text stringlengths 83 79.5k |
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H: Linearizing $\, y''+\sin(2x+\cos(2y'+y))+1-\sin(y+3y')=0$
I'd really love your help with the following problem. I need to make linearization and then to find a general solution of the accepted equation for the differential equation:
$$y''+\sin(2x+\cos(2y'+y))+1-\sin(y+3y')=0$$
I usually get a system of two equation... |
H: Number of well-ordering relations on a well-orderable infinite set $A$?
Given a well-orderable infinite set $A$, can we always say that the set $$\left\{R\subset A\times A:\langle A,R\rangle\, \text{is a well-ordering}\right\}$$ has cardinality $2^{|A|}$? How much Choice is required for the proof of this?
I believe... |
H: Rate of Change Question
I have this equation for a rate of change problem below
$$s = t^
4
− 4t^
3
− 20t^
2
+ 20t,\qquad t \geq 0$$
The question asks me
At what time does the particle have a velocity of 20 m/s?
How do i solve this? Basically what steps do I take to find at what time the particle has that veloci... |
H: Which number was removed from the first $n$ naturals?
A number is removed from the set of integers from $1$ to $n$. Now, the average of remaining numbers turns out to be $40.75$. Which integer was removed?
By some brute force, I got $61$. I want to know if there's any analytic approach?
AI: The average of the integ... |
H: Limit of $\prod_{i=1}^n (1-1/2^i)$
I am trying to find the limit of the sequence $$s_n:=\displaystyle \prod_{i=1}^n \left(1-\frac{1}{2^i} \right)$$
The sequence is decreasing and bounded below by $0$. I guess that the limit is $0$, is there any way to show this ? Or, is there any argument which shows that the limit... |
H: Proof $\lim\limits_{n \rightarrow \infty} {\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}=2$ using Banach's Fixed Point
I'd like to prove $\lim\limits_{n \rightarrow \infty} \underbrace{\sqrt{2+\sqrt{2+\cdots+\sqrt{2}}}}_{n\textrm{ square roots}}=2$ using Banach's Fixed Point theorem.
I think I should use the function $f(x)=... |
H: Opening Doors Puzzle
You are in a corridor which has N doors all of the doors are opened.
Every time someone passes through the corridor, he closes randomly
with equal probability a certain number of doors between 1 and the
number of remaining opened doors. What is the expected number of
people that need t... |
H: ${10 \choose 4}+{11 \choose 4}+{12 \choose 4}+\cdots+{20 \choose 4}$ can be simplified as which of the following?
${10 \choose 4}+{11 \choose 4}+{12 \choose 4}+\cdots+{20 \choose 4}$ can be simplified as ?
A. ${21 \choose 5}$
B. ${20 \choose 5}-{11 \choose 4}$
C. ${21 \choose 5}-{10 \choose 5}$
D. ${20 \choose 4}$
... |
H: Differentiable map conserving geodetic lines which is no isometry
I am looking for a differentiable map $f: S^n\rightarrow S^n$, which conserves the geodetic lines of the standard metric on $S^n$, but is no isometry.
The geodetic lines on $S^n$ should be the great circles, I think.
Unfortunately, I was only able to... |
H: Normed linear space question
Suppose $x_1, x_2, \ldots, x_n$ are linearly independent elements of a normed linear space $X$. Show that there is a constant $c>0$ with the property that for every choice of scalars $\alpha_1, \ldots, \alpha_n$ we have $$\|\alpha_1x_1+\cdots+\alpha_nx_n\|\geq c(|\alpha_1|+\cdots+|\alp... |
H: Find the sum of this series :$ \frac{1}{{1!2009!}} + \frac{1}{{3!2007!}} + \cdots + \frac{1}{{1!2009!}}$
Find the sum of this series :
$$\sum\limits_{\scriptstyle 1 \leqslant x \leqslant 2009 \atop
{\scriptstyle x+y=2010 \atop
\scriptstyle {\text{ }}x,y{\text{ odd}} }} {\frac{1}{{x!y!}}} = \frac{1}{{1!2... |
H: Exponentiating cubic functions (and so on)
The exponential distribution follows from exponentiating a linear function, while the normal distribution follows from exponentiating a quadratic function. Do the distributions that follow from exponentiating functions of 3rd or higher degree have names or uses?
AI: Such d... |
H: first order logic question model
suppose we have a model for a language in first order logic $ M=<D,I> $
such that D is the domain and I is the interpetation such that for every $ a \in D $ we have a closed noun (a noun with no free variables) $ t_0 $ such that $ I[t_0] = a $
prove or disprove the following:
a) if ... |
H: Integrability of the Newton potential
For $x\in\mathbb{R}^n$ we define the Newton potential as follows:$$N(x) = \begin{cases} \frac{\log|x|}{2\pi}, & n=2 \\[10pt] \frac{|x|^{2-n}}{(2-n)\omega_n}, & n>2\end{cases}$$
where $\omega_n$ denoted to the volume of the n-ball. Moreover, let $\chi_r$ denote to the charact... |
H: Finding $3$ distinct prime numbers $a| (bc+b+c)$, $b|(ac+a+c)$, $c|(ab+a+b)$
How to find $3$ prime numbers $a,b$ and $c$ such that:
$$a| (bc+b+c)$$
$$b|(ac+a+c)$$
$$c|(ab+a+b)$$
AI: There are no such primes:
Without loss of generality assume $a<b<c$. Clearly 2 cannot be one of the primes, so $a\geq 3$ , $b\geq 5$ a... |
H: Uniform convergence of $\sum\limits_{n=1}^\infty \frac{x}{x^2+n^2}$
How to prove that $\sum\limits_{n=1}^\infty \frac{x}{x^2+n^2}$ is uniformly convergent for every $x$?
I was trying all sort of ways, but it think the answer might be in solving the problem for $|x|<1$ and then for $|x|>1$.
Its easy to show that for... |
H: Convergence of sequence: $X_n = \frac{n}{n^2 + 1} + \frac{n}{n^2 + 2} + \frac{n}{n^2 + 3} + \cdots + \frac{n}{n^2 + n}$
What I am trying is finding $A_n < X_n < B_n$, proving that $A_n$ and $B_n$ converge, and then $X_n$ converges.
I first found that $A_n = \frac{n}{n^2 + 1}$ converges because the limit is zero.
I ... |
H: Does the specification of a general sequence require the Axiom of Choice?
Many results in elementary analysis require some form of the Axiom of Choice (often weaker forms, such as countable or dependent). My question is a bit more specific, regarding sequences.
For example, consider a standard proof of the boundedn... |
H: Non-$C^{*}$ Banach algebras?
It suddenly occurred to me almost every Banach algebra I know is actually a $C^{*}$ algebra. Several kinds of function algebras are definitely $C^{*}$ algebras. So is the matrix algebra. Although one gets a non-$C^*$ algebra by focusing on the upper triangular matrices, the norm still s... |
H: Parallel lines passing through $(d, 3)$ & $(-2,1)$ and $(5,d)$ & $(1,0)$.
There are two parallel lines with one passing through $(d, 3)$ & $(-2,1)$ and the second line passes through $(5,d)$ & $(1,0)$.
Find the two values for $d$.
I found one value would be $-4$. But how would I do this to find the second value?
AI... |
H: Polar equation of $y = 2$
Maybe I do not understand what is going on here but I cannot get the right answer.
$$y = 2 $$
$$y^2 + x^2 = r^2$$
$$4 + 0^2 = r^2$$
$$ r = 2$$
$$y = r \sin \theta$$
$$1 = \sin \theta$$
$$\theta = \pi/2$$
This is wrong but I do not see anything wrong with my logic.
AI: Just because $y=2$, i... |
H: Odd-dimensional complex skew-symmetric matrix has eigenvalue $0$
There is the standard proof using $$\det(A)=\det( A^{T} ) = \det(-A)=(-1)^n \det(A)$$
I would like a proof that avoids this. Specifically, there is the proof that for $A$ a $\bf{real} $ matrix, the transpose is the same as the adjoint, which gives (u... |
H: Polar equation of cartesian $y = 1 + 3x$
I have no idea at all what to do on this
I got $$\cos^{-1} \left(\frac{r\sin \theta+1}{3} \right) = \theta$$
Which can be
$$\cos^{-1} \left(\frac{r\sin \left(\cos^{-1} \left(\frac{r\sin \theta+1}{3} \right) \right)+1}{3} \right) = \theta$$
$$\cos^{-1} \left(\frac{r\sin \le... |
H: What is the meaning of "unitize a vector"?
The expression "to unitize a vector" is often use in computational geometry. What does it mean?
AI: Yeah, it's a perversion of normalize. If $v\not = 0$, we normalize v as follows
$$w = {v\over \|v\|}.$$
Why this ugly neologism is needed is beyond me. |
H: What does the letter $t$ stands for in "$t$-value of a curve"?
The "$t$-value of a curve" is the parameter of a point on a curve. For example if a curve's domain is form $0$ to $1$, then a t-value of $0.5$ would be a mid point. What does the letter $t$ stand for?
AI: $t$ is the underlying parameter. Any "nice" one-... |
H: A (potential) projection operator
Let $I$ be a 3-by-3 identity matrix and $\hat n$ be a unit vector orthonormal to some surface. What then does $I-\hat n\hat n$ mean geometrically? Also what does it mean to multiply this matrix/operator by a vector $v$? Some sort of projection?
Also, I don't understand what $\hat n... |
H: What are the $\{x,y,z\}$ values of a vector?
A vector is often described as $\{x,y,z\}$ similarly to a $3$D point's cartesian coordiantes in CAD tools which is quite confusing. What are the $x$, $y$ and $z$ values in the case of a vector?
AI: First, $\{x,y,z\}$ denotes a set of things with no order. Coordinates are... |
H: How does partial fraction decomposition avoid division by zero?
This may be an incredibly stupid question, but why does partial fraction decomposition avoid division by zero? Let me give an example:
$$\frac{3x+2}{x(x+1)}=\frac{A}{x}+\frac{B}{x+1}$$
Multiplying both sides by $x(x+1)$ we have:
$$3x+2=A(x+1)+Bx$$
whe... |
H: Translation request: theorem concerning automorphisms of $p$-groups
I want to translate the following theorem from German to English:
5.12 Satz. Sei $\mathfrak{P}$ eine $p$-Gruppe und $\alpha$ ein Automorphismums von $\mathfrak{P}$ von su $p$ teilerfremdder Ordnung.*
Für $p\gt 2$ lasse $\alpha$ alle Elemente der ... |
H: Comparing the time in the time-distance problem
The question is:
Two trains start at point A and B and travel towards each other at a speed of $50$km/hr and $60$Km/hr respectively. At the time of meeting the second train has traveled $120$ km more than the first train. Now the distance between them is:
Now I did ... |
H: Odd primes $\neq5$ divide a integer made up of all $1$s. Ditto for integers coprime to $10$.
Okay, so I have worked on this problem and even though I can see it is true I just don't know how to show it.
Here it goes.
Question:
Show that every odd prime except $5$ divides some number of the form $111 \ldots 11$ ($k$... |
H: Set of convergence is measurable.
Possible Duplicate:
pointwise convergence in $\sigma$-algebra
Problem: Prove that the set of points at which a sequence of measurable real functions converges is a measurable set. (I believe the problem means functions from the reals to the reals.)
Source: W. Rudin, Real and Com... |
H: Hensel lifting square roots $\!\bmod p\,$ to $\!\bmod p^2$
I've been working on this problem for a while, but hit a dead end.
Here's the problem:
Suppose $p$ is an odd prime. Also let $b^2 \equiv a \pmod p$ and $p$ does not divide $a$. Prove there exists some $k \in \mathbb{Z}$ such that $(b+kp)^2 \equiv a \pmod {p... |
H: Possible errors in my professor's notes, Abel summation
In my professor's notes he has written this:
$$\int_1^N \frac{\{t\} - \frac{1}{2}}{t}dt = \int_1^N\frac{1}{t}d \left(\int_1^t B(y)dy \right) = \int_1^t B(y)dy|_1^N + \int_1^N \frac{\int_1^t B(y)dy}{t^2}dt$$
Where $\{x\}$ indicates the fractional part of $x$ an... |
H: a) Prove that $f$ has a removable singularity if $f'$ does; b) Evaluate $\int_0^\infty\frac{\log x}{(1+x)^3}\,dx$
a) Let $\,f\,$ be an analytic function in the punctured disk $\,\{z\;\;;\;\;0<|z-a|<r\,\,,\,r\in\mathbb R^+\}\,$ . Prove that if the limit $\displaystyle{\lim_{z\to a}f'(z)}\,$ exists finitely, then $\,... |
H: How do I prove this equality?
A vector x $\in \mathbb{R}^3$ makes angles $a, b, c$ with the
three axes. Using the dot product with standard basis vectors, show:
$$\cos^2a + \cos^2b + \cos^2c = 1$$
I wasn't sure how to start this one. Conceptually, I can see what was going on, but I would think to create a numbe... |
H: Antiderivative of $x \mapsto \operatorname{tr}\{(\mathbf{A}x+\mathbf{B})^{-1}\mathbf{C}\}$
I am wondering how to integrate the function
$$x \mapsto \operatorname{tr}\bigl\{(\mathbf{A}x+\mathbf{B})^{-1}\mathbf{C}\bigr\}$$
In my case, the matrices $\mathbf{A}$, $\mathbf{B}$, $\mathbf{C}$ are (strictly) positive defin... |
H: a question about conditional expectation
Suppose that X and Y are random variables such that E(Y/X)=aX+b,how determine expressions for a and b in terms of E(X),E(Y),Var(X),and Cov(X,Y).assuming that Cov(X,Y) exists and Var(X)>0.
AI: Using the tower property, recall that $E[E[Y|X]] = E[Y]$. You should try to work th... |
H: Conditions for random variables to be negligible in their sums
The followings are from Kai Lai Chung's A Course in Probability Theory, on page 207
$X_{n,j}, j=1,\dots, k_n, n=1,\dots,$ are random variables, where $k_n \to \infty$ as $n\to \infty$.
For every $\epsilon > 0$:
(a) for each $j$, $\lim_n P(|X_{nj}|>\eps... |
H: Simplify $\frac{9}{2}(1 + \sqrt 5)\sqrt{10 - 2\sqrt 5} + 9\sqrt{5 + 2\sqrt 5}$
Simplify $\displaystyle{\frac{9}{2}(1 + \sqrt 5)\sqrt{10 - 2\sqrt 5} + 9\sqrt{5 + 2\sqrt 5}}$.
I get this when I was doing another Q,
but I don't know how to further simplify it.
Can anyone help me, please?
AI: Hint: Let's note $o:=\frac... |
H: Need help about $P\Gamma L_2(q)$, $q=4,3$
I am asking kindly,
For which values of $n$ we have $$S_n≅P\Gamma L_2(3),S_n≅P\Gamma L_2(4)$$ This may be correct if we replace $S_n$ by $A_n$.
Any help will be appreciated. :)
Edit (JL): Adding the definition of the group. Let $\Omega$ denote the set $GF(q)\cup\{\infty\... |
H: solving differential equation: $y''-2y'\tan x=\frac{1}{\cos^3x}$
I'd really love your help with solving we following differential equation $$y''-2y'\tan x=\frac{1}{\cos^3x}.$$
First I tried to do it with $z=y'$ but it's just impossible,$z$ is a big and not nice expression, and to integrate it would be very hard pr... |
H: Example of an infinite sequence of irrational numbers converging to a rational number?
Are there any nice examples of infinite sequences of irrational numbers converging to rational numbers?
One idea I had was the sequence: $ 0.1001000010000001\cdots,0.1101000110000001\cdots,\cdots,0.1111000110000001\cdots,$ etc.
... |
H: $\int_{\partial \Omega}\frac{\partial u}{\partial N}d\sigma $ if $\Delta u = 0$ in $\Omega$
Can you help me please with this problem?
What do you know about $\int_{\partial \Omega}\frac{\partial u}{\partial N}d\sigma $ if $\Delta u = 0$ in $\Omega$?
Is it always possible to solve an equation $u''=0$ in $[0,1]$ if... |
H: Volume of the rotation of the area between two curves
Suppose I have two function $f(x)$ and $g(x)$ such that for $x \in (\alpha, \beta )$ we have $f(x) \ge g(x) $.
I found an "exercise solution" that state that the volume given by the rotation of the area between $f$ and $g$ is:
$$\pi\int_\alpha^\beta (f(x) - g(x)... |
H: linear algebra: inverse of a matrix
The inverse of the matrix
$A=\left( \matrix{1 & \frac{1}{2} & \frac{1}{3} \\ \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ \frac{1}{3} & \frac{1}{4} & \frac{1}{5} }\right)$
is
$A^{-1}=\left( \matrix{ 9 & -36 & 30 \\ -36 & 192 & -180 \\ 30 & -180 & 180 } \right)$.
Then, perhaps the m... |
H: How could I calculate this sum of series
Could you please help me to calculate this finite sum ?
\begin{align}
S=\sum _{n=1}^{\text{Nmax}} n*P^n
\end{align}
where $P\leq 1$ and $\text{Nmax}>1$.
AI: $$S=P.\sum_{n=1}^N nP^{n-1}=P.\frac{d}{dP}\sum_{n=1}^NP^n=P.\frac{d}{dP}\frac{P(1-P^N)}{1-P}$$ So $$S=\frac{(1-P)(1... |
H: How to find the product of n terms of an arithmetic progression where common difference is not unity?
How can we find the product of $n$ terms of an arithmetic progression where common difference is not unity?
I just want to know the last $3$ digits of $7 \times 23 \times 39 \times \ldots \times 2071$ where common ... |
H: Deriving the Formula for Average Speed (Same distance).
Let me start of by specifying the question:
A and B are two towns. Kim covers the distance from A to B on a scooter at 17Km/hr and returns to A on a bicycle at 8km/hr.What is his average speed during the whole journey.
I solved this problem by using the fo... |
H: Fixed Block is an orbit?
Reviewing some of my old questions here, I am stuck at a comment in which Prof. Holt gave me an interesting example (A small one) about non-transitive $1/2-$transitive group. Here is the link {http://math.stackexchange.com/q/138937/8581}. Of course, he noted a complete answer by giving two ... |
H: A form of cumulative distribution
Let $f(x)$ and $g(x)$ be two probability density functions. Does the expression:
$$
C = 2\int _{-\infty}^{\infty}\left[f(x)\int _{-\infty}^{x}g(y)\,dy\right]\,dx
$$
have any meaningful graphical representation when $E[Y]>E[X]$, where $X$ has $pdf$ $f(x)$ and $Y$ has $pdf$ $g(x)$?... |
H: solve ODE solution with intial value
Consider the initial value problem $y' + \frac{2}{3}y = 1-\frac{1}{2}t, y(0) = y_0$ Find the value of $y_0$ for which the solution touches, but does not cross, the $t$-axis. Solving this equation with $\mu(t)=e^{\frac{2}{3}t}$ we get $y=\frac{21}{8}- \frac{3}{4}t + C e^{-\frac{2... |
H: A lemma about extension of function
Definition
Suppose that $f(M)$ is a $\mathcal C^n$-function whose domain is $\mathcal X$. If $f^*(M)$ is a $\mathcal C^n$-function whose domain is $\mathcal X^*$, and $f(M)=f^*(M)$ whenever $M\in\mathcal X\cap\mathcal X^*$, we call that $f^*$ is an ($\mathcal C^n$-)extension of $... |
H: Calculation of $ \frac{\partial^2 g(f) }{\partial x_1 \partial x_2} $
Let $f : \mathbb R^2 \to \mathbb R$. $ g : \mathbb R \to \mathbb R$. Assume that all partial derivatives exist. Then is this statement right? $$ \frac{\partial^2 g\circ f }{\partial x_1 \partial x_2} = \frac{\partial}{\partial x_1} \left( \frac{\... |
H: Finding area of triangle
if the sides of the triangle are given by 20 cm, 30 cm, and 60 cm find the area of the triangle.
I tried a long time.
Apparently, Heron's formula does not seem to work
$\sqrt{s(s-a)(s-b)(s-c)}$
where $s = (a+b+c)/2$
In the above problem $s=55$ and thus we end up with a negative number i... |
H: Show that $6m \mid (2m+3)^n + 1$ if and only if $4m \mid 3^n + 1$
Let $m$ and $n$ be positive integers. How to prove that $$6m \mid (2m+3)^n + 1$$ if and only if $$4m \mid 3^n + 1$$
AI: $\newcommand{\jaco}[2]{{\left(\frac{#1}{#2}\right)}}$I have tried to solve this - I hope I did not make too many mistakes there. ... |
H: APB is congruent to DPC. Find $x$
I wrote in $x+23$ for DPC but then I'm stuck heres the image since I can't embed:
APB is congruent to DPC.
APB is $x+23$
BPC is $x$
BPD = $4x - 39$
Which can be seen here:
AI: HINT: BPC + CPD = BPD. So we have that $2x + 23 = 4x - 39$. Solve for $x$. |
H: Failure of uniqueness for linear ODE
Let's say we have a linear ODE with polynomial coefficients $p_j(x)$:
$$
p_n(x) y^{(n)}(x)+\dots+p_1(x)y'(x)+p_0(x)y=0
$$
and let's say $x_0$ is a root of $p_n(x)$. What can be said about uniqueness of the IVP for this ODE at $x_0$? In particular, if I succeeded to prove that a... |
H: A problem about a matrix in $SL_2(\mathbb Q)$
If a matrix $A\in SL_2(\mathbb Q)$ has finite multiplicative order, then $\operatorname{tr}(A)\le 2$. Does anyone know a demonstration of this fact?
AI: Using the triangle inequality, if $|\lambda_1|,|\lambda_2|\le1$ then $|\mathrm{tr}(A)|=|\lambda_1+\lambda_2|\le|\lamb... |
H: What is the difference between the symbols $\cap$ and $\setminus$?
What is the difference between $\cap$ and $\setminus$ symbols for operations on sets?
AI: Their definition is different:
$A\cap B=\{x\mid x\in A\text{ and } x\in B\}$, we take all the elements which appear both in $A$ and in $B$, but not just in on... |
H: To prove an elementary statement
I have an elementary doubt, Sorry for disturbing you all. I have a statement of this sort. $$r^2-1=p^a(f(p))=(r+1)(r-1). \tag{1}$$
Where $r$ is an even number, and $p$ is an odd prime. $f(p)$ is a degree $n$ polynomial in $p$ with integer coefficients ( Courtesy : André Nicolas ) ,... |
H: What is the usual meaning of the notation $ \| f \|_{C^k (A)}$?
If $f \in C^k ( \mathbb R)$ and $A \subset \mathbb R$, what is the usual meaning of the notation below?
$$ \| f \|_{C^k (A)} $$
AI: Assuming you meant $A\subseteq\mathbb{R}$, the notation usually means
$$\|f\|_{C^k(A)}=\|f\|+\|f'\|+\cdots+\|f^{(k)}\|$$... |
H: What is the difference between direct product and direct sum of a finite number of group representations.
I have been reading in Fulton & Harris's book on representation theory and it talks about things like the decomposition of a direct product of representations $ V \otimes V $ into a direct sum of representation... |
H: The Determinant of Transition Matrices Between Orthonormal Bases
I am trying to show that if $A$ and $B$ are orthonormal bases of a real finite dimensional vector space $V$ and $P$ is the change of basis matrix between $A$ and $B$ then $\det(P) = \pm 1$. Here is my progress:
When working on this problem, I discover... |
H: linear algebra linear transformation
I would really appreciate if someone could help me with this question, I know that you have to row reduce each matrix and put it in vector parametric form to find $U_1$ and $U_2$. any help would be much appreciated. thank you
Define a real linear transformation $L_1 : \mathbb R... |
H: Moment Generating Function for Sum of Independent Random Variables
I'm taking a graduate course in probability and statistics using Larsen and Marx, 4th edition and looking specifically at estimation methods this week. I ran into a homework problem that is related to moment generating functions and I can't quite c... |
H: Have there been efforts to introduce non Greek or Latin alphabets into mathematics?
As a physics student, often I find when doing blackboard problems, the lecturer will struggle to find a good variable name for a variable e.g. "Oh, I cannot use B for this matrix, that's the magnetic field".
Even ignoring the many ... |
H: Analysis operator $T_\Phi$ is injective and has a closed range
Definition of the problem
Let $\mathcal{H}$ be a separable Hilbert space on $J\subset\mathbb{N}$
an index set. Let $\Phi:=\left(\varphi_{j}\right)_{j\in J}\subset\mathcal{H}$
be a frame for $\mathcal{H}$.
I have to prove that the analysis operator $T_{... |
H: Does a closed form formula for the series ${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$ exist.
$${n \choose n-1} + {n+1 \choose n-2} + {n+2 \choose n-3} + \cdots + {2n - 1 \choose 0}$$
For the above series, does a closed form exist?
AI: Your series is
$$\sum_{k=0}^{n-1}\... |
H: Is there any number in $\mathbb Z[\sqrt{5}]$ with norm equals 2?
Is there any number $a+b\sqrt{5}$ with $a,b \in \mathbb{Z}$ with norm (defined by $|a^2−5b^2|$) equal 2?
AI: No.
Since $a^2-5b^2\equiv a^2\pmod{5}$, the values of $a^2-5b^2$ must be congruent to either $0$, $1$, or $4$ modulo $5$, since those are the... |
H: System of differential equations.
\begin{equation}
\dot{x}(t)=\left(1-\frac{x(t)}{E_{0}}\right) x(t) - a\ y(t)
\end{equation}
\begin{equation}
\dot{y}(t)=\left(1-\frac{a\ y(t)}{x(t)}\right) y(t)
\end{equation}
I am trying to solve the system of differential equations above, but (I think) it is nontrivial to calcula... |
H: Hellinger distance between Beta distributions
I am interested in calculating the Hellinger distance $H(f,g)$ between two Beta distributions $f$ and $g$ of which I already know the parameters for. I am aware that you can calculate it directly using the 2-norm of discrete distributions. But it would be nicer to have ... |
H: Finding Limits of Trig Functions: $\lim_{\theta \rightarrow 0}\frac {\sin^2\theta}{\theta}$
I am asked find the following limit
$$\lim_{\theta \rightarrow 0}\frac {\sin^2\theta}{\theta}$$
I recognize that $$\lim_{\theta \rightarrow 0}\frac{\sin\theta}{\theta}=1$$
But because I have $sin^2\theta$ in the numerator, I... |
H: Solving $y'' - \frac{1}{x} y' + (1+\frac{\cot x}{x}) y = 0$ by rank reduction
With the substitution $$y(x) = \sin x \int u(x) \, dx\tag{*}$$ I managed to get to$$u'(x) = \left(\frac{1}{x}-2\cot x\right)u(x)$$ Solving which gave me $$u(x) = C_1 \frac{x}{\sin^2 x}$$
Inserting that back into $(*)$ $$y(x) = (\sin x) C_... |
H: Tensor contraction with multiple indices
I think I missed a rule somewhere, because I can contract the following expression in multiple ways.
$\epsilon^{\alpha \beta} \sigma_{\dot{\alpha} \alpha} \epsilon^{\dot{\alpha} \dot{\beta}} = \sigma_{\dot{\alpha} \alpha} \epsilon^{\alpha \beta} \epsilon^{\dot{\alpha} \dot... |
H: Name Drawing Puzzle
There is a party with 20 people, and everyone writes their name down
on a piece of paper and puts it into a bag. The bag is mixed up, and
each person draws one piece of paper. If you draw the name of someone
else, you are considered to be in his "group". What is the expected
number of g... |
H: Confusion about Completion of Metric Space Proof
I have started studying Functional Analysis following "Introduction to Functional Analysis with Applications". In chapter 1-6 there is the following proof
For any metric space $X$, there is a complete metric space $\hat{X}$ which has a subspace $W$ that is isometric... |
H: Largest number that divides $n^2(n^2 - 1)(n^2 - n - 2)$ for all $n$
Obtain the greatest natural that divides $n^2(n^2 - 1)(n^2 - n - 2)$ for all natural numbers $n$.
What should be the approach in these type of questions?
Should I equate with prime factorization $2^a 3^b 5^c \cdots$ etc?
AI: \begin{align}
f(n) & =... |
H: Lexicographical order - posets vs preorders
I found the following definition for lexicographical ordering on Wikipedia (and similar definitions in other places):
Given two partially ordered sets $A$ and $B$, the lexicographical order on the Cartesian product $A \times B$ is defined as $(a,b) \le (a',b')$ if and onl... |
H: Question about cardinals without GCH
Without assuming the Generalized Continuum Hypothesis, how to show that there exists a uncountable cardinal $\kappa$ such that, for every $\lambda < \kappa$, one have $2^\lambda < \kappa$. With assumption of GCH, for $\kappa = \aleph_{\omega}$ the affirmation holds. I would app... |
H: Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous, show that $\{x:f(x)\leq g(x)\}$ is closed in $X$
Let $Y$ be an ordered set in the order topology with $f,g:X\rightarrow Y$ continuous. Show that the set $A = \{x:f(x)\leq g(x)\}$ is closed in $X$.
I am completely stumped on this ... |
H: image of complex continuous function cannot be a curve
Let $ f : U \subset \mathbb C \to \mathbb C $ a continuous function where $ U $ is an open and connected set.
Prove that the image of $ f$ cannot be a curve on compex plane.
AI: Are you sure there are no additional constraints, e.g. the Cauchy-Riemann equation... |
H: Finding horizontal tangents for trig function
I am asked to find the points on the curve at white the tangent is horizontal, for the function:
$$y=\frac{\cos x}{2+\sin x}$$
To find the points at which the tangent is horizontal, I need to know what values will result in the derivative of the function equaling zero,... |
H: If $(G : H) = r$ and there exists $K \triangleleft G$ contained in $H$ such that $(G : K) = r!$
Possible Duplicate:
How to prove that if $G$ is a group with a subgroup $H$ of index $n$, then $G$ has a normal subgroup $K\subset H$ whose index in $G$ divides $n!$
I'm trying to prove the following: "If $G$ is an in... |
H: Is the inverse of a one-to-one, total function itself one-to-one and total?
The problem, from Boolos and Jeffrey's Computability and Logic, states the following definitions:
"If a function $f(a)$ (from $A$ to $B$) is defined for every element $a$ of $A$, then it is called total"
"A correspondence between sets $A$ ... |
H: How many k-digit ternary strings are there in which no digit occurs exactly twice? The occurrences do not need to be consecutive.
How many k-digit ternary strings are there in which no digit occurs exactly twice? The occurrences do not need to be consecutive.
AI: This can be computed using an inclusion-exclusion ar... |
H: Probability: selecting exactly one from each group
A coaching held separate workshops for training in three languages.
50 students participated in workshops and learnt one or more languages. 13 students learnt a single language, 25 students learnt 2 languages and 12 students learnt 3 languages.
If three students ar... |
H: Calculate dimensions of inner box
This is my first post here and also i'm bad with math so don't be mad at me :)
Here is my issue, i have a box and i have another box inside, now i want that box inside to be at exactly same distance from each side, no matter what size it parent is.
Now i know how to calculate the d... |
H: Chain Rule Confusion
Below i have a function that i need to use the chain rule on. My friend showed me his answer which was correct which was $-8x^7\sin(a^8+x^8)$.
$$y = \cos(a^8 + x^8)$$
I am really confused as how he got that. I know that in the chain rule you bring whats outside to the front. So why is $a^8$ n... |
H: Necessary and sufficient condition for being a covering map
Suppose I have a map $f:X\rightarrow Y$ continuous, $X$ is compact, connected and also $f$ is a local homeomorphism, what condition should we include in $X$ so that $f$ becomes a covering map? Am I making any sense?
AI: If $X$ and $Y$ are Hausdorff, then c... |
H: Inequalities involving some common functions
I often see the following inequality is used over and over again
$$
1−x⩽e^{−x}
$$
for $x \in \mathbb{R}$, for proving or deriving various statements.
As a layman, I haven't seen this inequality appearing in any class I have taken in my life. So it seems quite unnatural t... |
H: When will these two trains meet each other
I cant seem to solve this problem.
A train leaves point A at 5 am and reaches point B at 9 am. Another train leaves point B at 7 am and reaches point A at 10:30 am.When will the two trains meet ? Ans 56 min
Here is where i get stuck.
I know that when the two trains m... |
H: Poincaré Lemma Contractible Hypothesis
Poincaré's Lemma is often stated as saying that a closed differential form on a star-shaped domain is exact. More generally, it is true that a closed differential form on a contractible domain is exact.
What I am wondering is if there is an easy example of a closed differentia... |
H: A space that has a countably locally finite basis but not second countable
[Munkres, Ch40, p252] Find a nondiscrete space that has a countably locally finite basis but does not have a countable basis.
I solved this problem. My solution is $X=\mathbb{R} \cup \{0'\}$ with topology given by basis $\mathfrak{B}$ consis... |
H: Is the zero map (between two arbitrary rings) a ring homomorphism?
I was looking at the definition under Wikipedia, which states that for arbitrary rings $R,S$, a ring homomorphism $f:R\to S$ must satisfy $f(1)=1.$ Here, I assume they mean $1$ as the multiplicative identity. Certainly then, this implies the zero ma... |
H: For order-reversing maps between posets/lattices, is the inverse always order-reversing?
I am having trouble with the following lemma from J. Rotman's Galois Theory book:
Lemma $83$. If $L$ and $L'$ are lattices and $\gamma: L \rightarrow L'$ is an order reversing bijection $[a \leq b$ implies $\gamma(b) \leq \gam... |
H: Orthogonal Operators and Inner Products
This question is a follow-up/extension to this question. Suppose $P$ is an orthogonal operator on a finite dimensional inner product space $V$. By definition, this means that
$$
\langle Pu, Pv \rangle = \langle u, v \rangle
$$
for all $u,v \in V$. I want to show that $P^TP =... |
H: Count the number of solutions of the inequality $x + y + z \leq N$
Problem
Given $A, B, C $ and $N$. How many integer solutions are there of the following inequality:
$$x + y + z \leq N$$
where $0 \leq x \leq A, 0 \leq y \leq B, 0 \leq z \leq C$?
When $A + B + C \leq N$, the solution is obvious $(A + 1) \cd... |
H: Does $\int_{\mathbb R^n} (1 + |x|^2)^{-1} dx$ converge in the Riemann sense?
Recently I was reading an article and came across the following integral:
$$\int_{\mathbb R^n}\dfrac{1}{1+|x|^2}\ dx$$
Is this integral convergent in the Riemann sense?
AI: The integrand is positive and smooth so Riemann or Lebesgue ma... |
H: Square units of area in a circle
I'm studying for the GRE and came across the practice question quoted below. I'm having a hard time understanding the meaning of the words they're using. Could someone help me parse their language?
"The number of square units in the area of a circle '$X$' is equal to $16$ times the... |
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