All Questions
1,570,616
questions
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Should discrete and continuous time models give the same results?
Today i thought a lot about very simple population models, and there are still a few things that bug me.
Consider a simple discrete exponential growth function:
$$ n(t+1) = n(t) + n(t) b - n(t) d = n (...
- 86.4k
0
votes
0
answers
2
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Independence and independent increments
Let $X_1, X_2, X_3$ be three independent identically distributed random variables. Does this mean that
$$ X_3 - X_2, \ X_2 - X_1 $$
are also independent random variables? We can instead inspect the ...
0
votes
0
answers
3
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Determining color of ball from urn with condicion that the two next balls are white
One ball is removed from an urn containing $a$ white and $b$ black balls.
To determine the color of the removed ball, two more balls are drawn. What
is the probability that a black ball has been ...
- 1,650
0
votes
1
answer
12
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Explanation to the "Synergestic relationship" example in Osborne's "Introduction to game theory"
If you happened to read the book, please help me understand the statement in the example 37.1.
How to understand the payoff function written as $a_i(c+a_j-a_i)$? How to interpret such a notation?
...
0
votes
0
answers
5
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Applications of analytic set theory in probability theory
Im studying the theory of analytic sets in polish spaces. In Donald L. Cohn's "Measure Theory" there is mentioned that there are a lot of applications of this in probability theory. But I ...
- 81
-2
votes
0
answers
21
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Probability of random
In bridge each of the players (A,B,C,D) receives $13$ cards. Suppose A and C have $11$ of $13$ spades between them. What is the probability the remaining two spades are distributed so that B and D ...
- 19.5k
-1
votes
2
answers
69
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Find $\lim_{z\to 1+i}\left(\frac{z+2-i}{3}\right)^\frac{z-i}{z-1-i}$
I am trying to find the limit
$$ \lim_{z\to 1+i}\left(\frac{z+2-i}{3}\right)^\frac{z-i}{z-1-i} $$
If we plug in $1+i$ for $z$, we get the following
$$ \lim_{z\to 1+i}\left(\frac{z+2-i}{3}\right)^\frac{...
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votes
0
answers
2
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Trouble with converting the negation of a formula to CNF
I'm trying to convert the negation of the following formula to CNF:
(p → (q → r)) → ((p → q) → (p → r))
These are the steps I am following:
¬((p → (q → r)) → ((p → q) → (p → r)))
¬(¬(¬p ∨ (¬q ∨ r)) ∨ (...
17
votes
3
answers
632
views
Does there exist a bijective, continuous map from the irrationals onto the reals?
Let $\mathbb{P}$ be the irrational numbers as a subspace of the real numbers. $\mathbb{P}$ is homeomorphic to $\mathbb{N}^\mathbb{N}$, which is also called the Baire space.
It is well known, and ...
- 11.6k
2
votes
2
answers
156
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Name for integer "quotient" rounded up (ceiling) instead of down (floor), and its negative or complementary "remainder"
If $168$ cookies (dividend) are shared between $17$ people (divisor), that's almost $10$ cookies each but we're $2$ cookies "short"; alternatively we have slightly more than $9$ cookies each ...
- 1,347
1
vote
0
answers
13
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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
1
answer
10
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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, $R(\mathbf{x})=\frac{\mathbf{x}^{\...
- 17.3k
5
votes
1
answer
42
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$\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
$$
F_\nu(t) :=
\int_0^\infty \frac{u}{2\nu^2} \left( u+ \left( u^2+\nu^2 \right)^\frac{1}{2} \right)
\left(
e^{-u} - e^{-\left(u^2+\nu^2\right)^\frac{1}{2}}
\right) \sin \left( ut \right)...
- 9,064
1
vote
2
answers
2k
views
Finding the slope of a line that cuts an area in half
A region $A$ in the first quadrant is bounded by $y=x^2$, $y=25$ and the $y$-axis. Find the value of $m$ with the property that the line $y = mx$ divides $A$ into two regions with the same area.
CommunityBot
- 1
1
vote
3
answers
69
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How do we evaluate this REALLY tricky integral?
$$\int{\frac{\cos 9x + \cos 6x}{2 \cos 5x-1} dx}$$
The objective is to find the answer in terms of $\sin 4x$ and $\sin x$
I would like to share my attempt and then ask a conceptual doubt as usual but ...
- 550
1
vote
0
answers
12
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Property of Lebesgue Density Point
I have this claim concerning a Lebesgue density point that I need to understand rigorously: Let $0$ be a density point of a closed set $A\subset \mathbb{R}^n$. Then for any $x\in\mathbb{R}^n$ there ...
- 175
-1
votes
1
answer
18
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Probability every fair number before $6$ [closed]
What is the probability of throwing every fair number on a fair dice before throwing a $6$?
- 12.7k
4
votes
3
answers
67
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Evaluate Integral : $\int\;\frac {1}{\sqrt {1-e^{2x}}}dx$
How to evaluate Integral :
$$\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 {...
2
votes
1
answer
15
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Evaluation of a Trigonometric Limit...
$$\lim_{x\to\tfrac{\pi}{4}} \frac{(\cos x + \sin x)^3 - 2\sqrt2}{1 - \sin 2x}$$
I tried solving this question without using L'Hospital's Rule but couldn't find a satisfactory way of approach to solve ...
- 14.1k
0
votes
0
answers
9
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Efficient way to compute average of outer products.
I have a ser of vectors $v_i \in \mathbb{R}^d$, with $i \in \{1, ..., n\}$. I have to compute a weighted average of the outer products of these vectors:
\begin{equation}
\sum_{i=1}^{n} v_i v_i^T w_i
\...
- 11
-2
votes
0
answers
17
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Find all solution of $(n-1)\lfloor x \rfloor + n\lceil x \rceil = (n+1)\{x\}$
Given that $n$ is an integer. I have to find all values of $x$ that satisfy the equation.
- 15k
0
votes
1
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18
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Wrapping my head around hypergeometric distributions
I'm playing a game where I'm allowed to draw three cards without replacement and can select one option from the three. The objective is to walk away with the best possible outcome. Different suits are ...
- 121k
-2
votes
1
answer
47
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Rudin's RCA $5.22$ - An Abstract Approach to the Poisson Integral.
I don’t understand how does the inequality $(3)$ shows that the mapping $f$ $\to$ $f(x)$ is a bounded linear functional on $M$, of norm 1 ?
Any help would be appreciated.
- 841
3
votes
5
answers
212
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How to compute $\lim\limits_{x\to 0}\dfrac{e^x-1}{x}$? [closed]
How does one compute $\lim\limits_{x\to 0}\dfrac{e^x-1}{x}$ without using l'Hôpital's rule or knowledge about the derivative of $e^x$ ? $e^x$ denotes the exponential function with base $e$ (Euler's ...
- 4,901
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votes
0
answers
15
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Upper bound of the dot product of two real-valued vectors
I am interested in proving the following:
$ \sum_{i=1}^n a_ib_i \leq \max \{\sum_{i=1}^n a^2_i,\sum_{i=1}^n b^2_i\}$, where $a_i,b_i \in \mathbb{R}$.
I came up with a geometric proof and a proof by ...
- 1
8
votes
1
answer
72
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Prove that $f(s) = \sup\{x \in [a,b] : f(x) > x\}$, where $f$ is increasing
Given $f: [a,b] \rightarrow [a,b]$ increasing, where $f(a) > a$ and $A = \{x \in [a,b] : f(x) > x\}. \ $ If $s = \sup(A), $ prove that $f(s) = s$.
In order to show that $f(s) = s$, I was trying ...
- 11.6k
-2
votes
0
answers
8
views
Question on subspace/dimension
Is it possible to find two vector subspaces $V$ and $W$ of $R^3$ with $V \cap W=\{\vec{0}\}$ so that $\dim V=\dim W =2$?
2
votes
1
answer
22
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Is a directional derivative both a 1-form and a vector field simultaneously?
I am reading this book, GAUGE FIELDS, KNOTS AND GRAVITY by John Baez and J P Muniain. The author uses the notation $vf$ for the directional derivative along the vector $v$ of the smooth function $f$ ...
- 3,155
2
votes
1
answer
66
views
Possible connection between binary numbers and $\pi$
Here is the Desmos if you want to follow along: https://www.desmos.com/calculator/b4vtzruupm
In messing around with binary numbers, I created a function $f(x)$ in Desmos that generated a list of ...
- 33.4k
0
votes
0
answers
4
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Weierstrass-Erdmann condition in light reflection
I have a calculus of variations problem with discontinuities where the Weierstrass-Erdmann condition seems impossible to satisfy. Hoping someone can figure out what I did wrong.
I consider light ...
- 1
0
votes
0
answers
13
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Analogy of Archimedean Property of powers of rational numbers
For any $a,b \in \mathbb{Q}$, $a > 1, b> 0$, can we find an integer $n$ such that $a^n > b$? (Can we prove this without using any property of real numbers and hence functions like logarithm ...
- 309
2
votes
0
answers
66
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Quotient topology, product topology and subspace topology
Let $S$ be the set $\mathbb R^2\setminus\{(0, y) \mid y \neq 0\}$.
Let $\tau_1$ denote the subspace topology on $S$ induced from the usual topology of $\mathbb{R}^2$.
Now, consider the surjective map $...
- 4,883
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votes
0
answers
3
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Evaluate area bounded by curve S=0, its Latus Rectum and X-axis
Several equilateral triangles with respective sides 1, 2 ,3 ...n (n is a natural number) are placed end to end, starting from origin, one side of each lying on X-axis in the first quadrant. If the ...
- 250
1
vote
0
answers
5
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Doubly transitive action on upper half plane.
Can I transform the triangle in the right hand side to a triangle in the left hand side by a Mobius transform?
If $PSL(2, \mathbb{R})$ acts doubly transitive on $\mathbb{R} \cup \infty$, then do we ...
- 139
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0
answers
12
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Non-convex optimization problem transformation
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
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0
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46
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+50
Basis for a lattice
Let $p\in\mathbb{Z}^+$ and $\mathbf{n}\in(\mathbb{Z}^+)^n$. Consider the set
$$\Lambda_{\mathbf{n},p}:=\left\{\dfrac{a}{p}\mathbf{n}+\mathbf{b}:a\in\mathbb{Z},\mathbf{b}\in\mathbb{Z}^n\right\}$$
Then, ...
3
votes
4
answers
36
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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 ...
- 73.8k
-2
votes
1
answer
77
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Find the minimum length
Given rectangle $ABCD$ where $AB=CD=4$ and $AC=BD=2$. Point $E$ is in middle of $AC$. Point $X$ is somewhere in the line $AB$. Find polygonal chain $EXC$ of minimum length.
How do you find minimum ...
- 13
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votes
5
answers
8k
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Non-Metrizable Topological Spaces
What are some motivations/examples of useful non-metrizable topological spaces? I am trying to get a feel for what parts of math have topologies appear naturally, but not induced by a metric space. ...
- 1,128
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votes
0
answers
14
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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 ...
- 2,512
1
vote
0
answers
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
14
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Let $n\geq 3$. Is there a connected, planar, bipartite graph with $n$ regions and $n$ vertices?
The answer given is that according to a Corollary of Euler’s formula (Corollary 3 Section 10.7), such a graph has at most $2n − 4$ edges. Applying this to Euler’s formula ($r = m − n + 2$), there are ...
- 11.1k
2
votes
1
answer
24
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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 ...
- 139
3
votes
2
answers
167
views
Deducing the Gaussian Formula
How would one prove the following formula: For all $n \in \mathbb{N}$,
$$\sum_{m=0}^n (-1)^m \binom{n}{m}_q = \begin{cases} \displaystyle \mathop{\prod_{k \text{ odd}}}_{1 \le k\leq n} (1-q^k), & \...
- 65.1k
-3
votes
0
answers
38
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$a,b \in \mathbb Z$ and $\frac{ab}{a+b}=2$ How many possible values exist for $a$?
$a,b \in \mathbb Z$ and $\frac{ab}{a+b}=2$
How many possible values exist for $a$?
The question is from my cousin's high school test book. We couldn't figure out how to solve this, any help is ...
- 2,512
-2
votes
0
answers
41
views
Tighter upper bound of $\sum\limits_{k=1}^\infty\frac{\tanh'(\cosh k)}{\cosh(\sin k)}$
How do I find tight upper bounds of $$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 ...
- 3,540
5
votes
2
answers
1k
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Arcwise connected but not connected?
In his book "Geometry, Topology and Physics", Nakahara makes the following claim with regard to topological spaces:
With a few pathological exceptions, arcwise connectedness is practically ...
- 1,128
4
votes
1
answer
157
views
If $X$ is a totally disconnected space, then is $\beta(X)$ totally disconnected?
I know that when $X$ is a normal and totally disconnected space, the Stone-Cech compactification $\beta(X)$ is totally disconnected. But I can't find a counterexample when considering $X$ totally ...
- 1,128
8
votes
1
answer
318
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cardinality of a basis for a topology
Suppose X is a space of cardinality $\le \kappa$. I would like to claim that any topology on X has a basis of cardinality $\le \kappa$. Intuitively it's true since even the discrete topology has such ...
- 1,128
1
vote
4
answers
797
views
The perimeter is equal to the area, i.e. $2a+2b=ab$.
The measurements on the sides of a rectangle are distinct integers. The perimeter and area of the rectangle are expressed by the same number. Determine this number.
Answer: 18
It could be $4*4$ = $4+...
- 124k