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H: Statistics problem (normal distribution)
A factory produces roller stands (of cylindrical form) that has $4cm$. of diameter and $6cm$. of length. In fact, the diameters $X$ are normally distributed with a mean of $4cm$. and a standard desviation of $0,01cm$, being its density $f_{1}(x)$. By the way, the lengths $Y$... |
H: What is the $8^{th}$ term of $\left(3x-\frac{y}{2}\right)^{10}$?
What is the $8^{th}$ term of $\left(3x-\frac{y}{2}\right)^{10}$?
My solution: I'am not sure if I'am correct :)
$^{10}C_r (3x)^{10-r} \left(-\frac{y}{2}\right)^r$
where $r= 7$ since we start at $r=0$
$^{10}C_7 (3x)^3 \left(-\frac{y}{2}\right)^7$
$\left... |
H: Dot products in the context of linear algebra and matrix multiplication
I've been self-teaching myself linear algebra from Linear Algebra and its Applications 4th from D. Lay. I'm about 8 sections deep and I've had this bothersome feeling regarding the section describing the process of multiplying matrix $A$ and ve... |
H: Is there a noncompact operator from $\ell^\infty$ to a reflexive space?
It is well known that bounded operators from $c_{0}$ to a reflexive Banach space $X$ are in fact all compact. Indeed, since it can be shown that an operator is compact iff for any weakly convergent sequence in its domain, its image is convergen... |
H: Is $e^z + \overline{z}^2$ holomorphic?
Can a sum of a holomorphic and non-holomorphic functions be itself holomorphic? As I understand, $\overline{z} ^2$ is not holomorphic?
AI: The difference of two holomorphic functions is holomorphic. Is $(e^z + \bar z^2) - e^z$ holomorphic? |
H: If $10^{20} +20^{10}$ is divided by 4 then what would be its remainder?
If $$10^{20} +20^{10}$$ is divided with 4 then what would be its remainder?
AI: Since
$$
\begin{eqnarray*}
10^{20}+20^{10} &=&\left( 10^{10}\right) ^{2}+2^{10}10^{10} \\
&=&10^{10}\left( 10^{10}+2^{10}\right) \\
&=&2^{10}5^{10}\left( 2^{10}5^{... |
H: Algebraic manipulation with square roots
I have always had problems with the algebraic manipulation of square roots. For example, recently I encountered this in a problem I was working on:
$$\sqrt{\left(\dfrac{x-1}{2x}\right)^2 - \dfrac{y}{x}} = \dfrac{1}{2x} \sqrt{(x-1)^2 -4xy}$$
I still don't grasp why this is co... |
H: Simple question about Liouville formula
Liouville Formula
$\det X(t)=\det X(t_0)\exp\left( \int^t_{t_0}tr A(u) du\right)$
Why when $t_0=0$ we have $\det e^{tA}=e^{t(trA)}$
My book says this, but I couldn't understand why.
I need help.
Thanks a lot
AI: Presumably $X$ is the solution to the ODE $\dot{X} = A(t) X$, ... |
H: Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$
Let $f:[a,b]\to\mathbb R$. Evaluate $\lim_{n\to\infty}\int_a^bf(x)\sin(nx)\,dx$.
$f$ is continuously differentiable.
I'm told this can be done using basic calculus. It's difficult for me to see where I should begin. I'd like some hint... |
H: Does this variable have a HyperGeometric distribution?
David has 100 cards in his right pocket: 70 black, 20 blue and 10 white.
in each time David chose randomly one card, and move it to his left pocket.
What is the distribution of the number of blue cards that left in David's right pocket after $n$ times.
HyperGe... |
H: if $|D_n - \frac{n!}{e}| < \frac{1}{1+n}$, why is $D_n$ nearest integer?
Given $D_n$ is the number of derangements (i.e. permutations s.t. $\pi(i) \neq i$ for all $i$ in $\{1,...,n\}$, where $D_n$ is defined to be: $\frac{D_n}{n!} = \sum_{0 \leq q \leq n} \frac{(-1)^q}{q!}$
Then, supposing this inequality is true:
... |
H: Finding $\sin^6 x+\cos^6 x$, what am I doing wrong here?
I have $\sin 2x=\frac 23$ , and I'm supposed to express $\sin^6 x+\cos^6 x$ as $\frac ab$ where $a, b$ are co-prime positive integers. This is what I did:
First, notice that $(\sin x +\cos x)^2=\sin^2 x+\cos^2 x+\sin 2x=1+ \frac 23=\frac53$ .
Now, from what... |
H: Finding $A^k$ for a large $k$
If I have a matrix A that I have found a matrix $P$ for such that $P^{-1}AP$ is now diagonal, is it possible to calculate $A^k$ for a large $k$?
I assume it has something to do with the fact that $(P^{-1}AP)^k=P^{-1}A^kP$, but I'm not sure how to use it.
AI: If $P^{-1}AP$ is diagonal, ... |
H: How to find the height, given the volume of a cylinder and cone (conjoined together)?
Here is a picture for a diagram:
If water fills this "vase" up to half its capacity (NOT half its height), what will the height of the water be, starting from the bottom?
And if you could explain the steps leading up the answer, ... |
H: Growth of y with respect to time based on some given assumptions
I would be thankful if someone can help me with the following problem:
Given:
$\frac{\dot{N}(t)}{N(t)} = c_b - \frac{c_d}{y(t)}$
$\frac{\dot{A}(t)}{A(t)} = N(t)g - \delta $
$y(t) = A(t)\cdot(\frac{T}{N(t)})^{1-\alpha}$
$T$ is fixed.
Question:
How is ... |
H: Determining dimension given a parameter $a$
If I have a homgenous matrix with one of the entries being $a$ and I need to determine which values of $a$ will give the matrix a space of solutions that has dimension $1$ (or dimension 2), how would I go about doing that?
For example ( just making this up from the top of... |
H: All 1-tensors are alternating
This statement from page 155 of Guillemin and Pollack's Differential Topology. I would assume because 1-tensors can not alternate because they have nothing to alternate with, so they are alternating...?
AI: The statement is vacuously true: If $\sigma \in S_1$ is a permutation, then we ... |
H: Linear Independence by row space.
Let $(1,1,-1), (2,1,0)$ and $(-1,0,1)$ be vectors, show if they are independent.
I wrote each vector on the rows of the matrix $A$.
$A=\begin{pmatrix} 1 & 1 & -1 \\ 2 & 1 & 0\\ -1 & 0 & 1 \end{pmatrix}$
Then I put $A$ on an echelon form.
$A=\begin{pmatrix} 1 & 1 & -1 \\ 2 & 1 & 0... |
H: Definitions of connected space
I have seen several definitions of connected space, but I would like to discuss those from Wikipedia. I am concerned about these:
$X$ is disconnected, if it is the union of two disjoint nonempty open sets.
What about $[0,1] \cup [2,3] $ ? It is not the union of two disjoint nonempty o... |
H: Inverse of tangent map
Let $Tf$ be the tangent map of a one to one map $f$ and let $g$ be the function: $$g(x,v)=(f(x),Tf(x).v)$$
$g$ is one-to-one, but what is the expression of its inverse ?
Thanks !
AI: In general, if $f$ is one-to-one then $Tf$ is not necessarily one-to-one.
For example, if $f(x) = x^3$, then
... |
H: Moving only to the right and down, how many different pathways exist to get from A to Point B
The answer I think is 10 following the rules in my textbook am I correct?
AI: |
H: Counting letters
Arrange the letters IMAGINATORIUM where a 3 I's are separate and NAT appears as a subsequence.
My thinking:
Start by arranging the NAT. There are three spots in the string, so we need three letters. There is one N in the word, so to chose which N and which spot 1C1. Then for the A, there are 3 A'... |
H: Solving $A^TAx=A^Tb$ without using $A^TA$ or its inverse
Given $A=\pmatrix{2&1\\ 2&3\\ -2&1}$ and $b=\pmatrix{1\\ 1\\ 1}$, solve $A^TAx = A^Tb$ without calculating $A^TA$ or its inverse.
I solved a previous problem in this set where I calculated the QR factorization of $A$ using Gram-Schmidt orthogonalization. I ... |
H: Points in $\mathbb{R}^3$ are coplanar and Determining Area of Triangle in $\mathbb{R}^3$
I came across this question in my textbook while working on some problems:
Find the criterion for four points in $\mathbb{R}^3$ to be coplanar; then find the formula involving the cross product for the area of a triangle with ... |
H: Conditional compactness iff totally bounded
We say that a metric space $M$ is totally bounded if for every $\epsilon>0$, there exist $x_1,\ldots,x_n\in M$ such that $M=B_\epsilon(x_1)\cup\ldots\cup B_\epsilon(x_n)$.
A metric space in which every sequence has a Cauchy subsequence is said to be conditionally compact... |
H: A triangular "spot function"
z = (cos πx + cos πy) represents the classical "spot function", made by square cells, used in every laser printer's halftone screening. Does anyone knows the corresponding function to produce TRIANGULAR cells instead of squared ones?
AI: That is, you want a function such that the contou... |
H: Manifold being locally euclidean vesus Manifold being locally homeomorphic to an open set in $R^n$.
I was reading the definition of smooth manifold and i am little bit of confused. Informally it says
A smooth manifold is a topological manifold (i.e. a topological space locally homeomorphic to a Euclidean space) eq... |
H: Limit involving power tower: $\lim\limits_{n\to\infty} \frac{n+1}n^{\frac n{n-1}^\cdots}$
What is the value of the following limit?
$$\large \lim_{n \to \infty} \left(\frac{n+1}{n}\right)^{\frac{n}{n-1}^{\frac{n-1}{n-2}^{...}}}$$ In general what do limits of infinite decreasing numbers strung together in familiar w... |
H: Related rates shadow question
A $5$ meter lamp is casting a shadow on a $1.8$ meter man walking away at $1.2$ meters a second, how fast is the shadow increasing?
I have no idea how to do this, it feels like there is missing information. I know that this is a problem about triangles but there is some weird trick tha... |
H: Split by percentage
I need to give $n$ apples to three persons $A$, $B$ and $C$.
$A$ should get $50\%$,
$B$ should get $30\%$, and
$C$ should get $20\%$ of however much ever I give.
For example, if I have given $10$ apples, $A$ should have $5$, $B$ should have $3$, and $C$ should have $2$.
Apple A B C
1st ... |
H: What's the correct name in english for "Analysis in $\Bbb R^n$"?
Well, this question may seem silly and I fear it's even out of topic here. My motivation to ask that is to know the correct terminology when talking about that here in Math.SE. The point is, here in Brazil the topics covered in Spivak's Calculus on Ma... |
H: Use Euler's method to approximate $\int^2_0 e^{-u^2}du$
We learned Euler's method today there is one hw problem totally stunned my hat off. It says:
Use Euler's method to approximate $\int^2_0 e^{-u^2}du$. I know Euler's method is $y_{n+1} = y_n + hf(t_n,y_n)$, but this is for $y'$ right? somehow I can compute an i... |
H: Probability - how to satisfy an order with given probability?
The probability that randomly chosen shirt from a current production can be qualified as a premium sort shirt is $p = 0.8$. Find the probability that out of $n = 100$ shirts we will have at least $85$ premium sort shirts. How many shirts we have to pro... |
H: total differential of $f+g$, $fg$ and $\frac fg$
Let $f,g:\mathbb R^n\rightarrow\mathbb R$ be differentiable. We had in lectures that $f+g,fg,\frac fg$ are differentiable too.
As an exercise I want to prove this.
$f$ is differentiable in $x\in\mathbb R^n$ $\Leftrightarrow$ there is a lineare function $A$ such tha... |
H: wedge product and determinant
I don't really know what $[\phi_i(v_j)]$ really is. As far as I understand, $\phi_i$ is a linear transformation - a matrix; and $v_j$ is the column vector it eats. So $[\phi_i(v_j)]$ spit out column vectors, rather than $k \times k$ real matrix as the problem stated.
Again, thanks for... |
H: Need Help With These 2 Problems find k*x)=f(og)
Need someone to check If I'am write I think I have the write solution.
First question:
Given: $g(x)=2x^2+5$ and $f(x) = x^2+4x$, find $k(x) = (f\circ g)(\sqrt3)$.
My answer:
$k(x)=119+48\sqrt3$
Second question:
domain for $k(x)$ if: $d(x)=\sqrt{x-2}$ and $g(x) =... |
H: Critical number $y = \frac{1}{x^2 + 2}$
Seems pretty straight forward but my book seems to be giving an incorrect answer without any explanation to their magic.
$$y = \frac{1}{x^2 + 2}$$
I know that this has no 0 so that rule of finding a critical number can be discarded.
$$y = (x^2 + 2)^{-1}$$
$$y' = -1(x^2 + 2)^... |
H: $f$ is constant if derivative equals zero
Suppose $f'(x)=0$ for all $x\in (a,b)$. Prove that $f$ is constant on $(a,b)$.
This seems painfully obvious, but I can't prove it rigorously.
$f'(x)=0$ for all $x\in (a,b)$ means that for any $c\in(a,b)$, we have $$\lim_{x\rightarrow c}\frac{f(x)-f(c)}{x-c}=0$$ So for any... |
H: Is $X$ has a strong rank 1-diagonal?
Definition 1: A space $X$ has a strong rank 1-diagonal \cite{5} if there exists a
sequence $\{\mathcal U_n: n\in \omega\}$ of open covers of $X$ such
that for each $x\in X$, $\{x\}=\bigcap
\{\overline{\operatorname{St}(x, \mathcal U_n}): n\in \omega\}$.
Example 2: Let $Y=\b... |
H: how can i find best value of t in this equation?
I need to evaluate
$$\tan^{-1} (x) - x = O(x^t)$$
as $x$ approaches $0$, in order to find the best value of $t$.
Big O notation is described here.
I tried: $$\lim_{x \to 0} \frac{\tan^{-1}(x) - x}{x^t}.$$
AI: If you are familiar with the power series expansion for $\... |
H: Combinatorics: Options that do not involve a specific object
The question was posed as: Marissa is doing a Tarot reading in which she must pick 6 cards from a deck of 72. The order of their selection is not important. Marissa does not want to see the Fool card. How many of the possible readings do not feature the F... |
H: How can I definitively determine whether a universal/existential statement is true or false?
How can one look at a universal/existential statement and determine with absolute certainty whether it is true or false? For example, is
$$∀x\in\mathbb R\,∀y\in \mathbb R\,∃z\in \mathbb R\,\left(z = \frac{x+y}2 \right)$$ t... |
H: Using partial fractions to find explicit formulae for coefficients?
The set of binary string whose integer representations are multiples of 3 have the generating function
$$\Phi_S(x)={1-x-x^2 \over 1-x-2x^2}$$
Let $a_n=[x^n]\Phi_s(x)$ represent the number of strings in $S$ with length $n$. Using partial fraction ex... |
H: Is $E(xy) \geq E(x) E(y)$?
Is $E(xy) \geq E(x) E(y)$ always? Where $E$ means expectation.
Intuitively I feel like covariance should always be nonnegative, but I cannot prove why.
Thanks for your help
AI: That is not true. For example, let $x$ be $0$ or $1$ with equal probability of $0.5$, and let $y = 1 - x$. Then... |
H: Help with a conditional probability problem
There are 6 balls in a bag and they are numbered 1 to 6.
We draw two balls without replacement.
Is the probability of drawing a "6" followed by drawing an "even" ball the same as the probability of drawing an "even" ball followed by drawing a "6".
According to Bayes Theor... |
H: Help identify this operator
Except the whole operator is more compact. It is used between two fractions like.... x / y mystery operator x /y . I don't know how to write it on my computer.
~
=
text ~ text
---- = ----
text text
AI: $\cong$, perhaps? Your question is very hard to understand. $\propto... |
H: Need help in number theory
I wanted to know, how do I go about finding solutions to the equation $(x+1)(y+1) = z^3 + 1$ (integral solutions).
Any help appreciated.
Thanks.
AI: For any integer $z$ there will be a solution. If $z^3 + 1 \neq 0$, then just take $x+1$ to be a divisor of $z^3 + 1$ and let $y+1 = \frac{z^... |
H: How to break a quadratic form based on two complementary subspaces
I'd like to prove the following for my research, but I don't know how. It's from the paper An Elementary Proof of the Restricted Invertibility Theorem by Daniel A. Spielman and Nikhil Srivastava.
Let's say I have a positive semi-definite matrix $\m... |
H: estimating the error of $\sin(x) = x$ with Taylor's Theorem
I want to calculate the numerical error in approximating $\sin(x)=x$ with Taylor's Theorem. Furthermore, what values of $x$ will this approximation be correct to within $7$ decimal places?
Here is what I have done:
$\sin(x) = \sum\limits_{k=0}^n (-1)^k\dfr... |
H: The set of all subsequential limits of a bounded sequence is a non-empty compact set
Let $(x_n)$ be a bounded sequence and let $Y$ be the set of all subsequential limits of $(x_n)$. Prove that $Y$ is a non-empty compact set.
I think it's possible to solve this problem by proving that $Y$ is bounded (because if $Y$ ... |
H: Are these two quotient groups of $\mathbb{Z}^2$ isomorphic to each other?
I am trying to tell if two quotient groups of $\mathbb{Z}^2$ are isomorphic.
Let $H$ be the subgroup generated by $\{(1, 3),(1, 7)\}$ and $G$ the subgroup generated by $\{(2, 4),(2, 6)\}$. Are the quotient groups $\mathbb{Z}^2/H$ and $\ma... |
H: How prove this inequality $\sum\limits_{i=1}^{n}\frac{1}{x^{\alpha}_{i}+1}\ge\frac{n}{(n-1)^{\alpha}+1}$
let $n\ge 3,n\in N$, and $x_{1},x_{2},x_{3},\cdots,x_{n}$ are positive numbers,and such that
$$\sum_{i=1}^{n}\dfrac{1}{x_{i}+1}=1,$$
show that:
for any real numbers $\alpha\ge 1$,we have
$$\sum_{i=1}^{n}\dfrac{1... |
H: In a torsion module over a PID, is the annihilator of a sum of two elements the product of the annihilators?
Let $R$ be a PID, and $M$ a torsion $R$-module. Let $m,m'\in M$ such that $\mathrm{ann}(m)=(c)$ and $\mathrm{ann}(m')=(c')$ with $c$ and $c'$ coprimes. Is true that $\mathrm{ann}(m+m')=(cc')$?
If that's the ... |
H: How can I investigate the differentiability of this function?
I leanred, if all partial derivatives exist and all are continuous, then it is differentiable. Am I wrong?
I tried same way for this problem, I think it is differentiable because
all the derivative exist and are continuous.
However, it is not differentia... |
H: Is there a difference between abstract vector spaces and vector spaces?
I am following my Oxford syllabus and my next step is abstract vector spaces, in my linear algebra book I've found vector spaces.
I've searched a little and made a superficial comparison between both and found that they are the same thing. Is ... |
H: Recursion problem help
The following are the teachers example problems. The issue is that I don't understand the exact steps they took to go from $f(0)$ to $f(1)$ to $f(2)$ to $f(3)$. What I'm asking here is if someone could be so kind to show me how the answers for $f(0)\ldots f(3)$ are derived for each problem.
L... |
H: Finite Series - reciprocals of sines
Find the sum of the finite series
$$\sum _{k=1}^{k=89} \frac{1}{\sin(k^{\circ})\sin((k+1)^{\circ})}$$
This problem was asked in a test in my school.
The answer seems to be $\dfrac{\cos1^{\circ}}{\sin^21^{\circ}}$ but I do not know how. I have tried reducing it using sum to produ... |
H: How the generating function $P(s)=\mathbb E[s^X]$ uniquely determines probabilities $p_n$, $n=1,2,\ldots$
for determining the probabilities, it has been written on the book that:
$$p_n=\frac{\frac{d^n}{ds^n}P(s)|_{s=0}}{n!};\ldots(A)$$
But if i set $s=0$ then $p_n$ becomes $0$.
$$p_1=\frac{\frac{d}{ds}P(s)|_{s=0}}{... |
H: Euler's formula for triangle mesh
Can anyone explain to me these two facts which I don't get from Euler's formula for triangle meshes?
First, Euler's formula reads $V - E + F = 2(1-g)$ where $V$ is vertices number, $E$ edges number, $F$ faces number and $g$ genus (number of handles in the mesh). Now my book says
S... |
H: Subgroup with Finite Index of Multiplicative Group of Field
Let $F$ be an infinite field such that $F^*$ is a torsion group. We know that $F^*$ is an Abelian group. So every subgroup of $F^*$ is a normal subgroup.
My question:
Does $F^*$ have a proper subgroup with finite index?
AI: Let $F$ be the algebraic closure... |
H: Prove the image of separable space under continuous function is separable.
Let $f:X\to Y$ be continuous. Show that if $X$ has a countable dense subset, then $f(X)$ satisfies the same condition.
AI: HINT: This is a case in which the most obvious thing to try does work. $X$ is separable, so it has a countable dense s... |
H: $\hat{f}(x)=\frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty}{f(x)e^{-i\omega x}dx}$ and $\hat{f}(x)=\int_{-\infty}^{\infty}{f(x)e^{-2\pi ixy}}dx$
Fourier transform - difference between $$\hat{f}(x)=\frac{1}{\sqrt{2\pi}} \int^{\infty}_{-\infty}{f(x)e^{-i\omega x}dx}$$ and $$\hat{f}(x)=\int_{-\infty}^{\infty}{f(x)e^{-2\... |
H: limits of Surface area of revolution in polar co-ordinates.
My Question is Find the area of the surface generated by revolving the right-hand loop of the lemniscate $\;r^2=\cos2\theta\;$ about the vertical line through the origin (y-axis). I know the formula
$$S=2\pi \int_\alpha^\beta r \cos\theta\sqrt{r^2+\left(\... |
H: On the direct sum of rings
Let $A,B$ be rings. Suppose that
$$A\cong A\oplus B$$
Can I conclude that $B=0$, the trivial ring? If so, how can be proved?
Thanks
AI: No: take $A$ to be the ring of subsets of $\Bbb N$ with $\cap$ as product, $\triangle$ (symmetric difference) as sum, $\Bbb N$ as $1$, and $\varnothing$ ... |
H: Lax's proof of a property of principal minors
There's a section in Peter Lax's Linear Algebra (2nd edition) that I am struggling to understand. I think it involves at least one typo, so let me write it out here exactly as it is in the book. Lax has just established that if $\lambda$ is a simple root of $\chi_A(x)$,... |
H: How many parameters are required to specify a linear subspace?
A problem in Peter Lax's Linear Algebra involves looking at the family of $n\times n$ self-adjoint complex matrices and asking: on how many real parameters does the choice of such a matrix depend?
This made me ask:
On how many real parameters does a ch... |
H: Dimension of isometry group of complete connected Riemannian manifold
Given an $n$-dimensional geodesically complete connected Riemannian manifold $M$, we want to prove that the dimension of its isometry group is
$$\dim {\rm ISO}(M) \leq \frac{n(n+1)}2.$$
Does it suffice to say that, since Euclidean space $\mathbb{... |
H: Representation of an oriented manifold as a set of common zeroes of smooth functions
Let $M$ be an arbitrary oriented smooth manifold of dimension $m$. Is it always diffeomorphic to a sumbanifold in ${\mathbb R}^n$ (with some $n$) defined as a set $X$ of common zeroes of $n-m$ smooth functions $f_1,...,f_{n-m}$ (de... |
H: How to calc $ a^{2^n}$ mod $m$ in less than O(n) time?
a,m are positive integers. if needed, you can assume m is a prime.
Is there any fast algorithm?
I'm sorry for my not clear description.
AI: If $m$ is a prime, you can reduce the exponent $$ \large a^{2^n}\equiv a^{2^n \operatorname{mod} (m-1)} \pmod m$$ where... |
H: Integrating velocity field to get position
I feel silly for simply being brainstuck, but consider the following integral, physically it would be the solution of $\mathbf{p} = \tfrac{d\mathbf{v}}{dt}$ - the position of a given particle in space with respect to the time and a velocity vector field.
$$\mathbf{p}(x,y)... |
H: Check whether the given matrix is diagonalizable or not?
I am stuck on the following question:
This question carries only $1$ mark. So,without going into details how can I check whether the given matrix $M$ and $M^2=\begin{pmatrix}
1 &15 &45 \\
0&16 &45 \\
0&0 &81
\end{pmatrix}$ are diagonalizable ?
... |
H: find the least square solution for the best parabola
find the least squares solution for the best parabola going through (1,1), (2,1), (3,2), (4,2)
so to solve this problem I have 4 equation set up
a + b + c = 1
4a + 2b + c = 1
9a + 3b + c = 2
16a + 4b + c = 2
I found my $A$ to be
1 1 1
4 2 1
9 3 1
16 ... |
H: How's it possible for each element of the empty set to be even?
I was reading Pugh's Real Analysis:
I've found this in the beginning of the book:
A class is a collection of sets. The sets are members of the class. For example we could consider the class $\mathcal{E}$ of sets of even natural numbers. Is the set $\{... |
H: Riemann sum calculation
I would like to find out the sum of the following series
$$
\lim_{n\to\infty} \left[\frac{n^2}{{(n^2+1)}^{3/2}} + \frac{n^2}{{(n^2+2)}^{3/2}} + \dots + \frac{n^2}{{(n^2+(n+1)^2)}^{3/2}}\right]
$$
now it would be just upto $n$ then I could have calculate with Riemann's integral by narrowing i... |
H: Concise notation for the successor of a cyclic index
Very frequently we index "cyclic objects" using the integers. For instance, we might say that the vertices of a polygon are $x_1, x_2, \dotsc, x_n$, where the "next vertex" after $x_n$ is $x_1$. This discontinuity gets quite annoying if I make definitions that de... |
H: Straight Forecast Probability (Horse Racing)
I have probabilities on four horses winning a race. Horse A =0.52, B=0.33, C= 0.11 and D=0.04.
What I want to do is find the straight forecast probability of Horse A winning and Horse B finishing second. I know there are 2! combinations for this to happen, but I'm not s... |
H: Prime made from the digits of $\sqrt{22}$
Which is the smallest prime derived from the digits of $\sqrt{22}$, where the 4
before the comma is not considered ?
To be more precise :
$x:=\sqrt{22}-4$ , so $x = 0,690415...$
for every natural number n : a(n):=truncate(x*10^n) , a(n) gives the first n digits
of x
Which... |
H: Cantor's Diagonal Argument
Why Cantor's Diagonal Argument to prove that real number set is not countable, cannot be applied to natural numbers? Indeed, if we cancel the "0." in the proof, the list contains all natural numbers, and the argument can be applied to this set.
AI: The list contains all natural numbers, b... |
H: About $\sum_{n=1}^{\infty}\frac{(-1)^n}{n+(-1)^{n+1}}$
Determine if the following series is converges or diverges
$$\sum_{n=1}^{\infty}\frac{(-1)^n}{n+(-1)^{n+1}}$$
I suspecting that we need rewrite $ \frac{(-1)^n}{n+(-1)^{n+1}} $ somehow, but how?
Thanks!
AI: It often helps to write out a few terms to try to see... |
H: Urn, Expected Value and Covariance
How would you solve the next problem:
A Urn contains 10 balls; 7 white and 3 black. 2 balls are taken randomly (without replacement), say $X$ white and $Y$ black. $X+Y=2$. Find the Covariance of $X$ and $Y$.
AI: Because $Y = 2 - X$, we have
\begin{align*}
Cov(X, Y) & = Cov(X, 2 - ... |
H: Limit of $\frac1{n^{50}}\sum\limits_{k=1}^{n}(-1)^{k}k^{50}$ when $n\to\infty$
Evaluate $\lim_{n\to\infty}\dfrac{\sum_{k=1}^{n}(-1)^{k}k^{50}}{n^{50}}$.
Or can we get some formula when $50$ is replaced with $m$?
AI: Denote $T_m(n) = \sum\limits_{k=1}^{n} (-1)^m k^m$;$\qquad$
$S_m(n) = \sum\limits_{k=1}^{n} k^m$ $\... |
H: Uniform Convergence... What did I wrong for this function?
I want to know if this function is unif.conv. or not...
Actually it is one of the my mid-exem problems.
I got zero score for it....
What did I wrong in this problem?
and What is the easiest way to see whether it is unif.conv. or not?
AI: Evaluate $|f_n(x)-... |
H: Detail of a proof question about linear dependent vector
There is a lemma saying that if $v_1, ..., v_n$ are linearly dependent in a vector space $V$ and $v_1 \neq 0$ then there exists $j \in \{2, ..., n\}$ such that $v_j \in span(v_1,...,v_{j-1})$. The proof uses that in $a_1 v_1 + ....+a_nv_n = 0$ not all $a_i$ w... |
H: Fractional overlap of 1/2 and 1/3
Given a subset of the natural number sequence (positive integers starting from 1) we could say that $\frac12$ of the numbers in the set are divisible by 2.
e.g if the set were ${[1,2,3,4,5,6,7]}$ we could say that $3\frac12$ of the numbers in it are divisible by 2.
If we now wanted... |
H: What is a two point support in this lemma?
What is the terminology of two point support in this lemma?
AI: The meaning is that the random variable $T$ takes on the two values $r$ and $\bar{r}$ with
probabilities $p$ and $1-p$ respectively. Consequently, one of $T-r$ and $T-\bar{r}$ always
has value $0$ and so $E[(... |
H: Integration gamma and beta: $\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$
How can we evaluate the following integral? $$\int_0^4y^3\sqrt{64-y^3}\,\mathrm dy$$
I can't find anything to substitute because all of the trigonometric identities are in square form...
AI: $$\begin{align}\int_0^4 dy \, y^3 (64-y^3)^{1/2} &= \under... |
H: Continuity of max (moving the domain and the function)
Let $f:\mathbb{R}\times[0,1]\to\mathbb{R}$ continuous and $c:\mathbb{R}\to[0,1]$ continuous.
Consider
$$F:\mathbb{R}\to\mathbb{R},\ \ F(x)=\max_{t\in[0,c(x)]}f(x,t)$$
Is $F$ continuous? I believe it is true, but I've difficulties to prove it.
I managed to prove... |
H: $\frac {(x-y)}{x}$ Can it be simplified?
I appreciate anyone taking a look at this.
It's been ages since I've been in algebra/calculus and need to figure out if $\dfrac {(x-y)}{x}$ can be simplified or would it be $\left(1 - \dfrac yx\right)$?
Thank you,
Josh
Thanks to all the very speedy responses. I guess my alg... |
H: A simple proof for vector space of continuous function
Since my math skill/knowledge is limited (basic real analysis, linear algebra etc.) and I can only follow proofs based on this knowledge, I am looking for a simple proof of the following claim:
Let $C([a,b],\mathbb{R})$ denote the set of all of continuous real-... |
H: Does this system of congruences have a solution even if they are not relatively prime at first?
$$x \equiv 4\ (\textrm{mod}\ 15) \ \ \ \ \land\ \ \ \ \ x\ \equiv 6\ (\textrm{mod}\ 33)$$
Does this system of congruences have a solution even if they are not relatively prime at first?
If I try to break the congruences ... |
H: Continuous with no left derivative
I'm sure there are many examples for this, but I just can't find it. What is an example of a function continuous at point $a$, but has no left derivative at point $a$?
AI: Of course, simple absolute value functions won't work since you want the failure of a unilateral derivative.... |
H: Is there a series $\sum (a_n) $ that converges conditionally but $\sum (a_n -1/n) $ doesn't?
I'm studying for a test in calculus and have encountered a question I can't find a proof that contradicts the existence of such series.
Contradict the existence of the series such that:
$\sum(a_n) $ that converges conditio... |
H: Concavity and Convexity
A set $X \subseteq \mathbb{R}^n$ is said to be convex if $tx + (1-t)y \in X$ for all $x,y \in X$ and $t \in (0,1)$. Given a convex set $X \subseteq \mathbb{R}^n$, a function $f: X \to \mathbb R$ is said to be concave if $f(tx + (1-t)y) \ge tf(x) + (1-t)f(y)$ for all $x,y \in X$ and $t \in (... |
H: Stronger two-space formulation of Hurewicz theorem about homotopy and homology groups
The following theorem of Hurewicz holds (let $\cdot$ be the one-point space and $n\!\geq\!2$):
If $\pi_i(X)\cong\pi_i(\cdot)$ for $i\!<\!n$, then $H_i(X)\cong H_i(\cdot)$ for $i\!<\!n$ and $\pi_n(X)\cong H_n(X)$.
Does the followi... |
H: What's special about the first vector
My linear algebra notes state the following lemma: If $(v_1, ...,v_m)$ is linearly dependent in $V$ and $v_1 \neq 0$ then there exists $j \in \{2,...,m\}$ such that $v_j \in span(v_1,...,v_{j-1})$ where $(...)$ denotes an ordered list.
But if at least one $v_i$ is $\neq 0$ then... |
H: Meaning of $\int\mathop{}\!\mathrm{d}^4x$
What the following formula mean?
$$\int\mathop{}\!\mathrm{d}^4x$$
I know that this $\int f(x)\mathop{}\!\mathrm{d}x$ is the integral of the function $f$ over the $x$ variable, but the following $\int\mathop{}\!\mathrm{d}^4x$ leave empty the argument and also have the $\math... |
H: Irreducible representations over $\Bbb R$
How to prove that all irreducible representations over $\mathbb{R}$ of finite abelian group have dimension 1 or 2?
AI: For all elements $g$ in your finite abelian group $G$, you have an endomorphism on your vector space $V$ underlying your representation. Since your group i... |
H: Proof of $\lim_{x \to 0^+} x^x = 1$ without using L'Hopital's rule
How to prove that
$\lim_{x\to 0^+} x^{x} = 1$,
or
$\lim_{x\to 0^+} x\ln(x) = 0$
without using L'Hopital's rule.
AI: Another answer: when $x>0$, $x^x$ is a strictly monotonic function. So it suffices to show that $\lim_{n\to\infty}\left(\frac1n\rig... |
H: What is the next number in the sequence: $24, 30, 33 , 39 , 51,...$
What is the next number in the sequence:
$24, 30, 33 , 39 , 51,...$
Here all numbers are divisible by $3$,
difference between numbers are $6,3,6,12,...$
but I can't find common relationship in the sequence.
AI: Looks to me like each term is the pre... |
H: Existence of product in the category of pre-sheaves of abelian categories
Let $X$ be $Top(X)$ be the category of open sets of $X$ with inclusion maps as morphism. Let $\mathcal{C}$ be abelian category and $\mathcal{C}_x$ denote the category of contravariant functors from $Top(X)$ to $\mathcal{C}$. Let $\mathcal{F}$... |
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