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H: Best method to determine if a first order formula is logically valid?
I know there isnt a "standard method" to determine if a formula is logically valid, but i would like to know if there is something you can always try first of all to determine it.
Not all formulas look obviously valid like $\forall x P(x) \righta... |
H: Let $|\langle u,v\rangle|=\|u\| \cdot \|v\|.$ How to show that $u,v$ are linearly dependent?
Let $|\langle u,v\rangle|=\|u\|\cdot \|v\|.$ How to show that $u,v$ are linearly dependent?
Without loss of generality let $u\ne0,v\ne0.$
Then, $\displaystyle\left\langle\frac{u}{\|u\|},\frac{v}{\|v\|}\right\rangle=1=\le... |
H: A little confusion about extensions $E(-,-)$ and $\mathrm{Ext}(-,-)$
If we want to calculate $E(\mathbb{Z}/p\mathbb{Z},\mathbb{Z})$, i.e. equivalence classes of short exact sequences $\mathbb{Z}\rightarrow E\rightarrow\mathbb{Z}/p\mathbb{Z}$, we have $\mathbb{Z}/p\mathbb{Z}\cong\mathrm{Ext}(\mathbb{Z}/p\mathbb{Z},\... |
H: Find the number of way the books are kept together.
The number of ways that $5$ Spanish , $3$ Enligh & $3$ German books are arranged if the books of each language are to be kept together are ...
My Try :
$5!+3!+3!$ Which is wrong answer .
where i am getting wrong ?
AI: In enumeration problems like this, you shou... |
H: At what angle do $y=x^2$ and $y=\sqrt{x}$ intersect at $(1,1)$?
Problem: at what angle do $y=x^2$ and $y=\sqrt{x}$ intersect at (1,1)?
Solution: As far as I know angle of intersection of 2 curves is the angle of intersection of their tangent lines. Also, coefficient before x in tangent line equals tangent of the an... |
H: Probability that $n$ vectors drawn randomly from $\mathbb{R}^n$ are linearly independent
Let's take $n$ vectors in $\mathbb{R}^n$ at random. What is the probability that these vectors are linearly independent? (i.e. they form a basis of $\mathbb{R}^n$)
(of course the problem is equivalent of "taken a matrix at rand... |
H: How to find an area?
I have this question:
A farmer plans to enclose a rectangular pasture adjacent to a river. The pasture must contain $320,000$ square meters in order to provide enough grass for the herd. What dimensions will require the least amount of fencing if no fencing is needed along the river?
$$x = ... |
H: Evaluating $\int^1_0 \frac2{\sqrt{2-x^2}} dx$
$$\int^1_0 \frac2{\sqrt{2-x^2}} dx$$
using substitution $x=\sqrt 2 \sin \theta$
$$\int^{\pi/4}_0 \frac{2\cos \theta d\theta}{\sqrt{2-2\sin^2 \theta}} = \int^{\pi/4}_0 \frac{2\cos\theta d\theta}{\sqrt2 \cos\theta} = \int^{\pi/4}_0 \frac{2d\theta}{\sqrt2} = \int^{\pi/4}_... |
H: matrix representations of linear transformation
I have a indexing problem about the matrix representation of linear transformation.
Let $V$ be a $3$ dimensional vector space over a field $F$ and fix $(\mathbf{v_1},\mathbf{v_2},\mathbf{v_3})$ as a basis. Consider a linear transformation $T: V \rightarrow V$.
Then we... |
H: Probability of k triangles in a random graph
Let $G_{n,p}, n\in \mathbb{N}, p\in(0,1)$ be the binomial random graph, i.e. a graph on $n$ vertices where an edge is in $G_{n,p}$ with probability $p$ and denote as $V$ its vertex set.
Let $j\in \binom{V}{3}$ be a set of 3 vertices and denote as $\mathcal{E}_j$ the even... |
H: Power series in complex analysis
We derived from cauchy's integral formula that a holomorphic function converges locally in a power series. Now we had the Identity theorem and I wanted to know whether I can conclude from this that the power series of a function is uniquely determined and converges on the whole doma... |
H: Problem with the Analytic theorem
The analytic theorem claims that
If $f(t)$ is a bounded and locally integrable function on $t \geq 0$ and
$g(z) = \int_0^\infty f(t)e^{-zt}$ is analytic with $\Re(z) > 0$ and extends holomorphically to an analytic function with $\Re(z) \geq 0$
Then $\int_0^\infty f(t) dt$ exists an... |
H: Exercise of a basis in the product of two metric spaces
Let $(E_1,d_1)$, $(E_2,d_2)$ two metric spaces and $E=E_1\times E_2$ the metric space product. Consider the following metrics in $E$ \begin{eqnarray*}
d^{(\infty)}(\mathbf{x},\mathbf{y}) & = & \max(d_1(x_1,y_1),d_2(x_2,y_2)) \\
d^{(1)}(\mathbf{x},\... |
H: Quotient topology is homeomorphic to nonnegative reals
I'm having some difficult to solve this:
Consider the following equivalence relation on $\mathbb R^2$:
$(x_1,x_2)\sim (y_1,y_2)$ if $x_1^2 + y_1^2 = x_2^2 + y_2^2$
Show that the quotient space $\mathbb R^2/\sim$ is homeomorphic to the set $\mathbb R_+ := \{x \... |
H: Visualizing the Area of a Parallelogram
I've seen how to visualize the formula for getting the area of a parallelogram. The first picture shows 2 ways which give the same result of Area = base * height.
http://tinypic.com/view.php?pic=2na1kr7&s=5 (link to first pic)
However, the first time I thought about it before... |
H: How much slower is a Turing Machine if you only give it one end of the tape to work with?
Turing Machines start with the input string and tape head in the "middle" of a tape that extends infinitely in either direction. Suppose instead that the tape head starts at the "far left" of the tape: the tape extends infini... |
H: Roulette outcome probability
I have a question about the probability of a certain Roulette outcome.
In the American Roulette wheel with double zero pockets, what are the chances of rolling black and red alternately for a total of nine times consecutively (i.e. B,R,B,R,B,R,B,R,B)? And also how many rolls will it tak... |
H: What is an effective means to make divisibility tests a mathematical 'habit', particularly for algebra?
Divisibility tests are a useful problem-solving technique for particularly dealing with larger numbers (thousands etc) and algebraic problems. However, I have always found that many students will just reach for t... |
H: Is $\int_0^\infty \left (\int_0^\infty f(k) k \sin kr \, \mathrm dk \right) \mathrm dr = \int_0^\infty f(k) \, \mathrm dk$ correct?
I am a physicist, and as a physicist I have proved the following equality:
$$
\int_0^\infty \left (\int_0^\infty f(k) k \sin kr \,dk \right) dr = \int_0^\infty f(k) dk,
$$
where $f$ is... |
H: Can you explain to me the statement of this problem?
Let $n$ a natural number and $A=(a_{ij})$, where $a_{ij}=\left(\begin{array}{c} i+j \\ i \end{array}\right)$, for $0 \leq i,j < n $. Prove that $A$ has an inverse matrix and that all the entries of $A^{-1}$ are integers.
What I don't understand is the definition ... |
H: Find the four elements of $M(S)=\{\pi, \rho, \sigma, \theta\}$
Question from Marcel Finan's A Semester Course in Basic Abstract Algebra.
Consider the set $S=\{a,b\}$. Find the four elements of $$M(S)=\{\pi,\rho,\sigma,\theta\},$$
where $M(S)$ is the set of mappings from $S$ to $S$.
I'm just confused. So these mappi... |
H: Calculate $\sum\limits_{n=1}^{\infty}na(1-a)^{n-1}$, where $a \in (0,1)$.
Calculate $a\displaystyle \sum_{n=1}^{\infty}n(1-a)^{n-1}$
where $a \in (0,1)$.
AI: For fun (and this is probably not the solution you are expected to give) we give a mean proof. A certain coin has probability $a$ of landing heads, and probab... |
H: How do I prove this map is a two-sided inverse
I'm trying to solve this exercise:
I almost solved it, to prove the bijectivity, we have to show that $\theta^{-1}=f^{-1}$.
Since $f$ is an epimorphism, we have $\theta(\theta^{-1}(G))=\theta(f^{-1}(G))=f(f^{-1}(G))=G$, for any $G\in S_{M'}$.
My problem is to prove th... |
H: I need help solving a ODE converting it to system of linear different equations
$y'''+4y''+5y'+2y=10cos(t)$
Where it is subject to a condition
$y''(0)=3, y'(0)=0, y(0)=0$
Ok I'm having a very hard time to solving this b/c if one is going to solve using system of linear different equations don't we need a forcing ... |
H: Proving that $\Pr[|X| > T\sqrt{n}/2] \geq \Pr[|X-\mathbb EX| < T\sqrt{n}/2]$
I am reading a paper Revealing information while preserving privacy and I am stuck in a step in the proof of Lemma 4. I'll write the relevant details below so you do not need to extract them from the paper.
Let $x,d \in [0,1]^n$ be constan... |
H: For $a_n$ alternating and $b_n \to c \neq 0$, don't we have $\sum a_n b_n$ converges iff $\sum a_n$ converges?
I'm reading the following set of notes on Taylor series and big O-notation, written by a professor at Columbia: http://www.math.columbia.edu/~nironi/taylor2.pdf. He repeatedly refers to what he calls "lim... |
H: Permutation with Repetition
how many different four letter words can be formed from the word "ENHANCEMENT" if the first and last letters is A and T respectively.
i am confused.Do we need to include A and T in 2 letters?
possible positions: _ _
with repition so 11 * 11 [do we need to eliminate E-3 times N-3 times..... |
H: Manipulating integral with u substitution
I have one of those stupid homework questions that is obscure and not examples given.
$f$ is a continuous function and these values are known:
$$\int_0^1 f(x) dx = 5$$
$$\int_{-1}^1 f(x)dx = 3$$
$$\int_0^2 f(x)dx = 8$$
$$\int_0^4 f(x)dx = 11$$
Evaluate:
$$\int_2^3 x f(8-x^2... |
H: Prove that $(g \circ f)^* = f^* \circ g^*$
Assume $g: F \mapsto G$ is a linear map
Prove that $(g \circ f)^* = f^* \circ g^*$
My solution
$(g \circ f)^* = g^* \circ f^*$ by the properties of associativity in linear maps.
If we assume that $g^* \circ f^* = f^* \circ g^* $ then $g$ and $f$ are inverse functions of ea... |
H: Is there a shorter way to prove this?
Let $n$ a natural number and $A=(a_{ij})$, where $a_{ij}=\left(\begin{array}{c}i+j \\i\end{array} \right)$, for $0≤i,j<n$. Prove that A has an inverse matrix and that all the entries of $A^{−1}$ are integers.
I tried to prove this like this: (but I'm not sure if it's correct or... |
H: $x^4-3x^3-9x^2+2=0$, why does wolframalpha give complex solutions when they are real?
Although $x^4-3x^3-9x^2+2=y$ intersects with the $x$-axis $4$ times (this is shown in the graph) Wolframalpha gives me complex solutions. Why does this happen? Thanks.
AI: Algorithms for computing explicit roots to general polyno... |
H: Repetition in piece-wise function
In my pre-calculus course lesson, I have this word problem:
Amy's electric bill can be represented by the piecewise function:
$$\begin{cases}
8.25+0.0705x, &x≤400 \\
36.45+0.0605x, &x>400
\end{cases}$$
where $x$ is the number of kilowatt hours used. Use this function to dete... |
H: integration of differential equation for wind profile
I'm trying to calculate the vertical wind speed gradient with some equations and am having trouble with the integration. From the book I'm using, the wind profile is calculated according to
$$\frac{kz}{u_*}\frac{\partial u}{\partial z}=\phi_m(\zeta),$$
where k i... |
H: system of equations - when does it have a solution?
Find all values of $t$ for which the system of equations
$$\begin{array} 22x_1 + x_2 + 4x_3 + 3x_4 = 1\\
x_1 + 3x_2 + 2x_3 − x_4 = 3t\\
x_1 + x_2 + 2x_3 + x_4 = t^2 \end{array}$$
has a solution?
I was given a theorem, that system has a solution, when column vecto... |
H: Solutions for integer $n$, given $ \exp(m-\frac{2}{\pi}n)=n^{b/{\pi}}. $
Let $m, n$ be integers. Let $b \in \mathbb R$.
Solve the following equation for $n$.
$$
\exp(m-\frac{2}{\pi}n)=n^{b/{\pi}}.
$$
Thank you.
AI: Let's play around and see what happens.
We have
$\exp(m-\frac{2}{\pi}n)=n^{b/{\pi}}$,
or
$\exp(m)/\e... |
H: If $\lim_{x \rightarrow \infty} f(x)$ is finite, is it true that $ \lim_{x \rightarrow \infty} f'(x) = 0$?
Does finite $\lim_{x \rightarrow \infty} f(x)$ imply that $\lim_{x \rightarrow \infty} f'(x) = 0$? If not, could you provide a counterexample?
It's obvious for constant function. But what about others?
AI: Sim... |
H: Solution of Lagrangian
Could you give me advice how to solve the following Lagrangian?
$$L=x^3+y^3 - \lambda (x^2-xy+y^2-5)$$
$$ \left\{ \begin{array}{c}
\frac{\partial L}{\partial x} = 3x^2 - \lambda (2x-y) = 0
\\\frac{\partial L}{\partial y} = 3y^2 - \lambda (-x+2y) = 0
\\ \frac{\partial L}{\partial \lambda} = x... |
H: Proving that $f ^{*} $ is a linear map
Here's another that I want to see if my reasoning was correct.
Let $f:E \rightarrow F$ be a linear map and let $$f^* : \text{Hom}_K(F,K) \rightarrow \text{Hom}_K(E,K) $$ if we define $f^*(u) = u \circ f$, prove that $f^*$ is a linear map.
Originally my reasoning was that ... |
H: Interesting Ordinary Differential Equation Problem
consider the DE
$y'=\alpha x,x>0$
Where $\alpha$ is a constant. Show that :
1) if $\phi(x)$ is any solution and $\psi(x)=\phi(x)e^{-\alpha x}$ then $\psi(x)$ is a constant.
2) if $\alpha<0$, then every solution tends to zero as $x \rightarrow 0$
I tried the normal... |
H: Proving a projection is a linear map
OK, I think I am on the right track but I am trying to figure out if I am going wrong somewhere.
Let $V$ be a vector space over a field $K$. Let $B = \{x_1, x_2, x_3, \ldots , x_n\}$ be a base of $V$ over $K$. Then for all $x \in V$ there exist unique scalars $\lambda_1, \lambd... |
H: Unintentional Negative Sign in Limit Evaluation
I've been working on evaluating the following limit:
$$\lim_{x\to 0} \left(\csc(x^2)\cos(x)-\csc(x^2)\cos(3x) \right)$$
According to my calculator, the limit should end up being 4.
Though I've tried using the following process to find the limit, I continue to get -4:
... |
H: How to show that a set $A$ is closed iff it is covered by a family $\mathcal{P}$ of open sets $U$, where $A \cap U$ is closed in $U$?
How to show that a set $A$ is closed if and only if it is covered by a family $\mathcal{P}$ of open sets, where $A \cap U$ is closed in $U$ for each open set $U$ in $\mathcal{P}$? Th... |
H: Summing a series with a changing power
$$\sum_{r=0}^{k-1}4^r$$
Hi, I was wondering whether anyone could explain how to work this out. I know the end result is $\frac{4^k-1}{3}$, but I don't know why or how to get there.
Thank you :D
AI: Multiply it by $(4-1)$. Expand without turning it into $4-1=3$. a lot of power... |
H: Probability of getting a "double" in at least two throws of two dies
We play a game where we throw two distinct dies twice. A player wins $\$3$ if he gets at least one time a double between the two throws, and loses $\$1$ if he doesn't get a double in any throw of the two. What is the expected value of the player... |
H: Determining whether a coin is fair
I have a dataset where an ostensibly 50% process has been tested 118 times and has come up positive 84 times.
My actual question:
IF a process has a 50% chance of testing positive and
IF you then run it 118 times
What is the probability that you get AT LEAST 84 successes?
My gut... |
H: Finding the DE of family of curves
FInd the DE of the family of circles in XY plane passing through the points $(-1,1)$ and $(1,1)$
AI: Replace $(\pm1,1)$ by $(\pm1,0)$ for the moment.
By symmetry it is sufficient to consider a point $P=(x,y)$ in the (interior of) the first quadrant. The circle through $(\pm1,0)$ a... |
H: Is $2^{\aleph_0}=c$?
When I search about the cardinality of real number set in Wiki
(http://en.wikipedia.org/wiki/Cardinality_of_the_continuum)
I found: By the Cantor–Bernstein–Schroeder theorem we conclude that $c=|P(\mathbb{N})|=2^{\aleph_0}$
And the Cantor–Bernstein–Schroeder state that if $A\preceq B$ and $B\p... |
H: Surjective endomorphisms of Noetherian modules are isomorphisms.
I'm trying to solve this question:
I didn't understand why the hint is true and how to apply it. I really need help, because it's my first question on this subject and my experience on this field is zero.
I need some help.
Thanks a lot
AI: For any $m... |
H: A slight modification to the recursion theorem
I always come across this version of the recursion theorem that is frequently used without justification:
Given a function $g:A\longrightarrow A$ there exists exactly one funcion such that,
$f(0)=a_{0}, f(1)=a_{1},...,f(k)=a_{k}$
$f(n+k)=g(f(n))$
I guess of cour... |
H: Question regarding GWP (Kesten-Stigum setup)
Let $(Z_n)_{n\in \mathbb{N}}$ be a GWP with $Z_0=1$ and mean of offspring distribution $m\in (1,\infty)$. Define $W_n=Z_n/m^n$ and denote its limit by $W$ (i.e. setup as in the Kesten-Stigum theorem). I already know that $\{W>0\} \subset \{Z_n \rightarrow \infty\}$, but ... |
H: The difference between closed and open sets of the product topology
I was tackling with this problem from Munkres:
If Y is compact, then the projection map of $X \times Y$ is a closed map.
And I thought the same things as Akt904. After reading Brian M. Scott's comment I was nearly convinced but what about the open ... |
H: Can a basis for a vector space $V$ can be restricted to a basis for any subspace $W$?
I don't understand why this statement is wrong:
$V$ is a vector space, and $W$ is a subspace of $V$. $K$ is a basis of $V$. We can manage to find a subset of $K$ that will be a basis of $W$.
Sorry if my English is bad... and if ... |
H: In $\mathbb{C}[x]$ is it true that $F_{a,b}=\{p\in\mathbb{C}[x] : p(a)=p(b)\}$ for $a\neq b$ is a maximal subring?
The problem is in the title. It is clear that $F_{a,b}$ is a ring, but it is not so clear to me that it is maximal in $\mathbb{C}[x]$. I tried to consider it as a vector space and show that it has codi... |
H: When is there a ring structure on an abelian group $(A,+)$?
Given an abelian group $(A,+)$, what are conditions on $A$ that ensure there is or isn't a unitary ring structure $(A,+,*)$? That is, an associative bilinear operation $* : A^2 \to A$ with an identity $1_A \in A$.
A possible follow-up question would be co... |
H: Continuous extension of a Bounded Holomorphic Function on $\mathbb{C}\setminus K$
Let $f:\mathbb{C}\setminus K\rightarrow\mathbb{D}$ be a holomorphic map, where $K$ is a compact set with empty interior. My question:
Prove or disprove that: $f$ extends continuously on $\mathbb{C}.$
Remark: Observe that if $K$ is di... |
H: Contour integration of $\int \frac{dx} {(1+x^2)^{n+1}}$
I want to compute
$$
\int_{-\infty}^\infty \frac 1{ (1+x^2)^{n+1}} dx
$$ for $n \in \mathbb N_{\geq 1}$. If I let
$$
f(z) := \frac 1 {(z+i)^{n+1}(z-i)^{n+1}}
$$ then I see that $f$ has poles of order $n+1$ in $\pm i$. Initially I thought that
$$
[-R,R] \cu... |
H: Norms in extended fields
let's have some notation to start with:
$K$ is a number field and $L$ is an extension of $K$. Let $\mathfrak{p}$ be a prime ideal in $K$ and let its norm with respect to $K$ be denoted as $N_{\mathbb{Q}}^K(\mathfrak{p})$.
My question is this: If $|L:K|=n$, what is $N_{\mathbb{Q}}^L(\mathfr... |
H: $G$ is 2-vertex-connected graph if and only if every 2 vertices from $G$ lie on a simple cycle.
$G$ is $2$-vertex-connected graph if and only if every $2$ vertices from $G$
lie on a simple cycle.
$\implies$
If $G$ is $2$-vertex connected graph it means for every $ v \in V(G), \deg(v) \ge 2 $ because if exist a... |
H: How to get the radius of an ellipse at a specific angle by knowing its semi-major and semi-minor axes?
How to get the radius of an ellipse at a specific angle by knowing its semi-major and semi-minor axes?
Please take a look at this picture :
AI: The polar form of the equation for an ellipse with "horizontal" semi-... |
H: Prime Mean of Primes
1) Are there infinitely many primes $p_1,p_2$ such that $\frac{p_1+p_2}{2}$ is also prime?
2) What can we say about the more general problem :
Are there infinitely many primes $p_1,p_2,\cdots, p_n$ such that $\frac{p_1+p_2+\cdots+p_n}{n}$ is also a prime for $n\geq 1$?
$\mathbf{Remark}$: Not... |
H: How to find asymptotics of integrand?
Let $ f \in C ([0, \infty)) $ be s. t. $$f(x) \int_0^x f(t)^2 dt \to 1, x \to \infty.$$
How to prove that $f(x) \sim \left( \frac 1 {3x} \right)^{1/3} $ as $x \to \infty?$
AI: Define
$$
F(x)=\int_0^xf(t)^2\,\mathrm{d}t\tag{1}
$$
For any $\epsilon\gt0$ there is an $M$, so that ... |
H: An Orlicz norm is a norm
I had asked a question pertaining to Orlicz norms here. However, in the book I was reading, it said (and I paraphrase) "It's not difficult to show it is a norm on the space of integrable random variables and for which $\|X\|_{\psi}$ is finite". So I decided to prove this.
To recap, if $\psi... |
H: Kernel of a linear map
One often says that the kernel of a linear functional has codimension 1. Okay, consider $\delta_1\colon C[0,1]\to \mathbb{R}$ given by $\delta_1(f)=f(1)$. Then for each monomial $f_n(t)=t^n$ we have $\delta_1(f_n)=1$ and $\{f_n\colon n\in \mathbb{N}\}$ are linearly independent... What am I co... |
H: Example of a non-affine irreducible scheme
What are basic examples for irreducible schemes which are not affine? What happens if I also demand the scheme to be Noetherian and/or locally Noetherian?
AI: I think the plane minus the origin is an example, and it's Noetherian if you take it over a field:
It is covered b... |
H: Prove that $\frac{1}{2\pi}\frac{xdy-ydx}{x^2+y^2}$ is closed
I would like to prove that $\alpha = \frac{1}{2\pi} \frac{xdy-ydx}{x^2+y^2}$ is a closed differential form on $\mathbb{R}^2-\{0\}$ . However when I apply the external derivative to this expression (and ignore the $\frac{1}{2\pi}\cdot\frac{1}{x^2+y^2}$ fa... |
H: Metric on a set
Can someone provide a hint for solving the following.
Show that $d:(R^{\infty})^2\to R_+$ is a metric.
$$d(x, y)=\sqrt{\sum_{i=0}^{\infty}{(x_i-y_i)^2}}$$
I need a hint for showing that $d$ satisfies the triangle inequality
AI: $\mathbf{Hint:}$ Use Cauchy-Schwarz inequality. |
H: If $M/G_1$ and $M/G_2$ are Noetherians, then $M/(G_1\cap G_2)$ is Noetherian.
I'm trying to prove if $M$ is a module, $G_1$, $G_2$ submodules of $M$ and $M/G_1$ and $M/G_2$ Noetherians, then $M/(G_1\cap G_2)$ is Noetherian.
I've tried by brute force (writing down explicitly an ascending chain), by second fundament... |
H: Functions whose derivatives can be written as a function of themself?
What kinds of function $f: \mathbb{R} \to \mathbb{R}$ can be written as some function of itself? I.e. $f'(x) = g(f(x))$ for some function $g$?
If $f$ is given, can $g$ be solved in terms of the symbol $f$ (not in terms of specific $f$), if $g$ e... |
H: How do we know $p/q$ can be expressed as a terminating fraction in base $B$ only if prime factors of $q$ are prime factors of $B$?
On cs.stackexchange I asked a math question: How to demonstrate only 4 numbers between two integers are multiples of .01 and also writable as binary.
Yuval Filmus answered with a explan... |
H: Finitely many non-convergent ultrafilters
I am trying to prove that if a space $X$ has finitely many non-convergent ultrafilters, then every non-convergent ultrafilter $\mathcal U$ contains a set $A$ that is not contained in any of other non-convergent ultrafilters.
I honestly have no idea why this should be true, ... |
H: For real numbers $x$ and $y$, show that $\frac{x^2 + y^2}{4} < e^{x+y-2} $
Show that for $x$, $y$ real numbers, $0<x$ , $0<y$
$$\left(\frac{x^2 + y^2}{4}\right) < e^{x+y-2}. $$
Someone can help me with this please...
AI: From the well known inequality: $1+z\leq e^z$, we replace $z$ by $\frac{z}{2}-1$ to get $\f... |
H: In how many ways can $3$ balls be tossed into $3$ boxes?
$3$ balls are tossed into $3$ boxes. In how many ways can that be done? Well, I did in the following way.
We have $5$ objects: $3$ balls and $2$ walls of boxes. So a configuration, for example:
00|0|
meaning : $2$ balls are in the left box, one ball is in th... |
H: Proof Complex positive definite => self-adjoint
I am looking for a proof of the theorem that says:
A is a complex positive definite endomorphism and therefore is A self-adjoint. Does anybody of you know how to do this?
AI: In general, for $A:H\longrightarrow H$ bounded linear operator on a Hilbert space $H$, $A\geq... |
H: limit question: $\lim\limits_{n\to \infty } \frac{n}{2^n}=0$
$$ \lim_{n\to\infty}\frac n{2^n}=0. $$
I know how to prove it by using the trick, $2^n=(1+1)^n=1+n+\frac{n(n-1)}{2}+\text{...}$
But how to prove it without using this?
AI: Let's do something different!!
Note that the sequence $\{\frac{n}{2^n}\}_{n\geq 1}$... |
H: Show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$
The problem is : if $z$ lies on a circle with diameter having endpoints $z_1$ and $z_2$ then show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$ where $z, z_1, z_2 \in \mathbb{C}$.
The angle subtended by the diameter on any point on the circle is a right angle and thu... |
H: Angle limit problem
I have been trying to interpret orientation angle data retrieved from a sensor device.
It returns the angle in Radian units towards North that the device is measuring at the moment.
The problem I am having right now, is that these measurements are noisy, and need some filtering, so I use a movin... |
H: Are $p$-groups Engel groups?
A group $G$ is termed Engel if whenever $x, y \in G$, there exists an integer $n$ (depending on $x$ and $y$ such that $[x,y,\dots,y]=1$, where $y$ occurs $n$ times.
Is it true that every infinite $p$-group is Engel?
AI: Well, nilpotent groups are almost trivially $\;n-$ Engel, with $\;... |
H: Number of combinations when you can choose none or multiple options for a questions.
I would like to know both the formula and the math name for such combination. Simple example: 1 questions with 3 options, you can choose none, one or multiple options. How can I calculate the number of combinations in such case.
AI... |
H: Can $\Phi^{-1}(x)$ be written in terms of $\operatorname{erf}^{-1}(x)$?
Can the inverse CDF of a standard normal variable $\Phi^{-1}(x)$ be written in terms of the inverse error function $\operatorname{erf}^{-1}(x)$, and, if so, how?
This seems like an easy question, but I am struggling with it. I know that $\Phi(... |
H: If $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not ring-isomorphic to $n\mathbb{Z}$
Show that if $m$ and $n$ are distinct positive integers, then $m\mathbb{Z}$ is not
ring-isomorphic to $n\mathbb{Z}$.
Can I get some help to solve this problem
AI: Assume you have an isomorphism $\phi: m\mat... |
H: Set of permutation matrices
I'm stuck in this problem.
Prove the set $P$ of $n×n$ permutation matrices spans a subspace of dimension $(n−1)^2+1$
AI: Remark that if $M\in \mathit{Span}(P)$, then there is $\lambda$ such that for all line or column $x$ of $M$, the sum of elements of $x$ is $\lambda$.
Let $V$ be the sp... |
H: Justify conditional
If there is an interpretation $I$ in wich $A=\forall x (P(x)\rightarrow Q(x))$ is true, then $(P(x)\rightarrow Q(x))$ is not true ands is not false in $I$.
I need to justifiy this is false. So, if A is true for $I$ this means that every valuation on $I$ satisfies A. A little more formally, valua... |
H: Show that $\int_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}$
I'm looking for a clever way to show that
$$ \int\limits_{(0,1)\times (0,1)} \frac{1}{1-xy} dxdy = \sum_{n=1}^{\infty} \frac{1}{n^2}.$$
All suggestions will be appreciated!
AI: Hint: $$\frac{1}{1-xy}=1+(xy)+(xy)^2+....$$
th... |
H: Can any finite group be realized as the automorphism group of a directed acyclic graph?
We are given a finite group $G$ and wish to find a DAG (directed acyclic graph) $(V,E)$ whose automorphism group is exactly G (a graph automorphism of a graph is a bijective function $f:V\to V$ such that $(u,v)\in E \iff (f(u),f... |
H: ve that the perpendiculars to the sides at these points meet in common point if and only if $ BP^2 + CQ^2 + AR^2 = PC^2 + QA^2 + RB^2 $
$P, Q, R$ are points on the sides $BC,CA,AB $ of triangle $ABC$. Prove that the perpendiculars to the sides at these points meet in common point if and only if
$ BP^2 + CQ^2 + AR^... |
H: Block matrix and invariant subspaces
I was wondering what the exact relationship between invariant subspaces and a block matrix is?
Is it correct to say: Each diagonal block matrix "creates a vector space decomposition" and vice versa? If this is so, I would be interested in understanding how one gets from the diag... |
H: Show that the equation of a line can be given as ℑm(αz+β)=0
I've just started a non-Euclidean Geometry course and the book we are using has a very brief (and not-so-helpful) section on complex numbers that we sort of went over in class. One of the questions is this: Given that α and β are complex constants and z =... |
H: Finding limit function $\lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1)$
\begin{align}
f(x) &= \lim_{n \rightarrow \infty} n ((x^2 +x + 1)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n ((\infty)^{1/n} -1) \\&= \lim_{n \rightarrow \infty} n (1 -1)\\& =
\lim_{n \rightarrow \infty} n \cdot 0 \\&= 0
\end{align}... |
H: How can i solve this System of first-order differential Equations?
My Problem is this given System of differential Equations:
$$\dot{x}=8x+18y$$ $$\dot{y}=-3x-7y$$
I am looking for a gerenal solution.
My Approach was: i can see this is a System of linear and ordinary differential equations. Both are of first-order,... |
H: Property of Banach algebra with involution
Let $\mathcal{B}$ be a Banach algebra with involution *.
Is it always true that $\forall A \in \mathcal{B}: \| A \|^2 \geq \| A^* A \| $?
(motivation: I read a proof that bounded linear operators on a Hilbert space form a C*-algebra, but for the C*-property they only prove... |
H: Proof of Hilbert's Basis Theorem: won't $\deg (f_{i})$ be a strictly decreasing sequence?
Say we have an ideal $I\subset R[X]$. We select a set of polynomials $f_{1},f_{2},f_{3},\dots$ such that $f_{i+1}$ has minimal degree in $I\setminus (f_{1},f_{2},f_{3},\dots f_{i})$.
Can't $\deg (f_{i})$ be a strictly decreas... |
H: Maclaurin series problem
I can't seem to understand the full solution
Why did they post x=0.2?
thanks in advance
AI: The reason they plug $0.2$ into the function is because that is what gives them the answer they seek.
If you define $f(x) = \sqrt{1 + x}$, then $f(0.2) = \sqrt{1+0.2} = \sqrt{1.2}$, as desired. |
H: Lower bound on the probability that the maximum of a sequence of $n$ i.i.d. standard normal r.v.'s exceeds $x$
Let $X_{\max}=\max(X_1,X_2,\ldots,X_n)$ where $n$ is large and each $X_i$ is i.i.d. standard normal random variable, i.e. $X_i\sim\mathcal{N}(0,1)$. Is there a lower bound on the probability $P(X_{\max}\g... |
H: Show that the statements are equivalents
I have to prove that these statements are equivalents:
(i) $f: X \rightarrow Y$ is continuous
(ii) $f(A') \subset \overline {f(A)} , \forall A\subset X$
(iii) $Fr(f^{-1} (B)) \subset f^ {-1} (Fr(B)) , \forall B \subset Y$
I could only show (i) implies (ii).
I don't know what... |
H: divergence of $\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$
i ran into this question and im sitting on it for a long time.
why does this integral diverge:
$$\int_{2}^{\infty}\frac{dx}{x^{2}-x-2}$$
thank you very much in advance.
yaron.
AI: $$\frac1{x^2-x-2}=\frac13\left(\frac1{x-2}-\frac1{x+1}\right)\implies$$
$$\int\lim... |
H: Determine coefficients of a Fourier series
Given the $2\pi$-periodic function $f(t)=t^2$ such that $-\pi \le t \le \pi$,
I want to determine the coefficients $f_k$ of the fourier series of this signal.
Therefore I use
$$f_k = \frac{1}{2\pi} \int_{-\pi}^{\pi} t^2 e^{-ikt}\,dt$$
Is that true? Because I'm looking at ... |
H: Show that $\lim\limits_{n\to\infty} x_n$ exists for $0 \le x_{n+1} \le x_n + \frac1{n^2}$
Let $x_1, x_2,\ldots$ be a sequence of non-negative real numbers such that
$$ x_{n+1} ≤ x_n + \frac 1{n^2}\text{ for }1≤n. $$
Show that $\lim\limits_{n\to\infty} x_n$ exists. Help please...
AI: The sequence is bounded from abo... |
H: Converting Maximum TSP to Normal TSP
Consider the Travelling Salesman Problem:
Given N cities connected by edges of varying weights. Given a city A what is the shortest path for visiting all the cities exactly once that returns back to A
and the Max Travelling Salesman Problem:
Given N cities connected by edges of ... |
H: Prove there is no such analytic function
Please help prove that there is no analytic function on $z=0$ such that
$$ n^{-\frac3 2}<\left|f\left(\frac1 n\right)\right|<2n^{-\frac3 2}$$ for every natural $n$.
AI: By the continuity of $f$, we must have $f(0) = 0$. Clearly, $f$ cannot be identically $0$. Since $f(0) = 0... |
H: Vector in the line of intersection of two planes
In the context of Geometric Algebra, in $ \mathbb{R}^3 $:
Let $A$ and $B$ be bivectors (representing planes). Show that $ (\langle AB \rangle_2)^* $ is a vector in the line of intersection of the two planes, where the $ ^* $ represents the dual.
My approach: $ A^*, B... |
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