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H: Show that $\sum_{cyc} J(x,J(y,z))=0$.
Let $x,y,z$ be functions of $(u,v)$ and $J$ be the Jacobian matrix.
Show that $\sum_{cyc} J(x,J(y,z))=0$.
I expanded the thing and realized that the first term in the sum is $x_u(J(y,z_v)+J(y_v,z))-x_v (J(y,z_u)+J(y_u,z))$, but I don't know how to carry on then. Furthermore, ... |
H: Given three vectors, how to find an orthonormal basis closest to them?
I know Gram-Schmidt process but that is not what I am looking for. Given three vectors in $\mathbb{R}^3$, $\{v_1,v_2,v_3\}$, I want to find three vectors $\{w_1,w_2,w_3\} \subset \mathbb{R}^3$ such that
$$(w_i,w_j) = \delta_{ij}, \quad \forall i... |
H: What can we say about $\dim \operatorname{null}(AB)$ from knowing $p_A$ and $p_B$?
Say, there are two matrices $A, B \in \mathbb R^{3,3} $ such that their characteristic polynomials are $p_A(t) = t^3 − t^2 + 2t$ and $p_B(t) = t^3 − 7t^2 + 9t − 3$. What do we know about $\dim \operatorname{null}(AB)$?
Clearly, $t=0$... |
H: proving $\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$ when f is continous on [0,1]
$$\lim\limits_{n\to\infty} \int_{0}^{1} f(x^n)dx = f(0)$$
when f is continuous on $[0,1]$
I know it can be proved using bounded convergence theorem but,
I wanna know proof using only basic properties of riemann integral a... |
H: A new way of solving cubics?
I found this (from http://www.quora.com/Mathematics/What-are-some-interesting-lesser-known-uses-of-the-quadratic-formula):
So my question is: Can this be generalized to solve any depressed cubic [in effect, all the cubics]. Maybe not exactly this method, but some other repeated manipul... |
H: Fourier and differentiation operators
For a function $f:\mathbb{R}\rightarrow\mathbb{R}$ in the Schwartz class, define $$Tf(y)=\dfrac{1}{\sqrt{2\pi}}\int_\mathbb{R}f(x)e^{-ixy}dx$$
We can show that $T^2f(y)=f(-y)$, and $T^4f(y)=f(y)$. Also, define $$Af(y)=yf(y)+\dfrac{d}{dy}f(y)$$ For what value of $a$ is $TA=aA... |
H: Find the point on the parabola
Find the point on the parabola $y^{2}=2x$ that is closest to the point $(1,13)$.
What is the parabola? and please show the process
AI: the distance is:
$d^2=(x-x_0)^2+(y-y_0)^2$. In your case: $x_0=1,y_0=13$. So you get:
$$d^2=(x-1)^2+(y-13)^2$$ but you know: $y=\sqrt{2x}$.
So, you ha... |
H: Conditional probability questions?
Of a group of children, $0.4$ are boys and $0.6$ are girls. Of the boys, $0.6$ have brown eyes; of the girls, $0.2$ have brown eyes. A child is selected at random from the group.
(a) Find the probability that the child is a girl.
This is $.6$
(b) Find $P(brown eyes | boy)$.
This... |
H: Similar to Poincare inequality on Sobolev spaces
The following looks quite similar to Poincare's inequality:
Let $\displaystyle{ 1 \leq p < \infty}$ and $\displaystyle{ U \subset
\mathbb R^n}$ open and such that $\displaystyle{ U \subset \mathbb
R^{n-1} \times (0,L) }$ with $L>0$. Show that for $\displaystyle{ u... |
H: Shortest distance from point and line
We want to calculated a shortest distance form point $T(0,1,2)$ and from line of intersection of planes $x+y+z =0$ in $x-z+4=0$
I try this:
I have equate both equations
$x+y+z =x-z+4$
$y+2z =4$
I get 2 points:
$A(0,0,2)$
$B(0,2,1)$
$p= (0,0,2)+t(0,-2,1)$
$T_0=x_0 + \frac{\langl... |
H: Composite of two purely inseparable extensions is purely inseparable.
Let $L,F$ be extensions of the field $K$ and are contained in a common field. Prove that, if $L$ and $F$ are purely inseparable extensions over $K$ then $LF$ is also a purely inseparable extension over $K$. Is the converse true?
How do I prove it... |
H: card expectations
Beginning probability question
Imagine a standard card deck where cards have the values 2,3,4,...,11(J),12(Q),13(K),14(A).
The deck is shuffled so that every permutation is equally likely.
I draw cards from the deck,
one at a time without replacement, until I draw the queen of spades (Q), at which... |
H: Determine if these are equivalence relations
I would appreciate if someone could go through the task and the answers I've got and check if I've done it correct, if not please correct me.
Here is the task:
Below we have listed some relationships over The set of $\{a, b, c, d, e\}$. For each of these determine wheth... |
H: Evaluate $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 } } n^{-1/2}$
I know that $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 }\frac { 1 }{ n } =\ln(2) }$ .
How about the series $\sum _{ n=1 }^{ \infty }{ { (-1) }^{ n+1 } } \frac { 1 }{ \sqrt { n } }$
To what number does it converge?
AI: You are looking for the functi... |
H: Proof of a trigonometric expression
Let $f(x) = (\sin \frac{πx}{7})^{-1}$. Prove that $f(3) + f(2) = f(1)$.
This is another trig question, which I cannot get how to start with. Sum to product identities also did not work.
AI: Let $7\theta=\pi, 4\theta=\pi-3\theta\implies \sin4\theta=\sin(\pi-3\theta)=\sin3\theta$
$... |
H: Symmetric matrix under orthogonal transformation still symmetric?
is there any way to see that a symmetric matrix is still symmetric after applying an orthogonal basis transformation to it? I would say that a proof that refers to the entries of the matrix may be cumbersome, therefore I am asking here for clever way... |
H: Probability questions.
The following experiment is used to check a person who claims to have powers of mental telepathy. Six cards are numbered 1 through 6. Seat the person being tested on one side of a screen and a person who will select a card on the other side. The person with the cards shuffles them, selects on... |
H: 4 convex sets in a plane have a point in common
Let $X_1,X_2,X_3,X_4$ be four sets in the plane such that any three of them have a point in common. Do all four of them have to have a point in common? What if sets are convex?
Attempt: I think all of them should not have a point in common, but I dont how them being... |
H: Evaluate $ \sum\limits_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}$
The question was: Evaluate, ${\textstyle {\displaystyle \sum_{n=1}^{\infty}\frac{n}{n^{4}+n^{2}+1}}}.$
And I go, since $\frac{n}{n^{4}+n^{2}+1}\sim\frac{1}{n^{3}}$ and we know that ${\displaystyle \sum_{n=}^{\infty}\frac{1}{n^{3}}}$ converges. so ${\disp... |
H: find a formula for the sum of combination
$5^n {n\choose 0} - 5^{n-1} {n\choose 1} + 5^{n-2}{n\choose2} +...... \pm{n\choose n}$
I was trying to use binomial theorem, but it doesn't seem to work.
AI: $$5^n {n\choose 0} - 5^{n-1} {n\choose 1} + 5^{n-2}{n\choose2} +...... \pm{n\choose n}=$$
$$=\sum_{k=0}^n\binom nk(-... |
H: How to solve mechanics problem when acceleration depends on position.
I'm curious about how problems such as the following are typically solved analytically, or in computer simulations such as games engines for 2D physics. It seems a bit harder than the typical constant acceleration scenario.
Suppose a particle wit... |
H: Discontinuous function
Show that the function, $f:\mathbb{R^2}\rightarrow \mathbb{R}$,
$$ f(x,y) = \begin{cases} 1, & \|(x,y)\|\geq1\\
0, & \|(x,y)\|<1 \end{cases} $$
is not continuous ($\mathbb{R}$ and $\mathbb{R^2}$ have their usual Euclidean metrics).
Could someone check if my proof is correct. Feel like I migh... |
H: why the Vitali set has size $2^{\aleph_0}$?
why the Vitali set has size $2^{\aleph_0}$ and therefore there is a bijection between $\mathbb R$ and the Vitali set? thanks
AI: Note that there are $2^{\aleph_0}$ equivalence classes in $\Bbb{R/Q}$. Since a choice function from these classes is injective, this shows that... |
H: Why are group theory and ring theory a part of abstract algebra?
I have followed the courses Algebra 1, which was about group theory and Algebra 2, which was about ring theory. I don't think I really understand why those subjects are part of abstract algebra. What does "algebra" has to do with those both subjects ?... |
H: Notation for translating vectors
I'm completely new to vector geometry and recently encountered some new notation (and wholly unfamiliar) for the translation of vectors.
$$T:Z \mapsto A + Z$$
The above is described as
A translation by vector $A$
Again, I have no idea why they call it a translation by vector $A$ ... |
H: Fibonacci sequences
I have the following:
$$ f_3+f_6 + \dots+f_{3n} = \frac 12 (f_{3n+2}-1) $$
for $f_0=0$ and $f_1=1$
When I calculate $n\ge2$ and $f_n= f_{n-1}+f_{n-2}$, I get: LHS = 8 while RHS = 10.
LHS
$$f_6 =f_5+f_4 \\ f_5 = f_4+f_3 \\ f_4 = f_3 + f_2 \\ f_3=f_2+f_1 \\ f_2 = f_1 + f_0 $$
and so :
$$f_2=1 \\ ... |
H: Solving a bijective function task
How would you solve this task actually? This is not homework btw, i'm just trying to understand how you can solve tasks like this.
Let $S$ be the set $\{$$1,2,3,4,5$$\}$. How many functions from $S$ to $S$ are there and how many of these are bijective?
Thanks alot for any help.
A... |
H: Does sum of random variables remain independent?
Suppose we have random variables $X,Y,Z$ such that $X,Z$ are independent, and $Y,Z$ are also independent. Is $X+Y$ independent from $Z$ ?
I cannot find a counterexample, and I do not see how to prove it.
AI: Toss a fair coin three times. Let $X$ be $1$ if the number... |
H: Minimum and maximum values
I am not sure if my method for this question is correct:
Given the function $$f(x,y)=x^2+y^2+2x+y$$, find it's minimum and maximum values about a closed disc of radius 2 centred at the origin.
I took the partial derivatives, $$f_x=2x+2; f_y=2y+1$$ and solved for $x$ and $y$. The minimum... |
H: Find upper and lower bounds to the function $f(n)=1\cdot3\cdot5\cdot\ldots\cdot(2n-1)$ where $n\in\Bbb N$
Find upper and lower bounds to the function $$f(n)=1\cdot 3\cdot 5\cdot 7\cdot 9\cdot\ldots\cdot (2n-1)\;,$$ where $n\in\Bbb N$.
I got $$\left(\frac{2n-1}e\right)^{(2n-1)/2}$$ as lower bound and
$$\left(\frac{... |
H: On polynomial of prime degree.
Let $K$ be a field, $f(X)\in K[X]$ be a polynomial of prime degree. Assume that for all extension $L$ of $K$, if $f$ has roots in $L$ then $f$ splits over $L$. Prove that either $f$ is irreducible over $K$ or $f$ splits over $K$.
How do I prove it?
Thanks a lot.
AI: If $f$ has a root ... |
H: Help required: please evaluate the two given integrals
Question:
Evaluate the following integral.
My answer:
$=\displaystyle\cos (6t)+6t . 7t/7 = -\cos(6 . \pi/2) + (6\pi)/2 . (7.(\pi/2)) /7$
Correct answer:
$\frac{2}{3}.$
While you are at it could you also answer this one as well
(sorry I did not know how to w... |
H: Galois group of irreducible polynomial in a field of characteristic zero in which every element is a perfect square
This is one of the exercises during my reading of Ian Stewart's Galois Theory. Whether the following statement is true:
If $K$ is a field of characteristic zero in which every element is a perfect s... |
H: Using induction to prove $a_n >2^n$
For the sequence $a_n=2a_{n-1}+1$ where $a_0=1$
Show that $a_n>2^n$ using induction.
Use proof by contradiction (minimum counterexample).
Attempt: 1. I assume, that $a_n>2^n$ as my induction hypothesys. Now, I try to show that $$a_{n+1}>2^{n+1} \\ 2a_{n}+1>2^{n}*2 \\ a_n+1/... |
H: Drawing a hasse diagram of a poset.
I have the poset $(\{1,2,3,4,5,6,8,15\}, |)$, where $|$ denotes divisibility
and was able to come up with the Hasse diagram
I feel like I had multiple options to connect the $8$ too as well as the $6$. This is my first hasse diagram and I just wanted to post to make sure I did i... |
H: Max area of triangle -PHP
How do i prove that the maximum area that can be obtained among 3 random points in a square is half the area of the square?-
I need it to for the following question
" Show that among any 9 points inside a triangle of area 1 there are
three points which form a triangle of area at most 1/4."... |
H: Show discontinuity of $\frac{xy}{x^2+y^2}$
How to show this function's discontinuity?
$ f(n) = \left\{
\begin{array}{l l}
\frac{xy}{x^2+y^2} & \quad , \quad(x,y)\neq(0,0)\\
0 & \quad , \quad(x,y)=(0,0)
\end{array} \right.$
AI: $y=mx\Rightarrow f(x,y)=\frac{mx^2}{(1+m^2)x^2}=\frac{m}{1+m^2}$
Does th... |
H: How to prove or disprove finiteness?
How to prove or disprove that statement: a group is finite if the set of all its subgroups is finite?
AI: A finite group certainly has only finitely many subgroups, so the question really boils down to this: is there an infinite group with only finitely many subgroups?
HINT: Let... |
H: Help needed with an equivalence relation task on natural numbers
I'm having a bit difficulties understanding and solving this task. I would appreciate any help on how you can solve tasks like this.
Here is the task:
Let ~ be an equivalence relation on the natural numbers, and let $E$ be $[0]$, ie equivalence class... |
H: Showing a ring has no non-zero nilpotents via a ring homomorphism
Let $\theta\colon R\to S$ be a ring homomorphism. If each of $\ker{\theta}$ and $\operatorname{im}{\theta}$ has the property that its only
nilpotent element is $0$, show that the same is true for $R$.
AI: If $r \in R$ is nilpotent, then $\phi(r)$ is ... |
H: Why a non-diagonalizable matrix can be approximated by an infinite sequence of diagonalizable matrices?
It is known that any non-diagonalizable matrix, $A$, can be approximated by a set of diagonalizable matrices, e.g. $A \simeq \lim_{k \rightarrow \infty} A_k$. Why this is true?
Note: I was faced with it for the f... |
H: How to prove something about the order of element at a group
$G$ is a cyclic group. the order of $G$ is $n$ ($|G|=n$).
$m\mid n$,
I have to prove that there is $b\in G$ that $ord(b)=m$.
Know, here is one part of the proof:
if $m\mid n$ that means that $n=mk$. Lets mark:
$$b=a^k$$
then: $$b^m=(a^k)^m=a^n=e\Rightarr... |
H: Validity of notation from the aspect of function description
I have the following notation that should describe the nature of my function
$for \forall a \in A \exists f:A \rightarrow S, A \subset N, S \subset [0,1]^n,|S|=n$
Can anyone tell me is the notation correct for the descriptive definition below.
Function ... |
H: Find the flaw in the attempted proof of the parallel postulate given by J. D. Gergonne
The attempted proof:
Given $P$ not on line $l$, line $PQ$ perpendicular to $l$ at $Q$, line $m$ perpendicular to $PQ$ at $P$ and point $A \neq P$ on $m$. Then, let $PB$ be the last ray between rays $PA$ and $PQ$ that intersects ... |
H: Permutations on four consecutive digits yield $n$ such that $2013 < n < 10000$
2013 is the first year since the Middle Ages that consists of four consecutive digits. How many
such years are there still to come after 2013 (and before the year 10000)?
AI: HINT: The possible sets of four consecutive digits are $\{0,1,... |
H: Floor and ceiling function proof
I have the following to prove:
$$\lfloor 3x\rfloor = \lfloor x\rfloor + \left\lfloor x+\frac 13 \right\rfloor + \left\lfloor x+\frac 23 \right\rfloor $$
The definition of a floor function is: $ \lfloor x \rfloor = n \le x \lt n+1 $
So my first instinct was to do $ \lfloor 3x\rfloor... |
H: How find this limit $\lim_{x\to 0}\frac{1}{x^4}\left(\frac{1}{x}\left(\frac{1}{\tanh{x}}-\frac{1}{\tan{x}}\right)-\frac{2}{3}\right)=？$
Find this following limit
$$\displaystyle \lim_{x\to 0}\dfrac{1}{x^4}\left(\dfrac{1}{x}\left(\dfrac{1}{\tanh{x}}-\dfrac{1}{\tan{x}}\right)-\dfrac{2}{3}\right)=？$$
My try: since $$\... |
H: Is my calculation correct about a probability problem?
Suppose there are $300$ tickets in the pool, where $7$ of them belong to me. $20$ tickets are randomly taken out of the pool, and are declared as "winning tickets". What is the probability that exactly 4 of the winning tickets are mine?
When I tried to solve th... |
H: Uncountably many non-homeomorphic compact subsets of the circle
As the title says, the question is whether there are uncountably many non-homeomorphic compact subsets of the unit circle.
I'm assuming this is true, but I wouldn't mind an elegant proof.
AI: For each countable ordinal $\alpha$ let $X_\alpha=\omega^\a... |
H: What is the group of endomorphisms of $\mathbb{Q}/\mathbb{Z}$
As the question says, I'm trying to work out what $End_{\mathbb{Z}}(\mathbb{Q}/\mathbb{Z})$ is. These are just group homomorphisms. But so far all I can see is that its probably enough to see where elements of the form $1/n$ map to, but some hints would ... |
H: Showing $\gcd(2^m-1,2^n+1)=1$
A student of mine has been self-studying some elementary number theory. She came by my office today and asked if I had any hints on how to prove the statement
If $m$ is odd then $\gcd(2^m-1,2^n+1)=1$.
It's been a while since I took number theory and I'm not sure what to do. She sai... |
H: $X_{n}$ converges to $X$ in distribution iff $E\{f(X_{n})\} \to E\{f(X)\}$ for all bounded $ f \in C^{\infty}$.
Let $(X_{n})_{n\geq 1}$, $X$ be $\mathbb{R}$-valued random variables. Show that $X_{n}$ converges to $X$ in distribution iff $E\{f(X_{n})\}$ converges to $ E\{f(X)\}$ for all bounded $C^{\infty}$ function... |
H: Non trivial group homomorphism from $G$ to $H$
Actual Question is to check if there is :
Non trivial group homomorphism $\eta : S_3 \rightarrow \mathbb{Z}/3\mathbb{Z}$.
What I have tried so far is :
I take $(1 2)\in S_3$, this has order $2$.
So, order of $\eta(1 2)$ should divide $2$ but then, I have no element o... |
H: Find the area of triangle
There is a square $ABCD$ of side $a$, points $E,F$ lies at centre of respectively $AB,CD$.
Line $AE$ intersect with $DF$ at $G$ and $BD$ at $H$. Find area of $DHG$.
I don't know why I can't add a comment but thanks for hint, I have already known how to do it
AI: Hint: $\text{area of }\tr... |
H: Pigeonhole problem - salvaging my solution
A student is solving combinatorics problems. Each day he solves at least one problem. He solves no more than 500 problems a year. Prove that there is an interval of days in which he solves 229 problems.
This is my approach.Lets consider partial sums. A(x) is a sum of probl... |
H: How are these definite integrals equivalent? $ \int_0^\infty B'(x)S'(t-x) dx = \int_{-\infty}^t B'(t-x)S'(x)dx $
I have been told that the following integrals are equivalent but I cannot figure out how:
$ \int_0^\infty B'(x)S'(t-x) dx = \int_{-\infty}^t B'(t-x)S'(x)dx $
Are there some rules on changing limits that ... |
H: Defining the Greatest Common Divisor using Symbolic Notation
I am trying to write the definition of greatest common divisor using symbolic notation. Here is my current attempt:
$d = gcd(m,n) \Leftrightarrow d \in Z \wedge max(d | m \wedge d | n)$
Any help or hints are greatly appreciated! Thanks!
AI: Let's try to ... |
H: Calculation of $σ_u σ_u$ and $σ_u σ_v$
Accourding to the info which I posted, how can I calculate $σ_u σ_u=\vert\vert σ_u\vert\vert^2$ and $σ_u σ_v$ I am stuck with there. Please show me. Thanks.
AI: For example:
$$\sigma_u\cdot\sigma_v:=2\cosh u\cosh v\sinh u\sinh v$$
If you really meant the dot product you jsut... |
H: Prove using mathematical induction that for every positive integer n, $\sum_{i=1}^n ( i * 2^i ) = (n-1) 2^{n+1} + 2 $
Prove using mathematical induction that for every positive integer n,
$$\sum_{i=1}^ni2^i=(n-1) 2^{n+1} + 2$$
There is what i did so far :
AI: Your solution is good, except the final form should be p... |
H: Finding Laurent series in annulus $1<|z|<2$
I want to find the Laurent series for $f(z)=\dfrac{1}{(z-1)(z-2)}$ inside $1<|z|<2$.
I can apply the formula here to get $$f(z)=\sum_{k=0}^\infty a_kz^k+\sum_{k=1}^\infty b_kz^k$$ where $$a_k=\dfrac{1}{2\pi i}\int_{|z|=2}\dfrac{f(z)}{z^{k+1}}dz$$ and $$b_k=\dfrac{1}{2\pi... |
H: Markov Chain: starting at $i$ reaching $N$ before $0$
Starting at some state $i$, we have probability of going $P_{i,i+1} = p$ and probability $P_{i,i-1} = 1-p$ what is the probability I reach N before I reach zero?
Can I convert this to a gambler's ruin problem where I start with i in bank and reach N before I rea... |
H: How many 5 digits numbers are there, whose digits sum to 22?
How many 5 digits numbers are there, whose digits sum to 22? Of course the first digit has to be larger than 0.
AI: The objects you are counting may be placed into bijection with solutions to $$x_1+x_2+x_3+x_4+x_5=22$$
such that $0\le x_i\le 9$ and also ... |
H: Root of $f(z)$ inside $|z|<1$
Let $c\in\mathbb{R}$. A non-constant function $f(z)$ is holomorphic in $|z|<2$. Suppose $|f(z)|=c$ for all $|z|=1$. Show that $f(z)$ must have a root in $|z|<1$.
I'm thinking about the maximum principle, which says $f(z)$ cannot attain a maximum inside $|z|<1$. But that still doesn't ... |
H: How many possibilities in tinyurl
Looking at tinyurl, there is anywhere from 1 digit to 7 digits of I believe 36 choices (lowercase letters a to z and digits 0 to 9)
How do I calculate mathmatically the number of permutations of the string with 1 to 7 digits and 36 characters?
thanks,
Dean
AI: $$36^1+36^2+36^3+36^4... |
H: What is the expectation of the following random variable
Let $X_1,X_2,X_3,\ldots $ be an i.i.d. sequence of uniform random variables over $[0,1]$. Define
$$N= \min \{n \geq 1:X_1+\cdots+X_n>1\}$$
Find $P\{N>n\}$ and compute $E[N]$.
AI: $S_n=X_1+...+X_n \\
P(S_n \leq t),t<n=\text{The volume between the axises and th... |
H: Meaning of $x^m + y^n = z^r$ (mod $p_1$)
I'm trying to understand how to find a counter example to the Beal Conjecture. One site (here) says that,
According to The Prime Pages, the largest primes less than $2^{32}$ are $p_1 = 2^{32}-5$ and $p_2 = 2^{32}-17$. If $x^m + y^n = z^r$ (mod $p_1$) and $x^m + y^n = z^r$ ... |
H: sequences-show that $\inf\{x_{n}: n\in \mathbf{ N}\}>0$
Knowing that $x_{n}>0$ for each $n \in\mathbf{N}$ and $x_{n}\rightarrow x>0$. Let $B$={$x_{n}:n \in\mathbf{N}$}. How could I show that $\inf B>0$??
AI: Look at the definition of limit and fix $\varepsilon = x/4$. Then there is $n$ such that for all $m \geq n$ ... |
H: Solving one equation for two unknowns
Is there a theorem that states that you need $N$ equations to solve for $N$ unknowns?
If I had the following equality,
$$x^25^y = 10125$$
isn't it possible to deduce that $x = 9$ and $y = 3$ simply by looking at prime factors?
Are there multiple solutions when there are $N$ un... |
H: Let $\varphi $:$G\to H$ be an onto group homomorphism. Show that if $K \unlhd G$, then $\varphi(K)\unlhd H$.
Let $\varphi$: $G\to H$ be an onto group homomorphism. Show that if $K \unlhd G$, then $\varphi(K)\unlhd H$.
This problem was in my abstract book, but does not explain why it is true. It is provided as an e... |
H: Stability of Analytic Continuation
Let $f(z)$ be an analytic function in an open set $U\subset\Bbb{C}$. Recall that an analytic continuation of $f$ is a pair $(F,V)$ such that $U\subset V\subset\Bbb{C}$, $F$ is analytic on $V$, and $F(z)=f(z)$ for all $z\in U$.
My question is, how stable is this process? If $\... |
H: $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$
As the title say, I want to prove that $C_c^{\infty}(\mathbb R^n)$ is dense in $W^{k,p}(\mathbb R^n)$
i.e. $\displaystyle{ W^{k,p} (\mathbb R^n) = W_0^{k,p}(\mathbb R^n) \quad (\star)}$.
In a book I found that in order to prove it, we need the follow... |
H: The range of continuous functions and the Intermediate value theorem
Let $a,b\in \mathbb{R}$, $a<b$ and let $f$ be a continuous real valued function on $[a,b].$ Prove that if $f$ is one-to-one then $f([a,b])$ is either $[f(a),f(b)]$ or $[f(b),f(a)].$
Suppose $f(a)<f(b)$ then by the Intermediate Value Theorem (IVT) ... |
H: If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?
If A is invertible, prove that $\lambda \neq 0$, and $\vec{v}$ is also an eigenvector for $A^{-1}$, what is the corresponding eigenvalue?
I don't really know where to start with ... |
H: Need a hint for sequence convergence homework
Hello and thank you for spending time to help me in advance!
The following exercise/homework problem has been giving me a difficult time and I am wondering if there is something simple that I am just not seeing at the moment.
I thought I could find the maximum sequence... |
H: Prove that $\vec{v}$ is also an eigenvector for $A^{k}$(k = a positive integer). What is the corresponding eigenvalue?
Prove that $\vec{v}$ is also an eigenvector for $A^{k}$(k = a positive integer). What is the corresponding eigenvalue?
What I have started with is, $A=(CDC^{-1})$ which can be used to prove
$A^{2}=... |
H: Proving if a function $f$ is differentiable and $f'(x)\ne0$ at all $x$, then it is one-to-one
Here's what the problem reads:
Suppose that the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0$ for any $x \in (a,b)$. Prove that $f$ must be one-to-one.
This looked easy at first, ... |
H: Expected value - random sets cardinality
I've got a set $|A|=n$ and two random subsets $B,C\subseteq A$ and $|B|=|C|=k$. What is the expected value of $|B\cap C|$? My approach: I consider $B$ given and try to fill $C$. The probability of picking no common terms ($l=0$) is $\dfrac{n-k\choose k}{n\choose k}$, because... |
H: Question about flipping terms in matrix multiplication in proving that $h(N_n(\mu , K))=\frac{1}{2}\log(2 \pi n)^n |K|$
So in my book, it is written:
Let $X_1,X_2,...,X_n$ have a multivariate normal distribution with mean $\mu$ and covariance matrix $K$ and $\textbf{X}=(X_1,X_2,...,X_n)$
The above isn't really rele... |
H: Is it true that $E[X^2]-E[Y^2] = 0?$
Suppose that $X$ and $Y$ are identically distributed and not necessarily independent. Then clearly $E[X]=E[Y]$.
But is it also the case that $E[X^2]=E[Y^2]$?
Can you think of a counter-example? If not, can you give a quick proof?
Thanks
AI: If $X$ and $Y$ have the same distribut... |
H: Uniqueness of minimal elements in a totally ordered set
Let $R$ be a total order on set $S$. Prove that if $S$ has a minimal element, than the minimum element is unique.
I have difficulties with proofs. I know any graph of a total order is a straight line, which clearly has a minimal element. How doI tell when some... |
H: Differential Equation Separation
I having trouble solving this diffeq.
$$\frac{\text{d}P}{\text{d}t} = \frac{(r(t) - B)}{z} \cdot P(t) + c\cdot w$$
,where $c\cdot w$ is a constant. Normally I would just separate but I do not think I can do that here. Any ideas?
AI: The normal method for solving $$\frac{dP}{dt} = g(... |
H: Prove for a $7\times7$ matrix that the set of all eigenvectors is linearly independent.
Suppose $A$ is a $7\times7$ matrix, $\left\{\vec{v_{1}},\vec{v_{2}}\right\}$ is a basis for $\operatorname{Eig}(A,3)$, $\left\{\vec{v_{3}},\vec{v_{4}},\vec{v_{5}}\right\}$ is a basis for $\operatorname{Eig}(A,7)$ and $\left\{\ve... |
H: How to get the garage to work. (parking functions)
At McGeorge's garage every driver has a favourite parking spot. Parking spots are arranged in a line and are numbered 1 through n. A driver always goes to his favourite spot, if it's free he takes it. If it's not he goes to the lowest unoccupied spot after his favo... |
H: Suppose a, b and n are positive integers. Prove that (a^n) | (b^n) if and only if a | b.
Suppose $a, b$ and $n$ are positive integers. Prove that $a^n\mid b^n$ if and only if $a \mid b$.
I have:
$$a^n\mid b^n$$
$$\implies b^n = a^n \cdot k$$
$$\implies \sqrt[n]{b^n}=\sqrt[n]{a^n}\cdot k$$
$$\implies a=b\cdot k$$
$$... |
H: subring of rational numbers and its ideal
Let $p$ be a prime number. For any $p$ the subring $\mathbb{Q}_p$ of of the field of rational numbers is defined:
$\mathbb{Q}_p=\{\frac{a}{b}|a,b\mbox{ are integers, $p$ does not divide $b$}\}$
Let $P$ be a subring of $\mathbb{Q}_p$ that is the set of all elements that are ... |
H: Everything in the Power Set is measurable?
Im taking a class in graduate probability. My background is in engineering (very used to math in an applied sense). I am also taking an undergraduate class in real analysis along side (should have taken it before, but I couldn't) I have a couple of questions:
We're spend... |
H: Prove that set of all lines in the plane is uncountable.
Let $L$ be the set of all lines in the plane. Prove that $L$ is uncountable, but only countably many of the lines in $L$ contain more than one rational point.
Attempt: Well, I was trying to define $L$ using linear combinations of points since a line is a li... |
H: isomorphisms- subspaces in topology
Consider the following topological spaces: $(X_1,\tau_1)=(\Bbb R,\tau_u)$ and $(X_1,\tau_2)=(\Bbb R, \tau_{kol})$
So the product topology is the following: $(\Bbb R^2, \tau_u \times \tau_{kol})$
I have to describe the subspace topology defined by each subsets.
$A=\{(x,y)\in \Bbb... |
H: Finding points on graph with tangent lines perpendicular to a line
Find all points $(x,y)$ on the graph of $y=\frac{x}{x-3}$ with tangent lines perpendicular to the line $y=3x-1.$
My thoughts on this problem:
First I should find the slope of the given line and the tangent to the given curve. I'm unsure of how... |
H: Strange results on removing lim in wolframalph
Can anyone explain this wolframalpha result?
$ f(x)=\lim\limits_{h \to 0} \frac{-1}{(3x-2)^2} = \frac{-1}{(2-3x)^2}$
[lim ((-1)/((3x-2)^2)) as h->0] = [((-1)/((2-3x)^2)) ]
While this is not equal:
$ f(x)=\lim\limits_{h \to 0} \frac{-1}{(3x-2)^2} = \frac{-1}{(3x-2)^2}$... |
H: Prove that $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex
At this link there is a demonstration that for $f$ continuously differentiable on $C \subseteq \mathbb{R}^n$ convex, $f(x+h)-f(x) - \langle\nabla f(x), h\rangle\geq 0 \Rightarrow f $ convex. This argument uses an intermediate step r... |
H: Real Analysis: Show that the Taylor expression converges at every point.
Suppose that the function $F:\mathbb{R}\to\mathbb{R}$ has derivatives of all orders and that
\begin{cases}
F'(x)-F(x)=0, & \text{for all $x$} \\
F(0)=2.
\end{cases}
Find a formula for the coefficients of the $n$th Taylor polynomial for F at $... |
H: Prove that $f g$ is differentiable at $x_0$.
$\newcommand{\R}{\mathbb{R}}$Let $f,g : \R \to \R$. Let $f(x_0) = 0$, $f(x)$ differentiable at $x_0$ and $g$(x) continuous at $x_0$.
I need to prove that $fg$ is differentiable at $x_0$.
Any ideas or hints about how to begin?
Continuous doesn't mean differentiable... so ... |
H: about possion gamma and exponential distribution
about possion gamma and exponential distribution
can someone explain how to sub in the numbers?
i tried subbing in numbers and the outcome its not the same as the answer.
maybe I'm not doing it correct .
I'm getting 1/2 e^-1.5 for the first question .
AI: Exponen... |
H: Prove whether or not $H$ is a subgroup of $S_n$
$H$ is the set of permutations where $H$ = {$ID_{S_n}$,(12),(34),(12)(34),(13)(24),(14)(23),(1432),(1234)}.
Is $H$ a subgroup of $S_4$?
Is there a simpler way to do this than checking for combinations that may not be closed under the operation? (composition is the ope... |
H: Determining the Coordinates of the point on the x-axis that are equidistant
So I've been doing my homework without any trouble so far and I came across this question that I did not understand how to do.
Given the points A(-2,1,3) and B(4,-1,3), determine the coordinates of the point on the x-axis that are equidista... |
H: Simplifying this logarithm series
$$\sum_{i\; =\; 2}^{99}{\frac{1}{\log _{i}\left( 99! \right)}}$$
How would you evaluate (or at least simplify) this logarithm series?
AI: Note that $\log_a b = \frac{\log b}{\log a}$, for any choice of base of the logarithm. (Mathematicians tend to mean the natural logarithm when ... |
H: Numerical integration of $\int_0^2 \frac{1}{x+4}dx.$
I have homework problem. Determine the number of intervals required to approximate
$$\int_0^2 \frac{1}{x+4}dx$$ to within $10^{-5}$ and computer the approximation using (a) Trapezoidal rule, (b) Simpson's rule, (c) Gaussian quadrature rule. I think the phrase "w... |
H: Cauchy Sequence if $|s_{n+1} - s_n| < 2^{-n}$
Let $s_n$ be a sequence such that
$|s_{n+1} - s_n| < 2^{-n}$
for all $n \in N$. Prove $s_n$ is a Cauchy sequence and hence a convergent sequence.
Here's what I've started with:
Proof:
Take $\epsilon > 0$. Let $N = -\log_2{\epsilon/2}$. Thus,
$ |s_{n+1} - s_n| < 2^{\lo... |
H: Euler Totient Issues
I was skimming again through Dummit & Foote's Abstract Algebra and I came across this exercise:
Prove that for any given positive integer $N$ there exist only finitely many integers $n$ with $\varphi(n)=N$, where $\varphi$ denotes Euler $\varphi$-function. Conclude in particular that $\varphi(n... |
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