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Are there infinitely many primes of the form $4n^{2}+3$? |
We, know that there are infinitely many primes of the form $4n-1,4n+1,5n-1,\cdots, \text{etc}$. I saw these things in Apostol's *Introduction to Analytic Number theory* textbook. I would like to have an argument working for $n^{2}$. The first expression which came to my mind was $4n^{2}+3$, which gives a prime for $n=1... |
In many combinatorial enumeration problems it is possible to find a rational generating function (i.e. the quotient of two polynomials) for the sequence in question. The question is - given the generating function, how can we find (algorithmically) the values of the sequence, i.e. the coefficients of the corresponding ... |
Yes. Your polytope $P'$ is exactly the polar $P^*$ of $P$ with respect to the standard unit sphere. Choosing another conic instead of the unit sphere will in general give another polar. See [this wikpedia page](http://en.wikipedia.org/wiki/Pole_and_polar) for the two-dimensional case of polarity with respect to other c... |
I found the following in wikipedia page for euler class.
"If the rank r is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$.
Under the splitting principle, this corresponds to the square of the Vandermonde polynomial equaling the discriminant: the Euler class correspond... |
I came across the following theorem that for any class $K$ of truth assignments, $K\subseteq Mod(Th(K))$, and if $K$ is axiomatizable, then $K=Mod(Th(K))$. Together these seem to me to imply that there is some class $K$ such that $K\neq Mod(Th(K))$, however, I can't find any class $K$ that would verify my assumption.
... |
Is it possible for $Mod(Th(K))\neq K$ for some class $K$ of truth assignments? |
I found the following in the wikipedia page for [Euler class](http://en.wikipedia.org/wiki/Euler_class#Squares_to_top_Pontryagin_class).
"If the rank r is even, then this cohomology class $e(E) \cup e(E)$ equals the top Pontryagin class $p_{r/2}(E)$.
Under the splitting principle, this corresponds to the square of ... |
I've been using the sentence:
> If a series converges then the limit of the sequence is zero
as a criterion to prove that a series diverges (when lim != 0) and I can understand the rationale behind it, but I can't find a **formal proof**.
Can you help me? |
Let $X$ be a contractible space. If $x_0 \in X$, it is not necessarily true that the pointed space $(X,x_0)$ is contractible (i.e., it is possible that any contracting homotopy will move $x_0$). An example is given in 1.4 of Spanier: the comb space. However, this space is contractible as a pointed space if the basepo... |
Does contractibility imply contractibility with a basepoint? |
arccos[(1-x^2)/(1+x^2)] = 2arctanx for x>=0
I'm not even sure what kind of math to try? :( |
How can I prove this trigonometric equality? |
Let $X$ be a contractible space. If $x_0 \in X$, it is not necessarily true that the pointed space $(X,x_0)$ is contractible (i.e., it is possible that any contracting homotopy will move $x_0$). An example is given in 1.4 of Spanier: the comb space. However, this space is contractible as a pointed space if the basepo... |
Yes. See exercise 7 [here][1].
[1]: http://books.google.com/books?id=tru-TWQFDAIC&lpg=PP1&dq=using%20the%20borsuk%20ulam%20theorem&pg=PA7#v=onepage&q&f=false "here" |
Put $t=\arctan x$. Then $x=\tan t$.
You have to prove that $2t=\arccos(1-x^2)/(1+x^2)$.
This is more-or-less the same as $(1-x^2)/(1+x^2)=\cos2t$. If you put
$x=\tan t$ into $(1-x^2)/(1+x^2)$ what do you get? |
$\displaystyle\arccos[\frac{(1-x^2)}{(1+x^2)}] = 2\arctan{x}$ for $x \geq 0$.
I'm not even sure what kind of math to try? :( |
$$\arccos\left(\frac{1-x^2}{1+x^2}\right) = 2\arctan{x}$$ for $x \geq 0$.
I'm not even sure what kind of math to try? :( |
Mathematical concepts named after mathematicians that have become acceptable to spell in lowercase form (e.g. abelian)? |
I've got two questions:
(1). I would like to know if by definition the Sobolev space norm is defined only by differentials with respect to one variable i.e.
$\parallel f \parallel \; = \; \left[ \int_{\mathbb{R}} \left( \left| f \right|^p \; + \; \left| \frac{\partial f}{\partial x} \right|^p \; + \ldots + \; \le... |
In India, generally during the graduate years, we follow a course work pattern, unlike many places in the U.S where students are exposed to research during their undergraduate years itself.
As, a graduate student, we have basic courses like, Algebra, Analysis, Rings and Modules, Measure theory, Topology, function... |
In classical real analysis I've only seen absolute continuity defined for functions on compact interval $[a,b]$, where the two equivalent definitions are: $f:[a,b]\rightarrow\mathbb{R}$ is AC if
(1) Given $\epsilon > 0$ there is a $\delta > 0$ such that $\sum_{i=1}^n |f(y_i)-f(x_i)|< \epsilon$ for every finite coll... |
My goal here is to make my own custom ["polyhedral map"][1] of Earth. If you print out something from the ["Map Fold-outs"][2] page, you will have something almost exactly like what I'm trying to make.
I have 2 triangles. One is a spherical triangle drawn on a 3D globe. By definition, each edge of a spherical triang... |
the fact that Ramanujan's Constant $e^{\pi \sqrt{163}}$ is almost integer ($262 537 412 640 768 743.99999999999925...$) doesn't seem to be a coincidence, but has to do with the $163$ appearing in it. Can you explain why it's almost-but-not-quite integer in layman's terms (I'm not a mathematician)?
TIA
Steven |
Why is $e^{\pi \sqrt{163}}$ almost integer? |
In classical real analysis I've only seen absolute continuity defined for functions on compact interval $[a,b]$, where the two equivalent definitions are: $f:[a,b]\rightarrow\mathbb{R}$ is AC if
(1) Given $\epsilon > 0$ there is a $\delta > 0$ such that $\sum_{i=1}^n |f(y_i)-f(x_i)|< \epsilon$ for every finite coll... |
This is quite a challenge to express in "layman's terms", but the
reason is that
$$j\left(\frac{1+\sqrt{-163}}{2}\right)$$
is an integer where $j$ is the [$j$-function][1]. When you substitute
$(1+\sqrt{-163})/2$ into the $q$-expansion (see the wikipedia page)
of $j$, all terms save the first two are small, and th... |
If A,B non empty, upper bounded sets and **A+B={a+b}**, how can I prove that **sup(A+B)=supA+supB**? |
Let $c$ be a postive integer that is not prime. Show that there is some positive integer $b$ such that $b|c$ and $b \leq \sqrt{c}$.
I know this can be proved by contradiction, but I'm not sure how to approach it. Usually I write the proof in the form $P \to Q$, and then if we can prove $P \land \neg Q$ is false, $P ... |
I asked this question on mathoverflow, but it was deemed too simple, so I'm posting here instead --
Is there a nice way to characterize an orthonormal basis of eigenvectors of the following $d\times d$ matrix?
$$\mathbf{I}-\frac{1}{d} \mathbf{v}\mathbf{v}'$$
Where $\mathbf{v}$ is a $d\times 1$ vector of 1's. ... |
If A, B two non empty sets of real numbers and for every a from A and b from B
> **a < b**
how can I prove that
> **supA < infB**
? |
If A, B two non empty sets of real numbers and for every a from A and b from B
> **a < b**
how can I prove that
> **supA <= infB**
? |
If A, B two non empty sets of real numbers and for every a from A and b from B
> **a < b**
how can I prove that
> **supA ≤ infB**
? |
In the question here: http://math.stackexchange.com/questions/4467/a1-2-is-either-an-integer-or-an-irrational-number
There was an answer by Douglas S Stones:
"
If $\sqrt{a}=x/y$ where $y$ does not divide $x$, then $a=(\sqrt{a})^2=x^2/y^2$ is not an integer (since $y^2$ does not divide $x^2$), giving a contradict... |
With the edited problem: if $a\lt b$ for all $a\in A$ and $b\in B$, why is it that $\sup(A)\leq\inf(B)$? Is every element of $A$ a lower bound for $B$? If so, what does that tell you about the elements of $A$ relative to $\inf(B)$? And given that, what does _that_ tell you about $\sup(A)$? |
**The problem:** Let X be a product of equidimensional spheres of arbitrary dimension, say k, and G a finite group acting freely on X. Assume that the induced G-action on the Z_2-cohomology ring of X is trivial. We then have a Serre spectral sequence with the E_2-page given by the expression H*(G; H*(X; Z_2)) and conve... |
Why does this particular Serre spectral sequence collapse at the E_{k+2}-page? |
The following is an example given by my professor, but there is an equality that I don't understand.
Let a partially ordered set $(S,\preceq)$ is the union of three sets such that $S=X\cup Y\cup Z$ such that $x\preceq y$ and $x\preceq z$ for all $x\in X, y\in Y, z\in Z$, but no $y\in Y$ and no $z\in Z$ are comparab... |
Here they talk about [abelian and tauberian theorems][1] after (Niels-Henrik Abel and Alfred Tauber), however it is also common to use [Abelian theorem][2] and [Tauberian theorem][3]. Also, in the last paper, by Lennart Carleson, we can read about "this thread" -- that is about people that becomes adjectives.
... |
The result does follow from unique factorization, but the point being made here is that the stated claim - that if $y$ doesn't divide $x$, it's also true that $y^2$ doesn't divide $x^2$ - is **equivalent** to the problem statement, so it's circular to use it to prove the problem statement (or in any case it doesn't add... |
The following is a problem in an algebra textbook, probably a well-known fact, but I just don't know how to google it.
Let $K/k$ be a finite field extension. Then $K(X)/k(X)$ is also finite with the same degree as $K/k$.
Obviously if $v_1,...,v_n$ is a $k$-Basis of $K$, any polynomial in $K(X)$ can be written a... |
Why does K->K(X) preserve the degree of field extensions? |
Some years ago I was interested in the following Markov chain
whose state space is the positive integers. The chain begins at state "1",
and from state "n" the chain next jumps to a state uniformly
selected from {n+1,n+2,...,2n}.
As time goes on, this chain goes to infinity, with occasional
large jumps. In any ... |
The [infinite] Ramsey theorem states that
> Let $n$ and $k$ be natural numbers. Every partition $\\{X_1,\ldots ,X_k\\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, for every $F\colon [\omega]^n\to \\{1, . . . , k\\}$ there exists an infinite $H \subseteq \omega$ such that $F$ is con... |
I decided to elaborate on my comment above.
What is $x^\pi$ if $x$ is negative? Trying to approximate with $x^r$ for rational $r$ is problematic, depending on which rational numbers you use to approximate; should the answer be positive, negative, imaginary? Perhaps more reasonable would be to take $x^\pi=e^{\pi \l... |
If I want to take the derivative of $ax^n$, I will get $anx^{n-1}$. If I were to take the derivative again, I get $an(n-1)x^{n-2}$.
We can generalize this for integer k easily to get the kth derivative $a\frac{n!}{(n-k)!} x ^{n-k}$. But what about for a more general k?
Does this have some name? Has it been wide... |
Is it meaningful to take the derivative of a function a non-integer number of times? |
It seems to be an accepted belief based on decades of experience that naive algorithms are not
adequate to solve NP-complete problems in a reasonable amount of time. Even those who believe P = NP seem to be looking for an algorithm with a very clever representation, so far without success.
Can the observations abo... |
I'm just starting to introduce myself to first-order logic, and I find it very confusing. As such, I'm not entirely sure how well I constructed the following. For $L$ a first-order language, let $L$ be equipped with the following nonlogical symbols: a unary function $\dot{+}$, a binary function $\dot{1}$, and a 3-ary r... |
The word "quad" generally means 4. Quadratics don't have 4 of anything. Can anyone explain where the name comes from? |
Why are quadratic equations called quadratics? |
Some hints:
If $S_{n} = \sum_{k=1}^{n} a_{k}$
then what is
$\lim_{n \to \infty} S_{2n} - S_{n}$?
Now can you use the fact that $a_{n}$ is non-decreasing to upper bound certains terms of the sequence $na_{n}$ with $S_{2n} - S_{n}$?
|
Some hints:
If $S_{n} = \sum_{k=1}^{n} a_{k}$
then what is
$\lim_{n \to \infty} S_{2n} - S_{n}$?
Now can you use the fact that $a_{n}$ is non-decreasing to upper bound a certain term of the sequence $na_{n}$ with a multiple of $S_{2n} - S_{n}$?
|
I have data modelled to be Gaussian-distributed, and then displayed on a logarithmic scale and like to know some properties of the displayed data.
Effektively, I ignore that a Gaussian $X$ (with positive mean) may be negative and need mean, standard deviation and full width at half maximum of $\log X$. Unfortunately... |
Characteristiccs of a Gaussian on a logarithmic scale? |
Given a quadratic form with discriminant $D$, what does the class number of $\mathbb{Q}(\sqrt{D})$ tell us?
(This question is inspired by a comment on the question [here](http://math.stackexchange.com/questions/4489/proving-that-the-given-diophantine-equation-has-a-solution)) |
What does the class number tell us about a quadratic form? |
Characteristics of a Gaussian on a logarithmic scale? |
Please mention ONE problem that is NP-Hard but not NP-Complete? And, explain why. I see some papers assert **Degree Constrained Minimum Spanning Tree** is an NP-Hard problem and some say it's NP-Complete. Why so? Is it something that we don't have clear idea about?
Thanks |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
How do I find a function from a differential equation that is in terms of itself? |
Why are differential equations called differential equations? |
Why are differential equations called differential equations? |
I am trying to identify the general case algorithm for counting the different ways dice can add to a given number. For instance, there are [six ways to roll a seven][1] with two 6-dice.
I've spent quite a bit of time working on this (for a while my friend and I were using Figurate numbers, as the early items in the... |
How can I (algorithmically) count the number of ways n m-sided dice can add up to a given number? |
I know the [four colour theorem][1] was solved by a computer checking a large number of cases. What I don't understand is why there are only a large but finite number of cases. It seems like there should be infinitely many planar graphs... What is the intuition behind cutting this down to a finite number of configurati... |
Intuition behind the four colour theorem? |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
Yet, another method for solving this differential equation is to look at it as a *linear differential equation*, whose general form is:
$$
y'(x) = a(x) y(x) + b(x) \ , \qquad\qquad\qquad [1]
$$
where $a(x), b(x)$ are arbitrary functions depending on the variable $x$. In your case:
$$
x = t \ , y(x) = Q(t) \... |
There are a number of ways to describe this problem. I shall name a few.
**Submultisets**
Let $(M, f)$ be a multiset where $M = \{x_1, ... x_k\}, |M| = k$ and $f(x_i) = m_i$, i ranging from 0 to k inclusive. Find the number of submultisets $(S,g)$ of $(M,f)$ of size n, $|(S, g)| = n$.
**Integer Partitions**
... |
There are a number of ways to describe this problem. I shall name a few.
**Submultisets**
Let $(M, f)$ be a multiset where $M = \{x_1, ... x_k\}, |M| = k$ and $f(x_i) = m_i$, i ranging from 0 to k inclusive. Find the number of submultisets $(S,g)$ of $(M,f)$ of size n, $|(S, g)| = n$.
**Integer Partitions**
... |
I will answer the following part of the question:
> And, explain why. I see some papers assert Degree Constrained Minimum Spanning Tree is an NP-Hard problem and some say it's NP-Complete. Why so?
Some people say “the Degree Constrained Minimum Spanning Tree problem (DCMST) is NP-hard” for a reason, other people sa... |
I made an assertion in [What are some examples of theories stronger than Presburger Arithmetic but weaker than Peano Arithmetic?][1] that Q has higher consistency strength than Pres, Presburger arithmetic; i.e., Q proves the consistency sentence for Pres.
But in fact, I only know something weaker, that Q can formali... |
Can Robinson's Q prove Presburger arithmetic consistent? |
Some hints:
If $S_{n} = \sum_{k=1}^{n} a_{k}$
then what is
$\lim_{n \to \infty} S_{2n} - S_{n}$?
Now can you use the fact that $a_{n}$ is non-increasing to upper bound a certain term of the sequence $na_{n}$ with a multiple of $S_{2n} - S_{n}$?
|
EDIT: the proof contains an error.
The division is always possible, which can be proved by complete induction.
The case n=1 is trivial - one stick of length 1. Then we suppose the assumption holds for n and consider the case n+1. I.e. given a number of sticks of integral length >= n+1 whose lengths add to (n+1)(n... |
You might also do this the other way around. What if
$$\lim_{n\to\infty}na_n \not=0,$$
that is if the limit does not exist or if it is positive, what does this tell you about $a_n$? What about sub-sequences of $(a_n)$? |
If you were to flip a coin 150 times, what is the probability that it would land tails 7 times in a row? How about 6 times in a row? Is there some forumula that can calculate this probability? |
What is the probability of a coin landing tails 7 times in a row in a series of 150 coin flips? |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
I believe that Theorem 1 of Bezboruah and Shepherdson 1976 [1] covers your question, at least in spirit. Their theory $T_0$ is a finite theory extending $Q$. Quoting their paper:
>**Theorem 1.** Let $L$ be any formal system with a recursive set of axioms, a finite number of finitary and recursive rules of inference ... |
There are a number of ways to describe this problem. I shall name a few.
**Submultisets**
Let $(M, f)$ be a multiset where $M = \{x_1, ... x_k\}, |M| = k$ and $f(x_i) = m_i$, i ranging from 0 to k inclusive. Find the number of submultisets $(S,g)$ of $(M,f)$ of size n, $|(S, g)| = n$.
**Integer Partitions**
... |
It is a well known theorem that any doubly connected domain can be conformally mapped onto an annulus.
Consider the simpler version :
Suppose $D$ is a bounded domain whose boundary is two non-intersecting circles. Then $D$ can be conformally mapped onto an annulus.
I believe that the proof of this should be ea... |
In measure theory, we have "lambda systems" and "pi systems". Pearl's message passing algorithm has "labmda messages" and "pi messages". Is there a reason that lambda and pi go together? |
Why do lambda and pi go together? |
Could anyone give an example of a sequence of *differentiable* (real) functions that uniformly converges to a *differentiable* function, but the derivatives of which don't converge to the derivative of the limit function? |
This is from the book, Counterexamples in Analysis by Gelbaum and Olmstead. (Google books link: http://books.google.com/books?id=cDAMh5n4lkkC)
This is under:
**A sequence of infinitely differentiable functions converging uniformly to zero, the sequence of whose derivatives diverges everywhere.**
$$f_{n}(x) = \... |
I'll sketch a solution; details are left to you.
As you flip your coin, think about what data you would want to keep track of to see whether $7$ heads have come up yet. You'd want to know: Whether you have already won and what the number of heads at the end of your current sequence was. In other words, there are $8$... |
Here are some details; I will only work out the case where you want $7$ tails in a row, and the general case is similar. I am interpreting your question to mean "what is the probability that, at least once, you flip at least 7 tails in a row?"
Let $a_n$ denote the number of ways to flip $n$ coins such that at no ... |
I think I see where your confusion is rooted. The notation f(x|θ) is chosen in the first place because there's a implication of a causal relationship between theta (the inferred cause) and x (the resulting observations.) And also because we usually have simple pdf representing the density on x given θ. This function ... |
Here are some details; I will only work out the case where you want $7$ tails in a row, and the general case is similar. I am interpreting your question to mean "what is the probability that, at least once, you flip at least 7 tails in a row?"
Let $a_n$ denote the number of ways to flip $n$ coins such that at no ... |
Here are some details; I will only work out the case where you want $7$ tails in a row, and the general case is similar. I am interpreting your question to mean "what is the probability that, at least once, you flip at least 7 tails in a row?"
Let $a_n$ denote the number of ways to flip $n$ coins such that at no ... |
There are a number of ways to describe this problem. I shall name a few.
**Submultisets**
Let $(M, f)$ be a multiset where $M = \{x_1, ... x_k\}, |M| = k$ and $f(x_i) = m_i$, i ranging from 0 to k inclusive. Find the number of submultisets $(S,g)$ of $(M,f)$ of size n, $|(S, g)| = n$.
**Integer Partitions**
... |
Let $G$ be a [directed graph](http://en.wikipedia.org/wiki/Directed_graph) which contains N vertex's, and which satisfies the following condition: each vertex of $G$ has at least one incoming edge.
(For clarity, a vertex $V_1$ has an "incoming edge" if there is an edge $E$ which connects $V_1$ to some other vertex o... |
Does this solve the zero sum subset problem? |
Let $G$ be a [directed graph](http://en.wikipedia.org/wiki/Directed_graph) which contains N vertex's, and which satisfies the following condition: each vertex of $G$ has at least one incoming edge.
(For clarity, a vertex $V_1$ has an "incoming edge" if there is an edge $E$ which connects $V_1$ to some other vertex o... |
Let $G$ be a [directed graph](http://en.wikipedia.org/wiki/Directed_graph) which contains N vertex's, and which satisfies the following condition: each vertex of $G$ has at least one incoming edge.
(For clarity, a vertex $V_1$ has an "incoming edge" if there is an edge $E$ which connects $V_1$ to some other vertex o... |
Let $G$ be a [directed graph](http://en.wikipedia.org/wiki/Directed_graph) which contains N vertex's, and which satisfies the following condition: each vertex of $G$ has at least one incoming edge.
(For clarity, a vertex $V_1$ has an "incoming edge" if there is an edge $E$ which connects $V_1$ to some other vertex o... |
As an example, consider the polynomial f(x) = x^3 + x - 2 = (x - 1)(x^2 + x + 2) which clearly has a root x = 1.
But we can also find the roots using Cardano's method, which leads to
x = ((28/27)^(1/2) + 1)^(1/3) - ((28/27)^(1/2) - 1)^(1/3)
and 2 other roots.
It's easy to check numerically that this expression ... |
A polynomial continued fraction (p.c.f.) is a continued fraction (c.f.)
$b_{0}+\underset{j=1}{\overset{\infty }{\mathbb{K}}}\left( a_{j}|b_{j}\right) =b_{0}+\dfrac{a_{1}}{b_{1}+\dfrac{a_{2}}{b_{2}+\dfrac{a_{3}}{b_{3}+\cdots }}}\qquad \left( 0\right) $
whose elements $a_{n},b_{n}$ are polynomial sequences [D. Bowm... |
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