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a non-mathematician wants me to solve a PDE. The problem is that I don't know a lot of theory to solve PDE's except the fouriertransform. This is the PDE
$$A \partial_t w + B \partial_t\partial_x^4 w + C \partial_x^4 w + \partial_t^2 w = 0$$
for $(t, x) \in (0, \infty) \times (0, L)$. Moreover he told me that he has $$... |
I am trying to prove this statement is true for $k ≥ 4$. I don't know how to work with $k + 1$ factorial, so I'm a little lost on proving this.
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I don't want the solution at all, but I'm incredibly stuck, and I really need some (hopefully not much) help.
What I've considered:
Liouville's Theorem
Not applicable because f is not bounded.
Cauchy-Riemann Equations
Seems unlikely. We did not derive these for polar z in class, and the derivation is complex (for o... |
Let $f(x,y) = \frac{xy^\alpha}{x+y},\alpha\in(0,\infty)$. How to compute $$\sup_{(x,y)\in[a,b]\times [0,c]}\frac{xy^\alpha}{x+y},$$
with $b>a>0$?
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Prove that
$$16\cos^5A-20\cos^3A+5\cos A=\cos5A$$
My solution begins here;
$$
\begin{align}
\text{RHS} & =\cos5A \\
& =\cos(A+4A) \\
& =\cos A\cos4A-\sin A\sin4A \\
& =\cos A(2\cos^2 2A-1)-\sin A(2\sin2A\cos2A) \\
& =2\cos A\cos^2 2A-\cos A-2\sin A\sin2A\cos2A \\
& =2\cos A\cos^2 2A-\cos A-2\sin A(2\sin A\cos A)\c... |
So I am asked to find the positive, maximally extended solutions to this ODE.
$$u'(x) = \frac{x}{u(x)}$$
Now a solution is given by
$$u(x) = (\int_{y_0}^y t dt )^{-1}\circ \int_{x_0}^x s ds = G^{-1} \circ F$$
Now, I have to be carful here and consider a few things.
so first off,
$$u(x) = \sqrt{x^2-(x_0^2-y_0^2)}$$
and ... |
I've recently noticed a method of describing primes. As an example: $13=5*11-2*3*7$. This pattern must follow these rules: $x-y$ such that $x*y$ is the product of all previous primes (allowing powers of those primes as well) where $x$ and $y$ are relatively prime positive integers, and $1<x-y<z^2$ where $z$ is the larg... |
Say we have a disease and a test for it.
P(A :=a person has the disease)= 0.01 ( example)
P( B:=test is positive | A )=0.95
Is this enough information to calculate the probability that a person has the disease given that his test came back positive?
I've seen quite a lot of solved exercises taking Baye's theorem and as... |
I am a student of political science and have basic knowledge of mathematics. But I need to learn Mathematical Modeling if I want to continue a research of mine. I'm willing to spend even 1 or 2 years behind it. But I cannot understand where to start. Could anyone help me in this purpose?
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I am asked to find the area between ${y = 7}$ and ${x^2 -5x + 13}$
Combining these equations together I get ${-x^2 - 5x + 6 = 0}$.
Factorising into ${(x - 3)(x - 2)}$
I am taking ${y = 7}$ to be the top equation so I then find the integral by subtracting the first equation from the second:
$${\int_{2}^3 (7 - (x^2 -5x +... |
A while back I found the series $$\sum_{k=0}^n \binom n k (-1)^k (x+k)^n = (-1)^n n!$$ while messing around in Algebra class (specifically when $n$ is any natural number and $x$ is any real number) I wrote the formula on a card, stuck it in my "to prove" box. The formula arises when subtracting subsequent powers of na... |
Basic algebra problem I can't seem to figure out: $$ \frac{\frac{1}{x}+\frac{1}{y}}{\frac{1}{x^2}-\frac{1}{y^2}} $$ $x,y \in \mathbb{R}, x^2 \neq y^2, xy\neq0$.
Now I know the result is: $\frac{xy}{y-x}$, but I am not sure how to get it, I get into a mess like this: $=x+\frac{x^2}{y}-\frac{y^2}{x}-y=\frac{x(xy)+x^3-y^3... |
Can i get help solving the differential equation
$$y' = \frac{y^2+y }{x^2 -x}$$
I tried searching but could not find anything similar.
Thank you!
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We know that $$e^{\pi i}+1=0$$and $$e^{\pi i}=-1$$
So$$(e^{\pi i})^2=(-1)^2$$$$e^{2\pi i}=1$$
Because $1$ is the multiplicative identity,$$(e^{2\pi i})^x=1^x$$$$e^{2\pi xi} =1$$should also hold true.
But we also know that $$e^{xi}=\cos(x)+i\sin(x)$$and so$$e^{2\pi xi}=\cos(2\pi x)+i\sin(2\pi x)$$which does not equal 1 ... |
Let $V$ be a vector space of dimension $n$ over a finite field $\mathbb{F}$, and let $G$ be a subgroup of the finite group $\operatorname{GL}(V)$. Then $G$ acts on the graded algebra $\mathbb{F}(V)$ consisting of polynomial functions on $V$. Then the subalgebra of invariants by $\mathbb{F}[V]^G$ is known to be a finite... |
In some areas of mathematics it is everyday practice to prove the existence of things by entirely non-constructive arguments that say nothing about the object in question other than it exists, e.g. the celebrated probabilistic method and many things found in this thread: What are some things we can prove they must exis... |
If you had 2 items the sequence would go like this:
$$1,1,2,2,2,2,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5, \ldots$$
This is $\lfloor\log_2(n+2)\rfloor$.
What if I wanted to do for 3 items which goes like this: $$1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,... |
Does there exist a $k \in \mathbb{R}$ such that for all $n \in \mathbb{N}, n \ge 3$, $\text{gpf}(\lfloor n^{(\log{n})^k} \rfloor) \gt n$, where $\text{gpf}(x)$ is the greatest prime factor of $x$?
I am interested because I was trying to figure out if a polynomial-time algorithm for factoring $n$ implies the existence o... |
I've reading through this proof, I don't understand the last part: the claim $\tau \models \Sigma$.
Note: I'll use $AP(\varphi)$ and $\text{Var}(\varphi)$ interchangeably, to mean the variables that appear in the formula $\varphi$.
They show that if $X=\{v(p): p\in\text{Var}(\varphi)\}$, it is satisfiable, finite and ... |
Let $M_i,i\in\mathbb{N}$ be an inverse system of continous, discrete G-modules and let $M=\varprojlim M_i$. Under what conditions on $M$ and $M_i$ do we have $\varprojlim H^1(G, M_i) \cong H^1(G, M)$? I came across a paper that gives a criterion for commutativity with direct limits, but not inverse limits. Thank you!!
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If we assume $AC$ we can construct an uncountable well-ordered chain of subsets of $\mathbb{N}$ by well-ordering the reals and then using the same Dedekind's cut construction as in this question.
edit: as pointed out in the comments and Asaf's answer the construction above doesn't work
What if we don't assume $AC$? Can... |
Finding $y_{A,i}$ and $y_{B,i}$ in this geometric relationship problem.
I'm an high-speed aerodynamics student. I am studying a sweptback wing like in the figure below (in green). Notice that I divided the right half-wing in two regions I and II (yellow and orange respectively):
Notice that this wing has an infinite w... |
How to prove that any regular map $\phi : \Bbb P^1 \to \Bbb A^n(\Bbb C)$ maps $\Bbb P^1$ to point. Now $\phi=(F_1/G_1,...,F_n/G_n)$ where $F_i/G_i$ is a regular function.
Now how do I conclude?
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I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises.
If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $a$ and graded for 8%, so it's supposed to be pretty easy.
Use induction to prove that $(a+b)^n=a^n+na^{n-1}b+\dfrac{n(n-1)}{2!}a^{n-2}b^... |
I have three sets, $A:=\{a_1,\ldots,a_n\}$, $B:=\{b_1,\ldots,b_n\}$ and $C:=\{0\}$.
Let $D:=A\times B \cup C$. I do not know if this is a valid notation?
For example, Is $(0,b_2)\in D$? Or, is $(a_1,0)\in D$? Or, is $(0,0)\in D$?
I want only $(0,0)$ to be in $D$. How to represent it? Is $C=\{(0,0)\}$ enough?
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In the context of stochastic integration (when we define the space $L^2(M)$), we define the (possibly infinite) measure $$P_M = P \otimes [M]$$ by
$$E_M[Y] = E\left[\int_0^\infty Y_s(\omega) d[M]_s(\omega)\right]$$
where $M$ is a square integrable martingale.
My question is: why did they choose this notation? I know t... |
Let $X$ be a normed space and $f:X\rightarrow \mathbb R$ a linear function.
I saw an old post with this problem, but there is not a complete proof.
For beginning I have to consider that $\|f\|=\infty$.
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Firsly, I would like you to explain me what $\mathbb{Z}_5^*$ means. My teacher told me it is a group of units of $\mathbb{Z}_5$, but I'm not sure what a group of units is.
If $\mathbb{Z}_5^*=\{0,1,2,3,4\}$, what is $\mathbb{Z}_8^*$?
Now, what bothers me is what Cayley table tells us about the group. If $\cdot_5$ is a b... |
Prove that the sum of the vectors from the centre to the vertices of a
regular hexagon is 0
Let's call the centre $O$ and the vertices are $A, B, C, D, E$ and $F$. Therefore, the sum in the question is $OA + OB + OC + OD + OE + OF$. How do I prove with basic vectors formulae that this sum is equal to zero? I know y... |
Suppose I have topological spaces $X, Y$ and a continuous map $f: X \to Y$. Let $\mathbb{k}$ be a field, and $i \ge 1$ an integer.
If the induced linear map on homology $f_* : H_i ( X, \mathbb{k}) \to H_i ( Y, \mathbb{k}) $ is surjective, does it necessarily follow that the induced map
linear map on cohomology $f^* : ... |
Is there an integral domain $R$ with a prime ideal $\mathfrak{p}$ of
height $1$ which is not a principal ideal?
|
One can see that the solutions are $x=2, 4$ and $x=-0.77$(approximately) seen from the graph. I am posting this to find if there is a way to solve this and find solutions like polynomial equations. Thanks !!!
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Finding $n\in \mathbb{N}$ (with $n > 1$) such that $209$ divides $n^{180}-n^{20}-n^{36}+1$ is equivalent to solving $$ n^{180} - n^{20} - n^{36} + 1 \equiv 0 \mod 11 \quad \text{ and } \quad n^{180} - n^{20} - n^{36} + 1 \equiv 0 \mod 19. $$
Using Fermat Little Theorem one can further reduce the problem to $$ n^{6} ... |
Let $A_{n\times n}$ be a real matrix then:
if $A^4 = 8A$ then $A$ is not invertible.
if $A$ is diagonalizable over $\mathbb R$ then $A^2-4A+8I$ is diagonalizable over $\mathbb R$.
From $A^4 = 8A$ we get that $x(x^3-8)=0$ zero $A$ so $0$ can be one of its eigenvalues but it doesn't have to be, so that's false.
we ... |
To prove that $R(3,3,3)>16$, I need to build an example of 3-coloring of $K_{16}$ which contains no monochromatic triangles.
Can anyone help me finding it?
Note : in this link I found that it has to do something with Petersen graph.
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Given function $$f:\mathbb{R}^n_{+} \rightarrow \mathbb{R}, \ f(x) = \sum_{k=1}^K c_k x_1^{a_{1k}} x_2^{a_{2k}} \cdots x_n^{a_{nk}}$$
Show that for any $x, y \in \text{dom} \ f, \theta \in [0,1]$, and let $z : z_i = x_i^{\theta} y_i^{1-\theta}$, the following holds
$$ f(z) \leq f(x)^{\theta} f(y)^{1-\theta}$$
I know I ... |
This question is deceptively difficult. I feel like it's probably some classic example somewhere, but I'm not sure how to describe it in enough detail to get valid results in searching online.
Problem Statement
There is an elevator, which is pulled up by a rope. The rope wraps around the pulley (shown in the picture)... |
EDIT: I made a mistake in the beginning, the second condition has changed. Sorry for this.
I'm asked to count the number of sets of 4 elements that satisfy the two following conditions:
1) Each element of the set is a two-digit number (from 10 to 99).
2) There are no repeated digits in the set.
What I've done is separa... |
Let $F (x)$ be the real-valued function defined for all real $x$ except for $x = 0$ and $x = 1$ and satisfying the functional equation $F (x) + F ((x − 1)/x) = 1 + x$. Find $F (x)$.
This functional equation looks like I could do an inverse substitution. Meaning, let $x = \dfrac{x-1}{x}$ then we have $F \left(\dfrac{x... |
I'm doing some self study on Fulton's Algebraic Curves, and I've done a decent amount. But I'm stuck on a past question that's been bugging me.
Question : Let $K$ be a field, and $P,P'$ be points in $A^{2}(K)$. Suppose $L_1,L_2$ are lines through $P$ and $L_1',L_2'$ are lines through $P'$. Show that there is an affine ... |
Consider the sequence $\{\sqrt{n}|a_n-a|\}_n$ where $a_n, a \in \mathbb{R}$. Assume $\{\sqrt{n}|a_n-a|\}_n$ is bounded away from $0$ and $\infty$. Is this equivalent to or sufficient or necessary for
(1) $|a_n-a|=\frac{h}{\sqrt{n}}$ for some $0<h<\infty$?
(2) $|a_n-a|\in O(\frac{1}{\sqrt{n}})$?
|
I am doing some propositional logic and we learned about the natural deduction rule. Everything was going fine until the rule of introducing implication arose. I am slightly confused as to how it works. This is the basically how it was worked out. The thing that is confusing me is that the turnstile was introduced for ... |
Prove that in the extension of an infinite field $K$ by $K(x,y)$, the intermediate field $K(x)$ is Galois over K, but not stable (relative to $K(x, y)$ and $K$).
I know that if K(x) is algebraic it is stable, but I'm not clear on how to show its unstable when it's not algebraic.
|
Use the Gaussian elimination with partial pivoting manually
to reduce the following matrix to row echelon form:
$$
\begin{bmatrix}
1 & 0 & 0 & 0 & 1 \\
-1 & 1 & 0 & 0 & 1 \\
-1 & -1 & 1 & 0 & 1 \\
-1 & -1 & -1 & 1 & 1 \\
-1 & -1 & -1 & -1 & 1 \\
\end{bmatrix}
$... |
I have some math questions for a programming course where it says to provide closed-form solutions for a list of sums. I've never taken an algorithms course, so I'm not quite sure what I'm doing. I wiki'ed closed-form and I think I get it, but what's the procedure for actually solving for the closed-form solution? For ... |
Any ideas how to calculate this limit without using taylor?
$$f(x) = \lim \limits_{x \to 0} \left(\frac{\sin x}{x}\right)^{\frac1x}$$
|
Given a Dirichlet series with coefficients
$$ F(s)= \sum_{n=1}^{\infty}\frac{a(n)}{n^{s}},$$
is it then always possible (and how) to obtain the $ b(n) $ coefficients related to its reciprocal
$$ \frac{1}{F(s)}=\sum_{n=1}^{\infty}\frac{b(n)}{n^{s}},$$
can we get $ b(n) $ from $ a(n) $ ??
|
I'm working on an assignment and I'm more or less new to stats. It might be the wording of the questions that's getting me as well. It deals with a regular 6-sided die.
1.a) What is the mean number of throws between the appearances of a six?
So I think this means if you have already rolled a 6, how many rolls until you... |
Georg Cantor postulated a theorem that states that for any set (even if it's an infinite set) $A$, the power set of A ($\mathbb{P}(A)$) has cardinality greater than $A$.
Could this theorem also be extended to the idea that:
For any set $A$, if $A \subset B$ but $B \not\subset A$, then cardinality of B $>$ cardinality ... |
Consider an equilateral triangle $V_1V_2V_3$ with center at the origin, and vertex $V_1 = (0,1)$ and vertices $V_1, V_2, V_3$ in counterclockwise order. Consider the action of the symmetric group $S_3$ on {$V_1, V_2, V_3$} where $\pi \in S_3$ takes each vertex $V_i $ to $V_{\pi(i)}$. This extends to a unique (linear) a... |
Let $A$ be a $k$-algebra, and $M$ an $A$-module. If $\mathrm{Ext}_A^{1}(M,S)=0$ for every simple $A$-module $S$, then $M$ is projective.
I know that this is true if $A$ is finite-dimensional, but if $A$ is infinite?
|
Let $Y_n\sim \chi^2(n)$. What is the limiting distribution of $U_n= \dfrac{\sqrt{Y_n}-\sqrt{n}}{\sqrt{2}}?$.
What I know is that if $X_i\sim \chi^2(1)$, I can write $Y_n = \sum\limits_{i=1}^n X_i$. Then the Central Limit Theorem implies that
$$\dfrac{Y_n-n}{\sqrt{2n}/\sqrt{n}}=\dfrac{Y_n-n}{\sqrt{2}} \dot{\sim} N(0,1)... |
Statistical concepts like spaces of probability distributions, "metrics" like Fisher information or relative entropy, and convergence with respect to these quantities are needed in my research problems. I need to come up to speed on these tools quickly, but can't find any standard, rigorous treatment of them.
Can anyon... |
I have a function with the following shape:
$$
f(x) = \left|\frac{a_0 + a_1x+a_2x^2 + ... + a_nx^n}{b_0 + b_1x+b_2x^2 + ... + b_nx^n} \right|
$$
The constant $b_0=1$ (I don know if it matters)
When the function is plotted with an example set of values:
a: $$\begin{array}{rcccl} [ & 1.007619-0.003295j & -0.705018+0.005... |
How to expand $f(x)=1/(x-1)$ into the form $1/x+1/x^2+1/x^3+...+1/x^n$ for x>1
I know f(x) can be rewritten as $f(x)=\frac{(1-1/x)^{-1}}{x}$.
Next step is to expand $(1-1/x)^{-1}$ to $1+1/x+1/x^2+...$
Any ideas?
|
here is the task: One factory produces 3 types of cars: small, midsize and big. There are 6000 tons of steel and 60000 total time available. For each type of car produced, there must be 1000 cars of these types produced. Build an optimization model that maximizes profit.
$$\begin{array}{l|ccc}
& \text{Small} & \text{M... |
Prove: If the map $f_n$ is defined on the $n$-skeleton $X_n$ and you want to define it on an $n+1$-cell with attaching map $ϕ:S_n→X_n$, then you can do so if and only if $f_nϕ$ is nullhomotopic.
I have a good geometric intuition on what is going on: I define the map over the new cell while contracting simultaneously th... |
Please, can somebody help me? I was given the following definition of the LIMIT:
Let $I$ be a small category and $F:I \to C$ a covariant functor ( where $C$ is a category), $K \in Ob(C)$, $\sigma:k_{K}\to F$, where $k_{K}$ is a constant functor such that that diagram commutes. I dont understand that: that functor $k_{K... |
Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$.
Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in knowing how $R(f^n)$ behaves as $n \to \infty$.
Here is an example :
$g(x) = 0.5 x^5-0.5x-1$, with $n=1,2,3$
Here are... |
According to the book that I am reading,
A nonnegative, integer valued random variable T is a stopping time for the sequence {$Z_{n},n\geqslant0$} if the event T = n depends only on the value of random variables $Z_0 , Z_1,...,Z_n$. (It is known that the sequence $Z_1,Z_2,....,Z_n$ is a martingale).
Doubt 1: What does ... |
Using the definition of the limit sequence, find $\lim_{n\rightarrow\infty}\frac{n}{n^2-2}$ for $n=2,3,4,...$
I've tried to isolate n but its impossible to do.
|
I am having some trouble and looking for help on calculating the moment of inertia about the z axis of the region bound by the cone $z=\sqrt{3(x^2+y^2)}$ and the sphere $x^2+y^2+z^2=a^2$ if the density is inversely proportional to the distance from $z$ axis.
My thoughts:
I think it will come down to evaluating $$\iiint... |
I am currently looking at these online notes on PDEs, page 59.
How does it follow that if $f^R = \phi(x/R) f(x)$
$
\phi(x) = \left\{\def\arraystretch{1.2}%
\begin{array}{@{}c@{\quad}l@{}}
1 & \text{if $|x|\leq 1,$}\\
0 & \text{if $|x|\geq 2.$}\\
\end{array}\right.
$
then
$\partial^\alpha f^R=\phi^R\parti... |
There has been a snowstorm and Bob is trying to drive from A to C.
p and q are the probabilities that the two roads are passable.
What is the probability that Bob can get from A to C? Note that whether the 2 roads are passable/ not passable are independent
$ \:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:p\:\:\:\:\:\:\:\:\:\:\:\:... |
As the title says, given $x,y,z \in \mathbb{Z}$, where $8x-10y+27z=0$, prove that $(2x-y)(2y-z)(3z+2x)$ is divisible by $60$.
I tried to bring the formula in a format of $(\cdots)(8x-10y+27z) + 60(\cdots)$, didn't get anywhere.
|
I was trying to solve a problem that is as follows:
Find the minimum value of $$ (a+b+c+d+e)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}\right) ,\qquad a,b,c,d,e>0.$$
I have done this as :
\begin{align*}
& (a+b+c+d+e)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}\right)\\
= & \l... |
Let $(Y_n)_{n\in \mathbb N} $ be some sequence of independent random variables, and $(X_n)_{n\in \mathbb N} $ another sequence, defined recursively as follows:
$$X_{n+1} = \alpha X_n + \beta Y_n \qquad \forall n \ge 2$$
$$X_1 = Y_1$$
where $\alpha, \beta \in \mathbb R $.
Is there a general approach to dealing with such... |
$$\lim_{n\to\infty} \prod_{i=2}^n(1-{1\over{i+1 \choose 2}}) $$
This is a problem i have encountered in one of my textbooks.Solve it by using high school methods for real analysis.
|
$\lim_{s \rightarrow \infty} \zeta(s) = 1$
I have seen a proof using the fact $1 \leq \zeta(s) \leq \frac{1}{1-2^{1-s}}$ but this relies on proving the inequality first which is quite cumbersome.
I was looking for a proof which is short but easy to understand.
|
Let $X=\mathbb{R}[a,b]$. Is there any norm $\|\cdot\|$ on $X$ s.th. $X$ is a continuous image of some $F$-space. ($F$-space means that there exists complete metric s.th. $d(x+z,y+z)=d(x,y)$) ?
My idea is use fact that there is no complete norm on $X$. I tried to consider $A:=Y/\ker\varphi$, where $\varphi:Y\rightarrow ... |
denote by $X$ and $X_n$, $n\in \mathbb{N}$, random variables and $r\in\mathbb{R}_+$ with $E=\mathbb{E}\left[ e^{rX} \right] < \infty$ and
$E_n=\mathbb{E}\left[ e^{rX_n} \right] < \infty$ for all $n$. Assume that the $X_n$ converge to $X$ for $n\rightarrow\infty$, say almost surely. Apart from $X_n \nearrow X$ (monotone... |
As the title says, given $x,y \in \mathbb{R}$ where $3x+2y-1=0$ and $x \in [-1, 3]$, calculate $A = \sqrt{x^2+y^2+2x-4y+5} + \sqrt{x^2+y^2-6x+8y+25}$.
I tried using the given condition to reduce the complexity of the roots, but couldn't get rid of them.
|
For students in a first course in analysis or topology, proving that certain function are continuous can be very tricky. However, some proofs which are difficult for students to prove using the $\epsilon-\delta$ definition of continuity are much easier to prove using the topological definition that the pre-image of ev... |
I have seen it claimed online that:
Given two groups $G = \langle X \mid R \rangle$ and $H = \langle Y \mid S \rangle$ with some action $\phi\colon H \to \text{Aut}(G)$, then
$$
G\rtimes_\phi H \;=\; \langle X, Y \mid R,\,S,\,yxy^{-1}=\phi(y)(x)\text{ for all }x\in X\text{ and }y\in Y\rangle.
$$
Specifically I have... |
Let $\gamma_1$ be the semicircle above the x axis joining $-1$ and $1$.
Now divide this interval into $[-1,0]$ and $[0,1]$, and trace a semicircle joining $-1,0$ above the $x$-axis and another one joining $0,1$ below the $x$-axis, call this curve $\gamma_2$
Keep subdividing the interval into $2^n$ subintervals, procedi... |
The proof of the classification of finite simple groups is
thousands of pages which does not seem to be a human readable
proof because its too long (more than 10000 pages) to be read by one person and
" no-one in the world today completely understands the whole proof"
as quoted from
https://plus.maths.org/content/eno... |
If the stem of a mushroom is modeled as a right circular cylinder with diameter $1$, height $2$, its cap modeled as a hemisphere of radius $a$ the mushroom has axial symmetry, is of uniform density,and its center of mass lies at center of plane where the cap and stem join, then find $a$.
I really need help.
|
Today I have found that the Chinese mathematician Yitang Zhang has proven in 2013 that the sequence $d_n=p_{n+1}-p_n$ where $p_n$ is the $n$th prime has a finite inferior limit (and in fact, lesser than 70 millions).
I'm only a High School teacher, so I have not access to the article. I'm pretty sure that I can't under... |
Proof for $log\left(\sum_{n=1}^{\infty} \frac{1}{n}\right)$ diverging. I know that the harmonic series diverges. What is the quickest way to prove the logarithm of it diverges?
I have not used any logarithms in proving the harmonic series diverges.
|
My goal is to find a Mobius transformation that transforms $-1, i, 1+i$ onto the points
a) $0, 2i, 1-i$
b) $i, \infty, 1$
For part a, I know that the Mobius transformation $M$ will be such that $M(-1) = 0$. So, this means that it should have the form
$$M(z) = \frac {z+1} {az+b}$$
Therefore, what should happen is that ... |
Prove for any $a, b \in [-1, 1]$ that $$|a\sqrt{1-b^2}+b\sqrt{1-a^2}-\sqrt{3(1-a^2)(1-b^2)} +\sqrt{3}ab| \le2$$
I'm sure there is a solution using the Cauchy-Swartz inequality. Thus i tried to prove $$\Big(a\sqrt{(1-b)(1+b)}+b\sqrt{(1-a)(1+a)}-\sqrt{3(1-a)(1+a)(1-b)(1+b)}+\sqrt{3}ab\Big)^2 \le 4$$
Can we use C-S to p... |
Question: Convert $y = -\sqrt{1 - x^2} + 2$ to polar coordinates:
What I have done
$$ y = -\sqrt{1 - x^2} + 2 $$
$$ 2-y = \sqrt{1 - x^2} $$
$$ (2-y)^2 = (1-x^2) $$
$$ x^2 + y^2 -4y + 3 = 0 $$
$$ x^2 + y^2 -4y = -3 $$
$$ x^2 + (y^2 -4y + 4) = -3 + 4 $$
$$ x^2 + (y-2)^2 = 1 $$
Now I am stuck where do I continue from... |
Call the primitive moves on the rubik's cube "R,L,U,D,F,B" for right, left, up, down, front, and back respectively.
Let us say that I have a permutation of the stickers on the cube written as a word in the primitive moves, e.g. RUDU.
Is there a way to find the set of all equivalent words in the primitive moves (under ... |
I am trying to do the following question:
Number 24 and number 42 buses arrive independently at the corner of
Mayeld Road at a random rate of 3 and 4 per hour respectively. You
arrive at the bus stop at 10 a.m. What is the probability that it
takes at least 20 minutes for a bus (either the 24 and 42 bus) to
ar... |
Suppose that I have a sequence of random variables where $X_1, X_2, \ldots$ where the pdf of $X_n$ is equal to:
$$
f_n(x) =
\begin{cases}
(n-1)/2 & \quad \ -1/n < x < 1/n \\
1/n & \quad n < x < n+1\\
0 & \quad otherwise
\end{cases}
$$
I am trying to find $E(X_n)$. Here, I tried the computation ... |
A box with mass $m$ is attached to a spring with spring coefficient $k$. This system is then placed into a glass case filled with a liquid with drag coefficient $\alpha$.
Now I have the following equation set-up:
$m\ddot{x}+\alpha\dot{x}+kx=0$
Now the solution for this is with usage of $e^\left(\lambda t\right)$, but ... |
consider the following system on $\Bbb{R}^n$
$\dot{x} = f(x,t)+g(x,t) $$ $$ $$ $ $ (*) $
assume that f(0,t)=g(0,t) = 0 and
1. 0 is an exponentiolly stable equilibrium of $\dot{x}=f(x,t)$
2.$|g(x,t)| \leq \mu|x| $ $\forall x \in \Bbb{R}^n$
show that 0 is is an exponentiolly stable equilibrium of $(*)$
for small en... |
I came up with this function written as the following continued fraction (please correct me if my notation is incorrect): for $n\in\mathbb{N}$, let
$$f(x;n)=x+\operatorname*{\LARGE K}\limits_{i=1}^{2n}x$$
I'm not sure about the notation, so in an attempt to be clearer, it's the function given by this Mathematica syntax... |
Question: Show using the $\epsilon -\delta$ definition that
|
Here I'm assuming $M_{2 \times 2}(\mathbb{R}) \cong \mathbb{R}^{4}$. The definition of $\mathrm{SO}(2)$ is:
$\mathrm{SO}(2)=\{ \ A \in M_{2 \times 2}(\mathbb{R}) \ | \ \det(A)=1 \mathrm{\ and\ }AA^{T}=I\ \}$
With some work I have shown that this is equivalent to:
$\mathrm{SO}(2)=\left\{ \ A \in M_{2 \times 2}(\mathbb{R... |
I'm a bit confused on a problem. I've been given an $(n+1)\times(n+1)$ square matrix, which is written in the form of a block matrix with the following dimensions
$ \begin{bmatrix}
(1x1) & null \\
(nx1) & (nxn)
\end{bmatrix} $
where I'm assuming "null" means a "null row vector of size $1xn$". I nee... |
Use substitution and the reduction formula to find:
$$\int x^4e^{2x}\,\mathrm{d}x$$
|
I just started taking linear algebra and I am already confused about how to know whether linear combinations fill a line, plane, or $R^3$, my textbook simply says it you have one vector with a scalar multiple it fills a line, if you add two vectors with scalar multiples it fills a plane, and a 3 vector linear combinati... |
Suppose for every $y$, $f(x,y)$ is strictly convex in $x$. Further, $f(x,y)$ is continuous in $y$. Let $\mathcal X$ be compact (in my problem, $\mathcal X$ is an interval). Can anyone suggest any theorems or references for the problem of whether $x^*(y) = \text{argmin}_{x \in \mathcal X} f(x,y)$ is continuous in $y$?
|
$1+1+1+\cdots$ is clearly a divergent series, so you'd say that it tends towards infinity? Through analytic continuation of the zetafunction the value $-1/2$ could be assigned the sum, right? But if we consider the sum of all integers: $\cdots-2+-1+0+1+2+\cdots$ and group them up pairwise as follows: $0+(-1+1)+(-2+2)+(... |
It is a standard fact in probability that almost sure convergence is stronger than convergence in probability. I can only see the differences in the proof. However, is there a way to view it intuitively? Is it true that almost sure convergence has a tighter hold on the tails of a sequence of random variables than conve... |
Let $G$ be the set of all functions from $\mathbb{R}$ to $\mathbb{R}$ along with $+$. Show that $H$ defined by $H=\{f:f(x)=0 \text{ for all } x \in [0,1]\}$ is a subgroup.
I am able to show $H$ has inverses and that $H$ is closed. But how do I show $H$ is nonempty? Any argument I use would begin with "let $x\in H$..... |
Suppose that if G is a group, H is a subgroup of G, and K is a subgroup of H.
Show that if K is a normal subgroup of G, then K is a normal subgroup of H.
Ive tried writing out what is implied from all of the givens, but i guess I'm just having trouble seeing a connection.
K is a normal subgroup of G tells us that
aK=K... |
When parametrizing a curve how doe we obtain the range of $t$?
For example lets say we have the parametrization:
$x(t) = 1+3t$ and $y(t) = 2+5t$.
How do we find the range of t?
$t\to[?,?]$
|
How can I prove that the only prime number $p$, such that $ p,p+2,p+4$ are primes is 3?
|
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