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H: Approximating Coins Flips Problem
Approximate the probability of getting 500 heads out of a 1000 coin
flip of unbiased coins to be within 5% of its true value (without the use of a calculator).
I know that an exact probability is $$\binom{1000}{500}(.5)^{1000} = .02522...$$
I am unsure how one could simplify th... |
H: Change to standard basis
Let $A=\{a_1, a_2, a_3\}$ be a basis, of which each vector is aligned to a cartesian axis. Given a vector $v_{\langle A\rangle}$, how can I get transform it to the standard, canonical basis so it becomes $v_{\langle I\rangle}$?
I must apologize if this question is silly, I've completely fo... |
H: Some doubt on Linear Diophantine equation
We know $ax+by=c$ is solvable iff $(a,b)|c$ where $a,b,c,x,y$ are integers.
If $x=2$, $a=\dfrac{k(k+5)}{2}$, $y=k$, $b=k+3$ and $c=2k$, where $k$ is any integer, then
$$2 \frac{k(k+5)}{2} - k(k+3) = 2k.$$
So, $\left(\dfrac{k(k+5)}{2}, (k+3)\right) | 2$ for all integral val... |
H: Question about inner product and Cauchy Schwarz inequality
I'm wondering where the following equality came from: $$ \langle x , y \rangle = \|x \| \| y \| \cos \theta$$
where the thing on the LHS is the inner product and $\|\cdot\|$ is the norm induced by $\langle \cdot, \cdot \rangle$. Do we need the Cauchy Schwar... |
H: Conditions for two normal R.Vs to satisfy bivariate normal distribution
I have to Normally distributed Random Variables X and Y which are correlated. What conditions should they satisfy so that their joint distribution is a bivariate Normal distribution?
AI: A necessary and sufficient condition is that, for every $... |
H: Germs of continuous functions
Let $R$ be the ring of germs of continuous functions $\mathbb R \rightarrow \mathbb R$.
It is clear that this is a local ring with maximal ideal $\mathcal m$ consisting of those germs $f$ with $f(0)=0$.
What is not clear to me is why $(\mathcal m)^2 =\mathcal m$ should hold.
Perhaps so... |
H: A trivial problem in calculus
Try to compute $$\int\frac{dx}{x\ln x}$$
I compute it this way: first we have $x>0$.
\begin{align*}
\int\frac{dx}{x\ln x}
&=\int\frac{d(\ln x)}{\ln x}\\
&=\ln|\ln x|+C
\end{align*}
But the answer to the problem is $\ln\ln x+C$. Which one is right? Thanks!
Source Григорий Михайлович Фих... |
H: Direct limit and exact sequences of abelian groups
Suppose having a set of direct systems of abelian groups
$\ldots\{G_{\alpha}\}_{\alpha\in A}$, $\{G_{\beta}\}_{\beta\in B}$, $\{G_{\gamma}\}_{\gamma\in \Gamma}\ldots$
If there is a (long) exact sequence for certain indexes:
$$\cdots\longrightarrow G_\alpha\longrigh... |
H: Prove that a set defined by concave functions on $R^n$ is convex
I've been trying to prove this statement the whole weekend...
Let $g_1,\dots,g_m$ be concave functions on $\mathbb{R}^n$. Prove that the set $S=\{x:g_i(x)\geq{0},\ i=1,\dots,m\}$ is convex.
AI: Take $x,y\in S$ and $a\in [0,1]$ and $i\in\{1,\dots,m\}$... |
H: How to prove $f(x,y)=\sqrt {xy}$ is concave?
How can I prove (preferably elegantly) that $f(x,y)=\sqrt {xy}$ where $x≥0$ and $y≥0$ is concave in both $x$ and $y$?
AI: Take $ (x_0, y_0)$ and $ (x_1,y_1)$ and then you need to show
$ f( t( x_0 , y_0) + (1-t) (x_1 , y_1) ) \geq t f(( x_0 , y_0)) +(1-t) f((x_1 , y_1) )... |
H: Complex Numbers as a group under multiplication
While revising I came across this question:
Let $G$ be the group of complex numbers $\{1, i, -1, -i\}$ under multiplication.
Is it true that for every such homomorphism, there is an integer $k$ such that the homomorphism has the form $z\mapsto z^k$ ?
The answer is tru... |
H: Percentage increase in ratios..
The question is:
Seats for Math,Physics and biology in a school are in ratio 5:7:8. There is a proposal to increase these seats by 40%,50% and 75% respectively.What will be ratio of increased seats?
Apparently I am currently increasing 5 by 40% and 7 by 50% and 8 by 75% . But thi... |
H: How do I solve this pde (using finite differences)?
How do I solve the pde:
$\ -s_x(x,t) -p(x,t)s_t(x,t)=p(x,t)$
for s(x,t) when p(x,t)=2x,
subject to the condition s(0,t)=0?
Generally p(x,t) may not be analytical so I would like to use finite differences to transform this into a linear algebra problem:
$\ (-D_x-pD... |
H: Proof that convex open sets in $\mathbb{R}^n$ are homeomorphic?
This is an exercise from Kelley's book. Could someone help to show me a proof?
It seems very natural, and it is easy to prove by utilizing the arctan function in $\mathbb{R}^1$.
Thanks a lot.
AI: 3rd edit: I have now typed things up in a slightly mor... |
H: How to prove with Galois theory that a cube root is not geometrically constructible?
I could not find any proof on the Internet. I am looking for a formal proof with an explanation for the uninitiated (my knowledge of Galois theory is very basic).
With geometrically constructible I mean with compass and straightedg... |
H: Restoring a point after transformation
I am given a point $
\begin{bmatrix}
u & v
\end{bmatrix}^T
$ which I know is in form $\begin{bmatrix}
\frac{x}{f(r)} & \frac{y}{f(r)}
\end{bmatrix}^T$ where $f(r)$ is polynomial function, $f(r)=a_nr^n + \cdots + a_1r + a_0$ and $r=\sqrt{x^2+y^2}$.
I want to restore $x$ and $y$... |
H: Set Operations with Empty sets
I have a question regarding performing set operations on empty sets.
For example let A = ∅, Let B = {A, ∅}, Let C = {A, B}.
Would B = {∅, ∅} and C = {∅, {∅,∅}}? or would B = {∅} and C = {∅, {∅}} since {∅, ∅} reduces to {∅}?
Now if I wanted to do $A \cup C$ would the answer be $∅ \cup ... |
H: maximum possible value of $d$
We are given three integers $a , b,$ and $c$ such that $a , b, c,
a + b − c, a + c − b, b + c − a ,$ and $a + b + c$ are seven distinct primes. Let
$d$ be the difference between the largest and smallest of these seven primes.
Suppose that $800$ is an element in the set {... |
H: $X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$
$X = \log_{12} 18$ and $Y= \log_{24} 54$. Find $XY + 5(X - Y)$
I changed the bases to 10, then performed manual addition/multiplication but it didn't yield me any result except for long terms. Please show me the way.
All I'm getting is $$\frac{\lg 18\lg... |
H: While proving that every vector space has a basis, why are only finite linear combinations used in the proof?
Statement: Every vector space has a basis
Standard Proof:It is observed that a maximal linearly independent set is a basis.
Let $\mathscr{Y}$ be a chain of linearly independent subsets of a vector space ... |
H: Convergence of the sequence $\sqrt{1+2\sqrt{1}},\sqrt{1+2\sqrt{1+3\sqrt{1}}},\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1}}}},\cdots$
I recently came across this problem
Q1 Show that $\lim\limits_{n \rightarrow \infty} \underbrace{{\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+\cdots+n\sqrt{1}}}}}}}_{n \textrm{ times }} = 3$
After t... |
H: What do you call a generalised Fourier-like transform?
The Fourier series is a decomposition of an arbitrary function into a superposition of sinusoidal functions. Some time back, I asked a question about whether it is possible to decompose functions using other families of functions and indeed, it is.
Today's ques... |
H: How can " $\small {n \over \varphi(n) } \text{ is integer only if } n=2^r \cdot 3^s $ " simply be shown?
I've tried to answer the question concerning $\small { 3^n-2^n\over n } $ and wanted to show this simply by referring to the property that
$$ 3^{\varphi (n)}-2^{\varphi (n)}\equiv 0 \pmod n $$
and
$$ 3^{... |
H: Combinatorics thinking
I saw this question in a book I've been reading: in a group of four mathematicians and five physicians, how many groups of four people can be created if at least two people are mathematicians?
The solution is obtained by ${4 \choose 2}{5 \choose 2} + {4 \choose 3}{5 \choose 1} + {4 \choose 4}... |
H: proving there exists an integer $ k$ such that $ak^2+bk+c\equiv 0 \pmod{2^n}$ by induction
$n$ is a given positive integer.
how to prove by using mathematical induction that there exists an integer $ k$ such that $ak^2+bk+c\equiv 0 \pmod{2^n}$, where $b$ is odd and at least one of $a, c$ is even
AI: First check di... |
H: Computing large powers $\!\bmod n$ when $n$ has multiple prime factors.
How will the congruence modulo works for large exponents? What theorem/s may be used? For example to show that $7^{82}$ is congruent to $9 \pmod {40}$.
AI: Besides the methods mentioned by Andre, it's worth mentioning another more powerful tech... |
H: Ratios and Mixture
I can't seem to to find a way to solve this problem:
In a mixture $60$ litres, the ratio of alcohol and water is $2:1$. If this ratio is to be $1:2$ then the quantity of water further added is? Ans $60$ litres
Any suggestions?
AI: In the original 60 liter mixture, the ratio of alcohol to water... |
H: Something weaker than the Riesz basis
I have some function $f$, real valued and continuous.
I formed functions $\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ such that that $\mathrm{span}\{f_{m,k}, k \in \mathbb{Z}, m>0\}$ is dense in $L_p(R)$.
Now I would like to constract a Riesz basis from $\{f_{m,k}, k \in \mathbb{Z}, m>... |
H: Calculating the norm of an element in a field extension.
Given a number field $\mathbb{Q}[\beta]$, where the minimal polynomial of $\beta$ in $\mathbb[Z][x]$ has degree $n$, I would like to calculate the norm of the general element $$a_0+a_1\beta+\cdots+a_{n-1}\beta^{n-1}.$$
In particular, here is my attempt when $... |
H: How to prove $I + t X$ is invertible for small enough $ | t | ?$
Let $X \in \text{GL}_n(\mathbb{R})$ be an arbitrary real $n\times n$ matrix. How can we prove rigorously:
$$ \underset{b>0} {\exists} : \underset{|t|\le b} {\forall} : \det (I + t X) \neq 0 $$
If necessary, we could also assume that $t \ge 0.$
AI: No... |
H: Questions concerning a proof that $\mathcal{D}$ is dense in $\mathcal{S}$.
I am currently working through
this lecture notes
and on page 164, there it is said
The space of $\mathcal{D}(\mathbb{R}^n)$ of smooth complex-valued functions with compact support is contained in the Schwartz space $\mathcal{S}(\mathbb{R}... |
H: Covering $R^2$ by open half planes
Let $\{s_\iota, \iota\in I\}$ be any collection of vectors in $\mathbb{R}^2$ and let $U_\iota=\{\pi\in \mathbb{R}^2;\pi\cdot s_\iota >0\}$. Then, $U_\iota$ is either empty or an open half plane in $\mathbb{R}^2$ for any $\iota\in I$. Is it true, that we can find countably many $\i... |
H: Needs alternative algorithm for a graph problem
There is a network of connected electric bulbs each with a unique id. Each bulb has a switch. When the switch is clicked, the colour of the bulb changes and also the colours of connected bulbs change. The order in which the colour-change occurs is green-yellow-red-gre... |
H: Is it possible to combine two integers in such a way that you can always take them apart later?
Given two integers $n$ and $m$ (assuming for both $0 < n < 1000000$) is there some function $f$ so that if I know $f(n, m) = x$ I can determine $n$ and $m$, given $x$? Order is important, so $f(n, m) \not= f(m,n)$ (unles... |
H: Under what conditions does $(\frac{3}{p})(\frac{-1}{p})=1?$ Two ways, different results.
I have quite a problem, two methods, different results. something's wrong.
I'm trying to find under what conditions the Legendre symbol for $(\frac{3}{p})(\frac{-1}{p})=1$.
First Way: $(\frac{3}{p})(\frac{-1}{p})=(\frac{3}{p})\... |
H: Area of polar coordinate $r = e^{- \theta/4}$
$$r = e^{- \theta/4}$$
$$\pi /2 \leq \theta \leq \pi$$
I know the formula is
$$\int_a^b \frac{1}{2} r^2 d\theta$$
$$\int_{\pi/2}^{\pi} \frac{1}{2} (e^{- \theta/4})^2 d\theta$$
From here I cannot figure out an easy way to integrate this.
AI: Note that $(e^{-\theta/4})^2... |
H: Probability of the events
For the events of $A$ and $B$, probabilites are $\Bbb P(A) = 3/11$ and $\Bbb P(B) = 4/11$. Define the $\Bbb P(A \cap B )$ if:
a) $\Bbb P(A \cup B) = 6/11$.
b) events are indenpendent
I have done a task, and it's following
$$\Bbb P(A \cup B) = \Bbb P(A) + \Bbb P(B) - \Bbb P(A \cap B ) = 1/1... |
H: If $\int_0^1 f(x)x^n \ dx=0$ for every $n$, then $f=0$.
Possible Duplicates:
Nonzero $f \in C([0, 1])$ for which $\int_0^1 f(x)x^n dx = 0$ for all $n$
Slight generalization of an exercise in (blue) Rudin
What can we say about $f$ if $\int_0^1 f(x)p(x)dx=0$ for all polynomials $p$?
I found a nice problem I would l... |
H: Finding an imprimitive group on $12$ letters
By definition a permutation group $G$ acting on a set $\Omega$ is called primitive if $G$ acts transitively on $\Omega$ and $G$ preserves no nontrivial blocks of $\Omega$. Otherwise, if the group does preserve a nontrivial block then $G$ is called imprimitive.
Here I am... |
H: Probability game of independent events
In one game, probability that a gamer scores win is 1/4. If X is number of tries( games ) that gamer has done in order to achieve score:
(NOTE!: every try is independent from the other)
a) Define the probability that gamer scored a win in 3 tries.
b) Define the probability tha... |
H: Spectrum of a real number
Whilst reading Concrete Mathematics, the authors mention something which they refer to as the "spectrum" of a real number (pg. 77):
We define the spectrum of a real number $\alpha$ to be an infinite multiset of integers, $$\operatorname{Spec}{(\alpha)}=\{\left\lfloor\alpha\right\rfloor, \... |
H: Conditional probability
A girl goes to school by bus every day. If it doesn't rain, probability that she will be late for bus is 1/5. If it rains probability that she will be late is 2/3. Probability that it is raining is 1/4.
Girl forgot to pick up a bus. Define the probability that it was raining.
In this task I... |
H: Computing determinant of a specific matrix.
How to calculate the determinant of
$$ A=(a_{i,j})_{n \times n}=\left(
\begin{array}{ccccc}
a&b&b& \cdots & b\\
b& a& b& \cdots& b\\
\vdots& \vdots& \vdots& \ddots&\cdots\\
b&b&b & \cdots&a
\end{array}
\right)? $$
AI: We note that the sum of the elements of each column is... |
H: Unconditional and independent probability
Two archers, independently, target a mark firing one arrow at a time. Probability that one archer will hit a mark is 0.8 and another is 0.4. After contest it is determined that one hit got in a mark. Find a probability that first archer scored it.
I have no idea from where... |
H: Help in self-studying mathematics.
Is this a reasonable list for who seek to learn mathematics by "self learning" program?
and is it a well sorted list to follow?
http://www.math.niu.edu/~rusin/known-math/index/index.html
AI: Our very own Pete Clark, now of the University of Georgia faculty, was once upon a time a ... |
H: A good Open Source book on Analytic Geometry?
Hi my course specifically talks about :
Cartesian and Polar Coordinates in 3 Dim, second Degree eqns in 3 vars, reduction to canonical forms, straight lines, shortest distance between 2 skew lines, Plane, sphere, cone, cylinder, paraboloid, ellipsoid,hyperboloid of one ... |
H: Closed linear span of a frame in a Hilbert space $\mathcal{H}$ coincide with $\mathcal{H}$
Definition of the problem
Let $\mathcal{H}$ be a separable Hilbert space and $J\subset\mathbb{N}$
an index set. Let $\Phi:=\left(\varphi_{j}\right)_{j\in J}\subset\mathcal{H}$
be a frame for $\mathcal{H}$, i.e. $\exists A,B>0... |
H: What should I call this commutative monoid of order three?
I'm looking for a name for the monoid given by the following table:
$$
\begin{array}{c|ccc}&1&a&b\\ \hline
1&1&a&b\\
a&a&1&b\\
b&b&b&b
\end{array}
$$
Is there a name that would be understandable to an undergraduate student who hasn't read anything about sem... |
H: some references for regularity theory
I am gathering material that contains results of regularity for solutions of the equation
\begin{equation}
\mbox{div}(A(x) \nabla u) = f
\end{equation}
where the coefficints $a_{i,j}$ of $A$ are only measurable and $f \in L^p$ for some $p \in \mathbb{N}$ and
\begin{equation}
\... |
H: Area computed using polar coordinates
I have $r = \sqrt{\theta}$
http://www.wolframalpha.com/input/?i=cartesian+r+%3D+%5Csqrt%7Btheta%7D+
The graph given in the book ends at the first time it approaches to right side of the x axis (or 2pi). I attempted to set up an integral that cut each section of the graph so I h... |
H: How to reciprocal this imaginary exponent?
Assume $x\gt 0$, how does one simplify $$e^{(-x^2t)/i}\ ?$$
I don't understand how we could change the i under to the top so I could use Euler's formula
AI: Remember that $-i^2 = 1$ implies that $-i = \frac{1}{i}.$ Then apply Euler's formula as you mention. |
H: What will be the rate of inflation after n months?
Let's say I have a utility rate of 0.15 which inflates by 0.5% per year. I then want to convert that annual rate to a monthly rate and determine what the rate would be for month $n$.
The way I'm doing it now (which just feels incorrect) is:
$$\mathrm{utilityRate}\B... |
H: Prove diagonalizability of a continuous transform
We've been going over diagonalization in my linear algebra class, but we've only been dealing with matrices—nothing too complicated. All of a sudden this problem came along and blindsided me:
Let V be the vector space of continuous functions with basis {$e^t, e^{-t... |
H: lebesgue measure of $\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, 0 \leq y \leq 10, z \in \mathbb Z \} $
Find the lebesgue measure of the set:
$$ \Bigl\{ (x,y,z) \in \mathbb R ^3 : x \in \mathbb R, \quad 0 \leq y \leq 10, \quad z \in \mathbb Z \Bigr\} $$
I think is a null set but for some reason I have stuck ... |
H: Corollary of Cauchy's theorem about finite groups.
Cauchy's Theorem: Let $G$ be a finite group and let $p$ divide the order of $G$. Then $G$ has an element of order $p$ and consequently a subgroup of order $p$ (of course the cyclic subgroup generated by the aforementioned element of order $p$).
Corollary: Let $G... |
H: A question regarding sphere packing
The question of how many smaller spheres can be fit into a larger sphere is fascinating and has been examined extensively. I was curious, though, about the scenario of packing spheres of different radii into a larger sphere. For example: What is the maximum number of rigid sphere... |
H: Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$
Finding a splitting field of $x^3 + x +1$ over $\mathbb{Z}_2$.
Ok so originally I messed around with $x^3 + x +1$ for a bit looking for an easy way to factor it and eventually decided that the factors are probably made up of really messy nested roots. S... |
H: Is vector space a field? Or more than that?
As you all know, vector space is closed under scalar multiplication, scalar product, vector product and addition. If I take scalar product, vector space is a field, but if i take vector product, vector space is a group. Is there any term designating this kind of set? Plu... |
H: Simplifying to a desired expression structure
My book has this expression:
\begin{align}
((n(n+1)(2n+7))/6)+(n+1)(n+3)
\end{align}
And then the book simplified it, and ended up with the desired expression:
\begin{align}
((n+1)(n+2)(2n+9))/6
\end{align}
I tried to do such simplification. But I ended up with this:
\b... |
H: Is there a closed form to this expression?
Consider
$$2^{n-1} + 2^{n-2}\dfrac{(n - 1)!}{1!(n - 2)!} + 2^{n-3}\dfrac{(n - 2)!}{2!(n - 2 \cdot 2)!} +
2^{n-4}\dfrac{(n - 3)!}{3!(n - 2 \cdot 3)!} + 2^{n-5} \dfrac{(n - 4)!}{4!(n - 2 \cdot 4)!} + \ldots $$
Until $(n - 2 \cdot k)$ equals to either $0$ or $1$ (even/odd). ... |
H: Prove or provide a counterexample: Finite simple group with $|G|\geq 6$ cannot have an index-2 or index-3 subgroup.
Let $G$ be a finite simple group with $|G|\geq 6$. For each of the following statements, prove or provide a counterexample.
$G$ cannot have an index-$2$ subgroup.
$G$ cannot have an index-$3$ subgr... |
H: A Question about limit of function on real
First of all, assuming ONLY the knowledge of sequential characterization of limit and also the epsilon-delta formulation of limit, why is the following limit undefined?
$$\lim_{x\to 0} \sqrt{x}$$
This question is from Schroder's "Mathematical Analysis". My argument is foll... |
H: Equivalence of continuity of a functional on a locally convex space
Let $X$ be a locally convex space whose topology is defined by a family of seminorms $\mathcal{P}$. Let $f$ be a linear functional on $X$. Then, I am trying to prove that the following statements are equivalent.
$f$ is continuous.
There are semin... |
H: Prove this argument is valid: (~N v (~B*D), ~C --> ~D therefore ~(~C*N))
Prove the following argument is valid (and provide reasons):
~N v (~B*D)
~C --> ~D therefore ~(~C*N)
Our work (so far):
~N v (~B*D)
~C --> ~D therefore ~(~C*N)
D-->C (contrapositive of 2)
~N v (~B*C) (substitution 3 into 1)
~N v ~(B v ~C) (... |
H: if $f(n)$ is multiplicative prove that $f(n)/n$ is also multiplicative.
The question asks that if $f(n)$ is multiplicative to prove that $f(n)/n\qquad$ is also multiplicative.
This is what I have:
So, $f(n)\quad$ is multiplicative means that if $p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}\qquad$ is the prime-power decomposi... |
H: Messenger Riddle
A column of troops one km long is moving along a straight road at a
uniform pace. A messenger is sent from the head of the column,
delivers a message at the rear of the column and returns. He also
moves at a uniform pace and arrives back at the head of the column
when it has just covered i... |
H: What is the geometric meaning of singular matrix
Could anyone help explain what is the geometric meaning of singular matrix ? What's the difference between singular and non-singular matrix ? I know the definition, but couldn't understand it very well.
Thanks.
AI: A matrix can be thought of as a linear function from... |
H: If $\phi^{-1}(0)$ in $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a complete intersection, then is $\phi$ flat?
This is a simple question so I am hoping the answer is quite simple as well.
Suppose $\phi:X\rightarrow Spec\; \mathbb{C}[t]$ is a map such that the algebraic variety $\phi^{-1}(0)$ is a complete intersec... |
H: sequence of open intervals
Let $ \displaystyle{ \{ (a_n , b_n) : n \in \mathbb N \} }$ a sequence of open intervals on $\mathbb R$ such that $ \displaystyle{[0,15] \subset \bigcup_{n=1}^{n_0} (a_n ,b_n) }$ for some $ n_0 \in \mathbb N$
Prove that $ \displaystyle{ \sum_{n=1}^{n_0} \mu(a_n ,b_n) = \sum_{n=1}^{n_0} (b... |
H: Likelihood of continuing a discrete series
If I have a series like "aaabbbabaabaabbaa" and I'd like to know whether next element in series is "a" or "b", would a Poisson distribution be best solution for this?
If so, checking Poisson for "3" wouldn't be right since it can be >3 in the future. Would the the proper ... |
H: smooth maps between smooth manifolds - Jacobian coordinate independence
I have question about comment in Lee's Introduction to Smooth Manifolds - page 51.
Given smooth map $F:M\to N$ between smooth manifolds $M$ and $N$ we say that the total derivative of $F$ at $p\in M$ (given chart $(U,\varphi)$ around $p$ and $... |
H: Ball from platform with specific vertex
On earth in a vacuum. You throw a platon from a platform height
$h$ and want it to land at point $d$ distant. Note, h is absolutely fixed and d is absolutely fixed. It "must land" at point d, no matter what. You throw it with velocity expressed using $Vx$ and $Vy$.
Now,
(Pro... |
H: Behaviour of $\sum_{k=1}^n\left(\left(\frac{3}{2}\right)^k\ (\mathrm{mod}\ 1)\right)$
Using Mathematica I found that the relation
$$\sum_{k=1}^n\left(\left(\frac{3}{2}\right)^k\ (\mathrm{mod}\ 1)\right)\approx\frac{n}{2}$$
seems to hold. Actually, every fraction of the form $\frac{b}{a}$, with $b>a$ and $\mathrm{gc... |
H: All integer solutions for $x^4-y^4=15$
I'm trying to find all the integer solutions for $x^4-y^4=15$.
I know that the options are $x^2-y^2=5, x^2+y^2=3$, or $x^2-y^2=1, x^2+y^2=15$, or $x^2-y^2=15, x^2+y^2=1$, and the last one $x^2-y^2=3, x^2+y^2=5$.
Only the last one is valid. $x^2+y^2=15$ is not solvable since t... |
H: Changing domain of PDE (should be easy)
Suppose I have a PDE
$$u_t(x,t) + f(x,t)\cdot \nabla u(x,t) = 0 \quad \text{on $\gamma(t)$}$$
where $\gamma(t)$ is a curve for each fixed t in $\mathbb{R}^2$ and $f$ is given.
We have that $\gamma(t)$ is parametrised by the function $X(s,t)$ for each $t$. So $X(s,t) = x \in \... |
H: Finding last digit of numbers raised to large powers
This question came in a competitive exam I took recently.
The last digit of LCM of $3^{2003} - 1$ and $3^{2003} + 1$ is
is there any strategy by which we can quickly determine the answer? I am hoping there is, since the question came in a timed competitive test... |
H: What is the minimum $ \sigma$-algebra that contains open intervals with rational endpoints
What are the minimum $\sigma$-ring and $\sigma$-algebra on $\mathbb R$ which contain the open intervals with rational endpoints?
Is there a relation between this $\sigma$-algebra and Borels?
AI: Let $\mathcal C:=\{(a,b),a,b\i... |
H: proving that $2^{m-1}$ has reminder $1$ when divided by $m$
Let $$m = \frac{4^p - 1}{3}$$
Where $p$ is a prime number exceeding $3$. how to prove that $2^{m-1}$ has reminder $1$ when divided by $m$
AI: $2^{2p}=4^p=3m+1\equiv 1 \pmod m$ so the result follows if $2p\mid m-1$.
Since $m$ is odd $2\mid m-1$, and by Fe... |
H: A question regarding analysis.
Let $d\in\mathbb N$ and
$$
P_d := \left\{p : K \to \mathbb{R} :\quad p(x) = \sum\limits_{i=1}^d p_i x^i\quad\text{ where }\quad \{p_i\}_{i=1}^d \subset \mathbb{R}\right\},
$$
the set of all polynomials of degree at most $d$, where $K$ is a compact set. We have to show that $P_d$ isn'... |
H: Determine whether $\sum\limits_{i=1}^\infty \frac{(-3)^{n-1}}{4^n}$ is convergent or divergent. If convergent, find the sum.
$$\sum\limits_{i=1}^\infty \frac{(-3)^{n-1}}{4^n}$$
It's geometric, since the common ratio $r$ appears to be $\frac{-3}{4}$, but this is where I get stuck. I think I need to do this: let $f(x... |
H: Fraction of two binomial coefficients
In an exercise I was asked to simplify a term containing the following fraction: $${\binom{m}{k}\over\binom{n}{k}}$$
The solution does assume the following is true in the first step, without explaining why. I unfortunately cannot reconstruct the step:
$${\binom{m}{k}\over\binom... |
H: infinite number of irreducible polynomials in $\mathbb{Z}/2{\mathbb Z}[X]$
For $A= \mathbb{Z}/2{\mathbb Z}[X]$ ring of polynomials with coefficient in the field $\mathbb{Z}/2{\mathbb Z},$ I need to show that there are infinite number of irreducible polynomials in $A.$
How do I show that? I didn't come to any conclu... |
H: Probability question (two draws)
If I draw 2 balls from a bag which contains 2 pink, 3 blue and 4 orange balls, what's the probability that the first ball would be pink and the second blue - as a decimal? Thanks. :)
AI: Hint: what is the chance that you get a pink ball on the first draw? Given that one pink ball ... |
H: Function derivative in the vector direction
Let there be a function $z=x^3+xy+y^2$. Is there a vector $\vec U$ that in the point $(1,1)$ the function derivative in the vector direction is equal to 6?
Now I know that the $|\nabla F(1,1)|$ is $\sqrt{4^2+5^2} = \sqrt{41} = 6.4 > 6$.
So there is a vector $\vec U$ that ... |
H: Uniqueness of extension of zero measure
Let $(\Omega,\mathscr F)$ be a measurable space with two probability measures $\mu, \nu: \mathscr F\to[0,1]$ defined over it. Suppose that $\mathscr C\subset\mathscr F$ is some class of sets and
$$
(\mu-\nu)|_\mathscr C = 0.
$$
Which necessary and sufficient conditions are... |
H: Equivalences of $S^n$ vs. $\Omega^nS^n$
Let $H(n)$ be the group of self-homotopy-equivalences of $S^n$ preserving the basepoint. I read that $H(n)$ may be identified with ''two components of $\Omega^nS^n$''.
What does this mean and how can I see it?
AI: Maps $[S^n,S^n]$ are classified by degree. For each $z\in\mat... |
H: $\limsup$ and cluster points
Let $(x_n)$ be a sequence in $\mathbb R$. We call $y$ a cluster point of $(x_n)$ iff for every neighborhood $N$ of $y$ there are infinitely many $n$ such that $x_n \in N$.
Let $C$ denote the set of cluster points of $(x_n)$.
By definition, $\limsup_{n \to \infty} x_n = \inf_n \sup_{k \... |
H: Normal distribution of the dosage
I am having an issue with the following task:
For the dose of the MemPro medicine it is known that has normal distribution. Sample of 30 parameters has been taken into consideration and based on the obtained data, we know that mean of a dose of this medicine is 5grams. With possibi... |
H: Confidence interval of the car sample
In the sample of 25 cars of the VW Golf 5 recently sold in Augsburg, average cost was 9.000 euros with standard devation of 300 euros. Assume that the cost of all the cars of this type of vehicle are normally distributed. Construct 90% of confidence interval for the average cos... |
H: Question on meaning of a symbol: long thin C
I don't know how to write it in $\LaTeX.$ It is a tall skinny bold C. This is the context: A set is defined by:
where $\complement\atop{\smash \scriptstyle i}$ is the thing I don't understand. The $i$ is actually directly underneath the weird $C$ in this case. Can anyon... |
H: Seeking formula for a custom game mechanic
We are game developers currently working on a new game. We are looking to solve the following problem:
We start with the following variables:
$$x = 0, h = 1;$$
The variable x should always increase over time depending on $h$—the larger the $h$ the larger the rate that $x$ ... |
H: A question concerning on the axiom of choice and Cauchy functional equation
The Cauchy functional equation:
$$f(x+y)=f(x)+f(y)$$
has solutions called 'additive functions'. If no conditions are imposed to $f$, there are infinitely many functions that satisfy the equation, called 'Hamel' functions. This is considered... |
H: Finding the diagonal of a rhombus
I am trying to show that the diagonals of a RHOMBUS intersect each other at 90 degrees
However I need to find the length of the diagonal without using Trig. ratios. Any ideas how i could find that ?
The length of each side in the figure is 6.
AI: We give a traditional "ang... |
H: Probability that TV is defective
It is known that every 100th TV produced is defective. If we choose sample of 10 TVs from the tracks, find:
a) probability that one TV is defective.
b) probability that at most two TVs are defective.
I don't know where to start from. Should I use Poisson distribution here or not?
p... |
H: Studying this function: $y = \frac{x^2}{1+\log|x|}$
I'm still studying this function:
$$y = \frac{x^2}{1+\log|x|}$$
And now I'm dealing with the study of the monotony. So I got the first derivative, this:
$$y\,' = \frac{x(1+\log{x^2})}{(1+\log|x|)^2}$$
Then I put the first derivative $> 0$:
$$\frac{x(1+\log{x^2})}{... |
H: Is this mathematical definition iterative? If not, what does an iterative function look like?
I was debating with someone about iterative vs recursive in programming. I was defending the iterative side. He then said me that the true definition of Fibonacci number is this:
$$f(n) = f(n-1) + f(n-2);\space n > 2$$
wit... |
H: Give an example of a measure which is not complete
Give an example of a measure which is not complete ?
A measure is complete if its domain contains the null sets.
AI: The canonical example is the Borel measure on the Borel $\sigma$ algebra (the $\sigma$ algebra generated by the open intervals) on $\mathbb{R}.$ Th... |
H: Compact subsets of the real numbers
Let $C\subset \mathbb{R}$ be compact. I am wondering if
$$C=\bigcup_{i=1}^n[a_i,b_i]$$
then for some $a_i,b_i\in\mathbb{R}$, $a_1\le b_1 < a_2 \le b_2 \dots < a_n \le b_n$. By Heine-Borel, $C$ does indeed lie in some interval $[a,b]$, but is it the finite disjoint union of such... |
H: Some kind of relation between classical heat equation and Laplace .
If we have $k(x,t)= \frac {1}{(4t)^{\frac{n}{2}}} \exp\left(\frac{-|x|^2}{4t}\right)$ is the fundamental solution of heat equation. If we consider $n \ge 3 $, I would like to show that $\int_0^\infty k(x,t) dt$ is the fundamental solution of lLapl... |
H: Is a matrix with characteristic polynomial $t^2 +1$ invertible?
Given that $A$ is a square matrix with characteristic polynomial $t^2+1$, is $A$ invertible?
I'm not sure, but this question seems to depend on whether $A$ is over $\mathbb{R}$ or over $\mathbb{C}$. My reasoning is that if $A$ is over $\mathbb{C}$ the... |
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