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H: Small-angle approximation of $ \frac{\sin^2 x}{x^2 \sqrt{1-\frac{\sin^2 x}{3}}} $
I need to show the following:
$$ \frac{\sin^2 x}{x^2 \sqrt{1-\frac{\sin^2 x}{3}}} \approx 1-\frac{x^2}{6} $$ when $ x $ is small.
I think this problem is trickier than most other questions like it because in the original source there ... |
H: Infer the second isomorphism theorem from the first one
I'm trying to infer the second isomorphism theorem on groups from the first one. Could you please verify if my attempt is fine or contains logical mistakes?
Let $G$ be a group, $S \le G$, and $N \trianglelefteq G$. Then $(S N) / N \cong S /(S \cap N)$.
My a... |
H: Find $\theta$ such that $W = X \cosθ +Y \sinθ \text{ and } Z = X \cosθ −Y \sinθ$ are independent.
Find $\theta$ such that $W = X \cosθ +Y \sinθ \text{ and } Z = X \cosθ −Y \sinθ$ are independent. It is given that X and Y be jointly normal each with mean $0$ and variance $1$.
I have shown that
$$W \sim N(0, 1)$$
$$Z... |
H: Trace norm of a trace class operator exercise in Conway
Exercise in Conway's Functional Analysis book:
Let $T$ be a trace class operator on a Hilbert space ${\cal H}$.
Prove:
$$\sup\{|\mbox{tr}(CT)|:\ C\ \mbox{is compact}, ||C||\leq 1\}=||T||_1.$$
Here, $||T||_1=\mbox{tr}[(T^*T)^{\frac{1}{2}}]$ is the trace norm.
I... |
H: Albert Einstein's quotation on the nature of mathematics
How can it be that mathematics, being after all a product of human
thought independent of experience, is so admirably adapted to the
objects of reality$?$
The above quotation is by Albert Einstein. I was wondering what did he really mean by that$?$
So if we... |
H: Partition of $\mathbb{N}$ into APs
A continuation of this question :
How does one classify the set of all partitions (into disjoint parts) $\{AP_1, AP_2, AP_3\cdots\}$ of $\mathbb{N}$ into APs such that all the APs have a distinct common difference? One such example was given by @HagenVonEitzen in the linked post:
... |
H: Are the singularities of $f(z) = \frac{z^2+1}{z^2(z+1)}$ removable?
Looking at the function $f(z) = \frac{z^2+1}{z^2(z+1)}$, I have found the singularities to be at $z=0$ and $z=-1$.
My question is if they are removable.
I expanded this into the Laurent series $\frac{z^2+1}{z^2}-\frac{z^2+1}{z}+(z^2+1)-z(z^2+1)+...... |
H: Does $e^{-kx} = -ke^x$?
I have been working on a homework problem where I need to integrate $-ke^x dx$. I decided to factor out the $-k$ and get $e^x + C$ and then multiply $-k$ back in. However, I noticed that the back of the book gets $e^{-kx} + C$. I think that if one was to differentiate this, they would in fac... |
H: Absolutely continuous on $[-1,1]$ of a function
Show that $f(x)=x^2 \cos\left(\dfrac{\pi}{2x}\right)$ when $x\neq 0$, and $0$ when $x=0$, is absolutely continuous on $[-1,1]$.
I'm honestly not sure how to get this one off the ground. I thought about maybe trying to prove that it's Lipschitz, but 1) I'm not even c... |
H: Howard Eves' Introduction to the History of Mathematics -- editions and years published?
Can anyone point me towards where I may be able to find a list of all the editions of Howard Eves' Introduction to the History of Mathematics?
I know little about this book; I understand that it's used in some American universi... |
H: Euler ODE - particular solution problem
I have this ODE: $$x^2y'' -xy' +y = 6x\ln(x)$$
Once I solve the homogenous part I get that $$y_h = xC_1 + x\ln(x)C_2$$
But I am having problems with the particular solution. When I try to find it using variation of constants:
$$C_1'x + C_2'x\ln(x)=0$$
$$C_1' + C_2'(1+\ln(x))=... |
H: Find the singularities of $f(z) =\frac{1}{(2\sin z - 1)^2}$.
Find the singularities of $f(z) =\frac{1}{(2\sin z - 1)^2}$.
I am just learning about singularities and I was wondering if someone could give me feedback on my work. So I think, for this function, that there are singularities at $z=\frac{\pi}{6}+2k\pi,\fr... |
H: Sufficient to show the cases when $x = 0$, $y=0$, and $(x,y) \ne (0,0)$?
For a fixed $k \in \mathbb{N}$, define $f_k: \mathbb{R}^2 \rightarrow \mathbb{R}$ by:
$$
f_k(x,y)=
\begin{cases}
\dfrac{x^2(x+y^2)}{x^2+y^{2k}} &, (x,y)\neq (0,0)\\
0 &, (x,y)=(0,0)\\
\end{cases}
$$
Show that $f_1$ is not differentiable at $(... |
H: Increasing convergence of sequence bounded below.
Assume that you have a measurespace $(A,\mathcal{A},\mu)$. And you sequence of measurable functions $f_n \rightarrow \mathbb{R}$, that are increasing, and each function is bounded below by a common value $-M$.
Do we then have that $\lim\limits_{n \rightarrow \infty}... |
H: Convex function on closed bounded Interval Implies Lipschitz counterexample
I am considering a convex function $f:[a,b] \rightarrow \mathbb{R}$ and have been asked to show that $f$ is absolutely continuous on $[a,b]$. I've attempted to use the Chordal Slope Theorem to show that $f$ is Lipschitz on $[a,b]$, but have... |
H: MSE for MLE of normal distribution's ${\sigma}^2$
So I've known $MLE$ for ${\sigma}^2$ is $\hat{{\sigma}^2}=\frac{1}{n}\sum_{i=1}^{n} (X_{i} -\bar{X})^2$, and I'm looking for $MSE$ of $\hat{{\sigma}^2}$. But I'm having trouble to get the result.
What I tried goes like below:
By definition, $MSE$ = $E[(\hat{{\sigma}... |
H: Is it provable in ZF that there is no nontrivial elementary embedding $\pi: V\to V$?
Is it open whether it can be proved in $ZF$ alone that if $\pi:V\to M$ is a nontrivial elementary embedding, then $M\not=V$?
AI: The short version is that you're asking whether Reinhardt cardinals are consistent with $\mathsf{ZF}$,... |
H: Independence of two binomial variables
I am trying to figure out whether these two variables are independent or not, and why.
In the situation of the problem I was given, the number of clients $N$ follows a Poisson distribution of mean $c$. The number of successful clients $X$ has a probability $p$. The number of e... |
H: Composing trigonometric functions
Let $f(x)=\sin(x)$. If $g$ and $h$ are functions on $\mathbb{R}$ such that $g(f(x))= h(f(x))$,
can we conclude $g=h$ ?
Can we actually compare $g$ and $h$?
I am confused. Please, help me.
AI: The range of $sin(x)$ is only [-1,1], not all of $\mathbb R$ . It is possible to come up... |
H: How to integrate $\frac{\int _{-w}^w\:e^{-\frac{x^2}{w^a}}dx}{\int _{-\infty \:}^{\infty \:}\:e^{-\frac{x^2}{w^a}}dx}$
I have found $\frac{\frac{\sqrt{\pi }}{2}\text{erf}\left(\sqrt{w}\right)-\frac{\sqrt{\pi }}{2}\text{erf}\left(-\sqrt{w}\right)}{\pi ^{\frac{1}{2}}}$ for $\frac{\int _{-w}^w\:e^{-\frac{x^2}{w}}dx}{\... |
H: Geometric intuition about continuous and uniform continuous functions
I'm looking for a geometric way to identify functions that are continuous but not uniformly continuous without using the definition.
I can't really put my hands on a concrete difference between the two.
AI: If you do something like Robinson's non... |
H: Chcking if a function defined for a convergent sequence is Riemann Integrable
Let $\{a_n\}$ be a sequence of real numbers that converges to $1$. Define $f:[0,2]\to \mathbb{R} $ by
$$ f(x)=
\begin{cases}
1&\text{if }\, x\in \{a_n\}\\
0& \text{otherwise}\\
\end{cases} $$
Prove $f$ is Riemann integrable on $[0,2]$
I... |
H: Question about sequence convergent to a limit point and Axiom of (Countable) Choice
I read this question about how the Axiom of Countable Choice is both necessary and sufficient to show the following:
If a point $a$ in a metric space $X$ is a limit point of $A\subseteq X$, then there is a sequence of points in $A-... |
H: What exactly does the definition of a nilpotent group mean?
I'm studying nilpotent and solvable group and find it pretty hard to tell what the definition of a nilpotent group is after.
For example, a group is solvable iff it has a solvable series (that is, a subnormal series such that each factor is abelian). This ... |
H: 2 questions in Theorem 5 of Hoffman Kunze Linear Algebra
I am studying linear algebra from textbook Hoffman and Kunze and I have a question in a Theorem of Section 9.3 ( Positive Forms)
Image of Theorem:
Questions : why in 2nd paragraph g(X, X)$\geq$ 0 holds. Clearly, g(X, X) is a 1×1 matrix but I am unable to un... |
H: Vector addition and scalar multiplication
If $Q$ is the set of positive real numbers.
$Q^2 = \{(x,y)\mid x, y \in Q\} $
can be shown with operations of vector addition and scalar multiplication using the formulas
$(x_1, y_1) + (x_2, y_2) = (x_1x_2, y_1y_2)$ and $ c(x, y) = (x^c, y^c)$ where $c$, a real number, is a... |
H: How does $e^x\cdot e^X$ equal $e^{x+X}$?
I know that they equal each other, but when I'm trying to prove it, something doesn't match. Please mind the difference between the two equations, one is a lowercase $x$ and the other is an uppercase $x.$ I know that the formula to get $e^x$ is $\frac{x^n}{n!}$. So I apply o... |
H: Integer exponent equation
Show that $$(2^a-1)(2^b-1)=2^{2^c}+1$$ doesn't have solution in positive integers $a$, $b$, and $c$.
After expansion I got
$$2^{a+b}-2^a-2^b=2^{2^c}\,.$$
Any hint will be appreciated.
AI: $$2^{a+b}=2^a+2^b+2^{2^c}$$
Case $1$:
If $a=b$, then $$2^{2a}=2^{a+1}+2^{2^c}$$
If $a+1$ and $2^c$ a... |
H: Topology in an underlying set of $X.$
Let $X$ be a topological space and let $S$ be a subset $X$ fixed. Show that
$$\tau= \{A \cup (B \cap S) \mid A,B \text{ open at } X \}$$
determines another topology on the underlying subset of $X.$
My attempt
Using that $A \cup (B \cap S)= (A\cup B) \cap (A \cup S)$
where we... |
H: Solve $2x^2+y^2-z=2\sqrt{4x+8y-z}-19$
I am trying to solve the following equation.
$$
2x^2+y^2-z=2\sqrt{4x+8y-z}-19
$$
To get rid of the square root, I tried squaring both sides which lead to
$$
(2x^2+y^2-z+19)^2=16x+32y-4z
$$
which was too complex to deal with.
Also, I have tried some substitutions to simplify the... |
H: Find the turning points of $(x^{2} + y^{2})^{2} = x^ {2} - y^{2}$
Using implicit differentiation I found the derivative of this function. Here is my working:
but how would I evaluate the numerator as $0$ when $2x - 4x^3 - 4xy^2 = 0$?
I tried moving all the $x$ terms to one side and all the $y$ terms to the other b... |
H: Problem related to real monic quadratic polynomial
Let $f(x)$ be a real monic quadratic polynomial. If ${x_1},{x_2},{x_3},{x_4},{x_5}$ be the $5$ points where
$g(x) = |f(|x|)|$ is non-differentiable and $\sum_{i = 1}^5 {\left| {{x_i}} \right| = 8} $ then find the value of $\frac{1}{5}\mathop {\lim }\limits_{x \... |
H: Proof verification that $t(n+1)=t(n) + \pi$ using mathematical induction
I am beginning to learn how to write proofs and I would like some verification on this simple proof I have done for the sum of the interior angles of a polygon. I thought this would be a good one to prove since it is pretty basic. So, I have ... |
H: How to solve gaussian integral for $x^2e^{-\frac{x^2}{w}}$?
I am trying to find $\sigma=\displaystyle\sqrt{\int _{-\infty \:}^{\infty \:}x^2e^{-\frac{x^2}{w}}dx}$ for the function $f(x)=e^{-\frac{x^2}{w}}$. I have tried tabular integration by parts, but it quickly got messy and I stopped after the second integratio... |
H: Finding the length of intersected line given a square of 3 equal rectangles and a perpendicular line
This is probably very straight forward, but I'm drawing a blank..
I want to find the length of $A-I$. This isn't a trick question, if anything looks slightly off, it's just a bad drawing. It's essentially a square d... |
H: Can this distribution be expressed as a known distribution?
I have this density function
$$f_{\theta}(x)=\left\{\begin{array}{ll}\frac{1}{x^{2} \theta} e^{-\frac{1}{\theta x}} & \text { if } x \geq 0 \\ 0 & \text { other case }\end{array}\right.$$
Im trying to express it as a known distribution as i want to make so... |
H: Intersection of infinite sets -question
I am stuck at B . could any one help me ?
AI: Let $n\in \bigcap_{k=1}^{\infty}A_k$ such that $n\geq1$. Then $n\in A_k$ for all $k\geq 1$. That implies $n$ is divisible by every $k\geq 1$, which is not possible. So $\bigcap_{k=1}^{\infty}A_k=\{0\}=A_0$. |
H: Using algebra to solve a graph problem
I’m reading a textbook about graph theory and its application and suddenly I’m facing some problems understanding a part which needs algebra background.
I’m gonna skip the graph related part ;
The only think that I should mention is if $xy=yx$ then $x$ and $y$ are adjacent in ... |
H: 1 question in Theorem 10 of section Spectral Theory of Hoffman Kunze
I am self studying Linear Algebra from Textbook Hoffman and Kunze and I have a question in Theorem 10 of Chapter 9 .
Adding it's image:
How it's clear that for every $\alpha$ in $E_{j}V $ f(T) $\alpha$ = f($c_{j} $) $\alpha$ ? Can anyone please ... |
H: How to evaluate $\int_0^{\pi/2} \frac{\sin x}{\sin^{2n+1}x +\cos^{2n+1}x} dx$?
I have an exercise to evalute the following integral for all $n\geq 1 $
$$I(n)=\int_0^{\frac{\pi}{2}} \frac{\sin x}{\sin^{2n+1} x+\cos^{2n+1} x}dx$$
I attempted to find the closed form for the integral above in the following manner, wher... |
H: Determine which of the following sets is a null set as defined in the question.
A set $X\subseteq \Bbb R$ is said to be a null set if for every $\epsilon\gt0$ there exists a countable collection $\{(a_k,b_k)\}_{k=1}^\infty$ of open intervals such that $X\subseteq \cup_{k=1}^\infty\{(a_k,b_k)\}$ and $\sum_{k=1}^\in... |
H: How can I solve for $a$ in $0=\frac{\sqrt{\frac{a}{2}}-\sqrt{2a}}{a^{2}}+\frac{E_{\alpha}-E_{\beta}}{4}\exp\left(-\frac{a}{4}\right)$
I am trying to find the local maximum of $\frac{\sqrt{2a}}{a}-\left(E_{\alpha}-E_{\beta}\right)\exp\left(-\frac{a}{4}\right)$, where $E_{\alpha}$ and $E_{\beta}$ are constants:
$$\di... |
H: Does this ses $0\rightarrow \mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{R}/p\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z}\rightarrow 0$ split?
Does the following short exact sequence of $\mathbb{Z}$-modules split ?
$$0\rightarrow \mathbb{Z}/p\mathbb{Z}\rightarrow \mathbb{R}/p\mathbb{Z}\rightarrow \mathbb{R}/\mathbb{Z... |
H: Prove that $\tan^{-1}\frac{\sqrt{1+x^2}+\sqrt{1-x^2}}{\sqrt{1+x^2}-\sqrt{1-x^2}}=\frac{\pi}{4}+\frac 12 \cos^{-1}x^2$
Let the above expression be equal to $\phi$
$$\frac{\tan \phi +1}{\tan \phi-1}=\sqrt{\frac{1+x^2}{1-x^2}}$$
$$\frac{1+\tan^2\phi +2\tan \phi}{1+\tan^2 \phi-2\tan \phi}=\frac{1+x^2}{1-x^2}$$
$$\frac{... |
H: Simple question related to Correlation & Covariance
$X$ and $Y$ are two random variables with $\Bbb E[X] = \Bbb E[Y] = 1$ and $\Bbb E[X^2] = \Bbb E[Y^2] = 2$. Which of the following is not possible:
$\Bbb E[XY] > 0$
$\Bbb E[XY] < 0$
$\Bbb E[XY] = 0$
$\Bbb E[XY] \le 2$
I reached the following conclusions:
$\opera... |
H: Series + Number theory problem from JEE exam .
My approach : Used variables for given conditions and made some equations. I got the AP as 3,5,7 and GP as 1,3,27 but it is not matching with the condition given in the question. I would really appreciate your help.
Thank you.
AI: We have that
$$a_2=\frac{a_1+a_3}{2}... |
H: Does $F/N_{1}\cong F/N_{2}$ implies $N_{1}\cong N_{2}$ and vice versa?
It would be great for me to see an example of finite group $F$ and two epimorphisms $\varphi_{i}:F\twoheadrightarrow G$ from $F$ onto a group $G$, with $N_{1}\ncong N_{2}$, where $N_{i}=ker\varphi_{i}$, $i=1,2$ (e.g., finite group which contains... |
H: Prove that if ${(v_1- v_2) } \in U$ where $U$ is a subspace of $V$ then $v_1 + U$ = $v_2 + U$ where $v_1 , v_2$ belong in $V$
I assumed $v_1 - v_2$ to be equal to some $u \in U$.
Then I wrote $u$ = $u_2 - u_1$.
So now $v_1 - v_2 = u_2 - u_1$ for any $u_2,u_1 \in U$.
Therefore $v_1 + u_1= v_2 + u_2$ for any $u_2,u_... |
H: How to divide an interval into geometrically increasing sub-intervals
It'll be obvious when I see the answer, but I'm too tired/thick to figure it out.
If I have a certain time period, and I want to divide it into a given number of geometrically increasing (or decreasing) durations so that the ratio of interval(n)/... |
H: Find all positive integers $x$ and $y$ for which $\frac{1}{x} + \frac{1}{y} = \frac{1}{p}.$
Let $p$ be a prime number. Find all positive integers $x$ and $y$ for which $$\dfrac{1}{x} + \dfrac{1}{y} = \dfrac{1}{p}.$$
Multiplying the given expression by $xy$ results in $y+x = \dfrac{xy}{p} \Rightarrow p(x+y) = xy$.... |
H: Jordan normal form powers
Let $A$ be a $n\times n$ such that $A=PBP^{-1}$ where $B$ is in Jordan normal form with $\lambda_i(k)_j$ Where $i$ is the size, $k$ is the eigenvalue and $j$ the order.
If $A$ was diagonal($i=1$) then $A^n$ in Jordan form has $\lambda_1(k^n)_j$.
If the Jordan form has Jordan blocks bigge... |
H: Why is $\mathbb{R}-\mathbb{Q}$ an uncountable set and how can I prove it?
I am now starting to prepare for a discrete mathematics class. On a test, I came across the following question:
Which of the following sets are countable? $$\mathbb{Z},\mathbb{R}, \mathbb{R-Q}, \{31,2,2019\} $$
The only countable sets are: $\... |
H: Showing $\sum_{\alpha=0}^k \binom{k+15}{\alpha + 10}\binom{k}{\alpha}=\frac{2^{2k+15}(k+6)(k+7)\left(\frac{2k+15}{2}\right)!}{\sqrtπ (k+10)!}$
I want to simplify $\sum_{\alpha=0}^k \binom{k+15}{\alpha + 10}\binom{k}{\alpha}$ but this can't be directly simplified using Vandermonde Chu Identity. Wolfram shows a nice ... |
H: Holomorphic maps preserve Hausdorff dimension.
In a paper I read there is the following claim:
Let $f:\mathbb{C}\to \mathbb{C}$ be a non-constant entire transcendental function(essential singularity at infinity) and $A\subset \mathbb{C}$ a set in the complex plane. Then $f^{-1}(A)$, $A$ and $f(A)$ have the same Hau... |
H: Mapping a set with the function $z^2$ in the complex plane
I have the following set:
$G=\{ z \in \mathbb{C} : \Im{(z)}>0, \Re{(z)}<0 \}$
$f(z) = z^2$
I need to draw $ f(G) $ but I don't get a good answer using $ z=x+iy $ and trying to understand the complex plane with information on the cartesian plane.
How do I go... |
H: Power Series and Analyticity of a complex function
I was studying about the connection of analytic function and their power series representation.
Finally, I came to an understanding that, if I am given with an function, analytic at some point 'a', then I will be able to write a power series representation of that ... |
H: How does $A = P\cdot (1+r)^{n}$ becomes a graph function of $A = P\cdot (1+rt)$
Accumulated simple interest function is:
$A = P\cdot (1+rt)$
When I tried to graph it, it fails.
After I read some articles, it tells that to graph simple interest function is a different equation:
$A = P\cdot (1+r)^{n}$
where $A$ is on... |
H: Alternate way to solve $\lim\limits_{x \to 0} (\sin x) ^x$?
My solution:
$$ \lim_{x \to 0} (\sin x)^x = \lim_{x \to 0} e^{(x)(\ln \sin x)} = \exp \left( \lim_{x \to 0} (x) (\ln \sin x) \right)$$
Now we have $\lim_{x \to 0} (x) (\ln\sin x)$.
Now we can say that the limit is $0$ as $\ln$, $\sin x$ decreases more sl... |
H: Solving $\frac{x^4}{4}+\frac{1}{y}=c$ for $y$ is giving a wrong answer
I have a simple equation where I want to solve for $y$ but I am getting wrong answer using my steps with some slight sign changes. I know I am making a silly mistake somewhere but its frustrating that I can't find it. Kindly take a look.
NOTE: I... |
H: If $\operatorname{Spec} A＝\operatorname{Spec} B$, then $A$ is isomorphic to $B$
Proposition II.$3.2$ in Hartshorne
Regarding to this question,
I wonder why $\operatorname{Spec} A＝\operatorname{Spec} B$, then $A$ is isomorphic to $B$.
To be more precise,
Let $A,B$ are rings, and suppose the schemes $\operatorname... |
H: If $m\;|\;p^2k$ and $pk \lt m$, then $m$ must be $p^2k$.
Let $p$ be a prime and $1\le k$ be an integer.
If $m\;|\;p^2k$ and $pk \lt m$, how can I prove that $m$ must be $p^2k$?
I looks like there is no other option for $m$ since $p$ is a prime, but I can't write a formal proof.
Thanks for helping.
AI: This is false... |
H: A question about order and prime. (group)
Let $G$ be a group and let $a\in G$ have order $pk$ for some prime $p$, where $1\le k$.
Prove that if there is $x \in G $ with $x^p=a$, then the order of $x$ is $p^2k$.
My attempt:
If $m$ is the order of $x$, then $m\;|\;p^2k$, which means there is an integer $1\le n$ such... |
H: How to prove $|\Omega|^{-1/p}||u||_p\leq |\Omega|^{-1/q}||u||_q $?
How to prove $|\Omega|^{-1/p}||u||_p\leq |\Omega|^{-1/q}||u||_q $?
$u\in L^q(\Omega)$.
I guess using Holder inequality above inequality is true.
But I could not properly arrange term to get required?
Please give me a hint so that I can prove the a... |
H: Given the vertices of a convex polytope, calculate its centroid
I would like to calculate the centroid (center of mass in case of homogeneous materials) of a convex polytope (equivalent of polyhedron in $n-$dimensional space). The vertices are given and I can use a general programming language (without special math... |
H: Is the spectral radius of $DA$ less then the one of $A$?
Is the spectral radius of $DA$ less than the spectral radius of $A$ when $D$ is diagonal where all diagonal entries are nonnegative and less than 1?
This is true when $A$ is normal, since
$$
\rho(DA) \le \|DA\|\le \|D\| \| A\| \le \|A\| = \rho(A)
$$
My guess ... |
H: How do you generate a normal subgroup from relations?
From Rotman's Algebraic Topology:
A group $G$ is defined by generators $X = \{x_k \in K\}$ and relations $\Delta = \{r_j = 1 : j \in J\}$ if $G \cong F / R$, where $F$ is the free group on $X$ and $R$ is the normal subgroup of $F$ generated by $\{r_j : j \in J\... |
H: Calculate $\exp \left[\begin{smallmatrix} 4 & 3 \\ -1 & 2 \end{smallmatrix}\right]$
Given $$ A= \begin{bmatrix} 4 & 3 \\ -1 & 2 \end{bmatrix} $$ how do I calculate $e^A$?
I know the following formula:
If $X$ is a $2 \times 2$ matrix with trace $0$, then
$$e^X = \cos{\sqrt{\det(X)}} I_2 + \dfrac{\sin{\sqrt{\det(... |
H: If collection of subgroups of $G$ form a chain , then $G$ is cyclic?
If $G$ is a group such that for any two subgroups $H,K$ either $H\le K$ or $K\le H$ holds ,then $G$ is cyclic.
I am going to prove this is true for finite groups and false for infinite groups.
Proof (For finite group)
Let $G$ be a finite group sa... |
H: Convergence/divergence of the improper integral $ \intop_{1}^{\infty}\sin\left(x^{p}\right)dx $
I have to determine if the improper integral $ \intop_{1}^{\infty}\sin\left(x^{p}\right)dx $
convergent/divergent for any $ 0<p \in \mathbb{R} $
Here's what Ive done :
We can substitute $ x^{p}=y $
and then we'll get
$ ... |
H: Function series
I'm seeking assistance in correcting my inevitable mistakes and answers to my questions.
$\textbf{Problem}$
a) Let $f_{k}(x) = 1, k\leq x \leq k+1$ and $0$ for all other values of $x$. Show that $(f_{k})$ converges uniformly to $0$ on all intervals $0<t\leq x \leq T < \infty$
b) Is it true that $\li... |
H: Negation of the Universal Subset definition
I am trying to understand the use of quantifiers within the definition of a subset. The definition of a subset is:
$$ A \subseteq B \equiv \forall x(x \in A \rightarrow x \in B) $$
I am confused about, when you negate the statement of a subset, to an existential quantifie... |
H: Does automata theory have many interesting open questions?
I’m studying Automata Theory and I love it. My question is, is there much more research to discover in automata theory, aside from quantum automata? Also, as a side question, are quantum automata the place to be for research in this field?
AI: You will find... |
H: Distance between set and point, confused of partial derivatives.
Let $H = \{(x,y,z)\ \in \mathbb{R}^{3}: x^2+y^2 - z^2 + 4 = 0$
Compute the shortest distance between H and point $p=(2,4,0)$.
I am a bit confused because I tried a direct approach.
$$ x^2+y^2 + 4 = z^2$$
Let $D(H,p) = \sqrt{(2-x)^{2}+(4-y)^{2} + x^{2... |
H: Does collections of function on $\mathbb{R}_+$ means domain or range?
I do not fully understand a hint from Erhan Cinlar's Probability and Stochastics (its on page 67). This is the hint:
Let $F$ be the collection of all function $f$ on $\bar{\mathbb{R}}_+$ having the form $f(x)=\sum\limits_{i=1}^{n}c_ie^{-r_ix}$ f... |
H: Invariance and symmetric groups
My (naïve) idea of symmetry is that it requires some form of invariance under transformation, but I struggle to see how this survives the idea of the symmetric group $ S_n$.
Beyond the bijective property what, if anything, is invariant in a permutation? And if the answer is nothing, ... |
H: What is the indefinite integral of $\int \left(-\frac{1}{5x}\right) dx$?
Why does $\int \left(-\frac{1}{5x}\right) dx = -\frac{1}{5}\ln(x)+c$ and not $-\frac{1}{5}\ln(5x)+c$ instead?
When I differentiate both solutions I get the same answer, $-\frac{1}{5x}$.
Can anyone give me an explanation as to why? Thank you.
A... |
H: If $x$ and $y$ are integer variables that have value $0$ or $1$, then what does the expression $x + y - xy$ mean?
Answer Options:
$1.$ Logical AND
$2.$ Logical OR
$3.$ Nothing, it makes no sense
$4.$ Logical implies
I'm stuck between Logical OR and "Nothing, it makes no sense" because when I create a truth table fo... |
H: Let $T:\mathbb{R}^n\to\mathbb{R}^m$ be a linear transformation. Prove that the range of T is a subspace of $\mathbb{R}^m$ with dimension at most $n$
I'm having some trouble with the second part of this question. I tried using the result that any $n+1$ vectors in $\mathbb{R}^n$ are linearly dependent, but no luck he... |
H: A fiber bundle with trivial structure group is (isomorphic) to a trivial bundle.
How do I show that a fiber bundle with trivial structure group is (isomorphic to) a trivial bundle?
I say let $\phi$ and $\phi'$ be two charts over $U$. Then $\phi=\phi'$ since there is a continuous function $\theta_{\phi,\phi'}:U\to \... |
H: Relationship Between Moment-Generating Functions and Laplace Transforms
According to Wikipedia, the moment-generating function $M_X (t)$ of a probability distribution $f_X (x)$ is given by $M_X (t) = \int_{-\infty}^\infty e^{tx}f_X (x)\ dx$.
Is $t$ time? If so, why does it appear in the output of this transform r... |
H: Limit of $\lim _{\left(x,y\right)\to \left(0,0\right)}\left(\left(xy\right)\ln \left(x^2+y^2\right)\right)$
I want to calculate limit of $\lim _{\left(x,y\right)\to \left(0,0\right)}\left(\left(xy\right)\ln \left(x^2+y^2\right)\right)$ using Squeeze theorem or using definition of limit. please help
AI: HINT:
Note ... |
H: Minimum distance of intersection between two spheres along a specified vector
I have two spheres $(S_1, S_2)$ within a Cartesian space defined by their centroids $(p_1,p_2)$ and their radii $(r_1,r_2)$. $p_1$ is located at the origin of the coordinate frame such that $x_1^2 + y_1^2 + z_1^2 = r_1^2$. $p_2$ is locate... |
H: How $\{am + pn : m, n \in \mathbb{Z}\}=\langle 1 \rangle$?
I don't understand how $\{am + pn : m, n \in \mathbb{Z}\}$ is equal $\langle 1 \rangle$, doesn't $\langle 1 \rangle$ contains all integer of $\mathbb{Z}$? the passage I got it from -
Prime ideal property. If $p$ is prime and the ideal $\langle p \rangle$ c... |
H: Let $X$ be a Banach space and $E$ a sublinear subspace . Show there exists a surjective isometry $\phi : E^* \rightarrow \overline{ E}^* $
Let $X$ be a Banach space and $E$ a sublinear subspace of $X$ . Show there exists a surjective isometry $\phi : E^* \rightarrow \overline{ E}^* $
I think it could be a derived ... |
H: Question on showing invertibility of function $\Phi: \mathcal{L}(V,W) \to M_{m \times n}(F)$ in regards to establishing an isomorphism.
My question comes from a proof of establishing that the function $\Phi: \mathcal{L}(V,W) \to M_{m\times n}(F)$ is an isomorphism. The statement of the theorem comes from Linear Alg... |
H: $G= \langle a, b : a^{7} = b^{3} = 1,\ b^{-1}ab = a^{2} \rangle$ and commutator group
Consider the $a$ and $b$ the following permutation in $S_{7}$:
$$a = (1\ 2\ 3\ 4\ 5\ 6\ 7 ),\ b = (2\ 3\ 5)(4\ 7\ 6)$$
Consider the group $G = \langle a, b \rangle$. I know that $a^{7} = b^{3} = 1$ and $b^{-1}ab = a^{2}$. Moreover... |
H: Proving $\frac{1}{2^n}\sum_{z\in\{0,1\}^n} (-1)^{z\cdot (x\oplus y)}=\delta_{xy}$, where $x\oplus y$ is the bitwise sum
In quantum algorithms I often find this identity: $$\frac{1}{2^n}\sum_{z\in\{0,1\}^n} (-1)^{z\cdot (x\oplus y)}=\delta_{xy}$$ where $x\oplus y$ is the bitwise sum.
I am not able to prove in genera... |
H: Is there a formula $A(x_1,\ldots,x_k)$ with k variables that's true for a truth assignment o exactly when a prime number of variables are true under o
Answer Options:
Yes, such a formula exists in principle, but currently, no one knows how to construct it
Yes, one can construct such a formula, though it would get ... |
H: How do I find the local max And min
I'm stuck on this question and have no idea how to find the local max and min of $$\int_a^x \frac{t^2+2t-24}{1+\cos^{2}t}, dt$$ I'm not sure where to start with this one since the integral doesn't have an exact boundary.
AI: You can use the fundamental theorem of calculus which s... |
H: For which values the matrix $ B = \Big(\begin{matrix} A & -A\\ -A & \alpha A \end{matrix}\Big)$ results positive definite (strict)?
I'm trying to solve the next problem where given $A \in \mathbb{R}^{n \times n}$ symmetric positive definite I have to find the values for $\alpha \in \mathbb{R}$ such that this matrix... |
H: Show that $P(A\mid B) > P(A\mid B^{c}) \implies P(B\mid A) > P(B\mid A^{c})$
My initial thought is to start from $P(A\mid B)$ and $P(A\mid B^{c})$ and transform them to expressions involving $P(B\mid A)$ and $P(B\mid A^{c})$ respectively.
$P(A\mid B) = \dfrac{P(A)P(B\mid A)}{P(B)}$
$P(A\mid B^{c}) = \dfrac{P(A) - P... |
H: Construct a set of the real line such that nth derived set is empty.
I am self studying from the book Elementary Real and Complex Analysis but need some help with the following exercise.
Let $A'$ denote the set of all limit points of a given subset $A$ of a metric space $M$ and let $A^{(n)}=(A^{(n-1)})'$. Given an... |
H: What is the maximum possible value of $y$
Let $x$ be a prime number and $y$ be an integer.If following expression is an integer then find maximum possible value of$y$.
$$\frac{1}{xy}+\frac{2}{xy}+\frac{3}{xy}+\frac{4}{xy} + \cdots + \frac{54}{xy}+\frac{55}{xy}$$
My answer:$$\frac{1}{xy}+\frac{2}{xy}+\frac{3}{xy}+\f... |
H: Is there a transformation matrix for given mapping of vectors
I have vectors $v_1, v_2, v_3$ in $\Bbb{R}^3$ and vectors $w_1, w_2, w_3$ in $\Bbb{R}^4$. There are three mappings from $v_1$ to $w_1$, from $v_2$ to $w_2$, from $v_3$ to $w_3$.
The question is, is there a linear map $\phi$ that maps those vectors in tha... |
H: Trouble proving the equality when asked to compute the operator norm $\phi : \ell^{2} \to \mathbb R$ where $\phi(x)=\sum \frac{x_{n}}{n}$
Compute the operator norm $\phi : \ell^{2} \to \mathbb R$ where $\phi(x)=\sum\limits_{n \in \mathbb N} \frac{x_{n}}{n}$
My proof so far:
$\lvert \phi(x)\rvert=\lvert\sum\limits_{... |
H: Solving $\cos(z) = \frac{5}{2}$
I'm given $$\cos(z) = \frac{5}{2}$$ and I'm trying to solve for $z$ but I keep going in circles. I know $\cos z = 5/2 = 1/2(e^{iz}+e^{-iz})$ so then $e^{iz}+e^{-iz} = 5$ but then I'm stuck
AI: Taking $t=e^{iz}$ we get
$$t+\frac{1}{t}=5 \implies t^2-5t+1=0 \implies t_{1,2} = \frac{5 \... |
H: Spectral norm, eigenvalues range
I stumbled upon a property in solutions of some exercises which stated that if a hessian of a possibly non-convex function f(x) is bounded in spectral norm then its eigenvalues lie in the interval.
$$ ||\nabla^2f(x)||_2 \leq L $$
$$ eigenvalues \in [-L, L]$$
I fail to understand or ... |
H: Example of a ring with a unique two sided maximal ideal which is not a local ring (that is it has more than one left or right maximal ideals).
Let $R$ be a ring (possibly non-commutative).
Definition $R$ is called a local ring if it has a unique left(and equivalently right) maximal ideal.
I am looking for an exampl... |
H: Derivative of a matrix with respect to elements of another matrix
I am trying to take the "derivative" of a matrix $M = ABB^TA^T$ with respect to the inner matrix $B$, where $A$ is $n\times n$ and $B$ is $n\times m$ with $m \leq n$.
In other words, looking for a closed form solution to $\frac{\partial M_{ab}}{\part... |
H: What is the probability distribution of the maximum cycle length in a permutation game?
There is a "classic" counterintuitive scenario, in which you have $N$ boxes, $N$ players. Player $i$ has a dollar bill tagged with the number $i$. Each player places their dollar bill into a box at random, where each box is tagg... |
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