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H: recovering a representation from its character
The character of a representation of a finite group or a finite-dimensional Lie group determines the representation up to isomorphism.
Is there an algorithmic way of recovering the representation given the character and the conjugacy structure of the group?
AI: Below b... |
H: Show $(\log n)^{ (\log n) } = 2^{(\log n)(\log (\log n))}$
I am having difficulty understanding how this follows.
$$(\log n)^{ (\log n) } = 2^{(\log n)(\log (\log n))} = n^{\log \log n}$$
Which logarithmic identities are used to go through each equality?
e.g. how do you first go from
$$(\log n)^{ (\log n) } = 2^{(... |
H: When does a normal subgroup contain precisely one non-identity conjugacy class?
Every normal subgroup $N$ of a group $G$ is a union of conjugacy classes. Since every subgroup contains the identity, and the identity is in a class by itself, every normal subgroup already contains the conjugacy class of the identity.
... |
H: How to handle big powers on big numbers e.g. $n^{915937897123891}$
I'm struggling with the way to calculate an expression like $n^{915937897123891}$ where $n$ could be really any number between 1 and the power itself.
I'm trying to program (C#) this and therefor performance is key.
I tried
for(i <= 915937897123891... |
H: Asking a linear algebra problem in Berkeley Problems in Mathematics, Spring 1986.
Let $V$ be a finite dimensional vector space and $A$ and $B$ two linear transformations on $V$ such that $A^{2}=B^{2}=0$ and $AB+BA=1$.
1) Prove that if $N_{A}$ and $N_{B}$ are respective null spaces of $A$ and $B$, then $N_{A}=AN_{B... |
H: All real functions are continuous
I've heard that within the field of intuitionistic mathematics, all real functions are continuous (i.e. there are no discontinuous functions). Is there a good book where I can find a proof of this theorem?
AI: Brouwer proved (to his own satisfaction) that every function from $\math... |
H: Convergence of the sequence $f_{1}\left(x\right)=\sqrt{x} $ and $f_{n+1}\left(x\right)=\sqrt{x+f_{n}\left(x\right)} $
Let $\left\{ f_{n}\right\} $ denote the set of functions on
$[0,\infty) $ given by $f_{1}\left(x\right)=\sqrt{x} $ and
$f_{n+1}\left(x\right)=\sqrt{x+f_{n}\left(x\right)} $ for $n\ge1 $.
Pro... |
H: Distance Puzzles?
A man moves 1km east, 2km north, 3km west, 4km south, 5km east, 6km north, 7km west and so on until he travels total of 300km. So what will be the distance from origin?
AI: To travel 300 km, he would need to take 24 "steps".
For every 4 steps, he travels $2\sqrt{2}$ km south west. He will do this ... |
H: 4 by 4 Matrix Puzzle
I was solving the puzzle for the Company interview exam. I found this puzzle, I cannot come up with the solution. How to solve it and what is the correct answer?
Determine the number of $4\times 4$ matrices having all entries 0 or 1 that have an odd number of $1$s in each row and each column.
... |
H: Follow up on example computation of $\mathrm{Tor}_n$
I have a follow up question on this question of mine:
I can't reconstruct how I got $\operatorname{Im}{d_1^\ast} = 0$ from the following chain:
$$0 \to \mathbb Z \otimes_{\mathbb Z} (\mathbb Z / 2 \mathbb Z) \xrightarrow{d_1^\ast = \cdot 284 \otimes id} \mathbb Z... |
H: minimal polynomial of normal endomorphism with given eigenvalues
What's the minimal polynomial of a normal endomorphism $\phi$ with eigenvalues $2, 2, 1+i, 1+i, 1-i, 1-i, 3$?
It is $\mu_\phi | (t-2)^2(t-1-i)^2(t-1+i)^2(t-3)$ but is there any more I can derive from the fact that $\phi \circ \phi^*=\phi^* \circ \phi$... |
H: Conditional and joint probability manipulations when there are 3 variables
I'm having trouble verifying why the following is correct.
$$p(x, y \mid z)= p(x \mid y, z) p(y \mid z)$$
I tried grouping the $(x, y)$ together and split by the conditional, which gives me
$$p(x, y \mid z) = p(z\mid x, y) p(x, y)/p(z)$$
How... |
H: Formula for calculating the center of an arc
Is there a formula for calculating the point equidistant from the start point and end point of an arc given:
1) An arc is defined as: A center point $P$, a radius $r$ from the center, a starting angle $sA$ and an ending angle $eA$ in $radians$ where the arc is defined fr... |
H: Geometry Puzzle - Largest circle on Chess
The following puzzle was asked in company interview round.I have no idea ,how to do it?
What is the diameter of the largest circle that can be drawn on a
chessboard so that its entire circumference
gets covered by the black squares and no part of the circumference
falls o... |
H: Probability distribution with binomial
Could someone explain to my dumb head why we are seeking $P(X \leq2)$? Is it because that the "majority" of "five" is 3? And we want to find three correct transmissions?
AI: There are alltogether $5$ bits and you are interested in the total number of correct decisions for at ... |
H: polynomial, power series, radius of convergnce
Let $p(x)$ be a polynomial of degree $N$ then the radius of convergence of the power series $$\sum_{n=0}^{\infty}p(n)x^n$$
depends on $N$
is $1$ for all $N$
is $0$ for all $N$
is $\infty$ for all $N$
Radius of convergence $r=\lim\limits_{n\to\infty}\frac{p(n)}{p(n+1... |
H: When is a quasi-projective variety affine?
By an affine variety I mean a variety that is isomorphic to some irreducible algebraic set in $\mathbb A^n$ and by a quasi-projective variety I mean a locally closed subset of $\mathbb P^n$, with the usual Zariski topology and structure sheaf. (I am not quite familiar with... |
H: Keep getting generating function wrong (making change for a dollar)
Possible Duplicate:
Making Change for a Dollar (and other number partitioning problems)
I am working on the classic coin problem where I would like to calculate the number of ways to make change for a dollar with a given number of denominations.... |
H: Cantor's theorem
The theorem cardinality of set of real numbers is strictly larger than cardinality of set of natural numbers uses diagonal method in its proof.
Why can't we use the same argument for creating a natural number just like we do real number? i.e. take 1st digit of the first element, 2nd digit of the s... |
H: example of the vector (bounding expectation of a form in Rademacher functions)
Let $x\in R^{2m}$ such that $x_1+\ldots+x_{2m}=0$ and let $r_i, i=1, \ldots, 2m$ be Rademacher functions, i.e. $P(r_i=1)=P(r_i=-1)=1/2$.
I would like to find an example of the vector $x$ such that
$E(\sum_{i=1}^{2m}x_ir_i)^{2q}\geq C\sqr... |
H: Simpler expression for a certain determinant.
A question in elementary linear algebra, while considering the Cayley-Menger Determinant:
Given an $n\times n$ matrix $M$, consider $$\tilde{M}=\begin{pmatrix} M & (1,1,\cdots, 1)^\top \\ (1,1,\cdots, 1)& 0\end{pmatrix}$$ Is it possible to express $\det(\tilde{M})$ in... |
H: Upper bound on smallest prime $p$ needed to tell two numbers $\leq n$ apart modulo $p$
I'm going through this paper:
E. D. Demaine, S. Eisenstat, J. Shallit, and D. A. Wilson. Remarks on separating words. ArXiv e-prints, March 2011.
And on page 2, there is the following lemma:
Lemma 1. If $0 \leq i,j \leq n$ and $i... |
H: Inverse Hyperbolic Tangent type Series
Is there a name for this series?
$$\sum_{k=1}^{\infty}\frac{a^{2k}}{2k}.$$
I know that:
$$\tanh^{-1}(a)=\sum_{k=1}^{\infty}\frac{a^{2k-1}}{2k-1}.$$
AI: I don't believe there's a name for the series you have there, but,
$$\begin{align*}
-\log(1-z)&=\sum_{k=1}^\infty\frac{z^k}{k... |
H: Find (in terms of n) the shortest distance between lines L and M.
The line L has equation:$$r=(i-4k)+t(i+2j-2k)$$
The line M has equation $$r=(4i+nj+5k)+s(7i+3j-4k)$$
where n is a constant.
Find (in terms of n) the shortest distance between lines L and M.
Here's what I've tried:
The direction vector of the line per... |
H: Expected length of a game of Kings
The game of Kings is a drinking game played with a standard 52-card deck. The rules are irrelevant to the nature of this question; we only wish to calculate the expected length of a game of Kings. The game ends when all four kings have been drawn (without replacement of any cards)... |
H: Why the number e(=2.71828) was chosen as the natural base for logarithm functions?
Possible Duplicate:
What's so “natural” about the base of natural logarithms?
Why the number e(=2.71828) was chosen as the natural base for logarithm functions ? Mainly I am interested in knowing why is it called "natural " . The... |
H: Gradient of a function on the vertices of a graph
Let $G=(V,E)$ be a graph. The gradient of a function $f:V\longrightarrow R$ is defined on the edges of the graph, given by the discrete derivative
$$
\nabla f(e)=f(y)-f(x), \quad e=(x,y) \in E
$$
Take $V=S_{2n}$ the group of permutations $\pi$ of the set $\{1, \ldot... |
H: How many elements $a \in \Bbb{Z}_N$ such that $ax \equiv y \mod N$
Consider the ring $\Bbb{Z}_N$ of arithmetic modulo $N$: $\{0,1,2, \ldots ,N-1\}.$
Given $x,y \in \Bbb{Z}_N,$ how many of the elements of $\Bbb{Z}_N$ when multiplied with $x \pmod{N}$ result in $y$? And how can we calculate what they are efficiently?... |
H: A problem including the given norm of $u \in C_0^\infty (\mathbb R)$
Let $u \in C_0^\infty (\mathbb R)$, $v(x) := u(x) e^{-x^2 /2} $. And define the norm as $$ \| u \|_1^2 = \int_{\mathbb R} | u' (x) |^2 e^{-x^2} dx + \int_{\mathbb R} | u(x) |^2 e^{-x^2} dx $$
Then I want to prove that $$ \| u \|_1^2 = \int_{\m... |
H: A question of the norm calculation of Hermite polynomials
Define the (physicist's) Hermite polynomial $H_n (x)$ by $$H_n (x) = (-1)^n e^{x^2} \frac{d^n}{dx^n} e^{-x^2} $$ then prove that $$ \int_{\mathbb R} |H_n (x) |^2 e^{-x^2} dx = 2^n n! \sqrt{\pi}$$.
AI: We have the recursive relation
$$H'_n(x)=(-1)^ne^{x^2}\... |
H: Probability exercise
A couple has $3$ kids, and $1$ of them is known to be male.
What is the probability that $1$ (only $1$) of the other $2$ kids is male?
AI: in generally,if we consider $3$ child,and we know that only $1$ is male,then probability of being any choosen child male is $1/3$,but if our space is co... |
H: Reference request: Extremes of composite functions
Let $g : X \rightarrow Y$ be a homeomorphism, and let $f : Y \rightarrow \mathbb{R}$
be a continuous function. Then the extremes of $f$ in $Y$ are mapped to the extremes
of the composition $f \circ g $ in $X$, and if $f$ has a unique minimizer in $Y$, then
$f \ci... |
H: General solution for 3D line intersection
I've been trying to get the intersection point of 2 3D lines (in general, so i can code an algorithm) using the following equations:
$$ x_0 + k_0 a_0 = x_1 + k_1 a_1 \tag{1}$$
$$ y_0 + k_0 b_0 = y_1 + k_1 b_1 \tag{2}$$
$$ z_0 + k_0 c_0 = z_1 + k_1 c_1 \tag{3}$$
where $x,y,z... |
H: Stronger than ZF, weaker than ZFC
Can you please name axiom system that is strictly weaker than ZFC and strictly stronger than ZF? (such as DC, AC$_\omega$)
I searched for it but i could only find these two.
If there are statements equivalent to DC or AC$_\omega$ in ZF, please tell me or give me a link introducing ... |
H: Proving Quadratic Formula
purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become $\dfrac{b^2}{4a^2}$?
AI: $b/2a$ does NOT become $b^2/4a^2$. All that happens in the third row is t... |
H: What is a lift?
What exactly is a lift? I wanted to prove that for appropriately chosen topological groups $G$ we can show that the completion of $\widehat{G}$ is isomorphic to the inverse limit $$\lim_{\longleftarrow} G/G_n$$
I wasn't sure how to construct the map so I asked this question to which I got an answer ... |
H: Example of a proper homotopy between smooth functions on manifolds
Let $h:S^{n-1}\to S^{n-1}$ be $C^{\infty}$ map. How to prove that a function $F: S^{n-1}\times[0,1]\to S^{n-1}$ given by
$$F(v,t)=(\cos{\pi t})v+(\sin{\pi t})h(v)$$
is proper $C^{\infty}$ map?
AI: Assuming $F(v,t)\in S^{n-1}$ there is not much to sh... |
H: Approximation of bounded measurable functions with continuous functions
This is not homework. I was reading a paper where the authors showed a result for all continuous functions and then just proceeded to write "the usual limiting Argument gives the result for all bounded functions" - so I am asking myself what th... |
H: Trigonometric eigenvalue equation
In solving an eigenvalue problem, I've come to following equation ($\lambda=1$):
$$\begin{pmatrix}
\cos(\theta) & \sin(\theta) \\ \sin(\theta) & -\cos(\theta)
\end{pmatrix}\begin{pmatrix}
a \\ b
\end{pmatrix}=\begin{pmatrix}
a \\ b
\end{pmatrix}$$
Now, the solution says, "... |
H: On the intersection of closed sets
In a book on beginning measure theory, the following statement is made:
"It is clear that any intersections and finite unions of closed sets are closed."
However the intersection of two disjoint closed sets is the empty set which is open by definition. Is there something wrong wit... |
H: Global existence of solutions
How do I see whether or not these ODE's have a global solution:
$$x'=t^2+x^2$$
$$x'=t^2+x$$
Why?
AI: Applying the weaker form of the Cauchy theorem (probably known to many as Picard–Lindelöf theorem) to $$x'=f(t,x),$$ $\partial f/\partial x$ is required to be bounded (this implies it i... |
H: how is this series expansion $(\sum\limits_{i=1}^n{x_{i}})^2=(\sum\limits_{i=1}^n{x_{i}^2}+\sum\limits_{i
I'm reading my vector calculus text when I encountered below formula.
$(\sum\limits_{i=1}^n{x_{i}})^2=(\sum\limits_{i=1}^n{x_{i}^2}+\sum\limits_{i<j}{2x_{i}x_{j}})$
Is this a definition or there's a proof for a... |
H: Find endomorphism of $\mathbb{R}^3$ such that $\operatorname{Im}(f) \subset \operatorname{Ker}(f)$
I have a problem:
Let's find an endomorphism of $\mathbb{R}^3$ such that $\operatorname{Im}(f) \subset \operatorname{Ker}(f)$.
How would you do it?
The endomorphism must be not null.
AI: Well, you could always take... |
H: Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)
Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$,
is Cauchy (with respect to the operator norm) and that there is an
operator $L$, such that $Lx=\lim_{n\rightarrow\infty}T_{n}x$, ... |
H: Prove $\frac{1-\tan(x/2)}{1+\tan(x/2)}=\frac{1-\sin x}{\cos x}$
Can anyone offer please help me solve the following trig identity.
$$\frac{1-\tan(x/2)}{1+\tan(x/2)}=\frac{1-\sin x}{\cos x}=\frac{\cos x}{1+\sin x}$$
My work thus far has been on the left most side.
I did
$$\frac{\;\;1-\dfrac{\sin(x/2)}{\cos(x/2)}\;... |
H: Finding $a$ from $\lim\limits_{x\rightarrow0}(1+a\sin x)^{\csc x} =4$
The question is to find the value of $a$ from the following equation:
$$\lim_{x\rightarrow 0}(1+a\sin x)^{\csc x} =4 $$
AI: You have
$$
\lim_{x\rightarrow 0}\left(1+a\sin x\right)^\frac{1}{\sin x}=\\
\lim_{x\rightarrow 0}\left(\left(1+a\sin x\rig... |
H: Open mapping of the unit ball into itself
Does there exist a continuous open function $f:B^n\to B^n$ which is not injective?
(Here $B^n\subseteq\mathbb{R}^n$ is the open unit ball)
AI: Hint for your problem: $re^{ix}\mapsto re^{2ix}$. |
H: finding distribution $\mathbb{Z}$ in problem
suppose $X_1,X_2,\ldots,X_n$ is random sample of $Exp(0,\sigma)$. if $\mathbb{S_n}=X_1+X_2+\ldots+X_n$
and $\mathbb{Z}=\max\{n:\mathbb{S_n}\leq s\}$ how can find distribution $\mathbb{Z}$?
AI: Note that $(S_n)_{n\geqslant1}$ is the set of events of a Poisson process $(N_... |
H: Unramified extension is normal if it has normal residue class extension
Let $K/F$ be an unramified extension such that $\rho_K / \rho_F$ (the corresponding extension of residue classes) is normal. Prove $K/F$ is normal.
I guess I need to do some polynomial lifting, but if we take $f(x)$ the minimal polynomial of $... |
H: Norm of the sum of projection operators
Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an infinite-dimensional hilbert space ?
In finite dimensions this would be true, since a projec... |
H: Scalar product equals weighted sum of projection of the vectors onto the edges of a simplex
Given a triangle with the edges $e_1$, $e_2$, $e_3$, it seems (from numerical evidence) that there are coefficients $\alpha_i$ such that
$$
u^Hv = \sum_{i=1}^3 \alpha_i \, (u^He_i)\, (e_i^H v)
$$
holds for all vectors $u, v\... |
H: Does this sequence of operators in Hilbert space stop at $\text{rank}T+1$ steps?
Let $\left(T_{n}\right)_{n}$ be a sequence of operator in a infinitdimensional
Hilberspace $H$, defined as restrictions of an operator $T:H\rightarrow H$,
on smaller and smaller subsets, by the algorithm in this question (were also add... |
H: Is there a simple explanation why degree 5 polynomials (and up) are unsolvable?
We can solve (get some kind of answer) equations like:
$$ ax^2 + bx + c=0$$
$$ax^3 + bx^2 + cx + d=0$$
$$ax^4 + bx^3 + cx^2 + dx + e=0$$
But why is there no formula for an equation like $$ax^5 + bx^4 + cx^3 + dx^2 + ex + f=0$$
I'm not s... |
H: Ratio GRE question
Cashews cost 4.75 per pound and hazelnuts cost 4.50 per pound. What is
larger, the number of pounds of cashews in a mixture of cashews and
hazelnuts that costs $5.50 per pound, or 1.25? Alternatively, are they
equal, or is it impossible to calculate?
My answer:
I believe that 1.25 is larg... |
H: Why does a countably infinite dimensional space not have an uncountable chain of subspaces?
This question is motivated by Jyrki Lahtonen's comment in this question. It was an attempt at a short proof that for arbitrary field $k$, we have that $k^\mathbf N$ (as the full set-theoretic product with obvious linear stru... |
H: Taylor expansion on interval or at infinity
I'm trying to figure out how to get this result via taylor expansion as $x,y \rightarrow \infty$:
$f(x) = \sqrt{(x-1)y} = \sqrt{xy} - \frac{1}{2}\sqrt{\frac{y}{x}} + ...$
I've been told (on yahoo answers), that since you can't take taylor expansions at infinity, you do it... |
H: Factoring extremely large integers.
The question is about factoring extremely large integers but you can have a look at this question to see the context if it helps. Please note that I am not very familiar with mathematical notation so would appreciate a verbose description of equations.
The Problem:
The integer in... |
H: spectral gap of the graph / Markov chain
Let $G=(V,E)$ be a graph. The gradient of a function $f:V\longrightarrow R$ is defined on the edges of the graph, given by the discrete derivative
$$
\nabla f(e)=f(y)-f(x), \quad e=(x,y) \in E
$$
Take $V=S_{2n}$ the group of permutations $\pi$ of the set $\{1, \ldots 2n\}$ a... |
H: Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$
I'm working on a homework problem that is as follows:
Suppose that $n$ is a positive even integer with $n/2$ odd. Prove that there do not exist positive integers $x$ and $y$ with $x^2 - y^2 = n$.
It looked like a good candidate for p... |
H: Right Coset and the associated lemma from Herstein: Appreciating whats going on?
I.N. Herstein in Page 34 (last line) and Page 35 of "Topics in Algebra" book goes on to explain a definition of right coset and a lemma like this:
Def: If $H$ is a subgroup of G, and $a \in G$, then $Ha = \left \{ha|h\in H \right \}$;... |
H: Alternating sum of binomial coefficients
Calculate the sum:
$$ \sum_{k=0}^n (-1)^k {n+1\choose k+1} $$
I don't know if I'm so tired or what, but I can't calculate this sum. The result is supposed to be $1$ but I always get something else...
AI: Using the binomial theorem we have:
$$ (1 + (-1))^{n+1} = {{n+1} \cho... |
H: A recurrence relation for the Harmonic numbers of the form $H_n = \sum\limits_{k=1}^{n-1}f(k,n)H_k$
Working on Harmonic numbers, I found this very interesting recurrence relation :
$$
H_n = \frac{n+1}{n-1} \sum_{k=1}^{n-1}\left(\frac{2}{k+1}-\frac{1}{1+n-k}\right)H_k
,\quad \forall\ n\in\mathbb{N},n>1$$
My proof of... |
H: About a linear map from $ \mathbb R^n \to \mathbb R^n$
I want to prove the linear map $L : \mathbb R^n \to \mathbb R^n$ is onto if and only if $L$ is one to one.
AI: This is false. The trivial map $L : \mathbb{R}^n \rightarrow \mathbb{R}^n$ defined by $L(v) = 0$ is linear, but not one to one or onto.
For the new... |
H: If $f(x)$ is irreducible in $k[x]$, why is it also irreducible in $k(t)[x]$, for $t$ an indeterminate?
I've been thinking on this a few days, but I'm stuck.
Let $k$ be a field, $f(x)$ irreducible in $k[x]$. Why is $f(x)$ also irreducible in $k(t)[x]$, for $t$ an indeterminate?
I write $f(x)=c_0+c_1x+\cdots+c_nx^... |
H: Heine-Borel Theorem ($\mathbb{R}^k$) (in ZF)
Heine-Borel Theorem;
If $E \subset \mathbb{R}^k$, then $E$ is compact iff $E$ is closed and bounded.
I have proved 'closed and bounded⇒compact' and 'compact⇒bounded'.
(There exists $r\in \mathbb{R}$ such that for every $x\in E$, $|x|<r$)
The proof in Rudin PMA p.40 uses ... |
H: Can manholes be made in other shapes than circles, that prevent the cover from being able to fall down its own hole?
Circular manholes are great because the cover can not fall down the hole. If the hole were square, the heavy metal cover could fall down the hole and kill some man working down there.
Circular manhol... |
H: A simple question of finite number of basis
Let $V$ be a vector space. Define
" A set $\beta$ is a basis of $V $" as "(1) $\beta$ is linearly independent set, and (2) $\beta$ spans $V$ "
On this definition, I want to show that "if $V$ has a basis (call it $\beta$) then $\beta$ is a finite set."
In my definition, ... |
H: Pumping lemma - do I have to show every way to split string to have a complete answer?
In the pumping lemma, we have to split strings into $uvwxy$ (for example). Say the language was $a^n$$b^n$$a^n$$b^n$. We could it this way: $a^r$$a^s$$a^t$$a^u$$b^n$$a^n$$b^n$, with $uvwx$ all contained in the first $a^n$. We ... |
H: Expected values of Binomial?
A group of $d$ students tries to solve $k$ problems in the following way. Each student picks only one problem at random and tries to solve it. Assume that all the attempts have probability $p$ of success and are independent of each other and of the students choice. Let $X$ denote the n... |
H: How to find all solutions to equations like $3x+2y = 380$ using matrices/linear algebra?
I'm coming up blank on Wikipedia and other sources, though this seems elementary. I'd like to know what techniques or processes are used to find all (integer) solutions to an equation such as $3x+2y = 380$ using linear algebra.... |
H: Determine the number of factors for extremely large numbers.
An offshoot from a related question, is there a way to determine the number of possible factors (odd, even, prime, etc.) for extremely large integers without actually factoring them?
Even an estimation would help as long as it has some relevance to the nu... |
H: Why Strongly Continuous Representations?
When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi :G\rightarrow GL(V)$ be continuous with respect to the strong-operator topology ... |
H: $A' \cap (A' \cup B') \cap T= A' \cap T$
Let $A'$ denotes the complement of A with respect to $ \mathbb{R}$ and $A,B,T$ are subsets of $\mathbb{R}$. I am trying to prove $A' \cap (A' \cup B') \cap T= A' \cap T$, but I got some problems along the way.
$A' \cap (A' \cup B') \cap T= (A' \cap A') \cup (A' \cap B') \cap... |
H: A question of an orthonormal system
Let $H$ be an inner product space, $e_n\;(n \in \Bbb N )$ be the orthonormal system of $H$. Here I want to prove that for any $f \in H$ , $\langle f, e_n\rangle_H \to 0$ as $n \to \infty$.
The bracket means the inner product.
AI: Let $a_n = \langle f, e_n \rangle$. One has that ... |
H: a problem of trinomial distribution
$X_1$, $X_2$, $X_3$ are distributed according to the trinomial distribution with $n$($=X_1+X_2+X_3$) and $p_1$, $p_2$, $p_3$ ($p_1+p_2+p_3=1$). What is a correlation of $X_i$ and $X_j$? Is the conditional probability function $p(x_2,x_3|x_1)$ of $X_2$ and $X_3$ given that $X_1=x_... |
H: What is a good technique for solving polynomials?
Say for example:
$6x^{3}-17x^{2}-4x+3=0$
I sort of look at it and don't know where to start, other than just guessing what the first one would be and trying to do from there. Is there a good technique for approaching such polynomials?
AI: First, to minimize the lead... |
H: Are non-circular swords possible?
I was reminded of this by our recent discussion of the old chestnut about possible shapes for utility hole covers. Perhaps this question is less familiar.
A sword can be made in any shape at all, but if you want to be able to put it into a scabbard, only certain shapes will do. Fo... |
H: How to conceptualize conditional expectation inductively?
Attempting the solve the following problem
A fair die is successively rolled. Let $X$ and $Y$ denote,
respectively, the number of rolls necessary to obtain a 6 and a 5.
Find
$E[ X|Y = 5]$
I am confused about how to conceptualize such a problem?
I ... |
H: Find the limit without L'Hôpital's theorem
I'm trying to find $$ \lim_{ n\to\infty} { {n^2+1}\over {n^3+1}} \cdot {\frac {n} {1}}$$
I know the answer is $1$, but I can't remember how my professor found it so simply without using L'Hôpital's theorem.
Could you please show me the shortcut?
AI: First let's expand $$ ... |
H: A question about an uncountable summation.
Possible Duplicate:
The sum of an uncountable number of positive numbers
Consider $\sum_{\lambda \in \Lambda} a_{\lambda}$ . Here all $a_\lambda $ is non-negative.
Then I want to prove that if $\sum_{\lambda \in \Lambda} a_{\lambda} < \infty $ then there exists at most... |
H: Finite Subgroups of $GL(n,\mathbb{C})$
In the Artin's book on Algebra, the author stated a theorem (Ch.9, Thm. 2.2):
"A finite subgroup $G$ of $GL(n,\mathbb{C})$ is conjugate to a subgroup of $U(n)$.
Here, $U(n)$ is the unitary group, i.e. if $\langle \,\,, \rangle$ is the standard Hermitian inner product on $\mat... |
H: In mathematics, what is meant by induction?
I was going through MIT video lectures on "Introduction to Algorithms " . In order to solve recurrences by substitution the professor says that we can solve them by induction.
What is actually the principle behind induction?
AI: The most basic principle is this:
For any... |
H: For what values for m does $\sum \limits_{k=2}^{\infty}\frac{1}{(\ln{k})^m}$ converges?
For what values for m does $$\sum \limits_{k=2}^{\infty}\frac{1}{(\ln{k})^m}$$ converge?
What about
$$\sum_{k=2}^{\infty}\frac{1}{(\ln(\ln{k}))^m}$$ or more generally
$$\sum_{k=2}^{\infty}\frac{1}{(\ln(\cdots (\ln{k}))\cdots)^m... |
H: What is the correct way to solve $|2K^3-2K^4|$ determinant?
Given -
$$K_{3\times3} = \begin{bmatrix} 1&1&1 \\ 3&2&1 \\ 1&2&1 \end{bmatrix}$$
$$|K| = 2$$
Find -
$$|2K^3-2K^4|$$
I tried this:
Since $|A+B|=|A|+|B|$ ( $\Leftarrow$ This is the main mistake ) -
$$|2K^3-2K^4|=|2K^3+(-2K^4)|=|2K^3|+|(-2K^4)|$$
No... |
H: Complex and real forms of the Poisson integral formula
In my complex analysis book there is the expression
$$\frac{1 - |z|^2}{|1 - \bar z e^{it}|^2}$$
and it says that when $z = re^{it}$, we can write the above expression as
$$P_r(t) = \frac{1 - r^2}{1 - 2r\cos t + r^2} = \text{Re}\left( \frac{1 + z}{1 - z} \right)... |
H: "The space of all lines on a plane is an open Möbius Band."
I have come across this sentence when I was reading something about algebraic topology:
The space of all lines on a plane is an open Möbius Band.
I don't quite understand this, can anyone explain this to me? Thank You.
AI: From Wikipedia:
The open Möbiu... |
H: What does the letter epsilon signify in mathematics?
This letter "$\varepsilon$" is called epsilon right ? What does it signify in mathematics ?
AI: The greek letter epsilon, written $\epsilon$ or $\varepsilon$, is just another variable, like $x$, $n$ or $T$.
Conventionally it's used to denote a small quantity, lik... |
H: $T^2=S$ implies $\text{rank}T=\text{rank}S$?
Is it true, that if $T$ and $S$ are selfadjoint and positive operators in a finite-dimensional Hilbert space, such that $T^2=S$, that then $\text{rank}T=\text{rank}S$ ?
AI: This is true more generally if $T$ is diagonalizable (recall that if $T$ is self-adjoint, it is di... |
H: Using Trapezoid Rule to approximate $\int^{0}_{-2}\frac{1}{4+x^2}dx$
I need to use the trapezoid rule to approximate the following integral: $$\int^{0}_{-2}\frac{1}{4+x^2}dx$$
for a given step $n=5$.
I'm not exactly sure how to do a problem like this.
AI: http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/techniq... |
H: A certain family of continuous functions on $[0,1]^2$ the closure of which linear span is $\tilde{\mathcal{C}}([0,1]^2,\mathbb{R}))$
First of all I must apologize for the vague title and am open to suggestions.
This is not a Homework Assignment but something I once again encountered while reading a very compactly w... |
H: Sum of all elements in a matrix
The trace is the sum of the elements on the diagonal of a matrix. Is there a similar operation for the sum of all the elements in a matrix?
AI: I don't know if it has a nice name or notation, but for the matrix $\mathbf A$ you could consider the quadratic form $\mathbf e^\top\mathbf ... |
H: Convex hull approximated from inside by only finite number of elements?
In approximating the convex hull "from inside", i.e.
$$ \text{conv}S = \{ x \in \mathbb{R}^n \mid x= \sum_{i=1}^k \lambda_i x^i, x^i \in S, \lambda_i \geq 0, \sum_{i=1}^k \lambda_i= 1 \} \text{,}$$
in the case where $S$ is infinite - why can ... |
H: Three problems with binomial coefficients
I found three difficult problems for me, involving binomial coefficients. They are extremely interesting I think, but I don't know if I have enough knowledge to manage. Seem really hard, can you help me with them?
Prove that every $z\in\mathbb{N}$ we can represent (in one ... |
H: Fourier transform of the derivative - insufficient hypotheses?
An exercise in Carlos ISNARD's Introdução à medida e integração:
Show that if $f$ and $f'$ $\in\mathscr{L}^1(\mathbb{R},\lambda,\mathbb{C})$ and $\lim_{x\to\pm\infty}f(x)=0$ then $\hat{(f')}(\zeta)=i\zeta\hat{f}(\zeta)$.
($\lambda$ is the Lebesgue mea... |
H: Domain of an operator in functional analysis
I used to think that if we say $f$ is a function from a set $X$ to $Y$ then this implied that $f$ was defined on all of $X$. Because the definition of function is that it's a set $\{(x,y) \mid \text{ for every } x \in X \text{ there is exactly one } (x,y) \text{ where } ... |
H: Is $\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$ still valid for infinite limits?
Maybe this question is a stupid one but, let me ask it here just to be sure. :) We know that under continuity of function $f$ on an interval $[a,b]$ wherein $a<b$: $$\int_{a}^{b}f(x)dx=-\int_{b}^{a}f(x)dx$$ Now,
Does this equality remain ... |
H: Dimension of a quotient vector space of meromorphic functions
Let $U$ be an open set of the Riemann sphere, $z_i$ be $n$ distinct points of $U$, and $E$ the vector space of meromorphic functions on $U$ with poles of order no more than 2.
Let $F$ be the subspace of $E$ whose elements are holomorphic in a neighborhoo... |
H: yet another problem about trinomial
An enterprise has workers divided in 3 groups. $p_i$ is a proportion of the $i$-th group to all, $p_1+p_2+p_3=1$.
Select independently and equiprobably with replacement $n$ workers out of all, $X_i$ workers belong to $i$-th group. $n=X_1+X_2+X_3$. Use $Y_i=X_i/n$ as unknown prop... |
H: Rational points on singular curves and their normalization
Let $X$ be a curve over a field $k$. Assume that $X$ is geometrically connected, geometrically reduced and stable.
Let $Y\to X$ be the normalization. Is $Y(k) = X(k)$?
AI: The answer is no. Consider the curve $C: y^2=x^2+x^3,$ and let $\pi:\widetilde C \to ... |
H: Distribution of the sum of independent r.v.
Assume that $X_1$ and $X_2$ are independent random variables with given distribution $f(.)$ (say Normal distribution with $\mu_i$ and $\sigma_i$).
I am stuck with the calculation of:
$P(\{X_1 \leq a\} \; \cap\; \{X_2 \leq b\} \; \cap\; \{c \leq X_1 + X_2 \leq a+b\} )$
wh... |
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