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To what extent are submodules of $R^n$, for a ring $R$, describable by systems of linear equations? |
I've been struggling for a bit on this math problem from my homework.
$$
\int_{0}^{1}\int_{1}^{3}(8x^3-36x^2y^2)dydx
$$
From what I understand and have attempted, we'll integrate with respect to $x$ first, so something like:
$$
(2x^4-12x^3y^2)\vert_{1}^{3}=160-312y^2
$$
and then another integration, thi... |
Not an answer, but a progress into the understanding of the issue.
It is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation produced by GeoGebra has 2 branches : the straight line with equation $y=2x$ in blue and an hyperbola-look... |
This problem is from the book "Inside Interesting Integrals" written by Prof. Paul J, Nahin, which is stated as follows,
Imagine a circle (let’s call it $C_{1}$) that has radius $a$. We then chose at random (i.e., uniformly distributed) and independently, three points from the interior of that circle. These three po... |
> In the $xy$-plane a function is given: $f(x,y)=x+y$.
Let $A$ be the area that is in the $xy$-plane and is encapsulated by $x=0$, $x=\frac{\pi}{4}$, $y=x+\cos(x)$ and $y=x+\sin(x)$.
>
> a) Make a parametrization of $A$ and find its jacobi-function. Then calculate the area.
In my textbook the parametrization i ... |
I am having trouble understanding how to compute this integral $$\int_{0}^{\infty} \int_{0}^{y} \sqrt{x^2 + y^2} e^{-x^2 - y^2} \, dx \, dy$$
My idea is to consider the whole plane xy plane instead of the half of the positive quadrant. Then the integral would look like this. $$\frac{1}{8} \int_{-\infty}^{\infty} \int... |
Consider the cutting stock problem where $L$ is the length of the stock and $n$ is the total number of patterns. If customer demands are any width between $1, 2, \ldots, L$, why is it the case that $n \geq 2^{\sqrt{L}-1}$? |
The question is backwards. Not even non-finitists would argue that finite math is meaningless. The correct question is whether the non-finite is meaningful. Finitism and atheism are the positions that the posited entity, respectively infinity and god, makes no sense or does not exist. A finitist can no more prove infin... |
Is it true that for any projective variety $X$ that $\dim X = \text{trdeg}_k k(X) = \dim \mathcal O_{X,x}$ for $k$ an algebraically closed field of characteristic zero?
Here we have that $\mathcal O_{X,x}$ is the local ring of $X$ at $x$ defined by $k[X]_{I(x)}$.
I have learned that the equalities hold for standa... |
It's a fact that:
$\frac{a+b}{2} \ge \sqrt{ab}$
When a and b are non negative real numbers. What's the error in the following demonstration:
$\frac{a+b}{2} \ge \sqrt{ab}$ , then
$a+b \ge 2\sqrt{ab}$ , then
$a^{2}+2ab+b^{2} \ge 4ab$ , then
$a^{2}-2ab+b^{2} \ge 0$ , then
$(a-b)^{2} \ge 0$
the on... |
It's a fact that:
$\frac{a+b}{2} \ge \sqrt{ab}$
When a and b are non negative real numbers. What's the error in the following demonstration:
$\frac{a+b}{2} \ge \sqrt{ab}$ , then
$a+b \ge 2\sqrt{ab}$ , then
$a^{2}+2ab+b^{2} \ge 4ab$ , then
$a^{2}-2ab+b^{2} \ge 0$ , then
$(a-b)^{2} \ge 0$
the on... |
The axiom is on page 3 and is as follows:
**If two planes α, β have a point A in common, then they have at least a second point B in common**
Then on the same page a theorem is deduced and is as follows:
Theorem 1. Two straight lines of a plane have either one point or no point in
common; **two planes have no po... |
It's a fact that:
$\frac{a+b}{2} \ge \sqrt{ab}$
When a and b are non negative real numbers. What's the error in the following demonstration:
$\frac{a+b}{2} \ge \sqrt{ab}$ , then
$a+b \ge 2\sqrt{ab}$ , then
$a^{2}+2ab+b^{2} \ge 4ab$ , then
$a^{2}-2ab+b^{2} \ge 0$ , then
$(a-b)^{2} \ge 0$
discre... |
It's a fact that:
$\frac{a+b}{2} \ge \sqrt{ab}$
When a and b are non negative real numbers. What's the error in the following demonstration:
$\frac{a+b}{2} \ge \sqrt{ab}$ , then
$a+b \ge 2\sqrt{ab}$ , then
$a^{2}+2ab+b^{2} \ge 4ab$ , then
$a^{2}-2ab+b^{2} \ge 0$ , then
$(a-b)^{2} \ge 0$
discre... |
It's a fact that:
$\frac{a+b}{2} \ge \sqrt{ab}$
When a and b are non negative real numbers. What's the error in the following demonstration:
$\frac{a+b}{2} \ge \sqrt{ab}$ , then
$a+b \ge 2\sqrt{ab}$ , then
$a^{2}+2ab+b^{2} \ge 4ab$ , then
$a^{2}-2ab+b^{2} \ge 0$ , then
$(a-b)^{2} \ge 0$
discre... |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
>Find the generating function for the binary strings that do not contain $0110$ or $11010$ as a substring.
I am familiar with proofs similar to [this](https://math.stackexchange.com/questions/2464642/generating-series-for-binary-string-not-containing-a-substring) post, where the 'symbolic method' is used for countin... |
Given at least 3 points (xi, yi), how to confirm if all points lie on a sine wave? Or alternatively, how to determine the parameters of a sine wave that best fits the data (I'm guessing amplitude and phase would be sufficient?). Many thanks, Mike |
An answer, which needs at the end a complementary study to be fully rigorous.
First of all, it is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation of the implicit equation produced by GeoGebra has 2 branches : the straight line ... |
Consider linear evolution equations of the form
$$\tag{1}
x'(t)=r A x(t)
$$
Where $x$ is an $n$-tuple, $r$ is a scalar random variable and $A$ is an $n\times n$ matrix. The solution of (1) is
$$\tag{2}
x(t)=e^{rAt}x(0)
$$
You seek a differential equation for the average $\tilde{x}$ such that
$$\tag... |
Consider the cutting stock problem where $L$ is the length of the stock and $n$ is the total number of patterns. If customer demands are any length between $1, 2, \ldots, L$, why is it the case that $n \geq 2^{\sqrt{L}-1}$? |
Given the diagram, find SPNE and NE. Now, NE in my opinion will be (L1,L2) leading to the payoff (1,1) but SPNE by backward induction is coming out to be (R1,R2) leading to (2,2)?
[![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/LmKnj.jpg |
I'm an amateur mathematician and I found a property that I've never seen anyone mention before, I think I managed to demonstrate it below. I confess that I don't know if it's new or not, as I haven't found anything like it anywhere.
Theorem: Let $r > 0$, $c > 0$, and $n > 2$ be all integers, the sum $S_{n}$ of the f... |
Can any of you do something relevant with this mathematical property I found? |
In Appendix "Some Useful Definite Integrals" of Feynman & Hibbs "Quantum Mechanics and Path Integrals" they use some integral formulas that I'm struggling to derive. The integral is:
[![enter image description here][1]][1]
I've seen a question to solve an [almost similar integral][2], but the guy just gave a hi... |
I'm an amateur mathematician and I found a property that I've never seen anyone mention before, I think I managed to demonstrate it below. I confess that I don't know if it's new or not, as I haven't found anything like it anywhere.
Theorem: Let $r > 0$, $c > 0$, and $n > 2$ be all integers, the sum $S_{n}$ of the f... |
I am currently working on Spivak calculus 4th edition.
One of the problems asks the following:
Prove that if $0 < a < b$
$ a < \sqrt{ab} < (a+b)/2 < b$
Here is how I wrote it:
If $0<a<b$
Then $b-a \in P$ where $P$ is the set of all positive numbers
and $a \in P$
and $b \in P$
1) $(b-a)a... |
I am currently working on spivak calculus 4th edition.
One of the problem asks the following:
Prove that if $0 < a < b$
$$ a < \sqrt{ab} < \frac{a+b}{2} < b$$
Here is how I wrote it:
If $0<a<b$
Then $b-a \in P$ where $P$ is the set of all positive numbers
and $a \in P$
and $b \in P$
1) $... |
The round metric on $S^2$ with radius $\rho$ is given by $g_{S^2(r)} = \rho^2 \cdot \left(\begin{matrix} \sin^2(\phi) & 0 \\ 0& 1 \end{matrix} \right)$. Here the top-left entry is given by $g_{\theta\theta}$ from viewing $(X,Y,Z) = (\rho\sin\phi\cos\theta, \rho\sin\phi\sin\theta, \rho\cos\phi)$ of the inclusion of $S^2... |
An answer, which needs at the end a complementary study to be fully rigorous.
First of all, it is clear that setting $y=2x$ in the implicit equation gives $0$. Therefore, it is not surprizing that the following graphical representation of the implicit equation produced by GeoGebra has 2 branches : the straight line ... |
Consider linear evolution equations of the form
$$\tag{1}
x'(t)=r A x(t)
$$
Where $x$ is an $n$-tuple, $r$ is a scalar random variable and $A$ is an $n\times n$ matrix. The solution of (1) (which we will not need to evaluate) is
$$\tag{2}
x(t)=e^{rAt}x(0)
$$
You seek a differential equation for the av... |
Given at least 3 points $(x_{i}, y_{i})$, how to confirm if all points lie on a sine wave? Or alternatively, how to determine the parameters of a sine wave that best fits the data (I'm guessing amplitude and phase would be sufficient?). Many thanks, Mike |
Most of the time, any particular field of mathematics will be built on some kind of foundation, and the assumption is that the foundation is stable enough for the field to work.
At various times, someone constructs a paradox from some area of mathematics, and there's some work to find a system that avoids that probl... |
The reason that $\|(\sum_{j\ge0}|2^{js}\psi_j\ast f|^q)^{1/q}\|_{L^\infty(\mathbb R^n)}$ are not equivlant norms for different $\psi$ is mentioned in Triebel's "Theory of Function Spaces I" Section 2.3.2 Remark 4, where he refers to his book "[Spaces of Besov-Hardy-Sobolev type][1]" Section 2.1.4.
Nevertheless for $... |
In a previous question of mine (see [here][1] and [here][2]), I suggested the following general transformation for integrals involving $\tan{\left(\frac12\csc^{-1}{(x)}\right)}$ or $\tan{\left(\frac12\sec^{-1}{(x)}\right)}$ or both.
>$$\int f\left(\tan{\left(\frac12\csc^{-1}(x)\right)}, \tan{\left(\frac12\sec^{-1}(x... |
Given at least 3 points $(x_{i}, y_{i})$, how to confirm if all points lie on a sine wave? Or alternatively, how to determine the parameters of a sine wave that best fits the data (I'm guessing amplitude and phase would be sufficient?).
Context: I am processing images and attempting to identify sinusoidal features ... |
If a collatz sequence doesn't converge, how many odd numbers can its subsequence have? |
The following popped up when I was contemplating the completion of normed linear spaces.
**Question:** Let $E$ be an NLS (normed linear space) and $X$ a topological space. Let $E$ sit inside $X$ as a dense subset. Under what conditions can one ensure that there exists *at most one* NLS structure on $X$ that extends ... |
According to Aalipour et al. (2016), they defined the enhanced power graph of group $G$ as a simple undirected graph where the vertices are all elements of $G$ and two vertices, $x$ and $y$, are adjacent if $\langle x,y \rangle$ is belong to the same cyclic subgroup of $G$. Let $G$ be the dihedral group of order $2n$. ... |
How to prove the enhanced power graph of dihedral groups? |
Let a Collatz sequence of a natural number $x$ as follow:
$$\{x\}: \begin{cases}
x_0=x\\
x_{n+1} = \begin{cases}
\frac{3x_n+1}{2} \text{ if $x_n$ is odd}\\
\frac{x_n}{2} \text{ if $x_n$ is even}
\end{cases}
\quad\forall n\in\mathbb{N}^*
\end{cases}$$
The sequence is said to be convergent iff there exists k... |
It's well known that there are non-trivial bundles such as the Mobius strip. In these cases it is known the bundle $E$ with bundle map $\pi : E \to B$ over base space $B$ with fiber $F$ cannot be expressed as $E = B \times F$. This is because the topology of $E$ and $B$ might be such that the map $\pi$ is not continuou... |
Is there always a bijection between a fiber bundle and the cartesian product? |
everyone. I want to check this question and find it in an article.
[![enter image description here][1]][1]
so that \
$$s(r_{0},\theta_{0})~:=\left\{z\in\mathbb{C}\setminus\{0\}:|z|\geqslant r_{0}~~\text{ et}~~|\arg\left(z\right)\bigr|<\theta_{0}\right\}\cup B(0,r_{0}).$$
$$\displaystyle\Sigma_{r_{0},\theta_{0}}=\m... |
That's right ? $\operatorname{Re}(\sqrt{-z}\delta)>0~$ |
I was looking at Chapter 2 (Rings) in *Basic Algebra I* by Jacobson (1974). (Section 2.1 is Definition and Elementary Properties, 2.2 is Types of Rings, 2.3 is Matrix Rings (the ring $M_n(R)$ of $n\times n$ matrices over the ring $R$), and 2.4 is Quaternions.)
On page 96, Jacobson defines quaternions:
> We con... |
What is the meaning of this notation $\mathbb C_2$ in Jacobson's Basic Algebra I, or is it a typo.? |
This is a follow-up to this thread:
https://math.stackexchange.com/questions/2256360/proof-that-sum-over-autocorrelations-is-1-2
I am posting a new thread as that was posted 6 years ago. In that threat the stackexhange author (Kuhlambo) lists some equations from a paper found in https://doi.org/10.1515/ROSE.2009... |
What is the meaning of this notation $\mathbb C_2$ in Jacobson's Basic Algebra I? |
I was looking at Chapter 2 (Rings) in *Basic Algebra I* by Jacobson (1974). (Section 2.1 is Definition and Elementary Properties, 2.2 is Types of Rings, 2.3 is Matrix Rings (the ring $M_n(R)$ of $n\times n$ matrices over the ring $R$), and 2.4 is Quaternions.)
On page 96, Jacobson defines quaternions:
> We con... |
Helle, everyone. I want to check this question and find it in an article.
[![enter image description here][1]][1]
so that \
$$s(r_{0},\theta_{0})~:=\left\{z\in\mathbb{C}\setminus\{0\}:|z|\geqslant r_{0}~~\text{ et}~~|\arg\left(z\right)\bigr|<\theta_{0}\right\}\cup B(0,r_{0}).$$
$$\displaystyle\Sigma_{r_{0},\theta_... |
I was looking at Chapter 2 (Rings) in *Basic Algebra I* by Jacobson (1974). (Section 2.1 is Definition and Elementary Properties, 2.2 is Types of Rings, 2.3 is Matrix Rings (the ring $M_n(R)$ of $n\times n$ matrices over the ring $R$), and 2.4 is Quaternions.)
On page 96, Jacobson defines quaternions:
> We con... |
Let $\mathcal{S}(\mathbb{R}^n)$ be a Schwartz space and $T : \mathcal{S}(\mathbb{R}^n) \to \mathbb{C}$ be a continuous linear functional under the Frechet topology of $\mathcal{S}(\mathbb{R}^n)$. That is, $T$ is a tempered distribution.
Now, I wonder if there is any sufficient (or eqiuvalent, hopefully) criterion u... |
Is there any criterion under which a tempered distribution is (identified with) an ordinary function? |
Searching the $\pi$-Base can help with these sorts of questions:
1. https://topology.pi-base.org/spaces/?q=Separable%2B%24T_2%24%2BLocally+%24n%24-Euclidean%2B%7ELocally+Compact
All locally euclidian spaces are locally compact: https://topology.pi-base.org/theorems/T000332/
2. https://topology.pi-base.org/spac... |
Let $A$ be a commutative local noetherian ring and let $I$ be a proper ideal of $A$. Prove that $\bigcap_{n=1}^\infty I^n = 0$.
The first thing I tried was to see that
$$\displaystyle\bigcap_{n = 1}^\infty I^n \subseteq \displaystyle\bigcap_{n = 2}^\infty I^n \subseteq \displaystyle\bigcap_{n = 3}^\infty I^n \subs... |
I am exploring Taylor's Classical Mechanics and I am unsure about the mathematics when he derives the minimum of the action. We define this function $f(x(t),\dot{x}(t),t)$ to be a Lagrangian. We define the action to be
$$ S = \int_{t_{1}}^{t_{2}} f(x(t),\dot{x}(t),t)dt$$
Let some function $x$ satisfy the condition ... |
the intuition behind eigenbasis? |
Consider linear evolution equations of the form
$$\tag{1}
x'(t)=r A x(t)
$$
Where $x$ is an $n$-tuple, $r$ is a scalar random variable and $A$ is an $n\times n$ matrix. The solution of (1) (which we will not need to evaluate) is
$$\tag{2}
x(t)=e^{rAt}x(0)
$$
You seek a differential equation for the av... |
I'm reading "Introduction to Manifolds" 2nd edition by Loring W. Tu and have a question.
Remark 8.2 of this book says that
"If $U$ is an open set containing $p$ in $M$, then the algebra $C_p^{\infty}(U)$ of germs of $C^{\infty}$ functions in $U$ at $p$ is the same as $C_p^{\infty}(M)$. Hence, $T_pU$ = $T_pM$."
... |
From the definition of the Fréchet derivative and the linearity of the transpose it’s clear that the derivative of the vector transpose is the vector transpose itself.
$$f(x + h) = f(x) + D(x)h + o(\lvert\lvert h \rvert\rvert^2)$$
$$(x + h)^T = x^T + h^T + 0 = x^T + h^T + o(\lvert\lvert h \rvert\rvert^2)$$
Acc... |
There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to help me develop this model that I am thinking of...
Most geometric theorems are proven by putting several theorems to... |
I'm reading "Introduction to Manifolds" 2nd edition by Loring W. Tu and have a question.
Remark 8.2 of this book says that
"If $U$ is an open set containing $p$ in $M$, then the algebra $C_p^{\infty}(U)$ of germs of $C^{\infty}$ functions in $U$ at $p$ is the same as $C_p^{\infty}(M)$. Hence, $T_pU$ = $T_pM$."
... |
I am a little confused over a detail in the following result: [![enter image description here][1]][1]
[1]: https://i.stack.imgur.com/gakaW.png
In particular, my question is on the details of why the probability of a tie $X_i = X_j$ is $0$, the explanation for which I'm not satisfied by in the proof above. I c... |
In *Introductory Lectures on Convex Optimization* by Nesterov, Section 3.2.1, they construct a *difficult* function to show that all the schemes work badly on this function. Say, consider an unconstrained minimization problem $$\min_{x\in\mathbb R^n}f(x),$$ where $f(x)$ is convex and Lipschitz continuous on a bounded s... |
More precisely, if we know the proportion of 1's in a randomly chosen binary sequence of length $2N$ is $p$, then given an $\epsilon>0$ , what is the probability that the proportion of 1's in the subsequence consisting of the first $N$ digits is $p-\epsilon<p'<p+\epsilon$?
Answers with approximations and bounds are ... |
For an arbitrary binary sequence of length $2N$ with $n$ 1's, what is the probability that the first $N$ digits have a 'similar' proportion of 1's? |
How to ensure that integrals are remain within limits of domain of the variables? |
How do I solve this Diophantine equation, $a^{2n}+b^2=c^{2n}$, where $n$ is any postive integer $>1$ & $a,b,c\ne0$?
I tried it by applying the Pythagorean triplets generating formula, but unable to find any integer solutions & nor I'm able to prove that there are no integer solutions except $a,b,c=0$. Thanks in adva... |
I was asked to find the contour integral $dz/z$ of a triangle with vertices:
$1-2i$
$-2+i$
$1+i$
I am not sure how to begin this but my professor said that it would be:
$\int_{-1}^{2} \frac{-i}{1-ti} \, dt$
But this doesn't make sense to me. why would $z$ be equal to $1-ti$? I can see he may be wo... |
I was asked to find the contour integral $dz/z$ of a triangle with vertices $1-2i$, $-2+i$ and $1+i$.
I am not sure how to begin this but my professor said that it would be:
$$\int_{-1}^{2} \frac{-i}{1-ti} dt$$
But this doesn't make sense to me. why would $z$ be equal to $1-ti$? I can see he may be working f... |
From the definition of the Fréchet derivative and the linearity of the transpose it’s clear that the derivative of the vector transpose is the vector transpose itself.
$$f(x + h) = f(x) + D(x)h + o(\lvert\lvert h \rvert\rvert)$$
$$(x + h)^T = x^T + h^T + 0 = x^T + h^T + o(\lvert\lvert h \rvert\rvert)$$
Accordi... |
I guess this is a bit of a soft question. Suppose you have a hom of commutative rings with identity $R\to S$, and a hom of $S$ modules $f: M\to N$. Somehow it shouldn't matter whether you localize $f$ in the category of $S$ modules or in the category of $R$ modules (I realize I'm leaving vague the set you're using to ... |
In a previous question of mine (see [here][1] and [here][2]), I suggested the following general transformation for integrals involving $\tan{\left(\frac12\csc^{-1}{(x)}\right)}$ or $\tan{\left(\frac12\sec^{-1}{(x)}\right)}$ or both.
>$$\int f\left(\tan{\left(\frac12\csc^{-1}(x)\right)}, \tan{\left(\frac12\sec^{-1}(x... |
There is an idea that has been on my mind for a while, and I would like to share it so that it turns into a snowball. Perhaps it will be useful and attractive to engineering enthusiasts. I ask you to help me develop this model that I am thinking of...
Most geometric theorems are proven by putting several theorems to... |
Is this proof vigorous enough?
Proof:
Suppose every pair of vertices are contained in a common cycle. Thus, there are at least two paths between any two vertices. Note that at least two vertices from distinct paths must be removed to disconnect any pair of vertices. Hence, $G$ is 2-connected.
Let $G$ be a 2-co... |
At first, it should be noted that Linear Programming (LP) **has benefited from** many mathematical tools, for a review, see e.g., [Algebraic and Topological Tools in Linear Optimization][1] and [Integer Programming and Number Theory][2].
Moreover, due to its widespread usage, LP has **become a strong driver** for th... |
This is a homework problem from my adaptive control course:
> Given the plant $y_p = a_pu(t)$ ($a_p\neq 0$) and the reference model $y_m = a_mr(t)$, where $r(t)$ is bounded and continuous. Design a __differentiator free__ control $u(t)$ such that all signals are bounded and $|y_p(t) - y_m(t)|\to 0$. Consider two diffe... |
In *Introductory Lectures on Convex Optimization* by Nesterov, Section 3.2.1, they construct a *difficult* function to show that all the schemes work badly on this function. Say, consider an unconstrained minimization problem $$\min_{x\in\mathbb R^n}f(x),$$ where $f(x)$ is convex and Lipschitz continuous on a bounded s... |
Throughout studying a paper about finite $p$-groups,
I have the following question
Let $G$ be a finite $p$-group with nilpotency class 3 and $\gamma_i(G)$ denote
the i'th term of the lower central series of $G$. Let $\gamma_3(G)$ has prime order and $\displaystyle\frac{G}{\gamma_3(G)}\cong M(p^n), n\geq 3$,
wh... |
The quadratic formula seems to find intercepts correctly but not the right graph. What am I missing?
For example, if $0=2x^2+7x+5$, then it can be solved with the quadratic formula to give us $x=-5/2$ or $-1$.
When I graph $(x+5/2)(x+1)$ it looks like
[this][1].
However, when I graph $y=2x^2+7x+5$ it looks lik... |
Quadratic Formula False? |
I heard the following:
> For a smooth submanifold $M \subseteq \mathbb{R}^n$,
> given a local trivialization one can easily find a basis for
> the tangent space.
I want to know how.
So first I should maybe formalize the question:
Suppose $M \subseteq \mathbb{R}^n$ is a smooth $k$-dimensional submanifold and... |
I have started the introduction of calculus in high school, and I am confused about this problem.
Define $f(x) = -x$ for $x<0$. Then find out if $\lim_{x \to 0} f(x)$ exists or not. If it does, then state the value.
Here is my approach: The limit exists if $\lim_{x \to 0^{-}} f(x)$ = $\lim_{x \to 0^{+}} f(x)$.
... |
How to solve for thee angle $x$ ?($\tan(x)+ 4 \sin(x)=\sqrt (3)$) |
I haven’t studied much number theory at all, but I got asked this question by a family member who is a fan of recreational math, and I have no clue how to answer it, or even the subfield it precisely belongs to and keywords to search to find discussion of it:
Say that $n \in \mathbb{N}$ is unknown, but you have acce... |
Can you determine $n$ with oracle for primality of $n+m$ for given $m$? |
I have started the introduction of calculus in high school, and I am confused about this problem.
Define $f(x) = -x$ for $x<0$. Then find out if $\lim_{x \to 0} f(x)$ exists or not. If it does, then state the value.
Here is my approach: The limit exists if $\lim_{x \to 0^{-}} f(x)$ = $\lim_{x \to 0^{+}} f(x)$.
... |
Suppose we have a linear system $\dot{x} = A x$, where $x \in \mathbb{R}^n$ and $A$ is a constant matrix, such that all the trajectories of the system are bounded and bounded away from the origin (i.e., none of them converge to the origin). This I believe implies that all the eigenvalues of $A$ have 0 real parts and th... |
The code below is to use the simultaneous iteration method to calculate the eigenvalues of the 1d Laplacian operator. The exact eigenvalues are known analytically.
It is strange that there is no convergence even after 4000 iterations. Is it because of the degeneracy of the eigenvalues?
If the matrix $H $ is rep... |
simultaneous iteration fails for the 1d laplacian operator? |
How to solve for thee angle $x$ ?($\tan(x)+ 4 \sin(x)=\sqrt 3$) |
Would $v^TAv>0 forall $v$ v$ if A's eigenvalues are all > 0? |
Would $v^TAv>0$ forall $v$, $v$ if A's eigenvalues are all > 0? |
# Set up
Let $G$ be a complete, weighted, and directed graph with $N$ vertices as in the asymmetric Traveling Salesman Problem (TSP). Without loss of generality, let the the vertex set $V$ of $G$ be such that $V = \{1, 2, 3, \ldots, N\}$. Let $A$ be the adjacency matrix of $G$. Then the weight of the edge from ve... |
More precisely, if we know the proportion of 1's in a uniformly randomly chosen binary sequence of length $2N$ is $p$, then given an $\epsilon>0$ , what is the probability that the proportion of 1's in the subsequence consisting of the first $N$ digits is $p-\epsilon<p'<p+\epsilon$?
Answers with approximations and b... |
For a random binary sequence of length $2N$ with $n$ 1's, what is the probability that the first $N$ digits have a 'similar' proportion of 1's? |
I'm referring to [this version][1] of Kripke-Platek set theory minus axiom of infinity. Let $S$ be a Dedekind infinite set, and $f\colon S \to S$ be an injection with $f(S) \neq S$. A proof I know picks $s \in S \setminus f(S)$ and constructs function $g(n) = f^n(s)$ by recursion, and use replacement to obtain $\omega ... |
I am reading the Manfred Stoll, Introduction to real analysis, proof of 9.5.8 theorem and stuck at some statement.
> **9.5.8. THEOREM** Let $f$ be a continuous function on $[-\pi, \pi]$ with $f(-\pi)=f(\pi)$, and let $f'$ be piecewise continuous on $[-\pi,\pi]$. If $$f(x)=\frac{1}{2}a_0 + \Sigma_{n=1}^{\infty}(a_n... |
I was studying Schrödinger's equation for a school paper and wanted to solve it in a bit broader context than the potential well. I wanted to use a potential function $$V(x,y,z) = \chi(x) + \gamma(y) + \zeta(z)$$ for continuous (maybe continuously differentiable, don't know if that will matter later) functions $\chi(x)... |
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