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H: Proof of $\mathbf{x}^{T}A\mathbf{x}=0\Rightarrow A=0$
As far as I know, for any $A$:
$$\mathbf{x}^{T}A\mathbf{y}=0;\forall\mathbf{x},\mathbf{y}\in R^n\Rightarrow A=0$$
Does it mean that
$$\mathbf{x}^{T}A\mathbf{x}=0;\forall\mathbf{x}\in R^n\Rightarrow A=0$$
The condition of the first claim $\forall\mathbf{x},\mathb... |
H: Point addition on an elliptic curve
I have an elliptic curve $y^2 = x^3 + 2x + 2$ over $Z_{17}$. It has order $19$.
I've been given the equation $6\cdot(5, 1) + 6\cdot(0,6)$ and the answer as $(7, 11)$ and I'm unsure how to derive that answer.
I have $6\cdot(5, 1) = (16,13)$ and $6\cdot(0,6)=(0, 11)$ however when... |
H: $M$ be the subspace of $\Bbb R^3$ spanned by $(1,0,-1).$
I am stuck on the following problem:
Consider $\Bbb R^3$ with the standard inner product. Let $M$ be the subspace of $\Bbb R^3$ spanned by $(1,0,-1).$ Which of the following is a basis for the orthogonal complement of $M\,\,?$
$\{(2,1,2),(4,2,4)\}$
$\{(2,-... |
H: limit to infinity involving trig and root function
I was doing a ratio test for convergence and the final expression I got before applying limit to infinity was:
$\dfrac{(2+\cos(x) )}{\sqrt{x}}$, now I believe that this goes to zero, the $\dfrac{2}{\sqrt{x}}$ is trivially zero, but the $\dfrac{\cos(x)}{\sqrt{x}}$ I... |
H: SOA Exam P Question: Exponential Distribution
Here is an Exam P problem as I have it. That is, it was passed down to me from someone else and I am unsure if the wording is exactly as it was originally posted. I've tried searching for this problem on various sites but cannot find a similar one.
Problem: A company ... |
H: Understanding the support of a function
This is from exercise 5.5.A of Vakil's lecture notes.
Consider $f$, a function on $A: = \mathrm{Spec}(k[x,y]/(y^2, xy))$. Show that its support either empty, the origin or the whole space.
Now, I know that the support of any function $f$ must be closed. This comes from the fa... |
H: Approximation of DE
It depends on my previous question. Closed form solution of DE
I don't want to deal with Airy functions. How can I approximate this DE in continous domain $[0,1]$?
$$y''(x)+(x+1)y(x)=0\quad\text{ with the initial conditions}\quad y(0)=0\quad y'(0)=1$$
What if the conditions change to
$$y''(x)+(x... |
H: Complex integral on curve
I have to show that this integral is zero, but don't know how to evaluate it.
Consider a closed class $C^1$ curve $c:[a,b]\rightarrow\mathbb{C}\backslash \{0\}$ and show that $$\int_a^b\frac{\langle c(t),c'(t) \rangle }{\lVert c(t)\lVert^2}dt=0$$
It is necessary to consider $c$ as a curve ... |
H: Multiplication of nonsquare matrices
Could multiplication of non-square matrices result in square nonsingular matrix?
It's easy to show for square matrices via determinant. But what to do with non-square ones?
AI: Yes, indeed, this can happen: $A_{m\times n} \times B_{n\times m}$ may very well be non-singular (thou... |
H: is Lebesgue measure continuous?
Is Lebesgue measure continuous? Can someone prove it or attach a link to the proof?
I am trying to prove the existence of a plane in $\mathbb{R}^{3}$ that simultaneously divides 3 compact subsets of $\mathbb{R}^{3}$ into two peices of the same measure.
AI: One way to mathematically f... |
H: How many intersection points can two graphs have?
Let $F$ and $G$ be copies of the complete equipartite graph with each partition of size $v$. That is, $F,G:= K_{v,v,\dots ,v}$. Prove that if $F$ and $G$ intersect in at more than $v$ vertices, then they MUST share a common edge.
I am having some trouble proving thi... |
H: Unrotate/UnTranslate a Unit Vector
I have a Unit Vector (UV1) which I am transforming using a Rotation + Translation Matrix (MT1). The result of that Rotation and Translation is again normalized to create a Unit Vector (UV2).
Given UV1 and MT1 I can calculate UV2. Given only UV2 and MT1 is it possible to calculate ... |
H: How to find global extremum when constraint isn't compact set
Sometimes, the constraint is not a compact set. As a result, the local minimum may not be global.
For example, $ f=x^2+y^3$ subject to constraint $ x+y=4/3$.
Using Lagrange multiplier method, I calculated local minimum at $(x,y)=(\frac23,\frac23)$. But ... |
H: Hausdorff Distance between "Pure Black" and "Pure White" images
I am trying to use Hausdorff Distance to compare a pair of test images of equal dimensions. The images undergo some kind of threshold to obtain binary images. The Hausdorff Distance is calculated for the positions with non-zero pixels in those binary i... |
H: Identity as lower bound of sine
I'm struggling to rigorously proof
$$ \sin(2x) \geq x \qquad (0 \leq x \leq \pi/4) $$
Any ideas?
AI: Hint: Consider the function $f$ given by $f(x)=\sin (2x)-x$ for all $x$ in $[0,\pi/4]$. Differentiate it. |
H: What is $\frac{0}{0}$ and $\frac{\infty}{\infty}$? A question on indeterminate forms
I am wondering what is $\frac{0}{0}$ and $\frac{\infty}{\infty}$?
In my impression, both are undefined. But then I need to prove that
$$\lim_{n \rightarrow \infty} \frac{\int_{-n}^x g(t)dt}{\int_{-n}^n g(t)dt} = 1 \text{ when } x... |
H: Simple Double Summation
I understand how to sum a single sum, but I don't know how to solve a double sum without explicit limits. Please help guide me in the right direction to solve problems 3 through 5 in the included image. Thank you!!
$a_n = \sum\limits_{i=1}^n (2i-1)$
$a_n = \sum\limits_{i=1}^n (3i^2-3i+1)... |
H: differentiable prove of product functions
Let $E, F$ normed spaces and $f:A\rightarrow\mathbb{R}$, $g:A\rightarrow F$, with $A$ open set in $E$, and defined $h:A\rightarrow F$ by $h(x)=f(x)g(x)$. Suppose that $f$ es differentiable in $a\in A$, $f(a)=0$, and that $g$ is continuos at $a$. Prove that $h$ es differen... |
H: Question on polar decomposition of operators.
Suppose $\tau$ is an operator on a finite dimensional complex inner product space. I'm read the following,
If $\rho$ is the unique positive square root of the positive operatore $\tau^*\tau$, then
$$
\|\rho v\|^2=\langle \rho v,\rho v\rangle=\langle \rho^2 v,v\rangle=\l... |
H: Open set as a countable union of open bounded intervals
Can every nonempty open set be written as a countable union of bounded open intervals of the form $(a_k,b_k)$, where $a_k$ and $b_k$ are real numbers (not $\pm\infty$)?
If yes, can someone point me toward a proof?
If not, counterexample?
Note that this is not ... |
H: Proving that $x\in E^{o} \iff B_{r}(x)∩ E^{c}\not= \varnothing$
I know it is so easy proof. But I am confused.
Remark:
$x\in E^{o} \iff B_{r}(x)∩ E^{c}\not= \varnothing$
Proof
(İf) suppose $x\in E^c$ and $B_{r}(x)∩ E^{c}=\varnothing$
Then we have $B_{r}(x)⊆ E$ then, $x \not\in E^o$
But i cannot do and not if part.... |
H: Representing $m\times n$ matrix using ordered $n$-tuples and an $m$-tuple
Can a matrix, generally
\begin{bmatrix}
a_{1,1} &\cdots &a_{1,n} \\
\vdots &\ddots & \vdots \\
a_{m,1} &\cdots &a_{m,n}
\end{bmatrix}
be represented using ordered $n$-tuples inside an $m$-tuple, like this:
$((a_{1,1},...,a_{1,n}),.... |
H: Solution to $u_{n+1}=u_n/n+u_{n-1}/(n-1)$
What is the solution to the following recurrence relation $$u_{n+1}=\frac{u_n}{n}+\frac{u_{n-1}}{n-1}\ \forall n\geq 2$$ where $u_2=u_1=1$?
AI: Let $a_n=(n-1)!u_n$, and multiply the recurrence by $n!$:
$$\begin{align*}
a_{n+1}&=n!u_{n+1}\\
&=\frac{n!}nu_n+\frac{n!}{n-1}u_{n... |
H: Base and Independence proof
Adapted from Axler,
Could someone explain the last part for me? How does the unique representation imply all the constants are suddenly $0$? We are trying to show it is linearly independent, he doesn't know this yet, so why is he doing that?
Please excuse the length of this question.
... |
H: Non Identical Closure
I'm working on counterexample here. Can we construct two bounded non empty open sets $A,B$ with $A \subset B$ that are $\lambda(A)=\lambda(B)$ but $\overline{A}\ne\overline{B}$? Here $\lambda$ is the Lebesgue measure, thank you.
AI: No. If $A\subseteq B$, but $\operatorname{cl}A\ne\operatornam... |
H: How to set up a double integral with $x,y$ and $z$?
Use a double integral to find the volume of the solid bounded by graphs of the equations given by: $z=xy^3, Z>0,\; X>0,\; 5X<Y<5$
How would you set up this integral? please help me.
AI: Here is how
$$ \int_{0}^{1}\int_{5x}^{5} xy^3 dydx.$$
You should plot the gr... |
H: "uniquely written" definition
I'm having troubles with this definition:
My problem is with the uniquely part, for example the zero element:
$0=0+0$,
but $0=0+0+0$
or $0=0+0+0+0+0+0$.
Another example, if $m \in \sum_{i=1}^{10} G_i$ and $m=g_1+g_2$, with $g_1\in G_1$ and $g_2\in G_2$,
we have: $m=g_1+g_2$ or $m=g... |
H: Abelian subgroups of $GL_n(\mathbb{F}_p)$
Let $p$ be a prime number, and let $k=\mathbb{F}_p$ be the field of $p$ elements. Let $G=GL_n(k)$. We know that
$$|G|=\prod_{i=0}^{n-1}(p^n-p^i)=p^{\binom{n}{2}}\prod_{i=0}^{n-1}(p^{n-i}-1)$$
so that the Sylow $p$-subgroups of $G$ have order $p^{\binom{n}{2}}$. One such ... |
H: How to set up a triple intergal with $x, y,$ and $z$
Use a triple integral to find the volume of the solid bounded by $z=16xy$, $z\ge 0$, $0 \le x \le 5$, $0 \le y \le 4$. I know how to set up the integral for $x$ and $y$ it would be $0$ to $5$ for $x$ and $0$ to $4$ for $y$. How would you set up the integral for $... |
H: existence of the directional derivative of a function
Let
$$f:\mathbb R^2\rightarrow\mathbb R,(x,y)↦\begin{cases}1&(\exists z\in\mathbb R\setminus\{0\}:(x,y)=(z,z ^2)\\0 &(\textrm{else})\end{cases}$$
$f$ is obviously not differentiable in $(0,0)$.
But what about any directional derivative $v$ in $(0,0)$? So I have ... |
H: Injective map between same dimension implies bijectivity?
For an injective map between two spaces with the same dimension, does the map need to be linear in order to be bijective?
In other words, if this statement universally true:
For any function, injectivity between same dimension implies bijectivity.
or it is... |
H: Example of non G-delta set
An open set is clearly a $G_{\delta}$ set. A closed interval $[a,b]$ is a $G_{\delta}$ set as an intersection of the open intervals $(a-\frac1n,b+\frac1n)$ for all positive integers $n$. What is an example of a set that's not $G_{\delta}$?
AI: The set $\Bbb Q$ of rational numbers is not a... |
H: Proving the normed linear space, $V, ||a-b||$ is a metric space (Symmetry)
The following theorem is given in Metric Spaces by O'Searcoid
Theorem: Suppose $V$ is a normed linear space. Then the function $d$ defined on $V \times V$ by $(a,b) \to ||a-b||$ is a metric on $V$
Three conditions of a metric are fairly stra... |
H: Set containing another set but having same measure
A set of real numbers is said to be a $G_{\delta}$ set provided it is the intersection of a countable collection of open sets. Show that for any bounded set $E$, there is a $G_{\delta}$ set $G$ for which $$E\subseteq G\text{ and }m^*(G)=m^*(E)$$
I want to define ... |
H: First-grader problem in arithmetic
I found this problem in a text book on arithmetic for first graders (7 y.o.) of the former USSR* . The problem comes from the section that covers single-digit addition and subtraction. Here is the screenshot of the problem:
This is the entire problem: there is no textual descript... |
H: Trigonometric Anti-derivative
What is $$\int \frac{\sin(x)^2}{\cos(x) + 1}dx\;?$$ I've tried everything I can think of, but I can't get it into a form that I can solve.
AI: Note that $$\frac{\sin^2x}{1+\cos x}=\frac{1+\cos x}{1+\cos x}(1-\cos x)$$ since $1-\cos^2x=\sin ^2x$ and $1-y^2=(1-y)(1+y)$ |
H: Why $\int_{0}^{\pi}\arctan{\cos{x}}dx = 0$?
I saw this in Ron Gordon's answer to this question:
I need assistance in integrating $ \frac{x \sin x}{1+(\cos x)^2}$
Thank you!
AI: Odd around $\pi/2.$ That is, given $f(x) = \arctan \cos x,$ we have $f(\pi - x) = -f(x).$ Draw a graph. |
H: How many revolutions per minute does a wheel make if its angular velocity is 20π radians per second?
Note: This is a homework question, however, I am not asking for anyone to do it for me. I just need some direction in how I should go about solving it.
The question reads:
How many revolutions per minute does a wh... |
H: If $x^2+ax+b=0$ has a rational root, show that it is in fact an integer
I have tried as follows. Please help to double check the proof! Thank you!
Since $x=p/q$ ($p$, $q$ are integers), $(p/q)^2+(p/q)a+b=0$
So, $(p/q)^2=-b-a(p/q)$
then, $p^2=-bq^2-a(p/q)q^2$
and, $p = \dfrac {q(-bq-a(p/q)q)}{p}$
now, it is clear th... |
H: Is $\Bbb{Q}\cap [0,1]$ not connected?
Motivation: My textbook states that a connected subset of a normal space is mapped to all of $[0,1]$ if it has non-empty intersections with disjoint closed sets, one of which is mapped to $\{0\}$ and the other mapped to $\{1\}$. This is because the image of a connected set has ... |
H: deductions in a propositional calculus
Hope you're all doing well. I have a question about deductions in logical systems. Say we have a logic in the language of propositional logic. We can think of this as the set of tautologies of propositional logic, along with the inference rule modus ponents.
(See logic for a ... |
H: non-axiomatizable logics
Hope you're all doing well. My question is about non-axiomatizable logics. My understanding is that a "logic" (the mathematical structure) is just another word for a "propositional calculus" as in http://en.wikipedia.org/wiki/Propositional_calculus. (see the formal defn. under "general defi... |
H: Is there a better alternative to the phrase, 'it holds that'?
The following phrases abound in my writing:
There exists [whatever] such that [whatever].
For all [whatever] it holds that [whatever].
Lately, I've been feeling that the phrase 'it holds that' is overly long-winded. The only substitute I can think of i... |
H: Question about differentiability and sequences
Given $f\colon X\rightarrow \mathbb R$ differentiable in $a\in X \cap X'_+ \cap X'_-$.
If $x_n<a<y_n \;\forall n$, and $\lim x_n=\lim y_n=a$, prove that
$$\lim_{n\rightarrow +\infty}\frac{f(y_n)-f(x_n)}{y_n-x_n}=f'(a).$$
Does anyone know how to solve this problem?
O... |
H: How to show that for any real numbers $x,y,z$ $|x|+|y|+|z|\le|x+y-z|+|y+z-x|+|z+x-y|?$
How to show that for any real numbers $x,y,z$$$|x|+|y|+|z|\le|x+y-z|+|y+z-x|+|z+x-y|?$$
I'm don't know how to split RHS.
AI: Write $x=a+b$, $y=b+c$, $z=c+a$. Then the following equivalent inequality is clear.
$$|a+b|+|b+c|+|c+a|\... |
H: Bounded sets with finite minimum distance and sum of measures
Let $A$ and $B$ be bounded sets for which there is an $\alpha>0$ such that $|a-b|\ge\alpha$ for all $a\in A,b\in B$. Prove that $m^*(A\cup B)=m^*(A)+m^*(B)$.
We automatically have $m^*(A\cup B)\leq m^*(A)+m^*(B)$ for any sets $a,b$. Now we must prove t... |
H: If $y\in\mathbb R^+$ then $\exists~m\in\mathbb N$ such that $0<\dfrac{1}{2^m}
Using Archimedean Property how to show the following:
If $y\in\mathbb R^+$ then $\exists~m\in\mathbb N$ such that $0<\dfrac{1}{2^m}<y.$
AI: Hint: Start by showing that:
$$\forall n\in \mathbb{N}[n<2^n]$$ |
H: Inclusion and Exclusion how many n-strings with a,b,c and d
There are n different characters namely a, b, c, d. Use them to compose n-length string. Count the number of different n-length strings which at least contain one a, one b and one c.
My previous idea is like:
Set A contains strings with at least one a. Sim... |
H: relation between arithmetic series and `square` arithmetic series
For example:
$$1+2+\text{...}+n=\frac{n(n+1)}{2}~~~(1)$$
$$1^2+2^2+\text{...}+n^2=\frac{n(n+1)(2n+1)}{6}~~~(2)$$
In this equality, I sometimes recall by heart
$\frac{n(2n+1)(2n+3)}{6}$ or others.
Why I cannot memorize some formulas exactly over these... |
H: Is this formula logically valid
Is: $\exists x (P(x) \land Q(x)) \rightarrow \exists x P(x) \land \exists x Q(x) $ logically valid?.
I cant found an intepretation in wich the formula is false.
AI: Yes, indeed it is valid. No counterexample to be found.
If there exists an $x$ for which both ($P(x)$ and $Q(x)$) hold... |
H: Why does $\lim_{\lambda \to \infty} \frac{\cos(\lambda x) - \cos(\lambda y)}{\lambda} = 0$
Why does $$\lim_{\lambda \to \infty} \frac{\cos(\lambda x) - \cos(\lambda y)}{\lambda} = 0$$
Or I should really rephrase my question. Is this limit obvious?
Motivation comes from $$\lim_{\lambda \to \infty} \int_{a}^{b} \sin(... |
H: The same destination regardless of origin
When I was very little, I didn’t understand some basics of the space we live in. We always followed the same directions to get into town, to school, to the grocery store, and so on. So I figured that by following those directions, we would always arrive at the same destinat... |
H: Boundaries of finite intersections and unions of sets
I apologize if this is a duplicate - I looked but didn't find one.
This question is sort of a sanity check.
Let $A$, $B$ be sets and define the boundaries $\partial A$ and $\partial B$ as usual.
Is it true that both $\partial (A \cup B) \subseteq \partial A \cu... |
H: Math Parlor Trick
A magician asks a person in the audience to think of a number $\overline {abc}$. He then asks them to sum up $\overline{acb}, \overline{bac}, \overline{bca}, \overline{cab}, \overline{cba}$ and reveal the result. Suppose it is $3194$. What was the original number?
The obvious approach was modula... |
H: Subspace of a finite dimensional space is finite dimensional
I was reading a proof about subspaces of finite dimensional spaces are finite dimensional. The key was to add and remove vectors and then make a counting conclusion.
An excerpt (adapted from Axler) is given below
This is going to sound very stupid, but... |
H: Yet Another Monty Hall Question - Please advise if alternative scenario proves the same principle
Okay, I'm very embarrassed that there are already 71 questions (based on search of "monty hall") and I'm going to post another one. I read the first 5 before succumbing to choice-overload. I'll try to keep this short a... |
H: $T=U\Sigma V^T$ is the SVD of T. Given $\Sigma$ find T, U and V.
$T=U\Sigma V^T$ is the SVD of T.
$$\Sigma=\pmatrix{11.83&0\\0&0\\0&0}$$
The last two columns of $U$ are $[0.949,0,0.316]^T$ and $[-0.894,0.447,0]^T$
The first column of $V$ is $[0.316,0.949]^T$
a) what are the remaining values in $U$ and $V$?
b) iden... |
H: Problem in Cardinality and Order
Recently, I have read a Schaum's book. General Topology, and it introduced the concept of cardinality and order, but it points out something that I really don't understand.
If $A\preceq B$ and $B\preceq A$, then $A \sim B$.
If $X\supseteq Y\supseteq X_1$ and $X\sim X_1$, then $X\si... |
H: Convergence w.p. 1 vs convergence in probability: a "physical" example
I understand (proved) that convergence with probability one implies convergence in probability, and that the latter notion is indeed weaker; I've completed an exercise showing that a sequence of indicator variables on $[0,1]$ converged in probab... |
H: Approximation of differential equations
Can someone provide me a good reference about approximation techniques in continous domain (not piecewise nor numerical methods) for differential equations?
AI: There really would be different types of methods for this, particularly dependent on how and what your differential... |
H: Why does $(\cos \theta + i \sin \theta)^n =(\cos n\theta + i \sin n \theta)$
Is it the Euler identity $$ e^{i \theta} =(\cos \theta + i \sin \theta)$$
$$ e^{i n \theta} =(\cos n \theta + i \sin n \theta)$$
AI: Hints (sketches)
First Proof: Trigonometric identities + induction: $\;\;\;\;$ For $\,n=2\;$ :
$$(\cos +... |
H: Intersection of a countable collection of $F_\sigma$ sets
Let $\{f_n\}$ be a sequence of continuous functions defined on $\mathbb{R}$. Show that the set of points $x$ at which the sequence $\{f_n(x)\}$ converges to a real number is the intersection of a countable collection of $F_\sigma$ sets.
Continuity of $f_n$... |
H: Shortest way to travel to each point in a set of points exactly once, and return to starting point?
Is there a polynomial time algorithm that finds this?
Just interested.
Thanks in advance
edit: In this case you are given a set of cartesian co-ordinates that represents their physical distance from one another.
AI: ... |
H: Show that if the sequence$(x_n)$ is bounded, then $(x_n)$ converges iff $\lim\sup(x_n)=\lim\inf(x_n)$.
Show that if the sequence$(x_n)$ is bounded, then $(x_n)$ converges iff $\lim\sup_{x\to\infty}(x_n)=\lim\inf_{x\to\infty}(x_n)$.
The definitions that I’m using:
$$\begin{align*}
&\liminf_{n\to\infty}x_n=\lim_{n\... |
H: Are there any real life applications of the greatest common divisor of two or more integers?
I am looking for real life applications of gcd. I have found one with tiles but there must be many more of these type.
AI: Suppose that you have two sets of people of cardinalities m,n and you want to divide them into teams... |
H: $\ker A \cap\mathrm{Im} A = {0} \Rightarrow \ker A + \mathrm{Im} A = R^n$.
Suppose, we have a linear operator A on $R^n$ such that $n>1$. I'm trying to prove that if $\ker A \cap \mathrm{Im}(A) = {0}$, then $\ker A + \mathrm{Im}(A) = R^n$.
Generally, $$\dim (\ker A + \mathrm{Im}(A)) = \dim \ker A + \dim \mathrm{Im}... |
H: Continuous Functions - Topology
I'd like to prove the following.
A function $f:X \to Y$ is continuous if whenever $A$ is closed in $Y$,
$f^{-1}(A)$ is closed in $X$.
Proof. By definition, a function is continuous if the inverse image of every open set is open. Suppose that $A\in Y$ is closed. Then, $Y-A$ is ope... |
H: Evaluating $\int^1_0\sqrt{1 - x^2}$
How to Evaluate $\int^1_0\sqrt{1 - x^2}$
I know that this is just a $\frac14$ of unit circle, that is $\frac\pi4$, but I want to solve it algebraically.
AI: Hint: Use the trig substitution $x=\sin\theta$ so that $dx=\cos\theta d\theta$. This leads to:
$$
\int_0^1 \sqrt{1-x^2}dx ... |
H: Factor $x^6 +5x^3 +8$
I wanted to know, how can I factor $x^6 +5x^3 +8$, I have no idea. Is there any method to know if a polynomial is factored. Just some advice will do.
Help appreciated.
Thanks.
AI: Let $y = x^3$ to obtain $y^2 + 5y + 8 = 0$. This factors as $y = \frac{-5 \pm \sqrt{-7}}{2} = \frac{-5}{2} \pm \fr... |
H: Two questions on topology and continous functions
I have two questions:
1.) I have been thinking a while about the fact, that in general the union of closed sets will not be closed, but I could not find a counterexample, does anybody of you have one available?
2.) The other one is, that I thought that one could pos... |
H: Analytical Solution for Elastic Bar under applied end velocity
Say, a thin long rod is occupying the space $[0,L]$. It's isotropic, linear elastic, homogeneous. The partial differential equations for stress $\sigma(x,t)$ and displacement $u(x,t)$ are as follows ($E$ denotes the Young's Modulus and $\rho$ the densit... |
H: Express $\cos 6\theta $ in terms of $\cos \theta$
I think I'm supposed to use the chebyshev polynomials, as in $$ \cos n \theta = T_n(x) = \cos(n \arccos x)$$
But no idea what now?
AI: Since $$\cos 2a =2\cos ^{2}a -1, \qquad\sin 2a
=2\sin a\cos a,$$ $$\cos (a+b)=\cos a\cos b-\sin a\sin b,$$
and
$$
\begin{eqnarray*... |
H: Boundedness and finite limit of function
Suppose $f(x)$ is continuous on $[1,+\infty)$, differentiable on $(1,+\infty)$. If $f(x)$ is bounded on $[1,+\infty)$ and has finite $\lim_{x\rightarrow \infty} f'(x)$, then it has finite $\lim_{x\rightarrow \infty} f(x)$
I know it's false, but don't see why: if the function... |
H: Question about projection
If $B^T AB$ is not a projection, then either $B$ isn't orthogonal, or $A$ isn't a projection.
I understand that orthogonal $B$ and projection $A$ help transform the following: $B^T ABB^T AB = B^T AAB = B^T AB$. But I need to prove negation. How to deal with it?
AI: Hint: Prove the contrapo... |
H: A different way to write $(B\cap C)\cup(D \cap E)$
Let $A=(B\cap C) \cup (D\cap E)$ be a given set. I am looking for a different way to write this. I guess it is somehow possible to "reorder" this by using De Morgan's relations. Unfortunately, I am not successful. Does somebody see how one can write this differentl... |
H: Calculating the probability of an intersection
I have the following Venn Diagram,
Let:
$\color{red}A$ be the red circle
$\color{blue}B$ be the blue circle
$\color{green}C$ be the green circle
I know that $\dfrac{1}{6} = P(\text{(all three)}=x|\text{at least two}) = \dfrac{P(\text{at least two} \cap \text{all thr... |
H: Inequality between two sequences preserved in the limit?
Let $(a_n)_{n\in \mathbb{N}}$ and $(b_n)_{n\in \mathbb{N}}$ be two real sequences that satisfy $a_n\geq b_n, \forall n \in \mathbb{N}$ and converge to some $a,b$, respectively.
Is it always true that $a \geq b$?
AI: Yes. If you assume that $a < b$, you can ta... |
H: Subsets of $\mathbb{R}^2$ of which no $2$ are homeomorphic.
I'm reviewing general topology and I'm having trouble with this problem:
Let $Z_0 := \{ \frac{1}{i} \mid i = 1,2, \ldots\}$, $Z_1 := Z_0 \cup \{0\}$, $I_0 := (0,1)$, $I_1 := [0,1]$. Prove that no two of the following subspaces of $\mathbb{R}^2$: $Z_0 \tim... |
H: "such that" logical symbol
So, in the definition of what is a square root,
$\sqrt{x}$ are all numbers $y$ such that $y×y=x$.
are there any logical mathematical symbols so that the above definition can be written using logical operators only, and no natural language?
Where can I get some introductory or reference ... |
H: If $A = \tan6^{\circ} \tan42^{\circ},~~B = \cot 66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$
My trigonometric problem is:
If $A = \tan6^{\circ} \tan42^{\circ}$ B = cot$66^{\circ} \cot78^{\circ}$ find the relation between $A$ and $B$.
Working :
$$B = \cot 66^{\circ} \cot78^{\circ} = 1- \frac{... |
H: What is the splitting field of $x^3 - \pi$?
What is the splitting field of $x^3 - \pi$? Is it $\mathbb R(\sqrt[3] \pi, \xi_3)$
or $\mathbb Q(\sqrt[3] \pi, \xi_3)$? (where $\xi_3$ denotes the third root of unity)
It is a polynomial over $\mathbb R[x]$, so I guess it must be $\mathbb R(\sqrt[3] \pi, \xi_3)$, but I ne... |
H: Intersection of non-independent events
Let $A_1,\dots, A_n$ be not necessarily independent events. What can be said about the relation between $\mathbb{P}(\cap A_i)$ and $\prod_i^n \mathbb{P}(A_i)$?
AI: Not very much, in general. Consider a coin tossed just once. Let $A_1$ be the event that it's heads, and $A_2$ be... |
H: Is there a non-saturated measure?
Let $(X, \mathcal {M}, \mu) $be a measure space. A subset $E$ of $X$ is locally measurable, if for each $B \in \mathcal M$ with $\mu (B) < \infty$, we have $E \cap B \in \mathcal M$. The measure $\mu$ is saturated if every locally measurable set is measurable.
Is there a non-satura... |
H: How to prove this simple inequality?
Please help me to prove this inequality.
Suppose $X$ and $Y$ are independent and $EX=EY=0$, then we must have $E(|X|) \leq E(|X+Y|)$.
Thanks.
AI: The condition $EX=0$ is not necessary.
Since the absolute value function is convex, by Jensen's inequality for conditional expectatio... |
H: Prove $\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$ using real analysis techniques only
I have found a proof using complex analysis techniques (contour integral, residue theorem, etc.) that shows $$\int_0^{\infty}\! \frac{\mathbb{d}x}{1+x^n}=\frac{\pi}{n \sin\frac{\pi}{n}}$$ for $n\i... |
H: What is the expected number of ice creams that the saloon can still sell until the first customer who wants chocolate has to be dissapointed?
An ice saloon sells ice creams with one, two, or three scoops. Customers can choose from ten tastes, but you may also choose for more than one scoop with the same taste. The ... |
H: Describing a set using linear inequalities
I am having a hard time understanding the answer to the following exercise (which was taken from "Linear Optimization and Extensions: Problems and Solutions" by Padberg and Alevras).
My problem lies in the last sentence of the answer.
I know that:
$$\left| {x_j^ + - x_j... |
H: Complex Analysis Advice
Could anyone advise on this problem?
Let $g(z$) be an analytic function in punctured ball $B(z_1, R) - \{z_1\}$ and let $N$ be a fixed non-negative integer such that $\lim_{z\rightarrow\ z_1}(z- z_1) ^{m}g(z)=0$ $\forall m > N$, and $\lim_{z\rightarrow\ z_1}(z- z_1)^{n}g(z)= \infty$ $\forall... |
H: Properties of amenable groups
Let $G$ be an amenable countable group. Why does every subgroup and homomorphic image of $G$ is amenable? Further more, if $N$ is a normal subgroup of $G$, and both $N$ and $G/N$ are amenable, why does $G$ must be amenable?
AI: V. Runde, Lectures on Amenability. Springer, 2002 (Lecture... |
H: Bott periodicity and homotopy groups of spheres
I studied Bott periodicity theorem for unitary group $U(n)$ and ortogonl group O$(n)$ using Milnor's book "Morse Theory". Is there a method, using this theorem, to calculate $\pi_{k}(S^{n})$? (For example $U(1) \simeq S^1$, so $\pi_1(S^1)\simeq \mathbb{Z}$).
AI: In ge... |
H: T/F: $\vdash _{NDFOL}\forall x(B \to A) \to(\exists x B \to A)$
I need to decide whether the above holds or not.
I know that $\vdash _{HFOL}\forall x(B \to A) \to(\exists x B \to A)$, and if $T=\emptyset$, $T\vdash _{HFOL}\varphi$ then $T\vdash _{NDFOL} \varphi$.
Is it enought to say that $\vdash _{NDFOL}\forall x(... |
H: Closedness of $L^\infty_+$ in $\sigma(L^\infty,L^1)$
Let $L^\infty_+$ be the set of all $f\in L^\infty$ which are non negative. Our measure can be assumed to be finite. My goal is to prove that $L^\infty_+$ is closed in the weak-star topology $\sigma(L^\infty,L^1)$. Hence I view $L^\infty$ as the dual of $L^1$. We ... |
H: How many students were there if a total of 870 photographs were exchanged
After the graduation exercises at school, the students exchanged photographs with each others. How many students were there if a total of $870$ photographs were exchanged ?
My attempt:
I used the combination formula $^nC_r$ where $n = 870$... |
H: Integrate $\frac{x-1}{\sqrt{x}+1}$ by power-law
How to evaluate the following integral by power-law?
$$\int\dfrac{x-1}{\sqrt{x}+1}\,\mathrm dx$$
Here is my solution which is apparently wrong:
$$\begin{align}
\int\frac{x-1}{\sqrt x+1}\,\mathrm dx &= \int (x-1)(x^{1/2}+1)^{-1}\,\mathrm dx \\
&= \frac23 x\sqrt x+\fra... |
H: Hyperplane - explanation of theorem needed
For any subset $W$ of $\mathbb{R^n}$ the following are equivalent:
W is a hyperplane in $\mathbb{R^n}$;
There is a non-zero $a$ in $\mathbb{R^n}$ such that $W = \{x ∈ \mathbb{R^n} : x\cdot a = 0\}$;
There are scalars $a_1, . . . , a_n$, not all zero, such that
$W = \{(x_... |
H: Fibonacci sequence - how to prove $a_n=\frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$ without induction
How to prove that
$$a_n=\frac{1}{\sqrt{5}} ((\frac{1+\sqrt{5}}{2})^n-(\frac{1-\sqrt{5}}{2})^n)$$
without using induction?
AI: You can use generating functions. Let $F_n$ be the Fibbonacc... |
H: Finding vowels occupy places in word?
From the different words formed out of the letters in the world allahabad the number of words in which the vowels occupy the even places are ?
My Try :
$\frac{9!}{4!\times2!}=15120$
1 2 3 4 5 6 7 8 9
a l l a h a b a d
I found only one vowel which is A at the position 4,6,8
H... |
H: Integral $\int_0^\frac{\pi}{2} \sin^7x \cos^5x\, dx $
im asked to find the limited integral here but unfortunately im floundering can someone please point me in the right direction?
$$\int_0^\frac{\pi}{2} \sin^7x \cos^5x\, dx $$
step 1 brake up sin and cos so that i can use substitution
$$\int_0^\frac{\pi}{2} \sin^... |
H: How can one calculate this diferential equation
$y\ dy = 4x(y^2+1)^2\ dx \text{ and } y(0) = 1$
I'm trying:
$\dfrac{y\ dy}{ (y^2+1)^2 }= 4x\ dx$
but I can't figure out what to do now.
AI: Good start, you separated the variables. Now integrate.
For the integral of the left-hand side, make the substitution $u=y^2+1$... |
H: Disjoint union of random graphs again a random graph?
Let $G_{n,p}, n\in \mathbb{N}, p\in(0,1)$ be the binomial random graph, i.e. a graph on $n$ vertices where an edge is in $G_{n,p}$ with probability $p$. Also, let $q\in (0,1)$.
Can one regard $G_{n,q}$ as a disjoint union of many $G_{n,p}$?
AI: If one is willing... |
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