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H: Weak convergence definition
I have a following question, for example: let $f_n\in L^2$ be a bounded sequence of real functions. Then we know that there is some $f$ (in $L^2$?) such that $f_n$ converges weakly to $f$ in $L^2$ i.e..... and now I do not know if I understand it properly. Do weak convergence means:
a) f... |
H: Question about the second derivative of functions at a local max/min
Is the following true :If $f(x)$ is a twice differentiable function with a local maximum/minimum at a. The function $g(x)$=$d^2f\over d^2x$ will attain its local minimum/maximum at a
Q: is the above statement true if so how would you prove it?... |
H: If $S$ and $M$ are Positive Definitive Matrices, Why $S$ and $M^{-1/2}$S$M^{-1/2}$ share the same eigenvalues?
Following and argument to find the solution to the max of the generalized Rayleig Quotient, $\frac {x^tSx}{x^tMx}$, I found the stament:
if $S$ and $M$ be Positive Definitive, $S$ and $M^{-1/2}$S$M^{-1/2}... |
H: Gauss Lemma - Do Carmo's Riemannian geometry, use of parallel transport?
I was exactly having the same doubt as this question. I don't understand specifically why
$$
(d \exp_p)_v(v)=v
$$
I worked out exactly the same math as wikipedia and I ended up with
$$
(d \exp_p)_v(v) = \frac{d}{dt}\left. \left(\gamma((t+1),p,... |
H: Solve the following system of non-linear equations: $x^2+4xy+y^2=13$, $2x^2+3xy=8$.
Solve the following system of non-linear equations: $$\begin{align}x^2+4xy+y^2&=13\\ 2x^2+3xy&=8.\end{align}$$
I started off this problem by rearranging and substituting but I’m stuck. I think that I have to change this system t... |
H: True / false question about a Cauchy sequence in Real Analysis
I am solving assignments in Real analysis but I am unable to think about how I can solve this question.
Let $f: ( 0, \infty ) \to \mathbb{R}$ be a continuous function. Does $f$ maps any Cauchy sequence to a Cauchy sequence.
I tried by taking $ {x_n} ... |
H: image of differentiable manifold is again a manifold
Let $1\leq \mu <n,\, M\subset \mathbb{R}^n$ be a $\mu$-dimensional differentiable submanifold of $\mathbb{R}^n$ (that is, all the charts $\varphi$ are immersions: $\phi\in C^1$ and the rank of $D\varphi$ equals $\mu$). Now let $f\in C^1(\mathbb{R}^n,\mathbb{R}^n)... |
H: Does there exist an operation that could turn the set of all negative real numbers into an abelian group?
The answer is no for familiar operations of addition and multiplication. But could there exist any other operation that could turn the set of all negative real numbers into an abelian group. If yes, what is it?... |
H: True /False question about Compact sets and continuous functions
I am trying some assignment questions and I am unable to think on how can I solve this problem.
Question: Let K be subset of $\mathbb{R^{n} }$ such that every real valued continuous function on K is bounded. Then is K compact?
I think this statement... |
H: The smallest integers having $2^n$ divisors
Problem: For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\... |
H: Solve $y'=-3x^2\cdot y^2e^{-\frac{1}{y}}$
Solve the initial value problem $y'(x)=-3x^2\cdot y(x)^2e^{-\frac{1}{y(x)}}$ with $y(e^{1/3})=1$ and, for each solution $\varphi$, find the maximal interval such that $\varphi$ is defined and solves the differential equation.
I've tried to all the common methods like separ... |
H: Compact and convergent countable filter base
Prove that if a metric space $(S,d)$ is compact, then every countable filter base is included in a convergent one.
My efforts:
I suppose the "one" in the problem is also countable. Otherwise I can let it be a base for an ultrafilter and directly use the theorem about con... |
H: Change of speed problem (differential equations)
Problem
Let $V: \mathbb{R}^n \to \mathbb{R}^n$ be a Lipschitz continuous vector field, and let $k: \mathbb{R}^n \to (0,\infty)$ be a positive Lipschitz continuous function. Let ${x_t}$ be a solution to $\dot{x} = V(x)$ with initial condition $\xi$, and let ${y_t}$ be... |
H: Solving $x+x\ln(x)+\ln(x)=y$ for $x$
For $x,y\in\mathbb{R^+}$ , consider the equation:
$x+x\ln(x)+\ln(x)=y$
with constant $y$,
which is the same as
$x+\ln(x^{x+1})=y$
How do I solve for $x$?
AI: By differentiating, we can discover that $y$ is strictly increasing as a function of $x$, and it's easy enough to see tha... |
H: Question on linear span of subsets of $\Bbb{R}^3$
Let $V$ = $\Bbb{R}^3$. Consider
$S_1$ = {($x,y,z$)|$x^2$ + $y^2$ + $z^2$ = $1$} and
$S_2$ = {($x,y,z$)|$z$ = $0$}. Then choose the correct
$L$($S_1$ $\cup$ $S_2$) = $V$
$L$($S_1$) is a subspace of $L$($S_2$)
dim($L$($S_1$ $\cap$ $S_2$) = $2$
$L$($S_1$) $\cap$ $L$($... |
H: Show that if $HK$ is an abelian group ($H$,$K$ have coprime orders), then $H$ is in $N_G(K)$
My question stems form this excerpt from Dummet. I follow all of it, except the claim that $PQ$ is a subgroup of the normalizer. I feel like the solution is some simple algebraic one but i cannot seem to find it.
AI: Let $h... |
H: The meaning of "around" in the uniqueness and existence theorem
while learning on the uniqueness and existence theorem, I started to solve some examples and I noticed that the following is always repeated: "$f(x,y)$ and $\frac{df}{dy}$" are continuous around $(x_0,y_0)$
I do not understand what is the mathematica... |
H: Why do we need any test function to be infinitely many times differentiable?
I have started learning the very basic distribution theory and I encountered the idea of a test function and distribution. I am not entirely sure why the following definition of a test function is necessary.
A test function is a $C^{\inft... |
H: Is my proof of $|a| \leq b \iff -b \leq a \leq b$ correct?
Background
Hello, I'm teaching myself proofs, and am unsure whether or not my proof of $\forall a,b \in \mathbb{R}(|a| \leq b \iff -b \leq a \leq b)$ is correct. Your feedback is greatly appreciated.
Proof
Proof. $(\rightarrow)$ Suppose $a,b \in \mathbb{R... |
H: Getting into university - two variables distribution
If students want to get accepted to university, the chance to pass each stage is $0.5$. If you fail, you can't pass to the second stage and can't enter the program. If you fail the second stage, you can't enter the program too. We will assume a case of two people... |
H: Show that $\forall a,x,y \in G:ax=ay\Longrightarrow x=y$ with $(G,\cdot)$ being a group
Let $(G,\cdot)$ be a group, with $G$ being a finite set.
Show that $\forall a,x,y \in G:ax=ay\Longrightarrow x=y$
Since $(G,\cdot)$ is a group $a \in G \Longrightarrow a^{-1} \in G$ with $a^{-1}a=e$
So we now define the automo... |
H: Is this a vector?
Can anyone help me with this probability question?
Given two independent random variables $Y \sim N(\mu,\sigma)$ and $Z \sim U[0,3]$, we define
$X=Y+Z$
Prove or disprove: The vector $(Z,X)^T$ is a vector.
I would appreciate any help :)
Thanks
AI: The vector $V=(V_1,V_2)^T=(Z,-(-1)^I |Z|)^T=(Z,\pm ... |
H: $\mathbb{Z}\oplus \mathbb{Z}\cong \mathbb{Z}\oplus A \implies \mathbb{Z}\cong A$
Let $A$ be a $\mathbb{Z}$-module and suppose that $\mathbb{Z}\oplus \mathbb{Z}\cong \mathbb{Z}\oplus A$. Do we have $\mathbb{Z}\cong A$?
I know the result is true if $A$ is finitely generated as $\mathbb{Z}$-module. Clearly, $A$ can be... |
H: Topology of sequentially open sets is sequential?
Let $(X,\tau)$ be a topological space. The collection of all sequentially open subsets of $X$ (i.e. the complements of the sequentially closed subsets) is itself a topology $\tau_\text{seq}$, equal to $\tau$ if $X$ is a sequential space, and a strictly finer topolo... |
H: Finite group $G$ has $61$ Sylow $3$-subgroups
Suppose that the finite group $G$ has $61$ Sylow $3$-subgroups. Then I want to prove that there exist two Sylow $3$-subgroups $P$ and $Q$ satisfying $|P: P\cap Q|=3 $.
Since $n_{3}=61$, then the order of $G$ must be $61.3^{n}.p_{1}^{a_{1}}\dots p_{k}^{a_{k}}$. But I co... |
H: Root of Complex Number in Polar Representation with Negative "r"
A friend had been looking at, as an example, $z^3=-8cis(\frac{\pi}{2})$ and ran into a phenomenon he struggled with explaining to himself; he approached me for assistance and I wasn't sure, either.
If I look at the "-" as a $cis(\pi)$ and perform a co... |
H: Integrating $\int_{0}^{2 \pi} \cos^{2020}(x)dx$, $\int_{0}^{\pi/2} \frac{1}{\tan^{\sqrt{2020}}(x)+1}dx$, $\int_{0}^{\infty} x^5 e^{-x^4}dx$
I've been working through the 2020 MIT Integration Bee qualifier questions (20 total) for fun, and there are three that I haven't been able to crack yet. (The complete list of ... |
H: Do all finite groups contain an odd number of elements?
Hello ive got a maybe weird and primitive question, which purely comes from my status as a beginner. Thinking about groups and their properties, with $\forall a \in G\,\,\, \exists a^{-1} \in G:aa^{-1}=e$ and the existence of a neutral element $e \in G$, doesn... |
H: The derivative of $f(x)=\frac{3 \sin x}{2+\cos x}$
My solution:
The background
$$\begin{align} &3\frac{d}{dx}\left(\frac{\sin(x)}{2+\cos(x)}\right)\\
&=3\frac{\frac{d}{dx}(\sin(x))(2+\cos(x))-\frac{d}{dx}(2+\cos(x))\sin(x)}{(2+\cos(x))^2}\\
&=3\frac{\cos(x)(2+\cos(x))-(-\sin(x))\sin(x)}{(2+\cos(x))^2}\\
&=\frac{3+6... |
H: Proof verification: If factor group $G/Z(G)$ is cyclic then $G$ is Abelian
Let $G$ be a group. If $G/Z(G)$ is cyclic, then $G$ is Abelian.
I tried to prove it like this: I'll make use of the theorem that $G$ is Abelian $\iff Z(G)=G$.
Hence, it must be shown that $G=Z(G)$
Let $G/Z(G)=\langle gZ(G) \rangle$ for som... |
H: Does algebraic factorization imply the same 0 factoring pattern for non-polynomial expressions?
Suppose one has $e^{f(x)}(g(x)-e^{h(x)}) = 0.$ Can it then be inferred that either $e^{f(x)}=0$ or $g(x) - e^{h(x)} = 0$ makes this statement true, similar to how $x(1-x^2)$ would yield that $x=0$ or $x\pm 1$ are solutio... |
H: $90$ Degree Piping Cuts on Cylinders
I'm trying to find the math behind laying out piping cuts. For instance, when one piece of pipe tees (at $90$ degrees) into another piece of pipe, how do you cut the pipe? I'm trying to figure out the math of creating a flat template that I can wrap around a pipe to draw a line ... |
H: Hasse's Theorem: min and max values with or without point of infinity?
I have to calculate the min and max values of a field.
Min: $\lfloor{q+1-2 \sqrt{q}}\rfloor$
Max: $\lfloor{q+1+2 \sqrt{q}}\rfloor$
According to Hasse. BUT the exercise says that min and max should be found together with the point of infinity. So... |
H: Find basis to $\begin{bmatrix}1&-4&3&-1\cr2&-8&6&-2\end{bmatrix}$
I want to find the basis to
$$x_1-4x_2+3x_3-x_4=0$$
$$2x_1-8x_2+6x_3-2x_4=0$$
so I set up the matrix:
$\begin{bmatrix}1&-4&3&-1\cr2&-8&6&-2\end{bmatrix}$
to get $\begin{bmatrix}1&-4&3&-1\cr0&0&0&0\end{bmatrix}$
Then I would get $\begin{bmatrix}x_1\cr... |
H: Failed solution for solving $\cos(\theta) = -\sin(-\theta)$
I'm trying to solve $\cos(\theta) = -\sin(-\theta)$ on the interval $[0, 2\pi)$, but having trouble identifying what I'm doing wrong
$$\cos(\theta) = -\sin(-\theta)$$
By even-odd identities:
$$\sin(-\theta)=-\sin(\theta)$$
$$\cos(\theta)= -(-\sin(\theta)... |
H: In a given field $F$, if $ab=0$, is it true that at least $a$ or $b$ has to be zero?
The answer is true for integral domains, but I wonder if it is also true for any general field. Thanks.
AI: Without loss of generality, assume ${a\neq 0}$. Then ${a}$ has a multiplicative inverse. Multiply both left hand sides by t... |
H: Missing solutions from $\tan(\theta)=2\sin(\theta)$
I'm trying to solve $\tan(\theta)=2\sin(\theta)$ on the interval $[0,2π)$, but having trouble identifying what I'm doing wrong
$$\tan(\theta)=2\sin(\theta)$$
Using quotient identity: $$\tan(\theta)= \frac{\sin(\theta)}{\cos(\theta)}$$
$$\frac{\sin(\theta)}{\cos(\t... |
H: Need help understanding this recursion via pseudocode
Given the recursive algorithm in this pseudocode:
RTC(n)
Input: A nonnegative integer, n
Output: A numerator or denominator (depending on parity of n) in an approximation of
If n < 3
Return (n + 1)
If n >= 3
t: = RTC(n – 1)
If n is odd
... |
H: Find all the $x$ such that $N \mod x = 1$
Find all the $x$ such that $N \mod x = 1,$ where $x < N$ and $N$ is a big integer.
Please help me find a solution, or tell me if this is not a simple task to do.
AI: $N \bmod x=1$ means $x $ divides $N-1$,
so finding all the $x$ such that $N\bmod x=1$ is as hard as factorin... |
H: Does there exist a number that makes this work?
I am wondering if there is some kind of polynomial somewhere out there that makes the following functional equation true:
$y(x) = y(x-1) - \tfrac{1}{2}x.$ I don't normally work on this type of problem so I don't know what the standard procedure is. If I try testing sa... |
H: Proving limit by definition with absolute value
This is what I have:
$\lim_{x \to 0} x \cdot |x|=0$
And I know:
$$|x|=\begin{cases}x,&\text{if }|x|\ge 0\\-x,&\text{if }x <0\;.\end{cases}$$
Case 1:
$\lim_{x \to 0^+}=0$
$(0 < |x - 0| < \delta \Rightarrow |x-0|< \epsilon)$
since:
$o < x-0< \epsilon \Rightarrow \fbox{$... |
H: Will a tortoise moving $1$ unit at a time around a circle meet a hare moving $2$ units at a time?
Consider a circle with $100$ points numbered $1,2,3,4,\dotsc,99,100$. Now there is a tortoise at the point numbered $1$ and a hare at point numbered $2$. Both can only move in clockwise direction, and only on numbered... |
H: Invertibility of Matrix mapping $f(X) = DX + XD$
Consider the set of $n*n$ matrices.
If $D$ is a diagonal matrix, and the linear transformation from the set of $n*n$ matrices to itself is defined as
$$f(X) = DX + XD$$Is the mapping invertible?
Here are some observations I made. Obviously, if all diagonal entries ar... |
H: Evaluate using Residues $\int_0^{2\pi}\frac{d\theta}{1+a\sin\theta}$
I need to evaluate the following using residues: $\int_0^{2\pi}\frac{d\theta}{1+a\sin\theta}$ where $-1<\theta<1$.
I suppose the $a$ in front of $\sin\theta$ is throwing me off. I was thinking I could let $z=e^{i\theta}$ and so $\sin\theta=\frac{... |
H: When defining ordered pairs, are there any important distinctions between $\{\{a\},\{a,b\}\}$ and $\{a,\{b\}\}$?
The formal Kuratowski definition of ordered pair is that $\langle a,b\rangle = \{\{a\},\{a,b\}\}$.
While I think I understand the above definition well I wanted to check if below definition also works ju... |
H: $l_1$ has no infinite dimensional subspace that is reflexive.
How to show that $l_1$ does not contain an infinite dimensional subspace that is reflexive.
AI: See Schur's theorem and Eberlein-Smulian Theorem. Suppose that $Y \le \ell_1$ is a subspace, and let $B_Y$ be the closed unit ball of $Y$, i.e. $B_X \cap Y$. ... |
H: Convergence of the sum constructed by approximation of the integral of an $L^1$ function
Let $f \in L^1( \mathbb{R}^n)$. Does the sum
$$S(x) = \sum_{k \in \mathbb{Z}^n} f(k+x) $$
converge for almost every $x$?
Intuitively I'm approximating the integral (which is finite), so I think this should be true. Maybe $C||f|... |
H: Show that $|\sum_{i,j} a_{ij} x_i x_j|\le \max_i |x_i|\cdot \max_j |y_j|$ is equivalent to $|\sum_{i,j} a_{i,j} x_i y_j |\le 1$
Show that $|\sum_{i,j} a_{ij} x_i y_j|\le \max_i |x_i|\cdot \max_j |y_j|$ for all $x_i,y_j \in \mathbb{R}$ is equivalent to
$$
\bigg|\sum_{i,j} a_{i,j} x_i y_j \bigg|\le 1\quad \forall x_i... |
H: Does a statement of the form "for all $X>0$ there exists $x > X$ satisfying some condition" evaluate to "the condition must be true for all $x>0$"?
Question is essentially as the title states. I was inspired by the form which the negation of the Cauchy Criterion takes.
If I have a statement saying "$P$ is only true... |
H: If $T$ is an operator such that $T^2 = \lambda T$, and $\|Tx\| = \|x\|$ for some $x \neq 0$, then prove $\lambda = \pm 1$
Let $T$ be a linear transformation on $\mathbb R^n$ over $\mathbb R$ such that $T^2=\lambda T$ for some non zero $\lambda $ in $\mathbb R.$ If $\|Tx\|=\|x\|$ for some non zero vector $x$ in $\ma... |
H: Explain this math statement
Quoting a paragraph from Timothy Grover's The Princeton Companion To Mathematics,
(9) ∀n ∃m (m>n) ∧ (m ∈ P).
In words, this says that for every n we can find some m that is both
bigger than n and a prime. If we wish to unpack sentence (9) further,
we could replace the part m ∈ P by
... |
H: What is the definition of a variety in Mumford's red book?
In Mumford's red book, prevariety is defined (in II.3) as follows:
If $k$ is an algebraically closed field, a prevariety over $k$ is a reduced and irreducible prescheme of finite type over $k$.
Mumford then starts mentioning variety later on and I wasn't ... |
H: Query related to mathematical reasoning
I was studying mathematical reasoning and I guess my book did not explain the rules properly .
There is this statement-:
All natural numbers are even or odd.
This statement seems kind of correct because a natural number is even or odd. But if you try to break it in two compon... |
H: Show that an open interval $(a,b)$ is an open ball in $\mathbb{R}$
My attempt:
Let $(a,b)$ be the open interval. We probably need to show that $(a,b)$ is an open ball so something in this form $B(x,r)$.
So I chose $x=(a+b)/2$. Now let $y \in (a,b)$. Then, $(a-b)/2<(a+b)/2-y<(b-a)/2$.
Let $r = min[0,|\frac{a+b}{2}-b... |
H: Estimating probability of experiencing 100th percentile: is this a valid approach?
Say you work on a website. When someone accesses the website in their browser, then the website makes 5 more calls to different services to load the remaining content. The issue is, until all content is loaded, it takes a long time -... |
H: Does the least value of $\cos A + \cos B + \cos C$ exist, where $A$, $B$, $C$ are angles of a triangle?
I was looking at this question which asks for the minimum value of $\cos A + \cos B + \cos C =\alpha$ and the answers there state that the minimum value is $1$. This value exists for a degenerate triangle.
But in... |
H: On the two definition of weakly convergence of measures, which one is true?
Let $X$ be a metric space and $\mu, \mu_1,\mu_2,\ldots$ Borel probability mesures on $X$. I met the following two definitions of the weakly convergence of $\{\mu_n\}$:
1. We call $\{\mu_n\}$ converges weakly to $\mu$ if $\int f d\mu_n\to \i... |
H: Prove that $g$ is also continuous at $x=0$.
Let $f:\mathbb{R} \to \mathbb{R}$ be such that $f$ is continuous at $x=0$. For any $r\in\mathbb{R}$, define $g(x) = f(rx)\;\forall x \in \mathbb{R}$. Prove that $g$ is also continuous at $x=0$.
Proof:
If $f(x)$ is continuous at $x=0$, then
$$\forall \epsilon\;\exists \d... |
H: Find the P.D.F for these Variables(Jacobian transformation)
$Q)$ Let the continuous variables $X$ and $Y$ which are following the uniform distribution on $D=\{(x,y) \vert 0 < x < y < 1 \}$
Find the joint probability density function for variables $Y$ and $Z$ (Here the $Z = {X \over Y}$)
I tried this by Jacobi tran... |
H: How to check $f(x) = x^{-\alpha}, x \in (0,1]$ is Lebesgue integrable?
Let $f(x) = x^{-\alpha}, ~ x \in (0,1], ~ \alpha \in \mathbb{R}$, then how can we show that $f$ is Lebesgue integrable?
I can show that $f$ is measurable but I don't know how to proceed further. I think I need to use monotone convergence theorem... |
H: K(subset of C[0,1]) does not attain the minimal norm
Let $V=C[0,1]$ and $K\subset V$ be defined by $K= \{f \in V | \int_{0}^{1/2} f(t) dt - \int_{1/2}^{1} f(t) dt = 1\}$.
Show that $K$ does not admit an element with minimal norm.
AI: Hints:
Show that if $\|f\| < 1$ then $f \notin K$. Hence the distance is at l... |
H: Why the two methods give different answers?
Question
If Kinetic energy of the body is increased by $300\text{%}$, its momentum will increase by:
Method 1:Using proportionality
$$Ke=\frac{P^2}{2m}$$where Ke is Kinetic energy and P is momentum
Since mass is constant,
$$\frac{Ke_i}{Ke_f}=\frac{P_i^2}{P_f^2}$$
$$\frac{... |
H: A non-empty subset of integers bounded above has a maximum
Suppose the set $\mathrm A$ $\neq$ $\emptyset$ , $\mathrm A$ $\subseteq$ $\Bbb Z$ is bounded above. Then since $\Bbb Z$ $\subseteq$ $\Bbb R$, I know that by the completeness axiom there exists a $supremum$ for the set $\mathrm A$, say $s$ $=$ $sup$($\mathrm... |
H: Definition of Crossed homomorphism
Suppose a group $G$ is acting on an abelian group $M$. Then a mapping $\phi: G \rightarrow M$ is called a crossed homomorphism if it satisfies the condition: $\phi(gh)=\phi(g)(g\cdot \phi(h))$ for every $g,h\in G$. My question is, how we will specify the action of $G$ is left or r... |
H: PDE with change of variable
In my text book there is an example of PDE with change of variable like this:
$$\frac{\partial^2f}{\partial t^2}-c^2\frac{\partial ^2f}{\partial x^2}=0$$
with $\left\{\begin{matrix}
u=x+ct & \\
v=x-ct &
\end{matrix}\right.$
I can follow my text book until here $\frac{\partial f}{\parti... |
H: A question in proof theorem 12.9 of Tom M Apostol ( Mathematical Analysis)
While self studying Mathematical Analysis from Tom M Apostol, I am unable to deduce a statement in following proof :
Question: I am unable to deduce how to prove existance of $\delta$ such that x+tu belongs to S for all real t in that in... |
H: How to find the length if a segment given this problem?
Let a line with the inclination angle of 60 degrees be drawn through the focus F of the parabola y^2 = 8(x+2). If the two intersection points of the line and the parabola are A and B, and the perpendicular bisector of the chord AB intersects the x-axis at the ... |
H: Evalute $\lim\sup a_n=\left[1-(-2)^n\right]$
I am trying to evaluate the limit $$\lim\sup\limits_{n\to \infty}\sqrt[n]{|a_n|}$$
$$a_n=\left[1-(-2)^n\right]$$.
What I did here is notice that this sequence can be written as the following:$$ a_{2n+1}=3$$ $$a_{2n}=1$$
From here it is really clear that the $\lim\sup a_n... |
H: One of a math problems
For modules, let $M = M_1 ⊕ M_2$ and let $f :M→N$ be an epimorphism with $K = \ker f$ and $N = f(M_1) + f (M_2)$.
(1) Prove that if $K= ( K \cap M_1)+ (K \cap M_2)$, then this sum is direct.
Could someone give me hints about this question. I am having troubles about how to solve this question... |
H: show that if $\lim_{r\to\infty} \min_{|z|=r}|f(z)|$ converges to a positive limit, and it's holomorphic then it's constant
Given $f$ holomorphic in $\mathbb{C}$ and $$\lim_{r \to \infty} \min_{|z|=r}|f(z)|>0$$
show that the function is constant.
It is probably somehow related to Liouville's theorem.
I am able to sh... |
H: Let $G$ be an abelian group with elements $x, y$ of orders $m$ and $n$ respectively. Moreover, $\gcd(m,n)=1$. Then the order of $xy$ is $mn$
I'm trying to prove this special case of a well-known result. I'm actually very happy because the entire proof is by myself. Could you please confirm if it is correct?
Let $G... |
H: Compact normed real - spaces.
Is there a compact normed real space which contains more than one element?
Im trying to grab some intuition on this subject. Would love to get some intuition if possible.
AI: No, there is not. Such a space would have a vector $v\ne0$, and then the sequence $(nv)_{n\in\Bbb N}$ would hav... |
H: How to find the curve whose tangent line is always perpendicular to that of all parabolas centered at x=0 at the point of intersection?
Basically, I’m trying to solve the following differential equation, but for some reason I’m getting the wrong result and I don’t fully understand why.
$$ \frac{\mathrm{d} y}{\mathr... |
H: Spectral decomposition of a self-adjoint operator.
Let $\mathcal{H}$ be a Hilbert space, $A$ be a self-adjoint operator of $B(\mathcal{H})$. If $0\le A\le I$, how to show that there are mutually commutative projections $\{P_n\}_{n=1}^{\infty}$ such that $A=\sum_{n=1}^{\infty} \frac{1}{2^n} P_n$?
I know there is a d... |
H: Series convergence test, $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$
I'm trying to find all $x$ for which $\sum_{n=1}^{\infty} \frac{(x-2)^n}{n3^n}$ converges. I know I need to check the ends ($-1$ and $5$) but I'm not sure what to happen after that. I'm pretty sure I'd substitute the values of $x$ into the sums and... |
H: If $P$ is false, and $Q$ is true, then why does $P \implies Q$?
I have often seen truth tables similar to this one:
The first two rows, where $P$ is true, make sense to me. However, why does $P$ being true and $Q$ being false mean that $P \implies Q$?
To make the question clearer to myself, I came up with the foll... |
H: Floor function parity problem
Prove that for every natural k this expression is always odd $⌊(5+\sqrt{19})^k⌋=A^k$
Progress that I' ve done is:
I noticed $9^k<A^k<(9.5)^k$
Also I tried an induction approach, I used Binomial theorem, I rewrote the expression as $⌊(9+(\sqrt{19}-4))^k⌋$. But so far none of this approa... |
H: I solved but don't know if it is correct, can you help me? Showing $P(X\cup Y)\approx P(X)\times P(Y)$
Question:
$X, Y$ are infinite sets that are not empty, and $X\cap Y=\emptyset$.
Show $P(X\cup Y)\approx P(X)\times P(Y)$
Hi! I tried to solve the question I wrote above, but I don't know if it is correct. Can you... |
H: Finding The Tangent Line Of $y = \sin x$
Hello everyone I have a tangent line with slope $= a$ to $y = \sin x$ in 2 points $(u_1,v_1) , (u_2 ,v_2)$
How can I prove that $\tan a = a$?
My direction was to express $a$ with the points by $a = \frac{u_1-u_2}{v_1-v_2}$
and the tangent line equation is $y = ax -au_1 +v_... |
H: An ultraweakly continuous functional on the unit ball is ultraweakly continuous everywhere.
Given a von Neumann Algebra $\mathcal{M} \subset \mathcal{B}(\mathcal{H})$, one defines a normal linear functional on $\mathcal{M}$ to be an ultraweakly continuous linear functional. Usually it's proven that this property is... |
H: Why $\sum_{n=1}^{\infty}\frac{x^{\alpha}}{1+n^2 x^2}$ doesn't converge uniformly on $[0, \infty)$ for $\alpha > 2$?
I'm trying to understand why $\sum_{n=1}^\infty\frac{x^\alpha}{1+n^2 x^2}$ doesn't converge uniformly on $[0, \infty)$ for $\alpha > 2$.
My book says that $\frac{x^\alpha}{1+n^2 x^2}$ is monotonic and... |
H: Find the image of the circle under the Transformation?
Find the image of the circle $$x^2 + y^2 = 4$$ under the transformation $$T ((x,y)) = (x,y + 2x)$$
AI: Let $$T ((x,y))=(X,Y)$$$$\therefore X=x$$$$Y=y+2x$$$$\Rightarrow Y=y+2X$$ $$\Rightarrow y=Y-2X$$Plugging in the original equation $$X^2+(Y-2X)^2=4$$$$\Righta... |
H: Let $G$ be an abelian group with elements $x, y$ of orders $m$ and $n$ respectively. There exists $z \in G$ of order $\operatorname{lcm} (m,n)$
I'm trying to generalize my previous lemma to this well-known result. Could you please confirm if it is fine or contains logical mistakes?
Let $G$ be an abelian group with... |
H: Power series problems
How can I find the power series expansion of following functions. I don't have any idea, these seem intimidating. Please help how to proceed.
$(x+\sqrt{1+x^2})^a$
$\sqrt{\frac{1-\sqrt{1-x}}{x}}$
AI: The derivative of the first expression, call it $u(x)$, is
$$
u'(x)=\frac{a(x+\sqrt{1+x^2})^a... |
H: Volume enclosed by a paraboloid, cylinder and a plane by Spherical coordinates
I am stuck over this triple integral where i have to find following:
$\int\int\int_V \space z^2 \space dxdydz \space$ over a volume bounded by cylinder $x^2 + y^2 = a^2$, a paraboloid $x^2 + y^2 = z$ and a plane $z=0$.
I did it from cyl... |
H: Last 2 digits of $143^{101}$ in base 10.
Last 2 digits of $143^{101}$ in base 10.
I have to use Fermat's theorem or Euler's theorem but I dont know where can I started.
AI: As we need to find last two digits, we should divide by 100. Here is the full solution.
$$Find \space a^{x} \space mod \space n$$
$$a=143,x=10... |
H: Is there an improper subset that isn't equal to its superset?
Can there be a set $A$ and a set $B$ such that $A\subseteq B$ and $A\ne B$ ?
While trying to find a solution to this question, I've found this answer which states:
An improper subset (usually denoted as $A\subseteq B$) is such that $A=B$ is allowed (but... |
H: How do I show that these limits are equal?
I found this exercise in the Michael Spivak's calculus book. The author asked to "interpret precisely" and then prove that these are equal:
$$\lim_{x\to a}f(x)\space\text{and}\space\lim_{h\to0}f(a+h)$$
I interpret the first one as "the limit of f at a", but I don't really ... |
H: Question on linear independence of quotient space
Let $V$($F$) be a vector space of dimension $n$. Let $W$ is a subspace of $V$.
Consider $S$ = {$x_1, x_2,..., x_k$} be a subset of $V$, consider
$S'$ = {$W+x_1, W+x_2..., W+x_k$} be a subset of $\frac{V}{W}$.
Then I need to show the following:
Given $S$ is linearly ... |
H: How can I apply this easy derivative on this integral?
Im trying to understand a step in a certain proof, this is actually only easy calculus, but somehow I can't wrap my head around it. The equation is given as
$$ \frac{\partial}{\partial t} \int_0^t f(x+(s-t)b,s) \, \text{d}s = f(x,t) + \int_0^t -b \cdot \nabla f... |
H: Limit of $\frac{1}{t} \int_0^t f(x) dx$
A continuous function $f:[0,\infty) \rightarrow \mathbb{R}$ has a limit $\alpha = \lim_{x \to \infty} f(x) \in \mathbb{R}$.
(1) Let $ g(t) = \frac{1}{t} \int_0^t f(x) dx$. Show $\lim_{t \to \infty} g(t) = \alpha$.
(2) Let $ h(t) = \frac{1}{t^2} \int_0^t xf(x) dx$. Show the... |
H: Find the geometric position of all points satisfying the two equations $x^2+y^2+z^2=4$ and $x^2+y^2=1$
Find the geometric position of all points satisfying the two equations $x^2+y^2+z^2=4$ and $x^2+y^2=1$
I think the points for which their coordinates satisfies the two equations at the same time,are all on a circl... |
H: Example of a symmetric matrix which doesn't have orthogonal eigenvectors
I'm looking for an example of a symmetric matrix $A$ which doesn't have orthogonal eigenvectors.
Here's what I tried: I was able to prove that the eigenvectors corresponding to each distinct eigenvalue of a symmetric matrix are orthogonal. So,... |
H: proving the equation has at least one at least one real solution
If $g(x)=3^x+2x^{\frac{1}{2}}$. Then prove that the equation $g'(x)=21$ has at least one real roots
What i try:: given $g(x)=3^{x}+2x^{\frac{1}{2}}$
Then $\displaystyle g'(x)=3^x\ln(3)+2\cdot \frac{1}{2}x^{-\frac{1}{2}}=3^x\ln(3)+\frac{1}{\sqrt{x}}$... |
H: System of equations in $\mathbb{F}_p$
I have:$$\begin{matrix}X&+&Y&+&Z&=&3,\\& &Y&+&Z&=&2,\\&-&Y&+&Z&=&0.\end{matrix}$$
Let furthermore $p$ be prime. How can I solve this system with respect to $\mathbb{F}_p$?
AI: Continuing from the comments. If we are not in a field of characteristic $2$, we can solve the system ... |
H: Proving that this formula all over the positive integer gives us this sequence
Firstly, we have this sequence : $1,1,1,1,1,1,1,1,1,1,2,2,2,...$ which is the sequence of the number of digits in decimal expansion of $n$.
Secondly, we have this formula : $$a_n=\Bigl\lceil\log_{10}(n+1)\Bigr\rceil-\Bigl\lceil\frac{n}{n... |
H: How to carry out arithmetics with Bachmann–Landau notations?
Here's something from my class.
As $n \to \infty$, $|\mathcal{F'}| =
(1-o(1))\frac{n \choose t}{k \choose t}
+ o(1){n \choose t}
= (1 + o(1))\frac{n \choose t}{k \choose t}$
I can sort of see why this makes sense, and I guess writing out the expressio... |
H: Is is true that $|A+B|\ge \min(|A|+|B|-1,p)$ in $\mathbf{Z}_p$?
For two sets $A,B\subseteq \mathbf {Z}_p $, define
$$A+B:=\{a+b:a\in A,b\in B\}.$$
Note that "+" is performed under $\mathbf {Z}_p$.
For example, let $p=5$, $A=\{1,4\},B=\{2,4\}$. Then $A+B=\{0,1,3\}$.
Is it true that $|A+B|\ge \min(|A|+|B|-1,p)$ in $\... |
H: Calculate three dimensional integral with a delta function
Can anyone help to calculate the following integral: $$\int_a^\infty \mathrm dy_1\int_a^\infty\mathrm dy_2 \int_a^\infty \mathrm dy_3 \delta\left(y_1+y_2+y_3-3\right)$$ with a<0?
According to maple the result is: $\frac{9}{2}\left(1-a\right)^2$ but I don't ... |
H: A subsequence of $\{|\sin n|\}$ that converges to $0$ "fast"
It is well-known that for each $a\in [0,1]$ there is a subsequence of $\{|\sin n|\}_{n=0}^\infty$ that converges to $a$. I am curious about the speed of convergence. In particular, for each $\alpha>0$, is there a subsequence $\{|\sin n_k|\}_{k=0}^\infty$ ... |
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