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H: Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized
My professor mentioned something like "Laplace's equation is solved when the functional $E[u] = \int_{\Omega}|\nabla u|^2 $ is minimized."
I've been trying to understand this statement. If I say that $E[u+tv]$ has a mi... |
H: $\mathbb{R}$ - algebras in topological spaces
I´m reading an introduction to $\mathbb{R}$-algebras and in the text there is an observation that says:
If $X$ is a topological space, then the set of functions $f : X \to \mathbb{R}$ are a $\mathbb{R}$ - algebra and that the continuous functions are a subalgebra of the... |
H: Rationals with subspace topology from the reals
Suppose $\mathbb{Q}$ is endowed with the subspace topology of$\mathbb{R}$ Does it follow that $\mathbb{Q}$ is connected?
MY attempt: We can sue fact that $\mathbb{Q}$ is countable and so $\mathbb{Q} = \bigcup \{x\} $. And singletons are connected. Can we conclude that... |
H: Counting elements of $y^2 - y = x^3$ in finite fields
The problem I have to solve is the following:
Let $p$ be a prime number with $p \equiv 2$ mod $3$. Let $E$ be the elliptic curve given by $y^2 - y = x^3$. Show that $\#E(\mathbb{F}_p) = p+1$ and $\#E(\mathbb{F}_{p^2}) = (p+1)^2$.
I have solved the first part i... |
H: Number Theory Contest Problem
Given that $x, y$ are positive integers with $x(x + 1)\mid y(y + 1)$, but neither $x$ nor $x + 1$ divides either of $y$ or $y + 1$, and $x^2+ y^2$ as small as possible, find $x^2+ y^2$.
I have tried looking at the values, and it seems that neither $x$ or $x+1$ or the $y$'s are prime.
A... |
H: Show $g(z+1) = zg(z)$
This is for homework, and I am in need of a hint. Given the product
$$ g(z) = \prod_{k=1}^{\infty} \frac{k}{z+k}\left( 1 + \frac{1}{k} \right)^z, $$
I am trying to show that $g(z+1) = zg(z)$. Here is what I have so far.
I tried to get a better sense of $g$, and wrote
$$ g(z) = \frac{1}{z+1}\... |
H: For what prime $p$ is $x^2=-1\pmod{p}$ solvable?
This is essentially the same as the following question:
When $F_p[x]/(x^2+1)$ is a field?
I don't know much about number theory. I came up with such question when I doing the following exercise:
When
$$
R_p=\left\{\left(\begin{matrix}
a&b\\
-b&a
\end{matrix}\ri... |
H: I was trying to compute $\sum_{j=0}^{m} 3^j {m \choose j}$, but don't know where to start
compute $\sum_{j=0}^{m} 3^j {m \choose j}$. Then, use the binomial theorem to verify the result.
AI: $$4^m=(3+1)^m=\sum_{j=0}^m{m\choose j}\cdot3^j\cdot1^{m-j}=\sum_{j=0}^m{m\choose j}\cdot3^j$$ or $$\sum_{j=0}^m{m\choose j}\c... |
H: Given $f(x) = x\log_{2}x$, how do I compute $f^{-1}(10)$?
Let $f(x) = x\log_2 x$. Compute $f^{-1}(10)$ to at least three decimal places of accuracy. Explain how you did this.
Note: for a function $f:A\rightarrow B$ for which there is exactly one point $a$ that maps to each point $b \in B$, $f^{-1}(y)$ means: the un... |
H: Finding the order of a finite group
Let $x\in\mathbb{Z}/42$, and suppose that x has order $n\in\mathbb{Z^+}$. Without listing all of the subgroups of $\mathbb{Z}42$, determine all of the possible values that $n$ could be.
I'm having a hard time understanding the concept of the order of a group and don't know where ... |
H: Uniform Continuity Proof
$$f(x) = \frac{\sin x^3}{x+1}$$
If $f(x)$ is defined for $x \in [0,\infty)$, I can see that its derivative is bounded in the interval so it is just the matter of proving it. Im going for an $e-s$ proof, but I'm stumbling in terms of finding an appropriate $s.$ Also, would be nice if someon... |
H: can not find the proof that logarithms are the inverse of exponentials
I have been taught by the most powerful magic of mathematics - Hand waving - that the logarithms are inverses of the exponential. I have seen the graphs where each one is graph, showing they are reflective over the line $y=x$. I do not doubt t... |
H: A problem about class equation
Let $k$ be a finite field, where $|k|=q$ and $\operatorname{char}k\neq2$, and let
$$D=\{A\in\operatorname{SL}(2,k)\mid A \text{ is diagonalizable}\}.$$
Prove that $$|D|=2+\tfrac{1}{2}\cdot(q+1)\cdot q\cdot(q-3).$$
Can anyone help me? Thanks!
AI: We know that $|GL(2,k)| = (q^2-1)(q^2-q... |
H: Determinant of complex block matrix
Let $A$ be an $n\times n$ invertible matrix. Let $a \in \Bbb C$, let $\alpha$ be a row $n$-tuple of complex numbers and let
$\beta$ be a column $n$-tuple of complex numbers. Show that
$$(\det(A))^{-1}\, \det\left(\begin{bmatrix}a & \alpha\\
\beta & A
\end{bmatrix}\right)=\det\lef... |
H: What is the order of the sum of log x?
Let
$$f(n)=\sum_{x=1}^n\log(x)$$
What is $O(f(n))$?
I know how to deal with sums of powers of $x$. But how to solve for a sum of logs?
AI: Using Stirling's formula we have, for $n$ sufficiently large
$$
f(n)=\sum_{k=1}^n\log k=\log(n!)\simeq\log(\sqrt{2\pi}e^{-n}n^{n+1/2}).
$... |
H: Expressing logarithms as ratios of natural logarithms
$$\frac{\log_2 x}{\log_3 x}=\frac{\ln x}{\ln 2} \div \frac{\ln x}{\ln3}$$
Why can logarithms be written as ratios of natural logarithms?
Can you explain it abstractly, please?
Example of an abstract explanation: the logarithm function is an isomorphism from th... |
H: Prove that if a series converges to $a$, then that series to the power $k$ converges to $a^k$
Let $a_n$ be a sequence in the real numbers. Prove that:
$a_n\rightarrow a \implies (a_n)^k \rightarrow a^k$ $\forall k \in N$
I think I need to do this by induction. The base case is simple. When $k=1$:
$(a_n)^k = (a_n... |
H: The limit of $(s_j)$ when said $(s_j)$ is finite
Suppose that we have a sequence $(s_j)$ that is finite (where $j$ is an element of the natural numbers); it has $j$ terms. Can we say that $(s_j)$ has a limit? Also what about convergence? Can we say that $(s_j)$ converges to some number? I would think no on both cas... |
H: A form of the Gaussian integral
Consider the function $f(x,y)=ye^{-(1+x^{2})y^{2}}$ if $x\geq0$ and $y\geq0$ and $0$ otherwise. Integrate this function over $\mathbb{R}\times\mathbb{R}$ to show that $\int_{0}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}/2$.
Here is what I have done:
\begin{align*}
\int_{0}^{\infty}\int_{0}... |
H: Calculate time needed to solve problem
I have this question in an assignment and I was wondering if I could get help verifying whether my approach to this question is correct... The question is as follow:
Suppose that an algorithm uses 5n^2 + 3^n bit operations to solve a
problem of size n. Suppose that your mac... |
H: System of two ODEs with reverse sign
Consider the ordinary differential equations $$\dfrac{d}{dt}x_1(t)=x_2(t)$$ $$\dfrac{d}{dt}x_2(t)=-x_1(t)$$
for $t\in \mathbb{R}$. What are the solutions?
We have $\dfrac{d^2}{dt^2}x_1(t)=\dfrac{d}{dt} x_2(t)=-x_1(t)$ and also $\dfrac{d^2}{dt^2}x_2(t)=-\dfrac{d}{dt} x_1(t)=-x_2(... |
H: Why is conjugation by an odd permutation in $S_n$ not an inner automorphism on $A_n$?
I was reading about the outer automorphism group on wikipedia, and it mentions that conjugation by an odd permutation is an outer automorphism on the alternating group $A_n$. This suggests the automorphism defined on $A_n$ by
$$
\... |
H: How to graph absolute value equations that are not functions
Graph $|x + 1| + |y - 2| = 1 $
The graph should be a parallelogram, so it is not a function. How do you graph this?
AI: If we isolate $|y-2|$, then we get
$$|x+1|+|y-2|=1 \Rightarrow |y-2|=1-|x+1|.$$
Now we can apply the definition of absolute value,
$$... |
H: Allocation / Weighted Average question
I'm hoping someone can help with my problem. Forgive me as I don't even know what title to give this. I'm trying to come up with an allocation method for transportation expenses for multiple stops along a truck route based on a combination of mileage and cargo weight. I'm n... |
H: Let $p,q$ be 2 complex numbers with $|p|<|q|$.
I am stuck on the following problem that says:
Let $p,q$ be 2 complex numbers with $|p|<|q|$. Let $$f(z)=\sum\{3p^n-5q^n\}z^n$$ Then the radius of convergence of $f(z)$ is :
$|q|$
$|p|$
At least $\frac{1}{|q|}$
At most $\frac{1}{|q|}$
My Attempt: $f(z)=\sum(3p... |
H: Median for Continuous Probability Distribution
Consider a continuous random variable X with probability density function given by:
$f(x)=4x(1-x^2)$ for $0 \le x \le 1$
Find the median.
So to calculate the median, I calculated the CDF and then set that equal to 0.5 and solve for x:
$F(x)=2x^2-x^4$
$0.5=2x^2-x^4\ta... |
H: Solution to Hamilton-Jacobi differential equations
Let $H(x,y)$ be a $C^2$ function on $\mathbb{R}^2$ and let $(x(t),y(t))$ be a solution of the Hamilton-Jacobi equations $$\frac{dx}{dt}=\frac{\partial}{\partial y}H(x(t),y(t))$$$$\frac{dy}{dt}=-\frac{\partial}{\partial x}H(x(t),y(t))$$
Show that the function $H$ i... |
H: Well definedness of Lebesgue inner measure
This is for homework: if $A,A'$ are two elementary sets containing $E$, bounded set in $\mathbf{R}^d$, then $m(A)-m^*(A \backslash E)$ is equal to $m(A')-m^*(A \backslash E)$
So far my goal has been to show that they're both equal to the expression for $A \cap A'$ which is... |
H: Epsilon-Delta Continuity proof
Let $f_1, f_2$ be two functions from $\mathbb R\to\mathbb R$. Suppose that $f_1$ and $f_2$ are both continuous at $x_o∈ \mathbb R$. Let $g(x)=\min(f_1(x),f_2(x))$. Prove $g(x)$ is continuous at $x_o$. Has to be proven using the epsilon-delta definition of continuity: $∀ ε>0 ∃ δ>0$ su... |
H: Easy question concerning notation(Abstract Algebra)
I have a very easy question concerning some notation I have been coming across in Abstract Algebra (Dummit & Foote)
Context:
The relation between actions and homomorphisms may be reversed. Namely, given any nonempty set A and any homomorphism $\phi$ of the group ... |
H: Residues of Complex Functions
I need to find the residues of $f$ at the isolated singular points, namely $z=1,z=0$.
Where $f(z)=\dfrac{2z+1}{z(z+1)}$.
I already have that the residue at $z=0$ is $1$, and I know I need to do some slight of hand to get res at $z=1$.
I tried, $f(z)=\dfrac{2z+1}{z(z+1)} = (\dfrac{2z+1... |
H: Prove that if an integral is 0, the function is 0 across that interval (for $f(x) \geq 0$)
Assume $f:[a,b] \Rightarrow \mathbb{R}$ is continuous and $f(x) \geq 0$ for all $x\in[a,b]$. Prove that if $\int_a^b f dx = 0$, then $f(x) = 0$ for all x $\in [a,b]$.
My attempt at a proof a little obvious, but at the same ... |
H: Lie algebra homomorphism
I'm sure I'm missing something really obvious here. This seems too stupid.
On page 47 of Erdmann & Wildon's Introduction to Lie Algebras, we have the following set up. Let $L$ be a Lie subalgebra of $\mathfrak{gl}(V)$ of dimension $\geq 1$, and let $A \leq L$ be a maximal Lie subalgebra and... |
H: non prime generated cyclic numbers?
I recently watched a numberphile video on youtube talking about cyclic numbers. I was wondering if there was a number $1/a$, where $a$ wasn't prime, but $1/a$ turned out to be a cyclic number.
Or must $a$ always be prime?
AI: Indeed, if you want a number $\frac{1}{x}$, where $x$ ... |
H: equality for a measure $\mu(F\backslash E)= \mu(F)-\mu(E)$
Studying for Real Analysis I encountered this exercise and I am a bit confused about it.
Let $\mu$ be a measure on $(X,M)$, where $M$ is a $\sigma$-algebra on $X$. Show that if $E \subseteq F$ and $\mu(E)< \infty$ then $\mu(F\backslash E)= \mu(F)-\mu(E)$ (*... |
H: The closure of a product is the product of closures?
If $\{X_j:j\in J\}$ is a family of topological spaces and $A_j\subseteq X_j$, is it true that $\displaystyle\overline{\Pi_{j\in J}A_j}=\displaystyle{\Pi_{j\in J}\overline{A_j}}$? Is there an easy way to prove this?
Of course, we are considering in $\displaystyle{... |
H: Proof by cases, inequality
I have the following exercise:
For all real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \leq 1$ or $x \geq 4$.
I need you to help me to identify the cases and explain to me how to resolve that. Don't resolve it for me please.
AI: HINT:
If $(x-a)(x-b)\ge0$
Now the product of ... |
H: How to prove that $q^r$ and $q^s-1$ are relatively prime?
How to prove that $q^r$ and $q^s-1$ are relatively prime? ($q$ is prime or prime power)
AI: We do not need to assume anything about $q$, but we do need to assume that $s\gt 0$.
Let $d\gt 1$ be a divisor of $q^r$. Then some prime $p$ divides $d$, and hence $q... |
H: Simple linear regression - understanding given
The question is to fill out the missing numbers (A-L) of a simple linear regression model.
I am having problems with converting and interpreting the given table in terms of variables. Would it be possible for someone to confirm and clarify things for me.
The first tabl... |
H: calculate the loss to shop owner
A customer purchases clothes worth 200 Rupees from a shop. customer gives 1000 Rupee note. Since shop owner does not have change, he collects change (100 * 10 notes) from the neighbor shop and gives 800 rupees back to the customer.
After few hours neighbor shop keeper tells the sho... |
H: Is $(-\infty,\infty)$ a closed **interval**?
Note that we are working in the reals, not the extended reals. Now consider two opposing claims:
$(-\infty,\infty)$ is a closed interval, because a closed
interval is an interval that is a closed set.
$(-\infty,\infty)$ is not a closed interval, because a closed
interva... |
H: Interpreting inequalities
Two conditions are met for $x$:
$x \geq 4$ and $x \geq 1$
What's the final value of $x$? I believe it is $x \geq 4$, by simple logic if $x$ is said to be greater or equal than $1$ and greater or equal than $4$, then it means it is the greatest one..
However, what's the justification behind... |
H: $X$ a topological space. If $A$ lies inside a closed set. Does it follow that the closure of $A$ also lies inside this closed set?
PROBLEM: $X$ a topological space. If $A$ lies inside a closed set. Does it follow that the closure of $A$ also lies inside this closed set?
MY TRY:
Suppose $A \subseteq F$ where $F$ i... |
H: finding examples for a non negative and continuous function for which the infinite integral is finite but the limit at infinity doesn't exist
Question:
a. Find an example for a non-negative and continuous function s.t. $\int _0^\infty f(x)dx$ is finite but the following limit doesn't exist: $\lim_{x\to \infty} f(x)... |
H: Density of a set
Suppose $X$ is a topological space. We know by definition $A$ is dense in $X$ if $ \overline{A} = X $.
My question is. IS it enough that $\overline{A} \subseteq X$ to say that $A$ is dense in $X$ ??
AI: No. Take $X=[0,2]$ and $A=[0,1]$. Then $A= \overline{A} \subset X$, but clearly $A$ is not den... |
H: Proving the diameter is two times the radius
I am stuck on the following question:
Prove that each diameter is twice as long as each radius.
I drew a circle, with center O and diameter AB. Is there a theorem that could help me say that congruent segment AO and BO add up to form segment AB?
Or is there some other w... |
H: Tricky Puzzle!! Please help.
I stumbled upon a puzzle I can't crack. It goes like this:
In a certain Code language:
7321=6
5342=3
8645=15
Then 9312=?
The Answer is 9. But I can't seem to find the logic behind it??
AI: Cool Puzzle! Here is the pattern I found:
$7321:$ Take $7 \cdot 3 \cdot 2 \cdot 1 = 42$. Then $4... |
H: Let $f$ be a continuous function on $[a, b]$ such that $f(x)$ is rational for every $x$. What can be said about $f$?
can someone help me out with this question? I have been stuck on it for a while.
Suppose that $f$ is a continuous function on the closed interval $[a, b]$, and that $f(x)$ is rational for every $x$ ... |
H: Proving a random variable
Here is another question from the book of V. Rohatgi and A. Saleh. I would like to ask help again. Here it goes:
Let $\mathcal{A}$ be a class of subsets of $\mathbb{R}$ which generates $\mathcal{B}$. Show that $X$ is an RV on $\Omega\;$ if and only if $X^{-1}(A)$ $\in \mathbb{R}$ for all $... |
H: Proving the base of a gramian matrix with a defined form
Be $V$ n-dimensional in a $ \mathbb R $ vector space und be $q$ a nondegenerate quadratic form on $V$. To prove: It exists a base $B$ of $V$ in $ \mathbb R $, in which the gramian matrix of $q$ is defined as
$\begin{bmatrix}1 & & & & & \\ & ... & & &... |
H: Why does the method to find out log and cube roots work?
To find cube roots of any number with a simple calculator, the following method was given to us by our teacher, which is accurate to atleast one-tenths.
1)Take the number $X$, whose cube root needs to be found out, and take its square root 13 times (or 10 tim... |
H: ${{p-1}\choose{j}}\equiv(-1)^j \pmod p$ for prime $p$
Can anyone share a link to proof of this?
$${{p-1}\choose{j}}\equiv(-1)^j(\text{mod}\ p)$$ for prime $p$.
AI: $$\binom {p-1}j=\prod_{1\le r\le j}\frac{p-r}r$$
Now, $\displaystyle p-r\equiv -r\pmod p\implies \frac{p-r}r\equiv-1\pmod p$ |
H: Derive a formula of a specific curve
I have this curve
And I know that the first point is
$$A(0,5)$$
and the last point is
$$C(1650,9.5)$$
The point almost at the center where the curve changes (if you look close, you can see a green dot) is
$$B(1000,8.5)$$
There is a way to derive a generic f(x) so that I can pl... |
H: Show there is a closed interval $[a, b]$ such that the function $f(x) = |x|^{\frac1{2}}$ is continuous but not Lipschitz on on $[a, b]$.
Hi guys I was given this as an "exercise" in my calculus class and we weren't told what a Lipschitz is so i really need some help, heres the question again:
Show there is a clos... |
H: If given a regular language, how can we prove that a sub-language is regular?
This question has been quite confusing me.
$\sum = \{a,b,c\}, L \text{ is a regular language}$ and we have to prove that $L^{'} = \{w \in L : w\text{ containts at least one c} \}$ is regular.
What are the steps of the proof? What methods ... |
H: Help with a property of a convex function
I'm studying linear and nonlinear programming and on my book I bumped into the following statement:
$$\lim_{\alpha \to 0} \displaystyle \frac{f(\textbf{x}+\alpha (\textbf{y}-\textbf{x}))-f(\textbf{x})}{\alpha} = \nabla f(\textbf{x})(\textbf{y}-\textbf{x})$$
Could someone sh... |
H: $f,g$ continuous from $X$ to $Y$. if they are agree on a dense set $A$ of $X$ then they agree on $X$
Problem:
Suppose $f$ and $g$ are two continuous functions such that $f: X \to Y
$ and $g : X \to Y $. $Y$ is a a Hausdorff space. Suppose $f(x) = g(x)
$ for all $x \in A \subseteq X $ where $A$ is dense in $X$, t... |
H: Finding the median of a probability distribution
A gambler makes a long sequence of bets against a rich friend. The gambler has initial capital C. On each round, a coin is tossed; if the coin comes up tails, he loses 30% of his current capital, but if the coin comes up heads he instead wins 35% of his current capit... |
H: Multiple disjunctions with a Tableaux proof system
I am using the Tableaux proof system, and have a question about branching and disjunctions.
Normally the example on how to use the Tableaux proof system is to get the formula to CNF, and then start branching it. It can look like this:
$$
(A \lor B), \lnot B,\lnot A... |
H: Why are two definitions of ellipses equivalent?
In classical geometry an ellipse is usually defined as the locus of points in the plane such that the distances from each point to the two foci have a given sum.
When we speak of an ellipse analytically, we usually describe it as a circle that has been squashed in one... |
H: Optimisation of a rectangles area under a function curve
I have a questions asking for the dimensions of the rectangle with the largest area that has two bottom corners on the x axis and two top corners on the curve $y=12-x^2$.
I have plotted the curve and found it is a symmetrical parabola with a vertex of $x=0, y... |
H: Regular Language Operation
I need to show that the given regular language is closed under the following operation.
For example:
AllSuffixes(L) = {v : uv in L for some u in (0+1)* }
I do not have any idea about this question. I only know that regular languages are closed under + , . and * operations.
Do you have any... |
H: Probability on divisibility
Let S be the set of all 12-digit positive integers each of whose
digits is either 1 or 4 or 7 (for example, 477411171747 is a member
of S). What is the probability that a randomly picked member
of S is divisible by 12 ?
AI: HINT:
For a number to be divisible by twelve it must be divisibl... |
H: Questioning a Basis for $\mathbb{Q}[\sqrt[3]{2}]$ over $\mathbb{Q}$
Let $\omega = e^{2 \pi i /3}$ and $\alpha = \sqrt[3]{2}$. I'm seeing it claimed that $\mathcal{B} = \{\alpha, \alpha^2, \omega \alpha, \omega \alpha^2, \omega^2 \alpha, \omega^2 \alpha^2\}$ forms a basis for the vector space $\mathbb{Q}[\alpha, \o... |
H: $(A\cap B)\cup C = A \cap (B\cup C)$ if and only if $C \subset A$
I have a set identity: $(A \cap B) \cup C = A \cap (B \cup C)$ if and only if $C \subset A$.
I started with Venn diagrams and here is the result:
It is evident that set identity is correct. So I started to prove it algebraic:
1) According to distrib... |
H: Convergence of a series of random elements
Given the normally distribuited random variable $\nu(t)$ with $\mu=0$ and variance $\sigma$, I have to find if the series:
$$G(\sigma)=\sum_{k=1}^{\infty}\frac{1}{\exp\left(\nu(k)\right)}$$ where $\nu(k)$ is the random value of the $\nu(t)$ for $t=1,2,...$, is convergent a... |
H: Construct context-free grammar for $\{a^ib^jc^k : i\le j+k\}$
I'm looking through several of old exam sets in order to prepare for the exam and now I'm stuck on this exercise, where we have to construct a context-free grammar for the language:
$$L = \{a^ib^jc^k: i \le j+k \}$$
The best solution I've obtained so fa... |
H: Extensions of probability measures on fields
I am trying to solve exercise 3.3 in Billingsley's "Probability and measure" and I am not sure I correctly understood the text. I am not going to copy all the text, but I will ask directly some questions.
Some background:
Let P be a probability measure on the field $F_0$... |
H: Each player throws two dice
probability problem:
players a and b throws two dice and
a player wins if the sum for the first throw is 11 or 7 and Each player loses at once if it is 3, 2. For other
case,throw two dice is repeated.the probability that player a wins at the $k$th throw?
my attempt: $ p(\text{a win at t... |
H: Is the sum of positive definite matrices still positive definite?
I have two symmetric positive definite (SPD) matrices. I would like to prove that the sum of these two matrices is still SPD. Symmetry is obvious, but what about PD-ness? Any clues, please?
AI: A real matrix $M$ is positive-definite if and only if it... |
H: If $R,S$ are reflexive relations, so are $R \oplus S$ and $R \setminus S$?
Suppose $R$ and $S$ are reflexive relations on a set $A$. Prove or disprove each of these statements.
a) $R\oplus S$ is reflexive.
b) $R\setminus S$ is reflexive.
I think both of a) and b) are false, but I'm having trouble with coming up w... |
H: Is $\mathbb Z/p\mathbb Z$ a subfield of every finite field?
I translate this from a German book: "For every finite field
$K$ there exists a prime number $p$ such that $\mathbb Z/p\mathbb Z$ is a subfield of $K$"
But how is this possible? For example the field $K = \{0,1\}$ contains integers but $\mathbb Z/p\mathbb... |
H: Can one prove that a language is regular without having a regular expression?
I was wondering if one could prove that a language is regular without showing a DFA/NFA or a regular expression that expresses it.
For example: $L = \{w \in \Sigma^* : w \text{ has at least two identical letters} \}$
AI: In this case it’s... |
H: $-ia(1\pm \sqrt{1-1/a^2})$, $a>0$ inside unit circle?
Given $a>0$ I would like to know whether:
$\alpha=-ia(1+ \sqrt{1-1/a^2})$ and
$\beta =-ia(1- \sqrt{1-1/a^2})$ are inside the unit circle.
How can I check that?
AI: You have
$$\alpha\cdot\beta = (-i)^2a^2\left(1 - \sqrt{1-1/a^2}^2\right) = -a^2(1/a^2) = -1,$$
so ... |
H: Median for continuous distribution
Consider a continuous random variable X with probability density function given by $f(x)=cx$ for $1 \le x \le 5$, zero otherwise. Find the median.
First I calculate the CDF: $F(x)=cx^2/2$ for $1 \le x \le 5$, zero otherwise.
Now we have to solve for constant c by using the def... |
H: Painting $\mathbb R^+$ with two colors which sum of two same color numbers be the same.
Can any one paint $\mathbb R^+$ with two colors which sum of two numbers with the same color has the same color.
Additional condition: Both colors should be used.
I tried use Cauchy functions like ($f(x+y)=f(x)+f(y)$). But ther... |
H: Number of non negative Integral solutions
Need to find the non negative integral solutions for the equation
$x+y+xy=x^{3}+y^{3}$
I have tried various methods for simplifying the RHS and LHS but could not arrive at the solution, so any help will be appreciated
AI: First get a bound for possible solutions. Suppose wi... |
H: Does there exist a complement of a subgroup in a abelian group.
Let $G$ be an abelian group and $H$ subgroup of $G$.
Suppose that:
(i) $H$ has a complement in $G$.
(ii) $K$ is a subgroup of $G$ and K is isomorphic to $H$
Is there a complement of $K$ in $G$? If yes, what is the relation of complements of $H$ and co... |
H: Question regarding simple limit
Why is it, that:
$\lim_{x \to \infty} [x (1-\sqrt{1-\frac{c}{x}})] = \frac{c}{2}$
Link: Wolframalpha
and not $0$?
My (obviously incorrect) reasoning:
Since $c$ is an arbitrary constant, and as $x$ goes to infinity $\frac{c}{x}$ will practically equal $0$. thus $\sqrt{1-0} = \sqrt{1} ... |
H: How to document undergrad math knowledge?
If you did a degree which is low on math (read economics, psychology), but want to proceed to a more mathematically loaded master, how would you document your knowledge?
Are there standarized examinations that are widely recognized?
That would also be a way of discovering ... |
H: Pullbacks of monic morphisms.
I'm trying to prove that pullbacks of monics are monic.
Let
$\require{AMScd}$
\begin{CD}
X_1 @>f>> X_2\\
@V m' V V @VV m V\\
X_3 @>>g > X_4
\end{CD}
be a pullback square with $m'$ monic.
Let $h, k$ be parallel such that $m\circ h=m\circ k$. Let $x_{0}$ be the domain of $h, k$.
Suppos... |
H: How to find the domain of $f\left(g\left(y\right)\right)=\sqrt{\left(\frac{x+1}{x-1}\right)^3-27}$
Please help me find the domain of the following equation.
\begin{eqnarray}
\\f\left(x\right)=\sqrt{x^3-27},\space \space \space g\left(y\right)=\frac{x+1}{x-1},\space \space \space find \space f。g\\
\end{eqnarray}
T... |
H: Is $2^k = 2013...$ for some $k$?
I'm wondering if some power of $2$ can be written in base $10$ as $2013$ followed by other digits.
Formally, does there exist $k,q,r \in \mathbb N$ such that $$2^k=2013 \cdot 10^q+r \,\,\,; \,\,\,r<10^q $$
I'm not sure if it's true or not. I would go for a 'no', but I can't prove it... |
H: sum of square roots
I was wondering what the estimate for the value
$$
S= \sum_{j=1}^N \sqrt{j}
$$
is? Is there a way (or a formula) to estimate it well?
I am sure it is close to $\int_1^N \sqrt{t} dt$, but I guess I
was interested in estimating it better.
Thanks!
AI: Euler-Maclaurin series.
According to Maple, a... |
H: Negating statements / Finding $(A \cap B)',A \oplus B$ if $A=\{x \in\Bbb R \mid -3\le x\le0\}$ and $B=\{x \in \Bbb R\mid -1 < x < 2\}$
I am a bit new on this field and I am trying to solve some questions. I don't really think they are hard but there are some key points that I don't get it or I am stuck.
Lets see.
... |
H: Number of digits of $2^{1000}$
A friend asks me to find the number of digits of $2^{1000}$. I tried to look for a pattern by calculating the first powers of $2$ but I didn't find it. How should I proceed? Thanks.
AI: Hint: Solve for $x$ using logs:
$$10^x = 2^{1000}$$
Spoiler
$x \approx 301$ digits, so round up a... |
H: ${{p^k}\choose{j}}\equiv 0\pmod{p}$ for $0 < j < p^k$
$${{p^k}\choose{j}}\equiv 0\pmod{p}.\ \ \ \text{for $0 < j < p^k$ and p is prime}$$
I can show this for $k=1$ using the fact that in denominator all numbers are less than $p$. I need hint for proving this for $k>1$.
AI: $$\binom{p^k}j=p^k\frac{(p^k-1)(p^k-2)\... |
H: The calculation of roots of complex numbers.
How to calculate the roots of $x^6+64=0$?
Or how to calculate the roots of $1+x^{2n}=0$?
Give its easy and understanble solution method. Thank you. In general, the results of "exp" are obtained.
AI: $$x^{2n}=-1=\cos\pi+i\sin\pi=\cos(2k+1)\pi+i\sin(2k+1)\pi$$ where $k$ ... |
H: Differentiabilty of this function
I want to show that $(x^2+y^2)^{\alpha}$ is not differentiable for $\alpha\in(0,1)$. All other cases are pretty straightfoward.
AI: The non exstence of partial derivatives are enough to show that the function is not differentiable at $(0, 0)$. Namely,
$f_x(0, 0)=\displaystyle\lim_{... |
H: Is a set without limit points necessarily closed?
According to the definition on Rudin' Principles of Mathematicial Analysis, closed set is defined as:
$E$ is closed if every limit point of $E$ is a point of $E$.
Then I have a question: if a set has no limit point, is it necessarily closed? I think this idea doe... |
H: The ring of integers in $\mathbb{Q}[\zeta]$ is $\mathbb{Z}[\zeta]$
I am working on a proof in the lecture of Milne "Proposition 6.2 b)" but there is a step I don't get:
We have an inclusion $\mathbb{Z}\hookrightarrow \mathcal{O}_K$ that induces the following isomorphism $\mathbb{Z}/(p)\to \mathcal{O}_K/(1-\zeta)$.... |
H: Is the intersection of a chain of covers also a cover?
Let $\mathcal{F}$ be a family of sets covering some set $X$, so $\bigcup_{F \in \mathcal{F}} F=X$.
Assume we take an infinite chain $\mathcal{X}$ of nested families $\mathcal{F}''\subset \mathcal{F}'\subset \mathcal{F}$ so for every $\mathcal{F}_1, \mathcal{F}_... |
H: Check if a given limit of a sequence is valid
Using the definition of the limit with vicinities (rough translation from my native language) , prove that:
$$ \lim_{n\to \infty} \frac {2^n+3}{2^n+4^n}=0 $$
For convenience I will take $a_n=\frac {2^n+3}{2^n+4^n}$
For this to be true I need to prove that inside any vic... |
H: Show for a diagonalisable matrix that $\det A = \prod_{n=1}^n\lambda_i $
I am currently trying to solve this exercise:
Let A be a diagonalisable matrix with eigenvalues $\lambda_1,\lambda_2...\lambda_n.$ Show that $\det A = \prod_{n=1}^n\lambda_i $
I tried solved it for a $3x3$ matrix:
A diagonalisable matrix can b... |
H: What is the LCM of $3^{2001}-1$ and $3^{2001}+1$?
What is the LCM of $3^{2001}-1$ and $3^{2001}+1$?
I can not get whether the GCF is $2$ or more than that.
AI: Note that $b=3a+4$, where $a=3^{2001}-1$ and $b=3^{2002}+1$. So the greatest common divisor of $a$ and $b$ must divide $4$. So it is either $2$ or $4$. It i... |
H: How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$?
How to compute $\frac{t}{t+1}$ to the form $1-\frac{1}{t+1}$?
What else? Well.
Well can you use long division?
AI: HUGE EDIT IN RESPONSE TO OP CORRECTION:
$$\dfrac{t}{t+1} = \dfrac{t + 1 - 1}{t+1} = \dfrac{t + 1}{t+1} - \dfrac{1}{t+1} = 1 - \dfrac{1}{t+... |
H: A closed form for $\sum_{i=1}^{n} \prod_{k=1}^{i+2} (3k+2)$
I need to calculate the following expression. Is there any explanation to convert this expression into normal expression without those letters for sum and the product? Just normal expression.
$$ Z = \displaystyle\sum_{i=1}^{n} \displaystyle\prod_{k=1}^{i... |
H: Comparing determinants of an odd-dimension matrix
If $\mathbf{A}$ is a square matrix with odd dimensions, can $|\mathbf{A}|=|-\mathbf{A}|$?
This is true if $\mathbf{A}$ is over the field $\mathbb{Z}_2$, but are there any other situations?
AI: If $A$ is $k\times k$ with $k$ odd, then $\mid -A\mid = (-1)^k\mid A\mid ... |
H: Linear operator $T$ on a finite dimensional vector space
Question:
A linear operator $T$ on a finite dimensional vector space is
diagonalizable if and only if the multiplicity of each eigenvalue
$\lambda$ equals the dimension of $E_\lambda$.
According to me the answer should be $true$, but my book says otherw... |
H: Is $f: \mathbb{R}_l \to \mathbb{R}, f(x) = 1$ for $x\geq 0$ and $f(x) = -1$ for $x < 0$ continuous?
Let $\mathbb{R}$ be the set of real numbers with standard topology.
Let $\mathbb{R}_l$ is the set of real numbers with lower limit topology.
Is $f: \mathbb{R}_l$ $\to \mathbb{R} $ given by
$$
f(x) =
\begin{cases}
1,... |
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