Newest Questions
1,698,013 questions
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Chains intersecting both the subspace and its complement in relative homology
Let $X$ be top space and $A\subsetneq X$ nonempty. Then the group of relative chain complex of the pair is defined as $C_n(X, A):=C_n(X)/C_n(A)$. Suppose $u\in C_n(X)$ is a singular $n$-cube/simplex, ...
- 2,910
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6
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Representations of a separable Hilbert space as a Banach lattice
An infinite dimensional separable Hilbert space $H$ has at least
two non-equivalent representations as a Banach lattice: $\ell_2$
and $L_2(0,1)$.
Are there other non-equivalent representations of $H$ ...
- 234
1
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0
answers
8
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Simplify into reduced form
Simplify into reduced form the expression
$$\frac{1+2α+3α^2}
{(1+α)^{15}}
$$
, where the
minimal polynomial of $α ∈ \mathbb{C}$ over $\mathbb{Q}$ is $x^3 + x + 1$.
My Attempt: First, we can write $\...
- 354
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1
answer
13
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Compact operators in Hilbert space
I have tried to solve this exercise:
Let $H$ be a Hilbert space and let $T = T^* \in B_{\infty}(H)$ a compact operator. Given $\psi_0 \in H$ consider the equations:
\begin{align*}
(1)\quad &T\psi =...
0
votes
2
answers
21
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Does uniformly continuous functions apply to something like "sandwich theorem"?
Suppose $f,g$ are two uniformly continuous functions on $\mathbb R$, and $h$ is a continuous function on $\mathbb R$ that satisfies:$$f(x)\le h(x) \le g(x)$$Does that mean $h$ is a uniformly ...
- 1
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0
answers
12
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Angle between tangents of a hyperbola
Let a hyperbola with semi major axis length $a$ and shortest radius $r_p$ be given. For $r\geq r_p$ find angle $\gamma$ between the tangent at distance $r_p$ and the tangent at distance $r$ from the ...
- 335
-2
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answers
35
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Expressing $\tan\left(\frac{\arctan\frac12}{5}\right)$
I need an expression of $\tan\left(\frac{\arctan\frac12}{5}\right)$ for a project. Does one exist?
- 368
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2
answers
30
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Proof of the Third Isomorphism Theorem $(G/N_1)/(N_2/N_1)\cong G/N_2$
I tried to understand the concept that, for a group $G$ and two normal subgroups $N_1,N_2$ with $N_1\subseteq N_2$ it is $(G/N_1)/(N_2/N_1)\cong G/N_2$, but my following reasoning seems to suggest ...
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22
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Saddle Point / Steepest Descent for Bessel Functions
I am trying to understand how to approximate integrals with Bessel functions. In particular I have something like:
$$I_{\ell} = \int_{0}^{\infty} j_{\ell}(pr) dr = \frac{\sqrt{\pi} \Gamma[(1+\ell)/2] ...
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5
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Simplicial circle in motivic homotopy theory
I came up with a vauge question while reading this post. Does the simplicial circle $S^1$ have anything to do with the topology of $\mathbb{C}$-points of some variety? The linked question suggests ...
- 1,115
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7
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Problem with calculating projections of curl using rotation of coordinate system and contour
I tried to calculate projections of curl using rotation of coordinate system but I encountered with following problem.
Given:
$rot_xA=\frac{\partial A_z}{\partial y}-\frac{\partial A_y}{\partial z}=0$
...
- 1,159
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0
answers
11
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A family of lines with more than four points is 2-colorable
Let $E$ be a finite set of points and $\mathcal{F}$ be a collection of subsets of $E$ (lines) such that
each line has cardinal at least 4
two distinct lines intersects at exactly one point.
Show ...
- 41
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answers
11
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How to see $x_1\geq 0$ in the primal implies $p' A_1 \leq c_1$ in the dual, when there are no other constraints than just $x_1\geq 0$?
I understand why the constraint $x_1 \geq 0$ in the primal, implies $p'A_1 \leq c_1$ in the dual in the presence of a constraint involving an $A$ and a vector $b$ in the primal ($A_1$ being the first ...
- 1,367
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25
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What is this lattice property I'm looking at?
I'm looking at the lattice where elements represent a set of strings with a some prefix, or the empty set as $\bot$. The order is set inclusion, so $\mathrm{abc} \le \mathrm{ab}$ but $\mathrm{abc} \...
- 111
1
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answers
23
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Stochastic exponential and martingales
I have a Question regarding the chapter 5.5.(Girsanov Theorem) of the Book "Brownian Motion,Martingales, and Stochastic Calculus" from le Gall.
There is stated in Prop 5.21, that if D is a ...
- 71
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0
answers
19
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So we know about the Cartesian coordinate system, but how to naturally arrive at it when going on a quest to identify a location? [closed]
You would need to use the concept of distance and direction in order to derive the coordinate system with the three perpendicular axes from the first principles.
1
vote
1
answer
18
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Equality related to random variable and its conditional expectation.
Let $(\Omega,\mathcal F,\Bbb P)$ be a probability space and $X$ be a Banach space and $r_1,\cdots, r_N:\omega\to\Bbb R$ be random variables on the probability space, and $x_1,\cdots,x_N\in X.$
Suppose ...
- 318
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1
answer
51
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Deduce no integer solutions for $y^2 = x^3 + 7$ using prime factors. [duplicate]
I am working on an exercise that involves proving that the equation $$y^2 = x^3 + 7$$ has no integer solutions $(x, y)$. I am thinking of using modular arithmetic and congruency. Starting with LHS:
...
- 644
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15
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On the convolution identity of a sub arc of circle and the open set which is thickened epsilon amount of another subarc in circle.
Let $\sigma_I=\{e^{2i\pi t}:t\in I\}$ and $\sigma_J=\{e^{2\pi i t}:t\in J\}$ be two disjoint subarcs on the the first quadrant of unit circle of arc length $\theta$, where $I,J\subseteq [0,1/4]$ of ...
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1
answer
43
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Binomial sum without using Stirling numbers
I came across this binomial sum-
$p+q=99^{98}(1+99^{2})-\frac{99 \cdot 98^{98}}{1}(1+98^{2})+\frac{99 \cdot 98 \cdot 97^{98}}{1 \cdot 2}(1+97^{2})-\frac{99 \cdot 98 \cdot 97\cdot 96^{98}}{1 \cdot 2 \...
1
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0
answers
8
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Formula for number of unlabelled trees of n vertices [closed]
How do we prove that no closed-form expression exists for the number of non-isomorphic unlabeled trees of n vertices? How do we also prove that no closed-form expression can give the degree sequences(...
1
vote
1
answer
50
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Can we prove that the limit $e = \lim_{h \to 0^+} (h + 1)^{\frac1h}$ exists using the fact that the function in the limit always decreases for $h>0$?
(If you want to see the specific step that I just want to make sure is valid, it's the one at the very bottom; there is a comment in parentheses that goes before it.) I was trying to prove that the ...
- 305
1
vote
0
answers
42
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Why does the Generalized Collatz map ($3n+k$) with $k=3^x+2^x$ produce 1,023 cycles at $x=15$, but collapse to 1 cycle for $x \ge 21$?
I have been investigating the number of limit cycles (loops) in the Generalized Collatz Problem defined by the map:$$T_k(n) = \begin{cases} (3n+k)/2 & \text{if } n \text{ is odd} \\ n/2 & \...
- 11
1
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1
answer
42
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Function continuous nowhere whose domain and range are $[0,1]$
Find a function continuous nowhere, whose domain and range are both $[0,1]$.
My intuition was to start with $f(x)=x$ and exchange to $f(a)=b$, $f(b)=a$ for sufficiently many pairs of $(a,b)$. So I ...
- 4,882
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0
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38
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Proof of embeddibility of projective smooth $k$-scheme with dimension $d$ in $\mathbb{P}^{2d+1}_k$ ( Part 2, Gortz, Wedhorn ) [duplicate]
I am reading the Gortz, Wedhorn's Algebraic Geometry, proof on Theorem 14.132 and stuck at some statements. ( Can anyone who have the Gortz, Wedhorn's book help? )
EDIT : This post is not duplicate. ...
- 4,056
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0
answers
23
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Counting disjoint $k$-tuples of lines in $\mathbb F_q^n$
Let $k$ be a positive integer. How many $k$-tuples of disjoint lines are there in $\mathbb F_q^n$? Here two lines are disjoint if they do not share a point in $\mathbb F_q^n$.
I was wondering because ...
- 69
2
votes
0
answers
28
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A Principled(?) Way to Determine a Lie Algebra Automorphism from a Dynkin Diagram Automorphism (and invariant subalgebra)
I am wondering about how to write out an explicit form of a Lie Algebra automorphism from an automorphism of the associated Dynkin Diagram of the root system. Within this question are some other ...
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votes
2
answers
67
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One exponent for all numbers [closed]
We can say for given $a,b,c$ value,
$a^b=a^c$
Since base is equal, we can conclude
$b=c$
,but why this not valid for $a=1$,what is the intuitive idea?
1
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0
answers
19
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Every k-fold cover of the real line by intervals can be decomposed into k distinct covers.
A k-fold cover of the real line is a family of sets such that each point is contained inside k sets in the family.
I am trying to prove the following fact which i came across in Yufei Zhao's lecture ...
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0
answers
20
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Continuity of functions in b-metric spaces
Let $X$ be a $b$-metric space and let $B(x,\varepsilon)=\{y\in X: d(x,y)<\varepsilon\}$. Then we know that $\mathcal{B}=\{B(x,\varepsilon): \varepsilon>0, x\in X\}$
does not define a topology on ...
- 593
2
votes
0
answers
39
views
Can you find a great circle with only a compass?
Thinking about the construction of temari balls got me thinking about how one might begin the marking portion of the process, particularly how to build a great circle in the first place. To make the ...
- 53.4k
-1
votes
0
answers
53
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How applicable are the isomorphism theorems? [closed]
How broadly do the (four?) isomorphism theorems apply? Do they hold only of groups? What do they look like in set theory?
- 133
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votes
2
answers
93
views
How $\min(x,y)$ works in reasoning
I came across this proof from a reliable book, showing that the set $X=\lbrace x \in \mathbb{R} \mid x \gt 0 \text{ and }\:x^2\leq2\rbrace$ has a supremum $a$ which verifies $a^2=2$. We have proven ...
- 177
0
votes
0
answers
58
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Limit of the function satisfying $f(x)=x-f(x^2)$ as $x\to 1^-$
I guess $\lim\limits_{x\to 1^-} f(x) = 1/2$, where the function $f(x)$ defined by $f(x)=x-f(x^2)$ in $[0,1)$, or by the series:
$$
f(x) = x - x^2 + x^4 - x^8 + x^{16} - x^{32} + \cdots.
$$
I know $f(x)...
- 81
1
vote
1
answer
35
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Periodic Orbits of Arbitrarily Small Period for a Flow Without Fixed Points
Question: Let $\varphi_t: M \to M$ be a continuous flow with no fixed points on a compact metric space $M$. If $\varphi_t$ has a periodic orbit, is there a positive lower bound on the period of all ...
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39
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“Central limit theorem” for symmetric random variables with no finite mean
Let $(X_n)_{n\in\mathbb N}$ be a sequence of independent random variables with the same distribution. The common distribution $\mu$ is such that it is symmetric, that is, $\mu((-\infty,x])=\mu([-x,\...
- 24k
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0
answers
31
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Equivalent definitions of vector-valued Riemann integral
Let $X$ be a Banach space and $[a, b]$ a finite nondegenerate closed interval. I wish to define the Riemann integral of a function $f : [a, b] \to X$. There are two natural definitions that I think ...
- 7,759
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vote
0
answers
31
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How are defined these double complexes in Bott Tu, Section 14?
As in this question, I have trouble with this part of Bott and Tu's Differential Forms in Algebraic Topology :
The Spectral Sequence of a Double Complex. The starting point is a double complex $K=\...
- 11
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votes
1
answer
58
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$f: \mathbb{RP}^n \to \mathbb{RP}^m$ for $n > m$ induces trivial map on reduced cohomology
On a homework problem (Q8(i) from here) I am asked to show:
Show that any map $f: \mathbb{RP}^n \to \mathbb{RP}^m$ induces a trivial map on reduced cohomology if $n > m$.
Here's my attempt:
...
- 604
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0
answers
22
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Practical and historical role of Jordan measure
In my earlier questions, the proofs given by Asigan and D.R. showed that the Jordan outer/inner measure of the subgraph $[0,f]$ and the Darboux upper/lower integrals of $f$ are essentially the same ...
- 4,318
1
vote
1
answer
38
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Generalization of Cauchy's functional equation. What are the general solutions, $f$?
Consider a function $f : (0,\infty) \to (0,\infty)$ satisfying the identity
$$
f(x^a y^b) \;=\; f(x)^{1/a}\, f(y)^{1/b}
\qquad\text{for all } x,y>0 \text{ and all real } a,b\neq 0.
$$
This can be ...
- 1,189
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votes
1
answer
40
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Is $A_{\epsilon}=\{x \in X: d(x,A) \leq \epsilon\}$ a continuum metric space?
Let $(X,d)$ a continuum metric space and $A \subset X$ subcontinuum metric space. For all $\epsilon >0$ we define $A_{\epsilon}=\{x \in X: d(x,A) \leq \epsilon\}$. Is $A_{\epsilon}$ a continuum?
We ...
- 359
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votes
1
answer
73
views
How many odd numbers are there in one row in Pascal's triangle?
I was looking at the pattern of odd entries in Pascal’s triangle and noticed that every row contains an even count of odd numbers. This is easy to justify, but it led me to wonder how the exact count ...
- 9,035
1
vote
1
answer
56
views
Density of the set of positive triplets $(x,y,z)$ such that $x^{n_1}+y^{n_2}=z^{n_3}$ for positive integers $n_i$ in the halfopen unit cube $(0,1]^3$
Let
$$S:=\{(x,y,z)\in\mathbb{R}_{>0}^3:\ z<1,\ \exists n_1,n_2,n_3\in\mathbb{N}_{\ge1}\ \text{with }x^{n_1}+y^{n_2}=z^{n_3}\}.$$
Then the closure $\overline{S}=\{(x,y,z)\mid 0<x,y,z\le1\}$.
...
- 6,019
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votes
0
answers
17
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Asymptotic Expansion of Bessel Function using Sommerfeld Contour
On pages 291-294 of Bender & Orszag (Advanced Mathematical Methods for
Scientists and Engineers-Asymptotic Methods and Perturbation Theory) they
develop the full asymptotic expansion of $J_0(x)$ ...
- 368
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votes
0
answers
16
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Is a collinearity step missing in this Miquel point proof?
Problem:
Solution:
Question: The problem and solution are taken from the book A beautiful journey through olympiad geometry. The problem is from the chpater $19$, complete quadrilateral. In the ...
- 139
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votes
0
answers
19
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If a Stochastic Differential Equation is bounded [0,1], can we assume that models probability failure; [closed]
I want to model a system in terms of probability of failure. If I use a stochastic differential equation that is bounded [0,1], can I assume that it models probability failure? I know that failure ...
0
votes
1
answer
43
views
Computing Fourier transform for a real odd signal
Given a real and odd signal $x(t)$, such that $\vert X(\omega)\vert = e^{-\vert \omega\vert}$ is the magnitude of Fourier transform.
Question: Find the Fourier Transform $X(w)$.
My attempt:
We know, $...
- 123
1
vote
0
answers
37
views
How much less is the arithmetic mean than the max given the average deviation?
Given a finite (multi)set of elements $\{x_1, \ldots, x_n\}$ the arithmetic mean $\mathsf{AM}$ is less than or equal to the maximum element call it $\max$. In otherwords, $\mathsf{AM} \leq \max$. But ...
- 3,682
4
votes
2
answers
197
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Cutting a Möbius strip in thirds. Why are the resulting strips interlinked?
It is well known that cutting a Möbius strip "in half" down the middle results in a band with two twists, homeomorphic to a cylinder. See this question for example.
If instead, one begins ...
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