text stringlengths 83 79.5k |
|---|
H: basis functions do not lie in the space they form
For example, any continuous function in $\mathbb{L}^2(-\infty,\infty)$ space can be expanded by delta functions $\delta(x-a)$ or Fourier basis $e^{ikx}$. However, the basis functions, both $\delta(x-a)$ and $e^{ikx}$, are not square-integrable, thus not in $\mathbb{... |
H: number of possible matrix entries swaps
Let us say that there is a $n \times n$ matrix with entries defined/given as some natural number. (duplicates are fine.)
What would be the number of possible matrices that result from swapping/switching the entries in the original matrix that respects the entries in each row ... |
H: Introductory Treatment of Differential Geometry
I'm going to be taking a graduate course in differential geometry, this coming fall, but I am not prepared for it. Can anyone recommend a good introductory treatment of the background materials?
The list of topics in the course is:
Manifolds, Local Study of Manifolds... |
H: Solve $ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0$
I'm new to logarithms and I am having trouble solving this equation
$$ \left( \log_3 x \right)^2 + \log_3 (x^2) + 1 = 0.$$
How would I solve this? A step-by-step response would be appreciated.
Also, I know how to solve it with assigning $\log_3 (x)$ as $x$ a... |
H: How to solve this logarithm system?
I am new to logarithms and I am having trouble with this logarithm system.
\begin{align*}
\log_9(x) + \log_y(8) & = 2, \\
\log_x(9) + \log_8(y) & = 8/3.
\end{align*}
A step-by-step procedure would be highly appreciated.
Thanks in advance.
AI: Use the fact that $$\log_b(a)... |
H: Roots of a polynomial mod $n$
Let $n=n_1n_2\ldots n_k$ where $n_i$ are pairwise relatively prime. Prove for any polynomial $f$ the number of roots of the equation $f(x)\equiv 0\pmod n$ is equal to the product of the number of roots of each of the equations $f(x)\equiv 0\pmod{n_1}$, $f(x)\equiv 0\pmod{n_2}$, $\ldots... |
H: What is the meaning of this analysis problem and give some hint please?
What is the meaning of this analysis problem and give some hint please?
This problem was founded on Analysis 1 by Herbert Amann and Joachim Escher on page 100.
Determine the following subsets of $\Bbb R^2$ by drawing:
$$A = \{(x,y) \in \Bbb R^... |
H: Can the Coast Guard catch the thief, given speed and distance between them?
The problem is:
Given two boats, one, the coast-guard boat, stands on the point zero, the thief's boat, on the point $D$, the coast-guard boat travels in a speed of $Vg$ knots and the thief's boat, in $Vf$ knots, Given $D$, $Vf$ and $Vg$, s... |
H: Proof if $a \vec v = 0$ then $a = 0$ or $\vec v = 0$
I'm kicking myself over this one, but I just can't seem to make the argument rigorous. From Axler's Linear Algebra Done Right:
for a vector space $V$ with an underlying field $F$:
Take an element $a$ from $F$ and $\vec{v}$ from $V$. $a\vec{v}=\vec{0}\implies a=0 ... |
H: A continuous, injective function $f: \mathbb{R} \to \mathbb{R}$ is either strictly increasing or strictly decreasing.
I would like to prove the statement in the title.
Proof: We prove that if $f$ is not strictly decreasing, then it must be strictly increasing.
So suppose $x < y$.
And that's pretty much how far I go... |
H: Covariance question: A non squared matrix possible?
I have an academic economic paper that says the following:
$$q_r = \operatorname{Cov}(rx,v')\lambda$$
$$(14 \times 1)=(14 \times 4)(4 \times 1)$$
My vector $q_r$ is of size $14 \times 1$, my matrix $rx$ is size $T \times 14$, and my matrix $v$ is of size $T \times... |
H: Homework question regarding inverse function with a cosine
I'm given $f(x)= 3-4\cos(x-2)$
I've gotten to $(-x+3)/4 = \cos(2)\cos(y)+\sin(s)\sin(y)$
But I can't get the $y$ out to create an inverse ...
AI: Don't expand the $\cos(x-2)$. Solve the whole thing as you normally would, to get $x-2 = \cos^{-1}($something ... |
H: What is the volume of this 3d shape?
I'm wondering if there is an equation that represents the volume of an arbitrary 3d primitive
matching this description:
1.) Point at center of sphere
2.) Each edge is the length of the radius
3.) 3 flat sides, 1 arc side
Image:
So it's kind of a sector of a sphere, but instea... |
H: Proving Inequalities using Induction
I'm pretty new to writing proofs. I've recently been trying to tackle proofs by induction. I'm having a hard time applying my knowledge of how induction works to other types of problems (divisibility, inequalities, etc). I've been checking out the other induction questions on th... |
H: Given $(a_n)$ and $(b_n) \in \mathbb{R}$,if we have $|a_n - b_n| < \frac{1}{n} \forall n \in \mathbb{N}$
Given $(a_n)$ and $(b_n) \in \mathbb{R}$,if we have $|a_n - b_n| < \frac{1}{n} \forall n \in \mathbb{N}$, I think it is possible to find an $N$ such that $\forall n \ge N$, we have $|a_n-b_n| < \frac{\epsilon}{2... |
H: Fourth roots Complex analysis
I am trying to find the fourth roots of $8\sqrt2(1+i)$.
So then, I was deciding to convert $1+i$ to an $re^{i\theta}$, where $r = \sqrt2$ and $\theta = 45^\circ$ or $\pi/4$.
then; $z = \sqrt2e^{i\pi/4}$, but then we need the fourth roots so then, take $\sqrt2e^{i\pi/4}$ and raise it to... |
H: How are these two equal?
Which of these terms is greater ?
$2x-6y+1$ or $1$
if $x^4 + 3y^2=0$
According to the text they are equal ?How is that ?
AI: But since $x^2$ and $3y^2$ are positive, (assuming $x,y$ real) we have that $x^2+3y^2=0$ forces $x=y=0$ so the text is correct! |
H: Uncountable disjoint union of $\mathbb{R}$
I'm doing 1.2 in Lee's Introduction to smooth manifolds: Prove that the disjoint union of uncountably many copies of $\mathbb{R}$ is not second countable.
So first, let $I$ be the set over which we are unioning. Then I believe the disjoint union is just $\mathbb{R}\times I... |
H: Does there exist continuous map from $S^1$ to $\mathbb{R}$ such that $f(x)=f(y)$ for uncountably many $x,y$?
Does there exist continuous map from $S^1$ to $\mathbb{R}$ such that $f(x)=f(y)$ for uncountably many $x,y\in S^1$?
By the Borsuk-Ulam theorem, I know there is no injective map from $S^1\rightarrow \mathbb{R... |
H: RSA probabilistic decryption problem
I encountered this problem in an algorithms book and could not see how multiplicative could be used as a probabilistic algorithm.
Using the fact that RSA is multplicative:
$$P_A(M_1) P_A(M_2)\equiv P_A(M_1M_2)\pmod n$$
if someone could efficiently decrypt 1 percent of messages f... |
H: Nth derivative of $\tan^m x$
$m$ is positive integer,
$n$ is non-negative integer.
$$f_n(x)=\frac {d^n}{dx^n} (\tan ^m(x))$$
$P_n(x)=f_n(\arctan(x))$
I would like to find the polynomials that are defined as above
$P_0(x)=x^m$
$P_1(x)=mx^{m+1}+mx^{m-1}$
$P_2(x)=m(m+1)x^{m+2}+2m^2x^{m}+m(m-1)x^{m-2}$
$P_3(x)=(m^3+3m... |
H: Some method to solve $\int \frac{1}{\left(1+x^2\right)^{2}} dx$ and some doubts.
First approach.
$\int \frac{1}{1+x^2} dx=\frac{x}{1+x^2}+2\int \frac{x^2}{\left(1+x^2\right)^2} dx=\frac{x}{1+x^2}+2\int \frac{1}{1+x^2}dx-2\int \frac{1}{\left(1+x^2\right)^2}dx$
From this relationship, I get:
$2\int \frac{1}{\left(1+... |
H: quadratic polynomial investigation
in my mathematics textbook,i have found one interesting problem and i have one question.textbook asks following problem
deduce all possible value of $a$,for which equation
$4*x^2-2*x+a=0$ has roots in given interval $(-1;1)$
textbook used following method.it found axis of s... |
H: Is the following proof to Hölders inequality correct ?
Hölders inequality is $\int |fg|dx \le||f||_p||g||_q$
Define $F(x)= \frac{f(x)}{(g(x))^{q/p}}$ and $\nu dx =g(x)^q$
and $\Phi(t)= |t|^p$, $p\in (1,\infty)$
Now lets find $\Phi(\int F(x) d\nu)= \frac{(\int fg)^p}{||g||_q^{qp}}$
and similarly find $\int \Phi... |
H: Need help with a differential equation -like problem.
$\forall y \in \mathbb{R}, \int_{-\infty}^{\infty} f(x)f(x-y)dx=f(y)$
I also know that $\int_{-\infty}^\infty f(x) dx$ converges and that $f$ is symmetric about the origin. What does $f$ look like? Is it possible to identify a parametric set of solutions for $... |
H: Example of a function continuous at only one point.
Possible Duplicate:
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
I want to know some example of a continuous function which is continuous at exactly one point.
We know that $f(x)=\frac{1}{x}$ is continuous everywhere... |
H: Find a prime number $p$ so that $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ is divided by $x-\overline{2}$ in $\mathbb Z_p$
Let $f = \overline{3}x^3+ \overline{2}x^2 - \overline{5}x + \overline{1}$ be defined in $\mathbb Z_p$. Find a prime number $p$ so that $f$ can be divided by $g = x-\o... |
H: Insight of some concepts in commutative algebra
I really enjoyed the basic algebra course and wanted to teach myself a little more. So I am trying to learn commutative algebra from Atiyah-MacDonald and Eisenbud.
The department in our university is very good for analysis based subjects. I have enjoyed courses like ... |
H: stuck on a differential equation
let be the differential equation
$ x^{2} y''(x)+ y(x)(a^{2}+k^{2} _{n})=0 $
the boundary conditions are $ \int_{0}^{\infty}dx |y(x)|^{2} < \infty $ and $ y(0) $ must be finite (regular solutions near the origin )
here $ a^{2} >0 $ and $ k^{2} _{n}>0 $ these $ k_{n} $ are a discrete ... |
H: Let $S_n=\sum_{k=1}^{n}\frac{\sin\frac{k\pi}{25}}{k}$,how many positive $S_n$?
Let $S_n=\sum_{k=1}^{n}\frac{\sin\frac{k\pi}{25}}{k}$,how many positive $S_n$ are in $S_1,S_2,...S_{100}$
AI: I think it is positive for all values of n.As Prasad G mentioned $sin(\frac{k\pi}{25})$ is positive for the values between 25-5... |
H: General nonatomic measure that cannot be expressed as an integral
I read in a paper (Kingman — Poisson Processes, 2005) that:
In most cases the mean [of an inhomogenous Poisson process on a set
$A$] is given in terms of the rate function $\lambda(x)$ on $S$ by
\begin{align} \mu(A) &= \int_A \lambda(x)dx\qquad\te... |
H: "Connection Space"
Can we distill the idea of "connectivity" away from their topological context and study abstract properties of "connectivity"?
I define a connective space to be a set $X$ together with a collection $\gamma$ of subsets of $X$, which we define as "connected". $\gamma$ contains every singleton subse... |
H: Coherent Sheaves on Projective Space
I am having trouble proving the following claim and would be glad if someone could help me out.
Claim: Let $\mathbb P$ denote n-dimensional projective space, and let $F$ be a coherent sheaf on $\mathbb P$. Then there exists some integer k such that $F\otimes O(k)$ is generated b... |
H: Why aren't these loops homotopic?
Let $S^1 = \{z \in \mathbb{C} : |z| = 1\}$. Take the loops $f,g : [0,1] \rightarrow S^1$, $f(t) = 1$, $g(t) = e^{2\pi it}$. I know these represent different elements in $\pi_1(S^1, 1)$, but I don't see why $F(t,s) = e^{2\pi its}$ isn't a homotopy between $f$ and $g$.
AI: Loops ar... |
H: How do the $L^p$ spaces lie in each other?
Let $(S,\mathcal{B},\mu)$ be a measure space, $Y$ be a banach space and for $1\le p <\infty$ let $L^p(\mu;Y)$ be the set of all maps $f:S\rightarrow Y$ that are measurable and for which $|f|^p$ is integrable. Let $L^{\infty}(\mu;Y)$ be the of essentially bounded maps.
I wo... |
H: What is $d(y dx)$?
Let $x$, $y$ be 0-forms, thus $dx$, $dy$ are 1-forms. Since 1-forms compose an algebra over 0-forms ring, expressions like
$$y dx$$
make perfect sense. Now I ask what is
$$d(y dx)$$
I suggest it to be $y d(dx) = 0$, since $d$ is linear, however I feel it is likely to be wrong. Is there any other ... |
H: Unambiguous Way of Stating a Biconditional in Plain English
I am having a hard time understanding this section in Wikipedia's article on Logical biconditionals:
Colloquial usage
One unambiguous way of stating a biconditional in plain English is of the form "b if a and a if b". Another is "a if and only if b". Slig... |
H: How to calculate $\pi\int^1_0\sqrt{x(1-x)} \mathrm \, dx$ =?
I need to find the value of the following integral
$$\pi\int^1_0\sqrt{x(1-x)} \mathrm \, dx = ? $$
I tried parts integration, $a\sin x$ substitution and none of them seems work.
Can someone get me on the right path?
Thank you very much!
AI: Put $x=\sin^2... |
H: Finding $\sup$ of the given function in Ball of radius $1$ centered at origin.
I want to find the the supremum of $\dfrac{|x|^{2/3}-|y|^{2/3}}{|x-y|^{2/3}} $ in the unit ball centered at the origin . Here $x\neq y$, $x,y \in \mathbb R^n$.
How do I proceed ? Thank you for your help.
AI: Hint: By a modification of th... |
H: Solving $f(2011)=2012$, $f(4xy)=2yf(x+y)+f(x-y)$
How to find the all functions $f$ :$ \mathbb{R}\longrightarrow\mathbb{R}$ such that $f(2011)=2012$,for every $x,y\in\mathbb{R}$ then: $$f(4xy)=2yf(x+y)+f(x-y)$$
AI: If such a function exists, then $f(x)=f(0) \ne 0$ for every $x$, therefore $f(0)=2yf(0)+f(0)$ for ever... |
H: Chi Squared Distribution with $\mu = 0$, $\sigma^2 \neq 1$
Let $X_i$ be independent normally distributed random variables with zero mean and variance $\sigma^2 \neq 1$. What is the probability density function of the random variable formed by the sum of their squares?
Here is my attempt: Let $Y = \sum_{i=1}^{k}X_i^... |
H: Application of Pell's equation
Need to find $n,r$ (if any) for $121^r-2n^2=1$ where $n,r$ are natural numbers.
Observed that $n$ is odd then $n=2m+1$ (say).
But on replacement of $n$ by $2m+1$, increases the complexity of the problem.
AI: All solutions to $x^2-2y^2=1$ are given implicitly by
$$
x_k+y_k\sqrt{2} = ... |
H: Why is $\mathbb{C}^{g}$ the universal cover of any connected compact complex Lie group of dimension g?
This question came up while I was studying the book "Complex Abelian Varieties" by Lange/Birkenhake.
More precisely, the authors prove in Lemma 1.1 that every connected compact complex Lie group of dimension $g$, ... |
H: Oblique asymptotes
When we find oblique asymptotes, we divide the numerator by the denominator and take only the polynomial portion of the expression as the equation of the slant asymptote. Why? What is the proof that this equation is the slant asymptote?
AI: Because you're interested in the behaviour of the functi... |
H: How to prove $p$ divides $a^{p - 2} + a^{p - 3} b + a^{p - 4} b^2 + \cdots + b^{p - 2}$ when $p$ is prime, $a, b \in \mathbb{Z}$ and $a,b \lt p$?
If $p$ is a prime number and $a, b \in \mathbb{Z}$ such that $a,b \lt p$, then how could we prove that $p$ divides
$\left(a^{p - 2} + a^{p - 3} b + a^{p - 4} b^2 + \cdo... |
H: Less than infinity or Less or Equal to infinity
What is the difference between Less than infinity or Less or Equal to infinity?
We cannot substitute infinity anyway, and have to use limits. So does it make sense to write less and equal to infinity?
AI: In the context of the real numbers $\infty$ denotes "larger tha... |
H: Integration problem in matrix calculus
Let $\mathbf{A}=\begin{bmatrix} f(x_1,x_1), & \ldots,& f(x_1,x_n)\\
\vdots&\ddots& \vdots \\f(x_n,x_1),&\ldots, &f(x_n,x_n) \end{bmatrix} $, where $f:\mathbb{R}\times\mathbb{R}\rightarrow \mathbb{R}$. I want to calculate
$\int \mathbf{A}\mathrm{d}\mathbf{x}$, where $\mathbf{... |
H: How many triangles with integral side lengths are possible, provided their perimeter is $36$ units?
How many triangles with integral side lengths are possible, provided their perimeter is $36$ units?
My approach:
Let the side lengths be $a, b, c$; now,
$$a + b + c = 36$$
Now, $1 \leq a, b, c \leq 18$.
Applying mult... |
H: Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$
What is the number of Sylow 2-subgroups of the group $\mathrm{PSL}(2,q)$?
AI: When $q$ is a power of $2,$ we have ${\rm PSL}(2,q) = {\rm SL}(2,q)$ and a Sylow $2$-normalizer is a Borel subgroup of order $q(q-1).$ Hence there are $q+1$ Sylow $2$-subgroups as ${\rm S... |
H: What's the precise meaning of imaginary number?
The same to the title,what's the precise meaning of imaginary number? And on the other hand,how can the imaginary number be reflected in Physics?
AI: I'm unsure what you mean by the "precise" meaning of an imaginary number, but to me it seems best to talk about the pr... |
H: Spectrum Of The Laplacian Question
I'm given the following question (from Davies' book - Spectral theory of differential operators):
Use the theorem (*) below to prove that if $\Omega$ is a convex region in $ \mathbb{R}^2 $ , then:
$ \frac{1}{4} \int _\Omega \frac{|f|^2}{d^2} d^2 x \leq \int _\Omega | \bigtriangl... |
H: Pointwise convergence implies $L^p$ convergence?
Let $f: X \to [0, \infty) \subset \mathbb R$ measurable where $X$ is a measure space. Let $f_n : X \to [0, \infty) $ be simple functions (i.e. linear combinations of characteristic functions of measurable sets) such that for each $x \in X$, $f_n(x) \leq f_{n+1}(x)$ a... |
H: Compute $\lim\limits_{n\to\infty} \int_{0}^{2\pi} \cos x \cos 2x\cdots \cos nx \space{dx}$
Compute the following limit:
$$\lim_{n \to \infty}\int_{0}^{2\pi}\cos\left(x\right)\cos\left(2x\right)\ldots
\cos\left(nx\right)\,{\rm d}x$$
Today I was working on a W. L. Putnam competition's problem containing this
int... |
H: proving convergence of a sequence and then finding its limit
For every $n$ in $\mathbb{N}$, let: $$a_{n}=n\sum_{k=n}^{\infty }\frac{1}{k^{2}}$$
Show that the sequence $\left \{ a_{n} \right \}$ is convergent and then calculate its limit.
To prove it is convergent, I was thinking of using theorems like the monotone ... |
H: Could someone please confirm the contradiction in this equation?
I'm following an equation from a published paper in order to calculate probabilities using a Markov Chain. The equation says:
Construct a set $U$ that consists of all items that appear in the top-$k$ in at least one list.
For each pair of items $i$ an... |
H: Problem 18.1 in I. Martin Isaacs' Algebra
I am trying to prove the following:
Let $E/F$ be an arbitrary extensions. Show that $E/F$ is normal if and only if $E$ is the union of all those intermediate fields $K$ such that $K$ is the splitting field for some $f(X) \in F[X]$.
My attempt at the solution is the followin... |
H: Weak convergence and weak star convergence.
If region $\Omega$ is bounded and $u_n$ has weak star convergence in $L^\infty ( \Omega)$ to some $u\in L^\infty(\Omega)$ , does it imply that $u_n$ converges weakly in any $L^p(\Omega) $ ?
I think i got it : If $sup$ of a function is finite then integral over a bounded ... |
H: Convergence of $a_{n}=\frac{1}{\sqrt{n}}\sum\limits_{k=1}^{n}\frac{1}{\sqrt{k}}$?
For $n$ in $\mathbb{N}$, consider the sequence $\left \{ a_{n} \right \}$ defined by:
$$a_{n}=\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$$
I would like to prove whether this sequence is convergent, and if so what its limit is.... |
H: Solving a set of "circular" quadratic equations
$x_a'$ and $y_a'$ are unknown. What's the simplest way to solve it? Every time I tried, it grew into tremendous size or was unable to think out in reasonable amount of time due to it's complexity.
$\begin{align*}
(x_f-x_a')^2+(y_f-y_a')^2&=r^2\\
(x_a-x_a')^2+(y_a-y_a'... |
H: Different types of continuity for operators on Hilbert spaces
In chapter one of K-theory and $C^*$-algebras, a Friendly Approach, the author gives a very brief discussion about several types of continuity of operators between Hilbert spaces.
Let $T: \mathcal{H}_1\to\mathcal{H}_2$ be a linear operator between Hilber... |
H: Holoedric isomorphism?
While trying to read the following article
Schottenfells, Ida May. Two Non-Isomorphic Simple Groups Of The Same Order 20,160.
I found the term "holoedrically isomorphic". In an abstract for the article, I also came across the claim that "Holoedric isomorphism is the only isomorphism that ca... |
H: Is the Kleene/Brouwer ordering dense?
This question was motivated by a statement in Simpson's Subsystems of Second Order Arithmetic (second edition), p. 168.
It is straightforward to verify (in $\mathsf{RCA}_0$ for instance) that $\leq_{KB}$ is a dense liner [sic] ordering with no left endpoint and with the empty ... |
H: How to compute the min-cost joint assignment to a variable set when checking the cost of a single joint assignment is high?
I want to compute the min-cost joint assignment to a set of variables. I have 50 variables, and each can take on 5 different values. So, there are 550 (a huge number) possible joint assignment... |
H: derivative at a given point
I often see the following:
$$ \left. \frac{\partial q}{\partial \alpha} \right|_{\alpha = 0} $$
Where $q$ is a function $q(q', t, \alpha)$.
Is that just the same as that?
$$ \frac{\partial}{\partial \alpha} q(\alpha=0) $$
If they are the same, why do people write the first one? I find th... |
H: contour integral computations
Let $C$ be the boundary of the square whose vertices are $1+i$, $1-i$,
$-1 + i$ and $-1 -i$. Suppose that $C$ is oriented counterclockwise. How to compute
a) $$\int_C \frac{e^z}{z-1/2} \, dz$$
b) $$\int_C \ln(z+3) \, dz$$
c) $$\int_C \bar{z} \, dz$$
I seee that we can use $f'(z)z'$ and... |
H: Solving a Maximum Likelihood Estimation with an exponential distribution
I need someone's insight on applying a MLE for an exponential distribution. In a finance paper, I have the following:
$\displaystyle d_i \sim \frac{\epsilon_i}{\lambda_i}$ where $\epsilon_i$ is i.i.d. exponentially distributed with parameter $... |
H: Does every set have a well ordering with greatest element?
Does every non-empty set have a well ordering with greatest element?
It is well known that every set has a well ordering. But can we also assume that this well ordering has greatest element?
[Edited to remove an ambiguity revealed by the answers and comment... |
H: Why can/do we multiply all terms of a divisor with polynomial long division?
I'm trying to understand why polynomial long division works and I've hit a wall when trying to understand why we multiply all terms of the divisor by the partial quotient. Consider:
$$\frac{x^2 + 3x + 2}{x + 2}$$
During the first step we ... |
H: no simple group of order $945$
I need to show that there are no simple groups of order $945$.
I've tried the regular method using the Sylow theorems.
$$|G|=945=3^3\cdot5\cdot7 $$
If $G$ is simple then there should be 7 Sylow-3 groups ; 21 Sylow-5 groups and 15 Sylow-7 groups. Even if they would all intersect trivi... |
H: Quantative Comparison - Which is bigger
A quantitative comparison question states:
if $r<s<t$ and the average arithmetic mean of r , s and t is 90 . Which of the following is bigger a)The average of s and t or b)$90$.
The answer is a. However I cant quiet figure out how they got this:
I know that : $\frac{r+s+... |
H: Verify trigonometric equation $\frac{(\sec{A}-\csc{A})}{(\sec A+\csc A)}=\frac{(\tan A-1)}{(\tan A+1)}$
How Would I verify the following identity.
$$\frac{(\sec{A}-\csc{A})}{(\sec A+\csc A)}=\frac{(\tan A-1)}{(\tan A+1)}$$
I simplified it to
$$\frac{(\sin{A}-\cos{A})}{(\sin{A} \cos{A})}\div\frac{(\sin{A}+\cos{A... |
H: Differentiating the posterior distribution function
I am learning about Bayesian statistics and I'm currently doing loss functions. Let $f(\theta | \mathbf{x} ) $ be a posterior pdf . Let $F(\theta | \mathbf{x} ) $ be the associated distribution function. I want to differentiate $F(a - D| \mathbf{x} )$ with respect... |
H: In Taxicab Geometry, what is the solution to d(P, A) = 2 d(P, B) for two points, A and B?
Taxicab and Euclidean geometry differ a great deal, due to the modified metric function:
$$d_T(A,B)=|x_a-x_b|+|y_a-y_b|$$
(Note that this means when measuring distance, it is not the length of the hypotenuse, but the sum of th... |
H: Finding the derivative of a function using the Product Rule
I'm home teaching myself calculus because I'm 16 and therefore too young to take an actual class with a teacher, so I apologise if this seems simple.
I understand the definition of the Product Rule and its formula:
"If a function $h(x)=f(x)\times g(x)$ is... |
H: Compute: $\lim_{n\to\infty} n^{p+1} \int_{0}^{1} e^{-nx} \ln (1+x^p) \space dx $
Find the following limit for any $p$ natural number:
$$\lim_{n\to\infty} n^{p+1} \int_{0}^{1} e^{-nx} \ln (1+x^p) \space dx $$
If i'm not wrong, without much effort one may see that this integral may be rewritten as Gamma function and... |
H: $<$ in a preorder
The author of the book I am studying defines $<$ for a poset as
If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$.
From this, I can conceive of two definitions for $<$ for a preorder:
1) If $x, y \in X$, where $X$ is a preorder, then we shal... |
H: Embedding elliptic curves into the general linear group
Is it possible to embedd an elliptic curve $E:\;\; y^2=x^3+ax+b$, defined over an algebraically closed field $k$, into some $GL_n(k)$ ?
AI: If $E\hookrightarrow\mathrm{GL}_n$ is a closed immersion of $k$-schemes ($k$ any field) inducing an isomorphism with the... |
H: What is the relationship between integrals and areas?
Possible Duplicate:
Why is the area under a curve the integral?
Why does calculating an integral or an anti-derivatives represent an area? How do they figure it out? What is the relationship between them?
AI: This is basic level stuff covered in first course ... |
H: Distance Formula for n-dim Barycentric Coordinates
Assume that we are given every distance between each pair of points from a $n$-simplex $\triangle$. Given
$n$-dimensional barycentric coordinates (measured with respect to $\triangle$) of two points, how do we compute the distance between the two points?
A more de... |
H: A problem with definitions of rotation/reflection matrix/operator
I am a math undergraduate student taking a course called "Geometry
and symmetry" and I have something I don't understand with the definition
the lecture gave in class.
Definition: $T\,:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is called
a linear rotation oper... |
H: Which is the biggest integer that divides all integers that are the product of three consecutive odd numbers?
I read this problem from a high-school-math-problems-calendar, and I'm solving them in my spare time just for the fun of it (what in math is not about the fun? =) ), but this little one it's been hard for m... |
H: Relation between Diameter and Tangent of circle
A comparative question states:
One side of rectangle is the diameter of a circle. The opposite side of rectangle is tangent to the circle.Which is bigger a)The perimeter of rectangle or b)The circumference of the circle (Ans=$b$)
Now I know a tangent is perpendicula... |
H: The probabilty of a new arrangement of 52 cards deck?
I just read this article, it claims that if you just shuffle a 52 card deck, you will mostly be creating an arrangement that no human have ever seen before.
But this doesn't seem right since every time we create a new arrangement, this one is now a candidate to ... |
H: Convergence of $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$?
I need to prove the convergence/divergence of the series $\sum_{n=1}^{\infty }\frac{a_{n}}{1+na_{n}}$ based on the convergence/divergence of the series $\sum_{n=1}^{\infty }a_{n}$. It is given that $a_{n}> 0$, $\forall n\in \mathbb{N}$
If the series $\sum... |
H: PDE with series
Consider this PDE
The solution to the PDE is
So what I am having trouble is solving it using this method.
I am going to say that my $u(x,t) = \sum_{n=1}^{\infty} u_n(t) \sin(nx)$ and $x \sin(t) = \sum_{n=1}^{\infty}h_n(t)\sin(nx)$
The reason I chose sine for my inhomogeneous term is because my bo... |
H: Calculating Percentile Rank Using Relative Strength Ranking
I have a spreadsheet of stock quotes that contains Relative Strength Ranking (RSR) ranging from -45 to 65 and the count of ranks is 1500. Could someone please explain how I can calculate percentile ranks using these numbers? Is there a specific formula?
AI... |
H: Is $(x^3-x^2+2x-1)$ prime in $\mathbb{Z}/(3)[x]$?
This is somewhat of a follow up on this question: Why is $(3,x^3-x^2+2x-1)$ not principal in $\mathbb{Z}[x]$?
I'm curious, is $\mathbb{Z}[x]/I$ a domain, with $I=(3,x^3-x^2+2x-1)$? I know $I$ is not principal. Also, I took the sequence of epimorphisms
$$
\mathbb{Z}[... |
H: What role does $\mathfrak D$ play in the definition of the union and intersection of the complements of a collection of sets?
I'm puzzled by something that might be complete silly. Halmos writes in his "Naive Set Theory":
If $\mathcal C$ is a collection of subsets of a set $E$ (that is, $\mathcal C$ is a subcollec... |
H: Find undetermined coefficients in polynomial quotient and remainder
When a polynomial
$$P(x)=x^4- 6x^3 +16x^2 -25x + 10$$
is divided by another polynomial
$$Q(x)=x^2 - 2x +k,$$
then the remainder is
$$x+a.$$
I have to find the values of $a$ and $k$.
Can somebody tell me shortest way to get these values? Which t... |
H: Intuition about the Pixley-Roy topology
Let $R$ be Real line and let $F[R]$ be $\{x\subset R:\text{is finite}\}$ with Pixley-Roy topology.
Definition of Pixley-Roy topology is this: Basic neighborhoods of $F\in F[R]$ are the sets
$$[F,V]=\{H\in F[R]; F\subseteq H\subseteq V\}$$
for open sets $V\supseteq F$, see e.g... |
H: What is the relation between the average rate of change and the derivative?
A value in the range for any base polynomial function with a y-intercept of zero can be expressed as: $$f\left(x\right) = px$$ where $p$ is the average rate of change between $0$ and $x$. The average rate of change can be in turn expressed ... |
H: If $|f(z)| \geq |g(z)|$ for $z \in D$ and $E = \{ z \in D : |f(z)| =|g(z)| \}$ has a limit point, then $E=D$.
This is my problem:
Let $D := \{ z \in \mathbb{C} : |z| <1 \}$. Let $f$ and $g$ be analytic functions on $D$. Suppose $|f(z)| \geq |g(z)|$ for all $z \in D$. Define $E = \{ z \in D : |f(z)| =|g(z)| \}$. Sh... |
H: Ergodicity of the First Return Map
I was looking for some results on Infinite Ergodic Theory and I found this proposition. Do you guys know how to prove the last item (iii)?
I managed to prove (i) and (ii) but I can't do (iii).
Let $(X,\Sigma,\mu,T)$ be a $\sigma$-finite space with $T$ presearving the measure $\mu$... |
H: Sum of the thirteenth power of the roots of given polynomial
Find the sum of the thirteenth powers of the roots of $x^{13} + x - 2\geq 0$.
Any solution for this question would be greatly appreciated.
AI: Any root $r_i$ of $x^{13} + x - 2 = 0$ satisfies $r_i^{13} + r_i - 2 = 0,$ or $r_i^{13} = 2 - r_i.$ A polynomi... |
H: Why is the product of all units of a finite field equal to $-1$?
Suppose $F=\{0,a_1,\dots,a_{q-1}\}$ is a finite field with $q=p^n$ elements. I'm curious, why is the product of all elements of $F^\ast$ equal to $-1$? I know that $F^\ast$ is cyclic, say generated by $a$. Then the product in question can be written a... |
H: Local boundedness of continuous functions on a Banach space
Let $X$ be an infinite-dimensional Banach space and $f : X \to \mathbb{R}$ continuous (not necessarily linear).
Can $f$ be unbounded on the unit ball?
Of course, in a locally compact space these are impossible. Since $X$ is not locally compact one would... |
H: Working with phi function with larger numbers?
I've been recently learning about Phi function(Euler's totient function). I am attempting to efficiently find the $\phi(n)$ of higher numbers.
What I wanted to asked about was, if I have say:
$$\frac{30}{\phi(30)} = 3.75,$$
would I be able to now know that for every m... |
H: Segregation of countable sequence into two countable subsequences
This is I think a very simple question about infinite sequences. I thought I knew the answer but the manipulation described below worries me.
Suppose I divide the interval $(0,\frac{1}{4})$ into infinitely many subintervals $S_n = (\frac{1}{(n+1)^2}... |
H: How to simplify [3a(b-c)+5][-3a(b-c)-5] by using special product?
In simplifying $$[3a(b-c)+5][-3a(b-c)-5],$$ I used $$4(au+bv)(cu+dv)=acu^2+(ad+bc)uv+bdv^2.$$
I failed to apply the formula to the equation because $a=3a$, $b=-3a$, $c=5$, $d=-5$, $u=(b-c)$, $v=?$
There's no value of $v$ so I tried to find other sp... |
H: How to simplify $(3a-b^2-a)^3$ by using special product?
When I simplify $(3a-b^2-a)^3$, I used $(u±v)^3=u^3±3u^2v+3uv^2±v^3$
but I'm confused with $(b^2-a)$
AI: First of all, your cited identity is incorrect; the exponent on the $v$ was wrong in the last term. The correct version is
$$(u\pm v)^3=u^3\pm 3u^2v + 3uv... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.