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JoAnn is paid $14 an hour for the first 35 hours she works each week and $21 an hour for every hour after that. If she makes $574 one week, how many hours a week did she work?

Two cars travel in the same direction along a straightroad. When a stopwatch reads t=0.00h, car A is atdA=48.0km moving at a constant 36.0km/h.
Later,when the watch reads t=0.50h, car B is at dB=0.00kmmoving at 48.0 km/h. Answer the following questions: first,graphically by creating a position-time graph; second algebraicallyby writing down equations for the postion dA anddB as a function of the stopwatch time, t.
a) what will the watch read when car B passes car A?
b) at what position will the passing occur?
c) when the cars pass, how long will it have been since car Awas at the reference point, d= 0 km?

Leona Spent $57.95 at the mall. She bought a T-shirt and a book. The book cost $29.37. How much did the T-shirt cost?
A. Write an equation that models the problem.
B. Solve the equation.
C. Write your answer in the form of a sentence.

The bar chart below was created a survey by the U.S Departament of Energy's Energy Information Administration during the month of April 2004. If gives the average price of regular gasoline for the state of California on each Monday of the month. Use the information in the chart to fill in the table.

Given two integers. What can you claim about them, when their quotient is:
1. Positive?
2. Negative?
3. =0?

A temperature of 38 degrees Fahrenheit is equal to what in Celsius? Recall F = (9/5)C + 32

A manufacturer of 24-hr variable timers, has a mounthly fixed cost of $\mathrm{\$<!--\; \$\; -->}45,000$ and a production cost of $\mathrm{\$<!--\; \$\; -->}8$ for each timer manufactured. The units sell for $\mathrm{\$<!--\; \$\; -->}13$ each
a) Graph the cost and revenue functions.
b) Find the break-even point.
$(x,y)=?$
c) Graph the point function.
d) At what point does the graph of the profit function cross the x-axis?
$(x,y)=?$

There are eight furlongs in a mile. There are two weeks ina fortnight. The British cavalry traveled 2800 furlongsin a fortnight. How many miles per day did the cavalrytravel?

Conditional probability and entropy: How do I interpret given data?
As the title explains, I never could understand probabilities. It's one of those things that how much I try, I can't quite understand.
I have to do one homework exercise about entropy and I'm given a set of probabilities. I know how to calculate entropy but I don't know how to interpret the given data.
The alphabet is S={1,2} and the conditional probabilities are P(1|1)=0.8 P(2|1)=0.2 P(1|2)=0.6 P(2|2)=0.4 and P(1,2)=P(2,1)I've created this table (don't know if it is right or not):
| X = 1 | X = 2
Y = 1 | 0,8 | 0,2
Y = 2 | 0,6 | 0,4
I know that I need the probability of 1 and 2 to calculate the entropy. To get the probability of 1 is like this?
P(1)=P(1,2)P(2|1)
If so, How can I get P(1,2)?
I know that P(1)=0.75 and P(2)=0.25 but I don't understand how to get to this result

Probability of normal distribution: average daily temperature - interpretation
The City of Dallas has records indicating that the average daily temperature in the summer is 80 °F, which is normally distributed with a standard deviation of 3 °F.
Based on these records, what is the probability of a daily temperature
between 75 °F and 87 °F
between 83 °F and 85 °F
$P(X<75)=P(Z<-<!--\; -\; -->5/3)=0.0485$
$P(X<87)=P(Z<7/3)=0.9901$
$P(X<83)=P(Z<1)=0.8413$
$P(X<85)=P(Z<5/3)=0.9515$
Am I following the right steps?
How do I interpret my data?

Pascal's triangle, estimate row value by fixed row and maximum yields.
let res be the max yields. Let col be a fixed positive integer. Let $f(row)=(\genfrac{}{}{0ex}{}{\text{row}}{\text{col}})=A$, what is the largest value for row such that $f(row)\le <!--\; \le \; -->A$?
Standard Pascal's triangle expression:
$(\genfrac{}{}{0ex}{}{\text{row}}{\text{col}})=\frac{\text{row}!}{\text{col}!(\text{row-col})!}=\text{result}$
Update
I am archiving a regression program, which takes raw data of two-dimensional vertexes, and return a function correspond to the rule. It can be called which pass in x and return a y value.
In this procedure, generator yields different amount of sample data from the raw data. For example, in simplest linear regression
y=kx+b
which takes a minimum points of 2, [x1, y1], [x2, y2], for working out the coefficients k and b.
Here comes the part which is the most relevant to this question.
if the raw data contains n vertexes, then the amount of iteration equal to
$(\genfrac{}{}{0ex}{}{\text{n}}{\text{required}})=\frac{\text{n}!}{\text{required}!(\text{n-required})!}=\text{yields}$
for instance, for linear regression, required is equal to 2. for 10 dots:
$(\genfrac{}{}{0ex}{}{\text{10}}{\text{2}})=\frac{\text{10}!}{\text{2}!(\text{10-2})!}=\text{45}$
However, the original design of my program is to iterate all through the amount of yields, for example, when n=10, required=2 which res=45, do an iteration of 45.
The problem is:
n<20> required<2>: res<190>
n<40> required<2>: res<780>
n<60> required<2>: res<1770>
n<80> required<2>: res<3160>
n<100> required<2>: res<4950>
n<120> required<2>: res<7140>
n<140> required<2>: res<9730>
n<160> required<2>: res<12720>
n<180> required<2>: res<16110>
when n>=16, it freezes and overloads the machine, the fan is very loud.
Therefore, I seek for a way,
with known required, by limiting the largest amount of iteration, how many dots do I need to extract from raw data?
which the question can be briefed to:
$(\genfrac{}{}{0ex}{}{\text{n}}{\text{required}})=\frac{\text{n}!}{\text{required}!(\text{n-required})!}=\text{yields}$
with known the regression takes required vertexes, set a maximum iteration of yields, find a suitable n value.
Thanks very much for reading till the end. I am sorry if anything I have written doesn't make sense to you. Any help is appreciable.

What makes a theorem "fundamental"?
I've studied three so-called "fundamental" theorems so far (FT of Algebra, Arithmetic and Calculus) and I'm still unsure about what precisely makes them fundamental (or moreso than other theorems).
Wikipedia claims:
The fundamental theorem of a field of mathematics is the theorem considered central to that field. The naming of such a theorem is not necessarily based on how often it is used or the difficulty of its proofs.
So I am left wondering: what is the main criterion for a theorem to be considered fundamental?
Why, for instance, is there no "Fundamental Theorem of Complex Analysis" (say, de Moivre's formula, for example), or "Fundamental Theorem of Euclidean Geometry" (e.g. Pythagoras' theorem)?
Does anyone have a source of a more-rigourous definition of "fundamental", in this context?

I am reading this paper by Ian Goodfellow. Currently I have an undergraduate knowledge of probability/statistics/linear algebra/calculus/differential equations. I am trying to understand the formula below proposed in the paper but I have no idea it what it means.
$\underset{G}{min}\underset{D}{max}V(D,G)={\mathbb{E}}_{\mathbf{x}\sim <!--\; \sim \; -->{\mathit{p}}_{\mathit{d}\mathit{a}\mathit{t}\mathit{a}}\mathit{(}\mathbf{x}\mathbf{)}}<!--\; \; -->[\log <!--\; \; -->D(\mathbf{x}\mathbf{)}\mathbf{]}\mathbf{+}{\mathbb{E}}_{\mathbf{z}\sim <!--\; \sim \; -->{\mathit{p}}_{\mathit{z}}\mathit{(}\mathbf{z}\mathbf{)}}<!--\; \; -->\log <!--\; \; -->\mathbf{[}\mathbf{(}\mathbf{1}\mathbf{-<!--\; -\; -->}\mathbf{D}\mathbf{(}\mathbf{G}\mathbf{(}\mathbf{z}\mathbf{)}\mathbf{)}\mathbf{)}\mathbf{]}$
I can see that we have some value function(what is a value function? what topic does this fall under?) and it equals some expectation of x (what is x?) which is (~) similar to the probability of the data of x? I just have no idea how to start interpreting formulas like this when I begin reading papers. Any useful guide on what topics I need to study to understand the math in this paper is very appreciated.

I'm build an app for mountain biking, and I would like to estimate the ground slope using the phone's integrated accelerometer so that I can adapt my pedaling effort based on the current slope.
I took the Y-axis value of the accelerometer and converted it to degrees, using a notion abs(y-axis) * 90 with a offset. So, for example, if the accelerometer reads 0.5 or -0.5 on the Y-axis, that's converted to 45°. If I zero out the accelerometer, that 45° is now considered as 0°, and any frontal movement is negative and back movement is positive. I use that value for slope calculation.
Now, that works perfectly fine when standing still. When I angle the phone at 45° the slope is 100%.
But, problems arise in the real world. Since I'm using a accelerometer for my angle detection, readings when hitting potholes and bumps in the road go all over the place. One second I read 80%, the next -300%, and so on.
What can I do to smooth out the readings and make them semi-accurate?
I could try to discard any invalid value, such as anything +-20% of slope and average that over 10 seconds, but I feel that would be inaccurate.
I was looking into the Kalman filter, based on a thesis on slope detection using an accelerometer for Scania trucks, but the jerking of a truck is much less violent than that of a mountain bike.
What can I do to predict and smooth out sensor values over time?

Can you help me to figure out how this outout emerged?
A sequence of N data points as a list of comma separated integer values.
Task is generate N values representing the original values, but obfuscating their real values.
You should do this by selecting the highest and lowest value in the data set and creating N−2 data points numerically evenly spaced out between the two. Your output should be a comma separated list of N integer values ordered from low to high. Where the numbers are not integer, you should round them such that they end up integer.
Example 1:
Input: "3,2"
Output: "2,3"
Explanation: The lowest and highest values are 3 and 4 respectively, and there are no intermediate values that need to be interpreted
Example 2:
Input: "4,5,3,5"
Output: "3,4,4,5"
Explanation: The lowest and highest values are 3 and 5 and the intermediate values become 3.66 and 4.33, which when rounded to integers to make as 4 and 4.

I'm trying to solve the initial value problem $(i{\mathrm{\partial <!--\; \partial \; -->}}_t+{\mathrm{\Delta <!--\; \Delta \; -->}}_x)u(t,x)=0$ $u(0,x)=f(x)$ for the Schrödinger equation ($x\in <!--\; \in \; -->{\mathbb{R}}^n$, f Schwartz).
I know that a fundamental solution is given by $K(t,x)=(4\mathrm{\pi <!--\; \pi \; -->}it{)}^{-<!--\; -\; -->n/2}{e}^{i|x{|}^2/4t}$.
How do I interpret $\sqrt{i}$ here?
I'm trying to show that if I convolve the above fundamental solution K with the initial data f (convolution in the spatial variable x), then I obtain the solution to the initial value problem.
Specifically, how do I prove that $K\ast <!--\; \ast \; -->f\to <!--\; \to \; -->f$as $t\to <!--\; \to \; -->0$? More generally, what are the differences between this problem and the analogous problem for the heat equation $({\mathrm{\partial <!--\; \partial \; -->}}_t-<!--\; -\; -->{\mathrm{\Delta <!--\; \Delta \; -->}}_x)u(t,x)=0$?
[I know that the Schrödinger equation and fundamental solution are obtained from their heat counterparts via $t\mapsto <!--\; \mapsto \; -->it$
Why is the Schrödinger equation time reversible (i.e. why can it be solved both forwards and backwards in time), while the heat equation isn't? The total integral of the heat kernel (with respect to x) is 1; is the total integral of the "Schrödinger kernel" K also equal to 1?

I'm reading a book on linear algebra, where the author gives a method to test the handedness or chirality of a given set of 3 basis vectors.
if (v1×v2)⋅v3>0 then it's right-handed, while if it's less than 0, it's left handed.
What beats me is that numbers are just numbers, left or right handedness of a system depends on the viewer and how he interprets the given data.
Taking the canonical basis vectors $\stackrel{^<!--\; ^\; -->}{i},\stackrel{^<!--\; ^\; -->}{j},\stackrel{^<!--\; ^\; -->}{k}$ in both left and right handed systems $i\times <!--\; \times \; -->j=k$, thereby $k\cdot <!--\; \cdot \; -->k=\Vert <!--\; \Vert \; -->k{\Vert <!--\; \Vert \; -->}^2>0$ (always), then how does this test hold true?

Is hypothesis space a random variable?
I am reading where it states:
The previous statement is true for a given training set D. However, remember that D itself is drawn from Pn, and is therefore a random variable. Further, hD is a function of D, and is therefore also a random variable. And we can of course compute its expectation:
The definition of hypothesis space is given in the section Loss Functions.
There are typically two steps involved in learning a hypothesis function h(). First, we select the type of machine learning algorithm that we think is appropriate for this particular learning problem. This defines the hypothesis class H, i.e. the set of functions we can possibly learn. The second step is to find the best function within this class, $h\in <!--\; \in \; -->\mathcal{H}$. This second step is the actual learning process and often, but not always, involves an optimization problem. Essentially, we try to find a function h within the hypothesis class that makes the fewest mistakes within our training data. (If there is not a single function we typically try to choose the "simplest" by some notion of simplicity - but we will cover this in more detail in a later class.) How can we find the best function? For this we need some way to evaluate what it means for one function to be better than another. This is where the loss function (aka risk function) comes in. A loss function evaluates a hypothesis $h\in <!--\; \in \; -->\mathcal{H}$ on our training data and tells us how bad it is. The higher the loss, the worse it is - a loss of zero means it makes perfect predictions. It is common practice to normalize the loss by the total number of training samples, n, so that the output can be interpreted as the average loss per sample (and is independent of n ).
This is how I would rewrite the document:
We have a table where the last column are targets (predictions). This can be represented as a multivariate random variable or random vector X and a target random variable Y. The data distribution of the random variable D representing our dataset is: P(X,Y). When concrete dataset D is drawn from sample space Pn we get our dataset D with n rows.
But I am not sure of h? It is inside the "Loss Function" paragraph where actually to me it reflects the machine learning algorithm function. One of the infinitely many. Further hypothesis class may be a random variable H, and h just a realization.
1. How can dataset D be a random variable?In here when they refer to a dataset they use D.
2. How can hD be a random variable if this is a function associated to a concrete dataset, and most likely it means a function tryout (one of the many) that translate the input to the output?

Proving that a function is strictly monotonic knowing that $|f(x)-<!--\; -\; -->x^2|\le <!--\; \le \; -->2|x|$
Prove that the following real-valued function is strictly monotonic, knowing that $|f(x)-<!--\; -\; -->x^2|\le <!--\; \le \; -->2|x|$.
I can't actually interpret the given data in a way that would produce the required conclusion, any ideas?

Better understanding of the pre sheaf kernel of $\mathrm{\phi <!--\; \phi \; -->}$
I'm currently reading chapter II of Hartshorne's algebraic geometry and am comfortable with the definition of a sheaf. I am new to the idea of the presheaf kernel of $\mathrm{\phi <!--\; \phi \; -->}$ however.
Let $\mathrm{\phi <!--\; \phi \; -->}:\mathcal{F}\to <!--\; \to \; -->\mathcal{G}$ be a morphism of pre sheaves of abelian groups on a topological space X.
Am I correct in defining the kernel pre sheaf on φ to be the contra variant functor that consists of the data
For every open set $U\subset <!--\; \subset \; -->X$, we have an abelian group $\mathrm{\Gamma <!--\; \Gamma \; -->}(\ker <!--\; \; -->\mathrm{\phi <!--\; \phi \; -->}(U),\mathcal{F})$
For every inclusion $V\subset <!--\; \subset \; -->U$ open sets in X, we have a morphism ${\mathrm{\rho <!--\; \rho \; -->}}_{UV}:\mathrm{\Gamma <!--\; \Gamma \; -->}(\ker <!--\; \; -->\mathrm{\phi <!--\; \phi \; -->}(U),\mathcal{F})\to <!--\; \to \; -->\mathrm{\Gamma <!--\; \Gamma \; -->}(\ker <!--\; \; -->\mathrm{\phi <!--\; \phi \; -->}(V),\mathcal{F})$ that satisfies the usual three conditions (empty set, identity map and composition condition)?
Moreover,
Hartshorne remarks that while the pre sheaf kernel is a sheaf, in general, the pre sheaf cockernel and image are not sheaves. Is there any intuition for why this would be so?