instruction stringlengths 22 15.8k |
|---|
I'm not Even sure I'm using the correct terminology here, but I'm helping out my high school daughter with her algebra and was presented with the following rule:
$\sqrt3/\sqrt2 = \sqrt{3/2}$
Accepting that this is true (which a calculator did demonstrate), I should be able to step through a proof using the process for ... |
I am trying to prove the following assertion:
If an algebraic number field $K$ contains a complex root of unity, then the norm of every nonzero element of $K$ is positive.
I think this is supposed to be a simple application of the theory of geometry of numbers, but I can't see how to prove it.
Can someone please help m... |
Is there a way to perform these operations in a particular base b but 10? I can convert it back to base 10 and perform the operation, however, I think it's kinda odd by doing this way.
Thanks,
Chan
|
My professor gave us a puzzle problem that we discussed in class that I could elaborate on if requested. But I interpreted the puzzle and formed a recursive function to model it which is as follows:
$$f(n) = \frac{n f(n-1)}{n - 1} + .01 \hspace{1cm} \textrm{where } f(1) = .01\text{ and } n\in\mathbb{N}.$$
The question... |
Suppose you are given two oriented manifolds with boundary $M$ say $B, B'$ and $\partial B = M = \partial B'$. Identify the boundaries and form $C = B \sqcup_{Id: M \to M} B'$. I want to see why, with the orientation induced by being submanifolds of $C$, $B$ and $B'$ induce opposite orientations to their boundary $M$.... |
Among other things, the Fourier transform maps functions from $L^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$, $L^1(\mathbb{R}^n) \to C_0(\mathbb{R}^n)$ (continuous functions vanishing at infinity), and $\mathcal{S}(\mathbb{R}^n)\to\mathcal{S}(\mathbb{R}^n)$ (Schwartz space, or the space of rapidly decreasing functions).
I'... |
I am trying to show the completeness of R, using the LUB property.
Problem is that I don't know, given a Cauchy sequence,where the limit would come from; I can check if a sequence {an} converges to a specific value, but I don't know how to come up with the limit value that the sequence would converge to.
I imagine as t... |
I have been trying to derive this approximation but have been unsuccessful in doing so. Any help would be greatly appreciated.
|
The Stanford Encyclopedia of Philosophy's article on category theory claims that adjoint functors can be thought of as "conceptual inverses" of each other.
For example, the forgetful functor "ought to be" the "conceptual inverse" of the free-group-making functor. Similarly, in multigrid the restriction operator "ought... |
Let's consider this simple dice game: A coin is faked so it has p chance to land on heads, and 1-p chance to land on tails. Every round costs $1, and gives you $2 if you win (for a total of +$1).
Assume you're starting with $n. What are your odds to "go infinite" - be able to play the game forever? This sounds like Mar... |
Given x is a real number and c is a natural number. Prove that:
floor( floor(x) / c ) = floor( x / c )
When x > 0, I can prove it by let x = n + z where 0 < z < 1.
Then rhs = floor( n/c + z/c ) which equals to floor( floor(x) ) since z/c -> 0.
However, when x < 0, it's wrong when letting x = -( n + z ). Because it alw... |
I'm trying create a function that returns a matrix containing a variable "l" which is an independent variable to be swept for a plot later on.
I would calculate "phi" based on user inputs which include "n" and "d", then I would use "n", "d", and "phi" to find "a", "b", "c", and "d" to create a matrix "m" with. This ma... |
This is a little question inspired from Hartshorne's Geometry, which I've been juggling around for a while.
Suppose that $\Pi$ is the Cartesian plane $F^2$ for some field $F$, with the set of ordered pairs of elements of $F$ being the points and lines those subsets defined by linear equations. Let $\Pi'$ be the associ... |
I have attempted as follows:
$|f(x) - f(y)| = |x + 1/x - y - 1/y|$
$\leq |x - y| + |1/x - 1/y|$
Struck here.
Any help.
|
There are 5 questions in a quiz designated by:
a, b, c, d, e
Each of the questions can only have one of 4 possible answers:
Possible Answer #1: 5
Possible Answer #2: -5
Possible Answer #3: 7
Possible Answer #4: -7
-
Am I correct to say that there are 625 possible variations of answers that we can receive?
-
Can any o... |
We say $a_n$ converges slower than $b_n$ if there exist an $x$ such that for all $m>x$, $a_m>b_m$ and both $\sum a_n$ and $\sum b_n$ converges.
Ignoring constant factors, which type of function converges the slowest?
|
In the Eculidean Space, one can automatically define a norm if a specific scalar product is given. On the contrary, it's not always true. A p-norm is a scalar product if and only if p=2. My question is what condition do we need in order to move back?
|
Given a metric space, we can check what the largest number of equidistant points are (ie, such that the distance between any two of these points is the same). Of course, this might not be finite, as in the case of the discrete metric on an infinite space.
This is not an invariant of the induced metric, however, as we c... |
Suppose we don't believe the continuum hypothesis. Using Von Neumann cardinal assignment (so I guess we believe well-ordering?), is there any "familiar" ordinal number $\alpha$ such that, for non-tautological reasons, $\aleph_\alpha$ is provably larger than the cardinality of the continuum? I would hope not since it w... |
A triangle has rational side lengths and rational angles measured as degrees. Is such a triangle necessary equilateral?
|
I want to simulate a heat source (eg a cpu) connected to a heatsink without any cooling fans. The simulation will run indefinitely using small time steps.
What i want to measure at each time step is the temperature of the heat source and the temperature of the heatsink at each tick (of time).
The heat source will be g... |
What is closed form equation for the tail probability of multivariate normal distribution ?
y ~ $\mathcal{N}$($0$, $\sigma^2I_n$ ) ; y $\in\mathbb{R}^n$
Pr(y > $\gamma$) = $\int_\gamma^\infty$ pr(y) d y ; $\gamma\in\mathbb{R}$
-Sun
|
What are your favorite applications of integration by parts?
(The answers can be as lowbrow or highbrow as you wish. I'd just like to get a bunch of these in one place!)
Thanks for your contributions, in advance!
|
I'm totally ignorant about the Universal Coefficient Spectral Sequence (I used to work only with principal ideal domains, where the Universal Coefficient Theorem only amounts to a short exact sequence) but I need now to understand it. The book I was redirected to is the infamous User's Guide to SS where I have only bee... |
Let $f,g:\mathbb C \to \mathbb C$ be two analytic functions such that $f(z)(g(z)+z^2)=0$ for all $z$ .Then prove that either $f(z)=0$ or $g(z)=-z^2$.
|
We all know Dijkstra and so on.
But is there a way to extend this to get the N best shortest paths between 2 nodes? ...and still with "acceptable" complexity.
Thanks,
Arnaud
|
This is kind of a programming question, but really it's math in disguise!
I'm working on a little project that will look at the size of a file and make sure it's in within an acceptable range based on previous file sizes. I know how to get the average file size of previous files and all that.
I'm wondering if there is ... |
While I was solving a problem something interesting came up if we know the dot product of 2 vectors, and one of the vectors is known. Can we find the other? The interesting bit about this is that even though the dot product behaves pretty much like normal algebraic multiplication 1, but that does not extend to properti... |
Is there a notation for element-wise (or pointwise) operations?
For example, take the element-wise product of two vectors x and y (in Matlab, x .* y, in numpy x*y), producing a new vector of same length z, where $z_i = x_i * y_i$ .
In mathematical notation, there doesn't seem to be a standard for this, am I wrong?
The... |
How can I find the sum of the following series?
$$
\sum_{n=0}^{+\infty}\frac{n^2}{2^n}
$$
I know that it converges, and Wolfram Alpha tells me that its sum is 6 .
Which technique should I use to prove that the sum is 6?
|
I'm trying to find a set which is Lebesgue measurable but not Borel measurable.
So I was thinking of taking a Lebesgue set of measure zero and intersecting it with something so that the result is not Borel measurable.
Is this a good approach? Can someone give a hint what set I would take (so please no full answers, I w... |
Puzzle
In a country in which people only want boys, every family continues to have children until they have a boy. If they have a girl, they have another child. If they have a boy, they stop. What is the proportion of boys to girls in the country?
My solution (not finished)
If we assume that the probability of having a... |
What is the number of positive integers that, when $2004$ is divided by them, leave a remainder of $24$?
A. $036$
B. $020$
C. $022$
D. $021$
E. $014$
|
On a normal variety (possibly singular) one can make sense of the canonical class $K_X$ by computing it on the smooth locus and then extending over the singular part. I'm wondering how to actually do this in practice: for example, on the singular cone $C$ in $\mathbb P^3$ given by $y^2-xz=0$, what is the canonical cla... |
I am trying to prove the following:
Let $l$ be a prime and let $\zeta$ be a $l$th root of unity. Show that, in the order $\{ 1, \zeta, \ldots, \zeta^{l-2} \}$ of the field $\mathbb{Q}(\zeta)$, if a product $\alpha \beta$ is divisible by $1-\zeta$, then $\alpha$ or $\beta$ must be divisible by $1-\zeta$.
I know that ... |
There are two identities that have a seemingly dual correspondence:
$$e^x = \sum_{n\ge0} {x^n\over n!}$$
and
$$n! = \int_0^{\infty} {x^n\over e^x}\ dx.$$
Is there anything to this comparison? (I vaguely remember a generating function/integration correspondence)
Are there similar sum/integration pairs for other well-kn... |
This is a question I like to bring up in a multivariable calculus class from time to time:
A sphere of radius 4 is dropped into a bowl-shaped paraboloid given by $z = x^2+y^2$. How close will the sphere come to the vertex of the paraboloid?
The problem can be solved using standard optimization techniques discussed... |
If I'm given a curve as an implicit function, is there some algorithm for obtaining a parametrization of the curve?
For example, how would I parameterize
$\frac{2x}{y} - \frac{3y}{x} = 8$
I know I might use $\cosh$ and $\sinh$, but I'm wondering if there's a surefire way to parameterize a curve, when you have a formula... |
I have a metric space with a probability measure (implying that the measure is regular). What conditions should I ask of the measure or of the metric to get that
a function in $\mathcal{L}^2(d\mu)$ can be approximated in $\mathcal{L}^2$ by a uniformly continuous function?
Thanks
|
Let $f(x)$ defined on $[0,a]$ with the following property.
$f(0) = b$, $f(x)\geq 0$
$f(x)$ is a continuous and non-increasing function
Let $g(x) = xf(x) + k(ab - \int_0^a f(x) dx)$.
Does there exist a $m$, such that $\max(g(x)) \geq mab$ for every $f$? If so, how does $m$ related to $k$? I'm assuming $m = h(k)$ for s... |
Possible Duplicate:
Riddle (simple arithmetic problem/illusion)
Three men go to a shop and take some food which is cost $25. Waiter came and gave them bill. <pre>Each of them gave $10. So, waiter take total $10+$10+$10=$30. Waiter came back with $5</pre>. Three men took <pre>$1+$1+$1 and they decide to gave rest $2... |
A professor in my grad program gave me the impression that random graphs are an interesting and as of yet largely unexplored area of mathematics (feel free to correct this impression if I'm wrong). I've found the Erdos and Renyi papers from the late 50s/early 60s and nabbed a copy of Random Graphs by Bollobas. I was wo... |
I realize most people work in "convenient categories" where this is not an issue.
In most topology books there is a proof of the fact that there is a natural homeomorphism of function spaces (with the compact-open topology): $$F(X\times Y,Z)\cong F(X,F(Y,Z))$$ when $X$ is Hausdorff and $Y$ is locally compact Hausdorff.... |
This is a really basic question. Consider a statement:
A painted line is increased by length $x$
Can $x$ be a negative number just by the wording of the question? What would be a better way of rephrasing the question, to make it clear that $x$ can be negative?
(The reason I'm asking is that I'm programming and a met... |
I keep tossing a fair coin until I see HTHT appears. What is the probability that I have already observed the sequence TTH before I stop?
Edit:
Okay. This problem is the same as before. I am trying to think of the following. Could anyone please tell me if this is also the same as before, and if not, how to solve it.
Tw... |
Part of the reason I think algebraic topology has acquired something of a fearsome reputation is that the terrible properties of the topological category (e.g. the existence of space-filling curves) force us to work very hard to prove the main theorems setting up all of the big machinery to get the payoff we want (e.g.... |
I'm trying to prove that ${n \choose r}$ is equal to ${{n-1} \choose {r-1}}+{{n-1} \choose r}$ when $1\leq r\leq n$.
I suppose I could use the counting rules in probability, perhaps combination= ${{n} \choose {r}}=\frac{n!}{r!(n-r!)}$.
I want to see an actual proof behind this equation. Does anyone have any ideas?
|
It's possible that this question is trivial and I've overlooked something.
Let us impose the following constraint on the well-known PARTITION problem:
in all inputs of a new problem the number of elements is fixed and equals $k$?
Does such constraint make PARTITION polynomially solvable?
P.S. There is pseudopolynomia... |
Here is a problem which i have not been able to do for quite sometime.
For $n \geq 2$, show that $n \nmid 2^{n}-1$.
I have thought of proving this in two ways: One by using induction which didn't actually work. Next by Fermat's little theorem we have $2^{p-1} \equiv 1 \ (\text{mod} \ p)$, which actually says that for... |
I just read a very short paper on the De Bruijn-Erdos theorem. It essentially states that if we have $n$ points and $m$ lines, where any two points lie on a line, and each line has at least two points, then there are at least as many lines as points.
The proof is in this short paper. It is stated at the very top, and t... |
I've been looking at Janusz book Algebraic Number Fields. I can't seem to figure out the following correspondence:
Let $\mathfrak{p}$ be a finite prime of a field $K$ and $L/K$ a finite Galois extension. Then we get a valuation ring $R$ from $\mathfrak{p}$ by picking one valuation. Janusz then writes (p. 123) "Identify... |
Let's say I have many measures of a random variable - x.
Moreover, I have many measures of y and z.
Meaning I can approximate their PDF by the histogram of the measures.
How can I estimate the joint PDF of (x, y, z)?
I don't know whether they are uncorrelated.
Thanks.
P.S.
References to a practical demos and info on th... |
I'm reading the beginning of chapter 3 of Ahlfors where he starts to introduce analytic functions in regions, and I had a question on a part of Theorem 11:
Let $f=u+iv$ be an analytic function in a region $\Omega$. If the modulus of $f$ is constant, then $f$ must reduce to a constant.
His proof starts out as follows: ... |
This is probably not a very ethical question to ask but I need to have a fast introduction to a range of concepts about the representation theory of the $SO(n)$ and I would be happy to see some online references which will help me do this journey.
$\\$
I am listing below the specific concepts about it that I need to k... |
Why is a smooth connected scheme (say over a field) necessarily irreducible?
Intuitively it makes sense because we might very well expect points in the intersection of two irreducible components to be singular points.
But what is a proof? Feel free to add any extra hypotheses if needed (e.g., separated if that is requ... |
I have several numbers, say 3, 1, 4, 8, 5, and I wanted them to be mapped into a fixed domain [0.5, 3]. In this case, 1 should be mapped as 0.5 and 8 is 3. Then the rest numbers should be scaled down to their correspondences.
So, my question is what should I do to deal with this case? what's the name of this processing... |
I'm trying to wrap my head around this concept of extracting sets from a given set to show that a subset is indeed a subset.
Suppose I want to show that the set of all sets of two elements does not exist. I approach it like this.
Suppose such a set $V$ exists of all elements containing exactly the sets of two elements.... |
How to solve this trigonometric equation $x = \frac 1 2 \cos\left(\frac 2 3 \sin\left(\frac 3 4 x\right)\right) + 1$ ?
The iterative solution seems to be 1.417.
Can anybody suggest an algebraic solution ?
Does it really exist ?
|
In a previous question Summing something with random and non-random parts I asked how one could estimate (specifically I am interested in an upper bound).
$$\sum_{k=1}^n e^{-k B_k^2} $$
where $B_k$ is a uniformly distributed sequence on [0,1]. The response was to take the expected value of the summand. But what can I d... |
The Weierstrass M-test tells that given a function sequence $(u_{n}(x))$ where $x \in I$, if there exists a convergent series $\sum a_{n}$ such that $|u_{n}(x)|\leq a_{n}$ for all $n$ and $x\in I$, then $\sum u_{n}(x)$ converges uniformly in $I$.
What about the opposite of it?
If $\sum u_{n}(x)$ converges uniformly i... |
Given a property $P$, is there some rules that are sufficient or necessary to determine if there exists a set of all sets with property $P$?
|
Working on a pre-lab, some of the directions for drawing graphs is that the "x-axis" is for the non-variables, and the "y-axis" is for the variables (what you have measured).
Now, the first thing I have to do is to make a standard curve for "Glucose concentration" and "Absorbtion".
The next step is to measure the "Enzy... |
I am trying to compute Nth Squarefree numbers using Mathematica. What I am trying to utilize is the SquareFreeQ[i] function.
Here is my solution :
NSqFree[N_] :=
For[n = 1; i = 1, n <= N + 1,
If[SquareFreeQ[i], If[n == N, Print[i] ] n++] i++]
But I am supposed to compute NSqFree[1000000000] but seems like my appr... |
I am looking for software/algorithm for the smallest context free grammar describing a finite set of words (and no other words).
For a single word I found sequitur
Related to this seems: given a CFG what is the shortest functionally equivalent CFG?
I would prefer the size to be measured by the CNF (for a precise estima... |
Tried the squeeze theorem but it didn't get me anywhere since:
$$0 \leftarrow \frac{1}{n^{1-1/n}}=\frac{n^{1/n}}{n}\leq\frac{(1^n+2^n+...+n^n)^{1/n}}{n}\leq \ n^{1/n}\to 1$$
None of the other tools we studied seem to work. Any tips? Thanks.
|
I'm learning about transformation in mathematics in general. There are many of them like for example Jacobian (which I'm trying to understand now). The "how?" part is very easy. There are formulas and we use them but the "why?" part is what I'm looking for. I fail to understand the intuition behind it. Why are we using... |
After looking at the list of trigonometric identities, I can't seem to find a way to solve this. Is it solvable?
$$\cos(\theta) + \sin(\theta) = x.$$
What if I added another equation to the problem:
$$-\sin(\theta) + \cos(\theta) = y,$$
where $\theta$ is the same and $y$ is also known?
Thanks.
EDIT:
OK, so using the l... |
i'm studing probabilistic algorithms: the ones that - with a great gain in efficency - sometimes could return a false response.
They return the true response with a probability of $\frac{3}{4}$.
The the way to use them to lower the error probability is: runs $t$ times, compare results and gets the more frequent answer.... |
I am teaching a class in Number Theory and using a new textbook by James Pommersheim.
There is a proof in the first chapter that states:
Let $m$ and $n$ be integers. If both $m$ and $n$ are odd, then $mn$ is odd.
They then prove this by saying $m=2q+1$ and $n=2r+1$. There's a simple foil, and then you can factor and... |
Are there any infinite series about which we don't know whether it converges or not? Or are the convergence tests exhaustive, so that in the hands of a competent mathematician any series will eventually be shown to converge or diverge?
EDIT: People were kind enough to point out that without imposing restrictions on the... |
I think the title pretty much says it all. I'm getting confused in the subtle parts of a proof, and would appreciate some help.
|
As one can notice every integer greater than $1$ is a sum of two squarefree numbers.(numbers that are not divided by some prime square power). Can we prove that?
Edit: Can we have bounds for the length of these numbers? (meaning the number of the primes that divide it)
Chen's theorem asserts that for large enough even... |
Assume that a proof that pi is normal existed.
Is it then possible that starting at some finite position x in pi, from there on every p(n)'th digit is 0, where p(n) is the n'th prime?
I know probability arguments say no, but do they also prove that it is impossible?
Is there a way to disprove the statement?
Also, the d... |
Per now, I have basically come upon proofs of the irrationality of $\sqrt{2}$ (and so on) and the proof of the irrationality of $e$. However, both proofs were by contradiction.
When thinking about it, it seems like the definition of irrationality itself demands proofs by contradiction. An irrational number is a number ... |
I'm trying to factorise a 20-term polynomial of the 4th degree with four variables. Ideally, I'd like to do it by hand, but I get the idea that this is pretty improbable. Can anyone point me to some helpful resources on this subject?
Thanks
|
I have three points on a map: origin, position, and destination. For each point, I have a latitude and longitude. The position point includes the lat/long plus a heading. The heading is in degrees between $0^{\circ}$ and $360^{\circ}$. $0^{\circ}$ and $360^{\circ}$ both represent absolute North.
I want to draw an arrow... |
I'm searching for a good book or web page that has a good amount of problems and their solutions, at undergraduate level, of divisibility, inequalities, induction, etc.
Thanks in advance
|
I begin losing intuition when I start dealing with infinitely generated fields over $\mathbb{Q}$...
The naive guess is that it is noetherian of krull dimension 1. Is this correct?
A related question is what sort of morphism is $Spec($the integral closure of $\mathbb{Z}$ in $\overline{\mathbb{Q}}) \rightarrow Spec(O_K)$... |
If a function is a combination of other functions whose derivatives are known via composition, addition, etc., the derivative can be calculated using the chain rule and the like. But even the product of integrals can't be expressed in general in terms of the integral of the products, and forget about composition! Why i... |
If W is a subspace, is it also a linear space? If V is a linear space, is it also a subspace? I am having trouble wrapping my head around the difference between the two, as it seems that the way the book defines them is the following: both have to have a zero (neutral) element, both are closed under addition and scalar... |
The following expression:
$$\frac{\sqrt{4+h}-2}{h}$$
should be simplified to:
$$\frac{1}{\sqrt{4+h}+2}$$
(even if I don't agree that this second is more simple than the first).
The problem is that I have no idea of the first step to simplify that.. any help?
|
I was given this question by a friend:
Let $f(x)$ be 2 times differentiable in $R$ and $x_{0}$ a local extremum.
Show that there are $a, b, c \in R$ such that a function $g(x)=f(x+a)+b$ exists. And $g(x)=cx^2+h(x)$ where $\lim_{x\to 0}\frac{h(x)}{x^2}=0$.
I began by writing a first degree Taylor expansion of $f(x)$ a... |
I'd like to calculate a standard deviation for a very large (but known) number of sample values, with the highest accuracy possible. The number of samples is larger than can be efficiently stored in memory.
The basic variance formula is:
$\sigma^2 = \frac{1}{N}\sum (x - \mu)^2$
... but this formulation depends on know... |
The exercise asks to "Re-write completing the square": $$x^2+x+1$$
The answer is: $$\left(x+\frac{1}{2}\right)^2+\frac{3}{4}$$
I don't even understand what it means with "Re-write completing the square"..
What's the steps to solve this?
|
Given a symmetric positive definite matrix $A$ and a mostly-zeros non-negative diagonal matrix $D$, is there a way to cheaply update the eigenvalues and/or eigenvectors of $A$ to that of $A+D$? Ideally I'm looking for something akin to the Woodbury matrix identity.
|
This is a problem from an olympiad I took today. I tried but couldn't solve it.
Let $A$ and $B$ rectangular matrices with real entries, of dimensions $k\times n$ and $m\times n$ respectively. Prove that if for every $n\times l$ matrix $X$ ($l\in \mathbb{Z}^+$) the equality $AX=0$ implies $BX=0$, then there exists a mat... |
This is the other problem I couldn't solve in the olympiad test I took today.
Let $c_1, ... , c_n$ be complex numbers with unitary norm, and $S_k=\sum_{i=1}^n c_i^k$, $k\in \mathbb{N}$. Suppose $S_1, S_2, ...$ converges. Prove that $c_i=1$ for every $i=1, ..., n$.
By using De Moivre's formula, the problem is equivalent... |
The first step is clear -- the zero vector is in both, so it will be in the intersection, too. Scalar multiplication is pretty obvious too -- if the vector is in the intersection, it can be represented as some linear combination of vectors in $V$ = some linear combination of vectors in $W$. Multiplying both by $k$ pres... |
How would I evaluate these limits?
$$ \lim_{n \to \infty} \int_0^\infty \frac{n}{1+(nx)^2} \ dx$$
and
$$ \lim_{n \to \infty} \int_0^\infty \frac{(1+(nx)^2)}{(1+nx^2)^n} \ dx$$
|
Can someone explain in complete detail with the appropriate convergence arguments of the Gibbs Phenomenon for Fourier Series? I know that the overshoot near a jump does not die out as the frequency increases, but approaches a finite limit. We derived the overshoot in an engineering class and there was a lot of hand-wav... |
This has been annoying me for the past two days.
Suppose we have an algebra $A$ over some field and a subalgebra $B \subseteq A$. Suppose we also have that $A$ is a finitely-generated free module over $B$, so as a $B$-module $A =Bx_1 \oplus\dots\oplus Bx_n$ for some $x_1,...,x_n \in A$.
Can I assume $x_1=1$?
Or is it p... |
I'm reading lectures for the first time starting next week ^_^ The subject is calculus on manifolds I was told that the students I'll be lecturing are not motivated (at all), so I need to kick off the series with an impressive demonstration of what an cool subject it will be that I'll be able to periodically return to ... |
In the last two days, I got a bit obsessed with the following game, partly because I beat others most of the time I play it. The gamers are expected to have a sound logical reasoning coupled with the ability to analyze data.
Here is the rule of the game:
$(1)$ The game is for two players.
$(2)$ Each player holds a $4$-... |
This is question from Spivak's Calculus.
Question statement (paraphrased): For any even function $f$ there is infinite number of functions $g$ such that $f(x) = g(|x|)$
I have made attempt at proof, here is my work. My main concern is whether my proof that there are infinite such functions is correct.
Let $g$ be a fun... |
I'm having trouble solving the following exercise in Hatcher's Algebraic Topology(1.3 #5):
"Let $X$ be the subspace of $R^2$ consisting of the four sides of the square $[0,1] \times [0,1]$ together with the segments of the vertical lines $x=\frac{1}{2},\frac{1}{3},\frac{1}{4},\ldots$ inside the square. Show that for ev... |
Given a nowhere vanishing vector field, say $E_1$, on a manifold $M^n$, it should be possible to extend this locally to a basis of vector fields $E_1,\ldots , E_n$ so that $E_i = \frac{\partial}{\partial x_i}$ for some coordinates $x_i$. (For example, locally we can choose a chart where $E_1$ looks like $\frac{\partial... |
Do you know of a good book on linear programming? To be more specific, i am taking linear optimization class and my textbook sucks. Teacher is not too involved in this class so can't get too much help from him either,
Any help will be appreciated.
Thank you
|
This is from an exercise (1.2.2) in introduction to algorithms that I'm working on privately.
To find at what point a $n \lg n$ function will run faster than a $n^2$ function I need to figure out for what value $n$
$8n^2 > 64n \lg n$
(with lg here being the binary log)
after some elementary simplification we get
$n > 8... |
Initially the deck is randomly ordered. The aim is to sort the deck in order. Now in each turn the deck is shuffled randomly. If any of the initial or last cards are in sorted order then they are kept aside, and the process is followed with rest of the cards. The question now is what is the expected number of shuffles ... |
Claim: If $(x_\alpha)_{\alpha\in A}$ is a collection of real numbers $x_\alpha\in [0,\infty]$
such that $\sum_{\alpha\in A}x_\alpha<\infty$, then $x_\alpha=0$ for all but at most countably many $\alpha\in A$ ($A$ need not be countable).
Proof: Let $\sum_{\alpha\in A}x_\alpha=M<\infty$. Consider $S_n=\{\alpha\in A \mid ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.