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H: Inline elements denoted by {{}}? Cartesian product of these inline elements?
What does it mean when you have a set like A = {1, {2}, 3, 4}? Is {2} a subset within A?
Say I have the following:
A = {1, {2}, 3, 4}, B = {1, 2, {3}}, C = {1, 2}
How many elements are in $A \cap B \cap C$? 1 or 2?
What if I want to find $... |
H: Extended Hamming Code (8,4)
I just finished learning some basic concepts about Hamming code. Now I see this extended Hamming code and they say that you can "detect" up to 3 bits of error. Now, say, my actual codeword was $X=10100011$ and I received $Y=10011011$ (3 bits of error) and my transpose of parity check mat... |
H: Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.
Show that $\displaystyle\prod_{\Bbb{N}} \Bbb{R}$ with the box topology is Hausdorff but not metrizable.
$\Bbb{R}$ must be Hausdorff. For $x_1, x_2 \in \Bbb{R}$ (where $x_1 \not= x_2$), if $d$ denotes the dista... |
H: is it possible to proof that this number is not rational
It is an idea I had when reading the proof that $(0,1)$ is uncountable. There the numbers in $(0,1)$ are written into a list in decimal expansion and then the diagonal is modified and the resulting number is a number not on the list.
Now instead consider $S= ... |
H: Question regarding Lebesgue outer measure.
Given $m\geq1$, $0\leq s<\infty$, $0<\delta\leq\infty$ and $A\subseteq\mathbb{R}^{m}$ define: $$\mathcal{H}_{\delta}^{s}\left(A\right)=\inf\left\{ {\displaystyle \sum_{n=1}^{\infty}d\left(B_{n}\right)^{s}\,|\,\left\{ B_{n}\right\} _{n\in\mathbb{N}}}\:\mbox{is a covering of... |
H: Is this a hyperbolic PDE?
Is the PDE
$$
(1+x^2)^2\frac{\partial^2 u}{\partial x^2}-\frac{\partial^2 u}{\partial y^2}+2x(1+x^2)\frac{\partial u}{\partial x}=0\text{ in }\Omega:=\mathbb{R}^2
$$
hyperbolic?
To answer this I set
$a(x,y):=(1+x^2)^2, b(x,y):=0, c(x,y)=-1$, then $b^2-ac=(1+x^2)^2 >0$
and this ... |
H: Proving a predicate logic statement to be valid
I've been stuck on this question for the better part of the day, and I've succumbed to asking for help. I'm not sure how to go about it honestly. I've tried to do the contrapositive to prove it, but I get stuck and end up at a dead end.
The problem is as follows:
$... |
H: How do you simplify this big O sum?
I saw someone interpret $\sum_{i=1}^{n}\mathcal{O}\left(i^{k-2}\right)$ as $\mathcal{O}\left(n^{k-1}\right)$. Is this right? If so, can you explain?
AI: This is not quite true in this simplicity or at least depends on interpretation and context.
One might interprete the statement... |
H: Finding possibilities puzzle
The answer: 2
Hi guys, I'm doing the practice papers for an examination, and got stuck on this question. I ended up getting the right answer, but I'm not sure if my thinking is correct. I would love someone else's opinion on it.
My thinking -- >
"Statue to fountain, always pass the b... |
H: If a function is differentiable almost everywhere, can it be written as an integral?
Consider a function $f:\mathbb{R}^n \to \mathbb{R}$. If $f$ is differentiable with Lebesgue integrable derivative, we may write
$$
f(x+y) - f(x) = \sum_{i=1}^p \int_0^1 y_i \nabla f_i(x+ty)dt
$$
by the fundamental theorem of calcul... |
H: Are two Hilbert spaces with the same algebraic dimension (their Hamel bases have the same cardinality) isomorphic?
We know that two Hilbert spaces that have orthonormal bases of the same cardinality are isomorphic (as an inner product spaces).
My question is: what can we say when we know that their Hamel bases hav... |
H: How to work out sinh^2(x)
I just did a question where I had $sinh^2(x)$
I know this is simply $(sinh(x))^2$ however couldn't work out where the extra 2 came from when working out.
$sinh(x) = \frac{e^x-e^{-x}}{2}$
so
$sinh^2(x) = (\frac{e^x-e^{-x}}{2})^2$
which I figured
= $\frac{e^xe^x-e^{-x}e^{-x}}{4}$
= $\frac{... |
H: Why do my professors ignore my work?
I'm a high-school student finishing in December and about to pursue a career in mathematics.
In my free time, I like to ''research'' hard problems and come up with unique proofs or combine already-established proofs to come up with a beautiful solution. As a high-school student,... |
H: Probability, that the computer network will work
So a single computer blows up after turning it on with probability $0.05$. We have an order to create a network of $50$ computers, and we gathered $52$ computers for this purpose. What's the probability, that we can create a network with these computers?
My solution:... |
H: Is $f(n) = 2^{\frac{1}{2}(n^2-n)} / n!$ polynomially bounded?
The numerator counts the number of different adjacency matrices. I think Sterlings approximation helps to anwser my question but I fail to derive the answer.
So, is there a polynomial function $g(x)$ such that
$$f(x) \leq g(x)$$
AI: It's not polynomial... |
H: Changing an exponential function to logarithmic
I have a question stating that $P=75e^{-0.005t}$ and they want to get t by itself.
I used the example $y=2^x = x=log_2(y)$
To find that $-0.005t = 75ln(P)$
So $t=\frac{75ln(P)}{-0.005}$
However apparently this isn't correct. Can someone please show me where I went wro... |
H: Open set and sequences in $\mathbb{R}$
Let $A \subset \mathbb{R}$. Prove that $A$ is an open set if, and only if, the following condition is satisfied: " if a sequence $(x_n)$ converges for a point $a \in A$, then $x_n \in A$ for all $n$ sufficiently large".
I have doubts in the second implication $( \Leftarrow )$.... |
H: Find intersection points of two functions
I have
$f(x)=\sqrt{3x}+1$
$g(x)=x+1$
My thinking was that at the intersection points both will be equal to each other so
$\sqrt{3x}+1=x+1$
$\sqrt{3x}=x$
However I don't know where to go from here.
AI: $$\sqrt{3x} = x \implies 3x = x^2 \iff x^2 - 3x = x(x - 3) = 0$$
$$\imp... |
H: Coded language puzzle!!
Here is a puzzle I can't crack. It goes like this:
In a certain coded language
MANGO=3/5
ORANGE=2/6
APPLE=1/5
Then, POTATO=??
The answer is 5/6. I would like to know to arrive at the answer.
AI: HINT: It appears that the denominator is the number of letters in the word. The numerator also ap... |
H: Field and Algebra
What is the difference between "algebra" and "field"? In term of definition in Abstract algebra.
(In probability theory, sigma-algebra is a synonym of sigma-field, does this imply
algebra is the same as field?)
AI: An algebra is a ring that has the added structure of a field of scalars and a cohe... |
H: Percent spread in ratio
Say I have five solutions of various concentrations such as the below:
A = 16%
B = 12%
C = 12.5%
D = 17%
E = 5%
Their quantities do not matter, only their concentrations. Say I want to mix them and the final spread between them should follow a ratio of 1:2:1:2:6. Is this enough information ... |
H: The limit $\left( \sin x \right)^{x}$ when $x \rightarrow\; 0$
Take the limit $\left( \sin x \right)^{x}$ when $x \rightarrow\; 0$
I tried using $e^{\ln \left( \left( \sin x \right)^{x} \right)}=e^{x\cdot \ln \left( \sin x \right)}$ and then saying sinx → 0 but ln(0) is undefined. Stopped there.
AI: For $x$ near ze... |
H: Finding a,b,c,d in a quartic expression
Let $p(x)=x^4+ax^3+bx^2+cx+d$ where a,b,c,d are constants. If $p(1)=10$, $p(2)=20$, $p(3)=30$, compute $\frac {p(12)+p(-8)}{10}$. I have tried so far.
\begin{align}
a+b+c+d=&9\\8a+4b+2c+d=&4\\27a+9b+3c+d=&-51
\end{align}
Manipulating these, I got $6a+b=-25$. Now, $$\frac {p... |
H: geometric progression calculation
In a given geometric progression, a1 equals 30, q in absolute value is smaller than one, and the sum of all arguments in even positions is 11.25. what is the value if q?
attempt at a solution: for the even arguments, a1 is 30q, q is 2q and we know the sum in infinity is 11.25. p... |
H: Find integers $a$, $b$ and $c$ with $55a + 65b + 143c= 1$.
I'm not sure if this is a diophantine equation with three variables or not, but I can't find any resources for it. I am thinking there must be some sort of solving for two and then substituting the answer into another two variables. Unfortunately my notes a... |
H: Calculating $\operatorname{Res} \left(\frac{f(z)}{g(z)}, z=a\right)$ with $a$ a double zero of $g$.
I have to show that for $f,g$ analytic on some domain and $a$ a double zero of $g$, we have:
$$\operatorname{Res} \left(\frac{f(z)}{g(z)}, z=a\right) = \frac{6f'(a)g''(a)-2f(a)g'''(a)}{3[g''(a)]^2}.$$
The problem is ... |
H: Describe the kernel and the range, and any vector such that T(p)=y.
Let $T: \mathbb{P_2} \to \mathbb{R^2}$ be a linear transformation given by
$T(p)=[ p(0) ; p'(0)]$
Describe the kernel and range. That is to say, for what polynomials of the form
$p(t)= at^3 + bt^2 + ct + d$ does T map to zero?
Also, for wha... |
H: Proof that one large number is larger than another large number
Let $a = (10^n - 1)^{(10^n)}$ and $b=(10^n)^{(10^n - 1)}$
Which of these numbers is greater as n gets large?
I believe it is $a$ after looking at some smaller special cases, but I'm not sure how to prove it.
AI: Divide:
$$\frac{a}{b} = \frac{(10^n-1)^{... |
H: Equation of the tangent to the curve
I want to find the equation of the tangent to the curve:
$F(x) = \sqrt x$ at $x=4$. (Write answer in slope intercept form)
I am very confused on this and would need step by step directions. Thanks
AI: Given $f(x) = \sqrt x$.
"Step-by-Step Directions" (with "spoilers" so you can... |
H: Step in finding $\sin^{-1}z = w$ for a fixed complex $w$ and unknown complex $z$
This is in the section of the book preceding a general formula but I don't know how the author arrives to the second equation in the picture. The closes I have gotten to it is
$$2iz = e^{iw}(1-e^{-1})$$
but I don't see how I can get a... |
H: Limit of f(x). What does it tell about Limit of f(x)^n?
Suppose $\lim_{x\to\infty}f(x) = L$. Is it telling us something about $\lim_{x\to\infty}f(x)^n$?
AI: The function $\phi(x) = x^n$ is continuous, and $\lim_{x \to \infty} f(x) = L$, hence $\lim_{x \to \infty} \phi(f(x)) = \phi(\lim_{x \to \infty} f(x)) = \phi(L... |
H: How to find all perfect squares in a given range of numbers?
I need to write a program that finds all perfect squares between two given numbers a and b such that the range can also be a = 1 and b = 10^15 what is the best way I can do this, how do I list down all such square numbers, is there some abstract math hidd... |
H: Prove Multinomial Coefficient (Probability Theory)
Prove that the multinomial coefficient given by:
$$
\binom{n}{n_1}\binom{n-n_1}{n_2}\binom{n-n_1-n_2}{n_3}\cdots\binom{n-n_1-n_2-\dots-n_{k-1}}{n_k}
$$
equals the following expression
$$
\frac{n!}{n_1!n_2!n_3!\cdots n_k!}
$$
Thank you for any help you can give me
A... |
H: Path of an ellipse
A path is described by the position vector $\mathbf{r}$:
$$\mathbf{r}=a\cos(\omega t)\mathbf{\hat{i}}+b\sin{\omega t}\mathbf{\hat{j}}$$
I am asked to show that the path is the ellipse in the form of:
$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$$
Is this not converting from polar to cartesian using $x=\ma... |
H: Checking step by step proof of $\lim_{n\to\infty}(\sqrt2-\sqrt[n]2)^n=0$
$$\mathop {\lim }\limits_{n \to \infty } {(\sqrt 2 - \root n \of 2 )^n}$$
Is it right to say:
the limit of $\sqrt2$ is $\sqrt2$
the limit of $\root n \of 2$ is 1
then, $\sqrt2 - 1$ is between 0 to 1.
so, the limit of $(\sqrt2 - 1)^n $ must be... |
H: GRE counting problem
I have approached this question the following way.
For Quantity $A$: The number of ways to pick $3$ cards including $1$ is : $1\cdot 4\cdot 3=12$ ways. [$1$ is fixed]
For Quantity $B$: The number of ways to pick $3$ cards excluding $1$ is : $4\cdot 3\cdot 2=24$ ways resulting $24$ which mean... |
H: How to isolate y?
I've got an equation: $$-4y=12-3x$$
I want to simply isolate the y variable so i could get rid of the -4 coefficient of the y variable.
What can I do to isolate the y variable? I've done the following but i'm not sure:
$$ y = \frac{12}{-4}-\frac{3x}{-4} $$
Is that the right way for solving this ki... |
H: field extension-notation problem $\mathbb{Q}(\sqrt{2})=\mathbb{Q}[\sqrt{2}]$
let $K\subseteq L$ be a field extension and let $K=\mathbb{Q}$ and $L=\mathbb{C}$ also let $\alpha =\sqrt{2}$.
Then $\mathbb{Q}(\sqrt{2})=\mathbb{Q}[\sqrt{2}]$
What is the difference in the above notations? and why do we get them?
$\math... |
H: Stuck with Taylor expansion of $f(x+x')$
I know that the Taylor series of $f(x)$ around $a$ is given by:
$$f(x)=f(a)+f'(a)(x-a)+f''(a)\frac{(x-a)^2}{2}+\dots=\sum_{n=0}^\infty \frac{f^{(n)}(a) }{n!} (x-a)^n$$
In my textbook I see the following formula for
$f(x+x')$ which I however don't understand:
$$f(x+x')=f(x)+... |
H: roots of polynomial equation
How to find the roots of $x^5-2^5$ by hand. I see that we get a root of $x=2$ and 4 complex roots (should come in pairs). Not sure how to work out the complex roots. Do we need to convert to polar? Would that make it easier to see the other roots?
AI: Write as $x^5 - 2^5e^{2\pi ik} = 0$... |
H: integrating to clear differential
I'm unsure if that's the correct term.
I have an equation $v^2 = \frac{2ILB}{m}x$ and I need to find distance with respect to time (yes, physics.. but the silly math is what trips me up so I'm posting here)
Here's my attempt
$v^2 = [\frac{dx}{dt}]^2 = ...$
$\frac{dx}{dt} = \sqrt{\f... |
H: Stochastics with induction
prove that for all $n \in \mathbb{N}$:
$\sum_{r=0}^n \binom{n}{r}(-1)^{r} = 0$.
The base step is easy, i only get lots of problems when i try to mess with the sum boundries....
so far i've tried:
$\sum_{r=0}^{n+1} \binom{n+1}{r}(-1)^{r} = \sum_{r=0}^{n+1}(\binom{n}{r}+\binom{n}{r-1})(-1)^... |
H: Expansion of $ \frac{1}{|\vec r -\vec r'|} $
I would like to show that:
$$ \frac{1}{|\vec r -\vec r'|} =\frac{1}{r} + \frac{\vec r'\cdot r'}{r^3}+\frac{3 ((\vec r \cdot \vec r)^2 -\vec r^2 \vec r'^2 )}{2r^5} +\dots$$
What I derived so far is:
$$\frac{1}{|\vec r -\vec r'|}= \sum_{n=0}^{\infty} \frac{1}{n!} (-\vec r'... |
H: Find all positive/negative integers N for which $N^2+20N+11$ is a perfect square?
I know that there might be a duplicate of this. But I don't know where.
I tried equatin this to $X^2$ and and then bringing it to the other side and completing the square. What next? Is there a way to solve these types of questions?
A... |
H: Solutions to equation involving trigonometry and geometric series
$$\cos^2 x + \cos ^3 x +\dots = 1+ \cos x$$
I want to find values of $x$ between $0$ and $ 180$ degrees for which the above equation holds true.
Attempt at a solution: left side is a converging geometric progression, for which $a_1$ is $\cos^2 x$ an... |
H: Are there such things as non-extensional set theories?
I have always assumed that extensionality is a paradigmatic example of a property of mathematical objects (sets) which is essential to those objects--- if your set theory doesn't obey extensionality, it isn't set theory.
Given the existence of alternative set t... |
H: Show that $\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)}$
Show that $$\sum_{n=0}^\infty r^n e^{i n \theta} = \frac{1- r\cos(\theta)+i r \sin(\theta)}{1+r^2-2r\cos(\theta)},$$ where $0\leq r <1$.
Using this, prove that $\sum_{n=0}^\infty r^n \cos(n\theta)$ and ... |
H: Draw lattice for a join-semillatice
I am working with a set that I believe is a join-semilattice (all elements $x,y\in S$ have a least upper bound). I am a little bit confused as to how I would draw a lattice for this set $S=\{1,2,3,12,18,36\}$ when it is partially ordered by divisibility.
My confusion stems from h... |
H: Cardinality of sets discrete math
Given $A = \{\text{Vine}, \text{Tree}, \text{Shrub}\}$; $B = \{ \text{Tree}\}$; $C = \{ \text{Vine}, \text{Moss}\}$; $D = \{\text{Red}, \text{Green}\}$; $E = \{ \text{Red} \}$
what is $A \cup B \cup C$?
AI: $$A\cup B\cup C=\{\text{Vine, Tree, Shrub, Tree, Vine, Moss}\}$$ How many d... |
H: $x^6+x^3+1$ is irreducible over $\mathbb{Q}$
I have been trying to prove that $x^6+x^3+1$ is irreducible over $\mathbb{Q}$ (or $\mathbb{Z}$ since by Gauss' Lemma is the same), but I can't. Any idea of how to do so?
AI: HINT: Let $y=x-1$, and apply Eisenstein's criterion for $p=3$. |
H: Vector Calculus
I want to show that given $ax+by=c$ the vector $(a,b)$ is perpendicular to the line determined by the equation.
I was thinking using dot product.
AI: The line $\;ax+by=c\;$ can be given in parametric form as (assuming $\;b\neq 0\;$ to avoid trivialities)
$$\left\{\left(x,-\frac ab x+\frac cb\right)\... |
H: set identities discrete mathematics help?
$A-C\subseteq A-(B-C)$
how would you do this question?
and also b - c(with a line through b-c) $\subseteq C$
AI: I prefer the modern notation $A\setminus C$ to $A-C$, so I’ll use it. One way to attack the first problem is to try to prove that the statement is always true. ... |
H: Prove that the following element is an invertible element
Question in Abstract Algebra:
How can I prove that: $r+s \sqrt{2}$ , when $r$ and $s$ are rational, is an invertible element in $\mathbb{Q}(\sqrt{2})$?
(In fact I need to prove that's a number field but I only have problem with this.)
AI: Note that if $r+s\... |
H: Infinite Dyadic Rationals in any open interval $(a,b)$ where $a
Given that there exists at least one dyadic rational of the form $2^{-n}m$ between any two distinct real numbers $a<b$, show that there infinite such rationals between $a$ and $b$.
My Attempt
Given $a<b\in\mathbb{R}$ we have $$a<\frac{m_1}{2^{n_1}}<b... |
H: Permutations and combinations ! 9 different fruit pies divided between three
Different fruit pies are divided between 3 people so that each person gets and odd number of pies. Find the number of ways this can be done??
hint- so many combinations are added to get this answer ..
AI: We can do it by splitting into cas... |
H: Cardinality of set-Discrete math
Take $A=\{1,2,3\}$ and $B=\{1,2,5\}$.
If we unionized them together it would be $A\cup B=\{1,2,3,5\}$ and if we intersected them it would be $A\cap B=\{1,2\}$.
However, if we change $B$ to $B=\{\{1,2\},2,5\}$ would a set within a set change the previous answers and why?
AI: Yes, it ... |
H: A graph with a degree sequence 0,1,2,3,4
Please help me to prove whether or not a there exists a graph with the degree sequence 0,1,2,3,4. I do not know the formal way of writing the proof hence any advises and proofs will be much appreciated. Thank you.
AI: Hint There are 5 vertices. If one has degree $4$, it is c... |
H: KKT maximization problem
$x^2y \rightarrow$ max,
such that $x^2 + 4xy \leq 1, x \geq 0$ and $y \geq 0$.
I think I need to use the KKT conditions here. I did however not yet succeed in solving it, so could someone please give me an example of how this should be done? And should I include the constraints $x \geq 0$... |
H: Integral curves in the plane
Maybe this is a stupid question but i can not solve this mechanical problem...
How can I find the integral curves of the vector field
$$X_{(x,y)} = x \dfrac{\partial}{\partial x} − y\dfrac{\partial}{\partial y} = \begin{bmatrix}x \\ -y\end{bmatrix}\,.$$
AI: Well, the integral curves sa... |
H: Show why given set is not a frame
I am rather new to this material and an explanation of what is happening would be greatly appreciated. At first glance, it seems like the sum of squares is bounded at both ends but I guess I'm looking at it completely wrong. The question is as follows:
Consider the set $\Phi = \{... |
H: Limit of nth power of operator norm
I am given a compact operator $A$ which lives in a Banach algebra and whose spectral radius obeys $\rho(A)=\lim_{n\rightarrow\infty}||A^n||^{\frac{1}{n}}<1$. Now I want to prove that this implies that $||A^n||\rightarrow 0$ as $n\rightarrow \infty$ (if true).
I tried the followin... |
H: At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?
I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$.
I'm interested in the properties of
$$\log(1+x^2)=x^c.$$
Graphically, for small $c$, the function $x^c$ is concave an... |
H: Recursion problem.
$$A_{n+1}=A_{n}+\frac{G(n+1)-A(n)}{n+1}$$ $$A(n)=G(1)=80$$ $$G(2)=70$$
$$G(3)=60$$
$$G(4)=70$$
$$G(5)=100$$
My question is how did following equation is formed.
$$A(2)=A(1)+\frac{G(2)-A(1)}{2}=80-10/2=75$$
I want to know how it is done.
AI: It appears you are equating $A_n$ and $A(n)$, which is n... |
H: Is $2^n \mod m \equiv (2^{n/2} \pmod m ) ^ 2 \pmod m$?
I'm trying to write a procedure that solves (2^n - 1) mod 1000000007 for a given n.
n can be very large (~10^9).
Say m = 1000000007
So this is what I have so far:
func(n):
if n = 1 return 2
if n is even return ((func(n/2) mod m)^2)mod m
else return ((func... |
H: Did I take the derivative correctly? $x^y=y^x$
Need to differentiate following equation:
$$x^y=y^x$$
My attempt:
$$x^y\log(x) \cdot y' = y^x\log(y)\cdot1; $$
$$y'=\frac{y^x\log(y)}{x^y\log(x)}$$
Please tell me if I've made a mistake.
AI: Not quite. The idea is to take the $\log$ of both sides and implicitly differ... |
H: Showing a topology is not metrizable
Show $\prod_{N} \mathbb{R}$ with the box topology is not metrizable.
The Box Topology on $\prod_{j \in J} X_j$ ($X_j$ topological spaces) is generated by the basis $\left\{\prod_{j \in J} U_j \; \Big| \; U_j \text { is open in } X_j \right\}$.
I made an attempt, and found the... |
H: Ruler and compass construction of the unit-distance petersen graph embedding
The Petersen graph is a unit distance graph, and this embedding is shown below, where each edge of the graph is one unit in length.
Is there a ruler and compass construction for this embedding? If so, what is it?
It seems like the kind ... |
H: Does this general solution look right?
Here is the starting system:
\begin{cases}
-x+y+4z= -1 \\
3x-y+2z=2\\
2x-2y-8z=2
\end{cases}
I started by performing $2E1+E3\to E3$. This showed that they cancelled each other out and I was left with $0=0$ for $E3$ (this left me assigning $z$ to $t$). Then I performed $E1+E2... |
H: Proving something with Wilson's Theorem [continued.]
At first I asked this: Proving something with Wilson theorem.
Now I have to prove that if $p=4n+3$ it's impossible to represent $-1$ in the form $x^2$ modulo $p$. How can I prove it?
Thank you!
AI: It is generally true that $a$ is a quadratic remainder (i.e. is o... |
H: Determine run-time of an algorithm
Probably a stupid question but I don't get it right now. I have an algorithm with an input n. It needs n + (n-1) + (n-2) + ... + 1 steps to finish. Is it possible to give a runtime estimation in Big-O notation?
AI: Indeed, it is. Since we have $$n+(n-1)+(n-2)+\cdots+1=\frac{n(n+1)... |
H: radical of I is the entire ring R implies I=R?
Can anyone prove that:
radical of I (ideal) is the entire ring R implies I=R?
The ring has a unit and commutative.
Thanks...
AI: If $rad(I)=R$ then $1 \in rad(I)$. Since $1$ is idempotent, this implies $1 \in I$. Therefore $I=R$. |
H: Question About a Classical Root Appending Extension Field Proof
Let $F$ be a field and $f(x) \in F[x]$ s.t. $f(x)$ is irreducible and of degree $n \ge 1$. I've seen it proved that there exists an extension field $E$ of $F$ s.t. that there is a root $\alpha \in E$ of $f(x)$ and the degree of $E$ over $F$ is $n$. I... |
H: Quicker way to solve 10! congruent to x (mod 11)
I am new to modular arithmetic and solving congruences and the way I went about this was to write out $10\cdot 9\cdot 8\cdot 7\cdot 6\cdot 5\cdot 4\cdot 3\cdot2$, then multiply numbers until I get a number greater than $11$, replace it with a smaller number in its co... |
H: How do I calculate a weighted average purchase date?
A farmer bought 100 kg of seed on the following dates:
20 kg on 13-may-2007 (20%)
30 kg on 4-oct-2007 (30%)
50 kg on 31-jul-2008 (50%)
How do I calculate the weighted average purchase date? I don't want weighted average age because that changes daily. I'm think... |
H: How to find the height of a 2D coordinate on a four-sided 3D polygon plane?
How do I find the height of a given 2D coordinate on a four-sided 3D polygon plane? The polygon has no volume. I'm trying to match 3D terrain vectors to a 3D polygon. I'll always know that the 2D version of the 3D poly contains the 2D coord... |
H: Number Theory: Solutions of $ax^2+by^2\equiv1 \pmod p$
Assume $p$ is a prime number and $\gcd(ab, p)=1$. Show that the number of integer solutions $(x, y)$ of $ax^2+by^2 \equiv 1 \pmod p$ is
$$p - \left(\dfrac{-ab}{p}\right)$$ where $\left(\dfrac{x}{y}\right)$ is the Legendre symbol.
Ok, so here's my partial so... |
H: Logic Puzzle, Two Dollar Bills
I read the following online, and I can't seem to figure it out: A bank robber is planning a heist of a high-tech bank vault. In order to exploit a weakness in the bank's security system, he needs to set off his dynamite exactly 45 seconds after triggering the banks silent alarm - If h... |
H: How to interpret this formula from dsp?
Can you describe me what this formula does and what result in the end?
$$\delta(x)=\begin{cases}
0, & \text{if $x \neq 0$} \\
1, & \text{if $x = 0$} \\
\end{cases}$$
$$x(t)=\sum_{k=-\infty}^{\infty}x(t_k)\delta(t-t_k)$$
AI: $\delta(x)$ is also known as the Kronecker Delta ... |
H: Why must complex linear maps be of the form $h \mapsto ah$, $a \in \mathbb{C}$?
On page 4 of David Ullrich's "Complex Made Simple", he says that complex linear maps from $\mathbb{C}$ to $\mathbb{C}$ are precisely of the form $h \mapsto ah$ for some complex number $a$.
How do I prove that all complex linear maps mus... |
H: Subgroup of the centralizer
On pg.124 of Abstract Algebra by Dummit and Foote the observation is made that In any group $G$, $<\hspace{0.2mm}g \hspace{0.2mm}> \hspace{1mm} \le \hspace{1mm} C_G(g)$
I am having a difficult time proving this observation to myself. The book gives a particular example but does not provi... |
H: Is $\varphi$ a homomorphism?
Define $\varphi:\mathbb{Z}\times \mathbb{Z}\rightarrow \mathbb{Z}_6$ by $\varphi(a,b)=[a+b]_6$.
To show $\varphi$ preserves addition,
$\hspace{100pt} \varphi((a,b)+(c,d))=\varphi((a,b))+\varphi((c,d))$.
$\varphi((a,b)+(c,d))=\varphi((a+c),(b+d))=[a+c]_6+[b+d]_6$
$\hspace{73pt}$$=[a+b... |
H: Show that $\frac{|F(z)-F(a)|}{|F(z)-\bar{F(a)}|}\le\frac{|z-a|}{|z-\bar{a}|}$ if $z\in\Pi^{+}=\{z\in\mathbb{C}:Im(z)>0\}$
Consider $\Pi^{+}=\{z\in\mathbb{C}:Im(z)>0\}$ and let $a\in\Pi^{+}$. Suppose that $F:\Pi^{+} \rightarrow \Pi^{+}$ is holomorphic. Prove that for all $z\in\Pi^{+}$ we have:
$$\frac{|F(z)-F(a)|}{|... |
H: Harmonic function takes both positive and negative values
I am a little confused on the following question:
Suppose that $u$ is harmonic nonconstant on a $D(z_0,R)$ and $u(z_0)=0$. Is it true that on each circle $C(z_0,r)$, with $0<r<R$, the function $u$ will take both positive and negative values?
I think the foll... |
H: Probability of not picking a particular ball, $W1$, out of an urn on the 1st pick, and not picking $W2$ on the 2nd pick?
3 balls are chosen from an urn (without replacement) containing 5 white and 8 red balls. The white balls are numbered (that is, they are distinct), and the red balls are not.
Find $P\{Y_{1}=0,Y_... |
H: When is $1\vec{u} \neq \vec{u}$?
In a linear algebra textbook they define a vector space to be a nonempty set $V$ of objects that satisfy certain properties.
One of these properties is that $\forall\vec{u}\in V(1\vec{u}=\vec{u})$
The only way I can think of that property NOT holding would be if scalar multiplicatio... |
H: One to one and bijection in $\mathbb{Z}^2$
I have the following:
$f(m,n) = (3m+7n, 2m+5n)$ and I want to know if it is a bijection and if so, fine the inverse as well.
Here's my approach:
Suppose $f(m_1,n_1)=f(m_2,n_2)$
then:
$$ (3m_1+7n_1,2m_1+5n_1)=(3m_2+7n_2,2m_2+5n_2)$$
$$3m_1+7n_1=3m_2+7n_2 $$ and $$2m_1+5n_1=... |
H: $ 7^{50} \cdot 4^{102} ≡ x \pmod {110} $
The way I would solve this would be: $$ (7^3)^{15} \cdot 7^5 \cdot (4^4)^{25} \cdot 4^2 $$ and take it from there, but I know that this is most likely in an inefficient way. Does anyone have more efficient methods?
AI: Do prime factorization on $110$. It's $110 = 5 \cdot 2 ... |
H: Calculate $\sum\limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$
Calculate $$\sum \limits_{k=0}^{\infty}\frac{1}{{2k \choose k}}$$
I use software to complete the series is $\frac{2}{27} \left(18+\sqrt{3} \pi \right)$
I have no idea about it. :|
AI: Consider the function
$$f(x) = \frac{\arcsin{x}}{\sqrt{1-x^2}}$$
$... |
H: Neumann BV problem on disk (weak vs classical solution)
I am tring to solve
$\bigtriangleup u =-1$ such that the normal derivative vanishes at the boundary where the domain is the unit disc.
In polar coordinates I got
I got $u(r)=-1/4 r^{2} +1/2 \ln(r)$
as a solution. Does this qualify as a weak solution (since ... |
H: Clarification on a proof involving cluster point
Definition of cluster point- Let $A \subseteq \mathbb{R}$. A point $c\in\mathbb{R}$ is a cluster point of $A$ if for evert $\delta>0$ there exists at least one point $x\in A$, $x\neq c$ such that $|x-c|<\delta$.
Theorem- A number $c\in\mathbb{R}$ is a cluster point o... |
H: Changing operator to polar coordinates
Let $$\Delta=\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}$$ be the Laplace operator on the $(x,y)$-plane. Consider the polar coordinates with $x=r\cos\theta$ and $y=r\sin\theta$. Show that $$\Delta=\frac{\partial^2}{\partial r^2}+\frac1r\frac\partial{\partia... |
H: Orthogonal projection on vector space of intervals
Let $R_N$ be the set of $2^N$ intervals $$\left\{\left[0,\frac{1}{2^N}\right), \left[\frac{1}{2^N}, \frac{2}{2^N}\right),\ldots,\left[\frac{2^N-1}{2^N}, 1\right)\right\}.$$ Let $$V_N=\operatorname{span}\{1_I\mid I\in R^N\}$$ Let $P_N:L^2([0,1])\rightarrow V_N$ be ... |
H: Elimination of complex variable in integral
I have the equation: $$\frac{1}{\tau}\intop_{0}^{\tau}A\sin\left(\Omega t\right)\cdot A\sin\left(\Omega\left(t-\lambda\right)\right)\mathrm{d}t$$ for which the attempted solution is to convert the sine terms into complex natural exponents (engineering notation using j as ... |
H: Prove that the mapping $U(16)$ to itself by $x \rightarrow x^3$ is an automorphism
Prove that the mapping $U(16) = \{{1,3,5,7,9,11,13,15}\}$ to itself by $x \rightarrow x^3$ is an automorphism. What about $x \rightarrow x^5$ and $x \rightarrow x^7$? any generalization?
So far i have prove the first part.
Let $\psi... |
H: $f:U(\mathbb{Z}/(n)) \to Aut(G)$, defined by $f([s]_n) = \phi_s$ is surjective
Let $G = \left \langle {g} \right \rangle$ be a cyclic group of order $n \geq 2.$ Define a map $\phi_s : G \to G (x \to x^s)$ and $Aut(G): = \{f:G \to G | f\ is\ isomorphism\}. $ Then, $\phi_s$ is an isomorphism iff $gcd(s,n)=1. $
Coul... |
H: Can the Poisson Distribution be used to find the expected value of time of arrival given an expected arrivals per unit time?
My understanding of the Poisson Distribution is that its PMF $P(x=k) = \dfrac {\lambda^k e^{-\lambda}} {k!}$ refers to the probability of finding k events given an expected arrival expectancy... |
H: Finding the Hopf Algebra Coproduct coming from an Affine Group Scheme
I was wondering if anyone could help with how to, strictly from Yoneda's Lemma, obtain the coproduct map on the Hopf Algebra for an Affine Group Scheme. Particularly for something like $\text{SL}_2$
So if $G=\text{SL}_2$, let $m:G \times G \to G... |
H: very basic calculus doubt
Suppose $a,b \in \mathbb{R}^{\geq0}$. Let $\epsilon \in (0,1)$
$$ a \geq b \epsilon \implies a \geq b $$
My try: If $a < b $, then can find $n$ such that $b - a > \frac{1}{n} $. But I am stuck here. Maybe the result in not true??
if it is not true, why then is this book author assumes this... |
H: Monic and epic implies isomorphism in an abelian category?
Is it true that monic and epic implies isomorphism in an abelian category?
AI: The answer is yes, as a consequence of two facts. First, in any category (abelian or not), if the equalizer of a pair of maps is epic, then it is an isomorphism. (Proof: By defin... |
H: Role of differentiation in a polynomial
I have learnt that, to find maximum or minimum value of a polynomial $p(x)$, we take its derivative, equate it to zero, solve for x, find the maxima and minima, and then put the value of x in the original equation. But when we take the derivative of $p(x)$, what exactly are w... |
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