# All Questions

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### Permutations containing a given subsequence

Let $f(n)$ denote the number of $4n$-long strings formed from $2n$ a's and $2n$ b's, such that the string contains, as a (possibly non-consecutive) subsequence, a pattern containing $n$ a's and $n$ ...
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### Find $f'(2)$, where $f(x) =\frac{h(x)}{x}$.

Consider the function $h(x)$, for which $h(2) = 4$ and $h'(2) =-3$. Find $f'(2)$ for the function $f(x) = \frac{h(x)}{x}$.
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### Statistics - Unbiased estimators

Let $X_1, X_2, \ldots, X_N$ be an i.i.d. random sample of Bernoulli random variables, with $\mathbb{P}(X_i =1) = p$ and $\mathbb{P}(X_i = 0) = 1 − p$. I'm confused as to why $1 − X^{-}$ is an ...
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### “congruence modulo 7” is an equivalence relation on Z. Find three elements in the equivalence class [3].

“congruence modulo 7” is an equivalence relation on Z. Find three elements in the equivalence class [3]. so 3 is congruent to mod 7.. My attempt: a = bq + r = 7(1) + 3 = 10 , .. 7(0) + 3 = 3, .. ...
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### Show that H is dense or isomorphic to Z

Let be S a finite subset of $\mathbb{R}$, and let $H=\langle S \rangle$ the subset generated by S under the sum. a) Let $h_{min}=\inf\{h\in H|h>0\}$. Show that if $h>0$, then h is a element of ...
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### H is normal subgroup having index $3$, why is every $a^3$ in $H$?

Let $H$ be normal subgroup of a finite group $G$ such that $H$ has index $3$. Show that $a^3$ is in $H$ for every $a$ in $G$. Anyone can help me?
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### Suppose $S_1 =\{ u_1 , u_2 \}$ and $S_2 = \{ v_1 , v_2 \}$ are each independent sets of vectors in an n-dimensional vector space V..

Let us assume that every vector in S_2 is a linear combination of vectors in S_1. Question: Does that mean that S_1 and S_2 are bases for the same subspace of V? I know that the answer to this ...
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### .Prove that $\ker f$ is either dense or closed in

let $f$ be a linear functional on a normed linear space $X$ .Prove that $\ker f$ is either dense or closed in $X$. Two possibilities can occur i.e either $f$ is bounded or unbounded.If it is bounded ...
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### Disk - Surface Area

Can anyone tell me what is the formula to find out the surface area of a disk if it provide with thickness and diameter
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### What is the Fourier transform of $1/f(x)$?

Given $F(t) = \mathcal{F}\{f(x)\}$ is the Fourier transform of $f$, how can one express $\mathcal{F}\left\{\dfrac{1}{f(x)}\right\}$ in terms of $F(t)$? EDIT: To be more concrete, I want to compute ...
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### An example in which the Fubini theorem is inapplicable

This is example 8.9(a) in Rudin's Real and Complex Analysis, (alternatively, exercise 10.2 in Rudin's Principles of Mathematical Analysis). Let $X$ and $Y$ be the closed unit interval $[0,1]$, let ...
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### Prove integral belongs to dual space and find operator norm

Prove that $L: C([0,1])\to \mathbb{R}$ defined by $$L(f)=\int_{0}^{1}f(x)dx$$ belongs to $C([0,1])^{\star}$. What is its operator norm? ($C([0,1])^{\star}$ is the dual space/conjugate space).
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### Proving that a process is class DL

Let $(X_{t})$ be a stochastic process with $X_{t}\sim\mathcal{N}(\xi_{t},\sigma_{t}^{2})$ where $\xi_{t}\downarrow0$ and $\sigma_{t}^{2}\uparrow1/2$. What would be the most straightforward way ...
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### Complementary pair of sets

My book repeatedly uses the phrase "contains one of each complementary pair of sets" and I am wondering what do they mean by that exactly? An example, let $(Y,Z)$ be any partition of $X$. Then at ...
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### Proving uniform approximation by polynomials when sets are not compact

Here are two problems of the same flavor (and hence I posted them simultaneously) based on the Stone-Weierstrass Approximation Theorem. Let $f$ be continuous on $[1,\infty)$ with ...
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### Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?

Why does the complex equation $z=Ae^{it}+Be^{-it}$ represents an ellipse?, being $A,B \in \mathbb C$ How can it be described?
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### How is an infinitesimal $dx$ different from $\Delta x\,$?

When I learned calc, I was always taught $$\frac{df}{dx}= f'(x) = \lim_{h\rightarrow 0} \frac{f(x+h)-f(x)}{(x+h)-x}$$ But I have heard $dx$ is called an infinitesimal and I don't know what this ...
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### Inner product space related to pythagorean theorem.

So I understand that the an inner product space basically uses pythagorean theorem because it is similar to a distance formula. I'm still having trouble with this proof. I am a bit confused about ...
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### Show analytically that [on hold]

$$¬(¬(Z+(¬XY)¬(Y+W))=Y+W$$ How I can resolve analytically that equality please?
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### What algorithm can sort the first sqrt(n) elements of an array in O(n) time?

I want to partially sort an array of $n$ elements to get the first $\sqrt{n}$ elements sorted, and it has to be done in $O(n)$ time. The complexity $O(n)$ seems to imply that it is necessary to go ...
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### Random Variable independece! [on hold]

Can any one give me a hint on this question. I am new to probability. And what should be the value of a s that V and W and independent. Thanks
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### Finding equation of hyperbola with only foci and asymptote

This is a concept we learned in class today, which I still can't seem to grasp. I have no specific question that necessarily has to be done, so I will use one of the examples my book gives me: Given ...
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### Conditional expectations with identical marginals and positively dependent but unknown joint distribution

Let $A$ and $B$ be random variables, each with marginal distribution $% U\left( 0,1\right)$, but unknown joint distribution $H\left( a,b\right)$. Suppose $A$ and $B$ are each stochastically (weakly) ...
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### Calculating the area of an ellipse

I need to calculate the area of an ellipse described in polar coordinates by the following equation $$r=\frac{p}{1+\epsilon \cos{\theta}},\qquad |\epsilon| < 1$$ I need to so it by solving the ...
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### Finding the minimum value of $|a+b\omega+c\omega^2|$ if $a,b,c$ are unequal integers where $\omega^3=1$

My try 1: $$|a+b\omega+c\omega^2|\le\sqrt{|a+b+c||\underbrace{1+\omega+\omega^2}_0|}$$ Cauchy-Scwartz won't give us an upper bound since $a,b,c$ are nonequal integers. My try 2: ...
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### Prove that the following is a constant function

Let $f : R \rightarrow R$ $\lvert f(x)-f(y) \rvert \le (x-y)^2, \forall x,y \in R$ Any sort of help is appreciated! I know I am not suppose to ask for the entire solution, so I will ask for ...
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### Eliminating the parameter?

How would you eliminate the parameter where the x coordinate is in terms of t, but the t is squared. x= 3t - $t^2$ y= t + 1 I know to solve for y as a function of x, but I'm not sure how to do so ...
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### Continuity in a function defined only at one point

This might be a silly question, but if I have a function defined at only one point. Is the function continuous at that point?
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### stratified random sample

A report showed the number of homicides in each state. In Indiana, Ohio, and Kentucky, the number of homicides was, respectively, 380, 760, and 260. Suppose a stratified random sample with the ...
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### Prove that $T$ is one to one?

I'm very confused on how to do this one. I don't really understand how to find the length of a transformation. Also I'm confused on how that relates to being one to one. Any help appreciated.
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### How fast is this dot moving when the angle $θ$ between the beam and the line through the searchlight perpendicular to the wall is $π/6$?

A searchlight rotates at a rate of $4$ revolutions per minute. The beam hits a wall located $11$ miles away and produces a dot of light that moves horizontally along the wall. How fast (in miles per ...
### $f(x+1)=xf(x)$ and $g(x)=\log f(x)$, finding $g''(N+1/2)-g''(N)$
My try: $$f(x+1)=xf(x)\implies f(x+N)=x^Nf(x),N\in\mathbb N$$ Because: $$f(x+N)=xf(x+N-1)=x^2f(x+N-2)=...x^Nf(x+N-N)=x^Nf(x)$$ Now: \log f(x+N)=N\log x+\log f(x)\\ g(x+N)=N\log x+g(x)\\ ...