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H: Puzzle in Percentages
Okay, this is a real-time problem.
The following is a picture of Customer satisfaction rating, which was displayed next to an item in an online shopping website.
Satisfied customers click the vote-up button, and unsatisfied customers click the vote-down button. Every time a button is click... |
H: Trying to find more information about "Darboux's method/theorem" on coefficients of an analytic function
My supervisor briefly showed me a statement of something she called "Darboux's theorem," but I am having trouble finding more information about it on the internet. Here is what I have written down (some details ... |
H: $\varepsilon$-$\delta$ proof for inverse function
I have been struggling to prove the following claim without appealing any other theorem, except the definition of continuity. If any one give a step by step argument for it, that would be great. Many thanks!
Let $f$ be strictly increasing function on $[a,b]$. If $f... |
H: Combinatorial interpretation of this number?
It is straightforward to show that if $m,n\in\mathbb{Z}$ and $m\geq n$, then
$$m\mid \gcd(m, n)\binom{m}{n}.$$
I'm trying to find a combinatorial interpretation of this fact, but I can't seem to come up with one. My two proofs are formal, and do not give me any combinato... |
H: Why the matrix of $dG_0$ is $I_l$.
I am reading the proof of Local Immersion Theorem in Guillemin & Pallock's Differential Topology on Page 15. But I got lost at the following statement:
Define a map $G: U \times \mathbb{R}^{l-k} \rightarrow \mathbb{R}^{l}$ by
$$G(x,z) = g(x) + (0,z).$$
The matrix of $dG_0$ is $I_... |
H: Valid Alternative Proof to an Elementary Number Theory question in congruences?
So, I've recently started teaching myself elementary number theory (since it does not require any specific mathematical background and it seems like a good way to keep my brain in shape until my freshman college year) using Elementary N... |
H: differential equation with parameter
I've got problem with such an example:
Given equation $\frac{dx}{dt} = -x + x^7$ with initial condition $x(0) = \lambda$, where $x=x(t, \lambda)$. Find $\frac{ \partial x(t, \lambda)}{\partial \lambda} \mid _{\lambda = 0}$.
What I've got now is:
We can write: $x(t, \lambda) = ... |
H: Proof of Abel-Ruffini's theorem
From Galois Theory (Rotman): I wrote down the whole proof, but my question is only about the third paragraph.
There exists a quantic polynomial $f(x) \in \Bbb{Q}[x]$ that is not solvable by radicals.
Proof If $f(x)=x^5 -4x + 2$, then $f(x)$ is irreducible over $\Bbb{Q}$, by Eisenste... |
H: Minpoly and Charpoly of block diagonal matrix
I am currently struggling with an exercise where I have to treat a Block diagonal matrix (so it is a square matrix, where square block matrices are down the diagonal). Now I was wondering whether we can say something about the characteristic or minimal polynomial of the... |
H: Prime Numbers and Multiples?
Other than prime numbers are all numbers multiple of 2,3,5 and 7 (Other Prime numbers as well). Suppose like if we need 8 it's the combination of 2.2.2, and 15 as 5.3 etc.
AI: No:
$$11\cdot 13 = 143$$
$$13 \cdot 17 = 221$$
$$\vdots$$
The most we can say is what the Fundamental Theorem... |
H: Do every $3$ linearly independent vectors span all of $\mathbb{R}^3$?
I am given $3$ vectors that are linearly independent. I am trying to figure our if they span all of $\mathbb{R}^3$ to declare them as basis.
AI: Yes, because $\mathbb R^3$ is $3$-dimensional (meaning precisely that any three linearly independent... |
H: Proofs from the "Ugly Book"
There is a famous saying in mathematics from Paul Erdős: "You don't have to believe in God, but you should believe in The Book." "The Book" is an imaginary book in which God had written down the best and most elegant proofs for mathematical theorems.
If there is a book written down by Go... |
H: Prove $\sum_{n=2}^{\infty}{\frac{1}{n\ln (n)}}$ diverge
Prove that $\sum \limits_{n=2}^{\infty}{\dfrac{1}{n\ln (n)}}$ diverge
I know it's a well-known series, but all the proofs I've seen are based on the integral test and Cauchy condensation test. I need to prove it using only the following tests: Direct/limit Com... |
H: When $\;\text{FALSE}\implies P(x),\;$ is $P(x)\;$ false?
Say we know that $P(k) \implies P(k+3)$. Then if we know $P(1)$ is true, we know $P(4), P(8) \dots$ are also true.
However if we know $P(1)$ is false, does that mean $P(4), P(8) \dots$ are also false?
I'm a little confused on if we can trust an implication w... |
H: Proving $g(\omega)=\frac{1}{2\pi i}\int_{\gamma}\frac{zf'(z)}{f(z)-\omega}\, dz$ where $g$ is the inverse of $f$
I have the following exercise:
Let $G$ be an open subset of $\mathbb{C}$ and let $f$ be a one to one
function in $H(G)$ such that $f'(z)\neq0$ for all $z\in G$.
For each $\omega\in f(G)$ let $g(\omega... |
H: Better Proofs Than Rudin's For The Inverse And Implicit Function Theorems
I am finding Rudin's proofs of these theorems very non-intuitive and difficult to recall. I can understand and follow both as I work through them, but if you were to ask me a week later to prove one or the other, I couldn't do it.
For instanc... |
H: An odd integer minus an even integer is odd.
Prove or Disprove: An odd integer minus an even integer is odd.
I am assuming you would define an odd integer and an even integer. than you would use quantifiers which shows your solution to be odd or even. I am unsure on how to show this...
AI: An even number is an inte... |
H: Is $\binom{52}{n}\cdot\binom{52-n}{n}$ maximised by $n=\frac{52}{3}$? If so why?
I've been thinking a lot about cards, and recently about the combination of combination of hands....if you drew $n$ cards from a deck of $52$, and then drew another $n$ cards, how many combinations are there?
The number of combinations... |
H: Problem 3.1.2 in Liu -- Omission in problem statement?
Exercise 3.1.2 in Liu's Algebraic Geometry and Arithmetic Curves is as follows.
Let $f:X\rightarrow Y$ be a morphism of schemes. For any scheme $T$,
let $f(T):X(T)\rightarrow Y(T)$ denote the map defined by
$f(T)(g)=f\circ g$. Shows that $f(T)$ is bijecti... |
H: Finding the dual of this primal LP.
I am going over sample questions from a sample exam, and I got stuck on the following question. I need to determine the dual of this LP:
$min: c^Tx + d^Tu \\
s.t: Ax + Du = b\\
x \ge 0$
$A$ is an $m$ by $n$ matrix, D is an $m$ by $p$ matrix. So this is what is throwing me off whe... |
H: Self-adjoint operator on a finite-dimensional vector space
Let $V$ be a finite-dimensional inner product space and let $x,y\in V$ be nonzero vectors. If there is a self-adjoint operator $A:V\rightarrow V$ such that $A(x)=y$ and $\langle A(v),v\rangle\geq0$ for all $v\in V$, then $\langle x,y\rangle>0$.
I think we ... |
H: Change of limits of integration: $ \int_2^x \frac{\pi(x/u)}{\log u}du$
This is a change of variables that I do not see. $\pi(v)$ is the number of primes not exceeding $v.$
$$I_1 = \int_2^x \frac{\pi(x/u)}{\log u}du$$
becomes via $v = x/u$
$$I_2 = x\int_2^{x/2}\frac{\pi(v)}{\log x- \log v}\frac{dv}{v^2} $$
I see t... |
H: self studying advice on analysis
I am trying to learn analysis on my own but there are times when I can't solve
the problem or I get the solution wrong after looking it up, but I will only
look up the problems online after I am completely done trying the problems
out. The solutions I find to some of these problems ... |
H: Considering a sum of a monotonically increasing and decreasing sequence.
The following is the problem that I am working on.
Let $\{z_n\} = \{x_n\}+\{y_n\}$ be a sequence where $\{x_n\}$ is monotonically increasing, $\{y_n\}$ monotonically decreasing, and $\{z_n\}$ is bounded.
Is $\{z_n\}$ convergent ? What if $\{x... |
H: Why does a non-zero density function not imply infinitude of what it measures?
Consider the following density function for the twin primes:
Numbers $x-2$, $x-4$ are twin primes iff:
$x \ne 2,4 \ mod \ 2 $
$x \ne 2,4 \ mod \ 3 $
$x \ne 2,4 \ mod \ 5 $
$x \ne 2,4 \ mod \ 7 $
...
$x \ne 2,4 \ mod \ max prime < x-2... |
H: Comparing exponents.
Is there an easy way to compare $a^m$ and $b^n$ where $1 \le a, b, m, n \le 1000$ without raising the corresponding bases to their power (since it can easily cause an overflow)? Thank you.
AI: Take the logs to your favorite base and compare $m \log a$ and $n \log b$. This looks like it bears o... |
H: Lebesgue Measure and Symetric Difference
I need to know if this is true because I want to use it in a proof.
Let $\mu$ be the Lebesgue measure and $\Delta$ denote the symmetric difference. Is is true that $\mu(A\Delta B)-\mu(B\Delta D)-\mu(A\Delta C) \leq \mu(C\Delta D)$? I tried to apply the definition of $\Delta$... |
H: Frequency Change with Tape Speed
I recorded my own voice in an old tape recorder. When I put the device in fast forward mode my voice turned a little squeaky. I wonder if the fundamental frequency of the voice had changed? Is that possible?
AI: Yes, it is possible. If it is playing faster, that is outputting the a... |
H: Maximizing score in number-guessing game.
This is inspired by a puzzle (related to the two-envelopes problem) that I've seen in several places, including unbounded generalizations. The basic premise is that Alice chooses two real numbers from $[0,1]$ uniformly at random, and writes them down on slips of paper. ... |
H: Trying to understand set theory? Subset vs Member?
So I'm learning set theory and wanted to know if I'm understanding some of the basics correctly. I have a feeling 1 and 2 are correct, but I'm not sure about 3 and 4.
Z = {1,3}
W = {1,2,3,4}
1) Is Z ∈ W? No, Z is not a member or W, because Z is the set {1,3}, whi... |
H: Let E be the splitting field of $f(x)=x^4-10x^2+1$ over $\Bbb{Q}$. find $Gal(E/\Bbb{Q})$.
From Galois Theory (Rotman):
Let E be the splitting field of $f(x)=x^4-10x^2+1$ over $\Bbb{Q}$. find $Gal(E/\Bbb{Q})$. The roots of $f(x)$ are $\sqrt{2}+\sqrt{3}$, $\sqrt{2}-\sqrt{3}$, $-\sqrt{2}+\sqrt{3}$, $-\sqrt{2}-\sqrt{3... |
H: Confused on permutation cycles
I am a bit confused on how to interpret permutation cycle notation. I am going by the Wolfram definition. My initial interpretation of $(431)(2)$ was "4 moves to position 3, 3 to position 1, 1 to 4, 2 remains in place". which would leave me at ${3, 2, 4, 1}$ which apparently is incorr... |
H: Rudin's Construction of Lebesgue Measure
Self-studying Rudin's RCA, and I want to make sure I am understanding the intricacies of his construction of the Lebesgue measure on $\mathbb{R}^n$.
The uniform continuity of $f$ shows that there is an integer $N$ and
functions $g,h$ with support in $W$ such that: (i) $g,... |
H: The gradient of a function is an alternating one-tensor
I'm currently reading Spivak's Calculus on Manifolds and I seem to have hit a snag in Chapter Four: Integration on Chains. Spivak develops tensors, vector fields, alternating tensors and differential forms. I'm okay with these ideas however he makes the claim ... |
H: Why are the first few powers of $2^{10}$ a little more than those of 1000?
See the complete list here: http://en.wikipedia.org/wiki/Power_of_two#Powers_of_1024.
I'm wondering if there's a mathematical explanation for the relationship or if it's just coincidence.
AI: Since $2^{10}=1024$:
$$2^{10n}=(1000+24)^n=1000^... |
H: How prove this $|x_{p}-y_{q}|>0$
let $$x_{1}=\dfrac{1}{8},x_{n+1}=x_{n}+x^2_{n},y_{1}=\dfrac{1}{10},y_{n+1}=y_{n}+y^2_{n}$$
show that: for any $p,q\in N^{+}$
we have $$|x_{p}-y_{q}|>0$$
AI: Clearly all terms in both sequences are rational.
We will show by induction that if $y_n = \frac{ a_n}{b_n}$ where $\gcd(a_n,b... |
H: Probability Question about Tennis Games!
$2^{n}$ players enter a single elimination tennis tournament. You can assume that the players are of equal ability.
Find the probability that two particular players meet each other in the tournament.
I could't make a serious attempt on the question, hope you can excuse me t... |
H: Question regarding Hartshorne Example II.(6.5.2)
Let $k$ be a field, let $A=k[x,y,z]/\langle xy-z^2\rangle$ and let $X=\operatorname{Spec}A$.
Let $Y:y=z=0$ I want to know the divisor of $y$
In Hartshorne book, because $y=0 \Rightarrow z^2=0$ and $z$ generates the maximal ideal of the local ring at the generic point... |
H: Without appealing to choice, can we prove that if $X$ is well-orderable, then so too is $2^X$?
Without appealing to the axiom of choice, it can be shown that (Proposition:) if $X$ is well-orderable, then $2^X$ is totally-orderable.
Question: can we show the stronger result that if $X$ is well-orderable, then so too... |
H: Inequality in geodesic quadrupel
Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y \in AB$. Is it true that then $d(x,y)\geq \frac{1}{4}$ ? I have tried a lot of thin... |
H: A metric space is separable iff it is second countable
How do I prove that a metric space is separable iff it is second countable?
AI: HINT: One direction is pretty trivial; I’ll leave it to you, at least for now. The harder direction is to prove that a separable metric space is second countable. Suppose that $\la... |
H: What are the first 3 digits of the product of the first 1000 fibonacci numbers
What are the first 3 digits of the product of the first 1000 Fibonacci numbers?
Could anyone give me hints on how to start this problem? I haven't done a problem like this before and I am curious on how to approach such a problem.
AI: ... |
H: Upper Bound of Logarithm
For $1\leq x < \infty$, we know $\ln x$ can be bounded as following:
$\ln x \leq \frac{x-1}{\sqrt{x}}$.
Then what is the upper bound of $\ln x$ for following condition?
$2\leq x <\infty$
AI: Given $a>0$, if $a \le u < \infty$ then also $1 \le u/a < \infty$, and you can apply your inequa... |
H: recurrence relation: $x_{n+1} = x^2_n - 2x_n + 2$
$$x_0 = \frac32; x_{n+1} = x^2_n - 2x_n + 2$$
$$\Rightarrow x = x^2 - 2x +2 \Rightarrow x^2 - 3x +2 = 0 \Rightarrow x = {1;2} $$
How to determine which one is the limit, i.e. $\lim_{n\rightarrow \infty} x_n = 1$ or $\lim_{n\rightarrow \infty} x_n = 2$ ?
AI: HINT: Th... |
H: Is a semigroup $G$ with left identity and right inverses a group?
Hungerford's Algebra poses the question: Is it true that a semigroup $G$ that has a left identity element and in which every element has a right inverse is a group?
Now, If both the identity and the inverse are of the same side, this is simple. For, ... |
H: Equation with Parameter
I want to solve a seemingly simple equation:
$k+(k-2)*x = (2k+3)x-2x-3$
wolfram alpha says that k = -3 and x = 1 - but I don't see yet how I arrive
at this solution.
Thanks
AI: Expand the equation
$$k + kx -2x = 2kx + 3x -2x -3$$
$$k-kx =3x-3$$
$$k(1-x)=-3(1-x)$$
$$k(1-x)+3(1-x)=0$$
$$(k+3)... |
H: Work and Time related problem
A and B do a work together in 16 days. B and C can do the same work together in 20 days. How many days will it take for A, B and C to do the work together?
AI: Let $A,B,C$ represent their daily efficiencies in percentage, and the total amount of work to be $1$(think about $100\%$).
The... |
H: basis of space and subspace
I have a question which is pretty basic about basis and verctor spaces.
Generally, if I have a basis K of vector space V,
Why it is not a basis of a subspace W of V (real subspace)?
The vectors in basis K are linearily independent and for every vector of W I can find a linear combination... |
H: puzzle/equation with proportionality
We have three pumps filling a tank:
The first one fills the whole tank in a particular time, the second one is twice as fast as the first one, and the third one is three time as fast.
All the pumps together can fill the tank in two minutes.
How long would each pump take on its o... |
H: Let $p\colon X \to Y$ be a quotient map then if each set $p^{-1}({y})$ is connected, and if $Y$ is connected, then $X$ is connected.
Let $p\colon X \to Y$ be a quotient map. Show that if each set $p^{-1}({y})$ is connected, and if $Y$ is connected, then $X$ is connected.
I am totally stuck on this problem. Can I ... |
H: Measurability of projections
Let $(X,\mathcal{A})$ and $(Y,\mathcal{B})$ be measurable spaces. Consider the product space $X\times Y$ with the product $\sigma$-algebra $\mathcal{A}\otimes \mathcal{B}$ and $\pi:X\times Y \rightarrow X$ the natural projection.
Is it true that $Z \in \mathcal{A}\otimes \mathcal{B}$ im... |
H: Tricky Trigonometry Problem
I solved this problem by applying the law of sines to get the values for $\omega$ (74.61) and $Z$ (5.65) respectively. But now I'm told there is another combination of $\omega$ and $Z$ that will work. How could we find that?
AI: Hints:
$$\frac6{\sin\omega}=\frac4{\sin40^\circ}\implies\si... |
H: one-to-one correspondence between the divisors of $n$ and
Could any one help me to understand rigorously the statement from a book Elementary Number Theory by James J Tattersall
, Page $51$
"From the definition of
division and the fact that divisions pair up, it follows that, for any positive
integer $n$, there is ... |
H: Embeddings of 3 point metric spaces into ultra-metric spaces with distortion 2
I need to show that every 3 point metric space has an embedding into an ultra-metric space with distortion 2.
And then to show such an example.
How would I go about it?
Thank you.
Edit:
Distortion is defined as following:
An embedding $f... |
H: About functions and relations
A pure threoretical question here. I have the relation $\pm\sqrt x$. As far as I understand it, that is not a function, as 1 input can map to 2 outputs. If I have a relation, which is not a function, and I limit the inputs, so that every input maps to only one output, can the relation ... |
H: What are the set-theoretic foundations of classical calculus?
I am wondering what set theoretical foundation is needed for the development of classical results of, say, calculus, such as taught in first years undergraduate courses. More concretely, I wonder if and where the full power of ZFC is needed, if that's th... |
H: Euclidean Distance on a Sphere
I have that the Euclidean distance on the surface of a sphere in terms of the angle they subtend at the centre is
$(\sqrt{2})R\sqrt{1-\cos(\theta_{12})}$
(Where $\theta_{12}$ is the angle that the two points subtend at the centre.)
Why is this; what is the proof?
Cheers, Alex
AI: Cons... |
H: Well ordered sets
$(X,\le)$ is totally ordered. How do you prove that if every non empty countable subset of $X$ is well ordered then $(X,\le)$ is well ordered?
AI: Now you have the correct conditions, the proof goes like this:
Let $Y\subseteq X$ be non-empty. If it is not well ordered then it is uncountable. Let $... |
H: Fundamental volume of quotient group
I came across this rule in my old notes, but I need an explanation to how it could possibly originate:
The theorem says that for any lattice $L$ in $\mathbb{R}^n$, the order of the quotient group, $\lvert L/aL\rvert$, where $a$ belongs to $\mathbb{R}^n$, is given by $\text{vol}(... |
H: Fibonacci-like sequence
Today I have to deal with something which reminds Fibonacci sequence. Let's say I have a certain number k, which is n-th number of certain sequence. This sequence however is created by recursive formula that we know from Fibonacci $a(n) = a(n-1) + a(n-2)$, where $n \ge 2$ and $a(0) \le a(1) ... |
H: Sums of powers below a prime
Given a prime $p$ and a natural number $k$, such that $k$ is not divisible by $p - 1$, prove that $\sum_{i = 1}^{p - 1}i^k \equiv 0 \pmod p$.
I split the proof into two cases: one where $k$ is odd and another where it is even.
The case where $k$ is odd can be proven as follows:
$$\sum_{... |
H: to find number of distinct root of a three degree polynomial
Given that $a,b,c$ be three distinct real numbers then the number of distinct real roots of the equation $p(x)=(x-a)^3+(x-b)^3+(x-c)^3=0$ is
1
2
3
depends on $a,b,c$
what I did is $p'(x)=0$ which is two degree polynomial with three distinct root, so $p... |
H: (From Lang $SL_2$) Orthonormal bases for $L^2 (X \times Y)$
Lang $SL_2$ p. 13 :Let $\{\phi_i\}$, $\{\psi_i\}$ be orthonormal bases for $L^2(X)$ and $L^2(Y)$ respectively. Let $$\theta_{ij}(x,y) = \phi_i(x)\psi_i(y).$$ Then $\{\theta_{ij}\}$ is an orthonormal basis for $L^2(X \times Y)$. To see this, it is first cl... |
H: Relation between inverse and inverse of adjoint in a unital C*-algebra
Let $\mathcal{A}$ be a unital C*-algebra.
Is the following statement true?
$$A \text{ not invertible} \Leftrightarrow A^* \text{ not invertible}$$
Moreover, suppose $A\in \mathcal{A}$ has an inverse $A^{-1}\in \mathcal{A}$, what can I say about... |
H: Are the closed ordinals (apart from $0$ and $1$) precisely the regular cardinals?
Given a partially ordered set $P$, a collection of partially ordered sets, call it $\mathcal{Q}$, and an arbitrary function $f : P \rightarrow \mathcal{Q},$ we can form a new partially ordered set $\Sigma f$ as follows.
The elements ... |
H: Compare standard deviation with out using the standard formula
I very well know that Standard Deviation is the measure of spread of the data. If the data has higher deviation from its means then it has higher standard deviation.
What if the data has same mean and range and just a couple of values changed e.g.
Set1 ... |
H: Solving for X in a simple matrix equation system.
I am trying to solve for X in this simple matrix equation system:
$$\begin{bmatrix}7 & 7\\2 & 4\\\end{bmatrix} - X\begin{bmatrix}5 & -1\\6 & -4\\\end{bmatrix} = E $$ where $E$ is the identity matrix.
If I multiply $X$ with $\begin{bmatrix}5 & -1\\6 & -4\\\end{bmatri... |
H: Show that there are infinitely many solutions of distinct natural numbers $m,n$ such that $n^3+m^2\mid m^3+n^2$
Show that there are infinitely many solutions of distinct natural numbers $m,n$ such that $n^3+m^2\mid m^3+n^2$.
This question appeared in Round $2$ of British Math Olympiad $2007-08$. I have been tryi... |
H: If $f(-1) = 0$ and $f(2)=0$, and if $g(x)= 2x-1$, then find the value of $x$ for which $(f\circ g)(x) = 0$
I have found the following problem.
If $x = -1$ and $x=2$ then $f(x) = 0$. If $g(x)= 2x-1$, then find the value of $x$ for which $f\circ g(x) = 0$.
I have solved the above problem in the following way.
$f... |
H: How to solve equations with logarithms, like this: $ ax + b\log(x) + c=0$
I encountered an equation of type $$ ax + b\log(x) + c=0$$ Here a, b, and c are constants. Does anyone know how to solve these type of equations? I guess this way:
$$\log(x)= \frac{c-ax}{b}$$
$$x= 10^{(c-ax)/b}$$
But I do not even know how to... |
H: How to show $0$ is a point of closure of weak topology, but not a limit of weakly covergent sequence in a a subset of $l^2$
(von Neumann)For each natural number $n$, let $e_n$ denote the sequence in $\mathcal {l}^2$ whose $n$th component is $1$ and other components vanish. Define$$E = \{e_n + n \cdot e_m : n,m \in... |
H: different shaped open basis on $\mathbb{R}^2$
$J_1$ be the smallest topology on $\mathbb{R}^2$ containing the sets $(a,b)\times (c,d)\forall a,b,c,d\in\mathbb{R}$
$J_2$ be the smallest topology on $\mathbb{R}^2$ containing the sets $\{(x,y):(x-a)^2+(y-b)^2<c,\forall a,b\in\mathbb{R},c>0\}$
$J_3$ be the smallest top... |
H: How to find Perimeter and Area?
I have this question:
A rancher has 480 feet of fencing with which to enclose two adjacent rectangular corrals. What dimensions should be used so that the enclosed area will be a maximum?
x=....ft
y=....ft
Below is my work:
P=4x+2y
A=2xy
480=4x+2y
240-2x=y
A=2x(240-2x)
480x-4x^2
I ... |
H: Classical probability, need my work checked.
A kinder egg may contain a prize of type $A$, of type $B$, or be empty. The probability of a prize $A$ is $5$ times larger than of prize $B$. A box contains 12 kinder eggs, from which 7 are known to be empty. Three eggs are drawn at random.
a.) Find the prob. that you f... |
H: How does one combine proportionality?
this is something that often comes up in both Physics and Mathematics, in my A Levels. Here is the crux of the problem.
So, you have something like this :
$A \propto B$ which means that $A = kB \tag{1}$
Fine, then you get something like :
$A \propto L^2$ which means that $A = k... |
H: Manifolds with finitely many ends
In the article ' The structure of stable minimal hypersurfaces in $ R^{n+1} $ ( http://arxiv.org/pdf/dg-ga/9709001.pdf) of Cao-Shen-Zhu the remark 2 at page 3 contains a statement that i don't understand (actully it seems me false).
Let $ M $ be a manifold and let $ \{K_n\} $ be an... |
H: Derivative of $\sqrt[3]{x} e^x$
$$y=\sqrt[3]{x} e^x$$
I have no idea how to solved it.
AI: Start with writing $\sqrt[3]x=x^{\frac 13}$. Then use the product rule. |
H: Etymology of Tor and Ext Functors
The names of the derived functors $\operatorname{Tor}$ and $\operatorname{Ext}$ seem quite cryptic to me. Does anyone know what these abbreviations stand for? I would be glad if someone could tell me where these names come from.
AI: Ext stands for extension, as the group $\operator... |
H: How many different sets of $9$ questions can the student select ?
A history examination is made up of $3$ set of $5$ Question each and a student must select $3$ questions from each set . How many different sets of $9$ questions can the student select
?
I am in dilemma how i can use combination formula
AI: From ea... |
H: Inequality for finite harmonic sum and logarithm
How do you prove the inequality:
$|\sum_{k=1}^n 1/k - \log n | \leq 1$
?
AI: Hint:
Note that the expression is just the difference between the lower Riemann sum of the integral:
$$\int_0^n \frac{1}{x} dx$$
And the actual integral, $\log n$. |
H: $A$ be a complex $3\times 3$ matrix such that $A^3=-I$
Let $A$ be a complex $3\times 3$ matrix such that $A^3=-I$, then we need to find out which of the following statements are correct?
$A$ has three distinct eigenvalues;
$A$ is diagonalizable over $\mathbb{C}$;
$A$ is triangulizable over $\mathbb{C}$;
$A$ is non... |
H: Find $n$ such that $x^n \equiv 2 \pmod{13}$ has a solution
I am stuck on the following problem:
Consider the congruence $x^n\equiv 2\pmod{13}$. This congruence has a solution for $x$ if
$n=5$
$n=6$
$n=7$
$n=8$
Can someone explain in detail? Thanks in advance.
AI: Hint: $13$ is prime, so we know there is some pr... |
H: test for the convergence of the series
Test the convergence of summation $$\sum_{n=1}^\infty x_n$$ where $$x_{2n-1}=\frac{n}{n+1}\\ x_{2n}=-\frac{n}{n+1}$$
That is the series $$\frac 1 2-\frac 12+\frac 23-\frac 23 +-\cdots$$
what I did was let Sn be the partial sums of the series.Then
$$S_n=\begin{cases} 0 & \text{... |
H: Calculate x,y line terminiating point of section of a circle
I have a Cartesian plane running from -41 to 41 on the x and y axes and a circle centered on 0,0 with a radius of 41 divided up into a number of sections of different areas. I know the percentage area of each section (ie: section 1 is 16.1% of the total ... |
H: How do I solve $\; 3^{2x+1}-10\cdot 3^x+3=0 \quad?$
Solve the following equation for $x$ : $ \quad3^{2x+1}-10\cdot 3^x+3=0 $
I am baffled to solve this equation. With graphing I have found the answers to be x=1 and x=-1. I would like to know how to solve this equation though.
I have tried many different approache... |
H: Prime factors + number of Divisors
I know that one way to find the number of divisors is to find the prime factors of that number and add one to all of the powers and then multiply them together so for example
$$555 = 3^1 \cdot 5^1 \cdot 37^1$$
therefore the number of divisors = $(1+1)(1+1)(1+1) = 2 \cdot 2 \cdot 2... |
H: How to find the range of $f(x) = {e^x \over x-1}$
I want to find the range of the following function :
$$f(x) = {e^x \over x-1}$$
How do I find the range of the above function ? I have tried a lot , but do not have any idea to solve this.
AI: There is a discontinuity at $x=1$. Analyze both sides separately. Let's... |
H: Evaluate series of $\sum_{n=1}^\infty \frac{1}{n^2+n}$
I am trying to evaluate this sum, I know that $\sum\limits_{n=1}^\infty \dfrac{1}{n^2+n}$ is called telescopic series:
$$\sum_{n=1}^\infty \frac{1}{n(n+1)}$$
and I can show that as:
$$\frac{1}{k}-\frac{1}{k+1}$$ I would like to get some hint how I can evaluate ... |
H: Prove that for any invertable $n\times n$ matrix A, and any $b\in\mathbb{R}^n$, there exists a unique solution to $Ax=b$
I think I've got the two ideas needed to solve this, but it feels like they're not tied together properly. I'm not sure if I'm allowed to do something like this:
Let $A$ be an invertable $n\time... |
H: Expected Value for a Sequential Poisson Random Variable
The set up:
A mouse nest contains $n$ female mice. In a particular year, the number of female offspring that each female mouse produces has the following pmf:
$$
f(x) = \begin{cases} \frac{\alpha}{x!} & x = 0, 1, 2, \ldots\\[8pt] 0 & \text{else} \end{cases}
$$... |
H: What does the inverse Fourier transform of a constant non-zero function look like.
Worded another way, what does it look like to have all frequencies present at the same amplitude?
AI: It's a Dirac delta (details depending on which convention you use). |
H: Find the limit of the sequence $x_n=\frac{1}{n^2}+\frac{2}{n^2}+\cdots+\frac{n}{n^2}$.
Help me please, I don't know if I should treat this sequence as a series.
Thank you.
AI: Since $$1+2+\cdots+n=\frac{n(n+1)}{2},$$ we have that $$x_n=\frac{n(n+1)}{2n^2}=\frac{1}{2}\cdot\frac{n+1}{n}.$$
Can you solve it from here? |
H: Evaluation of $\sum_{n=1}^{\infty}\frac{2n-1}{2^n}$
I`m trying to evaluate this series and would like to get some advice how to do that.
What I need to find here to start with?
$$\sum_{n=1}^{\infty}\frac{2n-1}{2^n}= \frac{1}{2} + \frac{3}{4}+\frac{5}{8}+\dots$$
AI: Note that
$$\dfrac{2n-1}{2^n} = \dfrac{n}{2^{n-1}}... |
H: Why does Totally bounded need Complete in order to imply Compact?
Why does "Totally bounded" need "Complete" in order to imply "Compact"?
Shouldn't the definition of totally bounded imply the existence of a convergent subsequence of every sequence?
AI: No, total boundedness of $\langle X,d\rangle$ implies that ever... |
H: The general idea of prove openness.
I never really get the idea of proofs involves openness, here's an example:
Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a local diffeomorphism. Prove that the image of $f$ is an open interval.
So, is the general principle is to show:
Step n: For an arbitrary $y \in$ $f(U)$, ... |
H: Solve $\frac{dx}{dt} = x^3 + x$ for $x$
This is a seemingly simple first order separable differential equation that I'm getting stuck on. This is what I have so far:
$$\frac{dx}{dt} = x^3+x$$
goes to
$$\frac{dx}{x(1+x^2)} = dt$$
Now using partial fractions to integrate the left-hand side:
$$\frac{1}{x(1+x^2)} = \f... |
H: Why is $n+\omega=\omega$ for finite $n$?
I am trying to understand the following claim;
$n+\omega=\omega$ where $n$ is order type of a finite set and $\omega$ is the order type of $\left\{ 1,2,\dots, \right\}$ with the usual meaning of $<$.
My question is how is it possible, for instance, $\left\{1,2,\dots ,n,1,2,\... |
H: Understanding the definition of the represention ring
In Fulton, Harris, "Representation Theory. A first Course" there's the following paragraph which I don't really understand:
The representation ring $R(G)$ of a group $G$ is easy to define. First, as a group we just take $R(G)$ to be the free abelian group gener... |
H: Basic question on constructing a homomorphism between two cyclic groups
We want to find a non-trivial homomorphism between $Z_3$ and $Z_{24}$.
If we have $f:Z_3 \rightarrow Z_{24}$ then all I can really say form the properties of homomorphism that $f(0) = 0$ since the identity in $Z_3$ must be mapped to identity i... |
H: Distinguishing between the different eigenvalues
Consider the symmetric matrix
$$A=\begin{pmatrix} 2 & t & \cos t-1 \\ t & 2 & 0 \\ \cos t-1 & 0 & 2 \end{pmatrix}. $$
The (real) eigenvalues of $A$ can be found easily using the quadratic formula: they are $$\lambda_1,\lambda_2,\lambda_3=2,\frac{1}{2} \left(4-\sqrt{... |
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