-2
votes
0answers
6 views

?+?+?=30 (1,3,5,7,9,11,13,15)

?+?+?=30 Please slowe this question and reple answer
0
votes
0answers
2 views

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 $n$ ...
0
votes
1answer
15 views

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 $f(x) = \frac{h(x)}{x}$.
1
vote
1answer
10 views

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 ...
0
votes
1answer
13 views

“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, .. ...
0
votes
0answers
10 views

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 ...
-1
votes
1answer
10 views

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?
0
votes
0answers
7 views

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 ...
1
vote
1answer
11 views

.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 ...
-1
votes
0answers
11 views

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
1
vote
0answers
14 views

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 ...
1
vote
0answers
7 views

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 ...
1
vote
0answers
14 views

The norm $\|f_n-f\|_{L^1} \to 0$ but $f_n \not\to f$

A classmate and I are studying this following question from Stein-Shakarchi, Chapter 2, Exercise 12: Show that there are $f \in L^1(\mathbb{R}^d)$ and a sequence $\{f_n\}$ with $f_n \in ...
0
votes
0answers
2 views

Spectral radius and convergence of fixed point iteration

Let $F: \mathbb{R}^n \to \mathbb{R}^n$ be a differentiable map. Then, is it true that the spectral radius of the Jacobian $J_F(x)$ at $x = x^\star$ is less than 1 if and only if $x^\star$ is a fixed ...
1
vote
0answers
20 views

What would be an effective way to learn group theory on my own?

I've read the basics of this branch and I found it extremely interesing, and I would really love to learn more about it. I want to study as much as I can on my own, as my course doesn't have group ...
1
vote
2answers
14 views

Probability of sorting at least one correctly

If I have 5 balls label 1 through 5, to put one in each of 5 boxes also labeled 1 through 5. What is the probability of putting at least one ball in it's matching box? My first approach was to ...
0
votes
0answers
4 views

Finding Prime Implicants and Essential Prime Implicants for Boolean Functions

I am trying to solve a EE problem and am unsure whether I doing it correctly. The problem is: Find all the prime implicants for the following Boolean functions, and determine which are essential: ...
1
vote
0answers
10 views

Proof for the Rodrigues formula for Neumann's Spherical functions.

I've been trying to prove the Rodrigues formula for Neumann's Spherical functions. The Neumann's Spherical functions are: $$N_n(x))=-(-x)^n\left[\frac{1}{x}\frac{d}{dx}\right]^n\frac{\cos{(x)}}{x}$$ ...
0
votes
0answers
16 views

How to go about a “not divisible by..” proof

I need to show the following proof: For any integer x, x^2 + 4 is not divisible by 3. I was trying proof by contraposition, but I do not believe that is the most efficient way to go about this. ...
2
votes
1answer
28 views

Number of Hausdorff topologies on a set with $100$ elements.

Find the number of Hausdorff topologies on a set with $100$ elements. I know that number of topologies on a set with $2$ elements is $4$, with $3$ elements is $29$ , with $4$elements is $355$ , ...
0
votes
0answers
10 views

Long division for multipolynomial expression, little o notation

I have this expression: $$\mathrm{Exp}=\frac{1}{1-a}+\frac{d^3(-12a^4)+d^2(4a^4-16a^3)+d(4a^3-6a^2-a)}{d^3(-12a^4+12a^3)+d^2(4a^4-20a^3+16a^2)+d(4a^3-11a+7a)+(1-2a+a^2)}$$ Is there any way I can ...
0
votes
0answers
6 views

How many permutations of $x^5 + y^5 + z^5$ are possible given x, y, z are integers such that $1 \le x \le y \le z \le 180$?

I initially thought it would be $180^3$ possible permutations, but then quickly realized that something like $x=3, y=2, z=1$ would not be valid due to the constraints. How can I go about trying to ...
0
votes
1answer
19 views

Find max $a$ such that $\lim_{n\rightarrow \infty} n^{n^a} e^{-n} = 0 $

Find maximum value of $a$ such that $$\lim_{n\rightarrow \infty} n^{n^a} e^{-n} = 0 $$ Clearly $a < 1$ also I checked that $a> 1/2$. But how do I narrow it down?
0
votes
2answers
14 views

Inner Product spaces with functions?

I understand inner product space with vectors, but the conversion to functions is throwing me off. Also why do they use an integral here, I've always seen summations. I think I'm missing something ...
0
votes
1answer
11 views

Compute Galois Group of splitting field of $x^p-a$ over $\mathbb Q$

I am having trouble computing the Galois Group of the splitting field $E$ of $x^p-a$ (where $p$ and $a$ are prime) over $\mathbb{Q}$. Let $w$ be a $p^\text{th}$ root of unity, and $\alpha$ a root of ...
1
vote
1answer
19 views

Questions concerning elements in $F = \big\{f: \{1, 2, 3\} \to \{1, 2, 3, 4, 5\}\big\}$.

a) Find and simplify the number of functions $f \in F$ so that $f(1) = 4$. My attempt: there is $1$ choice for $f(1)$, and $5$ choices for $f(2)$ and $5$ choices for $f(3)$, thus $1\cdot 5\cdot 5 = ...
-1
votes
0answers
11 views

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).
0
votes
0answers
8 views

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 ...
0
votes
0answers
8 views

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 ...
0
votes
1answer
10 views

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 ...
0
votes
2answers
25 views

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?
5
votes
2answers
51 views

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 ...
0
votes
1answer
14 views

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 ...
-5
votes
0answers
21 views

Show analytically that [on hold]

$$¬(¬(Z+(¬XY)¬(Y+W))=Y+W$$ How I can resolve analytically that equality please?
0
votes
1answer
20 views

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 ...
-4
votes
0answers
23 views

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
0
votes
0answers
7 views

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 ...
0
votes
0answers
10 views

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) ...
3
votes
2answers
29 views

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 ...
0
votes
3answers
34 views

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: ...
1
vote
3answers
58 views

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 ...
1
vote
2answers
22 views

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 ...
1
vote
1answer
24 views

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?
-1
votes
0answers
7 views

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 ...
2
votes
2answers
35 views

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.
0
votes
0answers
17 views

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 ...
-1
votes
1answer
35 views

$f(x+1)=xf(x)$ 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)\\ ...
0
votes
2answers
39 views

Proving that $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable using an $ \epsilon $-$ \delta $ definition.

Problem. Show that a bounded function $ f: [a,b] \to \Bbb{R} $ is Riemann-integrable if for every $ \epsilon > 0 $, there exists a $ \delta > 0 $ such that for any partition $ \mathcal{P} = ...
0
votes
0answers
10 views

can any one tighten $|e_i^TXe_j|$?

Suppose we have the symmetric matrix $X\in R^{m\times m}$ with its 2-norm $\|X\|_2\leq m$. Then I can get that, for each entry of $X$, $|X_{ij}|=|e_i^TXe_j|\leq\|e_i\|_2\|X\|_2\|\|e_j\|_2=m$, where ...
0
votes
0answers
20 views

show linear transformation bijective

Can you please help me prove this? Let $T:\mathbb{R}^7\to\mathbb{R}^7$ be a linear transformation such that 9 is an eigenvalue of $T$ and $dim(E_9)=6$ Prove that either T-4I or T-5I is a bijection ...

15 30 50 per page