1007/978-3-642-39197-2.\n\n2. 2.\n\nMoiseev, K. D., Parkhomenko, Y. P., Gushchina, E. A., Kizhaev, S. S. & Mikhailova, M. P. Appl. Surf. Sci. 256, 435, https://doi.org/10.1016/j.apsusc.2009.06.094 (2009).\n\n3. 3.\n\nLorke, A. et al. Spectroscopy of nanoscopic semiconductor rings. Phys. Rev. Lett. 84, 2223, https://doi.org/10.1103/PhysRevLett.84.2223 (2000).\n\n4. 4.\n\nXu, H., Huang, L., Lai, Y.-C. & Grebogi, C. 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Matter 25, 485501, https://doi.org/10.1088/0953-8984/25/48/485501 (2013).\n\n41. 41.\n\nde Sousa, D. R., da Costa, D. R., Chaves, A., Farias, G. A. & Peeters, F. M. Unusual quantum confined Stark effect and Aharonov-Bohm oscillations in semiconductor quantum rings with anisotropic effective masses. Phys. Rev. B 95, 205414, https://doi.org/10.1103/PhysRevB.95.205414 (2017).\n\n42. 42.\n\nSomaschini, C., Bietti, S., Fedorov, A., Koguchi, N. & Sanguinetti, S. Concentric Multiple Rings by Droplet Epitaxy: Fabrication and Study of the Morphological Anisotropy. Nanoscale Res. Lett. 5, 1865, https://doi.org/10.1007/s11671-010-9699-6 (2010).\n\n43. 43.\n\nDavies, J. H. The physics of low-dimensional semiconductors: an introduction. (Cambridge University Press, Cambridge, 1998). https://doi.org/10.1017/CBO9780511819070.\n\n44. 44.\n\nŞahin, M. Photoionization cross section and intersublevel transitions in a one- and two-electron spherical quantum dot with a hydrogenic impurity. Phys. Rev. B 77, 045317, https://doi.org/10.1103/PhysRevB.77.045317 (2008).\n\n45. 45.\n\nCOMSOL Multiphysics, v. 5.2a. www.comsol.com. COMSOL AB, Stockholm, Sweden.\n\n46. 46.\n\nOñate, E. Structural Analysis with the Finite Element Method. Linear Statics. Lecture Notes on Numerical Methods in Engineering and Sciences (Springer Netherlands, 2009), 1 edn. https://doi.org/10.1007/978-1-4020-8733-2.\n\n47. 47.\n\nvon Neumann, J. & Wigner, E. P. Über merkwürdige diskrete Eigenwerte. Z. Physik 30, 467 (1929).\n\n48. 48.\n\nLandau L. D. & Lifshitz, L. M. Quantum Mechanics. (Pergamon Press, 1977).\n\n49. 49.\n\nBastard, G. Wave mechanics applied to semiconductor heterostructures. (Editions de Physique, Paris, 1990). https://doi.org/10.1080/09500349114551351.\n\n50. 50.\n\nHüttel, A. K., Ludwig, S., Lorenz, H., Eberl, K. and Kotthaus, J. P. Direct control of the tunnel splitting in a one-electron double quantum dot. Phys. Rev. B 72, 081310(R), https://doi.org/10.1103/PhysRevB.72.081310 (2005).\n\n51. 51.\n\nHüttel, A. K., Ludwig, S., Lorenz, H., Eberl, K. & Kotthaus, J. P. Molecular states in a one-electron double quantum dot. Physica E 34, 488, https://doi.org/10.1016/j.physe.2006.03.064 (2006).\n\n52. 52.\n\nDoty, M. F. et al. Hole-spin mixing in InAs quantum dot molecules. Phys. Rev. B 81, 035308, https://doi.org/10.1103/PhysRevB.81.035308 (2010).\n\n53. 53.\n\nLiang, S., Xie, W., Sarkisyan, H. A., Meliksetyan, A. V. & Shen, H. J. Phys.: Condens. Matter 23, 415302, https://doi.org/10.1088/0953-8984/23/41/415302 (2011).\n\n## Acknowledgements\n\nThe authors acknowledge the financial support from CONICYT-FONDECYT Postdoctoral program fellowship under grant 3150109, CONICYT-ANILLO ACT 1410, the Basal Program through the Center for the Development to Nanoscience and Nanotechnology (Grant No. CEDENNA FB0807), Yachay Tech startup funds and partial support from grant number SAF2014-58286-C2-2-R.\n\n## Author information\n\nAuthors\n\n### Contributions\n\nH.B.M., M.G.B., and D.L. equally contributed to the setting of the problem, calculation method and to the first version of the paper. H.B.M. did the numerical calculations. H.B.M., M.G.B., A.A.K., J.H.O., J.B., D.L. equally contributed to the final version of the paper.\n\n### Corresponding author\n\nCorrespondence to Henrikh M. Baghramyan.\n\n## Ethics declarations\n\n### Competing Interests\n\nThe authors declare no competing interests.\n\nPublisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.\n\n## Rights and permissions\n\nReprints and Permissions\n\n• ### Effective tuning of isotropic and anisotropic properties of quantum dots and rings by external fields\n\n• Manuk G. Barseghyan\n• , Aram Manaselyan\n• , Albert A. Kirakosyan\n• , Laura M. Pérez\n•  & David Laroze\n\nPhysica E: Low-dimensional Systems and Nanostructures (2020)\n\n• ### The transition from double to single quantum dot induced by THz laser field\n\n• M.G. Barseghyan\n• , H.M. Baghramyan\n• , A.A. Kirakosyan\n•  & D. Laroze\n\nPhysica E: Low-dimensional Systems and Nanostructures (2020)\n\n• ### Control of electronic and optical properties of a laser dressed double quantum dot molecule by lateral electric field\n\n• M.G. Barseghyan\n• , V.N. Mughnetsyan\n• , H.M. Baghramyan\n• , F. Ungan\n• , L.M. Pérez\n•  & D. Laroze\n\nPhysica E: Low-dimensional Systems and Nanostructures (2020)\n\n• ### Effects of external fields on the optical absorption of quantum multirings\n\n• S. Ghajarpour-Nobandegani\n• , V. Ashrafi-Dalkhani\n•  & M. J. Karimi\n\nInternational Journal of Modern Physics B (2020)\n\n• ### Binding energy and optical absorption of donor impurity states in “12-6” tuned GaAs/GaAlAs double quantum well under the external fields\n\n• E. Kasapoglu\n• , S. Sakiroglu\n• , H. Sari\n• , I. Sökmen\n•  & C.A. Duque\n\nPhysica B: Condensed Matter (2019)", null, "" ]

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[ "", null, "", null, "", null, "", null, "", null, "ISSN:\n1531-3492\n\neISSN:\n1553-524X\n\n## Journal Home\n\n• Open Access Articles", null, "All Issues\n\n## Discrete & Continuous Dynamical Systems - B\n\nNovember 2014 , Volume 19 , Issue 9\n\nSelect all articles\n\nExport/Reference:\n\n2014, 19(9): 2709-2738 doi: 10.3934/dcdsb.2014.19.2709 +[Abstract](1577) +[PDF](533.6KB)\nAbstract:\nWe study a special case of an optimal control problem governed by a differential equation and a differential rate--independent variational inequality, both with given initial conditions. Under certain conditions, the variational inequality can be reformulated as a differential inclusion with discontinuous right-hand side. This inclusion is known as sweeping process.\nWe perform a discretization scheme and prove the convergence of optimal solutions of the discretized problems to the optimal solution of the original problem. For the discretized problems we study the properties of the solution map and compute its coderivative. Employing an appropriate chain rule, this enables us to compute the subdifferential of the objective function and to apply a suitable optimization technique to solve the discretized problems. The investigated problem is used to model a situation arising in the area of queuing theory.\n2014, 19(9): 2739-2766 doi: 10.3934/dcdsb.2014.19.2739 +[Abstract](1099) +[PDF](615.4KB)\nAbstract:\nWe revisit our study of general transport operator with general force field and general invariant measure by considering, in the $L^1$ setting, the linear transport operator $\\mathcal{T}_H$ associated to a linear and positive boundary operator $H$ of unit norm. It is known that in this case an extension of $\\mathcal{T}_H$ generates a substochastic (i.e. positive contraction) $C_0$-semigroup $(V_H(t))_{t\\geq 0}$. We show here that $(V_H(t))_{t\\geq 0}$ is the smallest substochastic $C_0$-semigroup with the above mentioned property and provides a representation of $(V_H(t))_{t \\geq 0}$ as the sum of an expansion series similar to Dyson-Phillips series. We develop an honesty theory for such boundary perturbations that allows to consider the honesty of trajectories on subintervals $J \\subseteq [0,\\infty)$. New necessary and sufficient conditions for a trajectory to be honest are given in terms of the aforementioned series expansion.\n2014, 19(9): 2767-2783 doi: 10.3934/dcdsb.2014.19.2767 +[Abstract](1525) +[PDF](767.6KB)\nAbstract:\nThe translating and pulsating free surface Green function represents the velocity potential of a three-dimensional free surface source advancing in waves. This function involves singular wave integral, which is troublesome in numerical computation. In the present study, a regular wave integral approach is developed for the discretisation of the singular wave integral in a whole space harmonic function expansion, which permits the free surface wave produced by the fluid motion to be decomposed by plane regular propagation waves. This approximation gives rise to a simple and straightforward evaluation of the Green function. The algorithm is validated from comparisons between present numerical results and existing numerical data.\n2014, 19(9): 2785-2808 doi: 10.3934/dcdsb.2014.19.2785 +[Abstract](1377) +[PDF](1491.6KB)\nAbstract:\nThe present research paper proposes an extension of the classical scalar Auto-Regressive Moving-Average (ARMA) model to real-valued Riemannian matrix manifolds. The resulting ARMA model on matrix manifolds is expressed as a non-linear discrete-time dynamical system in state-space form whose state evolves on the tangent bundle associated with the underlying manifold. A number of examples are discussed within the present contribution that aim at illustrating the numerical behavior of the proposed ARMA model. In order to measure the degree of temporal dependency between the state-values of the ARMA model, an extension of the classical autocorrelation function for scalar sequences is suggested on the basis of the geometrical features of the underlying real-valued matrix manifold.\n2014, 19(9): 2809-2835 doi: 10.3934/dcdsb.2014.19.2809 +[Abstract](1428) +[PDF](453.6KB)\nAbstract:\nThe main objective of this article is to study the dynamic transition and pattern formation for chemotactic systems modeled by the Keller-Segel equations. We study chemotactic systems with either rich or moderated stimulant supplies. For the rich stimulant chemotactic system, we show that the chemotactic system always undergoes a Type-I or Type-II dynamic transition from the homogeneous state to steady state solutions. The type of transition is dictated by the sign of a non dimensional parameter $b$, which is derived by incorporating the nonlinear interactions of both stable and unstable modes. For the general Keller-Segel model where the stimulant is moderately supplied, the system can undergo a dynamic transition to either steady state patterns or spatiotemporal oscillations. From the pattern formation point of view, the formation and the mechanism of both the lamella and rectangular patterns are derived.\n2014, 19(9): 2837-2863 doi: 10.3934/dcdsb.2014.19.2837 +[Abstract](1307) +[PDF](540.3KB)\nAbstract:\nIn this paper, we study the water wave model with a nonlocal viscous term \\begin{equation*} u_t + u_x + \\beta u_{x x x} + \\frac{\\sqrt \\nu}{\\sqrt \\pi}\\frac{\\partial}{\\partial t} \\int_0^t\\frac{u(s)}{\\sqrt{t-s}} ds + u u_x = v u_{xx}, \\end{equation*} where $\\frac{1}{\\sqrt \\pi}\\frac{\\partial}{\\partial t} \\int_0^t\\frac{u(s)}{\\sqrt{t-s}} ds$ is the Riemann-Liouville half derivative. We prove the well-posedness of the equation and we investigate theoretically and numerically the asymptotical behavior of the solutions. Also, we compare our theoretical and numerical results with those given in for a similar equation.\n2014, 19(9): 2865-2887 doi: 10.3934/dcdsb.2014.19.2865 +[Abstract](1572) +[PDF](466.7KB)\nAbstract:\nWe study the spread of disease in an SIS model for a structured population. The model considered is a time-varying, switched model, in which the parameters of the SIS model are subject to abrupt change. We show that the joint spectral radius can be used as a threshold parameter for this model in the spirit of the basic reproduction number for time-invariant models. We also present conditions for persistence and the existence of periodic orbits for the switched model and results for a stochastic switched model.\n2014, 19(9): 2889-2913 doi: 10.3934/dcdsb.2014.19.2889 +[Abstract](1468) +[PDF](825.1KB)\nAbstract:\nSliding motion is evolution on a switching manifold of a discontinuous, piecewise-smooth system of ordinary differential equations. In this paper we quantitatively study the effects of small-amplitude, additive, white Gaussian noise on stable sliding motion. For equations that are static in directions parallel to the switching manifold, the distance of orbits from the switching manifold approaches a quasi-steady-state density. From this density we calculate the mean and variance for the near sliding solution. Numerical results of a relay control system reveal that the noise may significantly affect the period and amplitude of periodic solutions with sliding segments.\n2014, 19(9): 2915-2940 doi: 10.3934/dcdsb.2014.19.2915 +[Abstract](1647) +[PDF](2710.5KB)\nAbstract:\nA piece-wise epidemic model of a switching vaccination program, implemented once the number of people exposed to a disease-causing virus reaches a critical level, is proposed. In addition, variation or uncertainties in interventions are examined with a perturbed system version of the model. We also analyzed the global dynamic behaviors of both the original piece-wise system and the perturbed version theoretically, using generalized Jacobian theory, Lyapunov constants for a non-smooth vector field and a generalization of Dulac's criterion. The main results show that, as the critical value varies, there are three possibilities for stabilization of the piece-wise system: (i) at the disease-free equilibrium; (ii) at the endemic states for the two subsystems or (iii) at a generalized equilibrium which is a novel global attractor for non-smooth systems. The perturbed system exhibits new global attractors including a pseudo-focus of parabolic-parabolic (PP) type, a pseudo-equilibrium and a crossing cycle surrounding a sliding mode region. Our findings demonstrate that an infectious disease can be eradicated either by increasing the vaccination rate or by stabilizing the number of infected individuals at a previously given level, conditional upon a suitable critical level and the parameter values.\n2014, 19(9): 2941-2961 doi: 10.3934/dcdsb.2014.19.2941 +[Abstract](1665) +[PDF](702.0KB)\nAbstract:\nWe study a boundary value problem with an integral constraint that arises from the modelings of species competition proposed by Lou and Ni in . Through local and global bifurcation theories, we obtain the existence of non-constant positive solutions to this problem, which are small perturbations from its positive constant solution, over a one-dimensional domain. Moreover, we investigate the stability of these bifurcating solutions. Finally, for the diffusion rate being sufficiently small, we construct infinitely many positive solutions with single transition layer, which is represented as an approximation of a step function. The transition-layer solution can be used to model the segregation phenomenon through interspecific competition.\n2014, 19(9): 2963-2991 doi: 10.3934/dcdsb.2014.19.2963 +[Abstract](1611) +[PDF](561.3KB)\nAbstract:\nThis paper studies the second moment boundedness of solutions of linear stochastic delay differential equations. First, we give a framework--for general $\\mathrm{N}$-dimensional linear stochastic differential equations with a single discrete delay--of calculating the characteristic function for the second moment boundedness. Next, we apply the proposed framework to a specific case of a type of $2$-dimensional equation that the stochastic terms are decoupled. For the $2$-dimensional equation, we obtain the characteristic function that is explicitly given by equation coefficients, and the characteristic function gives sufficient conditions for the second moment to be bounded or unbounded.\n2014, 19(9): 2993-3018 doi: 10.3934/dcdsb.2014.19.2993 +[Abstract](1678) +[PDF](930.4KB)\nAbstract:\nThis paper is devoted to the mathematical analysis of a reaction-diffusion model of dengue transmission. In the case of a bounded spatial habitat, we obtain the local stability as well as the global stability of either disease-free or endemic steady state in terms of the basic reproduction number $\\mathcal{R}_0$. In the case of an unbounded spatial habitat, we establish the existence of the traveling wave solutions connecting the two constant steady states when $\\mathcal{R}_0>1$, and the nonexistence of the traveling wave solutions that connect the disease-free steady state itself when $\\mathcal{R}_0<1$. Numerical simulations are performed to illustrate the main analytic results.\n2014, 19(9): 3019-3029 doi: 10.3934/dcdsb.2014.19.3019 +[Abstract](1386) +[PDF](408.7KB)\nAbstract:\nThroughout this paper, we consider the equation $u_t - \\Delta u = e^{|\\nabla u|}$ with homogeneous Dirichlet boundary condition. One of our main goals is to show that the existence of global classical solution can derive the existence of classical stationary solution, and the global solution must converge to the stationary solution in $C(\\overline{\\Omega})$. On the contrary, the existence of the stationary solution also implies the global existence of the classical solution at least in the radial case. The other one is to show that finite time gradient blowup will occur for large initial data or domains with small measure.\n\n2018  Impact Factor: 1.008" ]

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[ "# 3.0.1.0: Atomic Spectroscopy and the deBroglie Wavelength (Problems)\n\n$$\\newcommand{\\vecs}{\\overset { \\rightharpoonup} {\\mathbf{#1}} }$$ $$\\newcommand{\\vecd}{\\overset{-\\!-\\!\\rightharpoonup}{\\vphantom{a}\\smash {#1}}}$$$$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\| #1 \\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\id}{\\mathrm{id}}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$ $$\\newcommand{\\kernel}{\\mathrm{null}\\,}$$ $$\\newcommand{\\range}{\\mathrm{range}\\,}$$ $$\\newcommand{\\RealPart}{\\mathrm{Re}}$$ $$\\newcommand{\\ImaginaryPart}{\\mathrm{Im}}$$ $$\\newcommand{\\Argument}{\\mathrm{Arg}}$$ $$\\newcommand{\\norm}{\\| #1 \\|}$$ $$\\newcommand{\\inner}{\\langle #1, #2 \\rangle}$$ $$\\newcommand{\\Span}{\\mathrm{span}}$$$$\\newcommand{\\AA}{\\unicode[.8,0]{x212B}}$$\n\nPROBLEM $$\\PageIndex{1}$$\n\nAssume the mass of an electron is 9.11 x 10-31 kg.\n\na. Calculate the wavelength of an electron traveling with a speed of 2.65 x 106 m/s.\n\nb. What would the wavelength of the helium atom be if it were traveling at the same speed as the electron in part a (1 amu = 1.6606 x 10-24 g)?\n\n2.74 x 10-10 m\n\n3.77 x 10-14 m\n\nPROBLEM $$\\PageIndex{2}$$\n\nWhat is the de Broglie wavelength of a 46 g baseball traveling at 94 mph?\n\n3.43 x 10-34 m\n\nPROBLEM $$\\PageIndex{3}$$\n\nWhich of the following equations describe particle-like behavior? Which describe wavelike behavior?\n\n1. $$c = λν$$\n2. $$E=\\dfrac{mν^2}{2}$$\n3. $$r=\\dfrac{n^2a_0}{Z}$$\n4. $$E = hν$$\n5. $$λ=\\dfrac{h}{mν}$$\n\ne. describes particle-like behavior\n\na&d describe wavelike behavior\n\nPROBLEM $$\\PageIndex{4}$$\n\nRGB color television and computer displays use cathode ray tubes that produce colors by mixing red, green, and blue light. If we look at the screen with a magnifying glass, we can see individual dots turn on and off as the colors change. Using a spectrum of visible light, determine the approximate wavelength of each of these colors. What is the frequency and energy of a photon of each of these colors\n\nRed: 660 nm; 4.54 × 1014 Hz; 3.01 × 10−19 J. Green: 520 nm; 5.77 × 1014 Hz; 3.82 × 10−19 J. Blue: 440 nm; 6.81 × 1014 Hz; 4.51 × 10−19 J.\n\nPROBLEM $$\\PageIndex{5}$$\n\nA bright violet line occurs at 435.8 nm in the emission spectrum of mercury vapor. What amount of energy, in joules, must be released by an electron in a mercury atom to produce a photon of this light?\n\n$$4.56\\times 10^{-19}J$$\n\nPROBLEM $$\\PageIndex{6}$$\n\nWhen rubidium ions are heated to a high temperature, two lines are observed in its line spectrum at wavelengths (a) 7.9 × 10−7 m and (b) 4.2 × 10−7 m. What are the frequencies of the two lines? What color do we see when we heat a rubidium compound?\n\nThe frequency of (a) would be 3.79 × 1014 s-1.\n\nThe frequency of (b) would be 7.13 × 1014 s-1.\n\nBecause (a) would be in the near-IR, the compound would appear purple/blue based on the wavelength of (b).\n\nPROBLEM $$\\PageIndex{7}$$\n\nThe emission spectrum of cesium contains two lines whose frequencies are (a) 3.45 × 1014 Hz and (b) 6.53 × 1014 Hz. What are the wavelengths and energies per photon of the two lines? What color are the lines?\n\nλ = 8.69 × 10−7 m; E = 2.29 × 10−19 J; red\n\nλ = 4.59 × 10−7 m; E = 4.33 × 10−19 J; blue" ]

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[ "", null, "## A planet moves around the sun. At point ‘A,’ it is closest to the sun at distance ‘d1’, and its spee….\n\nA planet moves around the sun. At point ‘A,’ it is closest to the sun at distance ‘d1‘, and its speed is ‘v1‘. At another point ‘B,’ if it is farthest from the sun at distance ‘d2‘. Its speed will be\n\n## Imagine a new planet having the same density as that of Earth, but it is 3 times bigger than Earth i….\n\nImagine a new planet having the same density as that of Earth, but it is 3 times bigger than Earth in size. If the acceleration due to gravity on the surface of Earth is g and that on the surface of the new plant is gp, then\n\n## A satellite orbiting Earth in a circular orbit of radius ‘R’ completes one revolution in ‘3h’. If or….\n\nA satellite orbiting Earth in a circular orbit of radius ‘R’ completes one revolution in ‘3h’. If orbital radius of geostationary satellites is 36,000 km, what is the value of R?\n\n## A satellite moves around Earth in a circular orbit of radius ‘R’ making a rev/day. A second satellit….\n\nA satellite moves around Earth in a circular orbit of radius ‘R’ making a rev/day. A second satellite moving in a circular orbit, moves around Earth once in 8 days. The radius of the orbit of the second satellite is\n\n## A body weighs 0.9 kgwt on the surface of Earth. What will be its weight on Mars, if Mars has a mass ….\n\nA body weighs 0.9 kgwt on the surface of Earth. What will be its weight on Mars, if Mars has a mass which is 1/9 times that of Earth and radius 1/2 times that of Earth?\n\n### Fill this form and get best deals on \" Coaching classes\"\n\n• Get immediate response from the institutes\n• Compare institutes and pick only the best!\n• Feel free to choose the institute you like, and rest will be taken care of\n\n## A planet is rotating around the sun in an elliptical orbit. If the lengths of semi-major and semi-mi….\n\nA planet is rotating around the sun in an elliptical orbit. If the lengths of semi-major and semi-minor axes are ‘a’ and ‘b’ respectively, then, the square of time period will be proportional to\n\n## The escape velocity from the surface of Earth is ve. The escape velocity from the surface of a plane….\n\nThe escape velocity from the surface of Earth is ve. The escape velocity from the surface of a planet whose mass and radius are three times that of Earth will be\n\n## It is given that radius of Earth is ‘R’ and length of a day is ‘T’, the height of a geostationary sa….\n\nIt is given that radius of Earth is ‘R’ and length of a day is ‘T’, the height of a geostationary satellite is\n\n## An asteroid of mass ‘m’ is approaching Earth, initially at a distance of 10Re with speed ‘vi’. It hi….\n\nAn asteroid of mass ‘m’ is approaching Earth, initially at a distance of 10Re with speed ‘vi‘. It hits the Earth with a speed of ‘vf‘ (Re and Me are radius and mass of Earth, respectively). In this case,\n\n## A particle of mass ‘m’ is placed at a certain point on the line joining two heavy particles of masse….\n\nA particle of mass ‘m’ is placed at a certain point on the line joining two heavy particles of masses ‘m1‘ and ‘m2‘, so that its distance from ‘m1‘ is ‘x1‘ and from ‘m2‘ is ‘x2‘. If a particle of mass ‘m’ experiences equal gravitational force due to ‘m1‘...\n\nVerify Yourself" ]

[ null, "https://www.facebook.com/tr", null ]

[ "# Seeking for advice on near real time object detection for mobile (detect garbage within images)\n\nI am trying to build a near-real time object detection model which should run on a mobile device. As I am new to this specific area of computer vision I would appreciate every advice on my current progress and feedback on what I could do differently to achieve the goal.\n\n### The goal\n\nThe goal is to detect garbage in images and classify them into one of the following disposal methods (3 target classes):\n\n• yellow sack/ can (German)\n• paper\n• glass\n\nIn addition to that the model should be lightweight so that it is possible to efficiently run it on a mobile device.\n\n### The dataset\n\nI am using the trashnet dataset which includes exactly 2527 images that are distributed among the classes: glass, paper, plastic, trash, cardboard, metal. Notable here is that there is only one item per image. Also the background of every image is the same (plain white).\n\n### The methodology\n\nQuiet frankly I am following the YouTube Tutorial from Sentdex on Mac'n'cheese detection and this medium post on gun detection. Therefore I am using Google Colab as my environment. Also I am trying to retrain a pretrained model (ssd_mobilenet_v2_coco_2018_03_29). Training the model and exporting the inference graph is done by using the provided methods from the tensorflow API (model_main.py and export_inference_graph.py). I am using the samples config from tensorflow for this model.\n\n### My steps so far\n\n1. I've set up my Google Colab environment similar to the Colab Notebook from the Medium post I mentioned before.\n2. I split the data into training and test data by 3/4 and 1/4 respectively.\n3. I labeled my data by using the popular labelImg tool so that every object has a bounding box.\n4. I deleted every image where the object fills the whole space or ranges out of the image since the bounding box wouldn't make that much sense.\n5. I created the label_map, csv and tfrecord files.\n6. I played around with the initial_learning_rate, the l2_regularizer > weight rate of the box predictor and feature extrator, set use_dropout=true and increased the batch_size=32.\n\n### My current results\n\nMost of the models I built had a very bad AP/AR, kinda high loss and tended to overfit. Also the model is only able to detect one object at a time within new images (maybe because of the dataset?).\n\nHere are some screenshots from my tensorboard. These were made after around 12k steps. I think this is also the point were the overfitting begins to show since the AP is suddenly rising and predicted images have an accuarcy around 90-100%.\n\nScalars:\n\nPredicted images:\n\n### Questions from my side\n\n1. Is it problematic that every image has only one object in it? Might this cause problems when running the model on a video stream?\n2. Are these enough images to build an accurate model?\n3. Does anyone of you guys have experience in this area and could give me advice on how to fine tune the pretrained model?\n4. I also ran the model on a video stream from my webcam but all models tend to classify the whole screen. So it seems that the model is detecting an object but draws the bounding box all over the screen. Might this be related to the nature of the dataset/ the poor model quality?\n\nThis has been a long post so thank you in advance for taking time to read this. I hope I was able to make my goal clear and provided enough details for you guys to follow my current progress.\n\nCurrent adjusted configuration for the pretrained ssd_mobilenet_v2_coco_2018_03_29 model:\n\nmodel {\nssd {\nnum_classes: 3\nbox_coder {\nfaster_rcnn_box_coder {\ny_scale: 10.0\nx_scale: 10.0\nheight_scale: 5.0\nwidth_scale: 5.0\n}\n}\nmatcher {\nargmax_matcher {\nmatched_threshold: 0.5\nunmatched_threshold: 0.5\nignore_thresholds: false\nnegatives_lower_than_unmatched: true\nforce_match_for_each_row: true\n}\n}\nsimilarity_calculator {\niou_similarity {\n}\n}\nanchor_generator {\nssd_anchor_generator {\nnum_layers: 6\nmin_scale: 0.2\nmax_scale: 0.95\naspect_ratios: 1.0\naspect_ratios: 2.0\naspect_ratios: 0.5\naspect_ratios: 3.0\naspect_ratios: 0.3333\n}\n}\nimage_resizer {\nfixed_shape_resizer {\nheight: 300\nwidth: 300\n}\n}\nbox_predictor {\nconvolutional_box_predictor {\nmin_depth: 0\nmax_depth: 0\nnum_layers_before_predictor: 0\n#use_dropout: false\nuse_dropout: true\ndropout_keep_probability: 0.8\nkernel_size: 1\nbox_code_size: 4\napply_sigmoid_to_scores: false\nconv_hyperparams {\nactivation: RELU_6,\nregularizer {\nl2_regularizer {\n#weight: 0.00004\nweight: 0.001\n}\n}\ninitializer {\ntruncated_normal_initializer {\nstddev: 0.03\nmean: 0.0\n}\n}\nbatch_norm {\ntrain: true,\nscale: true,\ncenter: true,\ndecay: 0.9997,\nepsilon: 0.001,\n}\n}\n}\n}\nfeature_extractor {\ntype: 'ssd_mobilenet_v2'\nmin_depth: 16\ndepth_multiplier: 1.0\nconv_hyperparams {\nactivation: RELU_6,\nregularizer {\nl2_regularizer {\n#weight: 0.00004\nweight: 0.001\n}\n}\ninitializer {\ntruncated_normal_initializer {\nstddev: 0.03\nmean: 0.0\n}\n}\nbatch_norm {\ntrain: true,\nscale: true,\ncenter: true,\ndecay: 0.9997,\nepsilon: 0.001,\n}\n}\n}\nloss {\nclassification_loss {\nweighted_sigmoid {\n}\n}\nlocalization_loss {\nweighted_smooth_l1 {\n}\n}\nhard_example_miner {\nnum_hard_examples: 3000\niou_threshold: 0.99\nloss_type: CLASSIFICATION\nmax_negatives_per_positive: 3\nmin_negatives_per_image: 3\n}\nclassification_weight: 1.0\nlocalization_weight: 1.0\n}\nnormalize_loss_by_num_matches: true\npost_processing {\nbatch_non_max_suppression {\nscore_threshold: 1e-8\niou_threshold: 0.6\nmax_detections_per_class: 1\nmax_total_detections: 1\n}\nscore_converter: SIGMOID\n}\n}\n}\n\ntrain_config: {\nbatch_size: 32\noptimizer {\nrms_prop_optimizer: {\nlearning_rate: {\nexponential_decay_learning_rate {\ninitial_learning_rate: 0.01\ndecay_steps: 800720\ndecay_factor: 0.95\n}\n}\nmomentum_optimizer_value: 0.9\ndecay: 0.9\nepsilon: 1.0\n}\n}\nfine_tune_checkpoint: \"PATH\"\nfine_tune_checkpoint_type: \"detection\"\n# Note: The below line limits the training process to 200K steps, which we\n# empirically found to be sufficient enough to train the pets dataset. This\n# effectively bypasses the learning rate schedule (the learning rate will\n# never decay). Remove the below line to train indefinitely.\nnum_steps: 200000\ndata_augmentation_options {\nrandom_horizontal_flip {\n}\n}\ndata_augmentation_options {\nssd_random_crop {\n}\n}\n}\n\ninput_path:\"PATH\"\n}\nlabel_map_path: \"PATH\"\n}\n\neval_config: {\nnum_examples: 197\n# Note: The below line limits the evaluation process to 10 evaluations.\n# Remove the below line to evaluate indefinitely.\n#max_evals: 10\nnum_visualizations: 20\n}\n\ninput_path: \"PATH\"\n}\nlabel_map_path: \"PATH\"\nshuffle: false" ]

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[ "", null, "", null, "", null, "", null, "", null, "", null, "", null, "Re: What am I doing wrong with this code?\n\n• To: mathgroup at smc.vnet.net\n• Subject: [mg82553] Re: What am I doing wrong with this code?\n• From: Bill Rowe <readnewsciv at sbcglobal.net>\n• Date: Wed, 24 Oct 2007 04:32:25 -0400 (EDT)\n\n```On 10/23/07 at 5:32 AM, sean_incali at yahoo.com (sean_incali) wrote:\n\n>I think I'm having a brain fart. Seriously...\n\n>Plot[{1/(1 + y^2), (1/y - 1)^(1/2)}, {y, 0, 5}]\n\n>What is wrong with that code?\n\nor y > 1, (1/y - 1) is negative and the square root becomes\nimaginary. Also, at y = 0, it blows up.\n\nUsing version 6 of Mathematica, there is no issue since Plot\nwill simply ignore the imaginary points and only plot the curve\nwhere it has real values. For earlier versions of Mathematica\nyou will get the error you saw since\n\nIn:= (1/y - 1)^(1/2) /. y -> 1.0000001666666667`\n\nOut= 0.+ 0.000408248 I\n--\nTo reply via email subtract one hundred and four\n\n```\n\n• Prev by Date: Re: Re: why no result & no error from this recursion\n• Next by Date: Problem with Differential Eq\n• Previous by thread: Re: What am I doing wrong with this code?\n• Next by thread: Re: What am I doing wrong with this code?" ]

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[ "## Amateur Extra Class Exam Question Pool- Subelement E-8\n\n###### Subelement E-6           Subelement E-7          Subelement E-8          Subbelement E-9          Subelement E-0\n\nSUBELEMENT E8 - SIGNALS AND EMISSIONS [4 Exam Questions - 4 Groups]\n\nE8A -  AC waveforms: sine, square, sawtooth and irregular waveforms; AC measurements; average and PEP of RF signals; Fourier analysis; Analog to digital conversion: Digital to Analog conversion\n\nE8A01\n\nWhat is the name of the process that shows that a square wave is made up of a sine wave plus all of its odd harmonics?\n\nA. Fourier analysis\n\nB. Vector analysis\n\nC. Numerical analysis\n\nD. Differential analysis\n\n~~\n\nE8A02\n\nWhat type of wave has a rise time significantly faster than its fall time (or vice versa)?\n\nA. A cosine wave\n\nB. A square wave\n\nC. A sawtooth wave\n\nD. A sine wave\n\n~~\n\nE8A03\n\nWhat type of wave does a Fourier analysis show to be made up of sine waves of a given fundamental frequency plus all of its harmonics?\n\nA. A sawtooth wave\n\nB. A square wave\n\nC. A sine wave\n\nD. A cosine wave\n\n~~\n\nE8A04\n\nWhat is \"dither\" with respect to analog to digital converters?\n\nA. An abnormal condition where the converter cannot settle on a value to represent the signal\n\nB. A small amount of noise added to the input signal to allow more precise representation of a signal over time\n\nC. An error caused by irregular quantization step size\n\nD. A method of decimation by randomly skipping samples\n\n~~\n\nE8A05\n\nWhat would be the most accurate way of measuring the RMS voltage of a complex waveform?\n\nA. By using a grid dip meter\n\nB. By measuring the voltage with a D'Arsonval meter\n\nC. By using an absorption wave meter\n\nD. By measuring the heating effect in a known resistor\n\n~~\n\nE8A06\n\nWhat is the approximate ratio of PEP-to-average power in a typical single-sideband phone signal?\n\nA. 2.5 to 1\n\nB. 25 to 1\n\nC. 1 to 1\n\nD. 100 to 1\n\n~~\n\nE8A07\n\nWhat determines the PEP-to-average power ratio of a single-sideband phone signal?\n\nA. The frequency of the modulating signal\n\nB. The characteristics of the modulating signal\n\nC. The degree of carrier suppression\n\nD. The amplifier gain\n\n~~\n\nE8A08\n\nWhy would a direct or flash conversion analog-to-digital converter be useful for a software defined radio?\n\nA. Very low power consumption decreases frequency drift\n\nB. Immunity to out of sequence coding reduces spurious responses\n\nC. Very high speed allows digitizing high frequencies\n\nD. All of these choices are correct\n\n~~\n\nE8A09\n\nHow many levels can an analog-to-digital converter with 8 bit resolution encode?\n\nA. 8\n\nB. 8 multiplied by the gain of the input amplifier\n\nC. 256 divided by the gain of the input amplifier\n\nD. 256\n\n~~\n\nE8A10\n\nWhat is the purpose of a low pass filter used in conjunction with a digital-to-analog converter?\n\nA. Lower the input bandwidth to increase the effective resolution\n\nB. Improve accuracy by removing out of sequence codes from the input\n\nC. Remove harmonics from the output caused by the discrete analog levels generated\n\nD. All of these choices are correct\n\n~~\n\nE8A11\n\nWhat type of information can be conveyed using digital waveforms?\n\nA. Human speech\n\nB. Video signals\n\nC. Data\n\nD. All of these choices are correct\n\n~~\n\nE8A12\n\nWhat is an advantage of using digital signals instead of analog signals to convey the same information?\n\nA. Less complex circuitry is required for digital signal generation and detection\n\nB. Digital signals always occupy a narrower bandwidth\n\nC. Digital signals can be regenerated multiple times without error\n\nD. All of these choices are correct\n\n~~\n\nE8A13\n\nWhich of these methods is commonly used to convert analog signals to digital signals?\n\nA. Sequential sampling\n\nB. Harmonic regeneration\n\nC. Level shifting\n\nD. Phase reversal\n\n~~\n\nE8B -  Modulation and demodulation: modulation methods; modulation index and deviation ratio; frequency and time division multiplexing; Orthogonal Frequency Division Multiplexing\n\nE8B01\n\nWhat is the term for the ratio between the frequency deviation of an RF carrier wave and the modulating frequency of its corresponding FM-phone signal?\n\nA. FM compressibility\n\nB. Quieting index\n\nC. Percentage of modulation\n\nD. Modulation index\n\n~~\n\nE8B02\n\nHow does the modulation index of a phase-modulated emission vary with RF carrier frequency (the modulated frequency)?\n\nA. It increases as the RF carrier frequency increases\n\nB. It decreases as the RF carrier frequency increases\n\nC. It varies with the square root of the RF carrier frequency\n\nD. It does not depend on the RF carrier frequency\n\n~~\n\nE8B03\n\nWhat is the modulation index of an FM-phone signal having a maximum frequency deviation of 3000 Hz either side of the carrier frequency when the modulating frequency is 1000 Hz?\n\nA. 3\n\nB. 0.3\n\nC. 3000\n\nD. 1000\n\n~~\n\nE8B04\n\nWhat is the modulation index of an FM-phone signal having a maximum carrier deviation of plus or minus 6 kHz when modulated with a 2 kHz modulating frequency?\n\nA. 6000\n\nB. 3\n\nC. 2000\n\nD. 1/3\n\n~~\n\nE8B05\n\nWhat is the deviation ratio of an FM-phone signal having a maximum frequency swing of plus-or-minus 5 kHz when the maximum modulation frequency is 3 kHz?\n\nA. 60\n\nB. 0.167\n\nC. 0.6\n\nD. 1.67\n\n~~\n\nE8B06\n\nWhat is the deviation ratio of an FM-phone signal having a maximum frequency swing of plus or minus 7.5 kHz when the maximum modulation frequency is 3.5 kHz?\n\nA. 2.14\n\nB. 0.214\n\nC. 0.47\n\nD. 47\n\n~~\n\nE8B07\n\nOrthogonal Frequency Division Multiplexing is a technique used for which type of amateur communication?\n\nA. High speed digital modes\n\nB. Extremely low-power contacts\n\nC. EME\n\nD. OFDM signals are not allowed on amateur bands\n\n~~\n\nE8B08\n\nWhat describes Orthogonal Frequency Division Multiplexing?\n\nA. A frequency modulation technique which uses non-harmonically related frequencies\n\nB. A bandwidth compression technique using Fourier transforms\n\nC. A digital mode for narrow band, slow speed transmissions\n\nD. A digital modulation technique using subcarriers at frequencies chosen to avoid intersymbol interference\n\n~~\n\nE8B09\n\nWhat is meant by deviation ratio?\n\nA. The ratio of the audio modulating frequency to the center carrier frequency\n\nB. The ratio of the maximum carrier frequency deviation to the highest audio modulating frequency\n\nC. The ratio of the carrier center frequency to the audio modulating frequency\n\nD. The ratio of the highest audio modulating frequency to the average audio modulating frequency\n\n~~\n\nE8B10\n\nWhat describes frequency division multiplexing?\n\nA. The transmitted signal jumps from band to band at a predetermined rate\n\nB. Two or more information streams are merged into a baseband, which then modulates the transmitter\n\nC. The transmitted signal is divided into packets of information\n\nD. Two or more information streams are merged into a digital combiner, which then pulse position modulates the transmitter\n\n~~\n\nE8B11\n\nWhat is digital time division multiplexing?\n\nA. Two or more data streams are assigned to discrete sub-carriers on an FM transmitter\n\nB. Two or more signals are arranged to share discrete time slots of a data transmission\n\nC. Two or more data streams share the same channel by transmitting time of transmission as the sub-carrier\n\nD. Two or more signals are quadrature modulated to increase bandwidth efficiency\n\n~~\n\nE8C -  Digital signals: digital communication modes; information rate vs bandwidth; error correction\n\nE8C01\n\nHow is Forward Error Correction implemented?\n\nA. By the receiving station repeating each block of three data characters\n\nB. By transmitting a special algorithm to the receiving station along with the data characters\n\nC. By transmitting extra data that may be used to detect and correct transmission errors\n\nD. By varying the frequency shift of the transmitted signal according to a predefined algorithm\n\n~~\n\nE8C02\n\nWhat is the definition of symbol rate in a digital transmission?\n\nA. The number of control characters in a message packet\n\nB. The duration of each bit in a message sent over the air\n\nC. The rate at which the waveform of a transmitted signal changes to convey information\n\nD. The number of characters carried per second by the station-to-station link\n\n~~\n\nE8C03\n\nWhen performing phase shift keying, why is it advantageous to shift phase precisely at the zero crossing of the RF carrier?\n\nA. This results in the least possible transmitted bandwidth for the particular mode\n\nB. It is easier to demodulate with a conventional, non-synchronous detector\n\nC. It improves carrier suppression\n\nD. All of these choices are correct\n\n~~\n\nE8C04\n\nWhat technique is used to minimize the bandwidth requirements of a PSK31 signal?\n\nA. Zero-sum character encoding\n\nB. Reed-Solomon character encoding\n\nC. Use of sinusoidal data pulses\n\nD. Use of trapezoidal data pulses\n\n~~\n\nE8C05\n\nWhat is the necessary bandwidth of a 13-WPM international Morse code transmission?\n\nA. Approximately 13 Hz\n\nB. Approximately 26 Hz\n\nC. Approximately 52 Hz\n\nD. Approximately 104 Hz\n\n~~\n\nE8C06\n\nWhat is the necessary bandwidth of a 170-hertz shift, 300-baud ASCII transmission?\n\nA. 0.1 Hz\n\nB. 0.3 kHz\n\nC. 0.5 kHz\n\nD. 1.0 kHz\n\n~~\n\nE8C07\n\nWhat is the necessary bandwidth of a 4800-Hz frequency shift, 9600-baud ASCII FM transmission?\n\nA. 15.36 kHz\n\nB. 9.6 kHz\n\nC. 4.8 kHz\n\nD. 5.76 kHz\n\n~~\n\nE8C08\n\nHow does ARQ accomplish error correction?\n\nA. Special binary codes provide automatic correction\n\nB. Special polynomial codes provide automatic correction\n\nC. If errors are detected, redundant data is substituted\n\nD. If errors are detected, a retransmission is requested\n\n~~\n\nE8C09\n\nWhich is the name of a digital code where each preceding or following character changes by only one bit?\n\nA. Binary Coded Decimal Code\n\nB. Extended Binary Coded Decimal Interchange Code\n\nC. Excess 3 code\n\nD. Gray code\n\n~~\n\nE8C10\n\nWhat is an advantage of Gray code in digital communications where symbols are transmitted as multiple bits\n\nA. It increases security\n\nB. It has more possible states than simple binary\n\nC. It has more resolution than simple binary\n\nD. It facilitates error detection\n\n~~\n\nE8C11\n\nWhat is the relationship between symbol rate and baud?\n\nA. They are the same\n\nB. Baud is twice the symbol rate\n\nC. Symbol rate is only used for packet-based modes\n\nD. Baud is only used for RTTY\n\n~~\n\nE8D -  Keying defects and overmodulation of digital signals; digital codes; spread spectrum\n\nE8D01\n\nA. Signals not using the spread spectrum algorithm are suppressed in the receiver\n\nB. The high power used by a spread spectrum transmitter keeps its signal from being easily overpowered\n\nC. The receiver is always equipped with a digital blanker\n\nD. If interference is detected by the receiver it will signal the transmitter to change frequencies\n\n~~\n\nE8D02\n\nWhat spread spectrum communications technique uses a high speed binary bit stream to shift the phase of an RF carrier?\n\nA. Frequency hopping\n\nB. Direct sequence\n\nC. Binary phase-shift keying\n\n~~\n\nE8D03\n\nHow does the spread spectrum technique of frequency hopping work?\n\nA. If interference is detected by the receiver it will signal the transmitter to change frequencies\n\nB. If interference is detected by the receiver it will signal the transmitter to wait until the frequency is clear\n\nC. A pseudo-random binary bit stream is used to shift the phase of an RF carrier very rapidly in a particular sequence\n\nD. The frequency of the transmitted signal is changed very rapidly according to a particular sequence also used by the receiving station\n\n~~\n\nE8D04\n\nWhat is the primary effect of extremely short rise or fall time on a CW signal?\n\nA. More difficult to copy\n\nB. The generation of RF harmonics\n\nC. The generation of key clicks\n\nD. Limits data speed\n\n~~\n\nE8D05\n\nWhat is the most common method of reducing key clicks?\n\nA. Increase keying waveform rise and fall times\n\nB. Low-pass filters at the transmitter output\n\nC. Reduce keying waveform rise and fall times\n\nD. High-pass filters at the transmitter output\n\n~~\n\nE8D06\n\nWhich of the following indicates likely overmodulation of an AFSK signal such as PSK or MFSK?\n\nA. High reflected power\n\nB. Strong ALC action\n\nC. Harmonics on higher bands\n\n~~\n\nE8D07\n\nWhat is a common cause of overmodulation of AFSK signals?\n\nA. Excessive numbers of retries\n\nB. Ground loops\n\nC. Bit errors in the modem\n\nD. Excessive transmit audio levels\n\n~~\n\nE8D08\n\nWhat parameter might indicate that excessively high input levels are causing distortion in an AFSK signal?\n\nA. Signal to noise ratio\n\nB. Baud rate\n\nC. Repeat Request Rate (RRR)\n\nD. Intermodulation Distortion (IMD)\n\n~~\n\nE8D09\n\nWhat is considered a good minimum IMD level for an idling PSK signal?\n\nA. +10 dB\n\nB. +15 dB\n\nC. -20 dB\n\nD. -30 dB\n\n~~\n\nE8D10\n\nWhat are some of the differences between the Baudot digital code and ASCII?\n\nA. Baudot uses 4 data bits per character, ASCII uses 7 or 8; Baudot uses 1 character as a letters/figures shift code, ASCII has no letters/figures code\n\nB. Baudot uses 5 data bits per character, ASCII uses 7 or 8; Baudot uses 2 characters as letters/figures shift codes, ASCII has no letters/figures shift code\n\nC. Baudot uses 6 data bits per character, ASCII uses 7 or 8; Baudot has no letters/figures shift code, ASCII uses 2 letters/figures shift codes\n\nD. Baudot uses 7 data bits per character, ASCII uses 8; Baudot has no letters/figures shift code, ASCII uses 2 letters/figures shift codes\n\n~~\n\nE8D11\n\nWhat is one advantage of using ASCII code for data communications?\n\nA. It includes built in error correction features\n\nB. It contains fewer information bits per character than any other code\n\nC. It is possible to transmit both upper and lower case text\n\nD. It uses one character as a shift code to send numeric and special characters\n\n~~\n\nE8D12\n\nWhat is the advantage of including a parity bit with an ASCII character stream?\n\nA. Faster transmission rate\n\nB. The signal can overpower interfering signals\n\nC. Foreign language characters can be sent\n\nD. Some types of errors can be detected" ]

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[ "# Tag Archives: Dancing p-values", null, "# A Critical Review of Cumming’s (2014) New Statistics: Reselling Old Statistics as New Statistics\n\nCumming (2014) wrote an article “The New Statistics: Why and How” that was published in the prestigious journal Psychological Science.   On his website, Cumming uses this article to promote his book “Cumming, G. (2012). Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. New York: Routledge.”\n\nThe article clear states the conflict of interest. “The author declared that he earns royalties on his book (Cumming, 2012) that is referred to in this article.” Readers are therefore warned that the article may at least inadvertently give an overly positive account of the new statistics and an overly negative account of the old statistics. After all, why would anybody buy a book about new statistics when the old statistics are working just fine.\n\nThis blog post critically examines Cumming’s claim that his “new statistics” can solve endemic problems in psychological research that have created a replication crisis and that the old statistics are the cause of this crisis.\n\nLike many other statisticians who are using the current replication crisis as an opportunity to sell their statistical approach, Cumming’s blames null-hypothesis significance testing (NHST) for the low credibility of research articles in Psychological Science (Francis, 2013).\n\nIn a nutshell, null-hypothesis significance testing entails 5 steps. First, researchers conduct a study that yields an observed effect size. Second, the sampling error of the design is estimated. Third, the ratio of the observed effect size and sampling error (signal-to-noise ratio) is computed to create a test-statistic (t, F, chi-square). The test-statistic is then used to compute the probability of obtaining the observed test-statistic or a larger one under the assumption that the true effect size in the population is zero (there is no effect or systematic relationship). The last step is to compare the test statistic to a criterion value. If the probability (p-value) is less than a criterion value (typically 5%), the null-hypothesis is rejected and it is concluded that an effect was present.\n\nCumming’s (2014) claims that we need a new way to analyze data because there is “renewed recognition of the severe flaws of null-hypothesis significance testing (NHST)” (p. 7). His new statistical approach has “no place for NHST” (p. 7). His advice is to “whenever possible, avoid using statistical significance or p values” (p. 8).\n\nSo what is wrong with NHST?\n\nThe first argument against NHST is that Ioannidis (2005) wrote an influential article with the eye-catching title “Why most published research findings are false” and most research articles use NHST to draw inferences from the observed results. Thus, NHST seems to be a flawed method because it produces mostly false results. The problem with this argument is that Ioannidis (2005) did not provide empirical evidence that most research findings are false, nor is this a particularly credible claim for all areas of science that use NHST, including partical physics.\n\nThe second argument against NHST is that researchers can use questionable research practices to produce significant results. This is not really a criticism of NHST, because researchers under pressure to publish are motivated to meet any criteria that are used to select articles for publication. A simple solution to this problem would be to publish all submitted articles in a single journal. As a result, there would be no competition for limited publication space in more prestigious journals. However, better studies would be cited more often and researchers will present their results in ways that lead to more citations. It is also difficult to see how psychology can improve its credibility by lowering standards for publication. A better solution would be to ensure that researchers are honestly reporting their results and report credible evidence that can provide a solid empirical foundation for theories of human behavior.\n\nCummings agrees. “To ensure integrity of the literature, we must report all research conducted to a reasonable standard, and reporting must be full and accurate” (p. 9). If a researcher conducted five studies with only a 20% chance to get a significant result and would honestly report all five studies, p-values would provide meaningful evidence about the strength of the evidence, namely most p-values would be non-significant and show that the evidence is weak. Moreover, post-hoc power analysis would reveal that the studies had indeed low power to test a theoretical prediction. Thus, I agree with Cumming’s that honesty and research integrity are important, but I see no reason to abandon NHST as a systematic way to draw inferences from a sample about the population because researchers have failed to disclose non-significant results in the past.\n\nCumming’s then cites a chapter by Kline (2014) that “provided an excellent summary of the deep flaws in NHST and how we use it” (p. 11). Apparently, the summary is so excellent that readers are better off by reading the actual chapter because Cumming’s does not explain what these deep flaws are. He then observes that “very few defenses of NHST have been attempted” (p. 11). He doesn’t even list a single reference. Here is one by a statistician: “In defence of p-values” (Murtaugh, 2014). In a response, Gelman agrees that the problem is more with the way p-values are used rather than with the p-value and NHST per se.\n\nCumming’s then states a single problem of NHST. Namely that it forces researchers to make a dichotomous decision. If the signal-to-noise ratio is above a criterion value, the null-hypothesis is rejected and it is concluded that an effect is present. If the signal-to-noise ratio is below the criterion value the null-hypothesis is not rejected. If Cumming’s has a problem with decision making, it would be possible to simply report the signal-to-noise ratio or simply to report the effect size that was observed in a sample. For example, mortality in an experimental Ebola drug trial was 90% in the control condition and 80% in the experimental condition. As this is the only evidence, it is not necessary to compute sampling error, signal-to-noise ratios, or p-values. Given all of the available evidence, the drug seems to improve survival rates. But wait. Now a dichotomous decision is made based on the observed mean difference and there is no information about the probability that the results in the drug trial generalize to the population. Maybe the finding was a chance finding and the drug actually increases mortality. Should we really make life-and-death decision if the decision were based on the fact that 8 out of 10 patients died in one condition and 9 out of 10 patients died in the other condition?\n\nEven in a theoretical research context decisions have to be made. Editors need to decide whether they accept or reject a submitted manuscript and readers of published studies need to decide whether they want to incorporate new theoretical claims in their theories or whether they want to conduct follow-up studies that build on a published finding. It may not be helpful to have a fixed 5% criterion, but some objective information about the probability of drawing the right or wrong conclusions seems useful.\n\nBased on this rather unconvincing critique of p-values, Cumming’s (2014) recommends that “the best policy is, whenever possible, not to use NHST at all” (p. 12).\n\nSo what is better than NHST?\n\nCumming then explains how his new statistics overcome the flaws of NHST. The solution is simple. What is astonishing about this new statistic is that it uses the exact same components as NHST, namely the observed effect size and sampling error.\n\nNHST uses the ratio of the effect size and sampling error. When the ratio reaches a value of 2, p-values reach the criterion value of .05 and are considered sufficient to reject the null-hypothesis.\n\nThe new statistical approach is to multiple the standard error by a factor of 2 and to add and subtract this value from the observed mean. The interval from the lower value to the higher value is called a confidence interval. The factor of 2 was chosen to obtain a 95% confidence interval.  However, drawing a confidence interval alone is not sufficient to draw conclusions from the data. Whether we describe the results in terms of a ratio, .5/.2 = 2.5 or in terms of a 95%CI = .5 +/- .2 or CI = .1 to .7, is not a qualitative difference. It is simply different ways to provide information about the effect size and sampling error. Moreover, it is arbitrary to multiply the standard error by a factor of 2. It would also be possible to multiply it by a factor of 1, 3, or 5. A factor of 2 is used to obtain a 95% confidence interval rather than a 20%, 50%, 80%, or 99% confidence interval. A 95% confidence is commonly used because it corresponds to a 5% error rate (100 – 95 = 5!). A 95% confidence interval is as arbitrary as a p-value of .05.\n\nSo, how can a p-value be fundamentally wrong and how can a confidence interval be the solution to all problems if they provide the same information about effect size and sampling error? In particular how do confidence intervals solve the main problem of making inferences from an observed mean in a sample about the mean in a population?\n\nTo sell confidence intervals, Cumming’s uses a seductive example.\n\n“I suggest that, once freed from the requirement to report p values, we may appreciate how simple, natural, and informative it is to report that “support for Proposition X is 53%, with a 95% CI of [51, 55],” and then interpret those point and interval estimates in practical terms” (p 14).\n\nSupport for proposition X is a rather unusual dependent variable in psychology. However, let us assume that Cumming refers to an opinion poll among psychologists whether NHST should be abandoned. The response format is a simple yes/no format. The average in the sample is 53%. The null-hypothesis is 50%. The observed mean of 53% in the sample shows more responses in favor of the proposition. To compute a significance test or to compute a confidence interval, we need to know the standard error. The confidence interval ranges from 51% to 55%. As the 95% confidence interval is defined by the observed mean plus/minus two standard errors, it is easy to see that the standard error is SE = (53-51)/2 = 1% or .01. The formula for the standard error in a one sample test with a dichotomous dependent variable is sqrt(p * (p-1) / n)). Solving for n yields a sample size of N = 2,491. This is not surprising because public opinion polls often use large samples to predict election outcomes because small samples would not be informative. Thus, Cumming’s example shows how easy it is to draw inferences from confidence intervals when sample sizes are large and confidence intervals are tight. However, it is unrealistic to assume that psychologists can and will conduct every study with samples of N = 1,000. Thus, the real question is how useful confidence intervals are in a typical research context, when researchers do not have sufficient resources to collect data from hundreds of participants for a single hypothesis test.\n\nFor example, sampling error for a between-subject design with N = 100 (n = 50 per cell) is SE = 2 / sqrt(100) = .2. Thus, the lower and upper limit of the 95%CI are 4/10 of a standard deviation away from the observed mean and the full width of the confidence interval covers 8/10th of a standard deviation. If the true effect size is small to moderate (d = .3) and a researcher happens to obtain the true effect size in a sample, the confidence interval would range from d = -.1 to d = .7. Does this result support the presence of a positive effect in the population? Should this finding be published? Should this finding be reported in newspaper articles as evidence for a positive effect? To answer this question, it is necessary to have a decision criterion.\n\nOne way to answer this question is to compute the signal-to-noise ratio, .3/.2 = 1.5 and to compute the probability that the positive effect in the sample could have occurred just by chance, t(98) = .3/.2 = 1.5, p = .15 (two-tailed). Given this probability, we might want to see stronger evidence. Moreover, a researcher is unlikely to be happy with this result. Evidently, it would have been better to conduct a study that could have provided stronger evidence for the predicted effect, say a confidence interval of d = .25 to .35, but that would have required a sample size of N = 6,500 participants.\n\nA wide confidence interval can also suggest that more evidence is needed, but the important question is how much more evidence is needed and how narrow a confidence interval should be before it can give confidence in a result. NHST provides a simple answer to this question. The evidence should be strong enough to reject the null-hypothesis with a specified error rate. Cumming’s new statistics provides no answer to the important question. The new statistics is descriptive, whereas NHST is an inferential statistic. As long as researchers merely want to describe their data, they can report their results in several ways, including reporting of confidence intervals, but when they want to draw conclusions from their data to support theoretical claims, it is necessary to specify what information constitutes sufficient empirical evidence.\n\nOne solution to this dilemma is to use confidence intervals to test the null-hypothesis. If the 95% confidence interval does not include 0, the ratio of effect size / sampling error is greater than 2 and the p-value would be less than .05. This is the main reason why many statistics programs report 95%CI intervals rather than 33%CI or 66%CI. However, the use of 95% confidence intervals to test significance is hardly a new statistical approach that justifies the proclamation of a new statistic that will save empirical scientists from NHST. It is NHST! Not surprisingly, Cumming’s states that “this is my least preferred way to interpret a confidence interval” (p. 17).\n\nHowever, he does not explain how researchers should interpret a 95% confidence interval that does include zero. Instead, he thinks it is not necessary to make a decision. “We should not lapse back into dichotomous thinking by attaching any particular importance to whether a value of interest lies just inside or just outside our CI.”\n\nDoes an experimental treatment for Ebolay work? CI = -.3 to .8. Let’s try it. Let’s do nothing and do more studies forever. The benefit of avoiding making any decisions is that one can never make a mistake. The cost is that one can also never claim that an empirical claim is supported by evidence. Anybody who is worried about dichotomous thinking might ponder the fact that modern information processing is built on the simple dichotomy of 0/1 bits of information and that it is common practice to decide the fate of undergraduate students on the basis of scoring multiple choice tests in terms of True or False answers.\n\nIn my opinion, the solution to the credibility crisis in psychology is not to move away from dichotomous thinking, but to obtain better data that provide more conclusive evidence about theoretical predictions and a simple solution to this problem is to reduce sampling error. As sampling error decreases, confidence intervals get smaller and are less likely to include zero when an effect is present and the signal-to-noise ratio increases so that p-values get smaller and smaller when an effect is present. Thus, less sampling error also means less decision errors.\n\nThe question is how small should sampling error be to reduce decision error and at what point are resources being wasted because the signal-to-noise ratio is clear enough to make a decision.\n\nPower Analysis\n\nCumming’s does not distinguish between Fischer’s and Neyman-Pearson’s use of p-values. The main difference is that Fischer advocated the use of p-values without strict criterion values for significance testing. This approach would treat p-values just like confidence intervals as continuous statistics that do not imply an inference. A p-value of .03 is significant with a criterion value of .05, but it is not significant with a criterion value of .01.\n\nNeyman-Pearson introduced the concept of a fixed criterion value to draw conclusions from observed data. A criterion value of p = .05 has a clear interpretation. It means that a test of 1,000 null-hypotheses is expected to produce about 50 significant results (type-I errors). A lower error rate can be achieved by lowering the criterion value (p < .01 or p < .001).\n\nImportantly, Neyman-Pearson also considered the alternative problem that the p-value may fail to reach the critical value when an effect is actually present. They called this probability the type-II error. Unfortunately, social scientists have ignored this aspect of Neyman-Pearson Significance Testing (NPST). Researchers can avoid making type-II errors by reducing sampling error. The reason is that a reduction of sampling error increases the signal-to-noise ratio.\n\nFor example, the following p-values were obtained from simulating studies with 95% power. The graph only shows p-values greater than .001 to make the distribution of p-values more prominent. As a result 62.5% of the data are missing because these p-values are below p < .001. The histogram of p-values has been popularized by Simmonsohn et al. (2013) as a p-curve. The p-curve shows that p-values are heavily skewed towards low p-values. Thus, the studies provide consistent evidence that an effect is present, even though p-values can vary dramatically from one study (p = .0001) to the next (p = .02). The variability of p-values is not a problem for NPST as long as the p-values lead to the same conclusion because the magnitude of a p-value is not important in Neyman-Pearson hypothesis testing.", null, "The next graph shows p-values for studies with 20% power. P-values vary just as much, but now the variation covers both sides of the significance criterion, p = .05. As a result, the evidence is often inconclusive and 80% of studies fail to reject the false null-hypothesis.", null, "R-Code\nseed = length(“Cumming’sDancingP-Values”)\npower=.20\nlow_limit = .000\nup_limit = .10\np <-(1-pnorm(rnorm(2500,qnorm(.975,0,1)+qnorm(.20,0,1),1),0,1))*2\nhist(p,breaks=1000,freq=F,ylim=c(0,100),xlim=c(low_limit,up_limit))\nabline(v=.05,col=”red”)\npercent_below_lower_limit = length(subset(p, p <  low_limit))/length(p)\npercent_below_lower_limit\nIf a study is designed to test a qualitative prediction (an experimental manipulation leads to an increase on an observed measure), power analysis can be used to plan a study so that it has a high probability of providing evidence for the hypothesis if the hypothesis is true. It does not matter whether the hypothesis is tested with p-values or with confidence intervals by showing that the confidence does not include zero.\n\nThus, power analysis seems useful even for the new statistics. However, Cummings is “ambivalent about statistical power” (p. 23). First, he argues that it has “no place when we use the new statistics” (p. 23), presumably because the new statistics never make dichotomous decisions.\n\nCumming’s next argument against power is that power is a function of the type-I error criterion. If the type-I error probability is set to 5% and power is only 33% (e.g., d = .5, between-group design N = 40), it is possible to increase power by increasing the type-I error probability. If type-I error rate is set to 50%, power is 80%. Cumming’s thinks that this is an argument against power as a statistical concept, but raising alpha to 50% is equivalent to reducing the width of the confidence interval by computing a 50% confidence interval rather than a 95% confidence interval. Moreover, researchers who adjust alpha to 50% are essentially saying that the null-hypothesis would produce a significant result in every other study. If an editor finds this acceptable and wants to publish the results, neither power analysis nor the reported results are problematic. It is true that there was a good chance to get a significant result when a moderate effect is present (d = .5, 80% probability) and when no effect is present (d = 0, 50% probability). Power analysis provides accurate information about the type-I and type-II error rates. In contrast, the new statistics provides no information about error rates in decision making because it is merely descriptive and does not make decisions.\n\nCumming then points out that “power calculations have traditionally been expected [by granting agencies], but these can be fudged” (p. 23). The problem with fudging power analysis is that the requested grant money may be sufficient to conduct the study, but insufficient to produce a significant result. For example, a researcher may be optimistic and expect a strong effect, d = .80, when the true effect size is only a small effect, d = .20. The researcher conducts a study with N = 52 participants to achieve 80% power. In reality the study has only 11% power and the researcher is likely to end up with a non-significant result. In the new statistics world this is apparently not a problem because the researcher can report the results with a wide confidence interval that includes zero, but it is not clear why a granting agency should fund studies that cannot even provide information about the direction of an effect in the population.\n\nCummings then points out that “one problem is that we never know true power, the probability that our experiment will yield a statistically significant result, because we do not know the true effect size; that is why we are doing the experiment!” (p. 24). The exclamation mark indicates that this is the final dagger in the coffin of power analysis. Power analysis is useless because it makes assumptions about effect sizes when we can just do an experiment to observe the effect size. It is that easy in the world of new statistics. The problem is that we do not know the true effect sizes after an experiment either. We never know the true effect size because we can never determine a population parameter, just like we can never prove the null-hypothesis. It is only possible to estimate population parameter. However, before we estimate a population parameter, we may simply want to know whether an effect exists at all. Power analysis can help in planning studies so that the sample mean shows the same sign as the population mean with a specified error rate.\n\nDetermining Sample Sizes in the New Statistics\n\nAlthough Cumming does not find power analysis useful, he gives some information about sample sizes. Studies should be planned to have a specified level of precision. Cumming gives an example for a between-subject design with n = 50 per cell (N = 100). He chose to present confidence intervals for unstandardized coefficients. In this case, there is no fixed value for the width of the confidence interval because the sampling variance influences the standard error. However, for standardized coefficients like Cohen’s d, sampling variance will produce variation in standardized coefficients, while the standard error is constant. The standard error is simply 2 / sqrt (N), which equals SE = .2 for N = 100. This value needs to be multiplied by 2 to get the confidence interval, and the 95%CI = d +/- .4.   Thus, it is known before the study is conducted that the confidence interval will span 8/10 of a standard deviation and that an observed effect size of d > .4 is needed to exclude 0 from the confidence interval and to state with 95% confidence that the observed effect size would not have occurred if the true effect size were 0 or in the opposite direction.\n\nThe problem is that Cumming provides no guidelines about the level of precision that a researcher should achieve. Is 8/10 of a standard deviation precise enough? Should researchers aim for 1/10 of a standard deviation? So when he suggests that funding agencies should focus on precision, it is not clear what criterion should be used to fund research.\n\nOne obvious criterion would be to ensure that precision is sufficient to exclude zero so that the results can be used to state that direction of the observed effect is the same as the direction of the effect in the population that a researcher wants to generalize to. However, as soon as effect sizes are used in the planning of the precision of a study, precision planning is equivalent to power analysis. Thus, the main novel aspect of the new statistics is to ignore effect sizes in the planning of studies, but without providing guidelines about desirable levels of precision. Researchers should be aware that N = 100 in a between-subject design gives a confidence interval that spans 8/10 of a standard deviation. Is that precise enough?\n\nProblem of Questionable Research Practices, Publication Bias, and Multiple Testing\n\nA major problem for any statistical method is the assumption that random sampling error is the only source of error. However, the current replication crisis has demonstrated that reported results are also systematically biased. A major challenge for any statistical approach, old or new, is to deal effectively with systematically biased data.\n\nIt is impossible to detect bias in a single study. However, when more than one study is available, it becomes possible to examine whether the reported data are consistent with the statistical assumption that each sample is an independent sample and that the results in each sample are a function of the true effect size and random sampling error. In other words, there is no systematic error that biases the results. Numerous statistical methods have been developed to examine whether data are biased or not.\n\nCumming (2014) does not mention a single method for detecting bias (Funnel Plot, Eggert regression, Test of Excessive Significance, Incredibility-Index, P-Curve, Test of Insufficient Variance, Replicabiity-Index, P-Uniform). He merely mentions a visual inspection of forest plots and suggests that “if for example, a set of studies is distinctly too homogeneous – it shows distinctly less bouncing around than we would expect from sampling variability… we can suspect selection or distortion of some kind” (p. 23). However, he provides no criteria that explain how variability of observed effect sizes should be compared against predicted variability and how the presence of bias influences the interpretation of a meta-analysis. Thus, he concludes that “even so [biases may exist], meta-analysis can give the best estimates justified by research to date, as well as the best guidance for practitioners” (p. 23). Thus, the new statistics would suggest that extrasensory perception is real because a meta-analysis of Bem’s (2011) infamous Journal of Personality and Social Psychology article shows an effect with a tight confidence interval that does not include zero. In contrast, other researchers have demonstrated with old statistical tools and with the help of post-hoc power that Bem’s results are not credible (Francis, 2012; Schimmack, 2012).\n\nResearch Integrity\n\nCumming also advocates research integrity. His first point is that psychological science should “promote research integrity: (a) a public research literature that is complete and trustworthy and (b) ethical practice, including full and accurate reporting of research” (p. 8). However, his own article falls short of this ideal. His article does not provide a complete, balanced, and objective account of the statistical literature. Rather, Cumming (2014) cheery-picks references that support his claims and does not cite references that are inconvenient for his claims. I give one clear example of bias in his literature review.\n\nHe cites Ioannidis’s 2005 paper to argue that p-values and NHST is flawed and should be abandoned. However, he does not cite Ioannidis and Trikalinos (2007). This article introduces a statistical approach that can detect biases in meta-analysis by comparing the success rate (percentage of significant results) to the observed power of the studies. As power determines the success rate in an honest set of studies, a higher success rate reveals publication bias. Cumming not only fails to mention this article. He goes on to warn readers “beware of any power statement that does not state an ES; do not use post hoc power.” Without further elaboration, this would imply that readers should ignore evidence for bias with the Test of Excessive Significance because it relies on post-hoc power. To support this claim, he cites Hoenig and Heisey (2001) to claim that “post hoc power can often take almost any value, so it is likely to be misleading” (p. 24). This statement is misleading because post-hoc power is no different from any other statistic that is influenced by sampling error. In fact,Hoenig and Heisey (2001) show that post-hoc power in a single study is monotonically related to p-values. Their main point is that post-hoc power provides no other information than p-values. However, like p-values, post-hoc power becomes more informative, the higher it is. A study with 99% post-hoc power is likely to be a high powered study, just like extremely low p-values, p < .0001, are unlikely to be obtained in low powered studies or in studies when the null-hypothesis is true. So, post-hoc power is informative when it is high. Cumming (2014) further ignores that variability of post-hoc power estimates decreases in a meta-analysis of post-hoc power and that post-hoc power has been used successfully to reveal bias in published articles (Francis, 2012; Schimmack (2012). Thus, his statement that researchers should ignore post-hoc power analyses is not supported by an unbiased review of the literature, and his article does not provide a complete and trustworthy account of the public research literature.\n\nConclusion\n\nI cannot recommend Cumming’s new statistics. I routinely report confidence intervals in my empirical articles, but I do not consider them as a new statistical tool. In my opinion, the root cause of the credibility crisis is that researchers conduct underpowered studies that have a low chance to produce the predicted effect and then use questionable research practices to boost power and to hide non-significant results that could not be salvaged. A simple solution to this problem is to conduct more powerful studies that can produce significant results when the predict effect exists. I do not claim that this is a new insight. Rather, Jacob Cohen has tried his whole life to educate psychologists about the importance of statistical power.\n\nHere is what Jacob Cohen had to say about the new statistics in 1994 using time-travel to comment on Cumming’s article 20 years later.\n\n“Everyone knows” that confidence intervals contain all the information to be found in significance tests and much more. They not only reveal the status of the trivial nil hypothesis but also about the status of non-nil null hypotheses and thus help remind researchers about the possible operation of the crud factor. Yet they are rarely to be found in the literature. I suspect that the main reason they are not reported is that they are so embarrassingly large! But their sheer size should move us toward improving our measurement by seeking to reduce the unreliable and invalid part of the variance in our measures (as Student himself recommended almost a century ago). Also, their width provides us with the analogue of power analysis in significance testing—larger sample sizes reduce the size of confidence intervals as they increase the statistical power of NHST” (p. 1002).\n\nIf you are looking for a book on statistics, I recommend Cohen’s old statistics over Cumming’s new statistics, p < .05.\n\nConflict of Interest: I do not have a book to sell (yet), but I strongly believe that power analysis is an important tool for all scientists who have to deal with uncontrollable variance in their data. Therefore I am strongly opposed to Cumming’s push for a new statistics that provides no guidelines for researchers how they can optimize the use of their resources to obtain credible evidence for effects that actually exist and no guidelines how science can correct false positive results." ]

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