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Here's a question whose answer turns out to be very useful: Given two vectors, what is the angle between them?
It may not be immediately clear that the question makes sense, but it's not hard to turn it into a question that does. Since vectors have no position, we are as usual free to place vectors wherever we like. If... |
Proceedings of the Centre for Mathematics and its Applications Proc. Centre Math. Appl. Workshop/Miniconference on Functional Analysis and Optimization. Simon Fitzpatrick and John Giles, eds. Proceedings of the Centre for Mathematical Analysis, v. 20. (Canberra AUS: Centre for Mathematics and its Applications, Mathemat... |
I'm implementing a interpreter for lambda calculus, and now I want to add the equality type. The introduction rule for it is easy, but the elimination rule is rather obscure for me. I found this stackoverflow thread, but it explains the J axiom only in one sentence. How can it be understood intuitively?
A complete unde... |
Now as for an intuitive explanation concerning Pete Clark's question:
I would be interested in, at least, a reference for the fact that
K(j(E),h(E[N])) contains the N-ray class field of K for arbitrary
orders.
Here is my intuitive point of view on it (I put it in another answer because I won't try to be perfectly rigor... |
Yes it would, by a few percent. It may or may not be a goal worth pursuing, but there is more to it.Different reactions would be slowed down to a different extent. Tiny as they are, these discrepancies suffice to disrupt the delicate biochemical machinery of the living cell. No life form more advanced than bacteria is ... |
When you first encountered the trigonometric functions it was probablyin the context of "triangle trigonometry,'' defining, for example,the sine of an angle as the "side opposite over the hypotenuse.''While this will still be useful in an informal way, we need to use amore expansive definition of the trigonometric func... |
We have already seen that a convenient way to describe a line in three dimensions is to provide a vector that "points to'' every point on the line as a parameter $t$ varies, like $$\langle 1,2,3\rangle+t\langle 1,-2,2\rangle =\langle 1+t,2-2t,3+2t\rangle.$$ Except that this gives a particularly simple geometric object,... |
Since the question about $U^c\times V^c$ is answered I answer just the other ones.
The projections $$p_X:X\times Y\to X,~p_X(x,y)=x\text{ and }p_Y:X\times Y\to Y,~p_Y(x,y)=y$$are continuous in the product topology. If $A\subset X$ closed, $U\subset X$ open, $B\subset Y$ closed and $V\subset Y$ open, then by continuoity... |
Set-theoretic arguments often make use of the fact that a particular property $\varphi$ is
local, in the sense that instances of the property can be verified by checking certain facts in only a bounded part of the set-theoretic universe, such as inside some rank-initial segment $V_\theta$ or inside the collection $H_\k... |
Answer
The solution set to this problem is $$\{\frac{\pi}{4},\frac{3\pi}{4},\frac{7\pi}{6},\frac{11\pi}{6}\}$$
Work Step by Step
$$(\csc x+2)(\csc x-\sqrt2)=0$$ over interval $[0,2\pi)$ 1) Solve the equation: $$(\csc x+2)(\csc x-\sqrt2)=0$$ $$\csc x+2=0\hspace{1cm}\text{or}\hspace{1cm}\csc x-\sqrt2=0$$ $$\csc x=-2\hspa... |
I have the following delay-differential-algebraic system:
\begin{align} c(t) =& \kappa\min (1, \tfrac{\beta}{\alpha} a(t)), \tag{1a}\\ a(t) =& \left\{ \begin{array}{lcl} \frac{\alpha}{\beta} & \text{if} & s(t) > 0,\\ \min(\frac{\alpha}{\beta}, c(t-\Delta)) & \text{if} & s(t) = 0 \end{array}\right. ,\tag{1b}\\ \frac{\te... |
I have
$$a(x,t+2) - a(x,t) = -\cos{\theta} [a(x+1,t+1) + a(x-1,t+1)]$$ Setting $t' = -t\cos(\theta)$ fetches under the continuum approximation $$2 \frac{\partial a(x,t')}{\partial t'} = a(x-1,t') - a(x+1,t')$$ which resembles Bessel functions's recurrence relations.
I used
RSolve but it is of no help.
In[1]:= RSolve[a[... |
Is there an "interesting" class of analytically-integrable, divergence-free vector fields over $\mathbb{R}^2$ and/or $\mathbb{R}^3$?
That is, I'm looking for a large class of vector fields given by $V(x;\eta)$ for $x\in\mathbb{R}^k$, $k\in\{2,3\}$, parameterized by a fixed set of values $\eta\in\mathbb{R}^n$ such that:... |
@JosephWright Well, we still need table notes etc. But just being able to selectably switch off parts of the parsing one does not need... For example, if a user specifies format 2.4, does the parser even need to look for e syntax, or ()'s?
@daleif What I am doing to speed things up is to store the data in a dedicated f... |
Let $N$ be a set for which $$\forall a\left(a\in N\iff a=\{\}\lor\exists b\left(b\in N\land a=\left\{b\right\}\right)\right)$$ Prove that $$\forall P\left(\left\{\right\}\in P\land \forall a\left(a\in N\implies\left(a\in P\implies\{a\}\in P\right)\right)\implies\forall a\left(a\in N\implies a\in P\right)\right)$$
I'm h... |
Answer
Please see the work below.
Work Step by Step
We know that $H=kA\frac{\Delta T}{\Delta x}$ We plug in the known values to obtain: $H=(1)(8.0\times 12)(\frac{20-10}{0.23})$ $H=4200W$
You can help us out by revising, improving and updating this answer.Update this answer
After you claim an answer you’ll have
24 hour... |
This relation can be obtained if we realize that Total Cost can be written as a function of output (which in turn is a function of input factors), but also, that Total Cost equals total payments to factors of production. In real terms,
$$TC=TC(Q) = (w/p)N + (r/p)K$$with the payment rates exogenous. But $Q = Q(N,K)$. Di... |
Strongly compact cardinal
The strongly compact cardinals have their origins in the generalization of the compactness theorem of first order logic to infinitary languages, for anuncountable cardinal $\kappa$ is
strongly compact if the infinitary logic $L_{\kappa,\kappa}$ exhibits the $\kappa$-compactness property. It tu... |
I am trying to solve this diffusion equation : $\dfrac{\partial D\dfrac{\partial f}{\partial x}}{\partial x}+S = \dfrac{\partial f}{\partial t}$ ($D$ is not constant and varies according to $x$) with the following BC: $f(x,0)=1 , f(0,t)=0, \dfrac{\partial f}{\partial x}(1,t)=0$
I am using a central finite difference sc... |
Worldly cardinal Every inaccessible cardinal is worldly. Nevertheless, the least worldly cardinal is singular and hence not inaccessible. The least worldly cardinal has cofinality $\omega$. Indeed, the next worldly cardinal above any ordinal, if any exist, has cofinality $\omega$. Any worldly cardinal $\kappa$ of uncou... |
I'm going through the paper
Weight Uncertainty in Neural Networks by Google Deepmind. In the final line of the proof of proposition 1, the integral and the derivative are swapped. Then the derivative is taken. But this somehow yields 2 derivatives of $f$ with respect to $\theta$. I thought that this was the result of a... |
It looks exactly like an integrator to me. Since $$y[k] = y[k-1]+x[k] = y[k-2] + x[k-1] +x[k] = \sum{x}$$
Applying the Z-transform gives \begin{align} Y(z) &= Y(z)\cdot z^{-1} + X(z)\\ \Rightarrow\frac{Y(z)}{X(z)} &= \frac{1}{1-z^{-1}} \end{align}
When I convert $\frac{1}{1-z^{-1}}$ into Lapace transform in Matlab usin... |
hw4 discussion (ECE438BoutinFall09)
Put your content here . . .
Shouldn't one of the bounds in the first part of question 1 be non-inclusive since if you plug in $ \frac{\pi}{2} $ for the top part, you get $ \frac{2}{\pi} - 1 $ instead of
0 ? - pclay The bounds are correct. The expression over the interval $ | \omega |... |
How do I express $\frac{1}{3^n}$, or $3^{-n}$ as a typical geometric series of the form: $ar^{n-1}$ ?
This is a small part of a larger question, in which I'm trying to find the convergence of $\sum_{n=0}^\infty \frac{1+2^n}{3^n}$.
I broke the series into two separate series', $\sum_{n=0}^\infty \frac{1}{3^n} + \sum_{n=... |
I get so frustrated with modular arithmetic. It seems like every example I look at leaves steps out. I am trying to solve this problem:
Solve the linear congruence equations for x:
$x \equiv 2 \mod 7$
$x \equiv 1 \mod 3$
Ok, so I start
We know that 1st equation has a solution when $7 \mid (x-2)$. So there exists an int... |
Can I be a pedant and say that if the question states that $\langle \alpha \vert A \vert \alpha \rangle = 0$ for every vector $\lvert \alpha \rangle$, that means that $A$ is everywhere defined, so there are no domain issues?
Gravitational optics is very different from quantum optics, if by the latter you mean the quant... |
Contents Practice Question on Computing the Output of an LTI system by Convolution
The unit impulse response h(t) of a DT LTI system is
$ h(t)= e^{-3t }u(t). \ $
Use convolution to compute the system's response to the input
$ x(t)= u(t). \ $
You will receive feedback from your instructor and TA directly on this page. O... |
Buchholz's ψ functions
Buchholz's functions are a hierarchy of single-argument ordinal functions \( (\psi _{\nu }:On\rightarrow On)_{\nu\le\omega}\) introduced by German mathematician Wilfried Buchholz in 1981.
Contents Basic Notions
Small Greek letters always denote ordinals. Each ordinal \(\alpha\) is identified with... |
I realize that there has already been an answer to this problem. But I want to know if my proof was correct. Thank you for your time.
Problem
Suppose A,B, and C are sets and $f: A \rightarrow B$
Suppose that C has at least two elements, and for all functions $g$ and $h$ from B to C, if $g \circ h = h \circ f$ then $g =... |
(One version of) the Whitehead theorem states that a homology equivalence between simply connected CW complexes is a homotopy equivalence. Does the following generalisation hold true?
Suppose $X,Y$ are two connected CW complexes and $f:X\to Y$ is a continuous map that induces isomorphisms of the fundamental groups and ... |
In «New Introduction to Multiple Time Series», page 90, we have the following formulas for the ML estimators of a stable Gaussian VAR$(p)$ process:
where $\tilde \alpha = vec(\tilde A_1,...,\tilde A_p)$.
And here is exactly where my doubt resides. If for $\tilde \alpha$ estimator, I need $\tilde \mu$ estimator, but the... |
While reading an introduction on elliptic curves, I stumbled upon something called reduction modulo $p$.
The definition states that we want to create a group homomorphism that maps an elliptic curve on the rational field to the integers modulo a prime $p$, i.e.: $$E(\mathbb{Q}) \rightarrow \tilde{E}(\mathbb{F}_p), \;\;... |
Page 552: Problem 20 $ \int_{0}^{\frac{1}{\sqrt{2}}}2x\sin^{-1}(x^2)dx $
Using 2x as u gets rid of it eventually, but the inverse sin just gets worse and worse. Actually the integral of the inverse sin is just the inverse sin minus some radical. So it just cycles through over and over. So using the inverse sin looks be... |
Variable Importance in Random Forests can suffer from severe overfitting Predictive vs. interpretational overfitting
There appears to be broad consenus that random forests rarely suffer from “overfitting” which plagues many other models. (We define
overfitting as choosing a model flexibility which is too high for the d... |
given: $$m_1+m_2+...+m_k=m$$ How can I find $\Theta(log(m_1)+...+log(m_k))$ as related to $m$?
I know that i can doing that: $O(log(m_1)+...log(m_k))=O(log(m)+...+log(m))=O(k \cdot log(m))$ , but can I find something without $k$ ?
Computer Science Stack Exchange is a question and answer site for students, researchers a... |
Question of the week came up in my schools logic club but there is not much information to it. Here is the question:
Show that
$$ \exists x\,[R(x)\wedge \lnot Q(x)],\ \forall x\,[P(x)\to Q(x)],\, \forall x[R(x)\to(P(x)\vee S(x))] \models \exists x\,[R(x)\wedge S(x)] $$ Describe in details all your work
Now I know that ... |
For probably twenty years, category theorists have known of some objects in the Platonic universe called "(weak) $\infty$-categories", in which there are $k$-morphisms for all $k\in \mathbb N$, with some weak forms of composition and associativity and .... It was recognized a bit later that it is worth recording two nu... |
So far we have been investigating functions of the form $y=f(x)$, with one independent and one dependent variable. Such functions can be represented in two dimensions, using two numerical axes that allow us to identify every point in the plane with two numbers. We now want to talk about three-dimensional space; to iden... |
The displacement of charge in response to the force exerted by an electric field constitutes a reduction in the potential energy of the system (Section 5.8). If the charge is part of a steady current, there must be an associated loss of energy that occurs at a steady rate. Since power is energy per unit time, the loss ... |
Aleksander from Gdynia Bilingual HighSchool No 3, Poland, David from Guilford County (is that UK, US orelsewhere?) Andrei from Romania and Chris from CSN used differentmethods to solve this problem, all of them leading to solving aquadratic equation. Chris used the observation that, removing threeresistors leaves the r... |
What is the difference between linearly and affinely independent vectors? Why does affine independence not imply linear independence necessarily? Can someone explain using an example?
To augment Lord Shark's answer, I just wanted to talk a little about the intuition behind it.
Intuitively, a set of vectors is linearly ... |
Answer
The current when the resistance is $10$ ohms is $11$ Amperes.
Work Step by Step
We have to write the relation between $ I $ and $ R $ as, $\begin{align} & I\propto \frac{1}{R} \\ & I=\frac{k}{R} \end{align}$ Where $ k $ is the constant of proportionality. Put the value of $ R=22$ and $ I=5$ in the equation to ge... |
You'll probably find a partial answer in this post. The reversible/irreversible tag refers to what is happening to the system, not the surroundings. Events in the surroundings are as a rule treated as reversible.
You should also keep in mind that heat is a signed quantity, and that there can be different conventions wi... |
Skills to Develop
Explain all the quantities involved in the ideal gas law. Evaluate the gas constant \(R\) from experimental results. Calculate \(T\), \(V\), \(P\), or \(n\) of the ideal gas law: \(P V = n R T\). Describe the ideal gas law using graphics. The Ideal Gas Law
The volume (
V) occupied by n moles of any ga... |
The difference between the "stabilising" and "destabilising" examples that you have chosen are that the "stabilising" examples are static molecules, whereas the "destabilising" examples are chemical reactions, in which a lot is happening. We'll look at the "stabilising" cases first, as these are easier to understand.
A... |
In addition to the moment of inertia, the product of inertia is commonly used. Here only the product of the area is defined and discussed. The product of inertia defined as \[I_{x_{i}x_{j}} = \int_{A} x_{i} x_{j} dA \tag{47}\] For example, the product of inertia for \(x\) and \(y\) axes is \[I_{xy} = \int_{A} x y dA \t... |
I dropped the $n\rightarrow\infty$ in the title as it was exceeding the character limit.
In the book I'm currently reading, the author claims that $\liminf\limits_{n\rightarrow\infty} E_{n_0+n} = \liminf\limits_{n\rightarrow\infty} E_{n}$ and $\limsup\limits_{n\rightarrow\infty} E_{n_0+n} = \limsup\limits_{n\rightarrow... |
The enthalpy of combustion $\Delta_\mathrm cH$ of a substance can be calculated from the enthalpy of formation $\Delta_\mathrm fH$. For a compound containing only carbon, hydrogen, and oxygen, the general combustion reaction is
$$\ce{C_$a$H_$b$O_$c$ + ($a$ + 1/4 $b$ – 1/2 $c$) O2 -> $a$ CO2(g) + 1/2 $b$ H2O(l)}$$
The c... |
Given two smooth projective schemes $X$ and $Y$ over some algebraically closed field $k$, we have $X\times Y$ with the projections $p$ to $X$ and $q$ to $Y$. Furthermore we have a "nice" sheaf of algebras $R$ on $X$, i.e. locally free and of global dimension at most dim($X$), e.g. Azumaya or something similar. Given tw... |
Let $M$ be a compact symplectic manifold equipped with a line bundle $\mathcal L$ with curvature $\omega$. Denote by $H^0(M,\mathcal L)$ the space of smooth global sections of $\mathcal L$ (this space is infinite-dimensional). Let $N$ parameterize complex structures on $M$ compatible with $\omega$. For $J\in N$, we may... |
As we briefly discussed in section 1.1, the most famous graph coloring problem is certainly the map coloring problem, proposed in the nineteenth century and finally solved in 1976.
Definition 5.8.1 A
proper coloring of a graphis an assignment of colors to the vertices of the graph so that no twoadjacent vertices have t... |
Let $x$ and $y$ be positive integers such that $xy \mid x^2+y^2+1$. Show that $$ \frac{x^2+y^2+1}{xy}= 3 \;.$$ I have been solving this for a week and I do not know how to prove the statement. I saw this in a book and I am greatly challenged. Can anyone give me a hint on how to attack the problem? thanks
Use the very t... |
Express $\sin e^i$ in $a + ib$ form.
$$\sin e^i = \frac{e^{ie^i}-e^{-ie^i}}{2i}$$
I feel like I can express $ie^i$ in a simpler way, but I'm not sure how.
Thanks for your attention.
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It onl... |
How to describe a $\operatorname{LL(1)}$ parsing algorithm for strings generated by a given grammar?
I have to design a parser for a specific grammar. Let $G$ be the grammar described as:
$$S \rightarrow aBC $$ $$B \rightarrow bB \mid cC $$ $$C \rightarrow c \mid S $$
My approach 1. Test if the grammar is $\operatornam... |
Ordinal numbers
An elegant formulation of the ordinal concept in ZFC was provided by von Neumann: an
ordinal is simply a transitive set well-ordered by the set membership relation $\in$. Equivalently, an ordinal is a hereditarily transitive set, meaning that it is transitive, and all of its elements are transitive.
The... |
I am learning finite element method(galerkin method) for solving ode/pde.
when searching this topic,I often see examples using the hat function
UnitTriangle[x] as the basis function of the galerkin approximation.
I understand that the Galekin method is a method of expressing the objective function by the sum of a basis... |
In Caramello,
Theories, Sites, Toposes, I read that
Every Grothendieck topos $\mathcal E$ admits a unique (up to isomorphism) geoemtric morphism $\gamma_{\mathcal E}:\mathcal E\to \bf Set$. The direct image of $\gamma_{\mathcal E}$ is the global section functor $Hom_{\mathcal E}:\mathcal E\to \bf Set$, while the invers... |
I would like to perform the numerical integration of an integral of the form $$ \int_{-\infty}^\infty e^{i \omega 0+} G(i \omega, \mathbf{v}) d \omega ,$$ or, using the symmetry $G(i\omega)^* = G(-i \omega)$, $$ \int_0^\infty \Re(e^{i \omega 0+} G(i \omega, \mathbf{v}) ) d\omega ,$$ where $\mathbf{v}$ is a vector of ad... |
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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 ... |
Published papers including conference proceedings
Year Title Reference Link 2019 Constraints on mediator-based dark matter and scalar dark energy models using 13 TeV pp collision data collected by the ATLAS detector JHEP 05 (2019) 142 2018 Search for four-top-quark production in the single-lepton and opposite-sign dile... |
This page looks at the relationship between orders of reaction and mechanisms in some simple cases. It explores what a mechanism is, and the idea of a rate determining step. It also explains the difference between the sometimes confusing terms "order of reaction" and "molecularity of reaction".
Reaction mechanisms
In a... |
Answer
The distance to the boat is approximately 845 feet.
Work Step by Step
We can express the angle in degrees: $\theta = 2^{\circ}10' = (2 + \frac{10}{60})^{\circ} = 2.17^{\circ}$ We can convert the angle to radians: $\theta = (2.17^{\circ})(\frac{\pi~rad}{180^{\circ}}) = 0.03787~rad$ Let $h$ be the height of the ma... |
25 2 Homework Statement Converting between wavenumber and wavelength Homework Equations None
By dimensional analysis, we have that the wavenumber: $$k = \frac{\text{radians}}{\text{distance}}$$
And the wavelength: $$\lambda = \frac{\text{distance}}{1 \text{wave}}$$ Then: $$\lambda k = \frac{\text{radians}}{\text{distan... |
Suppose that $y$ is a function of $x$, say $y = f(x)$. It is often necessary to know how sensitive the value of $y$ is to small changes in $x$.
Example 2.1.1 Take, for example, $\ds y=f(x)=\sqrt{625-x^2}$ (the upper semicircle of radius 25 centered at the origin). When $x=7$, we find that $\ds y=\sqrt{625-49}=24$. Supp... |
How can I write the Taylor series for $e^z\sin(z)$ at $z=0$ without making the procedure too complicated?
Isn't there an easier way than to compute it's derivatives and find a pattern?
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It ... |
LaTeX is a great tool for printable professional-looking documents, but can be also used to generate PDF files with excellent navigation tools. This article describes how to create hyperlinks in your document, and how to set up LaTeX documents to be viewed with a PDF-reader.
Contents
Let's start with a minimal working ... |
The Hypergeometric Distribution is usually explained via an urn analogy and formulated as the ratio of “favorable outcomes” to all possible outcomes:
\[
\displaystyle \boxed{P(x=a;N,A,n,a) = \frac{{A \choose a} \cdot {N-A \choose n-a} }{{N \choose n}}} \]
where \(N\) is the total number of balls in, \(A\) the number of... |
If $K(s)$ is the Kolmogorov complexity of the string $s \in \{0,1\}^*$,
Can we prove (or disprove) the following statement: "Every string $s$ is a prefix of an incompressible string; i.e. for every string $s$ there exists a string $r$ such that $K(sr) \geq |sr|$" ?
In a very informal (and perhaps not too meaningful) wa... |
Let's say we are evaluating two algorithms.
Algorithm 1 gets p1 = 80% conversion Algorithm 2 gets p2 = 85% conversion
The two sample sizes are
n1 and n2
respectively.
I'm trying to test the following hypothesis
$H_0:$ p1 = p2 $H_1:$ p1 $\neq$ p2
I though when the sample is large
$p1-p2$ ~ $N(p1-p2,\ \frac{p1(1-p1)}{n1}... |
"Abstract: We prove, once and for all, that people who don't use superspace are really out of it. This includes QCDers, who always either wave their hands or gamble with lettuce (Monte Zuma calculations). Besides, all nonsupersymmetric theories have divergences which lead to problems with things like renormalons, insta... |
Recap of Lecture 13
Last lecture continued the discussion of symmetry (and direct product tables for odd/even functions). We shows the quantum Harmonic Oscillator wavefunctions alternate between even and odd because of the Hermite polynomial component. This maps into the transition moment integral to predict that only ... |
I have a conic optimization of the form:
$$\min_x \langle c, x \rangle,\ \text{s.t.}\ Ax = b,\ x \in K.$$
where $x \in \mathbb{R}^{n}$, $A$ is an $m \times n$ matrix, $b \in \mathbb{R}^m$, $K$ is a self dual cone in $\mathbb{R}^n$ and $\langle~,~\rangle$ is the standard Euclidean inner product on $\mathbb{R}^n$.
I am l... |
Looking for this on the web I was provided with little information, only that as the pressure increases so does tension, and if there is a higher tension it results in a louder pop, how may that be expressed in the form of an equation?
It is true that tension increases when you inflate a balloon, but the pressure has a... |
Two Capacitors Are Better Than One
I was looking for a good reference for some ADC-driving circuits, and ran across this diagram in Walt Jung’s
Op-Amp Applications Handbook:
And I smiled to myself, because I immediately remembered a circuit I hadn’t used for years. Years! But it’s something you should file away in your... |
If I pass individual photons through a M-Z interferometer with equal arms I will observe interference (eg only one detector will respond). As I increase the path length of one arm I will observe the two detectors responding alternately as I pass through each phase cycle. Eventually I suspect that at a certain point, th... |
(This post is the third in a series on geometry. (A geometric series, har har har.) They all assume you know about differential forms and such things. The first was on Liouville geometry, also known as exact symplectic geometry, on surfaces. The second went from them to contact geometry. So I’m assuming you know what t... |
To make sense of this, you need to understand what the word “derivative” means here.
A function $f \colon \mathbf{R}^n \to \mathbf{R}^m$ is said to be
differentiable at the point $a \in \mathbf{R}^n$ if there is a linear transformation $L \colon \mathbf{R}^n \to \mathbf{R}^m$ such that$$f(a+h) - f(a) = L(h) + R(h)$$wit... |
Suppose to have a chain (of size $L$) with bosons, and $\hat{a}_i^\dagger$,$\hat{a}_i$ are the associated creation and annihilation operators at site $i$. A Fock state can be written as: \begin{equation} | n_1 \dots n_L \rangle = \prod_{i} \frac{1}{\sqrt{n_i!}} \left( \hat{a}_i^\dagger \right)^{n_i} |\rangle \end{equat... |
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... |
This will be a talk for the conference: Ouroboros: Formal Criteria of Self-Reference in Mathematics and Philosophy, held in Bonn, February 16-18, 2018.
Abstract. I shall give an elementary account of the universal algorithm, due to Woodin, showing how the capacity for self-reference in arithmetic gives rise to a Turing... |
I am taking a course on quantum field theory where there is some confusion regarding the renormalisation scheme we are using (and a corresponding one in my mind). Apparently the lecturer meant MS-bar when he said MS. Anyway, on with the question.
Consider the* 1-loop contribution to the 1PI 4-point correlation function... |
As we know perturbative expansion of interacting QFT or QM is not a convergent series but an asymptotic series which generally is divergent. So we can't get arbitrary precision of an interacting theory by computing higher enough order and adding them directly.
However we also know that we can use some resummation trick... |
Following (1) and (2), the Schwarzschild metric can be written as: $$ ds^2 = a(r)\cdot c^2 dt^2 - b(r)\cdot dr^2 - r^2\cdot(d\theta^2 + sin^2(\theta)\cdot d\varphi^2) $$ where: $$ a(r) = \begin{cases} \frac14\left(3\sqrt{1-\frac{r_s}{r_g}} - \sqrt{1-\frac{r^2 r_s}{r_g^3}}\right)^2 \qquad \text{if } 0\leq r \leq r_g\\ 1... |
Lethbridge Number Theory and Combinatorics Seminar: Adam Felix Date: 06/01/2015 Time: 11:00 KTH Royal Institute of Technology, Sweden
University of Lethbridge
How close is the order of a mod
p to p−1?
Let $a \in \mathbb{Z} \setminus \{0,\pm 1\}$, and let $f_{a}(p)$ denote the order of $a$ modulo $p$, where $p \nmid a$ ... |
The momentum equation can be simplified for the steady state condition as it was shown in example 6.3. The unsteady term (where the time derivative) is zero.
Integral Steady State Momentum Equation
\[
\label{mom:eq:govSTSF} \sum\pmb{F}_{ext} + \int_{c.v.} \pmb{g} \,\rho\, dV - \int_{c.v.}\pmb{P}\,dA + \int_{c.v.} \bold... |
Two representations of $R_z$ are equivalent if they are the same modulo only a global phase.
$$\begin{pmatrix}1 & 0\\0 & e^{i\theta}\end{pmatrix}=e^{+i\theta/2}\begin{pmatrix}e^{-i\theta/2} & 0\\0 & e^{i\theta/2}\end{pmatrix}$$
If you apply this gate to any state $|\psi\rangle$, the only difference in the outcome is a ... |
Electrochemical Impedance Spectroscopy: Experiment, Model, and App
Electrochemical impedance spectroscopy is a versatile experimental technique that provides information about an electrochemical cell’s different physical and chemical phenomena. By modeling the physical processes involved, we can constructively interpre... |
题目：Resolvent estimates for the Laplacian
报告人：Sanghyuk Lee (首尔大学) 时间：2019年2月28日 （星期四）下午2：00-5：00 地点：工商楼105教室
摘要： In this talk we are concerned with the resolvent estimates for the Laplacian $\Delta$ in Euclidean spaces:
\[ \| (-\Delta-z)^{-1}f \|_{L^q}\le C(z)\| f\|_{L^p}, \ \ z \in \mathbb C\setminus (0,\infty).\] The ... |
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What education websites do you people use other than Brilliant,AOPS,MITOCW,edX,Coursera,udacity??
Note by Eddie The Head 5 years, 5 months ago
Easy Math Editor
This discussion board is a place to discuss our Daily Challenges and the math and science related to th... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
It's hard to say just from the sheet music; not having an actual keyboard here. The first line seems difficult, I would guess that second and third are playable. But you would have to ask somebody more experienced.
Having a few experienced users here, do you think that limsup could be an useful tag? I think there are a... |
Let's define a sequence of families of sets inside $[0, 1]$ interval:
$$ F_n= \left\{ \left(\frac{j-1}{2^n},\frac{j}{2^n}\right] \mid \forall j \in \{ 1, \cdots, 2^n\} \wedge n \in \mathbb{N} \right\} $$
Now, we know that $[0, 1] = \{0\} \cup(0, 1]$. Now we are goint to define a new set $S_n$ as follow:
$$ S_n = \bigcu... |
Let $f(x), g(x) \in \mathbb Z[x]$ be polynomials of positive degrees $r$ and $s$ respectively, and let $\operatorname{Res}(f,g)$ denote their resultant. Further, let $H(f)$ denote the naive height of a polynomial $f$; that is, $H(f)$ is equal to the maximum of absolute values of its coefficients.
Now let us fix positiv... |
$\newcommand\Tr{\text{Tr}}$One of the intriguing lessons we have learned in the past half-century of set-theoretic developments is that there is a surprisingly robust connection between infinitary game theory and fundamental set-theoretic principles, including large cardinals. Assertions about the existence of strategi... |
The primary way ECNs determine if a liquidity taker's flow is 'toxic' or not is by looking at aftermath charts. The aftermath chart shows the average mark-to-market profit of trades done by the liquidity taker as a function of either time or number of top-of-book updates (optionally broken down by currency pair). The t... |
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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... |
marty cohen
Martin Cohen is a retired computer programmer who loves dancing (favorites are West Coast Swing, Waltz, Foxtrot, and Salsa), writing (but not revising) poems, and solving math problems (that's why I'm here). He is currently trying to learn improv at the Westside Comedy Theater (westsidecomedy.com) in Santa ... |
There are answers in -as I think- in mathoverflow for "what are physical interpretations of zeta at negative arguments" (or the like, I don't have the actual words in mind). Just browse a bit through MO or MSE using that terms as filter.
0) Need of a new concept - and a basic requirement: Make the general term explicit... |
Taylor series
A Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function’s derivatives at a single point
$$f(a) \approx \sum\limits_{n=0}^{\infty}{\frac{f^{(n)}(a)}{n!}(x-a)^n}$$
where $f^{n}(a)$ donates the $n^{th}$ derivative of $f$ evaluated at t... |
I am looking at the following image (from http://www.schoolphysics.co.uk/age16-19/Atomic%20physics/X%20rays/text/X_ray_spectra/index.html):
and I want to know, of the two curves, which one represents the element with the higher atomic number. That is, I understand that X-rays are scattered and that the peaks are charac... |
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