instruction stringlengths 22 27.2k |
|---|
I am wondering is my solution sufficient enough (or detailed enough) for the following question? or it is even a valid solution?
Question: Find a tight asymptotic bound ($\Theta$) in terms of the following code segment, where $n \ge 1$ is an integer. Give details of your analysis.
for i=1 to n do {
for j=1 to i do ... |
I have come across a problem I can't solve. Can anyone help? Here is the problem
Find least integer $N$ such that sum of the digits of both $N$ and $N+1$ is divisible by $7$.
|
Prove that to every $A\in L(\mathbb{R}^n,\mathbb{R}^1)$ corresponds a unique $\mathbf{y}\in \mathbb{R}^n$ such that $A\mathbf{x}=\mathbf{x}\cdot \mathbf{y}$. Prove also that $\Vert A \rVert=|\mathbf{y}|$.
Proof: Let $A\in L(\mathbb{R}^n,\mathbb{R}^1)$ and $\mathbf{x}=(x_1,x_2,\dots,x_n)$ then $\mathbf{x}=x_1e_1+\dots+x... |
Show that a metrizable space with a countable dense set has a countable basis.
My try:
Let $X$ be a metrizable space with a countable dense set $D$.
Consider for each $n\in \Bbb N;B(x,\dfrac{1}{n});x\in X$. Always we have that $B(x,\dfrac{1}{n})\cap D\neq \emptyset $. Let $a\in B(x,\dfrac{1}{n})\cap D$.
Claim $\scr... |
Let $\binom{n}{k}$ denotes the number of subsets with $k$ elements in $n$-elements set. Prove that
$$\sum_{k=0}^{n}(-1)^k\binom{n}{k}(n-2k)^m=\begin{cases}
0, & \text{ if } 0\le m \le n-1; \\
2^n \cdot n!, & \text{ if } m=n.
\end{cases}$$
|
I have been troubled by this:
$\tau(2^x \times 3^y)=m$
Being $x$, $y$ and $m$ positive integers
Then the number of solutions is $\tau(m)$
I already have done the proof for m prime however cant do it with m not prime, what can I do?
I am stuck with this equation basically:
$(x+1)(y+1)=m$
|
How many non-congruent triangles with perimeter 11 have integer side lengths?
I failed to solve it. Can anyone help?
|
We need to find the vector equation of the line of intersection of the above mentioned planes. I'm not that good with vectors so couldn't understand how to do it even though I had the answer in the mark scheme.
Here's the answer:
Edit: Turns out my answer is right. I will copy this and post it as an answer so if som... |
Let $X$ be a random variable such that $E|X|<\infty$ and $$P\left(X\ge\frac{1}{2}+x\right)=P\left(X\le\frac{1}{2}-x\right)\ \forall x\in \mathbb{R}.$$
Then
$E(X)=\frac{1}{2}$ and Median($X$)$=\frac{1}{2}$
$E(X)=\frac{1}{2}$ and Median($X$)$>\frac{1}{2}$
$E(X)<\frac{1}{2}$ and Median($X$)$=\frac{1}{2}$
$E(X)<\frac{1}{... |
I am a beginning learner of group theory.
Which of the following are isomorphic:$$\mathbb{Z_{24}}, D_{4}\times \mathbb{Z_{3}},A_{4}\times \mathbb{Z_{2}},\mathbb{Z_{2}}\times D_{6}, \mathbb{Z_{12}}\times \mathbb{Z_{2}},D_{12}, S_{4}$$
I know $\mathbb{Z_{24}}$ is cyclic and none of $\mathbb{Z_{12}}\times \mathbb{Z_{2}},D... |
Let $B_{k}$ the kth Bernoulli number, then using their asymptotic I can justify the absolute convergence of this series
$$\sum_{j=0}^{\infty}\frac{2^{2j-1}B_{2j}}{(2j)!},$$
since, if there are no mistakes $$\lim_{n\to\infty}\frac{c_{n+1}}{c_{n}}=\frac{1}{\pi^2}<1,$$
when $c_n=\frac{2^{2n-1}|B_{2n}|}{(2n)!}$.
Question.... |
This is an exercise from "Calculus on Manifolds" by Michel Spivack (first edition, p.100):
If $c$ is a singular $1$-cube in $\mathbb{R}^2-\{0\}$, with $c(0)=c(1)$, show that there is an integer $n$ such that $c-c_{i,n}=\partial c^2$ for some $2-$chain $c^2$.
Hint : First partition $[0,1]$ so that each $c([t_{i-1},t_... |
Problem statement: Let $G$ be a cyclic group of order $3$. The matrix $$A=\begin{bmatrix} -1 && -1 \\ 1 && 0 \end{bmatrix}$$ has order $3$, so it defines a matrix representation of $G$. Use the averaging process to produce a $G$-invariant form from the standard Hermitian product $\mathbf{x}^*\mathbf{y}$ on $\mathbb{C}... |
I was just reading a book called Proofs from the Book. It presented the proof given by George Polya to prove that two Fermat primes (numbers of the form $2^{2^n} + 1$) are always relatively prime, which in itself is a very elegant proof. But, to say that it implies there are infinitely many primes does not make sense t... |
If a matrix can be written as $A = D (I − N)$, where $D$ is diagonal
with nonzero entries and $N$ is nilpotent, then $A^{−1} = (I − N)^{−1}D^{−1}$. Use this to find inverse of:
$\begin{bmatrix}
2 & 2 & 4 &\\
0 & 2 &-2 &\\
\
0 & 0 & 3
\end{bmatrix} $
I am having trouble finding $D$ and ... |
Let $X_{1},\dots$ be a sequence of independent random variables. Suppose, for $k=1,2,\dots$ $$P\left(X_{2k-1}=1\right)=P\left(X_{2k-1}=-1\right)=\frac{1}{2}$$ and the probability density function of $X_{2k}$ is $$f(x)=\frac{1}{\sqrt{2}}e^{-\sqrt{2}|x|},\,\,\ -\infty<x<\infty$$ Then $\lim_{n\to \infty} P\left(\frac{X_{1... |
Without using Mathematical Induction, prove that $$\frac{1}{n+1} + \frac{1}{n+3}+\cdots+\frac{1}{3n-1}>\frac{1}{2}$$
I am unable to solve this problem and don't know where to start. Please help me to solve this problem using the laws of inequality. It is a problem of Inequality.
Edit: $n$ is a positive integer such th... |
I know that some integrals on solving by parts end up with the same integral on the right side, and then the integral is assumed to be $I$ or some variable, and then linearly solved. For example,
$$I = \int e^x \cos x \ dx$$
$$u = e^x,\ dv = \cos x \ dx, \ du = e^x \ dx,\ v = \sin x$$
$$\int u dv = uv - \int v \ du$$... |
Let $n \in \mathbb{N}$. May we have a closed form for the integral:
$$\mathcal{J}=\int_0^{\pi/2} \ln^n (\sin x) \, {\rm d}x$$
One obvious approach would be to go through beta functions and differentiate $n$ times but that sounds very tedious. My guess is, that is doable but a long shot.
Well, a relation that might help... |
let's assume we have a probability space $(\Omega , F , P)$.
and we have a random variable $X$ defined as : $X : \Omega \rightarrow \Bbb{R}$
and we also use a Borel set ($\mathcal{B}$).(making the alternative probability space $(\Omega , \Bbb{R} , \mathcal{B})$ )
now due to the "Measure-theoretic definition" of a rand... |
I have a quastion about product spaces in singular cohomology. I only know a formula for sinugular homology for product spaces from lecture, the universal coefficient theorem.
Let $R$ be a commutative unital ring, $(X, A), (X',A')$ pairs of topological spaces, such that $$H^*(X,A;R)\cong H^*(X',A';R).$$ Let $Y$ be a ... |
Let $H$ be a Hilbert space and $\{x_n\}_{n=1}^{\infty}$ given sequence of vectors from $H$. Suppose that for every $\{\alpha\}_{n=1}^{\infty}\in \ell^2$ series $\sum_{n=0}^{\infty}\alpha_nx_n$ is weakly convergent. How to show that for every $h\in H$ series $\sum|\langle h,x_n\rangle|^2$ is convergent and there exist $... |
Let a normal $A_{n\times n}\in \mathbb C^n $ matrix, then:
$\forall v \in \mathbb C^n:\lVert A^*v \rVert = \lVert Av\rVert $
$\forall v \in \mathbb C^n : \langle Av,v\rangle = \langle A^*v,v\rangle$
I thought to maybe use the Pythagorean theorem but it doesn't help, also the eigenvalues of the two matrices can ... |
if I denotes the principal congurence group of level 2 i.e. $I=\{ M \in SL(2,Z) ; \:M \:\:\text{congruent to I} \mod(2)\}$.
or I=
\begin{bmatrix}2\mathbb{Z}+1&2\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}
H be its subgroup \begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}
how to pro... |
An arithmetic progression in $\mathbb{Z}$ is a set
$A_a,_b=\bigg\{\dots,a-2b,a-b,a,a+b,\dots\bigg\}$ with $a,b\in\mathbb{Z}$ and $b\neq0.$ prove that the collection of arithmetic progressions
$B=\bigg\{A_a,_b|a,b\in\mathbb{Z}\ and\ b\neq0\bigg\}$ is a basis for topology on $\mathbb{Z}$
any hint
|
Let:
$$f(x) = \int \frac{x^2}{2\left(e^x+1\right)}dx $$
Is there a way to find $f(x)$?
I've tried through integration by parts, but that didn't work out. If substitution is the answer, I can't see an appropriate substitution to make.
Can anyone help?
|
The following is the quotient group of SL($2,\mathbb{Z}$).
Consider $(H/\{-1,1\} \cap H)$
where $H=\begin{bmatrix}2\mathbb{Z}+1&4\mathbb{Z}\\2\mathbb{Z}&2\mathbb{Z}+1\end{bmatrix}$
How do i find out its generators? I know it is a free group of rank 3 with generators
\begin{bmatrix}1&4\\0&1\end{bmatrix},\begin{bmatrix... |
Given that we know if the integral of $f(y)$ is $F(y)$ then we can say that $\frac{d}{dy} F(y) = F'(y) = f(y)$.
But what does $\frac{d}{dy} F(g(y),y)$ equal to? Can we say that it is $f(g(y),y)?$
|
I have to study the type of critical points of the function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$ and find minimum and maximum on the generic circle centered in $ (0,0) $ and radius $ r>1 $.
I did the partial derivates and tried to solve this system $$ \left\{\begin{matrix}
f_{x}=8x^3-6x+6xy^2=0\\
f_{y}=4y^3-4y+6xy^2... |
How do I prove $$\frac 12\leq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}...+\frac{1}{n+n}$$
without using induction? Note that clearly $n\neq 0$
Thanks for any help!!
|
(Q) is false since unitary matrix has modulus 1 eigenvalues.
I think (P) is true but I am not sure how to Prove or Disprove this.
Please suggest?
|
I read (page 8 here) that if $A$ and $B$
are rectangular matrices so that the product $AB$ is defined, then
$$(1)\quad||AB||_F^2\leq ||A||_F^2||B||_F^2$$
Does that mean that the inequality above also holds when the number of rows of $A$ is larger than the number of columns of $B$? The justification (Cauchy Swartz):
$$... |
How do I prove the following inequality
$$\frac 34\geq \frac{1}{n+1}+\frac {1}{n+2}+\frac{1}{n+3}+\cdots+\frac{1}{n+n}$$
without the help of induction?
Thanks for any help!!
|
Given a circle with r is equal 2cm, can we find the sector's angle?
|
I have a really big problem with this next task, determing the range of composite g(f(x)).
$f(x)=-2x+4$
$g(x)=\lbrace{x^2-4};|x| \le 2\rbrace$
$g(x)=\lbrace{4x^2-x^4};|x|>2\rbrace$
The result is supposed to be:
$-1 \le x \le 3$ for the first part of g(x) with $|x| \le 2$ and $x<-1$ and $ x>3$ for the second part of g(x... |
I want to prove that the weak-law of large numbers hold if $X_{k}$ has the distribution:
$$\mathbb{P}\left(X_k\leq x\right) = \int_{-\infty}^x c\left(1+t^2\right)^{-1}\left(\log(1+t^2)\right)^{-1}dt
$$
The exercise is from Vardhan test book (Chapter 3 3.6). I don't know how should I prove this.
|
What is the root $s_0$ of the equation $\zeta(s) = s$, where $\zeta(s)$ is Euler zeta function? This point $s_0$ has obvious property: the segment $(1,s_0]$ to the left of it is mapping on the half-line $[s_0, \infty )$ to the right. Is it expressed through Euler–Mascheroni constant $\gamma$?
|
I'd like to simplify this expression, which sums up the first $n$ triangular numbers:
$$\sum_{i=1}^{n}(\frac12i(i+1))$$
which is equal to:
$$\sum_{i=0}^{n}((n-i)(i+1))$$
|
Is the following always true?
$$
f:\mathbb{C}\to\mathbb{C},\ \operatorname{Re}\left(\int f(z)d z\right) = \int\operatorname{Re}(z)dz
$$
$$
\frac{d\operatorname{Re}(f)}{dz} = \operatorname{Re}\left(\frac{df}{dz}\right)
$$
Is it true for $\operatorname{Im}$?
|
I think the option $(Q)$ is true since $O(Q/\{-1,1\})= 8/2 = 4 = 2^2$.
Since order is $p^2$ thus $(Q)$ option is true.
Can anyone suggest about option $(P)$?
Thanks
|
$$I=\int_0^1\frac{\arcsin^2(x^2)}{\sqrt{1-x^2}}dx\stackrel?=\frac{5}{24}\pi^3-\frac{\pi}2\log^2 2-2\pi\chi_2\left(\frac1{\sqrt 2}\right)$$
This result seems to me digitally correct?
Can we prove that the equality is exact?
|
While solving inequality and finite series problem I often come across this series-
$$(n+1)(n+2)(n+3)...(n+n)$$.
Is there a general solution to this form of a series?
Thanks for any help!!
|
"Let P(n) be the property
n is divisible by a prime number.
We prove that P(n) is true for all integers n with n> 1.
Basis step. If n=2, then P(n) is true because 2 is a prime and every number divides itself.
Inductive step. Let k be some integer with k≥2 and assume, as induction hypothesis, that P(i) is tru... |
I know just the very basics on Category Theory and that's why I'm going to ask a stupid question.
I'm trying to get an intuition for direct limits for my course on Commutative Algebra. All the books I considered so far state the definition, maybe the construction in some category and the universal property. But why is... |
Let $S$ be the bounded surface of the cylinder $x^2+y^2=1$ cut by the planes $z=0$ and $z=1+x$
Then how to show that the value of the surface integral $∬3z^2 d \sigma $ over $S$ is equal to $\int_{0}^{2\pi}(1+cos \theta)^3d\theta$
I tried using $\int\int_{S}g d\sigma =\int\int_{R}g \frac{|\nabla f|}{|\nabla f . p|}dA$ ... |
Find a limit of sequence:
$$a_{n+2}=a^2_{n+1}+\frac{1}{6}\cdot a_n+\frac{1}{9}$$
$$a_1=0,a_2=0$$
I tried to prove that $a_n$ is bounded and monotonic, but I couldn't prove that $a_n$ is monotonic (by strong induction).
So please give a idea how to show that $a_n$ is monotonic.
|
A stunt kite has the shape in the diagram below:
How can I find the area using calculus integration.
Can anyone help me start this question, I am not looking for the full answer. I assume I only need to find half the area and multiply it by two. I know how to find the area between $2$ curves but this is different bec... |
Find the rank and the nullity of the following linear map $T : U \to V$ , and find bases of the kernel and image of $T$.
$U = \Bbb R^4 , V = \Bbb R^4$, $$T(α, β, γ, δ) = (α − γ, γ − δ, α − β, β − δ)$$
Attempt of the question: I have tried to find a basis of kernel, it is $(1,1,1,1)$. so $\text{nullity}(T)=1$
But then t... |
If $\sin\theta + \sin\alpha=x$ and $\cos\theta + \cos\alpha=y$, prove that ; $$\frac{\tan(\theta - \alpha)}{2} = \pm\sqrt{\frac{4-x^2-y^2}{x^2+y^2}}$$
Attempts:
Here $\sin\theta + \sin\alpha=x$
squaring both sides, we get
$$\sin^2\theta+2\sin\theta \sin\alpha+ \sin^2\alpha=x^2 \tag{1}$$
Again, squaring the... |
Consider the following subgroups of $\text{SL}(2,\mathbb{Z})$ :
$A$ the subgroup of matrices with determinant $1$ :
\begin{bmatrix}4\mathbb{Z}+1&8\mathbb{Z}\\4\mathbb{Z}&4\mathbb{Z}+1\end{bmatrix}
$B$ the subgroup of matrices with determinant $1$ :
\begin{bmatrix}2\mathbb{Z}+1&8\mathbb{Z}\\4\mathbb{Z}&2\mathbb{Z}... |
I am currently self-learning Calculus from an old and cheap edition of Calculus (by James Stewart) which contains both single and multivariable calculus.I am aware that Stewart's book is regarded as pretty simple, but for my measly purposes,i have to admit it does the trick,neither being bogged down in rigour nor compl... |
I can easily see a proof that shows its going to be an immersed submanifold . (I am removing the case if the vector field at that point is 0). I am not able to see if it's a embedded submanifold or not? Thank you.
|
($x^2+(y+1)^2=R^2$ should say $x^2+(y-1)^2=R^2$)
I am trying to derive a formula for the radius of the circle that is externally tangent to the internally tangent circles of the quarter-circle, and that is internally tangent to the quarter-circle.
Comment: this question is from a mathematics book, and it states that t... |
Prove that if $(X_1,d_1)$ and $(X_2,d_2)$ are metric spaces on $X_1\times X_2$ and metric $d:(X_1\times X_2)\times (X_1\times X_2)\rightarrow R$ is defined in following way:
(i)$d((x_1,x_2),(y_1,y_2))=((d_1(x_1,y_1)^2 +(d_2(x_2,y_2)^2)^{1/2}$
(ii)$d((x_1,x_2),(y_1,y_2))=d_1(x_1,y_1) +d_2(x_2,y_2)$
(iii)$d((x_1,x_2),... |
Using the Definition of a limit: [ Of form $\lim_{x\to a} \frac{f(x)-f(a)}{x-a}$]
Find $f'(x)$ when $x=9$ for $f(x)=\frac{2}{\sqrt{x}}$
I tried simplifying it but got jumbled when trying to multiply by the conjugate, does anyone have a solution?
After some working out i figured the answer.. however is this in the simpl... |
If $\sin\theta + \sin\alpha=m$ and $\cos\theta + \cos\alpha=n$, prove that:
$$\frac{\sec(\theta+\alpha)}{2}=\frac{\sqrt{m^2+n^2}}{2}$$
My attempt\
given: $$\sin\theta+\sin\alpha=m$$
$$2 \sin(\frac{\theta+\alpha}{2}) \cos(\frac{\theta-\alpha}{2})=m$$
Again, using the transformation formula for the s... |
In Derek Robinson, A Course in the Theory of Groups on page 224 he proves:
Let $G$ be a finite group and let $F$ be an algebraically closed field whose characteristic does not divide the order of $G$.
(i) $FG = I_1 \oplus I_2 \oplus \cdots \oplus I_h$ where $I_i$ is an ideal of $FG$ which is ring isomorphic with the r... |
How can I find the infimum and supremum in $\mathbb{R}^{2} $ of this function $$ f(x,y)=(2x^2+y^2-1)(x^2+y^2-1)+1 $$?
Thanks
EDIT:
Forgive me if I did not add my thoughts but I did not know where to start .
|
Let $T$ be a generalized function. I need to provide definitions for $T$ to be periodic and radially symmetric.
A function (on $\mathbb{R})$is said to be periodic if there exists a $p \in \mathbb{R}$ such that $f(x+p)=f(x)$ for all $x \in \mathbb{R}$. If $T$ comes from a function then $(T_f,\phi)=\int f(x) \phi(x) dx=... |
Let $$l^2=\left\{(x_n):\sum_{n=1}^{\infty}x_n^2<\infty\right\}$$
equipped with the norm
$$\|(x_n)\|=\left(\sum_{n=1}^{\infty}x_n^2\right)^{1/2}.$$
Prove that $l^2$ is complete with respect to the norm $\|\cdot\|$.
Can somebody please check my working?
Here is my working and some of my doubts:
We need to show that e... |
The problem:
There is a class. In each group of three students in the class there is a pair of friends and a pair of "enemies". Find the maximum number of students in the class.
I tried to play with combinatorics, to find numbers of pairs and triplets. Also there was an idea to assign 0 (enemies) or 1 (friends) to each... |
I'm solving an A Level paper, and came across this question. Basically, they have given plane $p$ has the equation $(\mathbf r-3\mathbf i)\cdot(2\mathbf i-3\mathbf j+6\mathbf k)=0$. Now, I can see that this means dot product, but I don't understand how to find the Cartesian equation (in the form $ax+by+cz=d$). I have n... |
Suppose $x,y$ are in the range $0 \leqslant x \leqslant 2,0 \leqslant y \leqslant 1$, I can use separation of variables to get $\frac{{{\partial ^2}u}}{{\partial {x^2}}} + \frac{{{\partial ^2}u}}{{\partial {y^2}}} = 0$ has a general solution of the form $\phi (x,y) = (A{e^{\lambda x}} + B{e^{ - \lambda x}})(C\cos \lamb... |
My book tells me the horizontal asymptotes of $\arctan2x$ is either at positive or negative $\frac{\pi}{2}$, yet the vertical asymptotes of $\tan2x$ occurs at positive or negative $x=\frac{\pi}{4}$, so obviously the horizontal asymptotes of $\arctan2x$ should be at either positive or negative $y=\frac{\pi}{4}$.
Can so... |
If $\alpha,\beta$ are roots of $x^2+px+q=0$ and also of $x^{2n}+p^nx^n+q^n=0$ and $\frac{\alpha}{\beta}$,$\frac{\beta}{\alpha}$ are the roots of $x^n+1+(x+1)^n=0$, then $n$ is
Odd
Even
Any integer
$\alpha+\beta=-p$ and $\alpha\beta=q$
The third equation is $$x^n+1+\binom{n}{0}+\dbinom{n}{1}x+\cdots+\dbinom{n}{n}x^n... |
the consept perfect group is new for me.
I was reading the following:
If a group $G$ is a FN-group and $G'=G$ then $G$ is finite.
I tried to prove this and here is my attempt:
let $G$ be a FN-group, then there existe a normal subgroup $N$ such that $N$ is finite and $G/N$ is nilpotent.
If we use the fact that if $G=G'$... |
How can I solve this system? $$ \left\{\begin{matrix}
8x^3-6x+6xy^2=0\\
4y^3-4y+6xy^2=0
\end{matrix}\right. $$
I do
$$ \left\{\begin{matrix}
8x^3-6x+6xy^2=0\\
4y^3-4y+6xy^2=0
\end{matrix}\right.
\Rightarrow
\left\{\begin{matrix}
2x(4x^2-3+3y^2)=0\\
2y(2y^2-2+3xy)=0
\end{matrix}\right.$$
Then $x=0 \Rightarrow y=0,... |
Let $A$ be a square real symmetric matrix with rank $1$ , then can all the diagonal entries of $A$ be $0$ ? I know that real symmetric matrices are diagonalizable . Also if all the diagonal entries be $0$ then sum of all the eigenvalues will be $0$ . But so what ? Please help . Thanks in advance
|
At the end of a lesson in Differential Geometry, my teacher said:
Fatto, che non dimostriamo, non è difficile ma il tempo scarseggia, se $G$ è compatto possiamo sempre trovare una metrica Riemanniana biinvariante su $G$, d'accordo? Tuttavia non è vero che $\varphi$, $\varphi$ invariante a sinistra non implica, d'accor... |
Given $((9) \text{inches})^{1/2} = ((0.25) \text{yards})^{1/2}$, then which of the following statements is true?
$((3) \text{inches}) = ((0.5) \text{yards})$
$((9) \text{inches}) = ((1.5) \text{yards})$
$((9) \text{inches}) = ((0.25) \text{yards})$
$((81) \text{inches}) = ((0.0625) \text{yards})$
My question is : Ca... |
I have just started learning about differential equations, as a result I started to think about this question but couldn't get anywhere. So I googled and wasn't able to find any particularly helpful results. I am more interested in the reason or method rather than the actual answer. Also I do not know if there even is ... |
Let $f$ be a generalized function on $\Omega$. Let $G$ be an open subset of $\Omega$. Assume that every point in $G$ has a neighborhood on which $f$ vanishes. Prove that $f$ vanishes on $G$
Let $\phi \in D(G)$. We show that $(f,\phi)=0$. Let $\text{Supp $\phi$} \subset K \subset G$, where $K$ is a compact set. For each... |
Let $I = [0,1]$ and $χ_I : \Bbb R → \Bbb R$ be the characteristic function on $I$. Prove that there is no continuous function $f : \Bbb R → \Bbb R $ such that $f = χ_I$ almost everywhere on $\Bbb R$.
[Note that for every $n ∈ \Bbb N$, one can construct a continuous function $g : \Bbb R → \Bbb R$ such
that $m\{x∈\Bbb R:... |
Prove that the points of contacts of tangents of the curve $y={\sin x}$ drawn from origin lie on the curve $\frac{1}{x^2} - \frac{1}{y^2} = -1$
|
At home I came across the exercise and had to compute: $\int_{0}^{2\pi} \sin^n (x) \cos^m (x) dx $ with $m$, $n \in \mathbb{N} $
My current set of tools for solving problems of that kind is rather limited: This problem can be easily done via integration by parts.
However I wonder, if one could also come up with a sol... |
If the equation $|x^2+4x+3|-mx+2m=0$ has exactly three solutions then
the value of m is equal to to ?
I drew the graph of $|x^2+4x+3|$.I found that that for the given condition $mx-2m$ must be a tangent to $-(x^2+4x+3)$.So using the concept of putting the discriminant to $0$, I'm getting $m=-8+2\sqrt{15}$ and $m=-8... |
Given an undirected connected simple graph with distinct edge weights. If we add a constant value to each edge of graph then : single source shortest path of new graph can be changed?
My attempt :
If question was for MST(minimum spanning tree) then it was not changed.
Can you explain for SSSP(single source shortest ... |
Given the matrix $A=\begin{pmatrix}
-1 &3 &3 \\
3& -1 & 3\\
3& 3 & -1
\end{pmatrix}$ then $A$ is invertible and $A^{-1}$ can be expressed with: $A^{-1}=bA+dI, b,d\in\mathbb R$
So it's easy to find that $|A|\neq 0$ so it's invertible, and from $AA^{-1}=I$ we get: $bx^2+dx-1= 0$ so if there are solutions to $bx^2+... |
I need to proof that this function is differentiable in all the points of its domain. I know that this is true if the function is a function $\in C^k$ and a function is $C^k$ if is composition of $C^k$ functions.
However considering
$$ f(x,y)=x^3+e^{x+y^2} $$
Surely I can say that it is a composition of $C^k$ functio... |
I am not very good at physics, so I don't understand how to solve the following problem with vector calculus.
A perfect incompressible fluid moves steadily under gravity around the outside of a fixed cylinder of radius $a$ and vertical axis $Oz$. The fluid particles move in horizontal circles with centres on $Oz$ and t... |
Let $V^{\bullet}$ be a graded vector space. Assume that $V^{i}=0$ for $i<1$. Then the tensor algebra has the structure of a connected Hopf algebra and in particular its Lie algebra of primitive $P(T(V))$ is naturally ismorphic to the free Lie algebra. This result is a consequence of the structure theorem for connected ... |
Given $A = \{0,1,2,3,4,5\}$, Write the appropriate equivalence relation of this quotient set:
$$A/_R = \{\{1,2\},\{3\},\{4,0,5\}\}$$
Well, if it was to compute $$A/_R = \{\{0\},\{1\},\{2\},\{3\},\{4\},\{5\}\}$$ I believe it would be simpler, something like:
$[0]_r = \{0\}$, $[1]_r = \{1\}$ , $[2]_r = \{2\}$ , $[3]_r = ... |
This is part of Exercise 5.13 from Undergraduate Algebraic Geometry by Reid:
Consider the Veronese surface $S$ defined by the map:
$$\phi: \mathbb{P}^2\rightarrow \mathbb{P}^5$$
where $\phi(x_0,x_1,x_2)=(x_0^2,x_0x_1,x_0x_2,x_1^2,x_1x_2,x_2^2)$.
The problem asks to show that a line in $\mathbb{P}^2$ is mapped to... |
How to get integer values for x and y that satisfy: $$54 x + 16 y = 2400$$
Someone told me that I can do it using Euclid-Wallis algorithm, but I don't understand it so, if there isn't any else solution, can you explain it to me? I'm elementary.
|
Let $V$ be a collectino of the search direction for the conjugate gradient applied on a quadractic minimisation problem. As a proof of orthogonality in conjugate gradient:
$$
V^T V = I
$$
Now supposed I have a vector $x$, and I want to express this as a sum of vectors that are parallel and perpendicular to the Krylov s... |
$$
F'(x)=(2ax+b)e^x+(ax^2+bx+c)e^x=(ax^2+(b+2a)x+c+b)e^x
$$
Can someone explain this? I can't understand the transition. What method is this?
|
The solution of differential equation $\displaystyle \left(\frac{dy}{dx}\right)^2-x\frac{dy}{dx}+y=0$
$\bf{My\; Try::}$ Let $\displaystyle \frac{dy}{dx} = t\;,$ Then Diferential equation convert into $t^2-xt+y=0$
So Its solution is given by $\displaystyle t=\frac{x\pm \sqrt{x^2-4y}}{2}$
So we get $$\frac{dy}{dx} = \f... |
I am studying discrete time martingale theory and came across the classical "You can't beat the system" theorem: given a martingale $M$ and a previsible process $C=(C_n)_{ n \ge 1}$ such that $C_n$ is limited for every $n$ the martingale transform $C \bullet M $ is a martingale. My question is: can we find a martingale... |
Consider:
\begin{equation}
\Big(e^x-1\Big)\mathbb{1}_{(x\geq0)} \leq \lambda_1+\lambda_2e^x + \lambda_3x^2
\end{equation}
where $\lambda_1$, $\lambda_2$, $\lambda_3$ are three unknown constants. By plotting these two functions, it seems that it is possible to find particular values for $\lambda_i$, $i=1,2,3$, such that... |
We know that if $(f_n)$ is a sequence of measurable functions on $E$, then $g = sup_n f_n$ defined as
$g(x) = sup f_n(x)$, $x ∈ E_ n$
is a measurable function.
Prove by an example that if $A$ is an uncountable index set and for each $α ∈ A$, $f_α$ is a measurable function on $E$, then $g = sup_{α∈A} f_α$ defined as
$g(... |
There are $10$ seats in the first row of the theater out of which $4$ are to be occupied. Find the number of ways of arranging $4$ people so that no two people sit side by side.
Making different cases would be very lengthy process. Is there any smarter way to approach this problem?
|
$\def\imp{\Rightarrow}$I have a theory $T=\{p \imp \neg q, \neg q, r \imp q, r \imp \neg p\}$ over $P=\{p, q, r\}$
I need to find all models of theory $T$.
My question is whether I could use any systematic approach? Instead of just "inventing" how particular models look.
My idea was to use the method of analytic tablea... |
From "Lecture Notes in Computer Science" by Christoph M. Hoffmann , on page 22-
Theorem 4
Let $X$ and $X'$ be two isomorphic graphs with vertex set $V = \{1 ..... n\}$
, Then the set of isomorphisms from $X$ to $X'$ is a right
coset of the automorphism group $Aut(X)$ in $S_n$.
Proof: Let$ X = (V,E)$ and $X' = (... |
The following is the proof taken from Lemma $13$.
Questions:
$(1)$: What is the Lipschitz-norm of $\phi_a$? The following is my attempt:
$\| \phi_a \|_{Lip} = \sup_{x \neq y}{\frac{|f(a) - \bar{g}(a)| |\tau(\| x - a \| - \tau(\| y - a \| |}{\| x - y \|}} = |f(a) - \bar{g}(a)| \sup_{x \neq y}{\frac{|\tau(\| x - a \| - ... |
I got interested by the first sentence of this wikipedia subsection. It claims that any linear operator $f:V\to V$ can be represented by an element of $V\otimes V^*$ in a very concrete way: the element $w\otimes h$ maps a vector $v$ to $h(v)w$.
Clearly such an object is a linear operator, but given a basis I wanted to... |
Take a differentiable manifold $M$. Define $\eta(M)$ as $\min\{\#\mathfrak{A} \mid \mathfrak{A} \text{ is an atlas for $M$}\}$. For example, if $M=S^n$, we have that $\eta(M)=2$, since $S^n$ is compact and $2$ charts (the stereographic projections) are enough to cover $M$.
Is $\eta(M)$ known for a wide range of manifol... |
I have converted this to a problem of modular arithmetic.
I seek to prove that $21n-3$ and $15n+2$ are never congruent to $0\pmod 4$ for the same value of $n$.
I observed that $21n$ is $\{1,2,3,0,1,2,3,0\}$ about $\pmod 4$, and that $21n- 3$ is $\{2,3,0,1,2,3,0\}$. Similarly, $15n+2$ is $\{1,2,3,0,1,2,3\}$.
Thus for th... |
During my research I came across the following problem. Intuitively this should be an easy one. However, the simplest version of it looks like this:
Let $C \geq \frac{1}{2}$ be some fixed universal constant and let $\varepsilon > 0$ be given such that
$$C \varepsilon > ac - bd > 0, \quad \text{where} \quad 0 < a,b,c,d... |
This is the context:
I tried substituting $y=3e^{2x}+4$ into $6e^{2y}$but I wasn't able to go any further. Does anyone what exactly is being done in the last step?
|
I need to find the volume of the solid that is formed when the (x>0, y< -1) region of the curve y= -1/x is rotated about the y-axis.
If I'm correct, this volume can be calculated by:
Evaluating the definite integral (upper bound = -1, lower bound = -infinity) of π*(1/y²) with respect to y.
In evaluating this integral... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.