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H: Sum of the number of unique factors of numbers from 1 to 100?
I want to compute
$\sum_{i\; =\; 1}^{i\; =\; 100}{d\left( i \right)}$
where d(i) is the number of factors of i. For example, for 1 it is 1, for 2 is is 2, for 3 it is 2, for 12 it is 6, and so on.
AI: HINT: $1$ is a divisor of every number in the set $[1... |
H: A question about rolls
A dodecagon labelled by 12 months at each edge is rolled in a game. One “turn” of the game is to roll it until one April appears, then the number of the rolls is recorded. What is the probability to have five consecutive turns with rolls no greater than 10?
Could you help me with this stuck?
... |
H: The king comes from a family of 2 children. What is the probability that the other child is his sister?
I've found this question on a book and I'd like a review in my answer.
The king comes from a family of 2 children. What is the probability that the other child is his sister?
AI: First of all, take note about @Mi... |
H: How to get the inverse of this function?
I have the function f(x) = (1+8x) / (3-3x).
I have been stuck on trying to get the inverse of this function by isolating for y. I ended up with:
x = (1+8y) / (3-3y) but I am not quite sure where to go from there.
AI: $$ y = \frac{1+8x}{3-3x}$$
$$(3-3x)y = 1+8x$$ (multiplying... |
H: Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$ then $R(T) \cap N(T) = \{0\}$
Let $V$ be a finite-dimensional vector space and let $T:V\to V$ be linear.
Prove that if $\operatorname{rank}(T) = \operatorname{rank}(T^2)$, then $R(T) \cap N(T) = \{0\}$.
I don't see this implication, at all. Please gi... |
H: Hiding Eggs Combinatorics
How can I figure out if I have 6 red eggs, 3 blue eggs, 1 green egg, and 2 yellow eggs. Aside from their color, they are identical. I have 12 different hiding spots, each big enough for 1 egg. How many ways are there to hide the eggs?
AI: HINT: How many ways are there to choose $6$ of the ... |
H: Simplifying euler exponent?
How would I go about simplifying and finding the exact value for this question:
$e^{6\ln(4)}$
I know that $e ^{\ln x} = x$ but how does the $6$ affect this answer?
AI: $$a^{bc} = \left(a^c\right)^b$$
If $a=e$ and $c=\ln(4)$, then $a^c = 4$. You should be able to do the rest yourself. |
H: Instantaneous rate of change
Let $f(x) = ax + b$.
Find the instantaneous rate of change of $f(x)$ at the following points: $1, 2, 4\text{ and }8$.
Is the instantaneous rate of change for all the points $a$, because the derivative for $f'(x)= a$?
AI: Yes, the instantaneous rate of change is $4$.
To see this (even... |
H: Rate of change of distance from particle (on a curve) to origin
A particle is moving along the curve
$y = 2\sqrt{4x + 9}$
As the particle passes through the point
(4, 10)
its x-coordinate increases at a rate of
3 units per second. Find the rate of change of the distance from the particle to the origin at this i... |
H: Using Logarithms
\begin{align*}
-2^{n-1} \ln2 &= -100 \ln 10\\
&\\
-100 \ln 10 &= -230\\
&\\
\dfrac{-230}{\ln (2)} &= -333\\
&\\
-2^{n-1} &> -333\\
&\\
(n-1) \ln(-2) &> \ln(-333)
\end{align*}
Here is where I am stuck.
I am not sure if this part is correct: $-n-1=8$. Then solving we would get $-9$.
AI: Log is an inc... |
H: Making first order logic statements
I'm working on an assignment that deals with predicate calculus, and I'm trying to put sentences into first order logic statements. I've got the hang of most of them, but I'm not quite sure how to do a couple of them:
Every X except Y likes Z, $$\forall m: \neg Y(m) \rightarrow ... |
H: For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap Q$ a clopen subset of the metric space $\mathbb{Q}$?
For which intervals $[a,b]$ in $\mathbb{R}$ is the intersection $[a,b]\cap Q$ a clopen subset of the metric space $\mathbb{Q}$?
My answer is : $[a,b]\cap\mathbb{Q}$ is a clopen subset if... |
H: Does the structure group of S^n homotopic to O(n+1)?
It is easy to show that Diff(S^1) is homotopic to O(2), but in the case of n bigger than 1 things become really complicated, I cannot see the conclusion directly.
AI: The proof for $S^2$ is in lecture notes by Lurie.
The proof for $S^3$ is due to Hatcher [added ... |
H: How to sample N points between 0 and R if they are exponentially distributed?
The density of my points x $\in$ [0,R] is exponential:
$\rho(x) \approx e^x$
How can I sample N points from there?
Thanks,
AI: This is an important class of problems, with many different sorts of solutions. We give a procedure that is ge... |
H: Finding a counterexample in model theory
I'm currently reading about models of Set Theory, and I'm working on exercises to better understand the concepts. In Kunen's most recent Set Theory text, he mentions that if we have a transitive model $M$ and if $\cap^M$ is defined, then $\cap$ is absolute. Then, he gives th... |
H: Parametrize the intersection of 2 planes.
Parametrize the intersection of $\frac {x^2} {3}+y^2+\frac {z^2} {10} = 1$ with $z=2$ (level curve) plane.
Here is what I did.
Plugged in $z=2$ into the plane $\frac {x^2} {3}+y^2+\frac 25=1$. I got $y^2=9-5x^2$ Then I substituted $y^2$ into the plane $\frac {x^2} {3}+9-5x^... |
H: If $A\subset B\subset C$, $A$ is dense in $B$, and $B$ is dense in $C$ prove that $A$ is dense in $C$.
If $A\subset B\subset C$, $A$ is dense in $B$, and $B$ is dense in $C$ prove that $A$ is dense in $C$.
Here's my answer:
$A$ is dense in $B$: $\bar{A}=B$
B is dense in $C$: $\bar{B}=C$
However $\bar{B}$ is the s... |
H: Find the probability function of random variable Y
Two balls are drawn at random form a box containing ten balls numbered 0,1,...9.Let random variable Y be their total.
Tabulate the probability function of Y if the sampling is without replacement.
I dont know how to solve this question.Please help.
Thank you very m... |
H: Why is the $\operatorname{diam}{(\mathbb Q)}$ infinite?
I was trying to find a counterexample to show that the $\operatorname{diam}{(A)}$ and the $\operatorname{diam}{(Int(A))}
$ may not be the same, where $A$ is the subset of the metric space $X$.
I chose the metric space $X$ = $\mathbb R$ , and chose $A$ = $\math... |
H: Sigma algebras...
"Let $\mathcal{C}$ = {A,B,C} be a partition of E. List the elements of the smallest sigma algebra containing $\mathcal{C}$."
Let me see if I got this right...because $\mathcal{C}$ is a partition of E, the smallest sigma algebra is the trivial one {$\emptyset$,E}?
What if $\mathcal{C}$ were not a p... |
H: Solving system of multivariable 2nd-degree polynomials
How would you go about solving a problem such as:
\begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix}
where $(x,y)\in\mathbb{C}^{2}$.
More generally, how would you solve any set of equations of the form:
\begin{matrix} { ax ... |
H: Finding Missing Observed Scores given standard deviation and mean
The question that i'm being asked it:
the mean of 10 observed scores was 20 and the standard deviation if 6.0. the observed scores are {16,11,20,24,29,24,16,20,,} what are the two missing scores
I have tried to figure this out and i keep ending up ne... |
H: Prove the probability is 0
Let E and F be independent with E = AUB and F=AB. Prove that either P(AB)=0 or else P(not A and not B)=0.
I dont know how to solve it. Please help.
Thank you very much.
AI: Since $E,F$ are independent, you have $P(E) P(F) = P(E \cap F)$. Note that $E \cap F = F$.
Hence we have $P(E) P(F) ... |
H: A couple has 2 children. What is the probability that both are girls if the eldest is a girl?
This is another question like this one. And by the same reason, the book only has the final answer, I'd like to check if my reasoning is right.
A couple has 2 children. What is the probability that both are girls if the el... |
H: If $x \in \overline{A}$ and $A \subset X$, $X$ first-countable, then there's a sequence of points in $A$ converging to $x$.
Supposedly this relies on first-countability of $X$.
Let $x \in \overline{A}$, then by definition there's a neighborhood $U_1$ of $x$ that contains some $x_1 \in A$. If this is the only nei... |
H: will the rank of a projector matrix be equal to the dimension of vector space it projects to?
Let the projector be the $N \times N$ matrix $A$. Let its rank be $r$. Let the dimension of the space for which it is the projector be $m$. Is $m==r$?
AI: Yes. The rank of matrix is the dimension of the image, and the imag... |
H: Counting edges in a specially defined graph
$G$ has vertices $\{0, \ldots, pq - 1\}$, where $p, q$ are different primes. There is an edge between $x$ and $y$ if $p \mid x - y$ or $q \mid x - y$. How many edges does $G$ have?
I looked at the cases where $p=2$ and $q=3, 5, 7, 11, 13, 17, 23$ and the formula I got was... |
H: Simplifying an inverse trig function?
I am trying to figure out how to simplify this expression but I am not quite sure how these inverses work. What sort of approach should I take for this equation?
$$\tan\left(2\cos^{-1}\left(\dfrac{x}{5}\right)\right)$$
AI: You will need two things. One is the identity for $tan(... |
H: Does two isomorphic lattices always have the same Hasse diagram?
I have done few examples of proving that lattices are isomorphic. I just draw their Hasse diagrams and if they are the same I say its isomorphic. Is there any case where two isomorphic latices don't have the same Hasse diagram?
AI: Two lattices (or, m... |
H: Showing that a function is not a homeomorphism
Consider the function $f: [0,1) \rightarrow \mathbb{C}$ given by $f(t) = e^{2\pi i t}$. I must show that the function $f^*: [0,1) \rightarrow \mathrm{im}(f)$ is not a homeomorphism, given the standard topologies on both sets, but I am not sure how to proceed.
Some wor... |
H: Keeping Focused: School
Mathematics is such a huge field, and I am just wondering how does a mathematician keep focused? For example to learn Stochastic Calculus you need some Analysis, Measure Theory, Probability, Set Theory, and all of those topics usually need some other branch of mathematics. So, how do you kn... |
H: What condition can make this event independent?
A population consists of F females and M males;the population includes f female smokers and m male smokers. If A is the event that the individual is female and B is the event he or she is a smoker , find the condition on f, m, F and M so that A and B are independent... |
H: Let $M$ be a metric space with the discrete metric or more generally a homeomorph of $M$.
Let $M$ be a metric space with the discrete metric or more generally a homeomorph of $M$.
I've proven that every subset of $M$ is clopen and every function defined on $M$ is continuous.
For the question, I answered:
Let $(x_m)... |
H: Finding the eqn. of a plane that passes through the line of intersection of 2 planes and is perpendicular to another plane.
Find the equation of the plane that passes through the line of intersection of the planes $4x - 2y + z - 3 = 0$ and $2x - y + 3z + 1 = 0$, and that is perpendicular to the plane $3x + y - z + ... |
H: Which are tangents
We are asked to see which are tangents and which aren't. I think B3, bottom left and bottom middle are not tangents
AI: A tangent to a curve is a straight line that just touches the curve, such that the slope of the tangent is exactly that as the slope of the curve. By that definition, all excep... |
H: If $|z| = 2$. Then Locus of $z$ Representing The Complex no. $-1+5z,$ is
If $|z| = 2$. Then Locus of $z$ Representing The Complex no. $-1+5z,$ is
$\underline{\bf{My\;\; Try::}}$ Let $z^{'} = -1+5z$, Where $z^{'} = x^{'}+iy^{'}$. So put $5z = z^{'}+1$ in $|z| = 2\Leftrightarrow |5z| = 10$
$|z^{'}-(-1+0\cdot i)| = ... |
H: Partial derivative paradox
Okay, perhaps not a paradox, but somewhat of a lack of understanding on my part.
Let $z$ equal some function of $x$ and $y$, i.e. $z = f(x, y)$ and take partial derivatives $\frac{\partial z}{\partial x} = f_x$ and $\frac{\partial z}{\partial y} = f_y$ all and good. But now say I do parti... |
H: The rate of change of the distance from the plane to the radar station
Problem statement:
A plane flying with constant speed of 4 km/min passes over a ground radar station at an altitude of 6 km and climbs at an angle of 35 degrees. At what rate, in km/min, is the distance from the plane to the radar station increa... |
H: polynomial with integer coefficients
Question: Let $\Pi_{j=1}^n (z-z_j)$ be a polynomial with integer coefficients. Is also $\Pi_{j=1}^n (z-z_j^k)$ for $k=1,2,3,\dots$ a polynomial with integer coefficients?
In fact, this is a question that someone asked 3 days ago, but the answer is not clear (I think)
For this, I... |
H: adjustment of Prof.,s in Round Table
In How many ways can $5$ Professors of Physics Including Prof. Hardy and $3$ Three Professors of Chemistry Including Prof. Julian be seated on a Round table, If Prof. hardy and Prof. Julian are not adjacent.
$\underline{\bf{My\;\;Try:}}$ First we will adjust $4$ Prof. of Physics... |
H: Question about property of circle
We know that equal chords are equidistant from the center.
However, I was curious if the lengths involved are proportional as well since the circle is a pretty symmetrical shape. Here's what I mean:
Let there be two chords,C1 and C2, in a circle at distances D1 and D2 from the cen... |
H: How to write $\frac{k}{k}$ using $\sum$ notation?
I want to use $\sum$ notation for this:$$\underbrace{\frac{1}{k}+\frac{1}{k}+\ldots +\frac{1}{k}+\frac{1}{k}}_{k\text{ times}}$$
I guessed$$\sum_1^k\frac{1}{k} ,$$but it equals $$1+\frac{1}{2}+\dots+\frac{1}{k-1}+\frac{1}{k}.$$
AI: You are getting confused because y... |
H: Equivalent definitions of algebra homomorphisms
I'm studying Atiyah-Macdonald's commutative algebra book and I'm trying to prove this equivalence:
One implication
If $h\circ f=g$. I can prove that $h(ax)=ah(x)$ but I have failed to prove that $f(x_1+x_2)=f(x_1)+f(x_2)$ (in order to complete the proof of A-module h... |
H: Permutations with repetition for some elements
Suppose we have $N$ slots, each of which can be filled with $X$ options, but $2$ of these slots can only be filled in $1$ way (out of $X$ ways), then what is the number of permutations possible ?
For example $N=4$,$\ X=\{a,b,c\}$ and one element must be $a$ and one mus... |
H: Why are these two statements about vector products equivalent?
Let $w_1, \dots, w_m \in \mathbb{C}^d$.
Condition (1) is:
$\sum_i |\langle v, w_i \rangle |^2 = \eta$ whenever $\|v\| = 1$.
Condition (2) is:
$\sum_i u_i u_i^* = I^d$, where $u_i = w_i / \sqrt{\eta}$
This paper claims that the two conditions are equival... |
H: Expected Value of Flips Until HT Consecutively
Suppose you flip a fair coin repeatedly until you see a Heads followed by a
Tails. What is the expected number of coin flips you have to flip?
By manipulating an equation based on the result of the first flip, shown at this link:
http://www.codechef.com/wiki/tutorial-expe... |
H: How to solve $x+(x\cdot\frac{20}{100})=600$?
I was solving a puzzle, and finally I got this equation as the result but I couldn't solve it.
$$x+\left(x\times\left(\frac{20}{100}\right)\right)=600$$
How to solve this equation?
Please provide a step by step solution.
AI: $$x+\left(x\times\left(\frac{20}{100}\righ... |
H: How many solutions are there to this equation involving the floor function: $(n+1)x-\lfloor nx \rfloor = c$?
How many solutions are there for this equation:
$(n+1)x-\lfloor nx \rfloor = c$
I can prove some basic properties of floors and ceiling, but here I'm stumped.
AI: Let $y_1(x) = (n+1)x$ and $y_2(x) = c + \lfl... |
H: Regarding continuity
We know that $\tau=\{G\subset \mathbb{N}: n\in G, m\mid n\Rightarrow m\in G\}$ is a topology on $\mathbb{N}$. I have shown that if $m\mid n\Rightarrow f(m)\mid f(n)$, then $f:(\mathbb{N},\tau)\to (\mathbb{N},\tau)$ is continuos. But I couldn't show the converse.
Let $f:(\mathbb{N},\tau)\to (\ma... |
H: Proof of the derivative of $x^n$
Can someone please explain why $(x^n)'=n\cdot x^{n-1}$?
Sorry for not writing it in math characters, I'm new here.
AI: Note that if $x>0$, then $\ln(x^n)=n\ln(x)$. Taking the derivative of each side, using $(\ln(t))'=\dfrac{1}{t}$, and the chain rule on the left side,
$$\dfrac{1}{x... |
H: 82% of an event in a persons life, what is the daily chance?
If there is an 82% chance that within your average lifetime, lets assume 70 years (25567 days), that "E" event will happen: what is the percent chance that it will happen on any given day?
I'm assuming that it'll be something like .02347% or something sma... |
H: Generalization of Fatou's lemma for nonpositive but bounded measurable functions.
Let $(f_n)^{\infty}_{n=1}$ be a sequence of measurable (not-necessarily $\ge 0$). Let $g \gt 0$ be a measurable function with $\int g d\mu < \infty$ (integrable) such that $f_n\ge -g$ a.e. relative to $\mu$ in $E\in S$ ($S$ $\sigma$-a... |
H: Find the values of $a,b$ so that the given limit equals $2$
$$\lim_{x\rightarrow 0}{(a\sin^2x)(b\log\cos x)\over x^4}={1\over 2}$$
My initial thought was to apply L'Hopital Rule so that at some stage a condition appears where i will have to set some value for $a,b$ to get the limit. This doesn't work though as the... |
H: Some questions about mathematics
This question is a soft one. Well, So far I have noticed stuff that is nice in math, particularly in algebra, topology and analysis. For instance, in algebra, there is theorem that says that we can think of groups just as some set of permutations. So, in other words, can we say all ... |
H: What factor has to be applied to $\phi(ab)\propto\phi(a)\phi(b)$ for non-coprime $a,b$?
For $a,b$ coprime, it is known that $\phi(ab)=\phi(a)\phi(b)$. But is there a connection between $\phi(ab)$ and $\phi(a),\phi(b)$ if they are not coprime?
AI: Note that Euler's product formula
$$\phi(n) = n\prod_{p\mid n}\left(1... |
H: closure and interior of subsets of a metric space
Suppose $X$ is a metric space and $S$ and $A$ are subsets of $X$.
If $S \subset A \subset Cl(S)$ , then $Cl(A) = Cl(S)$.
Also if $Int(S) \subset A \subset S$, then $Int(A) =Int(S)$.
What if, $Int(S) \subset A \subset Cl(S)$ , then would $Cl(A) = Cl(S)$ and $Int(A) ... |
H: partial derivatives continuous $\implies$ differentiability in Euclidean space
I am given this theorem:
If $f \in C^1(A,\mathbb R^m)$, i.e. every partial derivative of $f$ is
continuous on $A$, and $A$ is open in $\mathbb R^n$, then $f$ is
differentiable on $A$.
Is the following stronger assertion also true?
... |
H: Vanishing of a multivariable polynomial on a lattice
Let be $p(x_1,...,x_n) \in K[x_1,...,x_n]$ be a polynomial of degree $d$. Suppose there is a $n$-dimensional hyperbox $B = I \times \stackrel{n}{...} \times I = I^n$. Divide $I$ to $d$ segements by $d+1$ points. This creates a lattice of $(d+1)^n$ point on $B$. ... |
H: Members of equivalence classes in $Z(p^\infty)$
Let $p$ be a prime and let $Z(p^\infty)$ be the following subset of the group $\mathbb{Q}/\mathbb{Z}$:
$$Z(p^\infty)=\{\overline{a/b}\in\mathbb{Q}/\mathbb{Z}\mid a,b \in \mathbb{Z} \text{ and } b=p^i \text{ for some } i\geq0\}.$$
I am not sure if this is worth my ... |
H: Binomial Type sequence
If $p_0(n),p_1(n)\ldots$ is a sequence of polynomials satisfying
$$\sum_{k \geq 0}p_k(n) \frac {x^k}{k!}= \left ( \sum_{ k \geq 0}p_k(1) \frac {x^k}{k!} \right )^n$$
then
$$p_k(m+n)= \sum_{i=0}^{k} {k\choose i}p_i(m)p_{k-i}(n), \; k \geq 0$$
Any help would be appreciated.
AI: Hint : Try to wr... |
H: Compound quadratic problem
The first issue I have is that I am not sure why this is called a 'compound quadratic problem', but anyway to proceed:
Suppose that $x-y=14$ and $$(x+y)(x^2+y^2)(x^4+y^4)=a(x^b-y^b)$$
where $a$ and $b$ are constants. Find $a$ and $b$.
I am stuck as to how to approach this problem. I have ... |
H: Understanding cohomology with compact support
I am trying to understand the definition of (singular) cohomology with compact supports.
My understanding of singular cohomology goes like this. Let $X$ be a topological space. Define the singular chain group $C_n(X)$ to be the free abelian group generated by singular ... |
H: Fourier series of a square wave signal with a bias
Given a $f(t)$ of the kind:
$$f(t)=1, \{kt_0\le t\le kt_0+\tau\}$$
$$f(t)=a,\{kt_0+\tau\le t\le (k+1)t_0\}$$
with $a\lt 1$
what is the Fouries series development of f(t)?
Thanks
AI: $f$ is the sum of a constant and a multiple of a characteristic function of an inte... |
H: Centre of a simple algebra is a field
How can one show that the centre of simple algebra is a field?
I have tried it and proved that the inverse exists for every element of centre but cannot prove that inverse of every element belongs to centre. Please help me.
AI: If $ab=ba$ and $a^{-1}$ exists, then $a^{-1}b=a^... |
H: Countability over indexed families
I'm strugling with countability over indexed sets compared to ordinary sets. Basically we say that any set $A$ is countable if there is a bijective function $f$ such that $f:\mathbb{N}\longrightarrow A$. Here my problem is that I don't know if there is a similar standard definitio... |
H: Relation between quotients and subalgebras
If I have two algebras $A,B$, and one is the quotient of the other, i.e. there exists a surjective morphism $\phi : A \to B$. Then is $B$ isomorphic to some subalgebra of $A$? I think so, because I just need to select for each equivalence class $\phi^{-1}(b)$ one element ... |
H: How to solve matrix equation $AXH+AHX−BH=0$
How to solve matrix equation $AXH+AHX−BH=0$? All matrices are square, $A$, $B$ known constant matrices and invertible, $H$ can take any value, $X$ represent the solution to be found.
I have seen about the Sylvester Equation like in this post Solving a matrix equation $AX... |
H: Automorphisms of $\mathbb{G}_m$.
Let $\mathbb{G}_m$ be the multiplication group whose underlying set is $k^*$, where $k$ is a field. How to show that as an algebraic group there are only two automorphisms of $\mathbb{G}_m$? How many automorphisms are there of $\mathbb{G}_m$? I think that if $\phi: \mathbb{G}_m \to ... |
H: Uniform continuity?
Suppose that $f$ is continuous on the interval $I$, and $|f(x)|$ is uniformly continuous. Can we have the conclusion that $\sin^3f(x)$ is uniform continuous?
My intuition is that it is not. But I could not construct a counterexample. Would you help me out? Thank you.
AI: We have the conclusion ... |
H: Is there a continuous function on R such that $f(f(x))=e^{-x}$?
Is there a continuous function on R such that $f(f(x))=e^{-x}$? I have tried to take derivative of the two sides,but I can't get anything I want.what can I do?
AI: No. Hint: An injective continuous function is monotonic and for any monotonic $f(x)$ the... |
H: Represent $\frac{1}{3} \ln(x+2)^3 + \frac{1}{2}[\ln x - \ln (x^2+3x+2)^2]$ as a single logarithm
I am having some trouble trying to find the single logarithm for the following:
$$\frac{1}{3} \ln(x+2)^3 + \frac{1}{2}[\ln x - \ln (x^2+3x+2)^2]$$
I understand that I have to use the addition and subtraction rules but ... |
H: Ackermann function and primitive recursiveness
If we define $b_n(m) := a(n,m)$ for all $n$ and $m \in \mathbb{N}$. For which $n$ is the function $b_n$ primitive recursive and for which $n$ it is not a primitive recursive function? Can anyone please help me out with this?
AI: I assume that (as the title suggests)
$... |
H: A chess problem in Kanamori's "The Higher Infinite"?
Just after Corollary 21.17 (on p289) of Kanamori's The Higher Infinite, he outlines the direction in which he wants to take his discussion of iterated ultrapowers. However, immediately after he presents a chess problem. The passage reads as follows:
Having devel... |
H: Volume of a tetragonal pyramid
Express the volume $V$ of a regular tetragonal pyramid as a function of its altitude $x$ and the edge of a lateral face (lateral edge) $y$
The answer given by the book is $\frac{2}{3} (y^2 - x^2) x $. But,I've found the lateral edge is $2 \sqrt{y^2 - x^2 } $ and I thought that the a... |
H: Combinatorial Algebra with Variables
The problem is ${m+1} \choose {m-1}$. The answer is $\frac{m(m+1)}{2}$. I am stuck on solving this algebraically. If someone could tell me where the two comes from I would be helped, because I know how to get to m(m+1) just not two as the divisor.
AI: Here you go:
According to t... |
H: Solve $4^{9x-4} = 3^{9x-4}$
I am having some trouble trying to solve
$$4^{9x-4} = 3^{9x-4}$$
I tried to make each the same base but then I'm becoming confused as to what to do next.
These are the steps I took:
$$\begin{align}
4^{9x-4} &= 3^{9x-4} \\
\log_4(4^{9x-4}) &= \log_4(3^{9x-4}) \\
\end{align}$$
Where do I ... |
H: Help with proof of injection and surjection
For the record, I am sorry, I haven't yet learnt how to use LaTeX
I have a function $f(x) = 2x^3 - 1$
My proof of injection is as follows:
$f$ is one to one for all $x_1,x_2$ element $X$, if $f(x_1) = f(x_2)$ then $x_1 = x_2$
Proof
$f(x_1) = f(x_2)\\
2x_1^3 - 1 = 2x_2^3 -... |
H: Induced subgraphs (graph theory)
I have the following graph theory question that I am stuck on:
Prove or disprove:
For every graph G and every integer $r \geq \text{max} \{\text{deg}v: v \in V(G) \}$ , there is an r-regular graph H containing G as an induced subgraph. Thanks for any help.
AI: Can you construct an r... |
H: Finding critical points of $x^{(2/3)}(5-x)$
So I tried this out and got stuck with this:
$$0 = 3x^{(7/6)} + 2x - 10$$
I didn't think I could use a quadratic for this since its to the power of $7/6$
Here is the working I did:
We know its a critical point when f'(a) = 0
So I found the derivative of f(x) which is $$2*... |
H: Regression analysis on temperature/sensor data
Looking for a solution to what I thought should be an easy problem, but has me running in circles somehow...
I'm working with two sets of data: he first set is raw values from a sensor (accelerometer), and the second set is the temperatures at which these values were r... |
H: Standard notation for the set of integers $\{0,1,...,N-1\}$?
I was wondering if there exist a standard notation for the set of integers $\{0,1,...,N-1\}$. I know for example $[N]$ could stand for the set $\{1,2,...,N\}$ but what about the former, i.e. $\{0,1,...,N-1\}$?
AI: It depends on the context, but sometimes ... |
H: Using a function in Matlab
Very new to MATLAB and Im trying to use the FFT function. I got a video which showed me that a function and a normal m file is needed. Created that but now dont know how to call the function from the m file. Here's is the function:
function [ X,freq ] = positiveFFT( x,Fs )
%This function... |
H: How many ways can you distribute 3 types of candies to 8 children?
I have a big bag of candy. Peppermints, Chocolates, and Caramels. There are eight sweet children who deserve candy. One each, they are not that sweet. So I give each child a candy. How many ways are there for the candies to be distributed?
AI: I gue... |
H: Hamiltonian Graphs and connected graphs
Prove or disprove: There exists an integer k such that every k-connected graph is hamiltonian.
AI: Attempting to do these questions serve as good practice if you are doing a graph theory course.
An idea, I think look at $M_{2k+1}$ for all $k \geq 3$. |
H: Estimating the average number of passengers in cars in a parking lot.
All the workers at a certain company drive to work and park in the company’s lot. The company is interested in estimating the average number of workers in a car. Which of the following methods will enable the company to estimate this quantity?
... |
H: Solve using Pigeonhole principle
There are 45 candidates appear in an examination. prove that there are at-least two candidates in class whose roll numbers differ by a multiple of 44.
How can I prove this using pigeonhole principle?
AI: Consider the possible remainders mod $44$ as the boxes. There are $44$ of the... |
H: $L_2$ is of first category in $L_1$ (Rudin Excercise 2.4b)
We mean here $L_2$, and $L_1$ the usual Lebesgue spaces on the unit-interval. It is excercise 2.4 from Rudin. There's several ways to show that $L_2$ is nowhere dense in $L_1$.
But in (b) they ask to show that
$$\Lambda_n(f)=\int fg_n \to 0 $$
where $g_n =... |
H: What is the dot product between a vector of matrices?
There is a notation used in many sources (e.g. Wikipedia: http://en.wikipedia.org/wiki/Exponential_family) for the natural parameters of exponential family distributions which I do not understand, and I cannot find a description of.
With vector parameters and va... |
H: Differentiability of the function $f(z)=|z|^p z$($p>0$).
Suppose $p>0$ in a real number, is the function $f (z) =|z|^p z$ a differentiable function? Moreover, if $f (z) =|z|^p z$ is differentiable, dose $f$ belong to the space $C^{\left\lfloor p \right\rfloor,p-\left\lfloor p \right\rfloor}(\mathbb{C})$?
P.S. $\l... |
H: How to find support of functions
$\textbf{Support}$:$f$ is real valued function with domain $E^n$ the support of $f$ is the smallest closed set $K$ such that $f(x)=0$ for all $x$ is not in $K$
Find the support
$(1) f(x)=x-|x|$
$\displaystyle(2) f(x,y)=\frac{x}{e^{x^2+y^2}} $
$(3) f(x,y)=1$ if either x or y is a ra... |
H: distribution of the product of a poisson and a bolzmann
What is the distribution of the product of two variables for which each of them has its own distribution(specifically one poisson and one bolzmann)? I found on wikipedia that for the sum of the two you should use a convolution, but how is this for the product?... |
H: Given $ h(x)=f(x)+O(g(x)) $ estimate using asymptotic notation $\frac{1}{h(x)}$
Given $ h(x)=f(x)+O(g(x)) $ and knowing that $ \lim_{x \to \infty}=\frac{g(x)}{f(x)}=0$ (int other words $f(x)=o(g(x))$) find such F(x) and G(x), $\frac{1}{h(x)}=F(x)+O(G(x)) $.
Because $ f(x)=o(g(x)) $ that for any $e>0$ we can find $n... |
H: Coequalizer in $\mathsf{Sets}$
Assume $f,g:X \to Y$ are arrows in $\mathsf{Sets}$. Then the coequalizer is given by $c:Y \rightarrow Y/R$ where $R \subseteq Y\times Y$ is the smalles equivalence relation on $Y$ s.t. $\forall x \in X: (f(x),g(x)) \in R$. Given any $h:Y \rightarrow Z$ there is a unique $\overline h: ... |
H: Integral of the function $S(x)=\ln\left(1-\frac{x}{\exp(x)}\right)$
I have to check if the following series:
$$S(x)=\sum_{k=1}^{\infty}\frac{x^k}{k\exp(kx)}$$ gives a function of $x$
$$S(x)=-\ln\left(1-\frac{x}{\exp(x)}\right)$$
for which:
$$J=\left|\int_{0}^{+\infty}S(x)dx\right|\lt\infty$$
I used Maple and Mathem... |
H: Change of coordinates with Jacobian
We know that the change of variable in $\mathbb{R^n}$ with a $T: V \to U$ is a diffeomorphism of open sets in $\mathbb{R^n}$ and $f$ is an integrable function on $U$. Then
$$\int_U f dx_1 \cdots dx_k = \int_V (f \circ T) |\det(dT)|dy_1 \cdots dy_k.$$
How can i prove this in Manif... |
H: convergence in $L^p$ implies convergence of $p$-th moments
is there a simple way to prove that $X_n \rightarrow_{L^p} X$ implies that $\mathrm{E}(X^p_n) \rightarrow \mathrm{E}(X^p)$? the proof for $p=1$ is easy. but what about the case $p>1$? I would appreciate any comments. many thanks!
AI: If $0<a<b$ then, by the... |
H: basic linear system of equations
I am stuck with the following linear algebra problem:
given a basis $ \{e_{1} ... e_{n}\} $, I can define products $ \{p_{1} ... p_{m} \} $ as linear combination of the basis and the products itself:
$$
p_{i} = \sum_{j=1}^m u_{ij} p_{j} + \sum_{j=1}^n a_{ij}e_{ij}
$$
My goal is to ... |
H: Iteration of an operator
Let $f_0(x)$ be integrable on $[0,1]$, and $f_0(x)>0$. We define $f_n$ iteratively by
$$f_n(x)=\sqrt{\int_0^x f_{n-1}(t)dt}$$
The question is, what is $\lim_{n\to\infty} f_n(x)$?
The fix point for operator $\sqrt{\int_0^x\cdot dt}$ is $f(x)=\frac{x}{2}$. But it's a bit hard to prove ... |
H: Fibonacci Sequence Exercise
I need some help checking the following solution.
The Fib sequence is defined by $a_1 = 1, a_2 = 1$ and for all $n\geq 2$, $a_{n+1} = a_n + a_{n-1}$. Thus, the sequence begins:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...
Prove that for all $n\geq 1$, $a_n <\left(\frac{5}{3}\right)^n$.
So far... |
H: Difference between Probability and Probability Density
This question is from DeGroot's "Probability and Statistics" :
Unbounded p.d.f.’s. Since a value of a p.d.f.(probability density function) is a probability density, rather than a
probability, such a value can be larger than $1$. In fact, the values of the fo... |
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