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## Saturday, March 23, 2024

### Our world was designed from outside. Prove?

## Wednesday, February 14, 2024

### Negotiations

Negotiations are everywhere - at war, in trade, in personal relationship, even in relationships with yourself. Hereby I list some hints as they come to mind either internally or from external sources.

Negotiations process is a poker game. So same hints apply. Another science is mathematical game theory, however this one is way more sophisticated.

Don't trust. War is an art of lies - Sun Tzu.

Less talking, more questioning. Ask as many questions as you can, get as much information as you can.

Don't insult. Threat can also be an insult so use it only as a last remedy. If the other side isn't afraid to lose it can fight till the end just because it is insulted.

Less proposals, more inquiries. Let the other side make a proposal first so you know their limits and you have more information. Make your own proposal only in the end.

Plan B. Always have plan B for any step. Especially for such dangerous steps as proposals and threats. What if you proposal is rejected, what if they don't care about your threat.

Break down demands into small parts. If there are multiple demands, one of them is always major. So we need to find which one.

Don't be afraid. Turn off your fear. Fear prevents us from making good decisions. Use logic instead.

Don't take a decision under pressure. The other party might make a rush. We need to distinct serious ultimatum and alleged rush.

## Saturday, February 10, 2024

### Cruelty to animals

Vegans are concerned about cruelty to animals. They refuse to eat animal food because animals are killed in farms. Does it make sense? Yes and no.

Cruelty should be divided in 2 parts. First is cruelty for necessity. So it is somehow justified. The second is opposite - cruelty without necessity hence not justified.

Justified cruelty

Predators need to kill other animals to get food and survive. Wolf eats deer, fish eats other fish, insects eat other insects, even some plants are able to kill and eat small insects. Anyway, each one of us kill and eat bacteria and microorganisms even though we don't want and don't know about that.

Humans kill animals at farms to eat them. Even though I personally don't eat meat, this type of cruelty still can be considered as justified.

While refusing animals meat, vegans eat insects, seafood, plant roots and leafs. Plant doesn't want us to eat it's roots because it will die without roots. Roots is a crucial part of the plant body, it serve for digestion of water from the ground.

Neither plants want us to eat their leafs. Leafs play important role in the breathing process of the plant.

The only thing the plant wants us to eat is its fruit. Fruit serves for populating area and thanks to it seeds are spread on a long distance around the tree. So this is the only thing we should eat if we want to be truly vegans.

Non-justified cruelty

If cruelty is applied with no valid human reason it is not justified. Examples are circuses, zoo, sterilization. To understand the level of cruelty ask yourself: would you be ready to treat your child in the same way? What would you feel if someone treated you in the same way?

Just imagine your child is beaten by the circus trainer just to get skill to entertain public. Imagine your child to be hold in a dirty cage. This is even worse than a prison.

Imagine someone told you that you must be sterilized just to prevent overpopulation of similar creatures like you. What would you feel? Official excuse is homeless dogs contribute to overpopulation and deceases spread. However humans consume much more resources than dogs, humans harm our planet more, humans spread deceases either. Why don't sterilize humans then? Eugenics tried to do that in the beginning of the twentieth century. Today we recognize eugenics as racism and nazism. Why don't we recognize in the same way sterilization of animals?

This is the way we treat animals. At the same time we lie that we love them. It is not love.

We bring our children to zoo and circuses to entertain them. What a fun. We teach cruelty since the childhood. And this cruelty is not justified. It will never be.

## Thursday, February 8, 2024

### Society of excessive consumption

We complain about poverty and ecology problems, however at the same time we contribute to them by over consumption.

Examples are:

Cutting grass in the yard. Is there any value of doing that apart from that yard "looks cute and clean"? Greens produce oxygen. If greens are cut, they lay down and become rotten producing methane which destroys our atmosphere.

Cosmetics and perfume manufacturing. Any girl spending twenty bucks for a gym looks much more cute than the girl spending the same money on cosmetics. Cosmetics manufacturing harms ecology.

Gym and sports. I don't understand people paying for gym and taking taxi or using elevators at the same time.

Cleaning supplies, powders etc. Is there any value of using them apart from that clothes look cute and clean? If you want to kill all the microbes on the planet with chemicals, you need to kill all the life on the planet. Chemicals leftovers after washing machine go to rivers and oceans. They poison fish and we eat this fish afterwards.

Home renovation.

Spices, cooking food. These artificially stimulate our appetite which results in overeating, not good for health.

Rank and gender needs

Cosmetics, luxurious products, everything that make you look "beautiful" is the result of the need for higher status (rank). Males and females constantly fight for attention of each other - it is very natural.

To attract a partner of the opposite sex, one need to show its higher status (rank) in the pride. The higher your rank the more attractive you are for potential sex partners. We need to attract sex partners because it is our way to populate our genes over the planet, it is our instinct.

Big animals use power to demonstrate their status. Smaller animals like birds have bright colors of the body to attract opposite gender.

Humans didn't go far from that. Some of us use "fair" ways to demonstrate our attractiveness. We do sports to become more healthy, strong and hence attractive.

In the current society, not only physical gorilla power demonstrates our rank. Country president could be more attractive to women that a good athlete. The one's rank is determined by his or her position in the society, money power, carrier etc. That's why we study, fight for high paying jobs, make business.

However what the one could do if he or she has nothing to increase his or her rank in a "fair" way? What if you are not an athlete, you are not intelligent to get a good job?

Right. If the product is selling badly, invest in marketing and advertising. Doesn't matter what is inside the box, we will make the package look perfect. This is the easiest way to attract a potential buyer.

We will apply cosmetics, we will make photos with the background of luxury cars, we will dress on attractive dressings etc. All of these are redundant spendings for the society and these harm planet ecology. But these help to increase ranks of those species with lower statuses.

## Tuesday, February 6, 2024

### To the book "Visual group theory" by N. Carter: abstract algebra exercises, questions, ideas...

*Question 1.1. Can we apply any generator to any achievable Rubik's cube state?*

*Solution*

*Question 1.2.*

*Out of rules 1.6-1.8, is there any one redundant?*

*Question 1.3.*

*If generators are reversible, are combined actions reversible?*

*Solution*

Yes it does. Let's take some random action G bringing us from the node A0 to An. This action is the result of applying n generators as follows:

generator g0 brings us from A0 to A1,

generator g1 brings us from A1 to A2,

...

generator gn brings us from An-1 to An.

Let's denote reverse elements for generators as h.

Then

h1 is inverse for g1,

h2 is inverse for g2,

...

hn is inverse for gn.

Since every generator is reversible, we can apply h elements. Let's apply hn to An,...,h1 to A1. We can prove by mathematical induction that this gives us A0 in the end. Coming from An to A0 is an inverse action to G.*Question 1.4.*

*Does a group comply with commutative property?*

*Solution*

*Question 1.5.*

*Is chess game a group? Why? What are actions in this case?*

*Question 1.6.*

*What does really Rule 1.8 mean?*

*infra*solutions for Exercise 1.3, Exercise 1.5

*,*Exercise 2.17.

*Question 1.7. How many states can the object of Rubik's cube have?*

*Question 1.8. What terminology notation can we use for groups like Rubik's cube description?*

*Solution*

*Question 1.9. What is the strict definition of a generator?*

*Question 1.10. What does Rule 1.5 really mean?*

*supra*and Exercise 2.14 also.

*Question 1.11. How can we automate searching for Rubik's cube game solution?*

*Exercise 1.1*

*Exercise 1.2*

*Exercise 1.3*

*Exercise 1.4*

*Exercise 1.5*

*per se*. If we know what action is we then can count how many action does a group have.

*infra*.

*Exercise 1.6*

*supra*: quantity of actions equals to quantity of unique object states.

*Exercise 1.7*

*Exercise 1.8*

*Exercise 1.9*

*Exercise 1.10*

*Exercise 1.11*

*Exercise 1.12*

*Exercise 1.13*

*supra*.

*Exercise 1.14*

*Question 2.1. How to present Cayley diagram as a table (matrix)?*

*Solution*

g1 | g2 | ... | gm | |

s1 | ||||

s2 | ||||

... | ||||

sn |

*Question 2.2. Can we get from any point to any point in Cayley diagram of a group?*

*Solution*

*supra*we noted that we can get from the starting node to any node.

*Question 2.3. If we could get from any node to any other node does it mean that every action is reversible?*

*Solution*

*Question 2.4. What if we choose another node for start?*

*Question 2.5 What algorithm can allow us to search for all the possible object states in a given group?*

*Solution*

*Supra*in Question 2.1 we noted that Cayley diagram could be presented as a matrix: states x generators. Let's use this format for finding all the possible states.

g1 | g2 | ... | gm | |

s1 | ||||

s2 | ||||

... | ||||

sn |

g1 | g2 | ... | gm | |

s1 | ||||

s2 | ||||

... | ||||

sn | ||||

… | ||||

sn+k |

*Question 2.6. How can we automatically convert Cayley diagram to a matrix from Question 2.1 and back?*

*Question 2.7. Should Cayley diagram contain generators only or all the possible actions and why?*

*Exercise 2.1*

*Exercise 2.2*

*Exercise 2.3*

*Exercise 2.4*

*Exercise 2.5*

*Exercise 2.6*

*supra,*we draw generators only in our diagram. So we don't depict infinitely many arrows for all the possible actions.

*Exercise 2.7*

*Exercise 2.8*

*Exercise 2.9*

*supra*.

*supra*the diagram shows that h and b are sufficient to generate V4 because of the following: we can get from any point to any other point on the diagram with the current arrows only. See also Question 2.3 supra.

*Exercise 2.10*

*Exercise 2.11*

*Exercise 2.12*

*Exercise 2.13*

*supra*.

*Exercise 2.14*

*infra*.

*Exercise 2.15*

*Exercise 2.16*

*Exercise 2.17*

*supra*. Every node should have outbound arrows for all the generators. Otherwise, this Cayley diagram is not a group. E. g. like in Exercise 2.15

*supra,*this diagram is not in compliance with Rule 1.8:

*Exercise 2.18*

*supra*. The previous square rectangle game from Exercise 2.8 has the following properties:

(2) If possible, quantity of generators should be minimized. Like the book author did it in his solution for Exercise 2.8. Though there is no official requirement for minimization.

*supra*. Note we can form a group not complying with this optional requirement: say we have a rectangle puzzle from Section 2.2 with only one action of horizontal flip.

*-*Turn the triangle clockwise 120 degree around its centroid.

*supra*in Exercise 2.19(c).

*Exercise 2.19*

*supra*(like the book author proposed in Section 2.2).

*Or we can try to prove this conjecture for the common case which seems like more difficult. Anyway this question is still open..*

*supra*in Exercise 2.19(c).

*supra*we considered two cases for the conjecture:

*supra*. However we do not provide a proof for this case hereby. There are some difficulties. There is a situation when a polygon has even number of vertices. E. g. look at the square from Exercise 2.8. Flipping around two medians is not enough to cover all the possible positions of the placeholder shape.

*supra*.

*supra*.

*supra*. Any generator results in the polygon taking the same shape of the placeholder due to the symmetry of the polygon. So we can apply any generator to this shape again.

*supra*. However the question is open as whether we are able to generate all the possible positions of the figure in the placeholder shape as as per clause (3)

*infra*. Theoretically we could define a generator as a consequence of IF algorithm statements like for pentagon say: if it is iteration 1 then rotate, if it is iteration 6 then flip etc. However I assume it will violate Rule 1.7 in the book author's meaning. So the question bout compliance with this (2) clause is still open.

*supra.*Number of states for n-gon equals to 2n. We can get them as follows. By rotating the figure n times by 360/n degrees each time, we get first n positions.

*Question 3.1. What is the real practical use of groups?*

*Question 3.2. Should we improve the step 1 in Definition 3.1 so it becomes stricter and less vague?*

*maximum*number of equal parts. Parts are considered equal if (1) their figures are equal and (2) they are surrounded by the equal figures (spaces)". Namely:

*maximum*when counting the quantity of equal parts is necessary here. E.g. we can break equilateral triangle in two equal parts (laying on both sides of any median) as well as in three equal parts.

*maximum*equal parts. Also we can't take quarter leaf because the parts should be surrounded by

*equal*spaces.

*Question 3.3. Should we improve the step 2 in Definition 3.1 so it becomes stricter and less vague?*

*minimum*list of actions that cover all the possible shape positions within its placeholder". There is only one possible quantity number for the count of the minimum list of actions. However there could be multiple possible lists of actions complying with the given criteria. E. g. for Figure 3.2 (discussed by the author of the book in Section 3.1.1), first option is the action of rotating clockwise and second option is the action of rotating counterclockwise. Both of them give the same results in terms of possible positions sets.

*Question 3.4. How do we define symmetry?*

*Solution*

*Question 3.5. For Question 3.2 supra, how can we prove that the current quantity of parts is maximum possible?*

*Question 3.6. For Question 3.3 supra, how can we prove that the current quantity of generators is minimum possible?*

*Question 3.7. For every infinite pattern, can we create a finite analogue with similar generators?*

*Question 3.8. In Definition 3.1 step 1 how can we algorithmize the process?*

*Question 3.9. In Definition 3.1 step 2 how can we algorithmize the process?*

*Question 3.10.*

*Other than Definition 3.1, is there other way to create a group based on symmetry?*

*Question 3.11. Can the group be generated based on non-symmetric geometric figure?*

*Exercise 3.1*

*Exercise 3.2*

*Exercise 3.3*

*Exercise 3.4*

*Exercise 3.5*

*Exercise 3.6*

*Exercise 3.7*

*Exercise 3.8*

*Exercise 3.9*

*Exercise 3.10*

*Exercise 3.11*

*Exercise 3.12*

*Exercise 3.13*

*Exercise 3.14*

*Exercise 3.15*

*Exercise 3.16*

*Question 4.1. Can number of columns in a multiplication table be lower?*

*Question 4.2. How can creation of multiplication tables be automated?*

*infra*solution to Exercise 4.8 in part of Reference to Exercise 2.8. At least partially we could ease monkey job in that solution. However some is still remaining.

*Question 4.3. Are all 4 of the Definition 4.2 clauses necessary?*

*Question 4.4. Can we take any node as a starting point in the diagram of actions?*

*Question 4.5. Can 2 row headers of a multiplication table be the same?*

*Solution*

*Question 4.6. For the same given group with the same starting node, can there be 2 different multiplication tables?*

*Solution*

*infra,*references to Exercises 2.5 and 1.4(b). We can come to the 321 node via 2 different ways: lrl and rlr.

*Exercise 4.1*

*Exercise 4.2*

*Exercise 4.3*

*Exercise 4.4*

*Exercise 4.5*

*Exercise 4.6*

*(c)*

a | b | c | d | e | x | y | z | a2 | b2 | c2 | d2 | |

a | a | b | c | d | e | x | y | z | a2 | b2 | c2 | d2 |

b | b | a | d | c | x | e | z | y | c2 | d2 | a2 | b2 |

c | c | d | a | b | z | y | x | e | d2 | c2 | b2 | a2 |

d | d | c | b | a | y | z | e | x | b2 | a2 | d2 | c2 |

e | e | y | x | z | a2 | b2 | d2 | c2 | a | c | d | b |

x | x | z | e | y | c2 | d2 | b2 | a2 | b | d | c | a |

y | y | e | z | x | b2 | a2 | c2 | d2 | d | b | a | c |

z | z | x | y | e | d2 | c2 | a2 | b2 | c | a | b | d |

a2 | a2 | d2 | b2 | c2 | a | c | b | d | e | x | z | y |

b2 | b2 | c2 | a2 | d2 | d | b | c | a | y | z | x | e |

c2 | c2 | b2 | d2 | a2 | b | d | a | c | x | e | y | z |

d2 | d2 | a2 | c2 | b2 | c | a | d | b | z | y | e | x |

n | rrbr | b | rbrr | rr | rrb | rbr | brr | r | rb | brb | br |

*Exercise 4.7*

... | ... | -3 | -2 | -1 | 0 | 1 | 2 | 3 | ... |

... | ... | ... | ... | ... | ... | ... | ... | ... | ... |

-3 | ... | -6 | -5 | -4 | -3 | -2 | -1 | 0 | ... |

-2 | ... | -5 | -4 | -3 | -2 | -1 | 0 | 1 | ... |

-1 | ... | -4 | -3 | -2 | -1 | 0 | 1 | 2 | ... |

0 | ... | -3 | -2 | -1 | 0 | 1 | 2 | 3 | ... |

1 | ... | -2 | -1 | 0 | 1 | 2 | 3 | 4 | ... |

2 | ... | -1 | 0 | 1 | 2 | 3 | 4 | 5 | ... |

3 | ... | 0 | 1 | 2 | 3 | 4 | 5 | 6 | ... |

... | ... | ... | ... | ... | ... | ... | ... | ... | ... |

*Exercise 4.8*

pn | np | |

pn | pn | np |

np | np | pn |

123 | 132 | 213 | 231 | 312 | 321 | |||

n | r | l | lr | rl | lrl | |||

123 | n | 123 | 132 | 213 | 231 | 312 | 321 | |

132 | r | 132 | 123 | 312 | 321 | 213 | 231 | |

213 | l | 213 | 231 | 123 | 132 | 321 | 312 | |

231 | lr | 231 | 213 | 321 | 312 | 123 | 132 | |

312 | rl | 312 | 321 | 132 | 123 | 231 | 213 | |

321 | lrl | 321 | 312 | 231 | 213 | 132 | 123 | |

123 | 132 | 213 | 231 | 312 | 321 | |||

n | r | l | lr | rl | lrl | |||

123 | n | n | r | l | lr | rl | lrl | |

132 | r | r | n | rl | lrl | l | lr | |

213 | l | l | lr | n | r | lrl | rl | |

231 | lr | lr | l | lrl | rl | n | r | |

312 | rl | rl | lrl | r | n | lr | l | |

321 | lrl | lrl | rl | lr | l | r | n | |

*supra*. If it is about integers than the multiplication matrix is as follows:

... | ... | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | ... |

... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |

-5 | ... | -10 | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | ... |

-4 | ... | -9 | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | ... |

-3 | ... | -8 | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | ... |

-2 | ... | -7 | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | ... |

-1 | ... | -6 | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | ... |

0 | ... | -5 | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | ... |

1 | ... | -4 | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | ... |

2 | ... | -3 | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | ... |

3 | ... | -2 | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | ... |

4 | ... | -1 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | ... |

5 | ... | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ... |

... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... | ... |

1 | -1 | |

1 | 1 | -1 |

-1 | -1 | 1 |

1 | 2 | N | ||

3 | 4 | |||

1 | 3 | RH | ||

2 | 4 | |||

3 | 1 | R | ||

4 | 2 | |||

2 | 1 | H | ||

4 | 3 | |||

4 | 2 | RV | ||

3 | 1 | |||

4 | 3 | RR | ||

2 | 1 | |||

3 | 4 | V | ||

1 | 2 | |||

2 | 4 | RRR | ||

1 | 3 |

N | RH | R | H | RV | RR | V | RRR | |

N | NN | NRH | NR | NH | NRV | NRR | NV | NRRR |

RH | RHN | RHRH | RHR | RHH | RHRV | RHRR | RHV | RHRRR |

R | RN | RRH | RR | RH | RRV | RRR | RV | RRRR |

H | HN | HRH | HR | HH | HRV | HRR | HV | HRRR |

RV | RVN | RVRH | RVR | RVH | RVRV | RVRR | RVV | RVRRR |

RR | RRN | RRRH | RRR | RRH | RRRV | RRRR | RRV | RRRRR |

V | VN | VRH | VR | VH | VRV | VRR | VV | VRRR |

RRR | RRRN | RRRRH | RRRR | RRRH | RRRRV | RRRRR | RRRV | RRRRRR |

*supra*. So we shorten all the mentioned combinations from the table. We can do that because of associativity property of this group. Why this group is associative? (we don't list a proof here) Also we remove excessive N. We have:

N | RH | R | H | RV | RR | V | RRR | |

N | N | RH | R | H | RV | RR | V | RRR |

RH | RH | N | RHR | R | RHRV | RHRR | RHV | RHRRR |

R | R | RRH | RR | RH | RRV | RRR | RV | N |

H | H | HRH | HR | N | HRV | HRR | HV | HRRR |

RV | RV | RVRH | RVR | RVH | N | RVRR | R | RVRRR |

RR | RR | RRRH | RRR | RRH | RRRV | N | RRV | R |

V | V | VRH | VR | VH | VRV | VRR | N | VRRR |

RRR | RRR | H | N | RRRH | V | R | RRRV | RR |

N | RH | R | H | RV | RR | V | RRR | |

N | N | RH | R | H | RV | RR | V | RRR |

RH | RH | N | H | R | RR | RH | RRR | V |

R | R | V | RR | RH | H | RRR | RV | N |

H | H | RRR | RV | N | R | V | RR | RH |

RV | RV | RR | V | RV | N | RH | R | H |

RR | RR | R | RRR | V | RH | N | H | R |

V | V | R | RH | RR | RRR | H | N | RV |

RRR | RRR | H | N | RV | V | R | RH | RR |

*Exercise 4.9*

*Exercise 4.10*

*Exercise 4.11*

*Exercise 4.12*

*Exercise 4.13*

*Exercise 4.14*

*Exercise 4.15*

*Exercise 4.16*

*Exercise 4.17*

*Exercise 4.18*

*Exercise 4.19*

*Exercise 4.20*

*Exercise 4.21*

*Exercise 4.22*

*Exercise 4.23*

*Exercise 4.24*

*Exercise 4.25*

*Exercise 4.26*

*Exercise 4.27*

*Exercise 4.28*

*Exercise 4.29*

*Exercise 4.30*

*Exercise 4.31*

*Exercise 4.32*

*Exercise 4.33*

*Exercise 5.1*

*Exercise 5.2*

*Exercise 5.3*

*Exercise 5.4*

*Exercise 5.5*

*Exercise 5.6*

*Exercise 5.7*

*Exercise 5.8*

*Exercise 5.9*

*Exercise 5.10*

*Exercise 5.11*

*Exercise 5.12*

*Exercise 5.13*

*Exercise 5.14*

*Exercise 5.15*

*Exercise 5.16*

*Exercise 5.17*

*Exercise 5.18*

*Exercise 5.19*

*Exercise 5.20*

*Exercise 5.21*

*Exercise 5.22*

*Exercise 5.23*

*Exercise 5.24*

*Exercise 5.25*

*Exercise 5.26*

*Exercise 5.27*

*Exercise 5.28*

*Exercise 5.29*

*Exercise 5.30*

*Exercise 5.31*

*Exercise 5.32*

*Exercise 5.33*

*Exercise 5.34*

*Exercise 5.35*

*Exercise 5.36*

*Exercise 5.37*

*Exercise 5.38*

*Exercise 5.39*

*Exercise 5.40*

*Exercise 5.41*

*Exercise 5.42*

*Exercise 5.43*

*Exercise 5.44*

*Exercise 6.1*

*Exercise 6.2*

*Exercise 6.3*

*Exercise 6.4*

*Exercise 6.5*

*Exercise 6.6*

*Exercise 6.7*

*Exercise 6.8*

*Exercise 6.9*

*Exercise 6.10*

*Exercise 6.11*

*Exercise 6.12*

*Exercise 6.13*

*Exercise 6.14*

*Exercise 6.15*

*Exercise 6.16*

*Exercise 6.17*

*Exercise 6.18*

*Exercise 6.19*

*Exercise 6.20*

*Exercise 6.21*

*Exercise 6.22*

*Exercise 6.23*

*Exercise 6.24*

*Exercise 6.25*

*Exercise 6.26*

*Exercise 6.27*

*Exercise 6.28*

*Exercise 6.29*

*Exercise 6.30*

*Exercise 6.31*

*Exercise 7.1*

*Exercise 7.2*

*Exercise 7.3*

*Exercise 7.4*

*Exercise 7.5*

*Exercise 7.6*

*Exercise 7.7*

*Exercise 7.8*

*Exercise 7.9*

*Exercise 7.10*

*Exercise 7.11*

*Exercise 7.12*

*Exercise 7.13*

*Exercise 7.14*

*Exercise 7.15*

*Exercise 7.16*

*Exercise 7.17*

*Exercise 7.18*

*Exercise 7.19*

*Exercise 7.20*

*Exercise 7.21*

*Exercise 7.22*

*Exercise 7.23*

*Exercise 7.24*

*Exercise 7.25*

*Exercise 7.26*

*Exercise 7.27*

*Exercise 7.28*

*Exercise 7.29*

*Exercise 7.30*

*Exercise 7.31*

*Exercise 7.32*

*Exercise 7.33*

*Exercise 7.34*

*Exercise 7.35*

*Exercise 7.36*

*Exercise 7.37*

*Exercise 8.1*

*Exercise 8.2*

*Exercise 8.3*

*Exercise 8.4*

*Exercise 8.5*

*Exercise 8.6*

*Exercise 8.7*

*Exercise 8.8*

*Exercise 8.9*

*Exercise 8.10*

*Exercise 8.11*

*Exercise 8.12*

*Exercise 8.13*

*Exercise 8.14*

*Exercise 8.15*

*Exercise 8.16*

*Exercise 8.17*

*Exercise 8.18*

*Exercise 8.19*

*Exercise 8.20*

*Exercise 8.21*

*Exercise 8.22*

*Exercise 8.23*

*Exercise 8.24*

*Exercise 8.25*

*Exercise 8.26*

*Exercise 8.27*

*Exercise 8.28*

*Exercise 8.29*

*Exercise 8.30*

*Exercise 8.31*

*Exercise 8.32*

*Exercise 8.33*

*Exercise 8.34*

*Exercise 8.35*

*Exercise 8.36*

*Exercise 8.37*

*Exercise 8.38*

*Exercise 8.39*

*Exercise 8.40*

*Exercise 8.41*

*Exercise 8.42*

*Exercise 8.43*

*Exercise 8.44*

*Exercise 8.45*

*Exercise 8.46*

*Exercise 8.47*

*Exercise 8.48*

*Exercise 8.49*

*Exercise 8.50*

*Exercise 9.1*

*Exercise 9.2*

*Exercise 9.3*

*Exercise 9.4*

*Exercise 9.5*

*Exercise 9.6*

*Exercise 9.7*

*Exercise 9.8*

*Exercise 9.9*

*Exercise 9.10*

*Exercise 9.11*

*Exercise 9.12*

*Exercise 9.13*

*Exercise 9.14*

*Exercise 9.15*

*Exercise 9.16*

*Exercise 9.17*

*Exercise 9.18*

*Exercise 9.19*

*Exercise 9.20*

*Exercise 9.21*

*Exercise 9.22*

*Exercise 9.23*

*Exercise 9.24*

*Exercise 9.25*

*Exercise 9.26*

*Exercise 9.27*

*Exercise 9.28*

*Exercise 10.1*

*Exercise 10.2*

*Exercise 10.3*

*Exercise 10.4*

*Exercise 10.5*

*Exercise 10.6*

*Exercise 10.7*

*Exercise 10.8*

*Exercise 10.9*

*Exercise 10.10*

*Exercise 10.11*

*Exercise 10.12*

*Exercise 10.13*

*Exercise 10.14*

*Exercise 10.15*

*Exercise 10.16*

*Exercise 10.17*

*Exercise 10.18*

*Exercise 10.19*

*Exercise 10.20*

*Exercise 10.21*

*Exercise 10.22*

*Exercise 10.23*

*Exercise 10.24*

*Exercise 10.25*

*Exercise 10.26*

*Exercise 10.27*

*Exercise 10.28*

*Exercise 10.29*

*Exercise 10.30*