Newest Questions
1,677,671 questions
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Where and when are forms actually mandatory for differential geometry?
In the books that I've read (primarily Lee's Introduction to Smooth Manifolds, Tu's Introduction to Manifolds, and Arnol'd's Mathematical Methods of Classical Mechanics) I've seen cases where the ...
- 1,300
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3
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Wasserstein distances and weak* convergence proof
I'm looking at Figalli's invitation to optimal transport for a proof that the Wasserstein distances metrize weak convergence but am confused about a step.
I've put his proof of the direction in this
...
- 33
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votes
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answers
6
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Help identify 16-bit PRNG algorithm
I am trying to identify a 16-bit PRNG for which I have the whole sequence (period is 51445).
So far I have managed to identify that the algorithm does a bit swapping followed by an XOR:
...
- 141
1
vote
0
answers
8
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Proof of Jordan theorem for polygons
I am trying to prove the Jordan curve theorem for polygons without using topological notions, but only with the axioms of incidence and the axioms of order. In particular, I need to prove the ...
- 189
-1
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answers
8
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Pattern Analysis
I am stuck with couple of pattern finding questions to which i know solution of but not sure how to get it
Question 1-
4 , 5 , 7 , 8 = 168
9 , 3 , 7, 6 = 175
7 , 3 , 12 , 11 = ?
Answer-
231
Question ...
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7
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Intuition for stationary distribution of Markov Chains
Why aren't the matrix elements of the stationary distribution equal to weighted average of incoming probability?
For example, take this Markov chain:
In matrix form $\boldsymbol{P} = \begin{pmatrix} ...
- 81
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8
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Show that orthogonal matrix maps to Borel $\sigma$-algebra
Let be $L:\mathbb{R}^d\to\mathbb{R}^d$ an orthogonal matrix and $\mathcal{B}(\mathbb{R}^d)$ the Borel-$\sigma$-algebra. Show that if $A\in\mathcal{B}(\mathbb{R}^d)$ then $L(A)\in\mathcal{B}(\mathbb{R}^...
- 4,799
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25
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What are the modulus and the argument of a complex number equals to its own logarithm?
I’m looking for a complex number $z=re^{i\theta}$ equals to its own logarithm, i.e such that
\begin{equation}
z= \log z
\end{equation}
If so, what relations should satisfy the modulus $\vert z \vert=r$...
- 305
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1
answer
23
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Drunk Passenger Follow-up
A line of 100 airline passengers is waiting to board a plane. They each hold a ticket to one of the 100 seats on that flight. For convenience, let's say that the nth passenger in line has a ticket for ...
- 1,204
3
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1
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Is the inverse of an orthogonality-preserving bijection orthogonality-preserving?
Let $V,W$ be finite-dimensional vector spaces of a field $F$ equipped with non-degenerate symmetric bilinear forms $\langle \cdot,\cdot \rangle_V$ and $\langle \cdot, \cdot \rangle_W$. We call a map $...
- 23.4k
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27
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Does sequential compactness guarantee every convergent subsequence in S converges to a point in S
Let $E,F \subseteq \mathbb{R}^d$ be compact sets. Define
$$ d(E,F) := \inf_{x \in E, y \in F} |x - y|$$
Prove that there exists points $\hat{x} \in E,\hat{y} \in F$ such that
$$ d(E,F) = |\hat{x} - \...
- 235
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23
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polynomial $F$ divides $G$ if and only if the same holds for their homgenisations?
Let $F, G$ be polynomials in $k[x_1, .., x_n]$. Let $\beta(F)$ denote the homogenisation of $F$,
$$
\beta(F) = x_0^{\deg F} F( x_1/x_{0}, ..., x_n/x_0 ).
$$
Is it the case that $F|G$ if and only if $\...
- 2,953
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answers
7
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Expectation of MST of a fixed CRD model [closed]
Show that the expectation of mean square treatment of a fixed completely randomized design model is equal to $$\sigma^2+\frac{n\sum_{i=1}^{a}\tau_i^2}{a-1}$$
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23
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Show that the inclusion $\phi$ from $l^1$ to $c_0$ is not a topological homomorphism
Let $l^1$ be the space of absolute summable sequences, that is $(x_n)_{n \in \mathbb{N}}$ such that $\forall n \in \mathbb{N}, x_n \in \mathbb{C}$ and $\sum_{n=1}^{\infty} |x_n| \lt \infty$.
Let $c_0$ ...
- 345
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21
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Defining topological properties of functions without homeomorphisms
I started writing this thinking it was going to be more vague, but I ended up kind of half-attempting to answer my own question in the process and now my main question is whether or not my half-...
- 251
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29
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Compute the integral $\int_{-\pi}^{\pi} \frac{\cos^4 x + \frac{x \sin x}{1 + \cos^2 x}}{1 + e^{-x} } dx$ [closed]
Compute the integral
$$\int_{-\pi}^{\pi} \frac{\cos^4 x + \frac{x \sin x}{1 + \cos^2 x}}{1 + e^{-x} } dx$$
This is one of exercises in my real analysis textbook, and I am frustrated by it since I ...
- 31
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answers
13
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Minimize the result of matrix multiplication with a linear cost function
$\mathbf{A}$ is a square invertible matrix of order $n$. $\mathbf{y}$ is an $n$-vector (column orientation) where some elements are given and the rest are unknown. Find the unknown values of $\mathbf{...
- 131
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1
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How to "easily" calculate the limits and sections of convolution integral?
We started recently talking in my signal processing class about the convolution integral, and in theory, it sounds easy enough but now after a few exercises I realize I either don't know the technique ...
- 205
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answers
31
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Alternative proof of the Strong Law of Large Numbers
Wikipedia states that the Three Series Theorem together with Kronecker's Lemma can yield an alternative proof of the Strong Law of Large Numbers. I belive a proof like this must first use the Three ...
- 2,681
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20
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Different $\sigma$-algebras on the space of Borel probability measures on a metric space
This question is intended to distill the core components of this sister question on MathOverflow for increased visibility and transparency.
Let $X$ be a metrizable topological space and $\mathscr B_X$ ...
- 23.7k
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48
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Proving $||Cesaro sum|| \leq ||f||$
How do I show the sup norm of the Cesaro sum of the Fourier partial sums is less than or equal to the sup norm of f i.e.
(Sorry I'm new to StackExchange) I have the integral form of the Cesaro sum ...
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votes
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answers
10
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Involution of second kind in semisimple algebras
On page 20 of the book of Involutions, I have read something that I do not understand. Let $B$ be a simple $F$-algebra with centre $K$ and an involution of the second kind $\tau$.
The author said: &...
- 2,596
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11
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Each creative set is contained in some simple set
Prove that each creative set is contained in some simple set(Exercise 6.2.8 of Computability Theory of Barry Cooper)
These are the definitions:
$W_e = \{x| \phi_e(x) \ halt\}$ and $\phi_e$ is e-th ...
1
vote
3
answers
49
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Help with proving that a function is increasing and continuous
I'm working on an exercise and could use some guidance. The problem is as follows:
Let $f$ be a real-valued function that is continuous on $[0,1[$. I need to show that the function $\varphi$ defined ...
- 179
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12
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N-colorability bounds on grid maps
I currently have the problem of coloring a map with regions that are defined on a square grid.
Every region is a connected set of cells on this grid, such that the cells connect either horizontally or ...
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1
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65
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Combinatorial or geometric proof of formula for sum of consecutive squares? [duplicate]
The fact that
$$1^2 + 2^2 + \cdots + n^2 = \frac{n(n+1)(2n+1)}{6}$$
can be proven by a straightforward application of mathematical induction involving grinding through a bit of basic high school ...
- 2,487
2
votes
1
answer
43
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Does Forcing preserve $\mathsf{DC}$?
As title states, I'm curious whether plain-old Forcing is able to preserve $\mathsf{DC}$ (i.e. whether $\mathsf{DC}$ holds for a generic extension $M[G]$ of some previous model $M$ for which it did ...
- 624
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votes
2
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45
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Classical probability
I'm a student in high school:
I have come across the question
“Jacob buys $10$ packets of detergent of brand 'ALICE' having a ticket with a letter of brand name ...
- 29
2
votes
2
answers
43
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Is it valid to use an "infinite" number of universal/existential instantiations in a proof?
I was thinking about the statement that the sequential definition of a limit of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ and the "epsilon-delta" definition are equivalent. ...
- 2,322
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23
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How to deal with a complex function that has infinitely many poles and residues?
How do we solve a complex function that has infinitely many poles and residues when trying to do complex analysis on a real function?
For instance the answer to my question here. I looked it up in my ...
- 289
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answers
21
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Quaternionic groups $GL(n, \mathbb H)$ and $Sp(p,q)$
Let $\mathbb H$ be the quaternions. The general linear group $GL(n, \mathbb H)$ is defined as
$$GL(n, \mathbb H) = \{A \in GL(2n, \mathbb C) \mid JA = \bar{A}J \}$$
where
$$J = \begin{pmatrix} 0 & ...
- 305
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vote
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answers
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Determine the Galois group of a polynomial over $\mathbb{C}(x, y)$
Let $K = \mathbb{C}(x, y)=\mathrm{Frac}\left(\mathbb{C}[x, y]\right)$ be the field of rational functions with variables $x$ and $y$ over $\mathbb{C}$. Then
$$
f(R) = x \left( (1+R)^n - 1 \right) - y R ...
- 562
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33
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Proving the homotopy equivalence between the real projective plane and the adjunction space made of $D^2$ and $S^2$
The adjunction space is given by $S^2 \cup_f e^2$ where $f:S^1 \to S^1, z \mapsto z^2$ maps antipodal points of $S^1$.
Am I right to picture this as the sphere with the boundary of the disk glued to a ...
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-1
votes
0
answers
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Let the digits $a, b, c$ be in A.P. Nine digit numbers are to be formed with following restrictions. How many such numbers can be formed?
Let the digits $a, b, c$ (non-zero and distinct) be in A.P. Nine-digit numbers are to be formed using each of these three digits thrice such that three consecutive digits are in non- constant A.P. at ...
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answers
31
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A book suggestion on additive number theory
I'm looking for a book on additive Number Theory, which has somewhat introductory spirit. In particular, I'm interested in Erdös-Ginzburg-Ziv theorem. I prefer a book that includes exercises. Thanks!
- 5,036
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answers
13
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Tutte polynomial for the wheel graph on 6 vertices?
Say that you are sitting an exam and the first question of a problem is to find the Tutte polynomial of the wheel graph $W_6$ ( joining a vertex with the cycle $C_6$ ). The other questions being about ...
- 1,018
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Partial differential equation. Solve the problem using Green's function
Please help me find a way to decide which direction to move in.
Task:
\begin{equation}
\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} + f(x, t)
\end{equation}
\begin{equation}
u(x,t)...
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answers
17
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Satisfying many congruences with few points
This can be thought of as part II of another question I posted recently. I am interested in the following problem:
For a given positive integer $S$, consider the following set of conditions:
$$\...
- 561
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votes
1
answer
149
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What makes people say $\pi^{{\pi}^{{\pi}^{\pi}}}$ is an integer? [closed]
I know that $\pi^{{\pi}^{{\pi}^{\pi}}}$ being an integer is just a theory, but where does the theory come from? What evidence is there that tells us it could be an integer?
- 33
1
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Geometric interpretation of a theorem about localizations of Noetherian normal domain
Recall that a normal domain is an integral domain which is integrally closed in its field of fractions. In Eisenbud's Commutative Algebra with a View Toward Algebraic Geometry, he proves that if $A$ ...
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11
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ideals in a (generalized) number ring are cofinite
Let $R$ be a domain whose fraction field is a number field $K$. If $p$ is a prime number, why is $R/pR$ finite?
This mentioned in the paper "Arithmetic in Number Rings" (where by number ...
- 3,275
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answers
18
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Is it possible to make a 'sub' function for variables? [closed]
Non-relative sub set functions for a variable, where if you multiply them, you get the original variable. A theory of sub functions, such that basics variables like time or distance, a value, be ...
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answers
29
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Is there a name for the uncountable-chain-with-countable-supremum paradox?
[In this post, the power set of a set $S$ is denoted $\mathcal{P}(S)$. A "chain in $\mathcal{P}(S)$" is a collection of subsets of $S$ that is totally ordered by inclusion.]
An obvious fact ...
- 387
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25
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Expected Gain in a Repeated Probability Experiment with Success and Loss Conditions
I have a probability problem and am trying to calculate the expected gain from repeating an experiment with a probability of success $p$ in each trial. Here's the setup:
We start with $S_0$ points.
...
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answers
28
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What metric spaces have the property that, if we go nearer to two endpoints, we also go nearer to inner points?
Let $(X,d)$ be a geodesic metric space (informally, a space that contains a "shortest path" between any two points). Let $A, C$ be two points in $X$, and let $B$ be any point on a geodesic ...
- 11.1k
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39
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About open and closed sets of a topological space [closed]
Let $X$ be a topological space.
Let $F_1$ and $F_2$ be two closed and disjoint sets.
Then there exists two open and disjoint sets $A_1$ and $A_2$ such that $F_1 \subseteq A_1$ and $F_2 \subseteq A_2....
-2
votes
0
answers
27
views
Matrix decomposition
I have a matrix $K_{3x3}$ that I would like to convert into a matrix of the form:
$$
L' =
\begin{bmatrix}
a & 0 & 0 \\
b & 1 & 0 \\
c & 0 & 1 \\
\end{bmatrix}
$$
by ...
- 863
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votes
0
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18
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What is the minimum tiling probability before it becomes most likely that the plane will be tiled? [closed]
I'd be surprised if nobody thought to ask this question before I did, so there may be an answer out there already. I just have no idea what to look up because of the awkward phrasing of the problem. ...
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votes
1
answer
31
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Convex sets in general metric spaces
A set of points in a Euclidean space is called convex if it contains every line segment between two points in the set. An equivalent definition is:
a set $C$ is convex if for every two points $x,y \...
- 11.1k
-1
votes
2
answers
74
views
Is the identity element well-defined?
I was trying to find an identity element of the following structure:
Set: $S =\{ (k/2) + 1\mid k\in\Bbb Z\}$
Operation: $a*b = 2ab - 2a - 2b + 3$.
Identity element, when calculated, is $3/2$, which ...
- 23