text stringlengths 83 79.5k |
|---|
H: question about summation?
Are there any general rules to find
$???\leqslant \sum_{n=t}^{m}f(n)\leqslant ???$
when $m$ and $t$ $\in $ R
AI: A trivial, yet sometimes useful, inequality is $$(m-t+1)\min_{i\in\{t,t+1,...,m\}}f(i)\leq\sum_{n=t}^{m}\ f(n) \leq(m-t+1)\max_{i\in\{t,t+1,...,m\}}f(i)$$ |
H: Proper constants for $\alpha, \beta$
Here is the problem:
For what values of $\alpha$ and $\beta$, the function
$$\mu(x,y)=x^{\alpha}y^{\beta}$$
is an integrating factor for the OE $$ydx+x(1-3x^2y^2)dy=0.$$ I am working on it just knowing the definition. :(
AI: Let $$M(x,y)dx+N(x,y)dy=0$$
If exists $F$ such that ... |
H: Sequence $(a_n)$ s.t $\sum\sqrt{a_na_{n+1}}<\infty$ but $\sum a_n=\infty$
I am looking for a positive sequence $(a_n)_{n=1}^{\infty}$ such that $\sum_{n=1}^{\infty}\sqrt{a_na_{n+1}}<\infty$ but $\sum_{n=1}^{\infty} a_n>\infty$.
Thank you very much.
AI: The simplest example I can think of is $\{1,0,1,0,...\}$. If... |
H: Vectors that form a triangle!
I have a problem here.
How can I prove that sum of vectors that form a triangle is equal to 0 $(\vec {AB}+\vec {BC}+\vec {CA}=\vec 0)$ ?
AI: from the triangle law : $\vec {AB}+\vec{BC}=\vec{AC}$
$\vec {AC}$ will be resultant vector of addition of other two vectors.
$\vec {AB}+\vec{BC}=... |
H: Given that the family has at least one girl, determine the probability that the family has at least one boy.
Suppose that a family has exactly n children (n ≥ 2). Assume that the probability that any child will be a girl is 1/2 and that all births are independent. Given that the family has at least one girl, determ... |
H: Integrating: $\int_0^\infty \frac{\sin (ax)}{e^x + 1}dx$
I am trying to evaluate the following integral using the method of contour which I am not being able to. Can anyone point out what mistake I am making?
$$\int_0^\infty \frac{\sin ax}{e^x + 1}dx$$
I am considering the following contour. And function $\displays... |
H: Problem involving the computation of the following integral
I was solving the past exam papers and stuck on the following problem:
Compute the integral $\displaystyle \oint_{C_1(0)} {e^{1/z}\over z} dz$,where $C_1(0)$ is the circle of radius $1$ around $z=0.$
Here,$z=0$ is a pole of order $1$ and so Res$(f... |
H: sum of monotonic increasing and monotonic decreasing functions
I have a question regarding sum of monotinic increasing and decreasing functions. Would appreciate very much any help/direction:
Consider an interval $x \in [x_0,x_1]$. Assume there are two functions $f(x)$ and $g(x)$ with $f'(x)\geq 0$ and $g'(x)\leq 0... |
H: Formulas for calculating pythagorean triples
I'm looking for formulas or methods to find pythagorean triples. I only know one formula for calculating a pythagorean triple and that is euclid's which is:
$$\begin{align}
&a = m^2-n^2 \\
&b = 2mn\\
&c = m^2+n^2
\end{align}$$
With numerous parameters.
So are there othe... |
H: Question about limits with variable on exponent
So I have to find the following limit $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{1/n}.$$I said that this is $$\lim_{n\to\infty}\left[\left(1+\frac{2}{n}\right)^n\right]^{1/n^2}=\left(e^2\right)^{\lim_{n\to\infty}1/n^2}=1.$$Now I know that the final answer is corre... |
H: The restriction of a covering map on the connected component of its definition domain
Suppose $p:Y\to X$ is a covering map, $X,Y$ are manifolds and $X$ is connected. If $Z$ is a connected component of $Y$, I wonder if the restriction of $p$ on $Z$ is also a covering map? If not, what conditions should be added to g... |
H: Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
Let $G$ be a finite group with $|G|>2$. Prove that ${\rm Aut}(G)$ contains at least two elements.
We know that ${\rm Aut}(G)$ contains the identity function $f: G \to G: x \mapsto x$.
If $G$ is non-abelian, look at $... |
H: Multivariate normal distribution density function
I was just reading the wikipedia article about Multivariate normal distribution: http://en.wikipedia.org/wiki/Multivariate_normal_distribution
I use a little bit different notation. If $X_1,\ldots,X_n$ are independent $\mathcal{N}(0,1)$ random variables, $X=(X_1,\ld... |
H: Embedding of Tree
Q.
Proof for every Tree can be embedded into the plane.
Conditions.
We cannot use Euler Formula for Planar Graphs. We can use definition of tree, $V-E=1$, no-cycles, every edge is critical, there is a vertex of frequency at most one.
Attempt.
We prove this by induction. True for $V$=1. Trivial. ... |
H: Proving integrability in integration by parts in Rudin's text
Integration by parts, as stated in W. Rudin's Principles of Mathematical Analysis, Theorem 6.22, goes as follows:
Suppose F and G are differentiable functions in $[a,b]$, $F'=f\in \mathcal{R}$, and $G'= g\in \mathcal{R}$. Then $\int_a^bF(x)g(x)dx = F(b)... |
H: Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$
Find Aut$(G)$, Inn$(G)$ and $\dfrac{\text{Aut}(G)}{\text{Inn}(G)}$ for $G = \mathbb{Z}_2 \times \mathbb{Z}_2$.
Here is what I have here:
Aut$(G)$ consists of 6 bijective functions, which maps $G$ to itself,... |
H: Differentiation problem of power to infinity by using log property
Problem:
Find $\frac{dy}{dx}$ if $y =\left(\sqrt{x}\right)^{x^{x^{x^{\dots}}}}$
Let ${x^{x^{x^{\dots}}}} =t. (i)$ Taking $\log$ on both sides $ \implies {x^{x^{x^{\dots}}}}\log x = \log t$
This can further be written as $ t
\cdot\log x = \log t$
... |
H: Is the the number of generators of a group the number of different generators that one finds if one counts over every generating set of the group?
Consider the additive group of integers as an example as mentioned at the bottom of the Wikipedia article. There are two generating sets that are mentioned; The set cons... |
H: what is the diffrence between a term , constant and variable in first order logic languages ?
in the text , the author says that the language contains parenthises , sentintial connectives and n-place functions , n-place predicates , equality sign = , terms , constans and variables
i have two question ,
1- what is ... |
H: Sequence of continuous functions, integral, series convergence
Let $f_k$ be a sequence of continuous functions on $[0,1]$ such that $\int _0 ^1 f_k(x)x^ndx = \int _0^1 x^{n+k} dx$ for all $n \in \mathbb{N}$.
Is $\sum _{k=1} ^{\infty}f_k(x)$ convergent?
Could you tell me how to solve this? I would appreciate all th... |
H: For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I want to show is the following statement.
For every prime of the form $2^{4n}+1$, 7 is a primitive root.
What I get is that
$$7^{2^{k}}\equiv1\pmod{p}$$
$$7^{2^{k-1}}\equiv-1\equiv2^{4n}\pmod{p}$$
$$7^{2^{k-2}}\equiv(2^{n})^2\pmod{p}$$
Thus $(\fra... |
H: Combinations, Expected Values and Random Variables
A community consists of $100$ married couples ($200$ people). If during a given year, $50$ of the members of the community die, what is the expected number of marriages that remain intact?
Assume that the set of people who die is equally likely to be any of the ${2... |
H: Finding sequence in a set $A$ that tends to $\sup A$
I have been reading the book at http://www.neunhaeuserer.de/short.pdf, and have noticed that in the proof of the intermediate value theorem (Theorem 5.8 in the book), it seems to be quietly assumed that you can always find a sequence of points in a set that tend ... |
H: Last non zero digit of $n!$
What is the last non zero digit of $100!$?
Is there a method to do the same for $n!$?
All I know is that we can find the number of zeroes at the end using a certain formula.However I guess that's of no use over here.
AI: This question gets asked fairly frequently. There was originally a ... |
H: quadratic equation precalculus
from Stewart, Precalculus, 5th, p56, Q. 79
Find all real solutions of the equation
$$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{x^2-4}$$
my solution
$$\dfrac{x+5}{x-2}=\dfrac{5}{x+2}+\dfrac{28}{(x+2)(x-2)}$$
$$(x+2)(x+5)=5(x-2)+28$$
$$x^2+2x-8=0$$
$$\dfrac{-2\pm\sqrt{4+32}}{2}$$
$$\df... |
H: Euler lagrange equation is a constant
I'm working through exercises which require me to find the Euler-Lagrange equation for different functionals.
I've just come across a case where the Euler Lagrange equation simplifies to
$$1=0.$$
Please could someone explain what can be concluded about the set of extremals to... |
H: Finding the Fourier Series of $\sin(x)^2\cos(x)^3$
I'm currently struggling at calculation the Fourier series of the given function
$$\sin(x)^2 \cos(x)^3$$
Given Euler's identity, I thought that using the exponential approach would be the easiest way to do it.
What I found was: $$\frac{-1}{32}((\exp(2ix)-2\exp(2ix)... |
H: Is a Relationship Quadratic?
I have a relationship $y=f(x)$ for which I can obtain data through simulation.
I have good reason to suspect that this relationship is quadratic (rather than, say, exponential), and would like to provide evidence for this.
I was thinking of the following method, and I would like to ask ... |
H: Showing it is a joint probability density function
I have two random variables $X,Y$ with a joint density function $f_{X,Y}(x,y)=x+y$ if $(x,y)\in[0,1]\times [0,1]$ and otherwise $f_{X,Y}(x,y)=0$
I want to analyze this case in different cases, first of all, I want to show it is a probability density function. Well ... |
H: how would $f(T)$ look like if ...
I know the result that if $T:V_F\to V_F$ is a linear operator then for any polynomial $f(x)\in F[x],~f(T)$ is a linear operator. Now my question is how would $f(T)$ look like if
$f(x)$ is the zero polynomial
$f(x)$ is a constant polynomial other than $0$
AI: Then $f(T)$ is the ze... |
H: Prove that if $G$ is abelian, then $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$
Let $G$ be an abelian group. Prove that $H = \{a \in G \mid a^2 = e\}$ is subgroup of $G$, where
$e$ is the neutral element of $G$.
I need some help to approach this question.
AI: Hint: To show it is a subgroup you must show t... |
H: How to prove that: $\phi^{2}(0) \leq \|\phi\|^2_{L^{2}}+\|\phi'\|^{2}_{L^{2}}$
Let $\phi$ be a function and $\phi \in C^{\infty}(\mathbb{R}_{+},\mathbb{R})$ with compact support and $\mbox{supp }{\phi} \subset [0, \infty)$.
I want to prove that: $$\phi^{2}(0) \leq \|\phi\|^2_{L^{2}}+\|\phi'\|^{2}_{L^{2}}.$$
Someon... |
H: Area of a circle is $A = \pi r^2$. Is it possible that both $A$ and $r$ are perfect integers.
Can you produce an example where both the area of a circle and it's radius are integers?
AI: $$r\neq 0\;,\;\pi r^2\;,\;r\in\Bbb N\implies \pi\in\Bbb Q\;,\;\text{which is false}$$ |
H: Rationalizing quotients
I have
$$\frac{\sqrt{10}}{\sqrt{5} - 2}$$
I have no idea what to do, I know that I can do some tricks with splitting square roots up but pulling out whole numbers like I know that $\sqrt{27}$ is just $\sqrt{3*9}$ so I can pull out a nine which becomes a 3. Here though I have no such options... |
H: Trigonometrical Question
the question is solve the following equation in the interval
$$0<\theta\leq 360$$
$$\tan(\theta) = \tan(\theta)(2+3\sin(\theta))$$
I got 199.5 and 340.5 as my answers like so:
$\tan(\theta) = \tan(\theta)(2+3\sin(\theta))$
$1=2+3\sin(\theta)$
$\sin(-1/3) = 199.5$ and $340.4$
However in th... |
H: Proving an inequality: $|1-e^{i\theta}|\le|\theta|$
We have been using this result without proof in my class, but I don't know how to prove it. Could someone point me in the right direction?
$$|1-e^{i\theta}|\le|\theta|$$
I believe this is true for all $\theta\in\mathbb{R}$. It is easy to show that the left side is... |
H: limit of $e^z$ at $\infty$
What's the limit of $e^z$ as $z$ approaches infinity?
I am given that the answer is "There is no such limit."
Is this correct, and if so, am I correct to demonstrate this by showing that as $y$ tends to infinity along the $y$-axis, the magnitude of $e^z$ remains $1$, i.e. it doesn't have ... |
H: Proving a $Z- $transform
I am having trouble demonstrating the $Z$ transform of $a^{n-1}u(n-1)$ is $\frac{1}{(z-a)}$, as it says in this table.
I try using the definition of the z transform, but it comes out different than what the table says:
$$\sum_{k=-\infty}^{\infty}a^{k-1}u(k-1)z^{-k}=\sum_{k=1}^{\infty}a^{k-1... |
H: Central limit theorem - std dev away from mean
I was reading about the CLT and found something that I think people use interchangeably. On one hand I found that 68% of the means are 1 standard deviations from away and 95% are 2 std dev. On the other hand, if I take a look at the Z-table I found that 68% is approx 1... |
H: What is the perimeter of a sector?
I don't understand this.
So we have:
\begin{align}
r &= 12 \color{gray}{\text{ (radius of circle)}} \\
d &= 24 \text{ (r}\times2) \color{gray}{\text{ (diameter of circle)}} \\
c &= 24\pi \text{ (}\pi\times d) \color{gray}{\text{ (circumference of circle)}} \\
a &= 144\pi \text{ (... |
H: Integrate: $\int_0^{\infty}\frac{\sinh (ax)}{\sinh x} \cos (bx) dx$
Q: If $|a|< 1$ and $b>0$, show that
$$\int_0^{\infty}\frac{\sinh (ax)}{\sinh x} \cos (bx) dx = \frac{\pi \sin (\pi a)}{2 (\cos (\pi a)+\cosh (\pi b))}$$
I need to evaluate the above integral by method of contour. I tried to use this contour... |
H: If $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
Claim: if $f$ is differentiable at $x = x_0$ then $f$ is continuous at $x = x_0$.
Please, see if I made some mistake in the proof below. I mention some theorems in the proof:
The condition to $f(x)$ be continuous at $x=x_0$ is $\lim\limit... |
H: On boundedly invertible
Let $T:X\to X$ be a bounded and invertible linear operator. Show that $\inf_{x: \|x\|=1}\|T(x)\|\geq M$ if and only if $\sup_{x: \|x\|=1}\|T^{-1}(x)\|\leq N$, where $M,N\ge 0$.
.
AI: Let's show it in one direction:
$\displaystyle \sup_{x:\|x\|=1}\|T^{-1}(x)\|= \sup_{x:\|x\|>0}\frac{\|T^{-... |
H: Completing the square with simple polys
I am suppose to rewrite $x^2 + x + 1$ by completing the square. I don't really know what that means but I know that if I add 3 at the end of this I get
$$(x + 2) (x - 1) - 3$$ this is the same as the original now but the answer isn't right. What is wrong with what I did? It ... |
H: Math question efficiency
A solar collector has 1000 Btu/min of radiant energy available on a clear sunny day. The collector can transfer 450 Btu/min to a storage tank. What is the efficiency of the system?
I used n= energy-output/energy-input *100
I plugged it all in $\;\dfrac{450}{1000}*\times 100\%$.
I got $45\... |
H: How to sum numerator and denominator of a fraction?
I want to do sum over this. Can apply the summation to top and bottom separately?
$$\sum\limits_{i=1}^{n} \frac{-a(x_i-\mu)^2}{x_i}$$
$$=\frac{\sum\limits_{i=1}^{n}-a(x_i-\mu)^2}{\sum\limits_{i=1}^{n}x_i}$$
Is this correct?
Where can I find the rules to summati... |
H: Packing circles on a line
On today's TopCoder Single-Round Match, the following question was posed (the post-contest write-up hasn't arrived yet, and their explanations often leave much to be desired anyway, so I thought I'd ask here):
Given a maximum of 8 marbles and their radii, how would you put them next to eac... |
H: Differential equation (2nd order) with divergent coefficients.
I have this equation:
$$x(x-1)y''+6x^2y'+3y=0$$
I try to get the series for the solution around $x=0$, using Frobenius (however it's written). the first solution must be of the form:
$$y_1=\sum_{n=0}^\infty c_nx^{n+1}$$
If I try to get coefficients, I w... |
H: $\sum \frac{\ln(n)}{\sqrt{n^5}}$ test for convergence
Let $\sum a_{n}=\sum \frac{\ln(n)}{\sqrt{n^5}}$. To find if the serie is convergence or not, I had some difficult on finding the proper serie to test the given one.
After some work around, I found this sequence $b_{n}=\frac{1}{\sqrt{n^3}}$, whose serie converges... |
H: If $0
I have been really struggling with this problem ... please help!
Let a,b be real numbers. If $0<a<1, 0<b<1, a+b=1$, then prove that $a^{2b} + b^{2a} \le 1$
What I have thought so far:
without loss of generality we can assume that $a \le b$, since $a^{2b} + b^{2a}$ is symmetric in $a$ and $b$. This gives us $... |
H: Trigonometric problem
I'm trying to get the roots for a complex number $x^2+1$
$x^2+1=0\rightarrow x^2=-1 \rightarrow x = \sqrt{-1} \rightarrow i$
So, $w^2 = 0 + 1i$
$p = \sqrt{0^2+1^2} = 1$
$\theta = \tan^{-1} \left( \frac{1}{0} \right )$
But I don't know what I can do to get the $\tan^{-1}(\frac10)$
AI: $\theta =... |
H: What does $\vdash s \rightarrow (\neg s\rightarrow t)$ mean?
What does this statement mean $\vdash s \rightarrow (\neg s\rightarrow t)$?
And how can I prove it?
AI: Writing $T\vdash\varphi$ means that if we assume $T$ then we can prove $\varphi$. If $T$ is omitted then this means that without any assumptions we ca... |
H: How do you determine the particular solution to a non-homogeneous DE by undetermined coefficients?
I am asked to solve
$y'' +2y' = 2x + 5 -e^{-2x}$
I can find the general solution easily, but the particular solution in this case is hard to find. Here's the answer. I don't know why they got $Ax^2 + Bx + Cxe^{-2x}$
... |
H: Why the terms "unit" and "irreducible"?
I'm trying to understand why in a ring we choose the names unit to an invertible element and irreducible element in this definition
Maybe historical reasons?
For example, I suppose the second definition are named as prime elements because of the analogy to prime numbers.
... |
H: How to show that a valid inner product on V is defined with the formula $[x, y] = \langle Ax, Ay\rangle $?
Let $A \in L(V,W)$ be an injection and $W$ an inner product space with the inner product $\langle \cdot,\cdot\rangle $. Prove that a valid inner product on $V$ is defined with the formula $[x, y] = \langle Ax... |
H: Closed form for n-th anti-derivative of $\log x$
Is it possible to write a closed-form expression with free variables $x, n$ representing the n-th anti-derivative of $\log x$?
AI: $$\log^{(-n)}x=\frac{x^n}{n!}(\log x-H_n),$$
where $H_n$ is the harmonic number: $H_n=\sum_{k=1}^n k^{-1}=\,$$\gamma$$\,+\,$$\psi$$(n+1)... |
H: Definition of open set/metric space
On Proof Wiki, the definition of an open set is stated as
Let $(M,d)$ be a metric space and let $U\subset M$, then $U$ is open iff for all $y\in U$, there exists $\epsilon \in \mathbb{R}_{>0}$ such that $B_M(y;\epsilon) \subset U$
Then there's a remark that states
It is import... |
H: Is any compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?
Every compact metric totally disconnected perfect space is homeomorphic to a Cantor space.
Is every compact metric totally disconnected space homeomorphic to a compact subspace of a Cantor space?
In other words, ... |
H: Relationship between three matrices
I think this might be an odd question, and a little vague. But here goes.
This is related to coordinate transformations. Three matrices are given: $G_1 , G_2$, and $\Lambda$. $G_1$ and $G_2$ are symmetric. (They are metrics, actually.) $\Lambda$ is the matrix in question. T... |
H: Why is $\varphi$ called "the most irrational number"?
I have heard $\varphi$ called the most irrational number. Numbers are either irrational or not though, one cannot be more "irrational" in the sense of a number that can not be represented as a ratio of integers. What is meant by most irrational? Define what we m... |
H: $\iint f(x,y)\,dxdy$ and $\iint f(x,y)\,dydx$ exist but $f$ not integrable on $[0,1]\times[0,1]$
I want to look for a function $f(x,y)$, whose support is inside $[0,1]\times[0,1]$, such that $\int_0^1\!\int_0^1\!f(x,y)\,dxdy$ and $\int_0^1\!\int_0^1\!f(x,y)\,dydx$ both exist, but $f(x,y)$ is not Riemann-integrable ... |
H: How do I find the series expansion of the meromorphic function $\frac{1}{e^z+1}$?
in a theoretical physics book, the author makes the following claim:
$$\frac{1}{e^z + 1} = \frac{1}{2} + \sum_{n=-\infty}^\infty \frac{1}{(2n+1) i\pi - z}$$
and justifies this as
These series can be derived from a theorem which stat... |
H: Is the Dirac delta a function?
Is Dirac delta a function? What is its contribution to analysis?
What I know about it:
It is infinite at 0 and 0 everywhere else. Its integration is 1 and I know how does it come.
AI: To have a better understanding of what is the Delta Dirac "function", it is good to know what is a d... |
H: Proof on showing function $f \in C^1$ on an open & convex set $U \subset \mathbb R^n$ is Lipschitz on compact subsets of $U$
The question is as follows:
Given:
(1) function $f: U \subset \mathbb R^n ==> \mathbb R$
(2) $U$ is open and convex set
(3) $f \in C^1$ in $U$
Goal: Show that $f$ is Lipschitz on any compac... |
H: Calculating $\sqrt{28\cdot 29 \cdot 30\cdot 31+1}$
Is it possible to calculate $\sqrt{28 \cdot 29 \cdot 30 \cdot 31 +1}$ without any kind of electronic aid?
I tried to factor it using equations like $(x+y)^2=x^2+2xy+y^2$ but it didn't work.
AI: \begin{align}
&\text{Let }x=30
\\ \\
\therefore&\ \ \ \ \ \sqrt{(x-2)(x... |
H: Composition of Partial Isometry
Let $T$ be a linear operator in $H$, a Hilbert space.
An operator $T \in L(H)$ is said to be a partial isometry if the restriction of $T$ to $ker(T)^{\perp}$ is an isometry. I would like to prove that given a partial isometry $T$, then $T^{2}$ is a partial isometry if and only if $(T... |
H: Simplifying $\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$
How do I simplify:
$$\sqrt{\underbrace{11\dots1}_{2n\ 1's}-\underbrace{22\dots2}_{n\ 2's}}$$
Should I use modulos or should I factor them? Or any I suppose to use combinatorics? Any one have a clue?
AI: Nice question there!
Let $x... |
H: Integration of function help
I'm having problems integrating this function $\displaystyle E(X)=\int^ \infty_0 x\lambda e^{-\lambda x} dx$. I did the integration by parts and had $-xe^{-\lambda x}- \lambda e^{-\lambda x}$. However the solution gives $-xe^{-\lambda x} - \dfrac{1}{\lambda}e^{-\lambda x}$. I can't find... |
H: Proving that $\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$
How do I prove that:
$$\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{100}}<20$$
Do I use induction?
AI: Prove the following claim using induction on $n$:
$$\sum_{k=1}^n \dfrac1{\sqrt{k}} < 2 \sqrt{n}$$
In the induction, ... |
H: A basic doubt on Lebesgue integration
Can anyone tell me at a high level (I am not aware of measure theory much) about Lebesgue integration and why measure is needed in case of Lebesgue integration? How the measure is used to calculate the horizontal strip mapped for a particular range?
AI: The idea of (exterior) m... |
H: Order of operations (BODMAS)
$$40-20/2+15\times1.5\\\hspace{.1in}\\40-20/2+15\times1.5=\\ 40-10+22.5=7.5$$
I'm studying and this is from an example.
In BODMAS, aren't addition and subtraction have same level?
So, in the 3rd line, it should be from left to right, correct?
AI: $$40-10+22.5=52.5$$
You're correct, ... |
H: how to tell whether x and y are independent or not
Suppose that $f_{x,y}(x,y) = \lambda^2 e^{\displaystyle-\lambda(x+y)}, 0\leq x , 0\leq y.$ Find $\operatorname{Var(X+Y)}$.
I'm having trouble with this problem the way to find $\operatorname{Var(X+Y)} = \operatorname{Var(X)}+\operatorname{Var(Y)}+2\operatorname{Co... |
H: Having trouble using eigenvectors to solve differential equations
The question asked to solve $$\frac{dx}{dy} = \begin{pmatrix}
5 & 4 \\
-1 & 1\\
\end{pmatrix}x$$ ,where $$ x = \begin{pmatrix} x_1 \\
x_2 \\ \end{pmatrix}$$
I went ahead an found the determinant of matrix $$ |A - I\lambda| = ... |
H: Polynomials - The sum of two roots
If the sum of two roots of
$$x^4 + 2x^3 - 8x^2 - 18x - 9 = 0$$ is
$0$, find
the roots of the equation
AI: Generally, if there's two roots whose sum is zero, then it means that two factors are $x-a$ and $x+a$, which means that $x^2-a^2$ must be a factor. So clearly
$$
(x^2-a^2)(x^... |
H: Methods for determining the convergence of $\sum\frac{\cos n}{n}$ or $\sum\frac{\sin n}{n}$
As far as I know, the textbook approach to determining the convergence of series like $$\sum_{n=1}^\infty\frac{\cos n}{n}$$ and $$\sum_{n=1}^\infty\frac{\sin n}{n}$$ uses Dirichlet's test, which involves bounding the partial... |
H: Application of derivative - how to calculate change in error
Problem: If the error committed in measuring the radius of the circle is $0.05\%$ then find the corresponding error in calculating the area.
Solution: Let the error be denoted by $\delta r = 0.05\%$, therefore the corresponding error in calculating the a... |
H: Evalutate $\int\frac{dx}{x\sqrt{1-\frac{4}{3}x^4}}$
How to integrate
$$\int\frac{dx}{x\sqrt{1-\frac{4}{3}x^4}}$$
Mathematica found
$$\ln x-\frac{1}{2}\ln\left(1+\sqrt{1-\frac{4}{3}x^4}\right)$$
but I can't find a method to arrive at this solution.
AI: Try $u=\sqrt{1-(4/3)x^4}$, $u^2=1-(4/3)x^4$, $2u\,du=-(16/3)x^3\... |
H: Finding the MLE of a multinomial distribution (uneven probabilities)
I am trying to simulate loaded die where the face probabilities are:
$$
p_1=p_2=p_3=p_4=1/6+\theta\text{ and }p_5=p_6=1/6-2\theta
$$
And so using the multinomial distribution I have:
$$
\binom{n}{x_i}\prod_{i=1}^6 p_i^{\displaystyle x_i}=\binom{n}... |
H: Symmetric Matrices of $I_{2}$
Find $10$ symmetric matrices $ A = \begin{pmatrix}
a &b \\
c&d
\end{pmatrix}$ such that $A^{2}=I_{2}$
(I'm going to call matrix A the "square root" of $A^{2}$. If this is the incorrect name for it, may someone please tell me what it is actually called?)
My professor posed this que... |
H: Integration by parts disconnect
I'm trying to integrate $\displaystyle E(Y^2) = \int^\infty_0 y^2\lambda e^{-\lambda y} dy$
doing it by parts this is my logic.
$\displaystyle E(Y^2) = \int^\infty_0 y^2\lambda e^{-\lambda y} dy$ where $u=y^2$, $du=2y\,dy$, $dv=\lambda e^{-\lambda y}\, dy$ and $v = -e^{-\lambda y}$ s... |
H: Prove the edges of a multigraph may be oriented such that the net-degree of any vertex is $\leq 1$.
The net-degree of a vertex $v$, denoted $\text{netdeg}(v)$, in a digraph $G$ is defined by
$$ \text{netdeg}(v)=| ~ \text{outdeg}(v) - \text{indeg}(v) ~| $$
where $\text{outdeg}(v)$ and $\text{indeg}(v)$ are the out-d... |
H: Does $\,x>0\,$ hint that $\,x\in\mathbb R\,$?
Does $x>0$ suggest that $x\in\mathbb R$?
For numbers not in $\,\mathbb R\,$ (e.g. in $\mathbb C\setminus \mathbb R$), their sizes can't be compared.
So can I omit "$\,x\in\mathbb R\,$" and just write $\,x>0\,$?
Thank you.
AI: It really depends on context. But be safe... |
H: When two polynomials $f(x),g(x)$ over a field $F$ are said to be relatively prime?
When two polynomials $f(x),g(x)$ over a field $F$ are said to be relatively prime?
Following the definition given for the integers I guess when two of them have no factors in common other than $1.$ But if we follow such definition i... |
H: Group homomorphisms into a field
Let $G$ be a finite group, and let $k$ be a field, which should be algebraically closed, I think. How to describe all homomorphisms $G\rightarrow k^*$ (i.e. one-dimensional representations: $k^*=\mathrm{GL}_1$) or just find the number of them?
I've got the following:
First, we assum... |
H: Finding side of rectangle using given information
Really simple question but I am stuck. The following information is given:
$$BD=8,\quad AB = 6,\quad ED =5,\quad EF = EC$$
and we want to find $AF$.
If we have three $90^\circ$, what does that really mean, and how I can find $AF$?
AI: Hint:
$$\begin{align*}
AB=6,\... |
H: Composition of systems of equations
Suppose $$2x + 3y = u$$ $$x - 4y = v$$
and further that
$$3u - 5v = c$$ $$2u + 3v = d$$
Express c and d in terms of $x$ and $y$ by matrix multiplication.
It's quite easy by direct substitution but I can;t work out how to use matrix multiplication. Any ideas? Thanks in advance!
A... |
H: Proving that length of a curve is $\infty$
Let $f$ be a differentiable and continious function in $(0,1]$ and $lim_{x\to 0^+}f(x)=\infty$. Prove that the length of the curve on (0,1] is $\infty$.
Steps I tried: $L=\int_0^1 \sqrt{1+f\prime(x)^2}dx\geq\int_0^1\sqrt{f\prime(x)^2}=\int_0^1f\prime(x)=f(1)-lim_{x\to0^+}... |
H: Factorize in R[x]
I have the polynomial $x^8+1$, I know that there's no root for solve this in $\Bbb R[x]$ but i want to factorize this to the minimal expression. This is possible or this is irreducible?
AI: Fun fact:
$$x^4+1=(x^2+\sqrt{2}x+1)(x^2-\sqrt{2}x+1).$$
This can be derived by setting $x^4+1$ equal to a pr... |
H: How to find area of triangle from its medians
The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is
a) $48$
b) $144$
c) $24$
d) $72$
I don't want whole solution just give me the hint how can I solve it.Thanks.
AI: You know that medians divide a triangle to 6 equal... |
H: need to show antiderivative exist
Let $U$ be a simply connected open set and $z_1,\dots, z_n$ be points of $U$ and let $U^*=U\setminus \{z_1,\dots,z_n\},z_i\in U$ Let $f$ be analytic on $U^*$. Let $\gamma_k$ be a small circle centered at $z_k$ and let $$a_k={1\over 2\pi i}\int_{\gamma_k} f(\xi)d\xi$$ let $$h(z)=f(z... |
H: Definition of simple spectrum
From the book "Spinning Tops" by Audin, given Lax equation $[A_{\lambda},B_{\lambda}]$ where $\lambda$ is a parameter (so called spectral parameter), she claims that we have spectral curve $P(\lambda,\nu)=0$ where $P(\lambda,\nu)$ is a characteristic polynomial of $A_{\lambda}$. Then, ... |
H: When is it solvable:$10^a+10^b\equiv -1 \pmod p$
If $p$ is a prime, $(a,p)=1$,denote $ord(a,p)=d,$ where $d$ is the smallest positive integer solution to the equation $a^d\equiv 1 \pmod p$.We can prove that $$10^n\equiv -1 \pmod p\tag1$$ is solvable iff $ord(10,p)$ is even.
Now,consider this equation,$$10^a+10^b\e... |
H: Are there real numbers a and b such that $f(x,y,t) = x^a t^b$ satisfies the heat equation?
The question is in the title. The heat equation is as follows:
$$
\frac{\partial f}{\partial t} = k \left( \frac{\partial^2 f}{\partial x^2} + \frac{\partial^2 f}{\partial y^2} \right),\quad k\in\mathbb{R}
$$
Attempt at sol... |
H: Quadratic residues mod $n$ of $n-1$
While investigating something about square numbers, I started to wonder which numbers $n$ have quadratic residues of $n-1$. Obviously all $n$ of the form $m^2+1$ will work. But there are other numbers such as $13$, is there a pattern to them? Any insights on this would be great.
... |
H: Torus interior homeomorphic to torus exterior
Let $T^2 \subset \mathbb{R^3}$, then $X_i$ be its interior and $X_e$ its exterior. By computing homotopy groups of $X_i \cup T^2$ and $X_e \cup T^2$ and corresponding isomorphisms between, one could show with Whitehead's theorem that $X_i$ is homotopy equivalent to $X_e... |
H: $e^z$ is entire yet has an essential singularity (at $\infty$)
Is there no inconsistency? Or does the property of being entire exclude the point $z=\infty$?
p.s. following up from my previous question limit of $e^z$ at $\infty$
AI: The complex plane doesn't contain the point at infinity. (Infinity is *not*a number,... |
H: Set Theory Notation Crises
For those who are familiar with the following notation, could you explain it in plain English because I picked up a set theory textbook but the book assumes the reader is familiar with the notation without giving a formal explanation anywhere.
1) $$\{x:\mathscr{P}x\}$$
2) $$\{\mathscr{a... |
H: Level Sets Questions
1) In the following link, question 1:
http://mathquest.carroll.edu/libraries/MVC.student.14.01.pdf
Is it true that both partial derivatives are negative ? If so, can someone help me find an example of a contour plot with $f_x<0 , f_y>0$ in one of the points ? I just want to verify
2) In the fol... |
H: question about Riemann zeta $\zeta (0)$
i know that
$$\zeta (m)=\sum_{n=1}^\infty n^{-m}$$
so
$$\zeta (0)=\sum_{n=1}^\infty n^0=1+1+1+1+1+1+\cdots=\infty $$
but actually
$$\zeta (0)=-0.5$$
where is the wrong
please help
thanks for all
AI: I also struggled with this for a while. Your definition of the Riemann z... |
H: Using Semi-circle find side of triangle
The figure below above shown a bicycle path. If semicircular portion $ABC$ is $100$ $\pi$ and $CD$ is $100$$ft$ then what is $AD$?
I have tried to find the diamenter of the circle and the Pythagorean, I am not able figure out my answer.
AI: The ratio of a circle's circumfer... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.