Boost Your Math Modeling Skills

Why is modeling the joint distribution between many continuous random variables, obtains generalization more easily?
A fundamental problem that makes language modeling and other learning problems difficult is the curse of dimensionality. It is particularly obvious in the case when one wants to model the joint distribution between many discrete random variables (such as words in a sentence, or discrete attributes in a data-mining task). For example, if one wants to model the joint distribution of 10 consecutive words in a natural language with a vocabulary V of size 100,000, there are potentially $10000010-<!--\; -\; -->1=1050-<!--\; -\; -->1$ free parameters. When modeling continuous variables, we obtain generalization more easily (e.g. with smooth classes of functions like multi-layer neural networks or Gaussian mixture models) because the function to be learned can be expected to have some local smoothness properties. For discrete spaces, the generalization structure is not as obvious: any change of these discrete variables may have a drastic impact on the value of the function to be estimated, and when the number of values that each discrete variable can take is large, most observed objects are almost maximally far from each other in hamming distance.
Can someone explain, in simple words, why is that "When modeling continuous variables, we obtain generalization more easily"?

What is a formal model for equations in non-commutative ring?
The standard formal modeling for polynomials is the polynomial ring $R[X_1,...,X_n]$ which is a monoid ring R[Nn] over an rng R.
Under this construction, it is possible to commute $X_i$'s so that $X_i{X}_j=X_j{X}_i$.
However, over a non-commutative ring, we need more than this.
For example, say we want to find a solution $(x_1,x_2,x_3)$ of a equation $a{X}_1{X}_2^2{X}_3+b{X}_3{X}_1{X}_2=0$ over a non-commutative ring.
If we want to find 3-tuples in the center of the ring, then the standard polynomial ring can be used to find the solutions. However, in general, we don't want $X_i$'s to commute. For example, we want $(X_1{X}_2^2{X}_3)(X_3{X}_1)=X_1{X}_2^2{X}_3^2{X}_1$. Moreover, if there is a formal model for polynomials, it always form a ring.
Consquently, rather than ${\mathbb{N}}^n$, we need somewhat more complex structure.
Say, R[M] is a monoid ring over R and it is a correct model of polynomial rings.
Then, M should have information about how variables are ordered and what the powers of each variable in a term are. What would M be?
Is there a way to construct polynomials in non-commutative rings?

ODE water drop modeling question
I have been working on a ODE homework which involves modeling the velocity of a drop of water falling from the sky. The ODE that models its velocity is given by:
$$m{v}^{\prime }=k{v}^2-<!--\; -\; -->mg,k=\frac{1}{2}{C}_d{\mathrm{\rho <!--\; \rho \; -->}}_a,$$
$C_d$: friction, ${\mathrm{\rho <!--\; \rho \; -->}}_a$: air density, A: transversal section of the water drop.
I have had to find the theoretical velocity limit of the water drop as a function of A by solving directly the ODE and compare these results with Euler and Runge-Kutta IV methods on MATLAB. I have done all of that.
The last question of my homework is an open question: It asks to modify the ODE presented by imagining now that the water drop losses a fraction of mass (by evaporation) while it is falling. I will have to apply Euler and Runge-Kutta IV on this new ODE. So I am looking for suggestions to improve the equation. Mass m now is going to be a function of time, it has to decrease. I have been thinking to assume that the water drops are spheres and by using $density\ast <!--\; \ast \; -->volume=mass$

Modeling conditional probability
We are given three coins: one has heads in both faces, the second has tails in both faces, and the third has a head in one face and a tail in the other. We choose a coin at random, toss it, and the result is heads. What is the probability that the opposite face is tails?
Solution:
If $p=P(\text{Two headed coin was chosen | Heads came up})={\displaystyle \frac{\frac{1}{3}}{\frac{1}{2}}=\frac{2}{3}}$
New to probability, would like to see how this solution works in more detail.
My questions:
1.The sample space is {HH,HT,TT}. The condition that heads came up refers to the elements {HH,HT} and the fact that two headed coin was chosen refers to {HH}. We want the probability of $\{HH,HT\}\cap <!--\; \cap \; -->\{HH\}=\{HH\}$. The probability of choosing HH is ${\displaystyle \frac{1}{3}}$ by Uniform Probability Law. Is this correct?
2. When we calculate that $P(\text{Heads came up})={\displaystyle \frac{1}{2}}$, what's the sample space? Is it {H,T} or {HH,HT,TT}?

Modeling Combinations
"We are given n points in the plane such that no three are collinear and no four lie on the same circle. How many circles are there that contain three of these points?"
Now I attempted to solve it and I ended up having 3 circles as the answer. I got there by actually drawing possibilities. However, I feel it's wrong. Any correction or explanation is very much appreciated.

Modeling the Decay of a Pack of Cannibalistic Hyenas
A population of $p_0$ hyenas has run out of food in their ecosystem, and so sadly they have resorted to eating each other.
Hyenas need to consume one meal a day, and so exactly once per day, any given hyena will kill another hyena. The time at which this happens is random, meaning each hyena's mealtime is uniformly distributed throughout a set of 24 hours.
Assuming that the last hyena will get hungry and die trying to eat himself, after how long will the population of hyenas become extinct?
I'm can think of two different ways to answer this (neither of which I know how to solve). The first is simpler and less accurate (and will therefore merit less glory).

Sine and Cosine Models
This is a general question about modeling the seasons using sine and cosine functions; I am trying to use sine and cosine to model cyclic behavior in sales due to the seasons (spring, summer, fall and winter); I have the following seasonal factors so to speak:-
$t=0$ Start of year
$t=3$ Spring - $\frac{\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}t)}{12}$
$t=6$ Summer - $\frac{\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}t)}{12}$
$t=9$ Fall - $\frac{\sin <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}t)}{18}$
$t=12$ Winter - $\frac{\cos <!--\; \; -->(\mathrm{\pi <!--\; \pi \; -->}t)}{6}$
I want to get sales to be max in each period for a given product. For example I want sales for sandals to max in the spring and sales for cardigans to max in the fall.
I Would appreciate advice and guidance as am not sure if these are correct. But looking at the graphs, they seem reasonable.

Separable differential equation modeling salt dissolved in water
I have no idea what to do for this. The equation is supposed to model a solution having salty water dumped into a tank that leaks the solution.
? A tank contains 1000L of pure water. Brine that contains .05 kg of salt per liter of water enters the tank at a ra te of 5L/min. Brine that contains .04 kg of salt per liter of water enters the tank at a rate of 10L/min. The solution is kept thoroughly mixed and drains from the tank at a rate of 15L/min. From Stewart Calculus 7e
I am supposed to find the salt at t minutes and at one hour.
I try to simplify this problem by making it .07kg per lit of water at 5L/m and from here I have no idea how to make an equation for this.

Modeling following constraints in MILP
I want to know how I should formulate the following constraints in my MIP problem?
$$x=x_1{z}_1+\cdots <!--\; \cdots \; -->+x_n{z}_n\text{ and }{y}_1\le <!--\; \le \; -->y\le <!--\; \le \; -->y_n\text{ and }{z}_1+\cdots <!--\; \cdots \; -->+z_n=1$$
OR
$$y=y_1{w}_1+\cdots <!--\; \cdots \; -->+y_n{w}_n\text{ and }{x}_1\le <!--\; \le \; -->x\le <!--\; \le \; -->x_n\text{ and }{w}_1+\cdots <!--\; \cdots \; -->+w_n=1$$
x and y are continuous variables. $z_1,\dots <!--\; \dots \; -->,z_n$ and $w_1,\dots <!--\; \dots \; -->,w_n$ are binary decision variables. $x_1,\dots <!--\; \dots \; -->,x_n$ and $y_1,\dots <!--\; \dots \; -->,y_n$ are parameters.

Derive ordinary differential equation in physical modeling of a searchlight reflector
What shape should the reflector (a rotating surface) of the searchlight have to be, so that the light beam from the point source can be reflected as parallel wire bundles？
This problem is just about an ODE, and my biggest problem is that I don't know how to construct differential equations from physics. The answer provided by the author is
$$y^2=2C(x+\frac{1}{2}C)$$

Modeling the coin weighting problem
Suppose we have n coins with weights 0 or 1 and a scale for weighting them. We would like to determine the weight of each coin by minimizing the number of weightings.
The book that I am reading states that the above problem can be modeled in the following way. We say that the subsets $S_1,\dots <!--\; \dots \; -->,S_m$ of {1,…,n} are determing if any $T\subseteq <!--\; \subseteq \; -->\{1,\dots <!--\; \dots \; -->,n\}$ can be uniquely determined by the cardinalities $|S_i\cap <!--\; \cap \; -->T|$ for $1\le <!--\; \le \; -->i\le <!--\; \le \; -->m.$. The minimum number of weightings is then equivalent to the least m for which a determing sequence of sets exists.
My question is. How exactly does this reduce to the coin weighting problem?

$\models <!--\; \models \; -->A$ vs. $A\models <!--\; \models \; -->$
As far as I understand the notion of semantic consequence (denoted by $\models <!--\; \models \; -->$), $\models <!--\; \models \; -->A$ means A is a semantic consequence of the empty set. So the "empty space" on the left side of the double turnstile means "empty set".
However, when we take a look at $A\models <!--\; \models \; -->$, now that means A is a contradiction, i.e., everything is a semantic consequence of A. Now the empty space means "everything".
Why is that? Is there any explanation for that difference?

Modeling a Heat PDE
Consider a wall made of brick 10 centimeters thick, which separates a room in a house from the outside. The room is kept at 20 degrees. Initially the outside temperature is 10 degrees and the temperature in the wall has reached steady state. Then there is a sudden cold snap and the outside temperature drops to -10 degrees. Find the temperature in the wall as a function of position and time.
I am okay executing the separation of variables technique, but I can't really reason through how to model this scenario. The solution manual states that the Initial/Boundary Value Problem is...
$$u_t=k{u}_{xx},u(0,t)=20,u(10,t)=-<!--\; -\; -->10,u(x,0)=20-<!--\; -\; -->x$$
This question comes in a section before higher dimensional heat equations are introduced, but to me, it seems that this should be modeled as a three-dimensional heat equation, because thin walls are two-dimensional, and the bricks are prescribed thickness. How can I intuitively reason through this word problem to model it correctly?

How to model 3 glass problem to graph?
The barman gives you three glasses whose sizes are 1000ml, 700ml, and 400ml, respectively. The 700ml and 400ml glasses start out full of beer, but the 1000ml glass is initially empty. You can get unlimited free beer if you win the following game: Game rule: You can keep pouring beer from one glass into another, stopping only when the source glass is empty or the destination glass is full. You win if there is a sequence of pourings that leaves exactly 200ml in the 700ml or 400 ml glass.

What interger operation is modeled below?

Determine, to find out the time when I and my friend meet

Write an inequality to represent how many more people can enter the pool