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Fujimura's problem
Let [math]\overline{c}^\mu_n[/math] the largest subset of the triangular grid
[math]\Delta_n := \{ (a,b,c) \in {\Bbb Z}_+^3: a+b+c=n \}[/math]
which contains no equilateral triangles [math](a+r,b,c), (a,b+r,c), (a,b,c+r)[/math] with [math]r \gt 0[/math]; call such sets
triangle-free. (It is an intere... |
"$X$ and $Y$ are identically distributed" means that $P(X \in B)=P(Y \in B)$ for each Borel set $B$. Knowing that $F_X=F_Y$ means you know $P(X \in (-\infty,a])=P(Y \in (-\infty,a])$ for every real number $a$. So you prove the desired result in three steps:
$P(X \in U)=P(Y \in U)$ for each open $U$. If $P(X \in U)=P(Y ... |
I am trying to implement the following optimization (from this paper) in Matlab using fmincon:
$\min_\omega\omega'\Sigma\omega$ subject to $\min_Ur_p \geq r_0$
where $\Sigma$ is a positive definite matrix and $\omega$ is a vector of weights that sum to 1. Also, we have $r_p=\alpha'\omega$ and $U$ is the circle centred ... |
OpenCV 4.1.1
Open Source Computer Vision
In this tutorial you will learn how to:
Principal Component Analysis (PCA) is a statistical procedure that extracts the most important features of a dataset.
Consider that you have a set of 2D points as it is shown in the figure above. Each dimension corresponds to a feature you... |
This question already has an answer here:
I'm looking to evaluate whether the following is true or false:
For rotation of a rigid body about an arbitrary axis, the angular momentum always points along the axis of rotation.
Let:
$$r = x \hat i + y \hat j + z \hat k$$ and
$$ v = v_x \hat i + v_y \hat j + v_z \hat k$$
The... |
I recently asked about the Mahalanobis distance and I got pretty good answers in this post:
I think I got the idea, but what I still felt missing was the derivation of the formula for the Mahalanobis distance. So my question is: "How does one derive the formula for Mahalanobis distance?"
Why does the formula have the f... |
The aleph numbers, $\aleph_\alpha$
The aleph function, denoted $\aleph$, provides a 1 to 1 correspondence between the ordinal and the cardinal numbers. In fact, it is the only order-isomorphism between the ordinals and cardinals, with respect to membership. It is a strictly monotone ordinal function which can be define... |
Theorem. Let $T$ be an operator on the finite dimensional complex vector space $\mathbf{W}$. The characteristic polynomial of $T$ equals the minimal polynomial of $T$ if and only if the dimension of each eigenspace of $T$ is $1$.
Proof. Let the characteristic and minimal polynomial be, respectively, $\chi(t)$ and $\mu(... |
Home
Integration by PartsIntegration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
In the small amount of physics that I have learned thus far, there seems to be a (possibly superficial pattern) that I have been wondering about.
The formula for the kinetic energy of a moving particle is $\frac{1}{2}mv^2$.
The formula for kinetic rotational energy is $\frac{1}{2}I\omega^2$.
The formula for energy stor... |
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∫01(ln(1+x))2ln(x)ln(1−x)1−x dx=72ζ(3)(ln2)2−π26(ln2)3−π22ζ(3)+6ζ(5)−π448ln2\int_0^1\frac{(\ln(1+x))^2\ln(x)\ln(1-x)}{1-x} \, dx=\dfrac{7}{2}\zeta(3){(\ln 2)^2}-\dfrac{\pi^2}{6}{(\ln 2)^3}-\dfrac{\pi^2}{2}\zeta(3)+{6}\zeta(5)-\dfrac{\pi^4}{48}\ln2∫011−x(ln(1+x))2... |
I am master student and doing an assignment of Finite element method. In the instruction I could not understand the derivation of the weak form, which should not be difficult. I'm sorry for posting this easy and most likely not helpful question for other people.
So the derivation is about the weak form of the integral ... |
Let $G_1$ and $G_1$ be two semisimple algebraic groups defined over $\mathbb{Q}$, suppose we have a surjective homomorphism $f: G_1\to G_2$, with finite kernel contained in the center of $G_1$.
By congruence subgroup of $G_i$, for $i=1,2$, we means $K\cap G_i(\mathbb{Q})$, with $K$ compact open subgroup of $G_i(\mathbb... |
A curiosity that's been bugging me. More precisely:
Given any integers $b\geq 1$ and $n\geq 2$, there exist integers $0\leq k, l\leq b-1$ such that $b$ divides $n^l(n^k - 1)$ exactly.
The question in the title is obviously the case when n = 10. This serves as a motivating example: if we take b = 7 and n = 10, then k = ... |
I am running various models in R for sake of prediction. If I run a model and a specific variable is showing itself to be insignificant (say, at the alpha=0.05 level), would I want to simply discard this variable? Would my next step be to remove the variable from the model and re-run that same type of model, then see h... |
When you say "why aren't things being
destroyed", you presumably mean "why aren't the chemical bonds that hold objects together being broken". Now, we can determine the energy it takes to break a bond - that's called the "bond energy". Let's take, for example, a carbon-carbon bond, since it's a common one in our bodies... |
Automata Theory- Brzozowski Algebraic Method
A LaTeX typeset copy of this tutorial is below for download:
Brzozowski_Algebraic_Method.pdf
(126.4K)
Number of downloads: 1388
I. Introduction
Recall that a language is regular if and only if it is accepted by some finite state automaton. In other words, each regular expres... |
Difference between revisions of "Fujimura's problem"
(→n=6)
(→n=6)
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Therefore 222 is removed. There are six disjoint triangles 150-420-123, 051-321-024, 231-501-204, 132-402-105, 510-150-114, and 312-042-015. So 600, 411, 393, 114, and 006 are open. 600-240 open forces 204 to be removed and 600-150 o... |
We have this formula for centripetal acceleration - $a = v \frac{d\theta}{dt} = v\omega = \frac{v^2}{r}$
but in case of
usual acceleration i know that speed in $t_1 = v_0 + a \cdot (t_1 - t_0)$
but in circular case i don't understande the nature of acceleration, speed is always the same if motion is uniform, but accele... |
The "iterated" limit$$ \lim_{x\to 0} \lim_{y\to 0} x\sin \Bigl(\frac{1}{y}\Bigr)$$does not exists, because it does not exist the inner limit.
EDIT: my first treatment of the limit for $(x,y)\to (0,0)$ was not correct. I think that now all works well.
We have to be careful about the following limit in $\mathbb{R}^2$$$\l... |
If a matrix $A$ satisfies $x^TAx<0$ for some vector $x \neq 0.$ I wanna show that $\|A\| \neq 0$ for any matrix norm. Another one, if the spectral radius of $B, \rho(B)>1.$ I also want to show that $\|B\| \neq 0$ for any matrix norm.
I try like this: Since $x^TAx < 0$ for some $ x \neq 0,$ then $ 0 < \|x^TAx\| \leq \|x... |
시간 제한 메모리 제한 제출 정답 맞은 사람 정답 비율 2 초 512 MB 11 6 6 66.667%
Bessie's younger cousins, Ella and Bella, are visiting the farm. Unfortunately, they have been causing nothing but mischief since they arrived.
In their latest scheme, they have decided to mow as much grass as they can. The farm's prime grassland is in the shape ... |
Let X be a complex connected projective algebraic surface and let L be an ample line bundle on X. The maps associated with the pluriadjoint bundles t(K_X+L), t \geq 2, are studied by combining an ampleness result for K_X+L with a very recent result by Reider. It turns out that apart from some exceptions and up to reduc... |
The IPv6 (Internet Protocol version 6) has a very large address space. In fact, since an IPv6 address is $128$ bits long, the number of possible IPv6 addresses is: $$ \boxed{ N_{\textrm{IPv6}} = 2^{128} \approx 3.4 \times 10^{38} } $$ For comparison, the number of:
cells in a human body is about $100$ trillion ($10^{14... |
Thanks so much to everyone for trying out my transfinite epistemic logic puzzle, which I have given the name Cheryl’s Rational Gifts, on account of her gifts to Albert and Bernard. I hope that everyone enjoyed the puzzle. See the list of solvers and honorable mentions at the bottom of this post. Congratulations!
As I d... |
Let's light things up ( with an FPGA )!
Finally got around to learn the tools and implement the project on to the DE0 Nano. Check it out!
The Idea: Restate and test MILP model for determining the parameters of the clap algorithm.
With the help of GNU Glpk and PyGlpk, the mixed integer linear programming (MILP) model th... |
Ineffable cardinal
Ineffable cardinals were introduced by Jensen and Kunen in [1] and arose out of their study of $\diamondsuit$ principles. An uncountable regular cardinal $\kappa$ is ineffable if for every sequence $\langle A_\alpha\mid \alpha<\kappa\rangle$ with $A_\alpha\subseteq \alpha$ there is $A\subseteq\kappa$... |
We have written in our text book:
Let $(X_{1},\mathcal{A}_{1},\mu_{1})$ and $(X_{2},\mathcal{A}_{2},\mu_{2})$ be two $\sigma-$finite measures. Define $(X,\mathcal{A},\mu):=(X_{1}\times X_{2},\mathcal{A_{1}} \otimes \mathcal{A_{2}},\mu_{1}\otimes\mu_{2})$
Let $f: X \to \bar{\mathbb R}$ be $\mathcal{A}-$measurable
Then f... |
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1. Observation of a peaking structure in the J/psi phi mass spectrum from B-+/- -> J/psi phi K-+/- decays
PHYSICS LETTERS B, ISSN 0370-2693, 06/2014, Volume 734, Issue 37... |
Home
Integration by PartsIntegration by Parts
Examples
Integration by Parts with a definite integral
Going in Circles
Tricks of the Trade
Integrals of Trig FunctionsAntiderivatives of Basic Trigonometric Functions
Product of Sines and Cosines (mixed even and odd powers or only odd powers)
Product of Sines and Cosines (... |
I expect most readers of this blog to be familiar with the concept of lossy data compression. For instance, whenever you rip a song from a CD to a lossy format such as MP3 or Ogg Vorbis, you are effectively discarding some data from the song to make it smaller while still preserving most of the audio. A good compressio... |
By applying a voltage and a magnetic field on a (let's say metallic to keep things as simple as possible) sample, one is able to create the Hall effect and to obtain the Hall coefficient $R_H \sim 1/n$ where $n$ is the charge carrier density. But what is $n$, really? Is it the $n$ that appears in the conductivity formu... |
Sage wrongly suggests there are only trivial solutions, why?
I have a system of two quadratic equations which I want to solve. WolframAlpha shows me whole families of solution, but sage says there are only trivial solutions. The equations are
\begin{align} d\cdot\lvert\alpha\rvert^2 + \alpha^* \beta + \alpha\beta^* ==&... |
I've been thinking about ways to tackle an epidemic modelling problem I've been working on, and I've come up against a conceptual difficulty over the way survival analysis works. Here's a really simplified version with all the extraneous details stripped out.
There is a collection of $n$ individuals. At the beginning (... |
I think the picture can explain it better than words, but I'm wondering how to figure this out. Given three ratios of distances from corners, not lengths (in the picture I set the base, $base=1$, to be the distance from the top left corner, while the other two corners are of lengths $\alpha\cdot$$base$ and $\beta\cdot$... |
Probability Seminar Spring 2019 Thursdays in 901 Van Vleck Hall at 2:25 PM, unless otherwise noted. We usually end for questions at 3:15 PM.
If you would like to sign up for the email list to receive seminar announcements then please send an email to join-probsem@lists.wisc.edu
January 31, Oanh Nguyen, Princeton
Title:... |
In this post, we will prove that the set $\mathcal{L}_{\{0,1\}}$ of all languages over the set $\{0,1\}$ is not countable, i.e., we cannot enumerate the infinitely many languages in $\mathcal{L}_{\{0,1\}}$. A language over $\{0,1\}$ is a set of finite-length strings formed using only the symbols $0$ and $1$. For exampl... |
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@@ -6,7 +6,14 @@ title: Solver / Model Optimization
The solver orchestra... |
To the best of my knowledge, the main strategy to compute a moduli space $M(A, J)$ consists in:
finding a symplectic form $\omega$ on $M$ which (at least) tames $J$ (ideally, $J$ and $\omega$ are compatible) and for which $J$ is regular
i.e. a regular value for the projection map $\mathcal{M}(A, \mathcal{J}_{\omega}) :... |
I am studying entanglement entropy.
It's fullfilled for any local quantum system that the entanglement entropy of a region $A$ in a highly mixed state is extensvie,
$$ S_A \sim \frac{\text{Vol}(A)}{\epsilon^d} $$
where $\epsilon$ is the length between sites (or the UV cutoff for a regularized QFT). This is because
$$S_... |
Consider $N$ independent scalar fields $φ_i (x)$ in 4D space. Also consider a lagrangian density $$\mathcal{L} = \mathcal{L}(φ_i, \partial_μφ_i).$$
Suppose we perform the following infinitesimal transformations: $$x'^μ = x^μ + ε^α Χ^μ_α \tag{1} $$ $$φ_i '(x')=φ_i(x)+ ε^α Ψ_{iα} .\tag{2} $$
Let us denote $δφ = φ'(x')-φ(... |
8. Natural Deduction for First Order Logic¶ 8.1. Rules of Inference¶
In the last chapter, we discussed the language of first-order logic, and the rules that govern their use. We summarize them here:
The universal quantifier:
In the introduction rule, \(x\) should not be free in any uncanceled hypothesis. In the elimina... |
208 1 1. Homework Statement
Find basis and dimension of [itex]V,W,V\cap W,V+W[/itex] where [itex]V=\{p\in\mathbb{R_4}(x):p^{'}(0) \wedge p(1)=p(0)=p(-1)\},W=\{p\in\mathbb{R_4}(x):p(1)=0\}[/itex]
2. Homework Equations
-Vector spaces
3. The Attempt at a Solution
Could someone give a hint how to get general representation... |
J. D. Hamkins, “Tall cardinals,” Math.~Logic Q., vol. 55, iss. 1, pp. 68-86, 2009.
@ARTICLE{Hamkins2009:TallCardinals, AUTHOR = {Hamkins, Joel D.}, TITLE = {Tall cardinals}, JOURNAL = {Math.~Logic Q.}, FJOURNAL = {Mathematical Logic Quarterly}, VOLUME = {55}, YEAR = {2009}, NUMBER = {1}, PAGES = {68--86}, ISSN = {0942-... |
For your first question, the original paper (
J. Chem. Soc., Dalton Trans. 1984, 1349–1356) that described the geometry index $\tau_5$ defined it as an "index of trigonality". For example, they write for a compound with $\tau_5=0.48$
By this criterion, the irregular co-ordination geometry of $\ce{[Cu(bmdhp)(OH2)]2+}$ i... |
Let $A$ be a Metzler matrix, i.e. a real matrix (not necessarily symmetric) whose off-diagonal elements are all non-negative. Then, for $t\ge 0$, the matrix exponential $\exp(At)$ will have all non-negative elements.
Numerically, it seems that given a single off-diagonal element of $\exp(At)$, it is always a log-concav... |
Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:
Fact:If $\kappa$ is weakly compact, and $P$ has size $\kappa$ and is $\kappa$-c.c., then for any $P$-name $\tau$ for a set of ordina... |
You are here Guitar Speaker Power Handling
The power rating of a guitar speaker is an indication of how much power it can handle without being damaged thermally or mechanically. It is not an indication of how loud the speaker will sound in comparison to other speakers. Let us examine the basics of how speakers work, ho... |
Difference between revisions of "Higher-dimensional Fujimura"
(New page: Let <math>\overline{c}^\mu_{n,4}</math> be the largest subset of the tetrahedral grid: :<math> \{ (a,b,c,d) \in {\Bbb Z}_+^4: a+b+c+d=n \}</math> which contains no tetrahedrons <math>(a+...)
(→General n)
(3 intermediate revisions by 3 users not sh... |
The following question arised as a side-question in a geometric problem. It has a "feel" similar to problems in Ramsey-theory, but I have not found any mention of it (also I'm not very familiar with the field). Was this problem considered before? Does it have an easy answer?
Consider the set of grid points $[n] \times ... |
Integer programming algorithms minimize or maximize a linear function subject to equality, inequality, and integer constraints. Integer constraints restrict some or all of the variables in the optimization problem to take on only integer values. This enables accurate modeling of problems involving discrete quantities (... |
This question is based on exercise $2.14$ of chapter $2$ of Hartshorne.
Suppose $\varphi:S\rightarrow T$ is a graded homomorphism of graded (commutative, unital) rings such that $\varphi_d := \varphi|_d$ is an isomorphism for all $d$ sufficiently large. Then I want to show that the natural morphism $ f: $Proj $T \right... |
In the use of Bayes' Theorem to calculate the posterior probabilities that constitute inference about model parameters, the weak likelihood principle is automatically adhered to:
$$\mathrm{posterior} \propto \mathrm{prior} \times \mathrm{likelihood}$$
Nevertheless, in some objective Bayesian approaches the sampling sch... |
Football: Mirror Icosahedron
The surface of a classic soccer ball is composed of 12 slightly curved black regular pentagons and 20 white regular hexagons.
By the way, such a ball was not always considered ”classic”: this cut and colouring were first used for the official world cup ball in 1970 in Mexico. The black-and-... |
A Three Group Split Problem
Solution
Let's represent the three group split by three numbers "###," each standing for the amount of even numbers in a group. The order of the groups is of no consequence. $000$ is the starting configuration. There are nine open slots to place the first even number in. However it is done, ... |
Rotatable with your mouse after you click on “Read the rest of this entry“.
Most of randform readers might have heard that the socalled greenhouse effect is one of the main causes of global warming.
The effect is not easy to understand. There are two posts which give a nice intro to the greenhouse effect on Azimuth. On... |
I have read it a thousand times: "you only need local information to compute derivatives." To be more precise: when you take a derivative, in say point $a$, what you are essentially doing is taking a limit, so you only need to look at the open region $ (a-\delta,a+\delta) $.
Taylor's theorem seems to contradict this: f... |
Astrid the astronaut is floating in a grid. Each time she pushes off she keeps gliding until she collides with a solid wall, marked by a thicker line. From such a wall she can propel herself either parallel or perpendicular to the wall, but always travelling directly \(\leftarrow, \rightarrow, \uparrow, \downarrow\). F... |
Hi
is there any lower bound for $\Re\zeta(1+it)$.
I did try with computer until some ordinate and I saw $\Re\zeta(1+it)>0$.
If it is true, is there any reference to prove it. thanks
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.Sign up to join this communi... |
Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$: $$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$ where $h,R,w$ stand for the size of the class group, the regulator and the size of the subgroup of roots of unity.
On the other hand, we have the ... |
Trigonometry, Clocks, and YouWritten by Curtis Dyer Edit (9/28/2016): some major conceptual changes have been made to the section regarding explanations of the placement of the hour labels. Departing Euclidean Planes
Most people are familiar with the basics of graphing things like linear functions of the form $Ax + By ... |
Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity
1.
Institute of Mathematical Sciences, Renmin University, Beijing 100872, China
2.
Institut für Mathematik, Universität Paderborn, 33098 Paderborn, Germany
$\begin{equation} \left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\n... |
Research Open Access Published: The finite speed of propagation for solutions to stochastic viscoelastic wave equation Boundary Value Problems volume 2019, Article number: 121 (2019) Article metrics
206 Accesses
Abstract
In this paper, a class of second order stochastic evolution equations with memory
is considered, wh... |
As you said, the trend in your example data is obvious. If you want just to justify this fact by hypothesis test, than besides using linear regression (the obvious parametric choice), you can use non-parametric Mann-Kendall test for monotonic trend. The test is used to
assess if there is a monotonic upward or downward ... |
Learning Outcomes Conduct and interpret hypothesis tests for two population means, population standard deviations known
Even though this situation is not likely (knowing the population standard deviations is not likely), the following example illustrates hypothesis testing for independent means, known population standa... |
The Annals of Applied Probability Ann. Appl. Probab. Volume 24, Number 3 (2014), 1049-1080. Pathwise optimal transport bounds between a one-dimensional diffusion and its Euler scheme Abstract
In the present paper, we prove that the Wasserstein distance on the space of continuous sample-paths equipped with the supremum ... |
Difference between revisions of "Literature on Carbon Nanotube Research"
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This article can be found in our [[Archive|archive]].
This article can be found in our [[Archive|archive]].
+ + +
== In situ Observations of Catalyst Dynamics during Surface-Bound Carbon Nanotube Nucleation ==
== In situ Observat... |
A metal (or otherwise, suitably elastic) circle is cut and the points are slid up and down a vertical axis as shown:
How would one describe the resultant curves mathematically?
Physics Stack Exchange is a question and answer site for active researchers, academics and students of physics. It only takes a minute to sign ... |
You are asking for a condition on $f$ and $g$ such that the curve:$$C: f(y) - g(x) = 0$$has a uniformly bounded number of rational points.Note that if $f$ and $g$ are equivalent under an affine transformation,then $C$ is divisible by a linear factor and is not reducible. The converse is
almost true. Namely, as long as ... |
In a recent and fantastic collaboration between Jake Levinson and myself, we discovered new links between several different geometric and combinatorial constructions. We’ve weaved them together into a beautiful mathematical story, a story filled with drama and intrigue.
So let’s start in the middle.
Slider puzzles for ... |
(Sorry was asleep at that time but forgot to log out, hence the apparent lack of response)
Yes you can (since $k=\frac{2\pi}{\lambda}$). To convert from path difference to phase difference, divide by k, see this PSE for details http://physics.stackexchange.com/questions/75882/what-is-the-difference-between-phase-differ... |
I came across John Duffield Quantum Computing SE via this hot question. I was curious to see an account with 1 reputation and a question with hundreds of upvotes.It turned out that the reason why he has so little reputation despite a massively popular question is that he was suspended.May I ...
@Nelimee Do we need to m... |
Difference between revisions of "Geometry and Topology Seminar"
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Revision as of 22:39, 31 October 2016 Contents Fall 2016
date speaker title host(s) September 9 Bing Wang (UW Madison) "The extension ... |
Let $f:\mathbb R^n\to\mathbb R$ be a Morse function with uniform nondegenerate Hessian at critical points, i.e. for some $\delta>0$$$\forall x \in \{\nabla f=0\}\;\forall \xi\in\mathbb R^n: \quad |\langle \xi, \nabla^2 H(x)\xi \rangle | \geq \delta |\xi|^2 .$$
Edit: Liviu Nicolaescu pointed out a condition to ensure th... |
Overview:
When comparing two morphologically similar(i.e. geometrically similar) species that may differ in size it’s natural to ask under what circumstances their gaits might be similar. Given that the differences in size might be important-consider for example the difference in size between a cat and a rhinoceros-a d... |
Answer
The reduction formula is $$-\sin\theta$$
Work Step by Step
*Summary of the method: For a formula $f(Q\pm\theta)$ 1) See that $Q$ terminates on the $x$ or $y$ axis. If it terminates on the $x$ axis, go for Case 1. If it terminates on the $y$ axis, go for Case 2. 2) Case 1: - For a small positive value of $\theta$... |
It’s Tax Day here in the United States, and I spent a larger portion of the past weekend than I would have liked filling out the appropriate forms. But even among tax forms and legal documents you can find a mathematical gemstone or two!
The instructions for filling out, say, IRS form 1040, are rather peculiar in that ... |
Intersection of Subgroups is Subgroup/General Result Theorem
Let $\struct {G, \circ}$ be a group.
Proof
Let $H = \bigcap \mathbb S$.
Let $H_k$ be any element of $\mathbb S$.
Then:
\(\displaystyle a, b\) \(\in\) \(\displaystyle H\) \(\displaystyle \leadsto \ \ \) \(\, \displaystyle \forall k: \, \) \(\displaystyle a, b\... |
Definition:Antitransitive Relation Definition
Let $\mathcal R \subseteq S \times S$ be a relation in $S$.
$\mathcal R$ is
antitransitive if and only if: $\left({x, y}\right) \in \mathcal R \land \left({y, z}\right) \in \mathcal R \implies \left({x, z}\right) \notin \mathcal R$
that is:
$\left\{ {\left({x, y}\right), \l... |
Suppose $f(x) \in \mathbb{Z}[x]$ is such that $f(0)$ and $f(1)$ are odd. How do I show that $f(x)$ has no integer roots?
If $r$ is an integer root and $f(x)=\sum_{k=0}^na_kx^k$, $a_k\in\mathbb Z$ then $\sum_{k=0}^na_kr^k=0$.
If $r$ is even, then reducing modulo $2$ we get that $a_0\equiv 0[2]$ hence $f(0)$ is even, whi... |
Blow-up for the 3-dimensional axially symmetric harmonic map flow into $ S^2 $
1.
Instituto de Matemáticas, Universidad de Antioquia, Calle 67, No. 53–108, Medellín, Colombia
2.
Departamento de Ingeniería Matemática-CMM, Universidad de Chile, Santiago 837-0456, Chile
3.
Department of Mathematical Sciences University of... |
The points on the Elliptic Curves ($E$) over e field $K$ are forming a commutative additive group with the identity $\mathcal{O}$; the
point at infinity, also notated as $P_\infty$.
The
scalar multiplication $[k]P$ this actually means adding $P$, $k$-time itself. More formally;
let $k \in \mathbb{N}\backslash\{ 0\}$
\b... |
This is a countable family of first-order statements, so it holds for every real-closed field, since it holds over $\mathbb R$.
From a square matrix, we immediately derive that such a field must satisfy the property that the sum of two perfect squares is a perfect square. Indeed, the matrix:
$ \left(\begin{array}{cc} a... |
Definition:F-Homomorphism Definition
Let $R, S$ be rings with unity.
Let $F$ be a subfield of both $R$ and $S$.
Then a ring homomorphism $\varphi: R \to S$ is called an $F$-homomorphism if: $\forall a \in F: \map \phi a = a$ That is, $\phi \restriction_F = I_F$ where: Also see
The word
homomorphism comes from the Greek... |
In this tutorial section it is assumed that the parts Connection techniques I – naming policy ‘integrate’ and Connection techniques II – naming policy ‘encapsulate’ have been worked through.
In this section the use of ports in MOSAICmodeling is introduced. Ports are useful to define standardized input and output variab... |
Given a sequence of cadlag (right-continuous with left limits) martingales $X^n=(X^n_t)_{0\le t\le 1}$, we may use the well known criteria to determine whether it is weakly convergent, i.e. subtract a subsequence $(X^{n_k})_{k\ge 1}$ such that for all Skorokhod-continuous and bounded function $f$ one has
$$\lim_{k\to\i... |
On a (simply connected) domain $\Omega$ for a smooth vector field $F\colon \Omega \to \mathbb{R}^3$, when does $\nabla\times(\nabla\times F)=0$ imply $\nabla \times F=0$. I know that $n\cdot(\nabla\times F)=0$ on $\partial\Omega$ is sufficient, and also $t\cdot(\nabla\times F)=0$ is sufficient ($t$ the tangential). Is ... |
Basically 2 strings, $a>b$, which go into the first box and do division to output $b,r$ such that $a = bq + r$ and $r<b$, then you have to check for $r=0$ which returns $b$ if we are done, otherwise inputs $r,q$ into the division box..
There was a guy at my university who was convinced he had proven the Collatz Conject... |
I prepared $4\ \mathrm{M}$ solution of $\ce{NaOH}$, and the glass electrode I use cannot measure it the $\mathrm{pH}$ correctly. What are the other options do I have to exactly know the solution's $\mathrm{pH}$?
For $\mathrm{pH}>12$ and $\ce{Li+}$ or $\ce{Na+}$ concentrations greater than $0.1\ \mathrm{M}$, glass elect... |
Electronic Communications in Probability Electron. Commun. Probab. Volume 23 (2018), paper no. 87, 13 pp. A functional limit theorem for the profile of random recursive trees Abstract
Let $X_n(k)$ be the number of vertices at level $k$ in a random recursive tree with $n+1$ vertices. We prove a functional limit theorem ... |
I am using
Analysis on Manifolds by Munkres to study for a course and the following question comes from an early section in topology. Definition of limit point: Let $A \subset R^n$ and let $x_o \in R^n$. $x_o$ is a limit point of $A$ if, for every $r>0,B(x_o,r)$ contains a point of $A \setminus \{x_o\}$ Exercise: Let $... |
I would like to numerically solve the following system of coupled nonlinear differential equations:
$$ -\frac{\hbar^2}{2m_a} \frac{\partial^2}{\partial x^2}\psi_a + V_{ext}\psi_a + \left( g_a |\psi_a|^2 + g_{ab} |\psi_b|^2 \right)\psi_a = \mu_a \psi_a $$
$$ -\frac{\hbar^2}{2m_b} \frac{\partial^2}{\partial x^2}\psi_b + ... |
Fix an integer $h$ and let$$ S_h(x) = \sum_{t=0}^{h-1}\;\left(\frac{x+t}{p}\right) $$Where the $(\cdot/p)$ representes the Legendre symbol. We want to prove that$$ \sum_{x=0}^{p-1}S_h(x)^{2r} < (2r)^r p h^r + 4rh^{2r}\sqrt{p} $$Developping the powers in the left and inverting summations we obtain the sum $$\sum_{m_1\do... |
P(n): For all $n$, the number of straight line segments determined by $n$ points in the plane, no three of which lie on the same straight line,is: $\large \frac{n^2 - n}{2}$.
Inductive hypotheses: given $n = k$ points, assume $P(k)$ is true: $P(k) = \dfrac{k^2 - k}{2}$.
Proving $P(k+1)$ would require proving that for $... |
I am going to show in detail one unsupervised learning. The major use case is behavioral-based anomaly detection, so let’s start with that. Imagine you are collecting daily activity from people. In this example there are 6 people \(S_1 – S_6\) When all the data are sorted and pre-processed, then result might look like ... |
TextBlob Sentiment: Calculating Polarity and Subjectivity
Sunday June 7, 2015
The TextBlob package for Python is a convenient way to do a lot of Natural Language Processing (NLP) tasks. For example:
from textblob import TextBlobTextBlob("not a very great calculation").sentiment## Sentiment(polarity=-0.3076923076923077,... |
I was doing an exercise and during this exercise I had to solve the definite integral in order to calculate an area. The area was of $A=\{(x,z):x,z \geq 0, x+\frac{1}{2}z \leq 1, x^{2}+z^{2} \geq \frac{4}{5}\}$ The area is a triangle cutted by 1/4 of a circle,so the area i easy to evaluate and is $1-\frac{\pi}{5}$. I t... |
[1] Effect of Clustering
The msd of agents with and without clustering-effect was produced.
[2] Density-Dependency of Diffusion-Coefficient
A MC-simulation was made to check the density-dependency of the diffusion-coefficient.
Therefore, the "Pauli-effect" (a patch can only be occupied once), the clustering-effect (if ... |
It was an awkward problem, but application of a powerful general principle provided the breakthrough
This problem in contrast to the earlier one looked difficult. The challenge was on - would we be able to achieve its elegant solution? Not always we know about the end in the beginning of our journey, but finally concep... |
Reciprocation (pole and polar)
Like the Fano plane, the real projective plane is self-dual. How can we exchange the points and lines in such a way that collinearity becomes concurrency and vice versa? The answer lies in the pole of a line and the polar of a point.
Polar: Fix a circle $\omega$ in the Euclidean plane wit... |
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