text stringlengths 83 79.5k |
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H: Dirichlet problem on $[0,1] \times [0, \pi]$
Let $\Omega := [0, 1] \times [0,\pi]$. We are searching for a function $u$ on $\Omega$ s.t.
$$
\Delta u =0
$$
$$
u(x,0) = f_0(x), \quad u(x,1) = f_1(x), \quad u(0,y) = u(\pi,y) = 0
$$ with
$$
f_0(x) = \sum_{k=1}^\infty A_k \sin kx \quad, f_1(x) = \sum_{k=1}^\infty B_k ... |
H: Showing $x^3$ is not uniformly continuous on $\mathbb{R}$
Referring to the original source http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf
Prove that $f(x) = x^3$ is not uniformly continuous on $\Bbb R$.
I want to use a more elementary method, no mention of metric spaces. Their selection of $\delta$ co... |
H: Question on the relationships of two and three manifolds
The Question is: Let $W_c = \{ ( x,y,z,w) \in R^4 | xyz = c \}$ and $Y_c = \{ ( x,y,z,w) \in R^4 | xzw = c \}$. For what real numbers $c$ is $Y_c$ a three-manifold? For what pairs $(c1,c2)$ is $W_{c_1} \cap Y_{c_2}$ a two-manifold?
I think that we can say for... |
H: Matrices and vectors
I am wondering if you can write a 2x2 matrix
$$\left(\begin{array}{cc} a & b \\ c & d\end{array}\right)$$
as a one vector
$$\left(\begin{array}{c} a \\ b \\ c \\ d\end{array}\right).$$
AI: I think you can do this in two ways:
Row-major Order
Column-major Order
They both allow you to write a m... |
H: How can I measure the distance between two cities in the map?
Well i know that the distance between Moscow and London by km it's about 2,519 km and the distance between Moscow and London in my map by cm it's about 30.81 cm and the Scale for my map is 1 cm = 81.865 km but when i tried to measure the distance between... |
H: How to find the probability of damaged goods transported by air?
A company transports goods in three ways: air, sea and road. Based on the past records, of the total transported goods, 10% were by air, 30% by sea and 60% by road. It is found that 1% of goods transported by air were damaged, 5% of the goods transpor... |
H: Does this right adjoint of a geometric morphism preserve directed colimits?
Let $E$ be a sheaf topos $E=\operatorname{Shv}(C)$ and $\{x_i:\operatorname{Sets}\to E\}$ be a set of geometric morphisms $x_i:Sets\to E$ such that the morphism
$$
x^*\colon E\to \prod_i \operatorname{Sets}
$$
induced by the left adjoints $... |
H: How can i solve this second-order differential equation $y^{\prime\prime}=a\sqrt{1+y^{\prime}}$?
My problem is to solve this given equation: $y^{\prime\prime}=a\sqrt{1+y^{\prime}}$
My approach was: But i dont know how to handle $a$ and besides this fact, it is a second-order equation. so i thought i'll have to find... |
H: Regular space which is not Hausdorff
I know that normality in the absence of $T_{1}$ does not imply regularity (Sierpinski space being a counterexample as it is vacuously normal but not regular). I have the feeling that similarly regularity in the absence of $T_{1}$ does not imply Hausdorff. I tried thinking of a c... |
H: Outer measure fundamental question
Royce's introductory definition of outer measure. Let $A\subset R$. Then:
$$m^*(A)=\text{inf}\left\{\sum_{n=1}^{\infty}l(I_n):\,A\subset \cup_{n=1}^{\infty}I_n\right\},$$
where $I_n=(a_n,b_n)$.
However, in reading this, I suddenly question whether it is a fact that any subset of $... |
H: Evaluating $\frac{1}{\pi} \int_{0}^{\pi} e^{2\cos{\theta}} d\theta$
I ran into this integral when computing the volume of a family of polytopes and I'm not sure how to evaluate it analytically (I know Wolframalpha says 2.27...). Any ideas? I tried using complex analysis (Cauchy Integral Formula, Residue Theorem, et... |
H: permutations combinations for shirt
How many different ways can 6 identical green shirts and 6
identical red shirts be distributed among 12 children such that each
child receives a shirt?
Now what will be the answer if they get any number shirts?
EDIT :Actually I was supposed to solve the modified question, ... |
H: Collision between a circle and a rectangle
I am trying to create a simple model for collisions between a circle and a rectangle to be used in a computer game. The reason I am asking this question here rather than stack overflow is that the problem is not one of programming but rather of mechanics. The problem I am ... |
H: How to prove this statement on finite groups?
I Fulton and Harris Chapter 3.2 we have that if $\mathbb{C}^n$ is the permutation representation of $S_n$ (symmetric group) then we can write $\mathbb{C}^n=V\oplus U$, where $U$ is the trivial representation and $V$ is the standard representation. Then they claim that $... |
H: Question using $f(A) =\mathcal Sf(\lambda)\mathcal S^{-1}$
Given
$$A=\pmatrix{0&1\\-2&3}$$
I found $\lambda_1=1$ and $\lambda_2=2$
$$\mathcal S=\pmatrix{1&1\\1&2}$$
$$\mathcal S^{-1}=\pmatrix{2&-1\\-1&1}$$
Using the formula $\mathcal S\Lambda\mathcal S^{-1}$
Calculate:
a) $(A-I)^{4000}$
b) $e^{2A}$
c) $(3A-5I)(I+4A... |
H: For covariant tensors, why is it $\bigwedge^k(V)$, not $\bigwedge^k(V^*)$?
In learning the very basics of differential geometry, I have seen the exterior product defined a couple of ways: First, I have seen it as the image of the covariant tensors (which I believe are essentially the $k$-fold tensor product of $V^*... |
H: Customary layout of $\phi_i(v_j)$
What is the customary layout of $\phi_i(v_j)$? Is it $$\pmatrix{\phi_1(v_1)&\phi_1(v_2) & \cdots \\
\phi_2(v_1) & \phi_2(v_2) & \cdots\\
\vdots &\vdots & \ddots},$$
or$$\pmatrix{\phi_1(v_1)&\phi_2(v_1) & \cdots \\
\phi_1(v_2) & \phi_2(v_2) &\cdots \\
\vdots &\vdots & \ddots}?$... |
H: Why do we only consider quadratic domains as Euclidean domains with squarefree integers?
I have been reading "Introductory algebraic number theory" by Alaca and Williams, and in the opening chapters they use the quadratic domains $\mathbb{Z}+\mathbb{z}(\sqrt{m})$ for non-square $m$ and $\mathbb{Z}+\mathbb{Z}\Big(\f... |
H: How to find the base $n$ such that $2_{n}^{12_n}=2_{10}^{6_{10}}\cdot 5_{10}$?
I have met the following problem:
How to find the base $n$ such that $2_{n}^{12_n}=2_{\small10}^{6_{\small 10}}\cdot 5_{\small 10}$?
And until now, I have no idea of how to solve it. I could try the conversion with some bases but it wo... |
H: If $T(W)⊆W$ show $W$ spaned by eigenvectors.
Let $T$ be a linear transformation of a finite dimensional real vector space $V$ and assume that $V$ is spanned by eigenvectors of $T$. If $T(W)⊆W$ for some subspace $W⊆V$, show that $W$ spaned by eigenvectors. Any suggestion? Thanks.
AI: Note that the following are equi... |
H: $k \subset A \subset B$, $B\supset k$ f.g., $\text{codim}_k(A) < \infty$ $\Rightarrow$ $B \supset A$ f.g. module?
Does this hold?
Let $k \subset A \subset B$ where $k$ is a field and $A,B$ are commutative rings.
If $B$ is a finitely-generated ring over $k$ and $\dim_k(B/A) < \infty$ then $B$ is a finitely-generated... |
H: The implication of zero mixed partial derivatives for multivariate function's minimization
Suppose $f(\textbf x)=f(x_1,x_2) $ has mixed partial derivatives $f''_{12}=f''_{21}=0$, so can I say: there exist $f_1(x_1)$ and $f_2(x_2)$ such that $\min_{\textbf x} f(\textbf x)\equiv \min_{x_1}f_1(x_1)+ \min_{x_2}f_2(x_2... |
H: If $L$ is a chain, prove it is finite.
I need ideas on the following problem. Suppose $L$ is a poset and every subset $S$ of $L$ has a top and bottom element. Prove $L$ is a finite chain.
All I need to do is prove that $L$ is finite (I have already proved $L$ is a chain). Any ideas or suggestions on solving this pr... |
H: Closed form of $\sum\limits_{i=1}^n k^{1/i}$ or asymptotic equivalent when $n\to\infty$
Is there a "closed form" for $\displaystyle S_n=\sum_{i=1}^n k^{1/i}$ ? (I don't think so)
If not, can we find a function that is asymptotically equivalent to $S_n$ as $n\to\infty$ ?
AI: Cesaro (particular case of the Stolz-Cesa... |
H: Proof that there are infinitely many positive rational numbers smaller than any given positive rational number.
I'm trying to prove this statement:- "Let $x$ be a positive rational number. There are infinitely many positive rational numbers less than $x$."
This is my attempt of proving it:-
Assume that $x=p/q$ is t... |
H: Probability that random subspaces intersect
Given an ambient space $\mathbb{R}^d$ and two randomly oriented subspaces $A,B$ with dimensions $a,b$ respectively, how can I express the probability that $A$ and $B$ intersect non-trivially?
AI: Assuming a uniform independent distribution (say picking orthogonal vectors ... |
H: Complement vector subspaces and the direct sum.
From Advanced Linear Algebra (Roman):
Let $\dim(V) < \infty$ and suppose that $V = U \oplus S_1 = U \oplus S_2$. What can you say about the relationship between $S_1$ and $S_2$? What can you say if $S_1 \subseteq S_2$?
I couldn't completely answer it, but here are ... |
H: Confidence Interval for a Binomial
Having trouble with this question from my textbook. I was wondering if anyone could help me out.
The following set of $10$ data points are independent realizations from a Binomial model
$X$ ~ $\mathrm{Bin}(36,\pi)$
$$10,12, 7, 6, 6,11, 7,12, 9,10.$$
Compute numerically, showing al... |
H: What is the difference in the meaning of equality symbol "$=$" as a logical symbol vs. a parameter?
In my text, the author says that he considers $=$ to be a logical symbol, and he adds that this makes the translation of the equality symbol to English different from if $=$ were a parameter.
But the examples don't ... |
H: Mapping two disjoint intervals into one interval
I'm trying to find a continuous function that maps $[0,1] \cup [2,3]$ onto $[0,1]$. Could someone give me a hint? (Note I haven't taken topology.)
AI: One such function is: $$f(x)=\begin{cases}x&:\ x\in[0,1]\\ x-2&:\ x\in[2,3]\end{cases}.$$ We can think of this as fi... |
H: What is the column space in a $5\times 5$ invertible matrix?
If $A$ is any $5 \times 5$ invertible matrix, then what is its column space? Why?
I'm totally lost with column space. Any ideas?
AI: If $A$ is invertable, is there any vector $v \in \mathbb R^5$ which cannot be composed by a linear combination of the c... |
H: How to go About Undergraduate Research
I apologize in advance if this question is out of the scope or focus here.
I was just wondering about the whole prospect of researching as an undergraduate. How to do it? Who to talk to in my department (UCLA) and what to do in general? To give you an idea of where my knowledg... |
H: prove that $\cos x,\cos y,\cos z$ don't make strictly decreasing arithmetic progression
let $x,y,z\in R$,and such that $$\sin y-\sin x=\sin z-\sin y\ge 0 $$ show that:
$$\cos x,\cos y,\cos z$$ don't make strictly decreasing arithmetic progression
my idea:
we have $$2\sin y=\sin x +\sin z\cdots\cdots\tag 1$$
and a... |
H: How do I find out the coordinates of every point between two points?
Suppose all I am given, is the coordinates of two points like the following:
What are some ways I could go about finding the values of every point on this line segment? Like the y-value at 2.3, 2.4, 2.7 etc.
Any suggestions as to how I could go... |
H: Sum of two independent exponential functions
Hey, I need a little help with this question.
Let $X$ and $Y$ be independent, exponentially distributed random variables.
What is the distribution of $Z=\frac X {X+Y}$?
I just can't figure it out because of the sum on the denominator, thank you for your help.
AI: We will... |
H: $K$ is weakly-compact $\Longleftrightarrow$ $\Pi(K)$ is weak*-compact
Let $X$ be a Banach space and $K\subset X$.
$\displaystyle \Pi:X \longrightarrow X$** canonical injection
$\Pi(x)(f)=f(x)$
How can we prove that:
$K$ is weakly-compact $\Longleftrightarrow$ $\Pi(K)$ is weak*-compact
Any hints would be appreciate... |
H: Sum of Independent Folded-Normal distributions
Let $X$ and $Y$ be independent, normally distributed random variables.
How is $|X| + |Y|$ distributed?
Is it known to be $|Z|$, where $Z$ is distributed normally?
AI: For $\alpha > 0$,
$$F_{|X|+|Y|}(\alpha) = P\{|X|+|Y|\leq \alpha\}
= P\{(X,Y) \in A\}$$
where $A$ is a... |
H: How do I find the radius of convergence of these power series
Please help me find the radius of convergence of the following power series including the method of solving them.
$$\sum_{n=0}^{\infty}2^{n!}x^{n!}\tag{1}$$
$$\sum_{n=0}^{\infty}\frac{n^2}{2^n}x^{n^2}\tag{2}$$
AI: The formula for the convergence radius o... |
H: How this particular inequality deduced in finding the limit of a vector function?
There is a proposition in differential geometry which states:
Let $\vec{x}$ be a vector function from a subset of $\mathbb{R}$ into $\mathbb{R}^n$ that is defined over an interval containing $a$, through perhaps not at $a$ itself. S... |
H: Problem understanding proof for positive solutions of parabolic PDE in Friedman's textbook
This is Lemma 5 in the chapter on maximum principles in Friedman's
book Partial Differential Equations of Parabolic Type. I am having trouble understanding one of the steps in the proof.
Let
$$
\Omega_{0}\equiv\mathbb{R}^{n}\... |
H: approximate $[0, 1]$ continuous function with 2d basis.
everyone. I've been thinking of this problem when reading papers about Fourier series. I think I can state my question as follows:
in the interval $[0, 1]$, I want to approximate an unknown continuous function with maximum frequency $f_0$ with mean $0$ (don't... |
H: Finding rank of matrix
Suppose $B$ is a non-zero real skew-symmetric matrix of order $3$ and $A$ is a non-singular matrix with inverse $C$. Then rank of $ABC$ is:
(A) $0, 1, 2$
(B) definitely $1$
(C) definitely $2$
(D) definitely $3$
Here we are given $B^{T}=-B$ and $A$ is non-singular i.e. $A^{-1}$ exists and $A^... |
H: Proof that $n \in \mathbb{N}$ by combinatorial analogue?
(Disclaimer: I'm a high school student, and my highest knowledge of mathematics is some elementary calculus. This may not be the correct terminology.)
A while ago, I saw the following problem: prove, for natural numbers $a$, $b$, $c$, $d$ with $a \geq b + c +... |
H: Evaluating $\lim\limits_{x\to 0^{+}} \frac{x}{\ln^2 x}$
How can I find:
$$\lim_{x\to 0^+} \frac{x}{\ln^2 x} $$
I know that the limit is $0$. I tried sandwich theorem but I don't know what could be bigger.
Thanks in advance.
AI: HINT: You need only consider what happens when $x$ is close to $0$, and as long as $0<x\... |
H: How can i solve this Cauchy-Euler equation $x^{3}y^{\prime\prime\prime}+2xy^{\prime}-2y=0$?
My Problem is this given Cauchy-Euler equation: $$x^{3}y^{\prime\prime\prime}+2xy^{\prime}-2y=0$$
My Approach was: i can see this is a ordinary differential equation of third-order and i think its linear. I was told that th... |
H: How to distinguish convergence with limited fluctuation in a non-standard setting?
I'm reading Edward Nelson's Radical Probability theory, and got confused on two concepts, convergence and limited fluctuation in a non-standard setting.
On page 21-22(see here):
Let $T$ be a subset of $\bf R$, and let $\xi: T \to \b... |
H: Question about convergence in probability (topic confusion)
I'm taking second year stats and was introduced the below concepts
For the third one, we use that to estimate the mean squared error in the case where the estimator is a nonlinear function of the sample mean. However, I can't find it anywhere in mathemati... |
H: Showing that the functions $\sqrt{2x}\exp(2\pi inx^2)$ form a complete orthonormal system for $L^2([0,1])$
How do I show that the functions
$$g_n(x) :=\sqrt{2x}\exp(2\pi inx^2)$$
where $n$ is a integer, are a complete orthonormal set in $L^2([0, 1])$?
I am relatively new to this and need some help getting started.
... |
H: Inequality in $L^p$ involving integral
This is from Folland chapter 6. exercise 30.
Suppose that K is a nonnegative measurable function on $(0,\infty)$ such that
$\int_0^{\infty}K(x)x^{s-1}=\theta(s)<\infty$ for $0<s<1$.
If $1 < p <\infty,\ p^{-1} + q^{-1} = 1$, and $f, g$ are nonnegative
measurable functions o... |
H: Closed form for the definite integral $\int_3^5\exp\left[\frac{-3.91}{V}\left(\frac{\lambda}{0.55}\right)^{-q}R\right]\,\mathrm d\lambda$
I am stuck on the following definite integral:
$$\tau=\int_3^5\exp\left[\frac{-3.91}{V}\left(\frac{\lambda}{0.55}\right)^{-q}R\right]\,\mathrm d\lambda$$
Is it possible to solve ... |
H: If there is a continuous function between linear continua, then this function has a fixed point?
Let $f:X\to X$ be a continuous map and $X$ be a linear continuum. Is it true that $f$ has a fixed point?
I think the answer is "yes" and here is my proof:
Assume to the contrary that for any $x\in X$, either $f(x)<x$ or... |
H: Continuity of scalar product
In a Hilbert space $H$ with inner product and associated norm, why would if $\|x-x_n\| \longrightarrow 0$ and $\|y-y_n\| \longrightarrow 0$ also $\langle x_n,y_n\rangle \longrightarrow\langle x,y\rangle$?
I understand that by Cauchy-Schwarz $\lvert\langle x-x_n,y-y_n\rangle\rvert \leq ... |
H: Explicit formula for the series $ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $
I was wondering if there is an explicit formulation for the series
$$ \sum_{k=1}^\infty \frac{x^k}{k!\cdot k} $$
It is evident that the converges for any $x \in \mathbb{R}$. Any ideas on a formula?
AI: You can have the closed form
$$\sum_{... |
H: Complete metric space question
Suppose $(X,d)$ is a non empty and complete metric space and $f:X \to X$ is a contraction. Show that there exists exactly one $x \in X$ such that $f(x)=x$.
AI: It is called Banach fixed-point theorem or Contraction mapping theorem. For the proof see the link. |
H: Convert a line in $ \Bbb R^3 $ given as intersection of two planes to parametric form.
We have a line in $ \Bbb R^3 $ given as intersetion of two planes:
$$
\left\{
\begin{aligned}
A_1x+B_1y+C_1z + D_1 &=0 \\
A_2x+B_2y+C_2z + D_2 &=0 \\
\end{aligned}
\right.
$$
How to represent it in parametric form:
$$
\left\... |
H: Prove this inequality concerning integral average.
Let $f\in L^1([a,b])$, and extend $f$ to be $0$ outside $[a,b]$. Let
$$
\phi(x)=\frac{1}{2h}\int_{x-h}^{x+h}f
$$
How to prove
$$
\int_a^b\left | \phi\right | \leqslant\int_a^b\left|f\right|
$$
To @martini:
I asked this question because I find something strange: if... |
H: Basis of a space of upper triangular matrices with trace 0
What would a basis of a space of $n \times n$ upper triangular matrices with trace 0 be? Is it trivial?
AI: Hint:
trace A=0 then $\sum_{i=1}^na_{ii}=0 $then we have $a_{nn=}-a_{11}-...-a_{n-1 n-1}$
$$ A=\begin{pmatrix}
a_{11} & a_{12} & a_{13} & \cdo... |
H: Integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i]$
I am trying to compute the integral closure of $\mathbb{Z}$ in $\mathbb{Q}[i].$ I have managed to show that $\mathbb{Z}[i]$ is inside the integral closure, and I suspect it is the entire thing. Can someone give me a nudge in the right direction?
AI: $\mathbb{Z}[i]... |
H: The smallest positive integer in the set $\{24u+60v+200w : u,v,w \in \Bbb Z\}$is given by which of the following number?
I am stuck on the following problem:
The smallest positive integer in the set $\{24u+60v+200w : u,v,w \in \Bbb Z\}$is given by which of the following number?
The options are: $2,4,6,24$.
Since ... |
H: if $1=\int _{ 1 }^{ \infty }{ \frac { ax+b }{ x(2x+b) } dx } $ then $a+b=?$
I have the following question:
if $1=\int _{ 1 }^{ \infty }{ \frac { ax+b }{ x(2x+b) } dx } $ then $a+b=?$
a) $0$
b) $e$
c) $2e-2$
d) $1$
I tried finding it's anti-derivative but that doesn't seem to help.
I also tried to get it to $\int... |
H: The number of limit points of the set $\left\{\frac1p+\frac1q:p,q \in \Bbb N\right\}$ is which of the following:
I am stuck on the following problem:
The number of limit points of the set $\left\{\frac1p+\frac1q:p,q \in \Bbb N\right\}$ is which of the following:
$1$
$2$
Infinitely many
Finitely many
If I ta... |
H: Find the distance between two lines in $ \Bbb R^3 $
There are two lines in $ \Bbb R^3 $ given in parametric form:
$$
l_1:
\left\{
\begin{aligned}
x &= x_1 +a_1t\\
y &= y_1 +b_1t \\
z &= z_1 +c_1t \\
\end{aligned}
\right.
$$
$$
l_2:
\left\{
\begin{aligned}
x &= x_2 +a_2s \\
y &= y_2 +b_2s \\
z &= z_2 +c_2s \\... |
H: Poles of complex function
Let $f:\mathbb{C} \to \mathbb{C} $ be meromorphic and $\{ z_j \}$ be its poles. In the text I am reading $f$ also satisfies the identity
$$
f(z)^{-1} = \overline{f(\overline{z})} \qquad \text{for } \operatorname{Re}z >0.
$$
It also happens to satisfy the property that
$$
g(z) := \frac{f... |
H: An element of $L^2(0,T;V_n)$.
Let $V$ be Hilbert with basis $w_j.$ Let $V_n = \text{span}(w_1, ..., w_n)$. Is it true that every element $v \in L^2(0,T;V_n)$ can be written as
$$v(t) = \sum_{j=1}^n a(t)w_j?$$
I think so. But my doubt comes because I am told that $L^2(0,T;V_n)$ has basis
$$\{l_iw_j : i \in \mathbb{N... |
H: Trigonometric identity: $\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$
I have to prove the following result :
$$\frac {\tan\theta}{1-\cot\theta}+\frac {\cot\theta}{1-\tan\theta} =1+\sec\theta\cdot\csc\theta$$
I tried converting $\tan\theta$ & $\cot\theta$ into $\cos... |
H: Inegrable functions $f_k$ with $\frac 1 {2 \pi} \int_0^{2 \pi} |f_k(x)|^2 dx \rightarrow 0$ where $\lim_{k \rightarrow \infty} f_k$ does not exist
I am searching for integrable functions $(f_k)_{k=0}^\infty$ on the circle with
$$
\lim_{k \rightarrow \infty} \frac 1 {2\pi} \int_0^{2 \pi} |f_k(x)|^2 dx = 0
$$ and s.t... |
H: Show $A^*A$ is nonnegative operator
exercises from my lecturer too. I haven't found any idea.
Let $A$ be an arbitrary linear operator acting in unitary space $\mathbb{C}^n$ , and let $A^*$ be its adjoint. Prove that $A^*A$ is nonnegative operator. Prove that $A^*A$ is positive operator if $A$ is nonsingular.
Corre... |
H: Domain and Range in function composition
I just came out of an exam that included this question.
Let $$g(x)=\sqrt{x^2-1}, ~|x|\geqslant1,$$$$f(x)=\sqrt{x^2+1},~x\in \mathbb{R}.$$
Find $gf$ and $fg$, stating their domain and range.
My answer was $$gf=g(f(x))=\sqrt{(\sqrt{x^2+1})^2-1}=\sqrt{x^2+1-1}=|x|$$with $D_... |
H: Proof that restriction of hermitian operator to its invariant subspace is also hermitian
Proof that restriction of hermitian operator to its invariant subspace is also hermitian
What would be the most elegant way to prove this?
AI: Let $L$ be invariant subspace of $A$, then
$$
\begin{align}
A\text{ is Hermitian}&\L... |
H: Jacobbian Transformation Multiple Integral
The question says :
Sketch the region under the transformation of u=x+y and v=y for
$$R=\{(x,y): 0\leqslant x\leqslant 1 , 0\leqslant y \leqslant1\}$$
Find the Area of the region.
Given answer is $1$
I just need help on the calculation of the area. No need to draw the grap... |
H: Does removing random items effect probability?
Let us say you have a finite set of things (maybe playing cards?) called $S$. It has $n$ things. No let us say we want a thing from an arbitrary set $E$. First we remove $k$ number random things from S, were $0 \leq k<n$. Now I pick a random thing from $S$. Does the pr... |
H: Calculate the limit $\lim_{x \to \infty} \ \frac{1}{2}\sum\limits_{p \leq x} p \log{p}$ (here p is a prime)
Calculate the limit $\lim_{x \to \infty} \ \frac{1}{2}\sum\limits_{p \leq x} p \log{p}$
(here the sum goes over all the primes less than or equal to x) using the Prime Number Theorem.
I think I've managed to ... |
H: Recursive Properties As They Relate To Domains
I'm struggling with understanding what restrictions recursive properties generally place on domains. also on a more global level I don't think I fully understand how functions are described in symbolic logic.
I hope these two examples will illustrate these issues:
Say... |
H: Family of complemented subspaces
Let $X$, $Y$, $A$, $B$ be topological vector spaces. Given two jointly continuous families of linear injective maps $P: Y \times A \rightarrow X$ and $R: Y \times B \rightarrow X$, such that for $y=0$ we have the topological complementation $X = Im\, P_0 \oplus Im\, R_0$. Here and i... |
H: Projection of a vector
I am reading some material on the Householder reflection and it describes the projection of a vector $\vec{x}$, onto $\vec{v}$ in vector notation to be:
$\frac{\vec{x}\cdot \vec{v}}{||\vec{v}||^2}\vec{v}$
Where $a\cdot b$ is the dot product of vectors a and b.
So how is that?
I know if θ i... |
H: A group with two non trivial subgroups is cyclic
Let $G$ be a group. Suppose that $G$ has at most two nontrivial subgroups. Show that $G$ is cyclic.
Can anyone help me please to solve the problem?
AI: If $G$ has no nontrivial subgroups, it is clearly cyclic. If $G$ has exactly one nontrivial subgroup $H$, consid... |
H: Study , according to the values of x , the relative position of line & Curve
We have $F(x) = x + \ln \left(\frac{1+2e^{-2x}}{1+e^{-x}}\right)$ represents curve $C$ and a line $d: y=x$
In an exercise it is required to study the relative position of these two functions, I know we have to subtract them from each o... |
H: What is the summation notation for the Fibonacci numbers?
I learned about summation notation the other day, and I'm looking for a way to write the Fibonacci numbers with it. What would it look like?
AI: $$F_n=\sum_{k=n-2}^{n-1}F_k$$
Given the initial conditions:
$$F_0=0$$
$$F_1=1$$
It's trivial, but it does use the... |
H: Is it true in general that $\int \dots \int_{0 \le x_1 \le \dots \le x_n,\ 0 \le x_n\le1}dx_1\dots dx_n=\left(\frac{1}{2}\right)^n?$
I was looking at an example with the following integral:
$$\iiiint_{0 \le x \le y \le z \le t,\ 0 \le t \le \frac{1}{2}} 1 \,dx\,dy\,dz\,dt = \frac{1}{16}$$
Is it true in general tha... |
H: Integral Inequality Absolute Value: $\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$
Suppose we are given the following: $$\left| \int_{a}^{b} f(x) g(x) \ dx \right| \leq \int_{a}^{b} |f(x)|\cdot |g(x)| \ dx$$
How would we prove this? Does this follow from Cauchy Schwarz? Intui... |
H: Finding multiple integral on bounded area.
Today I just learn on multiple integral. Somehow this question quite confusing.
Find the area of the 1st quadrant region bounded by the curves y=$x^3$, y=2$x^3$ and x=$y^3$, x=4$y^3$ using subsitution method (Jacobbian method).
Let y=$u$$x^3$ and x=$v$$y^3$
Could someone... |
H: Amount of homomorphisms $S_5$ to $C_6$
I think I understand the amount of homomorphisms $C_6$ to $S_5$:
Because $C_6$ is cyclic, we only have to look where we can send $\langle a\rangle$, because $f(\langle a\rangle)$ must have order $1, 2, 3$, or $6$ we have:
-for $1$: trivial homomorphism.
-for $2$: the ten $(x... |
H: Show that $\sum\limits_{p \leq x} \frac{1}{p}$ ~ ${\log\log{x}}$ when ${x \to \infty}$ (here p is a prime)
I saw that some of you were upset over my last question, so I decided to ask a more interesting question:
Show that $\sum\limits_{p \leq x} \frac{1}{p}$ ~ ${\log\log{x}}$ when ${x \to \infty}$
(here the sum go... |
H: Big-$O$ inside a log operation
I would appreciate help in understanding how:
$$\log \left(\frac{1}{s - 1} - O(1)\right) = \log \left(\frac{1}{s - 1}\right) + O(1)\text{ as }s \rightarrow 1^+$$
I thought of perhaps a Taylor series for $\log(x - 1)$, but that would have an $O(x^2)$ term.
Also I would appreciate any r... |
H: $U_n= \int_0^1 \frac{e^{nx}}{e^x +1} \mathrm d x $
I have this integral in a sequence question
$$U_n= \int_0^1 \frac{e^{nx}}{e^x +1} \mathrm dx $$
how to solve it ?
AI: Try looking at $$U_{n+1}+U_n = \int_0^1\frac{e^{(n+1)x}+e^{nx}}{e^x+1}\,\mathrm dx$$ and see how that helps simplify the integral. |
H: How find $dy/dx$ $y= \frac{x^2+\sin2x}{2x+\cos^2 x}$
$$dy/dx\space\space y= \frac{x^2+\sin2x}{2x+\cos^2x}$$
AI: We need the quotient rule and the chain rule to find $\frac {dy}{dx} = y'$ given $$y= \frac{x^2+\sin2x}{2x+\cos^2x}$$
Given a quotient of functions: $f(x) = \frac{g(x)}{h(x)}$
$$f'(x) = \frac{g'(x)h(x) - ... |
H: Exponential distribution for the lifetime of an LCD screen: probability that it functions for $50{,}000$ hours
If the probability that an LCD screen functions for $x$ hours is defined by the density function:
$$f(x)=0.01*\exp(-x/100)I_{[0,\infty)}(x)$$
where $I$ is an indicator function and $x$ is measured in thou... |
H: Proving that $\int \frac{dx}{(1-x^2)}$ equals the inverse hyperbolic tangent of $x$
I prove the following fact
$$\int\limits \dfrac{dx}{(1-x^2)}=\tanh^{-1}x$$
I show, by integrating by substitution, that the integral equals
$$- \dfrac{\ln(1-x^2)}{2x}.$$
Setting $x=\tanh z$, we get $$-\dfrac{\ln\operatorname{sech} ... |
H: $f:(0,\infty)\to [0,\infty)$ concave implies $f$ bounded below?
Let $f:(0,\infty)\to [0,\infty)$ be a concave function such that $f$ is not indentically zero. It seems to me that the following statements are true:
I - For every $a>0$ fixed, $f_{|[a,\infty)}$ is bounded below by a positive constant depending on $a$,... |
H: Large exponential modular
Proof $2011^{2011^{2011}}-2011 \equiv 0 \mod 30030$
By Chinese Remainder Theorem this is equivalent to proving:
$2011^{2011^{2011}}-2011 \equiv 0 \mod 2$
$2011^{2011^{2011}}-2011 \equiv 0 \mod 3$
$2011^{2011^{2011}}-2011 \equiv 0 \mod 5$
$2011^{2011^{2011}}-2011 \equiv 0 \mod 7$
$2011^{20... |
H: Solving $L - L\sqrt{1-\frac{u^2}{C^2}} = u T$ for $u$
Hello how can i solve this for $u$ ($L,C,T$ are constants).
\begin{aligned}
L - L\sqrt{1-\frac{u^2}{C^2}} = u T
\end{aligned}
AI: $$\text{Rearranging: }L-uT=L\sqrt{1-\frac{u^{2}}{C^{2}}}$$
$$\text{Squaring both sides: }(L-uT)^{2}=L^{2}\left(1-\frac{u^2}{C^{2}}\r... |
H: Show that $\arg(\exp(z)) = y + 2\pi k$ for any $\arg(\exp(z))$ and some integer $k$.
The entire question is:
Show that $\operatorname{mod}(\exp(z)) = e^x$. Show that $\arg(\exp(z)) = y + 2\pi k$ for any $\arg(\exp(z))$ and some integer $k$.
I could do the first part. I do not know how to do the second.
Guidance p... |
H: Interpolating missing points in 3D data-set
Given the following x,y,z points (z is actually a signal strength indicator in dBm):
63 371 -21
142 371 -9
233 374 -18
288 371 -36
310 373 -38
349 374 -39
415 348 -44
507 334 -49
689 337 -56
635 254 -57
422 284 -42
380 278 -39
281 280 -39
214 299 -34
146 285 -30
81 302 ... |
H: A new question about solvability of a direct product
I had asked that if $G$ is a direct product of a $2$-group and a simple group, then is it possible that $G$ be a solvable group. That the answer is no!
But by a Remark on T.M. Gagen, Topics in Finite Groups, London Math. Soc. Lecture Note Ser., vol. 16, Cambridge... |
H: A helix problem
This equation
$$
x^2 + y^2 – \left(\tan^{-1} \frac{y}{x}\right)^2 = 0
$$
describes a helix.
What is the capacity of the first twist?
AI: Polar co-ordinates simplify the analysis greatly. When $r^2=x^{2}+y^{2}$ and $\tan(\theta)=\frac{y}{x}$, your equation becomes simply $r=\theta$, which is the we... |
H: $\mathbf{F} = \nabla f$ Get the function f, when given a vector field
For conservative vector fields F the following equation holds
$$\int_c \mathbf{F} \cdot d\mathbf{r} = \int_c \nabla f \cdot d\mathbf{r} = f(\mathbf{r}(b)) - f(\mathbf{r}(a))$$
Now I have a lot of trouble finding the function f so $\mathbf{F} =... |
H: How to compute $x$ and $y$
How can one find in an efficient way $x,y \in \mathbb{Z}$ with max$\{|x|,|y|\} > 0$ as small as possible such that $\mid \pi x + e y \mid < 10^{-4}$ ?
I have reduced the following lattice basis:
\begin{pmatrix} 1 & 0 \\ 10^4\pi & 10^4e \end{pmatrix}
but is this the right one? And how can ... |
H: Set of convergent sequences as a vector space, show associativity
I try to prove that a set of convergent sequences $S=\{(a_n) : (a_n) \text{ convergent sequence of natural numbers} \}$
is a vector space.
I'm guess I have to show that all 8 axioms are valid, but I have a problem how to show the associativity. Shoul... |
H: Normal at every localization implies normal
I'm having some trouble with basic ring theory. Let $A$ be an integral domain and $\alpha$ an element of its fraction field integral over $A$. I am trying to understand a proof that $\alpha\in A$ under the hypothesis that $A_{\mathfrak p}$ is normal for every prime ideal ... |
H: Closed-form expression for the exponent?
Assume we have a simple equation
$a^x = y, \quad a, x \in \mathbb{R}, \; a \neq 0$
from where $x$ needs to be evaluated. If we set a restriction $a > 0$, there
is a simple logarithm expression available
$x = \log_a y = \frac{\log y}{\log a}$.
Still I'm not sure how to d... |
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