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If $c$ is a singular $1$-cube in $\mathbb{R}^2 \backslash \left\{ (0,0)\right\}$ with $c(0)=c(1)$, show that there is an integer $n$ such that: $c-c_{(1,n)} = \partial (c^2)$ for some $2$-chain $c^2$.
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I am trying to work through an example from Jordan and Smiths book Nonlinear Ordinary Differential Equations. It's example $6.1$ on page $195$. The question reads:
Obtain an approximate solution of the family of equations $x^{\prime\prime} + x = ex^3$
with $x(e, 0) = 1$, $x^{\prime}(e,0) = 0$ and error $O(e^3)$ un... |
$$\lim_{\epsilon \to 0} \int \frac1{x^{1+\epsilon}} \mathrm dx$$
How should I go about evaluating this integral?
Does this integral converge to $\log_e x $ or to something else?
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If $f(x) = \sum_{j=1}^n c_j 1_{A_j}(x)$ ie a simple function. Why is $$\prod_{j=1}^n \exp(t[\exp(i(u,c_j))-1]\mu(A_j)) =\exp[t\int_A[\exp(i(u,f(x)))-1]\mu(dx)].$$ The notation $(u,c)$ is dot product, $1_{A}(x)$ is the indicator function and $i=\sqrt{-1}$.
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We know that the Mandelbrot set is derived from the iterations of z^2 + c.
Do anyone know something about magnet Mandelbrot? I found it in the software UltraFractal, and it is much more beautiful than the original Mandelbrot, in my opinion.
Do you know anything about it? What is it iterating?
EDIT: Here's a picture of ... |
In a boolean algebra, $0$ (the lattice's bottom) is the identity element for the join operation $\lor$, and $1$ (the lattice's top) is the identity element for the meet operation $\land$. For an element in the boolean algebra, its inverse/complement element for $\lor$ is wrt $1$ and its inverse/complement element for $... |
These operators are written in different forms in Cartesian,
cylindrical and spherical coordinates. For instance, in spherical
coordinate system, one has
$$\nabla \cdot \overrightarrow{F}=\frac{1}{r^{2}}\frac{\partial }{\partial r}
\left( r^{2}F_{r}\right) +\frac{1}{r\sin \theta }\frac{\partial }{\partial
\theta }\left... |
Suppose I have a binary vector $v$ that is copied twice by two separate machines $M$ and $N$, resulting in two new vectors $x$ and $y$.
Both machines are faulty in the sense that they both might fail to copy each character correctly. More formally, $M$ has an associated vector $e_M$ of length $|v|$, where $(e_M)_i$ is... |
In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, where $U$ is the universal class. So the (unary) intersection of the empty set is the class that contains all sets as elem... |
While proving that some normed spaces were complete, two questions came to my mind. They relate the topological and the metric structures induced by a norm.
Is it possible to find two equivalent norms $\|\cdot\|_1$ and $\|\cdot\|_2$ on a vector space $V \ $ such that $(V \ ,\|\cdot\|_1)$ is complete and $(V \ ,\|\cdot... |
I have an inequality:
$$ n \ge C\log^k n \times (A\log(n^r\log^l n) + B)$$
I want to turn it into an inquality
$$n > F(A,B,C,k,l,r)$$
that implies the first inequality, while making it $F$ as tight as possible.
How can I do that?
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Suppose I'm given a very slowly converging sequence $\sum_k a_k$. In the literature, the Levin u-transformation is mentioned as a good universal technique for convergence acceleration.
I have difficulty in understanding how this method actually works and what the transformation actually is. Can anyone help?
After und... |
Consider the function $Pt(n)$. It tells us how many primitive Pythagorean Triples there are (below $n$) when any argument $n \in \mathbb{N}$ is plugged in. Is there an 'exact formula'; i.e. an elementary function of even a combination of known special functions like the Gamma and Error Function, that describes $Pt(n)$ ... |
Suppose I have a variable $s$ that has a geometric distribution with success parameter x. So the probability of success on trial $s$ is
$p_s = (1 - x)^{s - 1} x$,
Consider the following function of $s$
$V(s) = \delta^s$,
where $\delta < 1 $. Then the expected value of this function is given by
$\sum_ 1^{\infty} V... |
I have a homework assignment to find the two smallest probable primes with $12$ digits, where a probable prime is defined as a number such that $a^{p-1} \equiv 1\ (\textrm{mod}\ p)$, where $a = 2, 3, 5, 7$.
I'm obviously not asking for this to be done for me, but I'm having trouble figuring out a way to do anything ot... |
I'm trying to find a factorization for $f(z) = z - c$ in inner and outer functions.
So, first I try to find for what $c$, $f$ is cyclic for the unilateral shift $M_z$. There is a theorem that states that if $A$ is a bounded operator on a Hilbert space $H$ then $x \in H$ is cyclic for $A$ iff zero is the only element th... |
Let $f : {\cal S}_+^n \mapsto \mathbb{R}$ be a function defined as
$f(Q) := {\rm tr} WQ^{-1}W + {\rm tr} Q$
where $W \in {\cal S}_{+}^{n}$ is a symmetric positive definite matrix.
How to compute a subdifferential of $f$ at a point $Q \in {\cal S}_+^n$ for which $f(Q) < \infty$?
Note that $f(Q) = \infty$ for $Q$ such ... |
Let $p(x)$ be a fixed distribution.
Let $A, C > 0$ be constants.
Let $\epsilon > 0$. Can we find an example of a distribution $q_{\epsilon}$ such that $\mathrm{KL}(p||q_{\epsilon}) < \epsilon$, but $E[(\log (p(x)/q(x)))^2] \ge A * \epsilon^C$?
If $p$ and $q_{\epsilon}$ would be a coin toss or a geometric distribution, ... |
If $x$ is an element in a standard convex linear optimization set constrained by $Ax = b, x \geq 0$, then how can I prove $d$ is a feasible direction only if $Ad=0$ and $di \geq 0$ for every $i$ where $xi=0$?
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I am looking for algorithms that compute the inverse of Euler's totient function.
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Construct a subset of [0,1] in the same manner as the Cantor set by removing from each remaining interval a subinterval of relative length $\theta$, $0<\theta <1$.
This is the first statement in a homework exercise. I don't know if my English parsing skills are lacking today or what, but it's unclear to me what this me... |
How do you prove the inequality
\begin{equation*}
|\sqrt[3]{x} - \sqrt[3]{y}| \leq \sqrt[3]{|x-y|}?
\end{equation*}
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I can't seem to understand Generalized Set Operations. Can someone explain it for me? For example, in Example 1.12, suppose k = 3 then set A(of k) = {3, 4, 5, 6 ...}, then how is {3,4,5 ...} = {1,2,3 ...}
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I'm facing a practical problem where I've calculated a formula that, with the help of some programming, can bring me to my final answer. However, the numbers involved are so big that it takes ages to compute, and I think it might not be necessary. I have the following formula,
$\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1... |
California issues license plates in numeric order (if we turn the letters into numbers). I have fun noticing the latest plate I have seen. I am interested in what you can derive from a series of these observations. I understand that sampling from $\{1,2,3...n\}$ the only useful data is the highest value you have see... |
I've been trying to work out the best way to understand why Fourier series converge, and it's a little embarrassing but I don't even know a rigorous proof. Can someone please help put me on the right track to thinking about these issue's in the proper way? I am especially interested any geometric ways to think about ... |
I'm trying to prove "the following generalization of Theorem 5 [ Th.5: if $a|bc$ and $(a,b)=1$ then $a | c$ ], which uses the same argument for its proof" (Sierpinski, The Theory of Numbers): if $a$, $b$, and $c$ are integers such that $b | ac$, then $b | (a,b)(b,c)$.
I haven't been able to prove it without any referen... |
Is there a simple explanation of what the Laplace transformations do exactly and how they work? Reading my math book has left me in a foggy haze of proofs that I don't completely understand. I'm looking for an explanation in layman's terms so that I understand what it is doing as I make these seemingly magical transfo... |
This is probably a silly question, but a couple of people that I have talked to have had different responses.
Does $C_0$ denote the set of continuous functions with compact support or the set of continuous functions which vanish at infinity?
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Consider a candy bag that contains $N=100$ candies. There are only two types of candy in the bag. Say the caramel candy and the chocolate candy. Nothing more is known about the contents of the bag.
Now, you are going to draw (randomly) one candy at a time from the bag until the first caramel appear. Suppose that the fi... |
The field of real numbers has a nice property - for a fixed $n\in \mathbb{N}$, every $r\in \mathbb{R}$ is either an $n$-th power (if $n$ is odd or if $r>0$) or -$r$ is an $n$-th power.
I want to generalize this - let $K$ be a field extension of $F$, then $a\in K$ is almost an $n$-th power if there is $b\in F$ and $c\in... |
Let $G$ be a group with generators and relations.
I know that in general it is difficult to determine what a group is from its generators and relations. I am interested in learning about techniques for figuring out the order of a group from the given information.
For example, I know that if the number of generators exc... |
Show that for any integer $n>1$, all the numbers $n!+2, n!+3, \ldots, n!+n$ are composite (i.e. not prime).
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Given $n$ smooth real functions $f_1, f_2, \dots, f_n$, define a composite function like this:
$$f(x) = \max(f_1(x), f_2(x), \dots, f_n(x)) - \min(f_1(x), f_2(x), \dots, f_n(x))$$
Is it possible to say anything useful in general about the shape of this function?
Intuitively, it seems like $f$ will be at least $C^0$ con... |
In previous question, I asked how one would simplify the following equation for the case where the variables are very big:
$\sum\limits^{k}_{i=m}(N-i)^{k-i}(\frac{1}{N})^k\frac{k!}{(k-i)!i!} \leq a$
This answer was basically to use an approximation like Stirling's formula. Having implemented this with some code, it sti... |
The standard approach for showing $\int \sec \theta \, \mathrm d \theta = \ln|\sec \theta + \tan \theta| + C$ is to multiply by $\dfrac{\sec \theta + \tan \theta}{\sec \theta + \tan \theta}$ and then do a substitution with $u = \sec \theta + \tan \theta$.
I like the fact that this trick leads to a fast and clean deriva... |
I have the following sequences:
$a = [0, 1, 2, 3, 4, 5]\\
b = [1, 2, 4, 7, 13, 26]\\
c = [1, 1, 2, 4, 7, 13]$
I'm trying to reverse-engineer an undocumented API and believe that the values in $a$ can be used to determine the corresponding values in $b$ and $c$. For instance, we could say that third element in $b$ ($4$)... |
Using stereoscopic cameras, I track a certain object through it's path in space. For every frame, I compute it's pose in 3D. I can represent it's pose by either a translation vector + rotation matrix or a 6 element parameter vector (X, Y, Z, roll, pitch, yaw).
The problem is that the output generated by the cameras is ... |
Let $G = \langle x , y \ | \ x^{12}y=yx^{18} \rangle$. I want to continue a discussion here on the normal closure of $y$ in $G$.
How can we determine a group presentation for $N$?
|
Rectangles will all be the same size, circle may vary.
I was discussing this with a friend and he said to just start placing them in the middle and moving outwards but it seems that you could free up more space for rectangles around the edges if you moved that center of mass to one side and maybe put some space betwe... |
Suppose x is the solution to a standard linear programming problem ($Ax=b$, $x>=0$) and the set $S$ is every $i$ where $x_{i} = 0$.
How can I show this is optimal only where
minimize $c'f$ subject to $Af=0, f_{i} \ge 0$, $i$ is in $S$
has a cost of zero.
|
Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to global elements $F(p): 1 \to Y$ (here $1$ is the terminal object of $\mathcal{E}$). Does there exist a (necessarily unique) arrow $f: X \to ... |
I'm playing around with a collection of subsets of $\mathbb{R}^n$ let's call them $X_i$. What I want to know is, is the following condition sufficient for some $X_i$ being measurable?
Almost all elements of $X_i$ are surrounded by a neighbourhood contained in $X_i$
This is motivated by the idea that non measurable sets... |
This may be a really basic question but the typical atan2 method isn't working when I write it in a program.
I need a camera object to face a certain target in 3D space.
I need to calculate the 3 Euler angles (yaw, pitch, roll) for the camera.
How can I do this?
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So let's say you have a curved mirror, $y=y(x)$ with this property:
Whenever a ray of light emanates from the origin, it reflects parallel to the x axis.
Find the equation of the mirror.
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I read that Smale recast classical mechanics in terms of symplectic geometry. I know a bit about classical mechanics but nothing about symplectic geometry.
Are there any writings from Smale on this which I could read?
And also what other books or texts are there which cover this?
Thank you!
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What are some lower bounds (if they exist) on $w^t A w$ in terms of $||w||_2^2$ for $A$ an matrix with all positive values? The lower bound can depend on $A$ and $\|w\|_2^2$. $w$ is $k \times 1$. Note when $A = I$ then we exactly get the squared norm of $w$.
If there are no such values, what in the case that $A$ is pos... |
If I have a bounded set $F$ in $N$ dimensional space and another set $G$ where every element $g$ in $G$ has $h'g=c$ and also must exist in $F$. $H$ is a vector in the $N$ dimensional space and $c$ is any constant $1\times 1$ matrix (scalar). $h$ is a vector of appropriate dimension.
How can I prove every extreme point ... |
I've been trying to simplify this equation:
$Y = (¬AB + ¬C)(¬A + ¬D)$
.. into this equation.
$Y = ¬A¬C + ¬C¬D + ¬AB$
Unfortunately I keep going in circles with expanding and minimizing the booleans. Any tips or advice? Thanks.
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The following is a question from the sample GRE Mathematics Subject Test
found on the ETS website:
Let $M$ be a $5\times 5$ real matrix.
Exactly four of the following five conditions on $M$ are equivalent
to each other.
Which of the five conditions is equivalent to NONE of the other four?
(A) For any two distinct colum... |
Note: This question has been posted on StackOverflow. I have moved it here because:
I am curious about the answer
The OP has not shown any interest in moving it himself
In the Communications of the ACM, August 2008 "Puzzled" column, Peter Winkler asked the following question:
On the table before us are 10 dots,
a... |
Originally I was just going to ask the problem on my practice math contest, which is asking how many ways there are to write a nine-digit number containing each digit $1-9$ so that the first digit is twice as big as the second and the last two digits are odd. (e.g. $846527913$).
I understood that in this case, the numb... |
It's a well known fact in probability theory that if $\lbrace X_i\rbrace$ is a family of random variables, $\phi : [0,\infty) \to [0,\infty)$ satisfies $$\lim_{x \to \infty} \frac{\phi(x)}{x} = \infty \quad (*) $$ and $\sup_i E\phi(|X_i|) < \infty$, then the family $\lbrace X_i\rbrace$ is uniformly integrable. I want ... |
Let $T:X \times X \to \mathbb{R}$ be a continuous bilinear operator defined on a normed linear space $X$ s.t.
$T(\alpha x + \beta y,z) = \alpha T(x,z) + \beta T(y,z)$) and $T(x,y) = T(y,x)$.
Does there exist a constant $C$ s.t. $||T(x,y)|| \leq C$ $||x||$ $||y|| \forall x,y$?
I know that the result is true if $X$ an... |
I'm trying to wrap my brain around adjoint functors. One of the examples I've seen is the categories $\bf IntLE \bf = (\mathbb{Z}, ≤)$ and $\bf RealLE \bf = (\mathbb{R}, ≤)$, where the ceiling functor $ceil : \bf RealLE \rightarrow IntLE$ is left adjoint to the inclusion functor $incl : \bf IntLE \rightarrow RealLE$. I... |
First of all, i'll confess i'm no math geek. I'm from Stackoverflow, but this question seemed more apt here, so i decided to ask you guys :)
Now, i know noone has discovered (or ever will) a Polynomial that generates Prime Numbers. But i've read about Curve Fitting (or Polynomial Fitting) so i was wondering if there wa... |
What is the difference between a complete metric space and a closed set?
Can a set be closed but not complete?
|
Let $g_{ij}$ be the components of a symmetric rank-2 positive definite tensor (metric on a Riemannian manifold). Let ${C^i}$ and ${ \beta ^i }$ be components of a vector field on it, the former of which is a functions of a variable $t$ and let $N$ and $M$ be two positive constants. Now let the following inequality hold... |
What is the continued fraction for $\displaystyle\sum_{i=1}^n\frac{1}{2^{2^i}}$
It seems to be "almost" periodic, but I can't figure out the exact way to express it.
|
I'm still making my way along in Niven's Intro to Number Theory, and the title problem is giving me a little trouble near the end, and I was hoping someone could help get me through it.
Now $x^8\equiv 16\pmod{2}$ is solvable with $x\equiv 0\pmod{2}$, so I assume $p$ is an odd prime. From a theorem earlier in the text, ... |
In measure theory there are three fundamentally related theorems about exchanging limits and integrals: Fatou's lemma, Lebesgue's Dominated Convergence Theorem, and Monotone Convergence Theorem. It is difficult to prove any of these from scratch, but once you have one, the others are easier.
My question is, for those w... |
Let $f: X \rightarrow Y$ be a local homeomorphism with X, Y connected, locally path connected, Hausdorff and with X also compact. Then f is also a covering with finite fibers.
I know how to show that the fibers are finite.
Given that f is a surjection, I know how to show that f is a covering map.
How do I show that f i... |
What is motivation behind the definition of a complete metric space?
Intuitively,a complete metric is complete if they are no points missing from it.
How does the definition of completeness (in terms of convergence of cauchy sequences) show that?
|
Sum of a set of sphenic numbers can't be equal to the sum of any other set of sphenic numbers.
By that I meant, Say S is the set of sphenic numbers. Let S$_1$ $\subset$ S.
Then there is no such S$_2$ $\subset$ S so that,
S$_1$ $\neq$ S$_2$ AND $\sum S_1$ $=$ $\sum S_2$
Question 1 : Is this statement correct?
Question... |
I need to calculate all the numbers in a certain row of Pascal's triangle. Obviously, this is easy to do with combinatorics. However, what do you do when you need to estimate all the numbers in, say, the 100000th row of Pascal's triangle?
Is there any way to estimate the number so that the costly multiplications and d... |
A high-school student named Erna Fekete made a conjecture to me via email three years ago,
which I could not answer. I've since lost touch with her.
I repeat her interesting conjecture here, in case anyone can provide updated
information on it.
Here is how she phrased it. Let $b(0) = 1$ and $b(n)= \tan( b(n-1) )$.
I... |
The Cantor set is uncountable so I expect countably many of them to be able to cover $\mathbb R$, but the set has measure $0$ so countably many of them also has set of measure $0$ and thus can't cover the real line.
Why/where is my intuition broken?
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A perfect number is a number that is the sum of its proper divisors:
$$28 = 1 + 2 + 4 + 7 + 14$$
Is there a name for numbers whose only proper divisors are $1$, $2$, and $N/2$? For example
$6$ is divisible by $1$, $2$, and $3$
$8$ is divisible by $1$, $2$, and $4$
$34$ is divisible by $1$, $2$, and $17$
Is there a name... |
I'm not quite sure what the mathematical term for what I'm asking is, so let me just describe what I'm trying to figure out. Let's say that I have two ordered sets of numbers $\{1, 2\}$ and $\{3, 4\}$. I'm trying to figure out the number of possible ways to combine these two sets into one without breaking the orderin... |
Do you think the following limits are correct?
$\displaystyle\lim_{d\to\infty}\frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{\varphi(N)^d}=0$
$\displaystyle\lim_{N\to\infty}\frac{\sum\limits_{k=1}^{d} {\varphi(N) \choose k} {d-1 \choose k-1}}{\varphi(N)^d}=c$
I plotted the equations and guesse... |
I'm trying to prove using induction that $9$ divides $n^3 + (n+1)^3 + (n+2)^3$ whenever $n$ is a non-negative integer. So far, I have:
Base case: $P(1) = (1) + (8) + (27) = 36, 36$ can be divided by $9$ so the base case is valid
Inductive step: let $P(n)$ be the statement $9$ divides $n^3 + (n+1)^3 + (n+2)^3$. Assume ... |
Let's call $S$ the infinite string that is made by concatenating the consecutive positive integers written down in base $10$.
Thus, $$S = 12345678910111213141516171819202122232425\ldots$$
Any number in $S$ occurs multiple times. The first occurrence of $3$ is in the third position of the series, the second occurrence i... |
This from an exercise in Aigner's book where one has to evaluate $\sum_{k\ge 0} (-1)^k \binom{n}{k}^2$ using sign reversing involutions. When $n$ is odd, the problem is trivial : let $[n] = \{1,2,\dots,n\}$ consider all pairs of subsets $\{ (A,B) \in 2^{[n]} \times 2^{[n]} : |A| = |B| \}$, and $(A,B)$ has sign $(-1)^{|... |
I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable set of reals. "Measurable" involves third-order concepts but I know that there's often a "coding trick" that gets around ... |
We know that the "Harmonic Series" $$ \sum \frac{1}{n}$$ diverges. And for $p >1$ we have the result that the series converges $$\sum \frac{1}{n^{p}}$$ converges.
One can then ask the question of testing the convergence the following 2 Series:
$$\sum\limits_{n=1}^{\infty} \frac{1}{n^{k + \cos{n}}}, \quad \sum\limits_{... |
Let $X_1,....,X_n$ be some positive random variables.
Is $A_{ij} = \frac{E[X_i*X_j]}{E[X_i]E[X_j]}$ a positive (semi) definite matrix?
(note that $B_{ij} = Cov(X_i,X_j) = E[X_i*X_j] - E[X_i]E[X_j]$ is PSD because it is a covariance matrix.)
|
Consider four $n$-sided polygons arranged so that their centers are the vertices of a square. The square is exactly as large as the diameter of the smallest circle enclosing each polygon. The polygons do not overlap. Polygons are oriented so that each one has either exactly two sides in common or exactly one side and o... |
Can anyone see a nice way to prove the following for $0\le x \le 1$?
$$\sqrt{1-x^2}\ge \operatorname{erf}(\sqrt{-\log x})$$
$\operatorname{erf}$ is defined as
$$\operatorname{erf}(z) = \frac{2}{\sqrt{\pi}}\int_{0}^{z} e^{-t^2} \, dt$$
|
I approached this exam question the wrong way apparently, help please?
Consider the word "mathematics". In how
many ways can you rearrange all the
letters so that the vowels are paired
and always apart?
|
Probably a simple question and possibly not asked very well. What I want to know is..
In binary, a decimal value of 1 is also 1.
It can be expressed as $x = 1 \times 2^0$
Question:
Why is two to the power of zero equal to one?
I get that two to the power of one is equal to two, or binary 10, but why is to the power of... |
12, 13, 11, 14, 10, 15, 9, 16... 0, maxNum
Start with x then alternate on each side of x until you reach 0 on one side and maxNum on the other. Need to do this in some programming code but can't nail it down. How would you write out the formula for that sequence?
|
What is the current state of the art in summing (where by 'summing', I mean 'representing in terms of already known constants and whatnot') series such as these:
$$\sum_{n=1}^{\infty} \frac{1}{\sqrt[3]{1+7^{n}}}$$
$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n!}}$$
$$\sum_{n=1}^{\infty} e^{-\sqrt{n}}$$
$$\sum_{n=1}^{\infty} \f... |
Consider that I have $3$ parametric equations as function of time and describe the motion of a body in space:
$x = f(t)$
$y = g(t)$
$z = h(t)$
These curves are pretty simple and can be modeled within a certain interval by a 2nd order polynomial.
Is it as simple as performing individual fitting on each function or that ... |
The problem is to find:
$\lim\limits_{x \to 0}\ \dfrac{\sin(\cos(x))}{\sec(x)}$
I rewrite the equation as follows:
$\lim\limits_{x \to 0}\ \dfrac{\sin(\cos(x))}{\dfrac{1}{\cos(x)}}$
And multiply by $\dfrac{\cos(x)}{\cos(x)}$, producing:
$\lim\limits_{x \to 0}\ \dfrac{\cos(x)*\sin(\cos(x))}{\dfrac{\cos(x)}{\cos(x)}}$
An... |
I am learning to work with inequalities. For an example, let's say I want to find the set of $(x, y) \in \mathbb{R}^{2}$ that satisfy: $|x \cdot y| \leq 2$.
|
The following is a lemma I read online, but I don't understand part of the proof.
Let $d$ be the maximum possible order among integers $a$ prime to $p$. Then for any integer $a$ not divisible by $p$, the order of $a$ divides $d$.
Proof: Let $b$ be an element of order $d$, let $k$ be the order of $a$, and let $g=\gcd(k... |
This is a weird logic problem that was posed to me today. This question isn't about the actual logic problem (or the trivial paradox) but in its implications, so read to the end!
A man has been sentenced to death. The executioner comes to his cell on May 31st and tells him that he will die before June 30th. He also tel... |
Given two complex varieties over a common base, I can take their fiber product in the category of varieties, or I can take their fiber product in the category of schemes and then take the reduced subscheme. I have heard, that these two operations yield the same.
Has someone a reference?
|
I'm interested, how is the dual input in a sensor fusioning setup in a Kalman filter modeled?
Say for instance that you have an accelerometer and a gyro and want to present the "horizon level", like in an airplane, a good demo of something like this here.
How do you actually harvest the two sensors positive propertie... |
I'm taking a course on machine learning and came across a model answer which I can't wrap my head around.
The whole model answer is the 2nd exercise here: https://noppa.tkk.fi/noppa/kurssi/t-61.3050/viikkoharjoitukset/T-61_3050_proposed_solutions.pdf
The part I'm specifically having trouble understanding is the last 2 ... |
I encountered a problem recently stated as below:
How many pieces of cheese we can obtain from a single thick piece by making five straight slices? (we can't move the cheese when slicing) If we want to maximize the number of pieces which is denoted by $P(n)$, is there any recurrence relation for $P(n)$, where $n$ is th... |
I often see notation/wording like "let $f(x)$ be a continuous function" or "let $f(x) \in C^0(\mathbb{R})$".
I would say that $\sin$ and $x \mapsto \sin(x)$ are functions, while $\sin(x)$ is a real number.
Are there any correct or best practices in this regard?
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This problem was asked in a test, couple of years ago. Looked interesting!
In a chess tournament, each pair of players plays exactly one game. No game is drawn. Suppose the $i^{th}$ player wins $a_{i}$ games and loses $b_{i}$ games. Show that $$\sum a_{i}^{2} = \sum b_{i}^{2}$$
I can't think of a better title. It would... |
When I first saw the title of this question, I forgot for a moment I was on meta, and thought it was asking about quick, catchy, attractive, informative one-or-two-liner summaries of various fields of mathematics. It turned out not to be… so here’s the question I thought it was!
There are lots of times when one wants ... |
If a quantity is successively increased by 20% for two times and another quantity is successively decreased by 10% for three times, the first quantity is 80% of the second. What percentage is the original first quantity as compared to the original second quantity?
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I'd like to understand the physical meaning of the Feynman's vector potential definition:
$$ A_{m}^{(b)}(x) = e_b \int \delta (xb_{\mu}xb^{\mu})db_m(b), \qquad m=0,1,2,3 $$
(component m of the vector potential of the particle b at the point x)
Here
$\delta$ is the Dirac delta function
$ xb_{\mu}xb^{\mu} $ is the 4-nor... |
A phase change is only possible in a physical system which obeys the laws of statistical mechanics if the infinite series for the partition function of that system converges non-uniformly (i.e. converges point-wise). This is because in order for a phase change to occur, the partition function must converge to two diff... |
A loan for $8\, 000$ must be repaid with $6$ year end payments at an annual rate of $11 \%$. What is the annual payment? I know that the present value of an annuity with end payments is $\frac{1-v^n}{i}$ where $v = 1/(1+i)$. Likewise, the future value is $(1+i)^{n}\frac{1-v^n}{i}$. How do I use this to solve this probl... |
I wanted to prove that $\varphi$ is multiplicative (that for $(a,b)=1$, $\varphi(ab)=\varphi(a)\varphi(b)$) using the following idea:
Define $\varphi'$ by $n = \varphi'(n) + \varphi(n)$.
Multiply out $\varphi(a)\varphi(b) = (a - \varphi'(a))(b - \varphi'(b)) = ab - \varphi'(a)b - a \varphi'(b) + \varphi'(a)\varphi'(b)... |
The Setting
Let $K$ be an Archimedean field. TFAE:
$K$ has the least upper bound property.
Every Cauchy sequence in $K$'s additive group converges.
Now proving that 1 implies 2 is easy, but the other direction is slightly harder. Not that that's a problem. Rather the problem is that I can't see a route that doesn't i... |
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