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I am working on a problem involving the maximization of a function subject to a normalization constraint, specifically in the context of quadratic forms and eigenvalues. The objective function is defined as:
$$ F(\mathbf{x}) = \mathbf{x}^T(w\mathbf{Q_1} + (1-w)\mathbf{Q_2})\mathbf{x} $$
where $\mathbf{x}$ is the ... |
I know that this obviously isn't true, but I can't find a gap in my "proof" and it's really bothering me. Here, "variety" means reduced, separated, finite type scheme over some field $k$.
If $X$ is some variety over $k$, then the morphism $X \to \text{spec} \, k$ is finite type, thus quasicompact. Thus the preimage ... |
Every Variety is Quasicompact? |
> 20.) How many terms of the sequence with general term $a_{n}=\dfrac{3n-72}{n}$ are integers?
> 21.) How many terms of the sequence with general term $a_{n}=\dfrac{n^{3}+4n^{2}+3n+1}{n+2}$ are integers?
I could not come up with a solution for both of the problems, so help is appreciated
|
I am reading the book "Engineering Optimization: Theory and Practice, 5th Edition" on Chapter 7 and page 409 and stumble upon this system of equation
$\begin{array}{*{20}{c}}
{\left\{ {\begin{array}{*{20}{c}}
{{{\left( {{x_1} + 1} \right)}^2} - 2r\left[ {\max \left( {0,1 - {x_1}} \right)} \right] = 0}\\
{1 - 2r\l... |
(for clarification, $mod(a,b)$ is the remainder of $a/b$, so $mod(5,2) = 1$)
Graphing $y=\int_0^1mod(x,g)dg$ in desmos gives a [continuous graph][1] (though it takes ages), so it seems like it should be properly integratable. Wolpham alpha gives a [visual representation][2], but not an answer
[1]: https://www... |
I am trying to prove that the Log-Sum-Exp function converges to the maximum function, i.e.
$$ lim_{\tau\rightarrow0}\tau\log\left(\frac{1}{N}\sum_{i=1}^N\exp\left(\frac{x_{i}}{\tau}\right)\right) = \max_{i}(x_1, \dots, x_N).$$
I saw that a possible direction is to solve the limit as $\rho=1/\tau\rightarrow \infty$ an... |
In a solution to an exercise I did they said:
<br/> Fr $y\in\mathbb{N}$, $y^2\equiv-1(mod23)$ is not possible because $23$ is a prime with $23\equiv3(mod4)$.<br/>
Can someone explain to me this argument? I dont quite get that yet |
I have a scenario where probabilities of multiple ship encounters has to be calculated. Let us say that there are 4 ships in an area. The ship A has a probability of 0.4 for hitting ship B and a probability of 0.8 for hitting ship C. What would be a round probability of this ship hitting vessels. How can I calculate th... |
How can the probability of multiple events be summed together? |
The function y = e^(1/ln(x)) has a singularity at x = 1 that partitions it into two separate continuous sections. Notice that it has reflective symmetry about a y=x axis, because it can be re-written as ln(x)ln(y) = 1.
When plotted, the 'left curve', (i.e. where x < 1), has a discreet length, is continuous between t... |
Function y=e^(1/ln(x)) has a singularity at x=1 which breaks it into two separate continuous parts. What is the minimum distance between these parts? |
I have a scenario where probabilities of multiple ship encounters has to be calculated. Let us say that there are 4 ships in an area. The ship A has a probability of 0.4 for hitting ship B and a probability of 0.8 for hitting ship C and probability 0 that it hits the ship D. What would be a round probability of this sh... |
(for clarification, $\bmod(a,b)$ is the remainder of $a/b$, so $\bmod(5,2) = 1$)
Graphing $y=\int_0^1\bmod(x,g)dg$ in desmos gives a [continuous graph][1] (though it takes ages), so it seems like it should be properly integratable. Wolpham alpha gives a [visual representation][2], but not an answer
[1]: https... |
Let $S$ be a smooth degree $3$ hypersurface in $\mathbb P^3$. Let $C$ be a smooth hyperplane section of $S$. Let $l$ be a line on $S$. It can be deduced that $h^0(\mathcal O_S(2C-2l)) \leq 4$. Can it happen that $h^0(\mathcal O_S(2C-2l)) = 4$?
In general, is there a recipe to compute $h^0(\mathcal O_S(nC-2l))$ for a... |
You have to think about **all** the possibilities. I think the logic is so that for each ship (B,C,D), ship A either hits it or not. Therefore, all possibilites are roughly summarized like so:
- A doesn't hit B; doesn't hit C and doesn't hit D
- A doesn't hit B; doesn't hit C and hits D (probability 0)
- A does... |
Let's say we have a vector space $X$ and a function $p:X\to \mathbb{R}^+$ such that $p(ax)=|a|p(x)$ for all $a\in \mathbb{R}$ and $x\in X$, and such that $p(x)=0$ iff $x=0$.
Then $p$ satisfies the triangle inequality (and is therefore a norm) iff it's a convex function. This is where our connection between the tria... |
Recently I am learning about homology theory from the point of view of $R$-modules. However, I can not find a satisfactory proof for it. I have found a proof using the Barratt-Whitehead Lemma but could not find the proof of this lemma. Also tried to generalize the Zig-zag Lemma for homology groups to $R$-modules doing ... |
I recently answered this StackOverflow question https://stackoverflow.com/questions/77854835/how-to-calculate-the-number-of-ways-to-choose-y-items-from-at-most-x-groups-effi/77855848#77855848 with a conjecture I constructed just from testing a bunch of values with a computer program I wrote to explore the problem. I ha... |
**Here's a hint if you want to try it yourself first**: what happens if you divide by $\exp(\rho x_j)$ where $x_j$ is a maximal element?
Suppose $x_j \geq x_i$ for all $i = 1, \ldots, N$. We then have
$$
\frac{\sum_i x_i \exp(\rho x_i)}{\sum_i \exp(\rho x_i)} = \frac{\sum_i x_i \exp(\rho (x_i - x_j))}{\sum_i \exp(... |
**Here's a hint if you want to try it yourself first**: what happens if you divide by $\exp(\rho x_j)$ where $x_j$ is a maximal element?
Choose $x_j$ so that $x_j = \max_i (x_1, \ldots, x_N)$. We then have
$$
\frac{\sum_i x_i \exp(\rho x_i)}{\sum_i \exp(\rho x_i)} = \frac{\sum_i x_i \exp(\rho (x_i - x_j))}{\sum_i... |
Is this an adequate definition of the natural numbers?
The term *number* is synonymous with *natural number* in this context. The terms *front* and *back* are used in order to avoid using numbers *first* and *second* to define numbers.
*$1$ is a number.*
*The set $\mathbb{N}$ of numbers is the exactly that ... |
We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then:
(i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ defined as
$u(x) := |x|^{−\alpha},$ for $x \neq 0 $
Find for what $p$'s and $\alpha$'s, it holds that $ u \in W^{1,p}(B... |
I recently answered this StackOverflow question https://stackoverflow.com/questions/77854835/how-to-calculate-the-number-of-ways-to-choose-y-items-from-at-most-x-groups-effi/77855848#77855848 with a conjecture I constructed just from testing a bunch of values with a computer program I wrote to explore the problem. I ha... |
The function $y = e^{1/\ln(x)}$ has a singularity at $x = 1$ that partitions it into two separate continuous sections. Notice that it has reflective symmetry about a y=x axis, because it can be re-written as $\ln(x)\ln(y) = 1.#
When plotted, the 'left curve', (i.e. where $x < 1$), has a discreet length, is continuou... |
Function $y=e^{1/\ln(x)}$ has a singularity at x=1 which breaks it into two continuous parts. What is the minimum distance between these parts? |
Considering the algebra $A=\mathbb{C}[z;\sigma]$ of the skew polynomials, which is like $\mathbb{C}[z]$ but with the multiplication of elements defined the following way: $xb=\sigma(b)x$ and extending it to $\sigma^n(b)$:
\begin{equation}
\sigma^n(b)=
\begin{cases}
b & \text{if $n$ is even} \\
\overline{b} & \text... |
For $p\in \omega$, let $p=\{0,\ldots, p-1\}$.
Fix a pattern $a=(a_0, \ldots, a_{n-1})$. Fix a permutation $\sigma:n\to n$ and note that this yields a way of assigning a type to a count of that type. That is, we can choose $a_{\sigma(i)}$ items of type $i$ for $i\in n$. However, some of these permutations will yie... |
We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then:
(i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ defined as
$u(x) := |x|^{−\alpha},$ for $x \neq 0 $
Find for what $p$'s and $\alpha$'s, it holds that $ u \in W^{1,p}(B... |
We want to prove that there exists a Sobolev function that is unbounded on each open set. Let $N \ge 2$, and $p \in [1, N )$. Then:
(i) For $α ∈ (0, ∞)$, consider the function $u : B(0, 1) → R$ defined as
$u(x) := |x|^{−\alpha},$ for $x \neq 0 $
Find for what $p$'s and $\alpha$'s, it holds that $ u \in W^{1,p}(B... |
I'm reading D.S.Jones' book, The theory of generalized functions, and in particular I'm studying theorem 3.18 on page 84, in the proof of which I don't understand a statement he makes.
Before expressing the specific doubt, I need to provide some context to explain myself better. Let's start with two definitions.
*A... |
**Context:**
My current work involves ellipsoidal cones - the intersection of a cone and ellipsoid where the origin of each is at $(0,0,0)$ and where I say it is in 'canonical position' when the cone axis is the $(0,0,1)$ vector and the ellipsoid is rotated accordingly.
For visualization, the pictures[![enter ima... |
The function $y = e^{1/\ln(x)}$ has a singularity at $x = 1$ that partitions it into two separate continuous sections. Notice that it has reflective symmetry about a $y=x$ axis, because it can be rewritten as $\ln(x)\ln(y) = 1$.
When plotted, the 'left curve', (i.e. where $x < 1$), has a discreet length, is continuo... |
Function $y=e^{1/\ln(x)}$ has a singularity at $x=1$, which breaks it into two continuous parts. What is the minimum distance between these parts? |
In a solution to an exercise I did they said:
<br/> For $y\in\mathbb{N}$, $y^2\equiv-1\pmod {23}$ is not possible because $23$ is a prime with $23\equiv3\pmod 4$.<br/>
Can someone explain to me this argument? I don't quite get that yet |
I have been asked to find the extreme values of the function $f(x,y) = x^2 - y^2 $ in the domain $D = \{(x,y) : x^2 + y^2 = 1\}$. When applying the method of Lagrange Multipliers, there is a problem because you get: $2x = \lambda (2x)$ and $-2y = \lambda(2y)$, which doesn't seem to make sense to me, because then $\lamb... |
Lagrange multiplier fails to work? |
I am studying a little bit of basis (for a Grothendieck topology), following MacLane's *Sheaves in Geometry and Logic*. Here they give an example as follows,
> Let $\mathbf{T}$ be a small category of topological spaces, which is closed under finite limits and under taking open subsets. Define a basis $K$ by taking $... |
I was reading the notes "Introduction to Complex Analysis" by Michael E. Taylor, and I am stuck on the following exercise about the Fourier Transform and the space $\mathcal{A}(\mathbb{R})$ defined as follows:
$$\mathcal{A}\left(\mathbb{R}\right):= \left\{ f \in L^{1}(\mathbb{R}) \cap C(\mathbb{R}) : \ f \ \text{bo... |
### Problem
Consider three positive numbers $a$, $b$ and $c$.
As the function $x \mapsto \sqrt{x}$ is concave, for any $(\alpha, \beta, \gamma)$ with $\alpha + \beta + \gamma = 1$ and $\alpha,\, \beta,\, \gamma \ge 0$:
\begin{equation}
\alpha \sqrt{a} + \beta \sqrt{b} + \gamma \sqrt{c} \le \sqrt{\alpha a + \... |
The function $y = e^{1/\ln(x)}$ has a singularity at $x = 1$ that partitions it into two separate continuous sections. Notice that it has reflective symmetry about a $y=x$ axis, because it can be re-written as $\ln(x)\ln(y) = 1$.
When plotted, the 'left curve', (i.e. where $x < 1$), has a discreet length, is continu... |
Function $y=e^{1/\ln(x)}$ has a singularity at x=1 which breaks it into two continuous parts. What is the minimum distance between these parts? |
**HINT**: Consider the geometric sequence $a,ar,ar^2,ar^3,\dots,ar^{2022}$. Choose any Pythagorean triplet $(x,y,z)$. Set $r=x/y$ and $a=y^{2022}$. Do you see what the individual terms become? Does it satisfy your required property? It becomes clear that there are infinitely many sequences that satisfy your given prope... |
Function $y=e^{1/\ln(x)}$ has a singularity at $x=1$ which breaks it into two continuous parts. What is the minimum distance between these parts? |
The formula for continuos compound interest is
A = Pe^(rt)
Where is the r is the interest rate per year.
Interestingly, even though the money is compounded continuosly , it still depends on the annual interest rate rather than instantaneous interest rate. Is there any intuition to understand this? |
I want to prove that $x \sin x + 2 \cos x \lt 2$ for $x \in (0, \frac{\pi}{2})$.
I have computed the derivative:
$$f'(x) = x \cos x - \sin x$$
And we have $f'(x) = x \cos x - \sin x = x \left( \cos x - \frac{\sin x}{x} \right)$. Since $\frac{\sin x}{x} \gt \cos x$ in that interval we have that $f'(x) \lt 0$ f... |
$$\sum_{n=1}^{k}\frac{1}{n^2+n}=\frac{k}{k+1}$$
I don't think that this summation requires too much context as this is a Q&A site, but I was just wondering why the summation is evaluated so nicely. I don't have too much knowledge and creativity in evaluating sums, so is there a bunch of equations that will make this... |
To get something out of the way: not all pairs of parallel epis in $\mathsf{Set}$ have a common section. A simple example is where $E = \{a, b, c, d\}$ maps epimorphically to $\{0, 1\}$ in two ways $f, g$, where $f^{-1}(0) = \{a, b\}$ and $g^{-1}(0) = \{c, d\}$.
The intuition I have for reflexive forks as such is f... |
Honestly, I am uncertain if I am interpreting *Brian Moehring*'s comments correctly, but here is how I think we can get around any logic-violations. It is certainly true that:
$f(x)=1+\frac{x}{2!}+\frac{x^2}{3!}+\cdots+\frac{x^n}{(n+1)!}+\frac{R_{n+1,0,\exp}(x)}{x}$
Also, using Spivak's formulation, which (in the... |
I have been studying stochastic differential equations (SDE) and came across the following questions.
Let $T \in (0,\infty)$.
Let also $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be any filtered probability space with a complete filtration and $B = (B_t)_{t\in [0,T]}$ be any $(\mathbb{F},\mathbb{P})$-Brownian... |
Let us consider data $(k \in \mathbb{Z}_{\ge 1}, (s_1, \ldots, s_k) \in \mathbb{R}^k_{\ge 0})$ such that $s_1 + \ldots + s_k = 1$. The *mesh* of a datum like that is $\max (s_1, \ldots, s_k)$. Given $p > 1$, what is a nice way to see that, as the mesh tends to zero, $s_1^p + \ldots + s_k^p$ tends to zero? |
I have been studying stochastic differential equations (SDE) and came across the following questions.
Let $T \in (0,\infty)$.
Let also $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$ be any filtered probability space with a complete filtration and $B = (B_t)_{t\in [0,T]}$ be any $(\mathbb{F},\mathbb{P})$-Brownian... |
In the two substructures case for the finite element tearing and interconnecting method (FETI) from "Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations" (pp 104, Smith 1996), the finite element stiffness matrix $A$ can be represented by representing the interior indices $I$ an... |
Simple block factorization? |
Problem statement: Assume $N=\sum_{i=-\infty}^{\infty} \delta_{\Gamma_i}$ is a homogeneous Poisson point process (hPPP) on the real line with positive rate $\lambda$. Let $N$ be independent of $(U_n)_{n\in\mathbb{Z}}$ which are i.i.d. uniform on the interval $\langle 0,1\rangle$. Now consider the process $N'=\sum_{\Gam... |
Not an answer, but too long for a comment.
You should give the conventions for the Fourier transform and write what $g_\epsilon$ is. Something like $(1/\epsilon) \exp(-x^2/\epsilon)$ I presume?
The difficulty in your approach is to show that the family of function $\hat{f}\hat{g_\epsilon}$ is bounded in $L^1$.... |
I am trying to solve exercise 28 from chapter 14 of Dummit & Foote's Abstract Algebra, which says the following:
----------
28. Let $f(x)\in F[x]$ be an irreducible polynomial of degree $n$ over the field $F$, let $L$ be the splitting field of $f(x)$ over $F$ and let $\alpha$ be a root of $f(x)$ in $L$. If $K$ is a... |
I am using a space-time discretization to discretize an initial value problem, using P1-FEM for space discretization and BDF-2 for time discretization.
To fix the notations: $\Omega\subsetneq\mathbb{R}^3$ is a domain, $\{\mathcal{T_h}\}_{h\ge0}$ a family of regular tetrahedral triangulations of $\Omega$, $\mathcal{S... |
I don't understand how to find the inverse element to 7 in the residue ring modulo 449. According to Euclid's algorithm, I get -64, although the answer should be the number 385. But at the same time -64 + 449 = 385. But how do I get 38 initially????Tell me please |
I am trying to use the least squares regression to fit a curve to a table of values representing a sine wave. Similar to the question:
https://math.stackexchange.com/questions/3926007/least-squares-regression-of-sine-wave
Except I want to fit a more general formula of: $a * sin(b*x + c) + d$
So given a function ... |
Find this limit:
$$ \lim_{n \to \infty } \frac{1 + n + \frac{n^2}{2!} + \cdots + \frac{n^n}{n!}}{e^n} $$
My friend managed to solve this through some probability theory (via Poisson limit theorem and Central limit theorem, but I didn't really understand the proof). He got an answer of $ \frac{1}{2} $. But is it pos... |
I have problems with this exercise: Let $f: [a, b] \to \mathbb{R}$ be continuous. Does $$
\lim_{x \searrow 0} \frac{\int_{x^4}^{2x^4} f(y) \ dy}{x^4}
$$
exist? If so determine the limits value.<br>
I actually have no idea how to approach this. Any help is appreciated. I think I am supposed to use the special case o... |
Does $\lim_{x \searrow 0} \frac{\int_{x^4}^{2x^4} f(y) \ dy}{x^4}$ converge? |
The formula for continuos compound interest is
$$A = Pe^{(rt)}$$
Where is the $r$ is the interest rate per year.
Interestingly, even though the money is compounded continuosly , it still depends on the annual interest rate rather than instantaneous interest rate. Is there any intuition to understand thi... |
In the two substructures case for the finite element tearing and interconnecting method (FETI) from "Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations" (pp 104, Smith 1996), the finite element stiffness matrix $A$ can be represented by representing the interior indices $I$ an... |
I got stuck on the proof of Zariski and Samuel's *Commutative Algebra*, Theorem 24 in Chapter 4.
The theorem says: (I modified the expression, expanding some terms)
>Let $R$ be a noetherian ring.
$\mathfrak{a}$ is an ideal admitting an irredundant primary decomposition
$\mathfrak{a}=\cap_{i=1}^n \mathfrak{q}_i$ ... |
For a principal $P$ and an annual rate of interest $r$ (compounded annually), the amount at the end of $t$ years is given by
$$A = P(1 + r)^t$$
**Part 1**
Before we move ahead, it's important to understand the meaning and significance of the phrase "compounded annually". The phrase implies that at the end of ... |
Let $A$ be a discrete time Markov chain, with states $1$ and $2$, and transition probability matrix
$$\begin{pmatrix}
\frac{1}{2}( 1 + \alpha ) & \frac{1}{2}( 1 - \alpha ) \\
\frac{1}{2}( 1 - \alpha ) & \frac{1}{2}( 1 + \alpha )
\end{pmatrix}
$$
where $-1 < \alpha < 1$.
Let $X(1)=R^{(1)}$ and $X(2)=R^{(2)}$... |
I want to evaluate cauchy principal value by complex contour integral, but I failed. Is this integral solved by contour integral?
$$PV\int_0^\infty\frac{1}{x^3-1}\,\mathrm{d}x$$ |
How to evaluate this by contour integral? |
Let $G$ be a finite group of order $n$ generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\rangle$ to the number of cycles of $g_i$?
The character of the permutation r... |
Let $G$ be a finite group of order $n$ generated by permutations $g_1,\dots,g_s$ such that $g_1g_2\cdots g_s$ is the identity permutation.
Consider the permutation representation obtained by embedding $G$ into $S_n$. Can we relate the character of the permutation representation of $G$ on the cosets $G/\langle g_i\r... |
I'm trying to understand the single-set category in Categories for the Working Mathematician , specifically the 2-category version, and I'm confused about the commutation of $s_0$ and $s_1$.
"Similarly a 2-category can be considered to be a single set X considered
as the set of 2-cells (e.g., of natural transforma... |
After experimenting with polynomial expansions for the Riemann $\Xi(t)=\xi\left(\frac12+it\right)$-function, I landed on these two equations:
[![enter image description here][1]][1]
with $M$ the [KummerM confluent hypergeometric funcion][2] and $L$ the [generalised Laguerre polynomial][3].
Both equations are e... |
Two slightly different polynomial expansions for $\Xi(0)$. Could a connection between these two derived? |
This question has been motivated by [origianl](https://math.stackexchange.com/questions/4870751/is-it-possible-to-approximate-any-nonnegative-schwartz-function-by-square-of-sch?noredirect=1#comment10384172_4870751) and comments therein.
Let $f$ be a nonngeative Schwartz function on $\mathbb{R}^n$ and let $\alpha \... |
$\sqrt{ f(x) + \alpha e^{-\lVert x \rVert^2}}$ is a Schwartz function for any non-negatvie Schwartz $f$ and $\alpha>0$? |
>Find the values of the parameters $A$ and $\phi, $ with $A> 0$ and $0\leqslant \phi< 2\pi, $ such that
$\begin{aligned}\cos t+\sqrt{3}\sin t=A\sin(t+\phi)\end{aligned}$
for all real values of $t.$
My first idea was to expand $\cos t+\sqrt{3}\sin t=A(\sin t \cos \phi + \cos t \sin \phi)$, but I am not sure what to... |
Problem 2-2 in Lee's *Introduction to Topological Manifolds* reads:
> Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that
> every topology on $X$ is homeomorphic to exactly one on your list.
Below is my attempted solution. Is it correct, or am I missing something? Also, is there a more efficient w... |
Recently I am learning about homology theory from the point of view of $R$-modules. However, I can not find a satisfactory proof for it. I have found a proof using the Barratt-Whitehead Lemma but could not find the proof of this lemma. Also tried to generalize the zig-zag Lemma for homology groups to $R$-modules doing ... |
Problem 2-2 in Lee's *Introduction to Topological Manifolds* reads:
> Let $X = \{1, 2, 3\}$. Give a list of topologies on $X$ such that
> every topology on $X$ is homeomorphic to exactly one on your list.
Below is my attempted solution. Is it correct, or am I missing something? Also, is there a more efficient w... |
This question has been motivated by [origianl](https://math.stackexchange.com/questions/4870751/is-it-possible-to-approximate-any-nonnegative-schwartz-function-by-square-of-sch?noredirect=1#comment10384172_4870751) and comments therein.
Let $f$ be a nonnegative Schwartz function on $\mathbb{R}^n$ and let $\alpha \... |
$\sqrt{ f(x) + \alpha e^{-\lVert x \rVert^2}}$ is a Schwartz function for any nonnegative Schwartz $f$ and $\alpha>0$? |
Does Denjoy's Probabilistic Interpretation actually "prove" that the Mertens function ratio between numbers with odd number of distinct prime factors and even number of prime factors is 1? Or does it just provide an estimate/gess based on a heuristic approach?
If not, then would one get credit if he/she actually for... |
Given a real number $x>1$ and a natural number $n$, what can we say about the integer part of $x^n$ in terms of $x$ and $n$?
For simplicity, let us assume $x<2$. For the first few values of $n$, $\left\lfloor x^n\right\rfloor =1$. But, once it crosses $1$, it starts growing quite rapidly. Is it possible to express $... |
Let's assume I have two normal distributions, X and Y with known expected values and standard deviations. How to calculate the distribution of greater values from random pairs of values from these distributions?
If it's not clear, step by step looks like that:
Based on E(X) and D(X) I sample one result, got 10.
B... |
Let $x$ be a vector of an $n$-dimensional subspace $\mathcal A_n$ with $P$ the projection operator onto this subspace. Then, let $A= P(A) = \langle A\rangle_r$ be an $r$-vector. Why is $\dot \partial (\dot x \cdot A) = A\cdot \partial x$?
|
**Starting out with the wrong characterization of the formula you present seems to have set you on the wrong path.** Your first formula gives the *probability* that a random set of five distinct cards from a 52-card Poker deck will meet the criteria for "two pairs" but not for any better hand. That is, that there wil... |
I'm trying to find whether
- $\frac{\sqrt{z^2+2z+2}}{z^2-4z+3}$
is analytic in $|z| < 2$ and / or $|z| > 2$.
So $z^2-4z+3 = 0$ when $z=1,3$ but doesn't this exclude both of the regions? Because $1<2$ and $3>2$ unless I'm missing something big
|
I am concerned with the following question: Let us assume we have some nice bounded domain $\Omega$
and $u\in W^{2,p}(\Omega)$ for some $1<p<2$. Let us further assume that we know that $-\Delta u\cdot u$ is in $L^1(\Omega)$ such that the integral
$$
\int_\Omega -\Delta u\cdot u <\infty
$$
is well-defined. Observe ... |
I am doing section 3-4 exercise 12 and 13 of Differential Geometry of Curves and Surfaces by Do Carmo and was stuck on those problems. Problem 12 is:
"Prove that if w is a differentiable vector field on a compact surface S and $\alpha(t)$ is the maximal trajectory of w with $\alpha(0)=p\in S$, then $\alpha(t)$ is d... |
Assume a discrete set $A=(1,2,\cdots,N)$.
I would like to model the following random phenomenon. Subsets of this set occurr at random with the following characteristics:
- Each element $i\in A$ has a probability $p_i$ of occurring in a subset
- The probability that the empty set occurs is zero.
So, for exampl... |
How to model this probability over a discrete set? |
I'm trying to find whether
- $\frac{\sqrt{z^2+2z+2}}{z^2-4z+3}$
is analytic in $|z| < 2$ and / or $|z| > 2$.
My attempt:
So $z^2-4z+3 = 0$ when $z=1,3$ but doesn't this exclude both of the regions? Because $1<2$ and $3>2$ unless I'm missing something big
This is where I'm stuck
|
The question is Exercise 3.89 from Advanced Modern Algebra, Rotman.
Stochastic group $\sum (2,\mathbb{F}_4) < \text{GL}_2(\mathbb{F}_4)$ is defined as following:
$\sum (2,\mathbb{F}_4) = \left\{ \begin{pmatrix} a & b \\ 1-a & 1-b \end{pmatrix} \bigg| a,b \in \mathbb{F}_4, a \neq b \right\}$
$A_4$ is the altern... |
Let $x$ be a vector of an $n$-dimensional subspace $\mathcal A_n$ with $P$ the projection operator onto this subspace. Then, let $A= P(A) = \langle A\rangle_r$ be an $r$-vector. Then, I know that $\dot \partial (\dot x \cdot A) = A\cdot \partial x$ if $r=1$, but why is this true for general $r$?
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When it comes to the concept of "equally likely," students are not told what makes two outcomes equally likely, but are instead shown what it means via examples. "When flipping a coin, heads and tails are equally likely" and "When rolling a die, the faces 1-6 are equally likely" are such common examples. However, this ... |
How to argue that two outcomes are equally likely? |
I have a circle of fixed radius $r$. I have a target that is $x$ units laterally separated from the center of the circle, and $y$ units vertically. I need to calculate the angle $θ$ which is the declination of the radius that connects that target to the upper tangent on the circle.
[![visual depiction of the above... |
How to solve following binomial equation to get the assortivity? |
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