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How many answers are there to the equation $|3^x-2^y|=5$ such that $x$ and $y$ are positive integers? Are there infinite? I've found $(2,2)$, $(3,5)$, and $(1,3)$. It seems to explode with larger values, but it's not a steady increase and there seems to be many dips. Do we KNOW that there are no large values for $x$ an... |
I understand there are two ways to determine validity of an argument. The first way is to construct a truth table and if the statement consisting of the premises combined together implying the conclusion is a tautology, then it is valid.
The other method is to use the fact that it no valid arguments have all premises t... |
Does $n\pi - \lfloor n\pi\rfloor$ have a subsequence that goes to zero? I was talking to a friend about it but neither of us were able to come up with anything. We're not sure if $\pi$ is essential here or if any irrational could be used instead, but we noticed that every term in the sequence is between $0$ and $1$, so... |
Claim: Let $A$ and $B$ be ideals of a commutative ring $R$ and $C$ a prime ideal of $R$. Suppose that the intersection of $A$ and $B$ is a subset of $C$. Prove either $A$ or $B$ is a subset of $C$.
My proof strategy so far is to show $a$ $\in$ $A$ and $b$ $\in$ $B$ implies $ab$ is in both $A$ and $B$ since both are id... |
Say for example, that I want the highest multiple of 36 below 100. Is there a function that allows me to generate this with two arbitrary numbers?
|
I have set up the following model:
Let $X_n$ be the number of heads in the $n$-th toss and $P(X_0=0)=1$. I can calculate the transition matrix $P$. Define
$$
T=\min\{n\geq 0\mid X_n=5\}.
$$
Then $P(X=1)=P(X_{T-1}=4)$. Noting that $X_n$ is a Markov chain and defining
$$
u_i=P(X_{T-1}=4\mid X_0=i)
$$
we have
$$
u_i=\su... |
What do those graphs look like which contain a copy of every tree on $n$ vertices and such that no proper subgraph has this property?
|
I've been having some trouble with proving or disproving tautologies. I am very new to proofs and am hoping I am on the right track. The question asks to show that:
If ψ → φ is a tautology, then ψ and φ are tautologies.
It also asks to show the converse:
If ψ and φ are tautologies, then ψ → φ is a tautology.
The q... |
I have the series: $$\sum_{i=1}^{i=10^N} \log_5 i$$ I'm trying to figure out how to get the closed-form solution to this problem. I entered it into WolframAlpha and got that it equals: $ (\log(\mathrm{Pochhammer}(1, 10^N)))/(\log(5))$
I have a few things I'm not understanding about this problem,
What procedure do you ... |
I have a series of random variables $X_1, X_2, \ldots$ where
$$ f(X_n) =
\begin{cases}
1/n & \quad \text{if} \ X_n = 1 \\
1-1/n & \quad \text{if} \ X_n = 0 \\
0 & \quad \text{otherwise}\\
\end{cases}
$$
I am trying to show that $X_n \to 0$ in probability. I have trying to use the definition o... |
I am trying to write a proof and was wondering if a then b, the converse if b then a might not be true. This leads me to wonder if the statement needs to be an if and only if statement if it can be proven by the contrapositive if not b then not a.
That is do I need a statement like this a if and only if b to prove that... |
Joe goes to a party with three friends. There is a drawing for four door prizes. Each person has an equal chance of wining a prize. No one can win more than one prize. If there are totally thirty people in the party, what is the probability that Joe and his friends each will win a prize?
|
I have a sequence of independent random variables $X_1, X_2, \ldots$ where
$$ X_n =
\begin{cases}
1 & \quad \text{with probability} \ 1/n \\
0 & \quad \text{with probability} \ 1-1/n
\end{cases}
$$
In my book, it claims that while $X_n \to 0$ in probability, it is not true almost surely. The book ... |
I apologize if this is something very trivial, but I couldn't find an answer to it anywhere:
I have a directed graph with
n = 280 nodes and
m = 64 edges
Now, my question is: If I want to compare this graph to a G(n,p) random graph, how do I calculate p?
If m = p*n(n-1), is it correct then to calculate p as
p = m/280(2... |
$\begin{bmatrix}\frac{1}{2} & -\frac{1}{2} \\ \frac{1}{2} & -\frac{1}{2}\end{bmatrix}\begin{bmatrix}x \\ y\end{bmatrix} = \begin{bmatrix}0 \\ 0\end{bmatrix}$
$\lambda_{1} = \lambda_{2} = -1$
The first vector is:
$v_{1} = \begin{bmatrix}x \\ x\end{bmatrix}$
$v_{1} = \begin{bmatrix} 1 \\ 1\end{bmatrix}$
How can I calcul... |
I want to prove the following equivalence:
Let $V$ an algebraic set, $K$ a field and $\overline K$ its algebraic closure. Then we say that $V/K$ ($V$ is defined over $K$) if $I_{V}$ (the ideal attached to $V$) is of the form $<S>$ where $S \subset K[x_1,...,x_n]$.
I want to prove that $V/K \Leftrightarrow I_V=I_{V/K}... |
The wikipedia entry for convex hull shows a 2-d example of a random set of points on x-y plane, and the "elastic band" solution that bounds the points with the convex hull solution. The definition of the solution seems to be the one that achieves a minimum perimeter length, although I don't think that is explicitly st... |
I just want to know more examples of equations that creates pseudorandom numbers. Right now I only know the Elliptic Curves.
$y^2 = x^3 - 3x + b \pmod p$
|
I'm doing homework for a programming class and came across this problem. There's no directions besides what I've shown, so I don't even know what it's asking me to do. What makes the most sense for what it's asking me to solve?
|
The question is: does the surjective, multilinear map $\pi: (\mathbb R^\infty)^n\to \Lambda^n(\mathbb R^\infty)$ defined by $(v_1,\cdots,v_n)\mapsto v_1\wedge \cdots \wedge v_n,$ define a fibre bundle? This would make geometric statements about linearly independent vectors.
|
i have this simple question that make me really confused :
let $\phi$ a smooth function with compact support and $p>0,t\ge 0$ and :
\begin{equation}
U_p=
\left\lbrace
\begin{array}{ccc}
0 & \mbox{if} & x<-pt\\
-2p & \mbox{if} & -pt\leq x<0\\
2p & \mbox{if} & 0\leq x<pt\\
0 & \mbox{if} & x\ge pt
\end{array}\right.
\e... |
Let $K \subset \mathbb{R}^n$ be compact, let $k \in \mathbb{N}$. Let $\{f_n\} \subset C^k(K)$ be a bounded sequence w.r.t $C^k$ norm. Does it have a convergent subsequence in $C^{k-1}(K)$. If so how can one prove this?
|
This is the question:
$X$ is a continuous random variable whose probability density function
is given by
$$f(x)=\begin{cases}
\frac{1}{9}x^2 & \text{if $0\leq x \leq 3$}.\\
0 & \text{otherwise}. \end{cases}$$
(a) what is the probability that $X$ is less than $1$?
(b) Write down the distribution funct... |
Let $z\in G$ where $G$ is a finite group, then is it always true that there exist elements $g,h\in G$ such that $z=gh^2$ where $|g|=2^k$ for $k \in \Bbb{Z_{\ge0}}$ and $|h|$ is odd?
|
Riemann zeta function at $\Re(s)=0$ and $\Re(s)=1$.
What happens to the zeta function at these points? For example $\sum_{n=1}^\infty \frac1{n^s}$ is defined for $\Re(s)>1$ and for $\Re(s)>0$ you have a different formula. But none of these include 0 or 1? Or does $\Re(S)>0$ include the 1? (or maybe it was defined wrong... |
In OSX Numbers I have a chart with these data points:
50 53
100 62
200 78
300 91
500 117
1000 192
2000 297
3000 412
5000 567
10000 990
Using the trending line option I see that an exponential formula works well and is given as:
35.907e^(0.3498x)
However when I run this formula using Y values I get nowhere near X. Wi... |
$$\int_C z e^ {-z} dz$$
with $C$ being any curve from $i$ to $1 + i$
I defined $C$ as
$$z(t) = t + 1 $$
$$i \leq t \leq 1 + i $$
Such that
$$\int_C f(z)dz = \int_{i}^{1+i} (t + 1)e^{-(t+1)}dt $$
$$=e^{-2-i}-e^{-1-i}$$
Does that seem correct?
Is the definition of C ok or it could be more general?
|
I understand there are two ways to determine validity of an argument. The first way is to construct a truth table and if the statement consisting of the premises combined together implying the conclusion is a tautology, then it is valid.
The other method is to use the fact that it no valid arguments have all premises t... |
I want to know number of solutions for following equation, where $r_k$'s are non-negative integers, and there is a constraint on $r_k$'s such that $r_1 \geq r_2 \geq \cdots \geq r_K$
\begin{equation}
r_1+r_2+\cdots+r_K=N
\end{equation}
|
In this question, a ring is defined to be with a unit distinct from the zero element, not necessarily a commutative ring though. Is the class of all such rings that can be embedded into fields an axiomatizable class? If so, what are the axioms?
|
This question was suggested to be placed in the math forum. 3 particles are at the corners of an equilateral triangle with side $a$. Assume that particle 1 is at $(0,0)$, particle 2 is at $(a,0)$ and particle 3 is at $(a/2, a\sqrt 3/2)$. They all start moving simultaneously with a velocity $v$ constant in modulus, but ... |
The question is to prove that:
$$\frac{f_1(x)+\cdots+f_n(x)}{n}$$ tends to $f(x)$ uniformly on $E$, as $n$ tends to infinity.
I am not sure how to do this generically without an actual sequence. (i.e $f_n(x) = x^n$)
|
I am trying to prove the following statement.
If a field $K$ is not algebraically closed, then any $K$-variety $V\subset\mathbb{A}^n$ can be written as the zero set of a single polynomial in $K[X_1,X_2,...,X_n].$
Suppose $V=V(f_1,f_2,...,f_n)$, and I would want to find a polynomial $g\in K[X_1,X_2,...,X_n]$ such tha... |
What would I have to multiply the following matrix ...
$$
\begin{bmatrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{bmatrix}
$$
by so that I get the following matrices below?
1.
$$
\begin{bmatrix}
d & e & f \\
a & b & c \\
g & h & i \\
\e... |
I want to show that every non-constant closed curve has positive period, but i'm not really sure how to do this.
A smooth curve $r(t): \mathbb{R} \to \mathbb{R}^n$ is $T$-periodic if $r(t+T)= r(t)$ for some $T \in \mathbb{R}$. The period is the smallest positive number $T$.
Since we are assuming the curve is closed, t... |
Say there's a bin of balls numbered 1 through 39, how many ways are there to pick 4 increasing numbers in a row?
First I figured that it would be a permutation without repetition problem, so I got that the number of ways of picking 4 different balls without considering increase is $$\frac{39!}{4!(39-4)!} = 82,250.9999... |
I'm reading Conway's complex analysis book and on page 79 he proved the following theorem:
I think the theorem he is mentioning is a corollary which says that each zero of $f$ has a finite multiplicity. I didn't understand why this implies the zeros of $f$ are isolated.
|
Let $G$ be a finite group and $P$ a Sylow $2$-subgroup. If $g$ is an element of $G$ with order a power of $2$ then it lies in some conjugate of $P$. I get this.
But if also $g\in C_G(P)$, then it is true that $g\in P$. Why? I can't see that immediately.
|
I feel like this might be really hard but I'm not sure. If you get this, you just might be a genius..
$X \sim \mathcal N(\mu_1,\sigma_1)$,
$Y \sim \mathcal N(\mu_2,\sigma_2)$,
$Z \sim \mathcal N(\mu_3,\sigma_3)$
They are independent! Seems easy so far, right?
You tell me, is this easy?
Compute
$$\mathbb E\left[\frac... |
An ODE (Ordinary Differential Equation) of order $n$ becomes a relation:
$$F(x,y,y^{(1)},...,y^{(n)})=0$$
Then $F(x,y,y^{(1)})=0$ defines an ODE of order one. In "basic standard texts", for purposes of simplicity, is assumed that some ODE of first order can take the form:
$$y^{(1)}=f(x,y)$$
for certain suitable $f$. He... |
We have not covered and thus it is not valid to use ideas such as Lebesgue integration, measure, etc. I was given a hint to use either squeeze theorem, or the criterion about if Riemann integrable functions are equal except on a finite set then they have the same integral.
id like to be able to show that the following ... |
I am reading mine's modular form note, proposition 1.12 states that the sum of residues of a differential form on compact Riemann surface is $0$. Then he states that applies this to $df/f$, then we can prove meromorphic function on compact Riemann surface has same zeros and poles.
I can not follows this, can someone he... |
How can I find a value from this pascal triangle given row and column number without calculating $^nC_r$?
For example, for row=$4$, column=$3$: value is $10$,
For row=$3$, column=$5$: value is $15$.
Is there any way to get this value without using $^nC_r$ explicitly?
|
I'm having trouble proving the following claim:
$\forall a, b, c \in \mathbb{R}_+: T(a, b, c) \Rightarrow [|a − b| < c$ and $|b − c| < a$ and $|a − c| < b]$
Where $T(a, b, c)$ is a predicate for a triangle with sides $a, b, c$.
Don't even know where to start. Can someonle please help me?
|
Let's say I have the matrix called Delta,
$$
\begin{matrix}
a & b & c \\
d & e & f \\
g & h & i \\
\end{matrix}
$$
What would I have to multiply the delta by to obtain the matrices below?
In brief, is there a way to backsolve matrices through $matrix inversion$ so I can easily f... |
If $z_0$ is a root of the equation $z^n\cos\theta_0+z^{n-1}\cos\theta_1+\cdots+\cos\theta_n=2$, then
$|z_0|<1/2$
$|z_0|>1/2$
$|z_0|=1/2$
Using triangle inequality,$$|z^n\cos\theta_0|+|z^{n-1}\cos\theta_1|+\cdots+|\cos\theta_n|\ge2$$
$|\cos\theta|\le1$
Inequality won't change if every cosine takes its maximum val... |
If I have for example, a system like this
$$\begin{matrix}
\dot{x}=f(x,y) & \\
\dot{y}=g(x,y) &
\end{matrix}$$
in which i have to prove stability using a Lyapunov function. Now, if i have function like $V(x,y)=x^2$ supposedly fulfilling all the requirements in the theorem, is this possible?.
Does a Liapunov functio... |
I want to see how to prove the functional completeness of NAND gate, but all the materials in the web I have reached just relies on the fact that the set $\{AND,NOT\}$ is complete and shows how to simulate those gates only using NAND gate. I want to see the proof or some list of references which contains the proof.
Tha... |
I am trying to convert these numbers to binary scientific notation, but I cannot figure out the process. Could someone please the process of going about solving this?
For IEEE 754 single precision floating point, what is the number, as written in binary scientific notation, whose hexadecimal representation is the foll... |
Show if $A^TA = I$ and $\det A = 1$ where
$ A =
\begin{bmatrix}
a & b \\
c & d
\end{bmatrix}
$, then $A =\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}$.
attempt:
Suppose $ A^TA =\begin{bmatrix}
a & c \\
b & d
\end{bmatrix}
\begin{bmatrix}
a & b ... |
Question is pretty much the title. It is pretty easy to show that $\zeta(2n)$ is irrational for all $n$ once you know that $\zeta(2n)$ is a rational multiple of $\pi^{2n}$ (and then also use the fact that $\pi$ is transcendental or some other related result). My question is, does anyone know of a proof that $\zeta(2)$ ... |
Let $\Omega \subset \mathbb{R}^n$ be open and bounded. What does it mean $f\in C(\bar{\Omega})$, i.e. what does it mean $f$ to be continuous at $x \in \partial \Omega$, maybe
$$\forall \epsilon >0 \quad \exists \delta >0 \quad \text{s.t.} \quad y \in \partial \Omega \cap B_\delta(x) \quad \implies |f(y)-f(x)| < \delta ... |
I've read two articles(1,2), but still have a question about structure of group, say $G$, as an one-object category. Let's call this category $C$.
I understand that morphisms of $G$, which is denoted by $Mor(G,G)$, forms a monoid, which is explained in the first article(1). However, how can we say that $Mor(G,G)$ is a... |
We have a triangle and within is a square with the side of $2\sqrt{3}$. What's the altitude of the triangle ABC? All 3 angles in the triangle are same (60). Pic: http://imgur.com/gallery/SAmhU7z/new
Now I know the result is $2\sqrt{3}+3$, but I am a bit lost on how to get it. Any help would be greatly appreciated.
|
While reading Boyd's paper on ADMM I encountered an issue.
Consider the following problem:
Problem. Minimize $f(u) + g(v)$ subject to $Au + Bv = c$, where $f$ and $g$ are closed, proper, convex and differentiable.
Denote by $\lambda$ the dual variable. On page 18 of the paper, it is stated that the necessary and suffic... |
My exercise is to write the twisted cubic as a graph and a level set. However, I am not sure what they mean by a graph and level set. Can anyone explain this please? Do they mean, for a graph, $(t,\gamma(t))$?
|
I'm looking for good examples of physical motivation for integrals over scalar field.
Here is an example I've found (source):
A rescue team follows a path in a danger area where for each position the degree of radiation is defined. Compute the total amount of radiation gathered by the rescue team along the path.
Doe... |
In this post a computer simulation to approximate $e$ is based on the mathematical knowledge that $E[\xi]=e$, where $\xi$ is the random variable defined as the minimum number of $n$ such that $\sum_{i=1}^n r_i>1$ and $r_i$ are random numbers from uniform distribution $X \sim [0,1]$.
It clearly works, but I am curious ... |
Let $A \subset \mathbb{R}^n$ a compact and $f : A \to \mathbb{R}^m$ a
continuous function. Let the graph of $f$ $$Gr(f) = \{(x,f(x) : x \in
A)\}.$$ Show that $Gr(f)$ is compact.
My proof :
As $A$ is compact et $f$ a continuous function, then f(A) is also compact.
The graph of $f$ is only the cartisian product of ... |
Let $f(x)=0, -1\le x\le0$ and $f(x)=x^4, 0<x\le 1$
If $$f(x)=\sum_{k=0}^n\frac{f^{(k)}(0)x^k}{k!}+\frac{f^{(n+1)}(\xi)x^k}{(n+1)!}$$ is the Taylor's formula for $f$ about $x=0$ with maximum possible value of $n$, then find the value of $\xi$ for $0<x\le 1$.
I know how to calculate remainder and I can do that. My questi... |
In linear algebra, what does it mean to go from $\mathbb{R}^1$ to $\mathbb{R}^2$ or $\mathbb{R}^2$ to $\mathbb{R}^3$?
|
In binary arithmetic, When you subtract 2 signed numbers you must discard the carry out. My question is, is it possible for overflow to occur and a carry out? So, on paper there would be two extra bits and you would have to discard one?
Also, if your subtracting 2 signed numbers, how do you differentiate between overfl... |
Let $f:[0, +\infty) \rightarrow \mathbb{R} $ be a bounded function in each bounded interval. If $$\lim_{x \to +\infty} [f(x+1) - f(x)] = L$$ then $$\lim_{x \to +\infty} \frac{f(x)}{x} = L$$
I tried using the definition on the first limit; then, I attempted to use that inequality to apply the triangular inequality to a... |
Show that the equation $x^2+y^2+z^2= (x-y)(y-z)(z-x)$ has infinitely many solutions in integers $x, y, z$.
It seems like if I find a set of $x,y,z$ that satisfy this for any values that will prove it. But how do I find that set?
|
I know that if something is odd then $\exists k \in \mathbb{Z}: a = 2k + 1$. But what I get is:
$n = 2k + 1$
$3n = 3(2k+1)$
$3n = 6k + 3 $
But i can't factor 6k + 3 to give me 2k + 1 !
Any suggestions?
|
I know that some quasigroups are not loops, meaning they don't have a two-sided identity. But are there non-empty quasigroups that don't even have one-sided identities?
|
Prove the following without using L'Hospital's Rule, integration or Taylor Series:
$$\lim_{n \to \infty} \frac{\ln(n)}{n}=0 $$
I began by rewriting the expression as: $$\lim_{n \to \infty}{\ln(n^{1/n})} $$
Since the text shows $$\lim_{n \to \infty}{n^{1/n} = 1} $$
I was wondering is the proof just as simple as stating:... |
Suppose that we have a sequence of probability measures $\{ \mathbb{P}_n \}$ converging weakly to a probability measure $\mathbb{P}$. Suppose that $M$ is a metric space with a compact subset $K$. I would like to show that for any bounded continuous function $f: M \to \mathbb{R}$,
$$ \int_K f \, d\mathbb{P}_n \to \int_... |
I need to take the derivative of $f(x) = \frac{x}{\cos(x)}$.
What I am doing:
$$f'(x) = \frac{d\ (x\cos(x)^{-1})}{d \ x} + (\frac{d\ (x\cos(x)^{-1})}{d\ (\cos(x))} * \frac{d\ \cos(x)}{d\ x})$$
$$f'(x) = 1 + (- x(\cos(x)^{-2}) * (-\sin(x))) $$
... is it correct?
|
I am highly interested in properties of digital root.
Digital Root: Digital root of a number is a digit obtained by adding digits of number till a single digit is obtained.
It's clear that Digital root Partition the set of Natural number in 9 Equivalence Classes.
When I was reading the proof that Prime Numbers are Inf... |
I wish to have some feedback on the following proof of the claim below, either if it is correct, what to fix, or other suggestions.
Claim: Let $\psi :[0,1] \times \mathbb{R} \to \mathbb{R}$ be a continuous function (or at least integrable). Then $F(x)=\int_a^b \psi(t,x(t)) \mathrm{d}t$ for any $x(t) \in C_{[a,b]}$ is ... |
Is there some literature about bipartite graphs whose minimal cycles all have length $4$? By that I mean that any cycle in the graph with length strictly greater than four can be divided into cycles of length four ($Z^d$ is an example of such graph).
Thank you for the answer
|
Materials in quantum information often emphasize that one and two bit classical reversible gates cannot achieve universality for the classical reversible computation, whereas universal quantum computing can be achieved only using one or two qubits gates.
I want to understand why classical reversible computing cannot be... |
I need to come up with:
two metric spaces ( X , d ) and ( Y , p )
A continuous function f: X → Y
A Cauchy sequence {xn} in X that isn't mapped to a Cauchy sequence in Y
My idea was to make the first metric space ℝ with the "usual" euclidean metric, and the second metric space would be ℝ with the discrete metric. Th... |
Given a submanifold $(S,\phi)$ of a manifold $M$, how do we find the subspace of $T_pM$ that is equal to $T_pS$ for $p\in \phi(S)$. I know how to do it for level sets. Is there a way for general submanifolds?
|
Let $X,Y$ be normed vector spaces over $\mathbb{C}$, and $L(X,Y)$ the space of all bounded linear maps from $X$ to $Y$. Its known that $L(X,Y)$ is a normed(operator norm) vector space.
Theorem:
The completeness of $Y$ implies the completeness of $L(X,Y)$ in the norm metric.
Proof:
Let $(f_j)_{j=0}^\infty$ be a Cauchy... |
Today, I just finda confusing quesiton. If I do this way:
$$\int { \frac { dx }{ x } } =x\left( \frac { 1 }{ x } \right) -\int { xd } \left( \frac { 1 }{ x } \right) =1+\int { \frac { dx }{ x } } $$
then, $0=1$
what's wrong with this calculation? in another words, why I cannot apply "integral by parts" method to th... |
Can we find a perfect cube like $111...111$(all digits are $1$), apart from the number $1$ itself?
It's easy to prove that there can't be anything like $111...11$ that is a perfect square besides $1$, but how to do this for perfect cube? Are there some new techniques to do this?
|
Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$
Attempted Solution:
We use induction. Additionally, we prove the stronger inequality omitting the floor function. That is, $$n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2^\frac{n}{2}}$$
The base case is clear. Suppose it holds f... |
So this is my first semester taking a Real Analysis class. We are using the book Introduction to Analysis by Gaughan 5th ed. This is my first real Math class and I'm really excited but I am having trouble following some of the most basic proofs.
So in theorem 0.21. Can you explain the part Hence there is $m\in A$ s.... |
Is it true that applying the Euler-Lagrange equation to the integral
$E(\gamma)=\int_{t_1}^{t_2} g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'\operatorname{d}\!t$ rather than the arc length integral $L(\gamma)=\int_{t_1}^{t_2} \sqrt{g_{\alpha\beta}(\gamma^{\alpha})'(\gamma^{\beta})'}\operatorname{d}\!t$ is just a... |
There are $5$ identical red balls and $6$ identical green balls. In how manys we can arrange them so that there are exactly two pairs of green balls.
Let red balls be $R,R,R,R,R$ and green be $G,G,G,G,G,G$ and since we need exactly two pairs of green balls, let $X=GG$ and $Y=GG$.
Now it means we need to arrange $R,R,R,... |
Show $SO_2(\mathbb{R}) \cong\{ A \in GL_2(\mathbb{R}) : A^tA = I, \det A = 1\}$, where $SO_2(\mathbb{R})$ is the group of rotations of the circle under the operation of composition.
Attempt: Suppose $A =\begin{bmatrix}
\cos\theta & -\sin\theta \\
\sin\theta & \cos\theta
\end{bmatrix}$.
I have verified th... |
Let $C \subset R^n$ with $C \neq \emptyset$ be a closed convex set. Consider some $x \in \mathbb{R}^n$ satisfying $x \notin C$. Prove that there exists some $y \in \mathbb{R}^n$ in the boundary of $C$ which implies the existence of a nonempty hyperplane defined by $\{z \in \mathbb{R}^n : (x-y)^T(z-y) = 0\}$.
Geometrica... |
I am trying to prove the limit I came up with:
$$\lim_{x\to\infty}\left(\csc\dfrac{m}{n+x}-\csc\dfrac{m}{x}\right)=\dfrac{n}{m}$$
This fact came from the double generalization of the special case $\,m=\pi,\;n=1\,$ which has exceptional connections to the longest length of a $2n$-gon of radius $1$, $\csc\left(\frac{\pi}... |
Let $R$ be a Noetherian ring and $I$ an $R$-ideal. The number $\operatorname{depth}_I R$ is the length of maximal $R$-regular sequence in $I$. It is well-known that If $\operatorname{depth}_I R = 0$, then any $x \in I$ is a zero divisor. Is the converse true?
If $I$ can be generated by zero divisors, then is $\operat... |
If $f(x)$ is continuous, it is known that $f(x+y)=f(x)+f(y)$ implies that $f(x)$ is linear, and non-continuous solutions are discussed in these links. (1,
2,3, 4)
However, what is wrong with this proof that all solutions to the Cauchy Functional Equation are of the form $f(x)=cx$?
If $x$ is rational, it is known that $... |
I have an exercise in complex analysis that begins,
If $U\subset \mathbb C$ is an open set and $f:U\to \mathbb C$ is real differentiable....
Later on, it allows me to assume $f$ is holomorphic.
It also mentions $$\frac{\partial f}{\partial x}.$$
I am unsure of what this is.
Is this
$$\frac{\partial f}{\partial x} = \... |
It's actually a question on finding the probability, but I am stuck on a different part of this question. There are $2^n$ players playing a tennis tournament. I have to find the total number of ways in which semifinalists are selected. The answer provided to me is $^{2^n}C_4$. But I think this is wrong.
Let the players... |
I am trying to express this formula in terms of N:
$$
A=\frac{a^2N} {a\cdot \tan{\frac{180}{N}}}
$$
I really don't know how to do this. I tried and got this:
$$
A \cdot a \cdot \tan{\frac{180}{N}} = a^2N
$$
$$
\tan{\frac{180}{N}}=\frac{a^2N}{Aa}
$$
$$
\frac{180}{N}=\tan^{-1}{\frac{a^2N}{Aa}}
$$
$$
N=\frac{180}{\tan^{-1... |
Show that every compact metrizable space has a countable basis.
My try:
Let $X$ be a compact space and metrizable. Now for each $n\in \Bbb N$; I can consider the open cover $\{B(x,\frac{1}{n}):x\in X\}$ of $X$ .As $X$ is compact we can find finite number of $x_i;1\leq i\le n$ corresponding to each $n$ .
$B_n=\{B(x_i,\... |
Definition of Open Ball
Let $(X, d)$ be a metric space and let $r\in\mathbb{R}^+$. Then the set,
$B_d(x, r) := \{y \in X : d(x, y) < r\}$
will be said to be the open ball of radius $r$ centered at $x$ in the metric space $(X, d)$.
Definition of Open Set
Let $(X,d)$ be a metric space and $U\subseteq X$. Then $U$ ... |
Seeing as Axler is very reluctant to talk about determinants and generally avoids computations and playing around with algebra, I'd like to get a book that will serve as a companion to Axler's unorthodox approach.
I'm not terribly interested in real-world applications. I don't much mind them, but the maths is what matt... |
We know that the integration of $\displaystyle\int\frac{1}{x}\,dx=\log\left(|x|\right)$+$c$ with $x\neq 0$ , but if we go by normal rule then it becomes $\infty$.
Is this a counterexample to the rule of $\;x^{n+1}\big/(n+1)\;$ or am I missing something? Now according to comments i amasking how to prove that derrivative... |
It is well known that :$\cos²\theta+ \sin²\theta=1$ for all $\theta$ real or complex ,I would like to ask about the general equality :$\cos^n\theta+ \sin^n\theta=1$ if there is others values of the positive integer $n$ than $n=2$ for which : $$\cos^n\theta+ \sin^n\theta=1$$ for all $\theta$ real or complex ?
probably ... |
The following theorem appears in the appendix of P.H. Rabinowitz monograph on Critical Point Theory:
Let $\Omega \subset \mathbb R^n$ be bounded. Let $g$ be such that
(i) $g \in C(\overline{\Omega} \times \mathbb R, \mathbb R)$
(ii) there exist constants $r,s \geq 1$ and $a_1,a_2 \geq 0$ such that
$$|g(x,\eta)| \leq a_... |
I'm trying to make sure I have correctly understood Bolzano-Weierstrass, which states that every compact subset of $\mathbb{R}^{n}$ is sequentially compact, which means that if $A$ is a compact subset of $\mathbb{R}^{n}$, then every sequence in $A$ must have a convergent subsequence (and hence a cluster point).
Consi... |
Positive integrals
$$\int_{0}^{1}\frac{2x(1-x)^2}{1+x^2}dx=\pi-3$$
and
$$\int_0^1\frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi
$$
(https://math.stackexchange.com/a/1618454/134791)
prove that $$3<\pi<\frac{22}{7}$$
Is there a similar argument for the following $\log (2)$ inequality?
$$\frac{2}{3}<\log(2)<\frac{7}{10}$$
|
Given a question like this:
Find the area between ${y = x^2 + 2x - 3}$ and ${y = 2x^2 -5x -3}$.
I know how to find the area ${\int y_1 - y_2}$ but how can I tell which one is the top curve? Are there any shortcuts to determining the top curve?
|
For $n\in \Bbb{N}$, $$\int_0^{\pi/2} (\log \sin x)^n\text{ d}x=\frac{1}{2^{n+1}}B^{(n)}\left(\frac{1}{2},\frac{1}{2}\right)$$
Can we extend that result a bit further, to $n\in \Bbb{Q},n\gt 1?$
|
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