text stringlengths 83 79.5k |
|---|
H: Do eigenvectors of a Toeplitz matrix form an orthogonal set?
It is true for a $2 \times 2$ Toeplitz matrix (put values $a$ and $b$ in the first row and $b$ and $a$ in the second and work out), but when I tried it for a $3 \times 3$, it turns out to be a bit difficult.
Any help or reference to my question is greatl... |
H: Definite Integral $\int_0^{\pi/4}\frac{\sqrt{\tan(x)}}{\sin(x)\cos(x)}\,dx$
$\displaystyle\int_0^{\pi/4}\frac{\sqrt{\tan(x)}}{\sin(x)\cos(x)}\,dx$
Needed a detailed solution with explanation. The answer is $2$.
AI: HINT:
$$\frac{\sqrt{\tan x}}{\sin x\cos x}=\frac{\sqrt{\tan x}}{\frac{\sin x\cos x}{\cos^2x}}\frac1{\... |
H: $S^1\times S^1$ diffeomoprhic to torus of revolution.
I am searching for an diffeomorphism between $$S^1\times S^1$$ and the torus of revolution $$\{(x,y,z)\in\mathbb{R}^3|z^2+(\sqrt{x^2+y^2}-a)^2=r^2\}.$$
I know it's true, I am just looking for an explicit formula. Thanks a lot!
I managed to wirte down a diffeomop... |
H: determinant calculation
This question is in my assignment. We are not allowed to use any symbol to represent any elementary row and column operations used in the solution. We must solve it step-by-step. Please help me to check my solution word by word including my spelling and grammar.
Question:
Given that
$$\begi... |
H: How to find all vectors so that a vector equation can be solved?
Unfortunately, my text book doesn't clarify this process at all. It's asking to find all all vectors [a b] so that the vector equation can be solved. The vector equation is:
$c1$ $[3,1]$ + $c2 [6,2]$=$[a, b]$
The linear system would look like:
$3c1+... |
H: Finding the fixed point and the suitable range
I have to find the fixed point of x$^3$-x$^2$-1=0.Then x=(1/x$^2$)+1 where I chose g(x)=(1/x$^2$)+1 .Then I tried to find a fixed point for g(x).Since I don't know the range of x,I chose x$\in$[1.3,1.6] as the suitable range because in this range 1.3
My question is my... |
H: Differential of a $C^\infty$ function from a manifold $M$ to $\mathbb{R}$
I am studying differentiable manifolds from Warner. In the book, the differential is defined as follows.
If $\psi:M\longrightarrow N$ is $C^\infty$, and if $m\in M$, then the differential of $\psi$ at $m$ is the linear map $d\psi:M_m\longrigh... |
H: What is the null vector for the vector space of continuous functions $f \colon \mathbf{R} \to \mathbf{C}$?
The set of all continuous complex-valued functions of real variable $x$ together with addition of two vectors $\boldsymbol{f} = f(x)$ and $\boldsymbol{g} = g(x)$ defined by
\begin{equation}
(f + g)... |
H: Intersection of Two Permutations
If you are generating 2 digits codes $[X, X]$ where you have 5 choices for $X$.
The number of choices for each code is:
Order matters so the total number of permutations is $n^r = 5^2 = 25$.
If for the first code the choices were [1,5], [4,8] (inclusive) and in the second code the ... |
H: Find an example of a positive function that its improper integral converge, but its series diverge
I need to find an example of a positive function f, and a constant a>0 such that the improper integral of f from 0 to infinity converge, but the series f(na) from 1 to infinity diverge.
AI: To expand on Jyrki's commen... |
H: solving $\frac{x}{3}+{[\frac{x}{3}]} = \sin(x) + [\sin(x)]$ for real x , in an efficient manner
$$\frac{x}{3}+{\left[\frac{x}{3}\right]}=\sin(x)+[\sin(x)]$$
I know the answer and the solution.
$$-1 \le \sin(x) \le 1\Rightarrow-2\le\sin(x)+[\sin(x)]\le 2 \Rightarrow -2 \le \frac{x}{3}+\left[\frac{x}{3}\right] \le 2 ... |
H: Percentile of CDF Function
I am trying to find the 0.5 (mean) percentile of a CDF function.
$$ F(X) = 1 - e^{-(x/3)^2} $$
In my book's example it says
$$ m = 3[-ln(1-0.5)]^{1/2} = 3\sqrt{ln2}=2.498$$
I am not sure how to get to the above equation even using the definition of a percentile of $$F(x_p) = p $$
AI: The ... |
H: Closed form for a fixed point of the exponential function?
Let $$e^z = \sum_{n=0}^{\infty} \frac{z^n}{n!} $$
denote the exponential function, which is defined on the entire complex plane.
There is a fixed point of this function at $w= a+bi$ where $a \approx 0.31813$ and $b \approx 1.33723$.
I am guessing there is n... |
H: Solving $y'' + (ax+b)y = 0$
This is a problem in quantum mechanics when one considers a linear potential; in physics-speak the equation would be written as
$$\frac{d^2\psi}{dx^2} + \frac{2m}{\hbar^2}(E-ax)\psi = 0,$$
with $V(x) = ax$.
I've been looking at it for a while and can't find a solution, and I thought I'd... |
H: If $X$ is a cone, show that $I(X)$ is homogeneous.
The exercise is 1.3(3) from HP Kraft, "Appendix A: Basics from Algebraic Geometry."
If a closed subset $X\subseteq \mathbb C^n$ is a cone, show that $I(X)$ is generated by homogeneous functions.
Closed means $X=\cap g_i^{-1}(0)$ the zero set for some $g_i\in\mathbb... |
H: Regular Expression that accepts a language L
Given, $L=\{ x \in\{0,\,1\}^* | x=0^n1^m \text{ and }n+m\text{ is a multiple of }3\}$ give a regexp that accepts the language.
My thoughts are: $(000)^*(\epsilon + 001 + 011)(111)^*$
Is this right?
AI: HINT: Divide the word into blocks of $3$; the critical point is whe... |
H: How to treat system of linear first order differential equations with trigonometric function coefficients?
I'm having trouble solving the following IVP:
$$x_1^\prime = -x_1\tan t + 3\cos^2t$$
$$x_2^\prime = x_1 + x_2\tan t + 2\sin t$$
where $x_1(0) = 4$ and $x_2(0) = 0$.
I'm not sure what to do when the coefficient... |
H: uniform continuous proof take a look
Hi take a look of my proof and fix if there is wrong or unnecessary part
$F(x) = x^2 , \alpha \in [0,\infty] $
and prove $F(x)$ is not uniformly continuous
anyway
proof
assume that this$ f(x)$ is uniformly continuous.
then there should be a some limit point(??) exist $L$,... |
H: Help find the MacLaurin series for $\frac{1}{e^x+1}$
What is the MacLaurin series up to $x^4$ for $\frac{1}{e^x+1}$?
My Attempt:
$$\begin{align}
\frac{1}{e^x+1} &=(1+e^x)^{-1} \\
&\approx 1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4 \\
\end{align}
$$
Since
$$
\begin{align}
e^x &\approx \frac{1}{24} \, x^{4} + \frac{1}{6} \, x^{... |
H: Recurrence sequence limit
I would like to find the limit of
$$
\begin{cases}
a_1=\dfrac3{4} ,\, & \\
a_{n+1}=a_{n}\dfrac{n^2+2n}{n^2+2n+1}, & n \ge 1
\end{cases}
$$
I tried to use this - $\lim \limits_{n\to\infty}a_n=\lim \limits_{n\to\infty}a_{n+1}=L$, so $L=\lim \limits_{n\to\infty}a_n\dfrac{n^2+2n}{n^2+2n+1}=L\... |
H: Something like a field but with 3 operations?
I know of Groups, and Rings, and Fields but what about tacking on a 3rd operation. Is there any use in considering some structure that consists of a field but with a 3rd operation (possibly less well behaved than the other two)?
The link in the comments is helpful, but... |
H: 2-digit combinations
I can count the number 2-digit combinations: each digit has 10 possibilities, so that gives 10*10 = 100 combinations.
But what if we write the combinations in one long string like this:
112131
This string is 6 long, but it gives us 5 combinations: 11, 12, 13, 21 and 31
How can I calculate the s... |
H: Where M is a matrix calculate a formula for M^n
Let $$M = \begin{bmatrix} -7 & 8 \\ -8 & -7 \end{bmatrix}.$$
Find formulas for the entries of $M^n$ where $n$ is a positive integer. (Your formulas should not contain complex numbers.) Your answer should be in the form of a matrix.
I diagonalized to the form $M = P ... |
H: Hölder's Inequality and step functions
Define functions $f(x) = \sum_{k=0}^\infty a_k \chi_{[k,k+1)}(x)$ and $g(x) = \sum_{k=0}^\infty b_k \chi_{[k,k+1)}(x)$ where $\chi_ {[k,k+1)}$ is the indicator function for the given interval. Let $f,g$ be zero for all negative numbers. Also assume that the $a_k, b_k$ are non ... |
H: Contrapositive Inquiry
Suppose you want to show that $A \implies B$.
This is equivalent to showing $\neg B \implies \neg A$.
But now suppose you show that assuming $\neg B$ it is POSSIBLE that $\neg A$, but not necessarily the case that $\neg A$.
Have you succeeded in showing that $A \implies B$?
AI: You have to pr... |
H: How to choose between poisson and binomial distributions
I don't get this thing... I know that binomial distribution is used to know the probability of a X v.a. that sounds like this: X = "the probability of having 4 blue balls doing 10 extraction from a chest containing 7 blue and 40 white", and I know that poisso... |
H: Functions that satisfy $f(x,z) = f(x,y) f(y,z)$
I am specifically looking for solutions that are NOT of the form: $f(x,y) = g(x)/g(y)$ since that is an obvious solution, as is $f(x,y) = 0$. I have a suspicion that there may be answers to this question that do not fall into these categories.
The functions don't nece... |
H: A non-equality and an inequality involving $y$ and $y_0$ from Spivak Calculus 4th ed.
It's Problem 22. from Chapter 1.
I'm given:
$y_0 \neq 0$
$|y - y_0| < \frac{|y_0|}{2}$
$|y - y_0| < \frac{\epsilon|y_0|^2}{2}$
and I must use them to prove that:
$y \neq 0$
$|\frac{1}{y} - \frac{1}{y_0}| < \epsilon$
I haven't real... |
H: On integrating a "gaussian-like" integral
Let the following "gaussian-like" integral:
$$
I = \int_{\Re^n}
\!
\frac{1}{(2\pi)^{n/2}|\Sigma|^{1/2}}
\exp
\left\{
-\frac{1}{2}
(\mathbf{x}-\mathbf{\mu})^T\Sigma^{-1}(\mathbf{x}-\mathbf{\mu})
\right\}
\mathbf{x}
\,\mathbf{d}\mathbf{x},
$$
where $\mathbf{x}=(x_1,\dots,x_n)... |
H: An Unexpected Circle...
I played around with
$$z=\frac{-1+e^{it}}{\phantom{-}2+e^{it}}$$
and found that, when I draw the real against the imaginary of $z$, it pretty much looks like a circle.
But neither ${\frak{R}} z $, nor ${\frak I} z$ look like $\cos $ or $\sin$. Is it due to a kind of transformed argument o... |
H: Regular $n$-gon in the plane with vertices on integers?
For which $n \geq 3$ is it possible to draw a regular $n$-gon in the plane ($\mathbb{R}^2$) such that all vertices have integer coordinates? I figured out that $n=3$ is not possible. Is $n=4$ the only possibility?
AI: This is impossible for $n=3$ as you mentio... |
H: Why does $\sum\limits_{i=1}^n \frac{i-1}{n} = \frac{n-1}{2}$?
I am trying to understand the accepted answer to this question: Find: The expected number of urns that are empty
And am stuck on the part I mentioned above. I understand that:
$\sum\limits_{i=1}^n \frac{i-1}{n}=\frac{0}{n}+\frac{1}{n}+...+\frac{n-1}{n}.$... |
H: How do i find the inverse laplace?
$$
F(s) = \frac{2s-1}{s^2(s+1)^3}
$$
If I try to use partial fractions, I end up with 8 constants to solve for!
Is there some shortcut I'm not seeing? Am I supposed to simplify it first? Am I even doing the partial decomposition right?
AI: You should have five.
$$\displaystyle F(s... |
H: Found transition matrix and state matrix
I tried to found a solution for this problem but I can't! Any suggestion?Thank you.
Given that A is a 2x2 matrix and that dx/dt=Ax(t)
suppose that x(0)=[1 ; -3] implies x(t)=[e^-3t ; -3e^-3t]
and that x(0)=[1 ; 1] implies x(t)=[e^t ; e^t]
find the transition matrix for the ... |
H: Finding the unknown cardinality of a Set given other information
How would you go about finding the number of elements in $A$ if you know that the number of elements in $A\cup B$ is $20$, the number of elements in $B$ is $7$ and the number of elements in $A\cap B$ is $3$?
I did it this way: I let $x$ be the number... |
H: measure of Cantor type set
Prove that the Lebesgue measure of the set $\{x\in[0,1]: \text{decimal expansion of $x$ contains only finitely many 7s}\}$ is zero.
I have thought that that if i can show that measure of $\limsup A_k$ is 1, then it is proven, where
$$A_k=\bigcup_i^{9^{k-1}} \left[\frac{10i+7}{10^k},\frac{... |
H: Exterior product of $\Bbb Z[x,y]$
Let $R$ be the polynomial ring $\Bbb Z[x,y]$ in the variables $x,y$. If $M=R$ (so we are considering the $R$-module $R$) then $\wedge^2 (M)=0$. This was an example in Dummit and Foote's Abstract Algebra, pg. 449. Can someone explain to me why $\wedge^2 (M)=0$? Could you explicitly ... |
H: Density of probability and Distribution function, how to turn one into the other
I know that to know the Distribution function I got to integrate the Density function from "-oo" to "x", but how to do the inverse?
AI: Differentiate.${}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}$ |
H: Finding general solution when given only particular solution
The differential equation is given by $$ty''+ (1-2t)y'+ (t-1)y = 0.$$
The particular solution give is $e^t.$
I haven't seen an equation like this so far in my class and so far haven't found anything of use by searching on google.
I've tried finding the ... |
H: A question in the proof of connectedness in $\mathbb{R}$
In my textbook there is a theorem saying that "If $S\subset\mathbb{R}$ is an interval, then $S$ is connected." I can follow most of the arguments provided there except the one indicated below. Can anyone help me understand it, please? To be complete, I briefl... |
H: How to know if a MacLaurin/Taylor Series expansion is good?
This question is motivated by this question.
So, given $\frac{1}{e^x + 1}$, the 4th order MacLaurin series $1 -e^x+(e^x)^2-(e^x)^3+(e^x)^4$, although correct in terms of the algebra manipulations, is not a good expansion.
In general, how do we know if a gi... |
H: Intuition for a physical real line vs. a physical "hyperreal line"
As a mathematical structure, I have no problem with the hyperreals. But I came across the following from Keisler's book "Elementary Calculus: An Infinitesimal Approach".
"We have no way of knowing what a line in physical space is really
like. It ... |
H: Strictly increasing continuous function
Prove that any onto strictly increasing map $f: (0,1) \to (0,1)$ is continuous.
Since its strictly increasing then for $x<y$ it implies that $f(x) < f(y)$. For continuity I must show that for any $y\in (0,1)$ there exists a $\delta>0$ such that for $\epsilon>0$ then $|x - y|... |
H: inequality for complex numbers
In general, from triangle inequality we have, for $|z_1+\epsilon~ z_2| < 1$ and $|\epsilon|=1$,
$|z_1+\epsilon~ z_2|\leq|z_1|+|\epsilon~| | z_2|=|z_1|+|z_2|$. But we can not guarantee that $|z_1|+|z_2|$ will necessarily be less than 1.
Following statement gives some general stateme... |
H: Are there any methods to exponentiate a real number with a number from an arbitrary field?
How can I take the following exponent, for some real-valued number a?
$$a^{3+2j-9k+3i}$$
over the field of quaternions, or any field for that matter? On wikipedia we are given the following formula,
which seems relatively st... |
H: Integer solutions of the equation $x^2+y^2+z^2 = 2xyz$
Calculate all integer solutions $(x,y,z)\in\mathbb{Z}^3$ of the equation $x^2+y^2+z^2 = 2xyz$.
My Attempt:
We will calculate for $x,y,z>0$. Then, using the AM-GM Inequality, we have
$$
\begin{cases}
x^2+y^2\geq 2xy\\
y^2+z^2\geq 2yz\\
z^2+x^2\geq 2zx\\
\end{c... |
H: Wave equation: show eventually $\int_{\mathbb{R}}u_x^2 = \int_{\mathbb{R}}u_t^2$
Suppose $u$ solves the wave equation in $\mathbb{R}$ and has compactly supported initial data $f(x) = u(x,0)$ and $g(x)=u_t(x,0)$. Show that the "kinetic energy" $\int_{\mathbb{R}}u_t^2$ eventually equals the "potential energy" $\int_{... |
H: Cantor-Lebesgue function and an increasing function are equal almost everywhere
Denote by $\varphi$ the cantor-lebesgue function and suppose $f$ is a certain increasing function defined on [0,1] and such that $f(x)=\varphi (x)$ for all $x\in[0,1]-C$ where $C$ is the cantor set. Prove that $f(x)=\varphi(x)$ for all ... |
H: How to graph aggregate portfolio performance?
Here is an example of a performance graph of a "motif" on Motif Investing. The motif is a portfolio containing Apple and Microsoft stocks weighted at 50% each. The graph represents the aggregate performance of the 2 stocks over time (the blue line). The green line is... |
H: Given $z=f(x,y)$, what's the difference between $\frac{dz}{dx}$ and $ \frac{\partial f}{\partial x}$?
Given $z=f(x,y)$, what's the difference between $\displaystyle\frac{dz}{dx}$ and $\displaystyle \frac{\partial f}{\partial x}$?
I got confused when I saw $dz/dx= \partial f/\partial x+\partial f/\partial y*dy/dx$... |
H: Prove or disprove: There exists a group $G$ and a normal subgroup $N$ such that $G$ is non-abelian, but both $N$ and $G/N$ are abelian.
Prove or disprove: There exists a group $G$ and a normal subgroup $N$ such that $G$ is non-abelian, but both $N$ and $G/N$ are abelian.
Can anyone give me some hint on this questio... |
H: name of curve of cluster of points of the form $(x,x^2...x^n)$ in $R^n$
what is the name of the curve made up of the points $(x,x^2...x^n)$ in $\mathbb {R}^n$ for all $x\in \mathbb R$??
For example: in $\mathbb R^2$ it would just be a parabola.
AI: This is basically the
rational
normal curve of degree $n$. ("Bas... |
H: Prove or disprove: There exists an element in the quotient group $\mathbb{R}/\mathbb{Z}$ of order 6.
Prove or disprove: There exists an element in the quotient group $\mathbb{R}/\mathbb{Z}$ of order 6. By counting formula, since $|\mathbb{R}/\mathbb{Z}|$ is infinite, it seems possible to have an element of order 6.... |
H: Let $F:\mathbb{R}^n\to\mathbb{R}^n$ a transformation of class $C^1$. Supose that $dF_{x_0}$ is inyective.
Let $F:\mathbb{R}^n\to\mathbb{R}^n$ a transformation of class $C^1$. Supose that $dF_{x_0}$ is inyective. Prove that there exist an $\varepsilon>0$ such that if $G$ is another transformation $C^1$ such that $||... |
H: H acting on G by left translation
$H$ is a subgroup of $G$ and acts on $G$ by left translation, describe the orbits.
Here is my take, at first I thought well, isn't that just left cosets? But, that seems too easy. By translations, does that mean by adding some element of $H$ on the left of $G$? Am I even close to ... |
H: Fraction exponential limit in complex plane
Let $n$ be an integer. I need to compute the limit $$\lim_{z\rightarrow 2n\pi i}\dfrac{e^z-1}{z-2n\pi i}$$ for complex number $z$. I think I can't use L'Hospital here since $z$ is complex. How can I do it?
AI: Hint: use the definition of the derivative at a point.
$$ f'... |
H: Can we express these sets as Cartesian products of two subsets of $\mathbf{R}$?
Let sets $A$ and $B$ be given as follows: $$A := \{ (x,y) \in \mathbf{R}^2 | \ \ x < y \ \ \} $$ and $$B := \{ (x,y) \in \mathbf{R}^2 |\ \ x^2 + y^2 < 1 \ \ \}.$$ Can we express $A$ or $B$ as a Cartesian product of two subsets of $\m... |
H: Why does the fiber coproduct in $\mathbf{Set}$ actually satisfy the universal property?
Suppose you have two morphisms $f\colon A\to B$ and $g\colon A\to C$. Then the fiber coproduct of $B$ and $C$ over $A$ exists and is the disjoint union $B\coprod C$ where we identify $fx$ and $gx$ for all $x\in A$.
I'm a little... |
H: Prove or disprove: There exists a ring homomorphism $\phi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$.
Prove or disprove: There exists a ring homomorphism $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$.
I think it is intuitive to try $\varphi: \mathbb{C}\rightarrow \mathbb{R}\times\mathbb{R}$ de... |
H: Combinatorial Proof of Multinomial Theorem - Without Induction or Binomial Theorem
I've been trying to rout out an exclusively combinatorial proof of the Multinomial Theorem with bounteous details but only lighted upon this one - see P2. Any other helpful ones?
$(x_1+\cdots+x_k)^n =$ Top of Page 39 from UNC:
$\s... |
H: Semicircle contour for integrating $t^2/(t^2+a^2)^3$
Let $a\in\mathbb{R}$. Evaluate $$\int_0^{\infty}\dfrac{t^2}{(t^2+a^2)^3}dt$$
The function is even, so the value of the integral is half of $\int_{-\infty}^{\infty}\dfrac{t^2}{(t^2+a^2)^3}dt$
I'm going to use countour integration along the semicircle in upper-ha... |
H: What is the center of fundamental groupoid?
$C$, $D$ are two categories. $F$, $G$ are functors between $C$ and $D$: $F, G: C\rightarrow D$. Let $Nat(F,G)$ be all the natural transformations between F and G.
Like what we do for groups, define the center of the a category $C$: $Z(C)=Nat(id_C,id_C)$.
Then what is the ... |
H: The forever moving billiard ball
Suppose I have a rectangular table, dimensions $x$ by $y$, and a billiard ball is positioned in the very center. For descriptive convenience, let us impose a coordinate system on this table with an origin (0,0) in the center of the table where I strike the ball.
Now say I strike th... |
H: Large Regression dataset
For a project I need a large regression (least squares) dataset:
If $n$ is the number of samples and $p$ the number of features, then I need $p < n$ and
$p,n$ both very large. For example $n=10,000,000$, $p=1,000,000$.
Does anybody know such a dataset or at least where I could get one? I wa... |
H: Question on polynomial.
Here is a question from Hoffman:
If $F$ is a field and $h$ is a polynomial over $F$ of degree $\ge 1$, show that the mapping $f \rightarrow f(h) $ is a one-one linear transformation of $F[x]$ into $F[x]$. Show that this transformation is an isomorphism of $F[x]$ onto $F[x]$ if and only if de... |
H: limit of an absolute sequence: ${b_n} = |{a_n} - 1|$
$$\eqalign{
& \mathop {\lim }\limits_{n \to \infty } {a_n} = 3 \cr
& {b_n} = |{a_n} - 1| \cr} $$
Hence,
$$\mathop {\lim }\limits_{n \to \infty } {b_n} = |3 - 1| = 2$$
Is it right to say that?
If so, is it sufficent for a proof?
AI: If you're asking, then it... |
H: how to integrate convolution $\int f*g$
I am stuck on this question:
$f$,$g$ are blocked, continuous functions and $\int_{-\infty}^\infty|f(x)|dx<\infty,\int_{-\infty}^\infty|g(x)|dx<\infty$.
show that:
$$\int_{-\infty}^\infty (f*g)(t)dt=\int_{-\infty}^\infty f(x)dx\int_{-\infty}^\infty g(x)dx$$
looking for some cl... |
H: Compactness and Convergence of Subsequences
Let $(X,\rho)$ be a metric space. Suppose that $(x_n)_{n\in\mathbb Z_+}$ is such a sequence in $X$ that any subsequence has a further subsequence that is convergent. However, the limits of these sub-subsequences are not necessarily the same.
Let $A$ be the image set of th... |
H: Calculating expected value for joint density function
$$f(x,y) = 2$$ $$0<x \le y < 1$$
I want to calculate $$E(XY)$$
In order to do that, I need to know the range for $x$ and $y$. Are they both $0$ to $1$?
AI: The range is
$$
\begin{align}
0<&x\leq y \\
0 <&y < 1
\end{align}
$$
If you're unsure about these things y... |
H: Combinatorics/Task Dependency
Here is a competitive programming question:
You have a number of chores to do. You can only do one chore at a time and some of them depend on others. Suppose you have four tasks to complete. For convenience, we assume the tasks are numbered from 1 to 4. Suppose that task 4 depends on b... |
H: Number of even and odd subsets -- wrong question?
The book on Discrete Mathematics I'm following poses the following problem:
Prove that a nonempty set has the same number of odd subsets (i.e., subsets with odd number of elements) as even subsets.
A short proof is also given, but I have a counter-example: Conside... |
H: Show an isomorphism between two quotient spaces
We are given 3 vector spaces $U \subseteq W \subseteq V$
We are asked to prove that there is an isomorphism between $V/W$ and $(V/U)/(W/V)$
What we tried (and it is false, I would like to know why):
We basically said that an element of $V/W$ is $v+W$ and that an eleme... |
H: finding an 'e' based limit for this sequence
i need to find the limit for
$$\lim_{n \rightarrow \infty} \left(1 + \frac{q}{n}\right)^n $$
where $q \in \mathbb{Q}$
how to i get this sequence to resemble
$$\lim_{n \rightarrow \infty} \left(1 + \frac{1}{n}\right)^n $$
so i can find its limit?
AI: $q = \frac{s}{t}$... |
H: Solving inequality. $5(y-2)-3(y+4)\geq 2y-30$
I solved this inequality, but at the end I got that $8 \geq 0$. Did I did this right. What does this mean.
This is how I solved it:
$$
5(y-2)-3(y+4) \geq 2y-30\\
5y-10-3y-12 \geq 2y-30\\
2y-22 \geq 2y-30\\
2y-2y-22+30 \geq 0\\
8\geq0\\
$$
Did I did it right?
Thank you v... |
H: Existence function $f$ which isn't uniform continuity, but $f^2$ is?
Existence function $f$ which isn't uniform continuity, but $f^2$ is? I don't have more asumptions about function $f$, so let $f: X \rightarrow Y$ where $X,Y \subseteq \mathbb{R}$ with euclidean metrice.
In my opinion if $f^2$ is uniform continuity... |
H: Multiplicative order
I tried to learn about the multiplicative order from different sources, but I want to get sure that I understand it well...
As I understand, if we define to numbers, $a \in \Bbb Z$, $n \in \Bbb N^+$, where $a$, $n$ are coprime, then $O_n(a)$ is the smallest positive integer $k$ such that $a^k\;... |
H: Linear Algebra - linear functionals
Let $S$ be a set of vectors in $V$. Define $S^0$ to be the set of linear functional that vanishes on $S$.
Show that $$S^0 = \langle S\rangle^0$$
I am really confused what is meant by $S^0$
AI: $S^0=\{f:V\to\Bbb R \mid \forall s\in S: f(s)=0\}$.
Show that, if $f\in S^0$ then $f\i... |
H: Fixed points through a general circle.
The circle $C: x^2 + y^2 + kx + (1+k)y - (k+1)=0$ passes through two fixed points for every real number $k$. Find $(i)$ co-ordinates of these two points and $(ii)$ the minimum value of the radius.
AI: HINT:
This is an arbitrary circle which passes through the intersection of $... |
H: Is every $T_2$ compact monotonically normal space separable?
I have a question about compact spaces:
Is every $T_2$ compact monotonically normal space separable?
Thanks ahead:)
AI: Hint: Every linearly ordered topological space is $T_2$ and monotonically normal.
Can you think of one which is compact and nonsepa... |
H: Formal proving of languages accepted by a finite automata.
Suppose $L_1 \cup L_2,L_1 \subseteq E^* $ are languages accepted by finite automata and $L_1\cap L_2 =\emptyset $. We need to prove that $L_2 $ is also accepted by a finite automaton.
So I've started with building an intuition, but I have a few questions.
... |
H: A Summation Question
Let $n$ be an odd integer. If $\sin (n)\theta=\displaystyle \sum _{r=0}^n b_r\sin (r\theta)$, for every value of $\theta$, then
$b_0=1, b_1=3$
$b_0=0, b_1=n$
$b_0=-1, b_1=n$
$b_0=0, b_1=n^2+3n+3$
How do I approach it? I am not able to find the values of $b_0$ or $b_1$.
AI: For example:
$$n=1\... |
H: Question about finding a limit with limit arithmetics
$lim_{n \to \infty}(\frac{4n^2}{(2n+1)(2n-1)})^{1-n^2}$
When I simplfy it I get: $\frac{1-4n^2}{(1-4n^2)^{n^2}}$
Now is it enough to use limit arithmetics on the denominator as if it's $\frac1{x^n}$ to show that it goes to 0 ?
Note: we can't use logs/lns/derive/... |
H: Finding the limit of $ \lim_{k \rightarrow \infty} \left(\frac{2^k + 1}{2^{k-1} + 3}\right) $
$$ \lim_{k \rightarrow \infty} \left(\frac{2^k + 1}{2^{k-1} + 3}\right) $$
I'm trying to prove that the limit of the sequence is $2$ using the squeeze theorem, but with no success.
Thanks
AI: HINT: Multiply the fraction by... |
H: check if 2 vectors in $\Bbb R^n$ = $\Bbb R^2$
I need to make sure that a system of any 2 vectors in $\Bbb R^n$ can make a lineair combination to $\Bbb R^2$.
The way I want to investigate is to row-reduce the matrix to enchelon form. This way I can check for consistentcy and see if the set has a single solution. Is ... |
H: How to calculate the rest of $2^{p^r-p^{r-1}+1}$ divided by $p^r$
I have the next problem:
$p$ is an odd prime number and $r$ is a natural number, $r>1$. How can I calculate the rest of the division of $2^{p^r-p^{r-1}+1}$ by $p^r$ ?
AI: Hint: $\varphi(p^r)=p^r-p^{r-1}$ and $2$ is coprime to $p^r$, so, by Euler-Ferm... |
H: Integrate over a curve (complex)
Given
$${\Gamma} = [{z(t) = t\sqrt{\frac2\pi\}}e^{it^2}}]$$ for $$0< t < \sqrt{\pi/2}$$
evaluate
$$\int_{\Gamma} ze^{z^2}dz$$
The usual process involves parametrization and then substituting z(t) into f(z) which in this case is the stuff under the integral and then multiplying by... |
H: Quantity of an object after proliferation
A small creature called "Charza" lives in blocks of an infinite square. An infinite number of them can live in a block. After one hour , one Charza is divided into 4 Charzas and each one moves into one of the adjacent blocks. we have only one Charza at the first . after 6 h... |
H: Kind find the limit of the following question
What is the limit of the following?
lim(h->0) ((2+h)^0.5 -(2)^0.5)/h
Note:
I'm struggling to make the denominator positive.
AI: First way:
$$\lim_{h\to 0}\frac{\sqrt{2+h}-\sqrt2}h=(\sqrt x)'|_{x=2}=\frac1{2\sqrt2}$$
Second way:
$$\lim_{h\to 0}\frac{\sqrt{2+h}-\sqrt2}h=\... |
H: sum of all coprimes of a number.
What is the sum of all coprimes to number less than that number?
I found a bit about it:
For example we have to find the sum of coprimes of 2016. Therefore, the required sum $S$ is:
$2016 = 2^5 * 7 * 3^2$
$S = \frac{2016}2 * 2016 * (1-\frac13)(1-\frac17)(1-\frac12) = 580608$
If this... |
H: the smallest value of $\alpha$
$f$ is a continuous function on the interval $[0,1] $ which satisfy:
1) $f(x) \leq \sqrt{5}$ for all $x \in [0,1] $
2) $f(x) \leq \frac{2}{x}$ for all $x \in [\frac{1}{2},1]$
Then find the smallest real $\alpha$ such that the inequality $ \int^1_0 f(x) dx \leq \alpha$ holds for any s... |
H: Can't find the limit of the following
I can't find the limit of the the following:
$$\lim_{p\to1} \frac{ p^{1/3} - 1 }{p - 1}$$
Any ideas?
AI: Putting $p^{\frac13}=q\implies p=q^3$ as $p\to1, q\to1$
$$\lim_{p\to1}\frac{p^{\frac13}-1}{p-1}$$
$$=\lim_{q\to1}\frac{q-1}{q^3-1}$$
$$=\lim_{q\to1}\frac{(q-1)}{(q-1)(q^2+q+... |
H: How does $\exp(0)=1$ follow from the definition $\exp(z):= \sum_{n=0}^\infty \frac{1}{n!} z^n$
We introduced the Exponential function as follows: $$ \exp: \begin{cases} \mathbb{C} & \longrightarrow \mathbb{C} \\z & \longmapsto \displaystyle \sum_{n=0}^{\infty} \frac{1}{n!}z^n \end{cases}$$
It might be a trivial thi... |
H: Best way to explain how the Infimum and Supremum of this function are obtained...?
I have the function $\;f(x)=\dfrac{x^{(1/2)}}{2+x}\;$ and I know that $\inf(f)$ does not exist and $\sup(f)=2$ but I don't know how to formally show this rigorously? Anyone got a formal way of showing this, it would be much appreciat... |
H: Is series convergent?
Does this series converge?
$$\sum_{n=1}^{\infty }\frac{{(-1)}^{n}(n+2)}{{n}^{2}+4}$$
How can i step-by-step calculate it?
AI: Check that
$$\frac{n+2}{n^2+4}\ge\frac{n+3}{(n+1)^2+4}\iff (n+2)(n^2+2n+5)\ge(n+3)(n^2+4)\iff$$
$$\iff n^3+4n^2+9n+10\ge n^3+3n^2+4n+12\iff$$
$$\iff n^2+5n-2\ge 0$$... |
H: Classification of groups of order 30
How do I find all the groups of order 30? That is I need to find all the groups with cardinality 30. I know Sylow theorems.
AI: Let $G$ be a group of order 30 and let $n_p$ be the number of its $p$-Sylow subgroups. By Sylow, $n_3\in\{1,10\}$ and $n_5\in\{1,6\}$. It is not possib... |
H: Is there a continuous function $f:S^1 \to \mathbb R$ which is one-one?
Is there a continuous function $f:S^1 \to \mathbb R$ which is one-one?
AI: Suppose such a function $f$ exists. $f(S^1)$ must be connected and compact. $f$ is one-one so it cannot be constant. It follows that $f(S^1) = [a, b]$ for $a \ne b$.
Pick... |
H: How to write this as a boolean expression?
How can I write the following sentence as boolean expression:
$$
\text{If two sides of triangle are the same, then two opposite angles are the same}
$$
I konw it should be something like this:
$$
a = \text{if two sides of triangle are the same}\\
b = \text{two opposite ang... |
H: find all the partial limits of this sequence
i need to find all the partial limits (is that the current terminology? i mean the limits of all possible subsequences for a given sequence)
$a_n = \sqrt[n]{4^2 + 2^n} $
find all partial sums,also find $\liminf_{n\rightarrow\infty} a_n$ and $\limsup_{n\rightarrow... |
H: calculating norm of the following matrix?
I have the following matrix and its norm
$$ \begin{aligned} A(s) &= \begin{bmatrix} 0 & e^{-sT} \\ e^{-sT} & 0 \end{bmatrix} \\
\lvert \lvert {A(s)} \rvert \rvert_{\infty} &= \sup_{w} \sqrt{\lambda_{\max} (A^{*}(jw)A(jw)) } = 1 \end{aligned}$$
I want to know how the norm... |
H: What does it mean for a value to be minimised?
I am trying to solve this problem, but I do not understand what is meant bythe following expression being minimised
$$ D = P_{k} + P_{j} $$
AI: A value is minimised when it becomes as small as it can.
I expect that your $P_k$ and $P_j$ are points. Then $\|P_k-P_j\|$ is... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.