# All Questions

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### Prove or Disprove the following statement. For any sets $A$, $B$, and $C$, we have $A \cup (B \& C) = (A\cup B) \cup (A\&C)$

Trying to figure this question out in my proofs class (tried venn-diagram the multiple set-notation signs are confusing me). Homework question in the fundamental sets unit.
1answer
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### Chance of Being runner up in the tournament

N cars take part in the race, N=2^K for given K > 0.Each car has unique ability that can be any integer from 1 to N. There are exactly K rounds in the tournament, 2^(K - i + 1) participants take part ...
0answers
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1answer
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### probability of the sum of i.i.d. RV with uniform distribution being $>x$

I am solving a question for applied stochastic processes homework and I am stuck on this part: Let $X_1,X_2,\cdots, X_n$ be independent identically distributed random variables with uniform ...
0answers
22 views

### SD of the sum of the scores? [on hold]

A card is drawn from a deck of $52$. The score equal to its rank unless it is a court card (Jack, Queen or King) with a score of 10, otherwise equal to its rank and Ace counts as one. I have found ...
2answers
30 views

### Injective function from a set of $n$ element to a set of $n$ element

How to determine number of Injective function from a set of $n$ element to a set of $n$ element and number of onto function on the same set to itself? Thank you for your help.
0answers
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### Suppose $|S|=n$, we need to determine number of operation on $S$

Suppose $|S|=n$, we need to determine number of operation on $S$ suppose $S=\{x_1,x_2,\dots,x_n\}$ well,A function from $\{x_1,x_2,\dots,x_n\}\times \{x_1,x_2,\dots,x_n\}\to\{x_1,x_2,\dots,x_n\}$ ...
1answer
33 views

### How to calculate the tensor product?

This question might be stupid for you.I ask because I have no clue about it. I don't really understand what is tensor product,although I know its definition. I have search what is tensor product,so I ...
1answer
33 views

### How can you measure out six liters of water?

You want to prepare a tub for your favorite game, dunking for apples. You have two buckets. One of the buckets will hold $4$ liters of water and the other will hold $9$ liters. There are no ...
5answers
70 views

### Why is $\{\emptyset\}\not\subseteq\{\{\emptyset\}\}$ true?

My book "Introduction to SetTheory" says $\{\emptyset\}\in\{\{\emptyset\}\}$ but $\{\emptyset\}\not\subseteq\{\{\emptyset\}\}$ When we say $\{\emptyset\}\not\subseteq\{\{\emptyset\}\}$, we mean ...
3answers
34 views

### Trigonometry Identity homework help

Could you please help me prove this: $${2\cos(\theta/2)-1-\cos\theta\over2\cos(\theta/2)+1+\cos\theta}={1-\cos(\theta/2)\over1+\cos(\theta/2)}$$
2answers
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### What does each term in the trinomial represent and their relation to each other?

I am trying to get a grasp or concept understanding what how trinomials answer questions other than answering questions in an algebra class. I. Looking for the practical application. What does the ...
0answers
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1answer
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### What are the pre- requisites required to learn Real Analysis?

I already have quite a solid foundation in Single and Multivariable calculus, however am I considered prepared to tackle real analysis? The reason I ask this is because before I get into the ...
1answer
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### What is pdf of Negative Binomial distribution

Exactly what is the definition of pdf of negative binomial distribution? So I have two textbooks one said: $$f_X(x)=\begin{pmatrix} r+x-1\\ x \end{pmatrix}p^rq^x$$ where as on the other it is defined ...
0answers
28 views

### Is the functor category of algebras into modules locally small?

Let $R$ be a ring. Is $[{}_R Alg, {}_R Mod]$ locally small?
2answers
50 views

### Let $f$ be a $C^2$ function on $\mathbb{R}$. Assume that $f$ is bounded with bounded second derivative. [on hold]

Let $A=\sup_{x}|f(x)|$ and $B=\sup_x|f''(x)|$. Prove that $\sup_x|f'(x)|\leq2\sqrt{AB}$.
2answers
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### Is there at least a continuous mapping from an arbitrary interval $[a,b]$ to $[0,1]$ with a periodic points of period $3$?

$$\Large-\textbf{Problem}-$$ Is there at least a continuous mapping from an arbitrary interval $[a,b]$ to $[0,1]$ with a periodic point of period $3$? $$\Large-\textbf{Thoughts and Ideas}-$$ Let ...
2answers
36 views

### Cartesian product of Riemann Surfaces [on hold]

I have a question regarding the Cartesian product of Riemann surfaces. Let $T^2$ denote the $2$ dimensional torus. $T^2=S^1\times S^1$. Also, let $T^4$ denote the $4$ dimensional torus. ...
1answer
46 views

### Ping Pong players

A and B play ping pong game multiple times. The person serving first has a probability p of winning that game. A serves the first game and thereafter the loser serves first. If p(n) = pbt that A ...
2answers
28 views

### Question about Integration by Parts

I have this problem here that is asking for me to use Integration by Parts. I solved it out but it seems as if it can be an ongoing function. I was told you can work with it if the second time you ...
1answer
23 views

### Borel measurable function

I'm struggling on the following question from a past paper: Suppose that $f:\mathbb{R}\rightarrow \mathbb{R}$ is a Borel measurable function and let $h:\mathbb{R}^2\rightarrow \mathbb{R}$ be defined ...
5answers
101 views

### variation of the binomial theorem

Why does: $$\sum_{k=0}^{n} k \binom nk p^k (1-p)^{n-k} = np$$ ? Taking the derivative of: $$\sum_{k=0}^{n} \binom nk p^k (1-p)^{n-k} = (1 + [1-P])^n = 1$$ does not seem useful, since you ...
1answer
41 views

### What time was it 237 hours ago if the starting point is 00:00 [on hold]

The title is the question, my girlfriend is completely lost at this point.
2answers
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### Applications of stochastic processes

What are stochastic processes? What are they used for? How can they be applied to real concepts? What is an example of a "stochastic process" problem?
1answer
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Consider the following vector function $y: \mathbb R^n \to \mathbb R$ $$y(\vec x) = y(x_1,x_2,\cdots,x_n)$$ Is it correct to state the following? $$dy = \sum_{i=1}^{n}{\left(\frac{\partial ... 0answers 41 views ### Partial fraction decomposition of \frac{1}{x^{2n}+a^{2n}} I came across a formula for the partial fraction decomposition of  \displaystyle \frac{1}{x^{2n}+a^{2n}}. It seems correct (at least for n=1,2, and 3). But how is it derived? ... 1answer 20 views ### Random photo generation? Well, how to calculate such value? Assume we took a picture (320x240px, 16 bit color) and want to know which chance of random generation it has. Exact this set of pixels, of course, to simplify our ... 0answers 22 views ### Maximize a function subject to the constraint x^2+y^2=R^2 Please help me how to deal with maximization of function$$f(x,y)=1-e^{-\pi x}+e^{\pi x}\left[1-\cos(\pi y)+\sin(\pi y)\right]$$subject to the constraint x^2+y^2=R^2. Thank you for reading. ... 2answers 40 views ### Use implicit differentiation to find derivative$$x\sin(4x+5y)=y\cos(x)$$I am trying to use implicit differentiation to find dx/dy for this problem but the answer i keep getting is$$4x\cos(4x+5y)=-y\sin(x)$$and I am stuck. 1answer 26 views ### Describe the fibers of \phi and that \phi is a homomorphism. Define \phi: \mathbb{R}^x \mapsto \{\pm1\} by letting \phi(x) be x divided by the absolute value of x. Describe the fibers of \phi and that \phi is a homomorphism. Need help getting a ... 0answers 17 views ### Locally complete intersection in a fiber Let Y be an affine noetherian scheme, Z = V_+(F_1, \ldots, F_r) a closed subscheme of \mathbb{P}^d_Y that is flat over Y. Let y_0 \in Y be a point such that Z_{y_0} is a complete intersection ... 1answer 49 views ### A plan to defeat a betting game where the odds of winning are 50/50. Help me understand why it's flawed. My friend has this plan where he implies that it's impossible to lose, as long as the odds of winning are 50/50 on each bet. His idea is that basically you keep doubling your bet until you win and ... 2answers 30 views ### simple Lie groups A Lie group is a group which is a smooth manifold such that the multiplication and inversion are smooth. When does a Lie group become simple? What is the difference between simple and semi-simple Lie ... 3answers 25 views ### Linear Algebra Vector Space matrix help Let M_{2\times2} be a vector space of all 2\times2 matrices. If the transformation from M_{2\times2} to M_{2\times2} is t(A)=A+A^T and A is a 2\times2 matrix with the top row a,b and ... 0answers 16 views ### Finding a base for a series to sum to a constant I'd like to find the value of r that solves the following equation:$$\sum_{n=1}^N r^{\frac{-1}{n}} = C \,, where $N$ and $C$ are positive constants. An approximate method would also work fine ...

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