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I'm taking an algebraic geometry course and I'm trying to solve the next problem:
Take the curve $V$ parametrized by $(t,t^3,t^4) \in \mathbb{R}^3$. Show that $V$ is an algebraic set and find $\mathbb{I}(V)$, where $\mathbb{I}(V)$ denotes the ideal of all polynomials that are zero over $V$.
So far, I've shown that ... |
The theater is configured so that in any row there is a wall, 6 seats, an aisle, 6 more seats, and then a wall. In total there are 2* 6! arrangements = 1400.
"How many ways can they sit if Jane and Mary refuse to sit next to each other?"
I have tried to approach this in 3 different ways. First that the number of a... |
Define an Interface as $I = <\theta,\phi>$, where $\theta$ defines a set of typenames and $\phi$ consists of function declarations $(f: t_1 \times ... t_k \mapsto t)$, that takes arguments of type $t$ and returns a value of type $t \in \theta$. If a function takes 0 arguments, it is said to be a constant. **Let $V$ be ... |
I would like to evaluate the mean and variance of the edge density for subgraphs obtained by repeatedly subsampling nodes.
Specifically, suppose we have an undirected graph $G$ with $N$ vertices and adjacency matrix $y$. We obtain a subset of nodes $S$ by [Bernoulli sampling](https://en.wikipedia.org/wiki/Bernoulli_... |
The theater is configured so that in any row there is a wall, 6 seats, an aisle, 6 more seats, and then a wall. In total there are 2* 6! arrangements = 1440.
"How many ways can they sit if Jane and Mary refuse to sit next to each other?"
I have tried to approach this in 3 different ways. First that the number of a... |
Let K be the largest value such that $(2023!)^{2023!}$ is divisible by $42^{\mathrm{K}}$. How many consecutive 0 digits are at the end of \mathrm{K}$?
I tried to find the highest power of $42$ that divides $2023!$ Since $42 = 2 \times 3 \times 7$, we need to count the occurrences of 2, 3, and 7 in the prime factori... |
I have iid random variables $Z_1,Z_2,...$ that are uniformly distributed on the set $\{1,2,...,n\}$. I define $X(n)=min\{k:Z_k=Z_j\ for\ some\ j<k\}$ and $Y(n)=min\{k:Z_k=Z_j,Z_i=Z_l\ for\ some\ j<k\ and\ i<l<k\}$ so X(n) is the first repeat of a number and Y(n) is the second repeat. I'd like to calculate the joint dis... |
Let K be the largest value such that $(2023!)^{2023!}$ is divisible by $42^{\mathrm{K}}$. How many consecutive 0 digits are at the end of \mathrm{K}$?
I tried to find the highest power of $42$ that divides $2023!$ Since $42 = 2 \times 3 \times 7$, we need to count the occurrences of 2, 3, and 7 in the prime factori... |
I have iid random variables $Z_1,Z_2,...$ that are uniformly distributed on the set $\{1,2,...,n\}$. I define $X(n)=min\{k:Z_k=Z_j\ for\ some\ j<k\}$ and $Y(n)=min\{k:Z_k=Z_j,Z_i=Z_l\ for\ some\ j<k\ and\ i<l<k\}$ so X(n) is the first repeat of a number and Y(n) is the second repeat (not necessarily a repeat of the sam... |
Suppose we have $A$ and $B$ as subsets of a metric space $(E, d)$. Is it true that $d(A, B) \leq d(A, C) + d(C, B)$ for some subset $C$ of $E$?
|
my school gave this question out and I couldn't think of a way to do it other than bashing by parts multiple times. Is there a clever solution to the question?
Given:
$$ I = \int_{0}^{4\pi} e^x (\sin^6 x + \cos^6 x) \, dx $$
$$ J = \int_{0}^{\pi} e^x (\sin^6 x + \cos^6 x) \, dx $$
Find the value of $$\frac{I}... |
Suppose we have $A$ and $B$ as subsets of a metric space $(E, d)$. Is it true that for any subset $C$ of $E$, $d(A, B) \leq d(A, C) + d(C, B)$?
|
I am reviewing some algebraic number theory, and found myself confused about two seemingly distinct notions of discriminant that are used.
Let $K$ be a field, and $L$ a finite separable extension. We can define the trace pairing $\mathrm{Tr}_{L/K}:L \times L \rightarrow K$ which sends $(\alpha, \beta) \to \alpha \be... |
I am having difficulties with this problem. \
Given a non-negative compact self-adjoint operator $\gamma$ i.e. $\langle \gamma u, u\rangle \geq 0$ for all $u$. Denote the cut-off function $\chi \in C^\infty_c\left(\mathbb{R}^d\right)$ satisfying $\chi$ is non-increasing, $\chi\left(\left|x\right|\right) = 1$ if $\left... |
I am reading complex analysis by Gamelin and I am having trouble understanding the square root function.
The principal branch of $\sqrt{z}$ ( $f_1(z)$ ) is defined as $|z|^{\frac 1 2} e^{\frac{i \operatorname{Arg}(z)}{2}}$ for $z \in \mathbb{C} - (-\infty,0]$ and $f_2(z)$ is defined as $-f_1(z)$
By this defin... |
I am having a hard time for understanding stopping time, especially on notations.
My book says:
Def. A stopping time for the filtration $(g_n)$ is a random variable
$T:\Omega \rightarrow \mathbb{N} \cup \infty \ \text{such that} $
\{ $T=n$ \} $ \in g_n \ \text{for all} \ n \ge 0$
It is difficult for me to ... |
I am solving a fairly basic non- homogeneous ODE, but I wanted to try using the differential operator, and a series expansion in order to find the particular solution. I am fairly convinced it is possible to solve an ODE using this method, however I an unable to understand how.
This is what I have done thus far:
... |
How does one solve this ODE using a differential operator and series expansions? |
How does one solve this ODE using a differential operator and a series expansion? |
I have the following equation
$$a=a_0+Mx,$$
where $a$, $a_0$, and $x$ are $N\times 1$ complex vectors, and $M$ is a $N\times N$ complex matrix.
For a given $a$, $a_0$, and $M$, I want to find a vector $x$ such that the absolute value of each element of $a$ is equal to 1.
Is there a nice algebraic way to fin... |
Despite the long physical introduction, I swear this is a mathematical question.
While introducing the motion of a point seen by two observers O and O' (with respective Euclidean spaces *E3* and *E3'*), my Professor of Analytical Mechanics stated that there is an isometry between the two spaces (which is a postulate o... |
I have this quantity:
$ T_n(x) = \exp\big[i\cdot n\arccos(x)\big] $
How can i represent the polynomials of first and second kind?
I tried substituting $n = 0$ and $n = 1$ but i don't know when to stop and plot the generative function
**EDIT using MathLaTeX**
$ T_0(x) = e^{i0\arccos(x)} = e ^ 0 = 1 $
$... |
I am having a hard time for understanding stopping time, especially on notations.
My book says:
Def. A stopping time for the filtration (\$g_n$) is a random variable
$T:\Omega \rightarrow \mathbb{N} \cup \infty$ such that $ \{ $T=n$ \} \in g_n $ for all $ n \ge 0$
It is difficult for me to understand what ... |
Is there a permutation of a given length in which every element divides sum of the elements before it? |
I am having a hard time for understanding stopping time, especially on notations.
My book says:
Def. A stopping time for the filtration $(g_n)$ is a random variable
$T:\Omega \rightarrow \mathbb{N} \cup \infty \ \text{such that} $
\{ $T=n$ \} $ \in g_n \ \text{for all} \ n \ge 0$
It is difficult for me to ... |
[![Question][1]][1]
I couldn't solve this question so I looked for hints. One method of solving was to use the Taylor Series expansion of each of the functions.
It was a bit long.
So another solution used the L'Hospitals Rule instead.
$$\lim_{x\to 0}{\cfrac{\alpha e^x+\beta e^{-x}+\gamma\sin(x)}{x\sin^2x}}$$
... |
Correct method to evaluate the limit $\lim_{x\to 0}{\cfrac{\alpha e^x+\beta e^{-x}+\gamma\sin(x)}{x\sin^2x}}=2/3$? |
**The question**
I am just exploring different properties of Steiner symmetrization and under continuities I have conjectured that for a given function $f:\mathbb{R}^n\rightarrow\mathbb{R}$ that is:
1. non-negative,
2. supported on a ball, and
3. of finite $L_1$ norm (for simplicity let's just say its range is $\... |
Is Steiner symmetrization Lipschitz continuous with respect to the direction? |
Transform the set of causes of the given formula are below:
$\overline{x}\lor y.....(1)$
$\overline{y}\lor x.....(2)$
$\overline{z}\lor w.....(3)$
$\overline{w}\lor z.....(4)$
$\overline{s}\lor t.....(5)$
$\overline{t}\lor s.....(6)$
$\overline{x}\lor \overline{z}\lor\overline{s}.....(7)$
$\o... |
I am stuck trying to understand part of the proof of Lemma 6.7 in Manin's *Logic for Mathematicians*. I am aware that notation may vary and parts of the question may rely on definitions made earlier in the book, but I would be grateful for any insight.
Here $\cal{E}$ consists of a set of consistent formulas of a lan... |
Hint: If you have a two digit number represented by $10a+b$, $b$ represents the units. What does $a$ represent?
>!(In general, a number can be represented by $10^a x_1 + 10^{a-1} x_2 + 10^{a-2} x_3 \dots 10^{a-(n-1)} x_{n-1} + 10^{(a-n)} x_n$, where $x_1, x_2, x_3, \dots x_{n-1}, x_n$ are the digits of that number... |
It a rotation matrix in 3D space a product of three basic rotations? |
Why are ultrametric spaces named as such? |
Knowing the equation of the tangent plane $\pi: 2x + y + 3z = 0$ at the point $(1, 1, 1)$ of a function $f(x, y)$, how can I find the values of $f'_x(1, 1)$ and $f'_y(1, 1)$?
I thought of this: since I know the equation of the plane, then I know it comes from
$$\pi: f(1, 1) + f'_x(1, 1)(x-1) + f'_y(1, 1)(y-1)$$
... |
I prefer to use the term "strong triangle inequality".
Anyway, the page
https://mathshistory.st-andrews.ac.uk/Miller/mathword/u/
says *ultrametric* is due to Krasner in 1944 "according to an Internet web page." I've never seen Jeff Miller's pages on words in mathematics use such a lazy reference before. Ima... |
Knowing the equation of the tangent plane $\pi: 2x + y + 3z = 0$ at the point $(1, 1, 1)$ of a function $f(x, y)$, how can I find the values of $f'_x(1, 1)$ and $f'_y(1, 1)$?
I thought of this: since I know the equation of the plane, then I know it comes from
$$\pi: z=f(1, 1) + f'_x(1, 1)(x-1) + f'_y(1, 1)(y-1)$... |
Is there a nice reference which explains in detail how composition works for the $A_\infty$-operad (associahedra operad)? It would be neat to have a visualization for low dimensions of what's happening (to get some intuition). I would be both happy with a detailed answer or a suggestion of where to find the proper defi... |
The way you stated the question demonstrates the confusion. You are trying to figure out how to map the points on the circumference to each other. But that isn't the amount of *turning*. In terms of distance rolled along, the ratio is indeed 3 as you would expect. But that isn't the amount of turning, its just the ge... |
I have the following equation
$$a=a_0+Mx,$$
where $a$, $a_0$, and $x$ are $N\times 1$ complex vectors, and $M$ is a $N\times N$ complex matrix.
For a given $a_0$ and $M$, I want to find a vector $x$ such that the absolute value of each element of $a$ is equal to 1.
Is there a nice algebraic way to find such... |
Consider a modified gamblers ruin Markov Chain on the set of natural numbers {0,1,2,3...}.
We start in state 1, and at each turn, for state i, we have a probability p to go to state i+1, and a probability q to go to state 0. Meaning that in this modification losing once means going bankrupt immediately, regardless of ... |
Up-front: Met the snobs on Math overflow and thought I'd try here instead. I have three STEM degrees including math, and this is for work not school; just trying to save hours googling.
This equation diverges so I can't just solve it. What I'd like to find is a way to sum a finite # of terms to a specific number of ... |
Let $V$ be a finite set consist of nonzero vectors $v=(v_1, \cdots, v_n)\in \mathbb{N}^n$, how to find a vector $w\in \mathbb{Z}^n$ such that the inner product $<w,v>$ is nonzero for any $v\in V$? |
I have asked this question once before, but I forgot about it and it's been a long time since then. Also, I have learned new things since then so my approach has changed. Therefore I am creating a new question. It goes something like this:
> Suppose a chess knight is positioned on some arbitrary square on a chessboa... |
I faced the following problem recently. When I was working on an electricity market clearing problem, it is hard for me to derive the dual problem of it, which is necessary to obtain the optimal price.
The original primal problem:
\begin{align}
&\mathop{Maximize}_{p_g^G, p_d^D, \theta_n}{\quad SW = \sum_d U_d p_d... |
Let $V$ be a finite set consist of nonzero vectors $v=(v_1, \cdots, v_n)\in \mathbb{Z}^n$, how to find a vector $w\in \mathbb{Z}^n$ such that the inner product $<w,v>$ is nonzero for any $v\in V$? |
The property:
$\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$
Derivation given in my textbook:
Let $t=a+b-x$. Then $d t=-d x$. When $x=a, t=b$ and when $x=b, t=a$.
Therefore,
$$
\begin{aligned}
\int_a^b f(x) d x & =-\int_b^a f(a+b-t) d t \\
& =\int_a^b f(a+b-t) d t \\
& =\int_a^b f(a+b-x) d x \
\end{aligned}
... |
How is this property of definite integral derived? |
I'm trying to solve this problem
$$\eqalign{
& {\text{Consider a sequence is given by}}: \cr
& {a_0} = 1,{a_{n + 1}} - {a_n} = \frac{{{a_n} + \sqrt {5a_n^2 - 4} }}{2} \cr
& {\text{Show that}}:{a_{2n}} - {a_{2n - 1}} = a_n^2 - a_{n - 1}^2 \cr} $$
I tried to used mathematical induction
$$\eqalign{
... |
I am solving a fairly basic non-homogeneous ODE, but I wanted to try using the differential operator, and a series expansion in order to find the particular solution. I am fairly convinced it is possible to solve an ODE using this method, however I an unable to understand how.
This is what I have done thus far:
$... |
The property:
$$\int_a^b f(x) d x=\int_a^b f(a+b-x) d x$$
Derivation given in my textbook:
Let $t=a+b-x$. Then $d t=-d x$. When $x=a, t=b$ and when $x=b, t=a$.
Therefore,
$$
\begin{aligned}
\int_a^b f(x) d x & =-\int_b^a f(a+b-t) d t \\
& =\int_a^b f(a+b-t) d t \\
& =\int_a^b f(a+b-x) d x \
\end{aligned... |
How is the satisfaction predicate usually defined? |
The property:
$$
\int_a^b f(x) \, dx=\int_a^b f(a+b-x) \, dx
$$
Derivation given in my textbook:
Let $t = a+b-x$. Then $dt = -d x$. When $x=a, t=b$ and when $x=b, t=a$. Therefore,
$$
\begin{aligned}
\int_a^b f(x) \, dx
&= -\int_b^a f(a+b-t) \, dt \\
&= \int_a^b f(a+b-t) \, dt \\
&= \int_a^b f(a+b-x) \, ... |
I'm attacking to solve the graph isomorphism problem using the adjacent matrices.
I would appreciate it if someone could show me whether following conjecture works or not. I expect this conjecture to work.
Let $A$ and $B$ be $n \times n$ symmetric matrices. Let the diagonal components of $A$ and $B$ are $0$, an... |
I am exploring options for using computer to extract the coefficient of $x^n$ from the Taylor series expansion of a function of $x$, where $n$ is kept as a symbolic entity. While Mathematica offers a function called `SeriesCoefficient` that achieves this, I am in search of an open source alternative that offers similar... |
Is There an Open Source Equivalent for Mathematica's SeriesCoefficient Function? |
I was trying to prove $L^\infty$ is complete with $(X,\mathcal{A},\mu)$ given and $\mu$ is $\sigma$-finite.
Outline of my attempted proof:
- (i) Assume Cauchy in norm - let $\{f_n\}$ be a Cauchy sequence which converges in $L^\infty$ norm. That is, let $\epsilon>0$, and assume $\left\Vert f_n-f_m \right\Vert_\inf... |
Previously, I asked [a question][1] about a treatment for establishing the limit of a Cauchy sequence in $L^\infty$-norm using the approach suggested by Munroe, Aliprantis and Burkinshaw, Wheeden and Zygmund, Folland, and so on to take care the measurability based on the dependency and countability of measurable subset... |
In general topology, it is very common to use open coverings and various other collections of sets. For instance, in the definition of [cover](https://en.wikipedia.org/wiki/Cover_(topology)) on Wikipedia. However, I found two ways of denoting these collections, one as $C :=\{ U_{\alpha}:a\in A \}$ or as $\{ U_{\alpha} ... |
Transform the set of causes of the given formula are below:
$\overline{x}\lor y.....(1)$
$\overline{y}\lor x.....(2)$
$\overline{z}\lor w.....(3)$
$\overline{w}\lor z.....(4)$
$\overline{s}\lor t.....(5)$
$\overline{t}\lor s.....(6)$
$\overline{x}\lor \overline{z}\lor\overline{s}.....(7)$
$\o... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
From observations, for quadratic functions, the average rate of change in $[a,b]$ is the **arithmetic** average of the derivative evaluated at $a$ and $b$.
$${\rm{Avg. ROC}_{[a,b]}}=\frac{f'(a)+f'(b)}{2}\qquad f(x)=c_0+c_1x+c_2x^2$$
So, motivated by [this video](https://www.youtube.com/watch?v=3r1t9Pf1Ffk) by Michael... |
Am I correct that this follows directly from two observations:
(1) The sum of the reciprocals of primes [diverges](https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes).
(2) The sum of the reciprocals of squares [converges](https://en.wikipedia.org/wiki/Basel_problem)
Here's m... |
Let $f:[0,\infty)\to[0,\infty)$ be a concave function with $p \in [0,\frac{1}{2}]$.
a) Show that $f$ is integrable on every interval $[a,b] \subset \mathbb R$.
b) Show that the function $F:[0,\infty)\to[0,\infty)$, $F(x)=\int_{0}^{x^{p}}f(t)dt$ is concave.
I know there is a theoren that says if a fu... |
In general topology, it is very common to use open coverings and various other collections of sets. For instance, in the definition of [cover](https://en.wikipedia.org/wiki/Cover_(topology)) on Wikipedia. However, I found two ways of denoting these collections, one as $C :=\{ U_{\alpha}:a\in A \}$ or as $\{ U_{\alpha} ... |
Isn't rolling with advantage unordered with replacement? |
Let’s play a game where you have ten chips, and you want to split them up into stacks such that you want to maximise your score, which is the product of the size of each stack. What’s the best you can do with 10? How about 100 chips? How about N chips arbitarily large?
I'm not sure how to go about this problem, bu... |
Maximum product of numbers that add up to 10? |
Am I correct that this follows directly from two observations:
(1) The sum of the reciprocals of primes [diverges](https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_reciprocals_of_the_primes).
(2) The sum of the reciprocals of squares [converges](https://en.wikipedia.org/wiki/Basel_problem)
Here's m... |
Suppose $\{f_n\}$ is a sequence of functions on $\mathbb R$ that is an approximate identity. In particular, it satisfies for every real $c>0$,
$$\int_{-\infty}^{-c} |f_n(x)|dx\to0\qquad\int_c^\infty |f_n(x)|dx\to0$$
Does it follow that for every $c>0$, $f_n\to0$ uniformly on $(-\infty,-c]\cup[c,+\infty)$?
My i... |
If $\forall c>0$, $\int_c^\infty|f_n|dx\to0$ and $\int_{-\infty}^{-c}|f_n|dx\to0$, then $f_n\to0$ uniformly on $(-\infty,-c]\cup[c,\infty)$? |
Suppose we are given a filtered probability $(\mathcal{F},\{\mathcal{F}\}_t,\mathbb{P})$ space fulfilling the "usual conditions" (.i.e. the filtration is right-continuous and complete). Let $\tau$ be a stopping time with respect to this space, let $a$ be a positive real number and assume $a<\tau$. Is it true that for a... |
I am learning parts of topos theory and it feels like I am missing something regarding indexed functors.
This comes from the definition of a locally connected geometric morphism $f : E \rightarrow F$ as being essential and the extra adjoint be F-indexed.
I am looking for a definition as I am unfamiliar with index... |
Suppose we are given a filtered probability $(\mathcal{F},\{\mathcal{F}\}_t,\mathbb{P})$ space fulfilling the "usual conditions" (.i.e. the filtration is right-continuous and complete). Let $\tau$ be a stopping time with respect to this space, let $a$ be a positive real number and assume $a<\tau$. Is it true that for a... |
Being rigorous, your proof is not by induction. When we are doing proofs by induction, we need to do "three steps". In order:
1. Prove the base, i.e., prove that the assertion is true for $n=0$ (or $n=1$ in some cases)
2. Suppose that the assertion is true for $n$.
3. Using the fact that is true for $n$, prov... |
Background
----
I am self-studying linear algebra, and I got stuck on some steps of the proof of the following theorem:
> **Theorem**$\quad$ *Let $A$ be an $n \times n$ complex matrix and $B$ be an $m \times m$ complex matrix. For any $n \times m$ complex matrix $C$, the Sylvester equation $AX+XB=C$ has a unique sol... |
I am learning parts of topos theory and it feels like I am missing something regarding indexed functors.
This comes from the definition of a locally connected geometric morphism $f : E \rightarrow F$ as being essential and the extra adjoint be F-indexed.
I am looking for a definition as I am unfamiliar with index... |
Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits can be settle to the following:
- The last $k$ digits of $a^n$ are given by $a^n\pmod{10^k}$, therefore the sequence of the $k$ last digits is eventually periodic and does not depend on subsequent digi... |
When are Kummer varieties isomorphic for abelian varieties? |
Background
----
I am self-studying linear algebra, and I got stuck on some steps of the proof of the following theorem:
> **Theorem**$\quad$ *Let $A$ be an $n \times n$ complex matrix and $B$ be an $m \times m$ complex matrix. For any $n \times m$ complex matrix $C$, the Sylvester equation $AX+XB=C$ has a unique sol... |
For human, in general, it's far easier to make an addition than a multiplication. One might argue this is just an effect of the particular way brains are wired.
But from what I find, multiplication is also more expensive on computers, where one could certainly arrange chips in an arbitrary way, or at least withou... |
Does any characterized property can give a solid account of why multiplication is a harder computation operation than addition? |
Let $a>1$ be an integer. I wish to analyze the digits of the power sequence $(a^n)_n$. The behave of extreme digits can be settle to the following:
- The last $k$ digits of $a^n$ are given by $a^n\pmod{10^k}$, therefore the sequence of the $k$ last digits is eventually periodic and does not depend on subsequent digi... |
Let
$$
f: \mathbb{R}^m \longrightarrow \mathbb{R}
$$
be a smooth, real-valued function. Then it holds that
$$
f(x)- f(0) = x\cdot \nabla f(0)+ x\ \mbox{Hess}f(0)\ x^T + \sum_{|\alpha|=3,\alpha\in\mathbb{N}^m} \frac{x^\alpha}{\alpha!} \int_{0}^{1}(1-t)^2\ {\partial^\alpha} f(tx)\ dt.
$$
Now if we assume tha... |
Let $p_n$ denote the $n$th prime throughout and define $p_n\# := p_n p_{n-1} \cdots p_1$ to be the $n$th "primorial".
Define the topology $\tau$ on $\Bbb{Z}/p_n\#$ to be that generated by the basis $B = \{ X_d : d \mid p_n\#\}$ and $X_d = \{ x \in \Bbb{Z}/p_n\#: x^2 \neq 1 \pmod q, \forall \text{ primes } q \mid d... |
I'm not sure if the last part is ok. Can you check my proof, please? Any observation wi'll be very valuable.
Let $A=\{(x_1,x_2)\in \Bbb R^2:\max(x_1,x_2)<5\}$, then show that $A$ is an open set in $\Bbb R^2$
So we want to show that $\forall y\in A (y\in \Bbb R^2), \exists \varepsilon_y\gt 0: B(y,\varepsilon_y)\s... |
> There's no smallest or biggest number in the open interval $\boldsymbol{(a, b)}$ ($a, b\in\mathbb{R}$).
Here's my proof:
*Proof*. Assume that $s$ is the smallest real number in the interval $(a, b)$. So obviously we know that $a<s$. Therefore:
$a<s\implies a\color{brown}{+s}<s\color{brown}{+s}\implies \boxed... |
Let $p_n$ denote the $n$th prime throughout and define $p_n\# := p_n p_{n-1} \cdots p_1$ to be the $n$th "primorial".
Define the topology $\tau$ on $\Bbb{Z}/p_n\#$ to be that generated by the basis $B = \{ X_d : d \mid p_n\#\}$ and $X_d = \{ x \in \Bbb{Z}/p_n\#: x^2 \neq 1 \pmod q, \forall \text{ primes } q \mid d... |
Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?
Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-\epsi... |
Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?
Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the unit sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-... |
[the question i am stuck on][1]
I have obtained the following answers:
c = 2
f(x) = sin(3*x)*cos(6*t)
g(x) = 28*sin(7*x)*(cos(14*t))
however these don't appear to make sense with the boundary conditions. What is the correct way to approach a problem like this?
apologies i am new here so please bear with me as... |
How would I approach a question on using d'Alembert's solution to find the solution of the initial boundary problem? |
Fix $k$. Asymptotically, how does the expected value of what you want behave as $n$ goes up?
Note that, if you take a small finite area (taking up a small fixed percentage $\epsilon$) of the unit sphere's surface and ask for the probability that all the vectors $X_1,X_2,\ldots,X_n$ miss that area, the answer is $(1-... |
Let $(x_j)_j$ be an ergodic and stationary sequence of random variables such that $E[x_j]=0$ (for all $j$) and:
$$\frac{1}{n}\sum_{j=0}^{n}f(x_j) \to E[f(x)] < \infty, \quad (n \to \infty)$$
for any function. We know that:
$$\log(1 + z) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k+1} z^{k+1} =z + \sum_{k=2}^{\infty... |
My question is closely related to this:
https://math.stackexchange.com/questions/614439/on-the-definition-of-a-geodesic-in-a-metric-space
I don't understand why in the definition of the geodesic there is the requirement of *constant speed*. As far as I understand, the *constant speed* is given by a particular par... |
One way to view your question is through the lens of computational complexity: why does the computational complexity of addition appear to be less than the computational complexity of multiplication? In particular, why can two $n$-bit integers be added with [$\Theta(n)$ time complexity](https://en.wikipedia.org/wiki/C... |
What is the outer automorphism group of the Pauli group? |
Let $(x_j)_j$ be an ergodic and stationary sequence of random variables such that $E[x_j]=0$ (for all $j$) and:
$$\frac{1}{n}\sum_{j=0}^{n}f(x_j) \to E[f(x)] < \infty, \quad (n \to \infty)$$
for any function. We know that:
$$\log(1 + z) = \sum_{k=0}^{\infty} \frac{(-1)^k}{k+1} z^{k+1} =z + \sum_{k=2}^{\infty... |
Use the function $$f(x) = \frac{1}{\operatorname{agm}(1 + x, 1 - x)}\tag{1}$$ which has the property that $$f(x) = \frac{1}{1 + x}\cdot f\left(\frac{2\sqrt{x}}{1 + x}\right)\tag{2}$$ and use the series expansion $$f(x) = 1 + a_{1}x^{2} + a_{2}x^{4} + \cdots\tag{3}$$ (note that the function $f$ is even) and find coeffic... |
Let $X$ be an equidimensional reduced scheme of finite type over an algebraically closed field $k$. Assume that $X$ has two irreducible components $X_1$ and $X_2$. Assume also that $X_1$ and $X_2$ are nonsingular varieties which meet transversally at $Y := X_1 \cap X_2$, so that $Y$ is a nonsingular variety of codimens... |
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