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H: Continiutity of a dense subset imply continuity of the set
I would like some help proving that if $f$ and $g$ are continous on $(a,b)$ and $f(x)=g(x)$ for every $x$ in a dense subset of $(a,b)$ then $f(x)=g(x)$ for all $x$ in $(a,b)$.
AI: Let $S\subset(a,b)$ be the dense subset in the question. Take any $x\in(a,b)... |
H: $\epsilon$-$\delta$ proof
I found an interesting problem in my textbook, it asks to prove the following statement:
If $f'(x_{0}) >0$, then there is a $\delta >0$ such that $f(x)<f(x_{0})$ if $x_{0}- \delta < x <x_{0}$, and $f(x) > f(x_{0})$ if $x_{0} <x < x_{0} + \delta$.
How would one prove this from the definit... |
H: Probability of coin toss there are 68% chance that % heads in the range 50% plus or minus SD?
A coin is tossed $2500$ times. There is about a $68\%$ chance that the percent of heads is in the range $50\%$ plus or minus ($0.5$, or $1$, or $1.5$, or $2$, or $2.5$)?
$\text{P(of coin tossed is 1/2)}=0.50 \pm \text{SD... |
H: Simplify $\sqrt n+\frac {1}{\sqrt n}$ for $n=7+4\sqrt3$
If $n=7+4\sqrt3$,then what is the simplified value of
$$\sqrt n+\frac {1}{\sqrt n}$$
I was taking LCM but how to get rid of $\sqrt n$ in denominator
AI: Hint: We have $\left(\sqrt n+\dfrac {1}{\sqrt n}\right)^2=n+2+\frac 1n$, then multiply top and bottom of... |
H: Finding specific alternative form of $\frac{(x-y)x+{y\over(y-z)}}{(x+y)z}=ab$
How does one approach; $$\frac{(x-y)x+{y\over(y-z)}}{(x+y)z}=ab$$ to find the form: $$-a b z (x+y) (y-z) = x^2 (-y)+x^2 z+x y^2-x y z-y$$
AI: $$\frac{(x-y)x+{y\over(y-z)}}{(x+y)z}=ab$$
take LCM on upper part
$$\frac{(x-y)(y-z)x+{y}}{(x+y... |
H: Showing uniqueness of Riemann's Integral
I am given the definition: Le $f$ be defined on $[a,b]$. we say that $f$ is Riemann Integrable on $[a,b]$ if there is a number $L$ with the following property: for every $\epsilon>0$, there is a $\delta > 0$ such that $\left\|P\right\|< \delta$ implies $| \sigma -L| < \epsil... |
H: Knowing that $n= 3598057$ is a product of two different prime numbers and that 20779 a square root of $1$ mod $n$, find prime factorization of $n$.
Knowing that $n= 3598057$ is a product of two different prime numbers and that 20779 is a square root of $1$ mod $n$, find prime factorization of $n$.
What I have don... |
H: When does a matrix $A$ with ones on and above the diagonal have $\det(A)=1$?
What conditions, if they're even necessary, must be placed on $\star$ so that the matrix
$$ \begin{pmatrix} 1 & & \huge{1} \\ & \ddots & \\ \huge{\star} & & 1 \end{pmatrix}, $$
so that $\det{(A)}=1$, where $\huge{1}$ denotes "all entries a... |
H: Expressing $\sqrt{n +m\sqrt{k}}$
Following this answer, is there a simple rule for determining when:
$$\sqrt{n +m\sqrt{k}}$$
Where $n,m,k \in \mathbb{N}$, can be expressed as:
$$a + b\sqrt{k}$$
For some natural $a,b$?
This boils down to asking for what $n,m,k \in \mathbb{N}$ there exist $a,b\in \mathbb{N}$ such th... |
H: Matrix $BA\neq$$I_{3}$
If $\text{A}$ is a $2\times3$ matrix and $\text{B}$ is a $3\times2$ matrix, prove that $\text{BA}=I_{3}$ is impossible.
So I've been thinking about this, and so far I'm thinking that a homogenous system is going to be involved in this proof. Maybe something about one of the later steps being ... |
H: How is Brownian motion predictable?
Could someone please explain how Brownian motion is predictable? My understanding is that a predictable process is one that depends on information up to time t say but not t itself, therefore W_t has to be observed at time t to determine its value.
AI: Brownian motion is continuo... |
H: Dual basis and annihilator problem
I think they're fairly simple but I really don't know where to start, the problems are these:
First one:
$V$ a vector space of dimension $n$ and $\phi \in V^* \setminus \{0\}$.
Prove that $\dim \ker \phi = n-1$.
Second one:
Let $B = \begin{bmatrix}2 & -2 \\ -1 & 1\end{bmatrix}$ ... |
H: Find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550)
I am attempting to find the angle between the 2 points (50.573,-210.265) and (117.833,-80.550).
Is my calculation correct because a program is giving me a different answer? It says the angle is 27'24'27.27 DMS
dx = x2 - x1;
dy = y2 - y1;
an... |
H: inner product space and gram matrix
I have a question from my proffesor that I can not figure it out.
V will be inner product system above R2.
Let E some basis with the gram matrix (E={v1,v2})
This is the gram matrix:
\begin{pmatrix}
2 & -1\\
-1 & 1\\
\end{pmatrix} =
\begin{pmatrix}
{(v_{1},v_{1})} & {(v_{1... |
H: Inclusions between $H^1_0(\Omega) \cap H^2(\Omega)$ and $H^2_0(\Omega)$
Let $\Omega$ be a bounded domain in $\mathbb{R}^n$. I would like to understand something about the relationships between these two spaces: $$H^1_0(\Omega) \cap H^2(\Omega)\quad \text{and}\quad H^2_0(\Omega).$$
Is there any inclusion? If there i... |
H: Critical points in multivariable calc
Find the critical points of
$z = x^{3} + 3xy^{2} - 3x^{2} - 3y^{2} + 7$
I understand if it was $f(x,y)$ but this z is really throwing me off..
I could take the partial derivs of x and y, but if I take the partial of z I get 0=0?
EDIT after comment. Ok, so I will take the parti... |
H: Proving $\frac{1}{2}(5x+4),\;2 < x,,\;\text{isPrime}(n)\Rightarrow n = 10k+7$
How is it possible to establish proof for the following statement?
$$n = \frac{1}{2}(5x+4),\;2<x,\;\text{isPrime}(n)\;\Rightarrow\;n=10k+7$$
Where $n,x,k$ are $\text{integers}$.
To be more verbose:
I conjecture that;
If $\frac{1}{2}(5x+4... |
H: Valid Proof that the Irrationals are Uncountable?
So I originally wanted to prove that the reals are uncountable, but the best solution I came up with was to prove the irrationals are uncountable so therefore the reals must be as well. I suppose my first question is, is this valid logic?
Next take any countable sub... |
H: What is the difference between regulator and stabilization
What is the difference between regulator and stabilization in control theory
don't they both minimize the disturbance to the system?
could answer be elaborated from the view of state and output?
AI: If I understand your question properly, we have two things... |
H: Quotient rings of Gaussian integers $\mathbb{Z}[i]/(2)$, $\mathbb{Z}[i]/(3)$, $\mathbb{Z}[i]/(5)$,
I am studying for an algebra qualifying exam and came across the following problem.
Let $R$ be the ring of Gaussian Integers. Of the three quotient rings
$$R/(2),\quad R/(3),\quad R/(5),$$
one is a field, one is ... |
H: what is the area of face of the cube in $m^2$
A fly is trapped inside a hollow cube. It moves from A to C along edges of cube, taking shortest possible route. It then comes back to A again along edges, taking longest route(without going over any point more than once). If the total distance traveled is 5040 meter , ... |
H: Are the graphs of these two functions equal to each other?
The functions are: $y=\frac{x^2-4}{x+2}$ and $(x+2)y=x^2-4$.
I've seen this problem some time ago, and the official answer was that they are not.
My question is: Is that really true?
The functions obviously misbehave when $x = -2$, but aren't both of them ... |
H: Find the minimum of $|a+\frac 2 {a-1}|$ where $|a|\leq2$.
Find the minimum of $|a+\frac 2 {a-1}|$ where $|a|\leq2$.
I tried using differentiation, but the absolute makes things troublesome...
Please help. Thank you.
AI: Find the minimum of $a+\frac{2}{a-1}$ on the interval $(1,2]$, and the minimum of $-(a+\frac{... |
H: Impossibility of polynomial approximation
This is exercise 12.6 in David Ullrich's Complex Made Simple. He has discussed many ways to prove the existence of polynomial approximations to functions in the complex plane, but not how to show such approximations are impossible in certain cases, which is the point of the... |
H: Is this elementary number theory proof correct?
Let $A(n)$ be the number of primes less than $n$, divided by $n$ (so for example, $A(n) \leq 1$, as there cannot be more primes less than $n$ as there are integers less than $n$). Suppose that $n$ is a positive multiple of the positive integer $q$. Show that $$A(n) \l... |
H: Solve a System with Variable
Given these matrices, how does one find two real solutions?
$dx/dt$ =
$\begin{bmatrix}
3 & -5\\
5 & 3
\end{bmatrix}x$
with $x(0) = \begin{bmatrix}
2\\
-3
\end{bmatrix}$
AI: Hints:
We are given the system:
$$x' = Ax = \begin{bmatrix}
3 & -5\\
5 & 3
\end{bmatrix}x$$
with IC:
$$x(0... |
H: $n$-linear alternating form with $\dim{V}
Prove that every $n$-linear alternating form on a vector space of dimension
less than $n$ is the zero form.
AI: Let $k$ be a field, and let $V$ be a vector space over $k$, with $\dim V <n$. Let $\omega$ be an alternating $n$-form on $V$. Then $\omega : V^n \to k$ is an al... |
H: Sufficient condition for irreducibility of polynomial $f(x,y)$
Suppose we have a non-constant polynomial $f(x,y)\in\mathbb{Q}[x,y]$ such that the following two conditions are satisfied:
1) For every $x_0\in\mathbb{Q}$, the polynomial $f(x_0, y)\in\mathbb{Q}[y]$ is irreducible.
2) For every $y_0\in\mathbb{Q}$, the p... |
H: How can I find these velocities without using the quadratic formula?
If a ball is thrown vertically upward with a velocity of $160 \text{ ft/s}$, then its height after t seconds is $s = 160t − 16t^2$.
a) What is the velocity of the ball when it is $384 \text{ ft}$ above the ground on its way up?
b) What is the v... |
H: Probability that we choose a two headed coin
We have a $501$ coins on the table, and assume that they have all been flipped onto that table (i.e., there is a mix of heads and tails). This also includes a two-headed coing.
Now if we pick up $1$ coin and its heads, what is the probability it is also the two-headed co... |
H: Find max and min of $IJ + FE + GH$
Let $D \in \triangle ABC$. Passing through D, contruct$\, FE \parallel AB, IJ \parallel AC, GH \parallel BC$. Find max and min of IJ + FE + GH
Can this problem be solved by AM-GM ? I tried $IJ + FE + GH \ge 3\sqrt[3]{\mid IJ\mid \cdot \mid FE\mid \cdot \mid GH\mid}$
and using $$... |
H: If a 3D-cake is cut by $n$ planes yielding the maximum number of pieces, then what is the number of pieces with the cake crust?
It is known that a 3D-cake can be cut by $n$ plane cuts at most into $N$ pieces, defined by Cake Number $N=\frac {1}{6}(n^3+5n+6)$. However, some of the pieces would have a crust of the ca... |
H: I'm having trouble with a definition of the upper and lower limits, and a theorem that follows it.
The following is the definition.
Let $\{s_n\}$ be a sequence of real numbers. Let $E$ be the set of numbers $x$ such that $s_{n_{k}}\rightarrow{x}$. This set $E$ contains all subsequential limits, plus possibly the n... |
H: Describing Domain of Integration (Triple Integral)
I'm really struggling to go about starting the following problem:
This question concerns the integral,
$\int_{0}^{2}\int_0^{\sqrt{4-y^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{8-x^2-y^2}}\!z\ \mathrm{d}z\ \mathrm{d}x\ \mathrm{d}y$.
Sketch or describe in words the domain of ... |
H: How to prove that $\int_0^{\infty} \log^2(x) e^{-kx}dx = \dfrac{\pi^2}{6k} + \dfrac{(\gamma+ \ln(k))^2}{k}$?
I was answering this question: $\int_0^\infty(\log x)^2(\mathrm{sech}\,x)^2\mathrm dx$ and in my answer, I encountered the integral
$$\int_0^{\infty} \log^2(x) e^{-kx}dx$$
which according to WolframAlpha for... |
H: Why does the differential equation $y' = y + 1$ have solution $y(x) = Ce^x - 1$?
I was watching a video on differential equations for a class that I'm taking. I took calculus so long ago that I can't seem to figure why the differential equation $y' = y + 1$ has solution $y(x) = Ce^x - 1$.
AI: $$\frac{dy}{dx}=y+1$$... |
H: Proof of a theorem with upper/lower limits.
Theorem: If $s_n \le t_n$ for all $n$ greater than a fixed integer $N$, then $$\lim_{n \to \infty} \inf s_n \le \lim_{n \to \infty} \inf t_n$$
I would like to prove this and it would be nice if someone could check my work.
Proof: Letting $$\lim_{n \to \infty} \inf s_n ... |
H: Strong characterization of $\mathbb C$ with respect to $\mathbb R$
$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ combined with $\mathbb R^2$ that make it a field?
Is ... |
H: Minimum number of coconuts
Three friends namely $A$, $B$ and $C$ collected coconuts with the help of monkey and fell asleep. At night, $A$ woke up and decided to have his share. He divided coconuts into three shares, gave the left out coconut to the monkey and fell asleep. In the same way in order $B$ and $C$ got u... |
H: probability density functions
Suppose $Y$ is a random variable pdf $f(y)=ky , y=3/n,6/n,9/n...,3n/n$
Find the value of the constant $k$ and write down $Y$'s cdf.
Find simple general expressions for $EY, \text{Var} \,Y, P(Y=3/2)$ and $P(Y>3/2)$
For the case $n=10$, evaluate $EY, \text{Var}\,Y,P(Y=3/2)$ and $P(Y>3/2... |
H: Relation between chords length and radius of circle
Two chords of a circle, of lengths $2a$ and $2b$ are mutually perpendicular. If the distance of the point at which the chords intersect,from the centre of the circle is $c$($c<$radius of the circle),then find out the radius of the circle in terms of $a,b$ and $c$.... |
H: Why is it meaningless for a closed set to be polygonal path connected?
My textbook (Complex Analysis by Saff & Snider) defines connectedness for open sets; the given definition of a connected open set is: a set in which every pair of points can be joined by a polygonal path that lies entirely in the set.
Using the ... |
H: Simple Math Equation find sum of 4 numbers and if greater then number X reduce all 4 numbers respectively
Im not the greatest at Math but i have the following problem:
impressions = 791.
watched 100 = 500
watched 75 = 383
watched 50 = 600
watched 25 = 700
The sum of all watched fields is 2183.
Since the sum of all... |
H: When speaking of neighbourhoods in complex analysis, are we always referring to circular neighbourhoods?
In Complex Analysis, does "neighbourhood" automatically mean "circular neighbourhood", or do non-circular ones exist?
AI: The term "neighborhood" of a point comes from topology, where it is taken to mean any ope... |
H: Does an irreducible $\mathbb CG$-module have a basis of the form $u,ug_1,\dots,ug_n$?
Suppose that $U$ is an irreducible $\mathbb CG$-module and $u\in U$. Let $\operatorname{span}(u_1,\dots,u_k)$ denotes the linear span of vectors $u_1,\dots,u_k\in U$.
I was thinking along these lines: there must exists some $g\in ... |
H: The closes point to a curve in space.
I am working on the following problem.
Find the point closest to the origin, of the curve of intersection of the plane $2y+4z =5$ and the cone $z^2 = 4(x^2+y^2)$
I was able to see that the intersection will be a tilted ellipse with coordinates $(x,y,4(x^2+y^2))$.
So I was thi... |
H: Series expansion with remaining $\ln n$
I'm studying the asymptotic behavior $(n \rightarrow \infty)$ of the following formula, where $k$ is a given constant.
$$ \frac{1}{n^{k(k+1)/(2n)}(2kn−k(1+k) \ln n)^2}$$
I'm trying to do a series expansion on this equation to give the denominator a simpler form so that it is ... |
H: Problem with Discrete Parseval's Theorem
I think I must be missing something obvious, but I can't for the life of me see what it is. The discrete version of Parseval's theorem can be written like this:
$\sum_{n=0}^{N-1} |x[n]|^2 = \frac{1}{N}\sum_{n=0}^{N-1} |X[k]|^2 $
Now, say you've got some function in time, lik... |
H: Open, closed and continuous
I have some troubles to understanding something:
We were asked to find a function that is open and continuous but not closed and actually I found such a function, but our tutor gave us this example
$ e^x$ since this function is continuous as a mapping $$\exp: \mathbb{R} \rightarrow \math... |
H: Measures on all subsets of $\aleph_0$
A theorem of Ulam says:
A finite measure $\mu$ defined on all subsets of a set of cardinality $\aleph_1$ must be $0$ for all subsets if it sends every $1$-element subset to $0$.
Will this statement hold if we substitute $\aleph_0$ for $\aleph_1$ and require the measure to be ... |
H: inner product space and polynomial
Let $V = \mathrm{span}\{1,x,x^2,x^3\}$ be a real inner product space with the
inner product defined by
$$
\langle f,g\rangle =\int\limits_{-1}^{1} fg
$$
Check that $T(f) = f(0)$ is a linear functional on this space and find the element of $V$ which represents it (corresponds to)... |
H: Is there an element in $^* \Bbb N$ is Dedekind-infinite?
One definition of a finite set is that it can be injected into an initial segment of $ \Bbb N$, thus any $n$ in $\Bbb N$ is finite.
Accordingly, if it's legitmate to define every element in $^* \Bbb N$ as its own intial segment, then every element $^* \Bbb ... |
H: Solving the domain and range of a region satisfying two inequalities?
The question I was provided was:
"Find the domain and range of the region satisfied by the following inequalities:
i) $y \ge (x-1)^2$
ii)$y \le2x+1$
Any help would be greatly appreciated. Would you recommend graphing or solving algebraically?
AI... |
H: Integrate: $\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz$
Q. Show that : $$\int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz = \frac{\sin pt}{p}$$
I considered the following contour
$$\int_\Gamma \frac{e^{tz}}{z^2 + p^2}dz + \int_{a - i\infty}^{a + i\infty} \frac{e^{tz}}{z^2 + p^2}dz = ... |
H: Why do these trig functions "overpower" each other?
For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine always increases/decreases faster than cosine. Why does sine overp... |
H: Solve a simple equation with log in it
I'm stuck with solving this equation,
$$2 \log x = \log 9 $$
This is how far I made it:
\begin{align}
\log x &= \log 4,5 \\
x &= ?
\end{align}
I'm a beginner at logarithms so I appreciate ways to solve it and not just an answer.
AI: If we start with $$2 \log x = \log 9,$$ the... |
H: if $p\implies q$ is the same as $\lnot p \lor q$, then...
If $p\implies q$ is the same as $\lnot p \lor q$, then what is $p\implies \lnot q$?
I'm not sure if this is $\lnot p \lor \lnot q$, or $\lnot p \lor q$.
I'm trying to figure this out, because i have a problem:
~(q v p) --> ~r). I use demorgans law on this to... |
H: Prove $\int_2^\infty{\frac{\ln(t)}{t^{3/2}}},\mathrm{d}t$ converges
Show, using a comparison test, that $\displaystyle \int_2^\infty{\frac{\log{t}}{t^{\frac32}}}\mathrm{d}t$ converges.
All the answers I've tried shows it diverges, taking $\log{t} \le t^{1/2}$ and $\log{t} \le t$.
Cheers
AI: Solving $\log t = t^{1/4... |
H: $p^{3}+m^{2}$ is square of a number.
Well i thought it is a nice problem so i will post it here.
1) Prove that for every natural numbers $m$, There is at most two primes $p$ where $p^{3}+m^{2}$ is a perfect square.
2) Find all natural numbers $m$ so that $p^{3}+m^{2}$ is the square of a number for exactly $2$ prime... |
H: very basic short exact sequence problem
Given a short exact sequence $0 \rightarrow A \rightarrow B \rightarrow C \rightarrow0$ and $f:A \rightarrow B, g: B \rightarrow C$, why is $C$ isomorphic to $B/A$? All I can show is that $C$ is isomorphic to $B/ Im (A)$. I've look up a few books but I still don't understand ... |
H: Complexified tangent vector, complex manifold
Consider a complex submanifold $M$ of a complex ambient vector space $X$. Suppose you have a base point $p \in M$ and a $C^1$ arc $\gamma(t)$ passing through $p$ and staying in $M$, with tangent vector at $p$ denoted by $u$ (the tangent vector is taken by considering $X... |
H: Counting Number of Objects - When to Add One Back
I know this might be a very basic question. Sometimes to count objects, we just subtract. For example -- If there are 5 apples and I take away 1, then remaining are $5 - 1 = 4$ apples.
But other times, we have to add one back.
For example -- the number of days from ... |
H: Another basic short exact sequence problem
In the following commutative diagram of R-modules, all of the rows and columns are exact. Prove that $K$ is isomorphic to $L$.
\begin{array}{ccccccccccc} &&&&&&&&0 &&\\
&&&&&&&&\downarrow &&\\ &&&&&& 0 & & L &&\\ &&&&&& \downarrow && \downarrow &&\\ &&&& M^{\prime\prime} ... |
H: distance travelled after nth bounce
A ball is thrown vertically to a height of $625$ meters from ground. Each time it hits the ground it bounces $\frac{2}{5}$ of the height it fell in the previous stage. How much will the ball travel during the first $20$th bounces? How can we derive a formula for finding this?
AI:... |
H: equality between the index between field with $p^{n}$ elements and $ \mathbb{F}_{p}$ and n?
can someone explain this?
$ \left[\mathbb{F}_{p^{n}}:\mathbb{F}_{p}\right]=n $
AI: $$|\Bbb F_p|=p\;,\;\;|\Bbb F_n^n|=p^n$$
amd since any element in the latter is a unique linear combination of some elements of it and scalars... |
H: Regularity of balanced binary strings
How can one tell which number of propositional variables is necessary
to express a Boolean function given as a sequence of 0s and 1s (a
binary string) of length $2^n$ as a Boolean formula?
The extremal cases are clear: if no 0s or no 1s are present, we need no variables a... |
H: How prove this $ab+bc+cd\le\dfrac{5}{4}$
let $a,b,c,d\in \Bbb R$ and $a,b,c,d>-1,a+b+c+d=0$
prove that
$$ab+bc+cd\le\dfrac{5}{4}$$
I have this solution
if $b\le c$, then
$$ab+bc+cd=a(b-c)-c^2\le -(b-c)-c^2=-(c-\dfrac{1}{2})^2+\dfrac{1}{4}-b\le\dfrac{1}{4}-b\le \dfrac{5}{4}$$
and then
$b>c$,as the same methods.I thi... |
H: Vector valued Mean value theorem: Norm for the gradient
The wikipedia article on the vector valued Mean value theorem, says
For $f:\mathbb R^n \to \mathbb R^n$, if the gradient is bounded,
$$
\| \nabla f \| \le M,
$$
then
$$
\|f(x)-f(y) \| \le M \|x-y\|.
$$
What is the norm used for the gradient $\| \nabla... |
H: Integral of a rational function: Proof of $\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}= {{\pi}\over{2}}$?
I suspect that
$$\sqrt{C}\,\int_{0}^{+\infty }{{{y^2}\over{y^2\,C+y^4-2\,y^2+1}}\;\mathrm dy}=
{{\pi}\over{2}}$$
for $C>0$.
I tried $C=1$, $C=2$, $C=42$, and $C=\frac{1}{1000}$... |
H: SO(2,1) not connected
I am trying to show that $SO(2,1)$ is not connected but I have no idea where to start really, I know that it is connected if there is a path between any two points. My definition of $SO(2,1)$ is:
$SO(2,1)=\{X\in Mat_3(\mathbb{R}) \mid X^t\eta X=\eta, \ \det(X)=1\}$ where $\eta$ is the matrix d... |
H: How to evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
AI: HINT:
As we are dealing with the limit in real numbers, $x>0\implies x\to+\infty$
Put $x=y^2$
$$\implies \sqrt{x+\sqrt{x}}-\sqrt{x-\sq... |
H: Looking for example of an order homomorphism that doesn't preserve joins.
I know that not every order homomorphism preserves joins. But, I can't think of an example!
Both minimal examples and 'natural' examples welcome.
AI: Let $S$ be the poset
a... |
H: Logarithmic equation. Need to know if i am teaching right
Two of my friends is studying for a test. They asked me about a simple question. But they told me that i was wrong on a question. I could be wrong. But i need you guys to make sure that they learn the right stuff. So if i was right. I then can tell them how ... |
H: Quick Conditional Probability Question
Been going through lecture notes (ones I made myself, so who knows how accurate they are - or rather - they most certainly are not) and can't seem to understand this one example
Example: Two horses - A and B. A wins with probability 0.5, B wins with probability 0.3. What is th... |
H: Using the hypothesis $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$ to prove something else
Assuming that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{1}{a+b+c}$$
Is it possible to use this fact to prove something like:
$$\frac{1}{a^{2013}}+\frac{1}{b^{2013}}+\frac{1}{c^{2013}}=\frac{1}{a^{2013}+b^{2013}+c^{2... |
H: General Solution of Diophantine equation
Having the equation:
$$35x+91y = 21$$
I need to find its general solution.
I know gcf $(35,91) = 7$, so I can solve $35x+917 = 7$ to find $x = -5, y = 2$. Hence a solution to $35x+91y = 21$ is $x = -15, y = 2$.
From here, however, how do I move on to finding the set of gen... |
H: A basic question on the definition of $E[X]$
My question is regarding the definition of $E[X]$ in a probability book. It starts with the definition in case of a simple random variable (a random variable which takes only finite number of values) where $E[X]$ is defined by :
$$E[X]=\sum_{i=1}^{n}x_i P(X=x_i)$$
Now a... |
H: Fermat Last Theorem for non Integer Exponents
We now that Fermat's last theorem is true so there are not positive integer solutions to
$$x^n+y^n=z^n$$
for $n\in\mathbb{N}$ and $n>2$.
But what about if $n\in\mathbb{R}$ or $n\in\mathbb{R}^+$?
AI: Suppose $z> \max(x,y)$ then $x^0+y^0 = 2 > z^0$ but there exists some ... |
H: What's more robust than a structural homomorphisms?
This question isn't category theory; but, category theoreticians tend to be interested in mathematical structure, so I thought the answer might exist within that knowledge base.
Given two mathematical structures $X$ and $Y$ with the same pattern of airities, there... |
H: Finding the Euler Lagrange equation - differentiation
I'm teachin myself the basics of Calculus of variations. So far I know how to calculate the Euler Lagrange equation for simple functionals.
I'm now stuck on how to compute the total differentiation of the following problem:
$$I[y]=\int_0^1 (y\frac{dy}{dx})^2 -\... |
H: Question about the definition of representability of a quadratic form
Say I have an integral quadratic form $q$ on $\mathbb{Z}^r$ and another integral binary quadratic form $\tilde{q}$. What does it mean for $q$ to primitively represent $\tilde{q}$? I can't seem to find the definition anywhere. Of course $q$ primit... |
H: Two random variable with the same variance and mean
Let $Y\in L^{2}(\Omega,\Sigma,P)$ and let $E[Y^2|X]=X^2$ and $E[Y|X]=X$. Could we prove that $Y=X$ almost surely.
My partial answer:
By the definition of conditional expectation we have $E[Y^2]=\int_{\Omega} E[Y^2|X] dP=\int_{\Omega}X^2 dP=E[X^2]$ and $E[Y]=\int_{... |
H: Prove that nearly all positive integers are equal to $a + b + c$ where $a | b$ and $b | c$, $a \lt b \lt c$
If a positive integer $n$ is equal to $a + b + c$ where $a | b$, $b | c$ and $a \lt b \lt c$, let it be called "faithful". Prove that nearly all numbers are faithful and list the non-faithful numbers.
Here'... |
H: What is the Sobolev Lemma?
In the paper I am reading the authors state that $|\nabla u|_\infty$ can be replaced by $|u|_3$ using the Sobolev Lemma. I am trying to find this lemma but its turned out to be very difficult.
The context is the following:
a smooth bounded domain $\Omega \subset \mathbb{R}^3$
$|\cdot|_s... |
H: Eigenvector Proof $(I+A)^{-1}$.
Show that the eigenvectors of the $n \times n$ matrix A are also eigenvectors of the matrix $$M = (I+A)^{-1} $$ Where I is the $n \times n$ unit matrix. Determine the eigenvalues.
My Work:
$$Mx=(I+A)^{-1}x = ???$$
AI: Hint: if you have
$$
Ax=\lambda x
$$
then what do you get when you... |
H: What is the meaning of 'columns have unit lengths'
What is the meaning of this?
In random projection, the original d-dimensional data is
projected to a k-dimensional (k << d) subspace through
the origin, using a random k × d matrix R whose columns
have unit lengths.
I have already searched around the intern... |
H: Write ‘There is exactly 1 person…’ without the uniqueness quantifier
During a lecture today the prof. posed the question of how we could write "There is exactly one person whom everybody loves." without using the uniqueness quantifier.
The first part we wrote as a logical expression was "There is one person whom ev... |
H: How to show this inequality?
Show that $$-2 \le \cos \theta(\sin \theta+\sqrt{\sin^2 \theta +3})\le2$$
Trial: I know that $-\dfrac 1 2 \le \cos \theta\cdot\sin \theta \le \dfrac 1 2$ and $\sqrt 3\le\sqrt{\sin^2 \theta +3}\le2$. The problem looks simple to me but I am stuck to solve this. Please help. Thanks in ad... |
H: Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators?
Why don't we consider the zero subspace (which is readily $T$-invariant) in the definition of direct sum of linear operators?
REF: Schaum's Outline of Linear Algebra
AI: Technically, you ... |
H: Cauchy’s functional equation for non-negative arguments
Function $f:[0,+\infty)\rightarrow\mathbb{R}$ satisfies $f(x+y)=f(x)+f(y)$ for every non-negative $x$ and $y$. It’s bounded from below with some non-positive constant $m$. Does it imply that $f$ has the form $f(x)=cx$ or is there another function satisfying th... |
H: Which topics of mathematics should I study?
I'm a first year econometrics student with a great interest in mathematics. I very much enjoy my study, but still I am interested to learn about more topics in mathematics which are not part of my study. Some of the main topics which I am already familar with or which wil... |
H: Comparing $\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$
Without the use of a calculator, how can we tell which of these are larger (higher in numerical value)?
$$\sqrt{1001}+\sqrt{999}\ , \ 2\sqrt{1000}$$
Using the calculator I can see that the first one is 63.2455453 and the second one is 63.2455532, but can we tell ... |
H: example of a flat but not faithfully flat ring extension
I am learning commutative algebra and there is a definition about faithfully flat modules or ring extensions. I can't think of an example of a flat but not faithfully flat ring extension or module. Can someone help me with it? thanks
AI: Take $f: A \rightarro... |
H: Minimal Polynomials Annihilating an Abelian Torsion-Free Group
Let $A$ be an abelian torsion-free group. Let $\theta \in\operatorname{Aut}A$. Assume that $\theta$ has a finite period in $\operatorname{Aut} A$, say $n$. Obviously $\theta^n-1$ annihilates $A$ (i.e. $A^{\theta^n-1}=\{0\}$, where $\theta^n-1$ is now th... |
H: How can I find all the solutions of $\sin^5x+\cos^3x=1$
Find all the solutions of $$\sin^5x+\cos^3x=1$$
Trial:$x=0$ is a solution of this equation. How can I find other solutions (if any). Please help.
AI: Hint: $ \sin^5 x\leq \sin^2 x$ and $ \cos ^3 x \leq \cos^2 x $.
Hint: Pythagorean Identity for trigonometric... |
H: Is a sequence of all the same numbers monotonic?
I'm wondering based on the definition of monotonicity:
A sequence where $a_n\geq a_{n+1}$ for all $n\in\mathbb{N}$ is monotonic.
So given that the sequence $a_n = 3$ is all the same numbers and is neither increasing or decreasing, is it monotonic?
AI: Yes, a const... |
H: The difference between $\mathbb{Z}$ and $\mathbb{Z}^2$
I know that $\mathbb{Z}$ is the set of integers. But, what does $\mathbb{Z}^2$ mean? How is it different from $\mathbb{Z}$?
Thanks.
AI: If $A$ is a set then $A^2$ is a shorthand for $A\times A$, which is the set of ordered pairs with elements from $A$. That is:... |
H: Non-commutative rings without identity
I'm looking for examples (if there are such) of non-commutative rings without multiplicative identity which have the following properties:
1) finite with zero divisors
2) infinite with zero divisors
3) finite without zero divisors
4) infinite without zero divisors
I'll be grat... |
H: When does equality holds in $A\subseteq P(\cup A)$
Note: $P$ is power set. It's easy to prove that this inclusion holds. But when is other inclusion true? I can't even think of one example...
AI: Here is one example: $A=\{\varnothing\}$.
Also is $A=V_{\alpha+1}$ for any ordinal $\alpha$, then $A=P(V_\alpha)$, there... |
H: Fourier Transforms of shifted sinc funtions
I would like to calculate the Fourier transform of the following functions:
$$\left(\dfrac{\sin(\pi x\pm\pi n/2)}{\pi x\pm\pi n/2}\right)^2$$
$$\dfrac{\sin(\pi x+\pi n/2)}{\pi x+\pi n/2}\cdot\frac{\sin(\pi x-\pi n/2)}{\pi x-\pi n/2}$$
with $n \in\mathbb{N}$
Any help wil... |
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