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H: Different log bases
I have done many $\log$ problems but I've never learned something such as $\log_ax-\log_by$. I know that to condense a logarithm you must have the same base: $\log_ax-\log_ay=\log_a\left(\frac{x}{y}\right)$. With that said,
Simplify the following expression:
$2\log_49-\log_23$ . Which now beco... |
H: Number of irreducible polynomials with degree $6$ in $\mathbb{F}_2[X]$
I'm looking for the number of irreducible polynomials with degree $6$ in $\mathbb{F}_2[X]$ with leading coefficient $1$.
First question: The leading coefficient $1$ is redudant because every polynomial of degree $6$ has leading coefficient $1$?
... |
H: Equivalent Definitions of the Operator Norm
How do you prove that these four definitions of the operator norm are equivalent?
$$\begin{align*}
\lVert A\rVert_{\mathrm{op}} &= \inf\{ c\;\colon\; \lVert Av\rVert\leq c\lVert v\rVert \text{ for all }v\in V\}\\
&=\sup\{ \lVert Av\rVert\;\colon\; v\in V\text{ with }\lVer... |
H: $n = a^2 + b^2 = c^2 + d^2$. What are the properties of a, b, c and d?
If n is a positive integer that can be represented as the sum of two odd squares in two different ways:
$$
n = a^2 + b^2 = c^2 + d^2
$$
where $a$, $b$, $c$ and $d$ are discrete odd positive integers, what properties can be deduced about $a$, $b$... |
H: Countably compact paracompact space is compact
The proof that I have seen for the result "countably compact paracompact implies compact" involves metacompactness, which follows from paracompactness.
I wonder if it can be proved without going through metacompactness.
AI: Perhaps the easiest argument uses the followi... |
H: $\mu$-recursive definition of ulam (3n+1) function
$\newcommand{\ulam}{\operatorname{ulam}}$
The ulam function is defined as
$$ \ulam(x) = \begin{cases} 1 & x = 1 \\ \ulam\left( \frac{x}{2}\right) & x \text{ even}\\ \ulam(3x+1) & x\text{ odd}\end{cases}$$
I want to show that $\ulam$ is $\mu$-recursive by using ... |
H: $-\iint_{A}(y+x)\,dA$ Evaluating
I am bit unsure about the following problem:
Evaluate the double integral:
$$-\iint_{A}(y+x)\,dA$$
over the triangle with vertices $(0,0), (1,1), (2,0)$
OK, so I figured here that I would do this by first evaluating the integral over the region bounded by the vertices $(0,0), (1,1),... |
H: Condensing logarithms
Simplify:
$2\log_{10}\sqrt{x}+3\log_{10}x^{\frac{1}{3}}$
I got to this: $2\log_{10}x^{\frac{1}{2}}+3\log_{10}x^{\frac{1}{3}}$.
Now, usually you bring the exponent the the front and that would yield:
$$\frac{1}{2}(2)\log_{10}x+\frac{1}{3}(3)\log_{10}x=\log_{10}x+\log_{10}x=2\log_{10}x... |
H: Motivation for definition of logarithm in Feynman's Lectures on Physics
I'm not sure if the title is descriptive enough; feel free to change it if you come up with something better.
I've been reading through Feynman's Lectures on Physics. In the first volume, he dedicates a chapter to just math. He starts with the ... |
H: Learning differential/Riemannian geometry for PDEs
I know there have been threads on which books to learn DG/RG from but hopefully this is sufficiently different to avoid closure.
Can anyone recommend a book to learn DG/RG (whichever is appropriate) so that I can do PDEs on manifolds? At the moment I am reading th... |
H: Verify trigonometry equation $\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$
How would I verify the following trigonometry identity?
$$\tan A - \csc A \sec A (1-2\cos^2 A)= \cot A$$
My work so far is
$$\frac{\sin A}{\cos A}-\frac{1}{\sin A}\frac{1}{\cos A}(1- \cos^2 A- \cos^2 A)$$
AI: $$\frac{\sin A}{\cos A}-\frac{... |
H: Is Vector arithmetic compatible between 2D and 3D Vectors?
As the title suggests, is Vector arithmetic (including Cross and Dot Products and Length Calculations) compatible between 2D and 3D Vectors where a "2D Vector" is a 3D Vector with a third parameter that is always one (1)?
That is, is $\vec{A}$(x, y, 1) comp... |
H: Verify trigonometry equation $\frac{\sin(A)}{\sin(A) + \cos(A)}=\frac{\sec(A)}{\sec(A)+\cos(A)}$
How would I verify the following trig equation?
$$\frac{\sin(A)}{\sin(A) + \cos(A)}=\frac{\sec(A)}{\sec(A)+\cos(A)}$$
My work so far is to write the RHS as
$$\frac{1/\cos(A)}{1/\cos(A) + \cos(A)}$$
But I am not sure wha... |
H: What is a separator object?
Let $S$ be an object of category $C$. We say $S$ is a separator object of $C$ if whenever
$$Y \stackrel{f_1} \longleftarrow X \stackrel{f_2}{\longrightarrow} Y $$
$(\forall x[S\stackrel{x}\longrightarrow X \Rightarrow f_1x=f_2x]) \Rightarrow f_1 =f_2$
This reminds of the definition of ... |
H: Countable sets?
Determine whether each of these sets is countable or uncountable. For those that are countably infinite, exhibit a one-to-one correspondence between the set of positive integers and that set.
1) Integers not divisible by $3$.
2) Integers divisible by $5$ but not by $7$.
I figured out the first o... |
H: $F, G \in k[X_1, \dots , X_n]$ homogeneous of degrees $r$ and $r+1$ $\implies$ $F+G$ is irreducible
I have a question about Exercise 2-34 from William Fulton's Algebraic Curves book. The exercise is as follows.
Suppose that $F, G \in k[X_1, \dots , X_n]$ are forms (i.e. homogeneous polynomials) of degree $r$ and ... |
H: Jointly distributed exponential random variables
Attempting to solve the following problem I am confused about what to use as the probability density function
Problem
The time that it takes to service a car is an exponential
random variable with rate 1.
If A.J. brings his car in at time 0 and M.J.
brings her car ... |
H: Why can you multiply out?
$$(a+b)(c+d)=ac+ad+bc+bd$$
Why is this? Is there a proof to this? And there is something similar in logic.
$$(A \land B)\lor (C \land D)=(A \lor C)\land(A \lor D)\land(B \lor C)\land(B\lor D)$$
Why is this? I used it all my life but I don't know why it works.
AI: It's called the distribut... |
H: spivak (sec. 11, question 49)
In exercise 49 of Spivak's Calculus, a function $h$ is termed to be increasing at any point $a$ if there exists a $\delta > 0$ such that
$$ a - \delta < x < a \implies h(x) < h(a) $$
$$ a < x < a - \delta \implies h(a) < h(x) $$
and the reader is asked to prove that a function which... |
H: Solve for $a$: $V=2(ab+bc+ca)$
Solve for $a$
$V=2(ab+bc+ca)$
$$\left(\frac{V}{2}\right)=ab+bc+ca$$ $$\left(\frac{V}{2}\right)-bc=ab+ca$$ $$\dfrac{\left(\frac{V}{2}\right)-bc}{b+c}=2a$$ $$\frac{\left(\frac{V}{2}\right)-bc}{2b+c}=a??$$ I honestly do not know what to do with this problem. And I think I may have ... |
H: Finding the inverse function, is there a technique?
I came across a way to find whether some number is inside a sequence of numbers.
For example the sequence (simple function for positive odd numbers):
$$a(n) = 2n + 1.$$
So the numbers inside it go: $1, 3, 5, 7, \ldots$
If I want to test if the sequence contains a ... |
H: calculus, self-study..recommendations?
I've been trying to educate myself in various areas of mathematics. I have been out of any formal math education for quite some time and so I brushed up on some basic (really, really basic) stuff including algebra and so forth. I'm working my way through some calculus and I'm ... |
H: Limit preserving metrics.
I need to prove that given a sequence of points $\{a_n\}$ in $(\Bbb R^k,d_m)$ that converges to $a$, then it converges to the same limit in both $(\Bbb R^k,d_e)$ and $(\Bbb R^k,d_t)$ (and conversely), where
$$ d_m(x,a)=\max_{1\leq i \leq k} \{|x_i-a_i|\} \;;\;\text{ the max metric}$$
$$d_e... |
H: Solve for $+r$ ; $A=2\pi r^2+2\pi rh$
Solve for $+r$
$$A=2\pi r^2+2\pi rh$$ Since $2\pi$ is common on both sides of the $+$ so I will take it out
$$A=2\pi (r^2+rh)$$
Now, divide both sides by $2\pi$
$$\dfrac{A}{2\pi}=r^2+rh$$
Then, we can divide by the $h$
$$\dfrac{Ah}{2\pi}=r^2+r$$
Then;
$$0=r^2+r-\dfrac{Ah}{2... |
H: Finding two numbers given their sum and their product
Which two numbers when added together yield $16$, and when multiplied together yield $55$.
I know the $x$ and $y$ are $5$ and $11$ but I wanted to see if I could algebraically solve it, and found I couldn't.
In $x+y=16$, I know $x=16/y$ but when I plug it bac... |
H: Solve $2x-2yd=y+xd$ for $d$
Solve for $d$
$$2x-2yd=y+xd$$ $$2(x-yd)=y+xd$$ $$2(x-yd)-y=xd$$ $$\frac{2(x-yd)-y}{x}=d$$ Now I think this is wrong considering I have a $d$ on the other side and I would have to reverse my work and end up at the beginning. Any small pointers or tips? Thanks!
AI: (When I said "... wri... |
H: Help Proving that $\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$ for $t\geq 1$
I'm trying to prove the asymptotic statement that for $t\geq 1$:
$$\frac{(1+\frac{1}{t})^t}{e} = 1 -\frac{1}{2t} + O(\frac{1}{t^2})$$
I know that $(1+\frac{1}{t})^t$ converges to $e$ and the right side looks like the ... |
H: Solve for $x$ ; $\dfrac{2x}{4\pi}+\dfrac{1-x}{2}=0$
Solve for $x$:
$$\dfrac{2x}{4\pi}+\dfrac{1-x}{2}=0$$
$$\dfrac{2x}{4\pi}+\dfrac{2\pi(1-x)}{2\pi(2)}=0$$ $$\dfrac{2x+2\pi (1-x)}{4\pi}=0$$ $$2x+2\pi (1-x)=0$$ $$2x+2\pi -2\pi x=0$$ $$2x-2\pi x=-2\pi$$ $$2x(1-\pi )=-2\pi$$ $$2x=\dfrac{-2\pi}{1-\pi}$$ $$\left(\f... |
H: For a defective matrix $B$, do $B$ and $B^*$ have the same eigenvalues?
From the definition of normal matrix, $AA^*=A^*A$, we know that $A$ and $A^*$ share the same eigenvectors, but my question is that do defective matrix $B$ and its conjugate transpose $B^*$ also have the same eigenvectors, although their eigenve... |
H: Please find the probability
Let X and Y be exponential random variables with parameters 1 and
2 respectively. Another random variable Z is defined as follows.
A coin, with probability p of Heads (and probability 1 − p of Tails) is
tossed. Define Z by
Z = X if the coin turns Heads
= Y if the coin turns Tails
Fin... |
H: Complete the square and write in standard form for $3x^2+3x+2y=0$
Standard forms: $y-b=A(x-a)^2$ or $x-a=A(y-b)^2$
$3x^2+3x+2y=0$
I honestly do not know how to start this problem. I have tried a lot of things and obviously not the right one. Can someone explain to me the first step and nothing more and I wil... |
H: Can't understand this remainder solution
The problem is:
$W$ is a positive integer when divided by $5$ gives remainder $1$ and when divided by $7$ gives remainder $5$. Find $W$.
Answer: Take the larger divisor , So Expression becomes $7k+5$. Now this number when divided by $5$ gives remainder $1$ so expression ... |
H: Classify all groups of order $1805.$
Classify all groups of order $1805.$ It may help to note that $\left(\begin{array}{c} 0 &-1\\ 1 & 4 \end{array}\right)$ has order $5$ in $GL_2(\mathbb{F}_{19}).$
My idea: Observe that $1805=5^{1}\times 19^{2}.$ So, $G\cong \mathbb{Z}_5\times \mathbb{Z}_{19^2}$ and $G\cong \mat... |
H: Integers that are a sum of two $k$th powers in $n$ different ways
Do there exist infinitely many $k$ such that for all $n$ we can find a sequence $x_i$ of distinct natural numbers
such that $x_1^k+x_2^k=x_3^k+x_4^k=\cdots=x_{2n-1}^k+x_{2n}^k$ ?
AI: This is an open problem. Little is known for $k>3$.
In particular, ... |
H: The line with equation $y=4x+c$ is a tangent to the curve with equation $y=x^2-x-5$
The line with equation $y=4x+c$ is a tangent to the curve with equation $y=x^2-x-5$. Find the value of $c$.
I did it
$y=x^2-x-5$
$4=2x-1$
$\frac{5}{2}=x$
$x=\frac{5}{2}$
$=(\frac{5}{2})^2-\frac{5}{2}-5$
$=-\fr... |
H: If $m\geq2$ is an integer, then $\sum\limits_{n=1}^{\infty}m^{-n^2}$ is irrational
Let $m \geq2$ be an integer. I want to ask how to prove that the sum of the following series is irrational:
$$\sum _{n=1}^{\infty} \frac{1}{m^{n^2}}$$
AI: To get a grasp on what is going on here, we start with a simple, seemingly un... |
H: If two polynomials are equal as functions, are they necessarily equal as polynomials?
Say you have a finite field $F$ of order $p^k$. Suppose that $f,g\in F[X_1,\dots,X_m]$, such that the degree of each $X_i$ is strictly less than $p^k$ in both $f$ and $g$. I'm putting this condition to avoid things like $f=X_1X_2X... |
H: What is the optimization formulation of this question?
I found the following puzzle on SO:
Puzzle:
A merchant has a 40 kg weight which he used in his shop. Once, it fell
from his hands and was broken into 4 pieces. But surprisingly, now he
can weigh any weight between 1 kg to 40 kg with the combination of
th... |
H: Compute $\lim\limits_{n\to{+}\infty}{{(2+n^3)}^{55-7n}}$
Find: $$\lim_{n\to{+}\infty}{{(2+n^3)}^{55-7n}}$$
According to Maple, that is equal to zero.
What theorem could I use?
AI: Put: $ y = {{(2+n^3)}^{55-7n}} $, then taking the natural log of both sides gives $$ \ln (y) = (55-7n)\ln(2+n^3) $$
Now, take the limi... |
H: Fourier transform of function composition
Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually?
I know you can do this for the sum, the product and the convolution of two functions. But I haven't seen a formula fo... |
H: Predict the height of a student whose weight is 60 kilograms.
The average height and weight of a group of students turned out to be
5 ft 6 inches and 65 kilograms respectively. The correlation between
heights and weights was found to be 0.6. Using the regression equation
for predicting weight from height, the estim... |
H: Name for a horizontal line function
What is the, or what are the, technical terms for a function that produces a horizontal line (all inputs map to the same output), like $f(x) = 5$?
AI: It is a constant function. ${}{}$ |
H: Log as the inverse of Exp in the complex plane
It is standard practice to define on $\mathbb{C}$, $$\operatorname{Log}(z) = \log(|z|) + i \operatorname{Arg}(z).$$ When composed with $\exp$, we get $\operatorname{Log} \circ \exp (z) = z$, the identity function, for all $z$ in the $2\pi $-wide strip $\{ z\, :\, 0 < ... |
H: Complete the square and write in standard form for $9y^2-6y-9-x=0$
Complete the square and write in standard form: $x-a=A(y-b)^2$
$9y^2-6y-9-x=0$
I do not know how to complete the square with $4$ terms. I started off like: $$9y^2-6y-9=x$$
I don't know whether to start trying to complete the square or if I wou... |
H: Factor $4x^3-8x^2-25x+50$ completely
Factor $4x^3-8x^2-25x+50$ completely
The highest numbers you can take would be $1$, $2$, or $4$. Neither of those apply to all. So let's try the $x$! But the last term $50$ doesn't have an $x$ attached. Anybody want to give a small hint please.
AI: Hint: Consider the first t... |
H: Limit inferior/superior of sequence of sets
Let $(\Omega, \mathcal{A}, \mu)$ be a measure space, where $\mu(\Omega)< \infty$. Further $(A_n)_{n \in \mathbb{N}}$ is a a sequence of $\mathcal{A}$-measurable sets. I want to prove, that
$$ \mu ( \liminf_{n \rightarrow \infty} A_n) \leq \liminf_{n \rightarrow \infty} \m... |
H: $\sigma$-algebra induced by $\{\{1\},\{2\},\ldots,\{n\}\}$ and the limit $n \rightarrow \infty$
Let $\Omega = \mathbb{N}$ be the natural numbers and $\mathcal{E}_n = \{\{1\},\{2\},\ldots,\{n\}\} \subset \Omega$. $\mathcal{A}_n = \sigma (\mathcal{E}_n)$ shall be the $\sigma$-algebra induced by $\mathcal{E}_n$. Clear... |
H: Where are good resources to study combinatorics?
I am an undergraduate wiht basic knowledge of combinatorics, but I want to obtain sound knowledge of this topic. Where can I find good resources/questions to practice on this topic?
I need more than basic things like the direct question 'choosing r balls among n' etc... |
H: A Proof About Annihilators and Linear Functionals
I'm following the proof of the fact that $ \mathrm{dim}(U) + \mathrm{dim} \mathrm{Ann}(U) = \mathrm{dim}(V) $ for $U \subset V$ and $\mathrm{Ann}(U)$ is an annihilator of $U$, in here (Proposition 2.20 (a)).
But I don't understand how $v^{'}(v_{i}) = c_{i}$.
AI: By... |
H: Additive inverse
Let $F$ be the set of $\alpha\subset \mathbb{Q}$ with following properties.
(I) $\alpha ≠ \emptyset$ and $\alpha ≠ \mathbb{Q}$
(II) $p\in \alpha$ and $q<p$ ⇒ $q\in \alpha$
(Notice that it's slight different from usual dedekind cut)
Define $\alpha < \beta$ iff $\alpha \subsetneq \beta$.
Then $F$ is ... |
H: Partition of a probability measure in a continuous and atomic part
Let $(\mathbb{R}, \mathcal{B}, \mathbb{P})$ be a probability space. I want to show that $\mathbb{P}$ can be written as $\mathbb{P} = \mu + \nu$, where $\mu$ is a continuous measure (no atoms) and $\nu$ an atomic measure ($\nu = \sum_i \epsilon_i \de... |
H: $k$ in trigonometric equality $\sin(a) =\sin(b)$
On a test there is the question: "Solve for $x$ on the interval $[-\pi,\pi]$ where $\sin(2x) = \cos(3x)$
I know that:
$\cos(x) = \sin(\frac12\pi - x)$
So you can rewrite the equation to:
$\sin(2x) = \sin(\frac12\pi - 3x)$
But then in the solution, the next step is... |
H: Finding the radical (or squarefree part) of an integer
Given a number $x = p_1^{e_1}\cdots p_n^{e_n}$ with different primes $p_i$ and exponents $e_i \ge 1$, is there an efficient way to find $p_1\cdots p_n$?
I ask this because for polynomials it's easy: with $K$ a field of characteristic $0$ and $$f = g_1^{e_1} \... |
H: Existence and finiteness of Lebesgue integral for: $f(x)=x^{-1}(e^{-x}-e^{-1/x})$
I think I am getting a little better at these MCT, DCT-type exercises. The issue is to show/prove the existence and finiteness (if they apply) to the following function:
$$f(x)=x^{-1}(e^{-x}-e^{-1/x})$$
Where applicable I want to show... |
H: An irreducible polynomial $f \in \mathbb R[x,y]$, whose zero set in $\mathbb A_{\mathbb R}^2$ is not irreducible
This is an exercise on Page 8 of Hartshone's Algebraic Geometry:
Give an example of an irreducible polynomial $f \in \mathbb R[x,y]$, whose zero set $Z(f)$ in $\mathbb A_{\mathbb R}^2$ is not irreducibl... |
H: autocorrelation of a random process calculation
I know if I want to calculate autocorrelation of a random process , I have this rule :
$ R_X (t_1 , t_2) = E \{ X(t_1)X^*(t_2) \} $ .
In my cource I had this example :
$ X (t ) = A cos(2πft + Θ) $
A: constant. Θ: uniform in [0, 2π].
Find the autocorrelation of X.
... |
H: Convergence in the absence of DCT, uniform integrability, and $\limsup E(X_n)$
This question is extended from Resnick's exercise 5.13 in his book A Probability Path.
Let the probability space be the Lebesgue interval:
$(\Omega=[0,1],\mathcal{B}([0,1]),\lambda)$ and define
$X_n:=\frac{n}{\log n}1_{(0,\frac 1n)}$
Sho... |
H: Choosing multiple elements multiple times to cover a set
A combinatorics problem:
There are $N$ balls in total, in every round we randomly pick without replacement $n$ balls ($n < N$), after $k$ rounds, what is the probability that all the $N$ balls are picked at least once? After each round, all the picked balls ... |
H: Convergence in expectation for: $X_n=\sum\limits_{k=1}^n\frac{(-1)^k}{k^2}x_k$
Here is another self-study exercise that I am struggling mightily with:
$X_n=\sum\limits_{k=1}^n\frac{(-1)^k}{k^2}x_k$ where $\omega=(x_1,x_2,...)$ is a series of Bernoulli (1/2) trials.
I am told that $X_n\to X$ a.s for some $X$, and am... |
H: Express each of the following expressions in the form $2^m3^na^rb^s$, where $m$, $n$,$ r$ and $ s$ are positive integers.
I just recently started relearning math as an adult, this should be easy but I have trouble understanding what the actual question is. I am not just looking for the answer to this, I merely wish... |
H: Uniform convergence and convergence in $S'(\mathbb{R}^n)$
Let
$$\hat{f_\epsilon}: \xi \mapsto \exp(-\epsilon |\xi|) \frac{\sin(|\xi|t)}{|\xi| t}$$
denote to the Fourier transform of $f$. How do I see
$\hat{f_\epsilon}$ converges uniformly on $\mathbb{R}^n$ to $\hat{f}=\frac{\sin(|\xi|t)}{|\xi|t}$ as $\epsilon \to ... |
H: How to place rectangle so it envelopes two circles, one circle in the very right end of the rectangle and one in the other end.
How can i place a rectangle so it envelopes two circles, one circle in the very right end of the rectangle and one in the other end.
Say i have two circles with the xy coordinates and a r... |
H: What is $\bigcup\limits_{n=1}^\infty [0,1-\frac{1}{n}]$?
This is probably a pretty dumb question, but I am confused by set theory again. The question is whether
$$\bigcup_{n=1}^\infty \left[0,1-\frac{1}{n}\right]$$
equals $[0,1]$ or $[0,1)$. However, I am looking for some explanation and not only the result, since... |
H: An entire function is identically zero?
I'm preparing for a PhD prelim in Complex Analysis, and I encountered this question from an old PhD prelim:
Suppose $f(z)$ is an entire function such that $|f(z)| \leq \log(1+|z|) \forall z$. Show that $f \equiv 0$.
Well, for $z=0$, $|f(0)| \leq 0$. On the other hand, for... |
H: How to determine the position of neighbor points?
I have two points (red points represented on the image below), they form a line that is described by a specific formula (y = mx + n), I need to determine the coordinates of 4 other points (blue points) that are places on the two sides of the red segment and on the p... |
H: Prove or Disprove $xa \equiv 1 \pmod{ n}$
If $a\in\mathbb{Z}, n\in\mathbb{N}$, then the equation $xa\equiv1\pmod {n}$ has a solution for some $x\in\mathbb{Z}$.
I'm not quite sure where to start. I know that $n|(xa-1)$, so $ns=xa-1$ for some integer $s$.
Should I start plugging in numbers to find one that makes it ... |
H: Show that $(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$ for $x\rightarrow \infty$
So I'm trying to show that for $x\rightarrow \infty$:
$$(x+1+O(x^{-1}))^x = ex^x + O(x^{x-1})$$
So these complicated big-Oh expressions are clearly going to be a recurring theme in my book, and I simply have no idea how to manipulate them i... |
H: Mathematics of change money
Do you know any results or articles about change money?
Something like the statistics of different value notes in a cash box. Or answers to questions which distribution of notes values is best for starting a day in a shop. I mean obviously you need more small value notes than large ones.... |
H: Is every forest with more than one node a bipartite graph?
This is a question from my exam today:
The definition of a bipartite graph is: "A graph with at least two nodes is bipartite if and only if there is no odd-length cycle in the graph."
We'll remember that in a forest, and a tree in particular, there are no ... |
H: Is plugging numbers into the ratio test allowed?
So I'm going through the notes for my online summer calculus class, and something struck me as odd about the ratio test: it used variables only, there were no numbers plugged in. For example, in the series $$\sum_{n=1}^\infty\frac{n}{4^n}$$ we have $a_{n+1}=\frac{n+1... |
H: Proving inequality on functions $x-\frac{x^2}{2}<\ln(1+x)
To prove: $$x-\frac{x^2}{2}<\ln(1+x)<x-\frac{x^2}{2(1+x)},\quad\forall x>0$$
I have used Taylor series expansion at 0 for both the inequalites. The greater than by expanding $\ln(1+x)$ and the less than by expanding $\int \ln(1+x)\,dx$ at 0.
Is there a clea... |
H: Books/lecture notes/videos on category theory for programmer
I want to learn category theory. I tried different books and had several problems with them:
Books are for mathematicians and they use a lot of examples with which I am not comfortable, like algebraic topology, advanced algebra, etc.
Book which simplify... |
H: Is the continued fraction of the square root of a base $\phi$ (golden ratio) number periodic when the continued fraction is expressed in base $\phi$?
I have been looking at concise ways to represent irrational numbers using only integers.
I was thinking about base $\phi$ (golden ratio base) and how it can represent... |
H: $d(x,A)=0\iff $ every neighborhood of $X$ contains a point of $A$
Mendelson, Introduction to Topology, p.52
$(8)$. Let $A$ be a non-empty subset of a metric space $(X,d)$. Let $x\in X$. Prove that $d(x,A)=0$ if, and only if, every nieghborhood $V$ of $x$ contains a point of $A$.
DEFINITION Given a subset $A$ of a... |
H: Find all real solutions to $8x^3+27=0$
Find all real solutions to $8x^3+27=0$
$(a-b)^3=a^3-b^3=(a-b)(a^2+ab+b^2)$
$$(2x)^3-(-3)^3$$ $$(2x-(-3))\cdot ((2x)^2+(2x(-3))+(-3)^2)$$ $$(2x+3)(4x^2-6x+9)$$
Now, to find solutions you must set each part $=0$. The first set of parenthesis is easy $$(2x+3)=0 ; x=-\left(\frac... |
H: $3\sin^2x=\cos^2x;$ $ 0\leq x\leq 2\pi$ Solve for $x$
$3\sin^2x=\cos^2x;$ $0\leq x\leq 2\pi$ Solve for $x$:
I honestly have no idea how to start this. Considering I'm going to get a number, I am clueless. I have learned about $\sin$ and $\cos$ but I do not know how to approach this problem. If anyone can go step-... |
H: Characteristic time?
Could somebody tell me the definition of a "characteristic time"? For example, what is the characteristic time for a function $f(t)=\operatorname{tanh}(t)$ to reach 1? I tried looking up a definition, but there seems not to be a universal definition. Is there a preferred definition?
Many thank... |
H: Calculate the average number of cards of a certain suit in my opponent's hand
Let's suppose that I am playing a card game with 3 other friends. One of my friends is on my team while the other 2 people are on the opposing team. The cards have just been shuffled and dealt so that each player now has 13 cards and the... |
H: Finding the angle between the negative y-axis and the cross product of two vectors
I'm trying to help a friend with the following homework question:
Vectors A and B lie in an xy plane. A has a magnitude of 8.00 units and an angle of 130 degrees; B has components Bx = -7.72 units and By = -9.2 units. Find the angle ... |
H: Solve $\ddot\theta +k\sin(2\theta)=0$ given initial value and constraints
How is it possible to deduce from the equation $$\ddot\theta +k\sin(2\theta)=0$$ where $\theta=\theta(t)$ and $\tan(\theta)={b(t)\over a(t)}$, $k$ is constant, and $a(0)=a_0$, $a(t)^2+ b(t)^2=a_0^2$.
that $a(t)=a_0\operatorname{sech}(c t)$ w... |
H: Closedness of sets under linear transformation
Let $Y$ be a closed subset of $\mathbb{R}^m$ (in fact $Y$ is convex and compact, but I think the extra assumptions are irrelevant). Let $A \in \mathbb{R}^{n \times n}$ be a non-singular matrix (so $A^{-1}$ exists). Let $C \in \mathbb{R}^{m \times n}$ be any matrix. Is ... |
H: What does $(B+I)/I\sim B/(B\cap I)$ tell us?
Let $A$ be a $C^*$-algebra in which $B$ is a $C^*$-subalgebra and $I$ is a closed ideal. In several books on $C^*$-algebras I have encountered the following:
$(B+I)/I$ is $*$-isomorphic to $B/(B\cap I)$.
It seems important, but none of the books I read gives a hint why... |
H: Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$
Solve for $x$; $\cos^2x-\sin^2x=\sin x; -\pi\lt x\leq\pi$
$$\cos^2x-\sin^2x=\sin$$
Edit
$$1-\sin^2x-\sin^2x=\sin x$$
$$2\sin^2 x+\sin x-1=0$$
$\sin x=a$
$$2a^2+a-1=0$$
$$(a+1)(2a-1)=0$$
$$x=-1,\dfrac{1}{2}$$
$$x=\sin^{-1}(.5)=30^{\circ}=\dfrac{\pi}{6}$... |
H: Let $f,g:X\rightarrow \mathbb{R}$ continuous functions .If $X$ is open set,then the following set is open:$A=\{x \in X;f(x)\neq g(x)\}$
Let $f,g:X\rightarrow \mathbb{R}$ continuous functions .If $X$ is open set,then the following set is open:$A=\{x \in X;f(x)\neq g(x)\}$.
And if $X$ is a closed set , then the foll... |
H: If $n = m^3 - m$ for some integer $m$, then $n$ is a multiple of $6$
I am trying to teach myself mathematics (I have no access to a teacher), but I am not getting very far. I am just working through the exercises at the end of the book's chapter, but unfortunately there are no solutions.
Anyway, I am trying to pro... |
H: Expressing the wave equation solution by separation of variables as a superposition of forward and backward waves.
(From an exercise in Pinchover's Introduction to Partial Differential Equations).
$$u(x,t)=\frac{A_0 + B_0 t}{2}+\sum_{n=1}^{\infty} \left(A_n\cos{\frac{c\pi nt}{L}}+ B_n\sin{\frac{c\pi nt}{L}}\right)\... |
H: Oblique asymptotes?
A rational function, $\frac{p(x)}{q(x)}$ has an oblique asymptote only when the degree of $p(x)=$ degree of $q(x) -1$.
What "causes" the "slant" of the asymptote? Most asymptotes are caused by a function approaching an undefined value - I assume this is the same, but why (unlike others) would... |
H: Sum of the series : $1 + 2+ 4 + 7 + 11 +\cdots$
I got a question which says
$$ 1 + \frac {2}{7} + \frac{4}{7^2} + \frac{7}{7^3} + \frac{11}{7^4} + \cdots$$
I got the solution by dividing by $7$ and subtracting it from original sum. Repeated for two times.(Suggest me if any other better way of doing this).
However ... |
H: Showing existence of a field extension of degree $n$ for a finite field $F$
EDIT: Just mentioning that this is a homework question.
This is my first time posting a question on math.stackexchange, so I hope you find it in your hearts to forgive any stylistic or rule transgressions I make. I have searched through qui... |
H: Finding the remainder from equations.
I am having problems solving this question :
When n is divide by 4 the remainder is 2 what will the remainder be when 6n is divided by 4 ? Ans=$0$
Here is what I have got so far
$\frac{n}{4} => Remainder ~ 2$ so we get $n=4q+2$
$\frac{6n}{4} => Remainder ~ ?$ so we get $6n=... |
H: Laplace transform of $ \int_1^\infty\frac{\cos t}{t}dt$
Is the result of the of Laplace transform of $\int_1^\infty\frac{\cos t}{t}dt$ equal to $\frac{\int_1^\infty\frac{\cos t}{t}dt}{s}$?
AI: Yes, it is. Note that you have a definite integral which, indeed, converges (it is a variant of the Cosine Integral). As su... |
H: $\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k
Show that $\phi(n)=\frac{n}{2}$ if and only if $n=2^k$ for some positive integer k. I think I have it figured and would like to see if I am on the right track. Thank you.
AI: Suppose that $n=2^k$ where $k$ is positive. Then the numbers in the ... |
H: infinite sums of trigonometric functions
Find the sum of the series:
$$\sum_{n = 1}^\infty \left( \sin \left(\frac{1}{n}\right) - \sin\left(\frac{1}{n+1} \right) \right).$$
By the convergence test the limit of this function is $0$ but I'm not sure how to prove whether or not this function converges or diverges.
AI:... |
H: Checking divisibility of an expression - Need Pointers
I would like it if someone could give me pointers on solving problems like these. And why was 4 the answer here ?
If $a=4b+26$ and $b$ is positive , then a could be divisible by all of following except
a)2 b)4 c)5 d)6 e)7
Edit:. I know by taking b=4 i... |
H: Help with a partial fraction decomposition
One of my homework problems last week was to find the inverse Laplace transform of the following:
$$F(s)=\frac{2s+1}{s^2-2s+2}.$$
The answer is $f(t)= 2e^t \cos t + 3e^t \sin t$.
Obviously once you have the decomposed fraction the remainder of the problem is simple but ... |
H: Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
Solve for $x$; $\tan x+\sec x=2\cos x;-\infty\lt x\lt\infty$
$$\tan x+\sec x=2\cos x$$
$$\left(\dfrac{\sin x}{\cos x}\right)+\left(\dfrac{1}{\cos x}\right)=2\cos x$$
$$\left(\dfrac{\sin x+1}{\cos x}\right)=2\cos x$$
$$\sin x+1=2\cos^2x$$
$$2\cos^2x-\s... |
H: Please help me integrate the following: $\int \frac{y^2 - x^2}{(x^2 + y^2)^2}dy$
I'm self-studying a Cramster solution and I came across this integral and I don't know what they've done with it. Help would be appreciated.
$$\int \frac{y^2 - x^2}{(x^2 + y^2)^2} ~dy.$$
AI: In an integral $dy$, $x$ is a constant. Rew... |
H: Continuous function $g$ satisfying $g(x + y) = 5g(x)g(y)$
Let $g$ be a continuous function with $g(1) = 1$ such that
$$g(x + y) = 5g(x)g(y)$$
for all $x$, $y$. Find $g(x)$.
AI: setting $y=1$ gets
$$g(x+1)=5g(x)g(1)=5g(x)$$
So every time you increase the argument by $1$, you multiply by $5$. Can you see what functio... |
H: Computing a Laurent series
Let $$f(z) = \frac{1}{(2z-1)(z-3)} $$. Compute the Laurent series about the point z = 1 in the annular domain $$ \frac{1}{2} < |z-1| < 2$$
My attempt:
I broke f(z) up into the partial fraction decomposition:
$$ -\frac{2}{5(2z-1)} + \frac{1}{5(z-3)} = -\frac{2}{5}*\frac{1}{(1-\frac{(z+\fra... |
H: How to solve this Pell's equation $x^{2} - 991y^{2} = 1 $
How to solve the following Pell's equation?
$$x^{2} - 991y^{2} = 1 $$
where $(x, y)$ are naturals.
The answer is $$x = 379,516,400,906,811,930,638,014,896,080$$
$$y = 12,055,735,790,331,359,447,442,538,767$$
I can't think of any way apart from brute force. ... |
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