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I came across this divergent sum-
$$\sum_{n=1}^\infty\frac{1}{n+1}$$
Now,a divergent sum does not a limit.So is it possible to get a maximum value for the sum or more specifically prove that the series is lesser than a particular value?
More specifically I got this idea from this answer-How do I prove $\frac 34\geq ... |
If $X \sim U (0,1)$ then if we define a new random variable $Y=-\log (1-X)$ then what will be distribution of $Y$.
Please explain.
|
I have an optimization problem in the form:
$$\max (a-\bar{a})(b-\bar{b}) \qquad \text{subject to} \qquad a+b=1.$$
Here $\bar{a}$ and $\bar{b}$ are known values and both of them are positive.
Let $a_{opt}$ and $b_{opt}$ are the solutions or optimal values. Then
$$a_{opt}-\bar{a}>0 \qquad \text{and} \qquad b_{opt}-\bar... |
Please help me with the following proof:
Suppose $\dot x=f(x(t))$ and suppose that we have:
$$ \frac{d}{dt}\left( x(t)^TPx(t) \right)\le -x(t)^TQx(t) $$
where $P$ and $Q$ are symmetric positive definite matrices and the
time derivatives are taken along solutions of the system. Prove that
under this condition the s... |
Let$$A=\begin{pmatrix} 0 & a \\ 1 & b \end{pmatrix}.$$
How to prove or disprove that the centralizer of $A$ is equal to $\langle A \rangle$ (matrices generated by A)?
For a matrix to be in the centralizer I think by its definition it must commute with $A$. I considered the matrix
$$C=\begin{pmatrix} c & d \\ e & f \end... |
Given the set of vectors $\{\mathbf{g}^{1}, \ldots, \mathbf{g}^{N-1} \}$ where $\mathbf{g}^{i} \in \mathbf{R}^M$. Assume that $N \leq M$ and elements of $\mathbf{g}^{i}$ follows normal distribution, i.e. ,$\mathbf{g}^{i}_{m} \sim \mathcal{N}(0,1)$.
I would like to compute the probability that a new vector $\mathbf{g}^{... |
Poisson with customer arrival to the shop rate given by $\lambda (t)=16-(t-4)^2$ Calculate
$P(N(5)-N(3)=40|N(4)=70)$ where $N(i)$ means the number of arrivals in
the first $i$ hours. The shop is open for 8 hours.
I thought about using the formula:
$$P(N(5)-N(3)=40)=\frac{e^{-(\Lambda (5)-\Lambda (3)}(\Lambda (5)-... |
True/False: Let G be a group with $|G| = p^n$ for a prime number $p$ and $n \in \mathbb{N}$, and let $\mathbb{K}$ be a field of characteristic $p$.
Is every representation of $G$ over $\mathbb{K}$ trivial?
attempt: False. Because we would need irreducibly.
Can someone please verify and provide some feedback? Thanks
|
If $f:[a,b] \to \mathbb R$ is such that
$$\lim_{n \to \infty} \frac 1n \sum_{k=1}^n[f(k/n)] = 1,$$
does that imply that $f$ is Riemann integrable on $[a,b]$ ?
Thank you.
|
What are the possible known methods for solving first order PDEs in higher dimensions? Is there anything else besides the method of characteristic curves?
In particular, I have four first order, linear, non-homogeneous PDEs in 8 dimensions which I (unfortunately) have to solve analytically.
|
Problem:
$f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be tangent to $f(x)$ at $P$, then find $∠XPC$ + $∠XPB$.
My attempt:
Other than brute force (which I'm not sure will even work... |
Claim: Let $L \subseteq \Sigma^*\{\#\}\Sigma^*$ be a context-free language, where $\# \notin \Sigma$. Suppose that for each $x \in \Sigma^*$, $\{y|x\#y \in L\}$ is finite. Then $\{y|\text{ for some } x\in \Sigma^*, x\#y \in L\}$ is a regular set.
I need to prove the above claim.
Since for each $x \in \Sigma^*$, $\{y|x\... |
This is probably a silly question and maybe in the end the answer is trivial but I can't see it.
The problem is the following.
Let $M$ be a Riemannian manifold, $\gamma \colon[0,l]\to M$ be a geodesic and let $X \in \mathcal{X}(M)$, where $\mathcal{X}(M)$ is the space of all the smooth vector fields on $M$. Assume that... |
Evaluate $$\int\int z^2\,dS,$$ where $S$ is the part of outer surface of cylinder $x^2+y^2=4$ between the planes $z=0$ and $z=3$.
The answer given in book is $\pi$ but I am not getting this answer.
|
suppose S is a metric space and $B(S)$ is the set of bounded functions and $C_b(S)$ is the set consisting of bounded continuous functions.
Prove that $C_b(S)$ is a closed subspace of $B(S)$.
I thought of looking at the complement $B(S) \backslash C_b(S) = \{f| \text{ f is bounded and not continuous}\}$ and proving this... |
The Chebyshev Polynomials of the second kind $U_n$ are the solutions of the differential equation
$$(1-x^2)U_n''(x)-3xU_n'(x)+n(n+2)U_n(x)=0$$
Alternatively they are defined inductively:
$$U_0(x)=1 \qquad U_1(x)=2x \qquad U_{n+1}(x)=2xU_n(x) - U_{n-1}(x)$$
My question is whether or not
$$\sum_n |U_n(x/2)|^2\tag{1}$$
di... |
Suppose we have a Hilbert space $\mathscr{H}$ and a bounded linear map $T\in\mathscr{B(H)}$ NOT necessarily self-adjoint. There seems to be loads of definitions of the essential spectrum of $T$. My question is, whether in the Hilbert space setting the following are equivalent:
$\lambda$ is such that $T-\lambda{I}$ is ... |
Suppose we have the following quadratic form:
$$
x^T(t)(A^TP+PA)x(t)\le-x^T(t)Qx(t)\quad\forall t
$$
where $P$ and $Q$ are symmetric positive definite matrices and $\dot x(t)=Ax(t)$. Why does the above imply that:
$$
A^TP+PA\le -Q\,\,?
$$
Thanks a lot!
|
Let $f:\;\mathbb{R}^n\to\mathbb{R}^n$ be differentiable. Suppose that
for all $x\in\mathbb{R}^n:$ $$\lVert
\mathrm{D}f(x)-\mathrm{I}\rVert\leq \frac{1}{2}$$ where
$\lVert\cdot\rVert$ is the operator norm. I need to show $f$ must be a
diffeomorphism.
By using the contraction mapping theorem I have shown that $... |
Find the centroid of the lamina described in polar coordinates as
$\left \{ \strut \left ( x,y \right )~|~0\leq r\leq 4 \cos\left ( \theta \right ),0\leq \theta \leq \frac{\pi}{3} \right \}$
Having a bit of a problem with this problem. Any hints to set me in the right direction would be ideal!
|
The sides of the rectangle $ABCD$ are parallel to the co-ordinate axes. If the coordinates of the vertices $B$ and $D$ are $(7,3)$ and $(2,6)$ respectively, find the coordinates of the vertices $A$ and $C$.
|
The integral can be represented as
$$
\int \frac{1+x}{\sqrt{1-x^2}}\mathrm dx=
\int \left(\frac{1+x}{1-x}\right)^{1/2}\mathrm dx
$$
Substitution $$t=\frac{1+x}{1-x}\Rightarrow x=\frac{t-1}{t+1}\Rightarrow dx=\frac{2}{(t+1)^2}dt\Rightarrow \int\limits \left(\frac{1+x}{1-x}\right)^{1/2}\mathrm dx=2\int\limits \frac{\sqr... |
If some topological space $X$ is locally compact, then each point of $X$ has at least one compact neighborhood. The book I'm reading now doesn't mention if such compact neighborhood of a point should contain an open neighborhood of the point. Would it be fair to assume that such compact neighborhood of a point contain ... |
A universe contains the three individuals $a,b$, and $c$. For these individuals, a predicate $Q(x,y)$ is defined, and its truth values are given by the following table:
\begin{array}{c|ccc}
x\backslash y&a&b&c\\\hline
a&T&F&T\\
b&F&T&F\\
c&F&T&T
\end{array}
Write each of the following expressions without quantifi... |
We start with the following function $g: (0,\infty)\rightarrow [0,\infty)$,
$$ g(x)=x+2x^{-\frac{1}{2}}-3.$$ From this function we need a 'smooth' square-root. Thus, we check $g(1)=0$,
$$g'(x)=1-x^{-\frac{3}{2}},$$ $g'(1)=0$ and
$$g''(x)=\frac{3}{2}x^{-\frac{5}{2}}\geq 0.$$
Therefore, we can define the function $f: ... |
This question is motivated by
Why is $\pi$ so close to $3$?,
Why is $\pi^2$ so close to $10$? and Proving $\pi^3 \gt 31$.
I. $\pi$ and $\pi^2$
There are series with all terms positive for $\pi-3$ and $10-\pi^2$
$$\pi-3=\sum_{k=1}^{\infty}\frac{3}{(4k+1)(2k+1)(k+1)}$$
(see Lehmer, http://matwbn.icm.edu.pl/ksiazki/a... |
I have two different functions $y_1=f_1(x)$ and $y_2=f_2(x)$, both invertible but quite complex. I am able to find their inverse functions numerically, i.e. $f^{-1}_1(x)$ and $f^{-1}_2(x)$, by solving for $x_1$ and $x_2$ two equations
$$
f_1(y_1) - x_1 = 0, \\
f_2(y_2) - x_2 = 0,
$$
for specific realizations of $y_1$ ... |
Say $X_i$ can have a real value in the range [1,100]. All $X_i$ are independent of each other and all values are equally likely.
So then $\mathbb{P}(X_1 < X_2) = \frac{1}{2}$, right?
But then, what is $\mathbb{P}(X_1 < X_2 \cap X_1 < X_3)$?
Hints would be greatly welcome.
Edit:
So $\mathbb{P}(X_1 < X_2 \cap X_1 < X_3)... |
In Section 6.8 of Undergraduate Algebraic Geometry by Reid, the author proved the following Theorem:
There is a natural isomorphism of vector spaces $(T_pV)^*\cong m_p/m_p^2$ where $^*$ denotes the dual of a vector space.
Here $T_pV$ is the tangent space to $V$ at $p$, where $V$ is a variety in $\mathbb{A}^n$. $m_p$ ... |
Consider an open disk of unit radius in the real (two dimensional) plane.If we want to define the Ricci curvature and bisectional curvature on that disc,what will be their equivalent forms and the significance?Also if our disc is considered to be in the complex one dimensional plane what will be the Holomorphic bisecti... |
Could I find the inverse function of the following integral equation?
I am going to write it as $h(x)=...$
The integral equation is:
$$R(i) = \int_{y\in S} {\frac{{w(y) g(y,i)_{}^\sigma }}{{\int_{x\in S} {h(x)g(x,y)_{}^\sigma f(x,y)_{}^\sigma dx} }}dy} $$
where S is a compact subset of a two-dimensional surface which h... |
Let $\mathbb{k}$ be an algebraically closed field. There are many interesting examples of $\mathbb{k}$-algebras $R$ which admit a large central subalgebra $Z_0$ such that $R$ is a free $Z_0$-module and $Z_0$ is an affine $\mathbb{k}$-algebra. For instance, this is the case if we consider the universal enveloping algebr... |
Find a continuous probability density function $f$ on $[0,6]$, such that $\mathbb{P}([0,2]) = 0.6$, $\mathbb{P}([1,4]) = 0.5$ and $\mathbb{P}([3,5]) = 0.2$.
After some calculations I came up with the piecewise constant function $$f(x)=\begin{cases}\frac{3}{10},&0\le x< 2\\\frac{1}{10},&2\le x\le6\\0,&\text{otherwise}... |
I am trying to evaluate $|\int_{\gamma}e^{z^2}dz|$ for $\gamma$ is a segment from $(-3-3i)$ to $3i$, and fail badly.
Tried to express $e^{z^2}$ as Taylor serie, but apparently it got me nowhere.
Any ideas?
|
From textbook: $f(x_1, x_2) = 2 e^{-x_2/x_1},$ where $ 0 < x_1 < 1$, and $ x_2 > 0.$ The marginal is $f(x_1) = 2x_1$, and accordingly $$f(x_2|x_1) = \frac{1}{x_1}e^{-x_2/x_1}.$$
My calculation $$E(X_2|X_1) = \int^{\infty}_{0} x_2 \frac{1}{x_1}e^{-x_2/x_1} = x_1.$$
Since the conditional is not given with $X_1 = x_1$, w... |
The elements of $A_n$ is either of the form $(a,b,c,...)...$ or of the form $(a,b)(c,d)...$
In both cases, the element is a product of an even number of transpositions, not pairwise disjoint in the first case. I thought I would just show that any product of two transpositions is a product of two 3-cycles.
Consider the ... |
Is $\tan^{-1}2$ an irrational number or a rational number? How to show that?
Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
|
Let the function $f$ be convex, $f :\Bbb R^n \rightarrow \Bbb R$ and let
$$S = \{x : f(x) \le b\}$$
The proposition states that the set $S$ is convex regardless of $b$. Can someone explain to me how this proposition holds for all $b$?
|
Question
Given $n,k$ find the number of integers between $1$ and $n$ that have odd number of divisors in {1,2,3,...,k}
Example
If $n=10$ and $k=3$, the numbers $1(1),5(1),6(1,2,3),7(1)$ have odd number of divisors each of which is less than or equal to $3$.
For instance 8 has 1,2 as its divisors so it's not qualified t... |
Suppose that $(X,d)$ is a compact metric space and $(E_n)$ is any sequence of nonempty closed subsets of $X$ with $E_{n+1}\subset E_n$ for all $n\in\mathbb{N}$. Show that $\cap_{n=1}^\infty E_n$ is nonempty.
What I know: Since $X$ is compact, we need to use the fact that every sequence in $X$ has subsequence which conv... |
I need help. I don't know if it is possible.
Example formula that uses English instead of math!
$f(x) = 3x$ + all previous values of $f(i)$ with $i$ from $0$ to $x-1$, where $x$ is a positive integer. To clarify,
$$f(0) = 0 + 0 = 0$$
$$f(1) = 3 + 0 = 3$$
$$f(2) = 6 + 3 + 0 = 9$$
$$f(3) = 9 + 9 + 3 + 0= 21$$
$$f(4) = 12... |
Let $X,Y,Z,Y',Z'$ be random variables where $Y\sim Y', Z\sim Z'$, $Y$ and $Z$ are independent, while $Y'$ and $Z'$ are, in the sense that we have
$p(X,Y,Z)=p(X|Y,Z)p(Y)p(Z)$
$p(X,Y',Z')=p(X|Y',Z')p(Y',Z')$
$p(X|Y,Z)=p(X|Y',Z')$
Is $H(X|Y,Z)=H(X|Y',Z')$?
It seems whether $p(Y,Z)$ factorises or not does not affect the ... |
Let $S = \sum_ {k=1}^\infty a_k $ where each $a_k$ is positive and irrational.
Is it possible for $S$ to be rational, considering the additional restriction that none of the $a_k$'s is a linear combination of the other ?
By linear combination, we mean there exists some rational numbers $u,v$ such that $a_i = ua_j + v$.... |
How does one show that Derived Algebra is nilpotent implies the lie algebra is solvable.
My attempt: Let $L$ be such a Lie-algebra then $[L,L]$ is nilpotent so it is solvable. So $[L,L]^{(n)}=0$ for some $n$. What can I do after this?
|
From Velleman's 'How to Prove it' book, there is one statement - written below - of which I don't know how to write the logical form of, and I'm wondering if somebody could write it out.
The statement is:
" Prove that there is a unique $A\in\mathscr{P}(U)$ such that for every $B\in\mathscr{P}(U), A\cup B = B)$ "
I know... |
I have a sample of n=1000. The sample covers cars being brought in for service after one year of ownership in my country. For each car, I know which defects it had when it was brought in. I'm trying to work out whether the occurrence of errors, which are specific for my country, are deviating from the global numbers o... |
Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{F}_p[x],$$ where $p$ is prime.
For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least
one root? And for which $\alpha, p \in \mathbb{Z}$ does $g(x)$
have exactly one root?
Supposing that $\beta \in \mathbb{Z}$ is a root of $g(x)$, then, by Euler's Theorem (or Fer... |
This is a question inspired by a recent MCMC paper and linked to an earlier question of Sam Livingstone that did not get any answer.
Given two Markov kernels $\mathfrak{K}$ and $\mathfrak{H}$ such that
$$\sup_{x\in\mathcal{X}}\vert\vert\mathfrak{K}(x,\cdot)-\mathfrak{H}(x,\cdot)\vert\vert_\text{TV}<\epsilon$$
where $$\... |
Please help me to solve this.
Prove that if $A$ and $B$ are $n\times n$ matrices, then $((AB)^{-1})^T=(A^{-1})^T (B^{-1})^T$.
a problem involve transpose and inverse of matrices. check the attachment
a product of two matrices ,the transpose of the inverse of AB
|
Show that the sign representation of $S_n$ is indeed a representation.
attempt: Recall the sign function of a permutation is given by $\mathrm{sgn}(\pi) = (-1)^k$.
Then recall a representation is a homomorphism.
So we we have to show $\mathrm{sgn}(\pi_1 \pi_2)$ = $\mathrm{sgn}(\pi)\mathrm{sgn}(\pi_2)$.
This is a 3 par... |
I can't figure out the answer for a question on my econometrics course. Somehow it seems simple, but still I can't seem to figure it out. Maybe I am thinking the wrong way about it. Could someone perhaps help me to figure this out? The question is a follows:
Let $E(X)=\mu$ and $\operatorname{Var}(X)=\sigma^2$. If $E(... |
The statement is
$e^x(1-e^x)\leqslant (1/4)$ for $x\lt 0$
I think the above statement is true by calculating approximate value at some points.
But how to prove it properly?
|
I have the following task:
Let $A\in \mathcal{M}_3(K)$ be a non-diagonalizable matrix where $K$
is a field and the characteristic polynomial of $A$ is
$c_A(\lambda)=-(\lambda-\lambda_1)^2(\lambda-\lambda_2)$, where
$\lambda_1,\lambda_2 \in K$ and $\lambda_1\neq \lambda_2$.
a) Show that $A=RBR^{-1}$ where $R$ i... |
In a national competition , ech school had to choose 10% of students who participated in the competition .
So my question is , why they didn't asked for $\frac{1}{10}$ of students who participated ?
For example for 200 students we have to choose 20 students ! For 120 we have to choose 12 . But for 20 we have to choose... |
Prof using the binomial theorem: for all integers $n ≥0$ and for all nonnegative real numbers $x$, $1+nx ≤(1+x)^n$.
Don't have a idea to start this one. I don't know how to use math induction yet, so I need a answer without that.
|
$p(x)=xe^{-x}$ for $x\geq 0$ or $0$ otherwise.
I tried to substitute $e^{-x}$ but then i found there is still a $x$ in front.
|
I want to compute the fundamental group of a space containing comb space along with boundary of $I \times I$ where I is $[0,1]$. Here is my attempt using transfinite induction and Van Kampen.
Let's call the given space $X$. Consider an open set $U_{1}$ say $(-1, 2/3) \times (-1, 2)$ and $U_{2}$ $(5/4, 2) \times (-1... |
I'm studying for a test and there's a question that I've tried and I don't understand:
Let $E$ be a binary relation on a set $A$; let a binary relation $F$ on $\mathcal P (A) \setminus \{\emptyset\}$ be: $(B,C) \in F \iff \exists b \in B, \exists c \in C : (b,c)\in E$. If $E$ is transitive is $F$ transitive?
(I need ... |
Let $M$ be a von Neumann algebra and $e,f$ be projections in $M$. For a given central projection $z\in M$, is the following true?
$$z(e\vee f)=ze\vee zf$$
|
I was given the following homework problem.
Let $f:X\rightarrow \bar{\mathbb{R}}$ a set function and $X$ be a measurable space whose $\sigma$-algebra is generated by a finite partition $E_1,\dots ,E_n$. Describe all such measurable set functions.
Since unions and intersections commute, it seems measurable sets are just... |
I've just been trying to understand taylor polynomials more intuitively from a visual perspective. But as soon as the terms start using second derivatives, it becomes unclear.
1, Some thought's I've had:
Consider the expression
$\frac{f(x)-f(a)-f'(a)(x-a)}{(x-a)^2}$
Using L'Hopital's rule, we know this goes to $\frac{1... |
Assume that the group of infinite order is also cylic. How would one prove that? I am quite stuck.
|
I have the following relation
$$X -1\quad 0 \quad 2 \quad 3$$
$$Y -8 \quad 3 \quad 1 \quad 12$$
I can define the relation as
$\begin{array}{rcrl}
11X-Y&=&-3&\text{ for $X$ between $-1$ and $0$}\\
X+Y&=& 3&\text{ for $X$ between $0$ and $2$}\\
11X-Y&=&21&\text{ for $X$ between $2$ and $3$}\end{array}$
Obtained these... |
Let $A$ be a commutative ring with unit, and $P$ a prime ideal.
My question is:
If $I$ is an irreducible ideal in $A$, is $(I+P)/P$ irreducible in $A/P$? If not, can you show a counterexample?
Of course, if $(I+P)/P$ is reducible in $A/P$, then $I+P$ is reducible in $A$, because
$$ \dfrac{I+P}{P} = \dfrac{J_1}{P} +... |
A definition of the derivative is that it is the slope of the tangent line. For example, $x^3$ has a quadratic derivative. How could the slope of the tangent line be non-linear?
|
Consider a linear partial differential equation
$$
(L u)(x)=f(x) \quad \forall x \text{ in } \Omega\\
u=g \quad\text{on }\partial\Omega.
$$
Assuming that $f$ and $L$ depend on a parameter $y\in\mathbb{R}$, $f=f(x,y)$, $L=\sum_{\alpha} a_\alpha(x,y)\partial^\alpha$ how is the regularity of $u$ with respect to $y$ relate... |
Have $\int_C(4z^2-2iz)dz$ integral.
Does it depend on choice of path?
Tried to express $f(z)=(4z^2-2iz)$, then $f(x+yi)=(4x^2-4y^2+2y)+i(8xy-2x)$
Then $\frac{\delta P}{\delta y}=-8y+2$
And $\frac{\delta Q}{\delta x}=8y-2$
so $\frac{\delta P}{\delta y}=-\frac{\delta Q}{\delta x}$
Does this mean, that integral doesnt dep... |
I have for example: $\int_{y_0}^{y} \frac{1}{2\eta} d\eta$ with $t_0$ a real constant
and the solution was: $\frac{1}{2}(ln|y|-ln|y_0|)$
But on other case I had: $\int_{y_0}^{y} \frac{1}{\eta} d\eta$
and the solution was: $ln(y)-ln(y_0)$
How can I know when I should use absolute value?
|
I have a real $ 4\times4$ matrix of the form
$$
C = \begin{pmatrix}
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
c_{31} & c_{32} & 0 & c_{34} \\
c_{41} & c_{42} & c_{43} & c_{44}
\end{pmatrix}
$$
The coefficients satisfy
\begin{align*}
c_{31}, c_{32}, c_{41} &\ge 0,\\
c_{42} &> 0,\\
c_{34}, c_{43}, c_{44} &\le 0,
\end{align*}
... |
I found a pdf file on the internet which gives you known expansions of Taylor's.
There is something I cant understand :
Why is the remainder of $\cos x$ is written like this?
$$\frac{\cos ^{(2n+2)}(c)x^{2n+2}}{(2n+2)!}$$
And not like this:
$$\frac{\cos^{(2n+1)} (c)x^{2n+1}}{(2n+1)!}$$
when the last element was :
$$\fr... |
Putting all vectors (matrices) in one gives
$$ \begin{bmatrix}
1 & 0 & 0 & m_1 & \cdots\\
0 & 1 & 0 & m_2 & \cdots\\
0 & 0 & 1 & m_3 & \cdots\\
\end{bmatrix}$$
The reduced row echelon form of this matrix is always that matrix.
Does this mean that the dimension of $\operatorname{sp... |
I am reading Douady and Hubbards "On the dynamics of polynomial-like mappings". I am relatively new dynamics of complex maps, and I would appreciate some help with aspects of the following.
Definition: Let $f:U'\rightarrow U$ and $g: V'\rightarrow V$ be two polynomial-like maps. They are termed quasi-conformally equiva... |
I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$.
I know this loop forms a triangle with all three sides changing with respect to the $x$ and $y$ axis. I understand how to do the integration part, but unfortunately I am really s... |
Playing around with the concepts of negativity and positivity, I came across the following equivalence relation defined for all elements $x,y$ of a field $\mathbb{F}$:
$$
x \sim y :\Longleftrightarrow x^2 = y^2.
$$
I would like to read more about it; but without a name, I cannot find appropriate references. A superfici... |
Suppose that $z+\frac{1}{z}=2cos\theta$, where $\theta$ is a real number. Then show that, for any integer $n$, $$z^n+\frac{1}{z^n}=2cos(n\theta)$$
My first thought was to use induction to achieve the conclusion but the manipulation does not reach the conclusion.
|
I am struggling with this problem for hours but it seems to be easy. Here is the problem:
Proof that every vertex $v$ in 2-connected graph $G$ has neighbour $u$ such that $G - v - u$ is connected.
Any help is highly appreciated! I think that I might be missing a trivial solution here =(
EDIT:
So far, I tried to create ... |
In an exercise I was asked to show that if $R$ is a ring with relatively prime ideals $I_1,I_2$ then $R/I \cong R/I_1 \oplus R/I_2$ where $I=I_1 \cap I_2$ and $\oplus$ is the direct sum. A follow on exercise asks to show, using the above result, that $\varphi(mn)=\varphi(m) \varphi(n)$ provided that gcd($m,n)=1$ and $\... |
I can intuitively understand that as I take more samples from a random variable $X$ (gaussian distribution), the mean would approach $E(X)$.
But what I don't get is if I look at it mathematically. Say I took two samples $x_1$, $x_2$. The mean would be $(x_1+x_2)/2$. But variances add, which is contrary to the previous ... |
I was in need to urgently solve this integral. I already know the result in the closed form, does anybody know how to solve it?
\begin{equation}
\int_{\mathbb{R}}e^{-\frac{x^{2}}{2}}\left(\cos\left(\pi nx\right)\right)dx=\sqrt{2\pi}e^{-\frac{n^{2}\pi^{2}}{2}},
\end{equation}
If anybody already knows it is welcome, ot... |
Consider the recurrence relation: $a_{n+1}a_{n-1} = 1 + a_n$ with initial values $a_1=x$ and $a_2=y$.
Is this an example of a homogeneous equation or just a linear one?
In any case does anyone have an idea how to solve this?
|
My task is: "Describe rotation $S \circ R$ by axis and angle, where $R$ is rotation around $(0,1,1)$ by 90 degrees, and $S$ is rotation around $(1,-1,0)$ by 90 degrees." I should use quaternion multiplication to solve that. How do I do that? The way I see it now, I represent $S$ with $e^{\pi/4 (j+k)}$ and $R$ with $e^{... |
Given any 40 people, at least four of them were born in the same month of the year.
Why is this true?
|
The norm
\begin{equation}
||u||^{2} = \int_{\mathbb R} 1 \cdot u(x) \cdot \overline{u(x)} dx
\end{equation}
Claim which should be correct
\begin{equation}
||u||^{2} = \int_{\mathbb R} 1 \cdot u(x) \cdot \overline{u(x)} dx \\
= /_{\mathbb R} x u(x) \overline{u(x)} - \int_{\mathbb R} x \left( u(x) \overline{u'(x)} + u'(x... |
I have been studying Reimann's functional equation (symmetric form) for Zeta(s) and wondering if a similar functional equation has been derived for the Hurwitz Zeta function?
|
Say you have three vectors $u,v$, and $w$ in $\mathbb{R}^3 $ that are linearly independent. Prove that the two vectors $u+w$ and $v+w$ have to be linearly independent.
(start by assuming $c_1(u+w)+c_2(v+w)=0$, and use the given information to prove that $c_1$ and $c_2$ must both be $0$.
So I understand that linear de... |
From Legendre's Theorem we know
$$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$.
Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for gamma function. Something like when $x$ is a real number,
$$\Gamma (x+1)=\prod_{p}p^{f_{p}(x) }$$
where i don't know what $f... |
So two independent analyses of a content in a water sample have been made using two different methods, both without systematical errors but with different standard deviations. Method $B$ is assumed to have twice the standard deviation as method $A$. $x_A$ and $x_B$ are the measured contents and $\mu$ is the real conten... |
Let $\Omega \subset \mathbb{R}^n$ be open and bounded, and let $Q \subset \Omega$ be a hypercube. Furthermore, denote by $D$ the $n$-dimensional unit cube $(0,1)^n$. Let $k \in \mathbb{N}$ be big, i.e., $k \gg 1$.
My question is the following:
How many translates of $D$ are completely contained in $k Q$?
In the pape... |
I have the following first order ODE to be solved via the integrating factor method:
$$\frac{\mathrm{d}z}{\mathrm{d}y}\pm z=-\frac12y\tag{1}$$
This is in the general form: $$\frac{\mathrm{d}a}{\mathrm{d}x}+P(x)a=Q(x)$$
Which can be solved with the use of the formula:
$$a e^I=\int Q(x)e^I\mathrm{d}x$$ where $$\begin{ali... |
Suppose that you have a bucket holds fiv-sev c, and one holds tw-one c. How
could you use them to measure out thre c of water?
|
For all $n \ge 1$,
$$\binom{n}{0} - \frac{1}{2}\binom{n}{1} + \frac{1}{2^2}\binom{n}{2} - \frac{1}{2^3}\binom{n}{3} + \cdots + (-1)^n\frac{1}{2^{n-1}}\binom{n}{n-1} = 0,$$ if $n$ is even
$$\frac{1}{2^{n-1}}, \text{ if } n \text{ is odd}$$... |
Find an upper triangular matrix $A$ such that $A^3=\begin{pmatrix}8&-57\\0&27\end{pmatrix}$.
I tried to solve this problem using Cayley–Hamilton Theorem, but I am unable to solve that.
|
It is well known that the four parameter logistic law has the following form
$$ F(x)=D+\frac{A-D}{1+\Big(\frac{x}{C}\Big)^B} $$
What characterise this curve is its four parameters.
A=starting point of the curve.
B= steepness of the curve.
C=inflection point.
D= Maximum asysmptote.
If I am stating the definition of t... |
I am trying to learn differential forms. I have read some scripts about differential forms and now I am trying to solve some problems.
So the problem is:
given $f: \mathbb{R}^2 \to \mathbb{R}^3, f(x_1,x_2):=(x_1+x_2, -x_1, -x_2)^T$
Now I have to calculate the basis representation of
$f^*(dx_1 \wedge dx_2 \wedge dx_3) ... |
I have a question about the derivation of the SVM algorithm (for example, page 3 here ). The question is about the math, so that's why I'm asking this here.
Suppose I have the following optimization problem:
$$min_{w, \xi} \frac{1}{2} ||w||^2 +C \sum_i\xi_i,\\ s.t. \ y_i w x_i \geq 1 - \xi_i, \ i = 1, ... m, \\ \xi_i... |
Let $M$ be a smooth manifold. As an example of a global property that arises from local data we know that if $(M,g)$ is a compact surface without boundary then the Euler characteristic is given by
$$
2\pi \, \chi(M) = \int_M K_p \; dA
$$
where $dA$ is the Area element associated to $g$ and $K_p$ is the scalar curvatur... |
Some time ago I saw this in my teacher's room. She called this picture in honor of some scientists (Lagrange,Lie or Liouville, or some other, but I don't remember). Please, name this picture. Thank you!
|
I need to find the volume of a torus-shaped object, but it which doesn't have space between the ring.
We can find the volume of the ring, but what about the inner part?
PS: What is that shape called?
EDIT:
I've edited the title. What I mean, is a Cylinder, with rounded sides.
|
I have solved this problem in a way, rather "inspired". I would like to have a solution found an easier way but I was unable so far.
Let $A,B,C$ the angles of a triangle $\triangle {ABC}$; prove that
$$(\sin^2 A+\sin^2 B+\sin^2 C)\le \frac14 \left(\frac{1}{\sin A}+\frac{1}{\sin B}+\frac{1}{\sin C}\right) (\sin A+\sin ... |
Let $(L_1,L_2,L_3)$ be an ordered triple of pairwise distinct plane in $K^3$. Prove that there is two possibles types of relative arrangement of such triples characterized by the fact that $\dim L_1\cap L_2\cap L_3 = 0$ or $1$.
In fact I imagine that I have to manipulate certains inequalities, like, taking two of these... |
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