All Questions
1,570,604
questions
0
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Finding limit including binomial coefficients
I recently came across a question from my calculus problem book and encountered a pretty good looking problem.
It goes as
Find $$ \lim_{n \to \infty}\sqrt[1/n]\frac{3n\choose n}{2n\choose n}$$
I tried ...
- 391
0
votes
3
answers
18
views
Determine p so that the line q does not have any point in common with the circle
I am preparing to take an entrance exam for a university in my country, which will happen soon. As part of my preparation, I have been practicing with some sample math tests provided by the university....
- 35
0
votes
1
answer
24
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To find the sum of all possible perimeters of a triangle
Here's the exact question:
Consider a triangle $ACD$. Point $B$ is on $AC$ with $AB = 9$ and $BC = 21$. Also $AD = CD$ , and $AD$ and $BD$ are integers. Let $S$ be the sum of all possible perimeters ...
- 43
-2
votes
0
answers
25
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show that $V = \operatorname{im} f + \ker f$ if and only if $\operatorname{im}f =\operatorname{ im}f ^2$
Let $V$ be a vector space of finite dimension $n$ and $f: V \to V$ is linear.
Show that $V = \operatorname{im}f + \ker f$ if and only if $\operatorname{im}f = \operatorname{im}f ^2.$
1
vote
1
answer
17
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Meaning of weak solutions in the class $C^{1,\alpha}$
There are quite a few papers who, e.g. for the p-Laplace equation, show that weak solutions under some assumptions are not quite in $C^2$ but at least in $C^{1,\alpha}$ (locally). But I am wondering ...
- 323
1
vote
0
answers
22
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Distance between unbounded closed convex subsets of Hilbert space
Suppose that $A$ and $B$ are nonempty closed convex subsets of the Hilbert space $H$.
Case I: If $A$ or $B$ is bounded, I can prove that there exist $a\in A$ and $b\in B$ such that
$$d(A,B) = d(a,b),$$...
- 545
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votes
0
answers
12
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Trigonometric function with a single root
Considering the following trigonometric function:
$f(x) = a \sin(2x+\phi_0) + b \sin(x) + c$
I need to find an explicit condition for the parameters $a, b, c, \phi_0$ such that this function has ...
- 33
1
vote
1
answer
28
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Suppose that $H$ and $K$ are subgroups of a group $G$. Show that if $Hg_1\subset Kg_2$ for some $g_1$ and $g_2$, then $H\subset K$.
I have been trying to prove this, but I am stuck, here's my try:
Suppose $h\in H$. So, we have $hg_1\in Hg_1\subset Kg_2$. So, $hg_1\in Kg_2$.
$\implies hg_1=kg_2$ for some $k\in K$.
So, $h=kg_2g_1^{-...
- 115
0
votes
2
answers
28
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Help with RSA cryptography
I need to find c of the encryption of m = 100 using RSA given " p = 89, q = 101 and e = 7 ". Then, i need to decypher the obtained c
I have already calculated n = p * q, which is 8,989 and φ(...
0
votes
0
answers
14
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The normalized unit group using GAP.
I want the structure of The normalized unit group using GAP for the group algebra $FD_{30}$, where $F$ is a finite field with characteristic $3$ and $D_{30}$ is the dihedral group of order $30.$ I ...
- 5,630
-3
votes
0
answers
27
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Prove that if a and b are both quadratic residues modulo p, then the quadratic congruence $ax^2 = b\bmod p$ is solvable. [closed]
Let p be an odd prime number. Prove that if a and b are both quadratic residues modulo p, then the quadratic congruence $$ax^2 = b\bmod p$$ is solvable. What happens if a and b are both quadratic ...
- 1
0
votes
0
answers
9
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Rewriting the Sum of indexes i and j across the ordered real numbers x(1), x(1) ... x(n) with a modulus consideration
Question is in image (sorry I don't know how to do the type set)
My attempt is halfway here and I got stuck.
Given LHS
= Sum (i=1 to i=n) for { Sum (j=1 to j=n) [x(i) - x(n)] where [u] denotes ...
1
vote
0
answers
20
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Degree of morphism between fibers
Let $k$ be a field, $X$ and $Y$ be geometrically integral projective $k$-schemes of finite type, and let $S$ be a $k$-scheme, such that $X$ and $Y$ are $S$-schemes as well. Consider an $S$-morphism $f:...
- 349
-1
votes
0
answers
18
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Transition Probability of Orstein-Uhlenbeck Process using Girsanov's Theorem.
How I can find the weak solution and transition distribution of Orstein-Uhlenbeck Process using Girsanov's Theorem. More specifically, suppose that
$$
dx_t = -\lambda \:x_t\:dt + \sigma\:d\beta_t,
$$
...
0
votes
0
answers
12
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Trying to show a claim about fibrations between smooth maps
I am trying to show that:
Any locally trivial fibration is a quotient map. (Definitions below).
My attempt:
First, let $U ⊂ Y$ be an open set. We need to show that $p^{-1}(U)$ is open in $X$. By the ...
- 497
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votes
1
answer
22
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Quotient of an abelian group by torsion subgroup provides a characteristic free abelian subgroup?
I have a follow up to this question: Rank of the quotient of an Abelian group by its torsion part?.
So my understanding is given a finitely generated abelian group $G$ and its torsion subgroup $T_G$ ...
- 11
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0
answers
22
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Given a tensor $T$, find a symmetric and an anti-symmetric tensor such that $T = T_a + T_s$
I am writing this because I think my professor made a mistake, but I do not feel confident enough in what I know to tell him. Yesterday we took a multilinear algebra test, and we were given a 2-...
- 147
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0
answers
29
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Algebra generated by a countable set is also countable
Let $M\subseteq\Omega$ be a countable set. I now want to show, that the Algebra $\mathcal{A}$ generated by $M$ is also countable.
Properties of an Algebra:
\begin{equation}
\mathcal{A}\neq\emptyset
\...
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-1
votes
0
answers
17
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Conformal transformation of entire complex plane without real axis
I am trying to find conformal transformation of entire complex plane without real axis ( C/{Im z =0}) to upper half plane (Im w >0). Any help would be appreciated.
- 1
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0
answers
9
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How to show the dual space of a normed space constructed by a sigma-finite measure space.
My question is as follows:
Given a $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$, let $$X=\{\{f_n\}_{n\in\omega}: f_n\in L^p(\Omega,\mu),\sum_{n=1}^\infty \|f_n\|_p^p<\infty\}$$And let the ...
- 63
1
vote
1
answer
44
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Solution of $u_{xx} + c^{2}u = 0 $ in terms of sinh.
Let $ c \in \mathbb{C}$ and
$$
u_{xx} + c^{2}u = 0 \ \ \text{in} \ \ (a,b)
$$
with $u(b) = 0$. Why
$$
u(x) = u(a)\dfrac{\text{sinh}(c(x-b))}{\text{sinh}(c(a-b))} ?
$$
I know the above equation can be ...
- 105
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votes
0
answers
23
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Equivalent definition of geometric vector bundles over schemes
The following is Hartshonre's definition of geometric vector bundles:
Let $Y$ be a scheme. A (geometric) vector bundle of rank $n$ over $Y$ is a scheme $X$ and a morphism $f:X\to Y$ together with ...
- 381
4
votes
1
answer
47
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How to manually infer that integral result?
I would like to simplify the formula manually as follows:
$$
\int_0^a{\sin \left( \frac{n\pi}{a}x \right) \frac{1}{x}\sin \left( \frac{m\pi}{a}x \right) dx}
$$
Discuss the cases where (1) $n=m$ and (2)...
- 151
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0
answers
17
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Arrangements of two sets of $n$ objects such that at any point, at least as many of the first set has appeared as the second set
Twenty children are queueing for ice cream that is sold at R5 per cone. Ten children have an R5 coin, the others want to pay with an R10 bill. At the beginning, the ice cream does not have any change. ...
0
votes
0
answers
13
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Relationship between steering angle and turning angle from centre of mass of a car
I'm aware of the Ackermann Steering Angle, however, I still struggle to come up with an equation that model the relationship between $\dot{x}, \dot{y}$ (the horizontal and vertical velocity of the ...
- 35
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0
answers
28
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How can I show that these two ellipses are the same?
Through testing, I have convinced myself that these two shapes are identical:
$$(x,y) = \frac{1}{\sqrt{a\cos^2\theta + b\sin^2\theta}}( a\cos\theta, b\sin\theta)$$
$$r(\phi) = \sqrt{\frac{ab}{a\sin^2\...
- 327
3
votes
0
answers
31
views
Converse of Dynkin's formula
It is a well known (see of theorem 19.21 of https://link.springer.com/book/10.1007/978-1-4757-4015-8 ) that any Feller stochastic process $X$ with infinitesimal generator $(\mathcal{L},\mathcal{D})$ ...
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-2
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0
answers
26
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For what values of 𝑝 p is the series convergent? [closed]
question is in this link
$$\sum_{n=1}^∞ \frac{(-1)^n(ln(n))^p}{n^2}$$
I have tried alternating series test and its conditions which the an part should be bigger than 0 and an+1 <= an and that it ...
- 1
-2
votes
0
answers
14
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Euclidean geometry + affine transformations [closed]
Good afternoon, I don't understand this exercise can anyone help me please?
Show the following are equivalent for any map F:E^n → E^n:
F is an affine transformation in some coordinate system,
F(ax + ...
- 1
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votes
2
answers
38
views
Open subset in $\Bbb R$ defintion question
Definition: A set $A \subseteq \Bbb R$ is said to be open if $\forall a \in A, \exists \epsilon >0$ such that $V_\epsilon(a) \subseteq A$.
Question: why there is $V_\epsilon(a) \subseteq A$ in ...
- 1
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votes
1
answer
26
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Homeomorphisms of unit squares satisfying properties
We were given the following problems in our topology course as fun, challenging problems (they are not homework, and we are incouraged to discuss with others)
Given $I=[0,1]$,
Problem 1: Find a ...
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0
answers
26
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Finding Burton's elementery number theory tough, any lower level book suggestion
I am studying number theory from David Burton's book elementary analysis and second chapter on primes, in that exercise I had no idea how to start attempting questions on primes. The first chapter was ...
- 103
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0
answers
10
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Prove if $S \subseteq Ass_N(M)$ then their exist a submodule $N$ such that $Ass_R(N)=S$.
$R$ is a commutative ring and $M$ is a $R$-module.
I am trying to prove,
If $S \subseteq Ass_R(M)$ then their exist a submodule $N$ such that $Ass_R(N)=S$.
I want use Zorn lemma to prove this problem. ...
- 137
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votes
1
answer
39
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Prove that $\operatorname{adj}(AB-BA) = \operatorname{adj}(AB) - \operatorname{adj}(BA)$ then $A, B \in M_{22}$
Prove or Disprove :
$$\operatorname{adj}(AB-BA) = \operatorname{adj}(AB) - \operatorname{adj}(BA)$$
where $A$ and $B$ are arbitrary $n \times n$ matrices, and $\operatorname{adj}$ is the adjugate(...
- 1,234
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votes
0
answers
22
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Bad reduction of normalization of singular curve $5y^2=x^4-17$
Let $C$ be a projective curve defined by $5y^2=x^4-17$ over $\Bbb{Q}$ and $C'$ be its normalization(N.B. $C$ is singular at infinity).
I want to determine in which prime $p$, $C'$ has bad reduction.
$...
- 4,275
1
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0
answers
25
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Quick exponentiation of bit matrices
Is there a method for quickly rising to a power a matrix with only 0s and 1s? I am aware of the diagonalization method. However, it is general and requires a lot of work. Due to the constraint I ...
- 996
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0
answers
33
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Use cases for $L^p$ and $l^p$ spaces where $p\neq 1,2,\infty$
Soft question: $L^1,L^2,$ and $L^\infty$ spaces all have many practical uses and an easy intuition behind them (Along with the $l^1$, etc. versions). Just for visualization, I was playing around ...
- 16.2k
-1
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0
answers
27
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Proof related to Chebyshev polynomials
Consider x_j (j=0,1,....,n) the roots of T_n+1(x)=2xT_n(x)-T_n-1(x). Prove that for x belonging to [-1;1], the maximum of the absolute value of the multiplicand of (x-xj), j=0,...n equals to 1/2^n.
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1
answer
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Given a $\sigma$-algebra $\mathcal{S}$, is $\mu:\mathcal{S}\to 0$ a valid measure?
Definition A real-valued function $\mu$ defined on a $\sigma$-algebra $\mathcal{S}$ is a measure if:
$\mu(\emptyset)=0$;
$\mu(A)\geq0$ for all $A\in\mathcal{S}$;
$\mu(\cup_k A_k)=\Sigma_k\mu(A_k)$ if ...
- 508
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votes
1
answer
7
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CGAL: rotate Nef Polyhedron
CGAL gives an example of how to rotate a Nef_polyhedron 90 degrees around the x-axis (https://doc.cgal.org/latest/Nef_3/Nef_3_2transformation_8cpp-example.html). I am not exactly sure what the ...
- 19
4
votes
0
answers
73
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What is elementary mathematics?
In Tao's exposition What Is Good Mathematics, he refers to the proof of Szemerédi's Theorem as "elementary" but at the same time "remarkably intricate and not easily comprehended". ...
- 41
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votes
0
answers
7
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Techniques to analyze locally ranked choices
Suppose that we have, say, 100 menus $M_1,\ldots,M_{100}$, with each menu having 4 choices $C_{i1},\ldots,C_{i4}$. Each choice is a vector of length $k$ and we can think of the components of the ...
- 10.3k
1
vote
0
answers
27
views
What is the density of integers whose prime factor exponents are in order?
Let $n = p_1^{a_1} p_1^{a_2} \cdots p_{\omega(n)}^{a_{\omega(n)}}$ be the prime factorization of $n$ for some prime $p_1, p_2, \ldots p_{\omega(n)}$ and exponents $a_1, a_2, \ldots, a_{\omega(n)}$. If ...
- 16.4k
0
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0
answers
29
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How do we solve this two variable equation (involving periodic functions)?
Is there a way to solve these sort of two variable system of equations:
$$x^2((\sin(Ax))^2+(\sin(Bx))^2) + My((\sin(Cy))^2+(\sin(Dy))^2) - E=0$$
where $A, B, C, D, E, M \in \mathbb{R}$. I could not ...
- 29
-1
votes
1
answer
25
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Can an unbounded series of functions converge pointwise to a bounded function?
I’m trying to find an example of a sequence of unbounded functions $f_n: [0,1] \rightarrow \mathbb{R}$ which converge pointwise to a bounded function $f: [0,1] \rightarrow \mathbb{R} $.
I can find ...
- 37
-2
votes
0
answers
15
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Curve 25519 and prime-order Ristretto group: given different group elements is it possible to get the same curve point? [closed]
Given two random elements $x_1$, $x_2$ from the prime-order Ristretto group and a Montgomery point y from Curve 25519, is it possible that values $s_1 = y^{x_1}$ and $s_2 = y^{x_2}$ will end up the ...
- 669
1
vote
1
answer
21
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Prove that any orientation preserving isometry preserving a symmetric, positive definite bilinear form belongs to $SO(n)$
Let $V = \mathbb R^n$, and $Q : V\times V\to \mathbb R$ be a symmetric and positive definite bilinear form. I define $SO(n)$ to be
$$ SO(n) := \left\{ A \in SL_n(\mathbb R) ~|~ A^T A = I_n \right\}. $$...
- 153
0
votes
1
answer
20
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For a distributive lattice, show $x ~\hat{\land}~ a = y ~\hat{\land}~ a$ and $x ~\hat{\lor}~ a = y ~\hat{\lor}~ a$ imply $x = y$.
Let $\hat{\land}, \hat{\lor}$ be binary operators denoting the infimum and supremum of two elements in a poset. I was given the following problem.
For a distributive lattice $\langle ~L, ~\hat{\lor}~,...
- 2,650
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votes
0
answers
19
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Terminology for collection of subsets of $G = \cup_i G_i$ (a partitioning) of subsets of $G$ which contain exactly one element of $G_i$ for any i
Let $G_1,\ldots,G_k$ form a partitonining of a finite set $G$, and let $\mathfrak S$ be the collection of subsets of $G$ which contain exactly one element from each $G_i$, i.e
$$
\mathfrak S := \{S \...
- 9,041
-1
votes
0
answers
11
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Finding Equation of Linear Transformation
Suppose the matrix representation of $T$ with respect to the ordered bases $B_1=\{(−2,1,0),(−2,0,1)\}$ for $V_1$ and $B2=\{(1,0,−2),(0,1,−2)\}$ for $V_2$ is given by $[(1,0),(0,0)]$. What's the ...
