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H: Prove $\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty$ for an increasing sequence $a_n$ of positive integers
The $a_n$'s are integers, positive, and increasing: $0< a_1 < a_2 < \cdots$, the problem asks us to prove that:
$$
\sum^{\infty}_{n=1} \frac{a_{n+1}-a_{n}}{a_{n}}=\infty
$$
While I have checked this... |
H: Find the basis and its dimention of a subspace
Given $S=\{x\in\Bbb R^4: x_1+x_2+x_3+x_4=0, x_2+x_4=0\}$
So (...)
If $x_2+x_4=0 \implies x_2=-x_4$ then $x_1+x_2+x_3+x_4=0\implies x_1+x_2+x_3-x_2=0\implies x_1=-x_3\implies x_1=x_2=x_3=x_4=0$
$X=(0,0,0,0)$ but $X$ is not linearly independent.
So $S$ is not a basis for... |
H: Closed form for $\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx$
I need to evaluate this integral:
$$Q=\int_0^1\frac{x^{5/6}}{(1-x)^{1/6}\,(1+2\,x)^{4/3}}\log\left(\frac{1+2x}{x\,(1-x)}\right)\,dx.$$
I tried it in Mathematica, but it was not able to find a closed form.... |
H: How do i find the exterior angles of an L-shaped polygon?
I'm trying to review exterior angles after many years. It's my understanding that the sum of a polygon's exterior angles must equal 360°. How would you find the exterior angles in this polygon?
AI: You've got most of it already.
An algorithm for drawing the ... |
H: How dividing a number with 5 gives no. of multiples of 5 from one till that number?
For example 60/5 = 12, which means there are 12 multiples of 5 between 1 to 60. What is the logic behind it ?
Is it the case with only 5 or dividing a number with any particular number x gives no. of multiples of x from one till tha... |
H: Compute all the directional derivatives of a trivariate polynomial function quickly
Given a trivariate polynomial $A\in\mathbb{R}[x,y,z]$, a direction $\vec v\in\mathbb{R}^3$ and a point $p\in \mathbb{R}^3$, what is the fastest way to compute the directional deriviatives $\nabla_{\vec v}A(p), \nabla_{\vec v}\nabla_... |
H: Solve discrete Math Problem using abstract algebra, postage problem?
The question I am looking at is not very hard:
Determine which amounts of postage can be written with $5$ and $6$ cent stamps.
To determine the amount, use a brute force way to solve it. Counting from $0$, see if each number can be written with $... |
H: Finding a formula for the sequence $10, \,110,\,1110,\,11110,\dotsc$
How can I find a formula for the sequence
$$10,\,110,\,1110,\, 11110,\dotsc$$
to make it ready for summation?
AI: If we call your numbers $a_1,a_2,a_3,\dots$, then
$$a_n=(10)\left(\frac{10^n-1}{9}\right).$$ |
H: Scaling function in $C^\infty(\mathbb{R})$
Let $f(x)\in C^\infty(\mathbb{R})$, and let $a>0$. Let $g(x):\mathbb{R}\rightarrow\mathbb{R}$ be defined by $g(x)=f(ax)$. Is it necessarily true that $g(x)\in C^\infty(\mathbb{R})$?
AI: $g$ is the composition of two $C^\infty$ functions, as such it is of class $C^\infty$. ... |
H: How do I calculate this second derivative?
Suppose $
\frac{dy}{dt} = \frac{dy}{dx}\frac{dx}{dt}
$. How do I calculate $\frac{d^2{y}}{dt^2}$?
AI: $\frac{d^2y}{dt^2} = \frac{d}{dt}\frac{dy}{dt} = \frac{d}{dt}\left[\frac{dy}{dx}\frac{dx}{dt}\right] = \frac{d^2y}{dx^2}\left(\frac{dx}{dt}\right)^2 + \frac{dy}{dx}\frac{d... |
H: Order of dodecahedron automorphism group.
I need to find out how many elements are in the automorphism group of a regular dodecahedron.
So Using Orbit-Stabilizer theorem I get $|G|=|G_x||O_x|$
If we pick a point on the wall of the dodecahedron we can rotate every wall $5$ different ways and the point on that wall ... |
H: Matrix representation of a orthogonal projection
Hey can some help me with these kinds of questions have a lot of them and I can't figure out how to do them thanks
Let $v = (cos \theta, sin\theta)^{T} \in R^2$ for some angle $\theta \in 2^{R}$, and let $P_{v}$ denote the orthogonal projection corresponding to v. Fi... |
H: Do the real numbers form a division ring with operations $a\oplus b=a+b+\frac12$ and $a\odot b=a+b+2ab$?
Is $(\Bbb R,\oplus,\odot)$ a division ring, where
$$a\oplus b = a+b+\frac12$$
and
$$a\odot b = a+b+ 2ab?$$
I have only issues with $\odot$. It doesn't work for inverse of $$a=\frac{-1}{2}$$ since $$a^{-1... |
H: Probability of weighted coin
I'm having trouble with the following question.
A weighted coin lands heads 2/3 of the time whereas it lands tails 1/3 of the time. If the coin is tossed 10 times what is the probability that it will land exactly 4 heads?
I would solve the problem doing (10 choose 4)(2/3)^4(1/3)^6
Kin... |
H: Algebraic intuition for Fourier inversion formula
I was thinking about the Fourier inversion formula, which says
$$f(x)=\dfrac{1}{2\pi}\int_{-\infty}^\infty\left(\int_{-\infty}^\infty f(z)e^{-iyz}dz\right)e^{ixy}dy$$
I know there is an issue about changing order of integration, but I want to know algebraically why ... |
H: Problem regarding filling squares inside a $n\times n$ grid.
Assuming a $n\times n$ square grid, what is the most number of squares that can be filled in such that there are no completed rows, columns, or diagonals?
Is there a formula to calculate this?
Real world example:
Given a Bingo sheet, what is the most ... |
H: Accumulation Points for $S = \{(-1)^n + \frac1n \mid n \in \mathbb{N}\}$
I was recently asked to find the accumulation points of the set $$S = \{(-1)^n + \frac{1}{n} \mid n \in \mathbb{N}\}$$
I answered that the accumulation points are $\{-1,1\}$, because despite the fact that $\frac{1}{n}$ diverges, we can still u... |
H: Calculation the Standard Deviation
I want to calculate the standard deviation of the following numbers:
30, 45, 45, 60, 75, 80, 90, 100, 110, 120.
As far as I know, that would be
$\sqrt{\frac{1}{10}((30-75.5)^2+(45-75.5)^2+(45-75.5)^2+(60-75.5)^2+(75-75.5)^2+(80-75.5)^2+(90-75.5)^2+(100-75.5)^2+(110-75.5)^2+(120-75... |
H: Equation of a plane containing 2 points
Find the equation of the plane containing the vectors $(1, 0, \sqrt{3})$ and $(1, \sqrt{3}, 0)$. The vectors are in standard position.
I first get the vector between the 2 points
$\vec{v}^{\ } = (1, 0, \sqrt{3}) - (1, \sqrt{3}, 0) = (0, -\sqrt{3}, \sqrt{3})$
Next I determine ... |
H: What does "non-decreasing" mean in relation to this definition about the prime factorization of numbers?
I'm reading a text on discrete math and came across a theorem which states:
"Every integer greater than 1 can be written uniquely as a prime or as the product of two or more primes where the prime factors are wr... |
H: Principal value of the singular integral $\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$
For a constant $0<A<\pi$, and natural $n$ I want to find the principal value of the integral:
$$\int_0^\pi \frac{\cos nt}{\cos t - \cos A} dt$$
First of all, I'm not certain what function in the complex plane I should look at. ... |
H: Does the dualizing process on vector spaces necessarily terminate?
It's well-known (assuming the axiom of choice) that the inclusion $\ell^1 \subset (\ell^1)^{**}$ is proper as a simple corollary of the Hahn-Banach theorem. But is this the end of the dualizing process; e.g., does $(\ell^1)^{**} = (\ell^1)^{****}$, ... |
H: Cosine and sine dense in unit circle
We may assume the following theorem:
Theorem: A real number $\lambda$ is irrational iff the set $\{m+\lambda n\mid m,n\in\mathbb{Z}\}$ is a dense subset of $\mathbb{R}$.
Assume $\lambda$ is irrational. How can we show that the set $\{(\cos(2\pi n\lambda), (\sin(2\pi n\lambda))... |
H: Schwartz class function convergence in $L^1$ and $L^2$
Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function in both $L^1(\mathbb{R})$ and $L^2(\mathbb{R})$. I want to show that there exists a sequence of functions $g_1,g_2,\ldots$ in the Schwartz class such that both $\|g_n-f\|_1\rightarrow 0$ and $\|g_n-f\|_2\rig... |
H: Understanding a proof: Eigenvalues of a real symmetric matrix are real
I'm trying to understand the following proof, but I have two questions about it that I hope someone could clarify to me:
1) How does the last equation ($\lambda ^t \overline{Z} Z = \overline{\lambda}^t Z \overline{Z}$) follow from the previous ... |
H: How to study abstract algebra
I am taking Abstract Algebra course at the university. We are doing chapters 1-20 from Gallian's abstract algebra text book.
I am just doing assigned homework everyweek ( About 5 questions from each chapter). Although I am getting an A in all the assignments and midterms, but I am real... |
H: Unbiased estimator of standard deviation
Let $X$ be a random variable of distribution $N(\mu , \sigma ^2)$, where we know the value of $\mu$ and we don't know the value of $\sigma$.
My task is to choose number $d$, such that random variable $ Y = d \sum_{i=1}^{n} |x_i - \mu| $ was unbiased estimator of $\sigma$.
So... |
H: Integration of $\int e^{\sin^2(x)+ \cos^2(x)}\,dx$
Calculate the indefinite integral
$$\int e^{\sin^2(x)+ \cos^2(x)}\,dx.$$
Not sure how to do this
AI: Well, it's not hard at all. $$\int e^{\sin^2(x)+\cos^2(x)}\,dx= \int e^1\,dx,$$ since $\sin^2(x)+\cos^2(x)=1$. So $e^1$ is a constant and you can pull this ou... |
H: Algebra 2 - Imaginary roots of Polynomials
Question:
One zero of $P(z) = z^3 +az^2 + 3z + 9$ is purely imaginary. If $a \in \mathbb{R}$, find $a$ and hence factorize $P(z)$ into linear factors.
What I've done:
I know that the $P(z)$ is real since its coefficients are all real. The imaginary root must be $bi$ an... |
H: Polynomial factorization to irreducible factors with respect to field
I have a question, I think I don't understand this material very well and could use an explanation / some help.
Basically we are asked to decompose $x^5-x$ to irreducible factors over $R,F2,F5,C$
But I don't understand how the field has anything ... |
H: Fourier transform on $\mathbb R^n$ of Gaussian function
Let $\displaystyle{K(x)= e^{- \pi |x|^2} \quad ,x \in \mathbb R^n}$ be
the Gaussian kernel on $\mathbb R^n$. Prove that its Fourier transform
is $$ \hat{K} (\xi) = e^{- \pi |\xi|^2} $$
I can prove this on $\mathbb R$ using the fact $\displaystyle{ \int_... |
H: Construct dense and disjoint sets of $\mathbb{R}^m$ so that every element of their Cartesian product has full rank
Or equivalently, can one construct sets $S_1 ,S_2 ,\dots ,S_n \subseteq \mathbb{R}^m$ so that
(i) the sets $S_i$ are dense and disjoint; and
(ii) if one picks from each set $S_i$ any element $u_i$, the... |
H: How to derive the marginal probability function of X?
Let $X$ and $Y$ be discrete random variables with joint probability function $f(x,y)=k\frac{2^(x+y)}{x!y!}$ for $x=0,1,2..$ and $y=0,1,2...$,where $k$ is a positive constant.
The answer is $k\frac{(2^x)(e^2)}{x!}$. I do not know how to get the $e^2$.
How to deri... |
H: Confusion regarding the Logarithmic function change of base formula
My textbook seems to be making a big leap when trying to prove the change of base formula for logarithms. If someone could help clear this up it would be very appreciated.
It starts with:
$b^{x \log_b(a)}$
and uses the power rule to get:
$b^{x ... |
H: Prove that this sequence diverges to infinity.
$\lim_{n\to \infty} (1 +\frac{1}{n}\tag{displayed})^{n^2} = \infty$
I don't know how to tackle this one. Knowing that it diverges to infinity and thus does not have an upper bound, should I try to prove that it's an unbounded subsequence, if so how? Is that sufficient ... |
H: Difference operation on factorials
Please how is the combination addition formula ${{t}\choose{r}}={{t-1}\choose{r}}+{{t-1}\choose{r-1}}$ useful in proving the difference equation $\Delta_{t}{{r+t}\choose{t}}={{r+t}\choose{t+1}}$?
Secondly, does ${{t}\choose{k}}={{t}\choose{t-k}}$ need verification? I thought defin... |
H: Determining the Fourier series of a given function (Verification)
Determine the Fourier series for the function $$f(x)=\begin{cases} &0 \quad -\pi \leq x \leq 0\\ &e^{x} \quad 0 \leq x\leq\pi \end{cases}$$
Here is what I have come up with; I first calculated (using integration by parts)
$$a_{n}=\frac{1}{\pi}\int_... |
H: Which is the mathematical theory that talks about these structures?
Let's define $\sigma(n)$ as the sum of the digits of the integer $n$ modulo $9$, having posed that $\sigma(9) = 9$.
Now consider 2 number $a$ and $b$ in the set $\{1, \cdots, 9\}$. Which is the value of $\sigma{(ab)}$?
Starting from this problem, o... |
H: Probability problem - 3 hunters and a boar
3 hunters fire simultaneously at a boar. One bullet hits the boar. What probability is for each hunter to be the one who hit the boar, when hunter A hits with accuracy of 0.2, B 0.4, C 0.6? I see 2 possible solutions: The first one is simple. When I take 10 average bullets... |
H: Cosets and Modulo
For any integer n greater than 1, psi(n) denotes the number of positive integers less than n and relatively prime to n. Prove that if a is any integer relatively prime to n, then a^psi(n)modn = 1
I was thinking of using Fermats Little Theorem, or some manipulation of U(n), but I cannot put togethe... |
H: In a right triangle, given slope and length of hypotenuse find length of legs.
Say I have a right triangle.
I know the slope and length of $c$, how do I find the length of $a$ and $b$?
AI: We have a right triangle, so there are two things we know:
Slope $\;m = \dfrac{a - 0}{b-0}=\dfrac ab\implies a = bm$.
And
... |
H: Matrix power and its product
If $A$ and $P$ are $n \times n$ matrix, when does $(P^{-1}AP)^n=P^{-1}A^nP$?
And how do I prove that the equation is true given the condition?
AI: $(P^{-1}AP)(P^{-1}AP)=(P^{-1}A)(PP^{-1})AP$ |
H: If a function $x\mapsto xf(x)$ has a derivative at $a \ne0$, then $f$ is differentiable at $a$
Prove if $x\mapsto xf(x)$ has a derivative at $a \neq 0$, then $f$ is differentiable at $a$.
The problem I'm encountering is with the setup. If I am trying to show
$$\lim_{x \to a}\frac{f(x)-f(a)}{x-a},$$
would I first... |
H: What am I doing wrong in attempting to solve this system of differential equations?
Problem:$$\quad x''+y''=t^{ 2 }\quad \quad x''-y''=4t\quad \quad x(0)=8\quad x'(0)=y(0)=y'(0)=0\\ \\$$ Attempt:$$\\ s^{ 2 }L\{ x\} -sx(0)-x'(0)+{ s }^{ 2 }L\{ y\} -sy(0)-y'(0)=2/{ s }^{ 3 }\\ { s }^{ 5 }L\{ x\} -8{ s }^{ 4 }+{ s }^{... |
H: determining the greatest $n$ for which $3^n$ divides $30!$
Determine the greatest integer $n$ such that $3^n\mid 30!$
I have no idea of how to approach this problem. I would first calculate $30!$ but obviously that number is way too large. Any help?
AI: How many factors of $3$ go into $30!$?
To answer this conside... |
H: For what values of $r$ does $y=e^{rx}$ satisfy $y'' + 5y' - 6y = 0$?
For what values of $r$ does $y=e^{rx}$ satisfy $y'' + 5y' - 6y = 0$?
Attempt:
$y' = [e^{rx}] (r)$
$y''= r^2e^{rx}$
AI: If you plug them in, you obtain : $$r^2+5r-6=0$$
Solving this equation you get $r=1$ or $r=-6$.
That means that the general solu... |
H: Matrix associated with a unitary operator
So, I'm looking at this proof, and it makes no sense to me at all.
Theorem: Let $V$ be a finite dimensional vector space over $\mathbb{R}$, with a positive definite scalar product. A linear map $A \colon V \to V$ is unitary if and only if
$$A^tA=I.$$
Proof: The operator $... |
H: Prove that in $S_n$ there are an equal number of even and odd permutations.
Prove that in $S_n$ there are an equal number of even and odd permutations.
$S_n$ is a group of all possible permutations on a set of $n$ elements. For this problem we can assume $n>1$.
I'm pretty sure I need to prove this by contradictio... |
H: Show that this sequence converges. (cauchy criterion)
Given $a_0 \geq 0$ and a sequence ($a_n)_{n\in\mathbb{N}}$
$$ a_{n+1}= \frac1{(2+a_{n})}.$$
for ${n\in\mathbb{N_0}}$.
Show that $(a_n)_{n\in\mathbb{N}}$ is convergent and determine the limit.
All I've got so far is that this sequence is not monotonous, but tha... |
H: How to find the inverse involving 2 variables
I am trying to solve this using what I have read from this site but I always hit a dead end.
Consider the function $$f: \mathbb R \times \mathbb R \to \mathbb R\times\mathbb R$$ defined by $$f(x,y) = (x+y, x-y)$$ Show that the inverse is
$$f^{-1}(a,b) = \left(\frac{a+... |
H: Find the value of the limit using l'Hôpital's Rule
What is $$\lim_{x \to 1^-} \frac{\ln(2x)}{\ln(x)}?$$ I tried evaluating and I got $\frac{2}{1}$ but I know it's wrong.
AI: One should not disturb the poor Marquis, he is busy enough with problems for which the Rule he bought from Bernoulli is relevant.
The top is... |
H: Is every prime ideal in $\Pi_{n=1}^{\infty}{k}$ maximal?
Suppose k is a algebraic closed field, is every prime ideal $\mathfrak{p}$ in the product ring $\Pi_{n=1}^{\infty}{k}$ maximal?
AI: Yes. More generally, define a commutative ring $R$ to be von Neumann regular if for every $x \in R$ we have $x^2 | x$. Clearly ... |
H: If $Y$ and $Y \cup X$ are connected. Must their be some $X$-component ($C$) where $C \cup Y$ is connected?
This is a question I had, while trying to solve a homework problem. My original approach was dependent upon the following statement being true.
If $Y$ and $Y \cup X$ are connected, then there is some connected... |
H: Convergent Sequence from Introduction to Analysis
Consider the sequence of real numbers
$$\frac 12, \cfrac 1{2+\cfrac 1 2}, \cfrac 1{2+\cfrac 1{2+\cfrac 12}}, \ldots.$$
Show that this sequence is convergent and find its limit by first showing that the two sequences of alternate terms are monotonic and finding t... |
H: Equation $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2013}$ in rationals
Consider the equation $\sqrt{x}+\sqrt{y}+\sqrt{z}=\sqrt{2013}$, where $x,y,z$ are rational numbers. Are there any solutions other than the trivial ones $(2013,0,0),(0,2013,0),(0,0,2013)$?
We can subtract $\sqrt{z}$ from both sides and square to get $x+y... |
H: Finding local max/min of a function
I am having difficulty finding the local maxima and minima on the function: $$f(x)=\frac{x^2}{x-a}$$ on the invterval $(0, \infty)$.
so far I have worked out $$f'(x)=\frac{x(x-2a)}{(x-a)^2}$$ using the quotient rule.
I have also found the critical points $x=2a$ and $x=a$
I ho... |
H: Easy exponents question
I have the GRE Friday... I got hung up on this easy exponents problem (I think it was these exponents, don't recall exactly)
$$\frac{6^{14}}{2^7 \times 3^5} = ? $$
The answer is $2^73^9$, but could anyone double check for me?
AI: It follows from:
$$
\frac{6^{14}}{2^7 3^5}=\frac{(2\times 3)^{... |
H: If $\gcd(a, b) = 1$ then $\gcd(ab, a+b) = 1$?
In a mathematical demonstration, i saw:
If $\gcd(a, b) = 1$ Then $\gcd(ab, a+b) = 1$
I could not found a counter example, but i could not found a way to prove it too either.
Could you help me on this one ?
AI: Hint:
If a prime $p \mid ab$, then $p$ divides either $a$ or... |
H: Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) could all be solve by the analytical geometry?
Is it true that all of the euclidean geometry problem in the IMO(international mathematical olympiad) or even generalize to say that all the plane geometry problem an... |
H: Etymology of 'finite place'
In study of algebraic number theory one often comes across the terms 'infinite' and 'finite' places, referring to the archimedean and non-archimedean valuations of your field, respectively - but I have no intuition as to why they're called that! What's the motivation for this terminology... |
H: Centralizer is nontrivial
If $|G| = p^n$ where p is a prime number then $Z(G) \ne e.$
I don't understand a couple of parts of my book's proof.
It says, if $a\in G$, since $N(a)$ is a subgroup of G, $|N(a)|$, being a divisor of $|G| = p^n$, must be of the form $|N(a)| = p^{n_a}$; $a\in Z(G)$ if and only if $n_a = n... |
H: Irreducibility and factoring of polynomials
Determine the irreducibility of $x^4 + x + 1$ in $\Bbb{Q}[x]$.
Well, suppose there is a rational solution $\frac{c}{d}$ to this polynomial. Then $c | \pm 1$ and $d | \pm 1$, implying the only possible factorization in $\Bbb{Q}[x]$ would be $(x - 1)(x^3 +\beta x + \rho)$... |
H: proving and Identity combinatorially
prove the Identity:
(n-k)$\binom nk$ = n$\binom {n-1}k$
I have proven it algebraically but now I need to prove it combinatorially ( count something in two ways).
Here is my attempt:
theorem: P(n,k) = $\frac {n!} {n-k!}$
theorem: C(n,k) = $\frac {n!} {k!(n-k)!}$
P(n,k) = C(n,k) *... |
H: Abstract Algebra Cosets
I don't know how to even approach this problem.
Let G be the group of rotations of a plane about a point P in the plane. Thinking of G as a Group of permutations of the plane, describe the orbit of a point Q in the plane.
AI: Since rotations are isometries that preserve distance, each image ... |
H: Polar Equation Conversion
Change the polar equation $\theta=\frac{\pi}{3}$ to rectangular coordinates.
How would I go about this question? I've tried $x=r\cos\theta$ and $y=r\sin\theta$, but I can't figure out $r$ since it's not provided. I also took into consideration the formula $\tan\theta=\frac{y}{x}$ and I k... |
H: Random Variable Problem w/ variance
Three zero mean, unit variance random variables X, Y, and Z are added to form a new random variable, W = X + Y + Z. Random variables X and Y are uncorrelated, X and Z have a correlation coefficient of 1/2, and Y and Z have a correlation coefficient of -1/2.
a) Find the variance o... |
H: Normalizer proof
If $a,x\in G$ show that $N(xax^{-1}) = xN(a)x^{-1}$?
I know that $N(a)$ is the set of all elements of G that commute with a. Thus I can get $N(a) = N(xax^{-1})$ so $N(xax^{-1})$ is also the set of all elements of G that commute with a, but how do I get $xN(a)x^{-1} = N(xax^{-1})$?
AI: $N(a) = \{y ... |
H: Can you have different integration constants for functions like $1/x^2$, one on each component of its domain?
We all learned back in calculus class that $\int \frac{1}{x^2}dx$ is $\frac{-1}{x}+C$ via the power rule for integrals. However, looking back at my calculus book, they define the indefinite integral of a f... |
H: Line integral Along curve C
For the Vector field, find the line integral along the curve $C$ from the origin to along the $x$-axis to the point $(3,0)$ and then counterclockwise around the circumference of the circle $x^2 + y^2 = 9$ to the point ($\frac 3{\sqrt{2}},\frac 3{\sqrt{2}}$).
$\vec G = (ye^{xy}+\cos(x+y))... |
H: variable with negative exponent in the denominator moved to nominator and vice versa
The top and bottom of the fraction both contain negative exponents. Since $c^{-3}$ on the bottom has a negative exponent, it is moved to the top of the fraction (numerator). Since the $d^{-3}$ on the top of the fraction has a negat... |
H: The number of worms (Moser's worm problem)
The Moser's worm problem [springer link] asks for the region of smallest area that can accommodate every plane curve of length 1. Curves can be rotated and translated and may be considered identical upto rotation and translation transforms.
What I ask is
How many such cu... |
H: Prove the sequence converges or diverges?
Is the sequence $\left\{\frac{n}{(n^2+1)}\right\}$ convergent? If so, what is the limit.
If limit exists prove it or prove that it is divergent.
I know that as $n\rightarrow \infty$ the limit is zero. My problem is proving it.
So how do I start this: Let $\epsilon>0$, then... |
H: When are square and curved brackets interchangeable?
Is it ever acceptable to interchange square and curved brackets? E.g. are the following both acceptable (and identical)? $$x = t(a + [b + c])$$ $$x = t(a+(b+c))$$
AI: They are absolutely identical. The addition of square brackets in algebra, or I have even seen ... |
H: Prove that $\det(kA) = k^n\det(A)$ for all $A \in M_{n\times n}(F)$
So I was just looking over an old homework problem, and my proof doesn't seem right. I got full credit for it, but it seems to be circular reasoning. Please tell me if this is actually a valid proof:
Suppose $A \in M_{n\times n}(F)$. Then $$\det(kA... |
H: What does the integer span of one irrational, and one (possibly irrational) real number look like in $\mathbb{R}$?
My title was rejected a few times, here is what it was initially:
If you take two real numbers- one irrational and one possibly irrational - how close does their $\mathbb{Z}$ span come to any given re... |
H: Bases for null space and range
Let $T: M_n(\mathbb F) \rightarrow \mathbb F$ defined by $T(A)=tr(A)$, where $tr(A)$ means the trace of $A$. Suppose that $T$ is a linear transformation.
I need to find bases for the $N(T)$, the null space of $T$, and for $R(T)$, the range of $T$.
I know that trace is just the sum o... |
H: What does 'sign' mean in an equation?
I'm curious what sign means in the context of mathematical notation. I'm reading a paper right now and it uses:
$$ sign \overrightarrow{\lambda} \cdot \overrightarrow{a} $$
Is that equivalent to $\pm$ ?
I've never seen this before.
Thanks.
AI: The value $\operatorname{sign}(x)$... |
H: Technique to solve this equation of 2 unkowns
I was solving a problem of single phase eletrical circuits where I had to find the inductor $L$ and resistance $R$. I managed to get two equations containing the two unknowns.
$$\frac{R}{R^2+(w*L)^2}=c_1$$
and
$$\frac{wL}{R^2+(w*L)^2}=c_2$$
where $w,c_1 \text { and } c... |
H: Pole of order $\ge 2 \; \Rightarrow \;$ not injective
Let $D \subseteq \mathbb{C}$ be open and $f : D \rightarrow \mathbb{C}$ meromorphic with a pole of order $\ge 2$ in $a \in D$. Then $f$ is not injective.
Is there an easy proof to this? This is not homework; it comes from user8268's answer in entire 1-1 functi... |
H: Counterexample to: if $1\le p
We know if $\mu(X)<\infty$, and if $1\le p<q<\infty$, then $L^q(X)\subset L^p(X)$ (can be proved by using Holder's inequality).
Is this still true if $\mu(X)=\infty$? Counterexample?
Thanks.
AI: Hint: Try some negative powers of $x$ on $[1,\infty)$. |
H: Laplace Transform or Characteristic Equation?
The proponents of the use of Laplace Transform in differential equations claim that is easier and faster but is that always the case? Often I have found out that solving an ODE through the characteristic equation and the use of undetermined coefficients is significantly... |
H: A correct logical representation for an iff expression
For every integer n, $n^{3}$ is even if and only if n is even
This is clearly an implication, the problem is the order of the statement confuses me.
$n^{3}$ is even $\Longrightarrow$ n is even
vs
n is even $\Longrightarrow$ $n^{3}$ is even
AI: If and only if me... |
H: An identity of 2-order complex matrice
Problem statement
$A,B,C$ are 2-order complex matrices.
Prove $A(BC-CB)^2=(BC-CB)^2A$.
What I have tried
I think the conditions 'complex' and '2-order' is to tell me that there're only two possibilities of their Jodan Cononial Forms,so I tried to write down JCF of $A$ but did ... |
H: What is $\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx$?
/A problem from the 2012 MIT Integration Bee is
$$
\int_0^1\frac{x^7-1}{\log(x)}\mathrm dx
$$
The answer is $\log(8)$. Wolfram Alpha gives an indefinite form in terms of the logarithmic integral function, but times out doing the computation. Is there a way to do it... |
H: Prove that $\lim_{x\to\infty} f(x) = 0$ if $\lim_{x\to\infty} xf(x) = L$
Question:
Let $f: (a,\infty) \to \mathbb{R}$ be such that $\lim_{x\to\infty} xf(x) = L$ where $L\in \mathbb{R} $.
Prove that $\lim_{x\to\infty} f(x) = 0$.
Attempt:
I see that in order for $\lim_{x\to\infty} xf(x) = L$, either $f(x)$ must be b... |
H: Find the three 2-Sylow subgroups of $S_3$ and find a 2-Sylow subgroup and a 3-Sylow subgroup of $S_4$
Find the three 2-Sylow subgroups of $S_3$ and find a 2-Sylow subgroup and a 3-Sylow subgroup of $S_4.$
I just learned Sylow' theorem at the moment and I don't know how to do these problems. I know a p-Sylow subgrou... |
H: calculus how fast is the milk's level rising at 3 seconds?
Suppose milk is being poured into a cylindrical bowl of radius 5 inches at a rate of 1 cubic inch per second. How fast is the milk's level rising at 3 seconds?
AI: Let V be the volume of the cylinder that is filled with milk.
Clearly $\frac {dV}{dt}$ will g... |
H: significant figure representation?
I was wondering:
Why does
$1.30 \times 10^3$ have $3$ significant figures
while $1300$ has $2$ significant figures
(they are both the same number)
Why is that distinction ?
When should I use each ?
AI: I found the following sentence in http://en.wikipedia.org/wiki/Significant_fi... |
H: Differential Equations for a Teardrop Shape
My research has led me to a nonlinear system of differential equations which should yield a teardrop shape in the $x-y$ plane. The equations, parameterized by $t$ are
$$\frac{x'''}{x'}=\frac{y'''}{y'}$$
$$x'^2+y'^2=1$$
Some obvious solutions to these equations are lines a... |
H: Existence of certain $\left\langle{\alpha_n | n \in \omega}\right\rangle$
Let $\beta$ be a countable limit ordinal. Prove $\exists$ sequence $\left\langle{\alpha_n | n \in \omega}\right\rangle$ with the following properties:
$(1): \alpha_0 = 0\;;$
$ (2): \forall n \in \omega [\alpha_0 \in \beta, \alpha_n <\alpha_{n... |
H: Maximize the angle subtended by the camera lens
A photographer is taking a picture of a four-foot painting hung in an art gallery. The camera lens is 1 foot below the lower edge of the painting. How far should the camera be from the painting to maximize the angle subtended by the camera lens?
I have no idea this... |
H: calculus where m is the slope and b is the y intercept
An equation to the tangent line of $h(x)=\tan (x)+\cos (x)$ is given by $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept. If $x=\frac{\pi}{4}$ find $m$
AI: If
$h(x) = \tan x + \cos x, \tag{1}$
then
$h'(x) = \sec^2 x - \sin x = 1 / (\cos^2 x) - \si... |
H: Riemann-integrability of $f(x)=2x\sin\frac{1}{x}-\cos\frac{1}{x}$ on $[0,1]$
Determine whether $\displaystyle f(x)=2x\sin\frac{1}{x}-\cos\frac{1}{x}$
is Riemann-integrable on $\displaystyle [0,1]$
Attempt: I can clearly see that $\displaystyle f$ is derivative of $\displaystyle g(x)=x^2\sin\frac{1}{x}$
$\display... |
H: Show that $\sum_{n=0}^\infty (order\ {S_n})q^n=\prod_{m\ge 1}(1-q^m)^{-1}$
Let $T=\mathbb (C^*)^2$ acts on $\mathbb C[x,y]$ via $(t_1,t_2)(x,y)=(t_1x,t_2y)$, let $S_n$ be the set of ideals $I$ of $\mathbb C[x,y]$ such that $TI=I$ and $\mathbb C[x,y]/I$ is $n$-dimensional $\mathbb C$-vector space. If $order\ S_{0}=0... |
H: Does the series $\sum_{n=1}^\infty n^{(-1)^n-2}$ converge?
Does the series $\sum_{n=1}^\infty n^{(-1)^n-2} $ converge?
I tried this way:
$$\sum_{n=1}^\infty n^{(-1)^n-2} = \sum_{n=1}^\infty \frac 1n - \sum_{n=1}^\infty \frac1{n+2} + \sum_{n=1}^\infty \frac1{n^3} -\sum_{n=1}^\infty \frac 1{(n+1)^3}$$
The first on... |
H: question about legendre symbol
I have a question about Legendre symbol or I think that we will use Legendre symbol I am not sure,because I tried to apply the definition of Legendre symbol, quadratic reciprocity and also the properties of Legendre symbol but I could not succeed... Perhaps that question needs somethi... |
H: calculus fine the position of the object at time t
Suppose an object is moving with acceleration $a(t)=\sin t+3t$, and at time $0$ the velocity $v(0)=0$ and the position $s(0)=2$.
Find the position of the object at time $t$.
AI: Note that:
$$v(t) = \int v'(t)dt = \int a(t)dt = \int\left(\sin t+3t\right)dt$$
$$\Righ... |
H: Convergence of $\sum_{n=1}^\infty\dfrac{\tan^{-1}n}{n+\sqrt{n}}$
How could I determine the convergence or divergence of this series?
$$\sum_{n=1}^\infty\dfrac{\tan^{-1}n}{n+\sqrt{n}}$$
AI: Hints:
For sufficiently big $n$
$$
\tan^{-1}n\geq\frac{\pi}{2}-1
$$
For all $n$
$$
\frac{1}{n+\sqrt{n}}\geq\frac{1}{n+n}
$$
T... |
H: Measure theory set construction
I have a typical problem concerning the measure theory. I would like to know if any one has a good general strategy to solve this kind of problem because I always don't have idea where have I to begin to prove this kind of things.
The problem which I have to solve is:
If $C\subset \m... |
H: Are "if" and "iff" interchangeable in definitions?
In some books the word "if" is used in definitions and it is not clear if they actually mean "iff" (i.e "if and only if").
I'd like to know if in mathematical literature in general "if" in definitions means "iff".
For example I am reading "Essential topology" and t... |
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