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How can i find the intersection points between
$$y=\frac{\pi}{4} x,\:\: y=\tan^3(x)$$
Thank you for your help.
I know that the first (the obvious one...) is $0$.
But how can i get another one?
|
I'm working from a text in which the coordinates of a point in $\mathbb R^n$ are denoted $(x^1,\dots,x^n)$. I'm wondering if there is a standard way to denote the sum of the squares of these coordinates. For example, should I write
$$
{x^1}^2 + \dots + {x^n}^2,
$$
which doesn't seem very clear, or should I write
$$
(x^... |
This was a question in our quiz today and no one in class knew how to answer it correctly or are not sure).
How many homomorphisms are there $t: \mathbb{Z}_3 \times U_8 \to S_5$, where $U_8$ is the group of invertible elements in $(\mathbb{Z}_8, \cdot)$.
I personally described it like this:
A homomorphism as such i... |
If K and Q are both groups and $h:Q\rightarrow \text{Aut}(K) $ is a homomorphism then the group operation for the semidirect product $K\rtimes_hQ$ is: $$(k_1,q_1)*(k_2,q_2)=(k_1h(q_1)(k_2),q_1q_2)$$
I am trying to prove that the inverse of $(k_1,q_1)\in K\rtimes_hQ $ is $(h(q_1^{-1})(k_1^{-1}),q_1^{-1})$.
I was able to... |
I have a set of data from my workplace where we note how many support calls we receive. I have been playing around with it in my spare time just to see if I could predict anything interesting.
I have been trying to fit the amount of calls to the poisson distribution and I know the MLE of the $\lambda$ parameter is the ... |
This is a terminology question from a non-native English speaker. Let $G$ be a group and $a,b\in G$ such that there exists $c\in G$ verifying :
$$b=cac^{-1} $$
I could say :
the element $a$ is conjugate to $b$.
the element $a$ is conjugate with $b$.
the element $a$ is conjugated to $b$.
the element $a$ is conjugated ... |
I have a formula for the Ewald method which can be used to speed up computations when working with periodic Green's functions. I will need to take the derivative of the function $G(x, y)$ with respect to $x$.
$$G(x, y) = - \sum_{n \in \mathbb{Z}^2} e^{i n \cdot \alpha}\int_0^E\frac{1}{4\pi t}\exp({\omega^2 t - \frac{|... |
I have been wondering about how to solve this recurrence but I don't get to any feasible solution. How can I do it?
|
Let $p,q : \Bbb{R} \to \Bbb{R^n}$ and $H:\Bbb{R^n}\times \Bbb{R^n} \to \Bbb{R} $
and the hamiltonian system:
$$ \begin{cases}
\dot p = - \frac{\partial H}{\partial q} \\
\dot q = \frac{\partial H}{\partial p}
\end{cases} $$
show that for p,q solutions of the system $ H(p,q)$ is constant over time
show that given th... |
Suppose $A$ is a principal ideal domain with every ideal of finite index (except the zero ideal). Must $A$ be a Euclidean domain?
If it's not known, are there any relevant partial results?
|
How can we find number of integer solutions of
$\frac{1}{x}+\frac{1}{y}=\frac{1}{2016}$
I want to ask what approach in general should be followed in such types of question?
|
Find all pairs of nonzero integers $(a,b)$ such that
$(a^2+b)(a+b^2)=(a-b)^3$
My effort
Rearranging the equation I have
\begin{array}
\space (a^2+b)(a+b^2)-(a-b)^3 &=0 \\
a^2(b^2+3b)+a(-3b^2+b)+2b^3 &=0 \\
\end{array}
Letting $a=x$, we have the polynomial $Q(x)$ such that $$Q(x)=x^2(b^2+3b)+x(-3b^2+b)+2b^3 $$
So I... |
Show that the lines $x^2+16xy-11y^2=0$ form an equilateral triangle with the line $2x+y+1=0$ and find its area.
--------________________________---------
My solution is here;
Here $x^2+16xy-11y^2=0$ represents a pair of straight lines.
Comparing with $ax^2+bx+c=0$ and using the quadratic formula i.e. $x={-b+/-√(b^2-4ac... |
I'm doing some independent work, and have managed to come across the following interesting 1-D autonomous ODE:
$\dot{x} = x(1-x) \log^2\left[\frac{x}{1-x}\right]$.
For the fixed points, i.e., where $\dot{x} = 0$, I know that the only valid one should be for $x = 1/2$ because of the Log function. However, technically, $... |
I came to the conclusion that sup$(A)= 3$ and inf$(A)=2$, but I am stuck at part $(a)$ showing it bounded above and below.
How do I solve the rest of $(a)$ and $(b)$ using the epsilon definition.
|
Suppose that $H$ is a separable Hilbert space and $(\psi_n)_{n=1}^{\infty}$ is a complete orthonormal sequence in $H$. We define a sequence $(\phi_n)_{n=1}^{\infty}$ by
$$
\phi_n=\psi_1+\psi_{n+1}\ ;\ n=1,2,3,...
$$
Indeed, $(\phi_n)_{n=1}^{\infty}$ is a complete linearly independent sequence in $H$ since for any $f\i... |
I have this assignment.
Let $X$ be a set and $f:X\to X$ a function. Let $A\subseteq X$. Determine whether each of the following statements is true in general, and give a proof of the correct statements. Give a counterexample for each of the statements that is not true in general.
a) $f(f^{-1}(A))\subseteq A$
b) $f(f^{-... |
Two graphs which contain the same number of graph vertices connected in the same way are said to be isomorphic. Formally, two graphs and with graph vertices are said to be isomorphic if there is a permutation of such that is in the set of graph edges iff is in the set of graph edges.
(so, now I have two graphs and eve... |
$f(r)=\frac{r+r^{-1}}{2}$
$f(re^{i \theta})=\frac{re^{i\theta}+r^{-1}e^{-i\theta}}{2}=\frac{r+r^{-1}}{2}\cos\theta+i\frac{r-r^{-1}}{2}\sin\theta$
Why is it then valid to set $r=e^t$, $-\infty≤t≤0$
in order to find
$f(re^{i\theta})=\cosh(t)\cos(\theta)+i\sinh(t)\sin(\theta)$
This is from:
http://people.reed.edu/~jerry/... |
Let $U$ be uniformly distributed on the interval [$\frac{1}{3},1$]. Let $X$ be a random variable such that the conditional distribution of $X$ given $U=p$ is Geometric with parameter $p$.
(a) Find expressions for $E(X|U)$ and $Var(X|U)$
(b) Using your answer for (a) explain why $E(X)=1.65$ and $Var(X)=1.64$
I'm gettin... |
I'm sorry for posting this here, but I would like to confirm my doubt about the correctness of the systems of equation in the textbook example. I enclosed an image.
|
Let $G$ be a linear algebraic group over a field $K$ of characterstic zero acting on a vector space $V$. Then does this action induce a representation :
$$\Gamma : Lie(G) \to gl(V)$$
If yes, how ? Please help me understand this. I would appreciate if the explanation is simple and from all prespectives like thinking of... |
I need to show $\lim_{n\to\infty}\frac{Li(n)}{\pi(n)} =1$, where $Li(n)$ = $\int_{2}^{n} \frac{dx}{ln(x)}$.
I know I need to use the Prime Number Theorem, and L'Hopital's rule. However, I am I can't seem to get it started correctly. Can you help?
|
How do you evaluate:
$$\displaystyle \int_{-\infty}^z e^{\frac{-t^2+2t\alpha\mu}{2\sigma^2\alpha^2}+\frac{\lambda t}{1-\alpha}} dt = ??$$
Many thanks.
|
Let $(E,\mathcal{E}),(F,\mathcal{F})$ and $(G,\mathcal{G})$ be measure spaces and $f:E\times F\rightarrow G$ a measurable function.
$X,Y$ are independent RV with values in $E$ and $F$. In my situation $f(X,Y)$ and $Y$ are independent.
Does there exists a measurable function $f':E\rightarrow G$ s.t. $f'(X)=f(X,Y)$ alm... |
I feel like there should be an easy example, but I can't think of one. So, specifically, I am looking for a Yes/No search problem that is not in the class NP. When you learn about P and NP, you get a lot of examples of problems in P, NP, NP-hard, NP-complete, co-NP, etc. I guess, halting problem would work, but I would... |
Let us be given a linear algebraic group $G$ over a field $K$ of characterstic zero. This group $G$ is defined as the common zeroes of a finite set of polynomials $\{f_1, \ldots ,f_r\}$ $\in K [T_1,\ldots,T_n]$with some additional properties like having the structure of a group along with 2 morphisms of varieties.
Now... |
Prove that
$\cos\left({\pi \over 11}\right)\cdot\cos\left({2\pi \over 11}\right)\cdot\cos\left({3\pi \over 11}\right)\cdot\cos\left({4\pi \over 11}\right)\cdot\cos\left({5\pi \over 11}\right)={1 \over 32}$
My solution starts here
LHS $= \cos\left({\pi \over 11}\right)\cdot\cos\left({2\pi \over 11}\right)\cdot\cos\left... |
There are a few very useful websites when it comes to either finding a specific object with certain properties (and maybe lacking other properties) or finding out which properties a certain object has.
For example, there are $\pi$-Base and Topospaces for topological spaces (the former having an amazingly powerful searc... |
I want to solve this primitive $$I=\int \frac{2}{x^2(x^2+1)^2}dx.$$
I substitute $u=x^2$ then,
$$I=\int \frac{2}{x^2(x^2+1)^2}dx=\int \frac{du}{u^{3/2}(u+1)^2}=\cdots$$
How do I use partial fraction decomposition? Is there another idea to compute this primitive?
|
What function can turn $z=x+iy$ into something involving $xy$?
What function takes the real parts of $z$ and then multiplies them?
Or would I perhaps need to consider the point $(x,iy)$, rather than the expression $x+iy$?
|
Find the minimum value of the following expression
$$ \frac {\sin \alpha \sin \beta}{\sin^2 \frac {\gamma}{2}}+ \frac {\sin \beta \sin \gamma}{\sin^2 \frac {\alpha}{2}}+ \frac {\sin \gamma \sin \alpha}{\sin^2 \frac {\beta}{2}},$$
where $\alpha, \beta$ and $\gamma -$ angles of some triangle?
|
Suppose that $X$ is a cadlag locally square-integrable martingale. Let $[X]$ denote the quadratic variation of $X$. My textbook claims, by Ito's formula that
$$ X^2 _t = X^2_0 + [X]_t + 2
\int_0^t X_{s^{-}}
\,d X_s, $$
which shows that $X^2 - [X]$ is indeed a local martingale. Let $
\Delta Y_t = Y_t - Y_{t^{-}}.... |
A square free word is a word that does not contain any subword twice in a row.
An infinite word is defined:
$$
w_i=\{ \textrm{the maximal natural j that } 2^j \textrm{ devides } i\}
$$
the first "letters" are:
$$w=01020103...$$
how can I prove $w$ is a square-free word?
|
A differential equations problem asked about the largest interval on which the solution was defined. The solution was defined except for $t=0$, which made me wonder whether the intervals $(-\infty, 0)$ and $(0, \infty)$ are the same size.
If these intervals are the same size, does that mean that $\mathbb{R}$ can be div... |
I saw in this post that
$\frac{d}{dt}\text{logm}(Z(t)) = \frac{dZ(t)}{dt}(Z(t))^{-1}$
Is this true to say:
$\frac{d}{{dU}}{\mathop{\rm logm}\nolimits} (A) = {A^{ - 1}}\frac{d}{{dU}}A$
where U is an m by n matrix and A is an m by m matrix which is a function of U ?
EDIT: A is not Symmetric and positive definite.
|
The two small circles (in black) are equal in radius, and tangential to the large circle. They also touch each other at the center of the large circle.
Now, I want to construct a circle (in orange) which is tangential to the inner two, and the larger circle.
How can I construct it?
|
Assume you have two subspaces $\mathcal{A}$ and $\mathcal{B}$. In general, these are matrices that the span of the columns makes the subspace.
Now: What is the best symbol for denoting the angle of these two subspaces?
I was about to use this $\phi=<\mathcal{A},\mathcal{B}>$, but that's only common for inner product o... |
For some reason I am having difficulty understand how to calculate the mle of a discrete rv.
The pmf is:
$$p(k;\theta) = \left\{\begin{array}{cl}
\dfrac{1-\theta}3&\text{if } k=0\\[5pt]
\dfrac{1}{3}&\text{if } k=1\\[5pt]
\dfrac{1+\theta}{3}&\text{if } k=2\\[5pt]
0&\text{otherwise}&\end{array}\right.$$
We're also told ... |
I am stuck in a proof in Davidson's "$C^*$ algebras by examples" book. In section II.3, he proves that any abelian Von Neumann algebra $N$ on a separable Hilbert $H$ has a non-zero projection with uniform multiplicity. In doing so, he starts as follows :
Choose a sequence of unit vectors $x_n$ and projections $Q_n$ (i... |
From a standard pack of cards $(\spadesuit, \heartsuit, \diamondsuit, \clubsuit)$ we are picking randomly one card in a each time (with replacement) until we get the first heart $\heartsuit$.
$(A.)$ Find the probability that there will be exactly $5$ cards picked.
$(B.)$ Find the probability that there will be more the... |
From the following:
$$\sum_0^\infty [ n(n-1)a_nx^{n-2} - n(n-1)a_n x^n -2na_nx^n + l(l+1)a_n x^n ] = 0$$ (a)
I'm trying to get to:
$$\sum_0^\infty [ (n+2)(n+1)a_{n+2} - [n(n+1) + l(l+1)]a_n]x^n = 0$$ (b)
Unfortunately, I'm not being very successful.
I can either get the first term ($(n+2)(n+1)a_{n+2} x^n$) by saying th... |
The Cauchy-Schwarz inequality states that
$$\left(\sum_{i=1}^n x_i y_i\right)^2\leq \left(\sum_{i=1}^n x_i^2\right) \left(\sum_{i=1}^n y_i^2\right).$$
The proof, with the discriminant argument, is easy to understand; however, it does not really (in my opinion) provide any intuitive justification as to why the inequali... |
What is the area of the shaded region of the given $8 \times 5$ rectangle?
Attempt: We know that the area of a kite is $\dfrac{pq}{2}$ where $p$ and $q$ are the diagonals. Thus, since the two lines in the figure intersect at the center of the square, the length of the long diagonal of one of the two shaded kites is ... |
I'm just starting out learning about optimization.
Not sure if this makes sense but...
Given a variable $x$ that is passed to two separate functions, $f(x)$ and $g(x)$, why do we add the derivatives $\frac{df}{dx}$ and $\frac{dg}{dx}$ to compute the total derivative of $x$?
Why is this derivative equivalent to $\frac{... |
Consider a group $G=S_4$. Let H={1, (123) ,(321), (12), (13), (23)}. Show that H is a subgroup of G.
What is the best way to do this.
I realised <(123)>={1, (123), (321)} but I don't think this helps much.
I do not want to compute every element * every other element to show closure.
|
I have been unable to solve the following question,
If $$\sin(2x) - \tan(x) = 0$$
Find $x$ , $-\pi\le x\le \pi$
So far my workings have been
Use following identity:
$$\sin(2x) = 2\sin(x)\cos(x)\\2\sin(x)\cos(x) - \tan(x) = 0\\2\sin(x)\cos(x) - \frac{\sin(x)}{\cos(x)} = 0\\
2\frac{\sin(x)\cos(x)}{1} - \frac{\sin(x)}{\... |
This is problem 5.3 and 5.4 in Armstrong's Basic toplogy. They are very much connected and i have solved problem 3.
3: Let $D$ be the disc bounded by $C$, i.e. $S^1$, parametrize $D$ using polar coordinates, and let $h : D \to D$ be the homeomorphism defined by
$h(0) = 0,\:\: h(r, \theta) = (r, \theta + 2\pi r)$.
F... |
When solving systems of linear equations one sometimes gets result like ${0x_n = 0}$ what does it mean for solving the system? Is it error on part of the solver or just feature of the assignment?
Edit: To extend the question. What would such result mean when looking for Nash equilibrium?
|
Show that if one of the lines given by $a_1x^2+2h_1xy+b_1y^2=0$ coincides with one of the lines of $a_2x^2+2h_2xy+b_2y^2=0$ then
$(a_1b_2 - a_2b_1)^2=4(a_2h_1 - a_1h_2)(b_1h_2-b_2h_1)$
Actually, I did not get any idea to start its solution. so please help me.
Help much appreciated.
Thanks.
|
show $b_1b_2b_3\cdots b_{\phi(m)} \equiv 1 \pmod{m}$ or $b_1b_2b_3\cdots b_{\phi(m)} \equiv -1 \pmod m$
where $b_1 < b_2 < b_3<\cdots< b_{\phi(m)}$ are the integers between $1$ and $m$ that are relatively prime to $m$.
Seems like a simple enough exercise, but i am running into a dead end on how to start. Suggestio... |
If $|A| \geq |B|$, then there exists an onto function $f: A \rightarrow B$.
If $|B| \leq |A|$, then there exists a one-to-one function $f: B \rightarrow A$.
My issue is that I don't think that $|A| \geq |B|$ implies $|B| \leq |A|$. If there is an onto function from A to B, there isn't necessarily a one-to-one function ... |
Peano's axioms for Natural numbers.
3rd axiom from Introduction to topology by Bert Mendelson -
There is one and only one object in $\mathbb{N}$ denoted by $1$, which is not the successor of an object in $\mathbb{N}$, i.e, $1 \neq s(x)$ for each $x \in \mathbb{N}$.
How can $1$ not be a successor, isn't $0$ a natural ... |
Show that up to isomorphism there exist only two groups of order 10: $C_{10}$ and $D_{10}$. I need some help on this question. I only know the basic definitions of isomorphism. Any hints?
|
Define $T : C[0,\infty)\to C[0,\infty)$ by $Tf(x) = \int_0^x f(y) dy$. Show that T is an integral operator, although with a discontinuous kernel k.
I'm not really sure where to start on this, any help would be appreciated.
|
Suppose recurrence is $a_{n+2}=a_{n+1}+6a_{n}$
Tried to solve it with solving $Fnc(n)=An^5+Bn^4+Cn^3+Dn^2+En+F$
Which gives $A = (-33/4), B = (365/4), C = (-1385/4), D = (2155/4), E = (-551/2), F = 5$
But $Fnc(0,1,2,3,4,5,6\dots)={5, 5, 35, 65, 275, 65\dots}$ and everything after 65, including it is definitely wrong.
R... |
A photographer takes a picture of his vehicle's instrument panel, and later realized that the odometer reading == his zip code. Upon making this observation, he asks (figure of speech, really), "What are the odds of that?"
A friend says the odds are 1:N, where N = the zip code. I am fairly certain that is wrong... N d... |
From Lecture 2, Algebra and Computation by V. Arvind, (page2,3), I understood below passage-
For our graph $G$, let $Aut(G) = H ≤ S_n$. We shall use Weilandt’s
notation where $i^\pi$ denotes the image of i under$\pi$. In this
notation, composition becomes simpler: $(i^\pi)^{\tau} = i^{\pi \tau}$
Define $H_i =... |
From what I understand, I am being asked to show that a function $f$ on an interval $[a,b]$ either has a root $c$ such that $f(c)=0$ or it does not have a root hence $|f(x)|\ge e$.
Should I apply the Intermediate Value Theorem for the first part?
|
I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems acceptable.
I am given a set $A = \{ (x,y) \in \mathbb R^2 \mid x^2 + y^2 \le 1 , x+y \ge 0 \}$ and a function $f(x,y) = (... |
If we have for example $a_n=1+\sqrt{a_{n-1}}$ and $\lim_{n \rightarrow \infty} a_n=L$ then can I say that $ L=1+\sqrt{L}$? If it's so, what's the proof?
|
I have number $-1e35$, and I'm supposed to convert it to binary. The answer is :
$-1.10101001010110100101101...\text{e}–117.$
I can't figure out how to get this! and how we can calculate numbers partially as you see here. Originally it is a programming practice, but what I care about is the mathematical part.
I tried ... |
Let
$X=\{(1,0),(0,0)\}\cup\bigcup_{n\neq1}\{(x,\frac{1}{n}):x\in\Bbb{R}\}$.
Determine whether or not $X$ is locally connected and find its
components.
Well, I know that a space $X$ is said to be locally connected at x if for every neighborhood $U\subset X$ of $x$, there is a connected neighborhood $V$ of $x$ su... |
I have three equations as follows (for a polytropic gas):
1) $\displaystyle\quad\frac{\partial\rho}{\partial t}+\nabla\cdot\rho \mathbf{u} = 0$
2) $\displaystyle\quad\rho \left( \frac{\partial \mathbf{u}}{\partial t}+\mathbf{u} \cdot \nabla \mathbf{u} \right) = -\nabla p$
3) $\displaystyle\quad \frac{\partial}{\partial... |
Related question: Regular polygon determined by three vertices
I have solved a problem that is related to the linked question. It boils down to the question "given three vertices of a regular polygon, what is the smallest possible area of a regular polygon these vertices are a part of?"
A possible solution is to find ... |
Let $G$ be a finite set and $*$ a binary operation on $G$ such that:
The operation $*$ is associative.
For all $x$, $y$, and $z$ in $G$, if $x*y=x*z$ then $y=z$ and if $y*x=z*x$ then $y=z$.
I must show that this set has an identity element $e$ in $G$ such that for all $x$ in $G$, $e*x=x$. I'm not sure how to start at... |
Given that $\lim\limits_{x\rightarrow1^-} \sum\limits_{k=0}^\infty a_kx^k = b \in\mathbb{R}$ for $|x|<1$. If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b$.
This is just a step in another proof that I am working on. If I can show this my proof is complete.
The trouble I am having is that I'm not sure ... |
I am trying to perform some computations in local coordinates on $\Bbb P ^n \Bbb C$ seen as a symplectic manifold, in order to get a better feeling of some facts. While I do know the coefficients of the symplectic form in the usual charts (where one projective coordinate is $1$), they do not lend themselves easily to t... |
This question will only require a simple explanation, however thus far I have not been able find this from my notes or the web.
The sum of an infinite series is:
$$a/(n)r$$
if the series is :$$1/(n)^2$$
then r = 1/2.
my question is
what is the derivation for $$n^2 == r = 1/2$$
Any help would be very much appreciated. ... |
The equation $x^{13}-e^{-x}+x-\sin{x}=0$ has
No real root
More than two real roots.
Exactly two real roots.
Exactly one real root.
I tried doing with the odd derivative and check whether the derivative change sign or remain same(positive or negative) so i find $f^{(15)}(x)=e^{-x}+cos(x)$ now i am not able to determ... |
Use logical quantifiers to write:
"Everybody loves somebody sometimes" (Where U=all people)
I came up with this but not sure how to type symbols in here.
$$\forall x \in U\,: \exists y\in U: x \text{ loves } y.$$
So... upside down A="For all"
Backwards E for "there exists"
curly little e for "belongs to"
My apologies a... |
I need to a maximal set of linearly independent columns of a matrix $A$. I've googled a lot and seen various solutions, but none of them seem to work for me. What I've seen so far is
1.- Using Cauchy-Schwarz inequality. But everything in this line simply finds non-parallel vectors, which is not what I need.
2.- Using t... |
Calculate the are enclosed by ${y = 4}$ and ${y = 12 - 2x - x^2}$
My first step is to let equation equal the ohter
${4 = 12 - 2x - x^2}$
This factorises into ${x + 4)(x -2}$
I then use these for the domain:
${\int_{-2}^{4} (16 -2x - x^2) - (- 4) dx}$
Taking the integral
${[16x - x^2 - {x^3\over 3}]}$ and plugging the d... |
If in a triangle $ABC$, $A \equiv (1,10)$, circumcenter $\equiv (-\frac13, \frac23)$ and orthocenter $\equiv (\frac{11}3, \frac43)$ then the coordinates of mid-point of side opposite to A is?
Here clearly point $Q$ is circumcenter and point $P$ is orthocenter. The only thing I see here is $AP ||DQ$. So their slopes ... |
Problem
Compute the integrals $$I=\iint_\Sigma \nabla\times\mathbf F\cdot d\,\bf\Sigma$$
And $$J=\oint_{\partial\Sigma}\mathbf F\cdot d\bf r$$
For $F=(x^2y,3x^3z,yz^3)$, and $$\Sigma:\begin{cases}x^2+y^2=1\\-1\le z\le 1 \end{cases}$$
I don't quite understand which kind of curve could be the boundary of such a cylinde... |
Can you please help me to show that if $\Omega\subset\mathbb R^n$ is a $C^2$ domain and $f$ is an application which belongs to $L^2(Ω)$ and $u$ is a weak solution of the linear elliptic equation:
$$\left\{\begin{array}{c}
−\Delta u = f & \text{on }\Omega \\
u(0)=u_0
\end{array}\right.,$$
if $f$ is in $C^b (Ω)$ with $b ... |
Find a general formula for $a_{n,k}$ , for $n,k\geq1$.
We have initial values $a_{1,1}=1$, and $a_{1,k}=0$ for $k>1$.
The recurrence relation is:
$a_{n+1,1}=-a_{n,1}$ , for $n\geq1$
and
$a_{n+1,k}=-ka_{n,k}+ka_{n,k-1}$ , for $n\geq1$ and $k\geq2$ .
|
I would like to ask the following. If we have the proposition $$\forall x\in\mathbb{R}^{+}(x^2>0)$$ and we wish to use as a domain the $$\mathbb{R}$$ instead. Is it correct that it will translate to: $$\forall x\in\mathbb{R}(x>0\land x^2>0)$$?
Similarly am I correct when from $$\exists x\in\mathbb{Z}^{+}(x^2=\sqrt2)$$ ... |
I am having hard time solving the following question:
Let $\gamma : \mathbb{R} \to \mathbb{R}^2$ curve in natural parameterization.
Suppose the curve has curvature $\kappa(s)=3s^2$
Then prove that $\gamma$ has symmetric axis, and that that $|\gamma|$ is bounded.
I tried using Frenet base to argue that :
$\ddot{\gamm... |
I have been reading some solution of a problem from abstract algebra and it says 'Take the maximum $n > 0$ of $\{v_2(|g|) : g \in G\}$, where G is group'. Does $v_2(|g|)$ means something? Because it is first mention at that point and I can't figure out what it means.
|
$\mathbf{9.35}\quad$ Theorem $\ \; $ If $[A]$ and $[B]$ are $n$ by $n$ matrices, then $$\det([B][A])=\det[B]\det[A].$$
$\mathbf{9.36}\quad$ Theorem $\ \; $ A linear operator $A$ on $R^n$ is invertible if and only if $\det[A]\neq 0$.
$\qquad\mathbf{9.37}\quad$ Remark $\ \; $ Suppose $\{\mathbf e_1,\ldots,\mathbf e_n\}... |
Let $F: \mathcal{A} \longrightarrow \mathcal{B}$, $G: \mathcal{B} \longrightarrow \mathcal{A}$ be two additive functors between abelian categories, such that $(F, G)$ is an adjoint pair.
I want to prove that, if there exist a cogenerator $B \in \mathcal{B}$ such that $G(B) \in \mathcal{A}$ is injective, then $F $ is ex... |
I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not integrable, I can't use the Theorem of Change of Variable. So, I'm trying to use residue formulae for some suitable holomorphic f... |
Let $k$ be an algebraically closed field. Let $V \subset \Bbb A^n$ be a nonempty variety. How to prove that $\dim_k k[V]< \infty$ implies $V$ is a point.
I am not getting the answer even I do not get what is $\dim_k k[V]$ here?
Because $k[V]$ is an integral domain... So is it the $k$ algebra dimension?
|
I'm looking at notes on the threshold for random 2-sat which is given as $r_{2}^{*}=1$.
In the first part of proving the threshold they claim that a 2-sat formula is satisfiable if and only if the graph of implications does not contain a cycle containing a variable and it's negation. In the calculation it says the foll... |
Prolog : The only compactification of a non-compact normal space $S$ is the one-point (Alexandroff) compactification IFF whenever $A,B$ are disjoint closed subsets of $S$, at least one of $A,B$ is compact.(Example. $\omega_1$ with the $\epsilon$-order topology.) In Engelking,General Topology, it is shown that $[(\omega... |
I am interested in a book that is about arithmetics but the presentation is not just the known to all formulas but the foundational logic behind it.
The closest example I can think about is the way Bertrand Russell has some books presenting such foundations in mathematics; but not in such a strong level of formality pr... |
Let $G = \{1,g,g^2\}$ be a cyclic group of order $3$. Consider the group ring $\mathbb Q[G]$. Then we have the two $G$-invariant simple subspaces
$$
I = \{ a_0 + a_1 g + a_2 g^2 : a_0 = a_1 = a_2 \}
$$
and $J = \{ a_0 + a_ g + a_2 g^2 : a_0 + a_1 + a_3 = 0 \}$. The first is generated by $1+g+g^2$, the second by $1-g, ... |
I can't understand one thing in one question of Arithmetic Progression. I have to find $21^{st}$ term from the set $\{12,2,-4,-10\}$ and this is where problem start.
Arithmetic Progression
Equation: $T_n=a+(n-1)d$
$Tn$ = Term
$n$ = Term Number
$a$ = First Number
$d$ = Difference between two series wise numbers
I need ... |
I have the following conjecture:
Let $k\in\mathbb{N}$ be even. Now $k+p$ is prime for infinitely many primes $p$.
I couldn't find anything on this topic, but I'm sure this has been thought of before. I tried to solve this using Dirichlet's theorem on arithmetic progressions and the Green–Tao theorem, but no luck wit... |
Find the disc of convergence:
$$\sum_{n=2}^\infty \frac{z^{n}}{n(log(n))^p};(p>0)$$
I have tried geometic series, ratio test, root test... What would be your thought on the best test to use?
|
Given that $\Sigma\vdash\phi \Leftrightarrow \Sigma\vDash\phi$, I want to prove: $\Sigma \text{ satisfiable} \Leftrightarrow \text{ every finite subset of } \Sigma \text{ is satisfiable}$.
I will post my idea below, but I can't quite finish the $\Leftarrow$ direction. I would need that the set which I will call $\Delta... |
I found the following question in my textbook:
(QUESTION) Let $\mathcal{H}$ a Hilbert space and $T: \mathcal{H} \rightarrow \mathcal{H}$ a compact operator. Show that exists $x \neq 0$ in $\mathcal{H}$, such that
$\|Tx\| = \|T\|\|x\|$
A few days ago, I found a similar question in another textbook, but I had $f: \mathc... |
As a programmer who hasn't had any higher mathematical training, I sometimes find mathematical equations described in books or online that I'd like to implement in my programs, but they have symbols in them that I'm unfamiliar with. Or they use symbols that I'm familiar with in an unfamiliar way. It's very frustrating,... |
Given any two regions in complex plane when can we say they are conformally equivalent? I mean does there exists some "complex-geometric" invariant which determines whether two regions are conformally equivalent or not. Obviously topological invariant won't work as both are at least diffeomorphic. Can you please provid... |
There are two urns $U_1$ and $U_2$. $U_1$ contains four white and four black balls, and $U_2$ is empty. Four balls are drawn at random from $U_1$ and transferred to $U_2$. Then a ball is drawn at random from $U_2$. Find the probability that the ball drawn from $U_2$ is white.
I did
Let $U_w$ be the event that the ball... |
A really simple question, but I thought I'd ask anyway. Does $n<x^n<(n+1)$ imply $\sqrt[n] n < x < \sqrt[n] {n+1}$?
Thank you very much.
|
If $\phi: V \to W$ is a polynomial map, and $X$ is an algebraic subset of $W$, show that $\phi ^{-1}(X)$ is an algebraic subset of $V$. If $\phi ^{-1}(X)$ is irreducible, and $X$ is contained in the image of $\phi$, show that $X$ is irreducible.
here $X$ being closed I think I can't write $X=X_1 \times \cdots \times X_... |
I have posted a similar question earlier and I guess this sounds naive to all of you, but nonetheless let me just ask:
Consider I have a simple and deterministic model $M$, with a number of input parameters. Given the fact that we do not have exact knowledge about the parameters, we can run $n$ simulations of $M$ with ... |
I'm using gradient descent with mean squared error as error function to do linear regression. Take a look at the equations first.
As you can see in eq.1, the prediction is done with a bias term b and a weight vector W. Eq.2 shows the error function (MSE) while eq.3 shows the partial derivatives used to update the wei... |
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