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H: How can i find the following probabilities?
Let $X$ be binomially distributed with $n = 60$ and $p = 0.4$. Now i have to compute
(a)$P(20\leq X$ or $X\geq40)$
(b)$P(20\leq X$ and $X\geq10)$
i know $P(x\leq X)=\sum_{k=x}^{60}\binom{60}{k}(0.4)^k(0.6)^{60-k}$
But i don't know how to compute the probability when i... |
H: $(X,\mathscr T)$ is compact $\iff$ every infinite subset of $X$ has a complete limit point in $X$.
Let $(X,\mathscr T)$ be a topological space. Given $A\subseteq X$, we say that $x$ is a complete limit point of $A$ if for every nbhd $N$ of $x$, $|N\cap A|=|A|$. I want to prove
Suppose $(X,\mathscr T)$ is compact.... |
H: Prove that $\frac{1}{(1+a)^2}+\frac{1}{(1+b)^2}+\frac{1}{(1+c)^2}+\frac{1}{(1+d)^2}\geq 1$
Let $abcd=1$ and $a,b,c$ and $d$ are all positive.
Prove that
$\dfrac{1}{(1+a)^2}+\dfrac{1}{(1+b)^2}+\dfrac{1}{(1+c)^2}+\dfrac{1}{(1+d)^2}\geq 1$
I am probably able to do this by assuming $a\geq b\geq c\geq d$
and by using de... |
H: $f(x)=x^{x}$ what happens when $x$ is a negative irrational number?
Just looking at negative numbers, $x^{x}$ is defined for all rational numbers (on the real plane) in all instances except whenever $x=\large \frac {2a+1}{2b}$ where $(a, b)$ are integers . However, what happens when $x$ takes a negative irrational ... |
H: Special operator on a normed space
Let $E$ be a normed space and $T \in L(E)$ with $\|Tx\|\lt\|x\|$ for all $x\ne0$ and $\|T\|=1$.
I want to prove the following:
$A=\{x\in E: \|Tx\|\ge1\}$ is closed.
There is no $x\in A$ with $$\inf_{y\in A} \|y\|=\|x\|.$$
Let $(y_n)_{n \in \mathbb{N}} \subset \mathbb{R}^+$ be con... |
H: On a proposition of Engelking's General Topology
Let $\mathcal{F}$ and $\mathcal{F'}$ be filters on a set.
We say that $\mathcal{F'}$ is finer than $\mathcal{F}$ if $\mathcal{F'} \supset \mathcal{F}$.
A point $x$ of a topological space $X$ is called a cluster point of a filter $\mathcal{F}$ if $x$ belongs to the cl... |
H: Differentiability of projection
Where $\pi_i:\Bbb R^n\rightarrow\Bbb R$ is projection onto the $i$th coordinate, the differentiability of $\pi_i$ at $X$ is given by:
$$\pi_i(X+H)-\pi_i(X)=\textrm{grad}\ \pi_i(X)\cdot H+||H||g(H)$$
Where $g$ tends to $0$ as its argument does. We deduce:
$$g(H)=\frac{h_i}{||H||}(1-x_... |
H: Prove exponent law $a^b\cdot a^c=a^{b+c}$ for all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$
For all $ a, b \text{ and } c \in \mathbb{R}$ and $a>1$, Prove that $a^b\cdot a^c=a^{b+c}$
I have come across this question and its bugging me. Its a basic property that we learn in HS and I was hoping someone can en... |
H: Showing $\max\limits_{|z|=r}|p(z)| \ge |a_n|r^n$, without Cauchy integral formula.
Let $p(z) = a_n z^n + a_{n-1}z^{n-1} + \cdots + a_0$.
My question is: Is there an elementary way to show that for all $r > 0$
$$ \max \limits _{|z| = r} |p(z)| \ge |a_n|r^n$$
without using complex analysis machinery that falls out o... |
H: Binomial-like Sum
We know that $\sum_{k=0}^n a^k \frac{n^{\underline k}}{k!} = (1+a)^n$. Is there a known (preferably closed) form for $\sum_{k=0}^n a^k/k!$ ?
This question was prompted by another recent question.
AI: It can be written in terms of the incomplete Gamma function: $\dfrac{e^a}{n!} \Gamma(n+1,a)$. |
H: Comparing Areas under Curves
I remembered back in high school AP Calculus class, we're taught that for a series:
$$\int^\infty_1\frac{1}{x^n}dx:n\in\mathbb{R}_{\geq2}\implies\text{The integral converges.}$$
Now, let's compare $$\int^\infty_1\frac{1}{x^2}dx\text{ and }\int^\infty_1\frac{1}{x^3}dx\text{.}$$
Of course... |
H: Area of a square with double the area of another square
There are 2 squares, one with an area of $a^2$ and another with an area of $b^2$, and when this is true: $(a+b)^2=2a^2$ and: $2b^2=c^2$ then: $b+c=a$.
My question is, is this called anything? I know this is pretty vague but I just want to know if this property... |
H: Finding the relation between function x,y,z - trigo problem
Problem :
For $\displaystyle 0 < \theta < \frac{\pi}{2}$ if
$$\begin{align}x &= \sum^{\infty}_{n =0} \cos^{2n}\theta \\ y &= \sum^{\infty}_{n =0} \sin^{2n}\theta\\ z &= \sum^{\infty}_{n =0} \cos^{2n}\theta \sin^{2n}\theta \end{align}$$
then
options are ... |
H: Change of Variables via Line Element Differential vs. Jacobian
If I had a double integral
$\int \int_R f(x,y) dxdy $
I would change variables to polar coordinates by expressing the position vector
$\vec{r} \ = \ \vec{r}(x,y) \ = \ x \vec{e}_x \ + \ y\vec{e}_y$
in terms of polar coordinates
$\vec{r} \ = \ \vec{r}(r,... |
H: Calculate the Lie Derivative
In trying to get to grips with Lie derivatives I'm completely lost, just completely lost :(
Is there anyone who could provide an example of calculating the Lie derivative of the most basic function you can, i.e. like in showing someone how to calculate the derivative you'd pick somethin... |
H: A die is rolled until a 6 comes up. Should the sample space of this experiment contain the set of all infinite sequences which do not contain a 6?
Is there a standard way to view this? The problem is,
In an experiment, die is rolled continually until a 6 appears, at which point the experiment stops. What is the sa... |
H: Question about the terms and operations in basic division
Let's pretend that I am a child and you want to teach me division. You demonstrate through an example division as repeated subtraction.
This is the simple algorithm the child learns from your lesson:
n=0
remainder=dividend
if(dividend == divisor) return 1... |
H: If the order of $x\in G$ is $p$, the smallest prime dividing $|G|$, and $h^{-1}xh=x^{10}$ for some $h\in G$, then $p=3$
Let $G$ be a group and let $p$ be the smallest prime dividing $|G|$. Let $x\in G$ be such that $|x|=p$. If $\exists h\in G$ such that $h^{-1}xh=x^{10}$, then show that $p=3$.
AI: Look at this one... |
H: Find the maximum of $xy(72-3x-4y)$?
$x$ and $y$ are positive. I have been stuck on this problem for a while now, any hints please?
AI: If $72-3x-4y\leq 0$, then the product is $\leq 0$, which clearly can't be the maximum.
Hint: Multiply by 12. Apply AM-GM to $3x, 4y, (72-3x-4y)$ (which are all positive)
$$\sqrt[3... |
H: Check Convergence of $\sum_{n=1}^{\infty}(-1)^n\cdot\sin(\frac{\pi}{n})$
I want to check the convergence of this series:
$$\sum_{n=1}^{\infty}(-1)^n\cdot\sin\left(\frac{\pi}{n}\right)$$
1) I can check the limit of the positive series and find that its equal to $0$.
after that I need to check more things? because mo... |
H: If $A$ and $B$ are positive definite, then is $B^{-1} - A^{-1}$ positive semidefinite?
I've found this while googling some properties of positive semidefinite matrices. (Unfortunately, I cannot remember where I've discovered it.) If this is true, it'll greatly save my time in my work. Is it true? How can you prove ... |
H: Notation of random variables
I am really confused about capitalization of variable names in statistics.
When should a random variable be presented by uppercase letter, and when lower case?
For a probability $P(X \leq x)$, what do $x$ and $X$ mean here?
AI: You need to dissociate $x$ from $X$ in your mind—sometimes ... |
H: Radius, Center, and Plane
How do you determine the radius, center, and the plane containing the circle $r(t)=7i+(12cos(t))j+(12sin(t))k?$ The way I tried it is using just the basic approach:
$$7^2+(12cos(t))^2=12sin(t))^2$$ $$ 49 + 144cos^2t=144sin^2t$$
$$144cos^2t-144sin^2t=-49$$
$$144sin^2t-144cos^2t=49$$
$$144(... |
H: Check Convergence of $\sum_{n+1}^{\infty}(-1)^n*\left(\sqrt{n+1}-\sqrt{n-1}\right)$
I want to check the convergence of this series:
$$\sum_{n+1}^{\infty}(-1)^n*\left(\sqrt{n+1}-\sqrt{n-1}\right)$$
what I did is:
$$\sum_{n+1}^{\infty}(-1)^n*\left(\sqrt{n+1}-\sqrt{n-1}\right)=$$
$$=\sum_{n+1}^{\infty} \frac{2}{\sqrt{... |
H: If $G$ is disconnected and the vertices $x,y$ are adjacent in $G$, then there is a vertex that isn't adjacent to $x$ and isn't adjacent to $y$.
I'm just starting graph theory and I'm trying to prove the following:
Let $G$ be a simple disconnected graph with vertex set $V(G)$ and edge set $E(G)$. If $x,y\in V(G)$ a... |
H: Check Convergence of $\sum_{n+1}^{\infty}(-1)^n*(\frac{e^n}{n!})$
I`m trying to check the convergence of this series:
$$\sum_{n+1}^{\infty}(-1)^n*(\frac{e^n}{n!})$$
what I decided to do is to use Delambre test, what I get is:
$$\sum_{n+1}^{\infty}\frac{e^{n+1}}{(n+1)!}*\frac{n!}{e^n} = \frac{e}{n+1}$$
1) the limit ... |
H: If $AA^T$ is the zero matrix, then $A$ is the zero matrix
Let $A$ be a $4 \times 4$ matrix. Show that if $A^TA$ or $AA^T$ is the zero matrix, then $A$ is the zero matrix.
I feel very close to solving the problem so far. I have said that
$$[0]_{ij}=\sum_{k=1}^4 [A]_{ik}[A^T]_{kj} =\sum_{k=1}^4 [A]_{ik}[A]_{jk} \qq... |
H: What does it mean to have an $L$-basis of $L\otimes_K V$?
Exercise: I have got a vector space $V$ over $K$, and $L$ a field extension of $K$. The task is to show that if $(v_1, ..., v_n)$ is a basis of $V$, then $(1\otimes_K v_1, ... ,1\otimes_K v_n)$ is an $L$-basis of $L\otimes_K V$.
Question: I know that I can c... |
H: Proving Continuity of a function
Prove that $$f(x)=\lim_{n\rightarrow \infty}{{x^{2n}-1}\over x^{2n}+1}$$
is continuous at all points of $\Bbb R$ except $x=\pm1$
AI: Hints:
$$|x|<1\implies \lim_{n\to\infty}x^{2n}=0$$
$$|x|>1\implies \lim_{n\to\pm\infty} x^{2n}=\infty$$
Now use arithmetic of limits to get (justify!)... |
H: Does every absorbing set of a Banach space contain a neighborhood of origin?
Let $X$ be a Banach space and $A$ be any absorbing subset of $X$. Does $A$ contain a neighborhood of the origin?
AI: We should expect the answer to this question to be "no", since being absorbent depends only on the algebraic structure of ... |
H: Application of Hahn-Banach
Let $(E,\mathcal{E},\mu)$ be a measured space with finite measure $\mu$. We denote with $K$ the space of all real valued functions on $E$, which are $\mu$-a.s. equal. This is a vector space. Now I have a function $\kappa:K\to \mathbb{R}$, which satisfies
$\kappa(\lambda f)=\lambda \kappa... |
H: Sequences of integers with lower density 0 and upper density 1.
It is possible to construct a sequence of integers with lower density 0 and upper density 1?
where lower and upper density means asymptotic lower and upper density (cf. References on density of subsets of $\mathbb{N}$)
EDIT: So, if this is true, then o... |
H: How to prove $XP = X'P$?
Two triangles $\triangle ABC$, $\triangle A'BC$ have the same base and the same height. Through the point $P$ where their sides intersect we draw a straight line parallel to the base; this line meets the other side of $\triangle ABC$ in X and the other side of $\triangle A'BC$ in $X'$. Prov... |
H: Complex integration: $\int _\gamma \frac{1}{z}dz=\log (\gamma (b))-\log(\gamma (a))?$
Let $\gamma$ be a closed path defined on $[a,b]$ with image in the complex plan except the upper imaginary axis, ($0$ isn't in this set).
Then $\frac{1}{z}$ has an antiderivative there and it is $\log z$. Therefore $\int _\gamma \... |
H: what does it mean when $\operatorname{E}[X^2]$ diverges?
is it possible for a random variable $X$, such that the expected value of $X^2$, $\operatorname{E}[X^2]$ is a divergent integral?
If it is impossible, does that mean the probability density function of $X$ is wrong? (the integral of the probability density fu... |
H: Determining Jordan form from ranks of matrix powers.
Suppose you're working over an algebraically closed field. If $J$ is a Jordan matrix, then one can determine the number of Jordan blocks and their sizes for any eigenvalue $\lambda$ by looking at the sequence of ranks $\operatorname{rank}(J-\lambda I)^k$ for $k=1... |
H: Choose two disjoint three-element sets, so the product is a set of five non-identical numbers
So I want to create two unordered sets $x_1=(a,b,c)$ and $x_2=(d,e,f)$ so that all possible products of a term from $x_1$ with a term of $x_2$ constitute five different numbers. The sets can't overlap and have to consist o... |
H: What is the difference between a reflexive relation and an identitive relation
Given a set $X$ and a relation $R$ over $X$, we say that $R$ is reflexive if
\begin{equation}
xRx\ \forall\ x\in X.
\end{equation}
What does 'identitive' mean? Is it the same as antisymmetry?
Seen in Struwe's book Variational Methods: A... |
H: Is it possible to find the term (variable) $c$ from the equation $a^2=\sqrt{b^2+c^2}$
If I have the equation $a^2=\sqrt{b^2+c^2}$. Is it possible to me to find the term (variable) $c$ from it ?
$c=\,?$
AI: Are you simply talking about re-arranging the equation?
$$\left(a^{2}\right)^{2}=\left(\sqrt{b^{2}+c^{2}}\rig... |
H: If $M$ is complete is the closed ball compact?
Let $M$ be a Riemannian manifold and $p,q \in M$. Let $\Omega=\Omega(M;p,q)$ be the set of piecewise $C^\infty$ paths from $p$ to $q$. Let $\rho$ denote the topological metric on $M$ coming from its Riemann metric. Let $S$ denote the ball $\{x \in M : \rho(x,p) \le \sq... |
H: Don't Gödel's completeness and incompleteness theorems contradict each other?
Gödel's completeness theorem: Given a set of axioms, if we cannot derive a contradiction, then the system of axioms must be consistent.
Gödel's incompleteness theorem:'Given any consistent, computable set of axioms, there's a true statem... |
H: Evaluating $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{2^{2n-1}}{(2n+1)\cdot 3^{2n-1}}}$
Calculate the summation $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{2^{2n-1}}{(2n+1)\cdot 3^{2n-1}}}$.
So I said:
Mark $x = \frac{2}{3}$. Therefore our summation is $\sum_{n=1}^{\infty} {(-1)^n \cdot \frac{x^{2n-1}}{(2n+1)}}$.
B... |
H: Continuity of an integral with respect to one variable
Let $V\subseteq \mathbb{R}^n$ and $f:V\to\mathbb{R}^n$. Consider the function
$$g(x_1,x_2,...,x_n) = \int_{x_2}^{x_1} {f(t,x_2,...,x_n)dt}$$
on $V$. What conditions will I need to conclude that $g$ is also continuous on $V$? (I would also appreciate if you give... |
H: Remainder problem using MOD
What's the remainder when $ 43^{101} + 23^{101}$ is divided by 66?
If we use the remainder obtained when $ 43^{101} + 23^{101}$ is divided by $66$, then it becomes,
$$13^{101}+23^{101}$$ then how can I use further MOD?
AI: Hint: $43\equiv -23\pmod{66}$. What happens when you raise $-1$ ... |
H: Show that $\frac {a+b+c} 3\geq\sqrt[27]{\frac{a^3+b^3+c^3}3}$.
Given $a,b,c>0$ and $(a+b)(b+c)(c+a)=8$. Show that $\displaystyle \frac {a+b+c} 3\geq\sqrt[27]{\frac{a^3+b^3+c^3}3}$.
Obviously, AM-GM seems to be suitable for LHS.
For RHS, $a^3+b^3+c^3=(a+b+c)^3-3(a+b)(b+c)(c+a)=(a+b+c)^3-24$, then I don't know what... |
H: Why in a totally disconnected seperable metric space, every open set is a disjoint union of clopen sets?
On page 14, Probability Measures on Metric Spaces,Parthasarath(1967):
We recall that a totally disconnected seperable metric space, every open set can be expressed as a countably disjoint union of closed and o... |
H: How to solve these three equations?
If α ,β ,γ are three numbers s.t.:
$\ α^ \ $ + $\ β \ $ + $ γ \ $ = −2
$\ α^2 \ $ + $\ β^2 \ $ + $ γ^2 \ $ = 6
$\ α^3 \ $ + $\ β^3 \ $ + $ γ^3 \ $ = −5,
then $\ α^4 \ $ + $\ β^4 \ $ + $ γ^4 \ $ is equal to ??
I tried out substituting the values of each equation to one oth... |
H: To what extent are morphisms required to be functions?
Just beginning category theory, and I am looking for clarification on the precise nature of morphisms. The most familiar categories, e.g. $\mathbf{Top}$, have morphisms that are functions in the traditional sense, i.e. subsets of $A \times B$ such that for $(a_... |
H: Measurability of restriction
Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces and $f:X\rightarrow Y$ a measurable function (that is, $f^{-1}(B)\in \mathcal{A}$ whenever $B\in \mathcal{B}$).
Let $A_1,...,A_n$ be measurable subsets of $X$ such that $X=\cup A_i$.
Is it true the following equivalence... |
H: Any way to tell when an algebraic expression takes on values that are a square?
Say I have the expression $256x^2 -480x$. As a polynomial this isn't a perfect square. However that doesn't stop it from taking real values that are a perfect square for given x, such as x = 8. Is there any way to determine what other v... |
H: Geometric progression problem
Please help me solve this:
A man decided to save 50 cents on the first day and on each successive day double the amount saved the previous day.
how long does it take for his total savings to be 1 USD million.
AI: This is simpler than it might seem. Each day, he more than doubles his to... |
H: Find for which values of $\alpha\in\mathbb R, f_n(x)$ converge uniformally
The question is
"Find for which values of $\alpha\in\mathbb R$ such that $ f_n(x)$ converge uniformally in $[0,\infty)$ where $f_n(x)=n^\alpha x \dot e^{-nx} $".
For $\alpha<0$, the sequence converges uniformally to $0$ (since $\forall x... |
H: How many times does the function $y=e^x $ meet $y=x^2$?
As you know $y=e^x$ and $y= x^2$ meet once on $x<0$.
But I want to know whether or not they meet on $x>0$.
Since $\lim_{x\rightarrow \infty } e^x/x^2=\infty$, if they meet once on $x>0$, they
must meet again.
To summarize, my question is whether or not th... |
H: Re-arranging the equation $t=\arctan\left(\frac{a}{b}\tan\theta\right)$ to find $\theta$
How can I re-array the equation
$t=\arctan\left(\frac{a}{b}\tan \theta\right)$
to find the equation of $\theta$.
$\theta=\,?$
Actually I tried this equation:
$\theta=\frac{\arctan\left(\tan t\right)}{\frac {a}{b}}$
but i... |
H: Simple probability problem
A bag contains $(2m+1)\,$ coins. It is known that
$m$ of these coins have a head on both sides and the
remaining coins are fair. A coin is picked up at
random from the bag and tossed.
If the probability
that the toss results in a head is $14/19 \,$, then $m$ is
equal to ?
How to go about... |
H: find length of semi major axes of ellipse
suppose that equation of ellipse is given by
$4x^2+3*y^2=25$
we should length of major axes ,first let us transform this equation into standard form or divide by $25$
$4*x^2/25+3*y^2/25=1$
if we compare it to $x^2/a^2+y^2/b^2=1$;we get that $a=5/2$ and $b=5/\sqrt{3... |
H: "Any finite poset has a maximal element." How to formalize the proof that is given?
For fun, I'm reading The Mathematics of Logic, and the author gives the following theorem
Theorem. Any finite poset has a maximal element.
and a proof thereof. But, I can't figure out how to formalize the proof. Help, anyone? I was ... |
H: Calculating this double integral in polar coordinates
Calculate the double integral $\iint_D {(1+x^2 + y^2)ln(1+x^2+y^2)dxdy} $ where $D = \{(x,y) \in \mathbb R^2 | \frac{x}{\sqrt3} \leq y \leq x , x^2 + y^2 \leq 4\}$.
I heard there is a way called Polar Coordinates but the more I looked and read about it the mor... |
H: Irreducible elements are not associates
I would like to know if every irreducible elements in a ring are not associates. I'm asking that because of this part in the page 61 of this book:
Thanks a lot
AI: Two primes are associates if one of them is a unit times the other. For example, in the integers $5$ and $-5$ ... |
H: Show that the limit exists and find it's value?
The Fibonacci series defined recursively by $x(1) = 1, x(2) = 2$ and
$x(n+1) = x(n) + x(n-1)$
Find$$\lim_{n\rightarrow\infty}\frac{x(n+1)}{x(n)}$$
AI: Let $L= \lim_{x\rightarrow \infty} \dfrac{x(n+1)}{x(n)}= L$. By the recursive definition we also have that $\lim_{x\... |
H: Re-arranging the equation $L=\sqrt{a^2\sin^2t+b^2\cos^2t\,}$ to find $\left(t\right)$?
How can I re-array the equation
$L=\sqrt{a^2\sin^2t+b^2\cos^2t\,}$
to find the equation of $\left(t\right)$ ?
$t=\,?$
I tried to solve it but I'm stuck at:
$L^2=a^2\sin^2t+b^2\cos^2t$
AI: Hint: $\cos^2 t+\sin^2 t = 1$
SOLUT... |
H: Where does the symbol $\mathcal O$ for sheaves come from?
Sheafs are often denoted by the letter $\mathcal O$. What does this O stand for? To me it seems that more natural choices of symbols for sheaves would be $\mathcal S$ or $\mathcal F$ (for the french faisceau).
AI: Quoting the historical footnote in Grauert/R... |
H: Determining the angle degree of an arc in ellipse?
Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ?
If I have an arc length at the first quarter of an ellipse and I want to know the angle of it, what is the data that I will need... |
H: Weak deformation retraction (exercise 0.4 from Hatcher) - Proof check
A deformation retraction in the weak sense of a space $X$ to a subspace $A$ is a homotopy $f_t:X\to X$ such that $f_0=Id_X$, $f_1(X)\subset A$ and $f_t(A)\subset A$ for all $t$. Show that if $X$ deformation retracts to $A$ in this weak sense, the... |
H: Integral of $\int \frac{dx}{\sqrt{x^2 -9}}$
$$\int \frac{dx}{\sqrt{x^2 -9}}$$
$x = 3 \sec \theta \implies dx = 3 \sec\theta \tan\theta d\theta$
$$\begin{align} \int \frac{dx}{\sqrt{x^2 -9}} & = \frac{1}{3}\int \frac{3 \sec\theta \tan\theta d\theta}{\tan\theta} \\ \\ & = \int \sec\theta d\theta \\ \\ & = \ln | \sec... |
H: Projection of closed set
Set $A \subset R^2$, set B is projection of A on x-axis. Do you know a counterexample to the statement: if A is closed, then B is closed.
AI: Hint: Consider the graph of $\tan(x)$ between two asymptotes. |
H: Conjecture on limit of $1-(n^{p-1}\mod p)$
Given $p \in \Bbb P$ prime, $n \in \Bbb N$ and
$$\mathcal V_p=1-(n^{p-1}\mod p)$$
let me conjecture that
$$\lim_{p\rightarrow \infty}\mathcal V_p = \operatorname{sinc}(2\pi \,n)$$
Question: Is this conjecture true?
AI: If $p$ does not divide $n$, we have $n^{p-1}\equiv 1\p... |
H: Why I cannot solve this limit problem this way?
$\lim_{x\to -1} \ \, \frac{1}{{1+x}} \left( \frac{1}{{x+5}}+ \frac{1}{{3x-1}} \right) $
As, the limit is not of the form$\ \frac{0}{{0}} $ so, put $\ x $ as $\ -1 $ we get Answer $\ 0$ .
What is wrong in this?
AI: If you write the limit as
$$
\frac{\frac1{x+... |
H: The orthogonal projection onto a plane - explanation
Could somebody explain, why orthogonal projection onto a plane with equation $x_1+x_2+x_3=0$ is given by $$y=(x_1,x_2,x_3)-\bigg( \frac{x_1+x_2+x_3}{3}\bigg)(1,1,1)$$
I don't understand, why we sum three coordinates and divide by $3$? I thought, we need to use th... |
H: solving arduous limit
How to simplify the following:
$$\lim_{x\rightarrow 0} \frac {\sin (\pi / 2 - 10 \sqrt x) \ln ( \cos (2x))}{(2^x -1)((x+1)^5-(x-1)^5)}$$
Here is what I've done:
$$ \sin (\pi / 2 - 10 \sqrt x) = \cos 10 \sqrt x$$
$$(x+1)^5-(x-1)^5 = 10x^4 +20 x^2 +2$$
So:
$$\lim_{x\rightarrow 0} \frac {\cos (10... |
H: Cover of a Grassmannian by an open set
I am reading this document here and in exercise 1, the author asks to show the Grassmannian $G(r,d)$ in a $d$ dimensional vector space $V$ has dimension $r(d-r)$ as follows. For each $W \in G(r,d)$ choose $V_W$ of dimension $d-r$ that intersects $W$ trivially, and show one has... |
H: A bounded holomorphic function
If $\Omega$ is a region which is dense in $\mathbb{C}$, $f\in H(\Omega)$ and is continuous on $\mathbb{C}$, moreover $f$ is bounded on $\mathbb{C}$, can we claim that $f$ is a constant?
AI: This answer is incorrect.
You can consider the function $g(x):=\sum_{k=0}^{\infty}\frac{x^{2^k}... |
H: First order linear differential equation, wrong result
I have a differential equation:
$y'= \frac{y}{x} + x^2$
I apply this formula:
$y(x)= e^{A(x)} \int {e^{-A(x)} b(x) dx} $
With $a(x) = \frac{1}{x}$ and $b(x)= x^2$, and $A(x)$ primitive of $a(x)$.
It gives:
$y(x)= e^{log|x|} \int{ e^{-log|x|} x^2 dx }$ = ... |
H: Finite extension of integrally closed ring again integrally closed
Let $S\subset R$ be a finite ring extension, i.e. $R$ is finitely generated as an $S$-module. Assume that $S$ is integrally closed. Does this imply that also $R$ is integrally closed (in its quotient field)?
AI: Take $ S = \Bbb{Z}$ and take $R = \Bb... |
H: Uses of the incidence matrix of a graph
The incidence matrix of a graph is a way to represent the graph. Why go through the trouble of creating this representation of a graph? In other words what are the applications of the incidence matrix or some interesting properties it reveals about its graph?
AI: There are ma... |
H: How to fit a formula to three data points?
I need a very basic formula that will be used to determine a CSS line-height based on a provided font pixel size.
So in essence, I need the formula to covert
13 == > 17
15 == > 22
19 == > 27
Been going crazy all morning trying to derive a formula to do this...
AI: Well, t... |
H: A basis such that $A$ is of a certain type.
Let $A$ be a $2\times 2$ real matrix without eigenvalues, and the roots of its characteristic polynomial be $\alpha+i \beta$ and $\alpha - i \beta$. Show that there exists a basis of $\mathbb{R^2}$ such that
$A=\begin{pmatrix} \alpha & \beta \\ - \beta & \alpha \end{pmat... |
H: How does $a^2 + b^2 = c^2$ work with ‘steps’?
We all know that $a^2+b^2=c^2$ in a right-angled triangle, and therefore, that $c<a+b$, so that walking along the red line would be shorter than using the two black lines to get from top left to bottom right in the following graphic:
Now, let's assume that the direct w... |
H: Testing polynomial equivalence
Suppose I have two polynomials, P(x) and Q(x), of the same degree and with the same leading coefficient. How can I test if the two are equivalent in the sense that there exists some $k$ with $P(x+k)=Q(x)$? P and Q are in $\mathbb{Z}[x].$
AI: The condition of $\mathbb{Z}[x]$ isn't requ... |
H: What does δA mean in differentiation?
To be more specific, I met this when doing analytical mechanics involving the principle of least action:
AI: The easiest way to understand this notation is as follows:
A variation is a transformation of the independent variables in a problem; it says that for each $\epsilon$ ... |
H: Using trig substitution to evaluate $\int \frac{dt}{( t^2 + 9)^2}$
$$\int \frac{\mathrm{d}t}{( t^2 + 9)^2} = \frac {1}{81} \int \frac{\mathrm{d}t}{\left( \frac{t^2}{9} + 1\right)^2}$$
$t = 3\tan\theta\;\implies \; dt = 3 \sec^2 \theta \, \mathrm{d}\theta$
$$\frac {1}{81} \int \frac{3\sec^2\theta \mathrm{ d}\theta}{... |
H: The indefinite integral $\int{\frac{\mathrm dx}{\sqrt{1+x^2}}}$
I need to solve this integral:
$$\int{\frac{\mathrm dx}{\sqrt{1+x^2}}}$$
First I thought it was easy, so I tried integration by parts with $g(x)=x$ and $g'(x)=1$:
$$\int{ \frac{x^2}{(1+x^2)^{\frac{3}{2}} }}\,\mathrm dx $$
But I've made it even more... |
H: Quadratic integer Programming
Would anyone mind helping me solve this problem
$$
\min\space f(x) = \frac12 x^\mathrm TQx + bx + c \qquad \text{s.t. } \sum_i x_i=\lambda
$$
where $x$ is a vector whose entries are positive integers and $Q$ is positive definite.
AI: These kinds of (mixed) integer quadratic programmin... |
H: Adjoint matrix eigenvalues and eigenvectors
I just wanted to make sure that the following statement is true:
Let $A$ be a normal matrix with eigenvalues $\lambda_1,...,\lambda_n$ and eigenvectors $v_1,...,v_n$. Then $A^*$ has the same eigenvectors with eigenvalues $\bar{\lambda_1},..,\bar{\lambda_n}$, correct?
AI: ... |
H: Proving the function f , which has zero first order parital derivatives, is constant
Let the function $f: \Bbb R^{2} \to \Bbb R$
The first order derivatives of f are zero.
i.e $f_x(x,y)$ = $f_y(x,y)$ = $0$
How can I prove that $f(x,y)$ is constant for all $(x,y)$
AI: If $\operatorname{f}_x \equiv 0$ then $\opera... |
H: Elementary set proof
On a statistics trial exam I encountered the following proof I was supposed to give but I have no idea how to start with this proof and solve it:
$P(A\cap B)$ $\geq$ $1 - P(A') - P(B')$ where $A'$ is the complement of A
If anyone could help me, that would be great!
AI: First of all, split up th... |
H: Mix GP and AP question
I appreciate your good help:
Two consecutive terms of a GP are the 2nd, 4th and 7th terms of an AP respectively. Find the common ratio of the GP.
AI: I’m going to assume that you meant that three consecutive terms of the geometric progression are the second, fourth, and seventh terms, respect... |
H: For non-negative $f$ such that $\int_1^\infty |f'(t)|dt < \infty$, $\sum f(k)$ and $\int_1^\infty f(t)dt$ converge or diverge together
Suppose that $f\in C^1([1, \infty))$, $f>0$, and $\int_{1}^\infty |f'(t)|dt < \infty$. I want to show that $\sum_1^\infty f(k)$ and $\int_1^\infty f(t)dt$ are either both convergent... |
H: If $\sum a_k^2 /k$ converges, then $1/N \sum_1^{N}a_k \to 0$
I want to show that if $\sum a_k^2 / k$ converges, then $1/N \sum_{1}^Na_k \to 0$.
Now, if $a_n \to 0$, then the result follows. But of course $a_n\to 0$ is not a necessary condition for $\sum a_{n}^2/n$ to converge. We might want to use Cauchy-Schwarz to... |
H: Proving a complex sum equals factorial
I have just stumbled across the equality that:
$$
\sum_{j=0}^{n}(-1) ^ {n + j} j ^ {n} \binom{n}{j} = n!
$$
How would I go about proving this equality?
Also, what is the left hand side equal to if the power of j is increased:
$$
\sum_{j=0}^{n}(-1) ^ {n + j} j ^ {n+k} \binom{... |
H: Formal Definition of Yang Mills Lagrangian
I have a question regarding the Lagrangian in non abelian gauge theory. Say, $G$ is the gauge group and $\mathfrak g$ the associated Lie algebra. The Lagrangian is often written as
$$
\mathcal L=-\frac {1}{4} \text{tr} (F_{\mu \nu} F^{\mu \nu}) + \overline \psi (i \tilde D... |
H: Solve $x\sqrt{10} = \prod\limits_{k = 1}^{90} \sin(k), x\in \mathbb Q$.
Can someone help me with this question? I've found a solution but it's not a very nice one. I used 6 times the relation $\sin(2\theta) = 2\sin(\theta)\cos(\theta)$. There's got to be a better way.
$x\sqrt{10} = \prod\limits_{k = 1}^{90} \sin(k)... |
H: Separability requirement for Measurable distance?
I was looking up Egorov's Theorem on Wikipedia. https://en.wikipedia.org/wiki/Egorov%27s_theorem
One of the conditions is that the functions attain values in a separable metric space M, and in the "Discussion of assumptions" section following the theorem, the follow... |
H: Simple algebra loss calculation
Kylie bought an item for $x$ and sold it for \$10.56. If Kylie incurred a loss of $x$ percent, find $x$.
The answer is apparently "12 or 88" but I cannot see how they got there. I have tried
$$\frac{10.56-x}{x}=x$$
But the result is no where near what the answer is.
AI: You want to s... |
H: Statements equivalent to $A\subset B$
Each of the following statements are equivalent to $A\subset B$:
(1) $A \cap B = A$
(2) $A \cup B = B$
(3) $B^{c} \subset A^{c}$
(4) $A \cap B^{c}= \emptyset$
(5) $B \cup A^{c}= U$
I only understand (1) and (4). Why is (5) $U$ and not $B$?
Can you explain/elaborate on the o... |
H: Line Integral where C is Line Segment
Evaluate the line integral
$$ \oint\limits_C xe^y \mathrm ds \;\text{where C is the line segment from (-1,2) to (1,1).}$$
My answer was $\sqrt{5}e(e-3)$. Did I get this right?
Thanks!
AI: I believe the answer is slightly different:
If you let $x=-1+2t$ and $y=2-t$ for $0\le t... |
H: Use of Multiple "if and only if" statements
I apologize in advance if this question is too basic to warrant a post. I just ran into the following question:
Let $f: A \to B$. If $A$ and $B$ are finite sets with the same number of elements, then $f: A \to B$ is bijective if and only if $f$ is injective if and only if... |
H: Bounded Derivate of a differentiable and Lipschitz function
Let $E, F$ normed spaces and $f:A\subseteq E\to F$ with $A$ open set, suppose that $f$ is differentiable at $a\in A$ and that $f$ is locally Lipschitz of constant $k>0$ in $a$. Show that $||Df(a)||\leq k$.
Note: $f$ is differentiable in $a$ iff exist $Df(a... |
H: Differential equation of a mass on a spring
I have the following differential equation which is motivated by the dynamics of a mass on a spring:
\begin{equation}
my'' - ky = 0
\end{equation}
I split this into a system of equations by letting $x_1=y$ and $x_2=y'$
\begin{equation}
x' = \begin{pmatrix} 0&1\\\dfrac{k}... |
H: Show that $|\lambda|\leq 1$ for each eigenvalue $\lambda$ of a partial isometry
Let $V$ be an inner product space. A linear map $U:V\rightarrow V$ is a partial isometry if there is a subspace $M\subset V$ such that $\parallel Ux\parallel =\parallel x \parallel$ for all $x\in M$ and $\parallel Ux\parallel =0$ for al... |
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