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H: Correctness of proof of division in ceiling function
My professor asked me to present a proof to my fellow students tomorrow that
$$\left\lceil\frac{n}{2^k}\right\rceil = \left\lceil\frac{\left\lceil\frac{n}{2^{k-1}}\right\rceil}{2}\right\rceil$$
I tried to prove it as follows, but I am not sure if my proof is corr... |
H: Show that $\lim_{\delta \to 0}(1-\lambda \delta)^{1/\delta} = e^{-\lambda}$
My professor said that
$$\lim_{\delta \to 0}(1-\lambda \delta)^{t/\delta}=e^{-\lambda t}$$
can be shown with L'Hospital's rule. I don't know what he meant. What is the best way to show this (or, more simply, $\lim_{\delta \to 0}(1-\lambda ... |
H: find equation of tangent and normal lines (differentiation)
What is the equation of the tangent and normal lines to this function at the point p
$f(x)$ = $x^3$ at the point $p = (2,8)$
AI: First, we take the derivative of $f(x)$
$f'(x) = 3x^2, \tag{1}$
the evaluate it at $x = 2$ to obtain the slope at $(2, 8)$:
$f'... |
H: Determine an equation for the tangent to the graph of f(x) at point P.
Determine an equation for the tangent to the graph of f(x) at point P.
Use of CAS tool allowed.
a) f(x)= 3/(1+√x) , P(4, 1)
b) f(x)= √(5-x^2 ) , P(1, 2)
AI: Since you are allowed to use a CAS tool, there is not much to do here.
Here is on... |
H: Trouble with caculus problem, parametric equations, I don't know what I'm doing wrong
When a mortar shell is fired with an initial
velocity of v0 ft/sec at an angle α above the
horizontal, then its position after t seconds is
given by the parametric equations
$x = (v0 \cos \alpha)t$ , $y = (v0 \sin \alpha)t − 16t^2$... |
H: Number of steps one needs to infer final height of a walk
Let us have a walk on $\mathbb Z$ of size $2^n$. To compute the final height of the walk, the trivial way is to sum $1$ for an ascending step and $-1$ for a descending step all along the walk. I would like to have some method to infer the final height in a m... |
H: How many different box combinations can you get?
I have 12 books, all different. Four of them are fiction, and eight of them are non-fiction. I want to send you a gift. I'm going to send you five books - two fiction, three non-fiction. You will get them in a big old box, all at once. How many possible gifts can you... |
H: Why is this defined even when divided by zero
so I've got $$\dfrac{x^3-4x^2+3x}{x^2-1}$$ and want to calculate the asymptotes. There's one a $x=-1$ since the function is not defined there. But the function seems to be defined for $x=1$. How come?
It should be undefined at $x=1$ since $f(1)=1^2-1 = 0$
AI: Factor th... |
H: Is connectedness $\implies $ local connectedness?
Let $X$ be a connected topological space.
Then I wonder that $X $ can be locally connected or not.
I think the title of my question is NOT correct considering a Euclidean space like $\mathbb R^n$.
Do you think so??
Also, conversely, if $X$ is a locally connected, an... |
H: Triangle area division
It is given that in a triangle ABC, a line from A to BC intersects BC at point D. If the ratio in which AD divides BC is given can we say anything about the ratio of areas of triangle ABD and triangle ADC ?
AI: Yes we can find the ratio of the areas of the two triangles.
Area of $\triangle AD... |
H: If X and θ are both random variables and θ is the parameter of the distribution of X, are X and θ independent?
The answer appears to be no because the distribution of X is defined conditionally by θ which is also assumed to have a distribution as opposed to be a constant. Essentially, the formulation of the probabi... |
H: Show that $x_{n+1}-\dfrac{x_n}{2}$ converges to zero implies that $(x_n)_n$ also converges to zero.
Let $(x_n)_n$ be a sequence of real numbers such that $$\lim\limits_{n\rightarrow \infty}\left( x_{n+1} - \dfrac{x_n}{2}\right)=0.$$ Show that $(x_n)_n$ must also converge to zero.
$\textbf{My Attempt:}$
My origin... |
H: Proof of $|\int _a ^b \mathbf f | \leq \int _a ^b |\mathbf f|$
Let $\mathbf f:[a,b]\rightarrow \mathbb R ^n$ be continuous. I'm trying to prove the following fact:
$$|\int _a ^b \mathbf f| \leq \int _a ^b |\mathbf f|,$$
where: $$|\mathbf x|^2=\sum _{i=1} ^n x_i ^2.$$
I understand it by intuition: if $\mathbf f$ is ... |
H: Help with two variable equation
I need help with this equation which was given to my son who is in 4rth grade.
Solve in positive integers the equation: $xy=10(x+y)$.
AI: $xy =10x+10y$ $\implies$ $y=\frac {10x}{x-10}$.
Now use hit and trial or you can easily see that y has an integral value when x equals an integral... |
H: Localizing the quotient at a prime
I was reading in Dummit and Foote when I came across this statement I don't believe it is true.
Let $R$ be a Dedekind domain (noetherian, integrally closed, domain of dimension 1). They say given a prime ideal $P\neq 0$ then
$$R/P^{m}\cong R_{P}/P^{m}R_{P}.$$
Why is this true... |
H: Triangle Trig and calculus
The square of the side opposite a MUNDANGLE in a triangle is equal to the sum of the squares of the other two sides added to the product of these two sides multiplied by $\sqrt{3}$. What is MUNDANGLE?
This was an extra credit problem on my test, but my teacher said we had to do it for hom... |
H: Something to the power of a logarithm
This is probably a very obvious question, but here goes...
An answer in my textbook claims that
$$3^{\log n} = n^{\log 3}$$
and that
$$4n^2 (3/4)^{\log n} = 4n^{\log 3}$$
Why, using more basic laws, is this the case?
(Unfortunately Google confuses this question with changing ba... |
H: Solving for a variable with square roots in equation
I'm working on the following equation:
$$ a= \frac{2}{t^2}[(d+l)^{1/2} - (d^{1/2})]^2 $$
I want to solve for l such that:
$$ l=\frac{1}{2}at^2+(\sqrt{2ad}) t $$
I can't get it in this form. Here's my attempt:
$$ \frac{at^2}{2} = [(d+l)^{1/2} - (d^{1/2})]^2 $$
$$ ... |
H: Property of the limit of function proof
If $c≠0$ and $\lim_{x→c}〖f(x)〗=L$, prove that $\lim_{x→1/c}〖f(1/x)〗=L$
I know that meaning for all $ε>0$, there exist a $δ>0$ such that $0<|x-c|<δ$ implies $|f(x)-L|<ε$. How can I use these to prove the conclusion.
AI: When you say |x-c| < $\delta$ you mean $\delta$ - c < ... |
H: Dividing square of 2013x2013
Is the square of 2013×2013 can be divided into rectangles 1×3 in such
a way that the number of rectangles arranged vertically differ by 1
from the number rectangles arranged horizontally? Prove your answer
I totally have no idea how to solve that. I need accurate solutions or good... |
H: Inequality and absolute value: $p + |k| \gt |p| + k$
Here is the problem I am confused about. The given relation is:
$p + |k| \gt |p| + k$
It is not mentioned whether $p$ and $k$ are integers.
I need to determine whether $p$ and $k$ are equal or one is greater than other. If the second is true which one is grea... |
H: Is this implication true?
Suppose that a real sequence $u_n$ is such that $$u_{n+1}-u_n \rightarrow0$$
That is not enough to prove that $u_n$ is convergent (take $u_n=ln(n)$)
Now what if $u_n$ is bounded ? I guess it does converge, but how to prove this ? I tried to show that it had only one accumulation point...
A... |
H: why is $a^nb^n$ context-free?
I am writing somthing about Ppumping Lemma. I know that the language L = { $a^nb^n$| n ≥ 0 } is context-free. But I don't understand how this language satisfies the conditions of pumping lemma (for context-free languages) ?
if we pick the string s = $a^pb^p$, |s| > p , |vxy| < p and |v... |
H: Multivariable Delta Epsilon Proof
I am trying to prove the following limits using the delta-epsilon method. Can you help me out?
$$ \lim_{(x,y)\to(a,b)}(x+y) = a+b$$
AI: Hint:
Since on $\mathbb R^n$ all norms are equivalent, chose the easiest: $\Vert\cdot\Vert_\infty$. This will leave you with two 1-D. $\epsilon$-$... |
H: Prove that x is rational
Let $x$ be a real number with the properties that $x^3+x$ and $x^5+x$ are rational.
Prove that $x$ is rational. Denote $a=x^3+x$; $b=x^5+x$. We can multiply and add them together until we get the desired result. I also know some non-elementary proofs of this, but have you some nice elementa... |
H: A question dealing with conditions for which $V_\alpha$ models $ZFC$
I've been reading through models of Set Theory in Kunen's most recent Set Theory text and practicing exercises. He mentions that $V_\alpha$ can be used to satisfy certain axioms of $ZFC$ when $\alpha$ is strongly inaccessible. Here, $V_\alpha$ is ... |
H: A proof about complex polar form
I need to proof that $e^{i\theta}e^{i\phi} = e^{i(\theta+\phi)}$ as my homework using Euler's formula. And then using trigonometric sum formulas, I guess.
I would appreciate if you gave me some hints.
AI: Note that
\begin{align}
e^{it} e^{is} &= (\cos{s} + i \sin s) (\cos t + i \si... |
H: Is it possible to get a 'closed form' for $\sum_{k=0}^{n} a_k b_{n-k}$?
This came up when trying to divide series, or rather, express $\frac1{f(x)}$ as a series, knowing that $f(x)$ has a zero of order one at $x=0$, and knowing the Taylor series for $f(x)$ (that is knowing the $b_i$ 's).
I write $$1=\frac1{f(x)}f(... |
H: Multiplication by $x^2$ linear maps
$T:P(\mathbb{R}) \mapsto P(\mathbb{R})$ defined by
$(Tp)(x) = x^2p(x)$
Verify that multiplication by $x^2$ is a linear map.
Additivity: $x^2(p+q) = x^2p+x^2q$
Homogeneity: $x^2(ap) = a(x^2p)$
Is this a correct verification?
AI: It seems that you've just written down the definitio... |
H: Find the sale price given the 40% markup over wholesale and later 20% discount
A retailer pays $\$86$ to a wholesaler for an article. The retail price is set using a $40\%$ rate of markup on selling price. To increase traffic to his store, the retailer marks the article down $20\%$ during a sale. What is the sale ... |
H: Complex power series expansion for $f(z) = \frac{e^z}{a-z}$
I have the following homework problem in my complex analysis class:
Find the complex power series expansion for the function
$$ f(z) = \frac{e^z}{a-z}$$
where $a \in \mathbb{C}$, and $z \ne 0$.
I know the power series expansions for the functions:
$$ g(z) ... |
H: Finding Revenue Function and Max Revenue
Studying for a midterm.
The demand function for a manufacture's product is $p=1000-\frac1{80} q$
Where $p$ is the
price (in dollars) per unit when $q$ units are demanded (per week) by consumers. Answer
the following questions.
1) Write the Revenue function $R(q)$ in terms of... |
H: mathematical induction proof 2 things that are equal
$\begin{align} \forall n \in \mathbb{Z}^+, \sum^n_{p=1} \frac{p^2}{4p^2-1}= \frac{n(n+1)}{2(2n+1)} \end{align}$
I have done mathematical induction proofs with only one phrase, but this one has two. Can someone assist me. The first meaningful case would be 2 righ... |
H: The convergence interval of the series
I want to prove whether $x=4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$
I used Raabe's test. But I got limit is 1. So the test is not valid. Please help me which test do I need to use ? Thank you
In fact, if you help me to find its conve... |
H: Natural Logarithm Taylor Series Expansion
f(x)=x$^3$ln(1+2x)
Write the first four non-zero terms of the Taylor Series for the above function with x centered at a=0.
Using this model:
ln(1+x) = Σ$\frac{(-1)^{k}(x)^{k}}{k}$
I get the following...
x$^{3}$ln(1+2x) => x$^3$Σ$\frac{(-1)^{k}(2x)^{k}}{k}$
k=0 -----> Canno... |
H: Proving |sv| = |s||v| and is it true for all dimensions
I've been sitting here quite a while trying to figure out how to prove
$$ |s\vec{\mathbf{v}}| = |s||\vec{\mathbf{v}}| $$
I was not told in what dimension to prove this is, so I am assuming a 3D plane. I think I have a solution but I'm not sure if my method is... |
H: Conditions necessary for commutators [A,B]=[B,A]?
I know that normally for commutators that [A,B]=-[B,A] where A and B are operators. But under what conditions does [A,B]=[B,A]?
AI: Since it is always true that $[A, B] = -[B, A]$, if we assume that $[A, B] = [B, A]$, we see that
$$[A, B] = -[A, B]$$
so that $[A, B... |
H: Show $e^{D}(f(x)) = f(x+1)$ where $D$ is the derivative operator
I would appreciate help showing $e^{D}(f(x)) = f(x+1)$
Where $D$ is the linear operator $D: \mathbb{C}[x] \rightarrow \mathbb{C}[x]$ where (in the context where this statement arose) $x \in \mathbb{N}$;
$f(x) \mapsto \frac{d}{dx} f(x)$
By the Taylor ... |
H: Combinatorics? HELP!
Find the generating function for the number of ways to distribute blank scratch
paper to Alice, Bob, Carlos, and Dave so that Alice gets at least two sheets, Bob
gets at most three sheets, the number of sheets Carlos receives is a multiple of
three, and Dave gets at least one sheet but no more ... |
H: Prove that lim sup $x_n$ for infinitely many n.
Let $\{x_n\}$ be a real sequence and $r$ a real number. Prove that $\lim \sup_{n\to\infty}x_n<r$ implies $x_n<r$ for $n$ large enough. Prove that $\lim \sup_{n\to\infty}x_n>r$ implies $x_n>r$ for infinitely many $n$.
I am not sure how to prove this. Could anyone giv... |
H: Examples of unit vectors that's in the same direction as vector, let's say $v=(1,2,-3)^T$
Can someone give me some examples of unit vectors that's in the same direction as vector, let's say $v=(1,2,-3)^T$ for:
(i) Euclidean norm
(ii) Weighted norm $||V||^2=2V_1^2+v_2^2+\frac13v_3^2$
(iii) The 1 norm
(iv) The infi... |
H: multiple function recursion
I'm working on a problem that ended up with a recursion in two functions,
$$f_n= f_{n-1}+ f_{n-2}+ 2g_{n-2}$$
$g$ also has a recurrence dependent on $f$,
$$g_n= f_{n-1}+ g_{n-1}$$
Initial conditions $f_0=0=g_0$ and $f_1=1=g_1$, so it appears there is a fibonacci relationship. I'm trying... |
H: Application of Stirling's theorem for the given series
I want to prove whether $x=-4/27$ is convergent or not for the series $$\sum_{n=0}^{\infty}\binom{3n}{2n}x^n$$
I applied alternating series test. But, while using this, I need to apply Stirling's theorem so as to show that $a_n(4/27)\to 0$ as $n \to \infty$.
... |
H: Let $f: W \rightarrow X$ & $g: W \rightarrow X$ be cont. functions. Prove that if $X$ is Hausdorff then $c(f,g)$ is closed in $W$.
Let $W, V$ be topological spaces, and let $f: W \rightarrow X$ and $g: W \rightarrow X$ be continuous functions. Define the coincidence set of $f$ & $g$ to be the subset $$c(f,g) = \{w... |
H: Prove that this limsup inequality is strict.
so they are non-negative and bounded.
how do i show this?
AI: I think that $x_n=1+\sin(n)$, $y_n=1+\cos(n)$ work. The limsup of the product is less than 3. The limsup of each separately is 2, so the product is 4. |
H: One-point-functors and the Yoneda Lemma
Let $\mathfrak{Sets}$ be the category of sets, and $\mathfrak{Sch}$ the category of schemes. For any scheme $X$, consider the functor $h_{X}(-)=\mathsf{Hom}_{\mathfrak{Sch}}(-,X):\mathfrak{Sch}\longrightarrow \mathfrak{Sets}$. Consider also the following lemma:
Lemma (Yoneda)... |
H: Can the intersection of two non subspace subsets be a subspace?
I feel as though it cannot, but I'm having difficulty explaining why.
AI: Within $\mathbb{R}^1$, let $S=\{0,1\}$ while $T=\{0,2\}$. Neither is a subspace, but their intersection is $\{0\}$, which is a subspace.
For a less silly example, in $\mathbb{R}... |
H: A discontinuous function such that $f(x + y) = f(x) + f(y)$
Is it possible to construct a function $f \colon \mathbb{R} \to \mathbb{R}$ such that $$f(x + y) = f(x) + f(y)$$ and $f$ is not continuous?
AI: "Fix a basis for $\mathbb R$ as a $\mathbb Q$-vector space" (which exists under the axiom of choice, but under w... |
H: About quadratic residue modular p(p is prime)
The question is this.
Suppose that p is a prime, $p\ge7$. Show that $(\frac np)=(\frac {n+1}p)=1$ for at least one number n in the set {1,2,...,9}.
I think seperating into two cases when n=1 and when n=4 will help me prove it. But I can't think further...
Plz, HELP ME... |
H: Prove that lim is 0.
Prove that $$\lim_{x\to0}\sqrt{|x|}\sin\left(\frac1x+x^{10}\right)=0.$$
How do I show in a rigorous way that this limit as $x\to 0$ equals $0$ ?
Any tips or suggestions would be great!
AI: Since
$|\sin(anything)| \le 1$,
$|\sqrt{|x|}\sin(anything)|
\le |\sqrt{|x|}|$,
and since
$\lim_{x \to 0... |
H: Understanding the proof that if a set is closed its complement has to be open.
The statement proven here is that: $A \text{ is closed} \implies A^{c} \text{ is open}$.
The proof given in class is this:
Suppose $A^{c}$ is not open, then it must be the case that for some $a \in A^c \implies \nexists N_{\delta}(a) \in... |
H: Boolean Algebra - Demorgan Laws
I am given the problem:
!((!A * B) * !((!B + C) * (!C * !D)))
where ! = NOT, * = AND, and + = OR and I tried simplifying it using only Demorgan Laws (no absorption) and I got:
(A + !B) + ((!B + C) + (C + D))
and I was just wondering if it was correct. Any help would be greatly appr... |
H: Complete and elementary proof that $(a^x - 1)/x $ converges as x goes to 0
Anybody who has taken a calculus course knows that
$$\lim_{x \to 0} \frac{a^x - 1}{x}$$
exists for any positive real number $a$, simply because the limit is by definition the derivative of the function $a^x$ at $x = 0$. However, for this ar... |
H: Show if $\phi$ is a ring isomorphism of $\mathbb{Z}\to\mathbb{Z}$, then $\phi$ is the identity mapping.
Show if $\phi$ is a RING isomorphism of $\mathbb{Z}\to\mathbb{Z}$, then $\phi$ is the identity mapping.
I don't really know where to start with this one. I know that since $\phi$ is an isomorphism, it is a homomo... |
H: Calculus Trigonometric Integral
I have been trying to solve this particular problem and can not figure out where I have gone wrong in my solution. This is a picture of my solution, if anyone could help point out where I have gone wrong it would be appreciated.
http://i.imgur.com/j0xGdM3.jpg
AI: First of all, your e... |
H: Unique of eigenbasis of self-adjoint operator.
Today I am reading an article: Eigenvalues and sums of Hermitian matrices there is an exercise copy from that article:
Exercise 1 Suppose that the eigenvalues $\lambda_1(A)>\cdots>\lambda_n(A)$ of an $n\times n$ Hermitian matrix are distinct. Show that the associated e... |
H: Question on types of continuity.
Determine if on the given interval, the function $f(x)=\frac{1}{\sqrt{x}}$, is a) continuous b) uniformly continuous, c) Lipschitz continuous, d) differentiable, and e) $C^1$
The interval $(0, 1]$ and $[1, \infty)$
My work: Let $\epsilon > 0$ given. We must show there exists $\delta... |
H: Integration trouble on a conservative vector field
I'm trying to integrate $\int_{\mathbf{x}}{\mathbf{F}\cdot{d\mathbf{s}}}$ where
$$\mathbf{F}=x^2\mathbf{i}+\cos{y}\sin{z}\mathbf{j}+\sin{y}\cos{z}\mathbf{k}$$and $$\mathbf{x}:[0,1]\rightarrow\mathbb{R}^3; \mathbf{x}(t)=(t^2+1,e^t,e^{2t})$$
I'm at the point where I... |
H: Good websites/books for geometry exercises?
I'm looking for exercises similar to those seen on putnam exams or olympiad exams, such as finding the area of polygons inscribed other polygons, finding certain angles, etc.
AI: I like Go-geometry, which covers the basics up to IMO level.
I'm not too certain what you wou... |
H: Generating functions word problem(balloons)
Find the generating function for the number of ways to create a bunch of n balloons selected from white, gold, and blue balloons so that the bunch contains
at least one white balloon, at least one gold balloon, and at most two blue balloons. How many ways are there to cre... |
H: Matrix of transformation
I have just finished understanding the topic of matrix of the transformation, and we just started T-invariant subspaces.
Can anyone please help me with these questions, because I don't know how to approach them. This is not homework by the way, it is just for my understanding.
1) Suppose $B... |
H: Is this absolute value notation or something else?
In this document, in Figure 1 (second to last page) there are several uses of $\| \;\;\|$:
Is this another notation for absolute value, or is this a notation for something to do with vectors/matrices?
AI: As the document says immediately following the introduction... |
H: Proving it doesn't exist a homeomorphism between $\mathbb R$ and $\mathbb R^n$, $n>1$.
I have to prove that for $n \ge 2$, there doesn't exist a homeomorphism between $\mathbb R$ and $\mathbb R^n$. Could anyone give me a hint on how could I prove this?
AI: The notion of connectdness is a topological property, meani... |
H: Selecting marbles from two jars
Jar $A$ contains $3$ red and $3$ black marbles, and Jar $B$ contains $4$ red and $6$ black marbles. If a marble is randomly selected from each Jar, what is the probability that the marbles will be the same color?
I'm having difficulty approaching this problem. It seems like I should... |
H: find a basis for the subspace $S$ = {$p \in \mathbb{P}_3$ |$ p$(5) = 0}
This is as far as I've gotten:
$p$($x$) = $a$ + $bx$ + $c$$x^2$ + $dx^3$ then $p$($5$) = $a$ + $5b$ + $25c$ + $125d$ = $0$
which gets the vector $(1, 5, 25, 125)$ but then I'm not sure how to a basis from this vector. I tried setting some othe... |
H: Creating a System Impulse Response in Matlab
Preface: I'm extremely new to Matlab.
Ok, so I have a sound file that I loaded in Matlab.
Two variables are loaded:
xx - the speech waveform - 16001 samples
fs - the sampling frequency of the speech waveform = 22050 Hz
Since fs = 1/T, T = 0.00004535147 seco... |
H: There are 2013 random points on the plain, prove exactly 8 lies within a circle
This was posed as a homework problem to me and I have made progress, but I don't think I have a solution.
So there are a lot of random points on the plane and we need to find a circle that contains exactly eight of them. The way I would... |
H: Show that if a and b are positive integers, and $a^3 | b^2$, then a | b
Show that if a and b are positive integers, and $a^3 | b^2$, then a | b.
If p is a prime divisor of a, and $p^r$ is the highest power of p dividing a. Then $p^{3r} | a^3$, and so $p^{3r} | b^2$. If $p^s$ is the highest power of p dividing b, th... |
H: Additional Law Word Problem
Forgive me if this is really basic:
Tammy is a general contractor and has submitted two bids for two projects (A and B). The probability of getting project A is 0.65. The probability of getting project B is 0.77. The probability of getting at least one of the projects is 0.90. What is t... |
H: What functions model this relationship?
I'm currently working a bit on an AI, and in order for it to function, it must be able to quickly predict where a point will be in space, given any distance. The movement of this point may be modeled as a linear graph 1 up until it reaches a maximum y value, at which point th... |
H: Trying to simplify $A'B'C'D + A'B'CD + A'BC'D + AB'CD + ABCD$
My solution so far:
$A'(B'C'D + B'CD + BC'D) + A(B'CD + BCD)$
$= A'(C'D(B' + B) + B'CD) + A(CD(B'+B))$
$= A'(C'D(1) + B'CD) + A(CD(1))$
$= A'C'D + A'B'CD + ACD$
$= D(A'C' + AC) + A'B'CD$
$= D(A\text{ xnor }C) + A'B'CD$
$= D[(A\text{ xnor }C) + A'B'C]\tex... |
H: In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.
In the field $\mathbb{Z}_7[x]/\langle x^4+x+1\rangle$, find the inverse of $f(x)=x^3+x+3$.
I know how to find the inverses of elements within sets, rings, and fields. I know what to do if the field was just $\mathbb{Z}_7$, bu... |
H: A set that is transitive but not well-ordered by $\in$?
I am trying to find a transitive set which is not well-ordered by $\in$. This question raises when I read Jech's Set Theory, in which an ordinal number is defined as a transitive and $\in$-well-ordered set.
Thanks!
AI: $V_\omega$ is a very easy example; more g... |
H: Prove that $\sup \{-x \mid x \in A\} = -\inf\{x\mid x \in A\}$
I need to prove that $\sup \{-x \mid x \in A\} = -\inf\{x \mid x \in A\}$ and am having trouble moving the $-x$ out of the $\sup$ to $\inf$. Another thing is that I don't quite know how to prove $b = \inf$ (see below). Any help would be appreciated! Her... |
H: How to solve $9\sin 2x-40\cos 2x=\frac{41}{\sqrt{2}}$
Solve the following question:
\begin{eqnarray}
\\9\sin 2x-40\cos 2x&=&\frac{41}{\sqrt{2}}\\
\end{eqnarray}
I know that there is a formula for solving the above question like :
\begin{eqnarray}
\cos a=\frac{A}{\sqrt{A^2+B^2}} , \space\space
\sin a=\frac{B}{\sq... |
H: For a Gaussian Random walk where $x_n$ is the sum of $n$ normal random variables, what is $P(x_1 >0, x_2 >0)$?
I know that the events $x_1 >0$ and $x_2 >0$ are not independent, but I can't think of a way to find a conditional probability so I can solve this.
Thanks!
AI: Presumably (but this ought to be in the quest... |
H: Why left continuity does not hold in general for cumulative distribution functions?
Definition: The c.d.f. $F$ of a random variable $X$ is a function defined for each real number $x$ as follows:$$F(x)=\Pr(X\leq x) \text{ for } -\infty<x<\infty$$
Let $$F(x^-)=\lim_{y\rightarrow x,\,y<x}F(y)$$ and $$F(x^+)=\lim_{y\r... |
H: If a group $G$ has odd order, then the square function is injective.
Suppose $G$ has odd order, show the function $f:G\rightarrow G$ defined by $f(x)=x^2$ is injective. This proposition is easily provable if we assume $G$ is Abelian, but I don't know how to start this without the assumption of being Abelian.
AI: Le... |
H: Computing an inverse modulo $25$
Supposed we wish to compute $11^{-1}$ mod $25$. Using the extended Euclid algorithm, we find that $15 \cdot 25 - 34 \cdot 11 =1$. Reducing both sides modulo $25$, we have $-34 \cdot 11 \equiv 1$ mod $25$. So $-34 \equiv 16 $ mod $25$ is the inverse of $11$ mod $25$.
im realy confuse... |
H: Why, precisely, is $\{r \in \mathbb{Q}: - \sqrt{2} < r < \sqrt{2}\}$ clopen in $\mathbb{Q}$?
Just going over some old notes and I realized I always took this for granted without actually fleshing out exactly why it is true.
The set of all r is closed in $\mathbb{Q}$, because the set of all r is just all of the rat... |
H: Any counterexample to answer this question on elementary geometry?
Question: See the figure below. If AB=BC and DE=EF, is the line DF parallel to the line AC?
This should be an elementary problem. But I can't construct a counterexample to disprove the above question. If the answer is negative, please give a counte... |
H: Writing real invertible matrices as exponential of real matrices
Every invertible square matrix with complex entries can be written as the exponential of a complex matrix. I wish to ask if it is true that
Every invertible real matrix with positive determinant can be written as the exponential of a real matrix. (W... |
H: Does this figure represent a cumulative distribution function?
Is this a c.d.f.?
I have no problem for random variable $X$ at $-\infty<X<x_2$. But if p.d.f. were continuous in interval $x_2\leq X<\infty$ , then c.d.f. should have been continuous. If probability mass function were discrete at $x_3$, then the c.d.f. ... |
H: A supposed incorrect theorem
Hi i'm new here and here's my question:
Suppose $A\subseteq$$P(A)$. Prove that $P(A)\subseteq$$P(P(A))$.
I am using the book How to prove it by Velleman and i was wondering if anyone had a link to the soluions manual pdf if there ever was one?? I've googled it but it seems that no one h... |
H: Finding the mistake in precalc equation
$$\sqrt{x+3}+\sqrt{x-1} = -1;\tag{1}$$ $$ \mathbb{D}=[1,\infty)\tag{2}$$
$$x+3=(-1-\sqrt{x-1})^2\tag{3}$$
$$x+3=1+2\sqrt{x-1}+x-1\tag{4}$$
$$x_{1/2} = \pm\sqrt{\frac 3 2}+1\tag{5}$$
$$\mathbb{D} -> x=\sqrt{\frac 3 2}+1\tag{6} $$
According to the solution there shouldn't be an... |
H: Adding weightage to number.
I have four numbers
x=50
y=30
z=20
sum of x+y+z=100
if i want to add a number n where
n=(x+y)/2
Edit:
n=45
such that x+y+z+n=100
now x+y+z=100.If any number (45 here) is added, then the sum should be always 100. i.e we need to subtract number(say m) from each x,y,z,n such tha... |
H: Conditional expected value of a product of two independent normal variables
I'm trying to work out the following conditional expectation:
$E[\epsilon_t z_t|\epsilon_t + z_t = k, \epsilon_{t-1}, z_{t-1}, \epsilon_{t-2}, z_{t-2},...]$
where $k$ is known and $\epsilon_{t}$ and $z_t$ are independent and normal distribu... |
H: What does the variable $v$ in characteristic function of a random variable signify?
The characteristic function of a random variable $X$ is given as:
$$E(e^{jvX}) = \int _{-\infty} ^{\infty} e^{jvx} p(x) dx $$
This is interpreted as either mean of function $e^{jvx}$ or fourier transform of pdf $p(x)$. I know that $... |
H: Verify proof that $\varnothing \in F$ implies $\cap F = \varnothing$
Hi so I want to know if my proof attempt is correct, so here's the question:
Suppose that $F$ is a family of sets. Prove that if $\varnothing\in F$ then $\cap F=\varnothing$.
Proof.
Suppose $\varnothing\in F$ and suppose $\cap F\ne\varnothing$. ... |
H: Question about $ m = \prod_i (p_i-1)^{b_i} $ and the unique factorization theorem.
The fundamental theorem of arithmetic says that every integer $n>1$ is of the form
$$ n = \prod_i {p_i}^{a_i} $$
where $p_i$ is the $i$ th prime and $a_i$ is a nonnegative integer.
My question is how many $m$ satisfy $ 1<m<n $ and
$... |
H: Derivative of f(x)=arth(lnx)
I'm struggling with finding derivative of f(x)=arth(lnx).
I've done following:
x=th(y)
f'(x)=(arth(lnx))'=1/(thlnx)'=1/(1/(xch^2(lnx)))=(x(ch^2(lnx)))/(ch^2x-sh^2x)=(ch^2(x(lnx)))/(ch^2(1-th^2x))=(x(lnx))/(1-x^2)
so in conclusion I get f'(arth(lnx))=(x(lnx))/(1-x^2)
but I think it's not... |
H: dominating function for $(1-\frac{x^2}n)^n(1+\sqrt{nx})$ in the dominated convergence theorem
Compute the following limit:
$\displaystyle\lim_{n\to\infty}\int_{-\sqrt{n}}^{\sqrt{n}}\Bigl(1-\frac{x^2}{n}\Bigr)^{n}(1+\sqrt{n}x)dx$
Hint: $\displaystyle\int_{-\infty}^{\infty}e^{-x^2}dx=\pi$
so I have a sequence $f_{n... |
H: Sturm-Liouville Problem --Suppose that the functions $p(x), p′(x), q(x)$, and $r(x)$ in $[p(x)y′]′ − q(x)y + \lambda r(x)y = 0, \quad (a < x < b)$
Suppose that the functions $p(x), p′(x), q(x)$, and $r(x)$ in $[p(x)y′]′ − q(x)y + \lambda r(x)y = 0, \quad (a < x < b)$. Question pasted as image given below:
AI: Here ... |
H: Evaluating the convolution using the convolution integral
I am having trouble evaluating the convolution of two signals using the convolution integral.I want to find the convolution of two signals x and h where,
$$
x(t) = \begin{cases}
e^{-at} & \text{$t > 0$} \\
0 & \text{$t < 0$ } \\
... |
H: Differentiation of function with chain rule
the following expression is part of a function I have to differentiate:
$y = \tan^3(5x^4-7)$
I tried using the chain-rule, so:
$ y' = 3\tan^2(5x^4-7)\cdot(20x^3)$
is this correct?
AI: Not quite correct; we need to also differentiate the "tangent" component.
In general, t... |
H: Functions satisfying $(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$
Let $f$ be a differientable real function such that $$(b-a)f'(\tfrac{a+b}{2}) = f(b)- f(a)$$
for all reals $a,b$.
Is $f$ polynomial of degree $\leq 2$ ?
AI: The answer is Yes.
The condition on $f$ is equivalent to
$$f(x+h)-f(x-h)=2f'(x)h\tag{1}$$
for al... |
H: Trying to simplify the expression $A'B'C'D' + A'BC'D' + A'BC'D + AB'C'D$
So far I've got:
$A'B'C'D' + A'BC'D' + A'BC'D + AB'C'D$
$= A'C'D'(B' + B) + C'D(A'B + AB')$
$= A'C'D'(1) + C'D(A \;\text{ XOR }\; B)$
$= C'[A'D' + D(A \;\text{ XOR }\; B)]$
Did I do this correctly? Is there a simpler solution?
Thanks
K
AI: Yes... |
H: when $p \mid ab \Rightarrow p \mid a$ or $p \mid b$?
I want to know when this statement is true ($p$ is a prime):
$p \mid ab \Rightarrow p \mid a$ or $p \mid b$ ?
AI: Given $p$ is a prime, then your statement$$p \mid ab \implies p \mid a\;\lor p\mid b$$ is always true.
But the proposition is not necessarily true w... |
H: How to estimate the value of this supremum?
Could you please guide me through how to estimate or calculate this supremum
$$\sup_{x\in\mathbb R}\left|x \cdot \arctan(nx)-x \cdot \frac{\pi}{2}\right|=?$$
AI: Use that
$$
\arctan x=\int_0^x\frac{dt}{1+t^2}\quad\text{and}\quad \frac{\pi}{2}=\int_\infty^x\frac{dt}{1+t^... |
H: Probability that coin will fall into a square
So the exercise is this:
We have and infinite chessboard and we have a coin. Every grid is of length and width $a$, whereas the coin has diameter $2 \cdot r<a$. We throw a coin into a chessboard and we want to know with what probability the coin will falll into the grid... |
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