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H: What is the name of the logical puzzle, where one always lies and another always tells the truth?
So i was solving exercises in propositional logic lately and stumbled upon a puzzle, that goes like this:
Each inhabitant of a remote village always tells the truth or always lies. A villager will only give a "Yes" or... |
H: How can one express $\sqrt{2+\sqrt{2}}$ without using the square root of a square root?
I was trying to review some analysis, and came across problem 3 from page 78 of Walter Rudin's Principles of Mathematical Analysis. As part of the problem, I wanted to try to write $\sqrt{2+\sqrt{2}}$ without using the square r... |
H: Map bounded if composition is bounded
Let $X,Y,Z$ Banach spaces and $A:X\rightarrow Y$ and $B:Y\rightarrow Z$ linear maps with $B$ bounded and injective and $BA$ bounded. Prove that $A$ is bounded as well.
If I knew that $B(Y)$ is closed I'd have a bounded linear map $B^{-1}:B(Y)\rightarrow Y$ by the bounded inve... |
H: Pythagorean Triplets with "Bounds"
I am interested in the algebraic/geometric way of finding the pythagorean triplets such that
$$a^2 + b^2 = c^2$$
$$a + b + c = 1000$$
I do the obvious
$$a + b = 1000 - (a^2 + b^2)^{1/2}$$
$$a^2 + b^2 = 1000^2 -2(1000)a - 2(1000)b +2ab + a^2 +b^2$$
$$2a + 2b - \frac{2ab}{1000} = 10... |
H: Legendre symbol, second supplementary law
$$\left(\frac{2}{p}\right) = (-1)^{(p^2-1)/8}$$
how did they get the exponent. May be from Gauss lemma, but how.
Suppose we have a = 2 and p = 11. Then n = 3 (6,8,10), but not
$$15 = (11^2-1)/8$$
n is a way to compute Legendre symbols from Gauss lemma:
$$\left(\frac{a}{p}\... |
H: Prove $\cos^2x\sin^4x = \frac{1}{32}(2-\cos(2x)-2\cos(4x)+\cos(6x))$
I need help with a problem some may consider odd but here it is.
I have the following trig identity I been working on and I managed to get it to.
$$\cos^2x\sin^4x=\frac{3}{16}-\frac{\cos(2x)}{4}+\frac{3\cos(2x)}{16}+\frac{1}{8}(1+\cos(4x))+\frac{1... |
H: Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$.
Simplify the difference quotient $\frac{f(x+h)-f(x)}{h}$ where
a) $f(x)=2x+3,$
b) $f(x)=\frac{1}{x+1},$
c) $f(x)=x^2.$
I believe that if anyone can help me out with the first one, the other two might come clearer to me. But I started out this probl... |
H: Map from $\operatorname{Ext}^1(M,M)$ to $H^1(G, \operatorname{End}(M))$
The setting is as follows: $(R,m)$ is a local ring (assume noetherian, complete, if you need) and $\rho\colon G\to \operatorname{Aut}(M)$ is a group representation on the free, finite-rank $R/m^n$-module $M$, for some $n\geq 1$.
Let $\operator... |
H: An exercise about convergence of series
Possible Duplicate:
Convergence/divergence of $\sum\frac{a_n}{1+na_n}$ when $\sum a_n$ diverges.
Let $a_n$ be a non-negative sequence such that the series $\sum a_n$ does not converge. Could the series $\sum a_n/(1+na_n)$ be convergent?
AI: Hint: Try $a_n=1$ if $n$ is a po... |
H: Local invariants of the discrete Galois module associated to a $p$-ordinary newform
Let $f=\sum_{n=1}^\infty a_nq^n$ be a $p$-ordinary newform of weight $k\geq 2$, level $N$, and character $\chi$, and let $\rho_f:G_\mathbf{Q}\rightarrow\mathrm{GL}_2(K_f)$ be the associated $p$-adic Galois representation, where $K_f... |
H: How do I integrate this distribution?
I have a multinomial multivariate normal distribution of the form:
$$\exp\left[-\frac{1}{2\sigma^2}(({\boldsymbol \beta}-\mu)^T\Sigma^{-1}({\boldsymbol\beta}-\mu)\right]$$
I wish to integrate with respect to $\boldsymbol \beta$.
I have found a form of the Gaussian integral from... |
H: Fermat's Last Theorem: rational solutions iff integer solutions
Problem Statement:
In Fermat's Last Theorem $$x^n + y^n = z^n$$ $x,y,z$ are considered integers. But upon closer inspection it is seen that it is also true for any rational numbers $x,y,z$. And that FLT is not applicable only when $x,y,z$ are irrationa... |
H: Kernel of Group Action
this is my first post here.
I have a question regarding a proof in Algebra by Hungerford:
Let $G$ be a group and $H$ a subgroup of $G$. Let $S$ be the set of all cosets of $H$, where $G$ acts on $S$.
Chapter II Collorary 4.9: "$\dots$ the kernel of $G \rightarrow A(S)$ is a normal subgroup... |
H: Dense subset of $C[0,1]$
In $C[0,1]$ the set $\{f(x): f(0)\neq 0\}$ is dense? I know only that polynomials are dense in $C[0,1]$, could any one give me hint how to show this set is dense?thank you.
AI: Yes. Take $f\in \mathcal{C}[0,1]$ so that $f(0) = 0$. Now define
$$f_n(x) = f(x) + {1\over n}, \qquad n\in\mathb... |
H: Finding positive integer solutions to $n = ax^2 +by^2 - cxy$
How can I find the positive integer solutions to $x$ and $y$, given that $n$, $a$, $b$ and $c$ are all positive integers, in an equation of the form:
$$n = ax^2 + by^2 - cxy.$$
Specifically, I want to find the positive integer solutions to the following e... |
H: I don' t understand why the ratio of two meromorphic functions is meromorphic
Let $D\subseteq\mathbb C $ be a connected set and consider $f,g\in\mathcal M(D)$.The ratio $r=\frac{f}{g}$ has the pole set $P(r)\subseteq P(f)\cup Z(g)$ (where $Z()$ is the set of all zeros). Why is $P(r) $ discrete in $D$ ? I think tha... |
H: For $L_1,L_2,L_3$ , $L_1 \cap L_2 = L_3$, if $L_1,L_2,L_3$ such that $L_1 \cap L_2 = L_3$, if $L_1$ and $L_3$ are CFLs, so $L_2$ is CFGLas well
I'm trying to answer this question:
Is it true that for three languages $L_1,L_2,L_3$ such that $L_1 \cap L_2 = L_3$, if $L_1$ and $L_3$ are context free languages, so $L_2... |
H: Lie algebra of a Lie subgroup
Let $G$ be a Lie group and $H$ a Lie subgroup of $G$, i.e. a subgroup in the group theoretic sense and an immersive submanifold. Let $\mathfrak{g}$ and $\mathfrak{h}$ be the associated Lie algebras.
Now, the Lie algebra $\mathfrak{h}$ is given by:
$$ \mathfrak{h} = \{ X\in \mathfrak{g}... |
H: Log likehood functions - Expected value
Let $X_1,X_2,\ldots,X_n$ be a random sample from a Bernoulli($θ$) distribution with probility function
$$P(X=x)= (θ^x)(1-θ)^{1-x},\qquad x=0,1;\ 0 < θ < 1.$$
$dl/dθ = [n \overline{x}/θ] \cdot (n-n\overline{x})/(1-θ)$ <-- Is it this that's wrong? :/ Got help with this too (Per... |
H: Number of field homomorphisms from an extension field of $\mathbb Q$ to $\mathbb C$
Take $\mathbb{Q}$ $\subset$ $K$ $\subset$ $\mathbb{C}$ with $[K:\mathbb{Q}]$ finite. How would you show that the number of field homomorphisms from $K$ to $\mathbb{C}$ is equal to $[K: \mathbb{Q}]$?
I guess it is clear that any e... |
H: Changing from quadratic formula to standard form.
The graph of a quadratic function has $x$-intercepts $-1$ and $3$ and a range consisting of all numbers less than or equal to $4$. Determine an expression for the function.
This is my problem. I know what the graph looks like, but I only know quadratic formula and... |
H: Showing elements with certain properties are in a normal subgroup
I'm preparing for an exam, and this little guy just had me stumped:
Let $G$ be a group, $N\subset G$ a normal subgroup of $G$, $x,y,z \in G$, and $x^3 \in N$, $y^5 \in N$, $zxz^{-1}y^{-1} \in N$. Show that $x,y\in N$.
What I want to do is to show for... |
H: Zero sections of any smooth vector bundle is smooth?
Could any one give me hint how to show that the zero section of any smooth vector bundle is smooth?
Zero section is a map $\xi:M\rightarrow E$ defined by $$\xi(p)=0\qquad\forall p\in M.$$
AI: Smoothness can be checked locally on $M$ and locally $E$ is trivial.
C... |
H: Applications of Operator Algebras to modern physics
I think that recently I've started to lean in my interest more towards operator algebras and away from differential geometry, the latter having many applications to physics. But while taking physics courses, it was also brought to my attention that operator theor... |
H: Find $a$ for which figure's area is maximum
Suppose that we have following interval $(-5,2)$,we should find such $a$, which takes all possible values from this interval,creates following inequality systems
$$5+a-|2y|\ge 0$$
$$|x|\leq \frac{|a-2|}{2}$$
we are working in $OXY$ cordinantes system,we have to find ma... |
H: Finding the number of factors of product of numbers
If $a,b,c,d\in\mathbb{N}$ be distinct. Each of which has exactly five factors, can we determine the number of factors of the product of $a,b,c,d$?
Edit
This is the solution given the in the back of the book I am reading. Does not make sense to me.
If a, b, c and ... |
H: Constructing sets from connected sets
I feel that this is probably really obvious, but I don't know how to get started. Is it true that every set in a metric space is the union of connected, pairwise-separated sets? And does this generalize to topologies easily?
AI: It does indeed generalize to arbitrary topologies... |
H: Play a slot machine for uncountable number of times
There is a slot machine. You insert one coin, it destroy the coin and return countable number of coins. Then you can pick any one of the returned coin and put in the slot machine again.
Formally, each coin is an ordinal. let $f(a)$ be the coin you insert into the... |
H: Prove: symmetric positive definite matrix
I'm studying for my exam of linear algebra.. I want to prove the following corollary:
If $A$ is a symmetric positive definite matrix then each entry $a_{ii}> 0$, ie all the elements of the diagonal of the matrix are positive.
My teacher gave a suggestion to consider the u... |
H: Summation equation for $2^{x-1}$
Since everyone freaked out, I made the variables are the same.
$$
\sum_{x=1}^{n} 2^{x-1}
$$
I've been trying to find this for a while. I tried the usually geometric equation (Here) but I couldn't get it right (if you need me to post my work I will). Here's the outputs I need:
1, 3, ... |
H: Inequality of weights on a graph
If $\sum_{j=1}^nx_j^2=1$ with $x_j\!\in\!\mathbb{R}$, why does it follow that $\sum_{j=1}^nx_j^4\geq\frac{1}{n}$. I'm trying to understand the following excerpt from Brandes & Erlebach's Network Analysis, p.407:
AI: By generalized mean inequality (see e.g. Wikipedia, PlanetMath or A... |
H: Simplifying $|a+b|^2 + |a-b|^2$
I want to simplify $|a+b|^2 + |a-b|^2$ where $a, b \in \mathbb{C}$. I've used Wolfram Alpha to get
$$
|a+b|^2 + |a-b|^2 = 2\left(|a|^2 + |b|^2\right)
$$
I'm trying to understand the steps involved in arriving at this result:
$$\begin{eqnarray*}
|a+b|^2 + |a-b|^2
&=& |(a+b)^2| + |(a... |
H: Does this sequence have this interesting property relating to the prime factorization of the index?
Define a sequence as $a_0 = 0$ and $a_n$ equals the number of divisors of $n$ (including 1 and $n$) that are greater than $a_{n-1}$. This is sequence A152188 in OEIS, by the way.
(For example, the first few terms are... |
H: What is the sum of $\sum\limits_{i=1}^{n}ip^i$?
What is the sum of $\sum\limits_{i=1}^{n}ip^i$ and does it matter, for finite n, if $|p|>1$ or $|p|<1$ ?
Edition :
Why can I integrate take sum and then take the derivative ? I think that kind of trick is not always allowed.
ps. I've tried this approach but I made mi... |
H: Sorting flags of different countries
Question: In how many ways can you sort 8 of 12 flags (4 flags for each country) so there will be at least one flag from each country? Final answer: 4620
I'm not sure whether it's possible, but I'd like to solve this question by reducing the impossible ways from the total option... |
H: Chain rule for multivariable functions confusion
Suppose $f=f(x,y(x))$.
Then applying the chain rule we get $\frac{\partial f}{\partial x}=\frac{\partial f}{\partial x}\frac{\partial x}{\partial x}+\frac{\partial f}{\partial y}\frac{\partial y}{\partial x}=\frac{\partial f}{\partial x}+\frac{\partial f}{\partial y}... |
H: Trace of the multiplication operator
Let $V$ be vector space, $\dim V=N$. Define the multiplication operator $L_{\mathbf{b}}$ as $L_{\mathbf{b}}:\omega\to \mathbf{b}\wedge\omega$, where $\omega\in\wedge V$ ($\wedge V$ is the entire exterior algebra) and $\mathbf{b}\in V$. I want to compute the trace of $L_{\mathbf{... |
H: Periodic solution of differential equation
let be the ODE $ -y''(x)+f(x)y(x)=0 $
if the function $ f(x+T)=f(x) $ is PERIODIC does it mean that the ODE has only periodic solutions ?
if all the solutions are periodic , then can all be determined by Fourier series ??
AI: No, it doesn't. But in in special cases it has... |
H: Evaluate $f(x_0)+f(y_0)$
Let $$f(x)=3(x-2)^{\frac{2}{3}}-(x-2),~0\leq x\leq 20$$ Let $x_0$ and $y_0$ be the points of the global minima and maxima, respectively, of $f(.)$ in the interval $[0,20]$. Evaluate $f(x_0)+f(y_0)$
Note that $$f'(x)=2(x-2)^{-\frac{1}{3}}-1=0$$ $$=>x=10$$ and $$f''(10)=-\frac{2}{3}(10-2)^{-... |
H: Two number partition problems
Let $p_k(n)$ be a number of ways to express $n$ as a sum of $k$ positive integers. For example $p_2(3)=1$.
Problem 1. Prove that following recurrences are correct:
$p_k(n)=p_{k-1}(n-1)+p_k(n-k)$
$p_k(n)=p_k(n-k)+p_{k-1}(n-k) + ... + p_1(n-k)$
Problem 2. Find $p_2(n)$ and $p_3(n)$.
Un... |
H: How to address mistakes in published papers?
I have recently discovered some mistakes in a published maths article. I have contacted the author pointing out politely my concerns, but I got no specific answer, just a "polite" one, that the aspects I am addressing are clarified in some of his other papers (without me... |
H: What is the sum of $\sum\limits_{i=1}^{n}i^k p^i$?
(I've asked similar question, but this is much more complicated, I think)
What is the sum of $\sum\limits_{i=1}^{n}i^k p^i$?
Interpretation (why is it important ) :
$f(k,n,p)=\sum\limits_{i=1}^{n}i^k p^i$
if n goes to plus infinity, then $f(1,n,1/2)$ is average le... |
H: A property of outer measure for bounded sets of real numbers.
I have a bounded set $E$ of real numbers. I'm in the process of showing that there is a set $G$ that is a countable intersection of open sets $G_i$ such that $E\subseteq G$ and $E,G$ have the same outer measure.
What I have so far:
I know that outer mea... |
H: Regular value: intuition about surjectivity condition
Let $f:M\rightarrow N$ be a smooth function between two smooth manifolds.
A $\textit{regular point}$ is a point $p\in M$ for which the differential $df_p$ is surjective.
What does the surjectivity condition for the differential mean intuitively? What is then so ... |
H: How can I get matrices for practicing Jordan normal form?
I would like to practice the algorithm for the transformation from a matrix to its jordan normal form (with change of basis).
To do so, I wrote this script that generates random $n \times n$ matrices, with $n \in \{2,3,4,5\}$
import random, numpy
n = random... |
H: Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$
This is an exercise from Kunen's book.
Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$
are equal.
What I've tried: I want to prove by using induction on $m$. For $m=0$, the ordinal exponentiation $n^m... |
H: Notations for groups of order $p^3$
Are there any relatively common notations for the two non-isomorphic nonabelian groups of order $p^3$ where $p$ is a prime number? I remember reading some notations like $p_+^{1+2}$ and $p_-^{1+2}$. Where can I find the definition of these notations?
AI: These groups are the so-c... |
H: Transforming an inhomogeneous Markov chain to a homogeneous one
I fail to understand Cinlar's transformation of an inhomogeneous Markov chain to a homogeneous one. It appears to me that $\hat{P}$ is not fully specified. Generally speaking, given a $\sigma$-algebra $\mathcal A$, a measure can be specified either exp... |
H: Calculate distance, knowing actual and perceived size
What's equation would I use to calculate distance to an object, if I know it's actual and perceived size?
Say there is a line, and I know it's actual length is 65 (units), I perceive it as 62 (units).
I found various angular size equations on wikipedia and calcu... |
H: Why is the probability that a continuous random variable takes a specific value zero?
My understanding is that a random variable is actually a function $X: \Omega \to T$, where $\Omega$ is the sample space of some random experiment and $T$ is the set from which the possible values of the random variable are taken.
... |
H: Need help with Unbiased estimator
Let $X_1,X_2,X_3,\ldots,X_n$ be a random sample from a $\mathrm{Bernoulli}(\theta)$ distribution with probabilty function
$P(X=x) = (\theta^x)(1 - \theta)^{(1 - x)}$, $x=0,1$; $0<\theta<1$.
Is $\hat\theta(1 - \hat\theta)$ an unbiased estimator of $\theta(1 - \theta)$? Prove or dis... |
H: Strongly complete profinite group
Let $G$ be a profinite group (or equivalently a compact and totally disconnected topological group ) with the property that all of its normal subgroups of finite index are open sets.
Does this imply that all of its subgroups of finite index are open sets ?
(if all subgroups of f... |
H: If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF)
I can prove this in ZFC, but don't know how to prove this in ZF.
Following is the argument of this in ZFC.
Fix $0<r\in \mathbb{R}$ and $x_0\in X$.
Let $A_i = \{x\in X\mid d(x,x_j)\ge r,\, j<i\}$.
Suppose $A_i≠\emptyset$ ... |
H: what is the easiest way to represent $ \sqrt{1 + x} $ in series
How to expand $ \sqrt{1 + x}$.
$$ \sum_{n = 0}^\infty {{\left ( 1 \over 2\right )!}x^n \over n! \left({1 \over 2 }- n\right )!} = 1 + \sum_{n = 1}^\infty {{\left ( 1 \over 2\right )!}x^n \over n! \left({1 \over 2 }- n\right )!}$$
How can I simplify $... |
H: Why does the bilinear form vanish between two direct factors?
I extract a problem from the book I am reading:
Let $R$ be a field, $A$ be a semisimple split $R$-algebra (associative with $1$). Let $A = \oplus_{n=1}^t A_n$ be a decomposition of $A$ into simple algebras. $(,): A \times A \rightarrow R$ is a non-degen... |
H: When a is less than c in $ \int_a^b \frac {dx} {x^4 - c^4} $
$$ \int_a^x \frac {dx} {x^4 - c^4} = \frac {1} {4c^3} \ln \left(\frac {x-c} {x+c} \right)_a^x - \frac {1} {2c^3} \tan^{-1} \Bigl(\frac {x} {c} \Bigr)_a^x $$
$$ =\frac {1} {4c^3} \Bigl[ \ln \Bigl(\frac {x-c} {x+c} \Bigr) - \ln \Bigl(\frac {a-c} {a+c} \Bigr... |
H: Maximal Separable Subextension is Finite?
Consider the following statement:
"Let $L/K$ be an algebraic field extension. Then the maximal separable sub-extension is finite."
Here is what seems to be a proof:
"Let $M/K$, $K \subset M \subset L$ be a finite separable subextension of maximal degree. Let $x \in L$ be se... |
H: Evaluating $\int_0^a \frac{\cos(ux)}{\sqrt{a^2-x^2}}\mathrm dx$
I believe this integral $$\int_0^a \frac{\cos(ux)}{\sqrt{a^2-x^2}}\mathrm dx$$
can not be computed exactly. However is there a method or transformation to express this integral in terms of the cosine integral or similar? I am referring to the integrals... |
H: Find the number of real roots of the polynomial
Find the number of real roots of the polynomial $$f(x)=x^5+x^3-2x+1$$
If I use Descarte's Rule then I get $$f(x)=x^5+x^3-2x+1$$ there can't be more than two positive real roots.
Again $$f(-x)=-x^5-x^3+2x+1$$ there can't be more than one negative real root.
AI: It's o... |
H: Winning strategy for a matchstick game
There are $N$ matchsticks at the table. Two players play the game.
Rules:
(i) A player in his or her turn can pick $a$ or $b$ match sticks.
(ii) The player who picks the last matchstick loses the game.
What should be the conditions on $N$ so that a winning strategy can be der... |
H: Prove that $\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0 $.
Given $x, y, z $ are 3 positive reals such that $xyz≥1$. Prove that
$$\frac{x^5-x^2}{x^5+y^2+z^2}+\frac{y^5-y^2}{x^2+y^5+z^2}+\frac{z^5-z^2}{x^2+y^2+z^5}≥0.$$
This question is so complicated. I failed many times to ... |
H: How many circles are needed to cover a rectangle?
TRUE OR FALSE
Suppose that a rectangle in $R^{2}$ can be covered by (allowing overlaps) $25$ discs of radius $1$, then it can also be covered by $101$ discs of radius $0.5$.
Of course, though it is a true or false question, I would like the logic on it and possible ... |
H: Boundary Question in $\mathbb{R}^{2}$ (Manifolds)
Given a subset $A$ of $\mathbb{R}^{n}$, a point $x \in \mathbb{R}^{n}$ is said to be in the boundary of A if and only if for every open rectangle $B\subseteq\mathbb{R}^{n}$ with $x\in B$, $B$ contains both a point of $A$ and a point of $\mathbb{R}^{n}\setminus A$.
M... |
H: Two hard number partition problems
For every positive integer $n$, let $p(n)$ denote the number of ways to express $n$ as a sum of positive integers. For instance, $p(4)=5$. Also define $p(0)=1.$
Problem 1.
Prove that $p(n)-p(n-1)$ is the number of ways to express $n$ as a sum of integers each of which is stric... |
H: Algorithm for planarity test in graphs
I am implementing a graph library and I want to include some basic graph algorithms in it. I have read about planar graphs and I decided to include in my library a function that checks if a graph is planar. I found on the web many efficient algorithms, but they all have the sa... |
H: No closed form for the partial sum of ${n\choose k}$ for $k \le K$?
In Concrete Mathematics, the authors state that there is no closed form for
$$\sum_{k\le K}{n\choose k}.$$
This is stated shortly after the statement of (5.17) in section 5.1 (2nd edition of the book).
How do they know this is true?
AI: The very ne... |
H: Holomorphic function in an annulus
I'm trying to do a question but I have a doubt on holomorphic functions, here is the problem.
Let $A = \{z ∈ \mathbb{C} : \frac{1}{R}< |z| < R\}$. Suppose that $f : A → \mathbb{C}$ is holomorphic and
that $|f(z)| = 1 $ if $|z| = 1$.
Show that $f(z) = {\left(\overline{f(\bar{... |
H: Kernel of adjoint of Lie algebra
Let $G$ be a Lie group and $\mathfrak{g}$ its Lie algebra.
The adjoint representation of the Lie algebra $\mathfrak{g}$ is defined as:
$$ \text{ad: } \mathfrak{g} \rightarrow \text{End}(\mathfrak{g}), X \mapsto [X,\cdot] $$
Now, it holds true that
$$ \text{ker ad} = \mathfrak{z}(\m... |
H: Solving linear system of equations when one variable cancels
I have the following linear system of equations with two unknown variables $x$ and $y$. There are two equations and two unknowns. However, when the second equation is solved for $y$ and substituted into the first equation, the $x$ cancels. Is there a w... |
H: Is my solution to this permutation question correct?
Question: Suppose there are m girls and n boys in a class. What is the number of ways of arranging them in a line so that all the girls are together? (Biggs, Discrete Mathematics 2nd ed, Exercise 10.7.5)
My solution
Say $m=4$ and $n=3$
The number of ways they ca... |
H: Expressing $\sin^4x-\sin^6x$ in another way
I am slightly confused on how one would subtract $\sin^4x-\sin^6x$.
I know that $\sin^2x=(1/2)(1-\cos2x)$,
so $\sin^4x$ would logically be $[(1/2)(1-\cos2x)]^2=(1/4)(1-2\cos(2x)+\cos^2(2x)$
However the value of $\sin^6x$ eludes me. Would it be $(1-\cos2x)^3$? I did that a... |
H: Exponentiation and Set of Functions Notation.
Given arbitrary sets $A$ and $B$, the notation $A^{B}$ is mostly clear from context to mean $A^{B} = \{f : f : B \rightarrow A\}$.
However, when these sets are ordinal or cardinals, especially $\omega$, the notation is not consistent even among subfields of logic.
For ... |
H: How likely is it for a randomly picked number to be larger than all previously chosen numbers?
Suppose we pick a uniformly distributed number on the range [a,b]. Then we continue to pick more numbers on the same range. Let n(t) be the number of times we have found a number bigger than any previously found, after sa... |
H: Generalized homomorphism theorem
Let $f: G \rightarrow G'$ be a group homomorphism with kernel $H$. Then we know as an elementary fact of abstract algebra that there is an injective group homomorphism $f^*:G/H \rightarrow G'$ such that $f^*(x+H) = f(x)$. More generally, let $\psi: G \rightarrow G/H$ be a surjective... |
H: Proof about Field conjugation isomorphisms
I'm having an awful time making sense of a proof and I was hoping someone could help.
Theorem: Let $\alpha$ and $\beta$ be algebraic over a field $F$ with $deg(\alpha, F) = n$, as elements of a field extension $E$ of $F$.
Define the map $\psi_{\alpha,\beta}:F(\alpha)\to F... |
H: Calculating $\lim_{n \to +\infty}\int_0^1 (n + 1)x^{n}(1 - x^3)^{1/5}\,dx$
This question is from a bank of past master's exams. I have been asked to evaluate $$\lim_{n \to +\infty}\int_0^1 (n + 1)x^{n}(1 - x^3)^{1/5}\,dx.$$
I did this problem in a hurried manner, but here's what I think. Since $x^n$ is decreasing i... |
H: Confused about which Hölder spaces are Banach
If $\Omega$ is an open set in $\mathbb{R}^n$, is the Hölder space $C^{k, \alpha}(\Omega)$ Banach? Or is it only that $C^{k, \alpha}(\overline{\Omega})$ is Banach, like with ordinary continuous functions? If not, why is that???
Norms are
$$|f|_{C^{0,\alpha}} = \sup_{x,y ... |
H: Uniform continuity of a function transforming Cauchy sequences into Cauchy sequences
Problem:
$f:(0,\infty )\rightarrow \mathbb{R}$ defined as: $f(x)=x^{2}$
Can anyone show me how to prove that $f$ transforms Cauchy sequences of elements of $(0,\infty )$ into Cauchy sequences, but $f$ is not uniformly continuous? T... |
H: Question on smoothness of the projective closure of a smooth variety
I was taught in the previous thread "Is the projective closure of a smooth variety still smooth?", that the projective closure $\bar X\subseteq \mathbb{P}^n$ of a smooth closed subscheme $X\subseteq \mathbb{A}^n$ (over a basefield) needs not to be... |
H: Limit of a function as accumulation point
In the context of this answer to another question about representing I thought of the following possible description of the limit of a function:
$\lim_{x\to a}f(x)=y$ iff $(a,y)$ is an accumulation point of $f$ (interpreted as a set of pairs) and there's no $y′≠y$ so that $... |
H: What kind of "mathematical object" are limits?
When learning mathematics I tend to try to reduce all the concepts I come across to some matter of interaction between sets and functions (or if necessary the more general Relation) on them. Possibly with some extra axioms thrown in here and there if needed, but the fu... |
H: Fourier transform of a measure
I'm a bit confused - How is the Fourier transform of a measure on a compact abelian group defined? specifically the Fourier transform of a measure on $\mathbb{T}$ the unit circle in the complex plain.
AI: If $\mu$ is a measure on the compact abelian group $G$ and $\gamma$ is in the du... |
H: What is the number of all possible values of $[Z^{6}]$?
Its given that $$[Z]=3$$
$$[Z^{2}]=11$$
$$[Z^{3}]=41$$
Then, what is the number of all possible values of $[Z^{6}]$ where $[\;\cdot\;]$ is floor function.
AI: Hint:
$$
[Z^k]=b\quad\Longleftrightarrow\quad
b\leq Z^k<b+1 \quad\Longleftrightarrow\quad
b^{1/k}\leq... |
H: Finite modules over finite local rings
Let $R$ be a finite commutative local ring with identity. If $M$ is a finite $R$-module it is necessarily projective?
AI: Let $p$ be a prime number.
Let $R = \mathbb{Z}/p^2\mathbb{Z}$.
Let $M = R/pR$.
Since the number of elements of $M$ is $p$, $M$ cannot be free.
Hence $M$ ca... |
H: Product of right cosets equals right coset implies normality of subgroup
I cannot see how to find a way to prove that if $H$ is a subgroup of $G$ such that the product of two right cosets of $H$ is also a right coset of $H,$ then $H$ is normal in $G.$
(This is from Herstein by the way.)
Thank you.
AI: Hint: if $Ha... |
H: Greatest common denominator of measurements
In a couple months, I'll do the Millikan experiment. Then, I'll end up with a number of charge measurements and their errors $$((q_i, \Delta q_i))_{i \in \mathbb N}.$$ The idea is that all those $q_i$ can be represented as a multiple of a fixed elementary charge $e$ like ... |
H: How to calculate the degree of this Gauss map?
In reviewing the familiar Poincare-Hopf theorem I come across the following question:
Suppose $x$ an isolated 0 of $V$. Pick up a disk around $x$ in its neighborhood. Calculate the degree of the map $$u:\partial D \rightarrow S^{m-1},u(z)=\frac{V(z)}{|V(z)|}$$
where $V... |
H: Dedekind Domains and Affine Varieties
Let $k$ be an algebraically closed field, and let $B$ be a finitely generated $k$ algebra that is also a Domain. Then $B$ is the affine coordinate ring of some affine variety $Y$; this part is straight out of Hartshorne and is not terribly difficult to understand. However, I ... |
H: For what values of $q$ would $3n - q^2 \equiv 0\pmod{4q}$?
For what values of $q$ would $3n - q^2 \equiv 0\pmod{4q}$, or for what values of $q$ would $3n - q^2$ divide by $4q$ and leave no remainder, where $n$ is a positive integer and $q$ is a positive divisor of $n$.
AI: If $n = q k$, $3n - q^2 = (3k - q) q$. So... |
H: Definition of quadratic equation?
What is a quadratic equation and what is its simplified and cannonic form?
AI: A quadratic equation (in one variable) is a polynomial equation $P(x)=0$, where $P(x)$ is a polynomial of degree 2.
The canonical form of a quadratic equation is $ax^2+bx+c=0$. The simplified form I'm no... |
H: Why is there no foliations of the 2-sphere, or a genus two surface?
I'm trying to see why there is no (one-dimensional) foliation of $S^2$ or an orientable surface of genus two. Originally I was thinking that such a foliation could give me a non-vanishing vector field, which would be a contradiction, but now I hav... |
H: Testing if a geometric series converges by taking limit to infinity
If the limit as n approaches infinity of a geometric series is not zero, then that means the series diverges. This makes intuitive sense to me, because it is an infinite series and we keep adding nonzero terms, it will go to infinity.
However, if ... |
H: How do I proceed with these quadratic equations?
The question is
$$ax^2 + bx + c=0 $$ and $$cx^2+bx+a=0$$ have a common root, if $b≠ a+c$, then what is $$a^3+b^3+c^3$$
AI: The value of $a^3+b^3+c^3$ is not determined. Just choose $a=c$. But leaving out the condition $a\ne c$ is probably an oversight, so assume fro... |
H: Writing inequalities in interval notation?
When writing the solution to an inequality in interval notation, say $x < 5$ as $(-\infty, 5)$, how can the $x$ be "involved"?
Is it correct to just write $(-\infty, 5)$, or should it be $x=(-\infty, 5)$, or perhaps $x\in(-\infty, 5)$?
AI: One defines the interval $(-\inft... |
H: Hellinger-Toeplitz theorem use principle of uniform boundedness
Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all $x,y\in\mathcal{H}$. Prove that $T$ is a bounded directly from the un... |
H: Sequences from the positive integer numbers $(a,b,c,d,e)$
How to find how many sequences from the positive integer numbers $(a,b,c,d,e)$,such that :
$$abcde \le a+b+c+d+e \le 10$$
AI: I would try to solve this logically. If 10 is the maximum number of a + .... + e, then that means the numbers must be limited to 6.... |
H: $x$-axis is meager set on $\mathbb{R}^2$
Subset $A$ of metric space $X$ is meager on $X$, iff $\text{IntCl}A=\emptyset$.
But, why $x$-axis is meager set on $\mathbb{R}^2$?
My attempt (please don't kill me):
$\text{IntCl}\mathbb{R}=\text{Int}\mathbb{R}=\mathbb{R}\neq \emptyset$
Thank you!
AI: The interior of $\mathb... |
H: Oscillation of $f$ (at $x_0$) equals zero iff $f$ is continuous.
I'm reading a proof of that claim mentioned in the title and I have some difficulties understanding it.
Statement:
$$O_f(x_0)=0 \iff f \ \ \text{is continuous on} \ \ x_0 $$
(Where oscillation, $O_f(x):= \displaystyle\lim_{n\to\infty}\text{diam}f(... |
H: Two corollaries in Lang's Algebraic Number Theory.
I'm having difficulty understanding the relationship between two corollaries in Lang's Algebraic Number Theory, on page 16 for those with the book. They can also be found in his Algebra.
The first is:
Let $A$ be a ring integrally closed in its quotient field $K.$ L... |
H: How to resize an image?
I am not sure about the title of this question, so if someone knows an appropriate one, please rename it. It's a programming related question (but doesn't involve any programming). I posted it on stack overflow but didn't get any responses so I am trying here.
I need to map a piece of rectan... |
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