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H: Can I prove that $A \Rightarrow \neg B$ is false given $A\iff(B∨C)$?
Let $A$, $B$, $C$ be statements. Given $A\iff(B∨C)$.
I am trying to prove (if possible) that the implication $A \Rightarrow \neg B$ is false.
The motivation for me to do this is that I am thinking of eliminating the ‘‘redundant’’ cases in an impli... |
H: Find the real numbers given their product
We have four real numbers $a,b,c,d$ and their six products should satisfy $\{ab,ac,ad,bc,bd,cd\}=\{2,2.4,3,4,5,6\}$. How do we find them?
The sequence of products is not necessarily in order. For example we don't know if $ab=2$ or $ac=2.4$.
AI: $a= \sqrt{1.2}$, $b= \sqrt{10... |
H: Why is $\frac{1}{\pi ^{1/4} \sqrt{a}}e^{-x^2/(2a^2)}$ not a gaussian function?
In an quantum mechanics exercise, we were asked to find the ground-state wavefunction of a perturbed harmonic system. The resulting wave-function is $$\psi_0(x) = \frac{1}{\pi ^{1/4} \sqrt{a}}e^{-x^2/(2a^2)}$$
After that, there is a true... |
H: Formula derivation of the Second derivative
I'm trying to prove the below equation:
The given function $f$ is twice differentiable. And the given formula is
$f''(x) = 2\lim_{h \rightarrow 0} \frac{f(x+h) -f(x) - f'(x)h } { h^2}$
I know this can be proved by l'Hospital's rule.
Since the numerator and denominator ... |
H: Prove that $\|R\|_2 = \|A\|_2^{1/2}$ where $A=R^* R$ is a Cholesky factorization of $A$
Prove that $\|R\|_2 = \|A\|_2^{1/2}$ where $A = R^* R$ is a Cholesky factorization of $A$.
In my book it says that I should use the Singular Value Decomposition.
I have that
$\rho(A)=\sqrt{\rho(A^*A)}=\sqrt{\rho(A*A)}=\sqrt{\rho... |
H: Showing properties of a specific maximization problem, as well as finding the maximum.
Let there be $p_1,p_2,..,p_n,q_1,q_2,...,q_n$ with $\sum_{i=1}^n p_i = 1 = \sum_{i=1}^nq_i$
For $M :=\{ x\in (0,\infty)^n: \sum_{i=1}^nq_ix_i = a \}$
With the following Maximization Problem.
$$(*) \sup \bigg\{\sum_{i=1}^np_i\ln (... |
H: If $X$, $Y$ and $Z$ are mutually independent random variables, is it true that $X+Y (X \cdot Y,X/Y,\dots)$ and $Z$ are independent?
I just wonder that given that RVs $X, Y, Z$ are mutually independent, how should I quickly determine whether a combination of $X$ and $Y$, e.g. $X+Y, X\cdot Y, X/Y, X^Y$, is independen... |
H: References on (more or less) explicit calculations of probability distributions of nonlinear transformations of random variables
Premise. After a former question and related answer, I searched for references on the calculation of the probability distributions of nonlinear functions of (one or more) random variables... |
H: Is $G^2$ necessarily a subgroup of $G$?
Let $G$ be a group and $H=\{g^2 : g\in G\}$ then which of the following is/are true?
$(1)H $ is always a subgroup of $G$
$(2)H$ may not be a subgroup of $G$
$(3)$ If $H$ is a subgroup of $G$, then it must be normal in $G$
$(4)H$ is a normal subgroup of $G$ only if $G$ ia abel... |
H: Closure of a subgroup is again a subgroup
Let $G$ be a topological group and $H$ a subgroup. Then $\overline{H}$ is again a subgroup.
Attempt: Let $x,y \in \overline{H}$. Choose nets $\{x_\alpha\}_{\alpha \in I}$ and $\{y_\beta\}_{\beta \in J}$ with $x_\alpha \to x, y_\beta \to y$ and these nets are in $H$. Then we... |
H: Finding the equation of the normal to the parabola $y^2=4x$ that passes through $(9,6)$
Let $L$ be a normal to the parabola $y^2 = 4x$. If $L$ passes through the point $(9, 6)$, then $L$ is given by
(A) $\;y − x + 3 = 0$
(B) $\;y + 3x − 33 = 0$
(C) $\;y + x − 15 = 0$
(D) $\;y − 2x + 12 = 0$
My attempt: Let $(h,k)... |
H: What is meant by $f : [a,b] \times D \rightarrow R^m$
I am working through Nonlinear Systems by Khalil.
Lemma 3.1 states the following
Let $f : [a,b] \times D \rightarrow R^m$ be continuous for some domain $D \subset R^n$. Suppose that $[\partial f/\partial x]$ exists and is continuous on $[a,b]\times D$. If, for a... |
H: Calculating Residue of $\frac{1}{\sin\left(\frac{\pi}{z}\right)}$
How do I calculate the residue of $f(z)=\frac{1}{\sin\left(\frac{\pi}{z}\right)}$ at the points $z=\frac{1}{n}$, $n\in\mathbb{Z}\setminus\{0\}$?
I know that the set $\{\frac{1}{n}:n\in\mathbb{Z}\setminus\{0\}\}$ is the set of isolated singularities o... |
H: Continuous Functions with countably many jump discontinuitites
Let $X$ be the set of functions from $\mathbb{R}$ to $\mathbb{R}$ which can be written as
$$
f = \sum_{i=1}^{\infty} f_i I_{[a_i,b_i]},
$$
where $a_i<b_i$, $f_i$ is continuous, but $f$ need not be continuous at $a_i$ (or $b_i$). What can be said about ... |
H: Implicit function theorem on vector valued function
Above is implicit function theorem, and here is a special case
In second case, $f(y)=(x,z)\in\mathbb{R}^2$, also $F\in\mathbb{R}^2$, how do we find $\partial_y f$ ? I know $\partial_y f=-\frac{\partial_y F}{\partial_y F}$,
but this only apply if $F\in\mathbb... |
H: when $a$ and $b$ are relatively primes, how is $ax - by= 1$ always possible?
If $a$ and $b$ are relatively primes, with any number of $x$ and $y$, you could always find a set of $x$ and $y$ which makes $ax-by=1$
How is it possible?
AI: Given $a,b$ with $\gcd(a,b)=1$, let $c$ be the smallest positive integer of th... |
H: A Square and two Quarter-Circles problem
Square $ABCD$ has side equal to $a$. Points $A$ and $D$ are centers of two Quarter-Circles (see image below), which intersect at point K. Find the area defined by side $CD$ and arcs $KC$ and $KD$.
Here's what I did: The darkened area can be found by Substracting area of f... |
H: What is wrong in the following method of obtaining the Maclaurin series of $\frac{2x}{e^{2x}-1}$?
$\frac{2x}{e^{2x}-1} = -2x(1-e^{2x})^{-1}$
We can obtain the binomial series expansion of $(1-e^{2x})^{-1}$:
$(1-e^{2x})^{-1} = \sum_0^\infty\begin{pmatrix}-1\\n\end{pmatrix}(-e^{2x})^{n} = \sum_0^\infty(e^{2x})^{n} = ... |
H: How is $(2+i\sqrt{2}) \cdot (2-i\sqrt{2})$ calculated?
What is $(2+i\sqrt{2}) \cdot (2-i\sqrt{2})$ ?
Answer:
(a) $4$
(b) $6$
(c) $8$
(d) $10$
(e) $12$
I calculate like this:
$(2+i\sqrt2),(2-i\sqrt2)\\(2+1.41421i),\;(2-1.41421i)\\3.41421i,\;0.58579i\\3.41421i+0.58579i\\4i$
Therefore, the answer is $4$.
But the corre... |
H: Countable sets bijective function
Suppose we have the infinite set $\{\ldots, - 1,-2,0,1,2,\ldots\}$
If the set is countable then there will be a bijective function which maps elements of this set to the set containing all natural numbers.
However, I can't find any such function. Does this prove the set is uncounta... |
H: From generalized eigenvector to Jordan form
I can't figure out the following part of Chen's Linear Systems book. How does he "readily obtain" $Av_2=v_1+\lambda v_2$?
AI: Note that the fact that we have a "chain of generalized eigenvectors" implies that $(A - \lambda I)v_i = v_{i-1}$. So, we have
$$
(A - \lambda I)v... |
H: Recursive to explicit form involving Fibonacci
I have a recursive formula for a sequence O: $ O_n = O_{n-1} + O_{n-2} + F_{n-1}$ where $F_n$ is the n-th Fibonacci number, $O_1 = 1$ and $O_2 = 2$.
After playing around with it, I found a new formula that might be easier to convert to the form I search: $ O_n = F_{n-3... |
H: Totalizing a complex in triagulated category
I am self-studying homotopy theory and trying to understand a proof in this paper on page 218
Let
$$ ... \to X_n \xrightarrow{{f_n}} X_{n-1} \xrightarrow{{f_{n-1}}}
... \xrightarrow{{f_2}} X_1 \to 0$$
be a sequence in trianguated category $\mathcal{I}$.
(sequence or mo... |
H: Number of ways to pick at least three of a kind in 5-card poker - what's wrong with C(49,2) for the last two cards?
In finding the number of ways to get a 5-card poker hand that contains at least three of a kind, what's wrong with the following
$$n=\binom{13}{1}\binom{4}{3}
\binom{49}{2}$$
So, we have 13 numbers to... |
H: Exercise 24(a) Chapter 3 Baby Rudin Proof Verification
Let $X$ be a metric space.
(a) Call two Cauchy sequences $\left\{ p_n \right\}$, $\left\{ q_n \right\}$ in $X$ equivalent if $$ \lim_{n \to \infty} d \left( p_n, q_n \right) = 0.$$ Prove that this is an equivalence relation.
Can someone let me know if my proo... |
H: Completion product is product of completions
Let $X,Y$ be metric spaces and $\tilde{X}, \tilde{Y}$ their completions. Is it true that $\tilde{X} \times \tilde{Y}$ is the completion of $X \times Y$? Here both these products have the product metric/topology. I guess this can be proven using the universal property of ... |
H: Let $a,b \in \mathbb{Z}$ and let $d = gcd(a,b)$. Show that $\{ ka + lb: k,l \in \mathbb{Z}\} = \{md : m \in \mathbb{Z} \}$
I know that given $d = gcd(a,b)$ that this also means $xa + yb = d$. Using this we get (showing from left to right side)
$$xa + yb = d$$ $$m(xa + yb) = md$$ $$xma + ymb = md$$
Now I am unsure ... |
H: How is the $ inf $ defined in a metric space?
In my lecture notes:
definition:
In a metric space $(X,d)$ the distance $d(p,E)$ from a point $p \in X$ and a subset $E\subseteq X$ is defined as: $d(p,E)=inf\{d(p,x)| x \in E\}$
Proposition: Let $(X,d)$ be a metric space and let $E\subseteq X$,
$p\in Cl(E)$ if and on... |
H: Math notation for modulo
I have a little trouble understanding how to write a mathematically notation for r = x%n. How should I write this in math notation if I want to get the remainder value, after dividing by $n$? $r$ is also not just an integer in this case, just the rest of $x$ (double) divided by $n$ (integer... |
H: Find the Sum of the Series $\sum_{n=0}^\infty \frac{3n^2 -1}{(n+1)!}$
Find the Sum of the Series $$\sum_{n=0}^\infty \frac{3n^2 -1}{(n+1)!}$$ I separated the Series in to the sum of $\sum_{n=0}^\infty \frac{3n^2}{(n+1)!}$ and $\sum_{n=0}^\infty \frac{-1}{(n+1)!}$. First i proceeded to find the sum of the Series $\s... |
H: Simplify $a.b+c.d$
Suppose that, $(S, +, \cdot)$ is a semiring, where the operations are defined as $x\cdot y=min(x, y)$ and $x + y=max(x, y)$.
Can we further simply the expression $a\cdot b+c\cdot d$?, where $a, b, c, d\in S$ and $a\leq d$ and $c\leq b$; also note that $(S, \leq)$ is a partially ordered set.
I c... |
H: Prove that $d(x,z) \leq d(x,y) + d(y,z)$ in $\textbf{R}^2$
You are potentially able to prove $d(x,z) \leq d(x,y) + d(y,z)$ in $\textbf{R}^2$ using the relation that $d(x + y, 0) \leq d(x, 0) + d(y, 0)$ where:
$0 = 0$-vector
$d(x, y) = \sqrt{(x_1 - y_1)^2 + (x_2 - y_2)^2}$ - the distance formula
I was able to find a... |
H: Decomposition of a locally free sheaf as tensor product of sheaves
The setting is as follows: Let $X$ be an algebraic surface, $\mathcal{F}$ a locally free sheaf of rank 2 on $X$ contained in $\Omega^1_X$, and $D$ a divisor on $X$ such that $\mathcal{F}\otimes\mathcal{O}_X(-D)$ has a non-zero global section. Then t... |
H: Prove that commuting matrices over an algebraically closed field are simultaneously triangularizable.
Given an algebraically closed field $\mathbb K$ and matrices $A, B \in \mathbb K^{n \times n}$ such that $A B = B A$, show that $A$ and $B$ are simultaneously triangularizable, i.e., show that there exists a matrix... |
H: Finding an element whose minimal polynomial is an Eisenstein polynomial
If we have an extension of number fields $L/K$ and $Q$ is a prime ideal of $L$ lying over $P$, do we know for certain if there exists an element $\pi \in Q \setminus Q^2$ whose minimal polynomial over $K$ is an Eisenstein polynomial?
If you don... |
H: How does $0 < 1/j$ and $1/k < 1/N$ imply that $| 1/j - 1/k | \leq 1/N$?
Synopsis
In Tao's Analysis 1, in his proof that the sequence $a_1,a_2, a_3, \dots$ defined by $a_n := 1/n$ is a Cauchy sequence, there is an inequality that I don't really feel comfortable with. I'll highlight this inequality below the proof an... |
H: Fourier transform of $\sqrt{f(t)}$
If the Fourier transform of $f(t)$ is $F(f)$, can you conclude that the Fourier transform of $\sqrt{f(t)}$ is $\sqrt{F(f)}$?
Probably this is not always the case, but what are the cases in which this is true?
AI: Counterexample: Let $f=\chi_{[-1,1]}.$ Then $F(f)(x)=2(\sin x)/x$. B... |
H: Find the mistake - 4th order homogeneous ODE with constant coefficients
Solve the following $4$th order ODE:
$\varphi''''+\varphi=0$
I've tried the standard approach and computed the zeros of $x^4+x$, which consist of $0,\cos(\frac{\pi}{3})+i\sin(\frac{\pi}{3}),-1,\cos(\frac{\pi}{3})-i\sin(\frac{\pi}{3})$.
This gi... |
H: A quiz question based on subspace topology i am unable to solve
While trying sample quiz papers I am unable to solve this particular question in topology.
It's image:
Questions:
(1) why 4th option is false. I think B can be deformed to a unit circle. So, it must hold.
I am unable to reason why 3rd option must hold ... |
H: Baby Rudin 2.17 Perfect Set?
I'm confused about the solution to 2.17 in Baby Rudin. Let be the set of all ∈[0,1] whose decimal expansion contains only the digits 4 and 7. Is countable? Is dense in [0,1]? Is compact? Is perfect?
The solution says that E is perfect. However, I don't see how E has a single limit ... |
H: How many maximum binary pairs are possible in a Poset?
Answer : $n(n+1)/2$
Maximum binary pairs is possible iff the poset is a toset.
A toset is reflexive so I don't have control over self loops like $(1,1),(2,2).$ They have to be there.
My approach to this was :
If we consider $n$ elements in a set, we have $n^2-n... |
H: Safe packing Constraint satisfaction problem - is it optimal?
Problem:
You need to pack several items into your shopping bag without squashing anything. The items are to be placed one on top of the other. Each item has a weight and a strength, defined as the maximum weight that can be placed above that item without... |
H: How to show that $|z_1+z_2|+|z_1-z_2| = |z_1+\sqrt{z^2_1-z^2_2} |+|z_1-\sqrt{z^2_1-z^2_2} | $?
How to show that $$|z_1+z_2|+|z_1-z_2| = |z_1+\sqrt{z^2_1-z^2_2} |+|z_1-\sqrt{z^2_1-z^2_2} | $$
Is this question correct?
I tried my best to prove it
But got nowhere near, Any help
AI: Let $a=z_1+z_2$ and $b=z_1-z_2$, the... |
H: System of equations of non-relativistic scattering in the laboratory system
Considering the system of equations of non-relativistic scattering in the laboratory system:
$$\begin{cases} \dfrac{1}{2} m_{1} v_{1}^{2} &=\dfrac{1}{2} m_{1} v_{2}^{2}+T_2 \,,\\
m_{1} v_{1} &=m_{1} v_2 \cos \psi_{1}+p_{2} \cos \psi_{2} \... |
H: The sum of these 9! determinants is?
(image is attached for those who think I have changed the statement of the question while copying from the book)
Chose any 9 distinct integers. These 9 integers can be arranged to from 9! determinants each of order 3. The sum of these 9! determinants is?
My approach
For any Δ,... |
H: Are all prime ideals of $\mathbb C[x,y]/(y^2-x^3+x)$ maximal?
In fact I'm trying to prove that $\mathbb C[x,y]/(y^2-x^3+x)$ is a Dedekind domain. Till now I believe I was able to show that it is a Noetherian integral domain (easy) which is an integrally closed domain. If I prove that all prime ideals of $\mathbb C[... |
H: Absolute convergence of a series with integral inside
Im having trouble solving the absolute convergence of this series. None of the common tests seem to work and so far couldn´t find any function to compare it to:
$\sum_{n=1}^\infty (-1)^{n} \int_{n}^{n+1}\frac{e^{-x}}{x}dx$
I would appreciate any suggestions.
AI:... |
H: If continuous images of $X$ are closed in every $Y$, is $X$ a compact space?
Suppose $X$ is a topological space. We have the following criterion for compactness:
Theorem. $X$ is compact if and only if for every space $Y$, the second projection $\pi_2: X\times Y \to Y$ is a closed map.
This property is known as bein... |
H: Evaluating an Infinite Limit that Wolfram doesn't like!
Evaluate
$$\lim _{n \rightarrow \infty} \ln (n +1) n(n+1)^{-n/(n+1)}- \ln (n)n^{1/n}. $$
According to Wolfram, this is equivalent to $0$, yet everything I've tried (like log-exponent) doesn't lead me to the answer. Could someone show why this is true? And why... |
H: the local extrema and the saddle point - i could not find the critical points
$$y=(e^y)-(ye^x)$$
I want to find the local extrema and the saddle point, what should I do?
i could not find the critical points.
what can I do?
AI: What one can do in a situation like this is to try and prove that no critical points exis... |
H: Is a Riemann integral of a real-valued function a number or a function?
For example, if we define $F(x)=\int^x_a f(t)dt$, where $f$ is Riemann integrable, then $F(x)$ is a function. Or for a 2 variables real-valued integrable function $f(x, y)$, $G(x)=\int^b_a f(x,y)dy$, then $G(x)$ is a function. But for $\int_a^b... |
H: Steps for computing Tor$(\mathbb{Z}, \mathbb{Z}\times\mathbb{Z})$
I'm reviewing algebraic topology, in particular the Kunneth Formula. I can't find online or in my book (by Hatcher) an explanation for how to calculate $\mbox{Tor}(G,H)$ for any two groups. My understanding is that Tor measures the failure of a ses t... |
H: Find the average value of $x^2 - y^2 + 2y$ over the circle $|z - 5 + 2i| = 3$.
Find the average value of $x^2 - y^2 + 2y$ over the circle $|z - 5 + 2i| = 3$.
Could someone please explain how to do this. I keep getting an answer of 17, but my professor says that is incorrect.
AI: First, recognize that the circle has... |
H: Show that these two diffeomorphisms cannot exist simultaneously
Let $d\in\mathbb N$, $x\in M\subseteq\mathbb R^d$ and $\psi^{(i)}:\Omega_i\to\psi^{(i)}(\Omega_i)$ be a diffeomorphism with $x\in\Omega_i$, $$\psi^{(1)}(M\cap\Omega_1)=\psi^{(1)}(\Omega_1)\cap(\mathbb R^k\times\{0\})\tag1,$$ $$\psi^{(2)}(M\cap\Omega_2)... |
H: Show a locally bounded Lipschitz function space is compact for sup metric
Just show $\Omega$ = {$f\in\text{Lip}(\alpha,M):|f(u)|\leq M$} is totally bounded and complete.
AI: Just show $\Omega$ = {$f\in\text{Lip}(\alpha,M):|f(u)|\leq M$} is totally bounded and complete.
Note that $\Omega$ is equicontinuous. Given a... |
H: Converting function containing summation into function without it.
Disclaimer: terminology and syntax may be incorrect as I do not use them often, please excuse the errors.
Function
The function I am trying to convert:
$ f(a,b,c) = \displaystyle\sum_{n=0}^{a} \left( \frac{b}{c} \right)^n$
$ a >= 0$
$ b,c > 0$
$ c >... |
H: Solving Modular System with 2 different Moduli
Is there any way to solve for $a$ and $b$ in:
$$ a*b \equiv s_0 \mod r_0 $$
$$ a - b - a*b \equiv s_1 \mod ( r_0 - 1) $$
I have the values of $r_0$, $s_0$, $s_1$ and would like to find the values for $a$, $b$? The $*$ is just normal integer mu... |
H: Let $G'\triangleleft G$ be a normal subgroup and $K
Let $G'\triangleleft G$ be a normal subgroup and $K<G$ a subgroup. Is there any relation between the normalizers $N_{G'}(G'\cap K)$ and $N_{G}(K)$?
I'm working with topological groups and I would like to know if $\dim N_{G'}(G\cap K)\leq \dim N_G(K)$.
AI: Short an... |
H: the equation $x^2-y^2 =a^2$ changes to the form $xy=c^2$ if the co-ordinate axes rotates through an angle (keeping origin fixed)
the equation $x^2-y^2 =a^2$ changes to the form $xy=c^2$ if the co-ordinate axes rotates through an angle
(keeping origins fixed) is
a) $ \frac \pi 2 $
b) $ - \frac \pi 2 $
c) $ \frac \pi... |
H: If $\forall n \in \mathbb Z_{\ge0} \ $ and $\forall x \in \mathbb R$, we know that $\big|f^{(n)}(x)\big|\le \big|p(x)\big|$, then $f=0$.
If $p(x)$ is an odd degree polynomial such as $\forall n \in \mathbb Z_{\geq 0}$ and $\forall x \in \mathbb R$ we know that $$\big|f^{(n)}(x)\big|\le \big|p(x)\big|\,.$$
I nee... |
H: Struggling To Follow How to Convert expression to Logarithmic Form | Binary Search Problem
I am reading "Problem Solving with Algorithms and Data Structures using Python" and the author is currently explaining the relation between comparisons and the Approximate Number of Items Left in an Ordered List.
I am struggl... |
H: Role of Tautologies in logic
I apologize if the title is inadequate. I am reading Loomis and Sternberg's Advanced Calculus textbook. After introducing the notation of a quantification and defining a tautology, they state:
Indeed, any valid principle of reasoning that does not involve quantifiers must be expressed ... |
H: Prove that $a_{n}$ converges for $a_{1} \ge 2$ and $a_{n+1} = 1+ \sqrt{a_{n}-1}$ and find its limit
Prove that $a_{n}$ converges for $a_{1} \ge 2$ and $a_{n+1} = 1+ \sqrt{a_{n}-1}$ and find it's limit. First of all, what i did is to find the first four terms of the sequence.
$a_{2} = 1+ \sqrt{a_{1}-1} \ge 2$
$a_{3}... |
H: If $f(x) \leq g(x)$ is it true that $\sup_I(f(x))\leq \sup_I(g(x))$?
If $f(x) \leq g(x)$ is it true that $\sup_I(f(x))\leq \sup_I(g(x))$ ?
I haven't been able to find this anywhere.I need to prove if this holds:
$\sup_I(e^{g(x)}|x^k|) \leq e$ with $x \in (-1,1)$ and $|g(x)|\in (0,x)$ , $k \in \mathbb{N}$
My tr... |
H: Moduli of complex numbers
In $\mathbb{C}$, a set of complex numbers, there are z and w, such that $|z|=|w|$.
How can I show that: $|z+w|^2+|z-w|^2=4|z|^2$?
I'm trying to do:
$|z+w|^2+|z-w|^2=\sqrt{(z+w)^2}+\sqrt{(z-w)^2}$
The problem is that solving the square of the binomial I cannot cut the elements inside the ro... |
H: If $\int_{-1}^1 fg = 0$ for all even functions $f$, is $g$ necessarily odd?
Suppose for a fixed continuous function $g$, all even continuous real-valued functions $f$ satisfy $\int_{-1}^1 fg = 0$, is it true that $g$ is odd on $[-1,1]$?
My intuition is telling me that this is correct, as I have not found any counte... |
H: proving that $D = \{(x,y) \in \mathbb R ^2: x^2 + y^2 < 1\}$ is opened
In a general topology exercise I have to prove the following:
Prove that the disk $D = \{(x,y) \in \mathbb R ^2: x^2 + y^2 < 1\}$ is opened in the euclidean topology.
This reminded me in how in multi-variable calculus we approximates the regio... |
H: Condition on $(x_n)$ equivalent to $\lim x_n \in U$
Let $(x_n)_{n=1}^\infty \subseteq \mathbb{R}$ be a Cauchy sequence of real numbers. Let $U \subseteq R$ be an open set - for simplicity, we can suppose $U$ is an open interval $(a,b)$. Is there a condition only involving the sequence $(x_n)$ that is equivalent to ... |
H: If $L=\lim_{n \to \infty} \sum_{i=1}^{n-1} {\left(\frac{i}{n}\right)}^{2n}$ what is $\lfloor \frac{1}{L} \rfloor$
If $$L=\lim_{n \to \infty} \sum_{i=1}^{n-1} {\left(\frac{i}{n}\right)}^{2n}$$What is $\lfloor \frac{1}{L} \rfloor$
I really am confuse. Can this be converted into Riemann sum? If not what do I do. Answe... |
H: Evaluating $\int_{-a}^a x^{2n+1}\mathrm{d}x$ for all non-negative integers $n$ simultaneously
My assumption would be
$$\int_{-a}^a x\ dx=0$$
Am I on the right track here? Also, for indefinite integrals
$$\int (f)x\ dx$$
would this be correct as well?
Background
My professor raised this question in his lecture and I... |
H: Why $i^{-3}$ equals to $i$?
What is $i^{-3}$?
Select one:
(a). 0
(b). i
(c). -1
(d). 1
(e). -i
I calculated like this:
$i^{-3}=\frac1{i^3}=\frac1{i^2\times i}=\frac1{-1\times i}=-\frac1i$
And therefore, $i$ is $-\frac1i$.
But the correct answer is: $i$.
How does it calculate?
AI: Because $\frac{-1}{i}=\frac{i^{2}}{... |
H: Primes And Quadratic Residues
Below is the question:
Let $p$ be a prime. Prove there exists an integer $1\le x\le9$ such that $x$ and $x+1$ are quadratic residues mod $p$.
Please include a proof
AI: If $2$ is a quadratic residue, then $1$ and $2$ are consecutive quadratic residues.
If $5$ is a quadratic residue, t... |
H: A confusion about the proof: every open set in $R^1$ is the union of an at most countable collection of disjoint segments.
I have met two types of solutions about this question, one is by the partition of the equivalence relation, and I understand that, and the other is the following:
I don't understand why the d... |
H: The solutions of $y'' = 2$ don't form a subspace
"The solutions of $y'' = 2$ don't form a subspace - the right side $b=2$
is not zero."
This is a quote on page 172 of Intro to Linear Algebra by Strang. What does this mean? Can someone explain how a differential equation relates to the idea of subspaces, and then ... |
H: Help finding the length of a line inside a rectangle.
I need the formula to calculate the length of the red line in the image attached. I always have the point that starts the line and the angle is always 45° but I don't know how to calculate the length.
check the shape here
I apologize that my description is not e... |
H: Questions about contraction
Let $X$ be a metric space and $f:X\to X$. What's true and what's false?
a. If $f$ is bijective and has a unique fixed point, then $f^{-1}:X \to X$ also has a unique fixed point.
b. If $f$ is bijective, then $f$ is a contraction iff $f^{-1}$ is a contraction.
c. $f:\mathbb {R}^2\to \mathb... |
H: Questions about distributing $k$ objects to $n$ recipients
Rule: Distributions of $k$ objects to $n$ recipients can be done in $n^k$ ways with no restrictions and $n!$ ways when each recipient receives exactly one object.
Obvious Examples:
In how many ways can we distribute $70$ computers to $6$ schools s.t. no tw... |
H: Why is the additional time added to faster expression in this uniform motion problem?
In the following and similar uniform motion problems, when equating unequal times, the additional time seems to be added to the faster object in order to solve the problem correctly.
However, to me this seems unintuitive as a $r =... |
H: Suppose that $0 < a < b$. Show that $\displaystyle 1 + \frac{1+a}{1+b} + \frac{(1+a)(1+2a)}{(1+b)(1+2b)} + \ldots$ converges.
Suppose that $0 < a < b$. Show that
\begin{align*}
1 + \frac{1+a}{1+b} + \frac{(1+a)(1+2a)}{(1+b)(1+2b)} + \ldots
\end{align*}
converges.
MY ATTEMPT
This is what I've tried:
\begin{align*}
S... |
H: A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder
A positive integer has $1001$ digits all of which are $1$'s. When this number is divided by $1001$ find the remainder.
I tried to think on it but couldn't get through. Please help.
AI: $$A=111111...... |
H: Show that there is a sequence $(m_{j})_{j=0}^{\infty}$ s.t. $m_{j}\to\infty$ as $j\to\infty$ and $\sum_{j=0}^{\infty}m_{j}a_{j}$ converges.
Suppose that $(a_{j})_{j=0}^{\infty}$ is a sequence of non-negative real numbers for which $\sum_{j=0}^{\infty}a_{j}$ converges. Show that there is a sequence $(m_{j})_{j=0}^{\... |
H: Is the resulting vector part of row/column space?
Per Wikipedia: In linear algebra, the column space (also called the range or image) of a matrix A is the span (set of all possible linear combinations) of its column vectors.
Wiki also gives an example:
Is the vector (c1, c2, 2c1) part of the column space as well? ... |
H: Finding $\sum_{r=1}^{\infty}\left(\frac{2r+5}{r^2+r}\right)\left(\frac{3}{5}\right)^{r+1}$
$\text{Find the value of}$ $$\lim_{n\to \infty}\left(\sum_{r=1}^{n}\left(\frac{2r+5}{r^2+r}\right)\left(\frac{3}{5}\right)^{r+1}\right)$$
$\text{Answer}: \frac{9}{5}$
Firstly I split "linear-upon-quadratic" term:
$$\frac{2r... |
H: True or False: If $$ is square and |$\det()|=1$, then $^{−1}$ has integer entries.
Thanks for the feedback on my most recent question. So it is not explicitly stated that A has integer matrices. Therefore I conclude that it is false. Can you please help me think of a counterexample? I am having a hard time doing so... |
H: Vectorizing Regularization in Linear Regression
I am wondering if someone could elaborate on the vectorized partial derivative of the MSE cost function. When writing code, I noticed that there seemed to be something wrong with the partial derivative terms that the class was outputting. I used the following formulas... |
H: What is wrong with my approach for CLRS 5.4-6 : Given n balls and n bins,find expected number of empty bins?
I am trying to find expected number of empty bins after n balls are tossed into n bins. And each toss is independent and equally likely to end up in any bin. Below is my approach.
My indicator variable is
... |
H: Selections with repetitions
suppose I have a list of numbers [1,2,3....20].In how many ways can i select three numbers with repetitions from this list
Can somebody explain how to solve this question .I thought about it and it seems to me that there are 20 identical objects of first type ,20 identical objects of se... |
H: Recursion relation for the moment of the normal distribution.
I am currently studying the Statistical Inference, 2nd, Casella & Berger.
On page 72, the authors asserts that for the normal distribution with mean $\mu$ and variance 1, $$EX^{n+1}=\mu EX^n - \frac{d}{d\mu}EX^n$$.
I cannot deduce it myself.
I know that ... |
H: Change in volume of sphere given change in radius
Finding the change in volume $$V=\frac{4}{3}\pi a^3$$ of a sphere when the radius change from $a_{0}$ to $a_{0}+da$
What I tried:
Using differential formula
$$\frac{\Delta V}{\Delta a}=\frac{d V}{da}=\frac{d}{da}\bigg(\frac{4}{3}\pi a^3\bigg)=4\pi a^2$$
$$\Delta ... |
H: Most general linear transformation of $|z|=r$ into itself using cross ratio
This question (without the cross ratio part) was asked earlier today, as well as a few times before. Here was the question that was asked earlier today: Find all Möbius transformations that map the circle $|z|=R$ into itself
Now, I am wond... |
H: Choosing the sign of determinant when taking a square root
Calculate the determinant $$\det(A)=\begin{vmatrix}a&b&c&d\\ \:\:\:-b&a&d&-c\\ \:\:\:-c&-d&a&b\\ \:\:\:-d&c&-b&a\end{vmatrix}$$
I found that $$\det(A)\det(A^T)=\det(A)^2=(a^2+b^2+c^2+d^2)^4$$
From this we get
$$\det(A) = \pm (a^2+b^2+c^2+d^2)^2$$
Now, how... |
H: Solve $x^{x^{x^{2017}}}=2017$
I have tried to use $\ln$, but couldn't solve:
\begin{equation}
\ln x^{x^{x^{2017}}}=x^{x^{2017}}\ln x=\ln 2017.
\end{equation}
I found that $x=\sqrt[2017]{2017}$ is a solution, and it is easy to check it. But how to find that solution without guessing and how to prove if it is the onl... |
H: Questions in study of Adjoint and inverse in Linear Algebra
While studying Linear Algebra from Hoffman Kunze, I am unable to understand few arguments given in lesson- Determinants. As my Institute is closed, so I have no help other than asking questions here :
It's image:
Questions : (1) How can formula (5-21) be ... |
H: False statement about a continuous and non negative function
It was asked in University of Hyderabad exam(2017).
I have shown 2nd and 3rd Statments to be true with the help of "Intermediate Value Theorem for Integrals".
But i can't reason why the statement 1 is false?
AI: The integral can be made as small as you pl... |
H: $rkA＋rkB＝n$, and A and B are diagonalizable
Let $A$ and $B$ be diagonalizable $n$-dimensional square matrices. Suppose each of $A$ and $B$ has no eigenvalues other than $0, 1$.Show that such $A$ and $B$ do not exist.
Any help would be appreciated, thank you.
P.S. Sorry, I missed important condition at first. I ... |
H: Topologies for graphs
Some of the basic definitions in Graph theory made me wonder if there is by any chance a way to give a graph $G$ a topology, such that these definitions can be understood as versions of analogous definitions given in topology. For example, is there a topology for $G$ such that the definition o... |
H: Prove that n^3 - n is a multiple of 6 for all positive integral values of n
Prove that
$$n^3 - n$$
is a multiple of 6 for all positive integral values of n
Does positive integral values of n refer to values of n once the expression is integrated to $$1/4n^4 - 1/2n + c$$
How do you deal with the constant of integr... |
H: Find all naturals numbers $φ(8n)< φ(5n)$ [Answer provided - Questioning for explanation]
Find all naturals numbers $φ(8n)< φ(5n)$
Answer:
then
Means $n$ is an odd number that is a multiple of 5
Question: Can I get an elaboration on how it was solved? also why did they use 4n?
AI: Note that for $n > 1$, $\varphi... |
H: Finding limits of complex functions using Taylor expansion
I am supposed to compute the following limit:
$$ \lim_{z \to 0} \frac{(1-\cos z)^2}{(e^z-1-z)\sin^2z} $$
I guess I have to use a Taylor expansion somehow, but I'm not sure what to expand and how, it looks a bit complicated. Any ideas?
AI: $$1-\cos(z)=\frac{... |
H: Exterior measure on $\mathbb{Q}$
I'm preparing myself for the final exam in real-analysis and I'm trying to solve this exercise. Show that there is no exterior measure $\mu^*$ on $\mathbb{Q}$ s.t. $\forall$ $p, q$ $\in$ $\mathbb{Q}$ with $p<q$
$$
\mu^*({x \in \mathbb{Q} : p\leq x\leq q})=q-p.$$
I was thinking to... |
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