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What is the minimum for this function of $x_1,x_2, \ldots, x_n$:
$$\sum_{i=1}^n c_i \log x_i + \lambda \; \sum_{i=1}^n d_i x_i, $$
where $\lambda$, $c$ and $d$ series are positive constants, $x_i \in (0.02,1]$ and $\sum x_i = 1$.
|
I'm looking to strengthen my understanding of the math that is directly useful to practical computer science, as opposed to unsolved computer science problems. In other words, the kind of math that would be useful in day to day algorithmic analysis, and come up during senior level interviews at software companies. My m... |
In Hatcher's Algebraic Topology, section 3.2, during the computation of the cohomology ring of a $n$-torus, the following assertion is made. Let $Y$ be a space and $R$ a commutative ring. Then the natural map in cohomology
$$ H^n(I \times Y, R) \to H^n( \partial I \times Y, R)$$
is a split injection. This is likely st... |
This is a follow-up (refinement?) of this question.
In learning some algebraic topology, I've learned to think of an affine scheme as spec $R$. (I've been told that this is a legitimate use of terminology because it provides an embedding of $Rings^{op}$ into a larger category.) One construction we can do, for example... |
In this post we saw isomorphism of vector spaces over $\mathbb{Q}$. Just came across this question:
Is $\mathbb{Q}(\sqrt{2}) \cong \mathbb{Q}(\sqrt{3})$?
I know these as $\mathbb{Q}$-Vector spaces, are isomorphic from the linked post. But as fields are they isomorphic? I neither know how to prove it nor how to dispr... |
I was reading John Lee's Introduction to Smooth manifolds, and I came across this question:
Let $M$ be a smooth manifold, and let $\delta : M \rightarrow \mathbb{R}$ be a positive continuous function. Using a partition of unity, show that there is a smooth function $\tilde{\delta} : M \rightarrow \mathbb{R}$ such that... |
I have a velocity vector on $X$, $Y$ and $Z$ axes. The values are represented as M/S (meters per second).
What algorithm should I use to convert this vector into a general MPH (miles per hour) value that takes into account all axes of the vector?
This may (or may not) be a very simple question. But I am far from being ... |
Let $f$ be in $L^{1}(\mathbb{R})$, $\mathbb{R}$ the real numbers. Show that for every $\varepsilon > 0$ there exists $A \subseteq R$ , measurable, such that $m(A) < \infty$ , $f$ is bounded on $A$ and $ \int_{\mathbb{R}} |f| < \int_{A} |f| + \varepsilon$.
If we take $A$ as the support of the simple function which appr... |
This is motivated by this question and the fact that I have no access to Timothy Chow's paper What Is a Closed-Form Number? indicated there by
Qiaochu Yuan.
If an equation $f(x)=0$ has no closed form solution, what does it normally
mean? Added: $f$ may depend (and normally does) on parameters.
To me this is equivalent... |
You are allowed to use +, -, / and * (plus, minus, division and multiplication) signs and bracketing. These signs you can put between the numbers
1963
to form mathematical expressions. You must put at least one sign between two numbers and – cannot be used as “negative”, thus -1+9+6+3 is not allowed, but 1-9+6+3 is all... |
The following inequality is from the proof that the $L^p$ norm is Gâteaux differentiable for $ 1 < p<\infty$ (from "Analysis" by Lieb and Loss).
Let $a$, $b\in\mathbb{C}$ and $-1\leq t\leq 1$, $t\not=0.$ Then
$$|a|^p-|a-b|^p\leq\frac{1}{t}(|a+tb|^p-|a|^p) \leq |a+b|^p-|a|^p.$$
I managed to prove the second inequality... |
I need to prove or disprove whether non-singular matrices commute. Can you help me and tell me how? What should i use cos i really don't know, I am beginner in this.
Thanks in advance.
|
According to the definition of stochastic process in Wikipedia:
Given a probability space $(\Omega, \mathcal{F}, P)$, a stochastic process (or random process) with state space $X$ is a collection of $X$-valued random variables indexed by a set $T$ ("time"). That is, a stochastic process $F$ is a collection
$\{ F_t : t... |
Is there a general formula to determine the probability of unbounded, cumulative dice rolls hitting a specified number?
For Example, with a D6 and 14:
5 + 2 + 3 + 4 = 14 : success
1 + 1 + 1 + 6 + 5 + 4 = 17 : failure
|
I'm having trouble with the following exercise in do Carmo's Riemannian geometry.
Let $X$ and $Y$ be differentiable vector fields on a Riemannian manifold $M$. Let $p \in M$ and let $c: I \to M$ be an integral curve of $X$ through $p$, i.e. $c(t_0) = p$ and $\frac{dc}{dt} = X(c(t))$.
Prove that the Riemannian connecti... |
The table of integrals says that
\begin{equation*}
\int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\arctan\frac{x}{a}+C
\end{equation*}
where $C$ is a constant. What's wrong with my proof?
$$
\begin{align*}
y &= \arctan\frac{x}{a} \\\
a\tan y&=x \\\
\tan y &= \frac{x}{a} \\\
\frac{d}{dx} \tan y &= \frac{d}{dx} \frac{x}{a} \\\
\... |
On a small island there are 25
inhabitants. One of these inhabitants,
named Jack, starts a rumor which
spreads around the isle. Any person
who hears the rumor continues
spreading it until he or she meets
someone who has heard the story
before. At that point, the person
stops spreading it, since nobody ... |
I see people like Terry Tao and others take extensive notes. But is this really necessary? When I do this I feel like I am rewriting a textbook.
|
I've read a few articles explaining the way to use the Minkowski reduced basis in a lattice in order to measure the uniformity of the output of a random number generator. However, I can't prove a conclusion drawn from the following statement.
Given a set of linearly independent vectors $\{v_{1}, v_{2}, ... ,v_{d}\}$ in... |
A powder can be compressed by packing it down. Each time it is packed down it loses $\frac{1}{2}$ then $\frac{1}{4}$ then $\frac{1}{8}$ ... etc. of it's total volume.
This powder is placed in to a container of unit volume and packed down. Then the remaining space is filled with fresh powder and packed down again. The p... |
After thinking about it for a while and consulting other students, no one seems to be able to find an example of the following:
Given the PDE
$\dfrac{\partial f}{\partial x} = 0 \quad $ on $U = { (x,y) \in \mathbb R^2 ; y>0, 1 < x^2 + y^2 < 4}$
I am looking for a solution $f$ that does not only depend on $y$.
How... |
$f$ is smooth (derivatives of all orders exist) except at $x_1$,$x_2$,$x_3$ where derivatives exist only upto orders of $k_1$,$k_2$,$k_3$ respectively, where $k_i$ is a natural number. Same is the case with another function $g$ which is smooth except at $t_1$,$t_2$,$t_3$ where derivatives exist only upto orders of $l_1... |
Imagine that we have something like the rubik's cube on $U=I^{3}=[0,1]^{3}$, and that we say that all the points on the xy, yz, and zx faces are black, and that all the points on the other three faces are white.
Define a legal move to be a rotation by an integral multiple of $\frac{\pi}{4}$ around one of the lines goin... |
Happened to get this from a friend.
How does one show that for all pairs of real numbers $x$ and $y$, there is $n \in \mathbb{N}$ such that $$ x < \frac{\Phi(n+2)}{\Phi(n)} < y$$ where $\Phi$ is the Euler's totient function.
$\textbf{Added.}$ This seems to be somewhat related to the problems given here: $\textbf{J 8}, ... |
How can I render the connected sum of $n$ topological spaces (or other quite strange topological operations) in a "product-like" manner? What I'm looking for is writing, for example
$$
{\Large{*}}_{i=1}^n G_i
$$
for the free product of n groups. How can I render this with the # operation in topological spaces? (Yes, I'... |
Prove that, where $a,b, \ldots, e$ are real numbers and $a \neq 0$, if $ax + by = c$ has the same solution set as $ax+dy=e$ then they are the same equation. What if $a=0$?
Note: If $a \ne 0$ then the solution set of the first equation is $\{(x,y) \mid x=c-by/a\}$.
|
How to solve this problem, I can not figure it out:
If we list all the natural numbers below 10 that are multiples of 3 or 5, we get 3, 5, 6 and 9. The sum of these multiples is 23.
Find the sum of all the multiples of 3 or 5 below 1000.
|
I have a simple formula for the n-th derivative of some expression. I'm suppose to prove that it's true for all N, by using induction. I set n=1, and show that the base case works.
The next step, were I assume that it works a K is fine. But how is that shown on paper? And the that I don't understand at all, is how to ... |
I understand that we use the chain rule to differentiate the $y$ part of it. Like if it stood $xy$, I would use the product rule, that say that the derivative of $y$ is $dy/dx$. But what if it's just $y$ alone or with a constant in front off it?
Thank you!
|
If I have a quadratic form like $y^2 - x^2 - x = k$ none of the techniques I know work because of the nasty $x$. Note that homogenizing doesn't work because a solution of $Y^2 - X^2 - X Z = k Z^{(2)}$ does not lead to a solution of the original equation in integers, at least as far as I have been able to determine.
How... |
I need to find an isomorphism between [0,1] to (0,1) can you help me with this please?
thanks. benny
|
I thought of writing this question Minimum for this function in a different way, if it helps.
I want to minimize
$$\sum_{i=1}^n a_ix_i + \nu \sum_{i=1}^n b_i 2^{x_i} ,$$
where $a_i \in [0,1]$, $b_i \in (0,\infty)$, and $x_i \in [x_{\min},0)$, and
$$ \sum_{i=1}^n 2^{x_i} = 1 .$$
$x_{\min}$, $\nu$, $a_i$ and $b_i$ a... |
I have $X_1,...$ Bernoulli variables with probability $p$ of success.
I want to get an $n$ such that with probability $\delta$
$P(\sum_{i=1}^n X_i \ge k) \ge \delta$
This $n$ would of course depend on $k$, $p$ and $\delta$. Can I get an upper bound for that $n(k,\delta,p)$? (meaning, I want some $n(k,\delta,p)$ for whi... |
Let $k$ be an algebraic closed field. Let $x$ be a point in $X=P_k^1$. What is $O_{X,x}$?
For example, if I have $x=(t-a)\in \text{Spec }k[t]$. Looking $x$ inside $P_k^1$, does $O_{X,x}=k[t]_{(t-a)}$? I'm confused when I have to deal with the sheaf of rings.
|
How to prove
$$\int_{0}^{\infty} \mathrm{e}^{-x^2}\, dx = \frac{\sqrt \pi}{2}$$
|
Let $R\subset A\times A$ a binary relation, and note $a\sim b$ for $(a,b)\in R$. We define
$$(a,b)\in R\circ R\Leftrightarrow \exists b\in A : a\sim b, b\sim c$$
and $R^n=R\circ\dots\circ R$.
My books says that $(a,b)\in R^n$ iff there is a path of length $n$ in the graph of $R$. Isn't it false ?
For example, if $(a,... |
I need to turn the equation L=12/x + x into a quadratic forumla to solve. i have been given the answers of 0.619m 0r 19.38m if L = 20m
what iv go so far
0= 12/x + x -20
but i dont know how to turn 12/x to a x^2 value
|
Say I have a symmetric matrix. I have the concept of 2-norm as defined on wikipedia. Now I want to prove (disprove?) that the norm of a symmetric matrix is maximum absolute value of its eigenvalue. I would really appreciate if this can be done only using simple concepts of linear algebra.
I am quite new to mathematics.... |
According to the solution manual:
$\int \frac{x}{\sqrt{1-x^{4}}}dx = \frac{1}{2}\arcsin x^{2}+C$
My solution doesn't seem to be working. I know another way of solving it (setting $u=x^{2}$) but the fact that this way of solving it doesn't work bothers me.
$$\text{set }u=1-x^{4}\text{ so } dx=\frac{du}{-4x^{3}}
$$
$$
\b... |
This should be an easy exercise: Given a finite odd abelian group $G$, prove that $\prod_{g\in G}g=e$. Indeed, using Lagrange's theorem this is trivial: There is no element of order 2 (since the order must divide the order of $G$, but it is odd), and so every element except $e$ has a unique inverse which is different f... |
Say I have a binomial coefficient $y=\binom{5n+3}{n+2}$ or $y=\binom{n^2+4}{3n}$ something of the sorts in terms of the variable $n$. How can I determine $f$ so that $y = O(f)$?
Is there a general method for this sort of thing. I have only ever seen $O$-notation in a computer science setting and its treatment was not ... |
We were given a challenge by our calculus professor and I've been stuck on it for a while now.
Show that the set of subsequence limits of $A_n=\sin(n)$ is $[-1, 1]$ (another way to phrase this would be: $\forall r\in [-1,1]$ there exists a subsequence of $A_n=\sin(n)$ that converges to $r$).
What would be a good way t... |
I'm reading page number 4 here. In particular the section where it deals with the case $p=\infty$, that is , showing that $L^{\infty}$ is complete.
http://www.core.org.cn/NR/rdonlyres/Mathematics/18-125Fall2003/5E3917E2-C212-463B-9EDB-671486133388/0/18125_lec15.pdf
Two questions:
1) Why is the convergence uniform? wher... |
Suppose that $S_1$ and $S_2$ are the vanishing sets of a system of polynomial equations in n variables over a field $\mathbb{k}$ (ideal in $\mathbb{k}[X_1,\dots,X_n]$) and a system of polynomial equations in m variables over a field $\mathbb{k}$ (ideal in $\mathbb{k}[Y_1,\dots,Y_m]$). We can give $S_1$ and $S_2$ the Za... |
I've been trying to google the answer to this question, but have had no luck, so I'm asking this question here. Let's say the origin is at (0, 0), body 1 with mass m1 is at (-r1, 0), and body 2 with mass m2 is at (r2, 0). How can I calculate the initial velocities v1 and v2 for the bodies so that they trace circular or... |
I was wondering if equilibrium distribution, steady-state distribution, stationary distribution and limiting distribution mean the same thing, or there are differences between them?
I learned them in the context of Discrete-time Markov Chain, as far as I know. Or do they also appear in other situations of stochastic pr... |
How to prove $\limsup(\{A_n \cup B_n\}) = \limsup(\{A_n\}) \cup \limsup(\{B_n\})$? Thanks!
|
Say I have the following numbers:
10
10
9
5
4
4
4
1
How do i go about calculating the percentile for each score? Is there a standard formula for figuring this out?
|
I am trying to reproduce the following bound:
$\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$,
for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on $n$).
I have tried to bound each term by $\frac{1}{|\log(1\pm 1/x)|}$, but this doesn't seem to be good enough. Splittin... |
Can someone simply explain to me how to calculate linear integral linke below?
$$\int_{L} 5y \mathrm{d}L$$
Where L is line segment from (0;0) to (0,2;0,2).
|
Is there some increasing function $f(n)$ that grows slower than $n^{c}$ for some $c > 1$ such that $\sum_{n=1}^{\infty} \frac{1}{f(n)}$ converges?
|
This is a bit of a silly question, but I've been puzzled on how this can be done, so I ask here.
You are given an implicit Cartesian equation like
$$4 x^4-4 x^3+8 y^2 x^2-27 x^2+12 y^2 x+4 y^4-27 y^2+27=0$$
or
$$x^6+3 y^2 x^4-3 x^4+3 y^4 x^2+21 y^2 x^2+3 x^2+y^6-3 y^4+3 y^2-1=0$$
and are told that you are not allowed t... |
In several "proof-theory-for-dummies"-type texts I have read, $I\Delta_0+Exp$ or a theory equivalent to it shows up as a "base" theory, though in what sense it is minimal is not clearly addressed.
I realise this is a vague question but if I had the knowledge to make it more specific, I wouldn't need to ask it in the fi... |
Consider a random walk on an undirected, non-bipartite graph. Let $\pi$ be the stationary distribution of this process, and let the hitting time $H(s,t)$ be the expected time until a walk beginning at node $s$ reaches node $t$.
I learned from Random walks on graphs: a survey, by L. Lovasz, that the quantity
$$ \sum_{... |
Further to my understanding What makes a proof, I'm doing exercises.
The purpose of this question is to compare the proof a certain textbook provides with the one that is logical to me, that I made up. If my proof is weak, why is it weak? Why is the proof from the book acceptable (other than it finds a way to involve... |
I had hoped that this would be easier.
I have power series that I'm using. I want to know what simple structure describes the operations that I'm performing on them. It would be better if I could find something that non-mathematicians (actually computer scientists) would understand, but I'd like to know the technical... |
In a Cartesian system, I've got the slope, start point and distance of a line segment. What's the formula to find the endpoint?
|
Suppose $A$ is a set of $2n$ positive integers { $a_{i}$ } with $a_{i} \geq 2$ which can be partitioned into at most $r \leq n$ coprime pairs. The following fact is true: For all $A$ and every $n \geq 1$,
$$
\prod_{i = 1}^{2n} (a_{i} - 1) \equiv 0 \pmod{2^{r}}.
$$
For example, the set {$2,3,5,6$} can be partitioned in... |
I encountered a set of linear equations with modulo in only $2$ variables
$$(a_{11}x + a_{12}y) \mod 8 = b_1$$
$$(a_{21}x + a_{22}y) \mod 8 = b_2$$
In the case of simple equations without modulo, the solution of $Ax=b$ is $x=A^{-1}b$, but how to handle this modulo? Any clue or pointers will be very helpful.
|
I am teaching a Calc II course and came across the following series when finding the interval of convergence for the Taylor series of $f(x)=\sqrt{x}$ centered at $x=1$:
$$
\sum_{n=2}^\infty \frac{1\cdot 3\cdot 5\cdot 7\cdots (2n-3)}{2^nn!}.
$$
I know that it doesn't converge, but how do you show it? I would like to be... |
Suppose a set S has 11 elements. How many subsets of S have an even
number of elements? Express your answer as a summation.
I'm not really sure how to approach this. I think the general formula for this type of question is n!/k!(n-k)! where n is 11 and k is the number we're choosing. So would we just take the summation... |
Where, or do there exist, good visualizations of sheaves, stalks, stacks, and/or schemes? I'm a better visual thinker than I am a symbolic thinker, and it would be easier for me to follow some of the algebraic geometry papers that have no pictures whatsoever if I had some kind of visual intuition of what these objects ... |
If $x$, $y$, and $z$ are natural numbers, how many solutions are there to $x+y+z=25$?
How would I figure this out? I can't even begin dissecting this problem. Where do I begin?
|
let $f$:R->N such that given any $x_1$,$x_2$ belong to R such that $x_1 < x_2$ then there exists a $x_3$ belongs to $(x_1,x_2)$ such that $f(x_3)> max(f(x_1),f(x_2))$. How many such mappings are possible ?
|
According to Wikipedia, the number of terms in $(x+y+z)^{30}$ is $496$. I'm assuming this is before like terms are added up. How many terms would there be if like terms were combined? How would I go about figuring that out?
|
What is the cardinality of set of all smooth functions belonging to $L^1$ or $L^2$ ? What is that of set of all integrable or square integrable functions ?
|
I have a function given implicitly, you know. X and Y on both sides. Then it says, assume y = y(x). That's fine. I should be able to find y'(0), but what about y''(0)? How do you treat the dy/dx parts when taking the second derivative?
Edit: I would also like to follow the tip in the book, that says when I'm after act... |
Suppose that we do not know anything about the complex analysis (numbers). In this case, how to calculate the following integral in closed form?
$$\int_0^\infty\frac{\cos x}{1+x^2}\,\mathrm{d}x$$
|
Let $f \colon (\mathbb{C}^{n},\mathbf{0}) \to (\mathbb{C},0)$ be a complex analytic function with isolated critical point at the origin. Define the singular hypersurface $V_{f, \kappa} = f^{-1}(\kappa)$ for small $\kappa > 0$. Let $\pi \colon \tilde{V} \to V$ be a resolution of $V = V_{f,0}$, and let $p_{g} = \dim H^{n... |
Let G be the group U(24) and H the cyclic subgroup generated by the element 7.
Exhibit all of the distinct cosets of H in G.
I am coming up with this.
(I created the Cayley table for U(24) - using the elements 1,5,7,11,13,17,19,23.)
<1> = {1}
<5> = (5*5 = 1) {5}
<7> = (7*7 = 1) {7}
and so forth since each element tim... |
what are some natural examples of integral domains $A$ together with sequences of elements satisfying $... | f_2 | f_1 | f_0$, but so that $f_i$ divides no power of $f_{i+1}$? So in the spectrum there will be an ascending chain of basic-open subsets.
Of course we could just take the union of the $k[y^{-n} x,y]$ in $k(x... |
In a calculus course I just completed I struggled to understand the Taylor and Maclaurin series. I got the basics but not some of the harder concepts. Could anyone point me in the right direction? Books, tutorials, etc?
Many thanks.
|
I have posted about induction on it's own so I think I understand that part. But I'm slightly confused when it comes to using induction to show that $n+1$ is also the derivative. The base case if fine, I just set $n=1$. And show that is equal to the derivative not using the formula. But how about $k+1$? Can I just choo... |
Say I pass 512 samples into my FFT
My microphone spits out data at 10KHz, so this represents 1/20s.
(So the lowest frequency FFT would pick up would be 40Hz).
The FFT will return an array of 512 frequency bins
- bin 0: [0 - 40Hz)
- bin 1: [40 - 80Hz)
etc
So if my original sound contained energy at say 115Hz, how ca... |
Please find a closed form for partial sum of a function
$$f(x)=a^{1/x}$$
I want it to be expressed in terms of bounded number of elementary functions and/or well known special functions.
No computer algebra systems I have tried so far could find a satisfactory solution. I believe that the expression can exist in the te... |
Suppose $h$ is a hash function, $h \colon \{ 0, 1 \}^* \rightarrow \{ 0, 1 \} ^n$ and for all $w_1$, $w_2$ it holds: $$h(w_1 \oplus w_2) =
h(w_1) \oplus h(w_2).$$
$\oplus$ is the XOR operation.
Why isn't $h$ a cryptographically good hash function?
|
Could you provide a geometric explanation of the Taylor series expansion?
|
I read a little remark in a topology text that the direct image of an open set under a continuous mapping is not necessarily open.
What is the definition of direct image in this case? I tried googling to no avail, and only found references to sheaf theory on wikipedia, which I don't believe to be what I'm looking for.... |
Question:
Determine all positive integers $x$ and $y$ that satisfy the equation
$x^2 + 2 = 5y$.
|
Given an $R$-algebra $S$, is $Hom_{R-Alg}(\prod_{i=1}^n R, S) = Hom_{S-Alg}(\prod_{i=1}^n S, S)$?
|
Question:
Show that $n^2 + 3n + 5$ is not divisible by $121$, where $n$ is an integer.
|
The Clifford product of a pair of vectors $a,b$ is an associative operation defined by
$$ ab = a \cdot b + a \wedge b.$$
In sufficiently low dimensions I am used to being able to define the Clifford product on arbitrary $k$-vectors by repeatedly applying the vector definition. For instance, suppose that I build a Clif... |
Why is every group the quotient of a free group by a normal subgroup?
|
If G is a finite nilpotent group, then every minimal normal subgroup of $G$ is contained in the center of $G$ and has prime order.
|
Let $g: \mathbb{C}\times[a,b]\to\mathbb{R}$ a continuous function and
$$g(t)=g(h_0,t)=\lim\nolimits_{h\to h_0} g(h,t) \ \forall t\in \mathbb{R}$$
Is the following result/reasoning correct ? If we have a sequence of functions, we need uniform convergence to pass the limit inside the integral, but it seems it works every... |
A set M $\in$ $\mathbb{Z}^2$ is defined by
(Rule 1) $(0, 1)$ $\in$ M
(Rule 2) If $(x, y)$ $\in$ M, then $(x + 1, y + 2x + 3)$ $\in$ M
(a) (2 points) Starting with $(0, 1)$, write out the six pairs with the smallest first coordinates.
(b) (2 points) State the (simple) relationship that holds between the first and second... |
Quite a few mathematical constants are known to arise in several branches of mathematics (more here). I have no doubt that they are useful to the understanding of some mathematical structures and add to the whole body of knowledge of mathematics as a whole. Sometimes, however, I find myself looking at these constants ... |
The usual symbol for the cardinality of the continuum (i.e. the real numbers) is Fraktur $\mathfrak{c}$. However, I recall some sources also using $\aleph$ (with no subscript). This usage is not mentioned in Wikipedia or Mathworld, but I found some support for it over Google.
Is the $\aleph$ notation standard?
|
The implicit expression $(b-a)=(a+b)^3$ looks like it could be written explicitly for $a$ as a function of $b$. The only region of interest is for $a,b>0$ Here is what the plot looks like:
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Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of course true).
I would appreciate some hints. Does this require the recursion theorem to construct the isomorphism?
|
Inspired by the recent post "Limit of integral", I propose the following problem (hoping it will not turn out to be too easy). Suppose that $g:[0,1] \times [0,1] \to {\bf R}$ is continuous in both variables separately. Is it true that, for all $x_0 \in [0,1]$,
$$
\lim \limits_{x \to x_0 } \int_0^1 {g(x,y)\,{\rm d}y} ... |
If $a$ and $b$ are positive real numbers, and if $f(x)$ has the following asymptotic property
$f(x) = \frac{a + O(\frac{1}{\sqrt{x}})}{b + O(\frac{1}{\sqrt{x}})}$
then is the following true?
$f(x) = \frac{a}{b} + O(\frac{1}{\sqrt{x}})$
This might look like homework but it isn't.
|
I have a table in my database that records, 48 times per day, the total users, total active users within 2 weeks, total users logged in within the past 24 hours , and total users that logged in within the last 24 hours that had visited the site prior to the last 24 hours.
I'm trying to use this data to calculate the c... |
I am interested in a method to find the roots of the following equation:
\begin{equation}
f(t) = \sum_{i=1}^n \alpha_i e^{\beta_i t} + \gamma t + \delta = 0.
\end{equation}
For my application, coefficients $\alpha_i$, $\beta_i$, $\gamma$, and $\delta$ are real. $n$ is typically a small integer, say 10. In particular ... |
We know that $2^4 = 4^2$ and $(-2)^{-4} = (-4)^{-2}$. Is there another pair of integers $x, y$ ($x\neq y$) which satisfies the equality $x^y = y^x$?
|
I need to write a coupon code system but I do not want to save each coupon code in the database. (For performance and design reasons.) Rather I would like to generate codes subsequent that are watermarked with another code.
They should like kind of fancy and random. Currently they look like this:
1: AKFCU2, 2: AKFDU2, ... |
Winding yarn into a ball suggests some mathematical questions:
Under some natural model, what paths should the yarn follow to achieve the most dense ball?
One model is that used by Henryk Gerlach and Heiko von der Mosel in their paper
"On sphere-filling ropes" arXiv:1005.4609v1 (math.GT).
(This is the same model I su... |
The unit doublet is a symbolic object whose convolution with a differentiable function is supposed to give the derivative:
$$(x * u_1)(t) = \frac{dx(t)}{dt}$$
See also: http://en.wikipedia.org/wiki/Unit_doublet
Can this equation be made rigorous, in a way that it would be valid for all $x\in C^1(\mathbb R)$? I.e., is ... |
I am trying to prove the following fact for a homework assignment in algebraic geometry:
Let R be a local ring, and X a prescheme. Show that there is a one-one correspondence between R-valued points on X and pairs (x, g) of points x on X and local homomorphisms $g: O_{X, x} \to R$.
Now, given f: Spec R -> X, and writin... |
Is there a way of enumerating necklaces, in the spirit in which the factorial number system ( http://en.wikipedia.org/wiki/Factorial_number_system ) can be used to enumerate permutations?
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