All Questions
1,570,603
questions
1
vote
0
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7
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Intersection of perpendicular tangent lines - generalization of directrix?
This is a funny little problem that I came up with.
For a differentiable function $f$, define a locus of points $P$ as follows:
Let $m$ be an arbitrary tangent line to $f$, and let $n$ be another ...
- 11
0
votes
0
answers
6
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What are the conditions for a non-Hermitian matric's Rayleigh quotient to be less than its maximum eigenvalue?
What I am trying to prove is that: for a row-stochastic matrix P (not necessarily symmetric), whose every row sums $\sum_{j=1}^nP_{ij}=1$, the Rayleigh quotient of P, $R(\mathbf{x})=\frac{\mathbf{x}^{\...
0
votes
0
answers
27
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$a,b \in \mathbb Z$ and $\frac{ab}{a+b}=2$ How many possible values exist for $a$?
$a,b \in \mathbb Z$ and $\frac{ab}{a+b}=2$
How many possible values exist for $a$?
The question is from my cousin's high school test book. We couldn't figure out how to solve this, any help is ...
- 9
0
votes
0
answers
10
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Expected smallest interval size when intervals are subdivided randomly $N$ times.
Each Iteration: Given a $[0,1]$ interval partitioned into sub-intervals $I_1\ldots,I_n$, choose $k\in[n]$ uniformly at random. Subdivide interval $I_k$ into two disjoint intervals of equal size.
Let $...
- 135
0
votes
1
answer
29
views
What is the generalized solution to $M = \int_0^1 (1-x^m)^n$?
What is the generalized solution to $$M = \int_0^1 (1-x^m)^n \,dx$$
(where it can be expressed in terms of n and m) and m and n are rational numbers if considering m and n as natural numbers makes ...
- 251
2
votes
4
answers
27
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If $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are b and c equal?
Is it true that, if $gcd(a,b)+LCM(a,b)=gcd(a,c)+LCM(a,c)$, then are $b$ and $c$ equal?
For coprime $(a,b)$ and $(a,c)$, it is trivial. It is also easy when $a|b$ and $a|c$. I am unable to construct ...
- 6,708
0
votes
0
answers
10
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Christoffel Symbol in terms of determinant of the metric
It is generally known that, if g is the metric,
$$ Γ^𝜇_{\mu\alpha}= \partial_{\alpha}(ln\sqrt{|g|}).$$
It is also known that Christoffels are symmetric on its second and third components, meaning ...
0
votes
1
answer
51
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How do we evaluate this REALLY tricky integral?
$$\int{\frac{\cos 9x + \cos 6x}{2 \cos 5x-1} dx}$$
The objective is to find the answer in terms of $\sin 4x$ and $\sin x$
I would like to share my attempt and then ask a conceptual doubt as usual but ...
0
votes
0
answers
8
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Showing that $\{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$ is a $T$ invariant set
Let $(X, \sigma, \mu)$ be a probability space, $T:X\to X$ a $\mu$-invariant map and $f\in L^1(\mu)$. Define $A := \{x \in X: \sup_n\sum_{i=0}^n f(T^i(x)) = \infty\}$. I would like to conclude that $A$ ...
- 810
0
votes
0
answers
8
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Markov chains: Prove vector of mean hitting times is minimal non-negative solution
Definitions: Let $H^A = \inf \{n \geq 0 : X_n \in A \}$ be the first hitting-time of the set $A$. Let $k_i^A = \mathbb{E}_i(H^A)$ be the mean hitting time of $A$ from $i$.
Theorem: The vector of mean ...
- 81
0
votes
0
answers
11
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Hadamard Gap Theorem and Lacunary Functions
Does anyone know any good reference or even a simple proof for Hadamard Gap Theorem or even just the fact that a lacunary function diverges at 1 (I mean the limit not just the evaluation). In fact, I ...
- 832
0
votes
0
answers
17
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Vector Newton's method without using matrix inverse?
For a set of non liner equations $f_i(\vec x)$ were $i\in \mathbb{N}$ was the index, one can construct a vector
\begin{equation}
F(\vec x)=
\begin{pmatrix}
f_1{\vec x} \\
f_2(\vec x)
\end{pmatrix}
\...
4
votes
2
answers
44
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Evaluate Integral : $\int\;\frac {1}{\sqrt {1-e^{2x}}}dx$
How to evaluate Integral :
$$\int\;\frac {1}{\sqrt {1-e^{2x}}}dx $$
My attemp:
I know that my answer is wrong, but I don't know exactly where the error is. I need help finding out?
$$I = \int\;\frac {...
- 522
0
votes
0
answers
13
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Is the following conjecture relating to the chromatic number correct
Define $\mu(G)$ as the minimum number $k$ so that we can partition $V$ in $n, 1<n<k+1$ sets $V_i$ and so that every set $V_i$ has less than $k$ edges between $V_i$ and $V_i^c$
Let $G=(V,E)$ be ...
- 183
0
votes
0
answers
9
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The linear system associated to a blow-up(of surfaces)
Take a point $p\in\mathbb{P}^2_k$ and blow it up. Then we get a morphism $f:X \rightarrow\mathbb{P}^2_k$ from the blow-up. This should then be describe by a linear system on $X$. What is that linear ...
0
votes
0
answers
12
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Find all subspaces, that are simultaneously invariant relative to linear transformations $A$ and $B$
Find all subspaces, that are simultaneously invariant relative to linear transformations $A$ and $B$, where
$A = \left( \begin{matrix}5 & -1 & -1\\ -1 & 5 & -1 \\ -1 & -1 & 5 \...
- 615
-1
votes
0
answers
15
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Probability of random
in bridge each of the players(A,B,C) reciving 13cards suppose A and c have 11of 13 spade between he what is the probaility of remaining +wo spade is distributed so that B and D have one spade piece
- 7
1
vote
2
answers
24
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Rudin PMA 4.20 - how can this function be unbounded ? Considering Rudin hasn't introduced "divergence" of functions yet in the chapter.
Here is the very beggining of Rudin's Principles of Mathematical Analysis 4.20 theorem:
Let $E$ be a noncompact and bounded set in $\mathbb{R}^1$. Then there exists a continuous function on $E$ ...
- 598
-2
votes
0
answers
42
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Can I write $\prod_{i=1}^{n}(y_i+\lambda z_i) = \prod_{i=1}^{n}y_i + \lambda \prod_{i=1}^{n}z_i$?
Is product notation distributive. As in, can I write the following? $$\prod_{i=1}^{n}(y_i+\lambda z_i) = \prod_{i=1}^{n}y_i + \lambda \prod_{i=1}^{n}z_i$$
- 965
-4
votes
0
answers
17
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I have a bag of N marbles and there are C different types of colors of marble, I take one marble out and note it, I then place it back in the bag.
How many times T should I repeat this in order to be 99% sure about the number of colors of marble? The answer T should be written in terms of N and C.
Addition:
The bag is of sufficiently large size ...
-1
votes
0
answers
13
views
Definite Integrals Without a function
Let
the integral from 0 to 2 of f(x) = 13
the integral from 0 to 3 of f(x) = -10
the integral from 0 to 2 of g(x) = 8
the integral from 2 to 3 of g(x) = 14
Use these values to evaluate the following ...
-4
votes
0
answers
20
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Steps for solving ODE
Which mechanism would i follow to solve: $y' = \frac{1}{x} + \frac{1}{y}$.
There is a explicit method to solve it?
-1
votes
0
answers
11
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Isomorphism of Elliptic Curves Over extension of finite fields
I am trying to understand the isomorphism between two elliptic curves $E_1$ and $E_2$ defined over a field $K$ (such as $Fq$ where q is a prime number) and $\#E_1(K) \neq \#E_2(K)$ as mentioned in ...
2
votes
1
answer
22
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Triply transitive action on upper half plane
I am calculating the area of a triangle in the upper half plane. Consider the following triangle in the upper half plane with the Poincare metric.
Can I transform this triangle to the following ...
- 129
0
votes
1
answer
20
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Is the set X perfect? Abbott 3.4.9
I have been solving a problem similar to Abbott's 3.4.9.
Let $(r_1,r_2,r_3,\dots)$ be an enumeration of the rationals. For $n \in \mathbb{N}$ let $\epsilon_n = \frac{1}{2^n}$. Define the set $Y$ as ...
- 145
0
votes
0
answers
6
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How to use Poission distribution to generate bursts of data packets
I’m doing some research on computer networking and trying to use a model for simulating burst of sending data packets. After searching around for a while it seems that the most common one for this ...
0
votes
1
answer
12
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Probability of every number on fair dice appearing before throwing a 6.
This is my first posing a question on stack exchange, spare me if the question is too basic.
1
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0
answers
26
views
How to integrate $\frac1{\sqrt{x^2+y^2+z^2}}$ in cylindrical coordinates
I’m trying to integrate this equation in the region W. I converted to cylindrical coordinates but I’m having trouble computing this integral. Is my setup correct? How do I compute this integral?enter ...
- 11
2
votes
1
answer
40
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Rationalising the equation $ \sqrt{A}+\sqrt{B}+\sqrt{C}+ \sqrt{D}+\sqrt{E}=0$
This is an exercise in Carlo Bourlet's textbook, LECONS D'ALGEBRE ELEMENTAIRE. Rationalising the following equation,
$$ \sqrt{A}+\sqrt{B}+\sqrt{C}+ \sqrt{D}+\sqrt{E}=0$$
I tried to square both sides ...
- 461
0
votes
1
answer
36
views
Is there a name for permutations of numbers 1 to n such that there is no case where $i\to j$ and $j\to i?$
Is there a name for permutations of numbers $1$ to $n$ such that there is no case where $i\to j$ and $j\to i?$
Such permutations would have to be derangements and would include circular shifts other ...
- 845
0
votes
0
answers
37
views
Tighter upper bound of $\sum\limits_{k=1}^\infty\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$
How do I find tight upper bounds of $$S=\sum_{k=1}^{\infty}\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$$ ? The derivative of $\tanh$ is $\text{sech}^2$, so using $$S=\sum_{k=1}^\infty\frac{1}{\cosh^2(\cosh ...
- 3,544
-2
votes
0
answers
12
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Rate of change of depth of a liquid in a spherical container
I'm just wondering about how you would do this question but I am unable to type it using LaTeX as for some reason it is not properly covnerting my code to the correct equation but here it is as an ...
-1
votes
0
answers
10
views
Fractional Brownian Motion and Infinite Crossings
It's well known that regular Brownian motion crosses any point infinitely many times.
Is this fact also true for fractional brownian motion?
In particular when the hurst parameter $H>1/2$.
- 1
2
votes
0
answers
29
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Constructing Smooth Manifolds
I’ve been trying to teach myself differential geometry using Will Merry’s notes (found here) and am struggling to prove Proposition 1.17 on page 7. Here’s the statement…
Let $M$ be a set. Suppose we ...
- 21
0
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0
answers
8
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Existence of a non-trivial linear combination of k+1 orthonormal linearly indep. vectors in the left nullpace of a matrix with k columns
Suppose $v_{1}, ..., v_{k+1}$ is a linearly independent, orthonormal set of vectors in $R^{n}$. Prove that for $Y \in R^{n \times k}$ there exists a non-trivial linear combination of these vectors $w$ ...
- 69
1
vote
1
answer
11
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Let $X_n$ be a simple random walk in $\Bbb Z^d$. For all $n,m,k\in\Bbb N,x,y\in\Bbb Z^d$ $\Bbb P(X_{n+k}=y\mid X_n=x)=\mathbb P(X_{m+k}=y \mid X_m=x)$
I wrote the following proof and wanted to know if there is anything I can improve in terms of maths but also writing :
Let $S_n$ be a simple random walk in $\Bbb Z^d$. For all $n, m, k \in \Bbb N$, $x,...
- 1,461
-2
votes
0
answers
20
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Proving convex set [closed]
Sorry for this homework type question, but I have been scratching my head on this question for quite long, can anyone give me the proof to this question:
...
1
vote
0
answers
28
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How do we split this integral?
The minimum value of $f(x)$=$\int_{0}^{4}{e^{|x-t|}} dt$ where
$x\in{[0,3]}$
I really have no idea on how to solve this integral. I started of by splitting the integral recognizing the absolute value ...
1
vote
1
answer
15
views
chromatic number when drawing edges between components in a graph
Is the following true? Let a graph consist of $k$ connected components , each being k-colorable. Now, if for each pair of components, we draw $k-1$ edges between them, then the resulting graph is ...
- 183
1
vote
0
answers
16
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Three equal segments on this angle, explanation for solution needed
We're given an angle $\angle BAC$. We want to construct $D \in \overrightarrow{AB}$ and $E \in \overrightarrow{AC}$ such that $CE=DE=DB$. This is Euclidea's problem 15.4 and I have the solution but I ...
- 1,065
0
votes
0
answers
13
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$2$-norm of a matrix obtained through discretization of the eigenvalue problem
Define $$A = \frac{1}{2h} \begin{pmatrix}
0 & 1 & & & & -1\\
-1 & 0 & 1 & & \\
&\ddots & \ddots & \ddots \\
& & & -1 & 0 & 1 \\
1 &...
- 1,693
0
votes
0
answers
14
views
Number of polyhedra covering a sphere
Let's assume we have a ball of radius $r$
$$\{(x,y) \in \mathbb{R}^2|x^2+y^2 \leq r\}$$
For now, we can fix $r=1$.
We further assume to have $n$-line equations
$$a_{i1}x+a_{i2}+b_i=0 \quad \forall i =...
- 417
0
votes
0
answers
14
views
A question related to an inequality involving a complex function
I'm trying to solve the following problem from one of the past qualifying exams.
Find an explicit formula for all meromorphic functions $g$ on
$\mathbb{C}$ such that $$|g(z)|\leq \frac{\log(2+|z|^2)}{...
- 418
4
votes
0
answers
33
views
$\int_0^\infty\frac{u}{2\nu^2}\left(u+\left(u^2+\nu^2\right)^\frac{1}{2}\right)\left(e^{-u}-e^{-\left(u^2+\nu^2\right)^\frac{1}{2}}\right)\sin(ut)du$
Evaluating
$$
F_\nu(t) :=
\int_0^\infty \frac{u}{2\nu^2} \left( u+ \left( u^2+\nu^2 \right)^\frac{1}{2} \right)
\left(
e^{-u} - e^{-\left(u^2+\nu^2\right)^\frac{1}{2}}
\right) \sin \left( ut \right)...
- 23
0
votes
2
answers
32
views
How do you suggest to solve this ODE analytically?
Consider the following ODE system
$\omega''(t)+\alpha\left(f(t) \omega'(t)-f'(t)\omega(t)\right)=0, $
$\omega(\infty)\rightarrow 0, \omega(0)=1,$
with
$f(t)=\mu+\beta t-2b\beta\left(1-e^{-t/b}\...
- 55
0
votes
1
answer
24
views
Advice on proof-based exercises and more theoretical math subjects
i am a soon to be graduate in physics and i am considering to do a master in the theoretical physics area, but i find myself a little bit "unfaithfull" of my mathematical skills.
Let me ...
- 3
0
votes
0
answers
8
views
Maximization optimization problem of possibly log-concave function?
I'm trying to solve a non-convex optimization problem, trying to figure if their are any tricks to transform it into an approximate convex form.
here's a simpler form of the problem I'm trying to ...
- 750
1
vote
0
answers
19
views
How is chirality represented in electroweak Lagrangian?
The electroweak Lagrangian before symmetry breaking is defined as:
$$L_{ew}=L_g+L_f+L_h+L_y$$
$L_{fermion}$ concerns the 4-component $\Psi$ vector fields of 12 fermion types $\{u,d,c,s,t,b,e,\nu_e,\mu,...
- 315
0
votes
0
answers
17
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Convergence of infinite series after integration
I am trying to compute the integral of the product between an infinite sum and a function:
$$
I(x) = \int_0^{\infty} \sum_{n=0}^\infty A_n(t) f(t) dt
$$
I know that the series converges and $\int_0^{\...
- 55
-3
votes
1
answer
41
views
How to prove that $\sum_{i=1}^{n} i^{-2} \leq 2 - \frac{1}{n}$ for all $n \geq 2$?
Prove that $$\sum_{i=1}^{n} i^{-2} \leq 2 - \frac{1}{n}$$ for all $n \geq 2$
The above series is coming to be like $1+1/4+1/9+1/16+...+1/(n^2)$. But how to prove the above inequality? I also tried to ...
