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I am sorry for the vague title - its because I don't know what I am talking about.
I have a function whose value is determined by considering the outputs of some other functions. For example,
F(x) = G(a,b) and H(x) and T(y)
As you can see, I am not sure how to write the "and" part. That is, I want to say: In order to ... |
Consider a reversible markov chain Xn whose steady state distribution is known, can we find the expected hitting time to a subset A of the states starting from some state i ? Additionally you can assume the the chain has no self transitions.
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Hopefully without getting too complicated, how is a norm different from an absolute value?
In context, I am trying to understand relative stability of an algorithim:
Using the inequality $\frac{|(x_0)- \tilde{f}(\epsilon, x_0)|}{|f(x_0)|} \leq \sigma_{rel} ||\epsilon|| + o(||\epsilon||)$
For which $\sigma_{rel}$ descri... |
I am working on my own version of a proof of RSA and have come do a conclusion based on these simplified statements.
Given: N = pq
X ≡ 1 (mod p)
X ≡ 1 (mod q)
X ≡ 1 (mod N) by chinese remainder theorem
I have the original expression:
MX ≡ M' (mod N)
When substituting X ≡ 1 (mod N) into this equation, is it correct to s... |
Given the following sequence $d(n) = x \cdot n - \lfloor x \cdot n \rfloor$ with $n \in \mathbb{N}$, $x \in \mathbb{R} \backslash \mathbb{Q}$ .
I want to show that for every $z \in [0,1)$ there is a subsequence of $d(n)$ that converges against $z$.
Notes:
I am having serious problems of finding such a subsequence, firs... |
Many courses and books assume that rings have an identity. They say there is not much loss in generality in doing so as rings studied usually have an identity or can be embedded in a ring with an identity. What then are the major applications of rings without an identity occurring naturally in mathematics?
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A paracompact space is a space in which every open cover has a locally finite refinement.
A compact space is a space in which every open cover has a finite subcover.
Why must the product of a compact and a paracompact space be paracompact?
I really have very little intuition about how to go about this question, so any... |
Let $K(s,t)$ be a real-valued function of two real variables, and let $T: L^2(\mathbb{R}) \to L^2(\mathbb{R})$ be defined by $(Tf)(s) = \int_\mathbb{R} K(s,t) f(t) dt$.
If $||K||_{L^2({\mathbb{R}^2})} < \infty$, can we say that $T$ is a compact operator?
I think this is true if we are looking at a bounded domain for $... |
Background
A function $ f: \mathbb{R}^n \to \mathbb{R} \ $ is linear if it satisfies
$$ f(x+y) = f(x) + f(y) \ \text {, and} \tag 1 \label 1 $$
$$ f(\alpha x) = \alpha f(x) \tag 2 \label 2 $$
for all $ x,y \in \mathbb{R}^n $ and all $ \alpha \in \mathbb{R} $.
A function satisfying only \eqref{2} is not necessarily line... |
I'm stuck on the following part of a proof:
Let $\phi: \mathbb R^m \to \mathbb R^n$ be a function such that $\gamma'(t) := \phi(\gamma(t))$ is smooth for every smooth function $\gamma: \mathbb R \to \mathbb R^m$.
I want to show that $\phi$ is smooth under these assumptions.
Could someone give me a pointer?
Thanks in a... |
My question can be read on many levels and so I welcome answers to any reading. The general question is:
What is the computational complexity of verifying a proof?
One way of looking at a computational complexity class (for decision problems) is that is is a set of theories (a theory being a set of theorems) whose t... |
I am reading the lecture notes of geometric representation theory: http://www.math.harvard.edu/~gaitsgde/267y/catO.pdf. I have a question on coroot. In general, if we have a root $\alpha$, then the corresponding coroot is $\check{\alpha}=2\alpha/(\alpha, \alpha)$. But on page 1 of the lecture notes, line -14, it is sai... |
Dealing with US Dollars...
Assuming we know the weights of half-dollars, quarters, dimes, nickels, and pennies.
Also assuming the weights remain constant (don't change from one quarter to another quarter).
If we have a bag of these coins (could include any coin), and we weigh them to get a total weight, can we know how... |
I'm working my way through Enderton's Elements of Set Theory, and one of the theorems from the book has a final note that is giving me some trouble. The theorem is found on page 60, and is as follows:
Theorem 3Q Assume that $R$ is an equivalence relation on $A$ and that $F\colon A\to A$. If $F$ is compatible with $R$,... |
Can someone give me, with proof, an example of a Noetherian ring which has Krull dimension one but is not a Dedekind domain?
I think it would also be instructive to see other "near misses."
|
Fix $\epsilon >0$. Suppose $p_{n}(z) = z^{n}+a_{n-1}z^{n-1}+ \cdots + a_{0} \in \mathbb{Z}[x]$ is irreducible and has all positive real roots. Show that independently of $n$ except for finitely many explicitly computable exceptions that $|a_{n-1}| \geq (2-\epsilon)n$.
The sum of the roots is $-\frac{a_{n-1}}{a_n}$ whic... |
Define $\mathrm{E}(\mathbb{Z}_{i})$ to be the group of invertible elements of the ring with unity $\mathbb{Z}_{i}$.
Show that $\mathrm{E}(\mathbb{Z}_{mn})$ is isomorphic to $\mathrm{E}(\mathbb{Z}_{m}) \times \mathrm{E}(\mathbb{Z}_{n})$ if and only if $m$ and $n$ are relatively prime.
I see that it is relatively easy ... |
I have been learning about sheaves and am thinking about the following problem. Let $F$ and $G$ be sheaves, say of abelian groups, on a space $X$. The sheaf $Hom(F, G)$ is defined by $Hom(F, G)(U)=Mor(F|_U, G|_U)$. Given a point $p \in X$ and an open set $U$ containing $p$, a morphism $\varphi: F|_U \rightarrow G|_U$ ... |
I need to show that for prime numbers of the form $p=4n+1$, $x=(2n)!$ solves the congruence $x^2 \equiv-1\pmod p$.
I then need to show this implies p isn't a gaussian prime.
I have started to solve this using Wilson's theorem that a number $z$ is prime iff $(z-1)!\equiv-1\pmod z$. Therefore the endpoint of my proof sh... |
Are there such numbers $a$ and $b$ that if $a < b$, then $a > b$ ?
Thanks.
|
how can I divide for example $\frac{x^2+1}{2x+1}$ in $\frac{\mathbb{Z}_3[x]}{x^3+1}$? It's like normal polynomial dividing but here I got in first step $\frac{x}{2} (\frac{x^2}{2x}=\frac{x}{2}$).What is the modulo value of this statement? I need it for solve a linear equation solve, it's part of the Euclidan algorithm.... |
I have a simple question on a Lipschitz flow. Recall that a flow on a non-empty set $X$ is a map $\Phi: \mathbb R \times X \to X$ satisfying
$$\Phi(0,x) = x \text{ and } \Phi(s, \Phi(t, x)) = \Phi(s + t, x)$$
Now let $X$ be a Banach space and $F$ a Lipschitz-function on $X$. Now it is well known that there exists for e... |
If $A$ can complete a job in $6$ days. $B$ can complete it in $10$ days.If they work on alternate days, in how many days the job would be completed ? Does it depend on who ($A$ or $B$) starts the job?
The solution suggested in my module is to find the work completed by both of them in $2$ days and if this is not an int... |
I am looking for hints on how to solve following problems:
1. Show that in every simple graph, there is a path of length at least $\frac{2m}{n}$.
2. In $G$ there is a cycle $C$ and a path of length $k$, that connects two vertices from $C$. Show that there is a cycle of length at least $\sqrt{k}$ in $G$.
3. For all $k$,... |
Quite simply, I need a function that will produce a graph similar to the one below. Important to note:
1) Y switches from negative to positive at X=1/3
2) Y ranges from -0.1 to +0.1, not to and from infinity
3) Y should be very close to 0 by X=1/6 and X=3/6.
4) It should have curves similar to those in the picture (Mod... |
$M_n$ is a $n\times n$ matrix with $M_{n+1}=\begin{pmatrix}M_n & a_n \\ b_n^T & c_n\end{pmatrix}$ and $a_n, b_n, c_n \to 0$ for $n\to \infty$. Is this sufficient to state $$ \lim_{n\to\infty}(M_n^{-1}) = (\lim_{n\to\infty}M_n)^{-1} ?$$
|
I am trying to calculate the radius of convergence of
$\sum_{n=1}^\infty \frac{n^n}{(n!)^2}z^n$
with $z \in \mathbb{C}$. According to WolframAlpha, both the root and the ratio test are conclusive, indicating convergence. However at the root test, I am stuck at:
(For convenience I drop the limes superior in the follow... |
Given a natural number $n$, can we completely characterize when $n$ and $2n$ are each a sum of two squares?
For example: $446,382,709=(13010)^{2}+(16647)^{2}$ and $892,765,418=2(446,382,709)=(3637)^{2}+(29657)^{2}$
Do these imply, perhaps, that 446,782,709 is the hypotenuse of a primitive pythagorean triple?
(I found t... |
Given a 2D picture of a face, how is it possible to measure the distance between two different points on the surface of the face?
Thanks
Joel
|
I have been thinking on and off about a problem for some time now. It is inspired by an exam problem which I solved but I wanted to find an alternative solution. The object was to prove that some sequence of functions converges weakly to zero in $L^2$.
I managed to show (with some help) that the limit $f$ (of a subseq... |
One of my homework questions was to prove, from the axioms of ZF only, that a countable union of countable sets does not have cardinality $\aleph_2$. My solution shows that it does not have cardinality $\aleph_n$, where $n$ is any non-zero ordinal (not necessarily finite). I have a sneaking suspicion that my solution i... |
Given a map $f:B^n \to S^n$, where $B^n$ is the unit ball and $S^n$ is the unit sphere, is it true that the degree of $f|_{S^n}$ is always 0, where $f_{S^n}$ is the restriction of $f$ to $S^n$? If so, why?
Thanks!
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Is it 'expected' that PSL(2,p) contains a cyclic subgroup of order $\tfrac{p+1}{2}$? I read such an element exists and generates a "nonsplit torus", but I'm not sure what that means.
Is there an easy explanation why such an element should exist and have a cycle structure $\tfrac{p+1}{2},\tfrac{p+1}{2}$ in its action o... |
I have a very noob questions about generators: what algorithm do I have to follow so I can prove that a finite subset is a generator?
Here is the background story (I'll tell it all because I suck at maths and I might have understood the whole concept wrongly):
I want to know how to compute the image of a vector space m... |
There are many summer research programs in the United States, targeted at good motivated undergraduate students majoring in mathematics.
The main aspects that characterize such programs are: (a) a great deal of lectures on specific topics; (b) the chance to gain some hands-on experience with research projects.
Most o... |
Find the values of $m$ in the 2nd degree equation $mx^2-2(m-1)x-m-1=0$ so that it has only one root between $-1$ and $2$.
Like in this almost identical question there are two ways to solve this, one is acknowledging that $f(-1)f(2) \lt 0$, the other applying the theorems for the two possible scenarios $x_1 \lt -1 \lt x... |
Let $\sum \limits_{n=1}^{\infty} a_{n}$ be a
convergent series when $\forall n, a_{n}\neq 0$.
Does $\sum \limits_{n=1}^{\infty} (1-\frac{\sin a_{n}}{a_{n}})$ converge if:
$\forall n, a_{n}\gt 0$.
$a_{n}\lt 0$ for infinitely many $n$'s.
For (1) I tried looking at the Maclaurian series for $\sin x$:
$$|(1-\frac{\sin ... |
I was trying some old problems and got stuck on this one. Then when I looked at the answer there was a step I could not understand. Perhaps you can explain it to me.
A-3 Find
$ \displaystyle \lim_{t \to \infty}\left[ e^{-t}\int_0^t \int_0^t \frac{e^x - e^y}{x - y}dx dy \right]$
or show that the limit does not exist.
... |
Inspired by the recent incident in which a professor was able to [detect that students in his class were cheating][1], I'm curious if there is a standard way to detect if a distribution is bimodal, or a standard measure of the 'bimodality' of a distribution.
Does such a test exist? What are caveats that I should be awa... |
The four-colour mapping theorem states that all maps can be four-coloured (adjacent regions receive distinct colours, and four different colours are used in total). However, the technical definition of "map" implies that regions must be contiguous, and, in fact, it's easy to construct counter-examples to a proposed "f... |
$9$ points are placed in a $3\times3$ array. If $3$ points are randomly selected, what is the probability that they are the vertices of a triangle?
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The expression $x^2-4x+5$ is a factor of $ax^3+bx^2+25$. Express the sum a+b as an integer.
Please give an explanation of how the answer
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May I refer you to:
taken from page 92 of Infinite Dimensional Analysis by Charalambos D. Aliprantis.
Why $G \subset f^{-1}(W)$ ? I don't see this inclusion. Can you please help?
|
Hi I am new here and have a calculus question that came up at work.
Suppose you have a $4' \times 8'$ piece of plywood. You need 3 circular pieces all equal diameter. What is the maximum size of circles you can cut from this piece of material?
I would have expected I could write a function for the area of the 3 circles... |
This is more of a textbook semantics issue. Is the alternative hypothesis always two-tailed when all that is known is a null hypothesis $H_0$ where $p$ equals some arbitrary figure, where an arbitrary sample proportion and significance level are also given (assuming all requirements are met)?
|
Let $c$ be the set of all convergent sequences and $c_0$ the set of sequences tending to zero. How would one find the codimension of $c_0$? In particular, how would one find $\text{dim}
\ c/c_0$?
So one needs to choose some arbitrary sequence in $c_0$ and represent it as a linear combination of elements in $c/c_0$?
|
How to find the inverse function of the following:
$f\left( x \right)=\sum_{i=0}^{x}{\frac{2^{i}}{i!}}$
|
As an exercise I was doing a proof about equicontinuity of a certain function. I noticed that I am always choosing the limits in a way that I finally get:
$\text{Expression} < \epsilon$
However it wouldn't hurt showing that
$\text{Expression} \leq \epsilon$
would it, since $\epsilon$ is getting infinitesimally small? ... |
Question : Prove that the number is never divisible by 5.
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Edited Question:
Let $\mu$ be some measure on $\mathbb{R}^d$ and $m$ the Lebesgue measure.
Define the maximal function $$ M(x) = \sup_{r>0} \; \frac{\mu(B(x,r))}{m(B(x,r))} .$$
Why is $M$ a measurable function?
Old Question:
In class we defined the "maximal function" similar to how it's defined here, but instead of tak... |
What is the general term $(a_n)$ of the alternating sequence $\displaystyle \cos \left( \frac{3n \pi}{2} \right)$ from $1$ to $\infty$, $n \in \mathbb{N}$ ?
|
I'm doing the exercises in "Introduction to commutive algebra" by Atiyah&MacDonald. In chapter two, exercises 24-26 assume knowledge of the Tor functor.
I have tried Googling the term, but I don't find any readable sources. Wikipedia's explanation use the the term "take the homology", which I don't understand (yet).
Ar... |
I have a point '$a$' with known coordinates, from which I have drawn a tangent to a circle with center '$c$' which is also known. What is the best way of finding the coordinates of point '$b$', the touching point between the tangent and the circle? Here is a diagram: https://i.stack.imgur.com/gcKYn.jpg
Thanks,
David
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How to show that:
$$f(x)= \lim_{k \to \infty} \Bigl(\lim_{j \to \infty} \cos^{2j}(k!\cdot \pi \cdot x)\Bigl)$$
is the Dirichlet's function.
|
If $x^3 + \frac{1}{x^3} = 52$, then what is $x^2 + \frac{1}{x^2}$?
I'm not sure which formulae or methods should I used to solve this problem, so could somebody show me a way? what should I be looking for?
|
it might be a stupid question but I was discussing with a colleague when the 3rd axiom of probability (sigma additivity) is really needed.
I argue that in the case of a discrete distribution, say a single die, the first two axioms are sufficient as this distribution has a finite number of events. Is this right?
And th... |
While trying to answer this question, I got stuck showing that
$$\sqrt[3]{26+15\sqrt{3}}=2+\sqrt{3}$$
The identity is easy to show if you already know the $2+\sqrt{3}$ part; just cube the thing. If you don't know this, however, I am unsure how one would proceed.
That got me thinking ...
If you have some quadratic surd ... |
I'm writing a software that does simple mathematical calculations for lazy people.
I need ideas.
I already wrote some pieces of code that calculates 'Quadratic formula', but I have no idea what to do next.
|
As the title says, am searching for a proof of
If $A,B \in \mathbb{R}^{n\times n}$ and
$AB=0$ then $\mathrm{rank}(A)+\mathrm{rank}(B) \leq n$
I am doing this as preparation for an upcoming exam and can't figure a way to start. Please just post small hints as answers. I will try to go from there.
Thank you
ftiaro... |
Why do you need logarithms?
In what situations do you use them?
|
I have some difficulty in understanding the concepts: line bundles, line bundles on a homogeneous space, and sections of line bundles. These concepts are on page 140 (the first paragraph of section 9.3) of the book Young Tableaux by Fulton.
I know that $R \times S^{1}$, where $S^{1}$ is the unit circle and $R$ is the... |
Is there a general way to encode if-then-else/ITE in arithmetic, i.e. using the usual mathematical operators?
Example: let $f(x) = x^2$ if $x < 10^{30}$ and $log x$ otherwise, shortly written as $f(x) = ITE(x<10^{30},x^2,log x)$.
|
I am trying to check why curvature $0$ implies that the horizontal distribution is involutive.
Let $\pi:P\to U$ be a principal $G:=GL_n$ bundle. Assume that $P$ is trivial and $\pi$ admits a section. Thus, $P\cong U\times G$. A connection on $P$ is a $G$ equivariant splitting of the short exact sequence of bundles $0\t... |
I am trying very hard to understand Gödel's Incompleteness Theorem. I am really interested in what it says about axiomatic languages, but I have some questions:
Gödel's theorem is proved based on Arithmetic and its four operators: is all mathematics derived from these four operators (×, +, -, ÷) ?
Operations such as l... |
Does anybody know of any techniques to solve the following sum (or a fast way (polynomial in N) to compute it):
$$\sum_{i=0}^{N} \binom{N-i}{A} \cdot \binom{\lfloor iP/Q \rfloor + 1}{B}$$
where $P$ and $Q$ are relatively prime positive integers, and $A$,$B$ are positive integers.
When $P = Q = 1$, the solution can be f... |
Source: Lovasz, Plummer - Matching Theory
I had a question related to the number of perfect matchings in a graph. While going through the 8th Chapter on Determinant and Matchings in the text I stumbled across the first exercise problem. So far, I have been unable to solve either of the directions stated in the problem ... |
I have found interesting problem in Gilbert's Strang book, ,,Introduction to Linear Algebra'' (3rd edition):
How many corners does a cube have in 4 dimensions? How many faces? How many edges? A typical corner is $(0,0,1,0)$
I have found the answer for corners:
We know, that corner is $(x_1,x_2,x_3,x_4)$. For every $... |
I am looking for a term used to describe an analog to Voronoi diagrams where instead of a single point defining a cell, a continuous set of points is used. For example, starting with five triangles on the plane, you could construct a diagram that contained regions where each point in that region would be closer to a c... |
In Rudin's Real and Complex Analysis (p. 97 in my 3rd edition), the following is stated as a corollary to Baire's Category Theorem:
"In a complete metric space, the intersection of any countable collection of dense $G_{\delta}$'s is again a dense $G_{\delta}$."
Proof: "This follows from the theorem, since every $G_{\de... |
Let $f \in C[0,1]$ and $f(0)=f(1)$.
How do we prove $\exists a \in [0,1/2]$ such that $f(a)=f(a+1/2)$?
In fact, for every positive integer $n$, there is some $a$, such that $f(a) = f(a+\frac{1}{n})$.
For any other non-zero real $r$ (i.e not of the form $\frac{1}{n}$), there is a continuous function $f \in C[0,1]$, suc... |
We have a problem the arises in asset management. I think (and hope) it raises interesting enough questions for this forum to consider it. We made pretty extensive searches of the literature and find things that talk around this question, but nothing directly dealing with the issue.
Background
We have time series dat... |
In an article by Tom Davis on Catalan numbers, Davis mentions several problems that are shown to be equivalent. One of them is the problem of determining the number of strings of $n$ pairs of balanced parentheses. Another is the problem of determining the number of "multiplication orderings" of $n + 1$ numbers. In both... |
Assuming the Axiom of Choice, (it seems that) there is a bijection between $\mathbb{R}$ and $\mathbb{N}$ that follows from any well-ordering of the reals. That is, given a well-ordering of $\mathbb{R}$, the nth real number in the ordering would correspond to the nth natural number.
On the contrary, if the reals are ass... |
I am so lost on this question I am not sure even where to start. I am not looking for an answer but more of a turn in the right direction.
Find all the eigenvalues and eigenfunctions of the following linear transformation: $T(f(x)) = f(-x)$ from $\mathbf{P}_2$ to $\mathbf{P}_2$. IS T diagonalizable?
All I know for... |
Suppose
$$\frac{d^2y}{dx^2} = \frac{e^{-t}}{1-e^t}.$$
After finding the second derivative, how do I find concavity? It's much easier to solve for $t$ in a problem like $t(2-t) = 0$, but in this case, solving for $t$ seems more difficult. Does it have something to do with $e$? $e$ never becomes $0$, but at what point is... |
Minkowski's inequality says the following: For every sequence of scalars $a = (a_i)$ and $b = (b_i)$, and for $1 \leq p \leq \infty$ we have: $||a+b||_{p} \leq ||a||_{p}+ ||b||_{p}$. Note that $||x||_{p} = \left(\smash{\sum\limits_{i=1}^{\infty}} |x_i|^{p}\right)^{1/p}$. This is how I tried proving it:
\begin{align*}
|... |
Given integer $n$ and real $0\leq d<1$, set $x := 0$, set $c := 0$.
set $y := n-x$, if $y < 1-d$, we are done
Randomly chose $l\in [1-d, \min(1+d,y)]$, set $x := x + l$, $c := c+1$. Go to 2.
So this is a program that count how many random numbers in $[1-d, 1+d]$ one can add until one reach $n$, except the program won... |
Let's say we have a 2D Polygon ( it can be concaved or convex), and on top of this polygon we are required to build a house roof. What is the algorithm that one can use for this?
The conditions:
The roof cannot be $0^o$ or $90^o$.
The created roof piece must be as smooth as possible ( smooth is defined as first order... |
We would like to generate simple mathematical formulas on the fly (e.g. $\sin^2 \theta$). The above function seems to be a MathJaX representation. Most of the solutions depend on writing mathematical equations using MathML or Latex or jsMath or MathJaX. I am assuming MathJaX would be the right solution to use if you ar... |
I have the following issue #26 from What Is the Name of This Book? of R. Smullyan:
There is a wide variety of puzzles
about an island in which certain
inhabitants called "knights" always
tell the truth, and others called
"knaves" always lie. It is assumed
that every inhabitant of the island
is either a ... |
I am given a vector space $V$, a basis $e_1,\dots,e_n$ and a vector $v\in V$ and I am asked to find $\lambda_1,\dots\lambda_n$ from the underlying field such that $v=\sum \lambda_i e_i$.
How can I do it?
Of course, if I know the basis is orthonormal then I can compute the inner product between $v$ and the $e_i$'s, but ... |
Let's $G=(U, V, E)$ be a balanced bipartite graph which $|U|=|V|=n$ and $|E|=n*(n-1)$; All nodes in $U$ are connected to all nodes in $V$ except $u_i$ to $v_i$ for $1\leq i \leq n$.
Definition1: Cross edges are two edges in $E$, one with two end points $u_i$, $v_j$ and the other with $u_j$, $v_i$.
Definition2: Good-p... |
If $a^2 = b^2 $ and $a,b \gt 0$ then we can answer "Is $a > b$?"
I know that $a^2 = b^2 \Rightarrow |a| = |b|$ but still I don't understand why the second condition is absolutely necessary.
|
What is exactly meant by
angle subtended by all sides of a polygon at its center
I encountered these while solving this problem:
How can we determine the number of sides of a polygon, given the angles subtended by all the side at its center are equal to $72 ^\circ$?
Could somebody please explain (with a figure) what ... |
I am trying to find the complex roots of: exp(4z)=i.
However I am confused as how to calculate tan-1(1/0) to find the argument.
Any help would be much appreciated. Thanks
|
Assume I have a universe of N elements.
The question is: How many sets of size $X$ are needed to assure that every set of K elements is a subset of (at least) one of these sets (where $K \ll X \lt N$). And also, how can these sets be chosen to obtain this minimum?
In particular, the sizes that interest me are:
\begin{... |
Let $U$ be unitary and $D$ be diagonal. Can anything be stated about the structure of $$U^\dagger D U$$ then?
Also, would this change for an infinite matrix, i.e. a discrete linear operator?
|
Consider the triangles with integer sides $a$, $b$ and $c$ with $a \leq b \leq c$.
An integer sided triangle $(a,b,c)$ is called primitive if $\gcd(a,b,c)=1$.
How many primitive integer-sided triangles exist with a perimeter not exceeding $10 000 000$?
I am trying to solve this on Euler project. I am wondering what is ... |
This is an old homework problem of mine that I was never able to solve. The solution may or may not involve the Baire category theorem, which I am terrible at applying.
Let $C[0, 1]$ denote the space of continuous functions $[0, 1] \to \mathbb{R}$. Suppose $|| \cdot ||$ is a norm on $C[0, 1]$ with respect to which th... |
I'm sure there must be lots of examples of three (infinite) groups $A$, $B$, $C$ where $A<B<C$ , $A\cong C$, $A\not\cong B$.
Here's one example with free groups: $\mathbf F(a^2,ab) < \mathbf F(a^2,ab,ab^{-1}) < \mathbf F(a,b)$, where A,C are a free group on 2 symbols and B is a free group on 3 symbols (in fact precise... |
How to sum $2^{2m}$ where $m$ varies from $0$ to $n$?
|
Here's a problem that I have pondered over many times without ever coming to a satisfactory solution:
Let's say that we have a series of random events: V(i) for I = 1 to n. Each of these events will have a result VR(i) of either 0 or 1, and a probability VP(i) (from zero to one) that represents the probability that VR... |
I just finished reading the proof that the average height of a random binary of given size $n$ is asymptotically $2\sqrt{\pi n}$.
I'm now searching for an intuitive, or geometric, or visual proof of this asymptotic equivalence. In other words, I'm trying to find an intuitive arguments that shows that this number grows ... |
Let M be a finitely generated R-module ,where R is a PID and N a submodule.Is there an epimorphism from M to N?
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How to find the sum of series, $$\sum_{k=1}^n 2^kC(n,k)$$
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I am a beginner in representation theory and algebraic geometry, so that references giving clear explanations of things like the tautological line bundle on $\mathbb P^n$, its dual, and the associated sheaves of sections $\mathcal O(-1)$ and $\mathcal O(1)$ would be especially welcome.
I have some difficulty in underst... |
The solution of the cubic equation is known in terms of a rational function of a cube root of a square root. If we just want to know the value it's easy to approximate it using a numerical method. I would like to know if the actual form (in terms of roots) of the solution of the cubic has any applications in number the... |
Given $f:(0,\infty) \rightarrow \mathbb{R}$ and $f''(x)>0\forall x \in (0,\infty)$. Is it correct to say that $f'(x)$ is a monotonically increasing function? Can I correctly assume that for any $a,b \in (0,\infty)$ $f'(a) > f('b)$ if $a > b$?
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$S$ is real symmetric. $T$ is real skew-symmetric. I have shown that $T\pm iS$ is skew-Hermitian. I am further asked to show that $U = (I+T+iS)(I-T-iS)^{-1}$ is unitary.
Denoting by $^\dagger$ the conjugate transpose, I have that
$U^\dagger = [(I+T+iS)(I-T-iS)^{-1}]^\dagger$
$= ((I-T-iS)^\dagger)^{-1}(I+T+iS)^\dagger$
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I was reading "Graph Theory" by Diestel and I tried to solve a problem from the chapter 3 (on connectivity). It seems at first sight easy (and really intuitive) but I have to admit that I can't work it out! Here is the problem: prove that every $k$-connected graph of order at least $2k$ contains a cycle of length at le... |
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