Boost Your Math Modeling Skills

Modeling differential equation
I am trying to model some differential equation which can explain buying some assets that can increase itself like a virus.
So, let's assume my money is y(t), and number of assets is n(t).
- y(t) to a certain level c, it will increase the n(t) by +1
- n(t) generate profit r %
so I can see something like
$$\begin{gathered} dy(t)/dt=y_0+r\cdot <!--\; \cdot \; -->n(t) \\ dn(t)/dt=If(y(t)\ge <!--\; \ge \; -->c,y(t)-<!--\; -\; -->20andn+1,n) \end{gathered}$$
However, I have no clue how I can model this discrete-continuous variable differential equation. Is there any way to model the system and get the solution?

Modeling with equations riddle
A father said that sevens years ago, he was eleven times as old as his daughter. Now he is four times as old as she is. How old is the father?
Can this be solved as a system of equations? I am stuck with this problem because I don't think that I am properly setting up the model.
$FathersAge-<!--\; -\; -->7=11daughtersage$
$4daughtersage=Fathersage$

Is there a relation between logical consequence, validity (tautology), and formula modeling?
I am new to propositional logic and I have noticed that these 3 things use the same symbol (double turnstile) and I am wondering if they are related to each other somehow.
to say that a formula B is a logical consequence of another formula A we write: $A\models B$.
also if we want to say that a valuation V models the formula A we write: $V\models A$.
and if we want to say that the formula A is valid we write: $\models A$.
so, here are my questions:
1. is there a reason they all use the same symbol?
2. can we define validity and formula modeling in terms of the logical consequence?
3. can a formula be a logical consequence of a valuation?

Variable naming convention in mathematical modeling
Is there a guide or source of inspiration I can use when picking variable and constant names for a mathematical formula? I'm in the process of converting a poorly written technical document into something cleaner, and I'm faced with garbage like this:
$HWP\mathrm{\_<!--\; \_\; -->}cu{r}_x=Cut\mathrm{\_<!--\; \_\; -->}cur\times <!--\; \times \; -->\underset{t=0}{\overset{ru\left(\frac{yr\mathrm{\_<!--\; \_\; -->}cur-<!--\; -\; -->start\mathrm{\_<!--\; \_\; -->}date}{Freq\mathrm{\_<!--\; \_\; -->}cu{r}_x}\right)}{\sum <!--\; \sum \; -->}}f(Decay\mathrm{\_<!--\; \_\; -->}cu{r}_x;yr\mathrm{\_<!--\; \_\; -->}cur-<!--\; -\; -->start\mathrm{\_<!--\; \_\; -->}dat{e}_x)$
I have a background in scientific computing, so I can appreciate an elegantly constructed formula, but I've always wondered how a good mathematician would choose variable names.

Question about problem linear programming math modeling
Consider points A(4.7,−4.1,−1.5),B(−0.4,−2.4,1.9),C(−0.3,−2.1,−6.5) and D(2.7,−3.6,4.0). How to discover if segment AB has intersection different of zero with the segment CD? Formulate this problem as a linear programming problem.

Exponential model for population
I am having trouble with a population modeling problem. The first part states:
Assume that Switzerland's population grows at a rate of 0.18 percent a year and that the 1988 population was 6.7 million.
I got the model to be:
$y(t)=(6.7\times {10}^6)e^{0.0018\ast <!--\; \ast \; -->t}$
I am having trouble with the second part of the problem:
If there is a net immigration of 10,000 people a year into Switzerland, write an expression for the population in year t.
I have tried multiple variations of adding 10000 and 10000t to different parts of the equation with no luck.
How do I account for the 10000 immigrants?

Modeling Integer LP
In the next semester, the school plans to replace the tutorials by self-organized student groups. Your task is to find the best possible partition of students into groups and to appoint a group leader for each group. To achieve this task, you are given a list of students students. Each student needs to be assigned to exactly one group, and each group needs to have a (student) leader. Every student can be a leader, but we can have only one per group. A leader needs to be assigned to the group he is leading. The number of groups needs to be at least g and at most G. The quality of the groups depends on two criteria. First, the collaboration quality of a student i in a group with leader j is given by collaboration[i,j]. Furthermore, each student i has a certain leadership quality given by leadership[j]. You have to maximize the overall collaboration quality plus the sum of the leadership qualities of the leaders.
I have come up with the following and was wondering if I'm missing something or if someone can show me in the right direction:
I used the variable $x_{i,j}$ if a student i is assigned to the group with leader j ($x_{i,j}=1$) or not ($x_{i,j}=0$). The variable yj shall denote whether student j is a leader ($y_j$) or not ($=0$).
I assumed that there are n students and m leaders. Then I came up with the following LP:
max $\underset{i=1}{\overset{n}{\sum <!--\; \sum \; -->}}\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_{i,j}\cdot <!--\; \cdot \; -->{\text{collaboration}}_{i,j}+\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{y}_j\cdot <!--\; \cdot \; -->{\text{leadership}}_j$s.t.
$\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{x}_{i,j}=1\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}$
$g\le <!--\; \le \; -->\underset{j=1}{\overset{m}{\sum <!--\; \sum \; -->}}{y}_j\le <!--\; \le \; -->G$
$x_{i,j}\le <!--\; \le \; -->y_j\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
$x_{i,j}\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}i\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,n\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
$y_j\in <!--\; \in \; -->\{0,1\}\mathrm{\forall <!--\; \forall \; -->}j\in <!--\; \in \; -->\{1,\dots <!--\; \dots \; -->,m\}$
This is as far as I have gotten. I know I'm missing the connection between $x_{i,j}$ and $y_j$. Also I'm not sure about my function I want to maximize. Help would be appreciated.

Confusion with modeling a trigonometric function with phase shift
I am studying trigonometry on Khan Academy and came across this problem:
The moon's distance from Earth varies in a periodic way that can be modeled by a trigonometric function.
When the moon is at its perigee (closest point to Earth), it's about 363,000 km away. When it's at its apogee (farthest point from Earth), it's about 406,000 k away. The moon's apogees occur 27.3 days apart. The moon will reach its apogee on January 22, 2016. Find the formula of the trigonometric function that models the distance D between Earth and the moon t days after January 1, 2016. Define the function using radians.
$D(t)=$
So the steps I took are:
Finding the amplitude: $(406000-<!--\; -\; -->36300)/2=21500$
Finding the midline: $21500+36300=384500$
Figuring out whether to use cosine or sine:
I figured that I can treat January 2nd as the beginning of the year. So I used cosine. Since at 0, a cosine function is at its max value.
4. The period: 1 year is a period so it must be $2\mathrm{\pi <!--\; \pi \; -->}/365$
5. The function without the shift is now: $21500\cos <!--\; \; -->(2\mathrm{\pi <!--\; \pi \; -->}/365t)+384500$
6. Now I must find the value of u in order to properly shift the function. I imagine that this must be $t-<!--\; -\; -->27.3$ since it is 27.3 days after January 1.
I feel like I must be missing something here or got one of the steps wrong. Please guide me in the right direction.

Logistic Regression and modelling probabiliy $\mathrm{\pi <!--\; \pi \; -->}$
If the link function is
$$g(\mathrm{\pi <!--\; \pi \; -->})=\log <!--\; \; -->(\frac{\mathrm{\pi <!--\; \pi \; -->}}{1-<!--\; -\; -->\mathrm{\pi <!--\; \pi \; -->}})=x^T\mathrm{\beta <!--\; \beta \; -->}$$
show that this is equivalent to modeling the probability $\mathrm{\pi <!--\; \pi \; -->}$ as,
$$\mathrm{\pi <!--\; \pi \; -->}=\frac{e^{x^T}\mathrm{\beta <!--\; \beta \; -->}}{1+e^{x^T}\mathrm{\beta <!--\; \beta \; -->}}$$
Again I think this is a simply inverse. But I'm not quite sure how to do it.

Consider a post office with two clerks. John, Paul, and Naomi enter simultaneously. John and Paul go directly to the clerks, while Naomi must wait until either John or Paul is finished before she begins service.
(a) If all of the service times are independent exponentially distributed random variables with the same mean 1/lambda, what is the probability that Naomi is still in the post office after the other two have left?
What if we change the assumption in (a) that the two clerks have different service rates lambda_1 and lambda_2?

Modeling the maximum number of moves in Tower of Hanoi problem
What would be the recursive algorithm for solving the Tower of Hanoi problem (with n disks and 3 pegs) in maximal number of moves (i.e. going through all possible disks/pegs combinations).

Mathematical modeling and operational research
I have a variable TE which $0\le <!--\; \le \; -->TE\le <!--\; \le \; -->180$
Also, I have a parameters L(k) and U(K), showing lower bound and upper bound for interval K $(k=1,2,3)$
The value of these parameters are as follows: $L(1)=0K(1)=60,L(2)=60K(2)=120,L(3)=120K(3)=180$
I need to find the related range of the variable TE. For example,
if `$TE=12$`, then `$K=1$`
if `$TE=150$`, then `$k=3$`
Which mathematical constraint(s) do this?

Violin String PDE Modeling
If you pluck a violin string, and then finger the string, fixing it precisely in the middle, the tone increases by one octave. In mathematical terms this means that the frequency is doubled. Explain why this happens.
Plucking the string without fingering should be represented as
$$u_{tt}=c^2{u}_{xx},u(0,t)=0,u(L,t)=0,u(x,0)=f(x),u_t(x,0)=0$$
Plucking the string and then fingering should be represented as
$$v_{tt}=c^2{v}_{xx},v(0,t)=0,v(\frac{L}{2},t)=0fort\ge <!--\; \ge \; -->t_1,v(L,t)=0,v(x,0)=f(x),v_t(x,0)=0$$
My game plan was to first solve for u, then find the first time at which $u=0$ at $\frac{L}{2}$ to use as $t_1$, and finally solve for v after $t_1$ is known. This means that the string is plucked from position f(x) and allowed to follow its natural trajectory only until the first time the string crosses the x-axis at $\frac{L}{2}$, and is then pinned on the x-axis forever. However, the PDE in v seems to have too many conditions for its order. Furthermore, the conditions of the PDE in v are a superset of the conditions of the PDE in u, so I think the solutions to the PDE in v should be a subset of the solutions to the PDE in u, even if it is somehow possible to apply this many conditions to the PDE in v. Are these PDEs possible to solve? Am I on the right track with this approach to the exercise?

Modeling simple linear equations
This should be pretty simple but I'm blanking on this. I need to model (graph) how path 1 becomes equally as efficient as path 2 as the distance of path 2 increases.
distance of path 1 (from A to B) $=10$
distance of path 2 (from A to C) $=y$
The path is a path taken by a forklift driver. The vehicle speed is 2.2m/s. The paths are measured in meters.
On path 1, there is an added flat time of 30 seconds (due to something he has to stop and do). On path 2, there is an added flat time of 10 seconds.
So in summary:
$$\frac{x}{2.2}+30=t_{\text{path }1}wherex>0$$
$$\frac{y}{2.2}+10=t_{\text{path }2}wherey>0$$
I know that once the distance of path $2=54$ meters, both routes take the same amount of time (34.545 seconds). But I can't figure out how to put this into equations that I can use to graph.

Modeling IP Problem
Hallco runs a day shift and a night shift. No matter how many units are produced, the only production cost during a shift is a setup cost. It costs $8,000 to run the day shift and $4,500 to run the night shift. Demand for the next two days is as follows: day 1, 2,000; night 1, 3,000; day 2, 2,000; night 2, 3,000. It costs $1 per unit to hold a unit in inventory for a shift. Determine a production schedule that minimizes the sum of setup and inventory costs. All demand must be met on time.
I am confused on how to model these sentences. It costs $1 per unit to hold a unit in inventory for a shift. What does it mean in mathematical language?

Continuous Hidden Markov Modeling
I am writing a speaker recognition program in Matlab using Mel Frequency Cepstral Coefficients, and although I have gotten the problem to work using discrete time wrapping, I was interested in try to successfully recognize speakers using Hidden Markov Modeling (HMM). I understand how to implement a HMM for simple examples (like urn and genie), but don't understand how it pertains to my problem exactly. For instance, what are actually the hidden states that the row vectors are trying to encode? And what does it mean to model these states as Gaussian mixtures? For more information on MFCCs, it gives for each speaker a matrix of numbers, where each row corresponds to the power spectrum at one given time.

Conditional Expectation of Response Variable given Predictor Variable in Statistical Modeling
Suppose we are given a set of data points $(X_i,Y_i),i=1,2,...,n$, where $X_i$ is the predictor variable, and $Y_i$ is the response variable.
- In the context of statistical modelling, we are interested in expressing Y as
$$Y_i=s(X_i)+{\mathrm{ϵ<!--\; ϵ\; -->}}_i$$
- This implies the assumption
$$\mathbb{E}(Y_i|X_i=x_i)=s(X_i)$$

Continuous-time versus discrete-time stochastic models
When modeling a dynamic phenomenon, (from a general point of view) people can use two type of models: (1) continuous-time models, (2) discrete-time models.
To be more precise, assume that we try to model a phenomenon observed in a time series form via dynamical systems. If the stochasticity involved in the system, we can use two type of models: continuous-time stochastic processes or discrete-time stochastic processes. Since the time-series data is discrete, it seems very natural to model the processes in discrete-time. However, in some application domains, like stochastic finance, continuous-time models are of great importance and interest.
Here are my questions:
1) Is there an advantage of continuous-time modeling? Since the calculus is more difficult (continuous-time models require Ito calculus), what is the computational or modeling advantage of continuous-time models?
2) If there is no analytical solution of a continuous-time model, we discretize it and solve it numerically as far as I know. So, is it so different from discrete-time modeling?

Modeling the probability of a uniform point on [a,b]
I was given the following exercise : Explain how to define a probability measure on R that corresponds to the intuitive notion of an uniform point on [a,b].
And this is the solution:
Let $f:\mathbb{R}\to <!--\; \to \; -->\mathbb{R}$ be defined as $f(x):=(b-a)x+a$. This function is an affine bijection between [0,1] and [a,b]. Let $\mathrm{\lambda <!--\; \lambda \; -->}$ be the Lebesgue measure on [0,1]. We can define a probability measure $\mu$ on R by setting $\mu (B):=\lambda (f^{-1}(B>\cap <!--\; \cap \; -->[a,b]))$ for any Borel measurable subset B of ${\mathbb{R}}^n$. The corresponding c.d.f. is given by $F(x)=0$ if $x>b$, $F(x)=1$ if $x>b$ and $F(x)=f^{-1}(x)=\frac{x-a}{b-a}$ if $x\in [a,b]$.
I do not understand the question the exercise is asking. What is "the intuitive notion of an uniform point on [a,b]" ? Are they looking for a probability where all points in [a,b] have the same probability ?

Modeling Question: What to make my decision variables
I was given the following problem, and I am having trouble determining what to make my decision variables to model it:
Your goal is to optimize your profit. Over a 10 week period the demands for your product are 85, 97, 120, 105, 84, 74, 116, 123, 104, 98. The item sells for 50 dollars. There is a 500 dollar shipping charge on each order and if you order 150 or fewer items, each item costs 30. If you order between 150 and 249, each item costs 27 dollars. If you order 250 or more items, then the cost is 24 dollars per item. Each item in inventory costs 5 dollars due to storage space, capital spent, etc. Each item with a shortage (demand exceeds inventory) costs 15 dollars due to a loss of customer satisfaction, these items are never sold. What is your ordering policy?
I've tried calling $x_{ij}$ the number of products bought in week i at quantity j, where i goes from 1 to 10 and j goes from 1 to 3. Then I need a binary decision variable. I said let $b_{ij}=1$ if $x_{ij}>0$ and 0 if it is 0. I have issues modeling the inventory and shortage costs, then. There has to be a better way to define my decision variables.