instruction stringlengths 12 30k |
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I'm trying to think of an example of a homomorphism of commutative rings $f:A\rightarrow B$ and ideals $I,J$ of $B$ such that $f^{-1}(I)+f^{-1}(J)$ is not a preimage of any ideal of $B$. I can't seem to come up with one... anyone know one?
Edit: To clear up some basic facts / head off some mistakes:
As Arturo po... |
How much do you know about vectors? You can take two points and find the difference between them, as a vector; this is the analogue of the slope of a line in $\mathbb{R}^2$. If the vector is $\mathbf{v}$ and one point is $a$, then points on the line are of the form $a+\lambda\mathbf{v}$, where $\lambda$ is real. (Un... |
What is an example of a lambda-system that is not a sigma algebra? |
What is an example of a lambda-system that is not a sigma algebra? |
This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The **determinant** in algebra has a lot in common with the **integral** in analysis. For example:
1. They are both applied to functions, the integral to integrable functions, ... |
This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The **determinant** in algebra has a lot in common with the **integral** in analysis. For example:
1. They are both applied to functions, the integral to integrable functions, ... |
Short question:
Can an element in the Kalman gain vector ever be over one?
|
I have a graph $G$ with a list of edges $E$, all the edges can form cycles, as shown below:
<img src="https://docs.google.com/drawings/pub?id=1Ux3b0FtuIMpVGX4e-AbblH05h2gIZII9u8pACRzymPE&w=960&h=720">
All the coordinates for the vertex $V$ **are known**.
Wat is the algorithm to find all the minimum cyc... |
I remember seeing something like the following problem in the past and would like to know if it has a solution (or if I can find a source for it).
**Problem** Given a finite set of points in the plane in general position (no three are collinear), does there exist a line that intersects no points? |
It is well-known that given two primes _p_ and _q_, _pZ + qZ = Z_ where _Z_ stands for all integers.
It seems to me that the set of natural number multiples, i.e. _pN + qN_ also span all natural numbers that are large enough. That is, there exists some _K>0_, such that
_pN + qN = [K,K+1,...)_.
My question is, giv... |
When do the multiples of two primes span all large enough natural numbers? |
For a ring $R$, an $R$-right-module $M$, an $R$-left-module $N$, and an abelian group $P$, one can use the universal property of the tensor product to construct maps
$$
M\otimes_R N\to P.
$$
It concrete cases, it is often easy to see that the constructed map is surjective by just writing down pre-images.
It seem... |
Is there a trick to prove injectivity for maps on tensor products? |
Imagine I have a company that makes widgets, where each widget costs me A dollars to make. Each month I can allocate money toward research and development with the aim of finding a new process that will allow me to build widgets for a cost of A/B dollars. Presume that I know that for each C dollars I spend on research ... |
This is probably an obvious parallel that most people are aware of, but I only just noticed it the other day and it made me quite excited. The **determinant** in algebra has a lot in common with the **integral** in analysis. For example:
1. They are both applied to functions, the integral to integrable functions, ... |
What is this general position stuff? Take any line outside the convex hull.
Maybe you're thinking of the <a href="http://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem">Sylvester-Gallai theorem</a>. |
I can not post comments, so I will post an answer.. Using $S_0 + B_t$ to model a stock price is not a very good idea, because $S_t$ may well turn negative. Have you considered using [Geometric Brownian Motion][1] ?
[1]: http://en.wikipedia.org/wiki/Geometric_Brownian_motion |
Since you asked for an intuitive way to understand covariance and contravariance, I think this will do.
First of all, remember that the reason of having covariant or contravariant tensors is because you want to represent the same thing in a different coordinate system. Such a new representation is achieved by a tra... |
How effective is this alternative to integration? |
I'd like to second one part of Qiaochu Yuan's answer: the recommendation to read Polya's book. Unlike many other books I've seen (albeit none of the others recommended above), it actually does contain guidance on how to construct a proof "out of nothing".
And that's one problem with the "practise, practise, practis... |
Is it true that for all proofs $\forall x \exists y : R(x, y)$, then $y = y(x)$?
A while back I remember reading a book on functional programming that was leading into some questions about what exactly mathematical proofs were and how they were related to functions. In fact, I just found the book and page: here on P... |
Is it true that for all proofs of the statement, $\forall x \exists y : R(x, y)$, then we can say $y = y(x)$? (Example given) |
Is there a trick to prove injectivity for maps out of tensor products? |
Simple example in order to clarify what I mean:
Sequence - {2 , -1, 4, 2, 9 }
Sum and product combination: { 16, -144 }
Question:
Is a combination of sum and product unique for any* sequence?
- "any sequence" except that one which consists only of zeros.
Thanks in advance. |
Is a combination of sum and product of any sequence unique? |
You seem to be asking: if for every $x$ there is a $y$ for which $R(x,y)$ holds, can we then define a function $f$ (in the usual sense) such that for every $x$, $R(x,f(x))$ holds?
From a set-theoretic point of view, that statement is in fact none other than the <a href="http://en.wikipedia.org/wiki/Axiom_of_Choice">... |
Is it true that for all proofs $\forall x \exists y : R(x, y)$, then $y = y(x)$?
A while back I remember reading a book on functional programming that was leading into some questions about what exactly mathematical proofs were and how they were related to functions. In fact, I just found the book and page: here on P... |
You seem to be asking: if for every $x$ there is a $y$ for which $R(x,y)$ holds, can we then define a function $f$ (in the usual sense) such that for every $x$, $R(x,f(x))$ holds?
From a set-theoretic point of view, that statement is in fact none other than the <a href="http://en.wikipedia.org/wiki/Axiom_of_Choice">... |
You seem to be asking: if for every $x$ there is a $y$ for which $R(x,y)$ holds, can we then define a function $f$ (in the usual sense) such that for every $x$, $R(x,f(x))$ holds?
From a set-theoretic point of view, that statement is in fact none other than the <a href="http://en.wikipedia.org/wiki/Axiom_of_Choice">... |
Simple example in order to clarify what I mean:
Sequence - {2 , -1, 4, 2, 9 }
Sum and product combination: { 16, -144 }
Question:
Is a combination of sum and product unique for any* sequence?
- "any sequence" except that one which consists only of zeros.
**UPD:** The answer to my initial question is "no".... |
Is a combination of sum and product of any sequence unique? |
Let's say we have a finite tuple (not set) of positive integers, $A = (a_1,a_2,\dotsb,a_n)$, with $\sum a_i = s$, $\prod a_i = p$. Consider the simplest prime factorization $p = p_1^{n_1} \dotsb p_k^{n_k}$ and construct $B = (1,\dotsb,1,p_1,\dotsb,p_1,p_2,\dotsb,p_k,\dotsb,p_k)$ where each $p_i$ occurs $n_i$ times and ... |
I'm trying to understand basic tensor analysis. I understand the basic concept that the valency of the tensor determines how it is transformed, but I am having trouble visualizing the difference between different valencies when it comes to higher order tensors.
I have this picture in my mind for the lower order ten... |
Is a combination of sum and product of any sequence unique? Updated. |
Simple example in order to clarify what I mean:
Sequence - {2 , -1, 4, 2, 9 }
Sum and product combination: { 16, -144 }
Question:
Is a combination of sum and product unique for any* sequence?
- "any sequence" except that one which consists only of zeros.
**UPD:** The answer to my initial question is "no".... |
Is a combination of sum and product of any sequence unique? Solved. |
For a ring $R$, an $R$-right-module $M$, an $R$-left-module $N$, and an abelian group $P$, one can use the universal property of the tensor product to construct maps
$$
M\otimes_R N\to P.
$$
It concrete cases, it is often easy to see that the constructed map is surjective by just writing down pre-images.
It seem... |
**Quick formulation of the problem:**
Given two points: $(x_l, y_l)$ and $(x_u, y_u)$
with: $x_l < x_u$ and $y_l < y_u$,
and given lower asymptote=0 and higher asymptote=1,
what's the logistic function that passes through the two points?
__Explanatory image:__
(Sorry I'm not allowed to post images, pl... |
Logistic function passing through two points? |
I am working on a game program.
I have an array of 8 points in 3d space (X,Y,Z) that are the 8 corners of a cube whose W=H=D. The 8 points are listed in no particular order. For the sake of determining this algorithm, The cube's center is at the world origin, (0,0,0) but the cube's orientation is not aligned with the ... |
This was suggested by a problem on FreeNode's #math a little while ago...
Construct a directed graph $\Gamma$ with vertex set the set of compact convex sets in $\mathbb R^2$, and an arrow $A\to B$ if $A$ and $B$ are disjoint and there is a point $a$ in $A$ and a $t>0$ such that $a+(t,0)$ is in $B$.
I guess there ... |
A tree on the set of convex sets? |
How does one go about solving the following quadratic congruence?
4x^2 ≡ 2 (mod 7) |
How does one go about solving the following quadratic congruence?
$4x^2 \equiv 2 \ (\text{mod} \ 7)$ |
This was suggested by a problem on FreeNode's #math a little while ago...
Construct a directed graph $\Gamma$ with vertex set the set of compact convex sets in $\mathbb R^2$, and an arrow $A\to B$ if $A$ and $B$ are disjoint and there is a point $a$ in $A$ and a $t>0$ such that $a+(t,0)$ is in $B$.
I guess there ... |
In a survey of political preference, 78% of those asked were in favor of at-least one of the proposals: I, II and III. 50% of those asked favored proposal I, 30% favored proposal II, and 20% favored proposal III. If 5% favored all the three proposals, what % of those asked favored more than one of the three proposals? |
How to solve this one ? |
Am I misunderstanding the conditions of the problem or is this a counterexample?
![Two ellipses in an X, and two circles between them][1]
[1]: https://i.stack.imgur.com/lFHHH.png |
This was suggested by a problem on FreeNode's #math a little while ago...
Construct a directed graph $\Gamma$ with vertex set the set of compact convex sets in $\mathbb R^2$, and an arrow $A\to B$ if $A$ and $B$ are disjoint and there is a point $a$ in $A$ and a $t>0$ such that $a+(t,0)$ is in $B$.
I guess there ... |
A tree on convex sets? |
A tree of convex sets? |
What is the sum of half of the choices (weighted)? |
For 3) if you are given an $\epsilon$, you can find $N$ such that $|a_n-a_m|<\epsilon \text{ if } n,m>N$ and similarly for $M$ and the $b$ series. Doesn't $\max{(N,M)}$ give a limit that $|(a_n,b_n)-(a_m,b_m)|<\epsilon$? |
Given are some positive integers $t_1, t_2, ...$ such that $\mathrm{gcd}(t_1, t_2, ...) = 1$. Now we create a linear combination of these: $k_1 t_1 + k_2 t_2 +\;...$ where $k_1, k_2, ...$ are nonnegative integers that you can choose to generate different numbers.
What is the smallest number $N$ such that all number... |
If gcd(a,b) = 1, x = 0 mod a, x = 0 mod b then prove that x = 0 mod ab
My attempt at answering the question:
x = 0 mod a
=> x is divisible by a
=> x = ma for some integer m
x = 0 mod b
=> x is divisible by b
=> x = mb for some integer m
x^2 = (ma)*(mb)
x^2 = (m^2)*(ab)
x = sqrt( (m^2)*(ab) )
x = (m)*... |
Given are some positive integers $t_1, t_2, ...$ such that $\mathrm{gcd}(t_1, t_2, ...) = 1$. Now we create a linear combination of these: $k_1 t_1 + k_2 t_2 +\;...$ where $k_1, k_2, ...$ are nonnegative integers that you can choose to generate different numbers.
What is the smallest $N$ such that all integers grea... |
Given are some positive integers $t_1, ..., t_n$ such that $\mathrm{gcd}(t_1, ..., t_n) = 1$. Now we create a linear combination of these: $k_1 t_1 + ... + k_n t_n$ where $k_1, ..., k_n$ are nonnegative integers that you can choose to generate different numbers.
What is the smallest $M$ such that all integers greate... |
Given are some positive integers $t_1, ..., t_n$ such that $\mathrm{gcd}(t_1, ..., t_n) = 1$. Now we create a linear combination of these: $k_1 t_1 + ... + k_n t_n$ where $k_1, ..., k_n$ are nonnegative integers that you can choose to generate different integers.
What is the smallest $N$ such that all integers great... |
Simple example in order to clarify what I mean:
Sequence - {2 , -1, 4, 2, 9 }
Sum and product combination: { 16, -144 }
Question:
Is a combination of sum and product unique for any* sequence?
- "any sequence" except that one which consists only of zeros.
**UPD:** The answer to my initial question is "no".... |
Is a combination of sum and product of any sequence unique? |
**Suppose n is an integer, such that the sum of the digits of n is 2, and $10^{10} \lt n \lt 10^{11} $. The number of different values for n is:**
Let me try to list them :
(1) 11000000000
(2) 10100000000
(3) 10010000000
...
(10)10000000001
So I am getting 10 possible values but the ans... |
**Suppose n is an integer, such that the sum of the digits of n is 2, and $10^{10} \lt n \lt 10^{11} $. The number of different values for n is:**
Let me try to list them :
(1) 11000000000
(2) 10100000000
(3) 10010000000
...
(10)10000000001
So I am getting 10 possible values but the ans... |
Say N is an integer then,
if N is odd $$\frac{a^N}{(a+1)} = a $$
if N is even $$\frac{a^N}{(a+1)} = 1 $$
Is it possible to proof this without induction ? |
Given are some positive integers $t_1, ..., t_n$ such that $\mathrm{gcd}(t_1, ..., t_n) = 1$. Now we create a linear combination of these: $k_1 t_1 + ... + k_n t_n$ where $k_1, ..., k_n$ are nonnegative integers that you can choose to generate different integers.
What is the smallest $N$ such that all integers great... |
**Suppose n is an integer, such that the sum of the digits of n is 2, and $10^{10} \lt n \lt 10^{11} $. The number of different values for n is:**
Let me try to list them :
(1) 11000000000
(2) 10100000000
(3) 10010000000
...
(10)10000000001
So I am getting 10 possible values but the ans... |
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13. What is the 10001st prime number? |
How to find next prime number? |
Here is a statement of [Farkas Lemma][1] from the Wikipedia. Let $A$ be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true:
1. There exists an $x \in \mathbb{R}^n$ such that $Ax = b$ and $x \geq 0$.
2. There exists a $y \in \mathbb{R}^m$ such th... |
In the proof of Rouche's theorem I saw in a book, there are two points I failed to understand, or failed to prove myself. (if you aren't familiar with the theorem, please try to look at the two statements here below and explain them regardless of the theorem):
1. "Consider the complex plane. It's not difficult to sh... |
I assume you want to program this? Then the question belongs in a CS forum. However, I will try to motivate how to program this using properties of prime numbers:
1) For a number you *could* try to divide it by every single positive integer smaller than the respective number (except for 1). This is stupid, as you ob... |
I was recently asked to help someone with the following question on their first year analysis course.
Recall that $\mathbb{N}$ is the set of all positive integers. Use the principal of induction to show that $n \ge 1$ for all $ n \in \mathbb{N}.$ [ Hint: Let $ S = \lbrace n \in \mathbb{N} | n \ge 1 \rbrace.$ ]
I... |
Is this analysis problem involving induction flawed? |
I was recently asked to help someone with the following question on their first year analysis course.
Recall that $\mathbb{N}$ is the set of all positive integers. Use the principle of induction to show that $n \ge 1$ for all $ n \in \mathbb{N}.$ [ Hint: Let $ S = \lbrace n \in \mathbb{N} | n \ge 1 \rbrace.$ ]
I... |
**How many even positive integers are there that are divisors of 720 ?**
I know how to compute the number of divisors but how to compute the number of **even or odd positive divisors of a number** ?
If we list the divisors of 720 (using mathematica) : **{1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, ... |
How many even positive integers are there that are divisors of 720 ? |
What should be added to $15^{81}$ to make it divisible by 7 ? Suggest a general technique to counter problems like this. |
What should be added to $15^{81}$ to make it divisible by 7? |
I'm a student and I'm struggling to understand the basics of discrete maths. If such a noob question wont offend you, please help me understand the equation and how to prepare a truth table for this?
> p-> (~q V ~r) ^ ~p ^q
If you use any substitution or such, explain how you derived them too .
|
1.What should be added to $15^{81}$ to make it divisible by 7 ? Suggest a general technique to counter problems like this.
2.For how many two-digit numbers, the sum of the digits is greater than the product of the digits? |
I'm a student and I'm struggling to understand the basics of discrete maths. If such a noob question wont offend you, please help me understand the equation and how to prepare a truth table for this?
$$p \rightarrow (\neg q \lor \neg r) \land \neg p \land q$$
If you use any substitution or such, explain how you d... |
I'm learning about continued fractions, and I've enjoyed them so far, but I'm unsure if I've done the following correctly. I have no real experience with analysis, so I'm not sure if my reasoning is formal enough, or correct. Any feedback would be appreciated.
Let $\xi$ be an irrational number with continued fractio... |
Adding integers to an infinite continued fraction expansion doesn't change the value? |
I was recently asked to help someone with the following question on their first year analysis course.
Recall that $\mathbb{N}$ is the set of all positive integers. Use the principle of induction to show that $n \ge 1$ for all $ n \in \mathbb{N}.$ [ Hint: Let $ S = \lbrace n \in \mathbb{N} | n \ge 1 \rbrace.$ ]
I... |
Suppose $S^n$ is the n-sphere with basepoint $x$ and consider a based map $f:S^n\rightarrow S^n$ as a map of pairs $f:(I^n,\partial I^n)\rightarrow (S^n,\{x\})$. If $f$ is such that the pull-back $f^{-1}(S^{n}-x)$ is connected in $I^n$ do we necessarily have that $deg(f)$ is either -1,0, or 1? |
How to prove that if you take any 400 consecutive Octobers then exactly 14 % of those years have five Fridays and Sundays? |
If 72% percent of new apps are reviewed within the last 7 days, then there is a 72% chance that any app is reviewed in the last week.
$$72\% = \sum\limits_{i = 0}^{i = 6} {x\left( {1 - x} \right)^i }$$
where $x$ represents the probability that an app is reviewed each day.
If $p = \left( {1 - x} \rig... |
Suppose $S^n$ is the n-sphere with basepoint $x$. If based map $f:S^n\rightarrow S^n$ is such that the pull-back $f^{-1}(S^{n}-x)$ is connected do we necessarily have that $deg(f)$ is either -1,0, or 1? |
You seem to be asking: if for every $x$ there is a $y$ for which $R(x,y)$ holds, can we then define a function $f$ (in the usual sense) such that for every $x$, $R(x,f(x))$ holds?
From a set-theoretic point of view, that statement is in fact none other than the <a href="http://en.wikipedia.org/wiki/Axiom_of_Choice">... |
Is it possible to divide an equilateral triangle into 5 equal (i.e., obtainable
from each other by a rigid motion) parts? |
I need to find the number of natural numbers between 1 and 1000 that are divisible by 3, 5 or 7.
I know that given a set of numbers, 1 ... n, the number of numbers divisible by d is given by $\lfloor \frac{n}{d} \rfloor$
So if we let $A$ be the set of numbers divisible by 3, $B$ be the set of numbers divisible by... |
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the _quality_ or _power_ of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.
One version of the $abc$-Conject... |
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the _quality_ or _power_ of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.
One version of the $abc$-Conject... |
In a problem my teacher did the following:
5a - 2b = 3m
5b - 2c = 3n +
-----------------
5a - 2c + 3b = 3m + 3n
I tried solving for b in the first equation and then plugging it in to the second but could not get the same result.
When is adding full equations like this valid?
Thanks! |
When can you add full equations together? |
You seem to be asking: if for every $x$ there is a $y$ for which $R(x,y)$ holds, can we then define a function $f$ (in the usual sense) such that for every $x$, $R(x,f(x))$ holds?
From a set-theoretic point of view, that statement is in fact none other than the <a href="http://en.wikipedia.org/wiki/Axiom_of_Choice">... |
A triple of positive integers $(a,b,c)$ is an $abc$-triple if $a$ and $b$ are coprime and $c = a + b$. Define the _quality_ or _power_ of an $abc$-triple as $P(a,b,c) = \frac{\log c}{\log \text{rad}(abc)}$, where $\text{rad}(k)$ denotes the product of distinct prime divisors of $k$.
One version of the $abc$-Conject... |
I want to solve a simple stochastic problem. Imagine there are n agents who want to access a resource, with a probability p at a given time t. What ist the probability that the resource will be free and not accessed by an agent. How does the result change with 2n and 2 available resources?
How do i solve this problem?... |
Is there a relation between the max of a Gaussian random walk of 10
steps vs the max of 10 Gaussian random walks? Specifics (in
Mathematica notation):
<pre><code>
(* a Gaussian random walk with standard deviation 1 *)
a[0] := 0
a[n_] := a[n-1] + RandomReal[NormalDistribution[0, 1]]
(* the max of the ... |
I agree that if we define $\mathbb{N}$ as a "set of positive integers", the question is redundant. However if we say that $\mathbb{N}$ is a set of natural numbers (and, after all, that is exactly what $\mathbb{N}$ stands for), defined by Peano axioms, then the question is no longer redundant.
As a side note, callin... |
Here is a statement of [Farkas Lemma][1] from the Wikipedia. Let $A$ be an $m \times n$ matrix and $b$ an $m$-dimensional vector. Then, exactly one of the following two statements is true:
1. There exists an $x \in \mathbb{R}^n$ such that $Ax = b$ and $x \geq 0$.
2. There exists a $y \in \mathbb{R}^m$ such th... |
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