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Interested in the following function:$$ \Psi(s)=\sum_{n=2}^\infty \frac{1}{\pi(n)^s}, $$where $\pi(n)$ is the prime counting function.When $s=2$ the sum becomes the following:$$ \Psi(2)=\sum_{n=2}^\infty \frac{1}{\pi(n)^2}=1+\frac{1}{2^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{3^2}+\frac{1...
Consider a random binary str... |
@Secret et al hows this for a video game? OE Cake! fluid dynamics simulator! have been looking for something like this for yrs! just discovered it wanna try it out! anyone heard of it? anyone else wanna do some serious research on it? think it could be used to experiment with solitons=D
OE-Cake, OE-CAKE! or OE Cake is ... |
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Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
When viewing cars that are driving along side of us, sometimes their wheels appear to be turning backwards even though they are traveling in the same direction as our car. Why do they look that way?
The issue appears to be rather complex, so I do not aim at providing an exhaustive answer.
At a toy model level it is rea... |
A somewhat different Newton solver (HP35s)
08-07-2016, 08:47 PM (This post was last modified: 08-07-2016 08:48 PM by Dieter.)
Post: #1
A somewhat different Newton solver (HP35s)
I think most of us will know the usual Newton method for finding the roots of a function. This requires the evaluation of the function f(x) as... |
This is a personal interest project that has come to a dead-end. I'm looking for comments and suggestions.
Looking for interesting ways to calculate PI, I seem to have take an approach similar to the one for Viète's formula. I've come up with this extensible expression:
$${pi} \approx {2}^{3}\cdot\sqrt{2-\sqrt{2+\sqrt{... |
The answer to this question should be obvious, but I can't seem to figure it out. Suppose we have a surface $F$, and a representation $\rho : \pi_1(F)\to SU(n)$. We can define the homology with local coefficients $H_*(F,\rho)$ straightforwardly as the homology of the twisted complex $$C_*(F,\rho):=C_*(\widetilde{F};\ma... |
Definition:Perfect Number Contents Definition
A
perfect number $n$ is a (strictly) positive integer such that: $\sigma \left({n}\right) = 2 n$
where $\sigma: \Z_{>0} \to \Z_{>0}$ is the sigma function.
Let $A \left({n}\right)$ denote the abundance of $n$.
$n$ is
perfect if and only if $A \left({n}\right) = 0$.
A
perfec... |
Definition:Naturally Ordered Semigroup/Axioms
Jump to navigation Jump to search
A
Definition
A
naturally ordered semigroup is a (totally) ordered commutative semigroup $\left({S, \circ, \preceq}\right)$ satisfying:
\((NO 1)\) $:$ $S$ is well-ordered by $\preceq$ \(\displaystyle \forall T \subseteq S:\) \(\displaystyle ... |
Equivalence of Definitions of Normal Subset/3 and 4 imply 2
This article has been proposed for deletion. In particular:
Please assess the validity of this proposal. (discuss)
Superseded by contents of Equivalence of Definitions of Normal Subgroup Theorem
Let $\left({G,\circ}\right)$ be a group.
Let $S \subseteq G$.
The... |
In the Quantum Operations section in Nielsen and Chuang, (page 358 in the 2002 edition), they have the following equation: $$\varepsilon(\rho) = tr_{env} [U(\rho \otimes \rho_{env})U^\dagger]$$
They show an example where
$$\rho_{env} = |0\rangle \langle0|$$ $$U = CNOT$$
And they claim the final solution is: $P_0\rho P_... |
Differential and Integral Equations Differential Integral Equations Volume 16, Number 6 (2003), 757-768. Positive solutions for classes of $p$-Laplacian equations Abstract
We study positive $C^1(\bar{\Omega})$ solutions to classes of boundary value problems of the form \begin{eqnarray*} -\Delta_{p} u & = & g(\lambda,u)... |
I think you need to understand at what point the magnetic field needs to begin collapsing in order to prevent the projectile decelerating and hence losing built-up momentum. I have done a simulation of a 44 mm long 10 mm radius air cored solenoid to enable to concepts to be more clearly seen. I have made it this shape ... |
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Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
What does it mean to take the gradient of a vector field? $\nabla \vec{v}(x,y,z)$? I only understand what it means to take the grad of a scalar field.
The gradient of a vector is a tensor which tells us how the vector field changes in any direction. We can represent the gradient of a vector by a matrix of its component... |
How can I show that the universal cover of $SO(n)$, for $n\ge 3$, is a double cover? And how does that reflect the fact that the fundamental group of $SO(n)$ has two elements? What is the relation between the fundamental group of a topological space and its universal cover? How come that $SU(2)$ is simply connected but... |
What are the usual assumptions for linear regression?
Do they include:
a linear relationship between the independent and dependent variable independent errors normal distribution of errors homoscedasticity
Are there any others?
Cross Validated is a question and answer site for people interested in statistics, machine l... |
I have that $\mathbf{x}_{i}=(x_{i1},\ldots,x_{ip})' \sim N_{p}(0,V)$ and I'm interesting in the variance of:
$$ S = \sum_{i=1}^{n} \mathbf{x}_{i}\mathbf{x}_{i}' $$
for the case when the vectors are correlated. In my case: $\operatorname{Cov}(\mathbf{x}_{i},\mathbf{x}_{j})= - \frac{V}{n-1}$. I now that:
$$ \operatorname... |
A morphism $h$ in a category is an
if it is right-cancellative, i.e. for all morphisms $f$, $g$ in the category $f\circ h=g\circ h$ implies $f=g$. epimorphism
A function $h:A\to B$ is
(or surjective ) if $B=f[A]=\{f(a): a\in A\}$, i.e., for all $b\in B$ there exists $a\in A$ such that $f(a)=b$. onto in a (concrete) cat... |
In the book Conformal Field Theory (authors: Philippe Di Francesco, Pierre Mathieu, David Senechal), a field $f(z)$ is primary if it transforms as $$f(z) \rightarrow g(\omega)=\left( \frac{d\omega}{dz}\right)^{-h}f(z)$$ under an infinitesimal conformal transformation $z \rightarrow \omega(z)$. The physical meaning of t... |
Essentially similar question to here Different boolean degrees polynomially related? (change being error condition $\epsilon\in(0,1)$).
Let $p$ be the minimum degree (of degree $d_f$) real polynomial that represents boolean function $f$ such that $f(x)=p(x)$.
Let $p_{0,\epsilon}$ be the minimum degree (of degree $d_{0,... |
On the DNA Computer Binary Code
In any finite set we can define a
, a partial order in different ways. But here, a partial order is defined in the set of four DNA bases in such a manner that a Boolean lattice structure is obtained. A Boolean lattice is an algebraic structure that captures essential properties of both s... |
Understanding the Paraxial Gaussian Beam Formula
The Gaussian beam is recognized as one of the most useful light sources. To describe the Gaussian beam, there is a mathematical formula called the paraxial Gaussian beam formula. Today, we’ll learn about this formula, including its limitations, by using the
Electromagnet... |
I have seen that if a set $K$ on an Hilbert space $H$ is convex and strongly sequentially-closed, it is weakly closed. The teacher said that if you take a convex and weakly lower semicontinuous functional $F$, using the fact that the sets $F^{-1}(-\infty, \lambda]$ are convex and that closure implies weak closure, it i... |
Consider the predicates
$M(x,y):$ "x has sent an email to y",
$T(x,y):$ "x has called y".
The predicate variable x, y take values in the domain D = {students in the class}. I need to express these statements using symbolic logic:
"There are at least 2 students in the class such that one student has sent the other an em... |
Why there is no monopole radiation in Electromagnetic field? I read somewhere that it is impossible because it violates charge conservation. I don't understand how? How charge conservation gets violated here?
In a multipole expansion of the electric potential, outside of some charge charge distribution $\rho(\mathbf r,... |
This question already has an answer here:
Consider the following recursion: $\begin{cases} T(n) = 2T(\frac{n}{2}) + \frac{n}{\log n} &n > 1 \\ O(1) &n = 1 \end{cases}$.
The master theorem doesn't work, as the exponent of $\log n$ is negative. So I tried unfolding the relation and finally got the equation: $T(n) = n[1 +... |
My teacher gave us this problem where we need to find:
$$\lim_{(x,y) \to (0,0)} \frac{1-cos(x+y)}{x+y}$$
My first gut instinct would have been to try sandwich theorem (after attempting a few paths and getting all $0$s). However, he gave us a solution using a Taylor approximation:
$$\lim_{(x,y) \to (0,0)} \frac{\frac1 2... |
There are
many properties that are equivalent to uniqueness of factorization in $\,\Bbb Z.\:$ Below is a sample off the top of my head (by no means complete). Each provides a slightly different perspective on why uniqueness holds - perspectives that becomes clearer when one sees how these equivalent properties bifurcat... |
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Kaon femtoscopy in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV
(Elsevier, 2017-12-21)
We present the results of three-dimensional femtoscopic analyses for charged and neutral kaons recorded by ALICE in Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 2.76 TeV. Femtoscopy is used to... |
I am just a relatively new user, so please correct me if I'm wrong.
If you read about the questions with most votes in MSE, you can find that until the $35$-th of question, the questions are all asked at least 4 years ago. Some of them are even about 8 years ago.
Does this mean new questions can't receive enough attent... |
I want to find out whether the series $\sum_{n=1}^\infty \frac{(-1)^n(2n)!!}{(2n+1)!!}$ convergent and I know the alternating series test. However, I don't know whether the absolute term converges to 0 or not. I already show that it is not absolutely convergent. Thanks.
One may observe that, as $n \to \infty$, by the u... |
Definition:Barycenter Definition
Let $p_1,\ldots,p_n \in \mathcal E$ be points.
Let $\lambda_1,\ldots,\lambda_n \in k$ such that $\displaystyle \sum_{i \mathop = 1}^n \lambda_i = 1$.
The barycenter of $p_1,\ldots,p_n$ with weights $\lambda_1,\ldots,\lambda_n$ is the unique point $q$ of $\mathcal E$ such that for every ... |
We study the distribution of singularities (poles and zeros) of rational solutions of the Painlevé IV equation by means of the isomonodromic deformation method.
Seminars
An important conjecture in knot theory relates the large-$N$, double scaling limit of the colored Jones polynomial $J_{K,N}(q)$ of a knot $K$ to the h... |
I can think of one extreme case, numbers much larger than you are using. Given a real number $\delta > 0,$ the exponent of some prime $p$ in the superior highly composite number associated with $\delta$ is$$ k = \left\lfloor \frac{1}{p^\delta - 1} \right\rfloor. $$For small primes these exponents are roughly proportion... |
Abbreviation:
Grpd
A
is a category $\mathbf{C}=\langle C,\circ,\text{dom},\text{cod}\rangle$ such that groupoid
every morphism is an isomorphism: $\forall x\exists y\ x\circ y=\text{dom}(x)\text{ and }y\circ x=\text{cod}(x)$
Let $\mathbf{C}$ and $\mathbf{D}$ be Schroeder categories. A morphism from $\mathbf{C}$ to $\ma... |
What's the meaning of random variables $X_i^2(A)$
For example:
Consider we are doing Bernoulli trials, $\omega =\{A, \text{not} A\}$ with $P(A)=p$ and $P(\text{not} A)=1-p=q$, Given $n$ independent random variables $X_1,X_2,\text{...},x_n$, each taking
$$\begin{align*}X_i(A)=1,X_i(\text{not} A)=0,\end{align*}$$
set
$$\... |
Abbreviation:
CloA
A
is a modal algebra $\mathbf{A}=\langle A,\vee,0,\wedge,1,\neg,\diamond\rangle$ such that closure algebra
$\diamond$ is
: $x\le \diamond x$, $\diamond\diamond x=\diamond x$ closure operator
Remark: Closure algebras provide algebraic models for the modal logic S4. The operator $\diamond$ is the
, and... |
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Let's start trying to understand enriched profunctors. We'll start with a very nice special case: 'feasibility relations'.
Definition. Suppose \( (X, \le_X) \) and \( (Y, \le_Y) \) are preorders. Then a feasibility relation from \(X... |
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astrophysics (66)biophysics (16)chemistry (18)electric field (58)electric current (61)gravitational field (64)hydromechanics (123)nuclear physics (34)oscillations (40)quantum physics (25)magnetic field (29)mathematics (75)mechanics of a point mass (219)gas mechanics (79)mechanics of rigid bodies (188)molecular p... |
Bernoulli Bernoulli Volume 23, Number 1 (2017), 249-287. Concentration inequalities in the infinite urn scheme for occupancy counts and the missing mass, with applications Abstract
An infinite urn scheme is defined by a probability mass function $(p_{j})_{j\geq1}$ over positive integers. A random allocation consists of... |
It looks like you're new here. If you want to get involved, click one of these buttons!
Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
I’m a student with a pure math background starting to work through Arnold’s “Mathematical Methods...” and I’m struggling right of the bat with Section 1.2 on Galilean Structure. (pg 4 - 6)
So we have this affine space $A^4$ accompanied by a space of displacements $\mathbb{R}^4$. Fine.
On page 5, Arnold defined Time as ... |
Nuclear Experiment New submissions 1-9]
[ showing up to 2000 entries per page: fewer | more ]
New submissions for Tue, 15 Oct 19 [1] arXiv:1910.06086 [pdf, other] Title: Time-based Reconstruction of Hyperons at PANDA at FAIRComments: 5 pages, 4 figures, Conference Proceedings for Connecting the Dots and Workshop on Int... |
Abbreviation:
MSet
An
is a structure $\mathbf{A}=\langle A,f_m (m\in M)\rangle$, where $\mathbf M=\langleM,\cdot,1\rangle$ is a monoid, such that $\mathbf M$-set
$f_1$ is the identity map: $1x=x$ and
the monoid action associates: $(m\cdot n)x=m(nx)$
Remark: $f_m(x)=mx$ is a unary operation called
. the monoid action by... |
Search
astrophysics (66)biophysics (16)chemistry (18)electric field (58)electric current (61)gravitational field (64)hydromechanics (123)nuclear physics (34)oscillations (40)quantum physics (25)magnetic field (29)mathematics (75)mechanics of a point mass (219)gas mechanics (79)mechanics of rigid bodies (188)molecular p... |
Let $(X, \scr{A})$ be a measure space. If $\mu, \nu$ are finite signed measure, then $$|\nu + \mu|(A) \leq |\mu|(A) + |\nu|(A)$$ where $|\mu| := \mu^+ +\mu^-$ the to tal variation of $\mu$, and $\mu^+, \mu^-$ are the Jordan-decomposition of $\mu.$
I think that it is quite clear that $\mu + \nu$ is a signed measure. Let... |
If I know how long one side of a regular hexagon is, what's the formula to calculate the radius of a circle inscribed inside it?
Illustration:
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up.Sign up to ... |
I have mentioned this elsewhere, but it bears repeating because it is such an important concept:
Sufficiency pertains to
data reduction, not parameter estimation per se. Sufficiency only requires that one does not "lose information" about the parameter(s) that was present in the original sample.
Students of mathematica... |
How to Find Unknown Variables by Cramers Rule?
The concept of the matrix determinant appeared in Germany and Japan at almost identical times. Seki wrote about it first in 1683 with his
Method of Solving the Dissimulated Problems. Seki developed the pattern for determinants for $2 \times 2$, $3 \times 3$,$4 \times 4$, a... |
I was thinking about how would capillary action change in a tube (classic example) and in a tube fitted inside another tube (considering water as the liquid involved).
Height of liquid column: where:
$\gamma$ = liquid-air surface tension
$\theta$ = contact angle
$\rho$ = density of liquid
$g$ = gravity acceleration
$r$... |
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can anybody elaborate me the lcm and hcf iam not talking about the method what is its meaning and complete discription
Note by Hoor Ulain 4 years, 2 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and scie... |
I am having trouble with the following problem. I keep on getting a long unmanagable result - so any suggestion as to where I've gone wrong/how to do this would be a lifesaver! Please?
Consider a Vector Field in $\mathbb R^3 $ Given By F($\bar{x}$)=$\bar{\varepsilon} \times \bar{x}$
Where $\bar{\varepsilon}$ is a fixed... |
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Now showing items 1-1 of 1
Production of charged pions, kaons and protons at large transverse momenta in pp and Pb-Pb collisions at $\sqrt{s_{NN}}$ = 2.76 TeV
(Elsevier, 2014-09)
Transverse momentum spectra of $\pi^{\pm}, K^{\pm}$ and $p(\bar{p})$ up to $p_T$ = 20 GeV/c at mid-rapidity, |y| $\le$ 0.8, in pp and ... |
Abbreviation:
BoolLat
A
is a bounded distributive lattice$\mathbf{L}=\langle L,\vee ,0,\wedge ,1\rangle $ such that Boolean lattice
every element has a complement: $\exists y(x\vee y=1\mbox{ and }x\wedge y=0)$
Let $\mathbf{L}$ and $\mathbf{M}$ be bounded distributive lattices. A morphism from $\mathbf{L}$ to $\mathbf{M... |
Rest Mass
The rest mass of a particle is the mass a particle has when it's speed is zero, and is labelled
\[m_0\]. The mass of a particle is not fixed. As the speed increases, so does the mass, according to the equation
\[m=\frac{m_0}{\sqrt{1-v^2/c^2}}\].
Because
\[\gamma = \frac{1}{\sqrt{1-v^2/c^2}} \ge 1\],
\[m_0 \le... |
In class this week we've been learning about the CFLs and their closure properties. I've seen proofs for union, intersection and compliment but for reversal my lecturer just said its closed. I wanted to see the proof so I've been searching for the past few days but all I've found is most people just say that to reverse... |
Other answers address the question of gravity, so I'll just expand on the atmosphere topic.
The Problem
The biggest physics issue with your proposed world is the atmosphere.
A star is formed when enough gas is present that the gravity from all of the gas is enough to collapse it down into a dense hot sphere. It would t... |
We have $k$ independent random variables with exponential distribution ($T_1, T_2, \ldots , T_k$), parameters of random variables are ($\lambda,\frac{\lambda}{2},\frac{\lambda}{3},\ldots,\frac{\lambda}{k}$), what is the distribution of new variable $T = T_1 + T_2 + \cdots + T_k $
you can use the main result here applie... |
Show that $d$ is a metric for $X$ then , $d'(x,y) =\frac{d(x,y)}{1+d(x,y)}$ is a bounded metric space that gives the topology of $X$.
In dbfin.com the solution reads as follows :
Now, we show that d′ induces the same topology as d . Since $f$ and $f^{−1}(y)=\frac{y}{1−y}:[0,1)→R^+$ are continuous, $d′=f∘d$ and $d=f^{−1... |
I apologize in advance for the length.
The equation $\sin x = (\log x)^{-1}$ has exactly one solution $x_n$ in the interval $(2\pi n,2\pi n + \pi/2)$ for $n \geq 1$, and the exercise (de Bruijn, Asymptotic Methods in Analysis, ch. 2) asks me to show that
$$ x_n = 2\pi n + (\log 2\pi n)^{-1} + O((\log 2 \pi n)^{-3}). $$... |
In signal processing, cross-correlation is a measure of similarity of two waveforms as a function of a time-lag applied to one of them. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recog... |
In the course of solving a certain problem, I've had to evaluate integrals of the form:
$$\int_0^\infty \frac{x^k}{1+\cosh(x)} \mathrm{d}x $$
for several values of k. I've noticed that that, for k a positive integer other than 1, the result is seemingly always a dyadic rational multiple of $\zeta(k)$, which is not part... |
Key Idea $\ $ Composite polynomials take composite values (except for finitely many values)
Indeeed, suppose that $\ f(x)\color{#c00}{\ne 0}\ $ is a composite polynomial: $\, f(x) = g(x)h(x)\,$ with $\ g,\,h\color{#c00}{\ne \pm1}.\,$ Then $\, f(n) = g(n)h(n)\, $ is a composite integer if $\,g(n),\,h(n)\,\neq\, 0,\,\pm1... |
The OP asks:
What am I doing wrong with this method?
When should I not use polar coordinates to find limits of multivariable functions?
The answer to the second question is somewhat unsatisfactory: If you find a limit, then you can. If you don't find a limit, then you can't.
So now, let's just leave that behind us and ... |
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And why? Please choose dispassionately.
Note by Patrick Engelmann 5 years ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they... |
I need some help. This can be a dumb question as I am not an electrical engineer nor an electronics engineer, but need to solve a problem as given below. I tried to solve it but not sure if it is true because spice simulations give different result.
Where is my mistake?
Problem
Choose Emitter Resistance \$R_E\$ for the... |
Serial of year 18
You can find the serial also in the yearbook.
We are sorry, this serial has not been translated. Tasks 1. Series 18. Year - S. kinematics of point mass
* The position of point mass in time in Cartesian coordinates is described by position vector $\vect{r}(t) =(R \cos\(\omega t\)$,R sin\(\omega t\),d)\... |
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Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
For a presentation, I am learning about the Cantor Set and how it is homeomorphic to the p-adic numbers. I was reading section two of this paper. In it it states that the Cantor Set has a vanishing Lebesgue measure.
Wikipedia says: Given a subset ${\displaystyle E\subseteq \mathbb {R} } $, with the length of interval $... |
Answer
a) $ \omega_f=2.5 \ rads/s$ b) $W=3.4 \times 10^{-3}J$
Work Step by Step
a) We know that angular momentum is conserved. Since the old angular speed was 2.3 radians per second, we use the new moment of inertia to find the new angular speed: $L_0=L_f \\ I\omega_f = I\omega_0 + m\omega_0r^2 \\ .0154\omega_f=.0154(2... |
This question is prompted by (comments at) another one. There, I was surprised to find that despite traditional claims to the contrary, Boltzmann himself
did once write his formula $S=k\log W$:
$\hspace{11em}$
That’s in his book (1898, §61, p. 172), with a pointer to (1896, §8, p. 60) where he says the same thing in wo... |
The states in quantum mechanics belong to some Hilbert space while the states in quantum field theory belong to a Fock space. For simplicity, let me stick to the Fock space emerging after the quantization of a real scalar field.
A Fock space is defined as a direct sum, $$\mathcal{F}=\oplus_n\mathcal{H}_n$$ of Hilbert s... |
Canonical correlation analysis (CCA) is a technique related to principal component analysis (PCA). While it is easy to teach PCA or linear regression using a scatter plot (see a few thousand examples on google image search), I have not seen a similar intuitive two-dimensional example for CCA. How to explain visually wh... |
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Last time we learned most of the tricks needed to assemble a co-design diagram from the smaller boxes inside:
Remember, we're really building a feasibility relation out of other feasibility relations. Each smaller box is a feasibili... |
Also read, Circle Formulas Circle Theorems Lines and Angles Straight Line Coordinate Geometry Conic Section Formulas:
Since we have read simple geometrical figures in earlier classes. We already know about the importance of geometry in mathematics. Here we will learn conic section formulas. Circles, ellipses, parabolas... |
I read the following solution for Showing that NP is closed under unionand they used the same $c$ for both the verifies $V_1$ and $V_2$.
Why is it correct?
Let $L_1$ and $L_2$ be languages in $NP$. Also, for $i = 1, 2$ let $V_i(x, c)$ be an algorithm that, for a string $x$ and a possible certificate $c$, verifies wheth... |
I've tried using KVL and KCL but I always end up with two or more variables, and I've got another question, how can I know if this transistor is in the saturation region?
Call the collector node \$C\$ and call the voltage there \$V_C\$. Call the base node \$B\$ and call the voltage there \$V_B\$. We know that \$V_B=700... |
31 3
I’m trying to derive the infinitesimal volume element in spherical coordinates. Obviously there are several ways to do this. The way I was attempting it was to start with the cartesian volume element, dxdydz, and transform it using
$$dxdydz = \left (\frac{\partial x}{\partial r}dr + \frac{\partial x}{\partial \the... |
I'm trying to understand the reasons why an electric current should occur in the following circumstances:
Given a magnetic field in space, and a neutral (not charged) body moving at some speed, in a direction so the magnetic force is not zero upon its charged particles:
Will the electric current be caused by:
1.Electro... |
We know that if we take two atoms/molecules, their interaction energy shows a short-range attractive part and a medium-range attractive part.
One popular way to schematize this interaction is to employ the Lennard-Jones potential:
$$U(r) = 4 \epsilon \left[ \left( \frac{\sigma} r \right)^{12} - \left( \frac{\sigma} r \... |
Abbreviation:
CRLSgrp
A
is a residuated lattice-ordered semigroup $\mathbf{A}=\langle A, \vee, \wedge, \cdot, \to\rangle$ such that commutative residuated lattice-ordered semigroup
$\cdot$ is
: $xy=yx$ commutative
Remark: This is a template. If you know something about this class, click on the ``Edit text of this page'... |
The $\mathbb{Z_5}$-vector space $\mathfrak{B}$ 3 over the field $(\mathbb{Z_5}, +, .)$ $\mathfrak{B}$ 3over the field $(\mathbb{Z_5}, +, .)$ 1. BackgroundThis is a formal introduction to the genetic code $\mathbb{Z_5}$-vector space $\mathfrak{B}^3$ over the field $(\mathbb{Z_5}, +, .)$. This mathematical model is defin... |
I am trying to prove this limit to be true: $$\lim_{x\to a}(x^2)=(a^2)$$ using the Epsilon Delta Limit Definition.
So far I can understand how it works but I got stumped on this inequality $$|x+a|<|2a|+1$$ if $|x-a|<1$
I saw this on this following link: https://www.ma.utexas.edu/users/nrauh/teaching/m408d/limits.pdf
I ... |
Your count of the total number of inversions is right. There are $\binom n2$ pairs of elements that can be inverted, each of them is inverted in half of all permutations, and there are $n!$ permutations, for a total of $\frac14n!n(n-1)$.
Given that $p(\sigma)=\frac{\def\inv{\operatorname{inv}}\inv\sigma}{\sum_\sigma\in... |
I dont understand completely a proof of the dominated convergence theorem stated in page 104 of
Analysis III of Amann and Escher. I will transcribe here the proof and comment about my thoughts.
In the next: $(X,\mathcal A,\mu)$ is a $\sigma$-finite measure space, $E$ is a Banach space and $\mathcal L_1(X,\mu, E)$ is th... |
I don't know how to get the second line from the first line in the following:
In the above case, $Y=(y_1, \dots , y_n)^T$ is a random sample from $N(\mu,\sigma^2)$.
My trouble is in simplifying $ E\left(\left\{\sum\limits_{i=1}^n (Y_i-\mu ) \right\}^2\right)$. What I've tried:
$$ \begin{align} E\left(\left\{\sum\limits... |
I am trying to simplify Leibniz Rule to the (first) Fundamental Theorem of Calculus (FTC) but believe I am doing so incorrectly. Leibniz rule can be written as:
$$\frac{d}{dt} \int_{f(t)}^{g(t)} A(t,\sigma) d\sigma = A(t,g(t))\dot g(t) - A(t,f(t))\dot f(t) + \int_{f(t)}^{g(t)} \frac{\partial}{\partial t} A(t,\sigma) d\... |
Let $X_1,\dots,X_n$ be a sample of iid exponential random variables with mean $\beta$, and let $X_{(1)},\dots,X_{(n)}$ be the order statistics from this sample. Let $\bar X = \frac{1}{n}\sum_{i=1}^n X_i$.
Define spacings $$W_i=X_{(i+1)}-X_{(i)}\ \forall\ 1 \leq i \leq n-1\,.$$ It can be shown that each $W_i$ is also ex... |
Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm and let $c,c_0$ be the subspaces of sequences that are convergent, resp. convergent to zero. Show that:
The linear functional $\ell_0\colon c\rightarrow \mathbb{C}$ defined for $x = (x_n) \in c$ by $$ \ell_0(x) = \lim_{n\rightarrow \infty} x... |
The
dividend discount model ( DDM) is a method of valuing a company's stock price based on the theory that its stock is worth the sum of all of its future dividend payments, discounted back to their present value. [1] In other words, it is used to value stocks based on the net present value of the future dividends. The... |
A field in the $(A,B)$ representation of the Lorentz Group has a propagator that scales as $|p|^{2(s-1)}$ for $|p|\to\infty$, where $s=A+B$ is the "spin" of the field (Ref.1 §12.1) . Therefore, the propagator is a decaying (or constant) function of $p$ if and only if $s=0,\,1/2,\,1$. Otherwise, the propagator grows in ... |
The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the Cartan–Dieudonné theorem.
Yeah it does seem unreasonable to expect a finite presentation
Let (V, b) be an n-dimensional, non-degenerate s... |
The first question we have to ask is: what is a one particle state in an interacting theory? It is reasonable to require that they are states that are both momentum eigenstates and energy eigenstates. (In fact, as the Hamiltonian and the momentum operator commute, these are not two different conditions.) Weinberg, in h... |
How can i show that the following long language is not context free using the pumping lemma?
$L=\left\{abc^{i_1}bc^{i_2}...bc^{i_{2m}}def^{j_1}ef^{j_2}..ef^{j_{2n}}ghq^{k_1}hq^{k_2}...hq^{k_o}\right\}$
Such that:
$m,n,o \geq 1;$
$m>n>o>0;$
$i_1,i_2,...,i_{2m} \geq 0;$
$j_1,j_2,...,j_{2n} \geq 0;$
$k_1,k_2,...,k_o \geq ... |
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I spent a lot of time in this course explaining two big ideas:
Adjoint functors. We've focused a lot on the simplest of categories: preorders. Pairs of adjoint functors between these are also called Galois connections, and we first ... |
There's a good function going the other way, \\(f^{\ast}: PY \rightarrow PX\\), the **preimage** function, defined by
$$f^{\ast}(S \in PY) = \\{x \in X: f(x) \in S\\} .$$
Claim: this is right adjoint to the image function \\(f_{\ast}: PX \rightarrow PY\\).
Proof: \\(f_{\ast}(S) \subseteq T\\) means that \\(S\\) maps in... |
Equations for Estimating Creatinine Clearance or GFR
Considerations and Variations of Creatinine Clearance
Cockcroft-Gault 1976
1
Particularly for renally dosing medications, the Cockcroft-Gault equation has been the long-standing gold standard for the estimation of creatinine clearance for decades. The original study ... |
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Now let's look at a mathematical approach to resource theories. As I've mentioned, resource theories let us tackle questions like these:
Our first approach will only tackle question 1. Given \(y\), we will only ask
is it possible to... |
X is an arbitrary , non empty set, B(X) the set of bounded functions $f:X\rightarrow \mathbb{R}$ and $||f||_\infty = \sup_{x\in X }|f(x)|$.
Is $(B(X),||.||_\infty )$ a Banach Algebra?
My attempt at showing that this is true:
Definition of a Banach Algebra: A normed space E with elements f,g,... is called normed Algebra... |
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