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I have the following constraint inequalities and equalities:
$$Ax \leq b$$
$$A_{eq}x = b_{eq}$$
The problem is that the objective function, which I am asked to minimized, is defined as
$$f=\sum\limits_{i=1}^{m}u([\sum\limits_{j=1}^{n}k_{ij}x_j]-c_{i})$$
where $u(x)$ is the Heaviside step function.
Strictly speaking, t... |
I've been stuck on this homework question for a couple of days. Any hints would be much appreciated.
Let $G$ be a finite group and $\rho : G \to \mathrm{GL}(V)$ be a faithful representation, with $V$ a vector space over the field $F$. Suppose $Z(G) = \{ e \}$, and $H \le G$ is a subgroup with $Z(H) \ne \{ e \}$. Then, ... |
According to wikipedia the group isomorphism problem is an undecidable problem.
When we restrict to (countable) abelian groups does it become decidable or does it remain undecidable?
In case it becomes decidable I'd love to have a reference to an algorithm deciding it.
|
You have a circular table with $N$ seats.$K$ bellicose guests are going visit your house of-course you don't want them to sit beside each other.As the host, you want to find out how many ways there are to choose $K$ seats such that none of them is adjacent to each other.
I noticed that there is a solution other than ... |
Is there any good way to approximate following integral?
$$\int_0^{0.5}\frac{x^2}{\sqrt{2\pi}\sigma}\cdot \exp\left(-\frac{(x^2-\mu)^2}{2\sigma^2}\right)\mathrm dx$$
$\mu$ is between $0$ and $0.25$, the problem is in $\sigma$ which is always positive, but it can be arbitrarily small.
I was trying to expand it using Tay... |
I have a PDE, which I need to calculate the Courant number for. Does anyone have any resources on that which can help I. I do want a general way of calculating the number if that is possible, so I can use it further.
|
I am currently preparing my next physics exam, and I got courious if there may be on the Net some sort of time scale of mathematical discoveries, so that I could compare discoveries and achievements in physics with the mathematic instruments available at the time.
While there is large abundance of such graphs for compu... |
I don't understand left coset and Lagrange's theorem... I'm working on some project so please write me anything what could help me understand it better
|
Given a sequence of N numbers, how can we find the minimum number of numbers to be inserted to make this sequence to an Arithmetic progression.(we can insert at any position of this sequence)
For example consider a sequence of $4$ numbers :$1,3,7,13$ Here we have to insert at-least $3$ more numbers ($5,9,11$) so that $... |
I just had a short look on the definition of a Vector Space and couldn't find any obvious reason why
$\lambda a = a \lambda$
where $\lambda$ is an element of the field $K$ and $a$ is an Element of the set $M$, should be true. Its kind of intuitive since all the vectorspaces I normally come across have this property (Po... |
A question goes like this:
Given that there 2 questions out of 3 questions to pass an exam and follow are details of number of people who attended the questions correctly, what is the maximum number of people who could have qualified the exam.
1. 2 2 2
The answer here is 3. Consider ab bc ca. It can also be derived as... |
Let $A$ be a commutative noetherian ring, and let $M$ be a finitely generated projective $A$-module. It is well known and easy to prove that $M$ is locally free in the sense that for every $p \in\operatorname{Spec} A$, the module $M_p$ is a free $A_p$-module.
Is it true that projectives are also locally free in the fol... |
Here is a question that I have had in my head for a little while and was recently reminded of.
Let $X$ be a (nonempty!) topological space. What are useful (or even nontrivial) sufficient conditions for $X$ to admit a group law compatible with its topology?
There are many necessary conditions: for instance $X$ must b... |
The problem is:
For a prime number p the set of co-primes less than or equal to it is
given by {1,2,3,4,...p-1} .
We define f(x,p) 0 < x < p = 1 if and only if all the
numbers from 1 to p-1 can be written as a power of x in modulo-p arithmetic .
Let n be the largest 12-digit prime number . Find the product of
all int... |
As the title says, I'm wondering if there is a continuous function such that $f$ is nonzero on $[0, 1]$, and for which $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 1$. I am trying to solve a problem proving that if (on $C([0, 1])$) $\int_0^1 f(x)x^n dx = 0$ for all $n \geq 0$, then $f$ must be identically zero. I presume ... |
I need to write a program to solve matrix equations Ax=b where A is an nxn matrix, and b is a vector with n entries using LU decomposition. Unfortunately I'm not allowed to use any prewritten codes in Matlab. I am having problems with the first part of my code where i decompose the matrix in to an upper and lower matri... |
I was discussing with a friend of mine about her research and I came across this problem.
The problem essentially boils down to this.
$f(x)$ is a function defined in $[0,1]$ such that $f(x) + f(1-x) = f(1)$. I want to find a condition on $f(x)$ so that I can conclude $f(x) = f(1)x$.
Clearly, $f \in C^{0}[0,1]$ alone is... |
how do I do the following:
Consider the matrix $\begin{bmatrix}
\cos x & -\sin x\\
\sin x & \cos x
\end{bmatrix}$
and the vector $\begin{bmatrix}
\cos y\\
\sin y
\end{bmatrix}$. Show that the product of the $2\times 2$ matrix with $2\times 1$ matrix $\begin{bmatrix}
\cos(x+y)\\
\sin(x+y)
\end{bmatrix}$ g... |
Two sets
x = 1,2,3,4,5...
y = 1,4,7,10,13...
I need to write a function
$f(x_n) = y_n$
I found that if I
take a number from x
double it
subtract 2
add the result to the original number
I get the corresponding number from y
Here's what I have so far (it's in ruby code), it works but is there a better way of doing ... |
Suppose that A is a subset of B. How can we show that B-complement is a subset of A-complement?
|
$16a + 23b + 12c + 34d -2e -37f + 109g -141h + 139i + 149j + 29k + 12l + 131m = 74608$
$13a + 31b -5c + 17d + 29e -67f -16g -101h -7i -201j + 32k + 17l + 171m = -4194$
$81a -12b -5c + 17d -9e + 17f + 39g + 49h -67i -7j -41k + 13l + 121m = -15793$
$91a -43b + 17c + 19d + 17e + 23f + 97g + 101h -101i + 173j -12k + 137l ... |
In Milnor and Stasheff, it is taken as part of the first axiom that all Stiefel-Whitney classes of a bundle vanish in dimensions greater than the rank of the bundle. However, in other sources this is omitted. The wikipedia article makes it seem like this is actually a result of the axioms. Is there a simple proof of... |
We have been told in our homework to implement the following program:
function A = Cholesky(A)
% Cholesky Decomposition of a Matrix A, so that the resulting matrix L gives A=L*L
N = size(A,1);
for n=1:N
if abs(A(n,n))
error('* ERROR * Cholesky Decomp doesn't exist');
end;
A(n:N,n) = A(n:N,n) -... |
The problem is from Marden, the first section:
The polynomial $g(z) = z^n + b_1 z^{n-1} + ... + b_n$ has at least $m+1$ zeros in an arbitrary neighborhood of a point $z = c$ if $|g^{(k)}(c)| \leq \epsilon$ for $k = 0,1,...,m$ and for $\epsilon$ sufficiently small and positive.
There is a hint provided: use Rouché's The... |
How can we show, rigorously, that if $f: \mathbb{R} \rightarrow \mathbb{R}$ is a continuous map then its graph and the inverse of the graph (i.e the inverse of the relation) is a nowhere dense set? I'm trying to show that the plane cannot be covered by a countable number of graphs of continuous maps and its inverses, s... |
I think people here could guide me in solving a problem related to anomaly detection. The term anomaly here refers to some undesired event occurring in the system like a virus infection.
I could get to know about it from more than one source. For example extracting value from two different data structure and if the val... |
Let $f : \mathbb R \to \mathbb R$ be a given function with $\lvert f(x) \rvert \le 1$ and $f(0) = 1$. Is there a nice simplified expression for $$\begin{align}F(x) &= f(x) f(x/2) f(x/4) f(x/8) \cdots \\ &= \prod_{i=0}^\infty f(x/2^i)?\end{align}$$ If there isn't a general solution, as seems likely, can anything useful ... |
In[11]:= Select[Table[Fibonacci[n], {n, 1, 10000}], PrimeQ[# - 1] &]
Out[11]= {3, 8}
Edit: Fibonacci[n]-1 is always composite for n>6. why?
$$\sum\limits_{i = 0}^n {{F_i}} = {F_{n + 2}} - 1$$
In[16]:= Select[Table[Fibonacci[n], {n, 1, 10000}], PrimeQ[# + 1] &]
Out[16]= {1, 1, 2}
Fibonacci[n]+1 is always composite for... |
I'm stuck on something that must be incredible simple to do. I calculate the second derivative of $f(x) = e^{\sin(x)}$ like so in Maple:
diff(diff(exp(sin(x)),x),x);
But now i want to calculate $D^2f(0)$, and can't seem to find a proper way to do this...
I tried:
fsolve(Deriv = 0, x);
with Deriv:=diff(diff(...)) bu... |
Perform a taylor expansion in 3 dimensions in time on the time compontent of of $T^{\alpha \beta}(t - r + n^{i} y_{i})$ given that $r$ is a contstant and $n^{i} y_{i}$ is the scalar product of a normal vector $n$ with a position vector $y$.
I believe the formula is $\phi (\vec{r} + \vec{a}) = \sum^{\infty}_{n=0} \frac{... |
I have the answer of $1$. But for the answer they split it up into $b_n = n/\sqrt{n^2+1}$ and $c_n =\sin n/\sqrt{n^2+1}$ which is fine.
Then for $b_n$ they divide evrything through by $n^2$ including everything inside the square root to give $1/\sqrt{1+1/n^2}$ and I thought you couldn't do this and checked with real nu... |
The background of this question is this: Fermat proved that the equation,
$$x^4+y^4 = z^2$$
has no solution in the positive integers. If we consider the near-miss,
$$x^4+y^4-1 = z^2$$
then this has plenty (in fact, an infinity, as it can be solved by a Pell equation). But J. Cullen, by exhaustive search, found that the... |
Stirling numbers of the second kind $S(n, k)$ count the number of ways to partition a set of $n$ elements into $k$ nonempty subsets. What if there were duplicate elements in the set? That is, the set is a multiset?
|
How much faster are various mathematical operations using a Trachtenberg method rather than a conventional method?
|
If $x$ is prime and $1000000000000 < x < 1000100000000$,
how can I detect if $1 / x$ will result in a terminal decimal or not?
I can use programming.
|
Let $X$ be a variety and $\mathbb{C}$ be the field of complex. Then $L = X \times \mathbb{C}$ is a trivial line bundle. The set of sections of this line bundle is $\Gamma(X, L)$ which consisting all functions $s: X \to L$ such that $\pi \cdot s = id$, where $\pi: L \to X$ is the projection map. Why $\Gamma(X, L)$ is th... |
I have a real-life situation that can be solved using Queueing Theory.
This should be easy for someone in the field. Any pointers would be appreciated.
Scenario:
There is a single Queue and N Servers.
When a server becomes free, the Task at the front of the queue gets serviced.
The mean service time is T seconds.
The ... |
I have this circuits and signals course where i have been asked to learn matlab all by myself and that its treated as a basic necessity
now i wanted some help as to where should i start from as i hardly have around less than a month before my practicals session start
should i go with video lectures/e books or which one... |
I can't do this exercise of Conway's Book: For $r>0$ let $A=\{w:w=\exp(1/z), 0<|z|<r\}$, determine the set $A$. Any hints?
|
Let $f : M \rightarrow \mathbb{R}$ be a differentiable function defined on a riemannian manifold. Assume that $| \mathrm{grad}f | = 1$ over all $M$. Show that the integral curves of $\mathrm{grad}f$ are geodesics.
|
Let $\{e_0, e_1, \dots, e_{N-1} \}$ be the Euclidean basis for $l^2(Z_N)$, and let $\{F_0, F_1, \dots, F_{N−1} \}$ be the Fourier basis( where $F_m(n)= \frac{1}{N} e^{2 \pi i m n/N}$ and $\hat{z}(m) = \sum_{n=0}^{N-1} z(n) e^{-2 \pi i m n/N}$).
How to show that $\hat{e}_m(k) = e^{−2πimk/N}$ for all $k$? Notice that $\... |
Hi I have some problem how to get count of elements in $\Bbb{Z}_7[x]/(3x^2+2x)$. I think there belong to only polynomials which are indivisible with $3x^2+2x$ ($\gcd=1$). I think it is so as far I know that for example in every $\Bbb{Z}_m$, $m$ prime, is the count of belonging elements eqauls to $\phi(m)$. But I really... |
I was reading the Wikipedia article for Mathematical_logic. When reaching Formal_logical_systems, I was curious about its definition and clicked into its own article Logical_system, which redirected me to the article for Formal_system, where I found the definition for formal system:
In formal logic, a formal system (a... |
Let P be a point, not the center, in the interior of a (round) disk D⊂ℝ² and let A and B be points on ∂D such that the line segments AP and BP have equal length. Choose an arc AB. What's the shape bounded by the arc and the two segments called?
|
Let $\{f_i\}$ be a system of linear equations in $X_1,...,X_n$ with coefficients in $R = \mathbb Z/p^e \mathbb Z$ (i.e. modulo a primepower).
Assume
there is a unique solution $a_i$.
$f_1 = u g_1$ where $u$ is a non-unit, i.e. $p\mid u$.
Is it true that $\{f_i\mid i>1\}$ permits only the same unique solution? Or equi... |
Hi I have another problem..Two polynomials a(x) and b(x) are asociated iff a(x)|b(x) and b(x)|a(x)….Right? And now my problem..And polynomials are indivisible when gcd is asociated with 1..And there it is..In which universe is it for example 2??As I know 1 is indivisible by 2?Is there something I am overlooking? yes an... |
I must find the area of the region that lies inside the first curve and outside the second curve for:
$ r^2 = 8cos2\theta, r = 2$
How do I find $a$ and $b$? My solution manual tells me that the curves intersect when $cos2\theta = 1/2$ but I don't see where that came from. I tried setting it equal to each other, but kee... |
This is a softmax probability distribution:
$$P(i\mid w_1, w_2, \ldots, w_n) = \frac{\exp(w_i)}{\sum_{i=1}^n \exp(w_i)}.$$
It known also as Boltzmann distribution. It is used in generalized Bradley-Terry model and in multinomial logistic regression. There are efficient minorization-maximization algorithms for infering ... |
(this question is about trying to use some combinatorics to simplify an algorithm and save memory)
Let $K_{2n+1}$ be a complete undirected graph on $2n+1$ vertices.
I would like to generate a Eulerian circuit of this graph (visit each edge exactly once).
One solution is to run the DFS-based algorithm that can find a Eu... |
Could somebody check if this is correct?
$$\lim_{n \to \infty} \frac {\log_{2}(\log_{2}(n))}{\log_{2}(n)}$$
I exponantiate the numerator and the denominator with 2
$$\frac {(\log_{2}(\log_{2}(n)))^2}{(\log_{2}(n))^2}$$
$$ = \frac {\log_{2}(n)}{n}$$
I extract the constant from the logarithm
$$ = \log_{2}(e) * \lim_{n \t... |
In Solving very large matrices in "pieces" there is a way shown to solve matrix inversion in pieces. Is it possible to apply the method in-place?
I am refering to the answer in the referenced question that starts with:
$\textbf{A}=\begin{pmatrix}\textbf{E}&\textbf{F}\\\\ \textbf{G}&\textbf{H}\end{pmatrix}$
And computes... |
I've tried solve this for hours, but I still stuck in the beginning. Here's my problem:
Let $R \subseteq P(\mathbb{N} \times \mathbb{N})^2$ be the following equivalence relation:
$\langle r,s\rangle \in R \Leftrightarrow (\exists \pi : \mathbb{N} \to \mathbb{N} \forall x,y \in \mathbb{N} (\langle x,y\rangle \in r \left... |
I need to show that:
a) v(u) = 1 <=> v is in E(R), where E(R) is the group of elements in R with an inverse
b) a|b => v(a)|v(b) => v(a) ≤ v(b) oder b=0
c) a~b <=> v(a)=v(b) and a|b
I think I have managed to solve part b. Here is my proof:
a|b => there exists x in R s.t. b=ax
=> v(b)=v(ax)
(v euclidean) => v(b) =v(a)... |
In Summer Wars the main character (he is a mathematician) calculates the day of the week of someone's birthday (19/07/1992 is Sunday). I know (very) basic modular arithmetic but I can't figure out how to do it. Can someone point me to the right direction? It seems fun to do.
|
The problem
Show that the sequence, $49, 4489, 444889, \dots$, gotten by inserting
the digits $48$ in the middle of the
previous number (all in base $10$), consists only of
perfect squares.
has become a classic. For some reason, I got curious as to who actually discovered this.
After doing some research, seems... |
I've run into a small problem while working through Enderton's Elements of Set Theory. I'm doing the following problem:
Call a natural number even if it has the form $2\cdot m$ for some $m$. Call it odd if it has the form $(2\cdot p)+1$ for some $p$. Show that each natural number number is either even or odd, but neve... |
Question: Show that $\pi_{1}({\mathbb R}^{2} - {\mathbb Q}^{2})$ is uncountable.
Motivation: This is one of those problems that I saw in Hatcher and felt I should be able to do, but couldn't quite get there.
What I Can Do: There are proofs of this being path connected (though, I'm not exactly in love with any of those... |
I know that I need to find an irreducible element that isn't prime.
I was wondering whether these examples work.
In $\mathbb{Z}[\sqrt{10}]$, $9=(3)(3)=(\sqrt{10} +1)(\sqrt{10} -1)$.
$9$ is not prime, but is irreducible.
In $\mathbb{Z}[\sqrt{-17}]$, $18=(6)(3)=(1-\sqrt{-17})(1+\sqrt{-17})$.
Is writing the number as a p... |
Is it easy to describe them?
$G = \operatorname{Alt}(2p)$,
$P$ a Sylow $p$-subgroup generated by $(1,2,...,p)$ and $(p+1,p+2,...,2p)$,
$N = N_G(P)$,
$M = \{g \in \operatorname{Alt}(2p) : g \text{ takes every orbit of }P\;\text{ to some orbit of }P\;\}$
Is it true that the unique subgroup strictly between $N$ and $G$... |
I asked a question a couple days ago so please let me know if I'm asking too frequently on here, I'm not sure what the etiquette generally is. In short, I'm looking to prove that for P, Q distinct points in the complex plane $\mathbb{C}$, $\mathbb{C}-\{P,Q\}$ is not conformally equivalent to either $\mathbb{C}$ or $\ma... |
I was recently working on a piece of software and needed to know the number of different ways that a particular word could be represented in mixed upper-and-lowercase letters. For example, the word "bar" could be represented as:
{bar, Bar, bAr, baR, BAr, bAR, BaR, BAR}.
Or, more generally, let 0 = lowercase and 1 = ... |
So if $X$ is a vector space, and you define a norm, $x \mapsto \| x \|$, on it, then the bounded subset, $V = \{ x \in X: \|x\| < \infty \}$ is automatically a subspace.
This follows from the definition of a norm, so for all $x,y \in V, c \in \mathbb{C}$, $\|x + c y\| \le \|x\| + \|cy\| = \|x\|+ |c| \|y\| < \infty$, s... |
Let $Q: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ be an operator that preserves all distances. Is this condition alone enough for us to say that $Q$ must be a linear operator?
If not, what are some counterexamples (whether simple or pathological), where an operator preserves all distances but is not a linear operator?
[E... |
I've 2 questions:
let be $W_s$ a standard Brownian motion:
using Ito's formula show that $\left( W_t,\int_0^t W_sds \right)$ has a normal distribution;
and calculate $ E\left[e^{W_t}e^{\int_0^t W_sds} \right] .$
For the first part, i know that $W_t$ and $\int_0^t W_sds$
have normal distribution with mean and varia... |
I've created a JavaScript-based fractal drawer which you can see here:
http://jsfiddle.net/xfF3f/12/
As you're probably all aware, a Mandelbrot Set is created by iterating over pixels as though they were coordinates on the real and imaginary planes. Each pixel then has a real and an imaginary part (like all complex num... |
Can you please give me examples of real analytic functions with finite number of zeros that are not polynomials? I know that there are $\mathrm{e}^x$ and $\operatorname{sh}x$, but I can't think of anything else.
It would be even better if there was a method of constructing a large class of such functions.
|
I am trying to solve this problem, and i think i did something, but finally i couldn't get the conclusion. The question is:
Let $(X,d)$ be a metric space and let $A,B \subset X$. If $A$ is an open dense subset, and $B$ is a dense subset, then is $A \cap B$ dense in $X$?.
Well, i think this is true. We have to show th... |
Is the P versus NP question asking "P = NP" or "ZFC |- P = NP" (or "|- P = NP" for that matter)? Because if I say P = NP, then I will be asked to prove it. But if the goal is "ZFC |- P = NP" then the result will not be useful because of the set theoretic assumptions of ZFC. So you may say the third choice matches our i... |
Suppose $f$ is increasing and there exist numbers $A$ and $B$ such that for all $x$, $A \leq f(x) \leq B$. Show that for each $\epsilon >0$ the number of jumps of size exceeding $\epsilon$ is at most $(B-A)/\epsilon$.
Let $\epsilon >0$. So the jump size is $f(x+)-f(x-)$. So it seems that we divide the maximum possible... |
one can obtain solutions to the Laplace equation
$$\Delta\psi(x) = 0$$
or even for the Poisson equation $\Delta\psi(x)=\varphi(x)$ in a Dirichlet boundary value problem using a random-walk approach, see e.g. Introduction to Brownian Motion.
Now, already this fascinating concept is not really clear to me. Nevertheless, ... |
Whenever I am not doing anything, I generally happen to see pages of some good Mathematical Institutes in India, so as to know more about the faculty members and see what they are working on.
While doing so, in one of the faculty webpage he has written this statement.
The prime number theorem has an one-line proof : T... |
Let X be the quotient space of $S^2$ under the identifications $x\sim-x$ for $x$ in the equator $S^1$. Compute the homology groups $H_i(X)$. Do the same for $S^3$ with antipodal points of the equator $S^2 \subset S^3$ identified.
This is probably related to cellular homology. Thanks.
|
I am looking for a way of getting some intuition for some Convection Diffusion Equation. I would like to play around with it, simulate it, change its IC/BCs, having some interactive graphs and stuff.
Do you know of any free software that does the job? Thank you!
|
So I want to find all antiderivaties of
$\frac{x}{x^3-1}$
Since the denominator is of a lesser degree than the numerator, partial fraction is to be used instead of long division.
I've started by doing:
$\frac{x}{x^3-1} = \frac{A}{x^2+x+1} + \frac{B}{x-1}$
Hence,
$x = A(x-1)+B(x^2+x+1)$
Then setting $x = 1$ to get $3B =... |
My pdf for $X$ is
$$f_X ( x ; \theta ) = (1/\theta) x \exp (-x/\theta).$$
I am unsure as to how to show that the mean value of $X$ is $\theta$?
|
- * / 15 - 7 + 1 1 3 + 2 + 1 1 =
- * / 15 - 7 2 3 + 2 + 1 1 =
- * / 15 5 3 + 2 + 1 1 =
- * 3 3 + 2 + 1 1 =
- 9 + 2 + 1 1 =
- 9 + 2 2 =
- 9 4 =
5
I need some guidance. I am trying to approach polish notation. ... |
In infix notation, (2/2(3+3)) means (2/(2*(3+3)). The multiplication sign is omitted.
In polish/prefix notation, is it possible to omit multiplication sign? For instance, is it acceptable to write ( * + 2 3 ( / 12 4)) as ( + 2 3 ( / 12 4))?
|
For example, if we have a conic:
$$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$
What does homogenising this equation with another line (say $ax + by + c = 0$ ) actually mean? As in, what are the graphical consequences of this? I've been told that it represents a pair of straight lines passing through the origin and the poi... |
Here's the setup. Consider a vector bundle $E$ over a manifold $M$ and let $\Omega^*(M, E)$ denote the space of $E$-valued differential forms (i.e. the space of sections of the vector bundle $\bigwedge^* T^*M \otimes E$). Let $\nabla$ be an affine connection on $E$, i.e. a map $\Omega^0(M, E) \to \Omega^1(M, E)$ sati... |
1)Complete study of the function
-monotony of the function
-concavity
-maximum and minimum
-Zero of the function
-if the function is ascending or descending
-inflection points
2)make the primitive and find the area “under the curve” in a closed “interval” (break)
-Primitive-make the derivative
-Calculate the area analy... |
So $i = \sqrt{-1}$
But why is $\sqrt{-x^2}=|x| i$
Google tells me it is.
|
This question was motivated by another question in this site.
As explained in that problem (and its answers), if $\displaystyle f$ is continuous on $\displaystyle [0,1]$ and $\displaystyle \int_0^1 f(x)p(x)dx=0$ for all polynomials $\displaystyle p$, then $\displaystyle f$ is zero everywhere.
Suppose we remove the rest... |
I think it does but I'm having trouble showing it.
Using the integral test: the function $f(x)=\frac{1}{\ln(x^x+x^2)}$ is decreasing but I'm having trouble integrating $\int_{1}^{\infty} \frac{dx}{\ln(x^x+x^2)}$, trying the substitution $t=x^x$ didn't seem to work.
The root test gives $\sqrt[n]{\left |\frac{1}{\ln(n^n... |
I know that there is a first-order sentence $\varphi$ such that
$\varphi$ is written in the vocabulary given by just two binary relation symbols $E_1$, $E_2$ (and hence, without the equality symbol*),
$\varphi$ is satisfiable in a model where $E_1$ and $E_2$ are equivalence relations,
$\varphi$ is not satisfiable in... |
So, I've been revising for an exam and I came up against the question " prove $4(9^n) + 3(2^n)$ is divisible by 7 for all $n>0$.
Now, I know how to do this. If I assume $n=k$ divisible by $7$, then I need to show that $n=k+1$ is divisible by $7$. The easiest way to do this is to state that if we label $n_k=k$ and $n_{... |
I'm trying to solve the following exercise from Munkres book. Can you please check it?
Let $X$ be a second countable space and let $A$ be an uncountable subset of $X$. Show that an uncountable number of points of $A$ are limit points of $A$.
"Proof".
Let $C = \{x \in A: \textrm{x is a limit point of A} \}$. We want to ... |
It is well known that for a matrix Lie group the Lie exponential map is $e ^z$. This maps a tangent vector $z$ at the identity to a group element.
On the other hand the general Riemannian exponential map centered at point $x$ is given by $\exp _x \triangle$ which maps a tangent vector $\triangle$ at point $x$ (not ne... |
Let $G$ be a group where every non-identity element has order 2.
If |G| is finite then $G$ is isomorphic to the direct product $\mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \ldots \times \mathbb{Z}_{2}$.
Is the analogous result
$G= \mathbb{Z}_{2} \times \mathbb{Z}_{2} \times \ldots $.
true for the case |G| is infi... |
Do there exist two Lie groups which are diffeomorphic as smooth manifolds, have isomorphic group structures, yet are not isomorphic as Lie groups?
Of course, for this to happen, any diffeomorphism would fail to preserve the group structure, and any group isomorphism would either fail to be smooth, or its inverse would ... |
Let $p : [0,1] \to \mathbb{R}^2$ be continuous and injective.
Does there always exists a continuous function $f : [0,1]\times \mathbb{R}^2 \to \mathbb{R}^2$ such that
For all $x\in \mathbb{R}^2$, $f(0,x) = x$
and
For all $t\in [0,1]$, $x\mapsto f(t,x)$ is a homeomorphism
and
For all $s\in [0,1]$, $f(1,p(s)) = \lan... |
everyone. I am afraid that my question is too trivial. But here it is. The Klein four group is the first counterexample to the the statement: "If all proper subgroups of a group are cyclic, then the group is cyclic." I am looking for other examples, if any. Are there?
Thanks in advance.
|
Let $G$ be a simple graph such that
$|V(G)|$ is even
$\delta(G) \gt \tfrac12|V(G)|+3$ where $\delta(G)$ is the minimum degree of a vertex in $G$.
The question is to prove there exists a $5$-regular spanning subgraph of $G$.
I'm trying to figure this out for at least 10 hours, but no avail. :(
The only thing I've ma... |
Is there a real function that is differentiable at any point but nowhere monotone?
|
Is there a continuous and monotone function that's nowhere differentiable ?
|
I have a system described an equation,
and I want to find an (DE) equation for z(x,t), in the limit as l->0.
First some definitions to simplify it some:
$Z1=z(x,t)-z(x+l,t)$
$Z2=z(x,t)-z(x-l,t)$
These are not constants, just to simplify expr.
c however, is a constant.
EQ:
$c*l^2*d^2z(x,t)/dt^2=(1-l/sqrt(Z1^2+l^2))*Z1... |
Einstein's theory of gravitation, general relativity, is a purely geometric theory.
In a recent question I wanted to know what the relation of Brownian motion to the Helmholtz equation is and got a very thorough answer from George Lowther.
He pointed out that there is, roughly speaking, a very general relation of semi-... |
Can someone help me with this problem?
Let $D$ be a divisor on an algebraic curve $X$ of genus $g$, such that $\deg D = 2g-2$ and $\dim L(D) = g$. Then $D$ must be a canonical divisor.
By Riemann-Roch, I see that $\dim L(K-D) = 1$ for any canonical divisor $K$, as must certainly be the case. I don't know if this is to... |
I was working out some problems from a entrance test paper, and this problem looked a bit difficult.
Does there exist an analytic function on $\mathbb{C}$ such that $$f\biggl(\frac{1}{2n}\biggr)=f\biggl(\frac{1}{2n+1}\biggr)=\frac{1}{2n} \quad \ \text{for all} \ n\geq 1?$$
I did apply $f$ repeatedly to get some more... |
Let $S_3$ be the symmetric group on the three objects $x_1, x_2, x_3$.
We are given a (countably infinite) sequence of permutations in $S_3$:
$\sigma_1 \ , \ \sigma_2 \ , \ \sigma_3 \ , \ \ldots \ , \ \sigma_n \ , \ \ldots \ \, .$
Is it true that there exists a positive integer $N$ such that the sequence of products
$\... |
Let's say I have numbers each taken in a set $A$ of $n$ consecutive naturals, I ask myself : how can I found what are all the unique multisets, which could be created with $k$ elements of this set $A$?
For example I've got $A=[1,2,\dots,499]$. If I wanted to create unique multiset of 3 elements, I would search all the ... |
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