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H: Why are the rows of a parity-check matrix linearly independent?
I've been reading about error-correcting codes, and I came across the following definition for a parity-check matrix:
''There is an $(n-k) \times n$ matrix $H$, called a parity check matrix for an $[n,k]$ linear code $\mathcal{C}$, defined by
$$\math... |
H: Partial derivatives chain rule
Let $ w=\frac{x^2}{y} $ with $ x=e^{-u^2}u $ and $ y=e^{-u^2}v $
$ \frac{\partial w}{\partial x} = (x^2)'\frac{1}{y} = \frac{2x}{y} $
$ \frac{\partial w}{\partial y} = x^2(\frac{1}{y})' = -\frac{x^2}{y^2} $
$ \frac{\partial y}{\partial u} = (e^{-u^2})'v = e^{-u^2}(-u^2)'v = -2e^{-u^2... |
H: Proving $(x^s)^t=x^{st}$
Suppose $x\in G$ and $|x|=n<\infty.$ If $n=st$ for some positive integers $s$ and $t$, prove that $|x^s|=t.$
Observe that if we prove that $(x^s)^t=x^{st}$ then $1=x^n=x^{st}=(x^s)^t$ and we are done. We did not prove that $(x^s)^t=x^{st}$ in class and it is another exercise in the book. ... |
H: Prove that any finite group $G$ of even order contains an element of order $2.$
Prove that any finite group $G$ of even order contains an element of order $2.$[ Let $t(G)$ be the set $\{g\in G|g\neq g^{-1}\}$. Show that $t(G)$ has an even number of elements and every nonidentity element of $G-t(G)$ has order $2$.]... |
H: How is this angle relation true?
Either I'm silly and I'm missing something very simple, or my text book is incorrect. I'm trying to verify a line in the text book which claims that sin(a) = s/r. I can't seem to prove this to myself and its infuriating.
AI: Let $x$ be the angle (in the triangle) that is opposite t... |
H: $f \in L^p(\Omega) \Rightarrow$ that exist a ball $B_R \subset \mathbb{R}^n$ that $\int_{\Omega \backslash B_R}|f|^p < \varepsilon$.
Let $f \in L^p(\Omega)$ and $\varepsilon >0$. Show that exist a ball $B_R \subset \mathbb{R}^n$ such that $\int_{\Omega \backslash B_R}|f|^p < \varepsilon$.
AI: Hint: $$\int_\Omega = ... |
H: Every odd composite $=$ prime ${}+ 2x^2$
I was looking through some project-euler questions and I came across one that said
Every odd composite number can be written as the sum of a prime and twice a square...This was proven false.
So, I solved the problem through brute-force in Java but I was wondering how I ca... |
H: How to prove the boundedness of the solutions of a nonlinear differential equation
I have the following differential equation:
$$
\ddot{x} = -\log(x) - 1
$$
and I need to prove that every solution of this equation is a bounded function. From the phase plane portrait, it is obvious that this is true:
How can I cons... |
H: Number of triples $(a,b,c)$ of positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{4}$?
What is the number of triples $(a,b,c)$ of positive integers such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{3}{4}$ is:
A) $16$
B) $25$
C) $31$
D) $19$
E) $34$
Note: Through trial and error I've conc... |
H: When can a series be integrated term by term?
I have a function that is defined as a harmonic series, I would like to integrate it over part of its domain. I have been doing this by integrating term by term and summing the result, but I seem to remember something in little Rudin that gave conditions under which th... |
H: what is the usage of combination $C(r,k)$ where $r$ extends to real number?
Combination is defined as $C(n,k) = \dfrac{n!}{k!(n-k)!}$, where $n$ and $k$ are non-negative integers.
Now, the definition can be extended to $C(r,k)$, where $r$ is real number and $k$ is an integer:
$$
C(r, k) = \cases{
\frac{r(r-1)\cdots... |
H: A basis for some topology on $\mathbb{R}^2$
I know that the collection
$$\mathcal{B}= \{(x,y): a<x\leq b, c<y\leq d\}$$
is not a basis for the standard topology on $\mathbb{R}^2$, but the collection of open rectangles in the plane certainly is. For which topology on $\mathbb{R}^2$ is $\mathcal{B}$ a basis? Would i... |
H: how can one find the value of the expression, $(1^2+2^2+3^2+\cdots+n^2)$
Possible Duplicate:
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
Summation of natural number set with power of $m$
How to get to the formula for the sum of squares of first n numbers?
how can one find the value of the expre... |
H: Whats the probability that a set of dunif random variables is strictly ordered?
Let $\{X_i\}$ be $n$ iid dunif(0, u) (discrete uniform) random variables with u>n. How do I compute the probability that $\{X_{i+1}\}$ > $\{X_i\}$ for all i?
AI: Because of a comment by the OP, I will interpret the uniform as having p... |
H: Diameter of coins with a given height and source volume?
We need to make $1.15\times10^5$ coins with a thickness of $2.0\times10^{-3}m(2mm)$ from a $0.12m^3$ block of silver. I'm trying to calculate what diameter the coins will end up being.
Did I set up the equation correctly where I'd get my answer from solving f... |
H: Show that the integral of this rational function is equal to an infinite alternating harmonic series
One of my friends gave me the following question from his review, I have little experience to dealing with these types of questions in Analysis so if you could help us just to get started it would be quite helpful.
... |
H: How can I solve for $n$ in the equation $n \log n = C$?
Believe it or not, this isn't homework. It's been many years since grade school, and I'm trying to brush up on these things. But my intuition isn't helping me here.
AI: If what you want is to solve for $n$, there is no simple way. The solution has $n=e^{W(C)}... |
H: Disconnectedness of the rationals with the subspace topology
I have tried to prove that $\mathbb{Q}\subset \mathbb{R}$ equipped with the subspace topology is a disconnected space. I'd like to make sure my attempted proof is correct since topological properties of $\mathbb{Q}$ seem like nice things to be familiar w... |
H: Why is $a^n - b^n$ divisible by $a-b$?
I did some mathematical induction problems on divisibility
$9^n$ $-$ $2^n$ is divisible by 7.
$4^n$ $-$ $1$ is divisible by 3.
$9^n$ $-$ $4^n$ is divisible by 5.
Can these be generalized as
$a^n$ $-$ $b^n$$ = (a-b)N$, where N is an integer?
But why is $a^n$ $-$ $b^n$$ = (a-b... |
H: Calculating probabilities in two overlapping continuous uniform random variables (with an added constraint)
Given X~unif(a, b) and Y~unif(c, d) with a < c < b < d.
What's the probability that Y>X and Y being realized in the interval (c, b)?
AI: Here we assume that $X$ and $Y$ are independent random variables with u... |
H: Calculate the border width?
I tried this with an area of $157m^2$ but got the wrong answer of $0.7334m$ when the correct answer was $1.138m$. Now I'm trying it with $140m^2$ I wanted to know what might be the issue with my steps below?
A garden plot must have a central planting area of length $13m$ and width $8m$. ... |
H: Intersection of infinitely many sets is the empty set
Problem:
For each natural number $n$, we define: $A_{n}=\left \{ (n+1)k:k\in \mathbb{N} \right \}$.
It seems obvious to me that $\bigcap_{n=1}^{\infty }A_{n}=\varnothing $ where $\varnothing$ is the empty set. However, I couldn't come up with a proof for that. A... |
H: How to compute the series which seems different with the questions I raised before?
How to compute the series which seems different with the questions I raised before?
$\lim_{N\rightarrow\infty}N\sum^{N}_{k=2}\left(\frac{k-1}{N}\right)^{N^2}$
AI: The biggest term in the sum $\sum_{k=2}^N((k-1)/N)^{N^2}$ is the one ... |
H: If prime $p \equiv 1\pmod 4$ and $b = ((p-1)/2)!$ then show that $b^2 \equiv -1\pmod p$.
This question is from Victor Shoup's book on number theory chapter 2. The problem statement is as mentioned in the title of the question. I haven't been able to crack this one till now. I focused on solving this using the follo... |
H: Applying a multidimensional variant of Taylor's Theorem
Given $f:R^n\rightarrow R^n$ continuously differentiable in some convex open set $D$ and $x,x+p\in D$, taylor's theorem is given as:
$$f(x+p)=f(x)+\int_0^1 J(x+tp)p \, dt,\text{ where }J\text{ is the jacobian of }f.$$
I'm trying to follow a given proof which... |
H: Calculating which $(2^x,2^y)$ coordinates intersect diagonally ($45^{\circ}$) with an arbitrary rectangle in 2D space.
Calculating which $(2^x,2^y)$ coordinates intersect diagonally ($45^{\circ}$) with an arbitrary rectangle in 2D space. The rectangle in this case is your computer monitor.
Consider navigating a 2D ... |
H: Prove an inequality involving positive real numbers Lagrange multipliers
This is from Apostol's Calculus, Vol. II, Section 9.15 #11:
Find the maximum of $f(x,y,z)=\log x + \log y + 3 \log z$ on that portion of the sphere $x^2+y^2+z^2=5r^2$ where $x>0,\,y>0,\,z>0$. Use the result to prove that for real positive num... |
H: How to find the laplace transform of $ |\sin (t)| $?
How to find the Laplace transform of $ |\sin (t)| $?
AI: $$
\mathcal{L}_s\left(\left| \sin t \right|\right) = \int_0^\infty \left|\sin(t)\right|\mathrm{e}^{-s t} \mathrm{d} t = \sum_{n=0}^\infty \int_{\pi n}^{\pi (n+1)} \left|\sin(t)\right|\mathrm{e}^{-s t} \m... |
H: Power series for the sawtooth wave
This wikipedia article described a Fourier expansion of the sawtooth wave. Does this wave have a power series expansion (around any point)? If so, what is it? Does every function with a Fourier expansion also has a power series expansion?
AI: If the sawtooth wave is defined by, sa... |
H: Finding equilibrium points of a differential equation
OK, this is a weak question, but I'm a bit rusty on differential equations, so here it is:
I need to find the general solution of x'=ax+3 and then find the equilibrium points as well as which points are equilibria sinks and sources (a is a parameter).
I found t... |
H: Proof that if a set $S$ has a proper subset which is infinite, then $S$ must be infinite.
I have always found issues providing formal proofs for theorems. This time I'm stuck with this proof. I know my question might seem quite naive to most of you but please help me out with constructing a formal proof for the abo... |
H: Defining the notion of a derivative on fields endowed with a metric
If we have a field $F$ and a metric $d$ on $F$, given a function $f:F \rightarrow F$, we could define the derivative of the function at $x \in F$, as being the limit (if it exists) $$\lim_{h \rightarrow 0} (f(x+h)-f(x))h^{-1},$$ where a limit is de... |
H: Solving a single simultaneous equation with 2 unknowns
I was thinking about how I might go about solving a problem of, given an ATM that dispenses $50's and $20's, find all the combinations of notes dispensed that would be valid for a given amount.
Say, for $160, that could be 8 $20's or 2 $50's and 3 $20's.
Now, I... |
H: Each automorphism of the matrix algebra is inner.
Exercise I.II.IV in the book Local Representation Theory by J.L.Alperin:
Demonstrate that any automorphism of the algebra $M_n(k)$ is inner by using the fact that $M_n(k)$ has a unique simple module.
I want only hints, as this seems elementary. Thanks for your... |
H: Given a directed set, how do I construct a net with positive values that converges to $0$.
Suppose I have a directed set in which there is no maximal or equivalently maximum element. Is there a way to construct a net based on that directed set, of strictly positive real numbers that converges to $0$? This would b... |
H: I know that, $S_{2n}+4S_{n}=n(2n+1)^2$. Is there a way to find $S_{2n}$ or $S_{n}$ by some mathematical process with just this one expression?
$S_{2n}+4S_{n}=n(2n+1)^2$, where $S_{2n}$ is the Sum of the squares of the first $2n$ natural numbers, $S_{n}$ is the Sum of the squares of the first $n$ natural numbers.
wh... |
H: Taylor Series expansion when derivative is 0/0
I have a tutorial question on Taylor series.
$$f(x,y,z) = e^{(\sqrt{x y}+\sqrt{x z}+\sqrt{z y})}$$
I managed to get the Taylor series at (1,1,1)
Is it possible to get the expansion at (0,0,0)? Because at (0,0,0) the derivative is 0/0 which is confusing...how do we eval... |
H: Real number in terms of infinite series
This is just continuation of my previous post.
$$ A = a_0 + \left(\frac{1}{a_1}\right)^k + \left(\frac{1}{a_2}\right)^k + \left(\frac{1}{a_3}\right)^k +\ldots$$
Where $i\ge1$ and the recurrence relation $a_{i+1}\ge a_i\ge2$.
I just raised for each term by $k$. here $k\ge1$ a... |
H: $P^1$ not a regular level surface of a $C^1$ function on $P^2$
I'm working through the first chapter of Morris Hirsch's "Differential Topology". On Chapter 1, section 3 exercise 11, I encountered the following question.
"Regarding $S^1$ as the equator of $S^2$, we obtain $P^1$ as a submanifold of $P^2$ (Hirsch use... |
H: What is the $(\lg n)$-th root of $n$?
I am looking for the answer of the $(\lg n)$-th root of $n$, that is, $\sqrt[\lg n]{n}$. What is the answer and what log property should I use here? Please assume base as $2$ and $n$ as a natural number.
AI: Take the $\log$ of this expression, to get
$$\log (n^{1/\log(n)}) = \f... |
H: Limsups of nets
The limsup on sequences of extended real numbers is usually taken to be either of these two things, which are equivalent:
the sup of all subsequential limits.
The limit of the sup of the tail ends of the sequence.
For the situation with nets, the same arguments guarantee the existence of the above... |
H: Continuity of a function involving dot product
Let $\lambda \in \mathbb{R}$ and Let $A$ be a linear operator on $\mathbb{R}^3$. Define $f:\mathbb{R}^3 \to \mathbb{R}$ by $f(x):=\langle A(x),x\rangle- \lambda \langle x,x\rangle $ where $\langle,\rangle$ is the usual dot product on $\mathbb{R}^3$.
Does $\lim_{x\righ... |
H: Overriding order relation while looking for lower bounds
Let $(\mathbb Z \times \mathbb Z^*, \pi)$ be a poset defined as follows:
$$\begin{aligned} (a,b) \pi(c,d)\Leftrightarrow (a,b) = (c,d) \text{ or } r(a,b) <r(c,d)\end{aligned}$$
whereas $r(x,y)$ is the reminder of the division of $x$ by $y$.
Let $X=\{(3,2),(1... |
H: Calculating :$((\sqrt{3} + \tan (1^\circ)).((\sqrt{3} +\tan(2^\circ))...((\sqrt{3}+\tan(29^\circ))$
What is the easiest way to calculate :
$$(\sqrt{3} + \tan (1^\circ)).((\sqrt{3} +\tan(2^\circ))...((\sqrt{3}+\tan(29^\circ)) $$
AI: $\sqrt3+\tan(30^\circ-x)=\sqrt3+\frac{\frac{1}{\sqrt3}-\tan x}{1+ \frac{1}{\sqrt3}\t... |
H: Proof With and Without Truth Tables
$(6)$
Use truth tables to determine whether or not the following argument is correct:
"If the tax rate and the unemployment rate both go up, then there will be a recession.
If the GNP goes up, then there will not be a recession. The GNP and taxes are both going
up. Therefore, the... |
H: When would this employee retire?
I cant seem to solve this question
A certain company retirement plan has "a rule of 70" provision that allows an employee to retire when the employees age plus years of employment with the company total at least 70. In what year could a female employee hired in 1986 on her 32and bi... |
H: Is it possible to express $e$ in terms of $\pi$ algebraically and vice-versa?
Am I right in thinking this is not possible since both are known to be transcendental?
Also, $e^{i\pi}+1=0$ suggests this is not possible - we can not isolate $e$ or $\pi$ from this since it involves taking a log at some point, thus "canc... |
H: Fréchet derivative
I have been doing some self study in multivariable calculus.
I get the geometric idea behind looking at a derivative as a linear function, but analytically how does one prove this?
I mean if $f'(c)$ is the derivative of a function between two Euclidean spaces at a point $c$ in the domain... then ... |
H: Is $x_n(t) = t^n$ for $n\ge 1$ dense in $L^1[0,1]$?
My question concerns the following problem
Let $x_n(t) = t^n$ for $n\ge 1$. Is $\mathrm{span}(x_n; n\ge 1)$ dense in $L^1([0,1])$?
By Weierstrass' approximation theorem, it suffices to check whether the constant function $1$ is contained in the $L^1$-closure of ... |
H: Use the definition of "topologically conjugate" to prove that $F(x)=4x^3-3x$ is chaotic.
Let $V$ be a set. $f: V \rightarrow V$ is said to be chaotic on $V$ if
1) $f$ has sensitive dependence on initial condition, i.e. there exists $\delta>0$ such that, for any $x\in V$ and any neighborhood $N$ of $x$, there exists... |
H: Inequality $(a+\frac{1}{b})^2+(b+\frac{1}{c})^2+(c+\frac{1}{a})^2\ge 16$
For every real positive number $a,b,c$ such that $ab+bc+ca=1$, how to prove that:
$$\left(a+\frac{1}{b}\right)^2+\left(b+\frac{1}{c}\right)^2+\left(c+\frac{1}{a}\right)^2\ge 16$$
AI: This is problem J57 of mathematical reflections, vol.2007, i... |
H: How ${\left(\frac 12 \right)}^{{\lg n}}$ = ${\frac 1n}$ for any natural number $n$?
Consider binary logarithm . How is the value of ${(\frac 12 )}^{{\lg n}}$ = ${\frac 1n}$?
I was going through this video of skiplists and the professor at 53:22 seconds make this claim .
AI: $$
\left(\frac{1}{2}\right)^{\lg n} = \... |
H: What percent of Juan's income is Mary's income?
For the following question
Mary's income is 60% more than Tim's income , and Tim's income is 40% less than Juan's income. What percent of Juan's income is Mary's income? (Ans=96%)
I am attempting to solve this using the following two equations, but I can't figure ... |
H: Whether there exists or not, a group of $2$ elements $a$ and $b$ with $\lvert b \rvert = 2 , \lvert a\rvert = 15$ and ...
Does there exist a group with two elements $a$ and $b$ with $\lvert b\rvert = 2, \lvert a\rvert = 15$ and $bab = a^4$?
AI: Yes. There is such a group. It is a semi-direct product of cyclic grou... |
H: Proving a certain determinant $\left|\det A\right|$ is complete square
Consider the following matrix
$$
A_{ij}=
\begin{cases}
1\quad\text{ if }\space (i+j)\space\text{ is prime,}\\
0\quad\text{ otherwise.}
\end{cases}
$$
How can one prove that $\left|\det A\right|$ is a complete square?
AI: It was a problem posed a... |
H: Line integral of $\sin x$.
How to evaluate:
$$\int \sqrt{1+\cos^2x} dx$$
Is a simple antiderivative known to exist?
AI: A simple antiderivative is given by
$$\tag{1}
\int_0^x\sqrt{1+\cos^2t}\,dt.
$$
This is an example of what is called an elliptic integral and it cannot be expressed in terms of other well-kno... |
H: How do we define the Lie bracket on the complexification $\mathfrak{g} \otimes_\Bbb{R} \Bbb{C}$?
I already know how to do the complexification of a real Lie algebra $\mathfrak{g}$ by the usual process of taking $\mathfrak{g}_\Bbb{C}$ to be $\mathfrak{g} \oplus i\mathfrak{g}$. Now suppose I take the approach of tryi... |
H: Reduced Gröbner Basis
If have computed this Gröbner Basis with Buchberger's algorithm for Degree-Lexicographic-Ordering:
$$\{ x^²y+x+1,xy^2+y+1,x-y \} $$
I want to to transform it into a unique representation form called Reduced Gröbner Basis. Therefore I remove the leading powers which are divided by other leading... |
H: $n$-th digit in the sequence of natural numbers
What's the $n$-th digit in the sequence $S$ of numbers formed by the natural numbers, i.e.,
$n$-th digit in the sequence 1 2 3 4 5 6 7 8 9 10 11 12... ? For example 11th digit in the sequence is 0.
AI: The first step is to find what decade you are in. There are $9$ ... |
H: For $n=pq$, show that there exist $\alpha, \beta \in Z_n^*$ such that $\alpha, \beta \notin (Z_n^*)^2$ and $\alpha \cdot \beta \notin (Z_n^*)^2$.
For $n=pq$, where p,q are distinct odd primes show that there exist $\alpha, \beta \in Z_n^*$ such that $\alpha, \beta \notin (Z_n^*)^2$ and $\alpha \cdot \beta \notin ... |
H: Solve $HG^T=0$ for matrix G
I have no idea how to get matrix $G$ out of $ H $:
$$ HG^T = 0. $$
I apologize for such an elementary question, but I have completely forgotten how to solve such problems. Does anybody know anything for a quick revision on solving matrix equations? Thank you in advance.
AI: The notation ... |
H: Calculating the chess problem, sum $\sum_{k=0}^{63}2^{k}$
Possible Duplicate:
Summation equation for $2^{x-1}$
I'm solving the classic problem of the inventor of chess, who according to legend sold the invention for one grain for the first square of the board, 2 for the second, 4 for the third, 8 for the fourth,... |
H: Quotient ring of a polynomial ideal with two variables
Given an ideal $I = \langle x-y,y^3+y+1 \rangle \subset \mathbb{C}[x,y]$ (this is a Gröbner basis w.r.t. degree-lexicographic order). I want to write $\mathbb{C}[x,y]/I$ as a $\mathbb{C}$-Basis and determire $\operatorname{dim}_{\mathbb{C}}\mathbb{C}[x,y]/I$. I... |
H: Name of this division property
Let us take two integers, $a$ and $b$. Let us then take $\lfloor a / b\rfloor = c$ and $a \bmod b = d$. Obviously, it follows that $a = bc + d$. Our professor claimed that this was called the "Fundamental Theorem of Arithmetic" at his high school. Clearly, this isn't the case anymore,... |
H: Lebesgue theory and axiom of choice
I have been told that the existence of non-Lebesgue-measurable sets on $\mathbb R$ is impossible without axiom of choice. Do any other well-known results in Lebesgue theory depend on the axiom of choice?
I have been told of Radon-Nikodym theorem in particular as a theorem which ... |
H: Is every group a Galois group?
It is well-known that any finite group is the Galois group of a Galois extension. This follows from Cayley's theorem (as can be seen in this answer). This (linked) answer led me to the following question:
What about infinite groups?
Infinite groups appear as Galois groups on infini... |
H: Derivatives as defined by a two variable difference quotient limit
I imagine this is basic analysis material which someone who isn't me knows off the top of their head:
Given a function $f$, we may define the difference quotient $q(x, y) = \frac{f(y) - f(x)}{y - x}$. The derivative of $f$ at some point $x$ is of co... |
H: Maximizing the expectation of a function with a constraint
I am interested in computing the maximum of the following function
$E[ ((XB) - E(XB))^TR ] $ subject to the constraint $E(XB)=1$
where $X$ is a $n\times m$ random matrix, $B$ is a $m \times 1$ variable vector, and $R$ is a $n\times 1$ random vector. $T$ is... |
H: Queues with arrivals of random sized groups
I am trying to solve the following problem.
Customers arrive at a service facility in groups. The groups arrive according to a homogeneous Poisson process at the rate of 5 per minute. The number N of individuals in each group is described by the distribution
$P_N(1) =0.3$... |
H: Definition of vector bundle
Everywhere i see definition of vector bundle as triple $(E, p, B)$, $B$ and $E$ are manifold and local trivialization condition holds. For example see the definition here. .
Local trivialization gives manifold structure on $E$, then why initial assumption on $E$ that $E$ is manifold... |
H: Norm of matrix $A$ with the Euclidean norm
Suppose that there is matrix $A = \begin{bmatrix} 3 & 2 \\ 2 & 0 \end{bmatrix}$. We want to figure out what $$\max_{||\hat{\mathbb{x}}|| = 1}||A\hat{\mathbb{x}}||$$ is with the Euclidean norm. (where $\hat{\mathbb{x}}$ is $\frac{1}{||\mathbb{x}||}\mathbb{x}$.)
Answer is st... |
H: Deformation retraction induced on quotient space
I'm trying to show that a punctured torus deformation retracts to a wedge of 2 circles. So I considered a punctured solid square (which eventually becomes a torus after identification of opposites sides), and make it deformation retract onto its border (the wedge of ... |
H: Determining when $1^N, 2^N, \ldots, k^N$ are divisors of $k!$
Suppose $N$ is a large integer and let $k$ be an integer s.t. $k > N$. Under what conditions can we conclude that $1^N, 2^N, \ldots, k^N$ are all divisors of $k!$?
AI: Trivially, the result is true for $N=1$, so consider the case where $N\geq 2$.
Suppose... |
H: Matrix Norm Inequality proof
I came across this inequality
$$\|ABC\|_F \leq \|A\| \|B\|_F \|C\|$$
for all matrices $A$, $B$ and $C$, where $\|\cdot\|$ is the operator norm (max singular value)
I do not know how to prove this, nor do I have it accessible. Is there a proof available online?
AI: The key fact is that $... |
H: Throwing the dice, sum of the points
We are throwing the die (original cube for the board games). How many are ways to get the sum of the points equal to $n$ ?
I've heard this problem today in the morning and still can't deal with it, which is tiring. The only way I see it, is that I am looking for the number of ... |
H: $N=\langle g^2\mid g \in G \rangle$ is normal in $G$
Let $G$ be a group and let $N=\langle g^2\mid g \in G \rangle$.
Show that $N \lhd G$ and $G/N $ is abelian.
Well, I've come up with a simple approach:
If $ n=x^2 \in N$ for some $ x \in G$ then $n^g = gng^{-1}=(gxg^{-1})^2 \in N$ and from here it is simple to... |
H: differentiable functions
Let $E$ be a closed subset of $\mathbb R$. I proved that there is a real continuous function $f$ on $\mathbb R$ whose zero set is $E$. Is it possible, for each closed set $E$, to find such an $f$ which is differentiable on $\mathbb R$, or one which is $n$ times differentiable, or even one w... |
H: What am I losing if I decide to perform all math by computer?
I solve mathematical problems every day, some by hand and some with the computer. I wonder: What will I lose if I start doing the mathematical problems only by computer? I've read this text and the author says that as techonology progress happens, we sh... |
H: Proving something is the basis of a quotient space
Let $k$ be a field which does not have characteristic 2. Let $M$ be the free $k$-vector space generated by two elements $\{ c, x \}$. Let $T(M)$ be the tensor algebra of $M$ and let $I$ be the ideal generated by the elements $$\{c^2 -1, x^2, cx + xc \}.$$
I am tryi... |
H: What is a non-degenerate module?
I know what a non-degenerate bi-linear form is, but what does it mean for say a left $R$-module $M$ to be non-degenerate? (Here $R$ is a ring without unit$)
I came across a module being called non-degenerate studying representation theory.
AI: Here is the definition which I have see... |
H: Why do we need to learn integration techniques?
After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc.
Integral calc started out okay, learning about Riemann sums and the Fundamental Th... |
H: Analysis Question regarding absolute value
If $a<x<b$ and $a<y<b$ show that $|x-y| < b - a$.
I am guessing that I can say y
$y-a<b-a$
Also we can say that $a-x<0$
So if we add these, we get:
$y-x<b-a$
WLOG, $x-y<b-a$ so $|x - y| < b - a$
Does this look ok or am I missing some steps?
Thank you for your help!
AI: W... |
H: A question dealing with Lindelöf spaces
I've been going over previous exams that I've had in topology to study for my comprehensive exam, and I noticed a problem that I missed. I was wondering if anyone could help me out with this problem:
Let $X$ be a Lindelöf topological space and let $A \subset X$. Suppose that ... |
H: Lemma from Gilbarg/Trudinger
This is from Gilbarg/Trudinger "Elliptic Partial Differential Equations of Second order" Lemma 4.1 (p.54 of the third edition) The lemma states that for a bounded integrable function $f$, the newtonian potential is $C^1(\mathbb{R}^n)$.
Basically, I don't really get how this inequalit... |
H: Probability of landing on a particular point in an infinite 1D random walk
$50\%$ of the time you walk right one unit, and $n$ units to the left o.w. The probability in question is ever landing one unit to the right of where you start at (as your number of moves goes to infinity).
I've read somewhere that the answ... |
H: How do you simplify fractions that have exponents?
I'm not sure how to simplify a fraction with a large exponent for example:
$$\frac{2^{2001} \cdot 3^{2003}}{6^{2002}}$$
AI: To simplify a fraction with powers in the numerator and denominator a possible method is to factor each power base into prime factors. With
p... |
H: Minimal rotation to avoid collision of circles
The discriminant of a certain quadratic equation which determines when two circles (hyperspheres?) will collide, provided they travel with a constant velocity is
$$(\Delta v\cdot\Delta s)^2-(\Delta v\cdot\Delta v)\times(\Delta s\cdot\Delta s-r^2)$$
where $\Delta v$ is ... |
H: Analysis e-neighborhoods
Show that if $a,b \in \mathbb{R}$, and $a \not= b$, then there exist e-neighborhoods $U$ of $a$ and $V$ of $b$ such that $U \cap V = \varnothing$
I am thinking that I may need to use the triangle inequality but I am not sure really how to get started. Is this the appropriate thing to do?
Th... |
H: In $C[0,1]$ prove that the subset of Lipschitz functions is dense
In $C[0,1]$ prove that the subset of Lipschitz functions is dense. I can't prove it.
AI: No need to pull out the heavy guns.. Suppose $f(x) \in C[0,1]$. For a given $n$ let $f_n(x)$ be the piece-wise linear function whose graph connects $(0,f(0))$ to... |
H: Which separation axiom?
Let $X$ be a topological space. Assume that for all $x_1,x_2 \in X$ there exist open neighbourhoods $U_i$ of $x_i$ such that $U_1 \cap U_2 = \emptyset$. Such a space, as we all know, is called Hausdorff. What would we call a space, and which separation axioms would the space satisfy, if $\ov... |
H: limit on infinity of trigonometric function Sin x
i am make the follow limit with definition limit $\epsilon - \delta$
$ \lim\limits_{x\rightarrow\infty}{(x + 10\sin x )}$
but i don't know find a $A\in \mathbb R$ that $\forall_{x>A}$ $f(x)>M$ for all $M>0$
how is the treatment with this functions ?
AI: Note ... |
H: Integration Substitution Problem 2
Let us assume a definite integral: say, $\int_0^{\frac{\pi}{2}} \frac{dx}{5+4 \sin x}$. Normally by substitution of $\tan\frac{\theta}{2}$ as t I can prove the answer is $\frac{1}{3} \ln 2$. But suppose I do not know any mathematics and I am foolish enough to substitute $(\sin x+\... |
H: "Sum" over logical and?
Given a continuous sequence of integers $(a, a+1, a+2, \dots, b)$ I want to write:
$P_a \wedge P_{a+1} \wedge P_{a+2} \wedge \dots \wedge P_b$
Where $P_i$ is some logical statement parametrized by $i$
I have been writing:
$\forall_{i=a}^{b}P_i$
for this.
Is there a more correct and commonly ... |
H: What is '+' in set notation
In set $T = \{(0+2)^{i+5}|\space i \in \mathbb{N}\}$, what is the meaning of the $(0+2)^{i+5}$?
AI: A likely possibility, given your comment, is that $(0+2)$ is to be interpreted as a regular expression, namely "either 0 or 2" and the apparent exponentiation would be read as concatenatio... |
H: How do i calculate the probability of erroneous transmission?
In information theory, how do I calculate the probability of an erroneous transmission? Let's take for instance a binary symmetric channel with an error probability $ 1-d=0.25 $ and send codewords of length 6 coded in a Hamming code able to correct up to... |
H: How I can prove this? (fixed points of $f(x)=(x^3+1)/3$)
the function f defined by $f(x)=(x^3+1)/3$ has three fixed points say α,β,γ where
$-2<α<-1$, $0<β<1$, $1<γ<2$.
For arbitrarily chosen $x_{1}$, define ${x_{n}}$ by setting $x_{n+1}=f(x_{n})$
If $α<x_{1}<γ$, prove that $x_{n}\rightarrow β$ as $n \rightarrow \in... |
H: Statistic: calculating error of order of two sequences of objects
I am trying to derive a meaningful statistic from a survey where I have asked the person taking the survey to put objects in a certain order. The order the person puts the objects is compared to a correct order and I want to calculate the error.
For ... |
H: Exponentials: Don't understand the simplification step
Can you explain how they go from the first line to the second? More specifically, what are the steps for making the negative exponential terms positive? i.e., how does the third term: $-e^{j(4\pi t + \pi/3)}$ become $e^{-j(2\pi /3)} e^{j4\pi t}$?
\begin{align*}... |
H: Tensor calculation on mean curvature flow
I have two questions about tensor calculation.
First question : In the book, Lectures on mean curvature flows written by Xi-Ping Zhu, there exists the equaility $g^{mn} \nabla_m \nabla_n h_{ij} = g^{mn} \nabla_m \nabla_i h_{jn}$. I do not understand this.
The situation is... |
H: Kelly criterion for multiple gambles
Kelly says you should invest $x\%$ of your bankroll in a gamble:
$$x = \frac{pE-1}{p-1}$$
where $p$ is the probability of winning and $E$ is the expected payoff multiplier if you win (i.e. $E$ times how much you bet).
But lets say you have two options to invest in: One that has ... |
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