text stringlengths 83 79.5k |
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H: Endomorphisms in a symmetric monoidal category
Let $\mathcal{C}$ be a symmetric monoidal category generated by one element $X$ such that $End(X)=G$ where $G$ is a finite group. Is it true that, for any object $A \in \mathcal{C}$, $End(A)$ is isomorphic to a wreath product $G \wr S_n$, $n \in \mathbb{N}$ ?
AI: No. T... |
H: Which of the following cannot be the length of triangle
I have a question regarding triangles which is puzzling me:
In triangle PQR , PR=7 and PQ=4.5 . Which of the following cannot represent the length of
QR ? a)2.0 , b)3 , c)3.5 , d)4.5 , e)5.0
Any suggestions ?
AI: The sum of the two smaller sides must ... |
H: Squared Series Fourier
Possible Duplicate:
Fourier 1st step?
How to find fourier transform of a series of the such form:
$$y_k=\left[f(x) \right]^{2},$$
but I am not sure of the step by step for going about this computation.
how is the first step??
thank you very much!!
AI: Integration by parts. You're doing a... |
H: Fourier transform of $y_k=\left[k-\frac{m-1}{2}\right]^{2},$
I am trying to find the discrete Fourier transform of
$$y_k=\left[k-\frac{m-1}{2}\right]^{2},$$
but I am not sure of the step by step for going about this computation.
how is the first step??
thank you very much!!
AI: I assume you want a DFT of $(y_0,y_1... |
H: Lindenbaum algebra is a free algebra
The following is a continuation of this question.
I would like to prove that the Lindenbaum algebra is a free algebra. Hopefully I would like to hear hints on how to proceed in the 'right' direction.
Let $X$ be a set of propositional variables, $M$ the set of all boolean expres... |
H: Solving $\frac{dP}{dt} = k(M - P)$
I am suppose to solve for P(t), to find an epxression for P(t) and I am suppose to find the limit.
I can't find anything.
$$\frac{dP}{dt} = k(M - P)$$
$$\frac{dP}{M - P} = k \, dt$$
$$\int \frac{dP}{M - P} = \int k \, dt$$
$$ \ln \frac{1}{M - P} = xk + c$$
$$ \frac{1}{M - P} = e^... |
H: What does "+ complete" mean?
I'm reading notes about Liapunov stability, and in the book of Abraham, Marsden and Ratiu I found the next definition:
Let $m$ be a critical point of $X$. Then
$m$ is stable (or Liapunov stable) if for any neighborhood $U$ of $m$, there is a neighborhood $V$ of $m$ such that if $m'... |
H: Does a closed form sum for this fourier series exist?
Continuing from an earlier question of mine: Fourier-Series of a part-wise defined function?
I now got a fourier series which I believe is the correct one:
$$\frac{\pi(b-a)}{2} + \sum\limits_{n=1}^{\infty} \frac{(a-b)(1-(-1)^n)}{n^2\pi}\cos(nx) + \frac{(-1)^n(b-... |
H: Existence of the Lebesgue integral using a variety of examples
Problem:
I am self-learning about Lebesgue integration, and am just starting to try and apply some examples of the existence of the integral.
For each of the following 5 examples, does the Lebesgue integral exist on $(0,\infty)$, and if it does, is it f... |
H: Calculating angles when sides are known - Without Trignometric ratios.
The question is:
ABCD is a parallelogram and BFDE is a square . If AB is 20 and CF is 16 what is the perimeter of the parallelogram.
The question is fairly simple and I know how to solve it. However how would I get the remaining angles of t... |
H: Alternative approach for result involving harmonic functions.
I encountered the following 2-part problem on a practice exam:
(a) Show that if $f:\Bbb C\to\Bbb C$ is entire and the real part of $f$ is always positive, then $f$ is constant. (b) Show that if $u:\Bbb R^2\to\Bbb R$ is a harmonic function with $u(x,y)>0... |
H: Complex Analysis Book
I want a really good book on Complex Analysis, for a good understanding of theory. There are many complex variable books that are only a list of identities and integrals and I hate it. For example, I found Munkres to be a very good book for learning topology, and "Curso de Análise vol I" by El... |
H: Continued fraction question
I have been given an continued fraction for a number x:
$$x = 1+\frac{1}{1+}\frac{1}{1+}\frac{1}{1+}\cdots$$
How can I show that $x = 1 + \frac{1}{x}$? I played around some with the first few convergents of this continued fraction, but I don't get close.
AI: Doesn't this immediately foll... |
H: Conditions stronger than differentiability or weaker than integrability
Let $f: [a,b] \to \mathbb R$. If $f$ is (Riemann-)integrable on $[a,b]$, then define $F: [a,b] \to \mathbb R$ by $$F(x) = \int_a^x f.$$
We have the following:
$$\begin{array}{ccccccc}
f \text{ differentiable} & \implies & f \text{ continuous} &... |
H: Integration of $\int\frac{1}{x^{4}+1}\mathrm dx$
I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}\mathrm dx$. Do I have to use trigonometric substitution?
Many duplicate posts link to this one as the target. (Those posts were merged into this one, which is the source of the many answers.)
AI: I t... |
H: Angles formed by intersection of two diagonals in parallelogram and square?
Initially I was of the opinion that if two diagonals in a parallelogram intersect then the angle formed at the point of intersection is 90 degrees (I came up with this conclusion by inserting values) , if this applied to a parallelogram the... |
H: Computing rank using $3$-Descent
For an elliptic curve $E$ over $\Bbb{Q}$, we know from the proof of the Mordell-Weil theorem that the weak Mordell-Weil group of $E$ is $E(\Bbb{Q})/2E(\Bbb{Q})$. It is well known that
$$
0 \rightarrow E(\Bbb{Q})/2E(\Bbb{Q}) \rightarrow S^{(2)}(E/\Bbb{Q}) \rightarrow Ш(E/\Bbb{Q})[2] ... |
H: How many times do these curves intersect?
When the curves $y=\log_{10}x$ and $y=x-1$ are drawn in the $xy$ plane, how many times do they intersect?
To find intersection points eq.1 = eq. 2
$$\begin{align*}
\log_{10}x &= x-1\\
10^{x - 1} &= x \tag{a}
\end{align*}$$
Answer would be no. of solutions (a) has. One of th... |
H: Derive a Laurent series for the function $2z/(z+j)$
First of all, I apologize for the none mathematical notations. I've only ever hanged around Stackoverflow, and never learnt how to type Mathmatical notations. It would be great if someone could teach me. The good news is my question isn't too long.
I'm a beginner ... |
H: Hyperplane in projective space and existence of point
This is problem $2.11$ page $133$ of Kenji Ueno's book.
Consider an irreducible hypersurface $V(F)$ where we assume that the homogeneous polynomial of degree $d$ satisfies the conditions $F(0,x_{1},..,x_{n}) \neq 0$ and $F(1,0,...0) \neq 0$.
Let $P=(1:0:...:0)$... |
H: Apply Cauchy-Riemann equations on $f(z)=z+|z|$?
I am trying to check if the function $f(z)=z+|z|$ is analytic by using the Cauchy-Riemann equation.
I made
$z = x +jy$
and therefore
$$f(z)= (x + jy) + \sqrt{x^2 + y^2}$$
put into $f(z) = u+ jv$ form:
$$f(z)= x + \sqrt{x^2 + y^2} + jy$$
where
$u = x + \sqrt{x^2 +... |
H: Dirichlet's Test Remark in Apostol
Dirichlet's Test is theorem $10.17$ in Apostol's Calculus Vol. $1$.
The theorem itself says that if the partial sums of $\{a_n\}$ (can be complex numbers, not just reals) form a bounded sequence and $\{b_n\}$ is a (monotone?) decreasing function converging to $0$, then $\sum a_n b... |
H: A transform function from $(-\infty, \infty)$ to $(t_0, \infty)$?
I want to convert an integral from $(t_0, \infty)$ (or $(-\infty, t_0)$) range to $(-\infty, \infty)$ range by change of variable. What is the best transform function to do this - one that is simple, monotonic with $f(-\infty)=t_0$ and $f(\infty)=\in... |
H: calculate standard deviation from percentage of mean occuring.
I'm not a math person, although I find it quite interesting. I'm a programmer but I've got a math problem I'm trying to figure out.
Lets assume I'm trying to create a program that will predict at what mile marker a car will run out of gas. I can take t... |
H: Evaluate the series: $ \sum_{k=1}^{\infty}\frac{1}{k(k+1)^2k!}$
Evaluate the series:
$$ \sum_{k=1}^{\infty}\frac{1}{k(k+1)^2k!}$$
AI: Partial fraction decomposition gives
$$\frac{1}{k(k+1)^2}=\left(\frac{1}{k}-\frac{1}{k+1}\right)\frac{1}{k+1}=\frac{1}{k}-\frac{1}{k+1}-\frac{1}{(k+1)^2}$$
Hence this series is
$$\su... |
H: Is $K=K∩G_\alpha N=G_\alpha$ an error in this context?
There is a problem in Problems in Group Theory by J.D.Dixon :
2.51 If a permutation group $G$ contains a minimal normal subgroup $N$ which is both transitive and abelian, then $G$ is primitive.
He answered it amazingly:
Let $G_\alpha $ be a stabilizer of $G... |
H: Elementary Probability Questions
Toss a coin three times, so event space $\Omega=\{HHH,HHT,HTH,HTT,THH,THT,TTH,TTT\}$.
We win $\$1$ if we flip a Head and lose $\$1$ for a Tail. Let $\mathbb{P}(H) = p$ and $\mathbb{P}(T) = q$. The change in our wealth after flip $i$ is the r.v.
$$X_i = \cases{+1 \text{ if }H \\ -1... |
H: $L^\infty $ bound in terms of given data.
Let $\Omega \subset R^n$ be bounded and open, $u\in C^2(\Omega)\cap C(\bar \Omega)$ be a solution of $-\Delta u=f$ in $\Omega$ , $u=0$ on $\partial \Omega$.
Prove that there exists a constant $C$, depending only on $n$ and $diam(\Omega )$ such that
$||u||_{L^\infty} \le C... |
H: Reference request in number theory for an analyst.
I am a confirmed mathochist. My background is in analysis, and fairly traditional analysis at that; mainly harmonic functions, subharmonic functions and boundary behaviour of functions, but I have for many years had an interest in number theory (who hasn't?) withou... |
H: Would nonmath students be able to understand this?
For a course, I am required to do a presentation. The topic could either be something mundane, like a career strategy report, or something more interesting, such as a controversial topic, or an exposition on something you find interesting. What I would like to do i... |
H: Prove a group generated by two involutions is dihedral
Prove a finite group generated by two involutions is dihedral
Is my following argument correct?
Let $G=\langle x,y\rangle$ be a group generated by involutions $x,y$. Let $n=\mathrm{ord}(xy)$ to get a presentation $G=\langle x,y\mid x^2=y^2=(xy)^n=1\rangle $ s... |
H: Constant Radon-Nikodym derivative
Let $(\Omega, F, \mu)$ be a complete measure space, $\mu(\Omega)=1$ and $\mu$ takes values 0 or 1.Let $\nu$ positive measura, $\sigma$-finite and absolutely continuous with respect to
$\mu$. Show that then $f=\frac{d\nu}{d\mu}$ is constant a.e. equal to $\nu(\Omega)$ that is a fin... |
H: Proving no anti-automorphisms exist in this matrix ring.
I've been playing with a certain matrix subring, but there is one step I am having trouble with.
Let $u=\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0\end{pmatrix}$ in $M_3(\mathbb{Q})$ and let $x=\begin{pmatrix} u & 0 \\ 0 & u^2\end{pmatrix}$ and $y=\be... |
H: What is the mistake in my reasoning?
I am working on the following problem trying to use strategy in this problem. I am trying to simplify the proof by working with $v_{i}=0,i\not=1$ case. But the result looks very different from what I expected, so I want to ask if there is something wrong in my computation. To be... |
H: Sum set estimates for cofinite integer sets
I am interested in the sum set operation on subsets of the integers $\mathbb Z$:
$$A + B = \{ x + y | x \in A, y \in B\}$$
One readily arrives at the following cardinality bounds:
$$|A| + |B| - 1 \leq |A + B| \leq | A |\cdot | B |$$ for $A, B$ non empty and finite
What h... |
H: How to show that $\mathrm{ord}_m a = \mathrm{ord}_m \overline{a}$?
Let $a \in Z$ and $m \in N$ such that $\gcd(a,m)=1$. How to show that $\mathrm{ord}_m a = \mathrm{ord}_m \overline{a}$, where $\overline{a}$ is the inverse of a modulo m?
Hint: Solution starts as follows:
$1 \equiv (a \overline{a})^{ord_m a} \eq... |
H: Unique minimal normal subgroup $\implies$ faithful irreducible representation.
I'm tasked with proving that if I have a finite group G with a unique minimal normal subgroup, and a field F with characteristic not dividing the order of G, then there exists a faithful irreducible F-representation.
I can be fairly cert... |
H: Coordinate ring of general linear group
Let $n$ be a positive integer and let $k$ be an algebraically closed field. What is the coordinate ring of $GL(n,k)$ (the set of all $n \times n$ matrices with entries in $k$)? Here we identify this set as a subset of $k^{n^{2}}$.
Would it suffice to say that the coordinate ... |
H: equation of lines goes through origin
suppose that we are require to write equation of lines,whose go through origin and distance from point $F(-4,3)$ to this line is $1cm$
first of all ,distance from point $F(x_0,y_0)$ to line $A*x+B*y+C=0$ is
$d={+,-}(A*x_0+B*y_0+c)/(\sqrt{(A^2+B^2)})$
because we have... |
H: Derivative of a composition
Let $f$ and $g$ two differentiable functions on $]a, b[$. Then $f \circ g$ is differentiable on the same interval and we have the expression :
$$(f \circ g)' = g' \cdot f' \circ g$$
How do you prove this ?
AI: If you are familiar with o-notation, here is an alternative proof: Note that w... |
H: How can I determine these 3 variables?
How can I determine the values of $x,y,z$ below?
$a,b,c$ - are given variables
$x,y,z$ - must be found
$$\begin{align*}
ax&=S_1\\
by&=S_2\\
cz&=S_3\\
S_1-(y+z)&>0\\
S_2-(x+z)&>0\\
S_3-(x+y)&>0
\end{align*}$$
AI: First of all, if the $S_i$ are known, you are done. If the $S_i$... |
H: Subset of $\mathbb{I}\cap [0,1]$ (irrationals in [0,1]) that is closed in $\mathbb{R}$ and has measure $\epsilon \in (0,1)$
Measure theory guarantees that every Lebesgue finite measurable set $E$ has a closed subset $F$ such that $m(E \backslash F)<\epsilon$ for small $\epsilon$.
But today I saw in some text that... |
H: How many bits needed to store a number
How many bits needed to store a number $55^{2002}$ ?
My answer is $2002\;\log_2(55)$; is it correct?
AI: The number of bits required to represent an integer $n$ is $\lfloor\log_2 n\rfloor+1$, so $55^{2002}$ will require $\lfloor 2002\; \log_2 55\rfloor+1$ bits, which is $11,57... |
H: Evaluating Integral with Residue Theorem
The integral in question is
$$\int_{_C} \frac{z}{z^2+1}\,dz,$$ where $C$ is the path $|z-1| = 3.$
The two pole of $f(x)$ where $f(x)=\frac{z}{z^2+1}$ is $-j$ and $j$
$${\rm Res}_{z=z_0}f(x)=\lim_{z\rightarrow\infty}(z-z_0)f(z)$$
For the first pole:
$${\rm Res}_{z=j}f(z)= \l... |
H: manifold structure on on a finite dimensional real vector space
I am reading Warner's Differentiable Manifolds I do not get one example which is
Let $V$ be a finite dimensional real vector space. Then $V$ has a natural manifold structure. If $\{e_i\}$ is a basis then the elements of the dual basis $\{r_i\}$ are t... |
H: find equation of triangle sides in cartesian system
it is know that one vertex of triangle is located at point $A(2,-4)$ and equation of angle bisector of two another angle is given
1.$x+y-2=0$
2.$x-3*y-6=0$
we have to find equation of sides of triangle
i have found point where this two line intersect ,got... |
H: Weak and pointwise convergence in a $L^2$ space
Let $I$ be a measured space (typically an interval of $\Bbb R$ with the Lebesgue measure), and let $(f_n)_n$ a sequence of function of $L^2(I)$.
Assume that the sequence $(f_n)$ converge pointwise and weakly. How to prove that the pointwise limit and the weak limit ar... |
H: $k$th power of ideal of of germs
Well,We denote set of germs at $m$ by $\bar{F_m}$ A germ $f$ has a well defined value $f(m)$ at m namely the value at $m$ of any representative of the germ. Let $F_m\subseteq \bar{F_m}$ be the set of germs which vanish at $m$. Then $F_m$ is an ideal of $\bar{F_m}$ and let $F_m^k$ de... |
H: Is there a mathematical symbol for "For every element"?
Is there a mathematical symbol that means "For every element"? I want the meaning to have similar function as an iteration, say for loop.
AI: From your comments, it seems that you want to take the elements of your index set in a specific order, as in an iterat... |
H: Estimating missing values in a dataset and averaging values
I have a statistical maths problem where I have a rather large dataset consisting of a timestamp (rows) and a quantity of - lets say detections - per each day (columns for each location). Currently I have two challenges with the data:
Some of the data is m... |
H: independence of random variables for finite subfamilies
Let $X$ be a random variable and $\{Y_j\}, j\in J$ a family of random variables. $J$ should be an index set, perhaps uncountable. My question is, if $X$ is independent to every finite subfamily of $\{Y_j\}$, i.e. for every $ I \subset J$ and $|I|\in \mathbb{N}... |
H: How to solve this quadratic congruence equation
How to solve $x^2+x+1\equiv 0\pmod {11}$ ?
I know that in some equations like $ax\equiv b\pmod d$ if $(a,d)=1$ then the equation has one and only one solution $x\equiv ba^{\phi(d)-1}\pmod d$. Any help will be appreciated. ;)
AI: You use the quadratic formula!
No, real... |
H: Perimeter of Triangle in a Right triangle.
I am having difficulty solving this problem:
The perimeter of a right triangle is 18 inches. If the midpoints of three sides are joined by line segments they form another triangle . What is the perimeter of this new triangle ? (Ans: 9 inches) .
Any suggestions on how ... |
H: General quadratic form of two variables
I was referring to this lecture http://www.stanford.edu/class/ee364a/videos/video04.html. and he gave an example of a generalized quadratic equation
f(x,y) = x'Ax + 2x'By + y'Cy
The functions is convex if the matrix
A B
B' C
is positive semidefinite and also the matrix C. ... |
H: What is the expected area of a polygon whose vertices lie on a circle?
I came across a nice problem that I would like to share.
Problem: What is expected value of the area of an $n$-gon whose vertices lie on a circle of radius $r$? The vertices are uniformly distributed.
AI: For $n$ even and $m=(n-2)/2$, the expect... |
H: How many transvections are in a maximal unipotent subgroup of a general linear group?
If G is a general linear group GL( n, q ) of characteristic p, and U is a Sylow p-subgroup of G, then how many elements of U are transvections?
The size of a centralizer of a transvection is relevant.
If $g = \left[\begin{smallm... |
H: Stability of equilibria of a differential equation (by Hale-Koçak)
Consider the differential equation
$$
x'=f(x)
$$
where
$$
f(x)=\begin{cases}
0 & x = 0 \\[12pt]
-x^3\sin\left( {\frac{1}{x}} \right) & x \ne 0
\end{cases}
$$
I have to study the equilibrium points. First, I've proved that $f(x) \in C^1(\mathbb{R})$.... |
H: Show $f(x)=\int_E x^tg(t)d\mu(t)$ is continuous when $\mu$ is a general measure
Define the function $f:[0,1] \to \mathbb{R}$ by
$$
f(x)=\int_E x^tg(t)d\mu(t)
$$
where $E \subset \mathbb{R^+}$,
$\mu$ is a nonnegative measure on $\mathbb{R}$
and $g:\mathbb{R} \to \mathbb{R}$ is a $\mu$-integrable function, that is ... |
H: short exact sequences and direct product
Let
$$0\longrightarrow L^{(i)}\longrightarrow M^{(i)}\longrightarrow N^{(i)}\longrightarrow 0$$
be a short exact sequence of abelian groups for every index $i$. Clearly if I take finite direct products, then
$$0\longrightarrow \prod_iL^{(i)}\longrightarrow\prod_i M^{(i)}\lon... |
H: infinite series involving harmonic numbers and zeta
I ran across a fun looking series and am wondering how to tackle it.
$$\sum_{n=1}^{\infty}\frac{H_{n}}{n^{3}}=\frac{{\pi}^{4}}{72}.$$
One idea I had was to use the digamma and the fact that
$$\sum_{k=1}^{n}\frac{1}{k}=\int_{0}^{1}\frac{1-t^{n}}{1-t}dt=\psi(n+1)+... |
H: Summing a exponential series
What is the appropriate way to simplify such an expression.
i am unsure of how to use the series i know to apply to this situation
$$\sum_{L=0}^{M}s^{L}L^{2}$$
do i modify such a series as power series, or is there a more efficient series to use here?
thank you very much!!
AI: Try to ma... |
H: Question on sequences
Exercise 37 in Apostol $10.20$ asks to find all complex $z$ such that $$\sum_{n=1}^{\infty} \frac{(-1)^n}{z+n}$$ converges.
I suspect this requires the use of either Abel's Test or Dirichlet's Test. My attempt so far has been to set $\{b_n\}=\frac{1}{n}$, which is a decreasing sequence of rea... |
H: $E[x\mid x>1]$ if $X \sim \exp(\lambda)$
I need to find $E[x\mid x>1]$ if $X \sim \exp(\lambda)$.
I first tried: $$f(x|x>1) = \frac{f(x)}{\int_{x=1}^{\infty}f(x) dx}.$$
AI: Hint: Use the memorylessness property of the exponential distribution. Given that you have waited $1$ hour, what is the distribution of your ad... |
H: How to arrive at Stokes's theorem from Green's theorem?
I would like to verify the identity
$$ \oint \vec F \cdot (\hat i dx + \hat j dy) + \oint \vec F \cdot (\hat i dx + \hat j dy) + \oint \vec F \cdot (\hat i dx + \hat j dy) = \oint \vec F \cdot (\hat i dx + \hat j dy + \hat k dz) $$
If it is incorrect the... |
H: Show that $|x|^{-\eta}$ is an "eigenfunction" for the Hardy Littlewood centered maximal operator
Let $\mu$ be the Hardy Littlewood centered maximal operator in $\mathbb{R}^n$
$$\mu (f)(x) = \sup_{r>0} \frac{1}{|B_r(x)|} \int_{B_r(x)} |f(y)|dy.$$
If $g(x)=|x|^{-\eta}$, com $\eta \in (0,n)$, how to prove that $\mu(g)... |
H: What is the difference between plus-minus and minus-plus?
Possible Duplicate:
What is the purpose of the $\mp$ symbol in mathematical usage?
Just as the title explains. I've seen my professor actually differentiating between those two. Do they not mean the same?
AI: If you write
$$
\cos(a \pm b) = \cos a \cos b ... |
H: number prime to $bc$ with system of congruences
Can you please help me to understand why all numbers $x$ prime to $bc$ are all the solutions of this system?
$$\begin{align*}
x&\equiv k\pmod{b}\\
x&\equiv t\pmod{c}
\end{align*}$$
Here $k$ is prime to $b$, and $t$ is prime to $c$.
AI: Suppose that $x\equiv k \pmod{b}... |
H: Are test/bump functions always bounded?
A bump function is a infinitely often differentiable function with compact support. I guess that such functions are always bounded, especially because the set where they are not zero is compact and because they are continuous they should attain a maximum value on that set. or... |
H: Counting - Colored Houses Question
Here is a question. I seem to have a hard time answering questions of this kind. I would appreciate it if you would not only help answer this, but carefully explain the process so I can understand it, and apply the same when I encounter questions of this kind.
Six houses in a ro... |
H: find the bases for the range of a linear operator and the null space
I need to find the bases for a linear operator.
Here is the question given to me:
Consider $V=\mathbb{C}_{1\times 2}$ as a vector space over the real numbers.
let the linear operator $\tau : V \rightarrow V $ be defined by $\tau (z_{1},z_{2})=(... |
H: Restriction maps for structure sheaf of Spec A
For the space $X = \operatorname{Spec} A$, we define the structure sheaf $\mathcal{O}_X$ as follows. For an open subset $U \subseteq X$, we let $\mathcal{O}_X(U)$ be the projective limit of the family $\{ A_f : f \in A, D(f) \subseteq U \}$ indexed with the partial or... |
H: Determining values where a function is not differentiable
Given
$$g(x) =
\begin{cases}
-1-2x & \text{if }x< -1,\\
x^2 & \text{if }-1\leq x\leq1,\\
x & \text{if }x>1,
\end{cases}
$$
determine at which values $g(x)$ is differentiable.
The approach I have taken with this question is to determine the values at whic... |
H: What is the value of this sum?
Possible Duplicate:
Value of $\sum\limits_n x^n$
I am interested in finding what this sum converges to:
$$\sum_{n=0}^{\infty}e^{-n}=1+\frac{1}{e}+\frac{1}{e^2}+\frac{1}{e^3}+\cdots$$
Does a closed form exist? If so, what is is?
AI: This is a classic geometric series. Letting
$$S=1... |
H: parametrization of surface element in surface integrals
I don't understand this
How $ dS = \sqrt{ \left ( \partial g \over \partial x\right )^2 + \left ( \partial g \over \partial y\right )^2 + 1 } \; dA \; \; $ ?? Is $ dA = dx\times dy$??
AI: The surface in question is given by $z=g(x,y)$.
The vector in the surfa... |
H: something that looks sort of symmetrical but also not
Given the set $S_0$ of finite binary strings whose digit sum is congruent to 0 mod 2 and the set $S_1$ of finite binary strings whose digit sum is congruent to 1 mod 2,
what are the implications of the fact that $F: \{s_1 \in S_1 : s_1 \mbox{ends in 1} \} ... |
H: Calculating Perpendicular and Base of Triangle. Suggestion
In this diagram AB and CD are both perpendicular to BE.If EC=5 and CD=4. What is ratio of AB to BE ?
How would i go about solving this triangle (without trigonometric ratios). I could only get DE=3 using Pythagoras theorem and was stuck after that. How... |
H: When am I allowed to use ln(x) when integrating functions?
In Mathematics, we know the following is true:
$$\int \frac{1}{x} \space dx = \ln(x)$$
Not only that, this rule works for constants added to x:
$$\int \frac{1}{x + 1}\space dx = \ln(x + 1) + C{3}$$
$$\int \frac{1}{x + 3}\space dx = \ln(x + 3) + C$$
$$\int \... |
H: Algorithm to find transform random pairs into polar coordinates
I have some pairs of real numbers $(\rho_1,\alpha_1),\dots (\rho_n, \alpha_n)$. I know that all my $\rho$'s are positive, but there is no constraints on my $\alpha$'s. I want to find a function $\phi$ such as $((\rho_1,\theta_1),\dots,(\rho_n,\theta_n)... |
H: Using binomial theorem find general formula for the coefficients
Using binomial thaorem (http://en.wikipedia.org/wiki/Binomial_theorem) find the general formula for the coefficients of the expantion:
$$
\left(\sum_{i=0}^{\infty}\frac{t^{2i}}{n^i6^ii!}\left(1-\frac{t^2}{6}+\frac{t^4}{120}\right)\right)^n
$$
Thank yo... |
H: On calculating $\sigma(n^2) \pmod 4$ if $n$ is odd
This will be my very first post in math.stackexchange, so please bear with me if I make any silly mistakes with my maths.
So, to proceed: I am trying to calculate $\sigma(n^2) \mod 4$, given that $n$ is odd.
If I let $n = \displaystyle\prod_{i=1}^{r}{{p_i}^{{\alph... |
H: Sanity check, is $\{(-9,-3),(2,-1),(7,7),(-1,-1)\}$ a function?
EDIT#2: Yes, I'm crazy! This IS a function. Thanks for beating the correct logic into me everyone!
I'm using a website provided by my algebra textbook that has questions and answers. It has the following question:
Determine whether the following rela... |
H: Is it true that a 3rd order polynomial must have at least one real root?
While solving a problem a friend said - this polynomial is $3^{rd}$ order ($ax^3+bx^2+cx+d$), with $\{a,b,c,d\}$ real coefficients, so it must have a real root. I didn't want to sound stupid and I said sure.
I can't figure out if he's right. I... |
H: $\varphi:M\to N$ continuous surjective and closed. Then $f$ continuous iff $f\circ\varphi$ continuous.
$\varphi\colon M\to N$ continuous surjective and closed. Then $f\colon N\to P$ continuous iff $f\circ\varphi\colon M\to P$ is continuous. (Topological spaces)
I think that this proposition is true like I noted i... |
H: Continuity of positive operators
How to prove that an positive linear operator $T:C[0,1]\to R $ in the sense that $T(f)\geq 0$ when $f\geq 0$ is bounded?
AI: Suppose $\|f\|_\infty \leq 1$. Then $-1\le f\le 1$ so
$-T(1) \le T(-1)\le T(f) \le T(1)$ so $\|T\| \le T(1)$. In fact equality is achieved, since
$\|1\|_\inf... |
H: Two sequences with convergent ratio
Let $ (b_{n})$ be a decreasing a sequence such that $0< b_{n}<1$ for all $n\geq 1$, and $b_{n}\to 0$ as $n\to \infty$. Is there any way to find another sequence $(a_{n})$ with $\frac{a_{n}}{b_{n}}$ converges to a nonzero constant, and $\frac{a_{n}}{b^{2}_{n}}$ is bounded.
AI: By ... |
H: Expressing $\widehat{MN}=\{x : x \mid mn\}$ as a product of $\widehat M$ and $\widehat N$.
Let $m,n$ be any two positive integers. Note $\widehat X$ the set of positive divisors of $x$.
$$\widehat X = \{ d : d \mid x\}$$
(do not confuse it with $\hat a = \{x : x \equiv a \mod m\}$)
Assume $(m,n)=1$. How could one p... |
H: Probably simple factoring problem
I came across this in a friend's 12th grade math homework and couldn't solve it. I want to factor the following trinomial:
$$3x^2 -8x + 1.$$
How to solve this is far from immediately clear to me, but it is surely very easy. How is it done?
AI: Hint: Use the quadratic formula. |
H: Is this sequence of abelian groups exact?
Let $A$ and $A'$ be abelian groups.
Let $f\colon A \rightarrow A'$ be a surjective homomorphism.
Let $B$ be a subgroup of $A$.
Let $B' = f(B)$.
Let $A_0 = Ker(f)$.
Let $B_0 = A_0 \cap B$.
Is the following sequence exact?
$0 \rightarrow A_0/B_0 \rightarrow A/B \rightarrow A'... |
H: Set defined by $xy-zw=1$
This should be an easy question, but I found it ungooglable and not obvious to visualize...
What geometric object is defined by the equation $xy-zw=1$ in $\mathbb R^4$? And what is the homotopy type of the complement?
AI: It's $\text{SL}_2(\mathbb{R})$, of course! As a "geometric object" it... |
H: Finding the limit
I need to find the limit of this problem. I pretty much know you have to multiply by the conjugate but I get lost after I do that.
$$\lim\limits_{x\to 1} \frac{(1 / \sqrt{x}) - 1}{1-x}$$
AI: You don't have to multiply by a conjugate. Hint: $1-x=(1-\sqrt{x})(1+\sqrt{x})$. |
H: Number of prime divisors of the order of $E_8(q)$.
I am trying to compute the number of prime divisors of the order of $E_8(q)$. I am interested in the general solution, but in particular, my problem calls for $q=p^{15}$ (for prime $p$) and $q\equiv 0,1,$ or $ 4 \mod 5$, if this helps at all.
So, the order is $|E_... |
H: Prove that $\tan^{-1}\left(\frac{x+1}{1-x}\right)=\frac{\pi}{4}+\tan^{-1}(x)$
The question is:
Prove that $\tan^{-1}\left(\frac{x+1}{1-x}\right)=\frac{\pi}{4}+\tan^{-1}(x)$.
It's from A-level further mathematics.
AI: The identity should read $$\tan^{-1} \left(\dfrac{x+1}{1-x} \right) = \tan^{-1}(x) + \pi/4$$
Let ... |
H: Find $P(b^2\ge4ac)$ given that $a,b,c\in\{-9,-8,\dots,8,9\},a\ne0$
I was doing some review on probability and came across the following exercise:
A quadratic equation $ax^2+bx+c=0$ is copied by a typist. However, the numbers
standing for a, b and c are blurred and she can only see that they are integers of one
... |
H: A question about harmonic form of trigonometric functions.
The question is:
i) Find the maximum and minimum values.
ii) the smallest non-negative value of x for which this occurs.
12cos(a)-9sin(a)
I think it should be changed into the form of Rcos(a+x) and it should be 15cos(a+36.87), and I get the answer i)+15 / -... |
H: What is the mutual information $I(X;X)$?
$X$ is a random variable with normal distribution, assume $Y=X$, what is the mutual information $I(X;Y)$?
I guess that $h(Y|X)=0$ since when $X$ is known, $Y$ is completely known, so $$I(X;Y)=h(Y)-h(Y|X)=h(Y)=\frac{1}{2}\log 2\pi e\sigma^2$$ nat.
But, I was told I was wrong!... |
H: Homomorphism from $\mathbb{Q}$ to an ordered field F
I know that there exists a unique injective function $\gamma : \mathbb Q →F$ for any ordered field F.
I don't understand why 'Prove $\gamma(r) = r•1_F$ for every $r\in \mathbb Q$' is an exercise.. Don't we just see $\gamma(r)$ as an element of $\mathbb Q$, hence ... |
H: Similar Matrices and Change of Basis
I'm trying to understand a little better change of basis matrices and how they relate to determining if two matrices are similar.
Given finite vector spaces $V,W$ such that $\textrm{dim} V=\text{dim} W$ and a linear transformation $T:W\rightarrow V$ and ordered bases $V_B$ and $... |
H: $M_m$ is naturally isomorphic to $(F_m/F_m^2)^{*}$
Let us denote $M_m$ be the set of tangent vectors to a manifold $M$ at point $m$ and is called tangent space to $M$ at point $m$ we denote $\bar{F_m}$ be the set of all germs at point $m$ and $F_m$ be the set of germs vanishes at $m$ In warner book there is a lemm... |
H: Multivariable limit $xy\ln(xy)$
Does anybody know how to prove that in $D=\{(x,y)\in\mathbb{R}^2:x>0\wedge y>0\}$ the following is true:
$$
\lim\limits_{(x,y)\to(0,0)}x\cdot y\cdot\ln{(x\cdot y)}=0
$$
I have to find a $\delta$ so that if $\|(x,y)\|=\sqrt{x^2+y^2}<\delta$, that $|x\cdot y\cdot\ln{(x\cdot y)}|<\epsi... |
H: Prove that $G$ abelian if $|G|= pq^2$.
Let $G$ be a group of order $pq^2$, where $p \neq q$ prime and $p$ does not divide $| Aut (G) |$. Show that $G$ is abelian.
AI: Firstly,we can consider a homomorphism $f:G\rightarrow Aut(G)$ such that:
$f(x)=t_x$,where $t_x:G\rightarrow G$ is defined by $t_x(g)=xgx^{-1}$.
Note... |
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