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MP Board Class 10th Science Solutions Chapter 2 Acids, Bases and Salts

MP Board Class 10th Science Solutions Chapter 2 Acids, Bases and Salts

MP Board Class 10th Science Chapter 2 Intext Questions

Intext Questions Page No. 18

Question 1.
You have been provided with three test tubes. One of them contains distilled water and the other two contain an acidic solution and a basic solution, respectively. If you are given only red litmus paper, how will you identify the contents of each test tube?
We can identify the contents of each test tube, with the help of red litmus paper. Consider the following observations:

1. If there is little change in the colour of red litmus (such as purple), then it is distilled water.
2. If the colour of red litmus remains the same, then it is acid.
3. If the colour of red litmus becomes blue, then it is base.

Intext Questions Page No. 22

Question 1.
Why should curd and sour substances not be kept in brass and copper vessels?
Curd and other sour substances are acidic in nature. So, when they are kept in brass and copper vessels, harmful products along with hydrogen gas are produced which spoil the food.

Question 2.
Which gas is usually liberated when an acid reacts with a metal? Illustrate with an example. How will you test for the presence of this gas?
Hydrogen gas is usually liberated when an acid reacts with a metal. Let us illustrate it with the following examples:

1. Add some pieces of zinc granules into 5ml of a dilute solution of sulphuric acid (H2SO4).
2. Shake it well.
3. Pass the produced gas into a soap solution.
4. Now, soap bubbles are formed in the soap solution and these soap bubbles contain hydrogen.
5. Bring a burning candle near a gas-filled bubble. A candle burns with a pop sound. So, the following reaction takes place:
H2SO4(aq) + Zn(s) → ZnSO4(aq) + H2
6. We can test the evolved hydrogen gas by its burning with a pop sound when a candle is brought near the soap bubbles.

Question 3.
Metal compound A reacts with dilute hydrochloric acid to produce effervescence. The gas evolved extinguishes a burning candle. Write a balanced chemical equation for the reaction if one of the compounds formed is calcium chloride.

Intext Questions Page No. 25

Question 1.
Why do HCl, HNO3 etc., show acidic characters in aqueous solutions while solutions of compounds like alcohol and glucose do not show acidic character?
HCl or HNO3 dissolve in water to form H+ or H3O+ ions in aqueous solutions which show their acidic character. The following reactions take place when HCl or HNO3 are mixed with water:
HCl(aq) → H+ + Cl
H+ + H2O → H3O+
On the other hand, when alcohol and glucose are mixed with water they do not dissolve to form ions due to the presence of hydrogen bonds and basic character. Hence, they do not show acidic character.

Question 2.
Why does an aqueous solution of acid conduct electricity?
Due to the presence of hydrogen (H+) or hydronium (H3O+) ions, an aqueous solution of an acid conducts electricity.

Question 3.
Why does dry HCl gas not change the colour of the dry litmus paper?
Dry HCl gas does not change the colour of the dry litmus paper because it has no hydrogen ions (H+) or hydronium (H3O+) ions in it.

Question 4.
While diluting an acid, why is it recommended that the acid should be added to water and not water to the acid?
It is recommended that the acid should be added to water and not water to the acid while diluting an acid because the process is exothermic and lots of heat is generated which may cause the mixture to splash out or cause burns.

Question 5.
How is the concentration of hydronium ions (H3O+) affected when a solution of an acid is diluted?
When a solution of an acid is diluted, the concentration of hydronium ions (H3O+) per unit volume decreases which also decreases the strength of the acid. The dilution increases the volume of the solution but the concentration of ions remain the same.

Question 6.
How is the concentration of hydroxide ions (OH) affected when excess base is dissolved in a solution of sodium hydroxide?
The concentration of hydroxide ions (OH) would increase when excess base is dissolved in a solution of sodium hydroxide.
NaOH $$\overset { { H }_{ 2 }O }{ \rightleftharpoons }$$ Na+ + OH
Being a strong base, hydroxide ions are formed on complete dissolution.

Intext Questions Page No. 28

Question 1.
You have two solutions, A and B. The pH of solution A is 6 and pH of solution B is 8. Which solution has more hydrogen ion concentration? Which of this is acidic and which one is basic?
Solution ‘A’ is acidic because pH of the solution A is 6 which is less than 7 while solution ‘B’ is basic because pH of the solution ‘B’ is 8 which is greater than 7. Solution ‘A’ has more hydrogen ion concentration in comparison to solution ‘B’ because solution ‘A’ is acidic.

Question 2.
What effect does the concentration of H+(aq) ions have on the nature of the solution?
If the concentration of H+(aq) ions is increased then the solution becomes acidic and if the concentration of H+(aq) ions is decreased then the solution becomes basic in nature.

Question 3.
Do basic solutions also have H+(aq) ions? If yes, then why are these basic?
Yes, the basic solution also has H+ ions. However, their concentration is less as compared to the concentration of OH ions that makes the solution basic.

Question 4.
Under what soil condition do you think a farmer would treat the soil of his fields with quick lime (calcium oxide) or slaked lime (calcium hydroxide) or chalk (calcium carbonate)?
If the soil is acidic and improper for cultivation, then to increase the basicity of soil, the farmer would treat the soil with quick lime or slaked lime or chalk.

Intext Questions Page No. 33

Question 1.
What is the common name of the compound CaOCl2?
Bleaching powder.

Question 2.
Name the substance which on treatment with chlorine yields bleaching powder.
Dry slaked lime or calcium hydroxide.

Question 3.
Name the sodium compound which is used for softening hard water.
Sodium carbonate.

Question 4.
What will happen if a solution of sodium hydrocarbon is heated? Give the equation of the reaction involved.
When sodium hydrocarbon is heated then sodium carbonate and water are formed along with the evolution of carbon dioxide gas.
2NaHCO3 → Na2CO3 + H2O + CO2

Question 5.
Write an equation to show the reaction between Plaster of Paris and water.
Plaster of Paris reacts with water to form gypsum.

MP Board Class 10th Science Chapter 2 Ncert Textbook Exercises

Question 1.
A solution turns red litmus blue, its pH is likely to be
(a) 1
(b) 4
(c) 5
(d) 10
(d) 10

Question 2.
A solution reacts with crushed egg-shells to give a gas that turns lime-water milky. The solution contains
(a) NaCl
(b) HCl
(c) LiCl
(d) KCl
(b) HCl

Question 3.
10 ml of a solution of NaOH is found to be completely neutralised by 8 ml of a given solution of HCl. If we take 20 ml of the same solution of NaOH, the amount of HCl solution (the same solution as before) required to neutralise it will be
(a) 4 ml
(b) 8 ml
(c) 12 ml
(d) 16 ml
(d) 16 m

Question 4.
Which one of the following types of medicines are used for treating indigestion?
(a) Antibiotic
(b) Analgesic
(c) Antacid
(d) Antiseptic
(c) Antacid.

Question 5.
Write word equations and then balanced equations for the reaction taking place when:
(a) dilute sulphuric acid reacts with zinc granules.
(b) dilute hydrochloric acid reacts with magnesium ribbon.
(c) dilute sulphuric acid reacts with aluminium powder.
(d) dilute hydrochloric acid reacts with iron filings.
(a) H2SO4(aq) + Zn(s) → ZnSO4(aq) + H2(g)
(b) 2HCl(aq) + Mg(s) → MgCl2(aq) + H2(g)
(c) 3H2SO4(aq) + 2Al(s) → Al2(SO4)3(aq) + 3H2(g)
(d) 2HCl(aq) + Fe(s) → FeCl2(aq) + H2(g)

Question 6.
Compounds such as alcohols and glucose also contain hydrogen but are not categorised as acids. Describe an activity to prove it.
Take solutions of alcohols, glucose in a beaker, take a cork and fix two nails on the cork up to the end. Connect the nails to two terminals of a 6-volt battery through a bulb and a switch. Switch on the current. You will see that bulb does not glow or the current does not pass through the circuit. This means no ions or H+ ions are present in the solution. This shows that alcohols and glucose are not acids.

Question 7.
Why does distilled water not conduct electricity, whereas rainwater does?
Distilled water cannot conduct electricity because it does not contain ions while rainwater conducts electricity as it contains ions due to the presence of dissolved salts in it.

Question 8.
Why do acids not show acidic behaviour in the absence of water?
Acids do not show acidic behaviour in the absence of water because the dissociation of hydrogen ions from an acid occurs in the presence of water only.

Question 9.
Five solutions A, B, C, D and E when tested with universal indicator showed pH as 4, 1, 11, 7 and 9, respectively. Which solution is
(a) Neutral?
(b) Strongly alkaline?
(c) Strongly acidic?
(d) Weakly acidic?
(e) Weakly alkaline?
Arrange the pH in increasing order of hydrogen-ion concentration.
(a) Neutral → Solution D with pH7.
(b) Strongly alkaline → Solution C with pH11.
(c) Strongly acidic → Solution B with pH1.
(d) Weakly acidic → Solution A with pH4.
(e) Weakly alkaline → Solution E with pH9.
The pH can be arranged in the increasing order of the concentration of hydrogen ions as: 11 < 9 < 7 < 4 < 1 or (c) < (e) < (d) < (a) < (b)

Question 10.
Equal lengths of magnesium ribbons are taken in test tubes A and B. Hydrochloric acid (HCl) is added to test tube A, while acetic acid (CH3COOH) is added to test tube B. Amount and concentration taken for both the acids are same. In which test tube will the fizzing
occur more vigorously and why?
The fizzing will occur strongly in test tube A, in which hydrochloric acid (HCl) is added. This is because HCl is a stronger acid than CH3COOH and therefore, produces hydrogen gas at a faster speed due to which fizzing occurs.

Question 11.
Fresh milk has a pH of 6. How do you think the pH will change as it turns into curd? Explain your answer.
The pH of milk is 6. As it changes into curd, the pH will reduce as during curd formation lactose (sugar) present in milk gets converted into lactic acid. Therefore, the curd is acidic in nature. The acids present in it decrease the pH.

Question 12.
A milkman adds a very small amount of baking soda to fresh milk.

1. Why does he shift the pH of the fresh milk from 6 to slightly alkaline?
2. Why does this milk take a long time to set as curd?

1. The milkman shifts the pH of the fresh milk from 6 to slightly alkaline because, in alkaline condition, milk does not set as curd easily. Hence, it does not get spoiled for longer period of time, in which he can sell it to make a profit.
2. Since this milk is slightly basic than usual milk, acids produced to set the curd are neutralised by the base. Therefore, it takes a longer time for the curd to set which is usually acidic.

Question 13.
Plaster of Paris should be stored in a moisture-proof container. Explain why?
The Plaster of Paris should be stored in a moisture-proof container as it absorbs water from moisture and turns into a hard substance (Gypsum) as shown in the following chemical equation:

Question 14.
What is a neutralisation reaction? Give two examples.
When an acid and base react with each other to give a salt and water, it is termed as neutralisation reaction i.e.,
Acid + Base → Salt + Water
For example:
NaOH + HCl → NaCl + H2O.
HNO3 + KOH → KNO3 + H2O.

Question 15.
Give two important uses of washing soda and baking soda.
Important uses of washing soda are:

1. It is used in glass, soap, and paper manufacturing industries.
2. It is used to remove the permanent hardness of the water.

Important uses of baking soda are:

1. It is used in making baking powder. Baking powder is a mixture of baking soda and a mild acid known as tartaric acid. When it is heated or mixed in water, it releases CO2 that makes bread or cake fluffy.
2. Therefore, used in baking.
3. It is used in soda-acid fire extinguishers.

MP Board Class 10th Science Chapter 2 Additional Important Questions

MP Board Class 10th Science Chapter 2 Multiple Choice Questions

Question 1.
The range of a pH scale is:
(a) 1 – 10
(b) 1 – 100
(c) 0 – 14
(d) 1 – 14
(c) 0 – 14

Question 2.
pH is defined as:
(a) The logarithm of hydrogen ion concentration
(b) The negative logarithm of hydrogen ion concentration
(c) Hydrogen ion concentration
(d) None of the above
(a) The logarithm of hydrogen ion concentration

Question 3.
Which of the following solution will have pH = 7?
(a) Tea
(b) The salt solution in distilled water
(c) Hydrochloric acid solution
(d) Water distilled with chlorine gas.
(b) The salt solution in distilled water

Question 4.
Which colour indicate neutral solution on a pH paper?
(a) Brown
(b) Green
(c) Purple
(d) White or transparent
(b) Green

Question 5.
A solution is acidic if:
(a) it releases H+ ions in the solution.
(b) it has a pH of less than 7.
(c) it has dark red, orange or greenish-yellow colour on a pH paper.
(d) all of the above.
(d) all of the above.

Question 6.
Which of the following solutions will have pH < 7?

Choose correct combination:
(a) (i) and (ii)
(b) (ii) and (iii)
(c) (i) and (iii)
(d) All of the above.
(b) (ii) and (iii)

Question 7.
“p” in pH stands for:
(a) Phosphorus
(b) Potenz
(c) Potential
(d) Polarity
(b) Potenz

Question 8.
Four different jars A, B, C, D contains hydrochloric acid, black coffee, ammonia and soap solution. Choose the order of decreasing acidic strength.

(a) A > B > C > D
(b) B > C > A > D
(c) D > C > B > A
(d) D = C > B > A
(a) A > B > C > D

Question 9.
How can we find the pH of a solution?
(a) By dipping pH paper in it.
(b) By dropping some solution over pH paper.
(c) By heating pH paper in vapours of solution.
(d) By pouring all the solution over pH paper.
(a) By dipping pH paper in it.

Question 10.
In a class, while doing practical on different solutions, four students give their observations. Which student/s has/ve given a correct explanation if their observation is as follows?

Choose the correct combination of students with wrong observations:
(a) A and B
(b) C and D
(c) A, B, C
(d) All students
(b) C and D

Question 11.
Lemon juice gives orange colour over pH paper. What is its nature?
(a) Strong Acid
(b) Basic
(c) Neutral
(d) Moderate acid
(a) Strong Acid

Question 12.
Pure water has a pH = 7, while distilled water has pH = 8 – 10, it represents that distilled water is:
(a) Slightly basic
(b) Strong base
(c) Mild acid
(d) Strong acid
(a) Slightly basic

Question 13.
On a pH paper, pH values of 1 and 8 are represented by colours:
(a) Yellow and Orange
(b) Purple and Greenish
(c) Red and Bluish Green
(d) Green and Red
(c) Red and Bluish Green

Question 14.
Which of the following acids gives a dark red colour?
(a) Lemon juice
(b) Hydrochloric acid
(c) Acetic acid
(d) Nitric acid
(b) Hydrochloric acid

Question 15.
How pOH can by represented?
(a) -Log [H+] = pOH
(b) -Log [H] = pOH
(c) -Log [pH] = pOH
(d) -Log [OH] = pOH
(d) -Log [OH] = pOH

Question 16.
If [H+] is 1.0 × 10-9 mole, what will be pH of solution?
(a) 10-9
(b) 1
(c) 9
(d) -9
(c) 9

Question 17.
What is the nature of citric acid?
(a) Basic
(b) Acidic
(c) Neutral
(d) Both a and b
(b) Acidic

Question 18.
What is the nature of Sodium Hydroxide?
(a) Basic
(b) Acidic
(c) Neutral
(d) Unpredictable
(a) Basic

Question 19.
pH > 7 represents?
(a) Basic solutions
(b) Acidic solutions
(c) Neutral solution
(d) All
(a) Basic solutions

Question 20.
pH < 7 represents?
(a) Basic solutions
(b) Acidic solutions
(c) Neutral solution
(d) All
(b) Acidic solutions

Question 21.
Hydrogen ion concentration for pure water is:
(a) 7
(b) 10-7
(c) 10-7 mole/litre
(d) 107 moles/litre
(c) 10-7 mole/litre

MP Board Class 10th Science Chapter 2 Very Short Answer Type Questions

Question 1.
How can pH be represented using log?
pH = – log [H+].

Question 2.
What is the range of pH on pH paper?
0 – 14.

Question 3.
What is the pH of a strong acid?
0 – 2.

Question 4.
Why is water neutral?
On dissociation, water has equal numbers of H+ and OH ions. So, it does not go through any change and remains neutral.

Question 5.
What is the pH of concentric HCl?
1 – 2.

Question 6.
What will be the colour of pH paper when coffee is poured over it?
Reddish as it is slightly acidic in nature.

Question 7.
Which is more acidic-lemon juice or baking powder?
Lemon juice.

Question 8.
What is the best medium to check any chemical’s nature?
Water.

Question 9.
What is the universal solvent?
Aqua regia.

Question 10.
What kind of reactions are neutralisation reactions?
Any reaction between an acid and a base to form salt and water is called neutralisation reaction.

Question 11.
Which solution is considered to be neutral?
Solutions with no acidity or alkalinity are neutral. Acids and bases are present in equal amounts.

Question 12.
Write the formula of brine and bleaching powder.
NaCl and CaOCl2.

Question 13.
Name two products which we can be obtained by chemical processing of common salt.
Baking soda and bleaching powder.

Question 14.
What is the common name of sodium hydrogen carbonate?
Baking soda.

Question 15.
Write the formula for washing soda.
Na2CO3.10H2O.

Question 16.
What is the source of naturally occurring acid lactic acid?
Curd.

Question 17.
Which acid is present in tomato?
Oxalic acid.

Question 18.
Name two olfactory indicators.
Vanilla and clove.

Question 19.
What is the colour of methyl orange in acidic solution?
Red.

Question 20.
What is the colour of phenolphthalein in basic medium?
Pink.

MP Board Class 10th Science Chapter 2 Short Answer Type Questions

Question 1.
How acids are different from bases when dissolved in water?
Acids on dissolving in water produce H+ ions while the base produces OHions.

Question 2.
Name two indicators and write their colour in different mediums.

 Name of indicator Colour in acidic medium Colour in basic medium Methyl Orange Red Yellow Pink Phenolphthalein Colourless Pink

Question 3.
Name any three hydrated salts.

1. Barium chloride, BaCl2. 2H2O.
2. Copper sulphate, CuSO4. 5H2O.
3. Ferrous sulphate, FeSO4. 6H2O.

Question 4.
Give an equation of neutralisation reaction.

Question 5.
What causes acidity in our body? How can it be cured?
Our stomach produces hydrochloric acid which helps in digestion of food. During indigestion, the stomach produces too much acid and this causes pain and irritation. This can be cured by using bases called antacids.

Question 6.
What is the result of the reaction between an acid and a metal?
Corresponding salt is formed with the evolution of hydrogen gas when a metal reacts with acid.

Question 7.
Write two important uses of pH in everyday life. Also, give an example.
pH balance and its particular range of maintenance are very important in nature because it affects animal and plant life very much.

For example:

1. Curd formation: Atmospheric bacteria change the pH of milk which causes the curd formation.
2. Aids in digestion: Slight acidic conditions in the stomach due to the presence of hydrochloric acid aids in the digestion of food.

Question 8.
Give examples of two acids and bases present in nature.

1. Acids: Citric acid, acetic acid.
2. Bases: Calcium carbonate, sodium hydroxide.

Question 9.
Discuss the various types of salts.
There are three types of salts:

1. Neutral salts: Salts formed by the mixing of strong acid and strong base, e.g., NaCl, K2SO4 etc.
2. Acidic salts: Salts formed by the mixing of a strong acid and weak base e.g., NH4Cl, CaSO4.
3. Basic salts: Salts formed by the mixing of a strong base and weak acid e.g., Na2CO3, CH3COONa etc.

Question 10.
Common salt acts as raw material for many important daily use chemicals. Name some of them and also write their chemical formula.
Sodium hydroxide (NaOH), Baking Soda (NaHCO3), Washing soda (Na2CO3. 10H2O) etc.

Question 11.
What are the products of Chlor-alkali process?
The products of Chlor-alkali process are chlorine and sodium hydroxide.

Question 12.
Name two uses of each of the given salts:

1. Bleaching powder
2. Baking soda

Use of given salts are:
1. Bleaching powder:

• It is used as an oxidising agent in chemical industries.
• It is used for disinfecting water to make it free of germs.

2. Baking soda:

• It is used in soda acid fire extinguishers.
• It is an ingredient in antacids.

Question 13.
What is the water of crystallisation?
The water of crystallisation is the fixed number of water molecules present in one formula unit of salt.

Question 14.
Give reaction to show the formation of sodium zincate?

Question 15.
Name the products of electrolysis of brine and also give on use of each.
Chlorine gas, H2 gas, and sodium hydroxide are the products of electrolysis of brine:

1. Use of chlorine gas: It is used as a disinfectant.
2. Use of H2 gas: It is used in the manufacture of ammonia.
3. Use of sodium hydroxide: It is used for the manufacture of soaps and detergents.

MP Board Class 10th Science Chapter 2 Long Answer Type Questions

Question 1.
Discuss the nature of the solution and its type in brief and also explain the strength of a solution.
Nature of the solution:
When a solute is dissolved solvent (generally water), it shows different kind of nature with regard to its reactivity and solubility. On the basis of removal of H+ ion or OH the solution is formed. The types of solutions are divided as follows:

Strength of solution is determined by the,

1. Speed of reactivity i.e., how fast the ions are found dissociated.
2. Amount of ions (H+ or OH ) released or their ion concentration.

Question 2.
Give one example in each case:
(a) a weak mineral acid.
(b) a base which is not an alkali.
(c) a hydrogen-containing compound which is not an acid.
(d) a basic oxide soluble in water.
(e) a basic oxide insoluble in water.

MP Board Class 10th Science Chapter 2 Textbook Activities

Class 10 Science Activity 2.1 Page No. 18

1. Collect the following solutions from the science laboratory – hydrochloric acid (HCl), sulphuric acid (H2SO4), nitric acid (HNO3), acetic acid (CH3COOH), sodium hydroxide (NaOH), calcium hydroxide (Ca(OH)2], potassium hydroxide (KOH), magnesium hydroxide [Mg(OH)2] and ammonium hydroxide (NH4OH).
2. Put a drop of each of the above solutions on a watch-glass one by one and test with a drop of the indicators shown in the table.
3. What change in colour did you observe with red litmus, blue litmus, phenolphthalein and methyl orange solutions for each of the solutions taken?
4. Tabulate your observations in the table.

Result:

Class 10 Science Activity 2.2 Page No. 18,19

1. Take some finely chopped onions in a plastic bag along with some strips of clean cloth. Tie up the bag tightly and leave overnight in the fridge. The cloth strips can now be used to test for acids and bases.
2. Take two of these cloth strips and check their odour.
3. Keep them on a clean surface and put a few drops of dilute HCl solution on one strip and a few drops of dilute NaOH solution on the other.
4. Rinse both cloth strips with water and again check their odour.
6. Now take some dilute vanilla essence and clove oil and check their odour.
7. Take some dilute HCl solution in one test tube and dilute NaOH solution in another. Add a few drops of dilute vanilla essence to both test tubes and shake well. Check the odour once again and record changes in odour, if any.
8. Similarly, test the change in the odour of clove oil with dilute HCl and dilute NaOH solution and record your observations.

Observations:

1. On putting the cloth strip into dilute.HCl solution, the red colour of cloth strip changes to pale red.
2. On putting the cloth strip into NaOH solution, the red colour of strip changes to green in colour.

Odour test:

1. Add vanilla essence in dilute NaOH.
2. Add vanilla essence in dilute HCl.
3. Add clove oil in to dilute HCl.
4. Add clove oil in dilute NaOH.

Result:

1. No smell found.
2. The smell of vanilla exists.
3. The smell of clove present.
4. The smell of clove exists.

Class 10 Science Activity 2.3 Page No. 19,20

Caution:

1. This activity needs the teacher’s assistance.
2. Set the apparatus as shown in Figure.
3. Take about 5 ml of dilute sulphuric acid in a test tube and add a few pieces of zinc granules to it.
4. What do you observe on the surface of zinc granules?
5. Pass the gas being evolved through the soap solution.
6. Why are bubbles formed in the soap solution?
7. Take a burning candle near a gas-filled bubble.
8. What do you observe?
9. Repeat this Activity with some more acids like HCl, HNO3 and CH3COOH.
10. Are the observations in all the cases the same or different?

Observations:

1. 5 ml of dilute sulphuric acid + zinc granules → On the surface of zinc granules + bubbles of hydrogen gas are formed.
2. On passing the gas evolved through a soap solution → bubbles are formed due to low surface tension of soap solution. On taking a burning candle near a gas-filled bubbles → gas burns with a pop sound.
3. All other acids HCl, HNO3 and CH3COOH show the same observation.

Class 10 Science Activity 2.4 Page No. 20

1. Place a few pieces of granulated zinc metal in a test tube.
2. Add 2 ml of sodium hydroxide solution and warm the contents of the test tube.
3. This activity needs the teacher’s assistance.
4. Set the apparatus as shown in Figure.
5. Take about 5 ml of dilute sulphuric acid in a test tube and add a few pieces of zinc granules to it.
6. What do you observe on the surface of zinc granules?
7. Pass the gas being evolved through the soap solution.
8. Why are bubbles formed in the soap solution?
9. Take a burning candle near a gas-filled bubble.
10. What do you observe?
11. Repeat this Activity with some more acids like HCl, HNO3 and CH3COOH.
12. Are the observations in all the cases the same or different?

Observations:
Reaction:
2NaOH + Zn → Na2ZnO3 + H2

Conclusion:
Bubbles of hydrogen gas are formed.

Class 10 Science Activity 2.5 Page No. 20

1. Take two test tubes, label them as A and B.
2. Take about 0.5 g of sodium carbonate (Na2CO3) in test tube A and about 0.5 g of sodium hydrogen carbonate (NaHCO3) in test tube B.
3. Add about 2 ml of dilute HCl to both the test tubes.
4. What do you observe?
5. Pass the gas produced in each case through lime water (calcium hydroxide solution) as shown below and record your observations.

Observations:
Test Tube A:
Reaction:
Na2CO3(s) + 2HCl(aq) → 2NaCl(aq) + H2O(l) +CO2(g)

Test Tube B:
Reaction:
NaHCO3(s) + HCl(aq) → NaCl(aq) + H2O(l) + CO2(g)
On passing CO2 gas through lime water it turns milky because insoluble white precipitate of CaCO3 is formed as shown below:

On passing excess gas through lime water, it becomes colourless.

Class 10 Science Activity 2.6 Page No. 21

1. Take about 2 ml of dilute NaOH solution in a test tube and add two drops of phenolphthalein solution.
2. What is the colour of the solution?
3. Add dilute HCl solution to the above solution drop by drop.
4. Is there any colour change for the reaction mixture?
5. Why did the colour of phenolphthalein change after the addition of an acid?
6. Now add a few drops of NaOH to the above mixture.
7. Does the pink colour of phenolphthalein reappear?
8. Why do you think this has happened?

Observations:

1. Add 2 drops of phenolphthalein solution into NaOH solution → colour of the solution is pink.
2. Now add dilute HCl solution drop by drop.
3. Now, reaction mixture changes to colourless.
4. This is because neutralisation of HCl and NaOH takes place.
5. Now, add NaOH (few drops) → Pink colour reappears.

Class 10 Science Activity 2.7 Page No. 21

1. Take a small amount of copper oxide in a beaker and add dilute hydrochloric acid slowly while stirring.
2. Note the colour of the solution. What has happened to the copper oxide?

Observations:

1. Add dilute HCl to copper oxide solution:
2. Colour of the solution turns into green and CuCl2 dissolves.
3. The blue-green colour formed due to formation of copper (II) chloride,
CuO + 2HCl → CuCl + H2O

Class 10 Science Activity 2.8 Page No. 22

1. Take solutions of glucose, alcohol, hydrochloric acid, sulphuric acid etc.
2. Fix two nails on a cork and place the cork in a 100 ml beaker.
3. Connect the nails to the two terminals of a 6-volt battery through a bulb and a switch, as shown in Figure.

1. Now pour some dilute HCl in the beaker and switch on the current. Repeat with dilute sulphuric acid.
2. What do you observe?
3. Repeat the experiment separately with glucose and alcohol solutions.
4. What do you observe now?
5. Does the bulb glow in all cases?

Class 10 Science Activity 2.9 Page No. 23

1. Take about 1g solid NaCl in a clean and dry test tube and set up the apparatus as shown in Figure.
2. Add some concentrated sulphuric acid to the test tube.
3. What do you observe? Is there a gas coming out of the delivery tube?
4. Test the gas evolved successively with dry and wet blue litmus paper.
5. In which case does the litmus paper change colour?
6. On the basis of the above Activity, what do you infer about the acidic character of:
• Dry HCl gas?
• HCl solution?

Note to teachers:
If the climate is very humid, you will have to pass the gas produced through a guard tube (drying tube) containing calcium chloride to dry the gas.

Observations:

1. Adding concentrated H2SO4 to test tube containing NaCl leads to the production of HCl gas. Now, testing this gas with litmus paper following were recorded,
2. HCl gas passes → Colour change
3. Dry litmus paper → No change
4. Wet litmus paper → Blue litmus turns red

Thus, only HCl solution release H+ ions and acidic property exist due to H+ ions.

Class 10 Science Activity 2.10 Page No. 24

1. Take 10 ml water in a beaker.
2. Add a few drops of concentrated H2SO4 to it and swirl the beaker slowly.
3. Touch the base of the beaker.
4. Is there a change in temperature?
5. Is this an exothermic or endothermic process?
6. Repeat the above Activity with sodium hydroxide pellets and record your observations.

Observations:

1. Add a few drops of concentrated H2SO4 to the water in a beaker → it becomes hot as the reaction is highly exothermic.
2. Now, add NaOH pellets to water → the beaker becomes hot, the reaction is exothermic.

Class 10 Science Activity 2.11 Page No. 26

1. Test the pH values of solutions given in the table.
3. What is the nature of each substance on the basis of your observations?

Class 10 Science Activity 2.12 Page No. 27

1. Put about 2 g soil in a test tube and add 5 mL water to it.
2. Shake the contents of the test tube.
3. Filter the contents and collect the filtrate in a test tube.
4. Check the pH of this filtrate with the help of universal indicator paper.
5. What can you conclude about the ideal soil pH for the growth of plants in your region?

Observations:

 pH of soil Nature of soil Effect on plant 7 Neutral Grow well < 7 Acidic Affected > 7 Basic Affected

Class 10 Science Activity 2.13 Page No. 28,29

1. Write the chemical formulae of the salts given below.
Potassium sulphate, sodium sulphate, calcium sulphate. magnesium sulphate, copper sulphate, sodium chloride, sodium nitrate, sodium carbonate and ammonium chloride.
2. Identify the acids and bases from which the above salts may be obtained.
3. Salts having the same positive or negative radicals are said to belong to a family. For example. NaCl and Na2SO4 belong to the family of sodium salts. Similarly, NaCl and KCl belong to the family of chloride salts. How many families can you identify among the salts given in this Activity?

Class 10 Science Activity 2.14 Page No. 28

1. Collect the following salt samples – sodium chloride, potassium nitrate, aluminium chloride, zinc sulphate, copper sulphate, sodium acetate, sodium carbonate and sodium hydrogen carbonate (some other salts available can also be taken).
2. Check their solubility in water (use distilled water only).
3. Check the action of these solutions on litmus and find the pH using a pH paper.
4. Which of the salts are acidic, basic or neutral?
5. Identify the acid or base used to form the salt.
6. Report your observations in the table.

Class 10 Science Activity 2.15 Page No. 32

1. Heat a few crystals of copper sulphate in a dry boiling tube.
2. What is the colour of the copper sulphate after heating?
3. Do you notice water droplets in the boiling tube? Where have these come from?
4. Add 2-3 drops of water on the sample of copper sulphate obtained after heating
5. What do you observe? Is the blue colour of copper sulphate restored?

Observations:

1. On heating blue crystals of copper sulphate, it becomes colourless or white and few drops of water are seen on test tube due to condensation of water of crystallisation.
2. On adding few drops of water to heated anhydrous copper sulphate, the blue colour of copper sulphate reappear.


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# How do you calculate density of a vapor only vapor pressure and temp.

1. Jan 22, 2014

### aleksbooker

Hello all,

The question given is:

Mercury and many of its compounds are dangerous poisons if breathed, swallowed or even absorbed through the skin. The liquid metal has a vapor pressure of 0.00169mmHg at 24 degrees Celsius. If the air in a small room is saturated with mercury vapor, how many atoms of mercury vapor occur per cubic meter?

I can't use Clausius-Clapeyron because I only have one pressure and one temperature to work with, and I can't use PV = nRT, because I don't know how many moles of mercury I have. If I arbitrarily increase the moles, I increase the volume just as quickly, which may not affect density, but either way I don't know the concentration of moles per liter. I even considered the pressure = force/area, but that's not relevant to volume.

I'm certain the fact that the air is saturated is somehow key, but I can't find anything in my textbook that seems to address this question. What am I missing?

Thanks,
Aleks

2. Jan 22, 2014

### Staff: Mentor

Hi Aleks. Welcome to Physics Forums!!

You can use the ideal gas law. You first need to find out how many moles of mercury there are in each cubic meter. From the ideal gas law, the molar density is $$\frac{n}{V}=\frac{P}{RT}$$. This has units of moles per cubic meter. You then multiply by Avagado's number to get the number of atoms per cubic meter. In this equation, P is the vapor pressure at 24 (room temperature) and you use T = 273+24.

Chet

3. Jan 23, 2014

### aleksbooker

Thanks, Chet!

It took some fishing, but I figured it out. I didn't consider that liters could be converted into cubic meters.

format: plain


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# The population of rabbits in an area is modeled by the growth equation P(t)=8e^0.26t, where P is in thousands and t is in years. How long will it take for the population to reach 25,000?

Jun 23, 2018

I tried this:

#### Explanation:

Let us set $P = 25$ we get:

$25 = 8 {e}^{0.26 t}$

rearrange:

${e}^{0.26 t} = \frac{25}{8}$

take the natural log of both sides:

$\ln \left[{e}^{0.26 t}\right] = \ln \left[\frac{25}{8}\right]$

simplify:

$0.26 t = \ln \left[\frac{25}{8}\right]$

$t = \frac{1}{0.26} \ln \left[\frac{25}{8}\right] = 4.38 \approx 4.4$ years corresponding to 4 years and 5 months (more or less)


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Mathematical explanation of problems behind time and space derivatives being second order

$\left(\nabla^2 - \frac{1}{c^2}\frac{\partial^2}{\partial t^2}\right)\phi = \frac{m^2c^2}{\hbar^2}\phi$

with the wave function $\phi$ being a relativistic scalar: a complex number which has the same numerical value in all frames of reference. The space and time derivatives both enter to second order. This has a telling consequence for the interpretation of the equation. Because the equation is second order in the time derivative, then by the nature of solving differential equations, one must specify both the initial values of the wave function itself and of its first time derivative, in order to solve definite problems. Because both may be specified more or less arbitrarily, the wave function cannot maintain its former role of determining the probability density of finding the electron in a given state of motion. In the Schrödinger theory, the probability density is given by the positive definite expression.

Can anyone explain this quote mathematically? Why is time and space derivatives being second order so problematic?

-

Specifying both $\phi({\bf x},t)$ and $\dfrac{\partial}{\partial t} \phi({\bf x},t)$ at $t=0$ determines $\dfrac{\partial}{\partial t} \int_{{\mathbb R}^n} |\phi({\bf x},t)|^2\ d{\bf x}$ at $t=0$. But if that is nonzero, $\int_{{\mathbb R}^n} |\phi({\bf x},t)|^2\ d{\bf x}$ won't be constant in time. If $|\phi({\bf x},t)|^2$ was a probability density, that integral would have to be $1$.

-
The meaning of that sentence is the following. Schroedinger equation admits a conserved current such that $$\frac{\partial}{\partial t}|\psi|^2=-\nabla\cdot\left(i\psi^*\nabla\psi-i\nabla\psi^*\psi\right).$$ So, we can integrate on a volume $V$ and obtain $\frac{\partial}{\partial t}\int_VdV|\psi|^2=0$. The quantity $\int_VdV|\psi|^2$ is positive and a probabilistic interpretation holds.
When you turn your attention to Klein-Gordon equation for a complex scalar field, the current gives the equation $$\partial_\mu(i\phi^*\partial^\mu\phi-i\partial^\mu\phi^*\phi)=0$$ that will yield on a finite volume $$\frac{\partial}{\partial t}\int_VdV\left(\phi^*\frac{\partial}{\partial t}\phi-\frac{\partial}{\partial t}\phi^*\phi\right)=0$$ but now the integral $\int_VdV\left(\phi^*\frac{\partial}{\partial t}\phi-\frac{\partial}{\partial t}\phi^*\phi\right)$ can take also negative values and a probabilistic meaning cannot be given to this. The reason is explained in quantum field theory and this integral is just the balance between positive and negative charges while you work in a second quantized framework.


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## Angular analysis of the B-0 -> K*(0) mu(+) mu(-) decay using 3 fb(-1) of integrated luminosity

An angular analysis of the B-0 -> K*(0) (-> K+pi(-))mu(+)mu(-) decay is presented. The dataset corresponds to an integrated luminosity of 3.0 fb(-1) of pp collision data collected at the LHCb experiment. The complete angular information from the decay is used to determine CP-averaged observables and CP asymmetries, taking account of possible contamination from decays with the K+pi(-) system in an S-wave configuration. The angular observables and their correlations are reported in bins of q(2), the invariant mass squared of the dimuon system. The observables are determined both from an unbinned maximum likelihood fit and by using the principal moments of the angular distribution. In addition, by fitting for q(2)-dependent decay amplitudes in the region 1.1 < q(2) < 6.0 GeV2/(c)4, the zero-crossing points of several angular observables are computed. A global fit is performed to the complete set of CP-averaged observables obtained from the maximum likelihood fit. This fit indicates differences with predictions based on the Standard Model at the level of 3.4 standard deviations. These differences could be explained by contributions from physics beyond the Standard Model, or by an unexpectedly large hadronic effect that is not accounted for in the Standard Model predictions.

Published in:
Journal of High Energy Physics, 02, 104
Year:
2016
Publisher:
New York, Springer
Keywords:
Laboratories:


text:
MORE IN Engineering Mathematics 4
Total marks: --
Total time: --
INSTRUCTIONS
(1) Assume appropriate data and state your reasons
(2) Marks are given to the right of every question
(3) Draw neat diagrams wherever necessary

1 (a) Obtain y(0, 2) using Picard's method upto second iteration for the initial value problem $\dfrac {dy}{dx} = x^2 - 2y \ \ y(0)=1$
6 M
1 (b) Solve by Euler's modified method to obtain y(1, 2) given $y' = \dfrac {y+x} {y-x} \ \ y(1)=2.$
7 M
1 (c) Using Adam Bash forth method obtain y at x=0.8 given $\dfrac {dy} {dx} = x -y^2, \ \ y(0)=0, \ \ y(0.2)=0.02, \ \ y(0.4)= 0.0795 \ and \ y(0.6) = 0.1762. 7 M 2 (a) Solve by 4th order Runge-Kutta method simultaneous equations given by \[ \dfrac {dx} {dt} = y-t, \ \ \dfrac {dy} {dt} = x+t \ with \ x=1 = y at \ t=0,$ obtain y(0.1) and x(0.1).
6 M
2 (b) Solve $\dfrac {d^2y}{dx^2} - x \left ( \dfrac {dy}{dx} \right )^2 + y^2 = 0, \ \ y(0)=1 , \ y'(0)=0$ Evaluate y(0.2) correct to four decimal places, using Runge method of fourth order.
7 M
2 (c) Solve for x=0.4 using Milnes predictor corrector formula for the differential equation y''+xy'+y=0 with y(0)=1, y(0.1)=0.995, y(0.2)=0.9802 and y(0.3)=0.956. Also Z(0)=0, z(0.1)=-0.0995, z=(0.2)=-0.196, z(0.3)=0.2863.
7 M

3 (a) Verify whether f(z)=sin2z is analytic, hence obtain the derivative.
6 M
3 (b) Determine the analytic function f(z) whose imaginary part is $\dfrac {y}{x^2+y^2}.$
7 M
3 (c) Define a harmonic function. Prove that real and imaginary parts of an analytic function are harmonic.
7 M

4 (a) Under the mapping w=ez, find the image of $i) \ 1\le x \le 2 \\ ii) \ \pi /3 < y < \dfrac {\pi } {2 }$
6 M
4 (b) Find the bilinear transformation which maps the points 1, i -1 from z plane to 2, i, -2 into w plane. Also find the fixed points.
7 M
4 (c) State and prove Cauchy's integral formula.
7 M

5 (a) Prove $J_n (x) = \dfrac {x} {2n} [ J_{n-1} (x) + J_{n+1} (x)]$
6 M
5 (b) Prove (n+1) Pn(x) = (2n+1) × Pn(x)- n Pn-1(x).
7 M
5 (c) Explain the following in terms of Legendre's polynomials.
x4+3x3-x2+5x-2.
7 M

6 (a) A class has 10 boys and 6 girls. Three students are selected at random one after another. Find the probability that i) first and third are boys, second a girls ii) first and second are of same sex and third is of opposite sex.
6 M
6 (b) If P(A)=0.4, P(B/A)=0.9, P(B/A)=0.6. Find P(A/B), P(A/B).
7 M
6 (c) In a bolt factory machines A, B and C manufacture 20%, 35% and 45% of the total of their output 5%, 4% and 2% are defective. A bolt is drawn at random found to be defective. What is the probability that is is from machine B?
7 M

7 (a) A random variable x has the following distribution:
 X: -2 -1 0 1 2 3 4 P(x): 0.1 0.1 k 0.1 2k k k

Find k, mean and S.D. of the distribution.
6 M
7 (b) The probability that a bomb dropped hits the target is 0.2. Find the probability that our out of 6 bombs dropped
i) Exactly 2 will hit the target
ii) At least 3 will hit the target.
7 M
7 (c) Find the mean and variance of the exponential distribution.
7 M

8 (a) A die is tossed 960 times and 5 appear 184 times. Is the die biased?
6 M
8 (b) Nine items have value 45, 47, 50, 52, 48, 47, 49, 53, 51. Does the mean of these differ significantly from assumed of mean of 47.5. (γ=8, t0.05=2.31).
7 M
8 (c) A set of 5 similar coins tossed 320 times gives following table.
 No. of head 0 1 2 3 4 5 Frequency 6 27 72 112 71 32

Test the hypothesis that data follows binomial distribution (Give γ=5, x20.05 = 11.07.
7 M

More question papers from Engineering Mathematics 4


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# Row- and column-major order

Illustration of row- and column-major order

In computing, row-major order and column-major order are methods for storing multidimensional arrays in linear storage such as random access memory.

The difference between the orders lies in which elements of an array are contiguous in memory. In a row-major order, the consecutive elements of a row reside next to each other, whereas the same holds true for consecutive elements of a column in a column-major order. While the terms allude to the rows and columns of a two-dimensional array, i.e. a matrix, the orders can be generalized to arrays of any dimension by noting that the terms row-major and column-major are equivalent to lexicographic and colexicographic orders, respectively.

Data layout is critical for correctly passing arrays between programs written in different programming languages. It is also important for performance when traversing an array because modern CPUs process sequential data more efficiently than nonsequential data. This is primarily due to CPU caching. In addition, contiguous access makes it possible to use SIMD instructions that operate on vectors of data. In some media such as tape or NAND flash memory, accessing sequentially is orders of magnitude faster than nonsequential access.[citation needed]

## Explanation and example

The terms row-major and column-major stem from the terminology related to grouping objects. A general way to order objects with many attributes is to first group and order them by one attribute, and then, within each such group, group and order them by another attribute, etc. If more than one attribute participate in ordering, the first would be called major and the last minor. If two attributes participate in ordering, it is sufficient to name only the major attribute.

In the case of arrays, the attributes are the indices along each dimension. For matrices in mathematical notation, the first index indicates the row, and the second indicates the column, e.g., given a matrix A , a1,2 is in its first row and second column. This convention is carried over to the syntax in programming languages,[1] although often with indexes starting at 0 instead of 1.[2]

Even though the row is indicated by the first index and the column by the second index, no grouping order between the dimensions is implied yet. We can then choose to group and order the indices either row-major or column-major. The terminology can be applied to even higher dimensional arrays. Row-major grouping starts from the leftmost index and column-major from the rightmost index, leading to lexicographic and colexicographics orders, respectively.

For example, for the array

${\displaystyle A={\begin{bmatrix}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\end{bmatrix}}}$

The two possible ways are:

Note how the use of A[i][j] with multi-step indexing as in C, as opposed to a neutral notation like A(i,j) as in Fortran, almost inevitably implies row-major order for syntactic reasons, so to say, because it can be rewritten as (A[i])[j], and the A[i] row part can even be assigned to an intermediate variable that is then indexed in a separate expression. (No other implications should be assumed, e.g., Fortran is not column-major simply because of its notation, and even the above implication could intentionally be circumvented in a new language.)

To use column-major order in a row-major environment, or vice versa, for whatever reason, one workaround could be to assign non-conventional roles to the indexes (using the first index for the column and the second index for the row), and another could be to bypass language syntax by explicitly computing positions in a one-dimensional array. Of course, deviating from convention probably incurs a cost that increases with the degree of necessary interaction with conventional language features and other code, not only in the form of increased vulnerability to mistakes (forgetting to also invert matrix multiplication order, reverting to convention during code maintenance, etc.), but also in the form of having to actively rearrange elements which has to be weighed against any original purpose of increasing performance.

## Programming languages and libraries

Programming languages or their standard libraries that support multi-dimensional arrays typically have a native row-major or column-major storage order for these arrays.

Row-major order is used in C/C++/Objective-C (for C-style arrays), PL/I,[3] Pascal,[4] Speakeasy,[citation needed] SAS,[5] and Rasdaman.[6]

Column-major order is used in Fortran, MATLAB,[7] GNU Octave, S-Plus,[8] R,[9] Julia,[10] and Scilab.[11]

A special case would be OpenGL (and OpenGL ES) for graphics processing. Since "recent mathematical treatments of linear algebra and related fields invariably treat vectors as columns," designer Mark Segal decided to substitute this for the convention in predecessor IRIS GL, which was to write vectors as rows; for compatibility, transformation matrices would still be stored in vector-major rather than coordinate-major order, and he then used the "subterfuge [to] say that matrices in OpenGL are stored in column major order".[12] This was really only relevant for presentation, because matrix multiplication was stack-based and could still be interpreted as post-multiplication, but, worse, reality leaked through the C-based API because individual elements would be accessed as M[vector][coordinate] or, effectively, M[column][row], which unfortunately muddled the convention that the designer sought to adopt, and this was even preserved in the OpenGL Shading Language that was later added (although this also makes it possible to access coordinates by name instead, e.g., M[vector].y). As a result, many developers will now simply declare that having the column as the first index is the definition of column-major, even though this is clearly not the case with a real column-major language like Fortran.

### Neither row-major nor column-major

A typical alternative for dense array storage is to use Iliffe vectors, which typically store elements in the same row contiguously (like row-major order), but not the rows themselves. They are used in (ordered by age): Java,[13] C#/CLI/.Net, Scala,[14] and Swift.

Even less dense is to use lists of lists, e.g., in Python,[15] and in the Wolfram Language of Wolfram Mathematica.[16]

An alternative approach uses tables of tables, e.g., in Lua.[17]

### External libraries

Support for multi-dimensional arrays may also be provided by external libraries, which may even support arbitrary orderings, where each dimension has a stride value, and row-major or column-major are just two possible resulting interpretations.

Row-major order is the default in NumPy[18] (for Python).

Column-major order is the default in Eigen[19] (for C++).

Torch (for Lua) changed from column-major[20] to row-major[21] default order.

## Transposition

As exchanging the indices of an array is the essence of array transposition, an array stored as row-major but read as column-major (or vice versa) will appear transposed. As actually performing this rearrangement in memory is typically an expensive operation, some systems provide options to specify individual matrices as being stored transposed. The programmer must then decide whether or not to rearrange the elements in memory, based on the actual usage (including the number of times that the array is reused in a computation).

For example, the Basic Linear Algebra Subprograms functions are passed flags indicating which arrays are transposed.[22]

The concept generalizes to arrays with more than two dimensions.

For a d-dimensional ${\displaystyle N_{1}\times N_{2}\times \cdots \times N_{d}}$ array with dimensions Nk (k=1...d), a given element of this array is specified by a tuple ${\displaystyle (n_{1},n_{2},\ldots ,n_{d})}$ of d (zero-based) indices ${\displaystyle n_{k}\in [0,N_{k}-1]}$.

In row-major order, the last dimension is contiguous, so that the memory-offset of this element is given by:

${\displaystyle n_{d}+N_{d}\cdot (n_{d-1}+N_{d-1}\cdot (n_{d-2}+N_{d-2}\cdot (\cdots +N_{2}n_{1})\cdots )))=\sum _{k=1}^{d}\left(\prod _{\ell =k+1}^{d}N_{\ell }\right)n_{k}}$

In column-major order, the first dimension is contiguous, so that the memory-offset of this element is given by:

${\displaystyle n_{1}+N_{1}\cdot (n_{2}+N_{2}\cdot (n_{3}+N_{3}\cdot (\cdots +N_{d-1}n_{d})\cdots )))=\sum _{k=1}^{d}\left(\prod _{\ell =1}^{k-1}N_{\ell }\right)n_{k}}$

where the empty product is the multiplicative identity element, i.e., ${\displaystyle \prod _{\ell =1}^{0}N_{\ell }=\prod _{\ell =d+1}^{d}N_{\ell }=1}$.

For a given order, the stride in dimension k is given by the multiplication value in parentheses before index nk in the right hand-side summations above.

More generally, there are d! possible orders for a given array, one for each permutation of dimensions (with row-major and column-order just 2 special cases), although the lists of stride values are not necessarily permutations of each other, e.g., in the 2-by-3 example above, the strides are (3,1) for row-major and (1,2) for column-major.

## References

1. ^ "Arrays and Formatted I/O". FORTRAN Tutorial. Retrieved 19 November 2016.
2. ^ "Why numbering should start at zero". E. W. Dijkstra Archive. Retrieved 2 February 2017.
3. ^ "Language Reference Version 4 Release 3" (PDF). IBM. Retrieved 13 November 2017. Initial values specified for an array are assigned to successive elements of the array in row-major order (final subscript varying most rapidly).
4. ^ "ISO/IEC 7185:1990(E)" (PDF). An array-type that specifies a sequence of two or more index-types shall be an abbreviated notation for an array-type specified to have as its index-type the first index-type in the sequence and to have a component-type that is an array-type specifying the sequence of index-types without the first index-type in the sequence and specifying the same component-type as the original specification.
5. ^ "SAS® 9.4 Language Reference: Concepts, Sixth Edition" (PDF). SAS Institute Inc. September 6, 2017. p. 573. Retrieved 18 November 2017. From right to left, the rightmost dimension represents columns; the next dimension represents rows. [...] SAS places variables into a multidimensional array by filling all rows in order, beginning at the upper left corner of the array (known as row-major order).
6. ^ "Internal array representation in rasdaman". rasdaman.org. Retrieved 8 June 2017.
7. ^ MATLAB documentation, MATLAB Data Storage (retrieved from Mathworks.co.uk, January 2014).
8. ^ Spiegelhalter et al. (2003, p. 17): Spiegelhalter, David; Thomas, Andrew; Best, Nicky; Lunn, Dave (January 2003), "Formatting of data: S-Plus format", WinBUGS User Manual (Version 1.4 ed.), Robinson Way, Cambridge CB2 2SR, UK: MRC Biostatistics Unit, Institute of Public Health, PDF document, archived from the original on 2012-03-03
9. ^ An Introduction to R, Section 5.1: Arrays (retrieved March 2010).
10. ^ "Multi-dimensional Arrays". Julia. Retrieved 6 February 2016.
11. ^ "FFTs with multidimensional data". Scilab Wiki. Retrieved 25 November 2017. Because Scilab stores arrays in column major format, the elements of a column are adjacent (i.e. a separation of 1) in linear format.
12. ^ "Column Vectors Vs. Row Vectors". Retrieved 12 November 2017.
13. ^ "Java Language Specification". Oracle. Retrieved 13 February 2016.
14. ^ "object Array". Scala Standard Library. Retrieved 1 May 2016.
15. ^ "The Python Standard Library: 8. Data Types". Retrieved 18 November 2017.
16. ^ "Vectors and Matrices". Wolfram. Retrieved 12 November 2017.
17. ^ "11.2 – Matrices and Multi-Dimensional Arrays". Retrieved 6 February 2016.
18. ^ "The N-dimensional array (ndarray)". SciPy.org. Retrieved 3 April 2016.
19. ^ "Eigen: Storage orders". eigen.tuxfamily.org. Retrieved 2017-11-23. If the storage order is not specified, then Eigen defaults to storing the entry in column-major.
20. ^ "Tensor". Retrieved 6 February 2016.
21. ^ "Tensor". Torch Package Reference Manual. Retrieved 8 May 2016.
22. ^ "BLAS (Basic Linear Algebra Subprograms)". Retrieved 2015-05-16.


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# A piece of Al weighing $2.7g$ is titrated with $75\;ml$ of $H_2SO_4$ (specific gravity = $1.18gmL^{-1}$ and $24.7%$ $H_2SO_4$ by weight). After the metal is completely dissolved, the solution is diluted to $400\;ml$. Calculate molarity of free $H_2SO_4$ in solution.

$(a)\;0.214\qquad(b)\;0.183\qquad(c)\;0.32\qquad(d)\;0.225$

Weight of Al = 2.7g
Equivalent weight of Al = $\large\frac{2.7}{9}$$=0.3 Molecular equivalent weight of Al = 0.3\times 1000=300 \therefore Weight of H_2SO_4 = 24.7g Weight of the solution = 100g \therefore Volume of the solution = \large\frac{100}{1.18}$$ml=84.75ml$
$N_{H_2SO_4}=\large\frac{24.7}{49\times \Large\frac{100}{1.18\times 1000}}=$ $5.95$
Now, molecular equivalent weight of $H_2SO_4$ in 75ml = $5.95\times 75=446.25$
Molecular equivalent weight of Al added = 300
$\therefore$ Molecular equivalent weight of $H_2SO_4$ left after the reaction = $446.25-300 =146.25$
After the metal is completely dissolved, the solution is diluted to 400ml.
$\therefore N_{H_2SO_4}$ left = $\large\frac{146.25}{400}$$=0.367 \therefore M_{H_2SO_4} left = \large\frac{0.367}{2}$$=0.183$
Hence (b) is the correct answer.
edited Mar 18, 2014


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### Home > APCALC > Chapter 7 > Lesson 7.1.2 > Problem7-18

7-18.

It just so happens that Hustling Harry’s grade in math class at any week $t$ during the semester is calculated by $g(t)=46\cos(\frac{t}{10})+10\sin(\frac{t}{2})+40$. At what point during the $18$‑week semester is Harry’s grade equal to his average grade for the semester?

This is an application of the Mean Value Theorem.

Show setup and steps. $t ≈7.630$ weeks

The Mean Value Theorem

 The Mean Value Theorem for IntegralsIf $f$ is continuous on $[a, b]$, then there exists at least one point $x = c$ in $\left(a, b\right)$ such that $\frac { 1 } { b - a } \int _ { a } ^ { b } f ( x ) d x= f(c)$. The Mean Value Theorem for DerivativesIf $F$ is continuous on $[a, b]$ and differentiable on $\left(a, b\right)$, then there exists at least one point $x = c$ in $\left(a, b\right)$ such that $F ^ { \prime } ( c ) = \frac { F ( b ) - F ( a ) } { b - a } = f ( c )$.


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# P, Q and R are 3 small pumps fitted to a tank. S is a large pump

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VP
Joined: 18 Dec 2017
Posts: 1034
Location: United States (KS)
GMAT 1: 600 Q46 V27
P, Q and R are 3 small pumps fitted to a tank. S is a large pump  [#permalink]

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09 Dec 2019, 10:26
3
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Difficulty:

65% (hard)

Question Stats:

46% (02:56) correct 54% (03:09) wrong based on 24 sessions

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P, Q and R are 3 small pumps fitted to a tank. S is a large pump fitted to the tank. Q is 50% more efficient than P. R is 100/3% is more efficient than Q. S is 50% more efficient than R. All of the pumps are used to fill the tank.
What is the ratio of the time taken by pumps P and R to fill the tank together to the time taken by pumps Q and S to fill the tank together ?

A. 3 : 2
B. 4 : 3
C. 2 : 3
D. 3 : 5
E. 3 : 4

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VP
Joined: 19 Oct 2018
Posts: 1295
Location: India
Re: P, Q and R are 3 small pumps fitted to a tank. S is a large pump  [#permalink]

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09 Dec 2019, 21:06
efficiency of P= x
efficiency of Q= $$\frac{3x}{2}$$
efficiency of R= $$\frac{3x}{2}*\frac{4}{3}$$=2x
efficiency of S= 2x*$$\frac{3}{2}$$=3x

E(P+R) : E(Q+S)= 3x : 4.5x= 2:3

T(P+R):T(Q+S)= 3:2

TheNightKing wrote:
P, Q and R are 3 small pumps fitted to a tank. S is a large pump fitted to the tank. Q is 50% more efficient than P. R is 100/3% is more efficient than Q. S is 50% more efficient than R. All of the pumps are used to fill the tank.
What is the ratio of the time taken by pumps P and R to fill the tank together to the time taken by pumps Q and S to fill the tank together ?

A. 3 : 2
B. 4 : 3
C. 2 : 3
D. 3 : 5
E. 3 : 4
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Joined: 28 Feb 2014
Posts: 279
Location: India
GMAT 1: 570 Q49 V20
GPA: 3.97
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Re: P, Q and R are 3 small pumps fitted to a tank. S is a large pump  [#permalink]

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09 Dec 2019, 21:29
TheNightKing wrote:
P, Q and R are 3 small pumps fitted to a tank. S is a large pump fitted to the tank. Q is 50% more efficient than P. R is 100/3% is more efficient than Q. S is 50% more efficient than R. All of the pumps are used to fill the tank.
What is the ratio of the time taken by pumps P and R to fill the tank together to the time taken by pumps Q and S to fill the tank together ?

A. 3 : 2
B. 4 : 3
C. 2 : 3
D. 3 : 5
E. 3 : 4

rate of Q = 3/2 (rate of P)
rate of R = 4/3 (rate of Q) = (4/3) (3/2) (rate of P) = 2 * rate of P
rate of S = 3/2 (rate of R) = (3/2) (2 * rate of P) = 3 * rate of P

required ratio = (rate of Q + rate of S) / (rate of P + rate of R)
= ( 3/2 + 3) / (1 + 2) = 3 : 2

A is correct
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Re: P, Q and R are 3 small pumps fitted to a tank. S is a large pump  [#permalink]

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09 Dec 2019, 21:54
TheNightKing wrote:
P, Q and R are 3 small pumps fitted to a tank. S is a large pump fitted to the tank. Q is 50% more efficient than P. R is 100/3% is more efficient than Q. S is 50% more efficient than R. All of the pumps are used to fill the tank.
What is the ratio of the time taken by pumps P and R to fill the tank together to the time taken by pumps Q and S to fill the tank together ?

A. 3 : 2
B. 4 : 3
C. 2 : 3
D. 3 : 5
E. 3 : 4

Avoid fractions by assuming a convenient value for the rate of P. Say it pumps 4 lt/sec.

P = 4 lts/sec
Q = 50% more efficient than P = 6 lts/sec
R = 100/3% more efficient that Q = 8 lts/sec
S = 50% more efficient than R = 12 lts/sec

Rate of P and R together = 4 + 8 = 12
Rate of Q and S together = 6 + 12 = 18

Ratio of rates of P and R : Q and S = 2:3 so ratio of time taken by them will be inverse i.e. 3 : 2.

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Re: P, Q and R are 3 small pumps fitted to a tank. S is a large pump   [#permalink] 09 Dec 2019, 21:54
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Algebra and Logic

 Short Title: Algebra Logic Publisher: Springer US, New York, NY ISSN: 0002-5232; 1573-8302/e Online: http://link.springer.com/journal/volumesAndIssues/10469 Original: Algebra i Logika Comments: Indexed cover-to-cover
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# Schedule for: 18w5002 - Around Quantum Chaos

Arriving in Banff, Alberta on Sunday, July 15 and departing Friday July 20, 2018
Sunday, July 15
16:00 - 17:30 Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
20:00 - 22:00 Informal gathering (Corbett Hall Lounge (CH 2110))
Monday, July 16
07:00 - 08:45 Breakfast
Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
08:45 - 09:00 Introduction and Welcome by BIRS Staff
A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.
(TCPL 201)
09:00 - 09:50 Jeff Galkowski: Concentration of Eigenfunctions: Sup-norms and Averages
In this talk we relate concentration of Laplace eigenfunctions in position and momentum to sup-norms and submanifold averages. In particular, we present a unified picture for sup-norms and submanifold averages which characterizes the concentration of those eigenfunctions with maximal growth. We then exploit this characterization to derive geometric conditions under which maximal growth cannot occur. Moreover, we obtain quantitative gains in a variety of geometric settings.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:20 Semyon Dyatlov: Lower bounds on eigenfunctions and fractal uncertainty principle
Let $(M,g)$ be a compact Riemannian manifold and $\Omega\subset M$ a nonempty open set. Take an $L^2$ normalized eigenfunction $u$ of the Laplacian on $M$ with eigenvalue $\lambda^2$. What lower bounds can we get on the mass $m_\Omega(u)=\int_\Omega |u|^2$? There are two well-known bounds for general $M$: (a) $m_\Omega(u)\geq ce^{-C\lambda}$, following from unique continuation estimates, and (b) $m_\Omega(u)\geq c$, where $c>0$ is independent of $\lambda$, assuming that $\Omega$ intersects every sufficiently long geodesic (this is known as the geometric control condition). In general one cannot improve on the bound (a) for arbitrary $\Omega$, as illustrated by Gaussian beams on the round sphere. I will present a recent result which establishes the frequency-independent lower bound (b) for any choice of $\Omega$ when $M$ is a surface of constant negative curvature. This bound has numerous applications, such as control for the Schrödinger equation, exponential decay of damped waves, and the full support property of semiclassical measures. The proof uses the chaotic nature of the geodesic flow on $M$. The key new ingredient is a recently established fractal uncertainty principle, which states that no function can be localized close to a fractal set in both position and frequency. This talk is based on joint works with Jean Bourgain, Long Jin, and Joshua Zahl.
(TCPL 201)
11:30 - 13:00 Lunch
Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
13:00 - 14:00 Guided Tour of The Banff Centre
Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.
(Corbett Hall Lounge (CH 2110))
14:00 - 14:20 Group Photo
Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!
(TCPL 201)
14:20 - 15:10 John Toth: Reverse Agmon estimates for Schrodinger eigenfunctions
Let $(M,g)$ be a compact, Riemannian manifold and $V \in C^{\infty}(M; \R)$. Given a regular energy level $E > \min V$, we consider $L^2$-normalized eigenfunctions, $u_h,$ of the Schrodinger operator $P(h) = - h^2 \Delta_g + V - E(h)$ with $P(h) u_h = 0$ and $E(h) = E + o(1)$ as $h \to 0^+.$ The well-known Agmon-Lithner estimates \cite{Hel} are exponential decay estimates (ie. upper bounds) for eigenfunctions in the forbidden region $\{ V>E \}.$ The decay rate is given in terms of the Agmon distance function $d_E$ associated with the degenerate Agmon metric $(V-E)_+ \, g$ with support in the forbidden region. Our main result is a partial converse to the Agmon estimates (ie. exponential {\em lower} bounds for the eigenfunctions) in terms of Agmon distance in the forbidden region under a control assumption on eigenfunction mass in the allowable region $\{ V< E \}$ arbitrarily close to the caustic $\{ V = E \}.$ I will explain this result in my talk and then give some applications to hypersurface restriction bounds for eigenfunctions in the forbidden region along with corresponding nodal intersection estimates. This is joint work with Xianchao Wu.
(TCPL 201)
15:10 - 15:40 Coffee Break (TCPL Foyer)
15:40 - 16:30 Lior Silberman: Scarring of quasimodes on hyperbolic manifolds
Let $M$ be a compact hyperbolic manifold. The entropy bounds of Anantharaman et al. restrict the possible invariant measures on $T^1 M$ that can be quantum limits of sequences of eigenfunctions. Weaker versions of the entropy bounds also apply to approximate eigenfuctions ("log-scale quasimodes"), so it is interesting to construct such approximate eigenfunctions which converges to singular measures. Generalizing work of Brooks (hyperbolic surfaces) and Eswarathasan--Nonnenmacher (hyperbolic geodesics on Riemannian surfaces) we construct sequences of quasimodes on $M$ converging to totally geodesic submanifolds. A diagonal argument then realizes every invariant measure are a limit of quasimodes of fixed logarithmic width. Joint work with S. Eswarathasan
(TCPL 201)
16:40 - 17:30 Etienne Le Masson: Quantum chaos in the Benjamini-Schramm limit
One of the fundamental problems in quantum chaos is to understand how high-frequency waves behave in chaotic environments. A famous but vague conjecture of Michael Berry predicts that they should look on small scales like Gaussian random waves. We will show how a notion of convergence for sequences of manifolds called Benjamini-Schramm convergence can give a satisfying formulation of this conjecture. The Benjamini-Schramm convergence includes the high-frequency limit as a special case but provides a more general framework. Based on this formulation, we will expand the scope and consider a case where the frequencies stay bounded and the size of the manifold increases instead. We will formulate the corresponding random wave conjecture and present some results to support it, including a quantum ergodicity theorem. Based on joint works with Tuomas Sahlsten, Miklos Abert and Nicolas Bergeron.
(TCPL 201)
17:30 - 19:30 Dinner
A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.
(Vistas Dining Room)
Tuesday, July 17
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 09:50 Boris Hanin: Nodal Sets and Eigenvalues for Small Radial Pertrubations of the Harmonic Oscillator
In this talk, I will present some recent results (joint with Tom Beck) about the behavior at infinity of nodal sets of eigenfunctions for small radial perturbations of the harmonic oscillator. For the unperturbed oscillator, separation of variables eigenfunctions are products of Laguerre functions and spherical harmonics. The angular momenta for a given \hbar that are present at fixed energy E = \hbar (n + d/2) are the spherical harmonics of frequency up to \hbar^{-1}. After a radial perturbation, the energy E eigenspace will break up into the different energies for different angular momenta. The radial rate of growth for the eigenfunctions is an increasing function of their energy E (namely r^{E-d/2}\exp{-r^2/2} in dimension d). Our results give precise information about which angular momenta give the biggest energies after perturbation and hence a good understanding of the size of the nodal set deep into the forbidden region.
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:20 Robert Chang: Log-scale equidistribution of nodal sets in Grauert tubes
Let $M$ be a compact real analytic negatively curved manifold. It admits a complexification in which the metric induces a pluri-subharmonic function $\sqrt{\rho}$ whose sublevel sets are strictly pseudo-convex domains $M_\tau$, known as Grauert tubes. The Laplace eigenfunctions on $M$ analytically continue to the Grauert tubes, and their complex nodal sets are complex hypersurfaces in $M_\tau$. Zelditch proved that the normalized currents of integration over the complex nodal sets tend to a single weak limit $dd^c\sqrt{\rho}$ along a density one subsequence of eigenvalues. In this talk, we discuss a joint work with Steve Zelditch, in which we show that the weak convergence result holds on small scale,' namely, on logarithmically shrinking Kaehler balls whose centers lie in $M_\tau \setminus M$. The main technique is a Poisson-FBI transform relating QE on Kaehler balls to QE on the real domain. Similar small-scale QE results were obtained in the Riemannian setting by Hezari-Riviere and Han, and in the ample line bundle setting by Chang-Zelditch.
(TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:20 Graham Cox: Nodal deficiency, spectral flow, and the Dirichlet-to-Neumann map
Courant's nodal domain theorem provides a natural  generalization of Sturm–Liouville theory to higher dimensions; however,  the result is in general not sharp. It was recently shown that the nodal  deficiency of an eigenfunction is encoded in the spectrum of the  Dirichlet-to-Neumann operators for the eigenfunction's positive and  negative nodal domains. While originally derived using symplectic  methods, this result can also be understood through the spectral flow  for a family of boundary conditions imposed on the nodal set. In this  talk I will describe this flow for a Schrödinger operator with separable  potential on a rectangular domain, and describe a mechanism by which low  energy eigenfunctions do or do not contribute to the nodal deficiency.  Operators on non-rectangular domains and quantum graphs will also be  discussed. This talk represents joint work with Gregory Berkolaiko (Texas A&M) and  Jeremy Marzuola (UNC Chapel Hill).
(TCPL 201)
14:20 - 15:10 Gregory Berkolaiko: Nodal statistics of graph eigenfunctions
We start by reviewing the notion of "quantum graph", its eigenfunctions and the problem of counting the number of their zeros. The nodal surplus of the n-th eigenfunction is defined as the number of its zeros minus (n-1), the latter being the "baseline" nodal count of Sturm-Liouville theory.  It appears from numerics that the distribution of the nodal surplus of large graphs has a universal form: it approaches Gaussian as the number of cycles grows.  We will discuss our recent progress towards proving this conjecture. When the graph is composed of two or more blocks separated by bridges, we propose a way to define a "local nodal surplus" of a given block. Since the eigenfunction index n has no local meaning, the local nodal surplus has to be defined in an indirect way via the nodal-magnetic theorem of Berkolaiko, Colin de Verdiere and Weyand.  By studying the symmetry properties of the distribution of the local nodal surpluses we show that for a graphs with disjoint cycles the distribution of (total) nodal surplus is binomial. Based on joint work with Lior Alon and Ram Band.
(TCPL 201)
15:10 - 15:40 Coffee Break (TCPL Foyer)
15:40 - 16:30 Ram Band: Neumann domains on manifolds and graphs
The nodal set of a Laplacian eigenfunction forms a partition of the underlying manifold or graph. Another natural partition is based on the gradient vector field of the eigenfunction (on a manifold) or on the extremal points of the eigenfunction (on a graph). The submanifolds (or subgraphs) of this partition are called Neumann domains. We present the main results concerning Neumann domains on manifolds and on graphs. We compare manifolds to graphs and relate the Neumann domain results on each of them to the nodal domain study. The talk is based on joint works with Lior Alon, Michael Bersudsky, Sebastian Egger, David Fajman and Alexander Taylor.
(TCPL 201)
16:40 - 17:30 Junehyuk Jung: Boundedness of the number of nodal domains of eigenfunctions
The asymptotic of the number of nodal domains of eigenfunctions on a manifold is closely related to the dynamics of the geodesic flow on the manifold. For instance, if a surface with non-empty boundary has an ergodic geodesic flow, then for any given Dirichlet eigenbasis, one can find a subsequence of density one where the number of nodal domains tends to +\infty. In this talk, I'm going to discuss what happens to the unit circle bundle over a manifold. When equipped with a metric which makes the Laplacian to commute with the circular action on each fiber, the geodesic flow never is ergodic. Recently I and Steve Zelditch proved that among such metrics the following property is generic: for any given orthonormal eigenbasis one can find a subsequence of density 1 where the number of nodal domains is identically 2. This highlights how underlying dynamics can impact the nodal counting. I will sketch proof when we are considering a unit tangent bundle of a compact surface with the genus \neq 1. I also will present an explicit orthonormal eigenbasis on the 3 torus where all of them have only two nodal domains.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Wednesday, July 18
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 09:50 Gabriel Riviere: Witten Laplacians and Pollicott-Ruelle spectrum
Given a smooth Morse function and a Riemannian metric on a compact manifold, Witten defined a semiclassical operator which is now referred as the Witten Laplacian. In light of the recent developpement towards the spectral analysis of hyperbolic dynamical systems, I will discuss some well-known properties and some new ones of these Witten Laplacians. Namely, I will explain that the spectrum of these operators converges in the semiclassical limit to the so-called Pollicott-Ruelle spectrum. This is a joint work with N.V. Dang (Univ. Lyon 1).
(TCPL 201)
09:50 - 10:20 Coffee Break (TCPL Foyer)
10:20 - 11:10 Frederic Faure: Some properties of hyperbolic dynamics from micro-local analysis
In hyperbolic dynamics (Anosov dynamics) each trajectory is strongly unstable and its behavior is unpredictable. A smooth probability distribution evolves also in a complicated way since it acquires higher and higher oscillations. Nevertheless using micro-local analysis, this evolution is predictable in the sense of distributions. It is similar to a quantum scattering problem in cotangent space as treated by Helffer and Sjöstrand using escape functions in (86'). In this talk we will use wave-packet transform (or FBI transform) and explain how to derive some spectral properties of the dynamics, as the existence of the intrinsic discrete spectrum of Ruelle resonances, a fractal Weyl law, estimates on the wave front set of the resonances, and band structure in the case of geodesic flow. Collaboration with Masato Tsujii.
(TCPL 201)
11:10 - 12:00 Martin Vogel: Resonances for large random systems
There have been many works studying resonances generated by compactly supported potentials and by potentials which decay sufficiently quickly near infinity. However, in the case of random potentials there are only very few results. In this talk I will give an overview over some recent results in this direction by J. Sjöstrand, A. Drouot and F. Klopp. In particular I will discuss results obtained in collaboration with F. Klopp on the distribution of resonances close to the real axis and their link to the eigenstates of a full random Schrödinger operator in the localized regime.
(TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 17:30 Free Afternoon (Banff National Park)
17:30 - 19:30 Dinner (Vistas Dining Room)
Thursday, July 19
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 09:50 Igor Wigman: Russo-Seymour-Welsh estimates for the Kostlan ensemble of random polynomials
Beginning with the predictions of Bogomolny-Schmit for the random plane wave, in recent years the deep connections between the level sets of smooth Gaussian random fields and percolation have become apparent. In classical percolation theory a key input into the analysis of global connectivity are scale-independent bounds on crossing probabilities in the critical regime, known as Russo-Seymour-Welsh (RSW) estimates. Similarly, establishing RSW-type estimates for the nodal sets of Gaussian random fields is a major step towards a rigorous understanding of these relations. The Kostlan ensemble is an important model of Gaussian homogeneous random polynomials. The nodal set of this ensemble is a natural model for a typical' real projective hypersurface, whose understanding can be considered as a statistical version of Hilbert's 16th problem. In this paper we establish RSW-type estimates for the nodal sets of the Kostlan ensemble in dimension two, providing a rigorous relation between random algebraic curves and percolation. The estimates are uniform with respect to the degree of the polynomials, and are valid on all relevant scales; this, in particular, resolves an open question raised recently by Beffara-Gayet. More generally, our arguments yield RSW estimates for a wide class of Gaussian ensembles of smooth random functions on the sphere or the flat torus. This is a joint with with D. Beliaev and S. Muirhead
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:20 Yaiza Canzani: Local universality for zeros and critical points of monochromatic random waves
In this talk we will discuss the asymptotic behavior of zeros and critical points for monochromatic random waves on compact, smooth, Riemannian manifolds, as the energy of the waves grow to infinity. This is joint work with Boris Hanin
(TCPL 201)
11:30 - 13:30 Lunch (Vistas Dining Room)
13:30 - 14:20 Damien Gayet: Percolation of random nodal lines
If a real smooth function is given at random on the plane,  what is the probability that its vanishing locus has a large connected component ? I will explain some recent answers we obtained with Vincent Beffara  to this question, for some natural models coming from algebraic geometry and spectral analysis.
(TCPL 201)
14:30 - 15:20 Melissa Tacy: Does it matter what we randomise?
The behaviour of quantum chaotic states of billiard systems is believed to be well described by Berry's random plane wave model $$u=\sum_{j}c_{j}e^{i\lambda x\cdot \xi_{j}}$$ where the $c_{j}$ are Gaussian random variables. However, in $\R^{n}$ there are many other candidate waves over which we could randomise. Some are easier to adapt to manifolds than others. In this talk I will discuss when  (in $\R^{n}$) we can replace the $e^{i\lambda x\cdot \xi_{j}}$ with other waves and how those can be adapted to manifolds.
(TCPL 201)
15:20 - 15:40 Coffee Break (TCPL Foyer)
15:40 - 16:30 Nicolas Burq: Rough controls for Schrödinger equation on the torus
In this talk I will present some results on the exact controllability of Schrödinger equation on the torus. In a general setting, these questions are well understood for wave equations with continuous localisation functions, while for Schrödinger, we only have partial results. For rough localisation functions, I will first present some partial results for waves. Then I will show how one can take benefit from the particular simplicity of the geodesic flow on the torus to get (for continuous localisation functions) strong results (works by Haraux, Jaffard, Burq-Zworski, Anantharaman-Macia). Finally, for general localisation functions (typically characteristic functions of measurable sets) I will show how one can go further, by taking benefit from dispersive properties (on the 2 dimensional torus), to show that in this setting the Schrödinger equation is exactly controllable by any $L^2$ (non trivial) localisation function (and in particular by the characteristic function of any set (with positive measure).
(TCPL 201)
16:40 - 17:30 Luc Hillairet: The wave trace on a flat surface with conical singularities
We study the contribution to the wave trace of diffractive periodic orbits on  Euclidean surfaces with conical singularities. Using a new description of the propagator near  the so-called geometrically diffractive rays, we are able to compute the leading term of any kind of  diffractive periodic orbit. Joint work with A. Ford and A. Hassell.
(TCPL 201)
17:30 - 19:30 Dinner (Vistas Dining Room)
Friday, July 20
07:00 - 09:00 Breakfast (Vistas Dining Room)
09:00 - 09:50 Jared Wunsch: Diffraction of semiclassical singularities by conormal potentials
Consider a semiclassical Schr\"odinger operator $P=h^2\Delta+V-E,$ where $V$ has a conormal singularity along a hypersurface.  The singular structure of $V$ affects the propagation of semiclassical singularities for solutions to $Pu=0,$ and in particular there is a `diffraction' of wavefront set by the interface: singularities are reflected as well as transmitted as they cross the interface transversely.  The reflected wave, however, is more regular, with the improvement depending on the regularity of the interface; moreover singularities cannot (for high enough regularity) glide along the interface.  (Joint work with Oran Gannot.)
(TCPL 201)
10:00 - 10:30 Coffee Break (TCPL Foyer)
10:30 - 11:20 Semyon Klevtsov: Geometry and large N asymptotics in Laughlin states
Laughlin states are N-particle wave functions, successfully describing the fractional quantum Hall effect (QHE) for plateaux with simple fractions. It was understood early on, that much can be learned about QHE when Laughlin states are considered on a Riemann surface. Mathematically, it is interesting to know how do the Laughlin states depend on the Riemannian metric, magnetic potential function, complex structure moduli, singularities -- for the large number of particles N. I will review the results, conjectures and further questions in this area, and relation to topics such as Coulomb gases/beta-ensembles, Bergman kernels for holomorphic line bundles, Quillen metric, zeta determinants.
(TCPL 201)
11:30 - 12:00 Checkout by Noon
5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.
(Front Desk - Professional Development Centre)
12:00 - 13:30 Lunch from 11:30 to 13:30 (Vistas Dining Room)