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H: What is the right solution for derivative of implicit function?
There's need to find y' of:
$$\arctan(y/x)=\ln\sqrt{x^2 + y^2}$$
Tried:
$\dfrac{1}{(1+(y/x)^2)}*(\dfrac{y}{x})'=(x^2+y^2)^\dfrac{-1}{2}*(\dfrac{1}{2})*(x^2+y^22)^\dfrac{-1}{2} * (2x+2y'*y')$
AI: $$\arctan(\frac{y}{x})=\ln\sqrt{x^2+y^2}$$
$$\frac{1}{1+... |
H: Evaluate $\sum\limits_{k=2}^n \frac{n!}{(n-k)!(k-2)!} $
Question is to Evaluate $$\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!} $$
What i have done so far is $$\sum_{k=2}^n \frac{n!}{(n-k)!(k-2)!}=n(n-1)\sum_{k=2}^n \frac{(n-2)!}{(n-k)!(k-2)!}=n(n-1)\sum_{k=2}^n \binom{n-2}{k-2}$$
I could see that $$\sum_{k=2}^n \binom{n-2}... |
H: Subnet vs. Subsequence
I'm looking for an example of a topological space $X$, a sequence $(x_n)_{n \in \mathbb{N}}$ in $X$ and a converging subnet $(x_i)_{i\in I}$ of $(x_n)$, but with the property that $x_n$ does not have any converging subsequence.
I have an examples of $X$ and $(x_n)$ such that there is convergi... |
H: Show that every open subset of a surface is a Surface.
Show that every open subset of a surface is a Surface.
I think, first of all, i need to define an atlas $\{\sigma_{\alpha}: U_{\alpha}\to \Bbb R^3\}$ for surface S, Where $U$ is an open set.
After there, how to continue the proof?
Please help me. Thank you.
A... |
H: Find the Characteristic polynomial
The characteristic polynomial of $A \in M_{4}(\Bbb R)$ is: $P(t)= t^4-t$
Find the Characteristic polynomial of:
$A^2, A^4$ ($A^4$ was easy but with $A^2$ I'm stuck)
Same question with the field $\Bbb F_3$ and $\Bbb F_2$
Thanks.
AI: $A$ has distinct eigenspaces with eigenvalues ... |
H: Show that the following set has the same cardinality as $\mathbb R$ using CSB
We have to show that the following set has the same cardinality as $\mathbb R$ using CSB (Cantor–Bernstein–Schroeder theorem).
$\{(x,y)\in \Bbb{R^2}\mid x^2+y^2=1 \}$
I think that these are the two functions:
$f:(x,y)\to \Bbb{R} \\f(x)=x,... |
H: Probability that successive $8$ heads in a trial of $10$ tosses.
Question is :
A fair coin tossed ten times. What is the probability that we can observe a string of $8$ heads in succession at some time.
I do not even understand this Question properly..
Probability of getting first head would be $\frac{1}{2}$ as w... |
H: If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is exact and $B\simeq A\oplus C$ as a $R$-module, does this sequence split?
Suppose $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact sequence that $B\simeq A\oplus C$ as a $R$-module. Does this short exact sequence split?
I think th... |
H: Is the image of $\mathrm{Spec}(\varphi)$, where $\varphi$ is a localization, always open?
Let $R$ be a commutative ring and $S \subseteq R$ multiplicatively closed subset. Consider $\varphi: R \rightarrow S^{-1}R$ the localization of $R$ by $S$. Then it can be shown that the corresponding map
$\mathrm{Spec}(\varphi... |
H: Floor equation, determine solutions
there is a equation $\lfloor\sqrt{6}x\rfloor=\lfloor2.5x\rfloor$, determine how many integer solutions and real solutions has the equation.
I know that $\{-1,0,1\}$ satisfies the equation, but I don't know if there isn't any other solution.
AI: Since $\sqrt6<2.5$, $0\le\left\lfl... |
H: Interior and closure of $\left\{ \frac{1}{n}: n \in \mathbb{N} \right\} $ in $\mathbb{R}$ and $\mathbb{R} \setminus \{ 0 \}$
What is the interior and closure of $A = \left\{ \frac{1}{n}: n \in \mathbb{N} \right\}$ in $\mathbb{R}$ and $\mathbb{R} \setminus \left\{ 0 \right\} $ ?
I suppose that in both cases $\mbox... |
H: $\lim\sup_{n\to\infty}$ of a sequence $\lt e$
Is there a sequence $ c_n $ such that $$ \lim\sup_{n\to\infty}(\frac{1+c_{n+1}}{c_n})^n \lt e $$?
I tried to get the above sequence into a form that would reveal more about the potential upper limit, but I couldn't really figure out anything useful. How do I tackle this... |
H: Quick way to determine if a rational function has a hole on the $x$-axis....
To sketch a rational function's graph, one step is to determine the sign $(+/-)$ of various intervals. I create intervals separated by the vertical asymptote (VA) and $x$-ints on a number line (since these are where a sign change can occu... |
H: How to solve this summation: $\sum_{k=1}^{n}\frac{k2^{k}}{(k+2)!}$
How do I solve this summation?
I have tried simplifying k on the numerator with the factorial in the denominator but it just gets me nowhere.
$$\sum_{k=1}^{n}\frac{k2^{k}}{(k+2)!}$$
Thanks!
AI: Hint
$$\frac{k2^k}{(k+2)!}=\frac{(k+2-2)2^k}{(k+2)!}... |
H: Show $\int_{\gamma}e^{iz}e^{-z^2}dz$ same value on every line parallel to $\mathbb{R}$
From an old qualifier: Show that $$\large\int_{\gamma}e^{iz}e^{-z^2}\mathrm dz$$ has the same value on every straight line path $\gamma$ parallel to the real axis. Justify the estimates involved.
My first thought was to draw a lo... |
H: A sufficient condition for irreducibility of a polynomial in an extension field.
Here is another problem of the book Fields and Galois Theory by Patrick Morandi, page 37.
Let $f(x)$ be an irreducible polynomial over $F$ of degree $n$, and let $K$ be a field extension of $F$ with $[K:F]=m$. If $\gcd(n,m)=1$, show t... |
H: Flip all to zero
I have a square grid of size $N$, with rows numbered from $0$ to $N - 1$ starting from the top and columns numbered from $0$ to $N - 1$ starting from the left.
A cell $(u, v)$ refers to the cell that is on the $u$-th row and the $v$-th column. Each cell contains an integer $0$ or $1$. I can pick an... |
H: Field extension, primitive root of unity
Let $\xi_5$ be a primitive fifth root of unity in $\mathbb{C}$, then we know the extension $\mathbb{Q}(\xi_5)\supseteq\mathbb{Q}$ has degree 4 so the Galois group is of order 4. I am trying to find all intermediate extensions and it turns out the only one is $\mathbb{Q}(\sqr... |
H: Series Expansion of $\arcsin\left(\frac{a}{a+x}\right)$
Can anyone think of a good approximation to:
$$
\arcsin\left(\frac{a}{a+x}\right)\
$$
accurate at $x = 0$? The Taylor series is not available...perhaps some other kind of method?
AI: To answer my own question, the most accurate I can get is by using
$$
\arcsin... |
H: complex logarithm
I have $e^{jk}$ and I want to take a logarithm from it, $\log(e^{jk})$ must be $jk$, right?
here some example I have tried to do with matlab.
$$\log(e^{j2})=j2$$
$$\log(e^{j3})=j3$$
but for $e^{j4}$ it gives me: $\log(e^{j4})=-j2.2832$, why it doesn't give me $j4$
note: $\log$ is natural logarith... |
H: Cut circle with straight lines
Problem: To cut a circle with $n$ straight lines into the greatest number of possible parts.
What is the greatest number and how can you prove it is optimal?
AI: the number of parts is $\binom{n}{2}+\binom{n}{1}+\binom{n}{0}$
I'll give you a hint first, consider what happens when you ... |
H: Find the sum of $\sum (n^2+n)x^n$ using integrals
I'm having a difficult to find $\sum_{n=1}^\infty (n^2+n)x^n$. the solution is $\frac{2x}{(1-x)^3}$.
This is my solution:
$$1. \space\space\space\space S(x) = \sum_{n=1}^\infty (n^2 +n)x^n =$$
$$2. \space\space\space\space \sum_{n=1}^\infty n(n+1)x^n.$$
$$3. \spac... |
H: L2 Norm of Gaussian Integer
I'm writing a Java program that deals with Gaussian Integers. In my program, I have to compute the L2 norm of the GI (Gaussian Integer) and return it as a float.
I've looked around but I cannot seem to find a formula to compute said value.
Is this the correct formula?
The l2 norm of a v... |
H: Unfamiliar with notation : $S \subseteq [d]$ where {$d, w_{1},w_{2}, ..., w_{d}$}
What does $S \subseteq [d]$ mean in the context of {$d, w_{1},w_{2}, ..., w_{d}$}? I don't get what [d] stands for.
AI: Almost certainly $[d]=\{1,2,\ldots,d\}$; this is a fairly standard notation. Check to see whether the context show... |
H: find the eigenvectors of the eigenvalue 1 - simple question
simple question but I seem to be having difficulties.
$A = \begin{bmatrix}1 & \gamma & 4\\0 & 2 & \beta \\ 0 & 0 & 1\end{bmatrix}$, find the eigenvectors of the eigenvalue 1.
What I did:
$\begin{bmatrix}1 & \gamma & 4\\0 & 2 & \beta \\ 0 & 0 & 1\end{bmatri... |
H: Why does the Kronecker Delta get rid of the summation?
I am working with spherical harmonics and the radial equation (part of Laplace's equation in spherical co-ords). The coefficients and equations with I am working with aren't important to my question.
I am using the orthogonality relation for spherical harmonic... |
H: If $X_n\rightarrow X$ almost surely and $Y_n\rightarrow Y$ a.s., then is $X_n/Y_n\rightarrow X/Y$ almost surely true?
If $X_n\rightarrow X$ almost surely and $Y_n\rightarrow Y$ a.s., then is $X_n/Y_n\rightarrow X/Y$ almost surely true?
Is there a theorem for this or is this not correct?
AI: Claim: Let $f : S \subse... |
H: How to show the following cool equality:
I am looking for a proof of the following relationship:
$\newcommand{\ds}[1]{\displaystyle{#1}}$
$$
\frac{\ds{\int_{0}^{\pi}\sin^{n-2}\left(t\right)\,{\rm d}t}}
{\ds{\int_{0}^{\pi}\sin^{n-3}\left(t\right)\,{\rm d}t}}
=
{\ds{\Gamma^{\,2}\left(\left[n - 1\right]... |
H: Limit of power series
Exercise 3.4.22 Let $f_n(x) = x^n$ for n ∈ N.
Show that the sequence $(f_n)_(n∈N)$ converges pointwise to the function f(x) = 0 on the interval (−1, 1)
The definition of pointwise convergence: We say that a function f : X → F is the pointwise limit
of the sequence $(f_n
)
_(n∈N)$
if, for every... |
H: System of Equations- using profits
Question: Keller industries' profits were up $20,000 this year over last year. This was an increase of 25%.
a. Let T represent the profit this year and L the profit from last year and write a system of equations that can be used to determine the profits.
b. Which method would b... |
H: Residues computation when we need power series
I'm trying to compute residues in situation where we need to manipulate power series to get it, but I can't find a good way. Indeed for the sake of example, consider the residue of the following function at $z=0$ which wolfram says it's zero:
$$f(z)=\csc z\cot z = \dfr... |
H: Why do we use the natural base for the logarithm in the Kullback–Leibler divergence?
Well known formula of KL divergence when we have a discrete probability distributions.
$$D_{KL}(P \parallel Q)=\sum\limits_i \ln \left(\frac{P(i)}{Q(i)}\right) P(i)$$
Can someone explain why the natural base of the logarithm? That ... |
H: Divergence-free vector field from skew-symmetric matrix
Let $[a_{i,j}(x_1,\ldots,x_n)]$ be a skew-symmetric $n\times n$ matrix of functions $a_{i,j}\in C^\infty(\mathbb{R}^n)$. Show that the vector field $$v=\sum\left(\dfrac{\partial}{\partial x_i}a_{i,j}\right)\dfrac{\partial}{\partial x_j}$$ is divergence-free.
... |
H: uniqueness of the solution of the beam equation
Does anyone know how to prove there exists at most one smooth solution $ u $ of the following problem for the beam equation?
$ u_{tt} + u_{xxxx} = 0 $ in $ (0,1) \times (0, T) $
$ u(0,t) = u(1,t) = u_x(0,t) = u_x(1,t) = 0 $ for all $ t \in (0,T) $
$ u(.,0) = g $ and $... |
H: Task with local extreme
I don't know where and what to start calculating in the following task:
Determine the value of parameters $a,b$, so that the function $f(x) = x^3- 2ax + b$ has a local extreme $y=5$ at $x=1$.
The solution is: $a=3/2$, $b=7$
I need to solve quite a few similar tasks as this and I would reall... |
H: Intermediate Analysis Book Suggestions
the title basically says it all. I am looking for real/functional analysis books that would be considered "intermediate level" by that I mean books harder than baby rudin, and easier than big rudin. In general, I'm looking for something with good coverage of non measure theor... |
H: If $\sum_n \|x\|< \infty$, how to show that $\sum x_n$ is convergent in the Hilbert space $H$.
Let $\{x_n\}$ be a sequence in a Hilbert space $H$. If $\sum_n \|x\|< \infty$, how to show that $\sum x_n$ is convergent in $H$?
There is no doubt that $x_n \rightarrow 0$ as $n \rightarrow \infty$ (right?) since we have ... |
H: Absolutely convergent sums in Banach spaces
Let's say a sum of elements in a Banach space is absolutely convergent if even the sum of the norms converges, i.e. $\sum_{i=1}^\infty ||x_i|| < \infty$. This condition implies that the sum of the $x_i$ can be reordered, and in fact that it can be represented in this way... |
H: Finite series help with an obvious fact
Hi everyone I apologize is the following question is too stupid but I cannot figure out how use the principle of induction in this obvious fact:
Definitions: (Finite series): Let $m,n$ be integers and $a_i \in \mathbb{R}$ for each $m\le i \le n$. The we define the finite sum... |
H: Extremal problem
The task is to calculate $a'$ of a square (you cut out) if the volume of a cube is maximum.
(you cut out white squares and put together grey squares so you get a cube without a cover/cap)
I don't know what I'm doing wrong, but when I try to calculate critical points I always get $0$, and there ... |
H: Compute the sum of the following series
$$\sum_{j=0}^{\infty} \left(e^{-jx}\right)\left(x^j\right) $$for $x$ on $[0, \infty)$.
I already proved that this series converges uniformly for $x$ on $[0, \infty)$, i.e. its partial sum converges uniformly to a function on $[0, \infty)$. But I just could not get a fair gues... |
H: How to show that if a 3x3 matrix A and it's diagonal B are in SO(3) that their product is in SO(3)
I can easily determine part a and c but am stuck when it comes to part b and d. Any help would be greatly appreciated. I apologize for any formatting errors I am new to math.stackexchange.com I promise I tried to do m... |
H: How to find Equation of motions and Hamilton Function for this Lagrangian?
$$L = \frac{m \dot{x}^2}{2} - \exp(|x|) $$
I would appreciate if you could explain the steps needed to get the answer
AI: I feel the discussion was getting too long in the comments. Since we've established the system has one degree of freedo... |
H: series implication
Let $(a_n)_{n \in\ \mathbb{N^*}}$ , $(b_n)_{n \in\ \mathbb{N^*}}$ a sequence of real numbers. Show that:
converges $\sum\limits_{n=1}^{\infty}a_n$ and is $(b_n)_{n \in\ \mathbb{N^*}}$ bounded and monotone, so converges $\sum\limits_{n=1}^{\infty}a_nb_n$.
$~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~... |
H: $R\cong\mathbb{Z}$ or $R\cong\mathbb{Z}/p\mathbb{Z}$ ($p$ prime)
Let $R$ be an integral domain and each subgroup of additive subgroup of $R$ forms an ideal of $R$. Prove that either $R\cong\mathbb{Z}$ or $R\cong \mathbb{Z}/p\mathbb{Z}$ ($p$ prime).
Help me.
AI: Let $n$ be the characteristic of $R$. If $n =0$, then ... |
H: How to multiply permutations
I didn't understand the rules of multiplying permutations.
I'll be glad if you can explain me...
For example: we have $f=(135)(27),g=(1254)(68)$. How do I calculate $f\cdot g$??
Thank you!
AI: I’m going to assume that you evaluate products from left to right; note that many people use ... |
H: Help with natural number problems
Suppose that $\mathbb{F}$ is an ordered field with identities $0$ and $1^{\star}$ (in this problem, $1 \in \mathbb{N}$ )
Define inductively $f: \mathbb{N} \rightarrow \mathbb{F}$ by:
$$f(1) = 1^{\star}\qquad f(n + 1) = f(n) + 1^{\star}$$
(Imprecisely, $f(n) = \underbrace{1^{\star} ... |
H: $x\in R$ is a unit iff $(x+RadR)$ is a unit in $R/RadR$
Let $R$ be a commutative ring. Prove that the element $x\in R$ is a unit iff $(x+RadR)$ is a unit in $R/RadR$. ($RadR$ is Jacobson radical of $R$)
Thanks in advanced.
AI: $\hat x$ is invertible in $R/J(R)$ implies $\hat x\hat y=\hat 1$ for some $y\in R$, that ... |
H: Prove $\lim_{x\to 3}\frac{1}{x+1}=\frac{1}{4}$ using $\epsilon$-$\delta$ definition of a limit
First of all, I'd just like to say I'm fairly new to proving limits using $\epsilon-\delta$ definition, so I apologize in advance if I ask a very obvious question, or make an elementary error.
Given $\displaystyle\lim_{x... |
H: Irreducible polynomial over a finite field?
I am trying to solve a problem about irreducible polynomials over a finite field and i would like to ask you for a little help or any idea how to make this proof. Here is the problem:
We have a finite field $\mathbb{F}_{q}$ and a prime number $p$. Let $q$ be the generator... |
H: Differential Calculus - Swapping a sup and a limit.
Well, I'm doing some exercises on differential calculus and I'm stuck.
(a) Let $U \subset \mathbb R^m$ and $f: U \rightarrow \mathbb R^n$ be a continuous function on a line segment $[x, x+h]\subset U$ and differentiable on $]x, x+h[$. Show that if $T:R^m\rightarro... |
H: What is this curve formed by latticing lines from the $x$ and $y$ axes?
Consider the following shape which is produced by dividing the line between $0$ and $1$ on $x$ and $y$ axes into $n=16$ parts.
Question 1: What is the curve $f$ when $n\rightarrow \infty$?
Update: According to the answers this curve is not a... |
H: Is $x^2$ always congruent to $(y-x)^2$ modulo $y$? How could you prove the cases where its true?
I've came across a problem recently which I've been thinking about, and am not sure if I am correct.
Is $x^2$ always congruent to $(y-x)^2$ modulo $y$? Note: $y$ and $x$ are any integers.
Since $y$ and $x$ are integers,... |
H: notation question re: function space
This is a quick notation question: when one writes $X: C[0,\infty) \to \mathbb{R}$, what does that mean exactly? Is $C[0,\infty)$ the space of continuous functions with a domain of $[0,\infty)$ and thus $X$ is a functional? If so, shouldn't the range be a set of functions as wel... |
H: Identify when $f(x) = 0$.
I have the following problem to solve.
My attempt
a. $\int_0^{\pi} x^n f(x) dx =0$ $\forall$ $n \ge 0$ gives $ x^n f(x) dx =0$ almost everywhere in $[0,\pi]$ $\forall$ $n \ge 0$. Putting $n = 0$ we shall get $f(x) = 0$ almost everywhere. As $f(x) \in C[0,\pi]$ we shall say $f(x) = 0$ $\fo... |
H: Complex number and conjugate
If $z$ is a complex number and $z^6=-16|z|^2$, how to prove that $|z|=0$ or $|z|=2$?
I tried to change $z$ into the form $|z|$ and I get $|z|^4=-16(e^{-6iθ})$
But I don't know how to continue.
AI: We start with $$ z^6= -16|z|^2$$ .$z=0$ is an obvious solution. Now, for the nonzero solut... |
H: Markov Chain: I don't understand this solution to a conditional probability problem after n state transitions...
What is the probability of being in state 4 after two steps, given that one is in state 5 after 8 steps?
Markov Chain is at top of link. The sample solution is part (f). I have no idea what the summation... |
H: Determining $a$ values for convergence in alternating series
So we have this series:
$$
\sum^{\infty}_{n = 1}\left(-1\right)^n\sin\left(a \over n\right)
$$
The task is to find all $a$ values, in which case the series converges and all $a$ values in which case the series absolutely converges.
My first question is th... |
H: Help with a 'simple' sum of linear operators and their adjoints acting on an orthonormal basis
Given an orthonormal basis $\{u_1,\cdots, u_n\}$ of a vector space $V$ I am asked to show that
$$
\sum_{k=1}^n \|T^*u_k\|^2=
\sum_{k=1}^n \|Tu_k\|^2
$$
for all $T\in \mathcal{L}(V)$ where $T^*$ represents the adjoint of... |
H: Finding formula for a cubic polynomial
So I need to find a formula for a cubic polynomial with the given conditions:
Local max at $x=1$
Local min at $x=3$
$Y$ intercept of $5$
$x^3$ term whose coefficient is $1$
So all I really have is $x^3+x^2+x+5$.
I know that $F'(1) = 0$ and $F''(1) < 0$. Also $F'(3) = 0$ and... |
H: A closed ball in $l^{\infty}$ is not compact
Definition of compact in Real Analysis, Carothers, 1ed said that:
In Example 8.1 (c), he claimed that closed ball $\{x: \|x\| \leq 1\}$ in $l^{\infty}$ is not compact. Why?
AI: compact implies limit point compact.
Hint : consider $\{(1, 0 , 0 , 0, \cdots) , \hspace{2mm}... |
H: PDF of $Y = \cos(\pi X)$
I've run into a practice problem where $Y = \cos(\pi X)$, and $X$ is uniform on $[0, 1]$, and I'm supposed to prove that the PDF of $Y$ is $1/(\pi\sqrt{1 - y^2})$. I know that for derived distributions, you plug in the the equation for $Y$ in terms of $X$ into the PDF of $X$, integrate up t... |
H: Notation for "parallel" morphisms in a diagram
Suppose $f\colon A\to B$ and $g\colon A\to B$ are possibly-distinct morphisms. How do I stick them both in a diagram (along with, e.g., their (co)equalizer) without suggesting that they are equal?
AI: In simple cases (such as for (co)equalizers) you can get away with s... |
H: DE : $\frac{\mathrm d^2(F)}{\mathrm dx^2} -S^2 \cdot (F) = K \cdot e^{-ax}$
I have this ODE that I need to solve and I am not sure where to begin:
$$\frac{d^2(F)}{dx^2} -S^2 \cdot (F) = K \cdot e^{-ax}$$
Does anyone know any techniques that I could use?
AI: Hints:
For the homogeneous, we have:
$$m^2 - s^2 = 0 \ri... |
H: sequence with infinitely many limit points
I am looking for a sequence with infinitely many limit points. I don't want to use $\log,\sin,\cos$ etc.!
It's easy to find a sequence like above, e.g. $1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots$
But how can you prove the limit points? The problem I am having is the recursion ... |
H: Proving that $(abc)^2\geq\left(\frac{4\Delta}{\sqrt{3}}\right)^3$, where $a$, $b$, $c$ are the sides, and $\Delta$ the area, of a triangle
Let $a$, $b$, $c$ be the sides of $\triangle ABC$.
Prove $$(abc)^2\geq\frac{4\Delta}{\sqrt{3}}$$
where $\Delta$ is the area of the triangle.
(Editor's note: As observed in... |
H: Minimizing/Maximizing triangle
I plugged in the Y point e^(-x/3) into the equation (1/2)XY, found the derivate and set it equal to zero.
But I am only getting one value of X=3 (max). How do I get the min?
AI: There is no local minimum. For absolute min, you just plug in the end point of the domain, 1 and 5. The on... |
H: Finding determinant using properties of determinant without expanding
show that determinant
$$\left|\matrix{
x^2+L & xy & xz \\
xy & y^2+L & yz \\
xz & yz & z^2+L \\
}\right|
= L^2(x^2+y^2+z^2+L)$$
without expanding by using the appropriate properties of determinant.
All i can do is LHS
$$x^2... |
H: How find this Fibonacci sequence sum $\sum_{k=0}^{\infty}\frac{1}{F_{2^k}}$
let sequence $\{F_{n}\}$ such $$F_{1}=1,F_{2}=1,F_{m+1}=F_{m}+F_{m-1},m\ge 2$$
Find this value
$$I=\sum_{k=0}^{\infty}\dfrac{1}{F_{2^k}}$$
My try: I know this
$$F_{n}=\dfrac{1}{\sqrt{5}}\left(\left(\dfrac{\sqrt{5}+1}{2}\right)^n-\left(... |
H: Definite Integral $\int_2^4\frac{\sqrt{\log(9-x)}}{\sqrt{\log(9-x)}+\sqrt{\log(3+x)}}dx$
How can I find the value of this following definite integral?
$$\int_2^4\frac{\sqrt{\log(9-x)}}{\sqrt{\log(9-x)}+\sqrt{\log(3+x)}}dx$$
AI: HINT:
Use $$I=\int_a^bf(x)dx=\int_a^bf(a+b-x)dx$$
and $$2I=\int_a^bf(x)dx+\int_a^bf(a+b... |
H: Prove that there does not exist a surjective function from the set of rationals to reals.
Prove that there does not exist a surjective function f: $\mathbb{Q}\rightarrow \mathbb{R} $.
I think a proof by contradiction would work which means we want to prove $$\neg (\forall{y}\in \mathbb{R}, \exists{x}\in\mathbb{Q}... |
H: If $S_n=\left[-\frac{n}{n+1},\frac{n}{n+1}\right]$ then $\bigcup\limits_{n\geq1}S_n=(-1,1)$
I was self reading Mathmatics for Economists by Simon and Blume.
Consider the closed sets $S_n=\left[-\frac{n}{n+1},\frac{n}{n+1}\right]$ for $n\geq1,n\in\mathbb N$. Then $$\bigcup_{n\geq1}S_n=(-1,1),$$ is an open interval.... |
H: Finding an affine combination of a point on a triangle
I have a problem involving affine combinations that I can't figure out how to solve.
Given the above picture, write q as an affine combination of u and w.
Now, I understand how to write the simpler affine combinations. I can figure out p or s as an an affine c... |
H: show H is a subgroup
Let G be a finite abelian group. Let $H=\langle a,b\rangle = \{a^{i}b^{j}\;\;\;i,j\in\mathbb{Z}\}$
Show that $H$ is a subgroup of $G$
My solution.
1) Need to show $H\neq\emptyset$
This is true as we can take $i=j=0$ then $a^{0}b^{0}=1\in H$
2) closure
let $a^{i}b^{j}\in H$ and let $a^{k}b^{r}\... |
H: $l^q \subset l^p$ for $q\leq p$ with counting measure
Suppose $\Sigma = \mathcal{P}(\Omega) < \infty$ and $\mu$ is the counting measure. I'm seeking to show that $l^q \subset l^p$ for $1 \leq q\leq p \leq \infty$. The main obstacle to a proof is showing $$ \left(\sum_{\omega \in \Omega} |\omega|^q\right)^{1/q} \leq... |
H: How do you prove that there does not exist a $3 \times 3$ matrix over $\Bbb{Q}$ such as $A^8=I$ and $A^4 \ne I$?
I am kind of stuck with the following problem:
Prove that there does not exist a $3 \times 3$ matrix over $\Bbb{Q}$ such as $A^8=I$ and $A^4 \ne I$.
I already tried some things and got that if $A^8=I$, ... |
H: interior of a nested increasing union over a sequence of sets
What are the weakest hypotheses on a topological space $X$ so that for every increasing sequence $S_n$ of subsets of $X$ we have that $\cup _{i=1}^\infty (S_n^o)=(\cup _{i=1}^\infty S_n)^o$
AI: The proposition is false in the case of the compact regular ... |
H: Integration by trig substitution - Why can I draw a right triangle and use that to go back to the original variable?
Suppose we are given an integral of the form $\int f(x) dx$ with $f(x)$ defined on $[a;b]$.
We can then use inverse substitution using a function $g(t)$ defined over an interval $[\alpha;\beta]$ suc... |
H: Composition of Differential Operators
If I have:
$A=\partial_x^2+u(x)$
$B=u(x)\partial_x$
How do I compose: $AB$ and $BA$?
AI: This is just usual composition of functions: for all $f$ in an appropriate domain
$(AB)(f)=A(B(f))=(\partial^2_x+u)(u\partial_x f)=\partial^2_x (u\partial_x f)+u^2\partial_x f=\cdots$ using... |
H: Guide to sketching graphs of basic functions
I'm taking a test soon, where I would be asked to sketch graphs. I wonder if there is any kind of guide or general tutorial on the net how to carry out sketching. For instance I would much like to know how would you sketch $f(x)=\frac{ln (x)} {x}$ and $g(x) = \frac{e^x}{... |
H: Find the $2013$th power of a given $3\times 3$ matrix
Question from my linear algebra homework I'm struggling with:
Let $D = \begin{bmatrix} -2 & 5 & 4 \\-1 & 0 & 0 \\0 & 4 & 3 \end{bmatrix}$
We are asked:
Find $D^5+3D^2-D+I$
Find $D^{2013}$
Write $D^{-1}$ as a polynomial of $D$
I solved questions 1) and 3) but c... |
H: Sequences not strictly increasing.
Let $d_n=a_{n+1}-a_n$ be the difference of two consecutive terms of a sequence of natural numbers.
We can easily construct sequences of natural numbers $a_n$ using trigonometric functions or the floor function which have the property:
*(1)*For infinitely many $n, d_{n+1}<d_n$... |
H: Why are $g(1) < 0, g(0) > 0$ in proof of Fixed-point property for continuous function on $[0,1]$.
Can someone please help me understand how, in the solution below, we can get to $g(1)<0$ and $g(0)>0$ from the fact that $g(1) \neq 0 $ and $ g(0) \neq 0 $?
Thanks.
Rudin Chapter 4, question 14 (Exercise 4.14)
AI: I ... |
H: Is this a partially ordered set?
I was wondering if partially ordered sets could have loops in their diagrams. For example isn't the $S=\{1,2,3\}$ and relation $R=\{(1,1),(2,2),(3,3),(1,2),(2,3),(3,1)\}$ a partially ordered set that has a cycle? $R$ is reflexive, antisymmetric and transitive.
AI: Transitivy fails f... |
H: Algebraic structures on proper classes
From time to time, I see proper classes being endowed with algebraic structure. The ordinals with addition is one example, but I've seen a lot more, most of which have been above my head. The standard definitions of standard algebraic structures impose the requirement that the... |
H: Given that $z= \cos \theta + i\sin \theta$, prove that $\Re\left\{\dfrac{z-1}{z+1}\right\}=0$
Given that $z= \cos \theta + i\sin \theta$, prove that $\Re\left\{\dfrac{z-1}{z+1}\right\}=0$
How would I do this?
AI: $$\frac{z-1}{z+1}=\frac{z-1}{z+1}\frac{\overline z+1}{\overline z+1}=\frac{|z|^2-1+2i\,\text{Im}\,z}{|z... |
H: Proof on ring isomorphism- irreducible
Consider the ring isomorphism $\phi: A \to B$
I have to prove that $a\in A $ is irreducible if and only if $\phi(a)$ is irreducible.
By definition, $a$ is irreducible in A if and only if:
1) $a$ is no cero
2) $a$ a is not a unit
3) If $a=bc$ where $c,b\in A$, then $b$ or $c$ ... |
H: Finding $\displaystyle \lim_{x\to 0} \frac{\ln(3^{x}+1)-\ln(2)}{x}$
I have to find the limit of $\displaystyle \lim_{x\to 0} \frac{\ln(3^{x}+1)-\ln(2)}{x}$ using only notable limits avoiding l'Hopital's method and derivatives.
I tried to use logarithm's properties but then I came in a bad situation. Then I tried to... |
H: Neighborhood in topological groups
Let $G$ be a topological group, $e$ the neutral element and $U$ a neighborhood of $e$.
Claim: Then there exists a neighborhood $V$ of $e$, such that $V^2 \subseteq U$.
This should follow easily from the continuity of the group multiplication. But how?
AI: The map $m\colon (x,y)\in... |
H: Why is a certain subset of a regular uncountable cardinal stationary?
this is an excerpt from Jech's Set Theory (page 94).
For a regular uncountable cardinal $\kappa$ and a regular $\lambda<\kappa$ let $$E^\kappa_\lambda= \{\alpha<\kappa:\mbox{cf}\ \alpha=\lambda \}$$
It is easy to see that each $E^\kappa_\lambd... |
H: What is measure of the following set?
Suppose $ D= \lbrace (x,y) \in [0,1]\times [0,1]: x-y \in \mathbb{Q} \rbrace $. Is the measure of $D$ zero? Thanks.
AI: The set $D$ is a subset of the union over $q\in\Bbb Q$ of the lines
$$D_q = \{(x,y) \in \Bbb R^2 : x-y=q\},$$
each of the $D_q$ having Lebesgue measure $0$ (i... |
H: Calculating the limit of some sum
I want to calculate the following limit
$$
\lim_{n\to\infty}\left(\frac{1}{n}+\frac{1}{n+1}+\ldots+\frac{1}{2n}\right)=?
$$
Obviously the result is bounded between $1/2$ and $1$, but how can I calculate the exact result?
AI: Rewrite the given sum as follows:
\begin{align}\sum_{k=0}... |
H: Order of a subgroup on elliptic curve over a finite field
May i ask you for a little help for a problem about elliptic curves?
Here's the problem:
Given an elliptic curve $E$ over the finite field $\mathbb{F}_{101}$. We know that there is a point of order $116$ on this curve. Find $\left | E(\mathbb{F}_{101}) \rig... |
H: How to calculate the ⌉ operator?
In a book about finance, the following formula appears:
$$\large a_{10 \,⌉\, 0.04}$$
What is this ⌉ operator and how to read / calculate it?
AI: This just a notation in Actuarial sciences
$a_{\overline{n|}i} := v + v^2 + \cdots + v^n = \frac{1-v^n}{i}$
, where $v:=\frac{1}{i+1... |
H: Finding the Eigenvector for eigenvalue 0
I tried finding the eigenvectors of the following transformation: $T:\mathbb R_2[X]\rightarrow \mathbb R_2[X] $, $T(f(x))=f'(x)$. I found the matrix of this transformation to be $$\begin{matrix} 0&0&0 \\ 2&0&0 \\ 0&1&0 \end{matrix}$$
Thus I found the eigenvalue to be 0. But... |
H: What is the equation for this graph?
Here is what y(x) = -x+100 graph looks like:
I am trying to change this equation, so that the graph becomes a curve which still crosses the axis Y at (0,100) and axis X at (100,0):
Can anybody give me a hint on this?
AI: Probably the easiest way is to choose a third point $(a,... |
H: Convergence of sequence defined with other sequence
$a_n$ is a sequence with $\lim\limits_{n \rightarrow \infty}{\frac{a_{n+1}}{a_n}}=L$, $a_n>0$.
So the task is to show that $c_n:=(a_n)^\frac{1}{n}$ converges and $\lim\limits_{n \rightarrow \infty}{c_n}=L$.
I've been working on this for hours now and don't have a... |
H: Is this matrix positive-semidefinite in general?
for the matrix written below I was wondering if one can show that it is positive-semidefinite for $n>3$ and $0< \alpha<1$. (Or not. For $n=2, 3$ it works by showing that all principal minors are non-negative.)
$$
C_{n,n} =
\begin{pmatrix}
1 & \alpha^1& \alpha^2 & ... |
H: Let $b>0.$ Evaluate $\lim_{n \to \infty} \int^{b}_{0} \frac{\sin nx}{nx}dx$
I'm stumped at the last part of this problem. Could anyone advise, please? Thank you.
Here is my proof.
Define $f: \mathbb{R} \to \mathbb{R}$ by $$
f(x) = \left\{\begin{aligned}
&\frac{\sin nx}{nx}&&, x \neq0 \\
&1 &&, x=0
\end{aligned... |
H: Prove that $d_{1}$ and $d_{2}$ induce the same topology
Suppose $d_{1}(x,y) = |x-y|$, $d_{2}(x, y) = |\phi(x) - \phi(y)|$, where $\phi(x) = \frac{x}{1 + |x|}$. Prove that $d_{1}$ and $d_{2}$ are metrics on $\mathbb{R}$ which induce the same topology.
It's easy to prove that $d_{1}$ and $d_{2}$ are metrics. Call t... |
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